ERCOFTAC Henri Bénard

∂ ' Alembert

& Sagaut, Passive scalar decay in homogeneous turbulence

Keywords: Spectral Modelling, EDQNM, Transport and Mixing, Homogeneous Turbulence . . . . . . 8.3 Structure functions in helical turbulence . . . . . . . . . . . . . . . . . . . . . . .

Modelling of Transport in Homogeneous Turbulence

Modelling is essential to understand and reproduce the dominant physical mechanisms occurring in natural turbulent flows such as atmospheric and oceanic ones. Indeed, the dynamics of geophysical flows results of multiple complex processes interacting with each others, at various scales, intensities, and on different characteristic times. The fine description of such flows is currently out of reach of direct numerical simulations, notably because of Reynolds numbers limitations.

Consequently, we address in this thesis the modelling of homogeneous turbulence, using the spectral formalism of the eddy-damped quasi-normal Markovian (EDQNM) approximation. This first allows us to obtain results rapidly in terms of computational resources at very large Reynolds numbers, and thus to investigate separately some of the fundamental mechanisms at stake in natural turbulent flows, namely shear, mean temperature gradient, stratification, helicity, and combinations of these processes.

In this framework, a two-step approach is considered: first, EDQNM is used to close the non-linear terms in the second-order moments equations, and anisotropy is then modelled through spherically-averaged tensors. This methodology is applied to the various configurations mentioned above, permits to propose new theoretical results, and to assess them numerically at large Reynolds numbers. Among the most important findings, we focused on (i) the prediction of the decay and growth laws of crucial one-point statistics such as the kinetic energy, the scalar variance, and helicity; (ii) the determination of spectral scalings; and (iii) the scale by scale distribution of anisotropy.

La modélisation est essentielle pour comprendre et reproduire les phénomènes physiques dominants ayant lieu dans des écoulements turbulents naturels (atmosphériques, océaniques). En effet, la dynamique des écoulements géophysiques résulte d'interactions complexes à des échelles et intensités variées, et sur des temps différents. La description précise de tels écoulements est pour le moment hors de portée des simulations numériques directes, surtout à cause des limitations en nombre de Reynolds.

C'est pourquoi dans cette thèse on s'attaque à la modélisation de la turbulence homogène avec le formalisme spectral de l'approximation EDQNM. Ceci nous permet d'obtenir des résultats rapidement en termes de ressources numériques à très grands nombres de Reynolds, et ainsi d'étudier séparément la plupart des mécanismes en jeu dans les écoulements turbulents naturels, à savoir le cisaillement, le gradient de température, la stratification, l'hélicité, et des combinaisons de ces éléments. On procède en deux étapes: tout d'abord, l'EDQNM permet de fermer les équations des moments d'ordre 2, et ensuite l'anisotropie est modélisée grâce à des tenseurs moyennés sphériquement. Cette méthode est appliquée aux différentes configurations mentionnées ci-dessus, nous permet de proposer de nouveaux résultats et de les valider numériquement à grands nombres de Reynolds. Parmi les points les plus importants, nous nous sommes concentrés sur (i) la prédiction des lois de croissance et décroissance de quantités telles que l'énergie cinétique, la variance scalaire et l'hélicité; (ii) la détermination des comportements spectraux; et (iii) la distribution d'anisotropie échelle par échelle.

Tout d'abord, je souhaite remercier mon directeur, Thomas, qui m'a accompagné pendant toute cette longue et passionnante aventure depuis mon stage de M2. Malgré la distance et quand le téléphone était insuffisant, ce fut toujours avec plaisir que nous avons pu nous retrouver autour d'un café pour discuter turbulence, de mes derniers travaux, et de la rédaction des articles que nous avions sur le feu. J'imagine qu'il a pu être difficile d'encadrer cette thèse depuis Lille à partir de la deuxième année, car il faut savoir que j'ai besoin de discuter, d'échanger, et que j'envoie au moins trois mails par jour (dans une petite journée). J'ai cependant la conviction qu'en dépit de la distance qui nous a séparés, nous avons accompli un travail sérieux et complet: Thomas m'a laissé de l'autonomie dans le choix des thématiques que je souhaitais aborder, et m'a toujours suivi et encouragé malgré les changements parfois radicaux de mes intérêts, et je l'en remercie chaleureusement.

Je remercie grandement Luminita Danaila et Jean-Pierre Bertoglio qui ont accepté d'être rapporteurs de cette thèse, et je n'ose imaginer la quantité de travail que cela demande de lire avec attention les près de 300 pages qui vont suivre. Je souhaite adresser un remerciement très particulier aux autres membres de mon jury: en premier lieu, Pierre Sagaut, grâce à qui, avec son cours d'initiation à la turbulence en M2, je me suis découvert une passion pour cette discipline, et avec qui j'ai eu le privilège de collaborer sur certains travaux par la suite. Ensuite, j'aimerais remercier Claude Cambon, que je considère comme l'expert EDQNM anisotrope, pour toutes les discussions fructueuses que nous avons pu avoir lors de ses passages à Paris, au téléphone, ou encore en conférences. C'est grâce en partie à son "erreur de jeunesse" comme il l'appelle, que ce sujet de thèse est né, et pour cela, je lui en suis extrêmement reconnaissant. Je voudrais aussi remercier Annick Pouquet pour avoir accepté d'être membre de mon jury: c'est un immense honneur pour moi de rencontrer une des pionnières, entre autres, de l'EDQNM. J'adresse également de sincères remerciements à Benoît-Joseph Gréa pour toutes les discussions que nous avons pu avoir à de multiples occasions, et pour avoir partagé avec moi son expertise de l'USHT, ce qui m'a été d'une aide très précieuse : c'est avec lui que je continuerai ma formation en post-doc, et je lui en suis très reconnaissant. Enfin, je remercie Régis Marchiano d'être membre de mon jury, d'avoir participé activement à l'évaluation de ma thèse dès la deuxième année, même si mon travail n'est pas exactement dans sa thématique de recherche. Je remercie très chaleureusement l'ensemble du jury pour les discussions très intéressantes que nous avons eues après la soutenance.

Je tiens à remercier sincèrement les doctorants, post-doctorants et "permanents" du laboratoire avec qui j'ai pu discuter de tout et n'importe quoi au cours de ces trois années: faire une liste exhaustive serait fastidieux, aussi je vous prie de m'excuser si j'en oublie, mais les concernés se reconnaîtront: Anca, Aurélie, Claire, Clara, Maya, Alessio, Arthur, Augustin, Barend, Bruno, Charles, Christian, Gounseti, Hervé, Hugo, Julien, Jan, Jean-Camille, Jean-François, Philippe, Raphaël, Thomas, Valentin, . . . Je remercie également Manasa, dont j'ai encadré le stage de M1, pour ces quelques mois très intéressants à travailler ensemble sur la turbulence stratifiée instable: je te souhaite bonne chance dans la recherche de ton sujet de thèse! Je voudrais adresser un merci très particulier à deux personnes: Sophie et Vincent, avec qui j'ai passé la majorité de mes pauses café et déjeuners, où les discussions à propos de Star Wars se sont enchaînées aux conversations scientifiques, entre des considérations politiques et des réflexions sur les superhéros et les séries du moment. Et à Vincent, notamment, un immense merci de m'avoir aidé au début de ma thèse dans ma lutte contre les développements EDQNM anisotropes: tu m'as passé le relais, et sans ton temps et ta bienveillance, j'aurais probablement mis beaucoup plus de temps à démarrer. Enfin, je remercie Olivier Joachim, mon professeur de physique de maths spé : si quelqu'un m'a donné envie de continuer les sciences et en particulier la physique (lire mécanique pour les âmes sensibles), c'est bien lui, et je n'aurais probablement pas songé à faire

II Transport and Mixing in Homogeneous Anisotropic Turbulence

Introduction "When you have eliminated the impossible, whatever remains, however improbable, must be the truth."

-Conan Doyle, Sherlock Holmes

The understanding of turbulence is a complex task of crucial importance since turbulent flows can be found in many natural flows, such as atmospheric or oceanic ones, and in various industrial applications as well. The complexity of turbulence arises from the fact that it gathers multiple scales, from the largest which contain the energetic eddies and the signature of production mechanisms, to the smallest dissipative scales. These different scales notably interact with each other in an intricate way because of the non-linearity of the Navier-Stokes equation. This non-linearity constitutes both the richness and the challenge of turbulence since it makes it at the same time incredibly varied and complicated to predict. Though the prediction of turbulence is still extremely difficult, it is of practical interest for various applications, with meteorology among them. It is worth noting that proving the existence and uniqueness of a general solution to the three-dimensional Navier-Stokes equations remains one of the six problems of the millennium.

Even by assuming that the turbulence is homogeneous, the objective to fully master the distinct features of natural turbulent flows is rather ambitious. Indeed, the understanding of the impact of large scales anisotropic production mechanisms on the global dynamics is made difficult by the numerous interactions and energy transfers between scales that exist in developed turbulence. Moreover, turbulence is also known to considerably improve mixing properties, so that the transport of a scalar field, such as concentration or temperature fluctuations, is a relevant feature to address and comprehend, for both theoretical and practical purposes. If one considers for instance an unbounded atmospheric flow, it may be subjected to rotation and shear, while the advected scalar field can additionally experience stratification through a mean temperature or concentration gradient. Even with the homogeneity assumption, the task of disentangling the dominant mechanisms remains complex since they all contribute diversely in the dynamics, at various scales and during characteristic times which may be distinct for the velocity and scalar fields.

Thus, a relevant method would be to investigate separately some of these mechanisms, to accurately determine their dominant properties. In this spirit, some authors have brought insightful answers with pioneering experiments and Direct Numerical Simulations (DNS): [START_REF] Warhaft | An experimental study of the decay of temperature fluctuations in grid generated turbulence[END_REF] studied the decay of a passive scalar field in grid turbulence, without any production mechanisms, showing that the algebraic decay rate of the scalar variance < θ 2 > strongly depends on the initial conditions. Later on, [START_REF] Warhaft | An experimental study of the effect of uniform strain on thermal fluctuations in grid-generated turbulence[END_REF] proved experimentally as well that the presence of a mean strain caused by an axisymmetric contraction was accelerating the decay of the scalar field. On the contrary, [START_REF] Sirivat | The effect of a passive cross-stream temperature gradient on the evolution of temperature variance and heat flux in grid turbulence[END_REF] added a mean temperature gradient by several means -heated grid, a mandoline (screen of heated wires), a toaster -to analyze the dynamics of a passive scalar when the fluctuations are sustained. In their famous experiment, [START_REF] Tavoularis | Experiments in nearly homogenous turbulent shear flow with a uniform mean temperature gradient. part 1[END_REF] (TC81) combined both shear and a mean temperature gradient and studied mixed velocity-scalar statistics, along with some crucial one-point quantities for modelling, such the turbulent Prandtl number and diffusivity tensor. Regarding early DNS, [START_REF] Rogers | The structure of the vorticity field in homogeneous turbulent flows[END_REF] analyzed the properties of a shear flow and the resulting global anisotropy between the streamwise and transverse directions, and later [START_REF] Rogers | An algebraic model for the turbulent flux of a passive scalar[END_REF] added a mean scalar gradient, which is the same configuration as TC81, that nevertheless exhibited some significant quantitative discrepancies.

An exhaustive list of the first DNS and experiments which greatly participated into our general understanding of homogeneous turbulence would be tedious, nevertheless the previous references illustrate that the idea of addressing separately the various fundamental mechanisms at stake in natural turbulent flows is not new. The review of such works reveals that there is a large discrepancy between quantities of primary importance, which goes against the "universal principles" postulated in Kolmogorov (1941b,a), which could be briefly reformulated as follows: at asymptotically large Reynolds numbers, small scales of a turbulent flow should be locally isotropic whatever the large scales anisotropic forcing mechanisms are. These small scales are uniquely determined by the kinematic viscosity ν and the kinetic energy dissipation rate . And finally, at any scale l larger than the dissipative scales, but smaller than anisotropic ones, the statistics of the velocity field are only given by l and .

To illustrate the breakdown of these universality assumptions, one can mention for instance the values of the velocity derivative skewness S, recently reported in [START_REF] Antonia | Boundedness of the velocity derivative skewness in various turbulent flows[END_REF], which are quite dispersed for various turbulent flows. The reasons for this scattering are very likely multiple and complex, but some of them are obvious: first of all, if the Reynolds number based on the Taylor scale Re λ is not large enough, the small scales do not "forget" the large scales forcing mechanisms. Thus, one has persistent small scales anisotropy, which consequently renders the results flow-dependent. Other reasons can be proposed, such as the strong variations from one experimental apparatus to another, or low resolution of small scales in DNS. This suggests an essential requirement for the "mechanism by mechanism" investigation mentioned above: one should use the same consistent approach to accurately compare the results and draw relevant conclusions. We will come back to this later, and now return to the idea of anisotropic small scales, which is one of the reasons why some authors in the turbulence community question the local isotropy hypothesis of Kolmogorov. This makes the return to isotropy (RTI) of small scales a crucial point of interest, and it is required to first distinguish the turbulent velocity and scalar fields.

Indeed, it seems that the persistence of anisotropy at small scales for the scalar field is even more complicated than for the velocity field, as reviewed by [START_REF] Warhaft | Passive scalars in turbulent flows[END_REF]. For the velocity field, it is more or less admitted that the small scales of the second-order moments return to isotropy, whereas higher order moments do not systematically, as observed in [START_REF] Pumir | Turbulence in homogeneous shear flows[END_REF]. Of particular interest is the (third-order moment) velocity derivative skewness in shear flows, which should be zero if small scales were completely isotropic. The review by [START_REF] Antonia | Boundedness of the velocity derivative skewness in various turbulent flows[END_REF] Figure 1: The ramp-cliff structures of a turbulent advected scalar field θ or mean field Θ forced by a mean scalar gradient g (from [START_REF] Holzer | Turbulent mixing of a passive scalar[END_REF]). The velocity field of integral scale L is sustained by a Gaussian forcing. The curves at right represent horizontal slices of the plane at left.

clearly illustrates that it is not the case for various kinds of turbulent flows, not necessarily homogeneous. Nevertheless, S is generally found to decrease with increasing Re λ in shear flows, even if the exponent is an open question: in the DNS of Schumacher et al. (2003a), S ∼ Re -1 λ is reported, whereas S ∼ Re -0.6 the primary motivation of the thesis: indeed, it appears to be crucial to determine clearly the asymptotic behaviours of turbulence, in the Kolmogorov's paradigm of large Reynolds numbers.

Even though this if out of reach of DNS for now 1 , the idea is not absurd if one thinks of modelling. The developments of multiple models in the past decades was not only an alternative to DNS great need of computational resources, but also a way to identify and deeply understand the dominant mechanisms of turbulence. As sketched in Fig. 2, there are roughly three methods available to address complex anisotropic flows: Reynolds-Averaged Navier-Stokes (RANS) models, which require the tuning of multiple constants and do not contain much information about small scales and details of the flow. Large-Eddy Simulations (LES), which necessitate the calibration of a turbulent viscosity and subgrid models to reflect the effects of the filtered small scales, very likely different depending of the flow considered. And finally DNS, which have all the information possible, as illustrated by the schematic signal, but which is the most limited in terms of Reynolds numbers. The complexity of a three-part signal, corresponding to RANS, LES and DNS, is also presented for illustration purposes. Illustrations for RANS and DNS were taken online, whereas the one for LES if from [START_REF] Chollet | Parametrization of small scales of three-dimensional isotropic turbulence utilizing spectral closures[END_REF].

for informative purposes, and state why EDQNM is chosen among the collection of existing models. First, one can start with [START_REF] Kraichnan | The structure of isotropic turbulence at very high reynolds numbers[END_REF], who developed the Direct Interaction Approximation (DIA). The DIA is a two-point two-time model, with rather complex analytical expressions due to the presence of equations for response-functions. Unfortunately, the DIA does not respect the Galilean Invariance and does not recover the k -5/3 inertial scaling of the kinetic energy spectrum derived from the Kolmogorov (1941b) phenomenology. These defects were further corrected in the Lagrangian version of the DIA, namely the Lagrangian History DIA (LHDIA) [START_REF] Kraichnan | Lagrangian history closure approximation for turbulence[END_REF]: in the LHDIA framework, Lagrangian correlation times are notably used to restore the build-up of triple correlations. However, the analytical complexity is increased by the additional presence of Lagrangian equations. Finally, let's mention the Test-Field Model [START_REF] Kraichnan | An almost-markovian galilean-invariant turbulence model[END_REF] which is perhaps the closest to EDQNM: basically, an additional transport equation of a compressible test-field is used to determine the characteristic time of the triple correlations, instead of prescribing it in EDQNM. Of course, other models exist, and variations of the previous ones as well, but for the sake of clarity, we choose to not go further in this description.

As illustrated in Fig. 2, EDQNM permits to reach large Reynolds numbers, but is mainly limited to HIT. It will be explained throughout the manuscript that even though EDQNM does not contain as much information as DNS, it nevertheless permits to have a statistical description of all scales for physical quantities of importance, unlike RANS and LES. For this reason, we choose the EDQNM as a good candidate to investigate and model homogeneous anisotropic turbulence, rather than LHDIA and TFM, which are most costly in terms of computational resources, and regarding the former, much more complex analytically even in HIT.

The complete EDQNM approximation was first developed for hydrodynamics homogeneous isotropic turbulence, notably by [START_REF] Orszag | Analytical theories of turbulence[END_REF]; [START_REF] Leith | Atmospheric predictability and two-dimensional turbulence[END_REF]; [START_REF] Orszag | The statistical theory of turbulence[END_REF] (see also [START_REF] Lesieur | Turbulence in fluids[END_REF] and references therein for a more precise overview). Basically, it consists of three ingredients: a quasi-normal procedure to close the non-linear term in the evolution equation of the one-time two-point second-order spectral velocity-velocity correlation; an eddy-damping term which reflects the departure of statistics from normal laws; and finally a Markovianization step to ensure the realizability of the kinetic energy spectrum E(k, t), which further strongly simplifies the time-integration. The EDQNM approximation has proven many times since its creation to be relevant and accurate in HIT [START_REF] Lesieur | Amortissement autosimilaire d'une turbulence á grand nombre de reynolds[END_REF][START_REF] Métais | Statistical predictability of decaying turbulence[END_REF][START_REF] Lesieur | 3d isotropic turbulence at very high reynolds numbers: Edqnm study[END_REF][START_REF] Meldi | On non-self-similar regimes in homogeneous isotropic turbulence decay[END_REF]. The EDQNM framework was also extended to the transport of passive scalar, which is relevant with regard to our problematic, by [START_REF] Herring | A comparative assessment of spectral closures as applied to passive scalar diffusion[END_REF], and then further applied to investigate the decay of the scalar variance in [START_REF] Lesieur | The decay of kinetic energy and temperature variance in threedimensional isotropic turbulence[END_REF]. In addition, the EDQNM results were used to develop and improve subgrid-models for LES, for instance in [START_REF] Chollet | Parametrization of small scales of three-dimensional isotropic turbulence utilizing spectral closures[END_REF].

Furthermore, it is appealing to remember that EDQNM was also extended to more complex cases than HIT to explore configurations unreachable by DNS at this time: after the discovery that helicity, the scalar product between velocity and vorticity < u i ω i >, is an inviscid invariant of the three-dimensional Navier-Stokes equations by [START_REF] Moffatt | The degree of knottedness of tangled vortex lines[END_REF], EDQNM was successfully used by [START_REF] André | Influence of helicity on the evolution of isotropic turbulence at high reynolds number[END_REF] to show that the helical spectrum H(k, t) scales in k -5/3 in the inertial range, similarly to the kinetic energy spectrum. More or less at the same time, [START_REF] Pouquet | Strong mhd helical turbulence and the nonlinear dynamo effect[END_REF] broadened the reach of EDQNM to magnetohydrodynamics (MHD) turbulence by additionally considering the magnetic energy and magnetic helicity spectra. For both helical and MHD turbulence, EDQNM was exploited for subgrid modelling (Baerenzung et al., 2008b,a) as well. In a different framework, more sophisticated methods involving EDQNM were also used to significantly improve RANS mixing models in stratified turbulence (Gréa et al., 2016b). Moreover, it is important to stress that at some points, the EDQNM approximation inherited from particular DIA techniques. Three examples can be emphasized: first, in the framework of weakly compressible turbulence, [START_REF] Bertoglio | Two-point closures for weakly compressible turbulence[END_REF] greatly enhanced the Markovianisation step of the EDQNM approximation by using the DIA equations of the response functions, in order to better take into account the time-history of the compressible part of the velocity field. Secondly, [START_REF] Bos | A single-time two-point closure based on fluid particle displacements[END_REF] proposed an elegant way to get ride of the adjustable constant in the eddy-damping part of the EDQNM approximation, by using an additional field, namely the velocity-displacement correlation, which recalls the idea of the TFM. Thirdly, some information can be learnt thanks to the TFM regarding the characteristic time θ kpq of the triple correlations in EDQNM [START_REF] Herring | A comparative assessment of spectral closures as applied to passive scalar diffusion[END_REF].

What about shear-driven flows? The first attempt to extend EDQNM to strongly anisotropic turbulence dates back to the pioneering work of [START_REF] Cambon | Spectral modelling of homogeneous non-isotropic turbulence[END_REF]: the concept relies on a two-step approach. The classical EDQNM is applied to close the non-linear transfer terms as in HIT, but the general tensorial equation of the spectral two-point velocity-velocity correlation is kept. The resulting expressions are then combined with an appropriate modelling for anisotropy.

In 1981, this second step involved the choice of an arbitrary constant, an issue which was solved recently in [START_REF] Mons | A spectral model for homogeneous shear-driven anisotropic turbulence in terms of spherically-averaged descriptors[END_REF] (MCS). In the latter reference, the modelling of anisotropy is done through a truncated expansion into spherical harmonics of the spectral second-order moments, where part of the anisotropic angular information is restored thanks to deviatoric sphericallyaveraged tensors. As such, MCS is the starting point of this thesis, from which we aim at extending the anisotropic EDQNM modelling to the transport of a passive scalar field in shear-driven turbulence, to active scalar dynamics as well with stratification, and also to helical turbulence.

Upstream to practical considerations such as the development of subgrid-models for shear flows [START_REF] Germano | A dynamic subgrid-scale eddy viscosity model[END_REF], the principal objective of this thesis is to tackle various configurations partially representative of atmospheric turbulence with the same consistent approach: by this, we mean that the anisotropic EDQNM model aims at being valid in multiple cases, without changing any constants. The only constant is the eddy-damping one, which will be set once and for all on the well-known and accepted isotropic value. This is fundamentally different from RANS and LES models which require to tune some adjustable constants depending on the flow, or from earlier spectral models as well [START_REF] Clark | A spectral model applied to homogeneous turbulence[END_REF][START_REF] Bos | Inertial range scaling of scalar flux spectra in uniformly sheared turbulence[END_REF]. In order to get ride of one significant issue mentioned above, the study will be performed in the framework of large Reynolds numbers. Addressing with the same method shear flows, passive scalar transport with a mean temperature gradient, and unstable stratification, is an important contribution in terms of modelling. The idea is the following one: if our model is reliable and accurate enough by comparisons with DNS and experiments at moderate Reynolds numbers, we can have confidence in the predictions we will make at larger Reynolds numbers while combining various mechanisms such as shear and mean scalar gradient. This is inherently distinct from what is usually done in DNS, LES and RANS models for instance, where different codes and settings are used depending on the flow considered.

Among the numerous features listed above, our purposes are rather fundamental, and the different aspects and method of the thesis could be listed as follows:

• Development of the model with analytical calculations and algebra specific to the spectral formalism.

• Assessment of the model by comparisons of the numerical results with DNS and experiments, implying a link between spectral and physical quantities.

• Establishment of theoretical predictions based on physical arguments, such as inertial scaling of spectra and time evolution of one-point statistics.

• Numerical simulations to test the predictions, and to investigate complex interactions, such as the scale by scale distribution of anisotropy resulting from nonlinear exchanges and production at large scales.

• Deduction of the most important and relevant mechanisms for a given configuration, and propose explanations for some specific issues, such as the impact of moderate Reynolds numbers effects.

By going through all these points, we wish to constitute a "database" of both high Reynolds numbers simulations for homogeneous anisotropic turbulence, and a detailed form gathering the main analytical calculations in the spectral formalism which might be used for other purposes than EDQNM as well.

The manuscript is organized in three parts, as follows. In Part 1, we expose the two essential components of the thesis, namely the transport of a passive scalar field in HIT, and the spectral anisotropic model for the velocity field. More precisely, in Chapter 1, the transport of a passive scalar field in decaying HIT is addressed, along with effects of strong and weak diffusivity. Secondly, in Chapter 2, the anisotropic EDQNM modelling for the velocity field in homogeneous turbulence is presented, along with the main evolution equations and the spectral formalism.

Various applications of this model are gathered in Chapter 3, where both sustained shear flows and freely decaying turbulence initially submitted to shear are tackled.

In Part 2, the anisotropic model is extended to deal with the transport of a scalar field. In particular, in Chapter 4, the modelling is consistently broadened to include passive scalar dynamics. Applications such as shear flows with a mean temperature gradient are gathered in Chapter 5, along with multiple successful comparisons with experimental and numerical studies. These different configurations are revisited in Chapter 6 for weakly and highly diffusive scalar, with the emphasis put on isotropic turbulence with a mean scalar gradient. Afterwards, the spectral modelling is further extended to the case of active scalar dynamics to deal with unstably stratified turbulence in Chapter 7. Homogeneous isotropic turbulence with helicity is the subject of Chapter 8, with some considerations about the additional presence of a mean scalar gradient.

Finally, all the appendices mainly contain details about the lengthy calculations necessary to develop the model, along with some additional theoretical considerations.

Part I

Passive Scalar in Isotropic Turbulence & Velocity Field in Anisotropic Turbulence

Chapter 1

Passive Scalar Mixing in Homogeneous Isotropic Turbulence

"Anyone who has never made a mistake has never tried anything new."

-Albert Einstein (or Theodore Roosevelt)

In this chapter, we begin the study of mixing and transport in homogeneous turbulence with the classical case where a passive scalar field is advected by a turbulent isotropic field, meaning that statistics are invariant under any translations, rotations and mirror symmetries. Since the objective of the thesis is to investigate and model the transport of a scalar field in homogeneous anisotropic turbulence (HAT), it makes sense to start with Homogeneous Isotropic Turbulence (HIT). The results of this chapter will serve as a point of comparison throughout the manuscript.

The equations of homogeneous isotropic turbulence

In homogeneous decaying isotropic turbulence, the kinetic energy K =< u i u i > /2 and scalar variance K T =< θ 2 > of the fluctuating velocity and scalar fields u i and θ respectively, where < • > is an ensemble average, evolve according to

dK dt = -(t), dK T dt = -T (t), (1.1)
where and T are the kinetic energy and scalar variance dissipation rates. These two equations come from the Navier-Stokes and scalar transport equations (2.1) and (4.2) for fluctuations, which will be detailed in the next chapters. These four quantities, K, K T , and T are obtained by integrating the kinetic energy and scalar variance spectra over the whole wavenumber space

K (T ) (t) = ∞ 0 E (T ) (k, t)dk, (t) = 2ν ∞ 0 k 2 E(k, t)dk, T (t) = 2a ∞ 0 k 2 E T (k, t)dk, (1.2)
where ν is the kinematic viscosity and a the scalar diffusivity. Since Kolmogorov (1941b), the inertial scaling of the kinetic energy spectrum is known, and more recent studies have investigated the large scales scaling, so that

E(k, t) = A(t) k σ for k < k L , K 0 2/3 k -5/3 for k L < k < k η , (1.3)
where the Kolmogorov constant is found to be K 0 1.3 with the EDQNM simulations, and where

k η = ν 3 1/4
(1.4) is the Kolmogorov wavenumber beyond which dissipation effects are dominant with regard to inertial ones, and k L is the integral wavenumber, corresponding to the scale that contains most of the energy in decaying turbulence. The shape at large scales (k < k L ) is referred to as the infrared range, and is not part of Kolmogorov pioneering work. The scaling E ∼ k σ is given by theoretical arguments which relate spectra and correlation functions in physical space (George, 1992a). Some important features should be mentioned about the infrared scaling: initial conditions with an infrared slope σ > 5, such as a sharply peaked energy spectrum around k L , result in k 4 , the so-called Batchelor turbulence, because of non-local interactions [START_REF] Lesieur | Amortissement autosimilaire d'une turbulence á grand nombre de reynolds[END_REF][START_REF] Lesieur | 3d isotropic turbulence at very high reynolds numbers: Edqnm study[END_REF]. A k 2 infrared scaling is predicted by [START_REF] Lumley | Stochastic Tools in Turbulence[END_REF] with energy equipartition arguments. These two configurations have physical meanings since they refer to the conservation of linear and angular momentum respectively. According to [START_REF] Llor | Comment on "energy spectra at low wavenumbers in homogeneous incompressible turbulence[END_REF], all real positive values of σ ≤ 4 are possible, and σ = 2 might be the most probable value for experiments. Furthermore, only the slope near the peak of energy k L (and not for k → 0) is important and leads the decay [START_REF] Meldi | On non-self-similar regimes in homogeneous isotropic turbulence decay[END_REF]Mons et al., 2014a).

When Batchelor turbulence is mentioned, it is important to clearly define the Permanence of Large Eddies (PLE) [START_REF] Eyink | Free decay of turbulence and breakdown of self-similarity[END_REF][START_REF] Meldi | On non-self-similar regimes in homogeneous isotropic turbulence decay[END_REF]: the infrared range of the kinetic energy spectrum is given by E(k < k L , t) = Ak σ . The PLE is said to hold in decaying turbulence if both A and σ remain constant throughout the decay. Consequently, in HIT, PLE holds for Saffman turbulence, whereas it is broken for Batchelor turbulence, because of strong non-linear transfers from small to large scales.

The crucial assumption behind the scaling of the kinetic spectrum in the inertial range (k ∈ [k L ; k η ]) is the local isotropy of small scales, even if large scales are made anisotropic by various production mechanisms. This is discussed for instance in [START_REF] Sreenivasan | Local isotropy and large structures in a heated turbulent jet[END_REF]; [START_REF] Sreenivasan | On local isotropy of passive scalars in turbulent shear flows[END_REF]; [START_REF] Warhaft | Passive scalars in turbulent flows[END_REF] and will be investigated in this thesis, at the level of second-order moments, in the next chapters. A large inertial zone, and thus a clear separation of scales, requires a high Reynolds number. The integral Reynolds number is defined as

Re L = K 2 ν , (1.5)
which is linked to the integral and Kolmogorov wavenumbers through

k η = Re 3/4 L k L .
(1.6) Kind of anisotropy Re λ (t = 0) P r k min /k L (t = 0) k max HSRT (High P r) 5.10 3 1 ( 1) 10 -7 10k η (10k B ) HSRT Low P r 5.10 4 1 10 -7 10k η HST (Low P r) 5 (100) 1 and 1 ( 1) 10 -10 10 5 k η HITSG (High P r) 5.10 3 1 ( 1) 10 -7 10k η (10k B ) HITSG Low P r 5.10 4 1 10 -7 10k η HSTSG 5 1, 1 and 1 10 -10 10 5 k η HHT and HHTSG 5.10 4 1 10 -7 10k η USHT 5 1 10 -10 10 5 k η Table 1.1: Main numerical parameters used for the simulations: when low Reynolds numbers are reached in decaying turbulence, k = 10 -16 k L (0). Some simulations, especially for comparisons, have different initial parameters. The meaning of the labels for the different kinds of anisotropy can be found in the Abbreviations list in page x: the conditions for HSRT and HIT are the same.

An additional Reynolds number is defined, based on the Taylor microscale Re λ = 20 3 Re L .

(1.7)

Finally, the evolution equations of the kinetic energy and scalar variance spectra, known as the kinetic and scalar Lin equations, read

∂ ∂t + 2νk 2 E(k, t) = S NL(iso) (k, t), ∂ ∂t + 2ak 2 E T (k, t) = S T,NL(iso) (k, t), (1.8)
where S NL(iso) and S T,NL(iso) are conservative spherically-averaged isotropic non-linear transfers.

For the sake of brevity and generality, the EDQNM procedure to obtain the explicit analytical expressions of these terms is detailed in Chapter 2 in the more general framework of HAT.

When it comes to the inertial scaling of the scalar variance spectrum E T (k, t), this is more complicated than for E(k, t): indeed, depending on the value of the Prandtl number P r = ν/a, different scalings can be observed. For a unit Prandtl number, E T (k, t) scales in k -5/3 in the inertial-convective range (ICR) between the scalar integral wavenumber k T and the Kolmogorov wavenumber k η , similarly to the kinetic energy spectrum [START_REF] Obukhov | Structure of the temperature field in turbulent flow[END_REF]Corrsin, 1951a,b), so that E T (k, t) = K CO T -1/3 k -5/3 , (1.9)

where the Corrsin-Obukhov constant is found to be, with EDQNM simulations, K CO 0.74. Within the EDQNM framework, it is possible to obtain other values for K 0 and K CO by changing the eddy-damping constants presented later in Chapters 2 and 4.

In what follows, the transport of a passive scalar field in HIT for P r 1 is firstly addressed. Then, the framework of P r 1 is considered and some new fundamental features are exposed regarding third-order statistics. Finally, the P r impact on the time evolution of scalar integrated quantities is investigated, with in particular the transition from large to low Reynolds numbers.

Numerical set-up

At this point, we briefly present the numerical set-up of the simulations. Since the conditions are always more or less the same in the different configurations that will be addressed, the main elements are gathered here and in Table 1.1. A third order Runge-Kutta scheme with implicit viscous term is used. The wavenumber space is discretized using a logarithmic mesh k i+1 = rk i for i = 1, . . . , n where n is the number of modes in the discretization. Typically, r = 10 1/f where f is the number of discrete points per decade. Simulations have shown that in most of the cases, statistics are not modified within more than 1% from f = 15: for security, we nevertheless choose f = 17. This mesh extends from k min to k max = 10k η T with k η T = √ P rk η if P r ≥ 1 or k η T = k η if P r ≤ 1. The time step ∆t is controlled by defining a constant CFL number. Moreover, the time step is obtained by considering the characteristic time scales of scalar and kinetic dynamics at large and small scales. In the presence of a mean-velocity or scalar gradient, the intensity of the mean-field is also taken into account. The values of the physical and numerical parameters for simulations at large and small Reynolds numbers are gathered in Table 1.1. If not mentioned otherwise, the initial kinetic energy and scalar variance spectra E(k, t) and E T (k, t) are isotropic and the expression is borrowed from [START_REF] Pope | Turbulent Flows[END_REF]; [START_REF] Meyers | A functional form of the energy spectrum parametrizing bottleneck and intermittency effects[END_REF] E(k, t = 0) = K 0 k -5/3 2/3 f L (kL) f η (kη) (1.10)

where f L and f η are shape functions for large and small scales respectively f L (x) = x (x 1.5 + 1.5 -σ/4) 2/3 5 3 +σ , f η (x) = exp -5.3((x 4 + 0.4 4 ) 1 4 -0.4) . (1.11) This corresponds to an initial energy spectrum with energy already at all scales.

1.2 The inertial scaling of E T for P r 1

In this part, the emphasis is put on the scaling of E T in the case of a highly diffusive passive scalar P r 1.

The contents of this section were published in: Briard & Gomez, "Passive scalar convective-diffusive subrange for low Prandtl numbers in isotropic turbulence", Physical Review E, 91, 011001(R) (2015)

The dynamics of a highly diffusive passive scalar is a very controversial topic. There are four different theories regarding the scaling of the inertial-diffusive range (IDR) of the scalar variance spectrum E T . [START_REF] Batchelor | Small-scale variation of convected quantities like temperature in turbulent fluid part 1. general discussion and the case of small conductivity[END_REF] proposed that in the IDR, for k > k CO where the Corrsin-Obukhov wavenumber reads k CO = P r 3/4 k η , one has

E T (k, t) = K 0 3 T a -3 2/3 k -17/3 .
(1.12)

Whereas [START_REF] Chasnov | Turbulence spectrum of strongly conductive temperature field in a rapidly stirred fluid[END_REF] found E T ∼ k -11/3 for a very rapidly stirred fluid. Moreover, [START_REF] Gibson | Fine structure of scalar fields mixed by turbulence. ii. spectral theory[END_REF] derived a E T ∼ k -3 scaling by considering convection effects when scalar gradients are very weak at small scales. Finally, [START_REF] Granatstein | Fluctuation spectrum of a plasma additive in a turbulence gas[END_REF] established a E T ∼ k -13/3 range based on experimental data in a plasma. Thanks to EDQNM simulations, a large range of Prandtl numbers can be explored at high Reynolds numbers. First, it allows to explain directly how the k -13/3 could have been obtained experimentally before: this subrange was observed for fluids with 0.01 ≤ P r ≤ 0.1 and Re λ ∼ 160. However, there is no IDR for P r = 0.1 as revealed in Fig. 1.1a. And for P r = 10 -2 , the IDR is not completely established: this is probably the reason why k -17/3 is not observed in [START_REF] Granatstein | Fluctuation spectrum of a plasma additive in a turbulence gas[END_REF]. The first thing to remark in Fig. 1.1a is that the k -17/3 IDR only clearly appears for P r ≤ 10 -3 . Then, for P r = 10 -6 , the ICR has almost disappeared, which is expected from a physical point of view since the Péclet number P e λ = Re λ √ P r is rather small. Furthermore, there is clearly a third subrange, located between the IDR and k η , where E T decreases much slower. This effect can be understood as a transfer lack toward small scales while approaching the Kolmogorov wavenumber k η . Indeed, in the IDR, the diffusive effects are stronger than the dissipative ones.

On the contrary, the new range is generated by small-scales convection, that was neglected in [START_REF] Batchelor | Small-scale variation of convected quantities like temperature in turbulent fluid part 1. general discussion and the case of small conductivity[END_REF], as mentioned by [START_REF] Gibson | Fine structure of scalar fields mixed by turbulence. ii. spectral theory[END_REF]. From this point, we defined k -1 CD as the characteristic length scale at which this new range starts, where CD stands for convectivediffusive.

In Fig. 1.1b, one can observe three distinct ranges between k L and k η for P r 1: (i) for k ∈ [k L , k CO ], the k -5/3 ICR, dominated by large scales convection: the designation "inertial" comes from the cascade of kinetic energy. Then, (ii) for k ∈ [k CO , k CD ], the k -17/3 IDR, where diffusive effects drive the dynamics, and convection by large scales is negligible. Finally, for k ∈ [k CD , k η ], convection from small scales, and more precisely from the Kolmogorov wavenumber k η , dominates. There, the kinetic field creates small scalar fluctuations that balance diffusion of the IDR. Consequently, this new range is called the inertial-balanced range (IBR), where "balanced" stands for an equilibrium between diffusion and convection by small scales.

From Fig. 1.1a, it is clear that the IBR grows in size with the IDR. Numerically, a good agreement with a k -11/3 range for P r ≤ 10 -4 is obtained by a least square fit. In Fig. 1.1b, the convectivediffusive wavenumber k CD is consistently located between k CO and k η , and clearly separates the k -17/3 scaling from the k -11/3 one. Let us take a closer look at k CD . Since we have a competition between diffusion and convection mechanisms in the IBR, a length scale taking into account these two effects is built. To this end, the characteristic diffusion time t * = k -2 CO /a is considered, based on Corrsin-Obukhov wavenumber and the diffusivity. Then, the convective length scale is obtained using Kolmogorov characteristic velocity u η = (ν ) 1/4 and t * . This eventually yields k CD = (t * u η ) -1 = a -1/2 ν -1/4 = √ P r k η .

(1.13)

The latter relation is similar to the one for the Batchelor wavenumber k B with P r 1 (see next section). This clearly underlines that convection effects are at the origin of the k -11/3 IBR.

The new k -11/3 IBR can be seen as the reconciliation of Batchelor and Chasnov theories. While Batchelor claims that for a strongly diffusive passive scalar, E T ∼ k -17/3 , Chasnov predicts a k -11/3 scaling in particular conditions where the fluid is rapidly stirred. What we observe here is that the two ranges coexist when the Reynolds and Prandtl numbers are respectively large and small enough, and that we can find physical and theoretical arguments to explain it. Firstly, let's introduce an eddy conductivity a t [START_REF] Chasnov | Turbulence spectrum of strongly conductive temperature field in a rapidly stirred fluid[END_REF][START_REF] Batchelor | Small-scale variation of convected quantities like temperature in turbulent fluid part 1. general discussion and the case of small conductivity[END_REF]

a t (t) = ∞ k 2E(k, t) 3n c (k, t) dk (1.14)
where n -1 c is a local characteristic time which depends on k. Using the eddy conductivity a t , the scalar dissipation rate T reads

T (t) = 2(a + a t (t)) k 0 k 2 E T (k, t)dk.
(1.15)

The integral from 0 to k takes into account the main contribution of dissipation since k 1. The influence of small scales dynamics on T is modelled through the eddy conductivity a t . Consequently, one can assume that d T /dk = 0 [START_REF] Chasnov | Turbulence spectrum of strongly conductive temperature field in a rapidly stirred fluid[END_REF]. If we derive (1.15) with respect to k and consider that E(k → ∞) = 0, one gets E T (k, t) = K 0 3

T 2/3 k -11/3 (a + a t ) -2 n -1 c .

(1.16)

In [START_REF] Batchelor | Small-scale variation of convected quantities like temperature in turbulent fluid part 1. general discussion and the case of small conductivity[END_REF], a t is introduced differently and the characteristic time n -1 c is the diffusion time (ak 2 ) -1 . This makes sense when diffusion dominates in the k -17/3 IDR. With (1.16), it is obvious that if n c does not depend on k, then E T ∼ k -11/3 . In [START_REF] Chasnov | Turbulence spectrum of strongly conductive temperature field in a rapidly stirred fluid[END_REF], it is justified that for a rapidly stirred fluid (fluctuations at all scales) n c is constant. A physical and general interpretation of a constant n c could be the following one: in the IDR, the characteristic time (ak 2 ) -1 decreases at small scales because the fluctuations produced by the kinetic field become weaker while approaching k η . At a certain point, when k ≥ k CD , small-scales convection plays a non-negligible role and thus balances small-scales convection and diffusion, so that n -1 c becomes constant. We have two candidates to determine the characteristic time n -1 c : the Kolmogorov time scale τ η = ν/ and the characteristic time based on u η and k CD , namely

τ CD = (u η k CD ) -1 = a = τ η P r -1/2 .
(1.17)

If we use the common assumption verified numerically that a a T , one can write (1.16) differently

n -1 c (k, t) = E T (k, t) E(k, t) 3 T (t)
k 2 a 2 .

(1.18)

In Fig. 1.1c, it is clear that in the IBR, for k ≥ k CD , one has n -1 c constant. In other words, the prediction of [START_REF] Chasnov | Turbulence spectrum of strongly conductive temperature field in a rapidly stirred fluid[END_REF] is recovered. The other point of interest is that the constant reached by n -1 c is really close to the Kolmogorov time scale τ η . This result is consistent with the characteristic time of the convection being given by τ η close to k η . Moreover, one can observe that in the IDR, n -1 c is proportional to k -2 , in agreement with [START_REF] Batchelor | Small-scale variation of convected quantities like temperature in turbulent fluid part 1. general discussion and the case of small conductivity[END_REF]. Finally, the scalar variance spectrum in the inertial-balanced range scales like

E T (k, t) = K 0 3 T 1/6 √ P r a -3/2 k -11/3 . (1.19)
1.3 Mixed-derivative skewness S T for P r 1

In this part, the case of a weakly diffusive passive scalar P r 1 is addressed, and we focus on scalar third-order statistics with the mixed-derivative skewness.

The contents of this section were published in: Briard & Gomez, "Mixed-derivative skewness for high Prandtl and Reynolds numbers in homogeneous isotropic turbulence", Physics of Fluids,28 (8), 081703 (2016) The case P r 1 is of particular interest for various reasons. It specifically corresponds to the framework of biological fluids [START_REF] Scalo | High-schmidt-number mass transport mechanisms from a turbulent flow to absorbing sediments[END_REF] (low temperature dissolved oxygen where Sc 1000, crucial for marine ecosystems), of chemical reactions (reduction of ferricyanide for instance, where Sc can exceed 10 4 ) and of experiments with tracers (such as disodium fluorescein where Sc 2000, or sulforhodamine 101 where Sc 5000). Beyond these practical considerations, the case of weakly diffusive passive scalars is challenging as it presents some difficulties in DNS when it comes to solve the very small scales of the scalar field beyond the Kolmogorov wavenumber k η . These small scales experience friction by the Kolmogorov scale velocity field, up to the Batchelor wavenumber k B = √ P rk η . This continuous friction creates the viscous-convective range (VCR) where the scalar variance spectrum E T (k, t) scales as

E T (k, t) = K 0 3 T ν k -1 , k η < k < k B .
(1.20)

The framework of HIT (with or without a mean scalar gradient) with P r 1 has already received some attention, especially numerically [START_REF] Yeung | Schmidt number effects on turbulent transport with uniform mean scalar gradient[END_REF]Schumacher et al., 2003b;[START_REF] Yeung | Simulations of three-dimensional turbulent mixing for schmidt numbers of the order 1000[END_REF][START_REF] Borgas | High schmidt number scalars in turbulence: Structure functions and lagrangian theory[END_REF], and the k -1 viscous-convective range has been assessed numerous times. However in DNS, with an increasing P r comes a diminishing Re λ . Furthermore, at moderate Reynolds numbers, the spatial resolution beyond the Kolmogorov wavenumber can be questioned. Notably, it has been pointed out in a recent work of forced isotropic turbulence [START_REF] Donzis | Resolution effects and scaling in numerical simulations of passive scalar mixing in turbulence[END_REF]) that both the Reynolds number and the resolution are of great importance: especially, at a given Reynolds number, a better spatial resolution, of order k -1 B , improves local isotropy. The same conclusion is made at constant resolution for an increasing Re λ . A scalar field with a low diffusivity has also been studied experimentally [START_REF] Buch | Experimental study of the fine-scale structure of conserved scalar mixing in turbulent shear flows. part 1. sc ≥ 1[END_REF][START_REF] Miller | Measurements of scalar power spectra in high schmidt number turbulent jets[END_REF][START_REF] Lavertu | Differential diffusion of high-schmidt-number passive scalars in a turbulent jet[END_REF], often with dye where Sc ∼ 10 3 , at higher Reynolds numbers, but the framework is hardly homogeneous and isotropic (jets, shear flows, ...). Therefore, the present study is performed in HIT with EDQNM, which has been used recently [START_REF] Bos | Reynolds number effect on the velocity increment skewness in isotropic turbulence[END_REF]Meldi & Sagaut, 2013a) to study third-order moments of the velocity field, especially the velocity derivative skewness S. Here, the emphasis is put on the mixed-derivative skewness S T , which is of great theoretical interest since it directly appears in the equation of the scalar variance dissipation rate [START_REF] Ristorcelli | Passive scalar mixing: Analytic study of time scale ratio, variance, and mix rate[END_REF].

The evolution equations of the kinetic and scalar dissipation rates can be obtained by multiplying (1.8) by 2νk 2 and 2ak 2 respectively, and then integrating over k

∂ ∂t = 2ν ∞ 0 k 2 S NL(iso) (k, t)dk -4ν 2 ∞ 0 k 4 E(k, t)dk, (1.21) ∂ T ∂t = 2a ∞ 0 k 2 S T,NL(iso) (k, t)dk -4a 2 ∞ 0 k 4 E T (k, t)dk. (1.22)
Using classical algebra, which can be found in [START_REF] Kerr | Higher-order derivative correlations and the alignment of small-scale structures in isotropic numerical turbulence[END_REF]; [START_REF] Ristorcelli | Passive scalar mixing: Analytic study of time scale ratio, variance, and mix rate[END_REF] and which is detailed in Appendix A, yields

∂ ∂t = - 7 3 √ 15 S(t) Re L + 7 15 G(t) 2 K = - 7 15 1 2 S(t)Re λ + G(t) 2 K , (1.23)
where S(t) and G(t) are the velocity derivative skewness and palinstrophy respectively

S(t) = < (∂u/∂x) 3 > < (∂u/∂x) 2 > 3/2 = - 3 √ 30 14 ∞ 0 k 2 S NL(iso) (k, t)dk ∞ 0 k 2 E(k, t)dk 3/2 ,
(1.24)

G(t) =< u 2 > < (∂ 2 u/∂x 2 ) 2 > < (∂u/∂x) 2 > 2 = 30ν 7 K ∞ 0 k 4 E(k, t)dk ∞ 0 k 2 E(k, t)dk
.

(1.25)

Similarly, for the passive scalar field, one gets

∂ T ∂t = - 5 3 S T (t) Re L + r 5 9 G T (t) T K = - 1 2 S T (t)Re λ + r 5 9 G T (t) T K , (1.26)
where r is the kinetic to scalar time scales ratio r = (K T )/(K T ). These evolution equations (1.23) and (1.26) have already been obtained in previous works [START_REF] Zhou | Transport equations for the mean energy and temperature dissipation rates grid turbulence[END_REF][START_REF] Ristorcelli | Passive scalar mixing: Analytic study of time scale ratio, variance, and mix rate[END_REF]Meldi & Sagaut, 2013a) in a similar manner. This numerical study focuses on the mixedderivative skewness

S T (t) = < (∂u/∂x)(∂θ/∂x) 2 > < (∂u/∂x) 2 > < (∂θ/∂x) 2 > = - 3 10 ∞ 0 k 2 S T,NL(iso) (k, t)dk ∞ 0 k 2 E(k, t)dk ∞ 0 k 2 E T (k, t)dk ,
(1.27) which directly appears in (1.26). Note that we obtain a factor 3/10, instead of 2/ √ 15 proposed by [START_REF] Antonia | Similarity of decaying isotropic turbulence with a passive scalar[END_REF]. The scalar palinstrophy reads

G T (t) =< θ 2 > < (∂ 2 θ/∂x 2 ) 2 > < (∂θ/∂x) 2 > 2 = 18a 5 K T T ∞ 0 k 4 E T (k, t)dk ∞ 0 k 2 E T (k, t)dk
.

(1.28)

The kinetic and scalar palinstrophy G and G T can be interpreted as the dissipation of the gradients of the velocity and scalar fields respectively [START_REF] Kerr | Higher-order derivative correlations and the alignment of small-scale structures in isotropic numerical turbulence[END_REF], and more specifically, G represents the dissipation of enstrophy < ω 2 >= /ν. Now that the theoretical aspects have been recalled, numerical results are presented at various Prandtl and Reynolds numbers. The use of EDQNM to study third-order statistics is validated by comparisons with a moderate Re λ experiment [START_REF] Zhou | Transport equations for the mean energy and temperature dissipation rates grid turbulence[END_REF] in Fig. 1.2a and with a DNS of forced HIT [START_REF] Gotoh | Velocity field statistics in homogeneous steady turbulence obtained using a high-resolution direct numerical simulation[END_REF] at higher Re λ in Fig. 1.4b. In the experiment, Re λ ∼ 50 and the decay exponents K ∼ t α and K T ∼ t α T are α α T -1.33. As a first approximation, this corresponds to infrared exponents σ = σ T = 3 (see next section for more details). The comparison between experiment and EDQNM is presented in Fig. 1.2a where the velocity derivative and mixed-derivative skewnesses S and S T are displayed. The agreement is better for S T than for S, whose values obtained experimentally are more dispersed. At higher Reynolds numbers (38 ≤ Re λ ≤ 460), the agreement for S between EDQNM and the DNS of [START_REF] Gotoh | Velocity field statistics in homogeneous steady turbulence obtained using a high-resolution direct numerical simulation[END_REF] of forced HIT is rather good, as revealed in Fig. 1.4b: the velocity derivative skewness is quantitatively recovered within 5% on a broad range of Re λ . Finally, Fig. 1.2b gathers various values of S T obtained in DNS and experiments for P r ≥ 1, and illustrates the noteworthy dispersion, probably due to the different kinds of forcing, whose consequences are amplified at moderate Reynolds numbers: furthermore, the DNS of [START_REF] Kerr | Higher-order derivative correlations and the alignment of small-scale structures in isotropic numerical turbulence[END_REF] suffers from a very low resolution. EDQNM results that will be discussed later are also displayed. Now, the impact of a high Prandtl number on the mixed-derivative skewness S T is investigated. Such a framework has been studied, notably in DNS. However, this has been done only at moderate (or low) Reynolds numbers. Indeed, the more P r increases, the more additional points are necessary to describe the very small scales of the scalar spectrum which behave as k -1 beyond k η and up to k B . Thanks to EDQNM, it is possible to reach high Reynolds and Prandtl numbers, as illustrated in Fig. 1.3a, where the viscous-convective range predicted by [START_REF] Batchelor | Small-scale variation of convected quantities like temperature in turbulent fluid part 1. general discussion and the case of small conductivity[END_REF] grows in size with increasing P r and spans on two decades for P r = 10 5 . Nevertheless, because of the logarithmic discretization, elongated triads are not taken into [START_REF] Kerr | Higher-order derivative correlations and the alignment of small-scale structures in isotropic numerical turbulence[END_REF][START_REF] Yeung | Schmidt number effects on turbulent transport with uniform mean scalar gradient[END_REF][START_REF] Antonia | Similarity of decaying isotropic turbulence with a passive scalar[END_REF] and experiments [START_REF] Zhou | Transport equations for the mean energy and temperature dissipation rates grid turbulence[END_REF]: thick lines for EDQNM at P r = 1 and P r = 10 4 . (--) indicates the asymptotic P r-state S ∞ T at very large Re λ and P r. For Yeung et al. (2002) (×): the values of S T presented are in the plane perpendicular to the mean scalar gradient, the Prandtl number is 1 ≤ P r ≤ 64, and the P r = 1 results are linked by a dash-dot (-•) line.

account. Consequently, it is necessary to add non-local contributions to the scalar non-linear transfers of (1.8). For the sake of clarity, non-local considerations are gathered and detailed in Appendix B.

The P r-dependence of the mixed-derivative skewness S T is investigated in Fig. 1.3b in the high Reynolds numbers regime to avoid transitional effects towards low Reynolds numbers. It is revealed that |S T | increases from P r = 1 to a critical Prandtl number P r c = 10 and then slightly decreases up to P r = 10 3 . Such variations of |S T | for 1 ≤ P r ≤ 10 3 have already been observed in DNS [START_REF] Yeung | Schmidt number effects on turbulent transport with uniform mean scalar gradient[END_REF][START_REF] Yeung | Simulations of three-dimensional turbulent mixing for schmidt numbers of the order 1000[END_REF]. The latter works, at moderate Reynolds numbers, indicate that the decrease of |S T | happens from P r c 1, which is smaller than in our high Reynolds numbers simulations where the decrease starts around P r c 10. Consequently, these observations suggest that the decay threshold for

|S T | is Reynolds dependent, with P r c ∈ [1, 10].
The remarkable feature is that for P r ≥ 10 3 at high Reynolds numbers, the mixed-derivative skewness saturates to a constant value |S ∞ T | 0.435, which does not depend on the Prandtl number anymore. The ∞ symbol refers to the saturated P r-state P r ≥ 10 3 . DNS performed at higher values of P r would be useful to confirm (or not) the saturation of S T from P r ∼ 10 3 . It is worth noting that in HIT, when the scalar field is forced with a mean scalar gradient, values of S ⊥ T (in the direction perpendicular to the gradient) are close to the present S ∞ T = -0.435: values of [START_REF] Yeung | Schmidt number effects on turbulent transport with uniform mean scalar gradient[END_REF] are gathered in Fig. 1.2b, and one can note that at P r = 1, S ⊥ T increases with Re λ (-• × line) similarly to the present EDQNM results.

Physically, this saturation of the mixed-derivative skewness means that the statistical mixing properties of the flow do not evolve anymore at a sufficiently high Prandtl number, for high Reynolds numbers. This can be interpreted in terms of small scales equilibrium (k > k η ), if one considers the spectral definition (1.27) of the mixed-derivative skewness S T . Indeed, considering a given Reynolds number, or equivalently a given dissipation rate of kinetic energy, increasing P r leads to an indefinite extension of the VCR of E T toward small scales, whereas its ICR remains unchanged. Therefore, the variations of S T when P r increases are mainly due to the variations of the two functions appearing in the scalar integrated terms of S T , namely k 2 S T,NL(iso) and k 2 E T for k ∈ [k η ; k B ]. These quantities represent respectively the production rate of mean-square temperature gradients and scalar dissipation at wavenumber k. Moreover, Fig. B.2 shows that the production is mainly a non-local mechanism unlike the scalar dissipation.

For a sufficiently high Prandtl number, P r ≥ 10 3 , these two integrals evolve similarly so that they balance each other. Therefore, for high Reynolds numbers, the convergence of S T to a constant value S ∞ T for increasing Prandtl numbers reflects an equilibrium, occurring in the viscous-convective range, between non-local production of mean-square temperature gradients and scalar dissipation by diffusion.

A similar independence with regard to P r can be found for the scalar palinstrophy G T : injecting classical scaling for E T in the spectral definition (1.28) of G T , and assuming that Re λ 1 and P r 1, yields rG T ∼ Re λ . Such a result was also found in [START_REF] Ristorcelli | Passive scalar mixing: Analytic study of time scale ratio, variance, and mix rate[END_REF]. Numerical simulations and experiments have shown that r ∼ α T /α is a relevant approximation for the time scale ratio when the turbulence decay is algebraic. Therefore, one has r 1 when the kinetic energy and scalar variance decay similarly, i.e. when σ = σ T for the initial spectra considered here: this is relevant since it will be shown in the next section that P r does not affect the asymptotic decay of scalar integrated quantities. Qualitatively, the independence of G T with regard to P r provides the same physical information as our numerical results on S T : there is an asymptotic convergence of the mixing properties of the passive scalar field only for a sufficiently high P r. As said before, a dependance on P r for moderate Prandtl numbers, say 1 ≤ P r ≤ 10 3 , is in agreement with DNS [START_REF] Yeung | Schmidt number effects on turbulent transport with uniform mean scalar gradient[END_REF][START_REF] Yeung | Simulations of three-dimensional turbulent mixing for schmidt numbers of the order 1000[END_REF].

Finally, the decay of the derivative skewness S(t) and mixed-derivative skewness S T (t) from high to low Reynolds numbers is investigated in Fig. 1.4a for Saffman (σ = σ T = 2) and Batchelor (σ = σ T = 4) turbulence. The main results are the following ones: (i) Both S and S T are constant for high Reynolds and Prandtl numbers, and independent of large scales initial conditions: indeed, the curves are identical for Saffman and Batchelor turbulence, except in the transition zone between the high and low Reynolds numbers regimes where a slight difference is observed. (ii) The transition toward the low Reynolds numbers regime for the scalar field starts after the one for the velocity field, which is expected since the Péclet number P e λ = √ P rRe λ is much larger than Re λ in the case P r 1. (iii) For very low Reynolds numbers, both derivative skewnesses S and S T are zero, consistently with the fact that for Re λ < 1, the flow is not turbulent anymore and thus there is no turbulent mixing at all. One also has to point out that both S and S T increase during the decay, i.e. when the Reynolds number decreases, in agreement with George (1992a). Moreover, it is stated in the latter work that at some point during the decay, S should behave as Re -1 λ according to dimensional considerations. Assuming that the Taylor micro-scale λ is the relevant similarity length scale, and using self-preserving functions E(k, t) = E s (t)f (η), S NL(iso) (k, t) = S s (t)g(η), and η = kλ, one obtains

S T , σ = 4 2.145Re -1 λ + S ∞ 0.735Re -1 λ + S ∞ T S ∞ = -0.569 S ∞ T = -0.435 (b)
S(t) ∼ λ -4 νu 2 (λ -2 u 2 ) 3/2 η 2 g(η)dη η 2 f (η)dη 3/2 ∼ Re -1 λ .
(1.29) But this scaling is not always clearly observed. We believe this might be the consequence of too low Reynolds numbers in DNS. A low Reynolds defect is in agreement with Schumacher et al. (2003a), where Figure 1 herein clearly shows that the Re -1 λ scaling is achieved for high Reynolds numbers only (10 2 ≤ Re λ ≤ 10 3 ).

In Fig. 1.4b, relevant correlations are presented (with constants determined by least square fit, set to match with the beginning of the transition) with a clear Re -1 λ dependency for both the velocity derivative and mixed-derivative skewnesses. These correlations S(t) = S ∞ + 2.145Re -1 λ and S T (t) = S ∞ T +0.735Re -1 λ , where S ∞ = -0.569 and S ∞ T = -0.435, capture well the beginning of the transition zone. Hence, the scaling S ∼ Re -1 λ seems relevant mainly for high Reynolds numbers. Moreover, an interesting result, never confirmed previously to our knowledge, is that the mixed-derivative skewness S T scales in Re -1 λ as well. This scaling is in agreement with George (1992b) where similarity assumptions were used for temperature fluctuations:

E T (k, t) = E s T (t)f T (η), S T,NL(iso) (k, t) = S s T (t)g T (η)
, and η T = kλ T . Using a classical result (George, 1992b;[START_REF] Zhou | Transport equations for the mean energy and temperature dissipation rates grid turbulence[END_REF] linking the ratio of the kinetic and scalar Taylor lengths λ and λ T = 6aK T / T yields

λ λ T 2 = 5 6 rP r, S T (t) ∼ aλ λ 2 T u η 2 T g T (η T )dη T η 2 f (η)dη η 2 T f T (η T )dη T ∼ rRe -1 λ .
(1.30)

Time evolution of scalar integrated quantities

In this final part, the effects of a Prandtl number different from unity on the time evolution of scalar integrated quantities such as K T , T and L T , is addressed.

The contents of this section were published in: Briard, Gomez, Sagaut, & Memari, "Passive scalar decay laws in isotropic turbulence: Prandtl number effects", Journal of Fluid Mechanics, 784, 274-303 (2015) A crucial step toward the understanding of the passive scalar dynamics is the study of decay laws at very high or very small Prandtl numbers. Firstly, the decay permits to get ride of the forcing parameter, and secondly, dimensional analysis can provide theoretical results and boundaries regarding time exponents of one-point statistics. Thus, for the passive scalar field, the comparison between the experimental and numerical decay exponents of integrated quantities -such as the scalar variance K T (t), integral scale L T (t), and dissipation rate T (t) -and the Comte-Bellot and Corrsin (CBC) theory could give interesting information on what are the main phenomena which drive the scalar decay.

Multiple experimental works and DNS [START_REF] Lin | Turbulence spectrum of a passive temperature field: results of a numerical simulation[END_REF][START_REF] Warhaft | An experimental study of the decay of temperature fluctuations in grid generated turbulence[END_REF][START_REF] Sreenivasan | On the skewness of the temperature derivative in turbulent flows[END_REF][START_REF] Danaila | Calibration of a temperature dissipation probe in decaying grid turbulence[END_REF][START_REF] Zhou | Transport equations for the mean energy and temperature dissipation rates grid turbulence[END_REF][START_REF] Zhou | Performance of a probe for measuring turbulent energy and temperature dissipation rates[END_REF][START_REF] Antonia | Similarity of decaying isotropic turbulence with a passive scalar[END_REF][START_REF] Lee | Scaling range of velocity and passive scalar spectra in grid turbulence[END_REF][START_REF] Antonia | Invariants for slightly heated decaying grid turbulence[END_REF] have focused on scalar decay exponents. However, these scalar decay exponents exhibit a significant dispersion, as shown in Fig. 1.5, whereas the experimental setups are designed to produce a very similar turbulent dynamics. There is up to 20% of discrepancy for similar Reynolds numbers and a fixed Prandtl number (P r ∼ 0.7 ). This scattering between the measured scalar decay exponents may be due to transitional effects towards low Reynolds numbers, as studied in the kinetic case in Meldi & Sagaut (2013a), and also probably to the experimental production mechanisms which do not permit to obtain a universal decay. A supplementary potential explanation for the scattering might be that the flow is not fully isotropic.

It is worth noting that the decay of the passive scalar has only been studied in experimental works and DNS in a small region of the (Re,P r) map given in Fig. 1.5. In this region (the small grey zone), the Prandtl number is close to unity (0.7 for air which is the most used fluid) and [START_REF] Rust | Turbulent temperature fluctuations in mercury and ethylene glycol in pipe flow[END_REF]; [START_REF] Granatstein | Fluctuation spectrum of a plasma additive in a turbulence gas[END_REF]; [START_REF] Yeung | Schmidt number effects on turbulent transport with uniform mean scalar gradient[END_REF][START_REF] Yeung | Simulations of three-dimensional turbulent mixing for schmidt numbers of the order 1000[END_REF]. On the zoom of the small grey region at P r ∼ 1, experimental scalar decay exponents of Fig. 1.7 are reported: dots, squares, stars, circles, triangles and crosses represent respectively works of [START_REF] Antonia | Invariants for slightly heated decaying grid turbulence[END_REF]; [START_REF] Lee | Scaling range of velocity and passive scalar spectra in grid turbulence[END_REF]; [START_REF] Antonia | Invariants for slightly heated decaying grid turbulence[END_REF]; [START_REF] Zhou | Transport equations for the mean energy and temperature dissipation rates grid turbulence[END_REF]; [START_REF] Antonia | Similarity of decaying isotropic turbulence with a passive scalar[END_REF]; [START_REF] Danaila | Calibration of a temperature dissipation probe in decaying grid turbulence[END_REF].

the Reynolds number based on Taylor scale is Re λ ≤ 70. The values of the decay exponents obtained in these works will be reported later on along with our numerical simulations. Other experiments and DNS performed for Prandtl numbers different from one, spanning from 10 -2 to 10 2 in Fig. 1.5 [START_REF] Rust | Turbulent temperature fluctuations in mercury and ethylene glycol in pipe flow[END_REF][START_REF] Granatstein | Fluctuation spectrum of a plasma additive in a turbulence gas[END_REF][START_REF] Watanabe | Statistics of a passive scalar in homogeneous turbulence[END_REF][START_REF] Yeung | Simulations of three-dimensional turbulent mixing for schmidt numbers of the order 1000[END_REF] mainly focus on the inertial scaling of the scalar spectrum E T (k, t) and not on the decay of the passive scalar itself. Therefore, they cannot be exploited for comparison purpose in the present study.

When the Prandtl number departs from unity, as described in [START_REF] Tennekes | A first course in Turbulence[END_REF], various theoretical arguments show that at large or small P r, the shape of the scalar spectrum E T (k, t) is significantly modified at small scales. Nevertheless, as illustrated in Fig. 1.5, the regions where P r 1 and P r 1 have not been much explored for the passive scalar decay issue. Hence, an interesting question could be: does the Prandtl number, in addition to the Reynolds number and initial conditions, modify the decay laws of scalar integrated quantities such as K T (t) and T (t)? This study aims at answering this question of interest for the understanding of the passive scalar dynamics, and at providing an explanation to the scattering of experimental scalar decay exponents.

To this end, this part focuses on two main approaches. The first one is based on the CBC dimensional analysis that is extended to the more general case of passive scalar transport, following the work of [START_REF] Meldi | On non-self-similar regimes in homogeneous isotropic turbulence decay[END_REF] in the kinetic case. The second approach relies on the EDQNM closure to perform numerical simulations of the turbulent mixing. The main

Reynolds Cases

α, α T n , n T n L , n L T n Re L n Re λ High Kinetic -2 σ-p+1 σ-p+3 -3 σ-p+5/3 σ-p+3 2 σ-p+3 -σ-p-1 σ-p+3 -1 2 σ-p-1 σ-p+3 Scalar -2 σ T -p T +1 σ-p+3 -σ-p+5+2σ T -2p T σ-p+3 2 σ-p+3 Low Kinetic -σ+1 2 -σ+3 2 1 2 -σ-1 4 -σ-1 8 Scalar -σ T +1 2 -σ+3 2 1 2
Table 1.2: Kinetic and scalar exponents for the extended CBC analysis. K and K T are the kinetic energy and scalar variance, and T the kinetic and scalar dissipation rates, and L and L T the kinetic and scalar integral scales. σ and σ T are the kinetic and scalar infrared slopes, and p and p T the kinetic and scalar backscatter parameters.

advantage of this method is its accuracy and low cost in investigating the turbulent dynamics for a broad range of Reynolds and Prandtl numbers. EDQNM simulations have already been used to study kinetic decay exponents [START_REF] Lesieur | 3d isotropic turbulence at very high reynolds numbers: Edqnm study[END_REF][START_REF] Meldi | On non-self-similar regimes in homogeneous isotropic turbulence decay[END_REF], 2013a) and passive scalar dynamics [START_REF] Lesieur | The decay of kinetic energy and temperature variance in threedimensional isotropic turbulence[END_REF]. Moreover, this method also allows to recover theoretical results regarding the scalar spectrum scaling when P r ≤ 1 and P r ≥ 1 that have been obtained experimentally and numerically (see the two previous sections). Hereafter, new theoretical and numerical arguments are proposed to understand how the decay of a passive scalar field is affected by a Prandtl number strongly different from unity.

The basics of the CBC dimensional analysis

In the CBC theory, the kinetic energy spectrum is given by (1.3). The original method of CBC is based on the concept of invariance of very large scale eddies (PLE) corresponding to k < k L which is notably valid for infrared slopes σ = 1, 2 and 3. In fact in the kinetic case, the value of the infrared spectral slope σ is time-independent and remains constant for σ = 1, 2 (Saffman), 3 and 4 (Batchelor). As for the coefficient A(t) of the infrared spectrum, it remains constant for values of σ ∈ [1, 2, 3], and evolves in time as A(t) ∼ L(t) p in the case σ = 4, where p is the backscatter parameter. In the case of high Reynolds numbers, when there is an inertial zone, one obtains

α(σ, p) = -2 (σ + 1 -p) (σ + 3 -p) , K(t) ∼ t α (1.31)
in which p = 0 if σ ≤ 3 and p ∼ 0.55 if σ = 4 as computed in [START_REF] Lesieur | The decay of kinetic energy and temperature variance in threedimensional isotropic turbulence[END_REF]; [START_REF] Eyink | Free decay of turbulence and breakdown of self-similarity[END_REF]; [START_REF] Meldi | On non-self-similar regimes in homogeneous isotropic turbulence decay[END_REF]. Other kinetic exponents such as n and n L can also be determined using (1.31) and are gathered in Table 1.2.

The relevant parameter to study the dynamics of the passive scalar is not only the Reynolds number, but also the (Taylor) Péclet number P e λ = Re λ √ P r. In this section, the emphasis is put on the case σ T = 4. Numerical simulations show that the scalar variance also decreases with time following a power law K T ∼ t α T . Furthermore, after a transient relaxation phase, the kinetic and scalar integral scales L and L T exhibit the same behaviour and their decay exponents n L and n L T converge toward the same value. Consequently, we consider that n L = n L T .

The scalar variance spectrum E T (k, t) scales similarly to E(k, t) in the infrared range, namely

E T (k < k T , t) = A T (t)k σ T , σ T ∈ [1, 4] ,
(1.32)

Infrared slope σ P r = 10 -4 P r = 10 -2 P r = 1 P r = 10 P r = 10 2 P r = 10 where k T 1/L T is the peak of E T . Proceeding similarly to the kinetic case, the coefficient A T (t) is assumed to vary as L p T T , where p T is the scalar backscatter parameter. By writing the continuity of the scalar spectrum in k = k T , one can use the expressions given in Table 1.2 of n L and n to conclude that

T ∼ t n T , n T (σ, σ T , p, p T ) = - σ -p + 5 + 2(σ T -p T ) σ -p + 3 . (1.33)
Moreover, since T is the time derivative of K T , the exponent of the scalar variance reads

α T (σ, σ T , p, p T ) = -2 σ T -p T + 1 σ -p + 3 . (1.34)
From the theoretical decay exponents α and α T , it seems that a decay in K (T ) ∼ t -1 [START_REF] Ristorcelli | Passive scalar mixing: Analytic study of time scale ratio, variance, and mix rate[END_REF] occurs only for σ = σ T = 1 and corresponds to a constant Reynolds number.

All extended exponents (with p and p T ) for both the kinetic and scalar fields are gathered in Table 1.2. It will be shown later in Chapters 3 and 5 that these exponents are still valid when the turbulence is initially submitted to a mean-velocity gradient, and then freely decaying, as summarized in Table 5.1. However, when a mean scalar gradient is added, the continuous production of scalar fluctuations significantly modifies α T .

Furthermore, simulations indicate that A T (t) varies with time when σ T = 4. More precisely, p T decreases when σ increases at a fixed P r, and increases when P r departs from unity: values are gathered in Table 1.3.

Finally, the emphasis is put on the dynamics at small Reynolds numbers. In this case, the inertial effects are rather small, and so inertial zone contributions to the kinetic and scalar spectra become negligible. Thus, the kinetic energy behaviour is assumed to be mainly determined by the contribution of the spectrum at large scales. By dimensional analysis L(t) ∼ √ νt and so n L = 1/2 . This leads to

K(t) = ∞ 0 E(k, t)dk k L 0 Ak σ dk = A σ + 1 ν -(σ+1)/2 t -(σ+1)/2 . (1.35)
From the kinetic energy K, the other important kinetic decay exponents can be deduced, all gathered in Table 1.2. One obtains the exponent of the scalar variance using the relation L T (t) ∼ √ at. Then, proceeding similarly, one gets

K T (t) k T 0 A T k σ T dk = A T σ T + 1 a -(σ T +1)/2 t -(σ T +1)/2 . (1.36)
The other scalar decay exponents derived from α T are also presented in Table 1.2. For instance, n T is simply computed using dK T /dt = -T . In what follows, all exponents are calculated using the EDQNM simulations for both σ and σ T in the set [1,2,3,4]. For the kinetic and scalar cases, there is an excellent agreement between the EDQNM results and the predictions of the extended CBC analysis.

1.4.2 Validation at large Reynolds numbers for P r = 1

The emphasis is now put on the case where the initial Reynolds number is sufficiently large to allow the kinetic and scalar spectra E and E T to decrease according to the extended CBC exponents given on the two first lines of Table 1.2. For the sake of brevity, only the case P r 1 is presented here since the results for P r 1 are very similar (for more details, see the complete paper). The initial Reynolds number is Re λ (0) 2.10 4 , high enough to ensure a large Péclet number, so that there is a clear separation of scales. Since E T (k, t) is located "under" E(k, t), local energy transfers dominate, unlike the case P r 1 where the viscous-convective range is "outside" the kinetic energy spectrum. Time exponents of T , L T and K T are investigated in Fig. 1.6 for σ = σ T = 2 and σ = σ T = 4. The scalar decay exponents clearly follow the extended CBC theory. The result is the same for any P r ≤ 1 as soon as the Péclet number is large enough. Hence, the extended scalar CBC exponents are valid at large Reynolds numbers for both P r 1 and P r 1.

Transition to low Reynolds and Péclet numbers

In this section, the transition from high to low Reynolds numbers with various Prandtl numbers is investigated. A detailed comparison with experimental results is performed to provide some explanations about the scattering between existing measured scalar decay exponents.

Validation of decay exponents for P r = 1: The numerical method based on EDQNM analysis allows to illustrate the transition from high to low Reynolds numbers. Several simulations are made until very low Reynolds numbers Re λ ∼ 10 -1 , starting from Re λ (t = 0) = 240. This Reynolds number is high enough to capture the beginning of the transition and all the previous decay exponents are accurately recovered. According to the theory for the case σ = 1, the same exponents for both large and small Reynolds numbers are found. It is only from the shape of the spectrum that these two cases can be distinguished. For infrared exponents σ ≥ 2 and σ T ≥ 2, the Reynolds number decreases over time and inertial ranges of both spectra disappear. This is the low Reynolds numbers regime and the decay exponents α and α T converge to the values expected by the extended CBC analysis given in Table 1.2. This transition grows more rapidly for higher values of the infrared spectral slopes.

10 0 10 1 10 2 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 Re λ α , α T , α exp T α(σ (T ) = 2) α T (σ (T ) = 2) CBC High Re CBC Low Re

Review of experimental results:

In experiments for the passive scalar, the fluid is often air with P r 0.7. In Fig. 1.7, several experimental results regarding the scalar decay exponent α exp T are gathered in order to compare them to experiments [START_REF] Danaila | Calibration of a temperature dissipation probe in decaying grid turbulence[END_REF][START_REF] Zhou | Transport equations for the mean energy and temperature dissipation rates grid turbulence[END_REF][START_REF] Zhou | Performance of a probe for measuring turbulent energy and temperature dissipation rates[END_REF][START_REF] Lee | Scaling range of velocity and passive scalar spectra in grid turbulence[END_REF][START_REF] Antonia | Invariants for slightly heated decaying grid turbulence[END_REF], and DNS [START_REF] Antonia | Similarity of decaying isotropic turbulence with a passive scalar[END_REF]. Firstly, it is important here to stress that in these experiments, the Reynolds number is rather low Re λ ≤ 70. EDQNM simulations show that such Reynolds numbers correspond to the beginning of the transition zone, between high and low Reynolds regimes. This could explain the large scattering of scalar decay exponents that have been measured in the past years. The Re λ is not high enough to completely match with the high Reynolds and Péclet regimes, and thus the α exp

T measured could be misrepresented by the transitional Re λ -state. Even with DNS, [START_REF] Antonia | Similarity of decaying isotropic turbulence with a passive scalar[END_REF] found a α T which is not the one predicted by the CBC theory. Once again, this might be because of the moderate Re λ and a too low resolution of large scales, which are determinant for the decay exponents. In addition to this, the infrared initial slopes σ and σ T cannot be fixed in grid turbulence: hence, it is impossible to compare rigorously α exp T with the CBC theory.

Nevertheless, interesting results have been obtained experimentally that deserve to be emphasized. First, the scattering of the measured scalar decay exponents α T is more important than for the kinetic ones α [START_REF] Lavoie | Effects of initial conditions in decaying turbulence generated by passive grids[END_REF][START_REF] Antonia | Invariants for slightly heated decaying grid turbulence[END_REF]. In addition to the two facts mentioned earlier (low Re λ and undefined infrared slopes), α exp T varies a lot depending on how the temperature fluctuations are generated. It has been shown [START_REF] Warhaft | An experimental study of the decay of temperature fluctuations in grid generated turbulence[END_REF][START_REF] Sreenivasan | On the skewness of the temperature derivative in turbulent flows[END_REF][START_REF] Zhou | Transport equations for the mean energy and temperature dissipation rates grid turbulence[END_REF] that the measured value of α exp T varies significantly depending on the experimental apparatus: the power used to create the temperature fluctuations or the influence of the measurements origin (for example the grid, corresponding to the kinetic integral scale L, or the heated screen, corresponding to the scalar one L T ). Recent experimental works [START_REF] Zhou | Transport equations for the mean energy and temperature dissipation rates grid turbulence[END_REF][START_REF] Zhou | Performance of a probe for measuring turbulent energy and temperature dissipation rates[END_REF][START_REF] Lee | Scaling range of velocity and passive scalar spectra in grid turbulence[END_REF][START_REF] Antonia | Invariants for slightly heated decaying grid turbulence[END_REF] have been performed on the scalar decay using similar parameters, which allows to make useful comparisons. Most of these experiments were done with a mesh size M = 24.76mm, an input speed U 6m.s -1 , a mandoline (screen of heated wires) located at x T = 1.5M and heated with 2kW that generates temperature fluctuations T ∈ [2K, 3K]. Scalar quantities are measured with the cold wires technique and the Reynolds number is such that Re λ ∈ [30,70]. However, as soon as Re λ remains low, there is an uncertainty due to the transient phase from large to low Reynolds numbers. Finally, since the temperature is a passive scalar, it is still submitted to the variations of the kinetic field induced by the kind of grid chosen for experiments (square, round, solidity, active, passive, ...). Therefore, in addition to the moderate Re λ effect, the various techniques used to create the turbulent kinetic and scalar fluctuations may be responsible for the scattering.

Transition for P r = 1

In the previous high Reynolds and Péclet numbers cases, it has been shown numerically that the Prandtl number does not affect the scalar decay exponents predicted by the CBC theory. The relevant question is now to determine if this is still valid for low Reynolds and Péclet numbers. It is worth noting that if the Reynolds number is low, it implies a low Péclet regime. In other words, the case of a kinetic field in low Reynolds regime with a scalar field in large Péclet regime does not exist, in terms of the CBC theory. Case P r 1: The transition towards low Reynolds numbers begins when the inertial k -5/3 range of E(k, t) tends to disappear. From this point, the scalar spectrum E T still contains a k -1 VCR where the scalar destruction is fairly weak. At the beginning of the transition, with the disappearance of the inertial range, the production of small vortices stops but the friction between small scales creates some temperature fluctuations. Because of this production of scalar variance, the scalar decay slows down and α T increases. Then with the disappearance of the viscous-convective zone, the destruction of scalar variance accelerates under the accumulated effects of diffusive and dissipative processes. Hence, K T decreases more rapidly. Such a behaviour is recovered in Fig. 1.8 where α T is investigated for Saffman and Batchelor turbulence at P r = 10 3 and P r = 10 5 . Moreover, two critical Reynolds numbers are observed for the scalar field: a first one from which α T increases, and a second one from which it decreases, corresponding respectively to the disappearance of the ICR and VCR. The second one is smaller than the one found in the case P r = 1 which was Re c λ T 12 in Fig. 1.7. Indeed, reaching a low Péclet regime with a large Prandtl number is longer than with P r = 1, because Re λ has to decrease more.

Case P r

1: In Fig. 1.9, scalar decay exponents follow, once again, the extended CBC theory. There is no particular behaviour of α T during the decay: indeed, all scales of the scalar spectrum E T are fully controlled by the kinetic one. In the case P r 1, the main difference with regard to the case P r = 1 in Fig. 1.7 is that the transition happens earlier: the scalar critical Reynolds number Re c λ T (P r 1) is higher than Re c λ T (P r = 1). Since the Prandtl number is very small, the Péclet number P e λ = Re λ √ P r tends faster to the low Péclet regime, and so the critical Reynolds number Re c λ T is larger than with P r 1. The conclusion is that the scalar time exponents provided by the extended CBC theory are still valid in the cases P r 1 and P r 1 for small Reynolds numbers and all values of σ and σ T . Only α T , before reaching its asymptotic limit, is temporarily modified when P r 1.

Study of the integral scales L and L T

The emphasis is now put on the kinetic and scalar integral scales L and L T , defined as

L(t) = 3π 4K(t) ∞ 0 k -1 E(k, t)dk, L T (t) = π 2K T (t) ∞ 0 k -1 E T (k, t)dk.
(1.37)

Evolution of L T : The aim of this part is to show that whatever the Prandtl number is, the relative initial position of the kinetic and scalar integral scales L(t = 0) and L T (t = 0) does not change the asymptotic dynamics of the decay. The law provided by [START_REF] Lesieur | The decay of kinetic energy and temperature variance in threedimensional isotropic turbulence[END_REF] gives the temporal evolution of the scalar integral scale

L T (t) = α α T 3/2 L(t) (1 + B t α-2 3 ) 3/2 , (1.38)
where B is a constant close to -1 evaluated thanks to initial conditions. At first approximation, at large times, one has L T /L = (α/α T ) 3/2 . Such an equation is obtained by dimensional analysis, assuming that in the inertial ranges, and T scale like = 2K 1/3 L -2/3 and T = 2K T 1/3 L -2/3 T . However, in the low Reynolds and Péclet regimes, there are no longer inertial and inertial-convective ranges. Therefore (1.38) is only valid when Re 1 and P e 1. In [START_REF] Lesieur | The decay of kinetic energy and temperature variance in threedimensional isotropic turbulence[END_REF], it is claimed that (1.38) is only valid in the case L T (t = 0) < L(t = 0), meaning that the scalar variance is injected at smaller scales than kinetic energy. Hereafter, it is shown that all the three different cases L T (t = 0) = L(t = 0), L T (t = 0) > L(t = 0) and L T (t = 0) < L(t = 0) collapse into the same evolution after a transient phase. The cases where L T (t = 0) = 10 2 and 10 -2 , for P r = 10 -2 and P r = 10 2 with L(t = 0) = 1 are investigated in Fig 1 .10. Despite the large final turn-over time (t ∼ 10 11 τ 0 ), the Reynolds number is still high enough to make sure that there is a clear inertial range (Re λ ≥ 300). Results before t = 10τ 0 are not shown for the sake of clarity as L/L T is too high. The first conclusion is that P r does not affect the asymptotic time evolution of L T , as predicted by [START_REF] Lesieur | The decay of kinetic energy and temperature variance in threedimensional isotropic turbulence[END_REF].

t/τ 0 L/L T L T (t = 0) = 1 L T (t = 0) = 10 -2 L T (t = 0) = 10 2 (b) Figure 1.10: Evolution of L/L T for the three different cases L T (t = 0) = L(t = 0), L T (t = 0) > L(t = 0) and L T (t = 0) < L(t = 0). (a) P r = 10 -2 ; (b) P r = 10 2 .
This is an asymptotic result since L and L T collapse for very large turn-over times only. Such large turn-over times are never reached in practice in experiments. Nevertheless, the three cases L(0) > L T (0), L(0) < L T (0) and L(0) = L T (0) are physically meaningful: the first case correspond to the apparatus where the velocity fluctuations are heated after the grid by a mandoline for instance. The second case correspond to a toaster : the laminar field before the grid is heated. The comparison of these two settings has been done by [START_REF] Sirivat | The effect of a passive cross-stream temperature gradient on the evolution of temperature variance and heat flux in grid turbulence[END_REF]. The latter case correspond to the heated grid [START_REF] Warhaft | An experimental study of the decay of temperature fluctuations in grid generated turbulence[END_REF].

Prediction of k L and k T : Here, a law able to predict the relative position of the peaks of both kinetic and scalar spectra, respectively k L and k T , is derived. This law is valid in high Reynolds and Péclet regimes, as soon as kinetic and scalar integrated quantities decay according to the CBC theory. We define the ratio β L = k T /k L . Even though the assumption β L = 1 is commonly made, simulations reveal that it is not exactly verified. Moreover, the fact that k T = k L has already been observed in experiments: [START_REF] Warhaft | An experimental study of the decay of temperature fluctuations in grid generated turbulence[END_REF]; [START_REF] Sreenivasan | On the skewness of the temperature derivative in turbulent flows[END_REF] noted that there was a link between the scalar decay exponent α T and the ratio β L . Furthermore, [START_REF] Zhou | Transport equations for the mean energy and temperature dissipation rates grid turbulence[END_REF] also made the observation and proposed a correlation. Nevertheless, since the infrared exponents are unknown in experiments, it is impossible to make relevant comparisons. This is the reason why an analytical law linking β L and α T in the high Reynolds and Péclet regimes is proposed here. Piecewise spectra are used C = 1/(σ -p) + 3/5, and C = 1/(σ T -p T ) + 3/5. Then, using L/L T = (α T /α) 3/2 , an explicit law for β L is obtained

E = A(t)k σ , k < k L , K 0 k -5/3 2/3 , k L < k < k η , E T =          A T (t)k σ T , k < k T K CO T -1/3 k -5/3 , k T < k < k T,max K B T ν k -1 , k η < k < k B , K 0 /3 T a -3 2/3 k -17/3 k CO < k < k η , (1.39) where k T,max is either k CO for P r ≤ 1 or k η for P r ≥ 1. The computation of the ratio L/L T yields L/L T = (3β L C C T )/(2C C T ), where C = 1/(1+σ -p)+3/2, C T = 1/(1+σ T -p T )+3/2,
β L = 2 3 σ -p σ T -p T 1 + σ T -p T 1 + σ -p α T α 3/2
.

(1.40)

The law (1.40) is valid as soon as there are inertial and inertial-convective ranges for E and E T .

In order to show the relevance of this formula, β L is computed in several cases with σ = σ T and with σ = σ T . All results are gathered in Table 1.4 where β num refers to the numerical results and β th to the theoretical ones coming from (1.40). This formula provides less than 1% error when σ = σ T ≤ 3 and a maximum of 6.64% when σ T = 4. Finally, (1.40) does not depend on P r: all cases in Table 1.4 give similar values for various P r, consistently with the fact that P r does not affect large scales.

Th β L -law (1.40) completes the work of [START_REF] Ristorcelli | Passive scalar mixing: Analytic study of time scale ratio, variance, and mix rate[END_REF] regarding the time scale ratio r: indeed in the latter reference, the ratio k T /k L is introduced in the analytical computation of r, but no explicit formula is provided.

Conclusions for a passive scalar field in HIT

In this first chapter dedicated to the transport of a passive scalar field in homogeneous isotropic turbulence, several results regarding the impact of a Prandtl number different from unity were proposed. There are summarized hereafter. It is recalled at this point that the next step is to extend the EDQNM approach to HAT in Chapter 2, in order to further combine anisotropic mechanisms and scalar transport from Chapter 4.

First, it has been shown that both theories of [START_REF] Chasnov | Turbulence spectrum of strongly conductive temperature field in a rapidly stirred fluid[END_REF] and [START_REF] Batchelor | Small-scale variation of convected quantities like temperature in turbulent fluid part 1. general discussion and the case of small conductivity[END_REF] can be merged into a single one: for a highly diffusive scalar, a new k -11/3 inertial-balanced range

appears for k ∈ [k CD , k η ],
where k CD = √ P r k η is the characteristic wavenumber based on diffusion and small-scales convection. This new range appears thanks to small-scales convection that balances diffusion from the k -17/3 inertial-diffusive range. This small-scales convection predicted by [START_REF] Gibson | Fine structure of scalar fields mixed by turbulence. ii. spectral theory[END_REF] comes from small scales eddies of order k -1 η . This new range appears conjointly with the k -17/3 scaling when both the Reynolds and the Prandtl numbers are respectively high enough (Re λ ≥ 2.10 4 ) and small enough (P r ≤ 10 -3 ). Finally, both the k and P r dependence of the scaling E T ∼ √ P r k -11/3 were assessed numerically.

Secondly, the mixed-derivative skewness S T has been investigated. The main results of this study are twofold. (i) At high Reynolds numbers and for P r ≥ 10 3 , S T saturates to a constant value S ∞ T = -0.435, independent of the large scales initial conditions σ and σ T , which means that statistical properties of the scalar mixing are converged, and can be interpreted as a small scales equilibrium in the viscous-convective range. (ii) The Re -1 λ scaling for S T (and S), coming from self-similarity theory, was numerically assessed. These numerical and theoretical results exhibit some robust asymptotic states at very large Reynolds and Prandtl numbers for scalar third-order statistics.

Finally, we characterized the decay of a passive scalar field in HIT by extending the Comte-Bellot and Corrsin (CBC) analysis and comparing it to EDQNM simulations. Namely, a scalar backscatter parameter p T was defined to take into account strong scalar inverse non-linear transfers when σ T = 4: p T is found to depend much more on σ than on P r. The important result is that the theoretical scalar decay exponents of the extended CBC theory are valid whatever the Prandtl number is in high and low Reynolds and Péclet regimes: indeed, a broad range of Prandtl numbers (10 -5 ≤ P r ≤ 10 5 ) was investigated. The main finding of this study is that the Prandtl number only affects small scales of the scalar spectrum E T , but not the asymptotic time evolution of scalar one-point statistics: indeed, the large scales (k < k L ) depend only on the infrared slopes σ and σ T . In addition, it was shown numerically that neither the Prandtl number nor the initial position of L T (t = 0) affect the asymptotic dynamics of the passive scalar decay as soon as the Reynolds and Péclet numbers are large enough. In other words, the problem of the passive scalar decay in HIT has been simplified, reducing the relevant parameters from (Re, P r, σ, σ T , L T (t = 0)) to (P e, σ, σ T ) at large Reynolds numbers. In the continuity of this study, a law able to predict the relative position of the peaks of both kinetic and scalar spectra k L and k T was proposed. The consistency of the results over a wide range of Prandtl and Reynolds numbers shows that the decay of the passive scalar is driven only by the most energetic large scales of the initial spectra.

Chapter 2

Spectral Modelling of the Velocity Field in Homogeneous Turbulence

In the previous chapter, the transport of a passive scalar field in HIT was investigated using classical EDQNM. The scalar field is left aside for now, and we focus on the modelling of HAT for the velocity field, in order to later combine both anisotropic features and scalar mixing.

The anisotropic EDQNM modelling for the velocity field, introduced in [START_REF] Cambon | Spectral modelling of homogeneous non-isotropic turbulence[END_REF], and recently improved in [START_REF] Mons | A spectral model for homogeneous shear-driven anisotropic turbulence in terms of spherically-averaged descriptors[END_REF], is presented: details about calculations are gathered in Appendix C. The model consists in two steps: first, a classical EDQNM procedure is used to close the non-linear terms of the exact evolution equations of the spectral second-order moments. Secondly, anisotropy is modelled through spherically-averaged descriptors, following an expansion into spherical harmonics of the spectral Reynolds tensor Rij , further truncated at the second-order for the sake of simplicity.

The spectral formalism is presented here, along with the main evolution equations, the basics of EDQNM, and the detailed expressions of the spherically averaged production terms and nonlinear transfers. New theoretical considerations with respect to [START_REF] Mons | A spectral model for homogeneous shear-driven anisotropic turbulence in terms of spherically-averaged descriptors[END_REF] are proposed as well regarding the expansion of Rij . In what follows, non-rotating flows are considered. Direct applications of the anisotropic EDQNM modelling for the velocity field are proposed in Chapter 3.

Equations in physical space

In turbulence, the Reynolds decomposition is used to represent a field as the sum of a mean value and a fluctuating one. Thus, the velocity field v i = u i + U i , where u i is the fluctuating velocity, verifies the Navier-Stokes equation with a non-zero mean field

U i ∂ ∂t + u j ∂ ∂x j u i + U j ∂u i ∂x j + u j ∂U i ∂x j = - ∂p ∂x i + ν ∂ 2 u i ∂x j ∂x j , (2.1)
where p is the fluctuating pressure and ν the kinematic viscosity. For the sake of clarity, the time dependence was omitted. The two-point velocity correlation, or Reynolds stress tensor,

R ij is now introduced R ij (x, r, t) =< u i (x, t)u j (x + r, t) >, (2.2)
where r is the distance between two points and < . > an ensemble average. In homogeneous turbulence, R ij only depends on the separation vector r, and all spatial derivatives of secondorder moments (and higher) are zero. Hence, the evolution equation of the one-point correlation

R ij (0, t) is ∂R ij ∂t = P ij (t) + Π ij (t) -ij (t), (2.3)
where P ij is the production tensor

P ij (t) = - ∂U i ∂x k R kj (t) - ∂U j ∂x k R ki (t), (2.4)
which arises directly from velocity gradients, Π ij is the pressure strain tensor given by

Π ij (t) =< p(t) ∂u i ∂x j + ∂u j ∂x i >, (2.5)
which will be investigated in the next chapter, and ij is the dissipation tensor

ij (t) = 2ν < ∂u i ∂x k ∂u j ∂x k > .
(2.6) Surprisingly, the evolution equation of ij is not often investigated, as pointed out in [START_REF] Piquet | Turbulent Flows: Models and Physics[END_REF]. Its equation is consequently derived and simplified in (A.21) for homogeneous turbulence in Appendix A. The mean velocity gradients are represented by the space-uniform matrix A ij

A ij = dU i dx j .
(2.7)

Then, one can develop U i (x, t) = A ij (t)x j + u 0 i where u 0 i expresses the effect of a solid-body motion. The kinetic energy K(t) is defined as

K(t) = 1 2 < u i u i >= 1 2 R ii (r = 0, t), (2.8)
and its evolution can be obtained from (2.3)

∂K ∂t = P ii 2 - ii 2 .
(2.9)

The evolution equation of R ii (r = 0), the so-called von Kármán-Howarth equation (von [START_REF] Von Karman | On the statistical theory of isotropic turbulence[END_REF], is addressed in Appendix A. Consistently, the kinetic energy dissipation rate is then defined as = ii /2. Finally, to follow the time evolution of global anisotropy, one uses the anisotropy indicator

b ij (t) = R ij (t) 2K(t) - δ ij 3 , (2.10)
which is the normalized deviatoric part of R ij . It will be shown hereafter that b ij contains in fact two types of anisotropy.

Spectral equations and transfers

In this section, the exact evolution equation of the spectral Reynolds tensor Rij is derived, and a decomposition in terms of directional and polarization anisotropies is presented. There are no assumptions, except homogeneity, in this part: the modelling begins in the next section.

Craya equation for Rij

The counterpart of (2.1) in Fourier space is

∂ ∂t -A ln k l ∂ ∂k n + νk 2 ûi (k) + A ij ûj (k) + ik j u i u j (k) = -ik i p(k), (2.11)
where ûi is the Fourier transform of u i and k is the wavenumber: ûi , and most of the spectral quantities studied in this work, verifies the Hermitian symmetry, i.e. û * i (k) = ûi (-k), where () * is the complex conjugate. For the sake of clarity, the time-dependence was omitted; u m u n (k) is the convolution product that can be written as

u m u n (k) = k=p+q ûi (p)û m (q)d 3 p.
(2.12)

Thanks to the incompressibility condition ûi k i = 0 in Fourier space, the pressure term can be erased by projecting (2.11) on the plane perpendicular to p. This further yields the so-called Craya equation for Rij

∂ ∂t -A ln k l ∂ ∂k n + 2νk 2 Rij (k) + M in (k) Rnj (k) + M jn (k) Rni (k) = T NL ij (k), (2.13)
where Rij in Fourier space is given by (2.14) and where (2.15) where P imn (k) is the Kraichnan's operator, and P ij the projector

Rij (k, t)δ(k -p) =< û * i (p, t)û j (k, t) >= 1 2π 3 δ(k -p) e -ikprp R ij (r)d 3 r,
M ij (k) = (δ in -2α i α n )A nj with α i = k i /k.
T NL ij (k, t) = P imn (k) S njm (k, p, t)d 3 p + P jmn (k) S * nim (k, p, t)d 3 p,
2P imn (k) = k m P in (k) + k n P im (k), P ij (k) = δ ij -α i α j , (2.16)
with S ijn is the spectral three-point third-order correlation

S ijn (k, p, t)δ(k + p + q) = i < ûi (q, t)û j (k, t)û n (p, t) > .
(2.17)

The non-linear total transfer T NL ij can be written in a form that includes a conservative part with zero integral over k, and a "slow pressure" term that is responsible for a return to isotropy (RTI) mechanism and interactions between components

T NL ij (k, t) = τ ij (k, t) + τ * ji (k, t) Conservative transfer + W ij (k, t)
Return to isotropy

= P in τ nj (k, t) + P jn τ * ni (k, t), (2.18) where τ ij (k, t) = k n S ijn (k, p, t)d 3 p, (2.19) and W ij = -α i α n τ nj (k, t) -α j α n τ * ni (k, t), (2.20) with W ii = 0 (because k j τ ij = 0, but k i τ ij = 0). The conservative part is τ ij (k, t) + τ * ji (k, t)
, meaning that its integral over k is 0. However, integral over k for W ij is different from 0 since it is the spectral counterpart of the slow-part of the pressure-strain tensor Π ij .

Craya-Herring frame -E -Z decomposition

An optimal decomposition of Rij results from a trace-deviator splitting on the plane perpendicular to the wavevector k [START_REF] Cambon | Anisotropic developments for homogeneous shear flows[END_REF][START_REF] Cambon | Third-order statistics and the dynamics of strongly anisotropic turbulent flows[END_REF]. Without helicity, which is the topic of Chapter 8, the spectral Reynolds tensor can be written as follows

Rij (k, t) = E(k, t) 4πk 2 P ij (k) Isotropic + E(k, t) - E(k, t) 4πk 2 P ij (k) Directional anisotropy + Z(k, t)N i (k)N j (k)
E(k, t) = S k E(k, t)d 2 k. (2.23)
The energy density E is the distribution of energy along the wavevector k, and E -E/(4πk 2 ) reflects the directional anisotropy, i.e. the difference between energy in one direction and the spherical average. Then, Z represents polarization anisotropy and reflects the difference of anisotropy between two components of the spectral Reynolds tensor

Z(k, t) = Rij (k, t) 2 N * i (k)N * j (k).
(2.24)

Both E and Z must verify the realizability condition

|Z(k, t)| ≤ E(k, t), ∀(k, t). (2.25)
Without helicity, Rij is real and thus Rij (k) = Rij (-k). In the isotropic case, Z = 0 and E = E 0 = E/(4πk 2 ). Finally, N j are the helical modes [START_REF] Cambon | Spectral approach to non-isotropic turbulence subjected to rotation[END_REF][START_REF] Waleffe | The nature of triad interactions in homogeneous turbulence[END_REF], perpendicular to k j , and linked to the Craya-Herring frame (e (1) , e (2) , e (3) ) and to the fixed reference direction n, illustrated in Fig. 2.1, through

e (3) i = k i k , e (2)
i = ijl e

(3) j e

(1)

l , e

(1)

i = ijl k j n l |k × n| = ijl k j n l k ⊥ , (2.26) N j (k) = e (2) j (k) -ie (1) j (k).
(2.27)

In the Craya-Herring frame, the fluctuating spectral velocity ûi is contained in the plane (e (1) , e (2) ) and can be decomposed into toroidal and poloidal components according to ûi (k) = û(toro) (k)e

(1)

i (k) + û(polo) (k)e (2)
i (k).

(2.28)

Note that unlike other spectral classical quantities, the toroidal component does not verify the Hermitian symmetry since e

(1)

i (-k) = -e (1)
i (k), so that û(toro) * (k) = -û (toro) (-k). The toroidal and poloidal potentials E (toro) and E (polo) are simply linked to E and Z through

E (toro) (k)δ(k -p) =< û(toro) (k)û (toro) * (p) > E (polo) (k)δ(k -p) =< û(polo) (k)û (polo) * (p) > , E(k) = E (polo) (k) + E (toro) (k) Z(k) = E (polo) (k) -E (toro) (k).
(2.29) At this point, it is of interest to mention that the general decomposition (2.21) could also be applied in magnetohydrodynamics (MHD): indeed, the magnetic spectral tensor, defined as Bij (k)δ(k -p) =< b * i (p) bj (k) >, is also real, symmetric, and solenoidal. This might be interesting to apply this spectral anisotropic formalism to strong MHD turbulence where a mean magnetic field makes the conductive flow axisymmetric [START_REF] Boldyrev | Spectral scaling laws in magnetohydrodynamics turbulence simulations and in the solar wind[END_REF].

Generalized Lin equations

Let's now write the evolution equations for E and Z, called here the generalized Lin equations. One can rewrite (2.21) as

Rij (k, t) = E(k, t)P ij (k) + Z(k, t)N i (k)N j (k) . (2.30)
With this decomposition and the definition of E, the E-Lin equation reads (2.31) where A + ij is the symmetric part of A ij , and where the directional transfer T E , is

∂ ∂t -A ln k l ∂ ∂k n + 2νk 2 E -A + mn α n α m E + A + mn (ZN n N m ) = T E ,
T E (k, t) = T NL ii (k, t) 2 = 1 2 τ ii (k, t) + τ * ii (k, t) .
(2.32)

Similarly, the Z-Lin equation reads (2.33) where Ω CH , linked to both the frame and the velocity gradients, is computed in Appendix C, and where the polarization transfer T Z is

∂ ∂t -A ln k l ∂ ∂k n + 2νk 2 Z + 2iZΩ CH + A in N * i (EN * n + ZN n ) = T Z ,
T Z (k, t) = T NL ij (k, t) 2 N * i (k)N * j (k) = 1 2 τ ij (k, t) + τ * ji (k, t) N * i (k)N * j (k).
(2.34)

The effects of a mean rotation on these equations are considered in Appendix C.

The closure problem

In this section, the eddy-damped quasi-normal Markovian (EDQNM) approach is briefly recalled: since details on the procedure can be found in many references, only the main steps are presented here, and all the calculations specific to homogeneous anisotropic turbulence are detailed in the appendices. The EDQNM procedure is applied to close the Craya equation and to compute analytically the directional and polarization non-linear transfers T E and T Z . This step is then combined in the next section with a modelling of anisotropy, so that the k-dependence of the spectral second-order moments is transformed into a k-one.

The EDQNM approximation

Similarly to HIT, there is here a need to model the transfer term T NL ij . In order to do so, the evolution equation of the three-point third-order velocity correlation S ijn , defined in (2.17), is investigated. After some algebra, one gets

∂ ∂t + ν(k 2 + p 2 + q 2 ) -A lm k l ∂ ∂k m + p l ∂ ∂p m S ijn (k, p, t) + M im (q)S mjn (k, p, t) + M jm (k)S imn (k, p, t) + M nm (p)S ijm (k, p, t) = T ijn (k, p, t), (2.35)
where T ijn will be submitted to the EDQNM approximation. So far, previous equations were exact. From this point, the modelling begins. Fluctuating velocity probability distributions are assumed to be close to normal distributions. Hence, one can express T ijn as the sum of a quasi-normal part, and a modelled departure from normal laws part, namely

T ijn (k, p, t) = T QN ijn (k, p, t) -µ 1 (k, t) + µ 1 (p, t) + µ 1 (q, t) S ijn (k, p, t).
(2.36)

The quasi-normal part T QN ijn is expressed as a function of Rij according to

T QN ijl (k, p, t) = 2 P imn (q) Rmj (k, t) Rnl (p, t)
+ P jmn (k) Rml (p, t) Rni (q, t) + P lmn (p) Rmi (q, t) Rnj (k, t) .

(2.37)

The second part µ 1 (k, t) + µ 1 (p, t) + µ 1 (q, t) S ijn (k, p, t) takes into account and models the departure from a normal law: this is the eddy-damping contribution where (2.38) as defined in [START_REF] Pouquet | Evolution of high reynolds number two-dimensional turbulence[END_REF]; [START_REF] Orszag | Analytical theories of turbulence[END_REF]. The constant A 1 was originally chosen to be A 1 = 0.355 and we keep this value, which provides a Kolmogorov constant so that K 0 1.4. Choosing a different value, such as A 1 = 0.49, yields K 0 1.6 [START_REF] Bos | Reynolds number effect on the velocity increment skewness in isotropic turbulence[END_REF]. Now, the evolution equation of S ijn (2.35) with (2.36) can be solved. The resulting expression of S ijn is then simplified using the Markovianisation step: the characteristic time of the eddydamping is very small with respect to the turbulence characteristic time. This constitutes the classical EDQNM closure. In the homogeneous isotropic turbulence (HIT) framework, the expression of Rij is rather straightforward so that the analytical expression of (2.37) remains quite simple. In homogeneous anisotropic turbulence, the quasi-normal expression T QN ijl is further combined with the decomposition (2.21) and with the modelling of anisotropy for E and Z, so that the full anisotropic EDQNM modelling contains more complex features than in HIT. Indeed, in HIT, there is only one EDQNM formalism [START_REF] Lesieur | Turbulence in fluids[END_REF]. In HAT on the contrary, three versions can be found [START_REF] Sagaut | Homogeneous Turbulence Dynamics[END_REF], so called EDQNM-1,2,3. For nonrotating turbulence and in the presence of strong production mechanisms, the present EDQNM1 [START_REF] Cambon | Spectral modelling of homogeneous non-isotropic turbulence[END_REF], in terms of spherically-averaged spectra, is relevant enough. However, rotating turbulence, among other configurations such as stably stratified turbulence for example, requires the use of the EDQNM2 or EDQNM3, which is out of the scope of the present work, and models the effects of interacting inertial dispersive waves on the dynamics of the three-point third-order correlations.

µ 1 (k, t) = A 1 k 0 x 2 E(x, t)dx,
More precisely, the Green's tensor of the rapid distortion regime is used to solve the linear operator in the equation for the three-point third-order correlations [START_REF] Cambon | Spectral approach to non-isotropic turbulence subjected to rotation[END_REF]. As a consequence, in the EDQNM2 framework, the modelled non-linear transfers contain linear terms (with respect to the mean-velocity gradient) coming from the third-order correlations equations. This is at variance with EDQNM1, where these terms are discarded: a review of various models of turbulence can be found in [START_REF] Cambon | Linear and nonlinear models of anisotropic turbulence[END_REF]. Then, for a complete match in the asymptotic case of vanishing non-linearity between wave turbulence theory and EDQNM, the EDQNM3 was derived, which only slightly differs from EDQNM2 [START_REF] Cambon | Energy transfer in rotating turbulence[END_REF][START_REF] Cambon | Advances in wave turbulence: rapidly rotating flows[END_REF]: the separation of rapid and slow variables is refined in the quasi-normal procedure, so that E is treated as a slow variable, and the rapid phase of Z is accounted for. These different approaches were recently discussed in [START_REF] Cambon | Anisotropic triadic closures for sheardriven and buoyancy-driven turbulent flows[END_REF] where it is for instance shown how to move from EDQNM2 to EDQNM1 for stratified turbulence.

In what follows, since rotation is not considered, only the EDQNM1 approximation is used. Thanks to the whole EDQNM procedure, it is possible to express τ ij , coming from (2.18), as function of the second-order moments and of the characteristic time θ kpq containing the eddydamping term:

τ ij (k, t) = k l θ kpq T QN ijl (k, p, t)d 3 p, (2.39)
where θ kpq is the characteristic relaxation time of the third-order correlations

θ kpq = 1 -e -µ kpq t µ kpq , µ kpq = ν(k 2 + p 2 + q 2 ) + µ 1 (k, t) + µ 1 (p, t) + µ 1 (q, t).
(2.40)

Even for HAT and for consistency with previous studies, an isotropic eddy-damping term is kept, meaning in particular that each component of Rij has the same θ kpq . Also, this avoids the introduction of arbitrary constants at this level of the modelling.

It is worth noting that within the EDQNM1 framework, the characteristic time θ kpq can be tuned to take into account different effects. For unstably stratified turbulence, addressed later in Chapter 7, the stratification frequency N (t) was added to the viscous and inertial terms to match better with DNS in Burlot et al. (2015a):

θ (USHT) kpq = θ kpq +a 1 N (t)
, where a 1 is a constant, of order 0.25, which depends on the flow. In isotropic magnetohydrodynamics turbulence, a magnetic correction was added in [START_REF] Pouquet | Strong mhd helical turbulence and the nonlinear dynamo effect[END_REF] to take into account the propagation of Alfvèn waves: θ

(MHD) kpq = θ kpq + 2/3k k 0 E B (x, t)dx,
where E B is the magnetic energy spectrum.

Directional and Polarization transfers T E and T Z

The aim of the EDQNM1 approximation is to provide an explicit formula for both the directional and polarization transfers T E and T Z given in (2.32) and (2.34). For this purpose, a more convenient frame (β,γ,α) must be used, attached to the planed formed by the triad k+p+q = 0, where γ is perpendicular to this plane. From now, the following notations are used: and refer to quantities expressed in p and q respectively. Useful vectors and angles are gathered in Fig. 2.2. a, b and c are the angles formed by p and q, q and k, and k and p. Finally, x = cos a, y = cos b and z = cos c. The new frame (β,γ,α) is obtained from Craya frame (e (1) , e (2) , e (3) = α) by rotations of angles λ, λ and λ around k, p and q. All the details of the computation of τ ij from (2.18) are given in Appendix C. The final results for the polarization and the directional transfers are

T E (k, t) = 2 θ kpq kp (E + X ) (xy + z 3 )(E -E) -z(1 -z 2 )( X -X) + X (1 -z 2 )(x X -y X ) d 3 p,
(2.41)

T Z (k, t) = 2 θ kpq kpe -2iλ (E + X ) (xy + z 3 )( X -X) -z(1 -z 2 )(E -E) + i(y 2 -z 2 ) X + i X (1 -z 2 ) x(E + X) -iy X d 3 p, (2.42) with E(k, t) = E, E(p, t) = E , E(q, t) = E , X = Z(k, t)e 2iλ
, X = Z(p, t)e 2iλ and X = Z(q, t)e 2iλ . The expressions of T E and T Z can also be found in [START_REF] Cambon | Energy transfer in rotating turbulence[END_REF]; [START_REF] Mons | A spectral model for homogeneous shear-driven anisotropic turbulence in terms of spherically-averaged descriptors[END_REF].

Figure 2.2: The triad k + p + q = 0 and useful vectors and angles

Spherically-averaged equations

The generalized E-Lin and Z-Lin equations, along with explicit directional and polarization transfers T E and T Z , can be solved. In order to considerably reduce computational time, spherically-averaged descriptors are used, which depend only on the modulus k of the wavevector k. The procedure is to integrate analytically the generalized Lin equations over a sphere of radius k. To do so, the expansion of Rij into spherical harmonics is truncated at the secondorder.

Spherically-averaged descriptors

The decomposition of Rij provided by [START_REF] Cambon | Anisotropic developments for homogeneous shear flows[END_REF] is now used. As seen in (2.21), one has

Rij (k) = R(iso) ij (k) + R(dir) ij (k) + R(pol) ij (k) = E 0 (k) + E (dir) (k) P ij (k) + Z(k)N i (k)N j (k) , (2.43
) where E 0 = E/(4πk 2 ) and E (dir) = E -E 0 . The complete expansion of E and Z into spherical harmonics at the second-order was done in [START_REF] Mons | A spectral model for homogeneous shear-driven anisotropic turbulence in terms of spherically-averaged descriptors[END_REF], and all the details and technical steps are gathered in Appendix C. Here, for theoretical considerations, the fourth-order is briefly presented, even though only the second-order will be used in the numerical simulations. After some algebra, one gets the fourth-order expansion of Rij into spherical harmonics

E(k, t) = E 0 1 -15H (dir) ij (k, t)α i α j + 945 12 H (dir) ijpq (k, t)α i α j α p α q , Z(k, t) = 1 2 E 0 5H (pol) ij (k, t) + 21 2 H (pol) ijpq (k, t)α p α q N * i (k)N * j (k).
(2.44)

(2.45)

The possibility of a third-order contribution in the expansion of Z is only discussed in the next chapter in section 3.4.3, since the results are not satisfactory for the time being. At this point, some words need to be said about the new fourth-order tensors. One can remark that at the fourth-order, the polarization part H (pol)

ijpq is contracted with both normalized wavevectors α p α q and helical modes N * i N * j , unlike the directional part

H (dir)
ijpq which is only contracted with α i α j α p α q , and is therefore fully symmetric in its indices, as reported in [START_REF] Rubinstein | Scalar and tensor spherical harmonics expansion of the velocity correlation in homogeneous anisotropic turbulence[END_REF]. The latter property of full-symmetry is thus a priori not verified by H (pol)

ijpq , but we nonetheless make this assumption. Indeed, the analytical calculations are already very complex and lengthy, and this approximation renders the developments a bit easier. In addition, we assume that both H (dir) ijpq and H (pol) ijpq are trace-free, meaning that any contraction of two indices yields zero. For H (dir) ijpq , this is verified according to [START_REF] Rubinstein | Scalar and tensor spherical harmonics expansion of the velocity correlation in homogeneous anisotropic turbulence[END_REF], but for H (pol) ijpq this is a supplementary assumption, again for the sake of simplicity. All these spectral anisotropy descriptors are defined as

2E(k, t)H (dir) ij (k, t) = S k R(dir) ij (k, t)d 2 k, 2E(k, t)H (pol) ij (k, t) = S k R(pol) ij (k, t)d 2 k, (2.46) 2E(k, t)H (dir) ijpq (k, t) = S k E(k, t) P ijpq (k) d 2 k, (2.47) 2E(k, t)H (pol) ijpq (k, t) = S k Z(k, t)N ijpq (k) d 2 k, (2.48)
where S k is the sphere of radius k, and where P ijpq and N ijpq are generalized operators

P ijpq = α i α j α p α q - 1 7 (δ ij α p α q + 5 perm.) + 1 35 (δ ij δ pq + δ ip δ jq + δ iq δ jp ), (2.49)
N ijpq = (N i N j α p α q + N p N q α i α j + 4 perm.) - 1 7 (δ ij N p N q + δ pq N i N j + 4 perm.).
(2.50)

Additional details about the fourth-order expansion, such as the evolution equations of H (dir)

ijpq , and H (pol) ijpq , their linear and non-linear transfers, can be found in Appendix C.

Obviously, the truncation of the expansion into spherical harmonics of the exact decomposition (2.21) provokes a loss of angular information about the anisotropy of the flow. Part of this information is nevertheless restored thanks to the spherically averageddescriptors H () ij and H () ijpq . However, it is complicated to quantify what is lost because of this truncation. It will be shown in Chapter 3 that a consequence is that the exponential growth rate of the kinetic energy in shear flows is too large compared to values obtained in DNS and experiments, and that taking into account the fourth-order expansion tends to reduce this exponential growth rate. On the contrary, the second-order expansion seems to be sufficient for quantitative comparisons in multiple configurations involving the transport of a passive (Chapter 5) and active (Chapter 7) scalar field.

From now, in the context of moderate anisotropy, only the second order expansion of Rij is kept. In this framework, all quadratic contributions H () ij H () ij are discarded: nevertheless, these second-order contributions in anisotropy have been computed in Appendix C and it is shown that they are negligible, at least in shear flows. Note that the same kind of second-order truncations will be performed for the modelling of a scalar field in Chapter 4.

The indicator of anisotropy defined in (2.10) can be expanded into b

ij = b (dir) ij + b (pol) ij thanks to the previous decomposition b (dir) ij (t) = 1 K(t) ∞ 0 E(k, t)H (dir) ij (k, t)dk, b (pol) ij (t) = 1 K(t) ∞ 0 E(k, t)H (pol) ij (k, t)dk.
(2.51) Finally, a limit can be derived from the realizability condition (2.25)

max i (L i ) ≤ 1 15 , (2.52)
where L i are eigenvalues of

H (dir) ij
. This condition, obviously valid when only the second-order expansion is considered, was shown to hold true in multiple configurations in [START_REF] Mons | A spectral model for homogeneous shear-driven anisotropic turbulence in terms of spherically-averaged descriptors[END_REF].

Spherically-averaged final Lin equations

In this part, the final spherically-averaged Lin equations of the main spectra, namely E, EH , are derived: details of the calculations are given in Appendix C. It is recalled that only the second-order expansion of Rij is considered, and that quadratic anisotropic contributions in the non-linear transfers are discarded. The following compact equations were derived in [START_REF] Mons | A spectral model for homogeneous shear-driven anisotropic turbulence in terms of spherically-averaged descriptors[END_REF], and read

∂ ∂t + 2νk 2 E(k, t) = S L(iso) (k, t) + S NL(iso) (k, t), ∂ ∂t + 2νk 2 E(k, t)H (dir) ij (k, t) = S L(dir) ij (k, t) + S NL(dir) ij (k, t), ∂ ∂t + 2νk 2 E(k, t)H (pol) ij (k, t) = S L(pol) ij (k, t) + S NL(pol) ij (k, t),
(2.53)

(2.54)

(2.55)

where S NL(iso) (k, t) is the classical non-linear spherically-averaged isotropic transfer term

S NL(iso) (k, t) = S k T E (k, t)d 2 k (2.56) = 16π 2 ∆ k θ kpq k 2 p 2 q(xy + z 3 )E 0 (E 0 -E 0 )dpdq, (2.57)
with ∆ k the domain where k, p and q are the lengths of the sides of the triangle formed by the triad. The non-linear spherically-averaged directional transfer S

NL(dir) ij (k, t) is S NL(dir) ij (k, t) = 1 2 S k T E (k, t)P ij (k) d 2 k - δ ij 3 S NL(iso) (k, t) (2.58) = 4π 2 ∆ k θ kpq k 2 p 2 qE 0 (y 2 -1)(xy + z 3 )(E 0 -E 0 )H (pol) ij + z(1 -z 2 ) 2 E 0 H (pol) ij dpdq + 8π 2 ∆ k θ kpq k 2 p 2 q(xy + z 3 )E 0 (3y 2 -1)(E 0 -E 0 )H (dir) ij + (3z 2 -1)E 0 H (dir) ij -2E 0 H (dir) ij dpdq.
(2.59)

And S NL(pol) ij (k, t) is the non-linear spherically-averaged polarization transfer

S NL(pol) ij (k, t) = 1 2 S k T Z (k, t)N i (k)N j (k) d 2 k (2.60) = 4π 2 ∆ k θ kpq k 2 p 2 qE 0 (xy + z 3 ) (1 + z 2 )E 0 H (pol) ij -4E 0 H (pol) ij + z(z 2 -1)(1 + y 2 )(E 0 -E 0 )H (pol) ij + 2z(z 2 -y 2 )E 0 H (pol) ij + 2xy(z 2 -1)E 0 H (pol) ij dpdq + 24π 2 ∆ k θ kpq k 2 p 2 qz(z 2 -1)E 0 (y 2 -1)(E 0 -E 0 )H (dir) ij + (z 2 -1)E 0 H (dir) ij dpdq. (2.61)
For the linear production terms, A + ij and A - ij are respectively the symmetric and antisymmetric parts of A ij . First, S L(iso) (k, t) is the linear spherically-averaged isotropic transfer

S L(iso) (k, t) = 1 2 S k A ln k l ∂ Rii ∂k n -2M in Rni d 2 k (2.62) = -2A + lm ∂ ∂k (kEH (dir) lm ) + E(H (dir) lm + H (pol) lm ) . (2.63) Then, S L(dir) ij (k, t) is the linear spherically-averaged directional transfer S L(dir) ij (k, t) = 1 4 S k A ln k l ∂ Rmm ∂k n -2M mn (k) Rnm (k) P ij (k)d 2 k - δ ij 3 S L(iso) (k, t) (2.64) = 2 15 A + ij E - 2 7 E A + lj H (pol) il + A + li H (pol) jl - 2 3 A + ln δ ij H (pol) ln - 1 15 A + ij ∂(kE) ∂k + 2 7 A + il ∂ ∂k (kEH (dir) jl ) + A + jl ∂ ∂k (kEH (dir) il ) - 2 3 A + lm δ ij ∂ ∂k (kEH (dir) lm ) - 1 7 E A + jl H (dir) il + A + il H (dir) jl - 2 3 A + lm H (dir) lm δ ij + E A - jn H (dir) ni + A - in H (dir) nj . (2.65) Finally, S L(pol) ij (k, t) is the linear spherically-averaged polarization transfer S L(pol) ij (k, t) = 1 4 S k A ln k l ∂ Rrs ∂k n -2M rn (k) Rns (k) N * r (k)N * s (k)N i (k)N j (k) d 2 k (2.66) = - 2 5 A + ij E - 12 7 E A + jl H (dir) il + A + il H (dir) jl - 2 3 A + lm H (dir) lm δ ij - 2 7 A + il ∂ ∂k (kEH (pol) lj ) + A + jl ∂ ∂k (kEH (pol) li ) - 2 3 δ ij A + lm ∂ ∂k (kEH (pol) lm ) + 1 7 E A + lj H (pol) il + A + li H (pol) jl - 2 3 A + lm H (pol) lm δ ij - 1 3 E A - jl H (pol) li + A - il H (pol) lj .
(2.67)

Return to isotropy -Spectral tensor

In order to investigate more precisely the return to isotropy (RTI) mechanism, a specific transfer term T (RTI) was introduced in [START_REF] Mons | A spectral model for homogeneous shear-driven anisotropic turbulence in terms of spherically-averaged descriptors[END_REF]: the slow-pressure terms (contained in the pressure-strain tensor Π ij ) are at the origin of the return to isotropy and drive this mechanism.

Thus, the non-conservative transfer term W ij can be linked to the RTI mechanism as follows

W ij (k, t) = -T (RTI) (k, t)[α i N j (k) + α j N i (k)] , T (RTI) (k, t) = α i τ ij (k, t)N * j (k), (2.68) so that T (RTI) (k, t) =2 θ kpq e -iλ p(xy + z) 1 -z 2 (E + X ) (E + X)(zk -qx) -k(z(E + X ) -i X ) d 3 p, (2.69)
with details given at the end of Appendix C. Regarding the spherically-averaged RTI transfer, defined as follows

S (RTI) ij (k, t) = - S k T (RTI) (k, t)[α i N j (k) + α j N i (k)] d 2 k, (2.70) similar calculations yield S (RTI) ij (k, t) = 16π 2 ∆ k θ kpq k 2 p 2 q(x + yz)E 0 -y(z 2 -x 2 )E 0 (6H (dir) ij + H (pol) ij ) + E 0 y(z 2 -y 2 )(6H (dir) ij + H (pol) ij ) -(xz + y)H (pol) ij dpdq. (2.71)
Now that all transfer terms have been defined, and that the formalism has been presented, it is convenient to introduce the spherically-averaged spectral tensor

φ ij (k, t) = S k Rij (k, t)d 2 k = 2E(k, t) δ ij 3 + H (dir) ij (k, t) + H (pol) ij (k, t) .
(2.72)

Because of the spherical-average, even if the fourth-order expansion was considered in Rij , the equation would be the same since the H () ijpq contributions vanish. The evolution equation of φ ij is then

∂ ∂t + 2νk 2 φ ij (k, t) = S NL(tot) ij + S L(tot) ij = S NL ij + S (RTI) ij + S L(tot) ij .
(2.73)

The total non-linear spherically-averaged transfer and can be expressed as

S NL(tot) ij (k, t) = S k T NL ij (k, t)d 2 k = S NL ij (k, t) + S (RTI) ij (k, t) (2.74) = 2 δ ij 3 S NL(iso) (k, t) + S NL(dir) ij (k, t) + S NL(pol) ij (k, t) , (2.75) with ∞ 0 S NL ij (k, t)dk = 0.
The total linear spherically-averaged transfer, which depends linearly on the mean-velocity gradient matrix, is The study of homogeneous anisotropic turbulence is of great interest for a deeper understanding of the different mechanisms that occur in anisotropic turbulent flows. The specific case of homogeneous shear flows has been particularly investigated since it exhibits different fundamental physical processes: anisotropic production of turbulent kinetic energy, interaction between linear and non-linear mechanisms, return to isotropy process... Since Kolmogorov (1941b), it is known that small scales should return to an isotropic state, meaning that even with a mean shear applied on large scales that strongly modifies their properties, there is a return to isotropy (RTI) mechanism of the small scales. This RTI process and the modelling of the so-called slowpart of the pressure-strain tensor Π (s) ij is a challenging issue. The total pressure strain tensor Π ij =< p(∂ j u i + ∂ i u j ) >, where p and u i are the fluctuating pressure and velocity, intervenes in the evolution equation (2.3) of the Reynolds stress tensor R ij =< u i u j >. Several models were proposed for the slow part Π (s) ij and have been improved in the past decades. The most popular are the LRR model [START_REF] Launder | Progress in the development of a reynolds-stress turbulence closure[END_REF], the one of [START_REF] Shih | Modeling of Pressure Correlation Terms in Reynolds Stress and Scalar Flux Equations[END_REF], the SSG model [START_REF] Sarkar | A simple nonlinear model for the return to isotropy in turbulence[END_REF], and an improved version of the SSG model by [START_REF] Warrior | An improved model for the return to isotropy of homogeneous turbulence[END_REF]. These models rely on a Taylor series expansion around the isotropic state of the dimensionless tensor Π (s) ij / , where is the kinetic energy dissipation rate. The small parameter in this expansion is the anisotropy tensor b ij , defined in (2.10). These models yield good results in the early times of the flow dynamics. Indeed the arbitrary parameters introduced in these models are tuned in order to fit experimental data. However, a weakness of these models is their lack of universality: values of their parameters strongly depend on the choice of the experimental data, often obtained at moderate Reynolds numbers. Consequently, the analysis of long-time behaviour of b ij in the RTI process remains an interesting and open question. In the asymptotic case of high Reynolds numbers, this mechanism should be universal and thus should not require any adjustable constants.

S L(tot) ij (k, t) = S k T L ij (k, t)d 2 k (2.76) = 2 δ ij 3 S L(iso) (k, t) + S L(dir) ij (k, t) + S L(pol) ij (k, t) . ( 2
In addition to the modelling of Π (s) ij and the RTI mechanism, a fundamental feature to investigate is the influence of anisotropy on the decay of integrated quantities such as the kinetic energy K(t). It follows, from the pioneering work of Corrsin (1951a); [START_REF] Comte-Bellot | The use of a contraction to improve the isotropy of a grid generated turbulence[END_REF] (CBC) relying on dimensional analysis, and self-preservation analysis George (1992a), that the kinetic energy decays in power laws in the isotropic framework, K(t) ∼ t α . This has been recovered recently for very large Reynolds numbers with a classical EDQNM closure for HIT [START_REF] Meldi | On non-self-similar regimes in homogeneous isotropic turbulence decay[END_REF], 2013a) and Chapter 1).

To quantify the impact of anisotropy on the decay regime, comparisons are usually made with HIT. Moreover, the case of axisymmetric contraction (or expansion), which is representative of grid turbulence, has already received some attention. Notably, it has been shown [START_REF] Chasnov | Similarity states of passive scalar transport in buoyancy-generated turbulence[END_REF][START_REF] Davidson | On freely decaying, anisotropic, axisymmetric saffman turbulence[END_REF]Mons et al., 2014b) for this configuration that an initial axisymmetry does not modify the decay exponent in the asymptotic regime, i.e. for Saffman turbulence K(t) ∼ t -6/5 . An original configuration to explore, which has not been investigated yet in direct numerical simulations (DNS) nor in experiments, is the case of a mean shear which is suddenly released. This case could be physically interpreted as a volume of fluid that experiences an intense shear, and which is then convected in an almost shearless region. Characteristic time scales in these two different phases are of great importance, as shown later on. Is the decay of kinetic energy modified in such a homogeneous shear-released turbulence (HSRT), with respect to HIT? This fundamental question is of theoretical interest, since HSRT, unlike axisymmetric turbulence, creates a purely anisotropic correlation R 13 =< u 1 u 3 >. The understanding of such a correlation dynamics could provide relevant information on anisotropy.

Another relevant case to focus on is the homogeneous shear turbulence (HST) where the mean velocity gradient is maintained throughout the evolution of the flow. In this configuration, partially representative of atmospheric flows, kinetic energy is continuously produced by the mean shear. This framework is commonly studied in experiments [START_REF] Tavoularis | Experiments in nearly homogenous turbulent shear flow with a uniform mean temperature gradient. part 1[END_REF]Tavoularis & Karnik, 1989;[START_REF] De Souza | The structure of highly sheared turbulence[END_REF] and in DNS [START_REF] Pumir | Persistent small scale anisotropy in homogeneous shear flows[END_REF][START_REF] Pumir | Turbulence in homogeneous shear flows[END_REF][START_REF] Gualtieri | Scaling laws and intermittency in homogeneous shear flow[END_REF][START_REF] Brethouwer | The effect of rotation on rapidly sheared homogeneous turbulence and passive scalar transport. linear theory and direct numerical simulation[END_REF]. Notably, it is found that the small scales of the velocity field second-order moments return to isotropy, and that kinetic energy grows exponentially [START_REF] Tavoularis | Asymptotic laws for transversely homogeneous turbulent shear flows[END_REF]George, 1992a;[START_REF] Sagaut | Homogeneous Turbulence Dynamics[END_REF] when the anisotropy indicators b ij have reached an asymptotic state. Despite all these works, some discrepancies still remain, whose origin is not completely understood. For instance, in the DNS of [START_REF] Brethouwer | The effect of rotation on rapidly sheared homogeneous turbulence and passive scalar transport. linear theory and direct numerical simulation[END_REF], the anisotropy tensor b ij does not reach an asymptotic state as required by theory [START_REF] Sagaut | Homogeneous Turbulence Dynamics[END_REF] when the exponential growth rate γ of the kinetic energy is evaluated. Indeed, the dimensionless time St, or accumulated anisotropy, where S is the shear rate, is not high enough (St 12 only). The issue is similar in most of the DNS and experiments, thus leading to a large dispersion of the growth rates (from γ = 0.07 to 0.33, see Table 3.1). Moreover, especially in experimental works, the question of homogeneity can be raised: inhomogeneous flows are not studied in the present chapter, but their influence on the growth rate of the kinetic energy is an open question and deserves further investigations. In addition, the mean velocity gradient strongly varies near the boundaries, possibly inducing a different phenomenology in the growth of the kinetic energy. Finally, in DNS, the finite size of the box could alter the kinetic energy growth rate in a manner difficult to quantify. These aspects could explain the current dispersion of growth rates.

In both HST and HSRT, the mean-velocity gradient matrix reads

A ij = dU i dx j , A ij = -Sδ 1i δ 3j , (3.1)
with S expressed in units of τ -1 0 , where τ 0 is the eddy turn-over time K(0)/ (0), so that S = Sτ -1 0 , where S is the dimensionless mean velocity gradient. The evolution equation of the kinetic energy in homogeneous shear turbulence reads

dK dt = SR 13 (t) -(t) = 2SK(t)b 13 (t) -(t), (3.2)
where K and its dissipation rate are linked to the kinetic energy spectrum E through

K(t) = ∞ 0 E(k, t)dk, (t) = 2ν ∞ 0 k 2 E(k, t)dk. (3.3)
The chapter is structured as follows. The case of homogeneous shear-released turbulence (HSRT) is firstly investigated: the return to isotropy, the modelling of the slow part of the pressure strain tensor and decay laws are addressed. Then, the emphasis is put on sustained shear flows (HST). In this part, the results of the present anisotropic modelling are discussed and put into perspective with review of various DNS and experiments. Finally, the most important points developed in this chapter are recalled in the concluding section, and some considerations about the fourth-order expansion are discussed.

Homogeneous Shear-Released Turbulence (HSRT)

In this section, the emphasis is put on HSRT: this is an original configuration where the shear S is non zero only in the early times. During this phase, linear transfers defined in (2.76) increase the anisotropy and produce kinetic energy. Then, after the release of the velocity gradients, the velocity field freely decays and there is a RTI mechanism: non-linear transfers defined in (2.74) tend to isotropize small scales, decreasing as a consequence both K(t) and R 13 (t).

This framework, firstly presented in [START_REF] Mons | A spectral model for homogeneous shear-driven anisotropic turbulence in terms of spherically-averaged descriptors[END_REF], has never been investigated in DNS nor in experiments. This section is divided into five parts. Firstly, rapid distortion theory (RDT) is used to validate the model at short times, when the linear effects are dominant. Then, classical scalings for the different spectra involved in shear-driven turbulence are addressed. Afterwards, the RTI mechanism, once the shear is released, is investigated, with a particular attention on the difference between Saffman and Batchelor turbulence. Then, a model is proposed for the slow-part of the pressure-strain tensor in the RTI phase. Finally, effects of infrared exponent σ and shear rate S, along with the case of anisotropy at low Reynolds numbers, are discussed.

Validation of HSRT with Rapid Distortion Theory

In this part, the anisotropic EDQNM modelling is assessed by comparisons with RDT as done in [START_REF] Mons | A spectral model for homogeneous shear-driven anisotropic turbulence in terms of spherically-averaged descriptors[END_REF]. The main calculations coming from RDT are given in Appendix D. In RDT, non-linear terms are discarded: this theory is valid for short times only, when linear processes dominate the flow. Moreover, at short times (for large Reynolds numbers), viscous effects are also negligible with respect to inertial ones. Two different simulations are presented in Fig. 3.1: one with an initial isotropic kinetic spectrum E(k, t) in which linear transfers produce energy and anisotropy. And a second one where these linear transfers are set to zero and the initial spectral tensor φ ij is analytically determined thanks to RDT. It is clear that the two different initial conditions collapse into the same behaviour when the shear is released, for both σ = 2 and σ = 4. This validates our spectral transfers. Another point of interest is that with RDT, φ 13 displays an inertial k -5/3 range instead of an inertial anisotropic k -7/3 range, which proves that non-linear transfers are responsible for the anisotropic range. 3.1.2 Kinetic energy spectrum E(k, t) and spectral tensor φ ij (k, t)

In Fig. 3.2a, diagonal components of the spectral tensor φ ij (k, t) (only φ 11 is shown) display a k -5/3 scaling in the inertial range, as in the purely isotropic case, from the integral wavenumber k L (t) 1/L(t) to the Kolmogorov wavenumber k η , where L(t) is the kinetic integral scale.

The cross-tensor φ 13 (k, t) is also presented, and exhibits a k -7/3 scaling in the inertial range. This scaling comes from E(k, t)H () ij (k, t) spectra exclusively since Eδ 13 = 0. The spectral scaling of the cross-tensor can be found by dimensional analysis, assuming that φ 13 (k, t) depends on the k, , and linearly on the shear rate S φ 13 (k, t) ∼ S 1/3 k -7/3 .

(3.4)

This result was firstly found by [START_REF] Lumley | Similarity and the turbulent energy spectrum[END_REF] and derived in a different way by [START_REF] Ishihara | Anisotropic velocity correlation spectrum at small scales in a homogeneous turbulent shear flow[END_REF]. The k -7/3 slope has also been obtained in DNS [START_REF] Shen | The anisotropy of the small scale structure in high reynolds number (r λ ∼ 1000) turbulent shear flow[END_REF][START_REF] Ishihara | Anisotropic velocity correlation spectrum at small scales in a homogeneous turbulent shear flow[END_REF][START_REF] Sukheswalla | Simulation of homogeneous turbulent shear flows at higher reynolds numbers: numerical challenges and a remedy[END_REF]. The total non-linear and RTI transfers are now investigated in Fig. 3.2b, and more precisely their corresponding flux, computed according to

k/k L φ ij (k, t) φ 11 (k, t = 0) φ 11 (k, t) φ 13 (k, t) k L k η k -7/3 k -5/3 (a)
Π NL() ij (k, t) = - k 0 S NL() ij (u, t)du. (3.5)
The total non-linear flux is not conservative (meaning that Π ij (k = ∞) = 0) because of the RTI mechanism, originating from "slow pressure" terms. Nevertheless, when the RTI flux is subtracted, a conservative non-linear flux is recovered, as illustrated in Fig. 3.2b. Such a test case is an accurate validation of the previous analytical calculations of Chapter 2 for the non-linear transfers.

Anisotropy descriptors

b ij (t) and H () ij (k, t)
In this part, the emphasis is put on the RTI mechanism in HSRT. The case of Saffman turbulence has been presented in [START_REF] Mons | A spectral model for homogeneous shear-driven anisotropic turbulence in terms of spherically-averaged descriptors[END_REF], and is compared here with Batchelor turbulence. Conclusions with regard to the permanence of large eddies are drawn.

The shear is maintained during a small number of turn-over times and is then released. For high Reynolds numbers, it is well known that the anisotropy tensor b ij defined in (2.10) reaches an asymptotic anisotropic state in the RTI process. This has already been observed in DNS [START_REF] Sarkar | A simple nonlinear model for the return to isotropy in turbulence[END_REF] and is recovered here in Fig. 3.3a for Saffman turbulence. The initial spectrum E(k, t = 0) being isotropic, one has b ij (t = 0) = 0. Because of linear shear effects, a strong departure from the isotropic state is observed: the |b ij | increase up to the shear release, and then reach constant values. The interesting result here at high Reynolds numbers is that the final state of the anisotropy tensor b ∞ ij , reached from t = 10 3 τ 0 , is not zero. This means that there is still some anisotropy left in the flow. The anisotropy tensor b ij (t) being an average in space, it hides where the remaining anisotropy is. Therefore, spectral anisotropy descriptors H () ij (k, t) are used to provide information on the localization of anisotropy in wavenumber space: Fig. 3.3b reveals that H () ij = 0 only at small scales. This shows that there is a complete RTI of small scales in Saffman turbulence, whereas large scales keep their anisotropy. This behaviour is in agreement with Kolmogorov (1941b) local isotropy theory and with results of DNS [START_REF] Sarkar | A simple nonlinear model for the return to isotropy in turbulence[END_REF] as well. Batchelor turbulence is now addressed. Simulations show that the b ij continuously return to zero, as illustrated in Fig. 3.4a: this means that anisotropy globally decreases over time, unlike Saffman turbulence. Spatial information about the localization of anisotropy is available in Fig. 3.4b: the spectral anisotropy descriptor H (pol) 13 (k, t) reveals that large scales anisotropy decreases with time for Batchelor turbulence, whereas it remains constant for Saffman turbulence. Other components of H (pol) ij and H (dir) ij behave similarly. Therefore, because of this continuous loss of anisotropy in Batchelor turbulence, a complete RTI of all scales is theoretically possible, even though physically unreachable. Indeed, it would require an infinite Reynolds number in order to stay in the high Reynolds numbers regime: with a larger Reynolds number comes a greater quantity of anisotropy to evacuate.

The large scales loss of anisotropy in Batchelor HSRT is due to the classical backscatter of energy that already occurs in Batchelor HIT [START_REF] Eyink | Free decay of turbulence and breakdown of self-similarity[END_REF][START_REF] Lesieur | 3d isotropic turbulence at very high reynolds numbers: Edqnm study[END_REF][START_REF] Meldi | On non-self-similar regimes in homogeneous isotropic turbulence decay[END_REF]. Indeed, strong inverse non-linear transfers, from small scales to large ones, tend to isotropize the large scales, which causes the anisotropy to decrease. This is consistent with non-linear mechanisms being responsible for the return-to-isotropy process. Moreover, these strong inverse non-linear transfers result into the breakdown of the PLE in Batchelor turbulence.

Modelling of the pressure-strain tensor Π (s) ij

The modelling of the pressure-strain tensor Π ij , which directly intervenes in the evolution equation (2.3) of R ij , is a challenging topic. Indeed, the velocity-pressure correlation is complex, and its prediction is of particular interest for the development of RANS models. This term is commonly divided into two parts: a slow one Π (s) ij , responsible for the redistribution of energy between components, and a rapid one Π (r) ij , linked to the linear effects of mean-velocity gradients. The latter part is rigorously zero when the shear is released. This is why the emphasis is put on the slow-part Π (s) ij which is at the origin of the RTI mechanism highlighted in the previous part.

The slow part of the pressure-strain tensor can be written [START_REF] Sarkar | A simple nonlinear model for the return to isotropy in turbulence[END_REF])

Π (s) ij (t) = -(t) C (1) RTI b ij (t) + C (2) RTI b il (t)b lj (t) - 1 3 b kl (t)b kl (t)δ ij . ( 3
db ij dt = - 2 3 A + ij -A ik b kj -A jk b ki + 2A kl b lk δ ij 3 + b ij + Π ij 2K + K b ij , (3.7)
which can be found as well in [START_REF] Sarkar | A simple nonlinear model for the return to isotropy in turbulence[END_REF]; [START_REF] Warrior | An improved model for the return to isotropy of homogeneous turbulence[END_REF]. The dissipation tensor ij was assumed to be isotropic, i.e. 3 ij = 2 δ ij . This is a reasonable assumption for moderately anisotropic flows: indeed, simulations at high Reynolds numbers show that the nondiagonal components of ij are negligible with respect to diagonal ones. Moreover, when the shear is released, A ij = 0 and only the slow part of the velocity-pressure correlation remains,

Π ij = Π (s) ij .
In addition, it has been shown in the previous part that the b ij reach an asymptotic state when the PLE is verified: this implies that db ij /dt = 0. Thus, the evolution equation

(3.7) of b ij becomes Π (s) ij (t) = -2 (t)b ij (t) = -C RTI (t)b ij (t).
(3.8)

Consequently, for large times, large Reynolds numbers, an initial moderate mean-velocity gradient which is then released, and when the PLE is verified, this model predicts C RTI = 2 as an -universal -constant of the RTI mechanism. C RTI will be shown hereafter to be independent of both the large scales initial conditions and of the mean flow gradient A ij . The case of pure rotation is not considered here, and this will be briefly justified later.

Existing models [START_REF] Shih | Modeling of Pressure Correlation Terms in Reynolds Stress and Scalar Flux Equations[END_REF][START_REF] Sarkar | A simple nonlinear model for the return to isotropy in turbulence[END_REF][START_REF] Warrior | An improved model for the return to isotropy of homogeneous turbulence[END_REF] are intrinsically different since there are designed to capture the short time dynamics of the flow when the mean-velocity gradients are active, i.e. when A ij = 0 and Π (r) ij = 0, whereas the emphasis in this work is put on a freely decaying turbulence initially submitted to meanvelocity gradients, at large times and Reynolds numbers. For instance, in the strongly nonlinear model of [START_REF] Shih | Modeling of Pressure Correlation Terms in Reynolds Stress and Scalar Flux Equations[END_REF] S = 10 -3 τ -1 0 , () 13 , shear, σ = 1 S = 0.1τ -1 0 , () 11 , dist., σ = 2 S = 10 -2 τ -1 0 , () 33 , axi., σ = 2 S = 10τ -1 0 , () 13 , shear, σ = 3

C RTI = 1.9

Shear release ij model for different shear rate S, infrared exponents σ, components () ij and kinds of anisotropy. The asymptotic value at t = 10 6 τ 0 is C RTI = 1.9.

In Fig. 3.5, the time evolution of C RTI (t), computed thanks to (3.8), is displayed. Π (s) ij is the integral over k of the RTI non-linear transfer S (RTI) ij defined in (2.71). To assess the relevance of the value C RTI = 2, several cases are presented. C RTI is evaluated with different mean velocity gradient intensities S, for infrared slopes σ = 1, 2 and 3, for various components of the tensor Π (s) ij , and several kinds of anisotropy: shear, axisymmetry (A 11 = A 22 = -A 33 /2) and distortion (A 13 = A 31 ). The main result is that in all these cases C RTI (t) → 1.9 which is very close to the theoretical value C RTI = 2 expected from the previous development. Hence, the model Π

(s) ij (t) = -2 (t)b ij (t)
is assessed numerically. Moreover, this model seems to be robust since it holds for various initial parameters and kinds of anisotropy. The case of Batchelor turbulence is not presented in Fig. 3.5 since db ij /dt = 0 which is an assumption of the model. Consequently, in Batchelor HSRT, C RTI continuously decreases. But, on a strictly quantitative point on view, the value obtained at t = 10 6 τ 0 is C RTI = 1.87, which is close to 2 as well.

The slight difference between the expected value 2 and the 1.9 obtained numerically for C RTI could be, at least partially, attributed to the isotropic approximation for the dissipation rate tensor ij . Such large times (t = 10 6 τ 0 ) are never reached in practice in experimental works, even though they are essential at high Reynolds numbers to make sure that the decay follows completely the theoretical decay exponents of the CBC theory. Nevertheless, C RTI = 2 remains a relevant value as all our different curves in Fig. 3.5, corresponding to various initial parameters, are almost equal to C RTI = 2 within 5% from t = 10 2 τ 0 , which is a reasonable time.

This model mainly relies on two aspects: (i) The isotropic shape of the dissipation tensor ij , which is well assessed numerically: indeed, ij strongly depends on small scales, see equation (3.12), which return to isotropy according to Fig. 3.3b, meaning that extra-diagonal components are very small with respect to diagonal ones. (ii) The asymptotic behaviour of the b ij which become constant for large times, which implies db ij /dt = 0. Numerically, constant values at large times for the b ij seem to be strongly associated to cases where the PLE is verified. It is worth noting that these two hypothesis are independent of the mean-velocity gradient intensity S and of the mean flow shape (shear, axisymmetry, distortion), which explains the consistency of the results of Fig. 3.5. These hypothesis are notably satisfied for any flows dominated by production mechanisms, such as shear flows. Consequently, there is no guarantee that this model would work for rotating mean flows for instance: indeed, such a configuration involves turbulent waves which alter the thirdorder correlations dynamics [START_REF] Cambon | Third-order statistics and the dynamics of strongly anisotropic turbulent flows[END_REF]. Therefore, further investigations are needed to fully understand the impact of rotation on the eddy-damping terms and its consequences on the RTI process.

Additional remarks on HSRT

Effects of infrared exponent σ and shear rate S: it has been shown that the global indicators b ij reach asymptotic anisotropic values for large times, except in the case of Batchelor turbulence σ = 4 where they continuously decrease. In Fig. 3.6a, the emphasis is put on b 13 . The first obvious remark is that the stronger the anisotropy, i.e. the larger the shear rate S (at constant accumulated anisotropy), the more the asymptotic value b lim ij is far from zero. This is expected: if the initial anisotropy is strong, the residues will be important. Finally, in Fig. 3.6b, one can note that the rapider the decay, i.e. the larger σ, the more b lim ij is close to zero. Once again, this is consistent with the fact that when the decay is faster, small scales have a more efficient RTI process. Anisotropy at low Reynolds numbers: an interesting behaviour is illustrated in Fig. 3.7a where anisotropy increases at low Reynolds numbers. This is expected since the kinetic energy spectrum E(k, t) no longer displays an inertial range when Re λ → 0: indeed, all anisotropy is gathered at large scales, and b ij being an integral over k, it eventually represents the initial anisotropy injected in the flow: at low Reynolds numbers, the asymptotic values of b ij and

H (dir) ij + H (pol) ij
coincide. The increase of b ij at low Re λ is in agreement with [START_REF] Davidson | On freely decaying, anisotropic, axisymmetric saffman turbulence[END_REF] where axisymmetric anisotropic Saffman turbulence is considered. And one can note that in Figure 1.e) therein, for the low Reynolds simulation (Run 5), anisotropy (observed through u 2 ⊥ /u 2 ) slightly increases. There is no explanation in the paper but this phenomenon is in good agreement with our simulations: anisotropy increases with diminishing Reynolds number. In Batchelor turbulence, because of the continuous loss of anisotropy, a complete return to isotropy is possible at low Reynolds numbers as revealed in Fig. 3.7b where H 3.2 Decay of K(t) and R 13 (t) in Saffman and Batchelor HSRT

In this section, the general decay exponent for the anisotropic correlation R 13 in Batchelor HSRT is investigated. First, it is recalled that in HIT, following the Comte-Bellot and Corrsin (CBC) theory [START_REF] Comte-Bellot | The use of a contraction to improve the isotropy of a grid generated turbulence[END_REF]Corrsin, 1951a), the kinetic energy and integral scale decay as

K(t) ∼ t α , α = -2 σ -p + 1 σ -p + 3 , L(t) ∼ t n L , n L = 2 σ -p + 3 , (3.9)
where the backscatter parameter p(σ = 4) = 0.55 and p(σ ≤ 3) = 0 takes into account the classical Batchelor breakdown of the PLE [START_REF] Eyink | Free decay of turbulence and breakdown of self-similarity[END_REF][START_REF] Meldi | On non-self-similar regimes in homogeneous isotropic turbulence decay[END_REF].

In Fig. 3.8, the theoretical decay exponent α of the kinetic energy is still valid in Saffman HSRT at high Reynolds numbers (and low Reynolds numbers as well, even though it is not presented). The fact that the kinetic energy decay exponent is not affected by anisotropy in Saffman turbulence has already been found in DNS in the case of homogeneous axisymmetric turbulence [START_REF] Davidson | On freely decaying, anisotropic, axisymmetric saffman turbulence[END_REF]. In the case of Batchelor HSRT, α is also recovered thanks to the parameter p, as in HIT. However, when it comes to the decay exponent α 13 of the correlation R 13 =< u 1 u 3 >, one has α 13 = α only in the case of Saffman HSRT. Indeed, in Batchelor HSRT, one has α 13 (σ = 4) = α(σ = 4). This difference is due to anisotropy which modifies the classical back transfer of energy through non-linear transfers. Because of strong inverse non-linear transfers which tend to isotropize large scales, there is a global loss of anisotropy. Thus, the decay of R 13 is accelerated in Batchelor HSRT: the correlation R 13 experiences pressure effects in addition of viscous dissipation, and consequently |α 13 | > |α|, as revealed in Fig. 3.8. It is proposed to model this phenomenon very specific to Batchelor HSRT.

0 2 4 6 8 -2 -1.5 -1 -0.5 log 10 (t/τ 0 ) α(t), α 13 (t) α(t) α 13 (t), S = 10τ -1 0 α 13 (t), S = 0.1τ -1 0 Classical CBC (a) 0 2 4 6 8 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1 log 10 (t/τ 0 ) α(t), α 13 (t) α(t)
A wise approach is to adopt the same method as in the isotropic case: in HIT, the parameter p is introduced to take into account the breaking of the PLE. This allows to recover the kinetic energy decay exponent α(σ = 4) = -1.38 obtained numerically (see Fig. 3.8). Without p, the analytical expression gives α(σ = 4, p = 0) = -10/7. A similar idea is to modify this parameter p into a new one p S that additionally takes into account effects of initial anisotropy for purely anisotropic quantities.

The kinetic energy K(t) and the anisotropic correlation R 13 (t) follow the evolution equations dK dt = SR 13 (t) -(t), dR 13 dt = SR 33 + Π 13 -13 .

(3.10)

When the shear is released, S = 0 and only the slow-part of Π 13 remains. It is clear that 13 is very weak because of the small scales return to isotropy: consequently, unlike K whose decay is driven by , the destruction of R 13 is led by the slow part of Π 13 , i.e. pressure effects. Purely anisotropic quantities

R 13 (t) = ∞ 0 φ 13 (k, t)dk ∼ t α 13 , (3.11) 13 (t) = 2ν ∞ 0 k 2 φ 13 (k, t)dk, (3.12) Π (s) 13 (t) = ∞ 0 S (RTI) 13
(k, t)dk, (3.13) have different decay exponents in Batchelor HSRT than the classical ones of HIT. To determine the theoretical expression of α 13 , the continuity of φ 13 (k, t) at the integral wavenumber is used, similarly to what is done to obtain α. The first assumption is that the integral scale L(t) is the same for all components of φ ij (k, t), which is reasonable as they all depend on E(k, t). Then, one has to determine the new scaling of φ 13 once the shear is released. Indeed, when S = 0 one has φ 13 = 0, which is in contradiction with (3.4). Nevertheless, the k -7/3 scaling is still observed after the shear release. The solution to reconcile the persistence of φ 13 (k, t) and the k -7/3 scaling is to replace S -1 by another time scale: S being a large scale quantity, the intuitive time scale is the non-linear one τ (k L ) = (k 2 L ) -1/3 evaluated at the integral wavenumber k L . Then, numerical simulations show that the destruction mechanism for φ 13 (k, t) is the pressure rather than viscosity. This is notably illustrated in Fig. 3.9 where the budget terms of the evolution equations of φ 11 and φ 13 are displayed after the release of the shear, at t = 10τ 0 , so that small scales have already returned to isotropy. This explains why kS (RTI) 11 is very small compared to the other contributions. Moreover, it appears that the viscous dissipation -2νk 3 φ 11 is the destruction mechanism for φ 11 , whereas it is pressure through kS (RTI) 13 for φ 13 . Indeed, the viscous term -2νk 3 φ 13 is negligible compared to kS (RTI) 13

. This also confirms that neglecting 13 in the previous part was a reasonable assumption. Hence, is accordingly replaced by Π (s) 13 which has the same dimension. This finally yields

φ 13 (k, t) ∼ Π (s) 13 2/3 k 2/3 L k -7/3 . (3.14)
Then, a new coefficient p S is introduced for the purely anisotropic quantities in Batchelor HSRT, to reflect the effect of anisotropy on the backscatter of energy. Consequently, the continuity of

φ 13 at the integral wavenumber k L yields k σ-p S L ∼ Π (s) 13 2/3 k -5/3 L
. Hence, one obtains straightforwardly the theoretical decay exponent of Π (s) 13 . The resulting expression for the anisotropic decay exponent is then

R 13 (t) ∼ t α 13 , α 13 = -2 σ -p S + 1 σ -p + 3 , p S = 0 , σ ≤ 3 0.279 , σ = 4.
(3.15)

Simulations show that α 13 does not depend on the initial shear rate S in Fig. 3.8, from S = 10 -2 τ -1 0 to S = 10τ -1 0 . This anisotropic decay exponent is found numerically to be α 13 = -1.464. This implies the strong result that p S does not depend on the shear intensity. The corresponding value of p S is deduced using (3.15): p S = 0.279. Moreover, p S is not only independent of S, but also independent of the kind of anisotropy considered. Indeed, for distortion, α 13 = -1.464 as well (grey dash-dot line in Fig. 3.8). All these decay exponents are gathered in Table 5.1.

The value p S = 0.279 is close to the value of the backscatter parameter of a passive scalar field in decaying Batchelor HIT (see Chapter 1). This could be interpreted as φ 13 being almost passively convected by the turbulent velocity field, which is consistent with the production terms being zero once the shear is released.

In conclusion, p S must be seen as a supplementary parameter for purely anisotropic quantities when the PLE is not verified, as in Batchelor turbulence. p S is valid in the general case where the mean-gradient matrix A ij has non-diagonal components. From a physical point of view, it has been shown that in Batchelor HSRT, purely anisotropic quantities, such as the correlation R 13 (t), decay faster than the isotropic ones, such as the kinetic energy K(t). This is because of the continuous loss of large scales anisotropy, induced by strong inverse non-linear transfers. Moreover, the decay of R 13 is driven by pressure effects rather than viscous ones.

Homogeneous Shear Turbulence

In this part, the case of homogeneous shear turbulence (HST) is addressed, where the shear is maintained throughout the simulation.

Exponential growth of the kinetic energy K(t)

In sustained shear flows, the kinetic energy grows exponentially [START_REF] Sagaut | Homogeneous Turbulence Dynamics[END_REF] as a consequence of anisotropy production and non-linear redistribution. Because of the exponential growth of the integral scale L(t), DNS are quite limited in accumulated anisotropy St. The evolution equation of K(t) in HST is given by (3.2). The dimensionless shear rapidity is now introduced

S R (t) = (t) SK(t) , (3.16)
which is the ratio of shear and non-linear characteristic times. The evolution equation of K can be written 1 KS

dK dt = 2b 13 - KS Constant for St 1.
= γ.

(3.17)

For St 1, anisotropy indicators b ij and S R reach an asymptotic anisotropic state, as in the case of HSRT. So, for large St (St ≥ 25), the quantity 2b 13 -/KS = γ becomes constant, as revealed in Fig. 3.10a. The existence of a transient regime of order St 30 has already been found in DNS [START_REF] Pumir | Persistent small scale anisotropy in homogeneous shear flows[END_REF][START_REF] Pumir | Turbulence in homogeneous shear flows[END_REF][START_REF] Gualtieri | Scaling laws and intermittency in homogeneous shear flow[END_REF]. From the previous equation, the exponential growth of kinetic energy

K(t) = K(0) exp(γSt), (3.18)
is assessed by the present anisotropic EDQNM modelling in Fig. 3.10b. The growth exponent is γ = 0.33. However, asymptotic values of b ij and S R are different from those of reported in [START_REF] Sagaut | Homogeneous Turbulence Dynamics[END_REF], certainly because St was not high enough and thus anisotropy indicators were not constant yet. This will be discussed later. The exponential growth of K(t) has also been assessed in the DNS of [START_REF] Brethouwer | The effect of rotation on rapidly sheared homogeneous turbulence and passive scalar transport. linear theory and direct numerical simulation[END_REF], where St max = 12 only, and so the γ is different from ours, probably because once again b ij is not constant yet; in experiments [START_REF] Tavoularis | Asymptotic laws for transversely homogeneous turbulent shear flows[END_REF][START_REF] De Souza | The structure of highly sheared turbulence[END_REF]; and in another spectral modelling [START_REF] Clark | A spectral model applied to homogeneous turbulence[END_REF], where γ = 0.332 is found as well. It is argued in [START_REF] Pumir | Persistent small scale anisotropy in homogeneous shear flows[END_REF]; [START_REF] Pumir | Turbulence in homogeneous shear flows[END_REF]; [START_REF] Gualtieri | Scaling laws and intermittency in homogeneous shear flow[END_REF] that for a finite domain, i.e. wall-bounded shear flows, there are kinetic energy bursts since the kinetic energy cannot grow exponentially for very large St. These bursts are periodic (every St 20) and result from the cyclic deformation and stretching of elongated structures in the flow. In our simulations, the shear applies on an infinite length, meaning that there are no boundaries, theoretically allowing K(t) to become infinite. It has been shown by [START_REF] Lee | Structure of turbulence at high shear rate[END_REF] that a very high shear rate creates streaks in the flow, as would do physical boundaries at a more moderate shear rate. However, high shear rates are not reachable with the present anisotropic EDQNM modelling, in the assumption of moderate anisotropy.

In Fig. 3.10b, it is also revealed that the growth rate exponent γ seems to be quite robust within the present EDQNM modelling: indeed, it appears that γ = 0.33 is both independent of the shear rate S and the infrared slope σ. In particular, the kinetic energy grows exponentially at the same rate in Saffman and Batchelor turbulence. Moreover, the b ij in Batchelor HST are very similar to those in Saffman HST.

In addition to the exponential growth of K(t), the behaviour of the dissipation rate (t), the integral scale L(t) and the anisotropic component R 13 (t) are investigated. They also grow exponentially as revealed in Fig. 3.11a. In addition to the exponential growth, it is possible to determine the growth rate γ and γ L . From the main equation (3.2), and K are linked through a time-derivative, then γ = γ. Finally, from the dimensional analysis L ∼ K 3/2 / , γ L = γ/2 is straightforward. This is recovered numerically in Fig. 3.11a. Finally, 13 strongly decreases, which is expected: indeed, at high Reynolds number, isotropization of small scales tend to strongly reduce 13 . In Fig. 3.11b, small scales of the velocity second-order moments have completely returned to isotropy, as in HSRT. Hence, our results are in agreement with [START_REF] Pumir | Persistent small scale anisotropy in homogeneous shear flows[END_REF]; [START_REF] Pumir | Turbulence in homogeneous shear flows[END_REF]; [START_REF] Shen | The anisotropy of the small scale structure in high reynolds number (r λ ∼ 1000) turbulent shear flow[END_REF]; [START_REF] Gualtieri | Scaling laws and intermittency in homogeneous shear flow[END_REF] where velocity second-order moments are found to be isotropic at small scales. This is consistent with the fact that at small scales non-linear processes dominate the dynamics. This also explains why 13 is destroyed instead of growing exponentially, as illustrated in Fig. 3.11a. Following the definition (3.12), k 2 φ 13 is larger at small scales: but 13 is a purely anisotropic quantity and since small scales return to isotropy, it is continuously destroyed.

K(t) R 13 (t) ǫ(t) L(t) e 0.33St
k/k L H (dir) ij (k, t) H (dir) 11 H (dir) 13 H (dir) 33 k η k L k S (b)
A brief comparison with the DNS of [START_REF] Isaza | On the asymptotic behaviour of large-scale turbulence in homogeneous shear flow[END_REF] for a sustained shear flow is presented in Fig. 3.12. The evolution of the shear parameter [3; 15; 27]. This is of particular interest since this parameter is discussed later on. The initial Reynolds number is Re λ (0)

S * = 2/S R is investigated for various initial values S * 0 =
20. An initial isotropic field is considered, and the infrared slope of the kinetic energy spectrum is E(k < k L , t = 0) ∼ k 2 . It is revealed in Fig. 3.12 that the evolution of S * (t) at moderate St and Re λ , which is the domain of accurate DNS, is well-captured in all of the three cases by the present anisotropic EDQNM modelling. A slightly higher final value for the case S * 0 = 3 is obtained here, but not significant.

Non-linear transfers and the shear wavenumber

The shear wavenumber, or Corrsin wavenumber, is defined as

k S = S 3 . (3.19)
For wavenumbers k ≤ k S , linear effects dominate, meaning that production processes lead the dynamics. In the opposite case, for k ≥ k S , non-linear processes become significant, distribute energy from the main direction to other components through the spectral tensor φ 13 , thus participating into the restoration of isotropy at small scales. Note that other similar wavenumbers exist which indicate the beginning of an efficient RTI, for instance the Zeman wavenumber in rotating turbulence, or the Ozmidov one in USHT, where S is accordingly replaced by the mean rotation or the mean stratification respectively.

In Fig. 3.13, budget terms at St = 50 are displayed, when the anisotropic asymptotic state is reached, along with k S . Before k S , linear transfers S L(tot) 13

dominates whereas for k ≥ k S pressure strain and non-linear transfers become strong. From the evolution equations of K(t) (3.2) and R 13 (t) (3.10), the exponential growth of the kinetic energy can be understood thanks to transfer terms. The flow is heading toward the main direction () 11 : thus φ 33 do not receive as much energy as φ 11 . But S (RTI) 33 being positive, it takes energy from φ 11 and φ 22 , allowing φ 33 to grow. The growth of φ 33 implies the growth of φ 13 and thus the growth of K(t). Hence, the exponential growth of the kinetic energy is the result of non-linear processes and redistribution of energy between components. Moreover, there is a non-negligible dissipation effect for φ 33 in Fig. 3.13. Similar dissipation processes are obtained for φ 11 and φ 22 whereas it does not appear for φ 13 . It justifies a posteriori the assumption that the dissipation is nearly isotropic.

Discussion on the scattering of integrated quantities in HST

In this section, explanations to the scattering of quantities measured in DNS and experiments in shear flows are proposed. The quantitative discrepancy between the γ = 0.33 obtained with the present anisotropic EDQNM modelling and the smaller common values is also discussed. One can note that EDQNM simulations are able to reach St = 50, which is much higher than DNS, even the most recent ones. This is one of the strength of the current modelling, i.e. investigating the asymptotic states of shear-driven flows.

As mentioned previously, weaker values of γ (roughly between 0.07 and 0.20) are found in DNS and experiments, with a noteworthy dispersion, which is now addressed. This could be, for some of these works, the consequence of a too low final St for which the b ij and /(KS) are not constant yet, resulting in a value of γ not converged, that reflects transitional effects of initial conditions. Indeed, in the numerical work of [START_REF] Brethouwer | The effect of rotation on rapidly sheared homogeneous turbulence and passive scalar transport. linear theory and direct numerical simulation[END_REF], the last dimensionless time is St = 12 which is not enough to ensure that b ij and S R are constant. Other low values of St are reported by [START_REF] Sagaut | Homogeneous Turbulence Dynamics[END_REF], along with various values of the b ij and γ coming from DNS and experiments. Therefore, moderate values of the final St could be an explanation for the scattering of the experimental and numerical measured γ. It is also argued in [START_REF] Isaza | On the asymptotic behaviour of large-scale turbulence in homogeneous shear flow[END_REF] that re-meshing, in older DNS, led to loss of kinetic energy and dissipation rate: this could be an explanation as well, rather difficult to quantify.

Several relevant quantities such as γ, b 13 , Re λ (0), (St) max , S * (0) and S * end , coming from different DNS and experiments, are gathered in Table 3.1 to illustrate the noteworthy dispersion of integrated quantities. Qualitatively, it is interesting to point out that for small initial S * (0), the final value of |b 13 | is higher, whereas for high initial S * (0), the final value of γ is higher: this is expected since it corresponds to a strong initial production of kinetic energy, which is recovered with our anisotropic EDQNM modelling in Fig. 3.10b. There is also a slight tendency to increasing |b 13 | and γ in average in more recent DNS, very likely because of the better spatial resolution of small scales. Nevertheless, the maximum values are γ = 0.18, still much lower than our γ = 0.33, and |b 13 | = 0.19, close to our 0.21. Recent DNS studies have focused on the influence of initial parameters, such as the Reynolds number Re λ (0) or the shear parameter S * (0), on the final state of the flow. Notably, it is reported that there is a tendency toward an almost independence with regard to Re λ (0) and a noteworthy sensitivity to S * (0). Hereafter, possible explanations for this dependence on initial conditions are proposed. Let's mention that in numerical works, the infrared slope σ is a supplementary initial condition that defines large scales. However, the infrared slopes are not often investigated nor reported, which makes the comparisons and discussions complicated. EDQNM simulations revealed in Fig. 3.10b that the growth rate γ does not depend on σ: therefore in what follows σ = 2 is chosen.

In Schumacher et al. (2003a); [START_REF] Schumacher | Relation between shear parameter and reynolds number in statistically stationary turbulent shear flows[END_REF]; [START_REF] Isaza | On the asymptotic behaviour of large-scale turbulence in homogeneous shear flow[END_REF]; [START_REF] Sukheswalla | Simulation of homogeneous turbulent shear flows at higher reynolds numbers: numerical challenges and a remedy[END_REF], the final value of S * seems to depend on the initial conditions. This is not necessarily in contradiction with our EDQNM results, as revealed in Fig. 3.14a, where the shear parameter S * = 2/S R is displayed for various initial S * (0). Indeed for St ≤ 30, S * strongly depends on its initial value for both DNS (see the standard deviation in Table 3.1) and EDQNM. Incidentally, the dispersion of the S * computed with EDQNM at moderate St (≤ 20) is comparable to the dispersion obtained in DNS and experiments. Then, for sufficiently high St ≥ 30, S * becomes independent of initial conditions. Therefore, one could conclude that an universal asymptotic state could be reached only for sufficiently high St, or equivalently at very high Reynolds numbers. Similar assessments are made in [START_REF] Isaza | On the asymptotic behaviour of large-scale turbulence in homogeneous shear flow[END_REF]. At moderate St and Re λ , the comparison of the present anisotropic model to the recent DNS in Fig. 3.12 shows that the early dynamics of S * is well captured.

Another point of interest in [START_REF] Isaza | On the asymptotic behaviour of large-scale turbulence in homogeneous shear flow[END_REF]; [START_REF] Sukheswalla | Simulation of homogeneous turbulent shear flows at higher reynolds numbers: numerical challenges and a remedy[END_REF] is that the average value of the measured γ seems slightly higher (with a maximum of 0.18) than older ones, very likely because of the better spatial resolution. In [START_REF] Sukheswalla | Simulation of homogeneous turbulent shear flows at higher reynolds numbers: numerical challenges and a remedy[END_REF], small scales had to be filtered. But, from the present study and [START_REF] Clark | A spectral model applied to homogeneous turbulence[END_REF], it appears that large scales are not determinant in the final value of γ (γ is found to be independent of σ). This directly means that the inertial range scales have a preponderant influence on γ. Hence, the low resolution of small scales in some DNS could result in slightly imprecise values for γ, partially responsible for the scattering. This interpretation is confirmed in Figure 18.a of [START_REF] Sukheswalla | Simulation of homogeneous turbulent shear flows at higher reynolds numbers: numerical challenges and a remedy[END_REF] where the non-filtered kinetic energy is displayed and evolves with a growth rate somewhat higher to those of filtered kinetic energies. In addition, one can note that the unfiltered value of S * from [START_REF] Sukheswalla | Simulation of homogeneous turbulent shear flows at higher reynolds numbers: numerical challenges and a remedy[END_REF] reported in Fig. 3.14a would give a value not far from ours (S * → 20) at a higher St. For the other S * reported in [START_REF] Schumacher | Relation between shear parameter and reynolds number in statistically stationary turbulent shear flows[END_REF]; [START_REF] Isaza | On the asymptotic behaviour of large-scale turbulence in homogeneous shear flow[END_REF], the (St) max is to low to conclude. The influence of the initial Reynolds number Re λ (0) is now discussed. It has been reported in [START_REF] Sukheswalla | Simulation of homogeneous turbulent shear flows at higher reynolds numbers: numerical challenges and a remedy[END_REF] that Re λ (0) had not much impact on the final state of the flow. This is recovered in Fig. 3.14b where the kinetic energy exponential growth rate γ is displayed at Re λ (0) = 10 and Re λ (0) = 100 for various initial shear intensities S. The important result is that a different initial Reynolds number changes very slightly the final growth rate exponent γ: indeed, γ 0.33 was obtained previously for Re λ (0) = 1. Here, for S ≤ 1τ -1 0 and both Re λ (0) = 10 and Re λ (0) = 100, one has γ 0.330, whereas γ 0.334 for S = 10τ -1 0 and Re λ (0) = 100. This underlines that for sufficiently high final St, or equivalently sufficiently high Reynolds numbers, an asymptotic state independent of initial conditions is obtained. This result is consistent with what is observed in DNS, i.e. the independence with regard to Re λ (0). Let's mention that the Re λ (0) = 100 chosen here is higher than common initial Reynolds numbers for DNS, as revealed in Table 3.1. Moreover, Fig. 3.14b reveals that at comparable St, our γ is much higher than common ones and almost constant. Hence, it is very likely that γ in DNS and experiments would not increase for higher St. Therefore, the moderate St reached in DNS and experiments can only explain the scattering around the average value γ av = 0.114.

S = 10 -2 τ -1 0 S = 0.1τ -1 0 S = 1τ -1 0 S = 10τ -1 0 Schumacher Isaza Sukheswalla (a) 0 10 20 30 40 -0.1 0 0.1 0.2 0.3 0.4 St γ(t) S = 0.1τ -1 0 , Re λ (0) = 100 S = 1τ -1 0 , Re λ (0) = 100 S = 10τ -1 0 , Re λ (0) = 100 S = 1τ -1 0 , Re λ (0) = 10 (b)
The results of this discussion, summarized hereafter, are twofold. Firstly, Fig. 3.14a and 3.14b exhibited interesting behaviours: firstly, an unique asymptotic value for the shear parameter S * = 2KS/ is obtained only at high values of St. For typical final values of DNS and experiments (St ≤ 20) at an initial moderate Reynolds number Re λ (0), it appears that S * still depends on initial conditions. Furthermore, in agreement with existing works, an independence with regard to the initial Reynolds number has been obtained. Thus, high St can limit the dispersion of the results by erasing effects of initial conditions. Secondly, it has been pointed out that the present anisotropic EDQNM modelling provides higher values for the growth rate γ than DNS and experiments do. Even though similar values are found in the spectral model of [START_REF] Clark | A spectral model applied to homogeneous turbulence[END_REF], our numerical γ = 0.33 is much higher than common ones, gathered around γ av = 0.114.

The practical input of this numerical work is notably indications for future DNS. According to the previous discussion and Table 3.1, it seems crucial to reach final values of the accumulated anisotropy 25 ≤ (St) max ≤ 30 to limit the scattering of the results and transitional effects from initial conditions, and to systematically investigate the dependence in (Re λ (0), S * (0)), as reported in Schumacher et al. (2003a).

Conclusion and perspectives

Firstly, the main features of this chapter are recalled hereafter: results regarding decay and growth laws in shear-driven flows are gathered in Table 5.1, along with similar results for the passive scalar field addressed in the next chapters. Secondly, we come back on the value of the kinetic energy exponential growth rate γ = 0.33. Finally, some perspectives are drawn for future works.

Conclusions on HST and HSRT

Homogeneous anisotropic turbulence has been investigated with the anisotropic EDQNM modelling in the particular case of shear flows, when mean-velocity gradients are both released (HSRT) and sustained (HST). In this framework, we have revisited classical phenomena and provided results obtained at high Reynolds numbers, qualitatively in agreement with existing ones in DNS and experiments.

In the shear-released turbulence configuration, a model was derived for the slow-part of the pressure strain-tensor Π (s) ij , which is responsible for the return to isotropy mechanism, valid once the mean-velocity gradients are released. This model is in agreement with our simulations and must be seen as complementary to existing ones, since it focuses on the asymptotic anisotropic state at large times for high Reynolds numbers. Then, the present model allows to understand deeply the RTI mechanism: spectral descriptors show that small scales of the velocity second-order moments completely return to isotropy in both Saffman and Batchelor turbulence, leading to a global partial return to isotropy, which is in agreement with experiments and DNS. Regarding large scales, they keep their anisotropy in Saffman turbulence, whereas they continuously evacuate anisotropy in Batchelor turbulence, because of strong inverse non-linear transfers. In addition, the decay of the kinetic energy K(t) and of the anisotropic correlation R 13 was investigated in both Saffman and Batchelor HSRT. The decay of the kinetic energy is not modified by the initial shear, whereas R 13 , which also decays in t -6/5 in Saffman HSRT, decays faster than K(t) in Batchelor HSRT, because of the RTI mechanism driven by pressure effects.

At last, the case of the sustained shear turbulence was addressed. The original aspect highlighted here is that with the present modelling the growth rate γ seems to not depend on large scales initial condition (the infrared slope σ) nor on the shear rate intensity S, provided a sufficiently high accumulated anisotropy St, or equivalently a high Reynolds numbers, is reached. Quantitative differences have been exhibited, especially for the exponential growth rate γ of the kinetic energy, which is found to be two to three times higher than existing values. Nonetheless, as for the shear parameter S * , its dynamics seems to be well described by the present model. The dispersion of integrated quantities obtained in DNS and experiments has been discussed, and the main result of this work regarding sustained shear flows lies in the explanations proposed: it has been shown notably that moderate values of the final accumulated anisotropy St may be responsible for the scattering of integrated quantities, reported in Table 3.1, and that higher values of St, or higher Reynolds numbers, could limit this dispersion by erasing initial conditions effects. Another parameter that could be of importance is the nature of the initial flow, that we choose to be isotropic for simplicity reasons. In experiments for instance, the initial condition is clearly not isotropic. EDQNM simulations with an initial condition slightly anisotropic, as in [START_REF] Davidson | On freely decaying, anisotropic, axisymmetric saffman turbulence[END_REF], show that for instance γ slightly varies by ±5%. This indicates a small dependence on initial anisotropy, but not significant.

Exponential growth rate γ

This part aims at answering two questions of fundamental interest for the modelling of shear flows: (i) Why is the exponential growth rate γ of the kinetic energy not depending on the infrared slope σ whereas it strongly does in unstably stratified homogeneous turbulence (see Chapter 7) ? (ii) Is the value γ = 0.33 predictable? Theoretical considerations about shear flows which were found after the publication of the contents of this chapter in Journal of Turbulence are thus presented here.

The independence of γ with σ in HST is not a consequence of the modelling of anisotropy, since in Chapter 7, the kinetic energy exponential growth rate in USHT strongly depends on σ. Assuming self-similarity of the kinetic energy spectrum and a linear dynamics of large scales in both HST and USHT, one has K ∼ exp(γSt), L ∼ exp(γSt/2), so that

E(k, t) ∼ k σ exp σ + 3 2 γSt . (3.20)
Following the method proposed in Poujade & Peybernes (2010); [START_REF] Soulard | Large-scale analysis of self-similar unstably stratified homogeneous turbulence[END_REF], the time evolution of E is also given at large scales by the largest eigenvalue of the linear operator of the generalized Lin equations system (2.53)-(2.55). The linear operator of HST verifies, dropping the non-linear and viscous terms,

∂ ∂t              E EH (dir) 11 EH (dir) 13 EH (dir) 33 EH (pol) 11 EH (pol) 13 EH (pol) 33              = S             0 0 2(σ + 2) 0 0 2 0 0 0 -2(σ+11) 21 0 0 2 21 0 (σ-1) 30 3-σ 7 0 -4+σ 7 1 7 0 1 7 0 0 2(10-σ) 21 0 0 2 21 0 0 0 4 7 0 0 2(σ+4) 21 0 1 5 6 7 0 6 7 3σ-2 21 0 3σ+5 21 0 0 4 7 0 0 2(σ-3) 21 0                          E EH (dir) 11 EH (dir) 13 EH (dir) 33 EH (pol) 11 EH (pol) 13 EH (pol) 33              . (3.21)
It is worth noting that the linear operator, because of the space-derivative ∂/∂k production terms, depends explicitly on σ, unlike the linear operator of USHT, whose maximum eigenvalue is 4/ √ 5. Consequently, the largest eigenvalue of the linear operator in HST almost balances the self-similar expression (3.20) for E(k, t), thus strongly reducing the dependence of γ with σ.

The maximum real eigenvalue Γ max (σ) of the linear operator being a rather lengthy expression, only the final growth rate is given here, obtained by equalizing (3.20) and E ∼ exp(Γ max St), so that

γ =          0.358 for σ = 1, 0.339 for σ = 2, 0.346 for σ = 3, 0.367 for σ = 4. (3.22)
This result answers the two questions (i) and (ii) of the beginning of this part. First, the numerical simulations presented so far recover a value around 0.33 ≤ γ ≤ 0.34 for all σ, which is close to the linear prediction (3.22): this means that the non-linear redistribution of energy through scales permits to maintain this exponential growth rate, obtained by self-similar arguments and the linear dynamics of large scales where anisotropic mechanisms dominate. Secondly, the fact that the maximum eigenvalue of the linear operator depends on σ explains why γ hardly varies with σ in HST, unlike USHT where the maximum eigenvalue is independent of the large scales initial conditions, so that the exponential growth rate in USHT (7.26) varies a lot with σ.

Perspectives

In this section, two perspectives for future works regarding shear flows are proposed. The first one, already introduced in Chapter 2, is the consideration of the fourth-order expansion for E and Z. The second one is to consider that the mean-shear intensity S(t) can vary with time, i.e. to model the retro-action of the fluctuating turbulent quantities on the mean-field, similarly to what is done at the end of Chapter 7 for a variable stratification frequency N (t).

Fourth-order expansion: In Chapter 2, the formalism of the fourth-order expansion into spherical harmonics has been presented, with details in Appendix C. The fourth-order contributions EH . In the end, the total linear directional and polarization transfers with the fourth-order expansions are S

L(dir) ij = S L(dir2) ij + S L(dir4) ij , and S L(pol) ij = S L(pol2) ij + S L(pol4) ij .
The main consequence of the fourth-order contributions is that the kinetic energy exponential growth rate is decreased in Fig. 3.15 from γ = 0.33 to γ = 0.28. Even though the value of γ remains rather large, the significant decrease by 15% with the fourth-order expansion of E and Z perfectly illustrates that taking into account more spherical harmonics goes into the good direction, i.e. diminishes γ towards smaller values obtained in DNS, as reported in Table 3.1. The joint result is, in Fig. 3.15 as well, the decrease of b 13 from 0.215 to 0.18, which is a noteworthy feature as well. One can further remark that on the contrary, the fourth-order contributions increase |b 11 | and |b 33 |, which is expected. Indeed, taking into account more harmonics reduces the loss of information due to the spherical integration by restoring part of the anisotropic angular information. As a consequence, the strong anisotropy of the shear flow between the streamwise and transverse directions is better captured. To better illustrate the impact of the fourth-order contributions, we investigate the detailed linear transfers in Figure 3.16, with the directional and polarization parts of the streamwise, transverse and cross components in Saffman turbulence. In the this figure, the black curves represent simulations with the fourth-order contributions, at Re λ = 9.10 3 . Whereas the grey curves indicate simulations with only the second-order expansion, as in MCS. Since the Reynolds number increases faster in the latter case, the results are presented at St = 43 where Re λ 9.10 3 as well.

One can remark that the effects are different for the directional and polarization parts: indeed, the fourth-order contributions tend to decrease the intensity of the directional linear transfers for the streamwise () 11 and transverse () 33 components, while increasing it for the () 13 component.

The opposite happens for the linear polarization transfers. In particular, the strongest difference is observed for the transverse directional transfer S

L(dir) 33

, which is positive without the fourthorder contributions, and becomes mostly negative with them.

The practical input of this section in terms of modelling is that it is much more important to improve the linear production terms through the fourth-order expansion than considering , (e) S L(pol) 13

, and (f) S L(pol) 33

. In each case, the grey curves indicate simulations without the fourth-order contributions at Re λ (St = 43) = 9.10 3 . the quadratic anisotropic contributions in the classical non-linear transfers (see Appendix C). Another approach could be to solve directly the evolution equations of E(k, t) and Z(k, t) with the exact linear terms, and to keep the modelled non-linear transfers with the second-order expansion: this is currently the topic of a PhD under the direction of Claude Cambon.

Third-order expansion: Some considerations are now presented about the third-order expansion of Z: even though they are not conclusive right now, they could be of interest for future works. There are two reasons why we wish to further consider odd-order terms in the expansion of Z: (i) Odd-order expansions could improve the modelling of the 2iZΩ CH term in the evolution equation (2.33) of Z; (ii) Recent results by Claude Cambon show that the main difference between the MCS model with an exact treatment of linear terms lies in the polarization anisotropy.

Up to the fourth-order, the expansion of Z can be written

Z(k, t) = 1 2 E 0 5H (pol) ij (k, t) + 7iH (pol) ijk α k + 21 2 H (pol) ijpq (k, t)α p α q N * i (k)N * j (k), (3.23)
where H

(pol) ijk is a tensor which verifies, for simplicity reasons as before, full symmetry under any change of indices, and is zero when two indices are equal. Note that the expression of the third-order contribution differs from the one in [START_REF] Mons | A spectral model for homogeneous shear-driven anisotropic turbulence in terms of spherically-averaged descriptors[END_REF] because of the imaginary number i. The latter is crucial, otherwise it can be shown that the third-order of Z never contributes. Two features are needed to prove this statement. First, one requires the following equation when computing the polarization part R(pol

) ij = [ZN i N j ]: N i N j N * p N * q = P ip P jq + P iq P jp -P ij P pq - 1 2 iα a P jq ipa + P ip jqa + P iq jpa + P jp iqa ,
where ijk is the Levi-Civita permutation tensor. The second one is that the spherical average of an odd number of normalized wavevectors α i is zero, as explained for instance in [START_REF] Pope | Turbulent Flows[END_REF].

Consequently, without the i, [ZN i N j ] has an odd number of α i so that it vanishes with the spherical-average. Therefore, the present expansion of Z (3.23) corrects the equation (3.15) of [START_REF] Mons | A spectral model for homogeneous shear-driven anisotropic turbulence in terms of spherically-averaged descriptors[END_REF].

Then, as for the second and fourth-order terms, one needs an operator which gives only the thirdorder contribution and erases the others. This operator is

N ijk = α k N i N j + α j N i N k + α i N j N k ,
and we further define

H (pol) ijk as 2E(k, t)H (pol) ijk (k, t) = S k Z(k, t)N ijk (k) d 2 k. (3.24)
Similarly to the fourth-order expansion, the third-order expansion of Z does not modify the spectral tensor φ ij , which is still expressed as function of

H (dir) ij and H (pol) ij only.
Finally, the third-order expansion of Z gives a new contribution in the modelled spectral Reynolds tensor Rij , which reads

Rz3 ij (k, t) = 7 2 E 0 (k, t)H (pol)
pql (k, t)α l α n ipn P jq (k) + jqn P ip (k) .

(3.25)

Remark: It is worth noting that the third-order expansion of polarization can be related to the stropholysis tensor of [START_REF] Kassinos | One-point turbulence structure tensors[END_REF], defined as

Q ijk = ipq α p α k Rjq (k)d 3 k = - ijp α p α k E(k)d 3 k + α k Z(k)N i (k)N j (k) d 3 k. (3.26)
The final expression of Q ijk using the second-order expansions can be found in [START_REF] Mons | A spectral model for homogeneous shear-driven anisotropic turbulence in terms of spherically-averaged descriptors[END_REF]. Interestingly, the symmetric stropholysis

Q * ijk = (Q ijk + Q ikj + 4 perm.
)/6 erases the directional anisotropy and depends only on polarization. More specifically, only odd-order terms of the Z expansion can contribute in Q * ijk , which further justifies the interest of odd-order contributions. Now, we determine the explicit expressions of the production terms linked to the third-order contributions in the expansion (3.23) of Z. As before, there are three different kinds of terms: the third-order contributions in the equations of EH ijk . There are no third-order contributions in the Lin equation for E, and for simplicity reasons, we further discard the third-order contributions in the evolution equations of EH ijk : thus there is no direct linear coupling between the third and fourth orders.

First, the contributions of the third-order expansion in the equations of EH (dir) ij and EH (pol) ij are, after some algebra,

S L(dir3) ij (k, t) = 1 6 A + ln E ilp H (pol) jnp + jlp H (pol) inp , (3.27) S L(pol3) ij (k, t) = 1 3 A - ln E lnp H (pol) ijp -ilp H (pol) jnp -jlp H (pol) inp - 1 6 A + ln ∂ ∂k kE( ilp H (pol) jnp + jlp H (pol) inp ) + E( ilp H (pol) jnp + jlp H (pol) inp ) . (3.28) Note that unlike S L(dir4) ij and S L(pol4) ij , S L(dir3) ij and S L(pol3) ij
depend on the antisymmetric part A - ln of the mean-velocity gradient. The linear polarization transfer S

L(pol) ijk of EH (pol)
ijk is defined as

S L(pol) ijk (k) = 1 4 S k A ln k l ∂ Rrs ∂k n -2M rn (k) Rns (k) N * r (k)N * s (k)N ijk (k) d 2 k. (3.29)
This production term is divided into two contributions resulting from the second and third order expansions. First, we determine the impact of the second-order contributions EH ijk . For this purpose, we introduce two trace-free and symmetric operators, namely

H 2,+ ijk [EH () ] = 5E A + il jln H () nk + kln H () nj + A + jl iln H () nk + kln H () ni + A + kl iln H () nj + jln H () ni -2EA + ln H () np E δ ij klp + δ ik jlp + δ jk ilp (3.30) H 2,- ijk [EH () ] = 5E A - il jln H () nk + kln H () nj + A - jl iln H () nk + kln H () ni + A - kl iln H () nj + jln H () ni -A - ln iln H () jk + jln H () ik + kln H () ij + 2EA - ln H () np E δ ij klp + δ ik jlp + δ jk ilp .
(3.31)

After complex calculations, one gets

S L(pol2) ijk = - 1 7 H (2,+) ijk [EH (dir) ] + 1 21 H (2,-) ijk [EH (pol) ] -H (2,+) ijk [EH (pol) ] + 1 42 H (2,+) ijk [∂ k (kEH (pol) )]. (3.32)
Now, we proceed similarly to determine the impact of the third-order expansion on the linear transfers of EH (pol)

ijk . After some algebra, one gets

S L(pol3) ijk = 1 3 A - il H (pol) jlk + A - jl H (pol) ilk + A - kl H (pol) ijl , (3.33)
where notably the symmetric part of the mean-velocity gradient matrix does not intervene. The third-order polarization non-linear transfer is defined as

S NL(pol) ijk (k, t) = 1 2 S k T Z (k, t)N ijk (k, t) d 2 k. (3.34)
For the sake of simplicity, quadratic anisotropic contributions are discarded, as for the second and fourth orders: therefore, only the third-order terms contribute in S NL(pol) ijk

. Furthermore, because H

(pol) ijk is symmetric and trace-free, it follows that third-order expansions vanish in

S NL(dir) ij and S NL(pol) ij
. Eventually, injecting the third-order expansion into the expression (2.42) of T Z gives the spherically-averaged non-linear polarization transfer ijk E 0 2(xy + z 3 )(2z 2 -1) -z(3z 2 -1)(y 2 -z 2 ) dpdq. (3.35) The evolution equation of the third-order anisotropic descriptor EH (pol)

S NL(pol) ijk = 4π 2 ∆ k θ kpq k 2 p 2 qE 0 H (pol) ijk (1 -z 2 ) 2z(1 -2y 2 )(E 0 -E 0 ) + xy(1 -3y 2 )E 0 -4(xy + z 3 )E 0 H (pol) ijk + H (pol)
ijk reads ∂ ∂t + 2νk 2 E(k)H (pol) ijk (k) = S L(pol2) ijk (k) + S L(pol3) ijk (k) + S NL(pol) ijk (k). (3.36)
Moreover, the Lin equations of EH

(dir) ij and EH (pol) ij
derived are modified accordingly into

∂ ∂t + 2νk 2 E(k, t)H (dir) ij (k) = S L(dir2) ij (k) + S L(dir3) ij (k) + S NL(dir) ij (k), (3.37) ∂ ∂t + 2νk 2 E(k)H (pol) ij (k) = S L(pol2) ij (k) + S L(pol3) ij (k) + S NL(pol) ij (k). (3.38)
Note that the retro-action of the third-order contributions on the second-order ones is uniquely done through the linear transfers S

L(dir3) ij and S L(pol3) ij
, and that inversely, the impact of the second-order contributions on the third-order ones is uniquely done through the linear transfers S L(dir2) ijk and S L(pol2) ijk .

Variable shear: We consider a free-shear mixing layer created by two parallel streams of different uniform speeds U h and U l in the direction x 1 , with U h > U l , both independent of time and space. The notations of [START_REF] Dimotakis | Turbulent free shear layer mixing and combustion[END_REF] are used: the characteristic mean velocity is ∆U = U h -U l , and the mean velocity at the center of the mixing layer is U c = (U h + U l )/2. In the turbulent mixing layer, the mean flow is in the streamwise x 1 -direction and varies along the vertical x 3 -direction, from U l to U h according to

U 1 (x 3 , t) = ∆U L S (t) x 3 + U l = S(t)x 3 + U l , (3.39)
where L S is the free-shear layer length. The total turbulent velocity field can be decomposed, in the shear layer, as u

(tot) i (x, t) = U 1 (x 3 , t)δ 1i + u i (x, t). (3.40)
One can further define the dimensionless mean streamwise velocity as Ũ1 = U 1 /∆U , so that ∂ 3 Ũ1 = 1/L S , and the definition of L S is analogous to the one of the mixing length L of an unstably stratified flow (see the end of Chapter 7)

L S (t) = 6 ∆U 2 +∞ -∞ (U 1 -U l )(U h -U 1 )dx 3 , (3.41)
and is proportional to the momentum thickness θ, with L S = 6θ. The evolution equation of the streamwise mean velocity reads

∂U 1 ∂t = - ∂P ∂x 1 - ∂R 11 ∂x 1 - ∂R 13 ∂x 3 + ν ∂ 2 U 1 ∂x l ∂x l . (3.42)
Assuming that the viscous term is negligible at large Reynolds numbers, that there is no horizontal mean pressure gradient, and that the variations along x 3 are stronger than along x 1 , one gets

∂U 1 ∂t - ∂R 13 ∂x 3 , (3.43)
which was also obtained in [START_REF] Galmiche | The formation of shear and density layers in stably stratified turbulent flows: linear processes[END_REF]. Further using (3.43) in the time derivative of (3.41), one gets

dL S dt = - 12 ∆U 2 (U c -U 1 )R 13 +∞ -∞ - 12 ∆U 2 +∞ -∞ R 13 ∂U 1 ∂x 3 dx 3 = - 12 L S ∆U +∞ -∞ R 13 dx 3 , (3.44)
The first term is zero since it is assumed that there is no turbulent fluctuations outside the shear layer of extent L S (t). Then, choosing a parabolic shape for the Reynolds stress tensor R 13 , with

x 3 ∈ [-L S /2; L S /2], one gets dL S dt = - 8 ∆U < u 1 u 3 >= - 8 SL S < u 1 u 3 > . (3.45) Finally, since ∆U = -S(t)L S (t) is constant, one gets dS dt = - S L dL S dt = 8 L 2 S < u 1 u 3 >= - ∂ 2 R 13 ∂x 2 3 . (3.46)
Since most of the experimental studies investigate the development of a spatial free-shear layer, it is worth noting that the spatial extent L S can be obtained with a Taylor frozen-flow assumption, which transforms the streamwise spatial coordinate into a temporal one, according to

x 1 → U c t, U c ∂ ∂x 1 → ∂ ∂t . (3.47)
The local spatial extent δ(x 1 ) of the shear layer could be assimilated to L S with a Taylor frozenflow hypothesis, i.e. δ(x 1 ) ∼ L S (t). In [START_REF] Dimotakis | Turbulent free shear layer mixing and combustion[END_REF], the growth rate of the mixing layer region, for two fluids of equal density ρ h = ρ l = ρ 0 , is given by

δ(x 1 ) x 1 = 1 -r 1 + r C δ , (3.48)
with the ratio r = U l /U h , and C δ is a constant. In the latter reference, a noteworthy scattering of the values of C δ are reported, 0.25 ≤ C δ ≤ 0.45, with possible values outside these bounds. Further, one gets

C δ = 1 + r 1 -r δ(x 1 ) x 1 = 2U c ∆U L S U c t = 2 S(t) t . (3.49)
Here, we choose to define the self-similar free-shear layer length as The linear dependence in time of the free-shear layer length L S is assessed in Fig. 3.17a, for Saffman turbulence (Batchelor turbulence can be hardly distinguished from Saffman turbulence, thus it is not presented). The mean velocity gradient S(t) = ∆U/L S (t) is also presented and obviously evolves in t -1 .

L S (t) = α S ∆U t. (3.50)
The parameter C δ is presented in Fig. 3.17b for various initial values of the shear intensity S(0).

It seems that at sufficiently large Reynolds numbers, in the self-similar regime where L S ∼ t, C δ does not depend on S(0): a similar conclusion is obtained if L S (0) is varied instead of S(0). The values of C δ are such that 0.5 ≤ C δ ≤ 0.6, higher than what is reported in [START_REF] Dimotakis | Turbulent free shear layer mixing and combustion[END_REF].

Finally, the growth rate α S can be simply evaluated as

α S = LS ∆U = - 8R 13 (SL S ) 2 , (3.51)
and is presented in Fig. 3.17c for both Saffman and Batchelor turbulence. It appears that there is a slight dependance of the free-shear layer length L S on the infrared slope σ: indeed, it is slightly larger for σ = 2 (α L = 0.29) than for σ = 4 (α L = 0.265).

Part II

Transport and Mixing in Homogeneous Anisotropic Turbulence

Chapter 4

Spectral Modelling of a Passive Scalar in Homogeneous Turbulence "I am among those who think that science has great beauty."

-Marie Curie

This chapter aims at modelling the transport of a passive scalar field and its associated scalar flux in homogeneous anisotropic turbulence (HAT), in order to address two configurations: homogeneous isotropic turbulence with a mean scalar gradient (HITSG), and homogeneous shear turbulent with a mean scalar gradient (HSTSG). Cases of shear-driven turbulence without a mean scalar gradient will also be studied. This part is an extension of the previous modelling dedicated to the velocity field. A similar two-step approach, called anisotropic EDQNM modelling, is consistently applied here. Firstly, dynamical equations that govern the passive scalar and scalar flux fields are closed using a quasi-normal approximation and an isotropic eddy-damping procedure without any assumption regarding anisotropy. Then, for moderately anisotropic flows, scalar spherically-averaged descriptors that depend only on the wavenumber modulus k are defined. In the end, the dynamics of the velocity, passive scalar and scalar flux fields is described by six generalized spherically-averaged Lin equations: three for the velocity field, two for the passive scalar, and one for the scalar flux. These equations are valid for arbitrary mean velocity and scalar gradients of moderate intensity.

The contents of this chapter and the following one were published in: Briard, Gomez, & Cambon, "Spectral modelling for passive scalar dynamics in homogeneous anisotropic turbulence", Journal of Fluid Mechanics, 799, 159-199 (2016) Details on the calculations are provided in Appendix E along with additional considerations about the modelling of the scalar flux, and basic results about passive scalar dynamics in HIT framework are recalled in Chapter 1.

Scalar and scalar flux generalized Lin equations

The Reynolds decomposition for the scalar field T reads

T = Θ + θ, < θ >= 0. (4.1)
The mean-scalar gradient vector is written λ i = ∂ i Θ so that Θ = λ i x i . In physical space, the evolution equation of the scalar fluctuation θ in the homogeneous framework is

∂θ ∂t + A jl x l ∂θ ∂x j + u j λ j + ∂ ∂x j (θu j ) = a ∂ 2 θ ∂x j ∂x j . (4.2)
The Fourier transform of the previous equation gives

∂ ∂t -A jl k j ∂ ∂k l + ak 2 θ(k) + λ j ûj (k) = -ik j θu j (k), (4.3)
with the convolution product

θu j (k) = k=p+q θ(p)û j (q)d 3 p. (4.4)
The spectral scalar-scalar correlation E T is defined as

< θ * (p) θ(k) >= E T (k)δ(k -p), (4.5)
which corresponds to the two-point correlation R T (r) =< θ(x)θ(x + r) > in physical space. Its evolution is given by the Yaglom equation (A.97), firstly derived in [START_REF] Yaglom | On the local structure of a temperature field in a turbulent flow[END_REF], and recovered in Appendix A. The correlation E T is real, satisfies E T (k) = E T (-k), and verifies the scalar Craya equation

∂ ∂t -A jl k j ∂ ∂k l + 2ak 2 E T (k, t) + 2λ j F j (k, t) = T T,NL (k, t). (4.6)
The total non-linear scalar transfer T T,NL reads

T T,NL (k, t) = 2k i S T i (k, p, t)d 3 p . (4.7)
where S T i (k, p, t) is the three-point third-order spectral velocity-scalar-scalar correlation

S T i (k, p, t)δ(k + p + q) = i < ûi (q) θ(k) θ(p) > . (4.8)
The spectral scalar-velocity correlation F i -or the scalar flux -is defined as

< û * i (p) θ(k) >= F i (k)δ(k -p), (4.9)
which corresponds to the two-point correlation R F i (r) =< u i (x)θ(x + r) > in physical space. The scalar flux F i is solenoidal and verifies the scalar flux Craya equation

∂ ∂t -A jl k j ∂ ∂k l + (ν + a)k 2 F i (k, t) + M ij (k)F j (k, t) + λ j Rij (k, t) = T F,NL i (k, t). (4.10)
The non-linear scalar flux transfer T T,NL (4.11) where S F nm (k, p, t) is the three-point third-order spectral velocity-velocity-scalar correlation

F reads T F,NL i (k, t) = P inm (k) S F nm (k, p, t)d 3 p + k j S F * ji (p, k, t)d 3 p,
S F nm (k, p, t)δ(k + p + q) = i < ûn (q) θ(k)û m (p) > .
(4.12)

The expression of the non-linear scalar flux transfer (4.11) has also been obtained in recent study for active scalar dynamics (Burlot et al., 2015a). Furthermore, T F,NL i can be written in a way similar to τ ij for the kinetic case, namely

τ F i (k, p, t) = k n S F ni (k, p, t)d 3 p, (4.13)
so that the non-linear scalar flux transfer is

T F,NL i (k, t) = τ F i (k, p, t) + τ * F i (p, k, t) True transfer + W F i (k, t) Pressure effects . (4.14)
The term

W F i (k, t) = -α i α m τ F m (k, p, t)
is responsible for the return to isotropy of the crosscorrelation F i , i.e. the destruction of the scalar flux since it does not exist in the isotropic framework. The generalized Lin equations for the passive scalar and scalar flux are then

∂ ∂t + 2ak 2 E T (k, t) = T T,NL (k, t) + T T,L (k, t), (4.15) ∂ ∂t + (ν + a)k 2 F i (k, t) = T F,NL i (k, t) + T F,L i (k, t), (4.16)
where T T,L is the linear scalar transfer and T F,L i the linear scalar flux transfer

T T,L (k, t) = A jl k j ∂E T (k, t) ∂k l -2λ l F l (k, t), (4.17) T F,L i (k, t) = A jl k j ∂F i (k, t) ∂k l -M ij (k)F j (k, t) -λ j Rij (k, t). (4.18)
4.2 EDQNM closure for E T and F i Now that the evolution equations of E T and F i have been derived, the next step is to close the non-linear terms with the EDQNM procedure described in Chapter 2. Then, in the following section, the resulting closed expressions of the non-linear terms will be combined with a consistent modelling for anisotropy.

The quasi-normal expressions for the passive scalar and scalar flux non-linear transfers T T,NL and T F,NL i are

T T,NL (k, t) = 2k i θ T kpq T T,QN i (k, t)d 3 p, (4.19) T F,NL i (k, p, t) = P imn (k) θ F kpq T F,QN nm (k, p, t)d 3 p + k j θ F,QN kpq T * F ji (p, k, t)d 3 p, (4.20)
where θ T kpq and θ F kpq are the characteristic times of the third-order scalar and scalar flux correlations respectively

θ T kpq = 1 -exp -(a(k 2 + p 2 ) + νq 2 + µ 2 (k) + µ 2 (p) + µ 3 (q))t a(k 2 + p 2 ) + νq 2 + µ 2 (k) + µ 2 (p) + µ 3 (q) , (4.21) θ F kpq = 1 -exp -(ak 2 + ν(p 2 + q 2 ) + µ 2 (k) + µ 3 (p) + µ 3 (q))t ak 2 + ν(p 2 + q 2 ) + µ 2 (k) + µ 3 (p) + µ 3 (q) . (4.22)
Both θ T kpq and θ F kpq are obtained by writing the evolution equation of the passive scalar and scalar flux third-order correlations defined in (4.8) and (4.12) respectively. Such an approach for the scalar case has already been performed by [START_REF] Bos | Reynolds number dependency of the scalar flux spectrum in isotropic turbulence with a uniform scalar gradient[END_REF] in the framework of homogeneous isotropic turbulence with a mean scalar gradient (HITSG). The eddy-damping terms µ 2 and µ 3 reflect departure from normal laws according to

µ i (k) = A i k 0 u 2 E(u, t)du, i = 2, 3, (4.23)
with A 2 = 0 and A 3 = 1.3. The setting of A 2 and A 3 is discussed in [START_REF] Herring | A comparative assessment of spectral closures as applied to passive scalar diffusion[END_REF]; [START_REF] Lesieur | Turbulence in fluids[END_REF], along with the choice of the eddy-damping terms µ 2 and µ 3 . These two constants, also based on experimental considerations, are set to recover the Corrsin-Obukhov constant K CO 0.75, and no new constants are necessary for the scalar flux. Nevertheless, a different choice, for instance the setting A 1 = A 2 = A 3 which will be addressed in Chapter 7 for USHT, would lead to similar results; an alternative configuration is discussed by [START_REF] Bos | Passive scalar mixing in turbulent flow[END_REF] to account for pressure effects in the damping. But here, with the present definition of the scalar flux third-order correlation (4.12), pressure effects are already taken into account: A 2 = 0 affects the equation involving u i (x 3 )u j (x 2 )∂ 1 θ(x 1 ) which does not contain the fluctuating pressure. As a result of the quasi-normal approximation, T T,QN i and T F,QN ij can be written

T T,QN i (k, p, t) = 2P imn (q)F n (k, t)F m (p, t) + F * i (q, t) k n F n (p, t) + p n F n (k, t) -k n Rni (q, t) E T (k, t) -E T (p, t) , (4.24) T F,QN ij (k, p, t) = k n Rni (q, t)F * j (p, t) + Rnj (p, t)F * i (q, t) + 2F m (k, t) P imn (q) Rnj (p, t) + P jmn (p) Rni (q, t) . (4.25)
With this closure, and using calculations similar to the kinetic case detailed in Appendix E, the non-linear scalar transfer becomes

T T,NL (k, t) = 2 θ T kpq kp(xy + z)(E + X )(E T -E T )d 3 p + 2 θ T kpq k n F * n (p m F m + k m F m ) + p m F m k n F n ky -px q d 3 p. (4.26)
Since the contributions of the velocity-scalar correlation F i are quadratic in anisotropy, they are neglected in what follows in the moderate anisotropy framework. These quadratic contributions of anisotropy in the non-linear transfers for the scalar and the scalar flux are nevertheless calculated in Appendix E. The modelling of F i (k) is developed with an appropriate decomposition in the following section.

Final spherically-averaged scalar Lin equations

This section presents the final step of the modelling: combining the modelling of anisotropy with the closed expressions of the transfers obtained by EDQNM. Spherically averaged evolution equations for the scalar variance spectrum, the scalar directional anisotropy and the scalar flux are derived from the previous generalized scalar Lin equations: these final scalar equations depend only on the modulus k of the wavevector k.

Modelling of E T and F i

A decomposition similar to the one of E in the kinetic case is used for the scalar correlation E T since both E and E T verify the same properties

E T (k, t) = E T (k, t) 4πk 2 1 -15H (T ) ij (k, t)α i α j = E T 0 + E (T,dir) , (4.27) with E (T,dir) = -15E T 0 H (T ) ij α i α j and E T 0 = E T /(4πk 2 ).
One can remark that the fourth-order expansion of E T would be similar to the one of E. The following expansion is chosen for the scalar flux

F i (k, t) = 3 2 E F j (k, t)P ij (k) + (Antisymmetric contribution). (4.28)
This decomposition is consistent with the scalar flux being a solenoidal field (k i F i = 0). Hermitian symmetry for the scalar flux, F i (-k) = F * i (k), is straightforward from the decomposition of a vector into helical modes. Notably, it implies that the vector E F j is purely real and that the antisymmetric contribution is purely imaginary. The antisymmetric part brings an imaginary contribution to the scalar flux, which is zero in isotropic turbulence with or without mean scalar gradient, and which will be discussed in Chapter 8. Consequently, without helicity, only the projection part of (4.28) is considered here. Using a helical decomposition for F i shows that (see Appendix E) the scalar flux has a poloidal structure. Moreover, one can derive a realizability condition for the scalar field, starting from the decomposition (4.27) of E T , analogous to (2.52) for the kinetic field max

i (L T i ) ≤ 1 15 , (4.29)
where

L T i are eigenvalues of H (T ) ij .
The decompositions (4.27) and (4.28) are exact in the framework of homogeneous isotropic turbulence with a mean scalar gradient [START_REF] Herr | Edqnm model of a passive scalar with a uniform mean gradient[END_REF]. Whereas they are truncations at the the second order of the scalar correlation E T and scalar flux F i expansions in shear-driven turbulence, consistently with the modelling for the velocity field of E and Z done in (2.44) and (2.45).

About the distinction between directional and polarization anisotropies: scalars admit directional anisotropy only. Consequently, H (T ) ij , which appears in the expansion of E T , reflects directional anisotropy, as H (dir) ij in the decomposition (2.44) of E. Then, since F i is a vector, E F j represents polarization anisotropy. Finally, a solenoidal second-order tensor such as Rij admits both contributions. This classification is summarized in Table 4.1.

The scalar variance spectrum is given by

E T (k, t) = S k E T (k, t)d 2 k.
(4.30)

The spectral scalar directional anisotropy descriptor

H (T )
ij and the spherically-averaged scalar flux

E F i verify 2E T (k, t)H (T ) ij (k, t) = S k E (T,dir) (k, t)P ij (k)d 2 k, (4.31) E F i (k, t) = S k F i (k, t)d 2 k, (4.32) where 4πk 2 E F i = E F i (k, t). The scalar anisotropy tensor b T ij is defined as b T ij (t) = 1 K T (t) ∞ 0 E T (k, t)H (T ) ij (k, t)dk, (4.33)
where K T is the scalar variance, Finally, the second-order spectral scalar tensor φ T ij can then be written

K T (t) =< θ 2 >= ∞ 0 E T (k, t)dk. ( 4
φ T ij (k, t) = 2E T (k, t) δ ij 3 + H (T ) ij (k, t) . (4.35)
One can note that these definitions for the passive scalar field are very similar to the kinetic one.

Spherical average of the passive scalar and scalar flux

The spherically-averaged scalar and scalar flux Lin equations are then The usual non-linear spherically-averaged isotropic scalar transfer term S T,NL(iso) is

∂ ∂t + 2ak 2 E T (k, t) = S T,L(iso) (k, t) + S T,NL(iso) (k, t), ∂ ∂t + 2ak 2 E T (k, t)H (T ) ij (k, t) = S T,L(dir) ij (k, t) + S T,NL(dir) ij (k, t), ∂ ∂t + (a + ν)k 2 E F i (k, t) = S F,L i (k, t) + S F,NL i (k, t).
S T,NL(iso) (k, t) = S k T T,NL (k, t)d 2 k (4.39) = 16π 2 ∆ k θ T kpq k 2 p 2 q(xy + z)E 0 (E T 0 -E T 0 )dpdq, (4.40)
in agreement with [START_REF] Lesieur | Turbulence in fluids[END_REF]. The non-linear spherically-averaged directional scalar transfer term S

T,NL(dir) ij reads S T,NL(dir) ij (k, t) = 1 2 S k T T,NL (k, t)P ij (k) d 2 k - δ ij 3 S T,NL(iso) (k, t) (4.41) = 4π 2 ∆ k θ T kpq k 2 p 2 q(xy + z)(y 2 -1)E 0 (E T 0 -E T 0 )H (pol) ij dpdq + 8π 2 ∆ k θ T kpq k 2 p 2 q(xy + z)(3y 2 -1)E 0 (E T 0 -E T 0 )H (dir) ij dpdq + 8π 2 ∆ k θ T kpq k 2 p 2 q(xy + z)E 0 (3z 2 -1)E T 0 H (T ) ij -2E T 0 H (T ) ij dpdq. (4.42)
The isotropic term S T,NL(iso) is a conservative transfer, meaning that its integral over k is zero. However, the integral of S T,NL(dir) ij is different from zero, as the directional transfer in the kinetic case. This means that there is a return to isotropy of the passive scalar. Nevertheless, it is not possible to extract an explicit RTI term from S T,NL(dir) ij , since this mechanism is led by the pressure field which is absent of the scalar equations. This means that the RTI of the scalar field is driven by the velocity field. The production terms depend linearly both on the mean velocity and scalar gradients: the linear spherically-averaged isotropic scalar transfer S T,L(iso) is

S T,L(iso) (k, t) = S k T T,L (k, t)d 2 k (4.43) = -2A + ln ∂ ∂k (kE T H (T ) ln ) -2λ l E F l . (4.44)
The linear spherically-averaged directional scalar transfer S

T,L(dir) ij is S T,L(dir) ij (k, t) = 1 2 S k T T,L (k, t)P ij (k)d 2 k - δ ij 3 S T,L(iso) (k, t) (4.45) = - 3 7 E T A + lj H (T ) il + A + li H (T ) jl - 2 3 A + ln δ ij H (T ) ln + 1 5 A + ij E T - 1 15 A + ij ∂ ∂k (kE T ) -E T A - lj H (T ) il + A - li H (T ) jl - 1 10 λ i E F j + λ j E F i - 2 3 λ l E F l δ ij + 2 7 A + il ∂ ∂k (kE T H (T ) jl ) + A + jl ∂ ∂k (kE T H (T ) il ) - 2 3 A + lm δ ij ∂ ∂k (kE T H (T ) lm ) . (4.46)
At first order in anisotropy, with the decomposition (4.28), the non-linear scalar flux transfer becomes

T F,NL i (k, t) = 3 2 θ F kpq kE 0 E F j 2px(α i + zα i )(α j + yα j ) + q(y + xz) 2α i (α j + yα j ) -P ij + kE F j (1 -z 2 )P ij + (α i + zα i )(α j + yα j ) d 3 p + 3 2 θ F pkq k qE 0 E F j (α i + yα i ) α j + zα j + 2y(α j + xα j ) + kE 0 E F j (α i + yα i )(α j + zα j ) + (1 -y 2 ) P ij -2α i (α j + zα j ) + p E 0 E F j (α i + zα i )(α j + yα j ) -(xy + z)E 0 E F j P ij d 3 p -3 θ F kpq kα i k(1 -y 2 )E 0 E F j (α j + zα j ) + qE 0 E F j (1 -z 2 -2y(y + xz))(α j + yα j ) d 3 p. (4.47)
Since quadratic contributions of anisotropy are discarded, but nevertheless computed in Appendix E, only the isotropic part of the kinetic field appears. The non-linear sphericallyaveraged scalar flux transfer is thus

S F,NL i (k, t) = S k T F,NL i (k, t)d 2 k (4.48) = 4π 2 ∆ k θ F kpq k 2 pqE 0 kE F i (1 + y 2 -z 2 -xyz -2y 2 z 2 ) -2q(y 3 + xz)E F i dpdq + 4π 2 ∆ k θ F pkq k 2 pq E 0 qz(2xy 2 + yz -x)E F i -py(x + yz)E F i + kE 0 (1 -y 2 + z 2 -xyz -2y 2 z 2 )E F i -2(1 -y 2 )E F i dpdq. (4.49)
One can extract from the non-linear scalar-flux transfer the spherically-averaged scalar flux RTI transfer

S F,RTI i (k, t) = -8 ∆ k π 2 θ F kpq k 3 pqE 0 E F i (1 -y 2 )(1 -z 2 )dpdq. (4.50)
The linear spherically-averaged scalar flux transfer reads

S F,L i (k, t) = S k T F,L i (k, t)d 2 k (4.51) = -2λ j E 1 3 δ ij + H (dir) ij + H (pol) ij - 1 5 A + ij 2E F j + ∂ ∂k (kE F j ) . (4.52)
As a conclusion, the whole dynamics of a passive scalar field in homogeneous anisotropic turbulence is driven by six spherically-averaged compact equations. Three for the velocity field coming from Chapter 2, two for the passive scalar and one for the scalar flux. The last three ones are original results of the present work. The different anisotropy descriptors are gathered in Table 4.1.

Spherically averaged descriptors k-vectors descriptors Isotropy Directional anisotropy Polarization anisotropy

E(k, t) E(k, t) H (dir) ij (k, t) 0 Z(k, t) 0 0 H (pol) ij (k, t) E T (k, t) E T (k, t) H (T ) ij (k, t) 0 F i (k, t) 0 0 E F i (k, t)
Table 4.1: Description of anisotropy at the velocity, passive scalar and scalar flux levels, using k-vectors descriptors and their corresponding spherically-averaged contributions.

Cospectrum for an uniform mean scalar gradient

In this section, the emphasis is put on homogeneous isotropic turbulence with a mean scalar gradient (HITSG). The scalar flux is created by an uniform mean gradient

λ = (0, 0, -Λ), Λ > 0, (4.53)
whereas the kinetic field remains fully isotropic, and thus decays with time. In this framework, which has been widely investigated notably by Bos and coworkers, turbulent eddies bring the hot fluid to the cooler parts of the flow (and the opposite), thus creating a heat flux. Some definitions are given before starting the numerical study in the next chapter. Firstly, when the kinetic field is isotropic, it tends to destroy the scalar flux, created by the scalar gradient. Given the expression of the production term S F,L i , only the third component of E F i is non-zero, and its sign is opposite to the one of Λ. Thus, the cospectrum is defined as

F(k, t) = E F 3 (k, t), (4.54)
the mixed velocity-scalar correlation as

K F (t) =< u 3 θ >= ∞ 0 F(k, t)dk, (4.55)
and the cospectrum dissipation rate as

F (t) = (ν + a) ∞ 0 k 2 F(k, t)dk. (4.56)
Finally, the time evolution of the velocity-scalar correlation

R F i (t) =< u i (t)θ(t) >, for r = 0, is given by dR F i dt + R ij λ j + A ij R F j =< p ∂θ ∂x i > -2 F i -a < θ ∂ 2 u i ∂x l ∂x l > -ν < u i ∂ 2 θ ∂x l ∂x l >, with F i (t) = (ν + a) < ∂u i ∂x l ∂θ ∂x l > .
(4.57)

The last three rhs terms of the equation for R F i simplify into -F i using homogeneity. The evolution equation of F i has not received much attention, and is therefore derived and simplified for homogeneous turbulence in Appendix A, and then for HITSG in (A.34). In the classical case of a (vertical) mean scalar gradient, R F 3 = K F and the previous evolution equation simplifies into dK

F dt = P F (t) -F (t) + Π F (t), (4.58) where Π F (t) = ∞ 0 S F,NL 3 (k, t)dk (4.59)
is the cospectrum destruction, or cospectrum pseudo return to isotropy, driven by the kinetic field. And P F is the cospectrum production term

P F (t) = ∞ 0 S F,L 3 (k, t)dk = 2 3 ΛK(t), (4.60)
which decays with time along with the kinetic energy.

Chapter 5

Dynamics of a Passive Scalar in Homogeneous Turbulence

In this chapter, the complete anisotropic EDQNM modelling is used to investigate the dynamics of a scalar field and its flux, passively advected by the homogeneous turbulent flow. The Prandtl number P r is set to unity, and cases of P r 1 and P r 1 are addressed in Chapter 6. The extension to active scalar dynamics in homogeneous unstably stratified turbulence is presented in Chapter 7.

The contents of this chapter and the previous one were published in: Briard, Gomez, & Cambon, "Spectral modelling for passive scalar dynamics in homogeneous anisotropic turbulence", Journal of Fluid Mechanics, 799, 159-199 (2016) The study of a passive scalar, such as small temperature fluctuations θ, convected by a turbulent velocity field u i , is of interest for several reasons. From a fundamental point of view: though HAT has been at the center of many theoretical, numerical and experimental works for almost 40 years, numerous questions still remain without clear answers. How does the energy, mainly produced at large scales by mean velocity and scalar gradients, affect the small scales dynamics? Is there a complete return to isotropy of small scales? Is the growth or decay of integrated quantities, such as the kinetic energy and the scalar variance, predictable?

Upstream to these fundamental questions, there are practical reasons to the investigation of HAT. Indeed, taking into account anisotropy created by non-zero mean fields is an important feature to describe real flows by comparison to the classical case of HIT. Notably, the deep understanding of homogeneous turbulence dynamics could provide further insights into the analysis of high Reynolds numbers natural flows such as atmospheric and oceanic ones. Such flows are complex for multiple reasons, one being that their Reynolds numbers are much higher than the ones currently reachable in DNS and experiments. For instance, Re λ can be of order 10 4 in atmospheric flows. Such large Reynolds numbers simulations without modelling would require huge computational resources to capture only the early stage of the dynamics, and would need a fine description of all scales, from the most energetic ones to the dissipative ones at the level of the Kolmogorov wavenumber k η . In addition to very high Reynolds numbers involved in atmospheric flows, the nature itself of such flows is complex since it contains many different physical phenomena. Indeed, a fine description of atmospheric dynamics would require to take into account rotation, helicity, stratification, shear, and mean scalar gradient from the ground to high altitude [START_REF] Wyngaard | Cospectral similarity in the atmospheric surface layer[END_REF].

Mean velocity and scalar gradients are also deeply associated to production mechanisms in the turbulence dynamics regardless of the flow type. Indeed, they play a fundamental role in the energy transfers through scales. Therefore, insights concerning the role of mean velocity and scalar gradients are of great interest for turbulent flows, and specifically the ones where departure from isotropy is generated by anisotropic forces or by solid walls giving rise to shearing stresses. Moreover, a better understanding of all these anisotropic turbulent flows could be obtained by making separate investigations of isolated mechanisms at high Reynolds numbers, which are still quite unreachable using DNS. This is the approach followed in this study. In order to achieve this objective, the dynamics of a passive scalar field θ and its flux < u i θ > in HAT is addressed with the use of the anisotropic EDQNM modelling developed in Chapters 2 and 4. It is worth noting that the present model is developed for arbitrary mean velocity gradients that produce energy: consequently it is not adapted to the case of purely rotating turbulence in which there is no energy production and where the dynamics is dominated by dispersive inertial waves interacting non-linearly, requiring even more complex tools [START_REF] Cambon | Spectral approach to non-isotropic turbulence subjected to rotation[END_REF][START_REF] Sagaut | Homogeneous Turbulence Dynamics[END_REF]. The emphasis is thus put on three different configurations: Homogeneous Isotropic Turbulence with a mean Scalar Gradient (HITSG), Homogeneous Shear Turbulence (HST), and finally, these two frameworks are combined into Homogeneous Shear Turbulence with mean Scalar Gradient (HSTSG) as notably encountered in atmospheric flows.

In HITSG, the mean scalar gradient produces scalar fluctuations so that the scalar variance < θ 2 > can increase whereas the isotropic velocity field is decaying. This mean scalar gradient creates an anisotropic flux < u 3 θ >, called the cospectrum in spectral space, which has received a lot of attention: with spectral closures [START_REF] Herr | Edqnm model of a passive scalar with a uniform mean gradient[END_REF][START_REF] Bos | On the behavior of the velocity-scalar cross correlation spectrum in the inertial range[END_REF][START_REF] Bos | Reynolds number dependency of the scalar flux spectrum in isotropic turbulence with a uniform scalar gradient[END_REF][START_REF] O'gorman | Effect of schmidt number on the velocity-scalar cospectrum in isotropic turbulence with a mean scalar gradient[END_REF], with DNS [START_REF] Pumir | A numerical study of the mixing of a passive scalar in three dimensions in the presence of a mean gradient[END_REF][START_REF] Overholt | Direct numerical simulation of a passive scalar with imposed mean gradient in isotropic turbulence[END_REF], theoretically [START_REF] Lumley | Similarity and the turbulent energy spectrum[END_REF], and experimentally [START_REF] Venkataramani | Statistical features of heat transfer in grid-generated turbulence: constant-gradient case[END_REF][START_REF] Warhaft | An experimental study of the effect of uniform strain on thermal fluctuations in grid-generated turbulence[END_REF][START_REF] Sirivat | The effect of a passive cross-stream temperature gradient on the evolution of temperature variance and heat flux in grid turbulence[END_REF][START_REF] Mydlarski | Passive scalar statistics in high-péclet-number grid turbulence[END_REF][START_REF] Mydlarski | Mixed velocity-passive scalar statistics in high-reynolds-number turbulence[END_REF]. In all these studies, the scaling of the cospectrum is uncertain in the inertial range, k -7/3 or k -2 : this point is addressed hereafter.

The case of a mean velocity gradient without mean scalar gradient, has been less studied: a rapid decrease of K T =< θ 2 > was observed experimentally [START_REF] Warhaft | An experimental study of the effect of uniform strain on thermal fluctuations in grid-generated turbulence[END_REF]Karnik & Tavoularis, 1989), and this has been confirmed theoretically [START_REF] Gonzalez | Asymptotic evolution of a passive scalar advected by homogeneous turbulent shear flow[END_REF]. Interestingly, in such a configuration, the evolution of the passive scalar field is completely different from the one of the velocity field.

Finally, when both mean velocity and scalar gradients are applied, there is a continuous production of kinetic energy K(t) which grows exponentially for large dimensionless times St. Consequently, thanks to interactions with the scalar flux, K T grows exponentially as well. The HSTSG configuration has been at the center of many works as well: with a classical EDQNM approach [START_REF] Bos | Inertial range scaling of scalar flux spectra in uniformly sheared turbulence[END_REF], with DNS [START_REF] Shirani | Mixing of a passive scalar in isotropic and sheared homogeneous turbulence[END_REF][START_REF] Rogers | An algebraic model for the turbulent flux of a passive scalar[END_REF][START_REF] Brethouwer | The effect of rotation on rapidly sheared homogeneous turbulence and passive scalar transport. linear theory and direct numerical simulation[END_REF][START_REF] Kassinos | The transport of a passive scalar in magnetohydrodynamic turbulence subjected to mean shear and frame rotation[END_REF] and experimentally [START_REF] Tavoularis | Experiments in nearly homogenous turbulent shear flow with a uniform mean temperature gradient. part 1[END_REF]Danaila et al., 1999b;[START_REF] Ferchihi | Scalar probability density function and fine structure in uniformly sheared turbulence[END_REF]. Even without rotation, the HSTSG configuration remains quite representative of atmospheric flows [START_REF] Wyngaard | Cospectral similarity in the atmospheric surface layer[END_REF]. Another configuration where the kinetic energy, the scalar variance and the mixed-correlation grow exponentially conjointly is analyzed in Chapter 7 for active scalar dynamics.

Besides, the small scales RTI for each of the three cases presented is of primary importance: indeed, according to Kolmogorov (1941b), small scales of the flow should return to isotropy whatever the large scales are. While the small scales RTI of second-order moments of the velocity field is well-admitted [START_REF] Sarkar | A simple nonlinear model for the return to isotropy in turbulence[END_REF][START_REF] Pumir | Turbulence in homogeneous shear flows[END_REF][START_REF] Garg | On the small scale structure of simple shear flow[END_REF][START_REF] Shen | The anisotropy of the small scale structure in high reynolds number (r λ ∼ 1000) turbulent shear flow[END_REF], the case of third-order moments -such as the velocity derivative skewness in shear flows -is still an open question. Some considerations about statistics in HST are proposed in A. Moreover, it appeared that the scalar case is even more complicated, since the conclusion is not clear regarding second-order moments: departure from isotropy are observed experimentally and numerically at small scales with a mean scalar gradient only, in shear-driven flows, in boundary layers and in jets [START_REF] Sreenivasan | Local isotropy and large structures in a heated turbulent jet[END_REF][START_REF] Sreenivasan | On the skewness of the temperature derivative in turbulent flows[END_REF][START_REF] Sreenivasan | On local isotropy of passive scalars in turbulent shear flows[END_REF][START_REF] Pumir | A numerical study of the mixing of a passive scalar in three dimensions in the presence of a mean gradient[END_REF]Danaila et al., 1999b).

Consequently, and in order to clarify the RTI of the scalar small scales, high Reynolds numbers anisotropic flows are investigated thanks to the present anisotropic EDQNM modelling. For the different configurations (HST, HITSG, HSTSG, HSRT), comparisons with DNS and experiments are proposed in order to validate the model. Then, new numerical and theoretical results at very high Reynolds numbers are presented. The evolution equation of the scalar variance K T (t) in homogeneous turbulence reads

dK T dt = -2λ j R F j (t) -T (t), (5.1)
where the scalar variance dissipation rate T is

T (t) = 2a ∞ 0 k 2 E T (k, t)dk = 2a < ∂θ ∂x l ∂θ ∂x l > .
(5.2)

The evolution equation of T in homogeneous turbulence is derived in Appendix A.

Homogeneous shear-driven turbulence

In this section, the effects of a mean shear on the passive scalar dynamics are studied. Firstly, the scalar variance spectrum E T (k, t) is briefly investigated. Then, HSRT is addressed. Finally, the emphasis is put on HST, which presents an interesting result regarding the different behaviours of the kinetic energy and the scalar variance. All results regarding the passive scalar decay and growth laws in HSRT and HST are gathered in Table 5.1.

Scalar spectrum E T (k, t) and non-linear transfers

It can be shown by dimensional analysis that the scalar spectral tensor linked to the shear extra-diagonal component φ T 13 also evolves as k -7/3 in the inertial range between the scalar integral wavenumber k T = 1/L T and the Kolmogorov wavenumber k η . One has to assume that φ T 13 depends on the kinetic energy dissipation rate , the scalar variance dissipation rate T , the wavenumber k and the shear rate S. Since the transport equation of a passive scalar is linear with u i , it is assumed that φ

T 13 (k a , b , c T , S d ) = φ T 13 (k a , b , c T , S). Dimension analysis yields φ T 13 (k, t) ∼ S -2/3 T k -7/3 . (5.3)
This k -7/3 range is recovered both in HSRT and HST. Only the case of HSRT is presented in Fig. 5.1a. It has been said in Chapter 4 that the spherically-averaged non-linear directional scalar transfer S T,NL(dir) ij has a non-zero integral over k because the RTI process is driven by the kinetic field only. This is illustrated in Fig. 5.1b along with the isotropic scalar transfer S T,NL(iso) which has zero integral over k. ) Flux of S T,NL(iso) / max(S T,NL(iso) ) and S T,NL(iso) .

k T k η (b)

Scalar decay laws and RTI in HSRT

The scalar decay exponent α T of the scalar variance K T is well-known thanks to the CBC theory. This decay exponent has been extended to the case of Batchelor HIT for a passive scalar field in Chapter 1, using a scalar backscatter parameter p T similar to p for the kinetic field

K T (t) ∼ t α T , α T = -2 σ T -p T + 1 σ -p + 3 , (5.4)
where p T (σ = σ T = 4) = 0.27 and p T (σ = σ T ≤ 3) = 0. This parameter p T slightly depends on the Prandtl number and much more on the kinetic infrared slope σ. In Fig. 5.2, both low and large Reynolds numbers scalar decay exponents are recovered for Saffman and Batchelor HSRT.

Then, the return to isotropy of the scalar field is driven by the kinetic one: the consequence of this is that no explicit scalar RTI transfer term can be derived. Nevertheless, the RTI mechanism can be observed thanks to the anisotropy indicators b T ij in Fig. 5.3a for Saffman turbulence. An asymptotic anisotropic state is reached, similar to the kinetic one. The final non-zero values of b T ij indicate that there is still anisotropy left after the release of the velocity gradients. Batchelor turbulence is not presented, but the b T ij are found to continuously decrease, like the kinetic case in Chapter 3.

As revealed in Fig. 5.3b, the anisotropy of the scalar field is mostly contained in large scales around the scalar integral wavenumber k T . Small scales have almost returned to isotropy (H (T ) ij 0), but not completely for the extra-diagonal component, as revealed by the zoom near k η . The kinetic indicator H () ij are zero at small scales, whereas there is some anisotropy left here in the scalar small scales. This is consistent with most of the numerical and experimental observations, as it will be discussed later on. A deeper investigation of local isotropy is proposed at the end of this chapter for HSTSG.

T ij (t). (b) H (T ) ij (k, t) at t = 10 6 τ 0 (Re λ = 800).
The case of an axisymmetric contraction was studied by [START_REF] Gylfason | Effects of axisymmetric strain on a passive scalar field: modelling and experiment[END_REF], where the temperature fluctuations are created by a mean gradient that increases during the effective contraction. Although no quantitative comparison is possible because of the "moderate anisotropy" limitation of the present model, interesting qualitative facts can be reported. The measure of anisotropy is done using the fluctuating covariance C ij (t) =< ξ i ξ j >, where ξ i = ∂ i θ, which brings comparable information as the b T ij . During the contraction, |C ij | increases, and at the exit of the contraction, it converges to a constant value, different from zero. This behaviour is similar to the one of b T ij , and the authors concluded that there is a partial return to isotropy, which is in agreement with the presents results: however, there is no spectral information in [START_REF] Gylfason | Effects of axisymmetric strain on a passive scalar field: modelling and experiment[END_REF] to locate the remaining anisotropy.

Sustained shear (HST)

In this part, the shear is maintained. It has been shown in Chapter 3 that there is an exponential growth of the kinetic energy K(t) due to non-linear transfers. Is there a similar growth of the scalar variance? In the evolution equation of K T (5.1), there are no production terms unlike the evolution equation of K(t). This means that anisotropy only produces energy for the kinetic field. Hence, there should be no growth of K T (t) even if the shear is maintained. Fig. 5.4a exhibits a remarkable behaviour: b T ij and the scalar shear rapidity

S T R (t) = T (t) SK T (t)
(5.5) reach constant values for St ≥ 30, as in the kinetic case. Moreover, it is revealed that the scalar variance K T decreases exponentially in Fig. 5.4b, with a decay rate γ T -0.52. Let's replace K T and T in (5.1) (with Λ = 0) by

K T (t) = K ∞ T exp(γ T St), T (t) = ∞ T exp(γ T St).
An analytical expression for γ T is obtained

γ T = - T SK T = -0.52, K T (t) ∼ K T (0) exp(γ T St).
(5.6)

The scalar exponential exponent found by plotting K T is in good agreement with the asymptotic value of S T R , which gives γ T = -0.52. The important result is that the value of γ T does not depend on the shear rate S nor on the infrared exponents σ and σ T (and neither does γ for the exponential growth of K(t)). The scalar dissipation T , also displayed in Fig. 5.4b, exponentially decreases with the same rate γ T = -0.52, which is consistent with the evolution equation (5.1).

/

Algebraic exponents ∀S in HSRT Exp. rates ∀(S, σ, σ T , P r) in HST The fact that anisotropy accelerates the decay of the scalar field has been observed experimentally by [START_REF] Warhaft | An experimental study of the effect of uniform strain on thermal fluctuations in grid-generated turbulence[END_REF] with a contraction. Moreover, such an exponential decrease of < θ 2 > has been found theoretically by [START_REF] Pierrehumbert | Tracer microstructure in the large-eddy dominated regime[END_REF]; [START_REF] Gonzalez | Asymptotic evolution of a passive scalar advected by homogeneous turbulent shear flow[END_REF] using a self-preservation analysis. In the latter reference, the decay rate of Karnik & Tavoularis (1989) is computed by fitting the experimental data K exp T ∼ exp(-0.037x θ /M ). From this, it is possible to determine the associated γ exp T according to

K(t) α = -2 σ-p+1 σ-p+3 , p = 0 , σ ≤ 3 0.55 , σ = 4 γ = 2b 13 -/KS (t) n = -3 σ-p+5/3 σ-p+3 γ L(t) n L = 2 σ-p+3 γ/2 R 13 (t) α 13 = -2 σ-p S +1 σ-p+3 , p S = 0 , σ ≤ 3 0.279 , σ = 4 γ 13 (t) n 13 = α 13 -1 Destruction K T (t) α T = -2 σ T -p T +1 σ-p+3 γ T = -T /K T S = -0.52 T (t) n T = -σ-p+5+2σ T -2p T σ-p+3 γ T L T (t) n L T = n L γ/2
K exp T (t) ∼ exp(γ exp T St), γ exp T = -0.037U c M dU 1 /dx 2 .
The parameters are U c = 13m.s -1 , M = 0.0254m and dU 1 /dx 2 = k s U c = 8.06s -1 with the shear generator parameter k S = 0.62m -1 . Hence γ exp T = -2.35. This value seems very large and may come from a too low dU 1 /dx 2 . Indeed, details about the shear generation are provided in [START_REF] Karnik | Generation and manipulation of uniform shear with the use of screens[END_REF] where velocity gradients dU 1 /dx 2 from 43.5s -1 to 84s -1 are reported, which would respectively give γ exp

Decay and growth laws for the passive scalar in HSRT and HST

The decay and growth laws of the kinetic and scalar fields, obtained theoretically and assessed numerically, valid not only in HSRT and HST, but also in HIT, are gathered in the following Table 5.1.

Isotropic Turbulence with a mean Scalar Gradient

The cospectrum F(k, t) is now investigated in the case of homogeneous turbulence submitted to a mean scalar gradient λ 3 = -Λ with an isotropic kinetic field (HITSG). The scalar gradient accounts for a production term of scalar fluctuations, and initially the cospectrum F = 0. In these conditions, the study of the cospectrum, the only non-zero component of the scalar flux, amounts to the investigation of another passive scalar.

Spectra and transfers

The inertial scaling of the cospectrum can be found by dimensional analysis [START_REF] Lumley | Similarity and the turbulent energy spectrum[END_REF].

One has to assume that F only depends on the scalar gradient Λ, the wavenumber k and the kinetic energy dissipation rate so that

F(k, t) = C F Λ 1/3 k -7/3 , (5.7)
where C F is the cospectrum constant, found to be C F 3 in the present work. Bos (2005) reported C F 1.5 whereas O'Gorman & Pullin (2005) computed C F = 3.5 with their model. The k -7/3 dependence is similar to the fully anisotropic spectral tensor φ 13 in shear-driven turbulence. If one assumes that F depends on , k and its dissipation rate F , then F ∼ -1/3 F k -5/3 , like a passive scalar. This would imply that F is conserved throughout the cascade and this cannot be satisfied due to the pressure effects.

In what follows for numerical simulations, one needs to define a dimensionless mean scalar gradient S θ . There are different possibilities to define a reference mean scalar gradient Λ ref , unlike the mean velocity gradient which is unambiguously defined [START_REF] De Souza | The structure of highly sheared turbulence[END_REF]. We choose S θ = Λ Λ ref .

(5.8)

The reference mean scalar gradient Λ ref is defined explicitly in the following comparisons. If not mentioned otherwise, S θ = 1 is chosen.

In Fig. 5.5a, the k -7/3 scaling clearly appears for the cospectrum. However, it requires a very high Reynolds number (Re λ ≥ 10 4 here). Without it, it is hard to distinguish the theoretical power law k -7/3 from k -2 , as revealed in figure 5.5b for the moderate Reynolds number case Re λ = 100, where the inertial range is rather narrow. The k -7/3 scaling has also been obtained experimentally by [START_REF] Mydlarski | Mixed velocity-passive scalar statistics in high-reynolds-number turbulence[END_REF], in DNS by O'Gorman & Pullin (2005); [START_REF] Watanabe | Scalar flux spectrum in isotropic steady turbulence with a uniform mean gradient[END_REF] or with EDQNM by [START_REF] Bos | Reynolds number dependency of the scalar flux spectrum in isotropic turbulence with a uniform scalar gradient[END_REF].

An interesting point to mention that has not been reported so far is the infrared range of the cospectrum; indeed, since F = 0 in the initial isotropic flow, one can wonder how it evolves at very large scales. The result is displayed in Fig. 5.6a: the cospectrum infrared exponent is the same as the kinetic one σ. Moreover, the k -7/3 scaling is recovered for all the σ presented. Finally, the linear and non-linear transfers associated to the cospectrum are presented in Fig. 5.6b: S F,NL 3 -S F,RTI 3 represents the conservative non-linear transfer with zero integral over k. S F,RTI 3 is the RTI term associated with the pressure effects. S F,NL 3 is the total non-linear transfer that corresponds to a non-conservative flux. Finally, S F,L 3 is the linear transfer responsible for production of anisotropy through the scalar gradient, that decreases along time with the kinetic spectrum E(k, t). This is in agreement with similar results presented in [START_REF] Bos | Reynolds number dependency of the scalar flux spectrum in isotropic turbulence with a uniform scalar gradient[END_REF]. and

E(k, t) E T (k, t) F (k, t) k η k L k -5/3 k -7/3 (a) -1 0 1 2 3 4 -4 -2 0 2 4 log(k/k L ) log F / log k Re λ = 2.10 4 Re λ = 100 0.5 1 1.5 -2.5 -2 -1.5 -7/3 -2 (b)
F (k, t) E(k, t) F (k, t) σ = 4 σ = 2 σ = 1 σ = 3 (

Comparisons with experimental and numerical results

This section aims at assessing the anisotropic EDQNM modelling in the HITSG framework by comparisons with one DNS and one experiment.

Overholt and [START_REF] Overholt | Direct numerical simulation of a passive scalar with imposed mean gradient in isotropic turbulence[END_REF]: in this part, the emphasis it put on the ratio of the cospectrum dissipation F and cospectrum production P F defined in (4.56) and (4.60). In the DNS of [START_REF] Overholt | Direct numerical simulation of a passive scalar with imposed mean gradient in isotropic turbulence[END_REF], it is shown that the cospectrum dissipation is not negligible at low Reynolds numbers even though it decreases with Re λ . The following power law is found The ratio is evaluated for Reynolds numbers such that the kinetic field decreases according to CBC theory. It is observed that for Re λ = 28 the kinetic field is in the transition towards low Reynolds numbers regime. This is why here the minimum value for the ratio is at Re λ = 40. This agreement with DNS regarding the Re λ power law in low Reynolds numbers regime partially validates the model for the cospectrum. The discrepancy for the numerical factor arises from the fact that initial conditions are different, and mainly because in the DNS the velocity field is forced, whereas it is freely decaying here. Nevertheless, the Re λ power law is recovered. As for the high Reynolds numbers regime, the Re -1 λ predicted by [START_REF] Bos | Reynolds number dependency of the scalar flux spectrum in isotropic turbulence with a uniform scalar gradient[END_REF] is recovered numerically in Fig. 5.7 as well. This scaling law can be obtained analytically, assuming that in high Reynolds numbers regime the dominant region of the kinetic spectrum and cospectrum is the inertial range

F (t) P F (t) = 3(ν + a) 2Λ ∞ 0 k 2 F(k, t)dk ∞ 0 E(k, t)dk ∼ kη k L 1/3 k -1/3 dk kη k L 2/3 k -5/3 dk .
(5.9)

Then, using classical relations such as

(k η /k L ) = Re 3/4 L
where Re L is the integral Reynolds number so that √ Re L ∼ Re λ , and νk

4/3 η = 1/3 , one finds F /P F ∼ Re -1 λ .
Sirivat and [START_REF] Sirivat | The effect of a passive cross-stream temperature gradient on the evolution of temperature variance and heat flux in grid turbulence[END_REF]: in this part, the results provided by the current model are compared with the experimental work of [START_REF] Sirivat | The effect of a passive cross-stream temperature gradient on the evolution of temperature variance and heat flux in grid turbulence[END_REF]. The case where the scalar gradient is created with a mandoline (a screen of thin heated wires) is chosen. The parameters of the experiment are the following ones: the input speed is U = 3.4m.s -1 and the meshsize M = 0.024m. For this configuration, the initial Reynolds number is Re λ (0) = 26.4 and the turn-over time τ exp = 1.14s. The scalar dissipation rate, written θ for the experiment, is

θ 10 -2 • C 2 .s -1 for a scalar gradient β = 1.78 • C.m -1 .
Assuming that for this experiment the Prandtl number is about 0.7, a reference fluctuating scalar gradient is computed as

∂θ ∂x ref = Λ ref = θ 3a ,
so that the dimensionless scalar gradient is S θ = β/Λ ref = 0.152. Temporal results are transposed to spatial ones through

x M = t τ 0 U τ exp M ,
where τ 0 is the kinetic characteristic time K/ evaluated numerically after two turn-over times, so that transition effects from the initial conditions are erased. The experimental decay rate of the kinetic field being α exp = -1.3, Saffman turbulence (σ = 2) is an appropriate large scales initial condition for the simulations. The cospectrum correlation

ρ u i θ = < u i θ > < u 2 i >< θ 2 > , ρ u 3 θ = ρ wθ , (5.10)
is well recovered in Fig. 5.8a, and ρ wθ → -0.7. The ratio of scalar production -ΛK F and dissipation T is also in agreement with the present results in Fig. 5.8c. The final value of the characteristic times ratio

R T = K T K T (5.11)
matches quite well experimental data in Fig. 5.8d. However, there is a slight discrepancy in Fig. 5.8b for the ratio of integral scales L T /L defined in (1.37) ( 0.7 with EDQNM, and 0.9 experimentally). Since no definitions are given in the experiment, there could be a difference in the definitions. Nevertheless, the fact that L T < L is recovered. Let's underline that initial conditions are isotropic, which is not the case in the experiment: as mentioned by [START_REF] Sirivat | The effect of a passive cross-stream temperature gradient on the evolution of temperature variance and heat flux in grid turbulence[END_REF], the initial fluctuating temperature field is slightly inhomogeneous, and because of the grid itself the kinetic field contains some anisotropy. But still, the "early times" of ρ wθ and -ΛK F / T are well captured by the anisotropic EDQNM modelling.

About the cospectrum correlation ρ wθ , it has to be pointed out that there exists a large scatter, since measured values span from -0.19 to -0.8, with an average around -0.65 [START_REF] Venkataramani | Statistical features of heat transfer in grid-generated turbulence: constant-gradient case[END_REF][START_REF] Sirivat | The effect of a passive cross-stream temperature gradient on the evolution of temperature variance and heat flux in grid turbulence[END_REF][START_REF] Overholt | Direct numerical simulation of a passive scalar with imposed mean gradient in isotropic turbulence[END_REF][START_REF] Mydlarski | Mixed velocity-passive scalar statistics in high-reynolds-number turbulence[END_REF]. It appears in EDQNM simulations that ρ wθ strongly varies for 5 ≤ Re λ ≤ 300, roughly from 0.6 to 0.71 for decreasing Reynolds numbers. This is the classical range of Reynolds numbers covered by experiments and DNS. Consequently, the reason for the scattering of ρ wθ could be moderate Reynolds numbers, where it becomes more sensitive to the intensity of the mean-scalar gradient. Another explanation is proposed in Chapter 6.

Decay and growth laws for the cospectrum and passive scalar

In this section, the anisotropic EDQNM modelling is used to address the high Reynolds numbers regimes of HAT. The satisfactory agreements obtained at moderate Reynolds numbers in the previous comparisons give confidence in the following investigation. The main contribution of this chapter is the new results presented in HITSG, which aim at predicting the decay and growth of the mixed-correlation and scalar variance. These results, gathered in Table 5.2, also exhibit the dominant mechanisms during the decay. Decay of K F (t) and F (t): the scalar flux F is destroyed by the classical decay of the kinetic field and consequently experiences a decay itself. Is it possible to derive theoretical decay exponents based on CBC theory for < u 3 θ >? Two assumptions based on physical arguments for high Reynolds numbers regime need to be made: (i) It has been shown in Fig. 5.6a that the cospectrum does not have a specific infrared exponent and is completely controlled by the kinetic field. Therefore, only the inertial k -7/3 range, starting at the integral wavenumber k L , should be taken into account into the cospectrum decay process. (ii) In the case of Batchelor turbulence, backscatter parameters p and p T are introduced for the kinetic and scalar fields respectively. Since the cospectrum F is the spectral counterpart of the velocity-scalar cross correlation, its backscatter parameter p F should contain both effects. Hence, the simplest form is chosen: p F = (p + p T )/2 for σ = 4, which gives p F = 0.4075 with P r = 1.

With these reasonable assumptions, one can write

K F (t) = ∞ k L F(k, t)dk ∼ k -4/3 L 1/3 .
(5.12)

Injecting in this equation decay exponents of kinetic integrated quantities recalled in Table 5.1, and using p F , one finds

K F (t) ∼ t α F , α F = - σ -p F -1 σ -p + 3 . (5.13)
It can be deduced from the scaling (5.12) that the dominant mechanism in the decay of F at high Reynolds numbers is not production, but inertial effects of the velocity field. This is not true anymore in the low Reynolds numbers regime, where the production term P F leads the dynamics.

From the evolution equation (4.58) of the mixed-correlation, only dK F /dt ∼ P F = 2ΛK/3 remains. The return to isotropy term Π F , or cospectrum destruction, is weak and F is also negligible: indeed, given the shape of the cospectrum dissipation (4.56), this term is strong in the inertial range, which does not exist anymore in the low Reynolds regime. This immediately yields

α F = - σ -1 2 .
(5.14)

These decay exponents do not depend on the scalar gradient Λ, and are assessed numerically in Fig. 5.9a in the high and low Reynolds numbers regimes. The agreement is excellent even for the particular case of Batchelor turbulence. This is the first time such a result is presented: indeed, in existing DNS, the kinetic field is forced or artificially frozen so that no decay can occur. From (5.13) and (5.14), it follows that K F does not decay for σ = 1. In this case, the Reynolds number Re λ remains constant and so the dynamics of the inertial range, on which is based α F , remains unchanged. These theoretical decay exponents, assessed numerically, give further insights into the prediction of high Reynolds numbers decay in HITSG: this is also an extension of an analysis previously applied to the passive scalar in HIT in Chapter 1.

Regarding the cospectrum dissipation rate: since the scalar flux is a purely anisotropic quantity, F is not a conserved quantity unlike and T . Therefore, it is not possible to express it under the shape of a power law. Nevertheless, as the inertial range disappears in the low Reynolds numbers regime, it is possible to compute the decay exponent n F of F from the evolution equation (4.58) so that

F (t) ∼ t n F , n F = α F -1 = - σ + 1 2 .
(5.15)

The agreement between this theoretical result and numerical simulations is displayed in Fig.

5.9b.

Growth of K T (t) and T (t): the effect of the mean scalar gradient Λ on the passive scalar itself is now addressed. Such a study was not performed in previous references. The scalar spectrum still displays a k -5/3 inertial-convective range despite the mean gradient, as obtained experimentally by [START_REF] Mydlarski | Passive scalar statistics in high-péclet-number grid turbulence[END_REF]. In the HITSG framework, the evolution equation (5.1) of K T is driven by both dissipation and production. In the presence of a mean scalar gradient Λ, the production term leads the dynamics of the passive scalar field. Using the previous results regarding the decay exponents of K F , one can compute the exponent of K T in presence of a mean scalar gradient α Λ T . One gets in high and low Reynolds numbers regimes respectively

K T (t) ∼ t α Λ T , α Λ T = 4 + p F -p σ -p + 3 , K T (t) ∼ t α Λ T , α Λ T = - σ -3 2 .
(5.16)

(5.17)

The agreement between these theoretical expressions of α Λ T and numerical simulations is presented in Fig. 5.10a for both high and low Reynolds numbers regimes. The exponents do not depend on the scalar gradient Λ. For high Reynolds numbers, the scalar variance grows in time whatever σ is, whereas for low Reynolds numbers, it decays for σ = 4. This can be explained with the following arguments. The theoretical prediction (5.17) of α Λ T is based on the fact that the dynamics of K T is driven by the production term 2ΛK F , and K F is stronger for smaller σ. Consequently, for a large infrared slope such as σ = 4, the spectrum E(k, t), and thus F(k, t), has less energy in large scales, resulting into a weak production term for the scalar variance that does not balance dissipation. This is consistent with HITSG experimental results at moderate Re λ , where the scalar variance can grow or decay depending on the intensity of Λ: with a weak Λ, K T still decays (nevertheless more slowly than in HIT). Therefore, there is a link between physical and spectral spaces: a strong mean scalar gradient Λ corresponds to a small infrared slope σ, or equivalently to energetic large scales.

Moreover, in the particular case of Saffman turbulence (σ = 2), the value α Λ T = 4/5 was already found by [START_REF] Chasnov | Similarity states of passive scalar transport in buoyancy-generated turbulence[END_REF]. Therein, the decay and growth laws of passive and active scalar fields, with and without mean gradient, are studied. Power laws for the active scalar fields were assessed by Large Eddy Simulations (LES). Hence, the present EDQNM simulations valid the power laws for the passive scalar field, with an explicit dependence on the initial large scales conditions σ. Consequently, this result can be seen as an extension of [START_REF] Chasnov | Similarity states of passive scalar transport in buoyancy-generated turbulence[END_REF] on the passive scalar.

Finally, the dissipation rate of the passive scalar field is investigated. From the scalar variance evolution equation (5.1), it can be deduced that T should evolve as K F , meaning

T (t) ∼ t n Λ T , n Λ T = α F (5.18)
in both high and low Reynolds numbers regimes. This is assessed in Fig. 5.10b. All these new and partially new results regarding decay and growth exponents for K T and K F and their dissipation rates are gathered in Table 5.2. They notably permit to explain why the cospectrum correlation (5.10) does not depend on σ in both high and low Reynolds numbers regimes: indeed, the computation using the previous exponents shows that the time evolution of ρ wθ does not depend on σ anymore for very large or small Reynolds numbers.

When a passive scalar is forced with a mean gradient, it is fully dominated by the decaying isotropic velocity field which completely leads the dynamics of the flow. Indeed, the infrared scalar exponent σ T has no influence on the decay and growth exponents.

A similar result was obtained by De [START_REF] De Marinis | Temperature dynamics in decaying isotropic turbulence with joule heat production[END_REF] in HIT where the passive scalar field experiences a Joule heat production. Therefore, it can be concluded that in a presence of a production mechanism, the velocity field completely dominates the passive scalar dynamics. This will not be true anymore for an active scalar in Chapter 7.

High Reynolds regime ∀(Λ, σ T , P r) Low Reynolds regime ∀(Λ, σ T , P r)

K T (t) α Λ T = 4+p F -p σ-p+3 , p F = 0 , σ ≤ 3 1 2 (p + p T ) , σ = 4 α Λ T = -σ-3 2 T (t) n Λ T = α F = -σ-p F -1 σ-p+3 n Λ T = α F = -σ-1 2 K F (t) α F = -σ-p F -1 σ-p+3 α F = -σ-1 2 F (t) not defined n F = -σ+1 2
Table 5.2: Decay and growth exponents of integrated quantities in HITSG for the cospectrum and scalar fields.

Remark on the scalar large scales in HITSG: the mean scalar gradient Λ results in a production term in the passive scalar equation. This scalar production is linked to the cospectrum, itself linked to the kinetic energy spectrum. Hence, the "minimum of energy" of the flow is imposed by the infrared range of the kinetic spectrum, i.e. imposed by σ. So, the scalar infrared exponent σ T changes if initially different from σ. There are two cases: (i) σ T (t = 0) > σ rapidly results into σ T = σ. Indeed, σ T > σ means K T < K: since the kinetic field imposes the minimum of energy, σ T decreases. For instance, if one has σ T (t = 0) = 4 and σ = 2, the self-similar regime is σ = σ T = 2. (ii) For σ T < σ, then σ T = σ but it takes more time, as revealed in Fig. 5.11. Without the scalar gradient, the scalar variance would decrease more slowly than the kinetic energy. The production term being proportional to K(t), it forces the scalar field to grow with the infrared slope σ T = σ.

-

4 -2 0 2 -6 -4 -2 0 2 4 6 log k log E T / log k t = 0 t = 10 -2 t = 0.1 t = 1 t = 10 (a) -4 -2 0 2 -6 -4 -2 0 2 4 6 log k log E T / log k t = 0 t = 0.1 t = 10 t = 10 3 t = 10 5 (b)

Return to isotropy in HITSG

The small scales RTI is briefly addressed for HITSG at the level of scalar second-order moments.

Since only the third component of the mean scalar gradient is non-zero, this is an axisymmetric configuration, meaning that the scalar anisotropy indicators verify 2H

(T ) 11 = 2H (T ) 22 = -H (T )
33 . In Fig. 5.12a, the b T ij are presented: they become constant both in Saffman and Batchelor turbulence. This is qualitatively the same behaviour as the b ij in a sustained shear flow. Then, in the low Reynolds numbers regime, the b T ij increase and reach a constant asymptotic value, meaning that there is anisotropy left in the flow. The fact that anisotropy increases in the low Reynolds numbers regime has already been observed for the velocity field in Chapter 3. The spectral anisotropy tensor H (T ) ij reveals that small scales of the scalar second-order moments completely return to isotropy in Fig. 5.12b. Moreover, it has been pointed in experiments and DNS [START_REF] Pumir | A numerical study of the mixing of a passive scalar in three dimensions in the presence of a mean gradient[END_REF]Danaila et al., 1999b) that at the level of scalar third-order moments, anisotropy remains in the small scales. This is not incompatible with small-scales isotropic second-order moments as shown recently by [START_REF] Bos | On the anisotropy of the turbulent passive scalar in the presence of a mean scalar gradient[END_REF]. An interesting analogy can be made between the velocity field in a sustained shear flow and the scalar field in HITSG. For both fields, (i) there is a complete return to isotropy of small scales; (ii) there are no significant differences between Saffman and Batchelor turbulence: in both cases b ij and b T ij reach constant values (see Fig. 5.12a for the b T ij and Fig. 3.10a for the b ij ). (iii) Anisotropy fills large scales and does not remain around the integral wavenumber as in HSRT.

Homogeneous Shear Turbulence with Scalar Gradient

This final part focuses on homogeneous shear turbulence with a mean scalar gradient (HSTSG). The emphasis is put on the impact of both mean velocity and scalar gradients on the scalar flux and the passive scalar.

Definitions and transfers

Previously, it has been shown that with a mean scalar gradient Λ, only the third component of the scalar flux is non-zero, namely the cospectrum F. With a mean velocity gradient only, no scalar flux appears at all, and the scalar variance decays exponentially. With both mean velocity and scalar gradients, the first component of the scalar flux is also non-zero. Thus, the streamwise flux is defined as

E F 1 (k, t) = F S (k, t), (5.19)
and arises only due to the combined presence of a both mean velocity and scalar gradients. The streamwise mixed-correlation reads

K S F (t) = ∞ 0 F S (k, t)dk.
(5.20)

In Fig. 5.13, both linear and non-linear spherically averaged transfers are presented for the cospectrum and the streamwise flux in HSTSG. One can note that they are very similar and slightly differ in intensity.

k/k L kS F,L i (k, t) kS F,L 1 (k, t) kS F,L 3 (k, t) k L k η (a)

Comparisons with experimental and numerical results

This part aims at assessing the anisotropic EDQNM modelling in HSTSG -a configuration which combines various mechanisms at stake in atmospheric flows -by comparisons to two DNS and one experiment.

Tavoularis and [START_REF] Tavoularis | Experiments in nearly homogenous turbulent shear flow with a uniform mean temperature gradient. part 1[END_REF]: EDQNM simulations are compared to the experiment of [START_REF] Tavoularis | Experiments in nearly homogenous turbulent shear flow with a uniform mean temperature gradient. part 1[END_REF]. Such a comparison has also been performed by [START_REF] Bos | Passive scalar mixing in turbulent flow[END_REF] and the conclusions will be discussed. The mean speed is U c = 12.4m.s -1 and the characteristic length is the shear generator one h = 0.305m. The mean velocity and scalar gradients are dU 1 /dx 2 = 46.8m.s -1 and dT /dx 2 = 9.5 • C.m -1 . From the data of the kinetic characteristic time written τ u = 2τ 0 = 0.26s, one has S = 6.19τ -1 0 . Then, from the scalar characteristic time τ θ and < θ 2 >, it is possible to evaluate the scalar dissipation rate θ = 0.128 • C 2 .s -1 at x 1 /h = 7.5, and thus to compute the reference scalar gradient (∂θ/∂x) ref =

θ /(2a) = 52.1 • C.m -1 so that S θ = 0.1823. It is worth noting that initial isotropic conditions are used here, which is clearly not the case in the experiment. Two final Reynolds numbers are given in [START_REF] Tavoularis | Experiments in nearly homogenous turbulent shear flow with a uniform mean temperature gradient. part 1[END_REF]: R λg = 160 scaled for an isotropic framework, and R λ 11 = 266 for inhomogeneous flows. The comparisons are presented in Fig. 5.14a to 5.14d. Data is available at three locations:

x 1 /h = 7.5, 9.5 and 11. Using the appropriate conversion in dimensionless time, written τ in [START_REF] Tavoularis | Experiments in nearly homogenous turbulent shear flow with a uniform mean temperature gradient. part 1[END_REF][START_REF] De Souza | The structure of highly sheared turbulence[END_REF], one has

τ = St = x 1 U c dU 1 dx 2 ,
which provides experimental information at St = 8.63, 10.94 and 12.66. There are satisfactory agreements in Fig. 5.14a for the cospectrum and streamwise flux correlations ρ uθ and ρ vθ . Similar values for ρ uθ and ρ vθ are reported in [START_REF] Ferchihi | Scalar probability density function and fine structure in uniformly sheared turbulence[END_REF] which once again confirms the relevance of the present EDQNM closure for the passive scalar and scalar flux. A satisfactory agreement is also obtained in Fig 5 .14b for the characteristic times ratio R T defined in (5.11). A first discrepancy is observed for

B(t) = Λ S 2K K T , (5.21)
which is underestimated in Fig. 5.14d, whereas it is overestimated in [START_REF] Bos | Passive scalar mixing in turbulent flow[END_REF]. B(t) seems to be very dependent on initial conditions, which could explain the discrepancy. One has to keep in mind that here initial conditions are isotropic, whereas in the experiment there is initial anisotropy in the flow, difficult to model. Finally, in Fig. 5.14c a difference is observed for the turbulent Prandtl number

P r T (t) = Λ S R 12 (t) K F (t) , (5.22)
where P r exp T 1.1 and P r EDQNM T 0.74. The value obtained experimentally seems quite large: indeed, atmospheric data and theoretical considerations suggest that one should obtain 0.6 ≤ P r T ≤ 0.8 [START_REF] Herring | A comparative assessment of spectral closures as applied to passive scalar diffusion[END_REF][START_REF] Lesieur | Turbulence in fluids[END_REF], in agreement with existing values [START_REF] Shirani | Mixing of a passive scalar in isotropic and sheared homogeneous turbulence[END_REF][START_REF] Rogers | An algebraic model for the turbulent flux of a passive scalar[END_REF]. The comparison with the results of [START_REF] Bos | Inertial range scaling of scalar flux spectra in uniformly sheared turbulence[END_REF] is not relevant here because a constant of their model for linear transfers is set so that P r T = 1.1 is recovered.

Rogers, Mansour and Reynolds (1989):

The comparison is made with the DNS of [START_REF] Rogers | An algebraic model for the turbulent flux of a passive scalar[END_REF]. There, the mean velocity gradient dU 1 /dx 2 = S = Sτ -1 0 is such that the dimensionless shear is S = 14.142. Three cases for the scalar gradient are performed, one in each direction x 1 , x 2 and x 3 , with S θ = 2.5. Comparisons are made with the diffusivity tensor defined as

D ij (t) = -< θu i > dT dx j -1
.

(5.23)

Each column of D ij refers to a different simulation where the direction of the scalar gradient changes. For instance, D 13 refers to the third case. The agreement between EDQNM simulations and DNS is revealed in Fig. 5.15a where D ij is normalized by D 22 . A difference is observed along the flow direction for D 11 /D 22 where DNS predicts a higher value. This discrepancy may come from the limited DNS resolution that alters the dynamics in the streamwise direction. For each simulation, the turbulent Prandtl number is defined by

P r T (t) = - R 12 (t) SD ii (t) , (5.24)
where D ii (no summation) is the turbulent diffusivity, with i = 1, 2 or 3 depending on the case considered. The agreement is rather good in Fig. 5.15b: the classical asymptotic value of P r T = 0.8 is recovered in the second case, whereas the general behaviour is captured for the first and third cases: P r

(1)

T < P r

(3)

T < P r (2)
T . Nevertheless, the present simulations seem less sensitive to the mean scalar gradient intensity than experiments. Moreover, for larger St, P r

(2) T would be quite smaller, which indicates that the present anisotropic EDQNM modelling slightly underestimates the turbulent Prandtl number. A possibility to correct these lower values of P r T is to set the eddy-damping constants as A 1 = A 2 = A 3 = 0.355. [START_REF] Kassinos | The transport of a passive scalar in magnetohydrodynamic turbulence subjected to mean shear and frame rotation[END_REF]: a last comparison is performed with the DNS of [START_REF] Kassinos | The transport of a passive scalar in magnetohydrodynamic turbulence subjected to mean shear and frame rotation[END_REF]. This work deals about MHD but the validation is made in the purely hydrodynamic case with the data of [START_REF] Brethouwer | The effect of rotation on rapidly sheared homogeneous turbulence and passive scalar transport. linear theory and direct numerical simulation[END_REF]. Hence, only the case where the magnetic field is zero and where there is no rotation is considered. The mean velocity and scalar gradients are along x 2 such that S = 8.95 and S θ = 1. The kinetic field is allowed to decay without any forcing before velocity and scalar gradients are applied at t 0 . There, the Reynolds number is Re λ = 45 with SK/ (t = t 0 ) = 18. The scalar fluctuations are set to 0 at t = t 0 : this is why initially ρ vθ (t = t 0 ) = -1. For EDQNM simulations, S and Λ are applied after two turn-over times and there SK/ = 13 and Re λ = 50. The two correlations ρ uθ and ρ vθ are presented in which characterizes the relative strengths of the velocity and scalar fluctuations. There is a good agreement for the asymptotic values of ρ uθ and ρ vθ . Our weaker value for ρ uθ at moderate St may be the consequence of a slightly too strong growth of R 11 =< u 2 1 >. This does not prevent to reach the correct value at larger St in the asymptotic state. As for β, EDQNM simulations slightly differ from the DNS in Fig. 5.16b, where β is over-estimated and has almost reached a constant value whereas it slightly decreases in [START_REF] Kassinos | The transport of a passive scalar in magnetohydrodynamic turbulence subjected to mean shear and frame rotation[END_REF]. Nevertheless, in both cases β ∼ 1 at large St, which indicates that the velocity and scalar fluctuations have a similar contribution to the anisotropic asymptotic state.

Growth of

K, K T , K F and K S F
In this part, the emphasis is put on the growth of the scalar variance and its interactions with the scalar flux. Some additional results about the passive scalar and the scalar flux are presented, which may be of interest for one-point modelling, such as negligible quantities at high shear rates. The scalar anisotropy tensors b T ij are presented in Fig. 5.17 along with the scalar flux shear rapidities

S F R (t) = F K F S , S F ,S R (t) = S F K S F S .
(5.26)

As in the HST framework without mean scalar gradient, the scalar indicators reach constant values for large St, and the ratio S θ /S impacts only the short time dynamics without modifying the asymptotic state. There is a noteworthy similarity with the behaviour of b ij in shear flows. An interesting feature is that S F R and S F ,S R tend to zero for large St, whereas their kinetic and scalar counterparts do not. This means that the linear effects of shear are preponderant over non-linear exchanges: this is in agreement with figure 5.13, where the scalar flux transfers of energy are gathered at large scales, dominated by linear mechanisms. The evolution of the scalar variance and mixed-correlations are given by

F /(K (S) F S) ǫ S F /(K S F S) ǫ F /(K F S) (b)
dK T dt = 2ΛK F -T (5.27) dK F dt = ΛR 33 +Π F -F (5.28) dK S F dt = ΛR 13 +SK F +Π S F -S F , (5.29)
Here is what happens simultaneously: the cross-correlation R 13 produces K S F through the mean scalar gradient Λ. Then, R 13 brings energy to the transverse component R 33 thanks to nonlinear redistribution, which causes K F to grow as well through Λ. Finally, K F provokes the growth of K T . It is possible to compute the rapid pressure parts of the scalar flux Π F and Π S F . Details are given in Appendix E. This gives for the cospectrum 5Π r F (t) = SK S F . The numerical factor 0.08 obtained by [START_REF] Bos | Inertial range scaling of scalar flux spectra in uniformly sheared turbulence[END_REF] is far from the present 0.2. However, the streamwise part, 5Π r,S F (t) = -4SK F , is closer from 0.62 of the latter reference.

The main result here is that the scalar variance K T , which was exponentially decreasing in HST, now grows exponentially in HSTSG, as revealed in Fig. 5.18a. Its scalar exponential growth rate γ T is identical to the kinetic one γ so that γ T = γ = 0.34. This is qualitatively in agreement with the experimental work of [START_REF] Ferchihi | Scalar probability density function and fine structure in uniformly sheared turbulence[END_REF]: they found that K and K T grow with the same rate in the presence of scalar and velocity gradients (with γ exp 0.085, which is far inferior to γ = 0.34, and lower than common experimental ones). Moreover, both K F and K S F grow exponentially with the rate γ as well. Growths of the mixed-correlations have been obtained numerically by [START_REF] Rogers | The structure of a passive scalar field with a uniform mean gradient in rapidly sheared homogeneous turbulent flow[END_REF], even though it is complicated to determine if the growth is algebraic or exponential due to the DNS limitation. The fact that all these correlations grow exponentially with the same rate is consistent with the constant scalar flux correlations ρ u i θ obtained experimentally and in DNS for sufficiently high St. From the equations (5.27)-(5.29), it is possible to determine the expression of the cospectrum, streamwise flux and scalar exponential growth rates γ F , γ S F and γ T . Using the fact that F /(K F S) → 0 and S F /(K S F S) → 0, one has

1 K F S dK F dt = Λ S R 33 K F + Π F K F S Constant for St 1 = γ F , (5.30) 1 K S F S dK S F dt = K F K S F (1 + P r T ) + Π S F K S F S Constant for St 1 = γ S F , (5.31) 1 K T S dK T dt = 2 Λ S K F K T - T K T S Constant for St 1 = γ T .
(5.32)

The agreement between the asymptotic values of these quantities and the 0.34 expected are presented in Fig. 5.18b. Moreover, simulations show that γ F , γ S F and γ T do not depend on large scales initial conditions σ, as for the kinetic rate γ. The definitions (5.30) and (5.31) are not convenient for calculations since the slow-part of Π (S)

F cannot be expressed explicitly. One could want to use one-point modelling, such as the ones proposed by [START_REF] Wikström | Derivation and investigation of a new explicit algebraic model for the passive scalar flux[END_REF]. The simplest model that can be found is (5.33) where c 1 = 3.2 is calibrated on experimental data. Injecting this model in (5.30) and (5.31), and using the previous explicit calculations of the rapid contribution, yields

Π F,slow i (t) -F i (t) = -c 1 K < u i θ >,
γ F = Λ S R 33 K F + 1 5 K S F K F -3.2 KS 0.2827 (5.34) γ S F = K F K S F 1 5 + P r T -3.2 KS 0.2122. (5.35)
The obtained values of γ F and γ S F are not too far from 0.34. However, γ F = γ S F , which shows that the simplest models cannot handle such complex flows.

Streamwise flux spectrum F S (k, t)

The inertial scaling of the streamwise flux F S has not received much attention yet. [START_REF] Wyngaard | Cospectral similarity in the atmospheric surface layer[END_REF] proposed a scaling in k -3 . However such a slope did not agree well with atmospheric data and they concluded that a k -5/2 range would be more appropriate. The k -3 slope is obtained by assuming that F S depends on , k, S and Λ, so that F S (k, t) ∼ ΛSk -3 .

(5.36)

This expression can also be found starting from F S ∼ Λ 1/3 k -7/3 and replacing 1/3 by its expression as a function of the shear scale defined in (3.19), 1/3 ∼ k -2/3 S. [START_REF] Bos | Inertial range scaling of scalar flux spectra in uniformly sheared turbulence[END_REF] derived a k -23/9 range both analytically and numerically based on tensorial arguments. 1/3 is replaced by 1/3 ij (k) so that the scalar flux is written

F i (k, t) ∼ λ j 1/3 ij k -7/3 , (5.37) with ij (k)E(k) = 3φ ij (k)
. This recovers the classical scaling for F and yields

F S (k, t) = C S F Λ S 1/3 2/9 k -23/9 .
(5.38)

In a recent paper, [START_REF] Knaus | On the effect of heat release in turbulence spectra of non-premixed reacting shear layers[END_REF] studied reactive and non-reactive scalar flux spectra with DNS and found that a k -7/3 range was a satisfactory scaling. Such a spectral behaviour can be recovered assuming that F S depends on , k, S and

S F F S (k, t) ∼ S -2/3 F k -7/3 . (5.39)
But this is not consistent with the fact that F i is not conserved (and consequently that the scalar flux does not exist in the isotropic framework). Older atmospheric measurements reported a k -5/3 scaling [START_REF] Antonia | Inertial range behavior of the longitudinal heat flux cospectrum[END_REF]. Numerically, it is revealed in Fig. 5.19 that there is a good agreement for the streamwise flux F S with the k -23/9 scaling predicted by [START_REF] Bos | Inertial range scaling of scalar flux spectra in uniformly sheared turbulence[END_REF]. Numerically, the constant is found to be C S F 1.5.

F S (k, t) F (k, t) k -7/3 k -23/9 k -5/3 k η k L (a) -2 0 2 4 6 -4 -3 -2 -1 0 1 2 3 log k/k L log E/ log k log E T / log k log F S / log k log F / log k k η k L -23/9 -7/3 -5/3 (b)

Return to isotropy in HSTSG

The small scales RTI is finally addressed for HSTSG at the level of the scalar second-order moments. In Fig. 5.20, the scalar anisotropy tensors

H (T )
ij are presented at large St for HSTSG, at two different Reynolds numbers: local isotropy is almost respected for second-order moments of the scalar field as in the inertial range one has

H (T ) ij (k → k η , t) → 0. Nevertheless, H (T ) ij
is not rigorously zero at small scales, especially the extra-diagonal component H (T ) 13 . This shows that in the presence of shear, and in agreement with most of the DNS and experiments, some anisotropy persists at the scalar small scales, even at the second-order moments level. Furthermore, the Reynolds number is found to have a non-negligible impact on the small scales anisotropy: indeed, small scales anisotropy reduces from Re λ = 2400, which is slightly higher than Reynolds numbers reached in DNS, to Re λ = 1, 5.10 4 . The shear wavenumber k S = S 3 / is displayed as well: for wavenumbers k > k S , non-linear effects are dominant, consistent with the RTI of small scales, whereas for k < k S , linear effects are stronger are carry most of the anisotropy. A last remark is that the presence of a mean scalar gradient seems to smooth the scalar large scales anisotropy: indeed, for shear-driven flows without mean scalar gradient, anisotropy is gathered around the scalar peak of energy k T and is weaker in the infrared range (see Fig. 5.3b). Whereas for HSTSG, anisotropy progressively increases from moderate to large scales.

Conclusions for the passive scalar at P r = 1

This chapter was an application of the anisotropic EDQNM modelling. Three different configurations were considered, whose comprehension is crucial to understand the dynamics of complex flows such as atmospheric ones: isotropic turbulence with a mean scalar gradient In HITSG, the scalar flux along the mean scalar gradient, the cospectrum F(k, t), is found to scale as k -7/3 in the inertial range for very large Reynolds numbers Re λ ≥ 10 4 . New results are proposed regarding the decay and growth of < u 3 θ > and < θ 2 >, gathered in Table 5.2: theoretical exponents are derived using physical arguments, and are then assessed numerically. Such results were not provided before and complete the work of [START_REF] Chasnov | Similarity states of passive scalar transport in buoyancy-generated turbulence[END_REF]. This theoretical contribution provides further insights into the prediction of high Reynolds numbers decaying turbulence. In HST, the exponential decrease of the scalar variance was recovered, and furthermore, algebraic decay laws for K T were provided in HSRT, unchanged with respect to HIT. All these results for shear-driven flows without a mean scalar gradient are gathered in Table 5.1. In HSTSG, the inertial scaling of the streamwise flux F S (k, t) in k -23/9 is recovered [START_REF] Bos | Inertial range scaling of scalar flux spectra in uniformly sheared turbulence[END_REF], and alternative scalings are briefly discussed. The interesting result of this part is the exponential growth of the scalar variance and mixed-correlations < u 1 θ > and < u 3 θ >, at a rate equal to the one of the kinetic energy.

In these three configurations, the small scales RTI of scalar second-order moments was investigated, and the conclusions are threefold. (i) Scalar small scales return completely to isotropy in HITSG, which is not surprising since the velocity field remains isotropic. (ii) On the contrary, when there is a mean velocity gradient (with or without an additional mean scalar gradient), some anisotropy persists in the scalar small scales even at high Reynolds numbers, which is consistent with DNS and experiments. This persistent small scale anisotropy for the passive scalar is nevertheless found to diminish with an increasing Reynolds number. (iii) When the anisotropy consists of velocity gradients only, anisotropy is gathered around the scalar integral wavenumber k T , whereas when a mean scalar gradient is present, anisotropy fills more the large scales.

Rotation was not investigated in this work, because its effects on triple correlations are not clear, and as mentioned earlier, interacting waves require additional theoretical tools, such as EDQNM2 [START_REF] Cambon | Spectral approach to non-isotropic turbulence subjected to rotation[END_REF][START_REF] Sagaut | Homogeneous Turbulence Dynamics[END_REF], to be properly captured. It is nevertheless a necessary step to the deep understanding of atmospheric flows: the DNS by [START_REF] Brethouwer | The effect of rotation on rapidly sheared homogeneous turbulence and passive scalar transport. linear theory and direct numerical simulation[END_REF] suggests that the effects of a rotating shear on the passive scalar transport and its flux are multiple and rather complex. Finally, the present modelling is further extended to unstably stratified homogeneous turbulence in Chapter 7, since stratification amounts only to additional linear transfers, much simpler than the ones induced by mean-velocity gradients.

In conclusion, the anisotropic EDQNM modelling seems promising since it recovers quite well previous experimental and numerical results, and additionally permits to explore large Reynolds numbers. It can predict the velocity and scalar fields dynamics for various kinds of anisotropy with the same consistent method and does not rely on adjustable constants, except the classical ones used in the eddy-damping terms.

Chapter 6

Prandtl Number Effects on Passive Scalar Dynamics

Here, the anisotropic EDQNM modelling assessed and used for P r = 1 in Chapter 5 is applied to the case of a Prandtl number different from unity: the frameworks of a highly diffusive scalar P r 1, and of a weakly diffusive scalar P r 1, are investigated. For these two configurations, the scalar variance spectrum E T (k, t) is known to scale differently at small scales in HIT (see Chapter 1). Consequently, one can wonder if these scalings for the scalar variance spectrum are modified when anisotropy appears at the velocity and scalar levels, and what happens for the scalar flux as well.

This chapter is divided into two parts: section 6.1 focuses on the HITSG framework, whereas section 6.2 addresses shear-driven flows. The first section is the main contribution of this chapter. For shear-driven turbulence, varying the Prandtl number while adding a shear seems to be a limit of the present spectral modelling, especially when P r 1. Nevertheless, some cases with mean velocity gradients in HST, HSRT and HSTSG are briefly presented afterwards.

The contents of this chapter for HITSG were published in: Briard & Gomez, "Prandtl number effects in decaying homogeneous isotropic turbulence with a mean scalar gradient", Journal of Turbulence, 18 (5), 418-442 (2017) For P r 1, it is recalled that the inertial-diffusive range (IDR) spans from k CO = P r 3/4 k η , where diffusion effects become dominant, to k η . One can define k CD = √ P rk η from which convection from small scales balances diffusion effects (see Chapter 1). And for P r 1, the viscous-convective range (VCR) spans from k η , the smallest active turbulent scale for the velocity field, to k B = √ P rk η . In this region, small scalar fluctuations are advected by the velocity field of the Kolmogorov scale. Then, beyond k B , scalar fluctuations are destroyed by diffusive processes. Interactions that are at the origin of the VCR are strongly non-local: indeed, the cascade of energy computed with EDQNM does not reach scales much smaller than k η because of the logarithmic discretization of the wavenumber space. Non-local transfers have been studied notably by [START_REF] Lesieur | Amortissement autosimilaire d'une turbulence á grand nombre de reynolds[END_REF]; [START_REF] Métais | Statistical predictability of decaying turbulence[END_REF]; [START_REF] Lesieur | Turbulence in fluids[END_REF], and the derivation of the non-local fluxes for the velocity and passive scalar fields is proposed in Appendix B.

Prandtl number effects in HITSG

Most of the papers dealing with a Prandtl number different from unity in HITSG were done at moderate Reynolds numbers, and focused on its effects on (i) high-order scalar statistics, and on (ii) the cospectrum F(k, t) and scalar variance spectrum E T (k, t) spectral scalings [START_REF] Chasnov | Simulation of the inertial-conductive subrange[END_REF][START_REF] Yeung | Schmidt number effects on turbulent transport with uniform mean scalar gradient[END_REF][START_REF] O'gorman | Effect of schmidt number on the velocity-scalar cospectrum in isotropic turbulence with a mean scalar gradient[END_REF][START_REF] Bos | Inertial range scaling of the scalar flux spectrum in two-dimensional turbulence[END_REF][START_REF] Yeung | Direct numerical simulation of turbulent mixing at very low schmidt number with a uniform mean gradient[END_REF]. The aim of the present study is to explore asymptotic regimes of HITSG, at very large Reynolds numbers and either very high or small Prandtl numbers, in order to predict the growth and decay rates of the scalar variance < θ 2 > and mixed-correlation < u 3 θ > of highly and weakly diffusive scalars, which is a new feature, and to verify the proposed spectral scalings as well. This is of theoretical interest since these regimes cannot be reached experimentally nor in DNS yet. In addition, this permits to analyze the combined effects of anisotropy that mainly affect large scales, and P r which dominantly modifies small scales of the spectra. Furthermore, it has been shown numerically in Chapter 1 that the Prandtl number did not affect the theoretical decay exponent of the scalar variance in HIT at large Reynolds numbers. Consequently, a natural extension of this work is to address effects of Prandtl numbers on the time evolution of < θ 2 > and < u 3 θ > in an anisotropic framework such as HITSG at large Reynolds numbers.

In the previous Chapter 5, only the case P r = 1 was addressed: theoretical decay and growth exponents for < u 3 θ > and < θ 2 > respectively were derived for HITSG, and assessed numerically. Therefore, the present work is an application of the anisotropic EDQNM modelling when the Prandtl number strongly departs from unity, basically from 10 -5 to 10 4 . Investigating an anisotropic configuration such as HITSG at large Reynolds numbers, with either very large or small Prandtl numbers, with an approach previously validated in more complex configurations such as shear-driven flows, is an important contribution in terms of modelling.

First, the theoretical spectral scalings of the cospectrum and scalar variance spectrum are derived in HITSG for P r 1 and P r 1, and four comparisons are performed to assess the relevance of the model when the Prandtl number strongly departs from unity: this part serves as a new and additional validation of the present anisotropic EDQNM modelling. Then, original numerical results are exposed. Effects of both very large and very small Prandtl numbers on the time evolution of < θ 2 > and < u 3 θ > are investigated. Afterwards, the normalized mixed correlation ρ wθ is studied as a function of the Reynolds and Prandtl numbers, and compared to results obtained in DNS. Furthermore, the effects of varying the Prandtl number on the small scales return to isotropy of the scalar second-order moments are analyzed. Finally, these different features are discussed in the concluding section.

Inertial scalings for E T (k, t) and F(k, t) -Comparisons

The emphasis is put on the inertial scaling of the scalar variance spectrum E T (k, t) and cospectrum F(k, t) when the Prandtl number is either very low or very large. These theoretical scalings are recovered analytically and numerically, and are then compared with recent numerical studies, such as DNS, LES and other spectral models. The fact that the present results are not always compared with DNS is because in most of the DNS, either the Reynolds number is not high enough, or the Prandtl number is too close to unity.

Highly diffusive passive scalar P r 1

Scaling of the scalar spectrum E T (k, t): in the inertial-diffusive range for HIT, the scalar spectrum scales as

E T (k, t) = K 0 3 T a -3 2/3 k -17/3 . (6.1)
It has been shown by [START_REF] Chasnov | Simulation of the inertial-conductive subrange[END_REF] that with a mean scalar gradient Λ, the scalar dissipation rate T should take into account this production effect, thus leading to

T → T + 2aΛ 2 , (6.2)
where 2aΛ 2 is a pseudo scalar dissipation rate arising from the mean gradient. The HIT scaling for E T given in (6.1) is thus modified in HITSG into Here, an alternative method is proposed, based on dimensional analysis and physical arguments that will be consistent with further developments.

E T (k, t) = K 0 3 T a -3 2/3 k -17/3 1 + 2 aΛ 2 T . ( 6
In the HITSG framework, the integration of the scalar Lin equation (4.36) yields

∂K T ∂t = -T -2λ 3 K F = -T 1 + 2λ 3 K F T .
The whole rhs term can be seen as a general scalar dissipation rate. Moreover, dimensional analysis gives K F ∼ λ 3 a which results into (6.2). The present simulations, at very low Prandtl numbers and very large Reynolds numbers, show that 2aΛ 2 / T 1, so that the classical scaling (6.1) is still relevant. This is consistent with the RTI of small scales in the IDR: this feature will be illustrated later on. However, when a moderate Reynolds number is combined with a very small P r, this ratio becomes greater than unity, so that the isotropic scaling is modified into E T (k, t) ∼ Λ 2 a -2 2/3 k -17/3 , (6.4) derived in [START_REF] Bos | Inertial range scaling of the scalar flux spectrum in two-dimensional turbulence[END_REF]; [START_REF] Yeung | Direct numerical simulation of turbulent mixing at very low schmidt number with a uniform mean gradient[END_REF], and is notably obtained by neglecting the non-linear contribution in the scalar Lin equation with respect to production and dissipation.

It is worth noting that in [START_REF] Yeung | Direct numerical simulation of turbulent mixing at very low schmidt number with a uniform mean gradient[END_REF], the Prandtl number is very low, and the Reynolds number moderate, so that very likely small scales are still anisotropic due to production mechanisms. Consequently, the general expression (6.3) should be kept.

The k -17/3 scaling of the scalar spectrum in low P r HITSG has been assessed recently in a DNS by [START_REF] Yeung | Direct numerical simulation of turbulent mixing at very low schmidt number with a uniform mean gradient[END_REF]. Present results are compared with the latter DNS in Fig. 6.1a. The final Reynolds number is Re λ = 240 after ten turn-over times. The Prandtl number is P r = 1/2048 and the initial integral scales are L(0) = 1.346 and L T (0) = 3.468. E T (k, t = 0) = 0 and scalar fluctuations arise from a unit mean scalar gradient. A good agreement is found for the scalar spectrum. Near the Kolmogorov wavenumber (kη = 1), the scalar spectrum slightly increases: this is due to small scale convection, as discussed in Chapter 1. This phenomenon increases with higher Reynolds numbers and lower Prandtl numbers. This does not happen in the DNS result, may be because small scales are not completely resolved beyond k η . Nevertheless, the k -17/3 is well recovered. Scaling of the cospectrum F(k, t): the starting point to determine the cospectrum scaling is the scalar flux Lin equation

∂ ∂t + (a + ν)k 2 F(k, t) = 2 3 ΛE(k, t) + S F,NL 3 (k, t). (6.5)
A reasonable assumption is to say that the diffusive timescale (ak 2 ) -1 is much smaller than the non linear time scale defined as

τ (k) = k 3 E(k) -1/2 = k 2 -1/3 = kE(k) . (6.6)
This is obvious at large k for high thermal diffusivity a. Therefore, non-linear contributions can be neglected, as previously mentioned for the scalar spectrum. Then, for scaling considerations, the time derivative is dropped off, so that

ak 2 F(k, t) = 2 3 ΛE(k, t),
which yields the IDR scaling for the cospectrum The key point being to neglect the non-linear transfers in both cases. Another approach is possible. Thanks to the previous work performed in Chapter 5, let's assume that the spectral cospectrum correlation is constant in the inertial range

F(k, t) = 2 3 K 0 Λa -1 2/3 k -11/3 . ( 6
ρ wθ (k) ρ wθ = F(k) E(k)E T (k) . (6.8)
Hence, using Kolmogorov scaling for E, and Batchelor (1959) scaling (6.1) for E T , one gets

F(k, t) ∼ a -3/2 1/2
T 2/3 k -11/3 . (6.9)

Moreover, since the scalar field as no retro-action on the velocity one, T should not appear in (6.9). Consequently, equalizing (6.7) and (6.9) gives T ∼ aΛ 2 . This is consistent with the additional scalar dissipation rate coming from mean scalar gradient effect explained previously in (6.2).

The k -11/3 inertial-diffusive scaling of the cospectrum for P r 1 is assessed in Fig. 6.1b: our EDQNM simulation is compared to the LES of O' Gorman & Pullin (2005) where P r = 2.10 -4 and Re λ = 1500 (after 10 turn-over times for EDQNM). The agreement with the k -11/3 scaling is rather good at this Reynolds number, and the agreement between EDQNM and LES is excellent in the inertial-diffusive range.

6.1.1.2 Weakly diffusive passive scalar P r 1

The case P r 1 is now considered: small scales of the scalar variance spectrum experience convection from the velocity field of the Kolmogorov scale, which results in a viscous-convective range from k η , the smallest active turbulent scale for the velocity field, to the Batchelor wavenumber k B = √ P rk η , where E T scales as [START_REF] Batchelor | Small-scale variation of convected quantities like temperature in turbulent fluid part 1. general discussion and the case of small conductivity[END_REF])

E T (k, t) = K B T ν k -1 , (6.10)
where K B is the Batchelor constant, found to be 2.5 in the present simulations. This value is close to the first proposal K B = 2 by [START_REF] Batchelor | Small-scale variation of convected quantities like temperature in turbulent fluid part 1. general discussion and the case of small conductivity[END_REF], and in agreement with predictions of [START_REF] Gibson | Fine structure of scalar fields mixed by turbulence. ii. spectral theory[END_REF]:

√ 3 ≤ K B ≤ 2 √ 3 for HIT.
Other values measured in the ocean are slightly higher (see [START_REF] Qian | Viscous range of turbulent scalar of large prandtl number[END_REF] and values reported therein) even though other mechanisms may play a non-negligible role in the ocean. Values obtained in DNS at moderate Re λ are also higher [START_REF] Yeung | Schmidt number effects on turbulent transport with uniform mean scalar gradient[END_REF].

The scaling of the cospectrum F(k, t) for a weakly diffusive passive scalar field has been discussed notably in O' Gorman & Pullin (2005) and it has been found that the spectral velocity-scalar correlation is not strongly modified in the framework P r 1, unlike the case P r 1. This is expected if one compares the cospectrum Lin equations (6.5) for P r = 1 where a = ν, so that the dissipative term is 2νk 2 F, and for P r 1, where a ν, which yields for the dissipative term only νk 2 F. Hence, for a weakly diffusive scalar, the cospectrum still scales in k -7/3 in the inertial-convective range.

Finally, two comparisons are proposed hereafter. Since in DNS when the Prandtl number increases the Reynolds number conjointly decreases for numerical resolution issues, we first propose a large Reynolds number comparison with the SDIP model [START_REF] O'gorman | Effect of schmidt number on the velocity-scalar cospectrum in isotropic turbulence with a mean scalar gradient[END_REF] at P r = 100 in Fig. 6.2a. However, since the SDIP is an asymptotic model, the Reynolds number is unknown. The agreement is acceptable, and the slight discrepancy may be attributed to the uncertainty for the Reynolds numbers, which is Re λ = 2.10 4 here with the present anisotropic EDQNM modelling.

Then, in Fig. 6.2b, the compensated scalar variance spectrum is compared with the low Reynolds number DNS of [START_REF] Yeung | Simulations of three-dimensional turbulent mixing for schmidt numbers of the order 1000[END_REF], where Re λ 8 and P r = 1024. Initially, the scalar variance spectrum is zero, and the Reynolds number is chosen so that after ten turnover times the Reynolds number is Re λ = 10. Our minimum wavenumber was decreased on purpose to match with the DNS configuration, and the reason for the slight discrepancy at large wavenumbers could be that the DNS is forced at large scales, whereas here we have a freely decaying Saffman turbulence. This does not prevent us from getting a very good agreement, both in the viscous convective range and further in the viscous-dissipative range, which validates our approach, even at low Reynolds numbers.

Spectral transfers and conclusions for the inertial scalings

The inertial scalings of the cospectrum F(k, t) and scalar variance spectrum E T (k, t) were investigated for both low and large Prandtl numbers in subsections 6.1.1.1 and 6.1.1.2, where a mean scalar gradient Λ sustains the fluctuations in a homogeneous isotropic decaying turbulence. The theoretical predictions were recovered analytically, and more importantly, assessed numerically over a wide range of Reynolds and Prandtl numbers, which illustrates the relevance of our anisotropic EDQNM modelling.

Finally, the budget terms of the evolution equation of E T (k, t) are analyzed in Fig. 6.4, for large (left column) and low (right column) Reynolds numbers, at high (top line) and small (bottom line) Prandtl numbers. The first observation is that for all four cases, the linear production term is more intense than the non-linear transfer at large scales, and then is negligible at smaller scales, meaning that production of scalar fluctuations through the mean gradient is dominant at large scales, where anisotropy is consistently gathered. In the very large Péclet number case (a) where P e λ = Re λ √ P r = 1.8.10 5 , there is a clear separation of three domains, in agreement with [START_REF] Yeung | Direct numerical simulation of turbulent mixing at very low schmidt number with a uniform mean gradient[END_REF], even though freely decaying turbulence is considered here: at low wavenumbers, one has almost -S T,NL(iso) 2ΛF, and the difference is due to the term ∂ t E T = 0; at intermediate wavenumbers, all three contributions are very small and of the same order; finally, at large wavenumbers, there is a balance between non-linear transfer and dissipation S T,NL(iso) 2ak 2 E T . In the opposite case (d) where P e λ → 0, non-linear transfers are small and production balances well dissipation, in agreement with the prediction of [START_REF] Yeung | Direct numerical simulation of turbulent mixing at very low schmidt number with a uniform mean gradient[END_REF]. Furthermore, in the two low Prandtl number cases (c) and (d), the insets show that the dissipation term is always more intense that non-linear transfers, even in the high Re λ configuration, in accord with the latter reference. For the two high Prandtl number cases (a) and (b), non-linear transfers are more intense around k η , and then dissipation takes over while approaching k B at larger wavenumbers.

k E T (k, t) and F(k, t) E T (k, t) F (k, t) k η k -5/3 k L k -7/3 k -11/3 k -17/3 k CO (a)
k E T (k, t), F(k, t) E T (k, t) F (k, t), P r = 1 F (k, t), P r = 10 4 k -5/3 k -1 k η k -7/3 k L k B (b)

Numerical results -Time evolution and anisotropy

In the previous section, the anisotropic EDQNM modelling was assessed for small and large Prandtl numbers in HITSG by investigating the inertial scalings of the scalar variance spectrum E T (k, t) and cospectrum F(k, t). In this part, effects of the Prandtl number P r on the time evolution of the scalar variance K T , the mixed-correlation K F , the normalized cospectrum correlation ρ wθ , and the Nusselt number N u, are analyzed, along with the small scales return to isotropy of the flow.

6.1.2.1 Prandtl effects on the decay and growth of < u 3 θ > and < θ 2 >

The growth of K T =< θ 2 > and decay of K F =< u 3 θ > are addressed for both highly and weakly diffusive passive scalars. In Cahpter 5, for P r = 1, one had p F = 0.4075. Here, for P r 1 and P r 1, it is found that p F slightly increases to p F 0.42 when P r departs from unity, consistently with the variations of the scalar backscatter parameter p T in HIT with P r. It makes sense that p F varies less with P r than p T since < u 3 θ > is a mixed correlation where the velocity field is not affected at all by a change in P r. In Fig. 6.5a and 6.5b, both theoretical predictions for α F and α Λ T given in (5.13) and (5.16) are recovered numerically. The Reynolds number Re λ is much higher for P r 1 than for P r 1 in order to keep a sufficiently high Péclet number. One can say from Fig. 6.5a and 6.5b that the respective decay and growth of K F and K T in HITSG is not affected by high or small Prandtl numbers at large Reynolds numbers. A similar result was obtained for scalar integrated quantities such as K T in decaying HIT in Chapter 1.

In addition, α Λ

T is presented in Fig. 6.5c at moderate Reynolds numbers, typical of DNS and experiments, for various Prandtl numbers. This figure should be compared to the case P r = 1 presented in Chapter 5, where a monotonic decrease of α Λ T was observed from the high Reynolds to the low Reynolds predictions (recalled in grey in Fig. 6.5c for Batchelor turbulence). Therefore, this figure clearly illustrates that even though a P r strongly different from unity does not modify the asymptotic theoretical predictions at very large Reynolds numbers, it significantly alters the decay of the scalar variance K T at moderate ones. For P r 1, the growth exponent α Λ T slightly increases before diminishing toward the low Reynolds numbers (Re λ ≤ 1) predictions of Chapter 5: this is because when Re λ decreases, the k -5/3 inertial range vanishes. However, the k -1 viscous range survives, thus slightly slowing down the decay.

Whereas for P r 1, the decrease of α Λ T starts at quite high Reynolds numbers, because the Péclet number is very small. The same observations are made for the decay exponent α F of the mixed-correlation.

Consequently, one could conclude from Fig. 6.5a, 6.5b and 6.5c that the Prandtl number does not affect the time exponent at very large Reynolds numbers, but at moderate ones. This is of importance because it could explain why in DNS there is a significant scatter of the normalized mixed-correlation ρ wθ .

6.1.2.2 Cospectrum correlation ρ wθ , pressure-scalar correlation Π F , and Nusselt number N u

The normalized correlation ρ wθ , defined in (5.10), is addressed in Fig. 6.6a. Some values of this quantity at P r = 1 were reported in Chapter 5: therefore, the emphasis is put here on the influence of P r on ρ wθ . The first feature to point out is that at large Reynolds numbers, either with a small or large Prandtl number, ρ wθ is constant: this can be obtained analytically by considering the expressions of the exponents α, α Λ T and α F . Then, ρ wθ diminishes with decreasing Reynolds numbers because of the joint decay of < u 3 θ > and growth of the scalar variance, both studied in the previous part. It is worth noting that the magnitude of ρ wθ strongly depends on P r at moderate Re λ , because the Prandtl number affects decay exponents in this region of moderate Reynolds numbers, as revealed previously in Fig. 6.5c.

In addition, several low P r values from [START_REF] Yeung | Direct numerical simulation of turbulent mixing at very low schmidt number with a uniform mean gradient[END_REF] are included in Fig. 6.6a, and there is a good quantitative agreement with the present anisotropic EDQNM modelling: the three simulations of [START_REF] Yeung | Direct numerical simulation of turbulent mixing at very low schmidt number with a uniform mean gradient[END_REF], for P r = 1/2048, P r = 1/512, and P r = 1/128, are almost all consistently contained within our EDQNM simulations at P r = 10 -4 and P r = 10 -2 . Moreover, at these moderate Re λ , it is recovered that ρ wθ increases in magnitude with the Reynolds number at a given P r. Furthermore, an interesting behaviour is recovered, which is the decrease in magnitude of ρ wθ when P r departs from unity, either for P r 1 or P r 1, at a fixed moderate Re λ : this notably confirms the DNS results of [START_REF] Yeung | Schmidt number effects on turbulent transport with uniform mean scalar gradient[END_REF]; [START_REF] Yeung | Direct numerical simulation of turbulent mixing at very low schmidt number with a uniform mean gradient[END_REF], and can be interpreted in terms of loss of phase alignment between spectral velocity and scalar fluctuations: indeed, for both P r 1 and P r 1, there exists a subrange in wavenumber space (the inertial-diffusive and viscous-convective ranges respectively) where the scalar variance spectrum strongly depart from the kinetic energy one. This phenomenon is much more visible for P r 1.

This decrease in magnitude of ρ wθ with a Prandtl number different from unity is of practical interest since it happens at moderate Reynolds numbers only, and this might be the reason for the scattering of the obtained values of the cospectrum normalized correlation, as already mentioned in Chapter 5. Furthermore, the scalar-pressure correlation Π F =< p∂ 3 θ > is investigated: this correlation has not received much attention, even though it is the destruction mechanism of the scalar flux, and was not addressed in Chapter 5. There, it was shown that one cannot define a decay rate for the dissipation F at large Reynolds numbers, because it is not conserved in the inertial range unlike and T . But it is possible for Π F : indeed, according to the evolution equation (4.58) of < u 3 θ >, Π F should evolve as the production mechanism, proportional to the kinetic energy K(t). This is confirmed numerically in Fig. 6.6b for Saffman turbulence: Π F is found to decay in t -6/5 , similarly to the kinetic energy. Furthermore, Fig. 6.6b once more illustrates that the theoretical decay rate does not depend on the Prandtl number.

Finally, the Nusselt number, defined as

N u = - < u 3 θ > aΛ , (6.11)
is investigated for various Prandtl numbers. N u is in fact another normalization of the mixedcorrelation < u 3 θ > which is of practical interest for heat transfers. The theoretical prediction for the Nusselt number, detailed and assessed in [START_REF] Gotoh | Scalar flux in a uniform mean scalar gradient in homogeneous isotropic steady turbulence[END_REF], is that it should vary as N u ∼ P e, where the Péclet number is P e = P rRe T , with the turbulent Reynolds number Re T = 3Re 2 λ /20. This scaling is successfully recovered in Fig. 6.6c for a wide range of Péclet and Prandtl numbers.

Return to isotropy of small scales

The anisotropy of the flow is briefly investigated here, at the level of the scalar second-order moments: it is recalled that in Chapter 4, it was shown that small scales completely returned to isotropy, which was expected since the velocity field remains isotropic in HITSG. Consequently, one can wonder if the Prandtl number has an influence on the scalar small scales return to isotropy. Not surprisingly, it is found that the impact of the Prandtl number on the global anisotropy is comparable to moderate Reynolds numbers effects for the kinetic field: indeed, the relevant dimensionless parameter for the scalar is not only Re λ , but the product Re λ √ P r, which could be called a Taylor Péclet number P e λ .

Thus, even with a large Re λ , if the Prandtl number is as small as 10 -4 , P e λ will be moderately small, so that scalar small scales may not be not completely isotropic, in addition to other issues, such as the lack of scale separation in the spectra [START_REF] Yeung | Direct numerical simulation of turbulent mixing at very low schmidt number with a uniform mean gradient[END_REF]. Consequently, very large Re λ are required for highly diffusive passive scalars. This is illustrated in Fig. 6.7a, where the Taylor Reynolds number is very large Re λ ∼ 10 5 , so that even at small Prandtl numbers of order ∼ 10 -4 , the Péclet number based on the Taylor scale is still sufficiently high P e λ ∼ 10 3 : this is an important condition for weakly diffusive passive scalars to obtain clear scalings, as underlined in [START_REF] Yeung | Direct numerical simulation of turbulent mixing at very low schmidt number with a uniform mean gradient[END_REF]. This allows to observe in Fig. 6.7a that there is a complete return to isotropy of scalar second-order moments small scales. It is worth noting that from the Corrsin-Obukhov wavenumber k CO , i.e. in the inertial-diffusive range, there is no more anisotropy: the non-linearity being much stronger in the inertial-convective range, for k L < k < k CO , the return to isotropy mechanism occurs dominantly in this region of the wavenumber space. For weakly diffusive scalar, analogous assessments leading to a similar conclusion are made in [START_REF] Yeung | Schmidt number effects on turbulent transport with uniform mean scalar gradient[END_REF], where it is shown numerically that even with a moderate Re λ , increasing the P r -which amounts to increase P e λ -allows to recover scalar isotropic small scales. It is proposed to illustrate this feature in Fig. 6.7b, where H (T )

33 is displayed for Saffman turbulence at Re λ = 100, for P r = 1 and P r = 10 4 . It is clear, notably with the zoom around the Kolmogorov wavenumber k η , that increasing the Prandtl number at a fixed Reynolds number participates into restoring isotropy at small scales.

Finally, it is worth noting that within our modelling, according to Fig. 6.7a and 6.7b, the large scales level of anisotropy seems to be independent of the Prandtl number in Saffman turbulence, and always very close to 1/15, with 1/15 ≥ H (T ) 33 . This value of 1/15 is interesting because it is the maximum level of anisotropy that the scalar field can handle according to the realizability condition (4.29). Using the axisymmetric relations

H (T ) 11 = H (T ) 22 = -H (T )
33 /2, one obtains that the largest eigenvalue is H (T ) 33 . Thus, in HITSG, the realizability condition is verified (otherwise one would get negative scalar spectra) and the important feature is that the anisotropy reaches its maximal value at large scales in Saffman turbulence. The case of Batchelor turbulence is a bit different: indeed, because of classical backscatter of energy, strong inverse transfers initiate a return to isotropy mechanism at large scales [START_REF] Eyink | Free decay of turbulence and breakdown of self-similarity[END_REF], so that the large scales level of anisotropy can decrease with time (or equivalently can decrease when Re λ decreases). This is more visible in the case P r 1 because the Péclet number is in general lower than in the case P r 1, as illustrated in Fig. 6.8. There, H (T ) 33 for P r = 10 -4 is displayed at different times during the decay, or equivalently at various decreasing Reynolds numbers, so that the Péclet number goes from P e λ = 193 down to P e λ = 15. Even though the large scales level of anisotropy remains close to 1/15, it nevertheless slightly diminishes.

Conclusions for P r = 1 in HITSG

Decaying homogeneous isotropic turbulence with a mean scalar gradient (HITSG) that sustains scalar fluctuations has been investigated numerically at large Reynolds numbers with the anisotropic EDQNM modelling. The present work is a direct application of Chapter 5 for a Prandtl number either very small or very large: first, four quantitative comparisons are proposed. The good agreement between the present results and DNS, LES and other models, permits to assess the relevance of the model at P r 1 and P r 1 over a wide range of Reynolds numbers. This notably confirms numerically theoretical spectral scalings for the scalar variance spectrum E T (k, t) and the cospectrum F(k, t). Hence, in HITSG, at large Reynolds numbers, it is notably recovered that for P r 1, E T and F scale respectively in k -5/3 and k -7/3 in the inertial-convective range, and then, from the Corrsin-Obukhov wavenumber k CO , in k -17/3 and k -11/3 in the inertial-diffusive range. Moreover, when P r 1, the k -1 viscous convective range beyond k η for E T is not modified with the presence of a mean scalar gradient. For both low and large Prandtl numbers, budget terms of the evolution equation of E T (k, t) were analyzed as well: it was shown that at large scales, the production is always stronger than non-linear transfers. At small scales for P r 1, even at large Re λ , dissipation is stronger than non-linear transfers: in the limit where the Péclet number tends to zero, dissipation balances production.

Secondly, the time evolution of the scalar variance < θ 2 > and the mixed-correlation < u 3 θ > was investigated at large Reynolds numbers: it was shown numerically that the theoretical predictions of Chapter 5 for the algebraic time exponents are still valid for P r 1 and P r 1, consistently with a similar result for the scalar variance decay in HIT: the Prandtl number does not affect the asymptotic time evolution of < θ 2 > and < u 3 θ > at large Reynolds numbers, only at moderate ones. In addition, it was shown numerically that the pressure-scalar correlation < p∂ 3 θ >, which is responsible for the destruction of the scalar flux, decays with the same rate as the kinetic energy, independently of the Prandtl number.

Afterwards, the Reynolds and Prandtl numbers dependence of the normalized cospectrum correlation ρ wθ was addressed as well: the present spectral modelling provides good quantitative results with respect to DNS. Notably, it was found that at a fixed moderate Reynolds number, say Re λ ∼ 100, ρ wθ decreases in magnitude when the Prandtl number either increases or decreases, in agreement with the prediction of [START_REF] Yeung | Direct numerical simulation of turbulent mixing at very low schmidt number with a uniform mean gradient[END_REF]. The linear dependence of the Nusselt number with the Péclet number is also recovered.

Finally, it was shown numerically that the small scales of the scalar second-order moments return to isotropy, provided the Péclet number is large enough. This notably implies, for highly diffusive passive scalars, the need to reach very high Taylor Reynolds numbers Re λ when one wants to obtain a clear k -17/3 inertial-diffusive scaling, which numerically requires that P r ≤ 10 -3 .

Prandtl number effects in shear-driven turbulence

In this section, HSRT, HST and HSTSG are addressed. The results regarding the impact of a Prandtl number strongly different from unity are less conclusive than in the previous section dedicated to HITSG. There are mainly two reasons for this: (i) First, for P r 1, huge initial Reynolds numbers are needed, which make the simulations really long when St increases. This is particularly true when there is no mean scalar gradient, i.e. in HST, since K T decreases exponentially whereas K increases exponentially. Nevertheless, a new result regarding the inertial-diffusive scaling of F S is proposed for HSTSG when P r 1. (ii) Secondly, when P r 1, it is not clear if the use of non-local transfers, detailed in Appendix B, to sustain the k -1 viscous-convective range remains appropriate when a sustained shear is applied. Indeed, the non-local transfers were first derived for isotropic turbulence: since in HITSG the velocity field remains isotropic, it seemed relevant enough to use the non-local expansions in the previous section. But a sustained shear is the most anisotropic case. Hence, it is not straightforward to quantify how the anisotropic non-linear transfers "disturb" the isotropic scalar non-local transfer in the presence of shear.

Whatever framework is considered, HSRT, HST, or HSTSG, the k -17/3 inertial-diffusive scaling when P r 1 and the k -1 viscous convective scaling when P r 1 for E T (k, t) are not modified by the presence of shear. For the cospectrum F(k, t), the k -11/3 inertial-diffusive scaling when P r 1 is also not modified by the presence of shear.

Homogeneous shear-released turbulence

In this section, the decay of K T in HSRT is addressed when either P r 1 or P r 1. It is revealed in Fig. 6.9 that the Prandtl number does not affect the decay of the scalar variance K T in HSRT, as previously shown in HIT in Chapter 1 for asymptotically large or small Reynolds numbers. Saffman and Batchelor turbulence are presented. It is notably found that K T decays faster with P r = 10 -4 than with P r = 1, which is consistent with a moderate Péclet number: the low Reynolds numbers regime is reached more rapidly than with P r ≥ 1. The scalar global anisotropy indicators b T ij are then displayed in Fig. 6.10 for both very large and very small Prandtl numbers, and they have the same behaviour as in the case P r = 1, i.e. an asymptotic state different from zero, meaning that there is still anisotropy left in the flow, gathered at large scales.

Sustained shear flow

The case of sustained shear flows is now addressed. It is revealed in Fig. 6.11 that even with P r 1 or P r 1, the scalar variance K T still decays exponentially with γ T = -0.52. This is consistent with the Prandtl number dominantly affecting the small scales of the scalar variance spectrum. Nevertheless, for P r 1, with an initial Re λ (0) = 1, the associated Péclet number would be too small and consequently the transitional state before reaching the asymptotic anisotropic state would be rather large: furthermore, a too low Re λ (0) causes some numerical issues. Therefore, for a sustained shear flow and P r 1, the initial Reynolds number is chosen much higher (Re λ (0) = 100) to ensure an initial moderate Péclet number. Whereas in the configuration P r 1, the initial Péclet number is sufficiently large so that no specific precautions have to be taken. Afterwards, the time evolution of the b T ij and of the scalar shear rapidity S T R = T /(K T S) are displayed in Fig. 6.12 for Saffman HST, for both P r = 10 -4 and P r = 10 4 . They all reach the same asymptotic anisotropic state, independently of the value of the Prandtl number, which is similar to the previous findings about anisotropy in HITSG. Finally, for illustration purposes in the high Prandtl case, the scalar variance spectrum E T is presented in Fig. 6.13a at St = 50 and displays a clear k -1 scaling in the VCR. This scaling is obviously not modified by anisotropy, because mean velocity gradients mainly apply on large scales. In addition, the scalar fluxes are displayed in Fig 6 .13b: the impact of the direct nonlocal part T + T of the non-linear scalar transfer, which results in the non-local flux Π + T after integration, is clear on the scalar non-linear isotropic flux Π NL(iso) T : energy is brought beyond the Kolmogorov wavenumber and the total resulting flux is found to be constant through k η . This non-local transfer allows to maintain, as in HIT, the k -1 viscous-convective range.

T E(k, t) E T (k, t) k -5/3 k -1 k η k B (

Homogeneous Shear Turbulence with a mean Scalar Gradient

The HSTSG framework is now studied. For P r 1, the scaling of the streamwise flux spectrum E F 1 = F S in the inertial range can be easily predicted using arguments similar to the ones of [START_REF] Bos | Inertial range scaling of scalar flux spectra in uniformly sheared turbulence[END_REF] in the case P r = 1. Starting from the scaling of the cospectrum for a highly conductive passive scalar field (6.7), and replacing by ij , which is justified by the presence of the main shear dU 1 /dx 3 , yields

E F i (k, t) = 2 3 K 0 a -1 λ j 2/3 ij (k)k -11/3 . (6.12)
Then, the scaling of the spectral tensor is used ij (k)E(k) = 3φ ij (k) , where ii = 2 . This gives in the inertial range ij (k) ∼ S 2/3 k -2/3 . Finally, for very low Prandtl numbers, the streamwise scalar flux is, in the inertial-diffusive range,

F S (k, t) = -C S F Λa -1 4/9 k -37/9 . (6.13)
In Fig. 6.14a, the k -37/9 scaling for the streamwise flux F S is assessed over two decades in the IDR (the two components of the scalar flux have been decreased for readability reasons), and the k -23/9 slope in the ICR is recovered as well. The scalar spectrum E T and cospectrum F display the same scaling as in HITSG and are not modified by shear, similarly to the case of P r = 1. For weakly diffusive passive scalars P r 1, E T still displays a k -1 slope in the VCR, whereas F and F S scale similarly to the case P r = 1 in HSTSG. This is also illustrated in Fig. 6.14b. Finally, the exponential growth of K, K T , K F and K S F at the rate γ = 0.34 is presented in Fig. 6.15. In the latter figure, K F and K S F can hardly be distinguished. There is no strong differences with the case P r = 1, except that the transitory state for the scalar variance and mixed correlations < u 1 θ > and < u 3 θ > is longer for the case P r 1, as noted previously in HST for K T . One can nevertheless remark that either in the case P r 1 or P r 1, the kinetic energy is always less intense than K T , K F and K S F , as in the case P r = 1.

F (k, t) F S (k, t) k η k -7/3 k CO k -11/3 k -37/9 k -23/9 k -5/3 k -17/3
F (k, t) F S (k, t) k -1 k η k B k -7/3 k -23/9 k -5/3 (b)

Conclusions about shear-driven turbulence for P r = 1

As specified at the beginning of this chapter, the results for highly and weakly diffusive passive scalars in shear-driven turbulence are less conclusive than in HITSG. The lack of accurate data for these regimes is flagrant since it was impossible to perform quantitative comparisons against DNS and experiments.

Therefore, this section should be considered as a guide for future works, and may be some of the results presented here will be of use. Nevertheless, the noteworthy findings are twofold: (i) the Prandtl number does not affect the exponential rate of the scalar and scalar flux fields in HST and HSTSG. (ii) For P r 1, a new IDR scaling was derived for the streamwise flux F S based on the arguments of [START_REF] Bos | Inertial range scaling of scalar flux spectra in uniformly sheared turbulence[END_REF] for P r = 1, which reads F S ∼ k -37/9 . On a theoretical point of view, as mentioned earlier, it is not clear if the use of isotropic nonlocal transfers for the scalar field in the case P r 1 remains relevant in the presence of shear. Anisotropic non-local transfers were derived (but not presented in Appendix B) following the methodology of [START_REF] Lesieur | Amortissement autosimilaire d'une turbulence á grand nombre de reynolds[END_REF], but they proved to be completely negligible with respect to the scalar isotropic transfer. What are the reasons for doubting of these non-local transfers? First, the direct non-local transfer T + T brings strongly "anisotropic scalar variance" from large scales to almost isotropic small scales, and this transfer only depends on E and E T , not on the anisotropic descriptors H () ij . Secondly, in some configurations, the inverse non-local scalar transfers T - T caused some numerical issue, probably because it only affects the isotropic spectrum E T at larges scales (by bringing small scales "isotropic scalar variance") and not

E T H (T ) ij .
Chapter 7

Spectral Modelling for Unstably Stratified Homogeneous Turbulence

In this section, the anisotropic EDQNM modelling is extended to the case of active scalar dynamics.

The contents of this chapter, except the variable stratification part, were published in: Briard, Iyer & Gomez, "Anisotropic spectral modeling for unstably stratified homogeneous turbulence", Physical Review Fluids, 2 (4), 044604 (2017) Unstably Stratified Homogeneous Turbulence (USHT) can be seen as a simplified approach for Rayleigh-Taylor instability [START_REF] Soulard | Inertial-range anisotropy in rayleigh-taylor turbulence[END_REF][START_REF] Gréa | The rapid acceleration model and the growth rate of a turbulent mixing zone induced by rayleigh-taylor instability[END_REF][START_REF] Soulard | Influence of the mixing parameter on the second order moments of velocity and concentration in rayleigh-taylor turbulence[END_REF], which is a phenomenon occurring for fluid of variable density. This instability can be found in various areas, such as geophysical, astrophysical and confined industrial flows: more specifically, the Rayleigh-Taylor instability can happen in natural flows such as atmospheric ones because of the mean vertical temperature gradient, when the heavy fluid, located above the lighter one, pushes it downward due to gravitational acceleration, which creates a mixing zone.

To investigate both numerically and theoretically a mechanism as complex as Rayleigh-Taylor instability, it is convenient to work in the framework of USHT, which notably discards inhomogeneity, uses the Boussinesq approximation to reflect the retro-action of the convected buoyant field on the velocity one, and assumes that the mixing length L is constant in time, and larger than the turbulent integral scale [START_REF] Soulard | Large-scale analysis of self-similar unstably stratified homogeneous turbulence[END_REF]Burlot et al., 2015a,b;Gréa et al., 2016a). It follows that the stratification frequency N is constant as well. One has to point out that frameworks different from USHT can be considered as well, for instance variable-density flows where the Boussinesq approximation is not used anymore [START_REF] Livescu | Buoyancy-driven variable-density turbulence[END_REF][START_REF] Chung | Direct numerical simulation and large-eddy simulation of stationary buoyancy-driventurbulence[END_REF].

The USHT framework has been addressed recently, thanks to the axisymmetric EDQNM model (Burlot et al., 2015a,b;Gréa et al., 2016a) to analyze the large Reynolds numbers regimes. As specified by its name, this spectral approach is dedicated to axisymmetric configurations and permits an accurate investigation of the scale-by-scale anisotropy distribution, and of the time evolution of one-point statistics such as the Froude number F r and the mixing intensity Λ, which will be defined later on. The strength of the axisymmetric EDQNM is that production terms, linear with N , are exactly treated, whereas the non-linear transfers are closed by a classical EDQNM procedure, which is the most costly step in terms of computational resources. However, this approach cannot handle, at least in the present form, shear flows, where there is no particular symmetry, unlike the present anisotropic EDQNM modelling. For this reason, the latter model is extended here to the case of active scalar dynamics, and compared to the results obtained with the axisymmetric EDQNM of Burlot and coworkers. Furthermore, throughout this chapter, USHT is qualitatively compared to results obtained with the anisotropic EDQNM modelling in the frameworks of HST and HITSG (Chapters 3 and 5). In addition, both the effects of large Schmidt numbers Sc on the inertial scaling of the scalar flux spectrum, and pressure spectra, are addressed on a fundamental point of view.

The extension of the anisotropic EDQNM modelling to unstable stratification is a step further towards the modelling and understanding of high Reynolds geophysical flows, such as atmospheric and oceanic ones. Indeed, under the assumption of homogeneity, such flows contain effects of shear, temperature and concentration gradients, stratification, rotation, and helicity. Shear and mean scalar gradient mechanisms have already been addressed with our model, and helicity is the topic of Chapter 8, so that stratification appears to be an natural extension, whereas effects of rotation were addressed independently with EDQNM2 [START_REF] Cambon | Third-order statistics and the dynamics of strongly anisotropic turbulent flows[END_REF].

Then, a step further toward the modelling of Rayleigh-Taylor instability is crossed by allowing the stratification frequency N (t) to vary with time, with a retro-action of the fluctuating quantities on the mean field. In such a configuration, the dynamics is completely different, and the mixing length L(t) is known to evolve in t 2 [START_REF] Soulard | Inertial-range anisotropy in rayleigh-taylor turbulence[END_REF][START_REF] Gréa | The rapid acceleration model and the growth rate of a turbulent mixing zone induced by rayleigh-taylor instability[END_REF][START_REF] Soulard | Influence of the mixing parameter on the second order moments of velocity and concentration in rayleigh-taylor turbulence[END_REF].

Evolution equations in USHT

In this section, the spectral anisotropic modelling developed in Chapters 2 and 4 for passive scalar dynamics is extended to the case of unstably stratified turbulence: the additional linear transfers linked to the Boussinesq approximation are presented hereafter.

Additional coupling terms

As commonly done for USHT, one has to scale the scalar fluctuations θ, which is usually a concentration, as a buoyant velocity v [START_REF] Soulard | Large-scale analysis of self-similar unstably stratified homogeneous turbulence[END_REF]

according to v = 2Agθ N , (7.1)
where N is the stratification characteristic time, or buoyancy frequency

N = 2Ag dV dx 3 , (7.2)
where V is the mean buoyant field, g the gravitational acceleration, A the Atwood number A = (ρ 1 -ρ 2 )/(ρ 1 + ρ 2 ), assumed to be small or the Boussinesq approximation, with ρ 1 and ρ 2 the densities of the heavy and light fluid respectively. The evolution equations of the fluctuating velocity and scalar fields are then

∂v ∂t + u j ∂v ∂x j = a ∂ 2 v ∂x j ∂x j + N u 3 , (7.3) ∂u i ∂t + u j ∂u i ∂x j = - ∂p ∂x i + ν ∂ 2 u i ∂x j ∂x j + N v δ i3 . (7.4)
The spectral counterpart of these equations are straightforward (see Appendices C and E). In what follows, one has to replace θ by v in the definitions of the spectral quantities introduced in Chapter 4. The evolution equations of the two-point second-order correlations Rij , F i and E T are then

∂ ∂t + 2νk 2 Rij (k, t) = T NL ij (k, t) + N P j3 (k)F i (k, t) + P i3 (k)F j (k, t)
Additional stratification coupling , (7.5)

∂ ∂t + (ν + a)k 2 F i (k, t) = T F,NL i (k, t) + N Ri3 (k, t) + N P i3 (k) E T (k, t)
Additional coupling , (7.6)

∂ ∂t + 2ak 2 E T (k, t) = T T,NL (k, t) + 2N F 3 (k, t).
(7.7)

Spherically-averaged Lin equations for USHT

The computation of the new linear transfers linked to the buoyancy frequency N amounts to only two additional production terms with respect to HITSG, as seen just before: the retro-action of the scalar field on the cospectrum, and the retro-action of the scalar flux on the kinetic energy spectrum. The resulting four new linear spherically-averaged transfers for unstably stratified turbulence are, for a vertical scalar gradient (along x 3 ),

S L,USHT(iso) (k) = N S k P i3 (k)F i (k)d 2 k = N F(k), (7.8) S L,USHT(dir) ij (k) = N 2 S k P l3 (k)F l (k)P ij (k)d 2 k - δ ij 3 N F(k) = N 20 E F i (k)δ j3 + E F j (k)δ i3 - 2 3 F(k)δ ij , (7.9) S L,USHT(pol) ij (k) = N 4 S k P l3 (k)F n (k) + P n3 (k)F l (k) N * l (k)N * n (k)N i (k)N j (k)d 2 k = 3N 10 E F i (k)δ j3 + E F j (k)δ i3 - 2 3 F(k)δ ij , (7.10) S F,L,USHT i (k) = N S k E T (k)P i3 (k)d 2 k = 2N E T (k, t) 1 3 δ i3 + H (T ) i3 (k) . (7.11)
Consequently, the spherically-averaged Lin equations for USHT are

∂ ∂t + 2νk 2 E(k, t) = S NL(iso) (k, t) + N F(k,
∂ ∂t + (ν + a)k 2 F(k, t) = S F,NL 3 (k, t) + N φ 33 (k, t) + 2N E T (k, t) 1 3 + H (T )
33 (k, t) .

(7.17)

The non-linear transfers are the same as in passive scalar dynamics, since the linear operators of the three-point third-order correlations equations are not taken into account in the nonlinear closure. Quadratic anisotropic contributions in the non-linear transfers for the scalar and cospectrum can be found in Appendix E along with some illustrations for USHT.

In the following sections, high Reynolds USHT is investigated and results are qualitatively compared with the axisymmetric EDQNM developed in Burlot et al. (2015a,b). Only the component along the mean gradient () 33 will be presented since one has () 11 = () 22 = -() 33 /2 because of axisymmetry. Furthermore, to be consistent with the development of the present spectral modelling, the same set of eddy-damping constants is kept: A 1 = 0.355, A 2 = 0 and A 3 = 1.3. Differences between this set of constants, and A 1 = A 2 = A 3 = 0.355 used in Burlot et al. (2015a), are illustrated hereafter in section 7.3.7.

First, inertial scaling of the kinetic energy, scalar variance and scalar flux spectra are addressed, along with some considerations about the large scales initial conditions σ and σ T . Then, the time evolution of one-point statistics is studied, such as the kinetic energy and its exponential growth rate, the Froude number, the mixing parameter, and global anisotropy indicators. Influence of large scales initial conditions σ and of the intensity of the stratification N on the asymptotic anisotropic states of the previous quantities is also analyzed. Afterwards, a quantitative comparison with Burlot et al. (2015b) is proposed, to illustrate that our model is able to recover, with satisfactory quantitative agreement, some features obtained by a model without any truncation of the expansion in spherical harmonics specific for axisymmetric turbulence.

The scale-by-scale repartition of anisotropy in spectral space is then addressed and some considerations on the structure of the flow are proposed. These different parts constitute a complete validation of our anisotropic EDQNM modelling. Furthermore, qualitative comparisons with the cases of passive scalar dynamics and shear flows, which have been addressed with the same consistent closure, are proposed. Finally, in section 7.5, two new applications are presented: first, the pressure spectrum, and more specifically its anisotropic part resulting from stratification, is investigated. Then, effects of large Schmidt numbers, notably on the inertial scaling of the cospectrum F, are revealed.

Spectral scaling and infrared dynamics

In this part, the emphasis is put on on the three main spectra of USHT: the kinetic energy spectrum E(k, t), the scalar variance spectrum E T (k, t) and the cospectrum F(k, t), which are investigated at large Reynolds numbers. First, the inertial scaling is discussed, and then, we focus on the infrared dynamics.

Spectral scaling of E, E T and F

Firstly, the scaling of the main spectra is investigated. As in HITSG, E and E T scale in k -5/3 in the inertial range, and F in k -7/3 , as revealed in Fig. 7.1. Moreover, the peaks of the three spectra evolve in k -3 with increasing N t, as assessed in Figure 11 of Burlot et al. (2015a). This can be shown easily, and the idea of the proof is inspired from a Rayleigh-Taylor analysis by [START_REF] Poujade | Rayleigh-taylor turbulence is nothing like kolmogorov turbulence in the self-similar regime[END_REF] Furthermore, the possibility of an anisotropic correction to the Kolmogorov spectra is discussed in Burlot et al. (2015b), and it is shown numerically in the latter reference that the anisotropic part of φ 33 (k, t) scales in k -3 (mostly at the beginning of the inertial range). Nevertheless, if one zooms in in the corresponding Figure 12, it appears that k -3 is steeper than the real inertial range slope. This is in agreement with our numerical simulations displayed in Fig. 7.2a, where the isotropic, directional and polarization parts of φ 33 (k, t) are presented. Clearly, the isotropic part φ

(iso) 33

scales in k -5/3 . Whereas one has for φ (pol) 33

(and φ (dir) 33 ) a spectral slope steeper than k -7/3 , but not as sharp as k -3 . Numerically, k -2.52 is found here, in good agreement with the recent DNS of Gréa et al. (2016a), where the spectral scaling of φ (pol) 33 is also closer to k -7/3 than k -3 . Furthermore, it is recovered, in agreement with Burlot et al. (2015b), that the polarization part is more intense at large scales than the directional one.

In Fig. 7.2b, for illustration purposes, the linear and non-linear transfers of kinetic energy are displayed for Saffman turbulence. It is revealed that at large scales, linear production mechanisms dominate over non-linear transfers which take energy from these large scales and bring it to smaller scales. This feature, that production is strong at large scales and thus that anisotropy is dominant at large scales, will be used later on. Now, we investigate in detail the inertial scaling of E(k, t), E T (k, t) and F(k, t), and more precisely the value of the Kolmogorov and Corrsin-Obukhov constants K 0 and K CO respectively. These constants are obtained by compensating the spectra with an adapted scaling. For the kinetic energy and buoyancy spectra, it is shown in Fig. 7.3a that the usual isotropic inertial scalings (7.20) are relevant and allow to recover classical values for the constants, K 0 = 1.31 and K CO = 0.76, similar to what is obtained for passive scalar dynamics. This is completely different when it comes to the cospectrum F. First, let's point out that there exist two different inertial scalings: the classical one proposed by [START_REF] Lumley | Similarity and the turbulent energy spectrum[END_REF] which was shown to work nicely for passive scalar dynamics in HITSG in Chapter 5, with a constant C F 3, and a more recent one, derived by Burlot et al. (2015b), which unlike Lumley's, takes into account the scalar dissipation rate T , which seems a priori relevant for an active scalar field

E inertial (k, t) = E(k, t) -2/3 k 5/3 , (7.19) E T,inertial (k, t) = E T (k, t) -1 T 1/3 k 5/3 ,
F Lumley inertial (k, t) = F(k, t) N -1 -1/3 k 7/3 , (7.21) F Burlot inertial (k, t) = F(k, t) N -1 K 0 1/3 + K CO T -2/3 -1 k 7/3 . (7.22)
The inertial scaling F Burlot inertial , first derived in [START_REF] Soulard | Inertial-range anisotropy in rayleigh-taylor turbulence[END_REF], is presented in Fig. 7.3b: the plateau of the compensated cospectrum only starts appearing at Reynolds numbers as large as Re λ ∼ 3.10 6 . Whereas for the passive scalar case, at a similar Reynolds number, F was displaying a clear plateau around C F 3 in the inertial range. Nevertheless, the scaling proposed by Burlot and coworkers F Burlot inertial seems to be more relevant than Lumley's: indeed, for F Burlot inertial , a plateau seems to appear around 3.7, whereas F Lumley inertial settles around 12, which is too high, and consequently not displayed in Fig. 7.3b. The fact the scaling (7.22) is better than F Lumley inertial is very likely because it takes the scalar dissipation rate T into account.

Infrared dynamics

Now that the inertial range of the spectra has been investigated, the infrared dynamics is considered. It is recalled that in HITSG for a passive scalar, if initially σ T is different from σ, greater or lesser it does not matter, it always results in σ T = σ after a few turnover times (see Chapter 5). In USHT, it is fundamentally different since because of stratification, all of the three main spectra E, E T and F are coupled through the linear production terms at large scales. Hence, the most energetic initial spectrum, the one with the smallest infrared slope, imposes the minimum of energy to the others, so that in the end one has always σ = σ T = min(σ(t = 0), σ T (t = 0)). This is completely different from HITSG where the passive scalar field has no effect on the kinetic field. Two examples are presented. First, in Fig. 7.4a, one has σ(t = 0) = 2 and σ T (t = 0) = 4. The minimum of energy is thus imposed by E so that the scalar infrared slope results very rapidly, within one dimensionless time N t, in σ T = 2. This is identical to what happens in HITSG. In the opposite case illustrated in Fig. 7.4b, one has σ(t = 0) = 4 and σ T (t = 0) = 2: the minimum of energy is this time imposed by E T so that after one N t, σ = 2. This case is completely different from HITSG where σ T would have changed to 4.

A last aspect is presented in Fig. 7.4c: because of strong backscatter of energy towards large scales when σ ≥ 4, if the slope is initially σ = 5, it eventually becomes σ = 4 in a few dimensionless times N t. The same mechanism of strong inverse transfers of energy occurs in decaying isotropic turbulence [START_REF] Lesieur | 3d isotropic turbulence at very high reynolds numbers: Edqnm study[END_REF].

From this analysis, one can choose, without any loss of generality, σ = σ T ≤ 4. This result for USHT infrared dynamics notably simplifies the study of asymptotic anisotropic states of the flow: in particular, we choose to investigate only integer values of the infrared slopes: σ = σ T = {1; 2; 3; 4}.

-4 -3 -2 -1 0 1 0 1 2 3 4 5 log(k) log E/ log k E E T N t = 0 N t = 1 (a) -4 -3 -2 -1 0 1 0 1 2 3 4 5 log(k) log E/ log k E E T N t = 1 N t = 0 (b) -4 -3 -2 -1 0 2 3 4 5 6 log(k) log E/ log k N t = 0 N t = 1 N t = 3 N t = 4 (c)

One-point statistics

Here, two important quantities of unstably stratified turbulence are firstly addressed, namely the Froude number

F r(t) = (t) K(t)N , (7.23)
which is the ratio of the stratification characteristic time 1/N on the inertial one K/ , and the mixing parameter

Λ(t) = K T (t) K(t) , (7.24)
which is the ratio of the scalar variance to kinetic energy. Since the scalar field is scaled as a buoyant velocity, Λ is dimensionless. Then, the exponential growth rate β of the kinetic energy K =< u i u i > /2 is analyzed, along with the time evolution of global anisotropy using the normalized deviatoric Reynolds stress tensor b ij . More precisely, the influence of N and σ on the asymptotic values reached by F r, Λ, β and b 33 at large N t and Re λ is studied. Finally, a quantitative comparison with the axisymmetric EDQNM (Burlot et al., 2015b) is proposed. Additional considerations about the modelling are also briefly presented.

The Froude number F r

The Froude number can be interpreted as a ratio of characteristic time scales. As such, it could be qualitatively compared to the shear rapidity S R = /(KS) in shear flows, notably addressed with the same anisotropic EDQNM modelling in Chapter 3. In Fig. 7.5a it is revealed that F r, unlike S R , depends on σ in the asymptotic anisotropic state. Final values of F r spans from 0.44 for σ = 1 to 0.66 for σ = 4. The smaller σ, the smaller F r: this is expected since for small σ, large scales contain more energy and consequently are more anisotropic, because of the production terms which act dominantly at large scales, as illustrated in Fig. 7.2b. With the production terms being dynamically dominant with a small σ, the characteristic time scale 1/N diminishes, thus making F r decrease. Nevertheless, the values reached here by F r are slightly higher than the values obtained in Burlot et al. (2015b) (F r = 0.3 for σ = 1). This means that the flow within the present approach is less sensitive to stratification than with the axisymmetric EDQNM. It is shown in section 7.3.7 that by changing the eddy damping constants, we can increase the impact of stratification and thus reduce the Froude number: asymptotic values of F r are gathered in Table 7.2.

However, in Fig. 7.5b it is shown that F r, similarly to S R , does not depend on the intensity of the mean gradient N , except at small N t in the transitory regime. At short times, the larger N , the smaller F r: this is consistent with a strong stratification intensity making production of buoyant fluctuations dominant initially.

N t F r N = 0.1τ -1 0 N = 1τ -1 0 N = 2τ -1 0 N = 5τ -1 0 (b)

The mixing intensity Λ

Now, the mixing intensity Λ is addressed. Similarly to F r, its asymptotic value depends on σ, but not on its initial value, except for short times of course, as revealed in Fig. 7.6a and 7.6b. The dependence with N is not presented since it is very similar to what happens for F r, i.e. a dependence on N only at short times. The final values of Λ are contained between 1.6 for σ = 1 and 1.4 for σ = 4, which is quantitatively in agreement with Burlot et al. (2015b): asymptotic values of Λ are gathered in Table 7.2.

One can remark that K T /K depends strongly on initial conditions at a fixed σ, such as the initial Reynolds number Re λ (0) and the stratification frequency N . Indeed, Λ initially decreases in Burlot et al. (2015b), whereas it initially increases here. The reason is the following: here, Re λ (0) 5 implies that linear production mechanisms dominate whatever the value of N is, roughly for N ≥ 0.1τ -1 0 . For the axisymmetric EDQNM, Re λ (0) 70, which requires at least N = 1τ -1 0 to make Λ increase initially, meaning that linear production overcomes non-linearity. This is illustrated in Fig. 7.6c.

Λ(0) = 1 Λ(0) = 2 (b) 0 1 2 3 4 0 0.5 1 1.5 2 N t Λ N = 1τ -1 0 N = 0.1τ -1 0 N = 1τ -1 0 N = 0.1τ -1 0 (c)

Growth of the kinetic energy K(t)

The emphasis is now put on the exponential growth rate β of the kinetic energy K(t) in USHT. It was assessed by Burlot et al. (2015b) that this growth rate strongly depends on the large scales initial condition σ, according to the theoretical prediction

K(t) ∼ K(0) exp(β Burlot N t), β Burlot = 4 σ + 3 , (7.25)
which comes from the more general work of [START_REF] Soulard | Large-scale analysis of self-similar unstably stratified homogeneous turbulence[END_REF] where the stratification frequency N can vary: (7.25) corresponds to the specific case where N is constant. Furthermore, (7.25) relies on the fact that the largest eigenvalue of the linear operator of the evolution equations of the axisymmetric EDQNM is 2N : indeed, equalizing the growth rate of the linear limit E ∼ exp(2N t) with the one coming from self-similar analysis E ∼ KL ∼ exp((σ + 3)βN t/2) directly yields (7.25). In our case, because of the anisotropy modelling through a truncated expansion into spherical harmonics, our evolution equations (7.12) to (7.17) are different from those of Burlot and coworkers. The largest eigenvalue of the linear operator associated to (7.12)-(7.17) is 4N/ √ 5 here, against 2N for Burlot et al. (2015a,b); Gréa et al. (2016a). This leads to a different theoretical prediction for the exponential growth rate of the kinetic energy

K(t) ∼ K(0) exp(β th N t), β th = 8 √ 5(σ + 3) . (7.26)
The present theoretical predictions and those of Burlot et al. (2015a) are gathered in Table 7.1, and it is worth noting that for a given large scales initial condition σ, our predictions yield a smaller growth rate than in the axisymmetric EDQNM: this is very likely due to our truncated expansion into spherical harmonics of the spectral correlations. Moreover, this is consistent with our flow being less anisotropic. We could conjecture that taking into account more spherical harmonics would increase the exponential growth rate of the kinetic energy up to the limit β Burlot .

Large scales initial condition σ = 1 σ = 2 σ = 3 σ = 4

Present prediction : In fact, the evolution equations (7.5) to (7.7) of Rij , E T , and F i reveal that contributions of the fourth order expansion for E, Z and E T bring no contribution in (7.6). A higherorder expansion for the scalar flux F i is required to modify the production terms and thus get closer to the exact value 2N of the maximal eigenvalue of the linear operator.

β
The predictions for the exponential growth rate of the kinetic energy are compared to our numerical results in Fig. 7.7a. The agreement is excellent: for σ = 1, 2, 3, the prediction β th is recovered within 1%. There is a slight difference in the case of Batchelor turbulence, where the numerical result is 5% higher than the prediction: this is very likely because of the strong inverse non-linear transfers which naturally occur in Batchelor turbulence [START_REF] Lesieur | 3d isotropic turbulence at very high reynolds numbers: Edqnm study[END_REF]. One could add a backscatter parameter, i.e. a correction for Batchelor turbulence, to the prediction (7.26) in the specific case σ = 4, as usually done for decaying HIT. Here for USHT, in order to adapt the theoretical prediction in Batchelor turbulence to our numerical result, a least square fit leads to the backscatter parameter p USHT = 0.37, so that

β th = 8 √ 5(σ -p USHT + 3) , p USHT = 0 for σ ≤ 3, p USHT = 0.37 for σ = 4. (7.27)
It is revealed in Fig. 7.7b that the scalar variance K T and the mixed correlation K F both grow at the same rate β, in agreement with Burlot et al. (2015a,b); Gréa et al. (2016a). This can be qualitatively compared with the case of a passive scalar field advected by a turbulent shear flow with an imposed mean scalar gradient, where K T and K F also grow exponentially with the same rate as K, as revealed in Chapter 5.

In conclusion, the kinetic energy exponential growth rate strongly depends on the large scales initial conditions σ in the asymptotic states of USHT. This is interesting, since for shear flows, K(t) was growing at the same rate independently of σ, at least within the same anisotropic EDQNM modelling (further explanations were provided at the end of Chapter 3). This illustrates two intrinsically different mechanisms of kinetic energy production in shear flows and USHT.

Global anisotropy

The time evolution of global anisotropy is now addressed: the scale-by-scale distribution of anisotropy is the subject of the next part. In Fig. 7.8a to 7.8d, b 33 and b T 33 first increase, which shows the departure from the isotropic state, and then decrease and reach a final nonzero value. This decrease is the signature of a return to isotropy of the small scales when the Reynolds number increases. More specifically, it is revealed that polarization anisotropy for b 33 is stronger than the directional one, in agreement with Burlot et al. (2015b). The strong anisotropy in the component () 33 furthermore shows that turbulent structures mainly align with the mean scalar gradient. In addition, the values reached by b T 33 are quite similar to the ones reached by b Hence, the main difference with the axisymmetric EDQNM is that the present anisotropic EDQNM modelling under-estimates the global anisotropy of the flow, so that asymptotic anisotropic states of b 33 are lower in our case: values are reported in Table 7.2. It is shown in part 7.3.7 that we can slightly increase the global anisotropy of the flow by changing the eddy-damping constants.

Finally, in addition to these global anisotropy indicators, the pressure-velocity Π 33 and pressurescalar Π F correlations are presented in Fig. 7 dk.

(7.29)

The return to isotropy at the level of the scalar flux is found to be more intense than the one of the velocity field: this is expected since the cospectrum is a purely anisotropic quantity, for which pressure is the destructive mechanism. Furthermore, in agreement with previous statements, the return to isotropy mechanism is stronger for Saffman turbulence than Batchelor turbulence, because large scales are less anisotropic in the latter case than in the former.

Comparison with Burlot et al. (2015b)

In this part, we compare quantitatively the results of our anisotropic EDQNM modelling to the axisymmetric EDQNM (Burlot et al., 2015a,b) specifically for two one-point statistics investigated in the previous sections: the Froude number F r, defined in (7.23), and the mixing intensity Λ defined in (7.24). First, it was observed previously that the asymptotic anisotropic states obtained with the present anisotropic EDQNM modelling are less anisotropic than the ones obtained in Burlot et al. (2015b): this was notably seen through F r and b 33 which were higher and smaller respectively. In order to provide here a meaningful comparison, and only for this part, we use the eddy-damping constants of the axisymmetric EDQNM, i.e. A 1 = A 2 = A 3 = 0.355. The impact on USHT dynamics of the two different settings of eddy-damping constants ([EDC1]:

A 1 = 0.355, A 2 = 0, A 3 = 1.3; [EDC2]: A 1 = A 2 = A 3 = 0.355) is discussed in section 7.3.7.
So, for the comparison, we use the setting [EDC2] in Saffman turbulence (σ = 2), an initial turbulent Reynolds number close to Re T = 833, with Re λ = 20Re T /3, and the initial peak of energy is k peak = 40k L (0). Even though it is stated in Burlot et al. (2015a) that the initial Froude number is F r = 1.2, it seems to not be the case in Figure 1(a) therein. Consequently, we choose the stratification frequency N = 4τ -1 0 so that the initial behaviour of F r is recovered: taking N = 1τ -1 0 would not have changed much.

The results are presented in Fig. 7.9. For the Froude number, the overall agreement is excellent: the transient regime is correctly captured and notably the strong decrease; the asymptotic value of F r is quite well recovered. For the mixing ratio Λ, the initial behaviour is quite well captured, with a very good agreement for the asymptotic value. In conclusion, there is a satisfactory agreement with the axisymmetric EDQNM if one changes the eddy-damping constants from [EDC1] to [EDC2]. The drawback is that, as illustrated in section 7.3.7, by doing so the Corrsin-Obukhov constant decreases.

Conclusions on one-point statistics

In this part, various one-point statistics of crucial importance in unstably stratified turbulence have been studied: the Froude number F r, the mixing intensity Λ, the growth rate β of the kinetic energy, scalar variance and scalar flux, and the velocity and scalar global anisotropy indicators b 33 and b T 33 . The different results could be summarized as follows: (i) All these quantities strongly depend on σ in the asymptotic anisotropic state at large N t, or equivalently at large Reynolds numbers. When σ increases, the large scales energy diminishes along with the quantity of anisotropy injected in the flow, so that F r increases, and Λ, β, b 33 and b T 33 decrease. (ii) On the contrary, the asymptotic anisotropic state does not depend on N : changing the intensity of the mean scalar gradient only impacts the short time dynamics.

Finally, the main difference between the two approaches is that the flow is less anisotropic in our case, probably due to the truncation of the spherical harmonics expansion of spectral correlations for the modelling of anisotropy. The principal consequences are (i) an exponential growth rate of the kinetic energy 10% lower than in Burlot et al. (2015a) one-point statistics analyzed so far, and obtained with both the present anisotropic EDQNM modelling and the axisymmetric EDQNM, are gathered in Table 7.2.

Furthermore, throughout this part, qualitative comparisons were made with the cases of passive scalar dynamics (HITSG) and shear flows. It notably appeared that the asymptotic anisotropic states in USHT and shear flows strongly differ: indeed, in shear flows, at least within the same anisotropic EDQNM modelling, the asymptotic anisotropic state does not depend on the mean-field gradient intensity, nor on the large scales initial conditions σ.

Eddy-damping constants

In this section, we briefly discuss the impact of changing the eddy-damping constants on the dynamics of USHT. First, we recall that for consistency with the development of the present anisotropic EDQNM modelling, the same eddy-damping constants are kept here for the extension to the case of active scalar dynamics, i.e. A 1 = 0.355,

A 2 = 0, A 3 = 1.3 [EDC1],
where A 1 is for the velocity field, and A 2 and A 3 for the scalar field. The setting [EDC1] was consequently kept so far, except for the quantitative comparison against the axisymmetric EDQNM: indeed, in the latter work, a different choice of eddy-damping constants was made, i.e. Burlot et al. (2015a), a correction to the eddy-damping term is added to match with DNS: this is not considered here, since it only slightly affects the early dynamics.

A 1 = A 2 = A 3 = 0.355 [EDC2]. Furthermore, in
With the present setting [EDC1], the Kolmogorov and Corrsin-Obukhov constants are consistent with those obtained for passive scalar dynamics, K 0 = 1.31 and K CO = 0.76, as presented before.

Choosing [EDC2] as in Burlot et al. (2015a,b), tends first to decrease K CO to values smaller than usual ones (K CO = 0.6), as revealed in Fig. 7.10a, whereas K 0 remains unchanged. Furthermore, with [EDC2], the flow is slightly more anisotropic in Fig.

Scale by scale anisotropy and structure of the flow

In this part, the scale by scale distribution of anisotropy is investigated for the velocity and scalar fields, at the level of the second-order moments, thanks to H

sin 2 γ T (k, t) = 1 E T (k, t) S k sin 2 θ k (k) E T (k, t)d 2 k = 2H (T ) 33 (k, t) + 2 3 , (7.30)
where θ k (k) is the angle between the vertical axis and the wavevector k. When the considered scales are isotropic, the value of sin 2 γ T is 2/3, whereas it is 0 for

H (T ) ij .
It was shown in Fig. 7.8a to Fig. 7.8d that in the asymptotic anisotropic state, the global anisotropy indicators b 33 and b T 33 are non-zero, meaning that there is anisotropy in the flow. First, it is revealed in Fig. 7.11a, that anisotropy is mainly gathered at large scales for the velocity field, where H are different from zero, unlike small scales which have returned to isotropy. One can remark that, as previously, polarization anisotropy is much stronger that directional one at large scales. An interpretation of this is provided a bit later. Similarly, for the scalar field in Fig. 7.11b, small scales have returned to isotropy (sin 2 γ T = 2/3), whereas anisotropy is gathered at large scales. This figure additionally illustrates further that Saffman turbulence is globally more anisotropic than Batchelor turbulence, because the linear production at large scales is stronger in Saffman than in Batchelor turbulence. Interestingly, for both the velocity and scalar fields, Fig. 7.11a and 7.11b show that from the longitudinal Taylor scale λ = 20νK/ , the scales have completely returned to isotropy: in particular, this indicates that even in USHT, isotropic statistics could be used for scales smaller than λ. The Ozmidov wavenumber, defined in Gréa et al. (2016a) k O = 2π N 3 / , is displayed as well: it is (a) Another possibility is to investigate the energy contained in the toroidal and poloidal modes, defined in (2.28), using the Craya-Herring frame illustrated in Fig. 2.1. First thing to do is to link the spherically-averaged spectra EH (dir) 33

σ = 4 σ = 2 2/3 k η k L k O 1/λ (b)
0 1 2 3 -1 0 1 2 3 θ k E (polo) (θ k ) and E (toro) (θ k ) E (polo) /E 0 E (toro) /E 0 π/2 (b)
and EH

(pol) 33

to θ k , the angle between the vertical axis (the mean scalar gradient direction) and the wavevector k. After some algebra, one gets a relation between directional and polarization anisotropies E (dir) and Z, and θ k

E (dir) (k, θ k ) = - 15 2 E 0 (k)H (dir) 33 (k)(3 cos 2 θ k -1), Z(k, θ k ) = 15 2 E 0 (k)H (pol)
33 (k) sin 2 θ k .

(7.31) Finally, this permits to obtain E (toro) and E (polo) , defined in (2.29), as functions of the spherically averaged spectra

E (toro) (k, θ k ) = E 0 (k) 2 1 - 15 2 H (pol) 33 (k) sin 2 θ k + H (dir) 33 (k)(3 cos 2 θ k -1) , (7.32) E (polo) (k, θ k ) = E 0 (k) 2 1 + 15 2 H (pol) 33 (k) sin 2 θ k -H (dir) 33 (k)(3 cos 2 θ k -1) . (7.33)
At this point, it is of interest to point out that in Burlot et al. (2015a), instead of considering the equations of E and Z, the equations of E (toro) and E (polo) are solved, along with those of E T and F 3 . The variations of both E (toro) and E (polo) as functions of θ k are displayed in Fig. 7.12b for a wavenumber located in the infrared range: this is where the anisotropy is gathered, as shown in Fig. 7.11a. For 0 ≤ θ k ≤ π, one has E (polo) > E (toro) , and the poloidal and toroidal potentials are maximum for θ k = π/2. The poloidal mode being more intense, this means that in the configuration θ k = π/2, spectral velocity fluctuations are preferentially aligned in the mean scalar gradient direction. This is an interesting feature: indeed, in the previous sections it was underlined that polarization anisotropy is stronger than directional one at large scales. This is assessed in Fig. 7.12b, where in the infrared range one has clearly Z > E (dir) . Consequently, the present results show that in USHT, a dominant polarization anisotropy corresponds to spectral velocity fluctuations mainly aligned with the mean scalar gradient, so that the principal component is the poloidal one, in agreement with Fig. 7.12a, where our simulation goes toward the (1C) state. Also, for θ k = 0 or θ k = π, in a plane perpendicular to the mean scalar gradient, E (toro) = E (polo) so that there is no polarization anisotropy.

Pressure spectra and high Schmidt numbers

So far, the dynamics of USHT at a unit Schmidt number Sc = 1 was addressed at large Reynolds numbers, and the strong dependence of the asymptotic anisotropic state on the infrared slope σ was recovered, with a good overall agreement with the axisymmetric EDQNM. In the present section, the anisotropic EDQNM modelling is applied to two new cases: first, the pressure spectrum is studied, and in particular its anisotropic part resulting from stratification, with a qualitative comparison to the pressure spectrum in shear flows. Then, the case of very large Schmidt numbers Sc 1, corresponding for instance to saltwater, is analyzed on a fundamental point of view, with the emphasis put on the scaling of the cospectrum.

Pressure spectra

The emphasis is put on the pressure spectra, which have not been investigated in USHT in previous references. This study is done in the spirit of the work by [START_REF] George | Pressure spectra in turbulent free shear flows[END_REF] who analyzed the anisotropic part of the pressure spectrum for shear flows. This is notably presented, within the present anisotropic EDQNM modelling, in Appendix D. The same method is applied here for USHT. The Poisson equation in USHT is obtained by taking the divergence of (7.4), which yields

-∆p = ∂ 2 u i u j ∂x i ∂x j + λ i ∂v ∂x i . (7.34)
Then, with the definition of the two-point second-order pressure correlation (D.19), one gets

E P (k, t) = 2α i α j α p α q k=p+q Riq (p, t) Rjp (q, t)d 3 p + α i α j k 2 λ i λ j E T (k, t). (7.35)
The isotropic part remains unchanged with respect to HST, only the anisotropic part is different.

The spherical average of this equation eventually gives

E P (k, t) = 16π 2 ∆ k kpq(1 -y 2 )(1 -z 2 )E 0 E 0 dpdq + E T k 2 λ i λ j δ ij 3 -2H (T ) ij , (7.36)
where we call the second-contribution E (USHT) P the turbulence-unstable-stratification interaction. One can note that the total pressure spectrum E P (k, t) for USHT is similar, in its structure, to the one in HST.

First, in Fig. 7.13a, the scaling of the isotropic and anisotropic parts of E P are presented. The turbulence-turbulence interaction spectrum scales in E (iso) P ∼ k -7/3 : the constant C P = 2.3 in Fig. 7.13b is close to the value obtained in shear flows which indicates some universality of the isotropic pressure spectrum. Then, the anisotropic part resulting from stratification is presented in Fig. 7.13a and scales in E (USHT) P ∼ k -11/3 . The k -11/3 , analogous to the anisotropic part in shear flow, is expected from the expression (7.36), because E T ∼ k -5/3 in the inertial range. And similarly to shear flows, the anisotropic part has a quadratic dependence on the meanfield gradient, given its expression (7.36). Thus, only the dependence on the dissipation rates remains to be determined. Since the scalar field is rescaled as a buoyant velocity, there are infinite possibilities of the form a b T , with a + b = 2/3, by dimensional analysis. One could choose the inertial scaling of E T , i.e. a = -1/3 and b = 1, but this yields in Fig. 7.13b (in grey) a constant quite small of order 0.25: this is not satisfactory since for shear flows the constant was very close to Kolmogorov. Given the similarities pointed out so far, we choose a = 2/3 and b = 0, as for E depending on E T . In the end, the scaling of the anisotropic part of the pressure spectrum in USHT reads Finally, the time evolution of the isotropic and anisotropic parts K (iso) P and K

E (USHT) P (k, t) = C (USHT) P N 2 2/3 k -11/3 , C (
(USHT) P of the pressure variance are displayed in Fig. 7.13c: it is found, similarly to shear flows, that the pressure variances grow exponentially at a rate 2β, where β is the exponential growth rate of the kinetic energy. Interestingly, the exponential growth rate of the pressure variance still depends on the infrared slope σ of the kinetic energy spectrum, even though the infrared slope of the isotropic pressure spectrum is

E (iso) P ∼ k 2 .

Cospectrum at high Schmidt numbers

In this section, the case of a weakly diffusive active scalar with Sc 1 is addressed (instead of considering the Prandtl number, the Schmidt number Sc is used, which is equivalent for a being the molecular diffusivity). This configuration is representative of unstably stratified water columns in the ocean generated by double diffusion mechanisms: at the ocean surface, hot salty water is on top of cooler and saltier water, so that the stratification is stable. But when the temperature drops off in the air layer above the ocean, the upper salty water cools down very rapidly, because heat transfers are much more efficient than mass transfers. In the end, one has a heavier fluid on top, causing unstably stratified water columns [START_REF] Sigman | Polar ocean stratification in a cold climate[END_REF].

For Sc 1, the scalar flux F was found to decrease sharply after k η in Chapter 6, similarly to the kinetic energy spectrum. It appears that in USHT, the behaviour of the cospectrum beyond k η is completely different, as revealed in figure 7.14a: indeed, unlike passive scalar dynamics, the scalar flux survives in the viscous-convective range. The buoyant spectrum E T still scales in k -1 , and there are also small scales fluctuations for E beyond k η , but they are much less intense than for E T and F, and are therefore neglected.

The scaling of F in the viscous-convective range is not clear, but it seems to be close to k -1 , slightly steeper, as revealed in the zoom in Fig. 7.14b. Around k η , the cospectrum seems to scale in k -3 , but this is very likely just a transition toward the viscous-convective scaling.

F (k, t) k L k η k -5/3 k -7/3 k -1 k B (
F (k, t) k -1 k -1 k -3 k η k B (b)
Nevertheless, both the k -1 and k -3 scalings are briefly justified using classical arguments.

The k -3 scaling can be obtained by assuming that at small scales, there is a balance between dissipation and production of buoyant fluctuations in (7.17), so that (7.38) In this expression, E T H (T )

(ν + a)k 2 F ∼ 2 3 N E T ⇔ F(k, t) ∼ N T √ ν k -3 .
33 was neglected compared to E T . For reasons which are explained hereafter, it is preferred to express this new scaling as follows (7.39) where the inverse of the Kolmogorov time scale ν/ appears, consistently with the dynamics of the viscous-convective range. Then, after k η , the scaling is slightly steeper than k -1 , but nevertheless the Batchelor scaling seems relevant if one assumes, as for E T , that the characteristic time scale of F in the viscous-convective range is also independent of k: since this new range exists only thanks to the small scales coupling through N , it makes sense to assume that it depends linearly on N , and also on and ν: this provides N ν/ as the characteristic time scale of the cospectrum in the viscous-convective range. Further assuming that F depends only on this time scale, k and T , yields

F(k, t) ∼ N ν T k -3 ,
F(k, t) ∼ ν T k -1 . (7.40)
It is worth noting that, unlike E T for which non-local transfers are at the origin of the viscous convective range, the new range for F beyond k η is created by local production of buoyant fluctuations through the term N E T . Direct non-local expansions (q k ∼ p) were performed for the non-linear transfers of F but they are negligible.

The change from the scaling in k -3 , around k η , to k -1 , just after k η , can be understood in terms of characteristic time scales: for the scaling (7.39), the characteristic time is ν/ , which is the classical characteristic time of the Kolmogorov scale. For smaller scales, viscous dissipation becomes more and more important, so that the characteristic time evolves from ν/ toward (νk 2 ) -1 , which directly yields (7.40). Then, the characteristic time scale saturates to N ν/ . These two scalings and their characteristic times are consistent with the Kolmogorov scale being the wavenumber around which the cospectrum changes from k -3 to k -1 : indeed, equating (7.39) and (7.40) yields k = k η .

About the one-point statistics: obviously, even with high Schmidt numbers, simulations show that the asymptotic anisotropic state still depend on σ and not on N , and the exponential growth rate β is not modified with respect to the case Sc = 1. Nevertheless, it is proposed to illustrate in figure 7.15 the impact of a large Sc on the early dynamics of the scalar anisotropy indicator b T 33 and the mixing parameter Λ (the effects are negligible for F r and b 33 ). The main result is that a large Sc does not change the asymptotic values with respect to the case Sc = 1. However, specifically for b T 33 and Λ which depend explicitly on the scalar field unlike b 33 and F r, a large Schmidt number strongly impacts the transient regime as well: Λ initially increases much more with Sc = 10 4 than with Sc = 1, whereas on the contrary, b T 33 is always smaller at Sc = 10 4 . At Sc = 10 4 , even if there is no inertial range initially for E T , the viscous convective range spans almost two decades: this is completely different from the case Sc = 1. This initial viscous-convective range thus contributes greatly to Λ because it gives large initial values of K T , which explains the strong increase at small N t. Then, when the Reynolds number increases, the inertial ranges of E and E T become dominant in the integrals for K and K T , so that eventually, the same asymptotic value as for Sc = 1 is recovered. Whereas for b T 33 , the viscous-convective range initially adds isotropic small scales, thus reducing the early global anisotropy over the whole wavenumber space.

As a conclusion, unlike passive scalar dynamics, the cospectrum survives in the viscous-convective range for USHT for large Schmidt numbers, and scales in k -1 , similarly to the scalar variance spectrum, after a transient k -3 subrange around the Kolmogorov wavenumber. Finally, large Schmidt numbers strongly affect the early dynamics of b T 33 and Λ, nevertheless without changing the asymptotic state.

Conclusion on USHT

Unstably stratified homogeneous turbulence (USHT) was investigated at large Reynolds numbers with the anisotropic EDQNM modelling extended to the case of active scalar dynamics. Moreover, since the present modelling was applied previously for different configurations -notably transport of passive scalar in an isotropic turbulence with a mean scalar gradient and shear flows -qualitative comparisons are also made with these cases and some interesting differences and similarities were found between shear-driven flows and unstably stratified turbulence.

The time evolution of the kinetic energy, scalar variance (or buoyancy) and scalar flux spectra E(k, t), E T (k, t) and F(k, t) were first addressed: the k -5/3 inertial scaling of E and E T was recovered, along with the k -7/3 inertial scaling of the cospectrum F. For the latter compensated spectrum, a plateau starts appearing at the highest Reynolds numbers reached here (Re λ ∼ 10 6 ). The k -3 time evolution of the peak of the three previous spectra was also recovered and justified analytically. For the infrared dynamics, it is found that because of the strong coupling between E, E T and F due to stratification, the spectrum with initially the smallest infrared slope σ imposes the minimum of energy to the others, which significantly differs from passive scalar dynamics.

Then, the effects of varying the stratification frequency N and the infrared slope σ on the asymptotic anisotropic states of one-point statistics in USHT were studied, specifically the Froude number F r, the mixing intensity Λ, global anisotropy indicators for the velocity and scalar fields b 33 and b T 33 , and the exponential growth rate β of the kinetic energy, scalar variance and mixed-correlation. The conclusion is, in agreement with Burlot and coworkers, that the asymptotic states of these quantities strongly depend on σ. This feature is completely different from shear flows where one-point statistics do not depend anymore on σ asymptotically, at least within the same modelling. However, for both shear flows and USHT, varying the mean gradient intensity impacts only short times. In particular, it is recovered that the more energy initially in large scales, i.e. the smaller σ, the more anisotropic the flow: furthermore, at large Reynolds numbers, anisotropy is gathered at large scales whereas small scales return to isotropy for both the velocity and buoyancy fields (at least at the level of second-order moments). A satisfactory agreement is found in the quantitative comparison with the axisymmetric EDQNM, but nevertheless one can point out some differences between the two approaches: (i) with our anisotropic EDQNM modelling, the flow is less anisotropic than in Burlot et al. (2015a,b); Gréa et al. (2016a), meaning notably that our anisotropy indicator b 33 is slightly smaller; (ii) a new theoretical prediction for the exponential growth rate β of kinetic energy is proposed, based on the linear operator of our evolutions equations, and assessed numerically. Whatever the large scales initial conditions σ are, our growth rate is 10% smaller than the one of Burlot and coworkers, consistently with our flow being less anisotropic; (iii) the Froude number is higher with the anisotropic EDQNM modelling, but can be decreased by changing the eddy-damping constants, as exposed in section 7.3.7.

To recover the exact theoretical prediction (7.25) for β, it is possible to multiply the production terms of the anisotropic EDQNM modelling for USHT, in equations (7.12)-(7.17), by √ 5/2, so that the maximal eigenvalue 2N of the linear operator is recovered. This procedure artificially corrects the exponential growth rate of the kinetic energy, nevertheless without increasing the global anisotropy of the flow since it is already maximal.

Finally, two applications of our anisotropic EDQNM modelling were proposed, which constitute new fundamental results. First, pressure spectra in USHT were investigated, and it was found that the anisotropic part, resulting from stratification, scales in k -11/3 in the inertial range, whereas the isotropic part scales in k -7/3 : these scalings are completely similar to pressure spectra in shear flows, investigated in Appendix D. It is also shown that the pressure variance grows exponentially at a rate 2β, where β is the growth rate of the kinetic energy. Then, high Schmidt numbers were considered: the scalar variance spectrum still scales in k -1 in the viscousconvective range beyond the Kolmogorov wavenumber, as in HIT. The main result here is that the cospectrum, which was strongly decreasing in passive scalar dynamics, now also displays a viscous-convective range with a scaling close to k -1 , after a transient k -3 subrange around k η . It is worth noting that large Schmidt numbers do not affect the asymptotic values of one-point statistics, nor the exponential growth rate β, but only the transient regime of Λ and b T 33 .

Perspective -Variable stratification N (t)

In this section, the mixing length L(t) is not fixed anymore, unlike USHT, which causes the stratification frequency N (t) to vary as well. The active scalar field is now a dimensionless concentration c, and we do not consider anymore the rescaled buoyant-velocity v. This part is thus a step further toward the modelling of Rayleigh-Taylor instability: the fluctuating quantities now impact the dynamics of the mean dimensionless concentration field C according to

∂C ∂t = - ∂ < u 3 c > ∂x 3 + a ∂ 2 C ∂x 2 3 Re λ 1 - ∂ < u 3 c > ∂x 3 . (7.41)
It is assumed that the Reynolds number is large enough to neglect diffusion effects, and in addition inside the mixing zone, one has ∂ 3 C = -1/L. Hereafter, a new prediction for the growth rate of the mixing length is derived within the anisotropic EDQNM modelling framework.

Evolution equations with variable stratification

In what follows, the Boussinesq approximation is still considered. In the self-similar state, the mixing length evolves according to [START_REF] Poujade | Rayleigh-taylor turbulence is nothing like kolmogorov turbulence in the self-similar regime[END_REF][START_REF] Gréa | The rapid acceleration model and the growth rate of a turbulent mixing zone induced by rayleigh-taylor instability[END_REF]) (7.42) where α RT is the Rayleigh-Taylor growth rate, whose theoretical prediction was derived by [START_REF] Poujade | Growth rate of rayleigh-taylor turbulent mixing layers with the foliation approach[END_REF] in the limit of small Atwood number A 1,

L(t) = 2α RT Agt 2 ,
α RT th = 1 (σ + 2)(σ + 3) . (7.43)
In what follows, it will be shown that this prediction relies on two crucial features and needs to be adapted to the anisotropic EDQNM modelling. The mixing length is defined as

L(t) = 6 +∞ -∞ C(1 -C)dx 3 . (7.44)
Assuming that x 3 = 0 is at the center of the mixing zone, and that < u 3 θ > has a parabolic evolution from -L/2 to L/2 [START_REF] Soulard | Large-scale analysis of self-similar unstably stratified homogeneous turbulence[END_REF], the time derivative of this equation becomes, with the equation of the mean field (7.41),

dL dt = L(t) = 8 < u 3 c > . (7.45)
There exists another possibility, proposed in [START_REF] Soulard | Influence of the mixing parameter on the second order moments of velocity and concentration in rayleigh-taylor turbulence[END_REF] for instance, where L = 12 < u 3 θ >, with . referring to the average along the inhomogeneous direction x 3 . In what follows, the first equation (7.45) is kept. Then, since the mixing length L(t) and the stratification frequency N (t) are linked through

N (t) = 2Ag(t) L(t) , Ṅ N = 1 2 ġ g - L L . (7.46)
If the gravitational acceleration is assumed to be constant, one further gets (7.47) so that the growth rate of the mixing zone can be computed according to

Ṅ (t) = - 4N (t) L(t) < u 3 c >,
α RT = ( L) 2 8AgL(t) = 4 < u 3 c > N (t)L(t) 2 . (7.48)
The evolution equations, in physical space, of the fluctuating velocity and concentration u i and c are

∂u i ∂t + u j ∂u i ∂x j = - ∂p ∂x i + ν ∂ 2 u i ∂x j ∂x j + 2Agcδ i3 , (7.49) ∂c ∂t + u j ∂c ∂x j = a ∂ 2 c ∂x j ∂x j + 1 L(t) u 3 . (7.50)
If one wants to work with the buoyant velocity v, the equations becomes more complex because an additional term, linked to the variable stratification, appears

∂v ∂t + u j ∂v ∂x j = a ∂ 2 v ∂x j ∂x j + N u 3 -v Ṅ N - ġ g .
(7.51)

Prediction of the growth rate α RT

In this section, a new prediction for the growth rate α RT is derived, like what was done before for USHT, where a theoretical exponential growth rate β th for the kinetic energy was proposed, based on the linear operator of our equations within the anisotropic EDQNM modelling. The prediction (7.43) by [START_REF] Poujade | Growth rate of rayleigh-taylor turbulent mixing layers with the foliation approach[END_REF] relies on two crucial features: (i) foliated average in the inhomogeneous direction. This foliated average notably causes the scalar variance < c 2 > to evolve in t 2 , like the mixing length L(t), whereas the classical scalar variance is constant in the self-similar regime of Rayleigh-Taylor turbulence, since it is bounded by 0 (light fluid) and 1 (heavy fluid); (ii) the fact that at large scales F 2 /(φ 33 E T ) = 1. This result is essential to link α RT to the infrared slope σ (in [START_REF] Poujade | Growth rate of rayleigh-taylor turbulent mixing layers with the foliation approach[END_REF], one has for the vertical foliated averaged spectrum 2E z ∼ φ 33 ). This ratio F 2 /(φ 33 E T ) = 1 is recovered in the present simulations.

Here are the main steps to derive a new prediction for the Rayleigh-Taylor growth rate. First, at large scales, the self-similar spectra can be written (7.52) where E 0 , E 0 T and F 0 are independent of time and space. At large scales, the evolution equations of φ 33 and E T (7.12) and (7.15) read, neglecting the non-linear transfers,

φ 33 (k, t) = E 0 k σ t n E , E T (k, t) = E 0 T k σ t n T , F(k, t) = F 0 k σ t n F ,
∂φ 33 ∂t 16 5 AgF(k, t), ∂E T ∂t 2 L(t) F(k, t), (7.53)
which directly yields n E = n F + 1 and n F = n T + 1, since E 0 , E 0 T and F 0 do not depend on time. The equation of F cannot be used similarly since the production terms depend on both the kinetic and scalar variance spectra. From these equations, one obtains

E 0 E 0 T = 16 5 (Ag) 2 α RT n T n E . (7.54)
The exponent n T is determined using the fact that < c 2 > is eventually constant in time (7.55) which provides n T = 2(σ + 1), and consequently n E = 2(σ + 2) and 2n F = n E + n T . Afterwards, from (7.53), one gets at large scales (7.56) This finally yields a new theoretical prediction for the growth rate

< c 2 >= ∞ 0 E T (k, t)dk 1/L(t) 0 E 0 T k σ t n T dk = E 0 T σ + 1 t n T -2(σ+1) ,
F 2 (k, t) φ 33 (k, t)E T (k, t) = 5 16 α RT n E n T .
α RT th = 4 5(σ + 1 -p USHT )(σ + 2 -p USHT ) , (7.57)
which gives values 33.3% higher than the expression of [START_REF] Poujade | Growth rate of rayleigh-taylor turbulent mixing layers with the foliation approach[END_REF]. One can remark that unlike the original prediction of the latter reference, a backscatter parameter was added here, in order to take into account strong inverse non-linear transfers in Batchelor turbulence. The noteworthy feature is that numerically p USHT = 0.37 is obtained, meaning that the intensity of the back transfers in Batchelor turbulence is similar between fixed (USHT) and variable stratification frequency. This new prediction (7.57) relies on < c 2 > being constant in the self-similar regime, unlike [START_REF] Poujade | Growth rate of rayleigh-taylor turbulent mixing layers with the foliation approach[END_REF] where < c 2 >∼ t 2 with the foliated average, and F 2 /(φ 33 E T ) = 1 at large scales.

The theoretical prediction (7.43) for α RT th by [START_REF] Poujade | Growth rate of rayleigh-taylor turbulent mixing layers with the foliation approach[END_REF] becomes, without the foliated-average,

α RT th = 1 (σ + 1)(σ + 2) , (7.58)
as in [START_REF] Soulard | Large-scale analysis of self-similar unstably stratified homogeneous turbulence[END_REF]; [START_REF] Griffond | Numerical investigation of self-similar unstably stratified homogeneous turbulence[END_REF]. This prediction can be recovered by multiplying the linear production terms by √ 5/2 in the Lin equations of USHT obtained with the anisotropic EDQNM modelling, as mentioned earlier for USHT.

Numerical results

In what follows, if not mentioned otherwise, the initial Reynolds number is Re λ (0) = 10, the mixing parameter is Λ(0) = 1, N (0) = N 0 = 1τ -1 0 and L(0) = L 0 = 1, so that A = 5, 1.10 -2 , in agreement with the assumption of small Atwood numbers. First, spectral scalings are addressed and are compared with the results obtained with the anisotropic EDQNM modelling for USHT. Then, one-point statistics are investigated.

Spectral scalings:

The inertial scalings of the three main spectra E(k, t), E T (k, t), and F(k, t), are firstly addressed. It has been shown that for USHT, the classical inertial scalings for E and E T , (7.19) and (7.20) respectively, are relevant, and this is still the case for variable N (t), as revealed in Fig. 7.16a. For the scalar flux, as for USHT, the scaling initially proposed by Lumley (7.21) is not well-suited, because in particular it does not take into account the concentration dissipation rate T . On the other hand, the scaling derived for Rayleigh-Taylor turbulence in [START_REF] Soulard | Inertial-range anisotropy in rayleigh-taylor turbulence[END_REF], and used in Burlot et al. (2015b) with the buoyant velocity (7.22) was shown to be satisfactory for USHT, and this is still valid for variable N (t), as revealed in Fig. 7.16b. To obtain this compensated cospectrum, the inertial scaling (7.22) is adapted because the scalar field is now a dimensionless concentration: in [START_REF] Soulard | Inertial-range anisotropy in rayleigh-taylor turbulence[END_REF], the square of the stratification velocity V 2 N appears; here, instead of V N = √ gL, V N = N L is chosen, so that the Rayleigh-Taylor inertial scaling becomes

F Soulard inertial (k, t) = F(k, t) L K 0 1/3 + K CO (N L) 2 T -2/3 -1 k 7/3 . (7.59)
Simulations also show in Fig. 7.16b that a simpler expression for the inertial scaling of F can also be considered, namely Since for F Soulard inertial the plateau approaches a value closer to the one obtained in USHT, this inertial scaling is kept for consistency.

F RT inertial (k, t) = F(k, t) N -2 L -1 -1 T 2/3 k 7/3 . ( 7
One-point statistics: It is shown in Fig. 7.17a for Saffman turbulence that the mixing length reaches the self-similar state L ∼ t 2 after a transient regime of about N 0 t 100. From the same point, the stratification frequency evolves as N ∼ t -1 , which is straightforward using (7.46). Moreover, it is shown that in the self-similar regime, the scalar variance < c 2 > is constant, in agreement with what was discussed earlier. In Fig. 7.17b, the prediction (7.57) is assessed numerically for various infrared slopes σ. It is worth noting that for σ = 4, the backscatter parameter p USHT = 0.37, previously introduced for USHT, is still relevant, which shows that unstably stratified turbulence and the variable stratification case share some infrared dynamics properties. -computed with (7.48); prediction (7.57).

L(t) < c 2 > t 2 t -1 (a)
The emphasis is now put on the kinetic energy K, the scalar variance K T and the mixedcorrelation K F =< u 3 c >, whose evolution equations are

∂K T ∂t = -T + 2 L < u 3 c >, (7.61) ∂K ∂t = -+ 2Ag < u 3 c >, (7.62) ∂ < u 3 c > ∂t = -F + 2AgK T + R 33 L . (7.63)
Since the scalar variance becomes eventually constant in the asymptotic state, it means that dissipation balances production, so that T = 2 < u 3 c > /L, and < u 3 c >∼ t. It then follows that the kinetic energy evolves in K ∼ t 2 similarly to L(t). These different time-dependencies are assessed in Fig. 7.18a. Consequently, because of the continuous production of kinetic energy, it is reasonable to assume that ∂ t K 2Ag < u 3 c >. Furthermore, in agreement with Fig. 7.18b where F r is displayed, stratification dominates over turbulence in the self-similar regime, so that with a rough approximation, ∂ t K ∼ KN . This provides an approximation of the kinetic energy in the self-similar regime

K(t) ∼ N (t)L(t) < u 3 c > . (7.64)
This approximation, which involves the characteristic stratification velocity V N = N L, is well assessed in Fig. 7.18a in grey. In addition, it is revealed in Fig. 7.18b that both b 33 and b T 33 , like F r, become constant at large Reynolds numbers. The asymptotic value of the global anisotropy for σ = 2 with variable N (t) is b ∞ 33 = 0.223, slightly lower than in USHT where b ∞ 33 = 0.239, meaning that the variable stratification tends to decrease the global anisotropy. This is expected since the mixing length grows in time, thus reducing the intensity of the mean concentration gradient, which is the source term of anisotropy in the equations.

Chapter 8

Dynamics of Helicity in Skew-Isotropic Turbulence

In this chapter, the transport of a scalar field is put aside to investigate the dynamics of helicity: this is of fundamental interest since helicity can be considered as the "smoothest" kind of anisotropy, since it breaks only mirror-symmetry with respect to HIT.

The contents of this chapter were published in: Briard & Gomez, "Dynamics of helicity in homogeneous skew-isotropic turbulence ", Journal of Fluid Mechanics, 821, 539-581 (2017) Helicity is a quantity of interest since it is an invariant of the 3D inviscid Navier-Stokes equations [START_REF] Moffatt | The degree of knottedness of tangled vortex lines[END_REF] and has been consequently at the center of a great amount of theoretical [START_REF] Brissaud | Helicity cascades in fully developed isotropic turbulence[END_REF][START_REF] Moffatt | Helicity in laminar and turbulent flow[END_REF][START_REF] Chkhetiani | On the third moments in helical turbulence[END_REF][START_REF] Gomez | Exact relationship for third-order structure functions in helical flows[END_REF][START_REF] Ditlevsen | Dissipation in helical turbulence[END_REF] and numerical [START_REF] André | Influence of helicity on the evolution of isotropic turbulence at high reynolds number[END_REF][START_REF] Polifke | The dynamics of helical decaying turbulence[END_REF][START_REF] Borue | Spectra in helical three-dimensional homogeneous isotropic turbulence[END_REF][START_REF] Chen | The joint cascade of energy and helicity in three-dimensional turbulence[END_REF]Baerenzung et al., 2008b;[START_REF] Biferale | Inverse energy cascade in three-dimensional isotropic turbulence[END_REF] studies. Nevertheless, despite all the attention helicity has received for more than forty years, it remains a quantity quite complex, whose effects on the transfers of energy are not completely understood, as stated in [START_REF] Chen | The joint cascade of energy and helicity in three-dimensional turbulence[END_REF]: indeed, helicity K H =< u i ω i > /2, unlike kinetic energy K(t) =< u i u i > /2, is not positive definite since it is the scalar product of the fluctuating velocity u i and vorticity ω i = ijk ∂ j u k , so that it can be either positive or negative. It is worth noting that inviscid 3D turbulence has two invariants, kinetic energy and helicity, and 2D turbulence has two as well, kinetic energy and enstrophy < ω i ω i >. Therefore, some authors have evoked the possibility of interpreting helicity as a 3D analogous of enstrophy, despite the fact that the latter quantity is positive definite. Since enstrophy is responsible for an inverse cascade of energy in 2D [START_REF] Kraichnan | Inertial ranges in two-dimensional turbulence[END_REF], it has been concluded that helicity could also be associated to inverse cascade mechanisms in 3D [START_REF] Brissaud | Helicity cascades in fully developed isotropic turbulence[END_REF][START_REF] Chen | The joint cascade of energy and helicity in three-dimensional turbulence[END_REF].

Consequently, since the pioneering work of [START_REF] Brissaud | Helicity cascades in fully developed isotropic turbulence[END_REF], the possibility of inverse energy cascades has been a crucial point of discussion. At this time, two different scenarios were proposed: (i) Joint cascades of helicity and energy towards small scales with non zero kinetic and helical dissipation rates, respectively and H , so that the kinetic and helical spectra scale in E(k) ∼ 2/3 k -5/3 and H(k) ∼ H -1/3 k -5/3 . (ii) A pure helicity cascade, with no energy transfer = 0, so that the kinetic and helical spectra scale in E(k) ∼

2/3

H k -7/3 and H(k) ∼ 2/3 H k -4/3 in the forward cascade. In such a configuration, there would be an inverse cascade of kinetic energy in k -5/3 . This second scenario was proven to be impossible in decaying turbulence by [START_REF] André | Influence of helicity on the evolution of isotropic turbulence at high reynolds number[END_REF] in the EDQNM framework. However, for instance in rotating turbulence with a non-vanishing helical forcing, and in other very specific configurations, an inverse energy cascade is observed [START_REF] Biferale | Split energy-helicity cascades in three-dimensional homogeneous and isotropic turbulence[END_REF]. Furthermore, one must point out that recently, it was shown that the Navier-Stokes equations intrinsically contain this inverse energy cascade mechanism [START_REF] Biferale | Inverse energy cascade in three-dimensional isotropic turbulence[END_REF]: indeed, when considering specific triadic interactions between only positive (or negative) helical modes, there is an inverse kinetic energy cascade E(k) ∼ 2/3 k -5/3 . Still, as soon as there is a single helical mode of opposite sign, this inverse cascade vanishes.

On a practical point of view, large Reynolds numbers helical flows can be found notably in atmospheric turbulence, where helicity is naturally present and may be the reason for the persistence of tornadoes [START_REF] Moffatt | Helicity in laminar and turbulent flow[END_REF][START_REF] Lesieur | Turbulence in fluids[END_REF]. On a theoretical point of view, the high Reynolds numbers regime is of interest since the classical scalings, phenomenology and mechanisms of turbulence were developed in this framework, where small scales should always be isotropic and forget the effects of large scales (Kolmogorov, 1941b). The review of the different studies involving helicity shows that, except the early and pioneering work of [START_REF] André | Influence of helicity on the evolution of isotropic turbulence at high reynolds number[END_REF], there were no further attempts to investigate the dynamics of the helical spectrum H(k, t) at very large Reynolds numbers (Re λ ≥ 10 3 ). In addition, it appears that the long-time decay of helicity has not been addressed. Yet, the knowledge of the decay rate of integrated quantities, such as the inviscid invariants, is crucial for the understanding and prediction of the turbulence dynamics in asymptotic regimes at large Reynolds numbers. Therefore, it could be interesting to have clear decay exponents for helicity: indeed, there were no studies providing decay exponents for helicity, except the theoretical one by [START_REF] Levshin | Decay of helicity in homogeneous turbulence[END_REF], which is not fully satisfactory as explained later on. Consequently, this chapter first focuses on two fundamental questions: is the decay of helicity predictable? How does helicity modify non-linear transfers and the decay of kinetic energy?

Since mean helicity can be created in homogeneous turbulence, from non-zero spectral helical modes [START_REF] André | Influence of helicity on the evolution of isotropic turbulence at high reynolds number[END_REF], the knowledge of its decay law is of great interest when it is initially present in the flow. This is why the authors choose to focus on a classical configuration at large Reynolds numbers, namely Homogeneous Helical Turbulence (HHT), which is basically a skew-isotropic turbulence, i.e. HIT without mirror symmetry. In particular, there are no magnetic fields, so that only the kinetic helicity is considered: this is precisely the framework addressed by [START_REF] André | Influence of helicity on the evolution of isotropic turbulence at high reynolds number[END_REF], and unlike recent studies, no distinctions are made here between positive and negative helical modes [START_REF] Biferale | Split energy-helicity cascades in three-dimensional homogeneous and isotropic turbulence[END_REF].

In such a fundamental configuration, several crucial theoretical results in physical space were derived regarding two-point third-order correlations: notably, [START_REF] Chkhetiani | On the third moments in helical turbulence[END_REF] proposed an inertial scaling for the triple velocity correlation S(r) =< u L u 2 u 3 >, where r is the distance between two points located in x and x = x + r, the prime refers to quantities expressed in x , and the () L to the component along r: S(r) appears in the evolution equation of the antisymmetric part of < u i u j >, and is found to scale, neglecting the temporal and viscous dissipation terms, as S(r) ∼ H r 2 /30. In addition, mixed velocity-velocity-vorticity structure functions were analyzed in [START_REF] Gomez | Exact relationship for third-order structure functions in helical flows[END_REF], and it was found that < δu L δu i δω i > -< δω L δu i δu i > /2 = -4r H /3 in the inertial range, where δu i = u i -u i . These two laws are equivalent, and result from the conservation of helicity in inviscid flows. This is why the second law is analogous to the "four-thirds" laws for the kinetic energy and scalar variance, which both come from conservation laws as well [START_REF] Antonia | Analogy between predictions of kolmogorov and yaglom[END_REF]. Whereas an equivalent of the first law for S(r) has been assessed in DNS [START_REF] Kurien | Isotropic third-order statistics in turbulence with helicity: the 2/15-law[END_REF], it is not the case for the helical "four-thirds" law. Both these fundamental relations are assessed numerically here at high Reynolds numbers, and statistics of helical flows will be further investigated, notably the evolution equation of the helical dissipation rate H and derivatives of skew-isotropic tensors.

Finally, a new configuration is addressed, combining both a mean scalar gradient and helicity. This case, which could be of interest for the modelling of atmospheric turbulence where these two features may be present, permits to illustrate the subtle effects of helicity on the scalar flux. Indeed, unlike a passive scalar field where there is no explicit contributions of helicity in its evolution equations, the coupling of helicity and mean scalar gradient creates the quadrature spectrum Q(k, t), linked to the imaginary antisymmetric part of the scalar flux. The appearance of this additional contribution parallel to the cospectrum is called "skew-diffusion" by [START_REF] Moffatt | Helicity in laminar and turbulent flow[END_REF].

Spectral modelling of helicity

In this part, the evolution equations of the kinetic and helical spectra are derived starting from the spectral counterpart of the Navier-Stokes equation. The EDQNM approach is presented as well. In the following, helicity is injected initially at large scales along with kinetic energy so that both decay freely: there is no forcing mechanism nor rotation, and no magnetic field. Historically, this is the framework investigated by [START_REF] Brissaud | Helicity cascades in fully developed isotropic turbulence[END_REF] and it will be shown that even in such a classical case, there are still some important open questions which are tackled in the following sections, such as the prediction of the helicity decay and its impact on kinetic energy transfers.

The E-H decomposition

With helicity, the spectral Reynolds tensor Rij has an imaginary part, and is consequently not symmetric anymore. In the framework of homogeneous helical turbulence (HHT), Rij can be decomposed as

Rij (k, t) = E(k, t)P ij + (Z(k, t)N i N j ) + i ijk α k H(k, t) k , (8.1)
following the formalism of [START_REF] Cambon | Spectral approach to non-isotropic turbulence subjected to rotation[END_REF]. For simplicity reasons, mean velocity gradients are not considered, which simplifies the previous expression into the E-H decomposition which reads

Rij (k, t) = R(iso) ij (k, t) + R(hel) ij (k, t) = E 0 (k, t)P ij + i ijk α k H(k, t) k (8.2)
where H is a pseudo-scalar, real, not positive-definite, and reflects the density of helicity, defined as

H(k, t)δ(k -p) = 1 2 < û * i (p, t)ω i (k, t) > (8.3)
where ω i = ijk ∂ j u k is the vorticity. Similar E-H decompositions were used by [START_REF] Borue | Spectra in helical three-dimensional homogeneous isotropic turbulence[END_REF]; [START_REF] Chen | The joint cascade of energy and helicity in three-dimensional turbulence[END_REF]. The inverse relation for the energy density is straightforward, E 0 = Rii /2, whereas the one for the helical density is more complex [START_REF] Moffatt | Transport effects associated with turbulence with particular attention to the influence of helicity[END_REF][START_REF] Cambon | Third-order statistics and the dynamics of strongly anisotropic turbulent flows[END_REF])

H(k, t) = - 1 2 ik m ijm Rij (k, t). (8.4)
The E-H decomposition could also be applied in isotropic MHD to the spectral secondorder magnetic correlation

R M ij (k)δ(k -p) =< â * i (p, t)â j (k, t) >
, with âi the magnetic potential, where the antisymmetric part would be linked to the magnetic helicity H M , related to R M ij through an equation analogous to (8.4).

Using the equations of the fluctuating spectral velocity and vorticity given in Appendix D, one obtains the helical Craya equation

∂H ∂t + 2νk 2 H(k, t) = T H (k, t). (8.5)
The non-linear helical transfer T H can be expressed as a function of the τ ij , defined in (2.39), using the relation (8.4) so that

T H (k, t) = - 1 2 i ijl k l τ ij (k, t) + τ * ji (k, t) . (8.6)
This expression (8.6) links the helical transfer to the total non-linear one defined in (2.18), similar to what was done for T E and T Z in (2.32) and (2.34) respectively. The helical spectrum is further defined as

H(k, t) = S k H(k, t)d 2 k = 4πk 2 H(k, t), (8.7)
and is linked to helicity and the helical dissipation rate through

K H (t) = 1 2 < u i ω i >= ∞ 0 H(k, t)dk, (8.8) H (t) = ν < ∂u i ∂x j ∂ω i ∂x j >= 2ν ∞ 0 k 2 H(k, t)dk. (8.9)
The time evolution of helicity is thus given by

dK H dt = 1 2 A ij < u i ω j > -< u j ω i > -H . (8.10)
Without mean-velocity gradients, A ij = 0 and consequently the helicity follows the same evolution equation as the kinetic energy in HIT.

Spherically-averaged helical Lin equations for E(k, t) and H(k, t)

In this part, the explicit spherically-averaged transfer terms for HHT are derived within the EDQNM framework, as in [START_REF] André | Influence of helicity on the evolution of isotropic turbulence at high reynolds number[END_REF]. Using (8.6) and the details provided in Appendix D, the non-linear spherically-averaged helical transfer reads (8.11) with

S NL H (k, t) = S k T H (k, t)d 2 k = S NL H1 (k, t) + S NL H2 (k, t),
S NL H1 (k, t) = 16π 2 ∆ k θ H kpq k 2 p 2 q(xy + z 3 )E 0 (H -H)dpdq S NL H2 (k, t) = -16π 2 ∆ k θ H kpq k 2 pz(x + yz)H (p 2 E 0 -k 2 E 0 )dpdq. (8.12)
Moreover, from the E-H decomposition, the kinetic non-linear transfers now contain a helical part coming from products of R(hel) ij : details of the calculations are given in Appendix D. The non-linear spherically averaged purely helical transfer is

S NL(hel) (k, t) = S k T E (k, t)d 2 k -S NL(iso) (k, t) (8.13) = -16π 2 ∆ k θ kpq k 2 pz(x + yz)H (H -H)dpdq, (8.14)
with S NL(iso) given by (2.57). These three new transfer terms, S NL H1 , S NL H2 , and S NL(hel) , are independently conservative. The characteristic time of the third-order correlations is the same for the kinetic and helical fields: it is shown later on that such an assumption is consistent with joint cascades for the kinetic and helical spectra, and can be a posteriori justified with physical arguments. Beyond these physical justifications, it seems relevant to choose θ kpq for both the kinetic and helical fields since the closure comes from the same evolution equation, which is the one of the spectral velocity third-order correlation (2.17). One can further remark that the form of S NL H1 is similar to the scalar non-linear transfer S T,NL(iso) . It will be shown numerically that S NL H1 corresponds in fact to a direct transfer, whereas both S NL H2 and S NL(hel) to inverse ones.

The spherically-averaged Lin equations for the kinetic and helical spectra are

∂ ∂t + 2νk 2 H(k, t) = S NL H (k, t), ∂ ∂t + 2νk 2 E(k, t) = S NL(iso) (k, t) + S NL(hel) (k, t) = S NL E (k, t). (8.15) (8.16)
Finally, the helical spectrum H(k, t) must satisfy the realizability condition

|H(k, t)| ≤ kE(k, t), (8.17)
which was derived by [START_REF] Kraichnan | Helical turbulence and absolute equilibrium[END_REF]. When this equation is an equality, this condition is called the maximal helicity condition.

Numerical results on the helical and kinetic fields

In this part, the kinetic energy and helical spectra E(k, t) and H(k, t) are investigated numerically at very large Reynolds numbers thanks to the EDQNM modelling presented in the previous section. After a short discussion on initial conditions, basic properties of homogeneous skew-isotropic flows are recovered and some features regarding inverse transfers are addressed. Then, non-local interactions are considered in the infrared range of the spectra (for wavenumbers smaller than the integral one k L ) and non-local expansions are made in order to study the large scales dynamics. These results are directly used to predict the decay of kinetic energy and helicity in homogeneous turbulence.

The importance of initial conditions H(k, t = 0)

The initial condition for the helical spectrum is (8.17). One has to be careful with this initial condition, which increases the infrared slope σ H of the helical spectrum, and thus accelerates the decay of helicity. Indeed, in Saffman HHT for instance, at large scales

H(k < k H , t) ∼ k 3 whereas E(k < k L , t) ∼ k 2 ,
where k H is the helical integral wavenumber, defined similarly to the kinetic one

L H (t) = 1 k H = 3π 4K H (t) ∞ 0 k -1 H(k, t)dk. (8.18)
The last point to define is the shape of E(k, t = 0), which is of primary importance. So far, the initial condition (1.10) was used, called (IC1) from now, which corresponds to a spectrum with energy at all scales at t = 0. In particular, (IC1) implies that helicity is initially present at all scales as well: this strongly minimizes the impact of helicity on the kinetic energy cascade and decay. Hence, the initial condition (IC2) is used instead

(IC2) : E(k, t = 0) = k σ exp - σ 2 k k L 2 . (8.19)
One can wonder if H(k, t = 0) = E(k, t = 0) is an acceptable initial condition, even if for k ≥ k L this breaks (8.17). Fig. 8.1a reveals that the latter initial condition is physically unacceptable: indeed, after a hundred turn-over times, one has kE -H ≤ 0 around the integral wavenumber k L . Whereas for the maximal helicity condition, for all k one has kE -H ≥ 0, for both (IC1) and (IC2). Consequently, from this point, all the computations presented are initialized with the maximal helicity condition.

Unlike the kinetic energy spectrum E(k, t), the helical spectrum H(k, t) is not restricted to positive values: it was notably found by [START_REF] André | Influence of helicity on the evolution of isotropic turbulence at high reynolds number[END_REF] that negative values appeared at the beginning of the dissipative range near k η . This is also observed here with the present EDQNM simulations in Fig. 8.1b, where the helical spectrum has negative values while approaching k η . Positive and negative values for H(k, t) were reported in DNS [START_REF] Polifke | The dynamics of helical decaying turbulence[END_REF][START_REF] Polifke | Statistics of helicity fluctuations in homogeneous turbulence[END_REF].

k kE(k, t) -H(k, t) (IC1): H(k, t = 0) = kE(k, t = 0) (IC1): H(k, t = 0) = E(k, t = 0) (IC2): H(k, t = 0) = kE(k, t = 0) k H k η k L (a)
k E(k, t) and H(k, t) E(k, t) H(k, t) k -5/3 k 2 k 3 k L k H k H η k η (b)

Helical spectrum H(k, t) and non-linear transfers

In this section, the inertial scaling of H(k, t) and the non-linear helical transfers are addressed. Fig. 8.1b reveals that after a few turn-over times the helical spectrum scales in k -5/3 in the inertial range. This scaling assesses the joint cascades mentioned earlier for kinetic energy and helicity. The k -5/3 scaling can be deduced from dimensional analysis. The main hypothesis is that the characteristic time in the inertial range τ (k, t) = (k 2 (t)) -1/3 is the same for both the kinetic and helical spectra. From this assumption results an "Obukhov-like" scaling, H (t) = kH(k, t)/τ (k, t), which directly yields in the inertial range

H(k, t) = C H H -1/3 k -5/3 . (8.20)
The constant C H = 2 is obtained by investigated the compensated spectra in Fig. 8.2a at high Reynolds number Re λ = 2.10 4 . The Kolmogorov constant K 0 = 1.3 remains unchanged with respect to HIT. It is worth noting that similarly to K 0 , the value of C H depends on the choice of the eddy-damping constant. The present value C H 2 is in agreement with [START_REF] André | Influence of helicity on the evolution of isotropic turbulence at high reynolds number[END_REF], somehow higher that C H 1 obtained in [START_REF] Borue | Spectra in helical three-dimensional homogeneous isotropic turbulence[END_REF].

One can remark that the inertial scaling (8.20) of H(k, t) is similar to the one of a passive scalar convected by a turbulent velocity field; this is the reason why it is often said that helicity cascades linearly with the kinetic energy. Such a scaling can also be obtained by considering that non-linear transfers in the inertial range are mainly local: k ∼ p ∼ q. By dimensional analysis and dropping all geometric factors in (8.11), this yields for the fluxes Π(k

) ∼ θ E(k) 2 k 4 and Π H (k) ∼ θ E(k) H(k) k 4
, where θ kkk = θ is the same for E and H, as assumed earlier.

Then, for high Reynolds numbers, one has in the inertial range Π and Π H H , which is well assessed in Fig. 8.2b. Thus, one has In addition, a specific wavenumber k H η is displayed in Fig. 8.1b. This wavenumber was derived theoretically by [START_REF] Ditlevsen | Dissipation in helical turbulence[END_REF]

θ = /(k 4 E 2 ) = τ (k) so that (t) H(k, t) ∼ E(k, t) H (t) → ∞ 0 dk → (t) K H (t) ∼ K(t) H (t). ( 8
E(k) k 5/3 ǫ -2/3 H(k) k 5/3 ǫ -1 H ǫ 1/3 k η 2 1.3 k L (
k H η = 3 H ν 3 2 1/7 , (8.22)
and is supposed to mark the end of the helical inertial range, which is clearly not the case here.

The helical inertial range spans from the helical integral wavenumber k H ∼ k L to k η , similarly to the kinetic energy spectrum. This was also assessed both numerically and theoretically by [START_REF] Chen | The joint cascade of energy and helicity in three-dimensional turbulence[END_REF]. A different interpretation of k H η is proposed in Appendix D.

The total non-linear kinetic and helical transfers S NL E and S NL H are now investigated in Fig. 8.3a to 8.3c for Batchelor turbulence (results are similar for Saffman turbulence). In the previous part, S NL E was decomposed into the sum of a purely kinetic contribution S NL(iso) , identical to the non-linear transfer in HIT, and a purely helical contribution S NL(hel) . It is found in the EDQNM simulations that the latter part corresponds in fact to a transfer of energy from small to large scales. This inverse transfer is nevertheless less intense than the direct one, so that the total kinetic transfer S NL E is direct, as observed in Fig. 8.3b. Moreover, whereas the direct non-linear transfer S NL(iso) spans all scales of the wavenumber space, the inverse transfer S NL(hel) is very localized at large scales, which creates a small region where S NL(iso) is positive at large scales, which is different from HIT. Nevertheless, the total kinetic energy transfer S NL E is completely similar to the one in HIT.

Similarly, S NL

H was decomposed into two contributions S NL H1 and S NL H2 , which both span the entire inertial range. It is revealed in Fig. 8.3a that these two parts correspond to direct and inverse transfers respectively. Once again, the inverse transfer is weaker than the direct one, so that the total transfer of helicity S NL H goes from large to small scales. One can observe in Fig. 8.3c that S NL H is negative around k η , and this explains the negative values of the helical spectrum H(k, t) at the dissipative scales observed in Fig. 8.1b: this can be interpreted as viscous production of helicity, since -2νk 3 H(k) is positive in this region. The fact that inverse transfers of kinetic energy and helicity are hidden in the total direct cascade is in agreement with the recent results of [START_REF] Alexakis | Helically decomposed turbulence[END_REF].

In this part, it was recovered that in freely decaying HHT, there is a joint cascade of kinetic energy and helicity towards small scales. The main assumption behind the k -5/3 inertial scaling is that the kinetic and helical fields have the same inertial characteristic time. Finally, it was shown numerically that despite a direct cascade of kinetic energy and helicity, some inverse non-linear transfers occur, less intense than direct ones.

Infrared dynamics and non-local transfers

This section focuses on the permanence of large eddies (PLE) in the presence of helicity, and on non-local interactions between small and large scales. The starting point is Fig. 8.4a and 8.4b, where the time evolution of the kinetic energy and helical spectra E(k, t) and H(k, t) is displayed for Batchelor turbulence. Two features need to be underlined: firstly, it appears that H experiences no backscatter in Fig. 8.4b, so that the PLE hypothesis is verified in Batchelor HHT for the helical spectrum, unlike E. Secondly, the backscatter for E in Fig. 8.4a is weaker in presence of helicity than in HIT, as revealed by the grey curve corresponding to a HIT simulation for E, at the same time and Reynolds number. These two features can be explained analytically, using non-local expansions. These important results, and especially the fact that the PLE hypothesis is verified for H even in Batchelor turbulence, are applied in the next section to determine theoretical decay exponents for K(t) and K H (t).

Firstly, [START_REF] Lesieur | Turbulence in fluids[END_REF] showed that the kinetic non-local transfers acting in the infrared range are

T (iso)-(k, t) = 14 15 k 4 ∞ k L θ 0pp E(p) 2 p 2 dp - 2 15 k 2 E(k) ∞ k L θ 0pp 5E(p) + p ∂E ∂p dp. (8.23)
These terms come from the space derivative ∂Π -/∂k, with Π (iso)-= Π -defined in (B.17), evaluated at the lowest order in k/k L , with the non-local parameter a = k/k L . The first rhs term is responsible for the backscatter of energy that breaks the PLE hypothesis. The second rhs term can be written under the eddy-viscous form -2ν t k 2 E, and represents a pseudo kinetic energy dissipation, i.e. the damping of large scales by turbulence. To understand why back transfers of energy on E(k, t) are decreased with helicity, one has to expand S NL(hel) when k p ∼ q, using calculations similar to those presented in Appendix B. This gives

Π (hel)-(k, t) = - 14 15 k 0 k 4 ∞ sup(k,k /a) θ k pp H(p) 2 p 4 dpdk (8.24) + 2 15 k 0 k 2 H(k ) ∞ sup(k,k /a) θ k pp p 2 9H(p) -p ∂H ∂p dpdk . (8.25)
The spatial derivative of Π (hel)-, with the same assumptions, yields

T (hel)-(k, t) = - 14 15 k 4 ∞ k L θ 0pp H(p) 2 p 4 dp + 2 15 k 2 H(k) ∞ k L θ 0pp p 2 9H(p) -p ∂H ∂p dp. (8.26)
The first rhs term modifies the backscatter of energy whereas the second one can also be interpreted as a pseudo helical dissipation term in -2ν H t k 2 H. Combining this expression with (8.23) reveals the impact of helicity on the total inverse non-local kinetic transfer in HHT

T - E (k, t) = 14 15 k 4 ∞ k L θ 0pp E(p) 2 p 2 1 - H(p) pE(p) 2 dp k 4 backscatter -2ν t k 2 E(k) -2ν H t k 2 H(k)
damping of large scales .

(8.27)

The second term indicates a change in the effects of viscous damping on large-scales, difficult to quantify. However, for the first rhs term, using the realizability condition 0

≤ |H|/kE ≤ 1, one has 14 15 k 4 ∞ k L θ 0pp E(p) 2 p 2 1 - H(p) pE(p) 2 ≤1 dp ≤ 14 15 k 4 ∞ k L θ 0pp E(p) 2 p 2 dp
Non-local transfer in HIT .

(8.28)

Consequently, the k 4 backscatter of energy of E(k, t) is decreased by helicity, with respect to HIT. Then, similar expansions of

S NL H give Π - H (k, t) = Π - H1 (k, t) + Π - H2 (k, t) = 14 15 k 0 k 4 ∞ sup(k,k /a) θ k pp H(p)E(p) p 4 dpdk - 2 15 k 0 k 2 H(k ) ∞ sup(k,k /a) θ k pp 5E(p) + p ∂E ∂p dpdk - 14 15 k 0 k 4 ∞ sup(k,k /a) θ k pp H(p)E(p) p 4 dpdk + 14 15 k 0 k 4 E(k ) ∞ sup(k,k /a) θ k pp H(p) p 2 dpdk . (8.29)
The two terms responsible for the k 4 backscatter cancel, which explains that in the end, H(k, t) does not experience any strong back transfer of energy, so that the PLE hypothesis holds true for the helical spectrum. The space derivative of the inverse non-local helical flux yields (8.30) This inverse non-local helical transfer T - H indicates that there is no strong k 4 backscatter. The first term can be written -2ν t k 2 H (and was found also in Baerenzung et al. (2008b)), and the second one is quite original since it makes intervene a pseudo-enstrophy dissipation and could consequently be written -2ν ω t k 4 E.

T - H (k, t) = - 2 15 k 2 H(k) ∞ k L θ 0pp 5E(p) + p ∂E ∂p dp + 14 15 k 4 E(k) ∞ k L θ 0pp H(p) p 2 dp.
The method used now to describe analytically the infrared dynamics of the kinetic energy and helical spectra in HHT -at first order -is inspired from [START_REF] Lesieur | Turbulence in fluids[END_REF], where a similar reasoning is performed for HIT. When the turbulence is fully developed, both spectra scale in E(k, t) ∼ A(t)k σ and H(k, t) ∼ A H (t)k σ H at large scales. In this infrared range, viscous dissipation is negligible, and inverse non-local transfers T - E and T - H dominate with respect to local ones S NL E and S NL H . Thus, the simplified Lin equations in the infrared range are

∂ t E = T - E and ∂ t H = T - H , which yields dA dt = 14 15 k 4-σ ∞ k L (...)dp -2ν t k 2 A(t) -2ν H t k 2+σ H -σ A H (t) (8.31) dA H dt = - 2 15 k 2 A H (t) ∞ k L (...)dp + 14 15 k 4+σ-σ H A(t) ∞ k L (...)dp. (8.32)
With the present initial conditions (8.19), one has σ H = σ + 1, but one could think of different initial conditions that would result in a more complex infrared dynamics for E and B: this is the object of section 8.2.5. For now, one has σ H = σ + 1, and in this case, whatever the infrared slope σ is, the dissipation terms involving the turbulent eddy viscosities ν t , ν dA H /dt is not rigorously zero since the non-local expansion is kept at the lowest order in k/p 1 in the infrared range, as done in [START_REF] Lesieur | Amortissement autosimilaire d'une turbulence á grand nombre de reynolds[END_REF], but is equal to some subdominant terms for the dynamics of A H , as assessed by Fig. 8.4b. This means that whatever the large scales initial conditions σ are, the PLE hypothesis holds for the helical spectrum

Expression

Physical meaning Appears in eqs for ...

15ν t = ∞ k L θ 0pp (5E(p) + p ∂E ∂p )dp Dissip. of K(t) and K H (t) E(k, t) and H(k, t) 15ν H t = ∞ k L θ 0pp p 2 (p ∂H ∂p -9H(p))dp Dissip. of K H (t) E(k, t) 15ν ω t = -7 ∞ k L θ 0pp H(p) p 2 dp Dissip. of enstrophy H(k, t)

H(k, t)

. Regarding E(k, t), for σ ≤ 3, the rhs terms of (8.31) are negligible, meaning that the PLE hypothesis holds true, whereas for σ = 4, A(t) truly depends on time since the first rhs term is stronger than the two others

-2ν t k 2 A and -2ν H t k 3 A H , so that dA dt 0, ∀σ ≤ 3, dA dt 14 15 ∞ k L θ 0pp E(p) 2 p 2 1 - H(p) pE(p) 2 dp, for σ = 4. (8.34) (8.35)
The latter result contains the fact that helicity decreases the backscatter of E(k, t) in Batchelor HHT. The different turbulent spectral viscosity introduced previously are summarized in Table 8.1.

Decay laws in helical flows

In this part, the emphasis is put on both the impact of helicity on the kinetic energy decay, and on the decay of the helicity K H itself. It is obvious that K H will decay faster than the kinetic energy, because of the large scales initial condition (8.19) σ H = σ + 1.

Remarks about the decay of helicity and kinetic energy: it has been said in the introduction that helicity is known to slow down transfers. This result comes from [START_REF] André | Influence of helicity on the evolution of isotropic turbulence at high reynolds number[END_REF], and was also observed in DNS [START_REF] Polifke | The dynamics of helical decaying turbulence[END_REF][START_REF] Polifke | Statistics of helicity fluctuations in homogeneous turbulence[END_REF][START_REF] Moffatt | Helicity in laminar and turbulent flow[END_REF]. One must be precise to characterize this phenomenon: indeed, this does not concern the theoretical decay exponent α of the kinetic energy K(t), but only the cascade of kinetic energy: with helicity, it takes more time for the non-linear transfers to fill in the small scales of the kinetic spectrum with energy coming from larger ones. This phenomenon, which cannot be observed with (IC1) since all scales already contained energy initially, is illustrated with (IC2) in Fig. 8.5 for Batchelor turbulence (the result is identical for Saffman turbulence). The kinetic energy is constant during the first turn-over times with and without helicity, which corresponds to the filling of the small scales. It is then clear that the filling of small scales is slightly longer in presence of helicity.

EDQNM has been intensively used to determine theoretical decay exponents in different configurations, at high Reynolds numbers and after a large number of turnover times: the decay of kinetic integrated quantities in HIT (Meldi & Sagaut, 2013a), the decay of the scalar variance in HIT [START_REF] Lesieur | The decay of kinetic energy and temperature variance in threedimensional isotropic turbulence[END_REF] and other scalar quantities in Chapter 1, the decay of the kinetic energy in a turbulence initially submitted to mean-velocity gradients (see Chapter 3), and finally the decay of the velocity-scalar correlation in an isotropic turbulence with a mean scalar gradient (see Chapter 5). The study of the helical case is therefore a natural extension, and our predictions are compared to those of [START_REF] Levshin | Decay of helicity in homogeneous turbulence[END_REF] later on.

Impact of helicity on the decay of K(t): The effects of helicity on the kinetic energy decay are firstly addressed. Simulations show that the decay exponent α of the kinetic energy, where K(t) ∼ t α , is not modified by helicity, except in the case of Batchelor turbulence because of the reduction of the non-local inverse transfers analyzed in the previous section. Hence, one would expect the decay of K(t) to be rapider in Batchelor HHT than in Batchelor HIT: indeed, the non-local inverse transfers bring back less energy to the large scales. This is recovered in Fig. 8.6b.

To analytically take into account the breakdown of the PLE hypothesis, the backscatter parameter p usually introduced in HIT is modified. In HIT, one has p(σ = 4) = 0.55 and p(σ ≤ 3) = 0: in particular, in Batchelor HIT K(t) ∼ t -1.38 . Here, in Batchelor HHT, K(t) ∼ t -1.417 is obtained, and a least-square fit provides a new backscatter parameter p H = 0.14 for HHT. Consequently, with respect to HIT, only the backscatter parameter changes from p to p H in HHT for the decay of kinetic energy

α = -2 σ -p H + 1 σ -p H + 3 , p H (σ = 4) = 0.14 p H (σ ≤ 3) = 0 . (8.36)
These decay exponents for the kinetic energy are assessed in Fig 8 .6b in Saffman and Batchelor turbulence: only the case σ = 4 differs from HIT, where here in HHT the decay of K(t) is slightly rapider. The fact that helicity does not influence much the energy cascade once the turbulence is fully developed is in good agreement with conclusions drawn by [START_REF] Polifke | Statistics of helicity fluctuations in homogeneous turbulence[END_REF]; [START_REF] Borue | Spectra in helical three-dimensional homogeneous isotropic turbulence[END_REF].

Decay of helicity: Helicity was shown to impact the kinetic energy decay only in Batchelor turbulence. The emphasis is now put on the decay of K H itself. The method to predict the decay of helicity is similar to the one of an advected passive scalar, and even more simple. Indeed, as revealed in Fig. 8.4b, the helical spectrum H(k, t) experiences no strong back transfers, so that the PLE hypothesis holds even in Batchelor turbulence. Therefore, there is no need to introduce a helical backscatter parameter. Then, it is reasonable to assume that the kinetic and helical integral scales L(t) and L H (t) decay similarly, so that their algebraic exponents n L and n L H are equal

L H (t) ∼ t n L H , n L H = 2 σ + 3 n L = 2 σ -p H + 3 . (8.37)
This assumption is completely assessed in Fig. 8.6a. Then, using either the continuity of H(k, t) in k = k H to determine the decay law for H , or dimensional analysis with K H ∼ K/L H , or (8.21), one finds

α H = -2 σ + 2 σ + 3 . (8.38)
Theoretical values of this expression for α H , gathered in the last line of Table 8.2, are in excellent agreement with simulations presented in Fig. 8.6b for Saffman and Batchelor turbulence. Interestingly, α H is equivalent to α T in HIT with σ T = σ + 1 without backscatter. The decay exponent of H is then simply α H -1. Moreover, Fig. 8.6b shows that the more σ increases, the more K H decays rapidly, similarly to K(t).

Comparison with [START_REF] Levshin | Decay of helicity in homogeneous turbulence[END_REF]: Our results for the decay of kinetic energy and helicity are now compared with the predictions of [START_REF] Levshin | Decay of helicity in homogeneous turbulence[END_REF] (LC13): this work contains vagueness since the initial conditions are not defined, and it is well-known that they are crucial since large scales are determinant for the decay rate. Indeed, in LC13, for a given decay exponent α of the kinetic energy, two different decay exponents α H of the helicity are proposed, which makes no sense (Tables 1 and2 therein). The present theoretical predictions for the decay of helicity are gathered in Table 8.2, along with the two propositions made by LC13. To fill in Table 8.2, a very reasonable assumption is made: as pointed out earlier, initial conditions are not defined in LC13, so that there is no infrared slope σ. Hence, when in LC13 a decay exponent for the kinetic energy is proposed, it is associated

Decay exponent of helicity α H σ = 1 σ = 2 σ = 3 σ = 4 LC13 Table 1 / -3/2 -8/5 -5/3 LC13 Table 2
-8/5 -5/3 -12/7 / (8.38) assessed by EDQNM -3/2 -8/5 -5/3 -12/7 here with its corresponding infrared slope according to the CBC theory:

K ∼ t -1 → σ = 1, K ∼ t -6/5 → σ = 2, K ∼ t -4/3 → σ = 3
, and K ∼ t -10/7 → σ = 4 (backscatter is not taken into account in LC13). The values of α H from LC13 are the same as the ones coming from (8.38) (-3/2, -8/5, -5/3 and -12/7), but there are not associated with the correct infrared slopes. For instance in Saffman turbulence, two laws are proposed in LC13 for helicity:

K LC13 H (t) ∼ t -3/2 and K LC13 H (t) ∼ t -5/3
. This is impossible: using our theoretical prediction (8.38), K LC13 H (t) ∼ t -3/2 implies that σ = 1, and K LC13 H (t) ∼ t -5/3 implies that σ = 3, whereas σ = 2 in Saffman turbulence. In conclusion, it seems that the results of [START_REF] Levshin | Decay of helicity in homogeneous turbulence[END_REF] correspond to infrared slopes of E and H chosen independently, without respecting the realizability condition (8.17) which fixes σ H once and for all as soon as σ is chosen for E: (8.17) forbids initial conditions such as σ H = σ and σ H = σ -1. In addition, helical decay exponents gathered in LC13 seems to be erroneously reported.

Robustness of the decay exponents -Altered infrared dynamics

In the previous section, for kinetic energy and helical spectra scaling in E = Ak σ and H = A H k σ H in fully developed turbulence, the evolution equations of A(t) and A H (t) were derived in (8.31)-(8.32) in the infrared range, assuming the dominance of inverse non-local transfers T - E and T - H on local ones. For the initial conditions (8.19) where σ H = σ + 1, it was notably found that the permanence of large eddies holds for Batchelor turbulence (σ = 4).

Nevertheless, one can wonder if other initial conditions would produce a different infrared dynamics, and this could be relevant to test the robustness of the decay exponents α and α H derived in the previous sections. One can remark that in (8.31), the term responsible for the backscatter of E is the one where the k () factor vanishes. Thus, it is legitimate to wonder if in (8.32) one could have 4 + σ -σ H = 0 in order to obtain d t A H = 0, and consequently to create backscatter for the helical spectrum. It is worth noting that 2 + σ H -σ = 0 is impossible in (8.31) because of the realizability condition (8.17).

In order to simplify this study and to point out one original case, the maximal helicity condition H = kE is kept, but the initial kinetic energy spectrum is now changed into a sharply peaked Gaussian one

E(k, t = 0) = C 1 exp - 1 (C 2 ) 2 ln k k L 2 , (8.39)
with C 1 so that one has a unit initial kinetic energy, and C 2 = 0.1. After a few turnover times, the kinetic energy spectrum scales in E ∼ Ak 4 in the infrared range [START_REF] Lesieur | 3d isotropic turbulence at very high reynolds numbers: Edqnm study[END_REF]. Unlike (8.19), the helical infrared slope σ H is now different from σ + 1. As a consequence of (8.32), it follows that σ H = σ + 4 = 8, and that the helical spectrum experiences backscatter with

σ = 4, dA H dt = 14 15 A(t) ∞ k L θ 0pp H(p) p 2 dp. (8.40)
The theoretical infrared scaling prediction H ∼ k 8 is assessed in figure 8.7a, along with the k 4 infrared scaling of E(k, t): in this case, the permanence of large eddies is not verified anymore for H(k, t) for the initial conditions (8.39). Nevertheless, it is shown in figure 8.7b as well that the previous theoretical decay exponents for kinetic energy and helicity are still valid, which implies that the backscatter of H(k, t) is negligible in the decay. This case further illustrates the robustness of the theoretical predictions for the decay exponents. Simulations not presented here show that for Saffman turbulence (E ∼ k 2 ), a k 6 infrared scaling for H could be created, but this is out of the maximal helicity framework: indeed, H(k, t = 0) should be Gaussian with E(k, t = 0) ∼ k 2 exp(-k 2 ), and the theoretical decay exponents are still verified.

Structure functions in helical turbulence

In the introduction, the main theoretical results for high Reynolds numbers HHT in physical space were recalled: notably two equivalent laws, found independently, which result from the conservation of helicity in inviscid flows. These two laws are the inertial scaling for the two-point triple velocity correlation [START_REF] Chkhetiani | On the third moments in helical turbulence[END_REF])

S(r) =< u L u 2 u 3 >= r 2 30 H , (8.41)
and the inertial scaling for the third-order velocity-velocity-vorticity structure function [START_REF] Gomez | Exact relationship for third-order structure functions in helical flows[END_REF])

D (uuω) (r) =< δu L δu i δω i > - 1 2 < δu i δu i δω L >= - 4 3 r H . (8.42)
The formalism of structure functions is detailed in Appendix A: the separation vector between two points located in x and x is written r = xx, and x j and x j are independent variables. The prime refers to quantities expressed in x , which should not be misunderstood with the prime of correlations functions, such as f (r), h (r), which is the spatial derivative ∂/∂r.

For HIT, the Kármán-Howarth and Yaglom equations for the third-order structure functions D LLL =< δu 3 L > and D LTT =< δu L δθ 2 > are recovered analytically in Appendix A, along with the equations that permit to compute D LLL and D LTT from spectral non-linear transfers. These two equations, and their multiple formulations, are very well-known and have been assessed numerous times, mostly in DNS [START_REF] Yeung | Schmidt number effects on turbulent transport with uniform mean scalar gradient[END_REF][START_REF] Watanabe | Statistics of a passive scalar in homogeneous turbulence[END_REF][START_REF] Yeung | High-reynolds-number simulation of turbulent mixing[END_REF][START_REF] Bos | Reynolds number effect on the velocity increment skewness in isotropic turbulence[END_REF][START_REF] Gotoh | Power and nonpower laws of passive scalar moments convected by isotropic turbulence[END_REF]. A numerical validation using high Reynolds EDQNM simulations is proposed as well in Appendix A. The helical "four-thirds" law (8.42) is similar to the "four-thirds" laws for the kinetic energy and scalar variance in HIT, since they all come from conservation laws [START_REF] Antonia | Analogy between predictions of kolmogorov and yaglom[END_REF].

In this section, formula that allow to compute helical structure functions from spectral quantities are derived, similarly to what is usually done for velocity and scalar statistics in HIT [START_REF] Monin | Statistical Fluid Mechanics: Mechanics of Turbulence[END_REF]. This further permits, using the EDQNM model presented in the previous sections, to assess numerically at high Reynolds numbers the two laws (8.41) and (8.42). In continuity of these developments in physical space, the evolution equation of the helical dissipation rate H is addressed: H itself and its production term are simplified, similarly to what is usually done for the kinetic energy dissipation rate in HIT. These analytical considerations provide further insights on the derivatives of helical correlations and skew-isotropic tensors, and leads to the definition of a helical Taylor scale and a helical derivative skewness.

Inertial scaling for S(r) and D (uuω) (r)

This part aims at recalling the main steps of the derivation of the laws (8.41) and (8.42), linking S(r) and D (uuω) (r), and finally assessing their inertial scalings at large Reynolds numbers.

The law of [START_REF] Chkhetiani | On the third moments in helical turbulence[END_REF]: The Reynolds tensor reads

R ij (r) =< u i u j >= u 2 f (r)δ ij + r 2 f (r) δ ij - r i r j r 2 + h(r) r ijl r l , (8.43)
where f =< u L u L > /u 2 is the second-order longitudinal correlation (see Appendix A for details), with u 2 = 2K/3, and h(r) = R 23 . The mixed velocity-vorticity correlation is defined as

R H ij =< u i ω j >= jln ∂R in ∂x l = u 2 ijl r l r f + r l 2 f -2δ ij h r + h r -h δ ij - r i r j r 2 , (8.44) with < u i ω i >= 2K H . Also, R H ii (r) = -4 h r -2h , h(r) = - 1 3 rK H . (8.45)
The latter expression shows that h(0) = 0. The two-point third-order velocity correlation contains an additional antisymmetric part with respect to HIT, so that

< u i u j u k >= k -rk 2r 3 r i r j r k -δ ij k 2r r k + 2k + rk 4r (δ ik r j + δ jk r i ) + S(r) r l r 2 ( ikl r j + jkl r i ), (8.46)
where k(r) =< u L u L u L > and S(r) =< u L u 2 u 3 >: in particular, < u 2 u 3 u L >= 0 and < u L u 3 u 2 >= -S(r), which gives S(0) = 0. Thus, from the evolution equation (A.75) of R ij (r), it is possible to compute the evolution equation of the antisymmetric part (R ij -R ji )/2 = h ijl r l /r linked to helicity. The spatial derivative ∂ r j erases the part which contains the third-order longitudinal correlation k(r), so that, after some algebra and using d t K H = -H , one gets

r 3 H = 2 r 3 ∂ ∂r (r 3 S) + 2ν - 2 r 2 h(r) + 2 r ∂h ∂r + ∂ 2 h ∂r 2 . (8.47)
Further neglecting the viscous effects in the inertial range and integrating over r, one obtains (8.41).

The law of [START_REF] Gomez | Exact relationship for third-order structure functions in helical flows[END_REF]: The equation for < δω i > is derived analogously to the one for δu i (A.85), starting from (D.27):

∂ t δω i + δu j ∂ ∂r j (δω i ) = δω j ∂ ∂r j (δu i ) + 2ν ∂ 2 ∂r j ∂r j
(δu i ).

(8.48)

Combining (A.85) and (D.27) yields

∂ < δu i δω i > ∂t + ∂ ∂r j < δu j δu i δω i > - 1 2 < δu i δu i δω j > = 2ν ∂ 2 < δu i δω i > ∂r j ∂r j -4 ν < ∂δu i ∂r j ∂δω i ∂r j > = H
.

(8.49) Then, < δu j δu i δω i >= r j < δu L δu i δω i > /r, and ∂ r j (D (uuω) r j /r) = ∂ r (r 2 D (uuω) )/r 2 . Further neglecting the time dependance and the viscous term in the inertial range, integration over r yields (8.42). Interestingly, the Gomez and Chkhetiani laws can be linked. Remarking that

< δu i δω i >= 2 < u i ω i > -2 < u i ω i >, ∂ t R H 11 = -2 H /3, with R H 11 = -2h/r, this provides 1 r 2 ∂ ∂r (r 3 R H 11 ) =< u i ω i >= R H ii , 1 r ∂ ∂r 1 r 2 ∂(r 3 R H 11 ) ∂r = 1 r 4 ∂ ∂r r 4 ∂R H 11 ∂r ,
one gets by identification

D (uuω) =< δu L δu i δω i > - 1 2 < δu i δu i δω L >= - 8 r 3 ∂ ∂r (r 3 S(r)), (8.50)
from which (8.41) immediately follows using (8.42).

Link between spectral and physical space: The equations to compute D LLL and D LTT from spectral quantities are well known, even-though the second-one is much less documented: see for instance [START_REF] Monin | Statistical Fluid Mechanics: Mechanics of Turbulence[END_REF] and Appendix A for details. Here, the equations to obtain both S(r) and D (uuω) from S NL H are presented. First, one needs to derive the evolution equations of < u i ω i > /2: this is done starting from (8.47). Since

< u i ω i > 2 = - 1 r 2 ∂(r 2 h) ∂r , r H 3 = ∂h ∂t , (8.51) one obtains ∂ ∂t < u i ω i > 2 = - 2 r 2 ∂ ∂r 1 r ∂ ∂r (r 3 S) - 2ν r 4 ∂ ∂r r 4 ∂ 2 h ∂r 2
. (8.52) This equations needs to be identified with the helical Lin equation (8.15), so that

- 2 r 2 ∂ ∂r 1 r ∂ ∂r (r 3 S(r)) = ∞ 0 S NL H (k) sin(kr) kr dk.
This yields, after some algebra,

S(r) = 1 2 ∞ 0 S NL H k 2 sin(kr) kr -3 sin(kr) (kr) 3 + 3 cos(kr) (kr) 2 dk, D (uuω) (r) = 4 ∞ 0 S NL H k sin(kr) (kr) 2 - cos(kr) kr dk. (8.53) (8.54)
The formula for D (uuω) is very similar to the ones for < δu L δq 2 > and D LTT , which is expected since they all refer to conservation laws. The relevance of the two previous formula is illustrated in Fig. 8.8a and 8.8b, where the compensated helical third-order correlations -D (uuω) /(r H ) and S/(r 2 H ) are displayed at high Reynolds numbers. The theoretical values 4/3 and 1/30 are almost recovered at Re λ = 3.10 4 : the slight difference is comparable to the difference observed for the -4/5 law in decaying turbulence [START_REF] Bos | Reynolds number effect on the velocity increment skewness in isotropic turbulence[END_REF][START_REF] Tchoufag | Spectral approach to finite reynolds number effects on kolmogorov's 4/5 law in isotropic turbulence[END_REF]. Interestingly, D (uuω) is closer to 4/3 than < δu L δq 2 > in HHT. Let's mention that an equivalent scaling for S (the 2/15 law) was already assessed in DNS [START_REF] Kurien | Isotropic third-order statistics in turbulence with helicity: the 2/15-law[END_REF]. But so far, to our knowledge, the scaling for D (uuω) was not verified numerically, at least for freely decaying turbulence: the present simulations show a very good agreement between EDQNM results and the theoretical expectations. Finally, it is revealed in Fig. 8.8c that S ∼ r 4 at small scales, which is straightforward using (8.50), unlike < δu 3 L >, < δu L δq 2 >, < δu L δθ 2 > and D (uuω) which scale in r 3 .

Evolution equation of H

In this part, the evolution equation of the helical dissipation rate is addressed. The objective is to simplify its equation, in a manner similar to what is usually done for the kinetic energy dissipation rate in HIT [START_REF] Pope | Turbulent Flows[END_REF]. More precisely, H itself and its production term are greatly simplified, and expressed as functions of the derivatives of the fluctuating velocity and vorticity fields. The final expressions (8.58) and (8.67) constitute one of the main new theoretical contributions of the present work, and applications are proposed as well. The calculations being rather lengthy, the intermediate steps are gathered in Appendix D for the sake of clarity. The evolution equation of H reads

∂ ∂t H ν + D[u, ω] = -2ν < ∂ 2 u i ∂x j ∂x l ∂ 2 ω i ∂x j ∂x l >, (8.55)
D[u, ω] =< ∂u i ∂x j ∂u l ∂x j ∂ω i ∂x l > + < ∂u i ∂x l ∂u l ∂x j ∂ω i ∂x j > -< ∂u i ∂x j ∂u i ∂x l ∂ω l ∂x j >, (8.56)
where the production term D[u, ω] contains contributions from spatial derivatives of skewisotropic tensors such as < u i u j ω l > and < ω i u j u l >. In what follows, both H and D[u, ω] are simplified. This procedure consists into two steps: first, expressing H and D[u, ω] as functions of the derivatives of h(r) and S(r) respectively, and then expressing these derivatives as functions of particular components of the fluctuating velocity and vorticity fields. Obviously, this is much more lengthy for D[u, ω] since it is a third-order moment, composed of three different terms.

Derivatives of R H ij (r): The first step to simplify H consists into expressing it as a function of the derivatives of h(r). To do so, one first needs to know the derivatives of R H ij , analogously to what [START_REF] George | Locally axisymmetric turbulence[END_REF] did for in axisymmetric turbulence. One has

∂ 2 R H ij ∂r p ∂r q = -< ∂u i ∂x p ∂ω j ∂x q >, H ν = - ∂ 2 R H ii ∂r j ∂r j r=0 =< ∂u i ∂x j ∂ω i ∂x j > . (8.57)
After some algebra gathered in Appendix D, one gets the general expression of ∂ 2 pq R H ij , from which one can obtain some relations between the derivatives of h(r). Then, using a Taylor expansion of h(r) for r → 0 (with h(0) = 0) yields the important theoretical result (8.58) This expression permits to determine the helicity dissipation rate with one term instead of nine, and is equivalent to = 15ν < (∂ 1 u 1 ) 2 > for the kinetic energy dissipation rate in HIT.

H = 10νh (0) = 15ν < ∂u 1 ∂x 1 ∂ω 1 ∂x 1 > .
The natural extension of the previous calculations for H is the definition of a helical Taylor scale λ H . Its expression is found analogously to what is usually done for the longitudinal Taylor scale λ [START_REF] Pope | Turbulent Flows[END_REF], i.e. considering the osculating curve P H (r) of h(r) in r = 0, which reads

P H (r) = h(0) + rh (0) + r 2 2 h (0) + r 3 6 h (0) = -r K H 3 + r 3 H 60ν , (8.59)
and λ H is further defined by P H (λ H ) = 0 and λ H = 0, which yields

λ H = 20νK H H . (8.60)
It is worth noting that this new expression is completely equivalent to the classical one for the velocity λ = 20νK/ . The relevance of this formula is illustrated in Fig. 8.9. Both λ and λ H are displayed, at large and moderate Reynolds numbers for Saffman turbulence. Both scales indicate the beginning of a region where viscous dissipation balances non-linear transfers. At high Reynolds numbers, λ and λ H cannot be distinguished, whereas λ H is a bit smaller at moderate Reynolds numbers. The wavenumber k H η , defined in (8.22) and proposed by [START_REF] Ditlevsen | Dissipation in helical turbulence[END_REF], is presented as well: it seems to have a similar physical meaning as λ H , at least at large Reynolds numbers, and some explanations are provided in Appendix D. Moreover, Fig. 8.9 illustrates the viscous production of helicity, already observed in Fig. 8.1b where the helical spectrum was negative around k η : indeed, -2νk 3 H is positive near k η here.

A secondary application of these calculations is to express analytically the impact of helicity on the second-order longitudinal correlation f (r → 0). Indeed, unlike HIT where all odd derivatives of f (r) for r = 0 are zero, one can show that f (0) is a priori different from zero in HHT (one has always f (0) = 0 because of homogeneity). Expressing ∂ 2 11 R H 23 and ∂ 2 33 R H 23 thanks to (D.33) yields

f (0) = - 1 2 < ∂u 2 ∂x 1 ∂ω 3 ∂x 1 >= -< ∂u 2 ∂x 3 ∂ω 3 ∂x 3 >, (8.61)
so that the Taylor expansion of f (r) reads

f (r) = 1 - r 2 2 15νu 2 + r 3 6 f (0) Additional HHT term + r 4 24 2 35u 2 ∞ 0 k 4 E(k)dk.
(8.62)

Determination of the production term D[u, ω]: The procedure used to determine H is now applied to the production term D[u, ω]: since this term is complex and involves derivatives of two different tensors < u i u j ω l > and < ω i u j u l >, the determination is divided into two steps. First, D[u, ω] is expressed as a function of the derivatives of S(r), and it will be shown that only S (iv) (0) remains. Then, S (iv) (0) is expressed as a function of the derivatives of the fluctuating velocity and vorticity fields.

k/k η Normalized budget terms -2νk 3 E -2νk 3 H kS NL E kS NL H k H η k η λ -1 λ -1 H (a) 10 -4 10 -3 10 -2 10 -1 10 0 -1 -0.5 0 0.5 1 k/k η Normalized budget terms -2νk 3 E -2νk 3 H kS NL E kS NL H k H η k η λ -1 λ -1 H (b)
As pointed out in [START_REF] Gomez | Exact relationship for third-order structure functions in helical flows[END_REF], the tensor < ω i u j u l > is much more complicated to handle than < u i u j ω l >, which can be linked easily to < u i u j u l > given in (8.46)

φ (uuω) ijl =< u i u j ω l >= lpq ∂ < u i u j u q > ∂r p , φ (ωuu) ijl =< ω i u j u l > . (8.63)
Calculations detailed in Appendix D first permit to link derivatives of φ (uuω) ijl and φ

(ωuu) ijl to D[u, ω] according to D[u, ω] = ∂ 3 ∂r j ∂r j ∂r l φ (uuω) ili + φ (ωuu) ili -φ (ωuu) lii r=0 , (8.64)
where the derivatives of φ (uuω) ijl and φ

(ωuu) ijl contain derivatives of S(r). Taylor expansions of S(r) for r → 0 further give

D[u, ω] =< ∂u i ∂x j ∂u l ∂x j ∂ω i ∂x l > + < ∂u i ∂x l ∂u l ∂x j ∂ω i ∂x j > -< ∂u i ∂x j ∂u i ∂x l ∂ω l ∂x j >= -35S (iv) (0). (8.65)
Now that D[u, ω] has been linked to S (iv) (0), the final step is to express the fourth derivative of S(r) as derivatives of the fluctuating velocity and vorticity fields, so that D[u, ω] can be evaluated in DNS for instance. For this purpose, the explicit sixth-order tensor ∂ 3 npq φ

(uuω) ijl is needed, and given in Appendix D. From this lengthy expression, one notably gets

∂ 3 φ (uuω) 111 ∂r 3 1 r=0 = -S (iv) (0), (8.66)
from which one finally obtains the second important result of this section

D[u, ω] = -35S (iv) (0) = 35 < ∂ 2 u 2 1 ∂x 2 1 ∂ω 1 ∂x 1 >, (8.67)
which permits notably to compute D[u, ω] with only one term, instead of eighty-one. Further proceeding as in [START_REF] Kerr | Higher-order derivative correlations and the alignment of small-scale structures in isotropic numerical turbulence[END_REF], i.e. identifying (8.55) with the spectral evolution equation of

H ∂ H ∂t = 2ν ∞ 0 k 2 S NL H (k, t)dk -4ν 2 ∞ 0 k 4 H(k, t)dk, (8.68)
provides

S (iv) (0) = 2 35 ∞ 0 k 2 S NL H (k, t)dk. (8.69)
Hence, one can define, analogously to the mixed-derivative skewness of a passive scalar in HIT, a helical derivative skewness as

S uuω =< ∂ 2 u 2 1 ∂x 2 1 ∂ω 1 ∂x 1 > < ∂u 1 ∂x 1 ∂ω 1 ∂x 1 > < ∂u 1 ∂x 1 2 > = - 3 √ 30 14 ∞ 0 k 2 S NL H dk ∞ 0 k 2 Edk ∞ 0 k 2 Hdk . (8.70) (8.71)
Interestingly, the numerical factor 3 √ 30/14 is identical to the one of the velocity derivative skewness in HIT.

The helical derivative skewness S uuω is displayed in Fig. 8.10 as a function of the Reynolds number Re λ . Only Saffman turbulence is presented, because the curve for Batchelor turbulence is identical at high Reynolds numbers, similarly to the velocity and mixed derivative skewnesses in HIT (see Chapter 1 and Appendix A). The initial oscillations at large Re λ correspond to the first turnover times of the simulation, when the turbulence is not fully developed yet. As for the velocity and mixed derivative skewnesses, S uuω is negative and reaches an asymptotic value at large Reynolds numbers S ∞ uuω = -0.141. This value S ∞ uuω is lower in magnitude than asymptotic values for the velocity and mixed derivative skewnesses in HIT which are around -0.5. The knowledge of the helical derivative skewness is of importance, for two reasons: it is of theoretical interest since it permits to have a strong analogy between the evolution equations of in HIT and H in HHT. On a more practical point of view, the previous developments which led to S uuω show that there exists, in homogeneous helical turbulence, a quantity which is constant at large Reynolds numbers: such a result could be used to improve RANS models for helical flows, where the production term would be linked to S uuω .

Effect of helicity on the scalar flux

In this section, the transport of a passive scalar field θ is addressed. However, since there are no explicit contributions of helicity in its evolution equation, a vertical mean scalar gradient λ = (0, 0, -Λ) is added. In purely isotropic turbulence with a mean scalar gradient, the wellknown cospectrum F is created (see Chapter 5). When both helicity and a mean scalar gradient are combined, a second spectrum is created, called the quadrature spectrum Q(k, t), linked to the imaginary antisymmetric part of the scalar flux F i (k). This quadrature spectrum was reported in [START_REF] Mydlarski | Passive scalar statistics in high-péclet-number grid turbulence[END_REF], and shown to be zero in non-helical turbulence with a mean scalar gradient in O' Gorman & Pullin (2005). Consequently, we choose here the framework of Helical Homogeneous Turbulence with a mean Scalar Sradient (HHTSG) in order to create this quadrature spectrum and analyze its properties. In the two next parts, it is first proposed to derive the evolution equation of Q(k, t) and its non-linear transfer terms within the EDQNM framework, and secondly to investigate its inertial scaling. Analogies with the effects of helicity on the kinetic energy spectrum are pointed out, and the decay exponent of helicity along with the helical Taylor scale given previously are used. A unit Prandtl number is considered, and simulations not presented here have revealed that the scalar variance spectrum still scales in k -5/3 in the inertial range despite the presence of helicity.

Modelling of the quadrature spectrum

When the decomposition of the scalar flux (4.28) was introduced in Chapter 4, it was mentioned that in presence of helicity there could be an additional antisymmetric contribution. Thus, without mirror symmetry, the new decomposition of the scalar flux reads

F i (k, t) = 3 2 E F j (k, t)P ij (k) + 3 2 iε ijn α n E Q j (k, t) k , (8.72)
which is quite similar to the decomposition (8.2) of Rij . Both E Q j and E F j are real vectors. The imaginary antisymmetric part is linked to the quadrature spectrum Q(k, t), which is zero in HITSG and arises only with the additional presence of helicity, and is defined as

Q(k, t) = 4πk 2 E Q 3 (k, t) = S k i 3jl k l F j (k, t)d 2 k. (8.73)
It appears that the quadrature spectrum was never analytically investigated, and is only reported to be zero in HIT and HITSG by [START_REF] Mydlarski | Mixed velocity-passive scalar statistics in high-reynolds-number turbulence[END_REF]; O' Gorman & Pullin (2005). The mechanism which creates an additional contribution to the scalar flux, parallel to the mean gradient in the presence of helicity, is called skew-diffusion by [START_REF] Moffatt | Helicity in laminar and turbulent flow[END_REF]. It is worth noting that from the decomposition (8.72) and (8.73), imaginary components of the scalar flux, perpendicular to the mean scalar gradient, are non-zero. Nevertheless, these components vanish after spherical averaging. Hence, in HHTSG, the evolution equation of the cospectrum given in (4.38) is modified into

∂ ∂t + (a + ν)k 2 F(k, t) = 2 3 ΛE(k, t) + S F,NL 3 (k, t) + S F,NL(hel) 3 (k, t) S NL F (k,t) , (8.74)
where S F,NL(hel) i

is the additional contribution arising from the presence of helicity, the nonlinear spherically-averaged helical scalar flux transfer

S F,NL(hel) i (k, t) = 4π 2 ∆ k θ F kpq k 2 (x + yz)H 2pzE Q i -kE Q i dpdq + 4π 2 ∆ k θ F pkq k(x + yz) H p 2 E Q i -q(q -2ky)E Q i -k 2 H E Q i dpdq. (8.75)
The remarkable feature about S F,NL(hel) i is that, unlike S F,NL i which contains a RTI part, it is a conservative transfer, with zero integral over the whole wavenumber space. Then, the evolution equation of the quadrature spectrum reads

∂ ∂t + (a + ν)k 2 Q(k, t) = 2 3 ΛH(k, t) + S NL Q (k, t), (8.76)
where the production term is linked to helicity, and

S NL Q = S Q,NL 3
is the non-linear sphericallyaveraged quadrature transfer

S Q,NL i (k, t) = 4π 2 ∆ k θ F kpq k 3 pq E 0 E Q i (xyz + 2z 2 -y 2 ) -2z(xy + z)E Q i + z(xy + z)H (2E F i -E F i ) dpdq + 4π 2 ∆ k θ F pkq k 2 q kpE 0 E Q i (xyz + 2z 2 -y 2 ) -2(1 -y 2 )E Q i -kpz(xy + z)H E F i + kE 0 E Q i p(1 -y 2 ) -E Q i q(x -yz -2xy 2 ) -pqH E F i y(1 -y 2 ) + E F i (xz + y 3 -y(1 -x 2 )) dpdq. (8.77)
Similarly to S F,NL 3 , S NL Q is not a conservative transfer. Some details are provided in Appendix E for S F,NL(hel) i and S Q,NL i . 8.4.2 Decay of < ω 3 θ > and inertial scaling of Q(k, t) For these numerical simulations, one has initially Q = F = E T = 0, E is given by (8.19), and H = kE. First, let's consider the evolution equation in physical space of the one-point mixed vorticity-scalar correlation

K Q (t) =< ω 3 θ >= ∞ 0 Q(k, t)dk, Q (t) = (ν + a) < ∂ω 3 ∂x l ∂θ ∂x l >= 2(ν + a) ∞ 0 k 2 Q(k, t)dk, (8.78) which reads ∂ < ω 3 θ > ∂t = 2 3 ΛK H + < θω j ∂u 3 ∂x j > -(ν + a) < ∂θ ∂x j ∂ω 3 ∂x j >, (8.79)
where < θω l ∂ l u 3 > is the destruction term of < ω 3 θ >, and is given by the integral of S NL Q over the whole wavenumber space. From this evolution equation, it directly follows that the decay exponent of K Q is α Q = α H + 1, because the production term linked to helicity is the one responsible for the creation of the quadrature spectrum. This yields

K Q (t) ∼ t α Q , α Q = - σ + 1 σ + 3 , (8.80)
which is assessed, for both Saffman and Batchelor turbulence, in Fig. 8.11a. The decay exponent α F , derived in (5.13) for HITSG is still valid, using the helical backscatter parameter p H = 0.14. For a given infrared slope σ, the decay of < ω 3 θ > is faster than < u 3 θ >, similarly to the decay of K H being faster than K.

Regarding the non-linear transfers: the impact of the quadrature spectrum on the cospectrum can be observed through the conservative non-linear transfer S F,NL(hel) 3

. This transfer is linked to an inverse cascade of < u 3 θ >, localized at large scales, between the integral and the helical Taylor scales. This can be qualitatively compared to the impact of helicity on the kinetic energy spectrum dynamics through S NL(hel) in Fig. 8.3b. One could conclude that helicity only slightly reduces the non-linear transfers of the cospectrum at large scales. Then, the quadrature non-linear transfer S NL Q itself is similar to the cospectrum one S F,NL

3

, but less intense.

Finally, in Fig. 8.11c, the quadrature spectrum is presented. In the infrared range, it scales in Q ∼ k 3 for Saffman turbulence, because the helical spectrum itself evolves in H ∼ k 3 : indeed, helicity, through the mean scalar gradient, is the production term of the quadrature spectrum.

In the inertial range, Q(k, t) is first positive for scales larger than the helical Taylor scale λ H , and scales in k -7/3 similarly to F. Whereas for scales smaller than λ H , the spectral slope is close to k -5/3 and the quadrature spectrum is negative: it is recalled that around k η , H(k, t) is also negative.

One can propose a theoretical inertial scaling for the positive region of the quadrature spectrum: assuming in (8.76) that ∂ t Q ∼ ΛH, with the characteristic inertial time (k 2 ) -1/3 , this gives

Q(k, t) ∼ Λ H -2/3 k -7/3 , k L < k < λ -1 H . (8.81)
For the negative region of Q(k, t), the quadrature spectrum should only depend on H , and not anymore on , since this is the negative small scales of H(k, t) which are responsible for this inertial-helical range of Q(k, t). Thus, one gets This change of slope, from k -7/3 for kλ H < 1, to k -5/3 for kλ H > 1, observed in figure 8.11c, and justified with dimensional and physical arguments, can also be interpreted in terms of a change in characteristic time scales, from τ = (k 2 ) -1/3 to τ H = (k H ) -1/3 . The latter time scale was notably proposed by [START_REF] Kurien | Isotropic third-order statistics in turbulence with helicity: the 2/15-law[END_REF] for an alternative scaling of H(k, t) at small scales. This characteristic time scale τ H is relevant for the inertial-helical scaling of Q when it is negative: indeed, for kλ H > 1, the equation (8.76) of Q can be written ∂ t Q ∼ S NL Q . Using the characteristic time scale τ H , the classical inertial scaling (8.20) of H and F, one gets for the non-linear quadrature transfer S NL Q ∼ Λk -4/3 2/3 H , so that (8.82) is recovered.

Q(k, t) ∼ Λ 1/3 H k -5/3 , λ -1 H < k < k η . ( 8
k Scalar flux spectra F (k, t) Q(k, t) -Q(k, t) k 2 k 3 k L k η 1/λ H k -5/3 k -7/3 (c)

Conclusion on homogeneous skew-isotropic turbulence

The classical framework of decaying homogeneous helical turbulence (HHT) where mirror symmetry is broken at large Reynolds numbers was addressed using the anisotropic EDQNM modelling.

Some existing results were recovered here for decaying skew-isotropic turbulence, in order to validate the use of EDQNM for HHT, which could be summarized in three features. First, when helicity is initially present at large scales, helicity cascades towards small scales along with the kinetic energy, creating a k -5/3 inertial range that extends up to the Kolmogorov wavenumber k η . The k -5/3 scaling of the helical spectrum H(k, t) is similar to the one of an advected passive scalar, also obtained with dimensional analysis by assuming that the inertial characteristic time τ (k) = (k 2 ) -1/3 is identical for both the kinetic and helical fields. Secondly, in the early stage of the decay, helicity slows down the filling of the kinetic energy spectrum at small scales. Consequently, there is an initial reduction of the kinetic energy transfers: this is a transitory effect, since once the turbulence is fully developed, the effects of helicity on the kinetic energy decay are rather weak. Finally, two-point third-order helical correlations were investigated: notably, the "four-thirds" law for helical structure functions, and the 1/30 law for the helical correlation S(r), were assessed with EDQNM at very high Reynolds numbers in decaying turbulence. It is worth noting that the two formula linking helical correlations in physical space and spectral non-linear helical transfers are new results of this work.

Then, EDQNM simulations were used to assess some new theoretical predictions of fundamental interest for helical turbulence. First, the infrared dynamics of the kinetic energy and helical spectra was investigated theoretically using non-local expansions in the non-linear transfers. It clearly appears that in Batchelor HHT, helicity reduces the back transfers of kinetic energy with respect to HIT: consequently, inverse non-local transfers are weakened and bring back less energy to large scales. Furthermore, the permanence of large eddies is shown to be verified for H(k, t), even in Batchelor turbulence, with classical initial conditions such that the kinetic and helical infrared slopes are σ H = σ + 1. These two features are assessed numerically with EDQNM. An original configuration, with different initial conditions, also exhibited a k 8 infrared scaling for H(k, t), along with some helical backscatter.

Secondly, as a direct application of the previous infrared dynamics analysis, the impact of helicity on the long-time kinetic energy decay was shown to be quite subtle: indeed, the decay of K(t) is not modified with regard to HIT, except in the case of Batchelor turbulence where it is slightly accelerated, because of the less-intense inverse transfers. In addition, in agreement with dimensional analysis, theoretical decay exponents for helicity were derived, and assessed numerically at large Reynolds numbers: in particular, it is found that helicity decays faster than the kinetic energy.

Thirdly, the evolution equation of the helicity dissipation rate H was studied, with a particular attention on the derivatives of second and third order skew-isotropic tensors such as < u i ω j >, < u i u j ω l > and < ω i u j u l >. As an important result of these analytical developments, H itself is expressed as a function of h (0) only, where h = R 23 (r), and its production term D[u, ω] as a function of S (iv) (0) only, where S =< u L u 2 u 3 >. It follows that H and D[u, ω] can be expressed with only one term, instead of nine and eighty-one respectively.

Fourthly, as a direct application of these fundamental results, a helical Taylor scale was defined λ H = 20νK H / H , whose expression is analogous to the longitudinal Taylor scale λ for kinetic energy. Such a scale is new for the helical field, and its relevance was illustrated numerically: λ H is the scale from which viscous dissipation of helicity becomes dynamically important, and at large Reynolds numbers, it is very close to λ. Moreover, a helical derivative skewness was defined and is negative and constant at large Reynolds numbers, similarly to the velocity derivative and mixed-derivative skewnesses in HIT.

Finally, it was shown that combining a mean scalar gradient and helicity produces the quadrature spectrum Q(k, t), linked to the imaginary antisymmetric part of the scalar flux. As a consequence, the large scales non-linear transfers of the cospectrum are slightly reduced. The main result is that Q(k, t) has two different scalings in the inertial range: for kλ H < 1, Q ∼ k -7/3 , and then for kλ H > 1, Q ∼ k -5/3 : this change in the spectral slope can be interpreted as the characteristic time evolving from (k 2 ) -1/3 to (k H ) -1/3 at smaller scales.

Chapter 9

General Conclusions and Perspectives

"Do. Or do not. There is no try."

-Master Yoda, Star Wars V

The main objective of this thesis was to understand the fundamental turbulent mechanisms occurring in natural flows at large Reynolds numbers, such as atmospheric and oceanic ones.

For this purpose, we established a methodology which could be summarized as follows: we worked in the framework of homogeneous turbulence, with the spectral formalism, and aimed at modelling the various processes at stake in turbulent anisotropic flows. The resulting model, called anisotropic EDQNM modelling throughout the manuscript, was assessed by multiple comparisons against DNS and experiments. Then, theoretical predictions were derived, based on both physical arguments and analytical calculations, which were successfully verified by the model.

The starting point of this complex task was the pioneering study of [START_REF] Cambon | Spectral modelling of homogeneous non-isotropic turbulence[END_REF], further developed and improved in [START_REF] Mons | A spectral model for homogeneous shear-driven anisotropic turbulence in terms of spherically-averaged descriptors[END_REF]. The general concept of the spectral modelling relies on two steps: (i) a classical EDQNM procedure to close the non-linear terms in the evolution equations of the two-point second-order moments; and (ii) a modelling of anisotropy through truncated expansions in spherical harmonics of the spectral second-order moments. The final model is not an end in itself, but rather a general and robust method to address complex flows, which eventually requires no more adjustable constants than the eddy-damping one, chosen once and for all on the well-known and accepted isotropic value. Consequently, the model is relevant to explore configurations unreachable by DNS and experiments.

Since detailed conclusions were provided at the end of each chapter, it would be redundant to recall them here. Instead, we prefer to briefly put the emphasis on the main findings of the thesis.

In the eight previous chapters, we tackled isotropic turbulence (HIT), isotropic turbulence with a mean scalar gradient (HITSG), isotropic helical turbulence (HHT), helical turbulence with a mean scalar gradient (HHTSG), turbulence initially submitted to shear (HSRT), sustained shear flows (HST), shear flows with a mean scalar gradient (HSTSG), and unstably stratified turbulence (USHT). For each of these configurations:

• Lengthy and complex analytical calculations have been performed to determine the nonlinear transfers and production terms of the anisotropic EDQNM model...

• ... which was compared, when possible, to DNS, experiments and other models.

• The infrared and inertial scalings of the main spectra were accurately investigated.

• The scale-by-scale distribution of anisotropy was analyzed.

• Theoretical time exponents, assessed numerically, were derived for one-point statistics such as the kinetic energy, the scalar variance, the mixed velocity-scalar correlation, and the helicity.

Among these numerous features, the three most important findings would be, very likely:

• The new algebraic decay exponents for < θ 2 > and < u 3 θ > in HITSG, and for < u i ω i > in HHT, and when combining both, the creation of the quadrature spectrum.

• The profound difference between the asymptotic anisotropic states of shear flows and unstably stratified turbulence: the former is almost independent of large scales initial conditions σ, whereas the latter strongly depend on them, and this was justified analytically.

• The effects of moderate Reynolds numbers on the scattering of global quantities in shear flows and on persistent small scales anisotropy at the level of the scalar second-order moments.

In addition to all the points mentioned above, detailed appendices are provided which contain all the calculations needed to understand the anisotropy modelling, the establishment of the quasinormal normal expressions of the non-linear transfers within the EDQNM procedure (along with all the tricky but essential geometrical relations), the spherical and λ integrations, the non-local expansions, and statistics of second and third order moments in homogeneous turbulence. These appendices are rather long on purpose, in order to allow this work to be continued conveniently.

Further theoretical considerations were also proposed, such as the pressure spectra, quadratic anisotropic contributions in the non-linear transfers, and last but not least, the third and fourth orders expansions of E and Z for the kinetic field. This last remark makes the transition with the possible perspectives that I can imagine:

• Pursue the work, started Chapter 2 and Appendix C, about the third and fourth orders expansions of E and Z for the velocity field, and find some configurations, in addition to the sustained shear flow, where it could be of of importance. This is a promising track since first results in Chapter 3 indicate that fourth-order contributions tend to reduce the kinetic energy exponential growth rate γ. The extension to the scalar field through E T would be rather straightforward. But more interestingly, some work needs to be done regarding the higher-order expansion of the scalar flux. Indeed, given the equations of USHT, it could improve the theoretical linear prediction for the kinetic energy exponential growth rate β, and consequently makes it closer to the exact value 2N .

• Since shear was already combined to a mean scalar gradient for passive scalar dynamics in Chapter 5, and helicity to a mean scalar gradient in Chapter 8, a natural extension could be to combine both shear and USHT. First results tend to indicate that stratification overcomes shear, with an exponential growth rate for K which depends on σ. At a comparable level, it would be interesting to go further with the variable mean-fields S(t) and N (t), with some details and equations already provided at the end of Chapters 3 and 7. This could extend the reach of the anisotropic EDQNM modelling to address mixing and free-shear layers within a homogeneous framework.

• For the long-term perspective, it would be of great theoretical interest to extend the concept of the modelled anisotropy through spherical harmonics expansion to EDQNM2, in order to deal with rotating turbulence. Of course, this is a complex task which would require significant analytical developments to take into account the linear operators of the third-order correlations into the non-linear transfers. But it could permit to model more accurately geophysical flows by combining shear, stratification and rotation.

• Finally, the case of MHD turbulence could be addressed again thanks to the present anisotropic EDQNM modelling, in line with the pioneering works of [START_REF] Pouquet | Strong mhd helical turbulence and the nonlinear dynamo effect[END_REF]; [START_REF] Grappin | Alfvenic fluctuations as asymptotic states of mhd turbulence[END_REF]. Indeed, the spectral two-point magnetic-magnetic correlation verifies the same properties as Rij , and thus can be decomposed as well into directional and polarization parts, which could be of use to address strong MHD, where a mean magnetic field breaks the isotropy of the flow. Before that, it appears to be essential to focus first on isotropic MHD, with the effects of kinetic helicity, magnetic helicity, and cross-helicity combined. In the last months of this thesis, I started studying isotropic MHD in presence of cross-helicity, without a mean magnetic field, and despite some analogies with the kinetic helicity in hydrodynamics turbulence, it is much more complex and there is still a great amount of work to be done.

A.2 Tensorial relations for homogeneous turbulence

In this part, the emphasis is put on homogeneous turbulence to provide general relations between sixth order tensors involving second and third order moments of the fluctuating velocity and scalar fields. In what follows, only homogeneity is assumed.

A.2.1 Dissipation and enstrophy < ω 2 >

Expanding < ∂ l (u i ∂ k u j ) = 0 > and < ∂ k (u i ∂ l u j ) = 0 > yields the important result detailed in George & Hussein (1991) < ∂u i ∂x l ∂u j ∂x k >=< ∂u i ∂x k ∂u j ∂x l > . (A.5)
Let's call this result the 2 nd order law, since it involves second-order moments of the velocity field through a rank-4 tensor. Then, multiplying the incompressibility condition ∂ i u i = 0 by ∂ 1 u 1 , ∂ 2 u 2 and ∂ 3 u 3 provides, after ensemble average,

< ∂ 2 u 1 ∂ 1 u 2 > + < ∂ 3 u 1 ∂ 1 u 3 >= -< (∂ 1 u 1 ) 2 >, < ∂ 2 u 1 ∂ 1 u 2 > + < ∂ 3 u 2 ∂ 2 u 3 >= -< (∂ 2 u 2 ) 2 >, < ∂ 3 u 1 ∂ 1 u 3 > + < ∂ 3 u 2 ∂ 2 u 3 >= -< (∂ 3 u 3 ) 2 > .
This gives the relation always valid in homogeneous turbulence

< ∂u 1 ∂x 2 ∂u 2 ∂x 1 > + < ∂u 1 ∂x 3 ∂u 3 ∂x 1 > + < ∂u 2 ∂x 3 ∂u 3 ∂x 2 >= - 1 2 < ∂u 1 ∂x 1 2 > + < ∂u 2 ∂x 2 2 > + < ∂u 3 ∂x 3 2 > , (A.6) so that < ω 2 >= ν =< ∂u i ∂x j ∂u i ∂x j >= 1 2 < ∂u i ∂x j + ∂u j ∂x i ∂u i ∂x j + ∂u j ∂x i >= 1 2 < s ij s ij > . (A.7)

A.2.2 Identities for the velocity field

Let's define the sixth order tensor

B ijkpqr =< ∂u i ∂x p ∂u j ∂x q ∂u k ∂x r > . (A.8)
This tensor appears notably in the numerator of the velocity derivative skewness S(t), and on the evolution equation of the dissipation rate (t). In a manner similar to [START_REF] George | Locally axisymmetric turbulence[END_REF] with the 2 nd order law (A.5), there is a need to find different relations involving B ijkpqr .

Firstly, the use of homogeneity, through the difference of < ∂ j (u i ∂ q u p ∂ q u p ) >= 0 and < ∂ j (u i ∂ q u p ∂ p u q ) >= 0, gives

< ∂u i ∂x j ∂u q ∂x p ∂u q ∂x p >=< ∂u i ∂x j ∂u q ∂x p ∂u p ∂x q > . (A.9)
This is similar to (A.5), but for third-order moments of the velocity field, and thus this result is called the 3 rd order first law. One can also derive a second relation for third-order moments of the velocity field: < ∂ p (u i u q ∂ 2 jq u p ) >= 0 and < ∂ q (u q ∂ p u i ∂ j u p ) >= 0 yield

< u q ∂u i ∂x p ∂ 2 u p ∂x j ∂x q >= -< u i ∂u p ∂x q ∂ 2 u q ∂x j ∂x p >= -< u q ∂u p ∂x j ∂ 2 u i ∂x p ∂x q >,
which allows, from < ∂ 2 pq (u i u p ∂ j u q ) >= 0, to obtain

< ∂u i ∂x p ∂u q ∂x j ∂u p ∂x q > + < u q ∂u p ∂x j ∂ 2 u i ∂x p ∂x q >= 0.
Injecting the two previous equations into the development of < ∂ q (u p ∂ j u i ∂ p u q ) >= 0, and using the fact that < ∂ j (u q ∂ p u i ∂ p u q ) >= 0, yields the 3 rd order second law

< ∂u i ∂x j ∂u q ∂x p ∂u q ∂x p >= 2 < ∂u i ∂x p ∂u p ∂x q ∂u q ∂x j > . (A.10) Then, combining < ∂ i (u k ∂ j u i ∂ k u j ) >= 0 and < ∂ 3 ijk (u i u j u k ) >= 0 = 4 < u k ∂ j u i ∂ 2 ik u j >, provides < ∂u i ∂x j ∂u j ∂x k ∂u k ∂x i >= 0, (A.11)
which is consistent with (A.10). This result is often used in the isotropic framework. Finally, it is possible to derive one more relation for B ijkpqr : let's consider the following system

[1] < ∂ p (u i ∂ q u j ∂ r u k ) >= 0, [2] < ∂ p (u i ∂ r u j ∂ q u k ) >= 0, [3] < ∂ q (u i ∂ p u j ∂ r u k ) >= 0, [4] < ∂ q (u i ∂ r u j ∂ p u k ) >= 0, [5] < ∂ r (u i ∂ q u j ∂ p u k ) >= 0, [6] < ∂ r (u i ∂ p u j ∂ q u k ) >= 0.
Each equation creates three terms, with one involving a particular permutation of B ijkpqr . Combining these six equations, in the specific order Let's now define another sixth order tensor

C ijkpqr =< ∂ 2 u i ∂x k ∂x p ∂ 2 u j ∂x q ∂x r > . (A.13)
This tensor appears notably in the numerator of the kinetic palinstrophy G(t), and on the evolution equation of the dissipation rate (t). Firstly, the equations

< ∂ 2 jk (u i ∂ 2 jk u i ) >= 0 and < ∂ 2 jj (u i ∂ 2 kk u i ) >= 0 provide < ∂ 2 u i ∂x j ∂x j ∂ 2 u i ∂x k ∂x k >=< ∂ 2 u i ∂x j ∂x k ∂ 2 u i ∂x j ∂x k >, (A.14) Then, < ∂ 2 ij (∂ k u i ∂ k u j ) >= 0 gives < ∂ 2 u i ∂x k ∂x j ∂ 2 u j ∂x k ∂x i >= 0. (A.15)
These two equations will be used in the HIT framework. Finally, with the difference of < ∂ 2 kk (u j ∂ 2 pp u i ) >= 0 and < ∂ 2 kp (u j ∂ 2 kp u i ) >= 0, a more general result can be obtained

< ∂ 2 u i ∂x k ∂x p ∂ 2 u j ∂x k ∂x p >=< ∂ 2 u i ∂x k ∂x k ∂ 2 u j ∂x p ∂x p > . (A.16)
A.2.3 Evolution equations of W ij and < ω 2 >

In the homogeneous framework, the vorticity tensor W ij evolution equation reads

∂W ij ∂t = W il A jl + W jl A il + < ω l ω j ∂u i ∂x l + ω i ∂u j ∂x l > -ω ij , (A.17)
where the vorticity dissipation rate is ω ij = 2ν < ∂ l ω i ∂ l ω j >. Using (A.5), one obtains directly

W ij = δ ij < ∂u q ∂x p ∂u q ∂x p > -< ∂u q ∂x i ∂u q ∂x j > - ij 2ν . (A.18)
It is possible to simplify < ω l ω j ∂ l u i >. Firstly, the expansion of this term yields

< ω l ω j ∂u i ∂x l >=< ∂u i ∂x p a qp a jq > + < ∂u i ∂u j ∂u q ∂x p a qp >,
and the second rhs term is zero thanks to (A.9). Finally, one obtains using (A.10)

< ω l ω j ∂u i ∂x l + ω i ∂u j ∂x l >=< ∂u i ∂x p ∂u q ∂x j ∂u p ∂x q > + < ∂u q ∂x i ∂u j ∂x p ∂u p ∂x q > -< ∂u i ∂x p ∂u q ∂x j ∂u q ∂x p > = -< ∂u q ∂x i ∂u j ∂x p ∂u q ∂x p > + 1 2 < ∂u p ∂x q ∂u p ∂x q s ij > -< ∂u i ∂x p ∂u q ∂x j ∂u q ∂x p > -< ∂u q ∂x i ∂u j ∂x p ∂u q ∂x p > . (A.19)
Furthermore, using (A.11) and (A.14), one has

< ω i ω j ∂u i ∂x j >=< a jq a qi ∂u i ∂x j >= -< ∂u i ∂x j ∂u j ∂x k ∂u i ∂x k >, 1 2ν ω ii =< ∂ω i ∂x j ∂ω i ∂x j >=< ∂ 2 u i ∂x p ∂x p ∂ 2 u i ∂x q ∂x q >,
so that the enstrophy evolution equation in homogeneous turbulence reads

∂ < ω 2 > ∂t = 2W ij A + ij -2 < ∂u i ∂x j ∂u j ∂x k ∂u i ∂x k > -2ν < ∂ 2 u i ∂x p ∂x p ∂ 2 u i ∂x q ∂x q > .
(A.20)

A.2.4 Evolution equation of ij

The evolution equation of the dissipation tensor ij is not often investigated, as pointed out in [START_REF] Piquet | Turbulent Flows: Models and Physics[END_REF]. The starting point to compute the evolution equation of ij = 2ν < ∂ k u i ∂ k u j > is to derive the evolution equation (2.1) of u i with respect to x k , and to multiply it by ∂ k u j . The same process is done for the evolution equation of u j and the two resulting expressions are summed. Finally, the ensemble average is applied. Some terms are not straightforward to simplify, and details are provided hereafter: this yields the evolution equation of the dissipation tensor in the homogeneous framework

∂ ∂t ij 2ν = -< ∂u i ∂x k ∂u j ∂x p ∂u p ∂x k > -< ∂u j ∂x k ∂u i ∂x p ∂u p ∂x k > + < ∂p ∂x k ∂s ij ∂x k > -2ν < ∂ 2 u i ∂x k ∂x k ∂ 2 u j ∂x p ∂x p > -2 < A pk ∂u i ∂x p ∂u j ∂x k > -< A ip ∂u p ∂x k ∂u j ∂x k > -< A jp ∂u p ∂x k ∂u i ∂x k > . (A.21)
The term arising from the non-linearity:

< ∂u i ∂x k ∂ 2 u j u p ∂x p ∂x k > + < ∂u j ∂x k ∂ 2 u i u p ∂x p ∂x k >= < ∂u i ∂x k ∂u j ∂x p ∂u p ∂x k > + < u p ∂u i ∂x k ∂ 2 u j ∂x k ∂x p > + < ∂u j ∂x k ∂u i ∂x p ∂u p ∂x k > + < u p ∂u j ∂x k ∂ 2 u i ∂x k ∂x p > . (A.22) Considering < u p ∂ p (∂ k u i ∂ k u j )
>= 0 by virtue of homogeneity and incompressibility yields

< u p ∂u j ∂x k ∂ 2 u i ∂x k ∂x p >= -< u p ∂u i ∂x k ∂ 2 u j ∂x k ∂x p >, (A.23) which simplifies (A.22) into < ∂u i ∂x k ∂ 2 u j u p ∂x p ∂x k > + < ∂u j ∂x k ∂ 2 u i u p ∂x p ∂x k >=< ∂u i ∂x k ∂u j ∂x p ∂u p ∂x k > + < ∂u j ∂x k ∂u i ∂x p ∂u p ∂x k > . (A.24)
The mean-gradient terms: the terms involving the mean-velocity gradient can be grouped together thanks to (A.5), which yields

-< A pk ∂u i ∂x p ∂u j ∂x k > -< A pk ∂u j ∂x p ∂u i ∂x k > -< A ip ∂u p ∂x k ∂u j ∂x k > -< A jp ∂u p ∂x k ∂u i ∂x k > = -2 < A pk ∂u i ∂x p ∂u j ∂x k > -< A ip ∂u p ∂x k ∂u j ∂x k > -< A jp ∂u p ∂x k ∂u i ∂x k >,
and the terms involving only the mean velocity are zero due to (A.23)

-< U p ∂u j ∂x k ∂ 2 u i ∂x p ∂x k > -< U p ∂u i ∂x k ∂ 2 u j ∂x p ∂x k >= 0.
The pressure terms: there are several steps: expanding < ∂ 3 jkk (pu i ) >= 0 and using <

∂ k (∂ k u i ∂ k p) >= 0, < ∂ k (∂ j u i ∂ k p) >= 0, and < ∂ j (∂ k u i ∂ k p) >= 0, gives < p ∂ 3 u i ∂x j ∂x k ∂x k > + < u i ∂ 3 p ∂x j ∂x k ∂x k >= 0. (A.25)
Then, the sum of < ∂ 2 kk (u i ∂ j p) >= 0 and < ∂ 2 kk (∂ j u i p) >= 0 provides, using (A.25)

< ∂p ∂x k ∂ 2 u i ∂x j ∂x k > + < ∂u i ∂x k ∂ 2 p ∂x j ∂x k >= 0. (A.26)
Finally, summing the previous equation with (i ↔ j) yields the simplification

-< ∂u j ∂x k ∂ 2 p ∂x i ∂x k > -< ∂u i ∂x k ∂ 2 p ∂x j ∂x k >=< ∂p ∂x k ∂s ij ∂x k > . (A.27)
The viscous terms: expanding < ∂ 2 kp (∂ k u j ∂ p u i ) >= 0 and using (A.16) yields

ν < ∂u i ∂x k ∂ 3 u j ∂x p ∂x p ∂x k > +ν < ∂u j ∂x k ∂ 3 u i ∂x p ∂x p ∂x k >= -2ν < ∂ 2 u i ∂x k ∂x k ∂ 2 u j ∂x p ∂x p > (A.28)
A.2.5 Evolution equations of < ξ i ξ j > and < ξ 2 >

The starting point to compute the evolution equation of the scalar covariance tensor < ξ i ξ j > is to derive the evolution equation (4.2) of θ with respect to x i , and to multiply it by ∂ j θ. Finally, the ensemble average is applied, which yields the evolution equation of the scalar covariance tensor < ξ i ξ j > in the homogeneous framework

∂ < ξ i ξ j > ∂t + λ l < ∂u l ∂x i ∂θ ∂x j > + < ∂u l ∂x j ∂θ ∂x i > + < ∂θ ∂x j ∂ 2 θu l ∂x l ∂x i > + < ∂θ ∂x i ∂ 2 θu l ∂x l ∂x j > + < A li ∂θ ∂x l ∂θ ∂x j > + < A lj ∂θ ∂x l ∂θ ∂x i > + < U l ∂θ ∂x j ∂ 2 θ ∂x i ∂x l > + < U l ∂θ ∂x i ∂ 2 θ ∂x j ∂x l > = a < ∂θ ∂x j ∂ 3 θ ∂x i ∂x l ∂x l > + < ∂θ ∂x i ∂ 3 θ ∂x j ∂x l ∂x l > . (A.29)
This equation can be simplified in an analogous manner to what was done for ij . First, < ∂ l (U l ξ i ξ j ) >= 0 which simplifies the two terms in U l . Then, using < ∂ l (u l ξ i ξ j ) >= 0 simplifies the fourth and fifth terms. Finally, the diffusion terms can be grouped remarking that < ∂ 2 ll (ξ i ξ j ) >= 0. This yields .30) This is notably recovered in [START_REF] Gylfason | Effects of axisymmetric strain on a passive scalar field: modelling and experiment[END_REF]. In the end, the evolution equation of < ξ 2 > in homogeneous turbulence reads

∂ < ξ i ξ j > ∂t + λ l < ∂u l ∂x i ∂θ ∂x j > + < ∂u l ∂x j ∂θ ∂x i > + < ∂θ ∂x j ∂u l ∂x i ∂θ ∂x l > + < ∂θ ∂x i ∂u l ∂x j ∂θ ∂x l > + < A li ∂θ ∂x l ∂θ ∂x j > + < A lj ∂θ ∂x l ∂θ ∂x i >= -2a < ∂ 2 θ ∂x j ∂x l ∂ 2 θ ∂x i ∂x l > . (A
∂ < ξ 2 > ∂t + 2λ j < ∂u j ∂x i ∂θ ∂x i > +2 < ∂θ ∂x i ∂u j ∂x i ∂θ ∂x j > +2 < A + ij ∂θ ∂x i ∂θ ∂x j > = -2a < ∂ 2 θ ∂x i ∂x j ∂ 2 θ ∂x i ∂x j > .
(A.31)

A.2.6 Cospectrum in isotropic turbulence with mean scalar gradient

Another quantity which has not been investigated and deserves some interest is the dissipation rate of the scalar flux < u 3 θ > in HITSG. The evolution equation of the derivative scalar variance in HITSG can be obtained from (A.31) by taking A ij = 0 and λ 3 = -Λ. The procedure to derive the evolution equation of < ∂ j u i ∂ j θ > is similar to what was done previously: one needs to derive the evolution equation (4.2) of θ with respect to x j , to multiply it by ∂ j u i , and to sum it with the equation (2.1) of u i derived by x j and multiplied by ∂ j θ. This yields .32) This equation simplifies using < ∂ j (u j ∂ l θ ∂ l u i ) >= 0 and < ∂ l (∂ l u i ∂ 2 jj θ) >= 0, and reads

∂ ∂t < ∂u i ∂x j ∂θ ∂x j > +λ j < ∂u j ∂x l ∂u i ∂x l > +A ij < ∂u j ∂x l ∂θ ∂x l > +A jl < ∂θ ∂x l ∂u i ∂x j > + < ∂θ ∂x j ∂u i ∂x l > + < u j ∂θ ∂x l ∂ 2 u i ∂x j ∂x l > + < u j ∂u i ∂x l ∂ 2 θ ∂x j ∂x l > + < ∂u i ∂x j ∂θ ∂x l ∂u j ∂x l > + < ∂u i ∂x l ∂θ ∂x j ∂u j ∂x l > + < U j ∂θ ∂x l ∂ 2 u i ∂x j ∂x l > + < U l ∂u i ∂x l ∂ 2 θ ∂x j ∂x l > = -< ∂θ ∂x l ∂ 2 p ∂x i ∂x l > +a < ∂u i ∂x l ∂ 3 θ ∂x j ∂x j ∂x l > +ν < ∂θ ∂x l ∂ 3 u i ∂x j ∂x j ∂x l > . (A
∂ ∂t < ∂u i ∂x j ∂θ ∂x j > +λ j < ∂u j ∂x l ∂u i ∂x l > +A ij < ∂u j ∂x l ∂θ ∂x l > +A jl < ∂θ ∂x l ∂u i ∂x j > + < ∂θ ∂x j ∂u i ∂x l > + < ∂u i ∂x j ∂θ ∂x l ∂u j ∂x l > + < ∂u i ∂x l ∂θ ∂x j ∂u j ∂x l >= -< ∂θ ∂x l ∂ 2 p ∂x i ∂x l > -(ν + a) < ∂ 2 u i ∂x l ∂x l ∂ 2 θ ∂x j ∂x j > . (A.33)
In HITSG, this equation further simplifies into

∂ ∂t F ν + a = Λ 3ν -< ∂u 3 ∂x j ∂θ ∂x l s jl > + < ∂p ∂x 3 ∂ 2 θ ∂x l ∂x l > -(ν + a) < ∂ 2 u 3 ∂x l ∂x l ∂ 2 θ ∂x j ∂x j > .
(A.34)

These considerations on the evolution equations of T and F were added in the Journal of Turbulence of 2017.

A.3 Homogeneous isotropic turbulence

The previous equations and tensorial relations are simplified when the homogeneous turbulence is in addition considered isotropic. This notably yields important results for the second and third order moments of the fluctuating velocity and scalar gradients. From now and for clarity, the non-linear transfers S NL(iso) and S T,NL(iso) are called T and T T respectively. Some of the results of this section were used in 1.

A.3.1 Spectral formalism

In HIT, the enstrophy evolution equation is .35) which is (A.20) without the mean velocity gradient. This expression for HIT has been found and developed in [START_REF] Wyngaard | The effect of velocity sensitivity on temperature derivative statistics in isotropic turbulence[END_REF]; [START_REF] Piquet | Turbulent Flows: Models and Physics[END_REF]; [START_REF] Lesieur | Turbulence in fluids[END_REF] notably. The spectral counterpart is .36) Identification between (A.20) and (A.36) provides straightforward equivalences that are detailed hereafter. In an analogous manner, one can write the evolution equation of the derivative scalar variance < ∂ i θ∂ i θ > in the HIT framework

∂ < ω 2 > ∂t = 2 < ω j ω i ∂u i ∂x j > -2ν < ∂ω i ∂x j ∂ω i ∂x j >, (A
1 ν ∂ ∂t = 2 ∞ 0 k 2 T (k, t)dk -4ν ∞ 0 k 4 E(k, t)dk. (A
∂ < ξ 2 > ∂t = -2 < ∂θ ∂x i ∂θ ∂x j ∂u i ∂x j > -2a < ∂ 2 θ ∂x i ∂x j ∂ 2 θ ∂x i ∂x j >, (A.37)
which is (A.31) without the mean velocity and scalar gradients. This equation has been found in [START_REF] Wyngaard | The effect of velocity sensitivity on temperature derivative statistics in isotropic turbulence[END_REF], and has to be identified with its spectral counterpart

1 a ∂ T ∂t = 2 ∞ 0 k 2 T T (k, t)dk -4a ∞ 0 k 4 E T (k, t)dk, (A.38)
Finally, B 1111 =< (∂u 1 /∂x 1 ) 2 >= a 2 /2 and B ijij =< ω 2 >= 15a 2 /2. This calculation is detailed in [START_REF] Pope | Turbulent Flows[END_REF] and reported in [START_REF] Ristorcelli | Passive scalar mixing: Analytic study of time scale ratio, variance, and mix rate[END_REF]; [START_REF] Piquet | Turbulent Flows: Models and Physics[END_REF].

Velocity derivative skewness S: the sixth order tensor B ijkpqr defined in (A.8) is considered to compute the numerator of the derivative skewness. In the isotropic framework, this tensor is the sum of 15 terms, which are products of 3 δ-functions. Nevertheless, with symmetries, some of these terms can be grouped together, which eventually gives Then, the incompressibility B ijkiqr = 0 gives a set of three equations: 3a 1 + 2a 2 + 2a 3 = 0, 3a 2 + 4a 5 = 0 and 3a 3 + 2a 4 + 2a 5 = 0. In addition, equation (A.11) for the homogeneity yields a 1 + 3a 2 + 9a 3 + 10a 4 + 12a 5 = 0, so that

B ijkpqr =
B ijkpqr = a 1 δ ip δ jq δ kr - 4 3 δ ip δ jk δ qr + δ ik δ jq δ pr + δ ij δ kr δ pq - 1 6 δ ip δ jr δ qk + δ ir δ jq δ pk + δ iq δ pj δ kr - 3 4 δ iq δ jr δ pk + δ ir δ jp δ kq + δ ij δ kp δ qr + δ ij δ kq δ pr + δ ik δ jp δ rq + δ ik δ jr δ qp + δ iq δ jk δ pr + δ ir δ jk δ pq . (A.43)
Finally B 111111 =< (∂u 1 /∂x 1 ) 3 >= a 1 and B iijjll = 35a 1 /2. This calculation was done in [START_REF] Pope | Turbulent Flows[END_REF] (without the details regarding the homogeneity simplifications) and is reported in [START_REF] Ristorcelli | Passive scalar mixing: Analytic study of time scale ratio, variance, and mix rate[END_REF]; [START_REF] Wyngaard | The effect of velocity sensitivity on temperature derivative statistics in isotropic turbulence[END_REF]; [START_REF] Piquet | Turbulent Flows: Models and Physics[END_REF]. The final expression of B ijkpqr yields results in agreement with the recent work of [START_REF] Vreman | Statistics of spatial derivatives of velocity and pressure in turbulent channel flow[END_REF].

Kinetic palinstrophy G: the sixth order tensor C ijkpqr defined in (A.13) is now used to compute the numerator of the kinetic palinstrophy. As previously, symmetries allow to group some of the 15 products of δ-functions, which gives C ijkpqr = a 1 δ ij δ kp δ qr + a 2 δ ij δ kq δ pr + δ ij δ kr δ pq + a 3 δ ik δ jp δ qr + δ ip δ jk δ qr + δ iq δ jr δ kp + δ ir δ jq δ kp + a 4 δ iq δ jk δ pr + δ iq δ jp δ kr + δ ir δ jk δ pq + δ ir δ jp δ kq + a 5 δ ik δ jq δ pr + δ ik δ jr δ pq + δ ip δ jq δ kr + δ ip δ jr δ kq Then, the incompressibility C ijipqr = 0 gives two equations: a 1 + 4a 3 + 2a 4 = 0, and a 2 + a 3 + a 4 + 4a 5 = 0. In addition, the homogeneity relations (A.14) and (A.15) provide respectively 6a 1 -6a 2 + 8a 3 -4a 4 -4a 5 = 0 and a 1 + 4a 2 + 8a 3 + 16a 4 + 6a 5 = 0, so that C ijkpqr = a 5 -6δ ij δ kp δ qr -6 δ ij δ kq δ pr + δ ij δ kr δ pq + δ ik δ jp δ qr + δ ip δ jk δ qr + δ iq δ jr δ kp + δ ir δ jq δ kp + δ iq δ jk δ pr + δ iq δ jp δ kr + δ ir δ jk δ pq + δ ir δ jp δ kq + δ ik δ jq δ pr + δ ik δ jr δ pq + δ ip δ jq δ kr + δ ip δ jr δ kq .

(A.44)

Finally C 111111 =< (∂ 2 u 1 /∂x 2 1 ) 2 >= -6a 5 and C iijkjk = -210a 5 . This result was used (and misreported) in [START_REF] Ristorcelli | Passive scalar mixing: Analytic study of time scale ratio, variance, and mix rate[END_REF] without any details. The final expression of C ijkpqr yields results in agreement with the recent work of [START_REF] Vreman | Statistics of spatial derivatives of velocity and pressure in turbulent channel flow[END_REF].

Mixed-derivative skewness S T : the tensor .45) is used to compute the numerator of the mixed-derivative skewness. Similarly, B T ijkl can be expressed as

B T ijkl =< ∂θ ∂x i ∂θ ∂x j ∂u k ∂x l > (A
B T ijkl = a 1 δ ij δ kl + a 2 δ ik δ jl + a 3 δ il δ jk ,
and symmetry B T ijkl = B T jikl directly yields a 2 = a 3 . Then, with incompressibility B T ijll = 0, one has 2a 2 = -3a 1 . Finally, B T 1111 =< (∂θ/∂x 1 ) 2 (∂u 1 /∂x 1 ) >= 4a 2 /3 and B T ijij = 10a 2 . This relation was used in [START_REF] Wyngaard | The effect of velocity sensitivity on temperature derivative statistics in isotropic turbulence[END_REF] without any details.

Scalar palinstrophy G T (t): the tensor .46) is introduced to compute the numerator of the scalar palinstrophy. C T ijkl can be expressed as

C T ijkl =< ∂ 2 θ ∂x i ∂x j ∂ 2 θ ∂x k ∂x l > (A
C T ijkl = a 1 δ ij δ kl + a 2 δ ik δ jl + a 3 δ il δ jk ,
and symmetry directly yields a 2 = a 3 . Then, with homogeneity, < ∂ 2 ij (∂ i θ ∂ j θ) >= 0 and < ∂ 2

ii (∂ j θ ∂ j θ) >= 0 which provides C T iijj = C T ijij and thus a 1 = a 2 . Finally, C T 1111 =< (∂ 2 θ/∂x 2 1 ) 2 >= 3a 1 and C T ijij = 15a 1 . This result was used in [START_REF] Ristorcelli | Passive scalar mixing: Analytic study of time scale ratio, variance, and mix rate[END_REF]; [START_REF] Wyngaard | The effect of velocity sensitivity on temperature derivative statistics in isotropic turbulence[END_REF] without any details.

A.3.3 Results for the velocity field

Using the results for B ijkpqr and C ijkpqr , along with (A.11), (A.14) and (A.15), gives

< ω i ω i > =< ∂u i ∂x j ∂u i ∂x j > -< ∂u i ∂x j ∂u j ∂x i >= 15 -0 < ∂u 1 ∂x 1 2 >= 15 < ∂u 1 ∂x 1 2 >, (A.47) < ∂ω i ∂x j ∂ω i ∂x j > =< ∂ 2 u i ∂x j ∂x k ∂ 2 u i ∂x j ∂x k > -< ∂ 2 u i ∂x j ∂x k ∂ 2 u j ∂x i ∂x k >= 35 < ∂ 2 u 1 ∂x 2 1 2 >, (A.48) < ω i ω j ∂u i ∂x j > =< ∂u i ∂x j ∂u l ∂x i ∂u j ∂x l > -< ∂u i ∂x j ∂u l ∂x i ∂u l ∂x j > -< ∂u i ∂x j ∂u i ∂x l ∂u j ∂x l > + < ∂u i ∂x j ∂u i ∂x l ∂u l ∂x j > = 0 - 35 2 - 35 2 + 35 2 < ∂u 1 ∂x 1 3 >= - 35 2 < ∂u 1 ∂x 1 3 > . (A.49)
The following identification process is more or less done in [START_REF] Piquet | Turbulent Flows: Models and Physics[END_REF]. The present results are in agreement with [START_REF] Kerr | Higher-order derivative correlations and the alignment of small-scale structures in isotropic numerical turbulence[END_REF]. Identifying (A.20) and (A.36) yields

< ∂u 1 ∂x 1 3 >= - 2 35 ∞ 0 k 2 T (k, t)dk, < ∂ 2 u 1 ∂x 2 1 2 >= 2 35 ∞ 0 k 4 E(k, t)dk. (A.50)
Using the previous calculations gives

< ∂u 1 ∂x 1 3 >= S 15ν 3/2 , < ∂ 2 u 1 ∂x 2 1 2 > 2K 3 = G 15ν 2 ,
where K = 3 < u2 1 > /2 is the kinetic energy. This results in

S(t) = - 3 √ 30 14 ∞ 0 k 2 T (k, t)dk ∞ 0 k 2 E(k, t)dk 3/2 , G(t) = 30ν 7 K ∞ 0 k 4 E(k, t)dk ∞ 0 k 2 E(k, t)dk . (A.51)
Finally, one can write the spectral evolution equation of the kinetic energy dissipation rate as

∂ ∂t = - 7 3 √ 15 S(t) Re T + 7 15 G(t) 2 K = - 7 15 1 2 S(t)Re λ + G(t) 2 K , (A.52)
where the turbulent (or integral) Reynolds number is Re T was defined in (1.7). The normalized palinstrophy G can be interpreted as the dissipation of enstrophy, and interestingly can be linked to the dissipation skewness S of [START_REF] Kerr | Higher-order derivative correlations and the alignment of small-scale structures in isotropic numerical turbulence[END_REF] through G = Re λ S /2 so that (A.52) becomes

∂ ∂t = - 7 30 Re λ S(t) + S (t) 2 K .
(A.53)

A.3.4 Results for the passive scalar field

Using the results for B T ijkl and C T ijkl gives

T = a < ∂θ ∂x i ∂θ ∂x i >= 3a < ∂θ ∂x 1 2 >, < ∂ 2 θ ∂x i ∂x j ∂ 2 θ ∂x i ∂x j >= 5 < ∂ 2 θ ∂x 1 ∂x 1 2 >, < ∂θ ∂x i ∂θ ∂x j ∂u i ∂x j >= 15 2 < ∂θ ∂x 1 ∂θ ∂x 1 ∂u 1 ∂x 1 > .
The following identification process is done in [START_REF] Kerr | Higher-order derivative correlations and the alignment of small-scale structures in isotropic numerical turbulence[END_REF]. Identifying (A.37) and (A.38) yields

< ∂u 1 ∂x 1 ∂θ ∂x 1 2 >= - 2 15 ∞ 0 k 2 T T (k, t)dk, < ∂ 2 θ ∂x 2 1 2 >= 2 5 ∞ 0 k 4 E T (k, t)dk. (A.54)
Using the previous calculations gives

< ∂u 1 ∂x 1 ∂θ ∂x 1 2 >= S T 15ν T 3a , < ∂ 2 θ ∂x 2 1 2 > 2K T = G T T 3a 2 ,
where K T =< θ 2 > /2 is the scalar variance1 . This results in

S T (t) = - 3 10 ∞ 0 k 2 T T (k, t)dk ∞ 0 k 2 E(k, t)dk ∞ 0 k 2 E T (k, t)dk , G T (t) = 18a 5 K T T ∞ 0 k 4 E T (k, t)dk ∞ 0 k 2 E T (k, t)dk .
(A.55) There is an error in [START_REF] Antonia | Similarity of decaying isotropic turbulence with a passive scalar[END_REF] regarding the numerical factor of S T . The present result is in agreement with [START_REF] Kerr | Higher-order derivative correlations and the alignment of small-scale structures in isotropic numerical turbulence[END_REF]. Finally, one can write the spectral evolution equation of the scalar variance dissipation rate, with r the kinetic to scalar time scales ratio r = (K T )/(K T ).

∂ T ∂t = - 5 3 S T (t) Re T + r 5 9 G T (t) T K = - 1 2 S T (t)Re λ + r 5 9 G T (t) T K .
(A.56)

A.4 Structure functions and auto-correlations

This section is dedicated to the investigation of structure functions in HIT. So far, the spectral formalism was preferred, for convenience with EDQNM. Nevertheless, correlation and structure functions which depend on r, the separation vector between two points located in x and x so that r = xx, are another fundamental aspect of the turbulence theory. For this reason, and to extend a bit the range if this thesis, basic results are recovered: notably, the von [START_REF] Von Karman | On the statistical theory of isotropic turbulence[END_REF] equation is derived, and then the famous 4/5 th law of Kolmogorov (1941a) is recovered. In what follows, the prime refers to a quantity expressed in x , the time dependence is sometimes omitted for clarity, and u refers to the rms of the fluctuating velocity. Part of the results of this appendix, notably the 4/3 rd laws for the structure functions inertial scalings , were included in the Journal of Fluid Mechanics for helicity.

A.4.1 Second-order longitudinal correlation and structure function

The Reynolds stress tensor, defined in (2.2), can be written

R ij (r, t) = u 2 f (r, t) -g(r, t) r 2 r i r j + u 2 g(r, t)δ ij , (A.57)
where f and g are the longitudinal and transverse correlations functions, linked to R ij (r), if one chooses r along the x 1 direction for instance [START_REF] Pope | Turbulent Flows[END_REF], as

f (r, t) = R 11 (r, t) u 2 , g(r, t) = R 22 (r, t) u 2 = R 33 (r, t) u 2 . (A.58)
The expression of R ij being symmetric, f and g are even functions of r, so that a Taylor expansion yields A.59) and similarly for g. The second and fourth derivatives of f for r → 0 can be linked to quantities previously investigated:

f (r) = 1 + f (0) r 2 2 + f (iv) (0) r 4 4! + . . . , (
-u 2 f (0) = -lim r→0 ∂ 2 ∂r 2 < u 1 u 1 >= -< u 1 ∂ 2 11 u 1 >= -< ∂ 1 (u 1 ∂ 1 u 1 ) > =0 -< (∂ 1 u 1 ) 2 > =< (∂ 1 u 1 ) 2 >,
where ∂ 1 = ∂/∂x 1 . This gives, following the results of section A.3, f (0) = -/(15νu 2 ). Then, for f (iv) (0), one needs to use < ∂ 1 (u 1 ∂ 3 111 u 1 ) >= 0, which gives

u 2 f (iv) (0) = lim r→0 ∂ 4 ∂r 4 < u 1 u 1 >=< u 1 ∂ 4 1111 u 1 >= < ∂ 2 11 (u 1 ∂ 2 11 u 1 ) > =0 -< (∂ 2 11 u 1 ) 2 > -2 < ∂ 1 u 1 ∂ 3 111 u 1 > =-<u 1 ∂ 4 1111 u 1 >
. and the value D LL (∞) = 2u 2 at large scales as well. Furthermore, the constant of D LL is found to be C 2 = 1.73 in Fig. A.1b, which is a bit less than in [START_REF] Bos | Reynolds number effect on the velocity increment skewness in isotropic turbulence[END_REF]. A.4.2 Third-order longitudinal correlation and structure function

The two-point third-order correlation reads [START_REF] Von Karman | On the statistical theory of isotropic turbulence[END_REF])

< u i u j u k >= -< u i u j u k >= T ijk = r i r j r k k(r) -h(r) -2q(r) r 3 +δ ij r k h(r) r + q(r) r (δ ik r j +δ jk r i ),
(A.68) and the incompressibility condition ∂T ijk /∂r k = 0 yields k(r) = -2h(r), q(r) = -h(r) -r 2 ∂h ∂r ,

(A.69) so that T ijk can only be expressed as a function of k or h. Of particular interest, k can be expanded, for r → 0, as A.70) and k (0) is linked to the mixed-derivative skewness S(t) analyzed earlier, using

k(r) = k (0) r 3 6 + . . . , (
< ∂ 1 (u 1 ∂ 1 u 1 ∂ 1 u 1 ) >= 0: k (0) = lim r→0 ∂ 3 ∂r 3 < u 1 u 1 u 1 >=< u 1 u 1 ∂ 3 111 u 1 >= < ∂ 1 (u 2 1 ∂ 2 11 u 1 ) > =0 -2 < u 1 ∂ 1 u 1 ∂ 2 11 u 1 > =-<(∂ 1 u 1 ) 3 > , so that k (0) =< ∂u 1 ∂x 1 3 >= S(t) 15ν 3/2 . (A.71)
Then, the two-point third-order longitudinal structure function D LLL can be linked to the third-order correlation k through Here are some details:

D LLL (r, t) =< δu 3 L >= 6 r i r j r k r 3 T ijk (r) = 6k(r
1 u 2 ∂R ik ∂t = r i r k r 2 ∂f ∂t - ∂g ∂t + δ ik ∂g ∂t , 1 u 2 ∂ 2 R ik ∂r j ∂r j = r i r k r 2 -6 r 2 (f -g) + 2 r ∂f ∂r - ∂g ∂r + ∂ 2 f ∂r 2 - ∂ 2 g ∂r 2 + δ ik 2 r 2 (f -g) + 2 r ∂g ∂r + ∂ 2 g ∂r 2 , ∂ ∂r j (T ijk + T kji ) = r i r k r 2 -4 r h + 4 ∂h ∂r + r ∂ 2 h ∂r 2 + δ ik -4 r h -6 ∂h ∂r -r ∂ 2 h ∂r 2
.

The equation with δ ik allows to determine ∂ t g, and further using (A.62) allows to recover the Kármán-Howarth equation

∂(u 2 f ) ∂t - 1 r 4 ∂r 4 k ∂r = 2ν u 2 r 4 ∂ ∂r r 4 ∂f ∂r .
(A.76) From this, it is possible to express the evolution equation of R(r), defined in (A.64), in agreement with Davidson ( 2010)

∂R ∂t - 1 2r 2 ∂ ∂r 1 r ∂ ∂r (r 4 k) = 2ν 1 r 2 ∂ ∂r r 2 ∂R ∂r .
(A. Even at Reynolds numbers such as 2.10 4 , the theoretical expectation 4/5 is not rigorously reached, as in [START_REF] Bos | Reynolds number effect on the velocity increment skewness in isotropic turbulence[END_REF] for freely decaying turbulence. It is noteworthy to remark that unlike D LL , the integration for D LLL requires quadruple precision, otherwise one gets strong oscillations as in [START_REF] Tchoufag | Spectral approach to finite reynolds number effects on kolmogorov's 4/5 law in isotropic turbulence[END_REF]. Furthermore, the small scale r 3 scaling for both < δu L δq 2 > and < δu 3 L > is recovered in Fig. A.2a, followed by the linear dependence in r. Invariants: Saffman and Batchelor turbulence are now briefly discussed in terms of invariants, as done in [START_REF] Davidson | On the decay of saffman turbulence subject to rotation, stratification or an imposed magnetic field[END_REF]. The expansion of the kinetic energy spectrum for very low wavenumbers yields

E(k → 0) = L k 2 4π 2 + I k 4 24π 2 , (A.81)
where L and I are the Saffman and Loitsiansky integrals respectively. For the Saffman integral L, associated to the conservation of linear momentum, one has

L = < u i u i > dr = 4π ∞ 0 2r 2 R(r)dr = 4π[r 3 u 2 f ] ∞ , (A.82)
meaning that when L = 0 initially (i.e. E(k → 0) ∼ k 2 ), the longitudinal function should decrease as f ∼ r -3 when r → ∞. Furthermore, it was shown by [START_REF] Saffman | The large-scale structure of homogeneous turbulence[END_REF] that r 2 Rdr, and so L, is an invariant of motion in freely decaying turbulence: this can be shown by integration of (A.77) multiplied by r 2 , 1 8π

dL dt - 1 2r ∂ ∂r (r 4 k) = 2νr 2 ∂R ∂r ,
and further neglecting the viscous term for high Reynolds numbers, so that

dL dt = 4π 1 r ∂ ∂r (r 4 k) . (A.83)
Since k ∼ r -4 , L is independent of time and is consequently an invariant of motion. For the Loitsiansky integral, linked to the conservation of angular momentum,

I = -r 2 < u i u i > dr = -4π ∞ 0 2r 4 R(r)dr = -4π ∞ 0 r 2 ∂ ∂r (r 3 u 2 f )dr ∼ -4π[r 5 u 2 f ],
(A.84) meaning that when L = 0 initially (i.e. E(k → 0) ∼ k 4 ), the longitudinal function should decrease as f ∼ r -5 when r → ∞.

Kármán-Howarth 4/3 rd equation: the Kármán-Howarth-Kolmogorov equation (A.78) can be written differently: instead of considering the longitudinal second-order structure function δu 2 L , the emphasis is put on the kinetic energy increment δq 2 = δu i δu i . The method to derive this equation is detailed in [START_REF] Antonia | Analogy between predictions of kolmogorov and yaglom[END_REF], and the procedure is very similar to the 4/3 rd law derived in [START_REF] Yaglom | On the local structure of a temperature field in a turbulent flow[END_REF] for passive scalar structure function, developed in the next section. For clarity, ∂ j = ∂/∂x j and ∂ j = ∂/∂x j . Subtracting the evolution equation of u i to the one of u i , one gets

∂ t δu i + δu j ∂ ∂r j (δu i ) + u j (∂ j + ∂ j )(δu i ) = -(∂ i + ∂ i )(δp) + ν(∂ jj + ∂ jj )(δu i ), (A.85)
where

δu j ∂ r j δu i + u j (∂ j + ∂ j )(δu i ) = ∂ j (u i u j ) -∂ j (u i u j ).
Multiplying the previous equation by 2δu i and using ensemble average yields

∂ ∂t < δq 2 > + ∂ ∂r j < δu j δq 2 >= 2ν ∂ 2 ∂r j ∂r j
< δq 2 > -4 ν < ∂δu i ∂r j

∂δu i ∂r j > = .

(A.86)

Both the u j (∂ j + ∂ j ) and pressure terms are zero because ∂ j = ∂ r j = -∂ j . Further neglecting the time derivative and writing -4 = -4/3∂ r j ( r j ), one gets A.87) where the result of von [START_REF] Von Karman | On the statistical theory of isotropic turbulence[END_REF] has been used: 2f /r + f = 0 ↔ f = 0, combined with ∂ 2 /(∂r j ∂r j ) = (2/r∂ r + ∂ 2 rr ). At high Reynolds numbers in the inertial range, this eventually gives after the use of the divergence theorem

< δu j δq 2 >= 2ν ∂ ∂r j < δq 2 > - 4 3 r j , (
< δu L δq 2 > (r, t) = - 4 3 r. (A.88)
It is possible to make a link between (A.80) and (A.88): using < δq 2 >= 4K -4R and further identifying the evolution equations of < δq 2 > and R yields 3r As remarked before, even at Re λ = 2.10 4 , the 4/3 is not exactly recovered. Interestingly, double precision is enough to compute < δu L δq 2 >, unlike D LLL .

A.4.4 Yaglom and Corrsin equations

The [START_REF] Yaglom | On the local structure of a temperature field in a turbulent flow[END_REF] equation can be obtained in a completely analogous manner, by defining the scalar increment, and the scalar two-points second and third order structure functions

δθ = θ -θ, D TT =< δθδθ >, D LTT =< δu L δθδθ > . (A.91)
Interestingly, the mixed derivative skewness can be expressed as .92) and simple dimensional arguments give

S uθ (t) = lim r→0 D LTT D TT D 1/2 LL , ( A
D TT (r, t) =      2θ 2 for r > L, C T 2 r 2/3 T -1/3 for L > r η, r 2 T /(3a) for r ∼ η, (A.93)
and this is illustrated in Fig. A.3, where the three different scalings are recovered. The constant C T 2 = 1.88 is slightly higher than C 2 , qualitatively in agreement with [START_REF] Watanabe | Statistics of a passive scalar in homogeneous turbulence[END_REF], despite the much highers constant in the latter reference. The present C T 2 is in reasonable agreement with [START_REF] Yeung | Schmidt number effects on turbulent transport with uniform mean scalar gradient[END_REF]. A procedure similar to the one done for the equation of < δq 2 > yields

∂ ∂t < δθδθ > + ∂ ∂r j
< δu j δθδθ >= 2a ∂ 2 ∂r j ∂r j

< δθδθ > -4a < ∂δθ ∂r j

∂δθ ∂r j > =-4/3∂r j ( T r j )

.

(A.94)

Neglecting the time derivative and using the previous result of von [START_REF] Von Karman | On the statistical theory of isotropic turbulence[END_REF] yields the Yaglom equation This result can also be obtained in a way much more similar to von [START_REF] Von Karman | On the statistical theory of isotropic turbulence[END_REF], as detailed in Danaila et al. (1999a). First, one starts with the Corrsin equation (Corrsin, 1951a) .97) The links between correlations and structure functions are

< δu j δθδθ >= 2a ∂ ∂r j < δθδθ > - 4 3 T r j , (
∂ ∂t < θθ >= 2 2 r + ∂ ∂r < u L θθ > +a ∂ ∂r < θθ > , (A
R T (r, t) =< θθ >=< θ 2 > -D TT /2, D LTT = 4 < u L θθ >, (A.98) and r j < u L θθ > /r =< u j θθ >. Let's point out that -R T (0) = -lim r→0 ∂ 2 ∂r 2 < θθ >= -< θ∂ 2 11 θ >=< (∂ 1 θ) 2 >= T 3a .
Then, since r 2 (2/r + ∂ r )[f ] = ∂ r (r 2 f ), (A.97) can be simplified: multiplying by r 2 , integrating, and then dividing by r 2 , with

∂ t θ 2 = -2 T yields 1 r 2 r 0 s 2 ∂D TT ∂t ds = -D LTT + 2a ∂D TT ∂r - 4 3 r T . (A.99)
Finally, the links between structure functions and scalar variance spectra, even though less documented, are simply

< θθ > (r, t) = ∞ 0 E T (k, t) sin(kr) kr dk, D TT (r, t) = 2 ∞ 0 E T (k, t) 1 - sin(kr) kr dk.
(A.100) As for the scalar non-linear transfer, it can be found starting from the Corrsin equation (A.97) and identifying with (1.8)

2 r 2 ∂ ∂r (r 2 < u L θθ >) = ∞ 0 T T (k) sin(kr) kr dk,
so that eventually

D LTT (r, t) = 2 ∞ 0 T T (k,
= 2a < (∂ i θ) 2 >).
Furthermore, the linear scaling in r in the inertial range is recovered, along with the r 3 scaling at small scales.

B.1 Non-local fluxes

It is recalled that

S NL(iso) (k, t) = ∆ k θ kpq (xy + z 3 ) q E(q) k 2 E(p) -p 2 E(k) dpdq = ∆ k S(k, p, q)dpdq, (B.2) S NL(iso) T (k, t) = ∆ k θ T kpq (xy + z) q E(q) k 2 E T (p) -p 2 E T (k) dpdq = ∆ k S T (k, p, q)dpdq. (B.
3)

The non-local transfers from large scales to very small ones are such that q k ∼ p. They are referred to as T + (k, t) and T + T (k, t), for the velocity and scalar fields respectively. The non-local transfers in the opposite direction are such that k p ∼ q and are referred to as T -(k, t) and T - T (k, t). For convenience, these non-local transfers are computed through their associated energy-conservative non-local fluxes, so that

Π Non-Local (T ) (k, t) = ∞ k T Non-Local (T ) (k , t)dk , (B.4) T ± (T ) (k, t) = - ∂Π ± (T ) (k, t) ∂k = - ∂ ∂k Π + (T ) (k, t) -Π - (T ) (k, t) , (B.5)
where the derivation is done numerically. Therefore, the non-local fluxes read [START_REF] Lesieur | Amortissement autosimilaire d'une turbulence á grand nombre de reynolds[END_REF])

Π + (T ) (k, t) = 2 ak 0 dq k+q k dk k k -q S (T ) (k , p, q)dp, Π - (T ) (k, t) = 2 k 0 dk ∞ sup(k,k /a) dp p p-k S (T ) (k , p, q)dq. (B.6)
The region of non-local transfers is the grey part of the rectangle ∆ k delimited by q = p + k and q = p -k in Fig. B.1. These regions being symmetric with respect to q = p, this justifies the factor 2. The condition (B.1) yields Π + (T ) :

q < ap q < ak,

Π - (T ) : p > k/a q > k/a. Since k < k for Π - (T )
, it could happen that k > k /a. This is the reason why sup(k, k /a) is chosen in the bound. Then, it has been shown [START_REF] Lesieur | Amortissement autosimilaire d'une turbulence á grand nombre de reynolds[END_REF] that

S (T ) (k, p, q) + S (T ) (p, q, k) + S (T ) (q, k, p) = 0. (B.7)
Because of (p, q) symmetry, one has S (T ) (p, q, k) = 0. This allows to compute the non-local flux in a symmetric way

S (T ) (k, p, q) = 1 2 S (T ) (k, p, q) -S (T ) (q, k, p) . (B.8)
The final expressions are gathered in [START_REF] Lesieur | Turbulence in fluids[END_REF], and in what follows, these expressions are recovered. Before that, let's discuss the conservation property (B.7), which relies on the (p, q) symmetry: starting from (2.15) for T NL ij , one has

T NL ii (k) = k n S nii (k, p)d 3 p + k m S * iim (k, p)d 3 p = s(k, p, q)d 3 p, with s(k, p, q) = k n < û n ûi û i > +k m < û m ûi û i > .
(B.9) Further using k n + q n = -p n and incompressibility so that p n u n = 0 yields s(k, p, q) + s(q, k, p) + s(p, q, k) = 0. (B.10)

A similar property can be written for the scalar field. Using (p, q) symmetry, one can write the term under the integral in (4.7) as B.11) so that for the same reasons s T (k, p, q) + s T (q, k, p) + s T (p, q, k) = 0. (B.12)

s T (k, p, q) = 2k j < û j θ θ > +k j < û j θ θ >, (

B.2 Expansions for q k ∼ p

The small parameter of the expansion is ζ = q/k. Firstly, p is expanded as

p = k 1 -yζ + 1 2 (1 -y 2 )ζ 2 , and p -1 = k -1 1 + yζ + 1 2 (3y 2 -1)ζ 2 .
Then, geometrical relations yield

z = 1 - 1 2 (1 -y 2 )ζ 2 , and x = -k 2 + p 2 + q 2 2pq = -y + (1 -y 2 )ζ + 3 2 y(1 -y 2 )ζ 2 .
This gives

xy + z 3 = (1 -y 2 ) 1 + yζ - 3 2 (1 -y 2 )ζ 2 , xy + z = (1 -y 2 ) 1 + yζ + 1 2 (3y 2 -1)ζ 2 , xz + y 3 = (1 -y 2 ) -y + ζ + 2yζ 2 , xz + y = (1 -y 2 )ζ(1 + 2yζ).
Then, a Taylor expansion provides

E (T ) (p) = E (T ) (k) -qy ∂E (T ) ∂k + 1 2 ζ 2 (1 -y 2 )k ∂E (T ) ∂k + y 2 k 2 ∂ 2 E (T ) ∂k 2 .
The method is the following one: the different quantities are expressed as a function of y (the cosine of the angle in front of p) and the small parameter ζ = q/k. Then, the p-integration is done by using y and the variable

y = k 2 -k 2 + q 2 2k q so that k k -p S (T ) (k , p, q)dp = 1 y k q p S (T ) (k , p, q)dy.
Finally, the last integration, assuming that θ

k 2 E(p) -p 2 E(k) = k 2 yζ 2E(k) -k ∂E ∂k + ζ 2 -E(k) + 1 -y 2 2 k ∂E ∂k + y 2 2 k 2 ∂ 2 E ∂k 2 , kq p (xy + z 3 ) q = (1 -y 2 ) 1 + 2yζ + 2(2y 2 -1)ζ 2 , kq p S(k, p, q) = θ kkq (1 -y 2 )E(q) ykq 2E(k) -k ∂E ∂k + q 2 (4y 2 -1)E(k) + 1 2 (1 -5y 2 )k ∂E ∂k + y 2 2 k 2 ∂ 2 E ∂k 2 .
At the lowest order in q, only the first rhs term remains. Secondly, kq p

(xz + y 3 ) p = ζ(1 -y 2 )(-y + (1 -2y 2 )ζ + y(5 -4y 2 )ζ 2 ), kq(xz + y 3 ) p 2 q 2 E(p)E(k) = (1 -y 2 )E(k) -yq 2 ζE(k) + q 2 ζ 2 (1 -2y 2 )E(k) + y 2 k ∂E ∂k , kq(xz + y 3 ) p 2 k 2 E(p)E(q) = (1 -y 2 )E(q) -kqy E(k) -qy ∂E ∂k + q 2 (1 -2y 2 )E(k) + 1 2 qy(5y 2 -3) ∂E ∂k - 1 2 qky 3 ∂ 2 E ∂k 2 .
Using equation (B.8), this yields

2 kq p S(k, p, q) = θ kkq (1 -y 2 ) E(k) yq 2 ζE(k) -q 2 ζ 2 (1 -2y 2 )E(k) + y 2 k ∂E ∂k + E(q) kqy E(k) -k ∂E ∂k + q 2 2y 2 E(k) + 1 2 (1 -3y 2 )k ∂E ∂k + 1 2 y 2 k 2 ∂ 2 E ∂k 2 .
At the lowest order

2 kq p S(k, p, q) = θ kkq y(1 -y 2 ) E(k) 2 q 2 ζ + kqE(q) E(k) -k ∂E ∂k .
Hence, the direct non-local kinetic flux is

Π + (k, t) = 2 15 k E(k) -k ∂E ∂k ak 0 θ kkq q 2 E(q)dq + 2 15 E(k) 2 k ak 0 θ kkq q 4 dq. (B.14)
Computation of Π + T (k, t): the calculation is very similar to the previous one. One has kq p

(xy + z) q = (1 -y 2 ) 1 + 2yζ + (4y 2 -1)ζ 2 , so that kq p (xy + z) q E(q)(k 2 E T (p) -p 2 E T (k)) = (1 -y 2 ) kqy 2E T (k) -k ∂E T ∂k + q 2 (4y 2 -1)E T (k) + 1 -5y 2 2 k ∂E T ∂k + 1 2 y 2 k 2 ∂ 2 E T ∂k 2 . (B.15) Then kq p (xz + y) p E(p)(q 2 E T (k) -k 2 E T (q)) = (1 -y 2 )(1 + 4yζ)q 2 E(k) E T (k)ζ 2 -E T (q) .
Using (B.8), one obtains the direct non-local scalar flux at the lowest order k

Π + T (k, t) = 2 15 k 2E T (k) -k ∂E T ∂k ak 0 θ T kkq q 2 E(q)dq + 1 4 E(k) ak 0 θ T kkq q 3 E T (q)dq - 1 4 E(k)E T (k) k 2
Π + (T ) (k, t) Π + (k, t) Π + T (k, t) k η k B (a) -2 0 2 4 6 -4 -2 0 2 4 log k log E/ log k log E/ log k log E T / log k k η -1 k B (b)

B.3 Expansions for k p ∼ q

The small parameter of the expansion is ζ = k/p. Firstly, q is expanded as

q = p 1 -zζ + 1 2 (1 -z 2 )ζ 2 , and q -1 = p -1 1 + zζ + 1 2 (3z 2 -1)ζ 2 .
Then, geometrical relations yield

x = 1 - 1 2 (1 -z 2 )ζ 2 , and y = k 2 -p 2 + q 2 2kq = -z + (1 -z 2 )ζ - 3 2 z(z 2 -1)ζ 2 .
This gives

xy + z 3 = (1 -z 2 ) -z + ζ + 2zζ 2 , xy + z = (1 -z 2 )ζ(1 + 2zζ), xz + y 3 = z(1 -z 2 ) 1 + 3zζ + 1 2 (15z 2 -7)ζ 2 , xz + y = (1 -z 2 )ζ (1 + zζ) .
Then, a Taylor expansion provides

E (T ) (q) = E (T ) (p) -kz ∂E (T ) ∂p + 1 2 ζ 2 (1 -z 2 )p ∂E (T ) ∂p + z 2 p 2 ∂ 2 E (T ) ∂p 2 .
The method is almost the same: the different quantities are expressed as a function of z (the cosine of the angle in front of q) and the small parameter ζ = k/p. Then, the q-integration is done by using z, which simplifies, at the first order, into p p-k S (T ) (k , p, q)dq = 1 0 k p q S (T ) (k , p, q)dz.

As previously, it is assumed that θ

(T ) k pp θ (T )
kpp .

Computation of Π -(k, t): firstly, kp q

(xy + z 3 ) q = ζ(1 -z 2 )(-z + (1 -2z 2 )ζ + z(5 -4z 2 )ζ 2 ), kp q (xz + y 3 ) p = ζz(1 -z 2 )(1 + 4zζ + 8(3z 2 -1)ζ 2 ).
In all the following calculations, the lowest order always simplifies. Hence

2 kp q S(k, p, q) = θ kpp (1 -z 2 ) (1 + 2z 2 )k 2 ζ 2 E(p) 2 -k 2 E(k)E(p) -z 2 k 2 E(k)p ∂E ∂p .
Then, using

1 0 (1 -z 2 )(1 + 2z 2 )dz = 14 15 , 1 0 (1 -z 2 )dz = 2 3 , 1 0 z 2 (1 -z 2 )dz = 2 15 ,
one finds the inverse non-local kinetic flux

Π -(k, t) = 14 15 k 0 k 4 ∞ sup(k,k /a) θ k pp E(p) 2 p 2 dp dk - 2 15 k 0 k 2 E(k ) ∞ sup(k,k /a) θ k pp 5E(p) + p ∂E ∂p dp dk . (B.17) Computation of Π - T (k, t): firstly, kp q (xy + z) q = ζ 2 (1 -z 2 )(1 + 4zζ), kp q (xz + y) p = ζ 2 (1 -z 2 )(1 + 2zζ).
This directly yields the inverse non-local scalar flux

Π - T (k, t) = - 4 3 k 0 k 2 E T (k ) ∞ sup(k,k /a) θ T k pp E(p)dp dk + 4 3 k 0 k 4 ∞ sup(k,k /a) θ T k pp E T (p)E(p) p 2 dp dk .

B.4 Applications of the isotropic non-local transfers

In this section, two brief applications of the non-local transfers, in addition to the classical case for weakly diffusive scalars, are presented. iso) ](k, t) referring to the difference between a simulation with the non-local expansions and the same simulation without them. This quantity ∆[kS NL(iso) ](k, t) decreases in intensity with a better spatial resolution. In addition, it appears that the non-local expansions are more important around the integral and Kolmogorov wavenumbers than in the inertial range, which is expected. Homogeneous Shear Flows: it is revealed in Fig. B.4 that the isotropic non-local transfers T ± (k, t) are negligible with respect to the local ones for high Reynolds numbers shear flows (Π - is very small compared to Π + ), where Π NL(iso) is the flux of S NL(iso) , following

Π NL(iso) (k, t) = - k 0 S NL(iso) (k, t)dk.
This justifies a posteriori why the non-local developments are used only for HIT and more specifically for a scalar field with P r 1. (k, t) In addition to the usual kinetic and scalar isotropic non-local fluxes presented in this appendix, non-local fluxes were also computed for the non-linear transfers of F i , and for the non-linear directional and polarization kinetic transfers. These three contributions revealed to be completely negligible with respect to the local ones, and much less intense than the isotropic non-local contributions.

Π NL(iso) Π NL(iso) + Π ± Π NL(tot) 33 Π NL(tot) 33 + 2Π ± /3 k L k η (b)
where P imn (k) is the Kraichnan operator 2P imn = k m P in (k) + k n P im (k) and

M ij (k) = (δ in -2α i α n )A nj +P in nlj Ω l
= ∂ t -A ij k i ∂ j so that d Rij dt + 2νk 2 Rij (k) + M in (k) Rnj (k) + M jn (k) Rni (k) = T NL ij (k). (C.6)
C.1.2 Generalized Lin equations for E and Z

The helical modes N j (k) = e

(2)

j (k) -ie (1) j (k). (C.7)
are linked to the Craya-Herring frame (e (1) , e (2) , e (3) ) through e

= k i k , e (3) i
= ijn e

(3) j e (1) n , e

= ijl k j n l |k × n| = ijl k j n l k ⊥ , (C.8) (1) i
where n is a fixed reference vector. By definition of these helical modes, one has N j k j = 0, N j N j = 0, N j N * j = 2 and N i N * j = P ij -i ijn α n . The generalized E-Lin equation is obtained by taking half the trace of (2.13) and replacing Rij by the E -Z decomposition (2.30), so that

dE dt + 2νk 2 E + M in EP in + (Z(k, t)N i (k)N j (k)) = T NL ii 2 . (C.9)
Using the incompressibility condition A ii = 0, the E-Lin equation presented in (2.31) is recovered. One can remark that rotation does not intervene directly in the evolution equation of E. The calculation for the Z-Lin equation is more complicated: (2.13) is firstly multiplied by N * i (k)N * j (k)/2. One has to compute Rnj N * j and M in N * i which can be done by using the previous relations on N j and the fact that ∀z ∈ C, (z) = (z + z * )/2. One finds Rnj N

* j = EN * n + ZN n and M in N * i = A in N * i . This yields dZ dt + 2νk 2 Z - Rij 2 d dt (N * i N * j ) + A in N * i (EN * n + ZN n ) = T Z , (C.10)
where the non-linear transfer linked to the polarization anisotropy T Z is defined in (2.34). Some detailed calculations yields

(ZN n N j )M in N * i N * j = -A + ln α l α n Z -2iZ 2α l Ω l + 1 2 lpq α l A - pq EP nj M in N * i N * j = A + ln N * l N * n E
Since the Craya-Herring frame is moving in space with time, the term d t (N * i N * j ) in (C.10) is evaluated using (2.30) and the previous remark on Rnj N * j , so that

Rij d dt (N * i N * j ) = -ZN j dN * j dt .
Finally, to compute dN * j /dt, one has to use

d dt = ∂ ∂t -A ij k i ∂ ∂k j , dk i dt = -A ji k j .
Moreover, similar calculations yield

∂α i ∂k j = P ij k , ∂k ∂k i = α i .
Finally,

N j dN * j dt = 2ie (1) p A lp e (2) l + n l k k ⊥ = -2iΩ CH , (C.11)
where Ω CH expresses the rotation of the Craya-Herring frame. The Z-Lin equation, accounting for rotation, is then

d dt + 2νk 2 Z -A + ln α l α n Z + A + ln N * l N * n E -2iZ 2Ω l α l Rotation part + 1 2 lpq α l A - pq -Ω CH = T Z , (C.12)
in agreement with [START_REF] Cambon | Third-order statistics and the dynamics of strongly anisotropic turbulent flows[END_REF]; [START_REF] Mons | A spectral model for homogeneous shear-driven anisotropic turbulence in terms of spherically-averaged descriptors[END_REF]. One can simplify this expression a bit, using a previous relation that gives

A ln N n N * l Z = -α l α n A + ln Z -i jnl α j A - nl , so that (2.33) is recovered. C.1.3 Evolution equation of S ijk (k, p, t)
From (C.3), the evolution equation of the third-order tensor S ijk , defined in (2.17), is obtained by writing (C.3) for ûj , û i and û k , and summing these three equations, previously multiplied by û i û k , ûj û k and û i ûj respectively. Since S ijk (k, p, t) does not depend on q, the term q l ∂ ∂qn has to be erased. This is done by using q l = -k l -p l and k and p being independent variables. Hence,

u i u k ∂u j ∂k n = u k ∂u j u i ∂k n , u j u k ∂u i ∂p n = u k ∂u j u i ∂p n , ∂ ∂q n = - ∂ ∂p n = - ∂ ∂k n .
Averaging and integrating to erase Dirac functions yields

∂ ∂t + ν(k 2 + p 2 + q 2 ) -A lm k l ∂ ∂k m + p l ∂ ∂p m S ijn (k, p, t) + M im (q)S mjn (k, p, t) + M jm (k)S imn (k, p, t) + M nm (p)S ijm (k, p, t) = T ijn (k, p, t).
Here, some details on the quasi-normal closure T QN ijn (2.37) are provided. For this purpose, the correlation that intervenes in the previous equation is defined as

T ijn (k, p)δ(k + p + q) = i < s i (q)û j (k)û n (p) >, (C.13) where s j (k) = -P jpq (k) k=r+s ûp (r)û q (s)d 3 r. (C.14)
Thus, the previous equation becomes

∂ ∂t + ν(k 2 + p 2 + q 2 ) S ijn (k, p, t) + ... = T ijn (k, p, t) + T jni (p, q, t) + T nij (q, k, t). (C.15)
and using the facts that α

i α i = -z, α i N i = e iλ √ 1 -z 2 and α i N * i = e -iλ √ 1 -z 2 one finds α m α l R ml = (1 -z 2 )(E + X ).
Similar calculations, with α i W * i = 1 -y 2 and W i W i = 0, yield

N * i N * j R ij = e -2iλ (1 + y 2 )(E + X ) -2E -2iy X , k n N * i R ni = -k 1 -z 2 e -iλ z(E + X ) -i X .
→ Now, the contribution of τ + ij to the polarization transfer is evaluated. Using the previous calculations and a symmetric writing, one finds

τ + ij N * i N * j = 1 4 k 2 e -2iλ (1 -z 2 )(E + X ) (1 + y 2 )(E + X ) -2E -2iy X + 1 4 k 2 e -2iλ (1 -y 2 )(E + X ) (1 + z 2 )(E + X ) -2E -2iz X - 1 2 k 2 e -2iλ (x + yz)(y(E + X ) -i X )(z(E + X ) -i X ). (C.27)
Let's simplify the geometric factor that affects (E + X )(E + X ). Using the following relations

(1 -z 2 ) = q p (x + yz) = q 2 p 2 (1 -y 2 ), k(2yz + x) = qz + py, 2xyz = 1 -x 2 -y 2 -z 2 , one finds k 2 (1 -2y 2 z 2 -xyz) = kp(xy + z 3 ) + kq(xz + y 3 ). (C.28) Thus, with p ↔ q symmetry k 2 e -2iλ 2 (1 -2y 2 z 2 -xyz)(E + X )(E + X ) = e -2iλ 2 kp(xy + z 3 ) + kq(xz + y 3 ) (E + X )(E + X ) = e -2iλ kp(xy + z 3 )(E + X )(E + X ). (C.29)
The term

- k 2 e -2iλ 4 (1 -z 2 )(E + X )2E + (1 -y 2 )(E + X )2E simplifies using k(1 -z 2 ) = q(xz + y) and k(1 -y 2 ) = p(xy + z), into - k 2 e -2iλ 4 (1 -z 2 )(E + X )2E + (1 -y 2 )(E + X )2E = -kpe -2iλ (xy + z)(E + X )E . (C.30)
The remaining imaginary term

ik 2 e -2iλ 2 X (E + X )(z(x + yz) -y(1 -z 2 )) + X (E + X )(y(x + yz) -z(1 -y 2 ))
can be simplified using p ↔ q symmetry and the following relations

k(1 -z 2 ) = q(xz + y), k(1 -y 2 ) = p(xy + z), k(x + yz) = p(y + xz) = q(z + xy). Therefore ik 2 e -2iλ 2 X (E + X )(z(x + yz) -y(1 -z 2 )) + X (E + X )(y(x + yz) -z(1 -y 2 ))
C.3 Spherically-averaged non-linear transfers

C.3.1 λ-integration

In the anisotropic framework, the difficulty is to solve the integral that depends on the orientation of the plane of the triad. Triple integrals simplify using the change of variable

f 1 (k, p, t)d 3 p = ∆ k pq k 2π 0 f 2 (k, p, q, λ)dλ dpdq, (C.37)
where λ as been defined in (C.16). ∆ k is the domain where k, p and q are the lengths of the sides of the triangle formed by the triad. In the isotropic case, the λ-integral amounts to a multiplication by 2π which is not true anymore in the anisotropic case. In this part, the main integrals that are useful for the calculations are

2π 0 α i α j dλ = π[(1 -z 2 )δ ij + (3z 2 -1)α i α j ], 2π 0 e -2iλ α i α j dλ = π 1 -z 2 2 N * i N * j , 2π 0 W i W j dλ = 2π 0 W * i W * j dλ = π(z 2 -1)(δ ij -3α i α j ), 2π 0 e -2iλ W i W j dλ = π (z + 1) 2 2 N * i N * j , 2π 0 e -2iλ W * i W * j dλ = π (1 -z) 2 2 N * i N * j .
Similar results regarding quantities are obtained by changing z to y. Then, λ-integrations of E and Z give 2π 0

E (dir) dλ = 15πE 0 H (dir) ij (1 -3z 2 )α i α j , 2π 0 E (dir) dλ = -30πE 0 H (dir) ij α i α j , 2π 0 e -2iλ E (dir) dλ = 15 2 πE 0 H (dir) ij (z 2 -1)N * i N * j , 2π 0 e -2iλ E (dir) dλ = 0, 2π 0 X dλ = 15 2 πE 0 H (pol) ij (1 -z 2 )α i α j , 2π 0 Xdλ = 0, 2π 0 e -2iλ X dλ = 5 4 πE 0 H (pol) ij (1 + z 2 )N * i N * j , 2π 0 e -2iλ Xdλ = 5 2 πE 0 H (pol) ij N * i N * j , 2π 0 e -2iλ X dλ = - 5 2 πE 0 H (pol) ij zN * i N * j , 2π 0 e -2iλ Xdλ = 5πE 0 H (pol) ij N * i N * j .

C.3.2 Spherical integration

In the context of moderate anisotropy, only the second order of the Rij expansion is kept. This means that all quadratic contributions H

() ij H ()
mn can be simplified, because there are negligible with respect to H

() ij . Remembering that H () ii = 0 and H () ij = H ()
ji , and using the following relations

S k α i α j d 2 k = 4πk 2 3 δ ij , S k α i α j α m α n d 2 k = 4πk 2 15 (δ ij δ mn + δ im δ jn + δ in δ jm ), S k α i d 2 k = S k α i α j α n d 2 k = 0, S k H () mn α m α n P ij d 2 k = - 8 15 πk 2 H () ij , S k H () mn N * m N * n N i N j d 2 k = 16 5 πk 2 H () ij ,
it is possible to integrate spherically the directional and polarization transfer terms T E and T Z given by (2.41) and (2.42). Let's define S NL(iso) , the non-linear spherically-averaged isotropic transfer is obtained by spherically averaging T E as defined in (2.56). One can note that

S k α i α j H () ij d 2 k = 0,
and thus the classical isotropic transfer term (2.57) is recovered. For the anisotropic transfer terms, the first thing to do is to discard terms such as E (dir) E (dir) or E (dir) X and so on, because they are quadratic in H () ij . Using the previous relations for λ-integration and spherical averaging, expressions (2.59) and (2.61) are recovered.

C.4 Spherically-averaged linear transfers

Now, the emphasis is put on the linear terms that contribute to the total transfer. It is possible to write the Craya equation (2.13) as

∂ Rij ∂t + 2νk 2 Rij = T NL ij + T L ij , (C.38)
with the linear transfer being (starting from E-Lin and Z-Lin equations is more complicated)

T L ij = 2A ln α l (α i Rnj + α j Rni ) + A ln k l ∂ Rij ∂k n -(A il Rlj + A jl Ril ). (C.39)

C.4.1 Spherical integration

The previous relations are still verified. The following ones are used as well

A ln S k α i α j α l α n d 2 k = 8πk 2 15 A + ij , S k A ln k l ∂E 0 α i α j ∂k n d 2 k = 8πk 2 15 A + ij 3E 0 + k ∂E 0 ∂k , S k A ln k l ∂H () pq α p α q ∂k n d 2 k = 8πk 2 15 A + ln k ∂H () ln ∂k + 3H () ln , A ln H () pq S k α i α j α l α n α p α q d 2 k = 8πk 2 105 2A + il H () jl + 2A + jl H () il + A + ln H () ln δ ij , S k A ln k l ∂H () pq α p α q α i α j ∂k n d 2 k = 8πk 2 105 (A li + A il ) k ∂H () lj ∂k + 3H () lj + (A lj + A jl ) k ∂H () li ∂k + 3H () li + A ln k ∂H () ln ∂k + 3H () ln δ ij , k ∂ ∂k (E 0 H () ij ) + 3H () ij E 0 = 1 4πk 2 ∂ ∂k (kEH () ij ).

C.4.2 Computation of T L ij

The method to compute the linear transfers is the following one: firstly, the linear isotropic term S L (k, t) is evaluated. Then, the total linear term, defined in (2.77), is computed. The linear directional transfer S L(dir) ij (k, t), without the isotropic part, is calculated. Finally, both the isotropic and directional parts are subtracted from the total linear transfer to obtain the polarization one S L(pol) ij (k, t). This process is easier than computing directly the polarization transfer. Hence, one has

S L(tot) ij (k, t) = S k T L ij (k, t)d 2 k = 2 δ ij 3 S L(iso) (k, t) + S L(dir) ij (k, t) + S L(pol) ij (k, t) . A convenient expression of Rij is used Rij = Re ij + Rz ij = E 0 P ij 1 -15H (dir) pq α p α q + 5E 0 P in P jm H (pol) mn + 1 2 P ij H (pol) pq α p α q , (C.40)
where Re ij accounts for isotropy and directivity, and Rz ij for polarization. In the following calculations, the velocity gradient A ij is decomposed into symmetric and antisymmetric matrices

A + ij = A ij + A ji 2 , A - ij = A ij -A ji 2 .
Computation of S L(iso) and S L(tot) ij

: Using P ii = 2 and Rz ii = 0, one recovers easily (2.63) for S L(iso) . To compute the total linear transfer, three contributions of (C.39) have to be calculated for Re ij and Rz ij , namely

S k 2A ln α l (α i Re nj + α j Re ni )d 2 k = 4 5 EA + ij - 12 7 E A + lj H (dir) li + A + li H (dir) lj - 2 3 δ ij A + lm H (dir) lm + 4E A - jl H (dir) il + A - il H (dir) jl , S k 2A ln α l (α i Rz nj + α j Rz ni )d 2 k = 12 7 E A + lj H (pol) li + A + li H (pol) lj - 2 3 δ ij A + lm H (pol) lm + 4 3 E A - jl H (pol) il + A - il H (pol) jl S k A ln k l ∂ Re ij ∂k n d 2 k = - 2 15 A + ij ∂kE ∂k + 4 7 A + il ∂ ∂k (kEH (dir) lj ) + A + jl ∂ ∂k (kEH (dir) li ) -3δ ij A + lm ∂ ∂k (kEH (dir) lm ) , S k A ln k l ∂ Rz ij ∂k n d 2 k = - 4 7 A + il ∂ ∂k (kEH (pol) lj ) + A + jl ∂ ∂k (kEH (pol) li ) - 2 3 δ ij A + lm ∂ ∂k (kEH (pol) lm ) , S k (A il Re lj + A jl Re il )d 2 k = 4 3 EA + ij + 2E A + il H (dir) jl + A + jl H (dir) il + A - il H (dir) jl + A - jl H (dir) il , S k (A il Rz lj + A jl Rz il )d 2 k = 2E A + il H (pol) jl + A + jl H (pol) il + A - il H (pol) jl + A - jl H (pol) il . Computation of S L(dir) ij and S L(pol) ij : the definition of S L(dir) ij is given in (2.64). Firstly, 1 4 A ln S k k l ∂ Re mm ∂k n P ij d 2 k = - 1 15 A + ij ∂ ∂k (kE) + 1 5 A + ij E + E A - jn H (dir) ni + A - in H (dir) nj + 2 7 A + il ∂ ∂k (kEH (dir) jl ) + A + jl ∂ ∂k (kEH (dir) il ) -3A + lm δ ij ∂ ∂k (kEH (dir) lm ) - 3 7 E A + jl H (dir) il + A + il H (dir) jl - 2 3 A + lm H (dir) lm δ ij ,
and Rz mm = 0. Similarly,

1 2 A ln S k Re ln P ij d 2 k = 1 15 EA + ij - 2 7 E A + lj H (dir) il + A + li H (dir) jl -3A + ln δ ij H (dir) ln , 1 2 A ln S k Rz ln P ij d 2 k = 2 7 E A + lj H (pol) il + A + li H (pol) jl + 5 3 A + ln δ ij H (pol) ln .
Summing the three previous terms and removing the isotropic part δ ij S L(iso) /3 gives (2.65). The linear polarization transfer is obtained by removing the directional and isotropic linear transfers from half of the total contribution, according to (2.77). Equation (2.67) is then recovered.

C.4.3 Return to isotropy

The following relations for the λ-integration and spherical integration are needed

2π 0 α i α j e -iλ dλ = -πy 1 -y 2 (α i N * j + α j N * i ), 2π 0 W i W j e -iλ dλ = π(1 + y) 1 -y 2 (α i N * j + α j N * i ), 2π 0 W * i W * j e -iλ dλ = π(y -1) 1 -y 2 (α i N * j + α j N * i ), 2π 0 E (dir) e -iλ dλ = 30πy 1 -y 2 E 0 H (dir) ij α i N * j , 2π 0 X e -iλ dλ = 5πy 1 -y 2 E 0 H (pol) ij α i N * j , 2π 0 i X e -iλ dλ = -5π 1 -y 2 E 0 H (pol) ij α i N * j .
The p quantities (with ) are obtained by multiplying by -1 the rhs term and replacing y by z. Here, the useful relation for spherical integration is

H () mn α m α i N * n N j d 2 k = 4πk 2 5 H () ij .
Then, from the definition of S (RTI) ij

given in (2.70), it is possible to integrate spherically W ij using the previous relations, and to find equation (2.71).

C.4.4 Rotation

The effect of rotation on the velocity field was detailed in section C.1. It was revealed that the evolution equation of Z only was affected by rotation. Rotation was already taken into account in the Craya equation through the tensor M ij , accordingly modified in (C.4). We call Mij the rotating part of this tensor. One has first to evaluate the corresponding total linear transfer SL(tot

) ij = - S k ( Min Rnj + Mjn Rni )d 2 k.
There are no contributions from Re ij : all terms like lmn α l α n are zero which tremendously simplify the calculations. In the end, the polarization transfer is simply half the total contribution, which yields

SL(pol) ij = 1 2 SL(tot) ij = - 1 2 S k ( Min Rnj + Mjn Rni )d 2 k (C.41) = - 4 3 E(k, t)Ω m imp H (pol) jp + jmp H (pol) ip .

C.5 Kinetic quadratic anisotropic contributions

In this part, some details are given about the calculations of the second-order spherically-averaged nonlinear transfer terms, which take into account quadratic contributions of anisotropy. The convenient notation is used

H (dir) li H (dir) lj + H (dir) lj H (dir) li - 2 3 H (dir) ln H (dir) ln δ ij = H (dir) , H (dir) ij .
Here are some useful relations

1 -y 2 1 -z 2 = (x + yz), 2xyz = 1 -x 2 -y 2 -z 2 , S k H ln H pq α l α n α p α q P ij d 2 k = - 16πk 2 105 H li H lj + H lj H li -3H ln H ln δ ij , S k H ln H pq α n α p δ lq + α n α q δ lp + α l α p δ nq + α l α q δ np P ij d 2 k = - 16πk 2 15 H li H lj + H lj H li -4H ln H ln δ ij , S k H ln H pq δ ip δ jq + δ iq δ jp -α n α p δ lq -α n α q δ lp -α l α p δ nq -α l α q δ np P ij d 2 k, = 16πk 2 15 H li H lj + H lj H li + H ln H ln δ ij , S k H ln H pq α l α n N * p N * q N i N j d 2 k = - 32πk 2 35 H li H lj + H lj H li - 2 3 H ln H ln δ ij , S k H ln H pq (α n N * l + α l N * n )(α p N * q + α q N * p )N i N j d 2 k = 96πk 2 35 H li H lj + H lj H li - 2 3 H ln H ln δ ij , - S k H ln H pq N * l N * n (α p N q + α q N p )(α j N i + α i N j )d 2 k = - 96πk 2 35 H li H lj + H lj H li - 2 3 H ln H ln δ ij , - S k H ln H pq α l α n (α p N * q + α q N * p )(α j N i + α i N j )d 2 k = - 16πk 2 35 H li H lj + H lj H li - 2 3 H ln H ln δ ij . Since H ()
ln and H () pq are deviatoric tensors, terms in δ ln and δ pq do not bring any contributions.

Non-linear directional and isotropic transfers: the three main parts to compute of the directional transfer (2.41) are

T 1 E = E 0 H (dir) ij α i α j + (H (pol) ij W * i W * j ) E 0 H (dir) pq α p α q -E 0 H (dir) pq α p α q , T 2 E = E 0 H (dir) ij α i α j + (H (pol) ij W * i W * j ) E 0 H (pol) pq W * p W * q -E 0 H (pol) pq W * p W * q , T 3 E = E 0 (H (pol) ij W * i W * j ) xE 0 H (pol) pq W * p W * q -yE 0 H (pol) pq W * p W * q .
For the λ-integration and spherical integration, here are some useful expressions 2π 0 α i α j α p α q dλ = -yzπ(x + yz)(α i α p δ jq + α i α q δ jp + α j α p δ iq + α j α q δ ip )

+ α i α j α p α q π 2y 2 z 2 + 3 4 (1 -y 2 )(1 -z 2 ) -y 2 (1 -z 2 ) -z 2 (1 -y 2 ) + 4yz(x + yz) + π 4 (1 -y 2 )(1 -z 2 ) δ ip δ jq + δ iq δ jp -α i α p δ jq -α i α q δ jp -α j α p δ iq -α j α q δ ip , 2π 0 α i α j α p α q dλ = π(3y 2 -1)α i α j α p α q , 2π 0 (W * i W * j )α p α q dλ = yzπ(x + yz)(α i α p δ jq + α i α q δ jp + α j α p δ iq + α j α q δ ip ) + α i α j α p α q π 3z 2 (1 -y 2 ) + 1 4 (1 + y 2 )(1 -z 2 ) - 3 2 (1 -y 2 )(1 -z 2 ) -4yz(x + yz) + π 4 (1 + y 2 )(1 -z 2 ) δ ip δ jq + δ iq δ jp -α i α p δ jq -α i α q δ jp -α j α p δ iq -α j α q δ ip , 2π 0 (W * i W * j )α p α q dλ = 3π(1 -y 2 )α i α j α p α q , S k 2π 0 T 1 E dλ P ij - δ ij 3 d 2 k = 16π 2 k 2 105 E 0 E 0 (1 + 3xyz) H (dir) li H (dir) lj + H (dir) lj H (dir) li - 2 3 H (dir) ln H (dir) ln δ ij + E 0 (3y 2 -1) H (dir) li H (dir) lj + H (dir) lj H (dir) li - 2 3 H (dir) ln H (dir) ln δ ij + E 0 (3 -6z 2 -3xyz) H (pol) li H (dir) lj + H (pol) lj H (dir) li - 2 3 H (pol) ln H (dir) ln δ ij + 3E 0 (1 -y 2 ) H (pol) li H (dir) lj + H (pol) lj H (dir) li - 2 3 H (pol) ln H (dir) ln δ ij , S k 2π 0 T 1 E dλ d 2 k = 16π 2 k 2 15 E 0 (3x 2 -1)E 0 H (dir) ln H (dir) ln + (3 -3x 2 )E 0 H (pol) ln H (dir) ln -(3y 2 -1)E 0 H (dir) ln H (dir) ln -3(1 -y 2 )E 0 H (pol) ln H (dir) ln , 2π 0 α p α q (W * i W * j )dλ = 2π(1 -y 2 ) α i α j α p α q + δ ip δ jq + δ iq δ jp -α i α p δ jq -α i α q δ jp -α j α p δ iq -α j α q δ ip , 2π 0 (W * i W * j ) (W * p W * q )dλ = -πyz(x + yz)(α i α p δ jq + α i α q δ jp + α j α p δ iq + α j α q δ ip ) + 2πα i α j α p α q 9 4 (1 -y 2 )(1 -z 2 ) + 2yz(x + yz) + 1 8 (1 + y 2 )(1 + z 2 ) + π 4 (1 + y 2 )(1 + z 2 ) δ ip δ jq + δ iq δ jp -α i α p δ jq -α i α q δ jp -α j α p δ iq -α j α q δ ip , 2π 0 (W * i W * j ) (W * p W * q )dλ = π 2 (1 + y 2 ) α i α j α p α q + δ ip δ jq + δ iq δ jp -α i α p δ jq -α i α q δ jp -α j α p δ iq -α j α q δ ip , S k 2π 0 T 2 E dλ P ij - δ ij 3 d 2 k = 16π 2 k 2 105 E 0 + E 0 (3 -6y 2 -3xyz) H (dir) li H (pol) lj + H (dir) lj H (pol) li - 2 3 H (dir) ln H (pol) ln δ ij -3E 0 (1 -y 2 ) H (dir) li H (pol) lj + H (dir) lj H (pol) li - 2 3 H (dir) ln H (pol) ln δ ij + E 0 (3 -6x 2 -9xyz) H (pol) li H (pol) lj + H (pol) lj H (pol) li - 2 3 H (pol) ln H (pol) ln δ ij -3E 0 (1 + y 2 ) H (pol) li H (pol) lj + H (pol) lj H (pol) li - 2 3 H (pol) ln H (pol) ln δ ij , S k 2π 0 T 2 E dλ d 2 k = 16π 2 k 2 15 E 0 E 0 (3x 2 + 3)H (pol) ln H (pol) ln + (3 -3x 2 )E 0 H (dir) ln H (pol) ln -3(1 -y 2 )E 0 H (dir) ln H (pol) ln -3E 0 (1 + y 2 )H (pol) ln H (pol) ln , 2π 0 (W * i W * j ) (W * p W * q )dλ = -π(x + yz)(α i α p δ jq + α i α q δ jp + α j α p δ iq + α j α q δ ip ) + πα i α j α p α q 4(x + yz) + yz + πyz δ ip δ jq + δ iq δ jp -α i α p δ jq -α i α q δ jp -α j α p δ iq -α j α q δ ip , 2π 0 (W * i W * j ) (W * p W * q )dλ = -πy(α i α j α p α q + δ ip δ jq + δ iq δ jp -α i α p δ jq -α i α q δ jp -α j α p δ iq -α j α q δ ip ), S k 2π 0 T 3 E dλ P ij - δ ij 3 d 2 k = 16π 2 k 2 105 E 0 -6xyE 0 H (pol) li H (pol) lj + H (pol) lj H (pol) li - 2 3 H (pol) ln H (pol) ln δ ij -yE 0 (3x + 9yz) H (pol) li H (pol) lj + H (pol) lj H (pol) li - 2 3 H (pol) ln H (pol) ln δ ij , S k 2π 0 T 3 E dλ d 2 k = 16π 2 k 2 15 E 0 -6xyE 0 H (pol) ln H (pol) ln -yE 0 (-6x)H (pol) ln H (pol) ln .
Non-linear polarization transfer: the three main parts to compute of the polarization transfer (2.42) are

T 1 Z = E 0 H (dir) ij α i α j + (H (pol) ij W * i W * j ) (E 0 H (pol) pq W * p W * q ) -E 0 H (pol) pq W * p W * q , T 2 Z = E 0 H (dir) ij α i α j + (H (pol) ij W * i W * j ) E 0 H (dir) pq α p α q -E 0 H (dir) pq α p α q , T 3 Z = iE 0 H (dir) ij α i α j + (H (pol) ij W * i W * j ) (H (pol) pq W * p W * q ), T 4 Z = iE 0 (H (pol) ij W * i W * j ) xE 0 (H (dir) pq α p α q + H (pol) pq W * p W * q ) -iyE 0 (H (pol) pq W * p W * q ) .
For the λ-integration and spherical integration, here are some useful expressions

2π 0 e -2iλ α i α j (W * p W * q )dλ = π 2 yz(x + yz)(α i N * j + α j N * i )(α p N * q + α q N * p ) + πα i α j N * p N * q - 1 4 (1 -y 2 )(1 + z 2 ) + 1 2 y 2 (1 + z 2 ) + 3π 4 (1 -y 2 )(1 -z 2 )α p α q N * i N * j , 2π 0 e -2iλ (W * i W * j ) (W * p W * q )dλ = - π 2 yz(x + yz)(α i N * j + α j N * i )(α p N * q + α q N * p ) + 3π 4 (1 -y 2 )(1 + z 2 )α i α j N * p N * q + 3π 4 (1 + y 2 )(1 -z 2 )α p α q N * i N * j , 2π 0 e -2iλ α i α j N * p N * q dλ = π(3y 2 -1)α i α j N * p N * q , 2π 0 e -2iλ (W * i W * j )W * p W * q dλ = 3π(1 -y 2 )α i α j N * p N * q , S k 2π 0 T 1 Z dλ N * i N * j d 2 k = 16π 2 k 2 35 E 0 E 0 (3xyz + 2z 2 -1) H (dir) , H (pol) ij -3(xyz + 1)E 0 H (pol) , H (pol) ij + 2(3y 2 -1)E 0 H (dir) , H (pol) ij + 6(1 -y 2 )E 0 H (pol) , H (pol) ij , 2π 0 e -2iλ α i α j α p α q dλ = - π 2 yz(x + yz)(α i N * j + α j N * i )(α p N * q + α q N * p ) + πα i α j N * p N * q - 1 4 (1 -y 2 )(1 -z 2 ) + 1 2 y 2 (1 -z 2 ) + πα p α q N * i N * j - 1 4 (1 -y 2 )(1 -z 2 ) + 1 2 z 2 (1 -y 2 ) , 2π 0 e -2iλ α i α j α p α q dλ = π 2 (1 -y 2 )α p α q N * i N * j , 2π 0 e -2iλ (W * i W * j )α p α q dλ = π 2 (1 + y 2 )α p α q N * i N * j , S k 2π 0 T 2 Z dλ N * i N * j d 2 k = 16π 2 k 2 35 E 0 E 0 (xyz + 2x 2 -1) H (dir) , H (dir) ij + (1 + y 2 )E 0 H (pol) , H (dir) ij + E 0 (3xyz + 2y 2 -1) H (pol) , H (dir) ij + (1 -y 2 )E 0 H (dir) , H (dir) ij , 2π 0 ie -2iλ α i α j (W * p W * q )dλ = - π 2 y(x + yz)(α i N * j + α j N * i )(α p N * q + α q N * p ) + π 2 z(1 -3y 2 )α i α j N * p N * q , 2π 0 ie -2iλ (W * i W * j ) (W * p W * q )dλ = π 2 y(x + yz)(α i N * j + α j N * i )(α p N * q + α q N * p ) - 3π 2 z(1 -y 2 )α i α j N * p N * q , S k 2π 0 T 3 Z dλ N * i N * j d 2 k = 16π 2 k 2 35 E 0 3(xy + z)E 0 H (pol) , H (pol) ij -(3xy + z)E 0 H (dir) , H (pol) ij , 2π 0 ie -2iλ (W * i W * j )α p α q dλ = -πyα p α q N * i N * j , 2π 0 ie -2iλ (W * i W * j )N * p N * q dλ = 0, 2π 0 e -2iλ (W * i W * j ) (W * p W * q )dλ = π 2 (x + yz)(α i N * j + α j N * i )(α p N * q + α q N * p ), S k 2π 0 T 4 Z dλ N * i N * j d 2 k = 16π 2 k 2 35 E 0 2xyE 0 H (pol) , H (dir) ij + 3y(x + yz)E 0 H (pol) , H (pol) ij .
Non-linear return to isotropy transfer: the three main parts to compute of the non-linear return to isotropy transfer (2.69) are

T 1 RTI = E 0 E 0 H (dir) ij α i α j + (H (pol) ij W * i W * j ) H (dir) pq α p α q + H (pol) pq W * p W * q , T 2 RTI = E 0 E 0 H (dir) ij α i α j + (H (pol) ij W * i W * j ) H (dir) pq α p α q + (H (pol) pq W * p W * q ) , T 3 RTI = iE 0 E 0 H (dir) ij α i α j + (H (pol) ij W * i W * j ) (H (pol) pq W * p W * q ).
For the λ-integration and spherical integration, here are some useful expressions

2π 0 e -iλ α i α j α p α q dλ = -πy 1 -y 2 α p α q (α i N * j + α j N * i ), 2π 0 e -iλ α i α j W * p W * q dλ = -πy 1 -y 2 N * p N * q (α i N j + α j N i ), 2π 0 e -iλ (W * i W * j )α p α q dλ = πy 1 -y 2 α p α q (α i N * j + α j N * i ), 2π 0 e -iλ (W * i W * j )W * p W * q dλ = πy 1 -y 2 N * p N * q (α i N j + α j N i ), - S k 2π 0 T 1 RTI dλ (α i N j + α j N i )d 2 k = 16π 2 k 2 35 E 0 E 0 y 1 -y 2 H (dir) , H (dir) ij + 6 H (dir) , H (pol) ij -H (pol) , H (dir) ij -6 H (pol) , H (pol) ij , 2π 0 e -iλ α i α j α p α q dλ = 2π 3y 2 -1 4 z 1 -z 2 α i α j (α p N * q + α q N * p ) + 1 -3z 2 4 y 1 -y 2 α p α q (α i N * j + α j N * i ) + 1 -y 2 8 z 1 -z 2 N * i N * j (α p N q + α q N p ) - 1 -z 2 8 y 1 -y 2 N * p N * q (α i N j + α j N i ) , 2π 0 e -iλ α i α j (W * p W * q )dλ = 2π 1 -3y 2 4 z 1 -z 2 α i α j (α p N * q + α q N * p ) - 3 4 y 1 -y 2 (1 -z 2 )α p α q (α i N * j + α j N * i ) - 1 -y 2 8 z 1 -z 2 N * i N * j (α p N q + α q N p ) - 1 + z 2 8 y 1 -y 2 N * p N * q (α i N j + α j N i ) , 2π 0 e -iλ (W * i W * j ) (W * p W * q )dλ = 2π 1 + z 2 8 y 1 -y 2 N * p N * q (α i N j + α j N i ) + 3 4 y 1 -y 2 (1 -z 2 )α p α q (α i N * j + α j N * i ) - 1 8 z 1 -z 2 6(1 -y 2 )α i α j (α p N * q + α q N * p ) + (1 + y 2 )N * i N * j (α p N q + α q N p ) , - S k 2π 0 T 2 RTI dλ (α i N j + α j N i )d 2 k = 16π 2 k 2 35 E 0 E 0 (y 1 -y 2 -z 1 -z 2 ) H (dir) , H (dir) ij + (3y 1 -y 2 + z 1 -z 2 ) H (dir) , H (pol) ij -(y 1 -y 2 + 3z 1 -z 2 ) H (pol) , H (dir) ij + 3(z 1 -z 2 -y 1 -y 2 ) H (pol) , H (pol) ij , 2π 0 ie -iλ α i α j (W * p W * q )dλ = 2π y 2 -1 8 1 -z 2 N * i N * j (α p N q + α q N p ) 3y 2 -1 4 1 -z 2 α i α j (α p N * q + α q N * p ) + 1 4 yz 1 -y 2 N * p N * q (α i N j + α j N i ) , 2π 0 ie -iλ (W * i W * j ) (W * p W * q )dλ = 2π - 1 + y 2 8 1 -z 2 N * i N * j (α p N q + α q N p ) 3 4 (1 -y 2 ) 1 -z 2 α i α j (α p N * q + α q N * p ) - 1 4 yz 1 -y 2 N * p N * q (α i N j + α j N i ) , - S k 2π 0 T 3 RTI dλ (α i N j + α j N i )d 2 k = 16π 2 k 2 35 E 0 E 0 ((2 -3y 2 ) 1 -z 2 -3yz 1 -y 2 ) H (dir) , H (pol) ij + 3(y 2 1 -z 2 + yz 1 -y 2 ) H (pol) , H (pol) ij ,
The quadratic anisotropic RTI transfer term is

Q NL(RTI) ij (k, t) = 120 7 ∆ k θ kpq π 2 kp 2 qE 0 2y(x + yz)(xy + z)(zk -qx)E 0 6 H (dir) , H (dir) ij -6 H (dir) , H (pol) ij + H (pol) , H (dir) ij -H (pol) , H (pol) ij -kz(xy + z)E 0 12(y(x + yz) -z(1 -z 2 )) H (dir) , H (dir) ij -(6y(x + yz) + 2z(1 -z 2 )) H (dir) , H (pol) ij + (2y(x + yz) + 6z(1 -z 2 )) H (pol) , H (dir) ij + (z(1 -z 2 ) -y(x + yz)) H (pol) , H (pol) ij + k(xy + z)E 0 (6y(y + xz) -4(1 -z 2 )) H (dir) , H (pol) ij + y(y + xz) H (pol) , H (pol) ij dpdq. (C.45)
The impact of quadratic anisotropic contributions on the non-linear transfers is revealed in Fig. C.1a. One can note that the global shape of the transfers is preserved and that there is a rather small change in intensity, mainly at large scales which is expected since this is where anisotropy is dominant. The isotropic and polarization parts of the transverse component () 33 are more affected by the quadratic contributions than the directional part. This might also be the consequence of the Re λ which is slightly higher, at St = 50, for the simulation with the quadratic anisotropic contributions than the one without.

Regarding the b ij in Fig. C.1b: the asymptotic values are almost the same with and without the quadratic anisotropic contributions. One can observe a slight increase of the streamwise anisotropy and decrease of the transverse one. Moreover, the kinetic spectra scalings and the exponential growth rate of the kinetic energy are not displayed since the difference when adding the second-order contributions is not distinguishable. In conclusion, the quite heavy analytical calculations, which led to the second-order anisotropic contributions in the non-linear transfers, do not provide significant changes with regard to the first order developed in Chapter 3: this is why they are not used. C.6 Fourth-order expansion for E and Z

Here, some details are given about the method to obtain the fourth order expansions of E and Z (2.44) and (2.45). Starting from

E(k, t) = E 0 1 -15H (dir) ij (k, t)α i α j + U (dir)4 ijpq (k, t)α i α j α p α q + . . . , (C.46) Z(k, t) = 1 2 E 0 5H (pol) ij (k, t) + U (pol)4 ijpq (k, t)α p α q + . . . N * i (k)N * j (k), (C.47)
and using the definitions of the generalized operators P ijpq and N ijpq given in (2.49) and (2.50), one gets

S k E(k) P ijpq (k) d 2 k = 24 945 E(k)U (dir) ijpq (k), S k Z(k)N ijpq (k) d 2 k = 4 21 E(k)U (pol) ijpq (k).
ijpq reads Rij = E 0 P ij 1 -15H (dir) pq α p α q + 945 12 H (dir) rspq α r α s α p α q Re2 ij + Re4 ij + 1 2 E 0 5H (pol) pq + 21 2 H (pol) pqrs α r α s 2P ip P jq + P ij α p α q Rz2 ij + Rz4 ij . (C.48)

C.6.1 Fourth order linear transfers

In this part, we aim at computing the linear directional and polarization transfers T L E and T L Z associated with the evolution equations of E and Z. The linear spherically-averaged directional and polarization transfers at the fourth-order are defined by : here are some useful formula for the spherical integration:

S L(dir) ijpq (k) = 1 2 S k T L E (k)P ijpq (k) d 2 k, S L(pol) ijpq (k) = 1 2 S k T L Z (k)N ijpq (k) d 2 k. (C.
S k α l α p α r α s A ln H () rspn d 2 k = 0, S k α i α j α p α q α m α n α r α s H () mnrs d 2 k = 24 945 4πk 2 H () ijpq , S k α i α j α p α l α r α s A ln H () rspn d 2 k = 6 105 4πk 2 H () ijln A + ln , S k α i α j α p α q α l α n A ln H () ijpq d 2 k = 0. S k α i α j α p α q α l α n α r α s H () pqrs A ln d 2 k = 24 945 4πk 2 H () ijln A + ln , S k A lj α r α s α p α q α i α l H () rspq d 2 k = 0.
Consequently, one has for the directional transfer

A ln S k Re4 ln P ij d 2 k = 2A + ln EH (dir) ijln , A ln S k k l ∂ Re4 mm ∂k n P ij d 2 k = -4A + ln 2EH (dir) lnij + ∂(kEH (dir) ijln ) ∂k , A ln S k Rz4 ln P ij d 2 k = - 1 3 A + ln EH (pol) ijln , A ln S k k l ∂ Rz4 mm ∂k n P ij d 2 k = 0.
For the total transfer, one gets

2A ln S k α i α l ( Re4 nj + Rz4 nj )d 2 k = -4A + ln EH (dir) ijln + 2 3 A + ln EH (pol) ijln , A ln S k k l ∂( Re4 ij + Rz4 ij ) ∂k n d 2 k = -2A + ln ∂(kEH (dir) ijln ) ∂k + 1 3 A + ln ∂(kEH (pol) ijln ) ∂k . One can then compute S L(pol4) ij = S L(tot4) ij /2 -S L(dir4) ij
. Finally, the additional contributions to S L(dir) ij and S

L(pol) ij

resulting from the expansions of E and Z at the fourth order are

S L(dir4) ij (k, t) = -3A + ln EH (dir) ijln -A + ln ∂(kEH (dir) ijln ) ∂k + 1 6 A + ln EH (pol) ijln , (C.50) S L(pol4) ij (k, t) = -A + ln EH (dir) ijln + 1 2 A + ln EH (pol) ijln + 1 6 A + ln ∂(kEH (pol) ijln ) ∂k . (C.51) Contributions of H (dir) ij and H (pol) ij in S L(dir)
ijpq : here are some useful formula for the spherical integration:

S k P ijpq d 2 k = 0, S k H () rs α r α s P ijpq d 2 k = 0, S k A ln α l α n P ijpq d 2 k = 0, S k A ln H () ls α s α n P ijpq d 2 k = 0.
Furthermore, a complex and lengthy calculation yields

S k E 0 A ln H () rs α l α n α r α s P ijpq d 2 k = 2 6615 E 8 5 A + ln H () ln δ ij δ pq + δ ip δ jq + δ iq δ jp -4 δ ij (A + lp H () lq + A + lq H () lp ) + δ pq (A + li H () lj + A + lj H () li ) + A + lp (H () lj δ iq + H () li δ jq ) + A + lq (H () lj δ ip + H () li δ jp ) + H () lp (A + jl δ iq + A + il δ jq ) + H () lq (A + jl δ ip + A + il δ jp ) + 14 A + ij H () pq + A + pq H () ij + A + ip H () jq + A + iq H () jp + A + jp H () iq + A + jq H () ip = 2 6615 H (2,e) ijpq [EH () ].
Then, one obtains

A ln S k Re2 ln P ijpq d 2 k = 30 6615 H (2,e) ijpq [EH (dir) ], A ln S k Rz2 ln P ijpq d 2 k = 5 6615 H (2,e) ijpq [EH (pol) ], A ln S k k l ∂ Re2 mm ∂k n P ijpq d 2 k = 60 6615 5H (2,e) ijpq [EH (dir) ] -H (2,e) ijpq [∂ k (kEH (dir) )] .
Hence, the contribution of

H (dir) ij and H (pol) ij to S L(dir) ijpq is S L(dir2) ijpq (k, t) = 1 441 - 1 6 H (2,e)
ijpq [EH (pol) ] + 4H

(2,e)

ijpq [EH (dir) ] -H

(2,e)

ijpq [∂ k (kEH (dir) )] . (C.52) Contributions of H (dir) ij and H (pol) ij in S L(pol)
ijpq : here are some useful formula for the spherical integration:

S k H () rs N * r N * s N ijpq d 2 k = 0, S k A nl H () ls N * n N * s N ijpq d 2 k = 0, S k A + ln N * l N * n N ijpq d 2 k = 0,
and moreover, lengthy calculations yield

S k E 0 A + ln H () rs α l α n N * r N * s N ijpq d 2 k = S k E 0 A ln H () rs α l α s N * r N * n N ijpq d 2 k = 4 2205 E 4A + ln H () ln δ ij δ pq + δ ip δ jq + δ iq δ jp -10 δ ij (A + lp H () lq + A + lq H () lp ) + δ pq (A + li H () lj + A + lj H () li ) + A + lp (H () lj δ iq + H () li δ jq ) + A + lq (H () lj δ ip + H () li δ jp ) + H () lp (A + jl δ iq + A + il δ jq ) + H () lq (A + jl δ ip + A + il δ jp ) + 35 A + ij H () pq + A + pq H () ij + A + ip H () jq + A + iq H () jp + A + jp H () iq + A + jq H () ip = 4 2205 H (2,z) ijpq [EH () ].
Then, one obtains

A rl S k Rz2 ls N * r N * s N ijpq d 2 k = - 2 441 H (2,z) ijpq [EH (pol) ], A ln S k k l ∂ Rz2 rs ∂k n N * r N * s N ijpq d 2 k = 5 S k A + ln N * r N * s α l α n k ∂E 0 H (pol) rs ∂k -2A ln α l α s E 0 H rs N * r N * n N ijpq d 2 k = 4 441 -5H (2,z) ijpq [EH (pol) ] + H (2,z) ijpq [∂ k (kEH (pol) )] , A ln S k k l ∂ Re2 rs ∂k n N * r N * s N ijpq d 2 k = 0, A rl S k Re2 ls N * r N * s N ijpq d 2 k = - 12 441 H (2,z) ijpq [EH (dir) ].
Hence, the contribution of S k

H (dir) ij and H (pol) ij to S L(pol) ijpq is S L(pol2) ijpq (k, t) = 1 441 6H (2,z) ijpq [EH (dir) ] -4H (2,z) ijpq [EH (pol) ] + H (2,z) ijpq [∂ k (kEH (pol) )] . (C.
E 0 A + ln H () abrs N * l N * n α a α b α r α s N ijpq d 2 k = - 48 385 H (4) ijpq [EH () ], S k E 0 A + ln H () abrs N * r N * s α a α b α l α n N ijpq d 2 k = 32 1155 H (4) ijpq [EH () ].
Antisymmetric contributions arise from

S k E 0 A ln α l α b α r α s H () nbrs P ijpq d 2 k = 42 6615 H (4) ijpq [EH () ] + E A - il H () jpql + A - jl H () ipql + A - pl H () ijlq + A - ql H () ijlp , S k E 0 A ln α a α b N * l N * s H () abns N ijpq d 2 k = 2 21 H (4) ijpq [EH () ] + E A - il H () jpql + A - jl H () ipql + A - pl H () ijlq + A - ql H () ijlp , S k E 0 A ln α n α s α a α b N * l N * r H () abrs N ijpq d 2 k = 2 35 9 11 H (4) ijpq [EH () ] + E A - il H () jpql + A - jl H () ipql + A - pl H () ijlq + A - ql H () ijlp .
Then, one gets

A + ln S k Re4 ln P ijpq d 2 k = - 4 11 H (4) ijpq [EH (dir) ], A + ln S k Rz4 ln P ijpq d 2 k = - 6 55 H (4) ijpq [EH (pol) ], A rl S k Re4 ls N * r N * s N ijpq d 2 k = - 108 11 H (4) ijpq [EH (dir) ], A rl S k Rz4 ls N * r N * s N ijpq d 2 k = - 8 55 H (4) ijpq [EH (pol) ] + 2 5 E A - il H (pol) jpql + A - jl H (pol) ipql + A - pl H (pol) ijlq + A - ql H (pol) ijlp , A ln S k k l ∂ Re4 mm ∂k n P ijpq d 2 k = 8 11 H (4) ijpq [∂ k (kEH (dir) )] - 12 11 H (4) ijpq [EH (dir) ] + 4E A - il H (dir) jpql + A - jl H (dir) ipql + A - pl H (dir) ijlq + A - ql H (dir) ijlp , A ln S k k l ∂ Rz4 rs ∂k n N * r N * s N ijpq d 2 k = 8 5 2 11 H (4) ijpq [∂ k (kEH (pol) )] - 3 11 H (4) ijpq [EH (pol) ] + 2E A - il H (pol) jpql + A - jl H (pol) ipql + A - pl H (pol) ijlq + A - ql H (pol) ijlp , so that the contributions of H (dir)
ijpq and H

L(dir4) ijpq (k, t) = 1 11 2H (4) ijpq [∂ k (kEH (dir) )] -H (4) ijpq [EH (dir) ] + 3 5 H (4) ijpq [EH (pol) ] + E A - il H (dir) jpql + A - jl H (dir) ipql + A - pl H (dir) ijlq + A - ql H (dir) ijlp , (C.54) S L(pol4) ijpq (k, t) = 1 11 4 5 H (4) ijpq [∂ k (kEH (pol) )] - 2 5 H (4) ijpq [EH (pol) ] + 54H (4) ijpq [EH (dir) ] + 3 5 E A - il H (pol) jpql + A - jl H (pol) ipql + A - pl H (pol) ijlq + A - ql H (pol) ijlp . (C.55)

C.6.2 Fourth order non-linear transfers

The non-linear spherically-averaged directional and polarization transfers at the fourth-order are defined by

S NL(dir) ijpq (k) = 1 2 S k T E (k)P ijpq (k) d 2 k, S NL(pol) ijpq (k) = 1 2 S k T Z (k)N ijpq (k) d 2 k. (C.
S k P rs α i α j α p α q H () ijpq d 2 k = 0, S k α i α j N * p N * q N r N s H () ijpq d 2 k = 0.
For the λ-integration to compute S NL(dir) ijpq

, one needs

2π 0 α i α j α p α q H () ijpq dλ = π 4 α i α j α p α q H () ijpq (35z 4 -30z 2 + 3), 2π 0 α i α j (W * p W * q )H () ijpq dλ = 5π 4 (1 -y 2 )(7y 2 -1)α i α j α p α q H () ijpq .
This yields, after spherical-integration,

S NL(dir) ijpq (k, t) = 2 ∆ k θ kpq π 2 k 2 p 2 q(xy + z 3 )E 0 E 0 H (dir) ijpq (35z 4 -30z 2 + 3) + H (dir) ijpq (35y 4 -30y 2 + 3) -E 0 8H (dir) ijpq + H (dir) ijpq (35y 4 -30y 2 + 3) dpdq + 2 3 ∆ k θ kpq π 2 k 2 p 2 qE 0 (xy + z 3 )(1 -y 2 )(7y 2 -1)(E 0 -E 0 )H (pol) ijpq + z(1 -z 2 ) 2 (1 -7z 2 )E 0 H (pol) ijpq dpdq.
(C.57)

For the λ-integration to compute S NL(pol) ijpq

, one needs

2π 0 e -2iλ α r α s (W * p W * q )H () rspq dλ = π 2 α r α s N * p N * q H () rspq (1 -6y 2 + 7y 4 ), 2π 0 ie -2iλ α r α s (W * p W * q )H () rspq dλ = π 2 α r α s N * p N * q H () rspq z(5 -7z 2 ), 2π 0 e -2iλ α r α s α p α q H () rspq dλ = - π 2 α r α s N * p N * q H () rspq (1 -8y 2 + 7y 4 ),
and for the spherical integration

S k α r α s α p α q N * l N * n N i N j H () rsln d 2 k = 4 63 4πk 2 H () ijpq , S k α r α s N * l N * n N ijpq H () rsln d 2 k = 8 21 4πk 2 H () ijpq .
This gives where F ij is the Cauchy matrix, or the displacement matrix, which depends on the kind of anisotropy.

S NL(pol) ijpq (k, t) = 4 ∆ k θ kpq π 2 k 2 p 2 qE 0 z(5 -7z 2 )(y 2 -z 2 )E 0 H (pol) ijpq + xy(5 -7y 2 )(1 -z 2 )E 0 H (pol) ijpq + (E 0 -E 0 )H (pol) ijpq z(z 2 -1)(1 -6y 2 + 7y 4 ) + (xy + z 3 ) E 0 H (pol) ijpq (1 -6z 2 + 7z 4 ) -4E 0 H (pol) ijpq dpdq + 60 ∆ k θ kpq π 2 k 2 p 2 qE 0 z(1 -z 2 ) (1 -8y 2 + 7y 4 )(E 0 -E 0 )H (dir) ijpq + (1 -8z 2 + 7z 4 )E 0 H (
The initial Green's function is G ij (t 0 , t 0 ) = δ ij -α i α j = P ij (t 0 ), so that at each time k i G ij = 0. Finally, the second order spectral tensor is given by

φ ij (k, t) = G in (k, t, t 0 )G jm (k, t, t 0 )φ nm (k(t 0 ), t 0 ). (D.6)
For the sake of clarity, the following notations are used

K i = k i (t 0 ), K 2 0 = K 2 1 + K 2 2 + K 2 3 , K 2 ⊥ = K 2 1 + K 2 2 .
Turbulent shear flows: For a pure shear flow with gradient matrix A ij , the associated Cauchy matrix

F ij is A ij =   0 0 S 0 0 0 0 0 0   , F ij =   1 0 -St 0 1 0 0 0 1  
where S is the shear rate. The wavenumbers are k 1 (t) = K 1 , k 2 (t) = K 2 , and k 3 (t) = K 3 -StK 1 , with k 2 = k 2 1 + k 2 2 + k 2 3 . The three coupled equations to solve are consequently da 1 dt = S(2α 1 α 1 -1)a 3 (t) da 2 dt = 2Sα 1 α 2 a 3 (t) da 3 dt = 2Sα 1 α 3 a 3 (t). (D.7)

After some algebra, one gets Q 1 (t)a 3 (t 0 ) + a 1 (t 0 ), (D.9)

a 3 (t) = K 2 0 K 2 ⊥ + (K 3 -K 1 St
a 2 (t) = K 1 K 2 K 2 ⊥
Q 2 (t)a 3 (t 0 ) + a 2 (t 0 ), (D.10)

with

Q 1 (t) = - K 2 2 K 1 K ⊥ arctan StK ⊥ K 2 0 -StK 1 K 3 + StK 2 1 (K 2 0 -2K 2 3 + StK 1 K 3 ) K 2 0 k 2 (t) , (D.11) Q 2 (t) = K 2 0 K 1 K ⊥ arctan StK ⊥ K 2 0 -StK 1 K 3 + St(K 2 0 -2K 2 3 + StK 1 K 3 ) k 2 (t)
. (D.12)

Then, φ ij is computed thanks to (D.6) with the Green's function

G =   1 0 Q 1 K 2 0 /K 2 ⊥ 0 1 Q 2 K 1 K 2 /K 2 ⊥ 0 0 K 2 0 /k 2   (D.13)
and with the initial value φ ij (k(t 0 ), t 0 ) = E(k, t 0 ) 4πk 2 P ij .

D.2 Homogeneous Axisymmetric Turbulence

In this part, homogeneous axisymmetric turbulence (HAxT) is briefly addressed. Axisymmetric expansions (or contraction) are quite representative of grid turbulence: this is why this configuration has received some interest. Furthermore, the axisymmetric case is less restrictive than the isotropic one, but still presents some interesting symmetries for tensorial developments: see for instant the local axisymmetry theory of [START_REF] George | Locally axisymmetric turbulence[END_REF]. The case of a maintained axisymmetric turbulence is nevertheless not addressed: indeed, as discussed in [START_REF] Sagaut | Homogeneous Turbulence Dynamics[END_REF], when this kind of anisotropy is forced, the turbulence becomes 1D or 2D (depending on the compression or dilatation case), and the present 3D modelling cannot handle such singular flows. In general, the flow experiences a contraction (or expansion) and then freely decays, thus progressively returning to isotropy. In the expansion case, the mean velocity gradient matrix is The initial conditions are detailed in the appendix of [START_REF] Davidson | On freely decaying, anisotropic, axisymmetric saffman turbulence[END_REF]: runs 11 and 12 and considered here (initial isotropic turbulence submitted to a contraction and expansion respectively). The streamwise direction is () 33 : R 33 is noted u 2 and R 11 = R 22 = u 2 ⊥ . The initial Reynolds number Re ⊥ (0) is based on the integral scale l ⊥ , and the corresponding Taylor Reynolds number is computed with Re λ = 20Re ⊥ /3: simulations show that a slightly higher or lower Re λ (0) has no significant influence on the results. The characteristic time is defined as T = 1/( q 2 k L (0)). Firstly, the ratio u 2 /u 2 ⊥ is presented in both cases of expansion and contraction. A good agreement is obtained in the transition zone t < 50T , and for the asymptotic values as well, where the relative error is 3.5%. Then, the decay of the streamwise and spanwise energies u 2 and u 2 ⊥ is investigated for the expansion: despite a slight discrepancy at small t/T , the t -6/5 decay of Saffman turbulence is well-recovered.

A exp ij =   S 0 0 0 S 0 0 0 -2S   .

D.3 Homogeneous Plane Distortion

Here, the case homogeneous distortion-released turbulence (HDRT) is highlighted. In this framework, the mean-velocity gradient matrix reads

D.4 Pressure fluctuations in HAT

In this section, the pressure field is investigated. The main equations are derived in spectral space, and then applied in both HIT and HST. The contents of this part were included in the publication in Physical Review Fluids.

-∆p = ∂ 2 u i u j ∂x i ∂x j + 2A ij ∂u j ∂x i .
(D.17)

The Fourier transform yields p(k, t) = -α i α j u i u j (k, t) + 2i k A ij α i ûj (k, t).

(D.18)

The spectral two-point second-order pressure correlation is defined as The spectral pressure correlation is computed according to pp * = α i α j α p α q u i u j u p u q * + 4 α i α p kp A ij A pq ûj û * q + 2i α p p A pq û * q α i α j u i u j -

E P (k,
α i k A ij ûj α p α q u p u q * .
The latter term will bring no contribution during the spherical integration and is thus discarded from here. Ensemble average gives

E P (k)δ(k -p) = 4 α i α p kp A ij A pq Rjq δ(k -p)
+ α i α j α p α q < ûi (r)û j (s)û * p (v)û * q (w) > δ(k -rs)δ(pvw)d 3 rd 3 vd 3 sd 3 w.

The quasi-normal procedure is then used: the integral gives three terms that are products of two spectral Reynolds tensors. After integration over r and p (the latter erases the Dirac functions) only two terms remain that are equal, so that E P (k, t) = 2α i α j α p α q k=p+q Riq (p, t) Rjp (q, t)d 3 p + 4 α i α p k 2 A ij A pq Rjq (k, t). (D.21)

For the λ-integration, relations of Appendix C are used. At first order in anisotropy, five terms remain from the integration. Four of them are like α i α j H () ij and thus bring no contribution to the spherical average. The explicit expression of the pressure spectrum is then

E P (k, t) = 16π 2 ∆ k kpq(1 -y 2 )(1 -z 2 )E 0 E 0 dpdq + 4 E k 2 1 5 A + ij A + ij + 1 3 A - ij A - ij -H (dir) il 6 7 A + ij A + jl + 2A - ij (A - lj + 2A + lj ) + 8 E k 2 H (pol) il 3 7 A + ij A + lj + A - ij A - lj - 2 3 A + lj . (D.22)
The first contribution is purely isotropic and is therefore referred to as E (iso) P

(also called the turbulenceturbulence interaction). Whereas the second contribution E (S) P arises from velocity gradients and is quadratic in A ij (also called turbulence-mean-shear interaction). E (iso) P is in agreement with the one derived by [START_REF] George | Pressure spectra in turbulent free shear flows[END_REF]. There may be a factor 2 missing in the more recent work of Meldi & Sagaut (2013b) regarding E (iso) P .

D.4.2 Spectrum and pressure variance

First, the case of HIT is addressed. From the expression of E (iso) P , or dimensional analysis, it directly follows that E 1.44, which is close to the Kolmogorov constant, and this is expected since E (S) P scales in E/k 2 in (D.22). Furthermore, the value C (S) P 1.44 is in good agreement with the prediction [START_REF] George | Pressure spectra in turbulent free shear flows[END_REF], where the constant would be C (S) P = 16K 0 /15 = 1.40.

The compensated isotropic pressure spectra E (iso) P k 7/3 -4/3 for shear flows and USHT are presented in Fig. D.7b and 7.13b: the plateau settles around 2.5 for shear and 2.3 for USHT, which is quite close and proves some universality of the isotropic pressure spectrum between two completely different flows. Furthermore, these values are in good agreement with the prediction of [START_REF] George | Pressure spectra in turbulent free shear flows[END_REF], where the constant would be C P = 1.32K 2 0 = 2.27, close to our result.

Finally, the pressure variance

K P (t) = ∞ 0 E P (k, t)dk = K

D.5 Details on helical turbulence

In this section, details are provided regarding (i) the computation of the non-linear transfers involving the helical spectrum, (ii) the simplification of the evolution equation of the helical dissipation rate H , and (iii) the wavenumber k H η defined in (8.22). Before that, the equation of the physical and spectral fluctuating vorticity are given explicitly. A careful attention has to be given to the order of the index for the spectral Reynolds tensor which is not symmetric anymore. Then, the products of Reynolds tensors generate imaginary and real parts.

Only the imaginary part is computed here since the other terms bring no contributions to the spherical integration. The different contributions are

2k l k n ijl P ipq R pj R qn = ik 2 p E 0 H k (z(x 2 -1) -(xy + z 3 )) + E 0 H q (x + yz) , 2k l k n ijl P jpq R pn R qi = 2ik 3 E 0 H p z(1 -2y 2 ) -xy , 2k l k n ijl P npq R pi R qj = ik 2 p E 0 H k (z(1 -x 2 ) -(xy + z 3 )) + E 0 H q (x(2z 2 -1) + yz) .
The different relations used intensively to obtain these compact forms, in addition to the ones presented in Appendix C, are α j P jp = 0, ijl P ij = 0, α i α n P in = 1 -z 2 , α i α p P in = 1 -x 2 , α i α n P in = 1 -y 2 , α n α i P in = -(y + xz), α j α p P jp = -(x + yz), α i α q P iq = -(z + xy), 2xyz = 1 -x 2 -y 2 -z 2 .

Finally, the key relation to use in order to obtain helical transfers similar to the classical isotropic kinetic ones, is for the E 0 H /p term. One can show that k 2 (y -xz -2yz 2 ) = q 2 (xz + y 3 ) -p 2 z(x + yz). (D.31)

This allows, combined with p ↔ q symmetrization, to gather the three contributions. For a skew-isotropic flow, the λ and spherical integrations reduce to This gives 2k l P imn R mi R nl = q p HH (x + yz) = p q HH (x + yz),

2k l P imn R ml R ni = -k 2 H H pq (x + yz), 2k l P lmn R mi R ni = - p q HH (x -yz -2xz 2 ).
The key relations to use are

y + xz = k p (x + yz),
k 2 H H pq (x + yz) = 2H H k q z(x + yz) (using p ↔ q symmetry).

This results into (8.14).

D.5.3 Details on the evolution equation of H

In this part, details about the calculations and algebra used in section 8.3.2 are gathered. The evolution equation of H is obtained starting from the equations of u i and ω i (2.1) and (D.27) in HHT, so that

∂ ∂t < ∂u i ∂x j ∂ω i ∂x j > + < ∂ 2 u l ω i ∂x j ∂x l ∂u i ∂x j > + < ∂ 2 u l u i ∂x j ∂x l ∂ω i ∂x j > -< ∂ 2 u i ω l ∂x j ∂x l ∂u i ∂x j > D[u,ω] = -< ∂ 2 p ∂x i ∂x j ∂ω i ∂x j > +ν < ∂ 3 ω i ∂x j ∂x l ∂x l ∂u i ∂x j > + < ∂ 3 u i ∂x j ∂x l ∂x l ∂ω i ∂x j > .
To simplify the dissipative term, one uses < ∂ 2 ll (∂ j u i ∂ j ω i ) >= 0. The pressure term is zero since ω i is solenoidal and < ∂ i (∂ j ω i ∂ j p) >= 0. The term D[u, ω], arising from the non-linearity, is the production term of H . One needs to expand < ∂ l (u l ∂ j u i ∂ j ω i ) >= 0 and < ∂ l (ω l ∂ j u i ∂ j u i ) >= 0, which gives where the two expressions are linked using homogeneity and the definition of ω i . Then, a Taylor expansion of h(r) for r → 0 in (D.34) gives H = 10νh (0), with h (0) = 0 since H is finite. Finally, Taylor expansions in (D.35) give, using only h(0) = 0 since the h (0) and h (0) terms vanish, δ il r j r n r p r q + δ jl r i r n r p r q + 1 r 2 2 S r 2 -10 S r 3 + 16 S r 4 r i r j P

<
<
(3) lnpq + r i r l P

(3) jnpq + r j r l P

(3) inpq + r l r p P

(3) ijnq + r l r q P

(3) ijnp + r l r n P

(3) ijpq + r i r p (δ jn δ lq + δ ln δ jq ) + r i r q (δ jn δ lp + δ ln δ jp ) + r j r p (δ in δ lq + δ ln δ iq )+ r j r q (δ in δ lp + δ ln δ ip ) + r i r n (δ jp δ lq + δ lp δ jq ) + r j r n (δ ip δ lq + δ lp δ iq )+ δ ij (r p r n δ lq + r q r n δ lp + r p r q δ ln ) -1 r 2 S r -2 S r 2 -2 S r 3 + 8 S r 4 r n r p (δ il δ jq + δ jl δ iq ) + r n r q (δ il δ jp + δ jl δ ip )+ r p r q (δ il δ jn + δ jl δ in ) + r j δ il (r n δ pq + r p δ qn + r q δ pn ) + r i δ jl (r n δ pq + r p δ qn + r q δ pn ) + 2 S r 3 -4 S r 4 δ in (δ jq δ lp + δ jp δ lq ) + δ jn (δ iq δ lp + δ ip δ lq ) + δ ij (δ lq δ pn + δ lp δ qn ) + δ ln P Using as before a Taylor expansion of S(r) when r → 0, the S (0), S (0) and S (0) terms vanish, and with S(0) = 0, this yields (8.66). Furthermore, using (D.44), one obtains

∂ 3 φ (uuω) 111 ∂r 3 1 r=0 = 2 < ∂u 1 ∂x 1 2 ∂ω 1 ∂x 1 > +2 < u 1 ∂ 2 u 1 ∂x 2 1 ∂ω 1 ∂x 1 >=< ∂ 2 u 2 1 ∂x 2 1 ∂ω 1 ∂x 1 > . (D.50)
The two previous equations give (8.67).

D.5.4 Re-interpretation of the helical viscous cutoff k H η

In this section, another interpretation of the wavenumber k H η , defined in (8.22), is proposed. This wavenumber was originally derived by [START_REF] Ditlevsen | Dissipation in helical turbulence[END_REF] as a helical viscous cutoff. However, it was revealed in Fig. 8.1b that k H η is not a wavenumber equivalent to k η for the helical spectrum, since both E and H have a k -5/3 inertial range which extends up to k η .

Here are some elements, based on the demonstration of [START_REF] Ditlevsen | Dissipation in helical turbulence[END_REF], to explain why k H η cannot be a helical viscous cutoff. First, it is assumed in the latter reference that H(k, t) scales as kE(k, t) in the spectral definition of H , and that the dominant contribution of the integral comes from the largest wavenumber, which gives (D.51) and thus recovers (8.22). However, a scaling like H ∼ kE would imply that H ∼ k -2/3 in the inertial range, which is not the case as illustrated in Fig. 8.1b. Hence, the assumption H ∼ kE in the inertial range is wrong, and moreover, it is worth noting that if the scaling (8.20) is used in the previous integral, one recovers k H η = k η , as shown by [START_REF] Chen | The joint cascade of energy and helicity in three-dimensional turbulence[END_REF].

H = 2ν ∞ 0 k 2 H(k, t)dk ∼ νk H η 4 E(k H η , t) = νk H η 7/3 2/3 ,
Instead, it is shown hereafter analytically that k H η can be seen as the wavenumber at which viscous dissipation of helicity balances non-linear helical transfers. In the kinetic and helical Lin evolution equations (8.15) and (8.16) η is shown to be quite relevant in Fig. 8.9 at large Reynolds numbers: indeed, from k H η , there is a balance between -2νk 3 H and kS NL H . However, this is much less relevant at moderate Reynolds numbers: this is expected since in the previous demonstration, inertial scalings were used, which are valid only at large Reynolds numbers.

As a conclusion, the original helical viscous cutoff k H η proposed by [START_REF] Ditlevsen | Dissipation in helical turbulence[END_REF] was reinterpreted in terms of high Reynolds numbers balance between the viscous dissipation of helicity and non-linear helical transfers. E.1.2 EDQNM closure for E T Some details about how to obtain the closure (4.24) for T T,QN i (k, p) are given. Firstly, the fourth-order correlation is defined

T T jl (k, p, t)δ(k + p + q + v) = -i < ûj (q) θ(k) θ(p)û l (v) > . (E.1)
The process is slightly different from the purely kinetic one since the relation is not symmetric. The third-order scalar correlation S T j (k, p) evolution equation can be written as There is no need to go further for the evolution equation. Then, the quasi-normal approximation is used, consisting into neglecting the fourth order cumulants. The first rhs term gives k=r+s F l (r)F j (p)δ(r + s)δ(p + q)d 3 r + k=r+s E T (p) Rjl (q)δ(r + p)δ(s + q)d 3 r + k=r+s F j (r)F l (p)δ(r + q)δ(p + s)d 3 r = k l δ(k + p + q) E T (p) Rjl (q) + F * j (q)F l (p) .

The second term yields p=r+s F l (r)F j (k)δ(r + s)δ(k + q)d 3 r + p=r+s E T (k) Rjl (q)δ(r + k)δ(s + q)d 3 r + p=r+s F j (r)F l (k)δ(r + q)δ(k + s)d 3 r = p l δ(k + p + q) E T (k) Rjl (q) + F * j (q)F l (k) . Finally, using relations such as p l R jl = -k l R jl , one recovers the previous expression of T T,QN i (k, p). The main term to compute is then k i T T,QN i . To do so, the same method as in the purely kinetic case is applied: using k l R ln k n = kp(xy + z) E + X , and relations such as k i α i = -ky, q n F n = -p n F n and q n F n = -k n F n , one gets

k i T T,QN i = 2kp(xy + z)(E + X )(E T -E T ) + k n F * n (p m F m + k m F m ) + p m F m k n F n ky -px q .
The non-linear scalar transfer T T,NL (k, t) of (4.26) is thus recovered.

E.1.3 Spherically-averaged scalar Lin equations

Now, as in the kinetic case, spherical integrations are performed on this non-linear scalar transfer term to transform the (k, t) dependence into a (k, t) one. All quadratic contributions such as E E T , X E T , ... are discarded in the moderate anisotropy framework. Finally, products of the cospectrum flux in (4.26) like F * i F j and F i F j are neglected as well: indeed, since F i is zero in the isotropic case, it is a purely anisotropic quantity and thus quadratic contributions can be neglected.

The definition of the non-linear isotropic scalar transfer is given by (4.39). From (4.26), only six terms remain after the λ-integration. The two relations of use are

S k α i α j H () ij d 2 k = 0, S k H () mn α m α n P ij d 2 k = - 8 15 πk 2 H () ij .
There is only one term left from the λ-integration. The non-linear isotropic scalar transfer (4.40) is recovered with the first of this formula. The non-linear directional scalar transfer is defined by (4.41). The λ-integration also gives six terms: one of them is simplified when the isotropic part is subtracted. One has to use the second formula of spherical average to obtain (4.42).

The linear isotropic scalar transfer S T,L(iso) defined in (4.43) is computed using previous relations of Appendix C such as

E.1.4 Scalar quadratic anisotropic contributions

The quadratic anisotropic contributions in the non linear scalar transfer (4.26) are computed analytically.

The calculations involve expressions given in Appendix C for the kinetic case. After some algebra, one gets the scalar second-order isotropic term Q T,NL(iso) (k, t) = 12

∆ k θ T kpq π 2 k 2 pq(x + yz) E F i (kxE F i -pyE F i ) + zE F i E F i (ky -px) dpdq + 120 ∆ k θ T kpq π 2 k 2 p 2 q(xy + z)E 0 2H (dir) ij E T 0 H (T ) ij (3x 2 -1) -E T 0 H (T ) ij (3y 2 -1) -H (pol) ij E T 0 H (T ) ij (1 -x 2 ) -E T 0 H (T ) ij (1 -y 2 ) dpdq, (E.2)
and the scalar second-order directional term where

E F , E F ij = E F i E F j + E F j E F i - 2 3 E F l E F l δ ij .

E.2 Scalar-velocity correlation F

In this section, the spectral velocity-scalar correlation is addressed: the quasi-normal procedure along with the calculations of the linear and non-linear transfers are detailed. An alternative modelling for the scalar flux is proposed as well, and details are provided for the additional contributions linked to helicity in HHTSG.

E.2.1 Craya equation for the cospectrum flux

The scalar-velocity correlation is defined in (4.9). Its evolution equation is obtained by multiplying (4.3) by û * i (p) and summing it to the evolution equation of û * i (p) multiplied by θ(k). After ensemble average and integration over the whole domain to simplify δ(k -p), one has E.2.2 Quasi-normal approximation for F i Some details on how the closure (4.25) is obtained are given. Firstly, the fourth-order correlation is defined as T F ijl (k, p, t)δ(k + p + q + v) = -i < ûi (q) θ(k)û j (p)û l (v) > .

∂ ∂t -A jl k j ∂ ∂k l + (ν + a)k 2 F i + M ij F j -k j λ l ∂ Rij ∂k l = T F,NL
(E.4)

The process is similar to the scalar one. The third-order scalar flux correlation S F ij (k, p) evolution equation can be written as

∂ ∂t
+ ak 2 + ν(p 2 + q 2 ) θ(k)û j (p)û i (q) + ... = -i k l k=r+s θ(r)û j (p)û l (s)û i (q)d 3 r + P jmn (p) p=r+s θ(k)û n (s)û m (r)û i (q)d 3 r + P imn (q) q=r+s θ(k)û m (r)û n (s)û j (p)d 3 r , which becomes, after ensemble average and convolution rules ∂ ∂t + ak 2 + ν(p 2 + q 2 ) S F ij (k, p, t)δ(k + p + q) + ... = k l k=r+s T F ijl (r, p)δ(k + p + q)d 3 r + P jmn (p) p=r+s T F imn (k, r)δ(k + p + q)d 3 r+ + P imn (q) q=r+s T T mjn (k, p)δ(k + p + q)d 3 r.

The quasi-normal approximation yields calculations very similar to the scalar case. The three terms are respectively δ(k + p + q) k l F * j (p) Ril (q) + F * i (q) Rjl (p) + 2P jmn (p) Rin (q)F m (k) + 2P imn (q) Rjn (p)F m (k) .

E.2.3 Computation of the non-linear transfers of F i

The non-linear scalar flux transfer is defined by (4.14). The following calculations allow to recover (4.47).

Computation of τ F i (k, p): the starting point is

τ F i (k, p) = k n k j R nj F * i + R ni F * j + 2F m k j P jmn R ni + P imn R nj . (E.5)
Symmetry for the second rhs term can be used thanks to θ F kpq = θ F kqp . Useful relations are k l R ln k n = kp(xy + z) E + X , k n R ni = kE 0 (α i + zα i ), q -ky = px, k 1 -y 2 1 -z 2 = kq p (1 -y 2 ) = k(x + yz) = q(xy + z), α l α n R ln = (1 -z 2 )(E + X ).

The terms to compute are, at first order in anisotropy

k n k j R nj F * i = 3 2 kp(xy + z)E 0 E F j P ij , k n k j R ni F * j = 3 2
k 2 E 0 (α i + zα i )E F j (α j + yα j ),

k j F m P jmn R ni =
3 4 kE 0 (α i + zα i )E F j (α j + yα j )(q -2ky), (α i + yα i )(α j + zα j ) + (1 -y 2 ) P ij -2α i (α j + zα j ) .

k j F m P imn R nj = 3 4 kqE 0 E F j (α i + zα i )(
Computation of W F i (k, p): one has W F i = -α i α j τ F j . The two first terms can be grouped together

α i α j k n k l ( R nl F * j + R nj F * l ) = 3k 2 (1 -y 2 )E 0 E F j α i (α j + zα j ).
With symmetrization, there is only one term left to compute 2α i α j F m k l P jmn R nl = 3 2 kqE 0 E F j α i 1 -z 2 -2y(y + xz) (α j + yα j ).

E.2.4 Spherically-averaged cospectrum Lin equations

The useful parts of the λ-integration are the following ones

P ij dλ = π(1 + z 2 )δ ij -πα i α j (3z 2 -1).
The term in E 0 E F j of W F i brings no contribution to the λ-integration. For the spherical integration, the different terms are the following ones

S k M ij F j d 2 k = 2 5 A + ij E F j , S k A jl k j ∂F i ∂k l d 2 k = - 1 5 A + ij ∂ ∂k (kE F j ), S k P ij E F j d 2 k = 8πk 2 3 E F i .
The "rapid-pressure" part is given by

S k 2α i α n A nj F j d 2 k = 3 5 A + ij E F j + A - ij E F j .

E.2.5 Alternative modelling for F

The modelling for the scalar flux is now based on a helical decomposition and reads

F i (k, t) = E F j (k, t)P ij (k, t) = φ + (k, t)N i (k, t) + φ -(k, t)N * i (k, t). (E.6)
The scalar flux is solenoidal so that the φ ± functions read (k, p) = k n Rni (q)F * j (p) + Rnj (p)F * i (q) + 2F m (k) P imn (q) Rnj (p) + P jmn (q) Rni (p) .

φ + (k, t) = 1 2 E F j (k,
Additional useful results are 1 -y 2 1 -z 2 = (x + yz), kx -qz = y(qx -kz), 4kyz + 2kx -2qz = 2py,

The different parts of the computation are the following ones

k n k j R nj F * i N * i = -k 2 (1 -y 2 )E 0 e -iλ (Y * -+ zY * + ), k n k j R nj F * i N i = k 2 (1 -y 2 )E 0 e iλ (Y * --zY * + ), k n k j R ni F * j N * i = k 2 z(x + yz)E 0 e -iλ Y * + , k n k j R ni F * j N i = k 2 z(x + yz)E 0 e iλ Y * + , 2F m k j R ni P jmn N * i = k(2ky -q)z(x + yz)E 0 e -iλ Y + , 2F m k j R ni P jmn N i = k(2ky -q)z(x + yz)E 0 e iλ Y + , 2F m k j R nj P imn N * i = -k 2 (1 -z 2 )E 0 e -iλ (Y + + Y -) + k 2k(x + yz) -qz (x + yz)E 0 e -iλ Y + , 2F m k j R nj P imn N i = -k 2 (1 -z 2 )E 0 e iλ (Y + -Y -) + k 2k(x + yz) -qz (x + yz)E 0 e iλ Y + .
Hence, the p ↔ q symmetry for the E 0 part -valid thanks to θ F kpq = θ F kqp -gives Computation of τ * i (p, k): here, no symmetry can be performed because of the θ F pkq . The different parts of the computation are the following ones

p n k j R nj F i N * i = -k 2 (1 -y 2 )E 0 e -iλ (Y + + Y -), p n k j R nj F i N i = -k 2 (1 -y 2 )E 0 e iλ (Y + -Y -),
T F + (k, t)N i (k, t) + T F -(k, t)N * i (k, t) dλd 2 k = ∆ k 8 3 π 2 θ F kpq k 2 pqE 0 k -2(1 -z 2 )E F i + (1 + y 2 -z 2 -xyz -2y 2 z 2 )E F i + 2py(x + yz)E F i dpdq + ∆ k 8 3 π 2 θ F pkq k 2 pq kE 0 (1 -y 2 + z 2 -xyz -2y 2 z 2 )E F i -2(1 -y 2 )E F i -pE 0 y(x + yz)E F i + z(x 2 -y 2 )E F i dpdq. (E.16)
Moreover, k(1 -z 2 ) = q(y + xz) and p(x + yz) = q(1 -y 2 ) so that -2k(1 -z 2 )E F i + 2py(x + yz)E F i = -2q(xz + y 3 )E F i . In addition, p(x 2 -y 2 ) comes from p(1 -z 2 -2(y 2 (1 -z 2 ) + yz(x + yz))) which can be written q(x-yz -2xy 2 ). Consequently, the non-linear transfer (4.49) computed directly with F i ∼ P ij E F j is recovered. Here are some remarks on this modelling:

• The helical decomposition is not sufficient to completely compute the scalar flux non-linear transfer: indeed, a model is required for φ + and φ -.

• The calculations are more complicated since they involve helical modes instead of projectors.

• All calculations are made twice (once for T F + and once for T F -which are eventually equal) whereas only a single process is needed when directly injecting F i = P ij E F j in the closure. • The helical decomposition does not permit to compute easily the pressure part of the non-linear transfers.

• It brings information on the toroidal-poloidal structure of the scalar flux: any solenoidal field, such as the scalar flux, can be decomposed, in the Craya-Herring frame (e 1 i , e 2 i ), into toroidal and poloidal parts F i (k, t) = F tor (k, t)e 1 i (k, t) + F pol (k, t)e 2 i (k, t). (E.17) Thus, using (E.6) and the definition (2.27) of the helical modes, the scalar flux helical decomposition reads

F i (k, t) = i φ -(k, t) -φ + (k, t)
Ftor(k,t)

e 1 i (k, t) + φ + (k, t) + φ -(k, t)

F pol (k,t)
e 2 i (k, t).

(E.18)

The previous calculations showed that φ + and φ -led the same contribution to the non-linear transfer (T F + N i + T F -N * i = 2T F + N i ). Consequently, one can conclude that the scalar flux has a poloidal structure only. Finally, since E ∼ (toroidal) + (poloidal) and Z ∼ (poloidal) -(toroidal), a single quantity only is needed to describe anisotropy at the scalar flux level.

E.2.6 Scalar flux quadratic anisotropic contributions

Here, the second-order contributions in anisotropy of the non linear scalar flux transfer are computed analytically. The calculations are quite lengthy and involve some expressions gathered in Appendix C for the kinetic case. Afterwards, some illustrations for USHT are proposed. Here are some additional useful formula for the λ-integration: 2π 0 α l α p α q α i H () pq dλ = α i α l α p α q H () pq z 3 -5z 2 + 2z(z 2 -1)α i α p H () pl , 2π 0 α l α p α q α i H () pq dλ = α i α l α p α q H () pq -5yz 2 + y -2xz + 2z(x + yz)α i α p H () pl , 2π 0 α i α p α q α l H () pq dλ = α i α l α p α q H () (W p W q )α l α i H () np dλ = α i α l α p α q H () pq 5yz 2 -3z + 2xy -2yα i α p H () pl x + yz .

pq
The total second-order transfer for the scalar flux can be divided into two parts according to 2F m k j P imn R nj = -

Q F,NL i (k, t) = Q F,cons i (k, t) + Q F,RTI i (k,
3 2 p q H -E Q i (x + yz) + α i E Q l (xα l + yα l -2zα l ) + 2α i E Q l (xzα l + (x + yz)α l -z 2 α l ) + α i E Q l (zα l + α l ) .
Computation of τ F * i (p, k): no p ↔ q symmetry is used here

p n k j R * nj F i = 3 2
kp q H α i E F l (yα l -zα l ) -α i E F l (yα l + α l ) + α i E F l (zα l + α l ) ,

p n k j R * ni F j = 3 2 pH E F i (x + yz) + α i E F l (-xα l + yα l ) + α i E F l (zα l + α l ) , 2F * m k j P jmn R * ni = - 3 2 qH E F i (x -yz -2xy 2 ) + α i E F l (-xα l + zα l ) + α i E F l (-yα l + (1 -2y 2 )α l ) + 2yα i E F l (-xα l + zα l ) , 2F * m k j P imn R * nj = - 3 2 k 2 q H E F i (x + yz) + α i E F l (-xα l + zα l ) + α i E F l (yα l + α l ) .
Computation of i ijl k l τ F j (k, p)

i ijl k l k n k p R np F * j = 3 2 k 3 p (1 -y 2 )E 0 (α i α l E Q l + zE Q i ), i ijl k l k n k p R nj F * p = 3 2 k 3 q zE 0 α i E Q l (-yα l + xα l ) -E Q i (x + yz) -α i E Q l (zα l + α l ) + 3 2 k 3 p H E F l (α l + yα l )(zα i + α i ), 2i ijl k l k r F m R nj P rmn = - 3 2 kqE 0 E Q i (xz -y + yx 2 + y 3 ) + α i E Q l (α l (2y -xz -2x 2 y)
+ α l (yz + 2xy 2 ) + 2y 2 α l ) + α i E Q l (α l (1 -x 2 + y 2 ) + α l (2xy + z) + 2yα l ) + 3 2 k 2 q p H E F l zα i (yα l + α l ) + 2yzα i (yα l + α l ) + α i (yα l + α l )(1 -2y 2 ) , 2i ijl k l k r F m R nr P jmn = -+ α l 2z(xy + z) + yzα l ) + α i E Q l (α l (1 -x 2 + z 2 ) + α l 2(xy + z) + yα l ) + 3 2 k 2 p q H E F l α i (-2zα l (y + xz) + α l (y + 2xz) + zα l ) -(yα i + α i )(α l + zα l ) + α i (α l (y -2xz -2yz 2 ) + α l 2(x + yz) + α l ) + α i (α l z(2z 2 -3) + α l (1 -2z 2 ))

Computation of i ijl k l τ F * j (p, k)

i ijl k l p n k p R * np F j = 3 2 kp(xy + z)E 0 (α i α l E Q l -E Q i ) + 3 2 k 2 p q H E F l α i (yα l -zα l ) -α i (yα l + α l ) + α i (zα l + α l ) , i ijl k l p n k p R * nj F p = 3 2 kpHE F l (α i + zα i )(α l + yα l ) - 3 2 k 2 p q E 0 α i E Q l (xα l -yα l ) -α i E Q l (zα l + α l ) -E Q i (x + yz) , 2i ijl k l k p F * m R * nj P pmn = 3 2 kqHE F l (yα i + α i )(α l + (2xy + z)α l + 2yα l ) + 3 2 k 2 q p E 0 α i E Q l (xα l -zα l ) + α i E Q l (yα l + (2y 2 -1)α l ) + α i (2xyα l -2yzα l ) + E Q i (-x + yz + 2xy 2 ) , 2i ijl k l k p F * m R * np P jmn = 3 2 k 3 q H E F l (yα i + α i )(α l + zα l ) + 3 2 k 3 p E 0 (1 -y 2 )(zE Q i + α i α l E Q l ) + y(α i E Q l (xα l -zα l ) -α i E Q l (yα l + α l ) -E Q i (x + yz) .
For W F i (k, p): there is no contribution after the λ-integration for S F,NL(hel) i

, and the RTI transfer vanishes for S Q,NL i . Finally, all the formula for the λ and spherical integrations have been previously given so that the final steps are straightforward. One useful relation is kpq 2 (x -yz -2xy 2 ) = kpq(x + yz)(q -2yk).

  equations of homogeneous isotropic turbulence . . . . . . . . . . . . . . . . . 9 1.2 The inertial scaling of E T for P r 1 . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Mixed-derivative skewness S T for P r 1 . . . . . . . . . . . . . . . . . . . . . . 1.4 Time evolution of scalar integrated quantities . . . . . . . . . . . . . . . . . . . . 1.4.1 The basics of the CBC dimensional analysis . . . . . . . . . . . . . . . . . 1.4.2 Validation at large Reynolds numbers for P r = 1 . . . . . . . . . . . . . . 1.4.3 Transition to low Reynolds and Péclet numbers . . . . . . . . . . . . . . . 1.4.4 Transition for P r = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.5 Study of the integral scales L and L T . . . . . . . . . . . . . . . . . . . . 1.5 Conclusions for a passive scalar field in HIT . . . . . . . . . . . . . . . . . . . . . 2 Spectral Modelling of the Velocity Field in Homogeneous Turbulence 33 2.1 Equations in physical space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Spectral equations and transfers . . . . . . . . . . . . . . . . . . . . . . . .

Figure 2 :

 2 Figure 2: Schematic view of the different methods available to study turbulence. The Reynolds numbers reachable by simulations roughly increases from DNS to LES, RANS and EDQNM.The complexity of a three-part signal, corresponding to RANS, LES and DNS, is also presented for illustration purposes. Illustrations for RANS and DNS were taken online, whereas the one for LES if from[START_REF] Chollet | Parametrization of small scales of three-dimensional isotropic turbulence utilizing spectral closures[END_REF].

Figure 1 . 1 :

 11 Figure 1.1: Scalar variance spectrum E T (k, t) for small Prandtl numbers in Saffman turbulence (σ = 2), along with the scalar integral, Corrsin-Obukhov, Convective-Diffusive, and Kolmogorov wavenumbers k T , k CO , k CD , and k η . The wavenumber k CD is defined in (1.13). (a) P r = 10 -n , n = [0, . . . , 6]; (b) P r = 1 and P r = 10 -4 with the different ranges; (c) Characteristic time n -1 c of the IBR, defined in (1.18), for P r = 10 -4 .

Figure 1

 1 Figure 1.2: (a) Comparison of S and S T between EDQNM (lines) and experiment of Zhou et al. (2000) (symbols) at Re λ 50 and P r = 0.7. (b) Review of different values for |S T | obtained in DNS[START_REF] Kerr | Higher-order derivative correlations and the alignment of small-scale structures in isotropic numerical turbulence[END_REF][START_REF] Yeung | Schmidt number effects on turbulent transport with uniform mean scalar gradient[END_REF][START_REF] Antonia | Similarity of decaying isotropic turbulence with a passive scalar[END_REF] and experiments[START_REF] Zhou | Transport equations for the mean energy and temperature dissipation rates grid turbulence[END_REF]: thick lines for EDQNM at P r = 1 and P r = 10 4 . (--) indicates the asymptotic P r-state S ∞ T at very large Re λ and P r. For Yeung et al. (2002) (×): the values of S T presented are in the plane perpendicular to the mean scalar gradient, the Prandtl number is 1 ≤ P r ≤ 64, and the P r = 1 results are linked by a dash-dot (-•) line.

Figure 1

 1 Figure1.3: (a) Scalar spectrum E T (k, t) for various Prandtl numbers P r = 1, 10 3 and 10 5 , at Re λ = 10 3 in Saffman turbulence. The inertial convective k -5/3 and viscous-convective k -1 ranges are displayed as well, along with the integral, Kolmogorov and Batchelor wavenumbers k T , k η and k B for P r = 10 5 . (b) Absolute value of the mixed-derivative skewness S T for various Prandtl numbers from 1 to 10 5 in Saffman turbulence. Because of the high-P r saturation, the P r = 10 4 and P r = 10 5 curves are hardly distinguishable.

Figure 1 . 4 :

 14 Figure 1.4: Velocity derivative and mixed-derivative skewnesses S and S T from high to low Reynolds numbers in the saturated P r-state at P r = 10 4 . (a) In black for Saffman turbulence σ = σ T = 2, and in grey for Batchelor turbulence σ = σ T = 4. (b) Batchelor turbulence, along with correlations in grey that capture well the high Reynolds numbers regime and the beginning of the transition zone. ×: values of S from forced turbulence DNS of Gotoh et al. (2002).

Figure 1

 1 Figure1.5: Schematic view of the (Re λ , P r) combinations used for the passive scalar decay analysis. On the main figure, triangles, squares, crosses and circles respectively refer to works of[START_REF] Rust | Turbulent temperature fluctuations in mercury and ethylene glycol in pipe flow[END_REF];[START_REF] Granatstein | Fluctuation spectrum of a plasma additive in a turbulence gas[END_REF];[START_REF] Yeung | Schmidt number effects on turbulent transport with uniform mean scalar gradient[END_REF][START_REF] Yeung | Simulations of three-dimensional turbulent mixing for schmidt numbers of the order 1000[END_REF]. On the zoom of the small grey region at P r ∼ 1, experimental scalar decay exponents of Fig.1.7 are reported: dots, squares, stars, circles, triangles and crosses represent respectively works of[START_REF] Antonia | Invariants for slightly heated decaying grid turbulence[END_REF];[START_REF] Lee | Scaling range of velocity and passive scalar spectra in grid turbulence[END_REF];[START_REF] Antonia | Invariants for slightly heated decaying grid turbulence[END_REF];[START_REF] Zhou | Transport equations for the mean energy and temperature dissipation rates grid turbulence[END_REF];[START_REF] Antonia | Similarity of decaying isotropic turbulence with a passive scalar[END_REF];[START_REF] Danaila | Calibration of a temperature dissipation probe in decaying grid turbulence[END_REF].

Figure 1

 1 Figure 1.6: Exponents α T (a)-(c), n T (b)-(d) and n L T (c)-(f) for P r = 10 -4 . Top line, σ = σ T = 2; Bottom line σ = σ T = 4. Symbols for the CBC predictions: high Reynolds numbers; •: low Reynolds numbers.

Figure 1

 1 Figure 1.7: Experimental scalar decay exponents at P r ∼ 1 in comparison with high Reynolds CBC theory and EDQNM simulations at P r = 1. List of symbols: • Zhou et al. (2000, 2002) ; Antonia & Orlandi (2004); Lee et al. (2012) Sq35; × Danaila et al. (2000); Antonia et al. (2013) Sq35 and Rd44w; * Lee et al. (2012); Antonia et al. (2013) Rd35. (a) σ = σ T = 2; (b) σ = σ T = 4.

Figure 1

 1 Figure 1.8: Evolution of the kinetic and scalar decay exponents for Re λ (t = 0) = 240 at P r = 10 3 and P r = 10 5 . (a) σ = σ T = 2; (b) σ = σ T = 4.

Figure 1 . 9 :

 19 Figure 1.9: Evolution of the kinetic and scalar decay exponents for Re λ (t = 0) = 1620 at P r = 10 -3 and P r = 10 -4 . (a) σ = σ T = 2; (b) σ = σ T = 4.

Figure 2

 2 Figure 2.1: Craya-Herring frame (e (1) , e (2) , e (3) ) in blue, defined in (2.26); in red the wavevector k. The fluctuating spectral velocity û is contained in the plane (e (1) , e (2) ).

Figure 3 . 1 :

 31 Figure 3.1: Comparison of b 13 with RDT initial conditions (S L(tot) ij = 0) and with isotropic initial conditions with St = 0.1. (a) For σ = 2. (b) For σ = 4.

Figure 3

 3 Figure 3.2: (a) Spectral tensors φ 13 (k, t) and φ 11 (k, t). (b) Total non-linear flux and RTI flux. Both for σ = 2 with St = 1.

Figure 3

 3 Figure 3.3: Anisotropy for σ = 2 with S = 0.1τ -1 0 and St = 1. (a) Anisotropy tensor b ij (t). (b) Various spectral anisotropy indicators H () ij (k, t) at t = 100τ 0 , along with the integral and Kolmogorov wavenumbers k L and k η .

Figure 3

 3 Figure 3.4: (a) Anisotropy tensor b ij (t) for σ = 4. (b) Spectral anisotropy indicator H (pol)13 (k, t) for σ = 2 (grey lines) and σ = 4 (black lines), at t = 10τ 0 and t = 10 6 τ 0 .Both with S = 0.1τ -1 0 and an injected anisotropy St = 1.

  .6) EDQNM simulations at high Reynolds numbers show that powers of b ij are much lower than b ij alone. This would imply, at first order in anisotropy, that Π b ij . Such a linear relation between the normalized pressure-strain tensor and the anisotropy indicator b ij can be recovered starting from the evolution equation of R ij . Replacing R ij by its expression as a function of b ij , given in (2.10), yields

  , C RTI is not constant and depends on b ij through C RTI = 2+(terms in b ij b ij and b ik b kj b ij ), and in Sarkar & Speziale (1990); Warrior et al. (2014), one has 3.1 ≤ C RTI ≤ 3.4. The modelling for Π (s)ij proposed here is consequently complementary to existing models, and investigates the asymptotic RTI mechanism when the mean-velocity gradients are released.

Figure

  Figure 3.5: Constant of the Π (s)

Figure 3

 3 Figure 3.6: Asymptotic anisotropic state for b 13 . (a) With σ = 2 for various shear rates S at constant St = 10. (b) With S = 1τ -1 0 , St = 10 for various σ.

Figure 3

 3 Figure 3.7: (a) b ij : transition towards low Reynolds numbers for σ = 2 with St = 1. (b) H () 13 for different Reynolds numbers with σ = 4.

Figure 3

 3 Figure 3.8: Decay exponents of K(t) and R 13 (t) for various shear intensity S. : Classical CBC exponents; : Extended CBC exponents. (a) Saffman HSRT: α = α 13 = -6/5. (b) Batchelor HSRT: α = -1.38 is recovered. α 13 = -1.464 with p S = 0.279. The grey dash-dot curve -• corresponds to a distortion simulation with S = 1τ -1 0 .

Figure 3

 3 Figure3.9: Budget terms of the evolution equation of the spectral tensor after the release of the shear:∂ t φ ij = -2νk 2 φ ij + S (NL) ij + S (RTI) ij, at t = 10τ 0 for Saffman HSRT, where

Figure 3

 3 Figure 3.10: (a) Anisotropy indicators b ij and S R in Saffman HST with S = 1τ -1 0 . (b) Evolution of K(t) for various S and σ. The grey line indicates exp(0.33St). From top to bottom, the different cases are: σ = 2 and S = 10τ -1 0 ; σ = 4 and S = 1τ -1 0 ; σ = 1 and S = 0.1τ -1 0 ; σ = 2 and S = 0.1τ -1 0 ; σ = 3 and S = 10 -2 τ -1 0 .

Figure 3 .

 3 Figure 3.11: (a) Time evolution of K(t), (t), L(t), R 13 (t) and 13 (t) for σ = 4 with S = 1τ -1 0 . For clarity, L(t) has been increased by a factor 1000. (b) Anisotropy descriptors H (dir) ij for σ = 2 at St = 50.

Figure 3

 3 Figure 3.12: Evolution of the shear parameter S * (t): comparison to the sustained shear flow DNS of Isaza & Collins (2009) with three different initial values S * 0 . Lines and symbols represent respectively the present EDQNM simulations and DNS: +, S * 0 = 3; •, S * 0 = 15; and , S * 0 = 27.

Figure 3

 3 Figure 3.13: Budget terms with the shear wavenumber k S at St = 50 with σ = 2. S (NL) ij represents the conservative part of the total non-linear transfer. (a) For φ 13 . (b) For φ 33 .

Figure 3 .

 3 Figure 3.14: (a) Evolution of the shear parameter S * = 2/S R for various S with σ = 2. The lines correspond to EDQNM simulations with Re λ (0) = 1 and symbols to DNS. Crosses refer to Sukheswalla et al. (2013), run 256 30 without filtering; triangles to Isaza & Collins (2009); squares to Schumacher (2004), run 4. (b) Evolution of the kinetic energy exponential growth rate γ for various shear intensity S. The initial Reynolds number is either Re λ (0) = 10 or Re λ (0) = 100 and σ = 2. The average exponential growth rate is γ av = 0.114.

  ijpq have their own evolution equations (C.59) and (C.60): their non-linear transfers depend only on fourth-order contributions, whereas their linear production terms are modified by the second-order ones. The impact of EH (dir) ijpq and EH (pol) ijpq on the secondorder spectra EH (dir) ij and EH (pol) ij is uniquely done through the linear transfers S L(dir4) ij and S L(pol4) ij , which add to the previous ones (2.65) and (2.65), now written S L(dir2) ij and S L(pol2) ij

Figure 3

 3 Figure 3.15: Effects of the fourth-order expansion on the growth of the kinetic energy K(t) and the anisotropy tensor b ij . (a) K(t) for σ = 2 and σ = 4. (b) b ij for σ = 2.

Figure 3

 3 Figure 3.16: Effects of the fourth-order expansion on the linear transfers of EH (dir) ij and EH (pol) ij ,

  finally the second-order contributions in the equations of EH (pol)

  the fourth-order contributions in the equation of EH(pol)

Figure 3

 3 Figure 3.17: (a) Shear length L S (t) and shear rate S(t) for Saffman turbulence (σ = 2). (b) C δ , defined in (3.49), for σ = 2 (black), L S (0) = 1 and various S(0); and for σ = 4 (grey).

  .34) The global anisotropy indicators b T ij have a function analogous to the kinetic ones b ij . Similar anisotropy descriptors were introduced by Kassinos et al. (2007) with a different convention.

Figure 5

 5 Figure 5.1: (a) Scalar variance spectrum E T (k, t) and spectral tensors φ T 13 (k, t) and φ T 33 (k, t) for σ = 2 with St = 1. (b) Fluxes of S T,NL(dir) 13

Figure 5

 5 Figure 5.2: Scalar and kinetic decay exponents α T and α in both high and low Reynolds regimes, with St = 1, where symbols represent theoretical predictions. (a) σ = 2; (b) σ = 4.

Figure 5

 5 Figure 5.3: Scalar anisotropy indicators for σ = 2 with St = 1. (a) b T ij (t). (b) H

Figure 5

 5 Figure 5.4: (a) Scalar anisotropy indicators b T ij along with S T R for σ = 2. (b) Kinetic energy K, scalar variance K T , and scalar dissipation rate T , for σ = 2 and σ = 4.

T=

  -0.435, closer to the present value, and γ exp T = -0.235. Whatever it be, |γ exp T | < |γ T |.

Figure 5

 5 Figure 5.5: (a) Cospectrum, kinetic energy and scalar variance spectra F, E and E T for Re λ = 2.10 5 , along with the integral and Kolmogorov wavenumbers k L and k η . (b) Effect of low Reynolds numbers on the scaling of F, with a zoom on the narrow inertial range for the case Re λ = 100. Both for σ = 2.

Figure 5

 5 Figure 5.6: (a) Large scales behaviour of the cospectrum F for different infrared kinetic exponents σ. (b) Linear and non-linear transfers of the cospectrum for σ = 2, at Re λ = 2.10 5 .

  61Re -0.769 λ , reported in Fig. 5.7, from Re λ = 28 to 185. Results coming from EDQNM simulations are also presented for comparison purposes F P F EDQNM = 11.6Re -0.760 λ .

λ

  Figure 5.7: Comparison of the ratio of cospectrum dissipation and production with the DNS of Overholt & Pope (1996) for σ = 2 in high and low Reynolds numbers regimes.

Figure 5

 5 Figure 5.8: Comparisons with the experiment of Sirivat & Warhaft (1983) with the mandoline configuration, for σ = 2 and S θ = 0.152. (a) Cospectrum correlation ρ wθ defined in (5.10). (b) Scalar to kinetic integral scales ratio L T /L. (c) Ratio of production and dissipation of the passive scalar -ΛK F / T . (d) Kinetic to scalar time scales ratio R T defined in (5.11).

Figure 5

 5 Figure 5.9: Decay exponents for the scalar flux for various σ; Symbols represent theoretical predictions, for large Reynolds numbers and • for low Reynolds numbers. (a) Mixed-correlation decay exponent α F in high and low Reynolds numbers regimes. (b) Cospectrum dissipation rate decay exponent n F in the low Reynolds numbers regime.

Figure 5

 5 Figure 5.10: Time exponents for the scalar field for various σ; Symbols represent theoretical predictions, for large Reynolds numbers and • for low Reynolds numbers. (a) Scalar variance growth exponent α Λ T . (b) Scalar dissipation rate decay exponent n Λ T .

Figure 5 .

 5 Figure 5.11: Evolution of the scalar infrared slope σ T with Λ = -1. (a) σ T (t = 0) = 4 and σ = 2. (b) σ T (t = 0) = 2 and σ = 4.

Figure 5

 5 Figure 5.12: (a) Scalar anisotropy indicators b T ii (no summation) from high to low Reynolds numbers regimes, for both Saffman (black) and Batchelor (grey) turbulence. (b) H (T ) ii (no summation) at Re λ = 10 3 for σ = 2, along with the Kolmogorov and scalar integral wavenumbers k η and k T .

Figure 5

 5 Figure 5.13: Transfers for the cospectrum (grey) and streamwise flux (black) for σ = 2 at St = 50 where Re λ = 2.10 4 . (a) Linear transfers. (b) Non-linear transfers.

Figure 5

 5 Figure 5.14: Comparisons with the experiment of Tavoularis & Corrsin (1981), for σ = 2, S = 6.19τ -1 0 and S θ = 0.1823. (a) Scalar flux correlations ρ uiθ defined in (5.10). (b) Inverse of the time scales ratio defined in (5.11). (c) Turbulent Prandtl number P r T defined in (5.22). (d) Relative strength of the fluctuations B defined in (5.21).

FigFigure 5

 5 Figure 5.15: Comparisons with the DNS of Rogers et al. (1989), with σ = 2, S = 14.142 and S θ = 2.5. (a) Normalized diffusivity tensor, defined in (5.23), for the three orientations of the mean scalar gradient. (b) Turbulent Prandtl number P r T , defined in(5.24), for these three cases.

Figure 5

 5 Figure 5.16: Comparisons with the DNS of Kassinos et al. (2007), for σ = 2, S = 8.95 and S θ = 1. (a) ρ uθ and ρ wθ , defined in (5.10). (b) β defined in (5.25).

Figure 5 .

 5 Figure 5.17: (a) Scalar anisotropy indicators b T ij and scalar shear rapidity S T R . (b) Scalar flux shear rapidities S F R and S F ,S R . Both for σ = 2 and S = 10 -2 τ -1 0 .

Figure 5

 5 Figure 5.18: (a) Exponential growth of the kinetic, scalar, cospectrum and streamwise flux correlations. The cospectrum and streamwise flux dissipation rates F and S F are displayed in grey. (b) Exponential growth rates γ F , γ S F and γ T . Both for σ = 2 and S = 10 -2 τ -1 0 .

Figure 5 .

 5 Figure 5.19: (a) Scalar variance, cospectrum and streamwise flux spectra E T , F and F S . (b) Associated spectral scalings: the horizontal dashed lines correspond to -5/3, -7/3, -23/9. Both at St = 60 with σ = 2 and Re λ = 4.10 4 .

  Figure 5.20: Scalar anisotropy tensors H (T ) ij with Kolmogorov, shear and scalar integral wavenumbers k η , k S and k T for σ = 2 at two different Reynolds numbers. (a) St = 40 where Re λ = 2400. (b) St = 50 where Re λ = 2.10 4 . The zoom represents the small scales before k η where the persistence of anisotropy is clear.

  .3) Such a result was recovered analytically by O'Gorman & Pullin (2005) with their Sparse Direct-Interaction Perturbation (SDIP) model.

Figure 6

 6 Figure 6.1: (a) Scalar spectrum E T obtained with EDQNM compared to Yeung & Sreenivasan (2014): with P r = 1/2048 and σ = 2 at Re λ = 240. (b) Cospectrum F compared with O'Gorman & Pullin (2005), with P r = 2.10 -4 , Re λ = 1500 and σ = 2.

  .7) A similar process was performed by Bos et al. (2009); Yeung & Sreenivasan (2014). O'Gorman & Pullin (2005) obtained this result with other analytical considerations.

Figure 6

 6 Figure 6.2: (a) Compensated cospectrum compared with the SDIP model (O'Gorman & Pullin, 2005) at Re λ = 2.10 4 and P r = 100. (b) Compensated scalar variance spectrum compared with the DNS of Yeung et al. (2004) at Re λ 8 and P r = 1024.

Figure 6 . 3 :

 63 Figure 6.3: Scalar variance spectrum E T (k, t) and cospectrum F(k, t) at large Reynolds numbers for σ = 2. (a) Re λ = 2.10 5 and P r = 10 -5 , along with the integral, Corrsin-Obukhov and Kolmogorov wavenumbers k L , k CO and k η . (b) Re λ = 10 4 and P r = 10 4 , along with the integral, Kolmogorov and Batchelor wavenumbers k L , k η and k B . The cospectrum at P r = 1 at the same Reynolds numbers is displayed in grey as well.

Figure 6

 6 Figure 6.4: Budget terms of the evolution equation (4.36) of the scalar variance spectrum E T (k, t), along with the previous characteristic wavenumbers, for σ = 2. -Non-linear term kS T,NL(iso) ; --Production term 2kΛF; -• Dissipation term -2ak 3 E T . The insets represent the ratio S T,NL(iso) /2ak 2 E T . (a) Re λ = 1800 and P r = 10 4 , (b) Re λ = 23 and P r = 10 2 , (c) Re λ = 2.10 4 and P r = 10 -4 , and (d)Re λ = 400 and P r = 10 -4 .

Figure 6 . 5 :

 65 Figure 6.5: Scalar decay exponent α Λ T defined in (5.16) (black lines), and cospectrum decay exponent α F defined in (5.13) (grey lines). Simulations σ = 2 (-) and σ = 4 (--), with the theoretical predictions, for (a) P r = 10 -4 , and (b) P r = 10 4 . (c) α Λ T at intermediate Reynolds numbers, for both σ = 2 and σ = 4 and various P r (in black); the case σ = 4 and P r = 1 is recalled in grey.

Figure 6

 6 Figure 6.6: (a) Normalized correlation ρ wθ for σ = 4 at various Prandtl numbers as a function of Re λ . Symbols correspond to the DNS of Yeung & Sreenivasan (2014): P r = 1/2048; P r = 1/512; × P r = 1/128. Black and grey lines are respectively for small and large Prandtl numbers. (b) Pressure-scalar correlation Π F for σ = 2 at various Prandtl numbers. The grey dashed line corresponds to the theoretical prediction. The zoom focuses on small times to illustrate the difference with varying P r. (c) Nusselt number N u, defined in (6.11), as a function of the Péclet number P e, for various P r.

Figure 6

 6 Figure 6.7: Effects of P r on H (T ) 33 for σ = 2 along with the integral, Kolmogorov, Corrsin-Obukhov and Batchelor wavenumbers k L , k η , k CO and k B . (a) P r = 10 -4 and Re λ = 2, 1.10 5 so that P e λ = 2, 1.10 3 . (b) P r = 1 in black and P r = 10 4 in grey, both atRe λ = 100, so that the Péclet number varies from P e λ = 10 2 to 10 4 .

  Figure 6.8: Zoom at large scales for H (T ) 33 in Batchelor turbulence (σ = 4) for P r = 10 -4 , along with the integral wavenumber k L . The four different curves are at P e λ = 193, 87, 36 and 15.

Figure 6

 6 Figure 6.9: Scalar variance decay exponent α T , with σ = 2 in black and σ = 4 in grey, with St = 1; : high Reynolds predictions; •: low Reynolds predictions. (a) From high to low Reynolds numbers regimes for P r = 10 -4 . (b) For P r = 10 4 .

Figure 6

 6 Figure 6.10: b T ij for σ = 2 and St = 1. (a) P r = 10 -4 . (b) P r = 10 4 .

Figure 6

 6 Figure 6.11: Scalar variance K T in Saffman HST. (a) For S = 10 -2 τ -1 0 , Re λ (0) = 100 and P r = 10 -4 . (b) For S = 1τ -1 0 and P r = 10 4 .

Figure 6

 6 Figure 6.12: Scalar global anisotropy indicators b T ij (t) and scalar shear rapidity S T R (t). (a) For S = 10 -2 τ -1 0 , Re λ (0) = 100 and P r = 10 -4 . (b) For S = 1τ -1 0 and P r = 10 4 .

Figure 6

 6 Figure 6.13: (a) Normalized kinetic energy and scalar variance spectra E(k, t) and E T (k, t). (b) Scalar fluxes: non-linear isotropic Π NL(iso) T and non-local (direct) Π + T contributions. Both with the Kolmogorov and Batchelor wavenumbers k η and k B , at P r = 10 4 , σ = 2 and at St = 50 (Re λ ∼ 10 4 ).

Figure 6 .

 6 Figure 6.14: Scalar variance, cospectrum and streamwise flux spectra E T , F and F S in Saffman HSTSG. (a) For P r = 10 -5 and S = 1τ -1 0 , at Re λ = 2.10 5 , along with the Corrsin-Obukhov and Kolmogorov wavenumbers k CO and k η . (b) For P r = 10 4 and S = 10 -2 τ -1 0 , along with the Kolmogorov and Batchelor wavenumbers k η and k B .

Figure 6

 6 Figure 6.15: Kinetic energy K, scalar variance K T , and mixed-correlations K F and K S F , in Saffman HSTSG. (a) For P r = 10 -5 and S = 1τ -1 0 . (b) For P r = 10 4 and S = 10 -2 τ -1 0 .

Figure 7 . 1 :

 71 Figure 7.1: Evolution of the spectra in Saffman USHT. The denotes the peak of the spectra, which are represented at three dimensionless times N t = 19, N t = 22 and N t = 25, and Re λ (N t = 25) = 3.10 4 . (a) Kinetic energy spectrum E(k, t); (b) Scalar variance spectrum E T (k, t); (c) Cospectrum F(k, t).

Figure 7

 7 Figure 7.2: (a) Spectral slope of the isotropic, directional and polarization parts of the spectral tensor φ 33 (k, t). (b) Normalized linear and non-linear transfers of kinetic energy. Both in Saffman USHT at N t = 25 where Re λ = 3.10 4 .

Figure 7

 7 Figure 7.3: Compensated spectra in the inertial range for Saffman USHT, with the integral and Kolmogorov wavenumbers k L and k η . (a) Compensated kinetic energy spectrum (7.19) and scalar variance spectrum (7.20) at Re λ = 3.10 4 , along with the Kolmogorov constant K 0 = 1.31 and Corrsin-Obukhov constant K CO = 0.76. (b) Compensated cospectrum with Burlot scaling (7.22) at Re λ = 3.10 6 . The Lumley scaling (7.21) is not displayed since the curve is similar, but the plateau is located around 12.

Figure 7 . 4 :

 74 Figure 7.4: Infrared dynamics of E(k, t) and E T (k, t) with N = 1τ -1 0 . (a) σ(t = 0) = 2 and σ T (t = 0) = 4: black curves at N t = 0 and grey ones at N t = 1. (b) σ(t = 0) = 4 and σ T (t = 0) = 2: black curves at N t = 0 and grey ones at N t = 1. (c) Initially σ(t = 0) = 5, and then σ varies from N t = 0 to N t = 4.

Figure 7 . 5 :

 75 Figure 7.5: Influence of initial parameters σ and N on the Froude number F r = /(KN ). (a) Various σ at N = 1τ -1 0 . (b) Various N for σ = 2.

Figure 7 . 6 :

 76 Figure 7.6: Mixing intensity Λ = K T /K for: (a) various σ at N = 1τ -1 0 ; (b) various initial values Λ(0) for σ = 2 and N = 1τ -1 0 ; (c) various N and Re λ (0) with σ = 2: Re λ (0) = 70 in black and Re λ (0) = 5 in grey.

Figure 7

 7 Figure 7.7: (a) Kinetic energy exponential growth rate β for σ = 1, 2, 3 and 4. Straight lines indicate the numerical results, and the theoretical prediction (7.26), without the correction p USHT . (b) Exponential growth of K, K T and K F for σ = 2.

  important feature is to study the influence of initial conditions, such as σ and N , on the final state of anisotropy. The same conclusions as for F r and Λ are drawn for b 33 and b T 33 : as shown in Fig. 7.8d, varying N affects only the short time dynamics of b 33 and b T 33 , whereas increasing σ decreases the asymptotic values of the velocity and scalar anisotropy indicators. The latter feature is expected because by decreasing σ, one diminishes the large scales energy and consequently the amount of anisotropy of the flow. The results obtained here are quite different from shear flows, where the asymptotic anisotropic state of b ij does not depend anymore on σ. Whereas for both shear flows and USHT, varying the mean gradient intensity impacts only short times of b ij .

Figure 7

 7 Figure 7.8: Global anisotropy indicators for the velocity and scalar fields. (a) Polarization and directional anisotropy parts of b 33 for N = 1τ -1 0 and σ = 2. (b) b 33 at N = 1τ -1 0 for various σ. (c) b T 33 at N = 1τ -1 0 for various σ. (d) b 33 at σ = 2 for various N (the same behaviour is observed for b T 33 ). (e) Return to isotropy: normalized pressure-velocity Π 33 and pressure-scalar Π F correlations in Saffman (black) and Batchelor (grey) turbulence.

Figure 7 . 9 :

 79 Figure 7.9: Comparison of the present anisotropic EDQNM modelling with the axisymmetric EDQNM of Burlot et al. (2015b): σ = 2, Re λ (0) 70, N = 4τ -1 0 , and k peak = 40k L (0). (a) Froude number F r. (b) Mixing intensity Λ.

  7.10b: indeed, b 33 increases a bit from[EDC1] to[EDC2]. The main difference is observed on F r, which is reduced with [EDC2]: this means that the latter choice of eddy-damping constants enhance the importance of stratification in the dynamics, without increasing significantly the global anisotropy. Hence, changing the eddy-damping constants from [EDC1] to [EDC2] slightly increases the global anisotropy of the flow, and reduces F r, which makes our results closer toBurlot et al. (2015a,b). But the counterpart is a decrease of the Corrsin-Obukhov constant, which is another reason why [EDC1] is preferred here. Moreover, whether [EDC1] or [EDC2] is chosen, it does not improve the plateau for the cospectrum F nor change the exponential growth rate β of the kinetic energy.

Figure 7

 7 Figure 7.10: Comparisons of the eddy-damping constants settings in Saffman turbulence: A 1 = 0.355, A 2 = 0 and A 3 = 1.3 [EDC1] (black lines) and A 1 = A 2 = A 3 = 0.355 [EDC2] (grey lines). (a) Kinetic energy and scalar variance compensated spectra with Re λ (N t = 20) = 3.10 4 . Straight line (-) for Ek 5/3 -2/3 , dashed line (--) for E T k 5/3 -1 T 1/3 . (b) F r, Λ and b 33 .

  scale-by-scale dimensionality parameter, another indicator of anisotropy often used

Figure 7 .

 7 Figure 7.11: Spectral anisotropy indicators, along with the integral, Ozmidov, and Kolmogorov wavenumbers k L , k O , k η ; the Taylor scale λ is displayed as well. (a) H (dir) 33 and H (pol) 33 for σ = 2 at Re λ (N t = 25) = 3.10 4 . (b) sin 2 γ T for σ = 2 and σ = 4 at different N t so that for both Re λ = 3500.

Figure 7 .

 7 Figure7.12: (a) Lumley triangle: grey lines correspond to the boundaries between the isotropic, two-components axisymmetric (Axi.2C) and one-component (1C) configurations, and the black line to a EDQNM simulation. (b) Normalized potentials E (toro) /E 0 and E (polo) /E 0 , for σ = 2, at a wavenumber located in the infrared range.

  seems convenient because it is close to the Corrsin-Obukhov constant, consistently with E (USHT) P

  scaling is a new fundamental result, interesting for two reasons: first, it is strongly analogous to the inertial scaling of the anisotropic part of the pressure spectrum in shear flows ( 2/3 k -11/3 ); secondly, C (USHT) P is close to the Corrsin-Obukhov constant, similarly to C (S) P being close to the Kolmogorov one (see Appendix D).

Figure 7

 7 Figure 7.13: Pressure spectra in USHT, for σ = 2 at Re λ = 2.10 4 . (a) Isotropic and anisotropic pressure spectra E (iso) P and E (USHT) P , along with the pressure integral wavenumber k P and the Kolmogorov wavenumber k η , at Re λ 3.10 4 . (b) Compensated pressure spectra E (iso) P k 7/3 -4/3 and E (USHT) P k 11/3 -2/3 /N 2 in black, and E (USHT) P k 11/3 1/3 /( T N 2 ) in grey. (c) Isotropic and anisotropic parts K (iso) P and K (USHT) P of the pressure variance, along with the kinetic energy K: the grey lines indicate exp (βN t) and exp (2βN t).

Figure 7 .

 7 Figure 7.14: Saffman USHT for Sc = 10 5 at Re λ = 10 4 , along with the integral, Kolmogorov and Batchelor wavenumbers k L , k η and k B . (a) E, E T and F, along with the k -5/3 inertial scaling for E and E T , the k -7/3 inertial scaling for F, and the k -1 viscousconvective scaling for E T . (b) Zoom in the viscous-convective range for E T and F, with different scalings explained in the text.

Figure 7

 7 Figure 7.15: Saffman USHT for Sc = 1 and Sc = 10 4 . (a) Scalar anisotropy indicator b T 33 . (b) Mixing intensity Λ.

Figure 7

 7 Figure 7.16: Compensated spectra at Re λ (N 0 t = 10 4 ) = 1, 6.10 5 for σ = 2. (a) Compensated kinetic energy and scalar variance spectra, given by (7.19) and (7.20). (b)Compensated cospectrum, given by (7.59) and (7.60).

Figure 7

 7 Figure 7.17: (a) Stratification frequency N , mixing length L, and concentration variance K T =< c 2 > for σ = 2, and Re λ (N 0 t = 10 3 ) = 5.10 3 . (b) Growth rate α RT for various σ:-computed with (7.48); prediction (7.57).

Figure 7

 7 Figure 7.18: One-point statistics for σ = 2. (a) Time dependence of K and K F ; N L < u 3 c > is displayed as well in a grey dash-dot line. (b) F r, b 33 and b T 33 : b T 33 is increased by a factor 10 for readability.

Figure 8

 8 Figure 8.1: (a) kE -H for three different initial conditions, with σ = 4 at Re λ = 1400. (b) Kinetic and helical spectra E(k, t) and H(k, t) with σ = 2 at Re λ = 2.10 4 . k H η is defined in (8.22). Both along with the kinetic and helical integral wavenumbers k L and k H , and the Kolmogorov wavenumber k η .

Figure 8

 8 Figure 8.2: (a) Compensated kinetic and helical spectra. (b) Normalized kinetic and helical fluxes. Both with σ = 2, at Re λ = 2.10 4 , and along with the integral and Kolmogorov wavenumbers k L and k η .

Figure 8

 8 Figure 8.3: (a) Helical non-linear transfers S NL H1 and S NL H2 . (b) Purely helical and isotropic nonlinear transfers S NL(hel) and S NL(iso) . (c) Non-linear kinetic and helical transfers S NL E and S NL H . All for σ = 4 at Re λ = 5.10 3 , along with the integral and Kolmogorov wavenumbers k L and k η .

Figure 8 . 4 :

 84 Figure 8.4: Decaying spectra in Batchelor turbulence with Re λ (0) = 3400. (a) Kinetic energy spectrum E(k, t): at t = 10 6 τ 0 , E(k, t) for HIT is displayed as well in grey. (b)Helical spectrum H(k, t).

Figure 8

 8 Figure 8.5: Kinetic energy K(t) in HIT (grey line) and in HHT (black line) along with helicity K H (t) for σ = 4.

Figure 8 . 6 :

 86 Figure 8.6: Algebraic laws for the kinetic (-) and helical (--) fields, in Saffman (black) and Batchelor (grey) turbulence. Symbols refer to the theoretical predictions: (8.36) for α, (8.38) for α H , and (8.37) for L and L H . (a) Growth exponents of the kinetic and helical integral scales L and L H . At a given σ, the kinetic and helical theoretical exponents cannot be distinguished. (b) Decay exponents of the kinetic energy and helicity K and K H , where • and refer to kinetic and helical theoretical exponents.

Figure 8

 8 Figure 8.7: (a) Helical spectrum H(k, t) (black) for the initial condition (8.39) (--) at various times t/τ 0 = 10, 10 3 and 10 5 ; the kinetic energy spectrum E(k, t) is displayed as well (grey) at t/τ 0 = 10 5 . (b) Decay exponents α (-) and α H (--), where • and refer to the kinetic and helical theoretical predictions (8.36) and (8.38) respectively.

Figure 8

 8 Figure 8.8: Third-order helical correlations D (uuω) and S, for σ = 2 at Re λ = 3.10 4 , along with the integral and Kolmogorov scales L and η. (a) -D (uuω) /(r H ) and -< δu L δq 2 > /(r ). (b) S/(r 2 H ). (c) The different scalings of D (uuω) and S.

Figure 8 . 9 :

 89 Figure 8.9: Taylor scales for kinetic energy and helicity λ and λ H , along with the corresponding spectral viscous fluxes -2νk 3 E and -2νk 3 H and the non-linear transfers kS NL E and kS NL H for σ = 2. The wavenumbers k H η (8.22) and k η are displayed as well. The black curves are for the kinetic field, and the grey ones for the helical one. (a) Re λ = 10 4 . (b)Re λ = 400.

Figure 8

 8 Figure 8.10: Helical derivative skewness S uuω (t) in HHT for σ = 2.

Figure 8 .

 8 Figure 8.11: (a) Decay exponents α Q and α F of < ω 3 θ > (--) and < u 3 θ > (-) respectively, for σ = 2 (black) and σ = 4 (grey); theoretical predictions, for α Q (8.80), and • for α F (5.13). (b) Cospectrum and quadrature non-linear transfers, for σ = 2 at Re λ = 5.10 3 , along with the integral, helical Taylor and Kolmogorov wavenumbers k L , 1/λ H and k η . (c) Cospectrum F(k, t) and quadrature spectrum Q(k, t); -Q(k, t) is displayed in grey. Same configuration as (b).

  [2]+[3]+[5]-[1]-[4]-[6], yields the permutation law B ijkpqr + B ijkqrp + B ijkrpq = B ijkprq + B ijkqpr + B ijkrqp . (A.12)

Figure

  Figure A.1: Longitudinal structure function D LL (r, t) for σ = 2 at Re λ = 2.10 4 . (a) The different scalings. (b) Compensated D LL to obtain C 2 = 1.73.

  77)Using the relations between f , k and D LL , D LLL , and d t u 2 = -2 /3, one obtains the Kármán.78) by r 4 , and integrating from 0 to r yields[START_REF] Saffman | Calculation of velocity structure functions for vortex models of isotropic turbulence[END_REF] derivative and viscous dissipation at high Reynolds numbers in the inertial range yields the 4/5 th law D LLL (r, t) =assessed in Fig.A.2b, along with the scaling for < δu L δq 2 > derived hereafter.

Figure A. 2 :

 2 Figure A.2: Kinetic third-order structure functions < δu 3 L > and < δu L δq 2 > for σ = 2 at Re λ = 2.10 4 . (a) The different scalings. (b) Compensated kinetic structures functions.

Fig

  Fig. A.2b, along with the r 3 small scale scaling of < δu L δq 2 > in Fig. A.2a.As remarked before, even at Re λ = 2.10 4 , the 4/3 is not exactly recovered. Interestingly, double precision is enough to compute < δu L δq 2 >, unlike D LLL .

Figure A. 3 :

 3 Figure A.3: Structure function D TT (r, t) for σ = 2 at Re λ = 2.10 4 . (a) The different scalings. (b) Compensated D TT to obtain C T 2 = 1.88.

Figure A. 4 :

 4 Figure A.4: Scalar structure function D LTT (r, t) for σ = 2 at Re λ = 2.10 4 . (a) The different scalings. (b) Compensated scalar structure function D LTT .

  Computation of Π + (k, t): firstly, one has

  local fluxes are displayed in Fig. B.2 at P r = 10 4 . It is clear that they bring energy beyond Kolmogorov wavenumber k η to sustain the k -1 viscous-convective range.

Figure

  Figure B.2: (a) Non-local fluxes Π + and Π + T . (b) Slopes of the kinetic and scalar spectra E and E T . Both with the Kolmogorov and Batchelor wavenumbers k η and k B , at P r = 10 4 and σ = 2 at Re λ = 10 3 .

  Spatial resolution: in Fig. B.3, the number of points per decade f is changed. With a better spatial resolution in wavenumbers, a = 10 1/f -1 decreases and thus, according to (B.1), the influence of the non-local transfers should decrease: this is assessed in Fig. B.3 where three different resolutions are tested, with the quantity ∆[kS NL(

Figure

  Figure B.3: Influence of the spatial resolution on the non-local transfers in HIT at P r = 1, σ = 2 and Re λ = 2.10 3 .

Figure B. 4 :

 4 Figure B.4: Influence of the isotropic non-local transfers in shear flows at St = 50 (Re λ = 10 4 ) with S = 0.1 and σ = 2. (a) Budget terms for the isotropic part and the component () 33 . (b) Associated fluxes.

Figure C. 1 :

 1 Figure C.1: Quadratic anisotropic contributions in the non-linear transfers at St = 50, with σ = 2. (a) Budget terms along with the integral and Kolmogorov wavenumbers k L and k η : grey curves represent the classical transfers without the quadratic anisotropic contributions. (b) b ij and shear rapidity /KS, where grey curves are without the quadratic anisotropic contributions.

  ) 2 a 3 (t 0 ). (D.8) Then, using the change of variable T = (K 3 -StK 1 )/K ⊥ and arctan a-arctan b = arctan ((a -b)/(1 + ab)),

ForFigure D. 1 :

 1 Figure D.1: Anisotropy tensor b ij with St = 0.1: (a) σ = 2; (b) σ = 4.

  Figure D.2: (a) Spectral tensor φ 33 . (b) Decay exponent α. Both for St = 0.1 and σ = 2 in HAxT.

Figure D. 3 :

 3 Figure D.3: Comparison with Davidson et al. (2012): grey lines correspond to the DNS and black ones to EDQNM. For the expansion, Re λ (0) = 36 and for the contraction Re λ (0) = 25. (a) Ratio u 2 /u 2 ⊥ in axisymmetric expansion (plain line) and contraction (dashed line). (b) u 2 (plain line) and u 2 ⊥ (dashed line) decay in axisymmetric expansion only.

Figure

  Figure D.4: (a) Anisotropy tensor b ij with St = 10. (b) Decay exponents of K(t) and R 13 (t). Both for σ = 2.

D

  .4.1 Evolution equation of the pressure correlation E P The pressure fluctuations satisfy the Poisson equation obtained by taking the divergence of the Navier-Stokes equation (2.1)

  Figure D.6: (a) Pressure spectrum E P (k, t) in isotropic Saffman turbulence. (b) Decay of the pressure variance K P in Saffman and Batchelor turbulence: • high Reynolds predictions, low Reynolds predictions. (c) Infrared slope of the pressure spectrum E P with initial σ = 4.

  into isotropic and anisotropic parts. Both the isotropic K (iso) P and shear K (S) P parts of the total pressure variance K P grow exponentially at a rate γ P = 2γ as revealed in Fig. D.7c, and in agreement with theoretical predictions by George et al. (1984) (K P and K (S) P cannot be distinguished).

Figure D. 7 :PP

 7 Figure D.7: Pressure spectra in shear flows for σ = 2 at Re λ = 2.10 4 . (a) Isotropic and anisotropic pressure spectra E (iso) P and E (S) P , along with the integral and Kolmogorov wavenumbers k L ∼ k P and k η , at Re λ 10 4 . (b) Compensated pressure spectra E (iso) P k 7/3 -4/3 and E (S) P k 11/3 -2/3 /S 2 . (c) Isotropic and anisotropic parts K (iso) P and K (S) P of the pressure variance K P , along with the kinetic energy K for comparison: grey lines indicate the curves exp (γSt) and exp (2γSt).

  -linear purely helical transfer Helicity creates a purely helical transfer in the evolution equation of E(k, t), coming exclusively from products of R(hel) ij (products of R(iso) ij and R(hel) ij have zero contribution to the spherical integration). Then, transfers are computed with the spherical-integration of 2T E = τ ii + τ * ii . Considering only the products of imaginary parts, one has τ ii = τ * ii . The three terms to compute are thenk l τ ii = 2k l P imn R mi R nl + P imn R ml R ni + P lmn R mi R ni .

  k -rk )(δ ip r j r q + δ jp r i r q ) + lpq 4r(2k + rk )(δ iq δ jp + δ jq δ ip )+ -2k + 2rk + r 2 k r 2 (δ iq r j r p + δ jq r i r p ) + S 2 r i r j r l r 3 -δ il thelongitudinal component, i.e. r 1 = r, and N is either the second or third component, with r 2 = r 3 = 0 as usual. The second tensor used in Gomez et al. (2000) reads φ B, C, D, E, F and G are functions of r only. Unlike φ (uuω) ijl which is expressed as a function of k(r) and S(r), φ (ωuu) ijl depends on unknown functions, and is not symmetric in is two first indices.Nevertheless, some words can be said about φ (ωuu) ijl : using incompressibility ∂ r l φ (ωuu) iil = 0 and some algebra given in von[START_REF] Von Karman | On the statistical theory of isotropic turbulence[END_REF], one obtains A + B + C + 3D = 0, which notably implies that φ relations such as < u L u L ω L >= -< u L ω L u L >, one obtains an expression already given in[START_REF] Gomez | Exact relationship for third-order structure functions in helical flows[END_REF],

  4 S (iv) + 7r 3 S + 3r 2 S -6rS + 6S = 2∂ 3 φ (uuω) ili∂r j ∂r j ∂r l . (D.46) It is worth noting, afterwards, that only derivatives of φ (uuω) ijl are necessary to compute D[u, ω]. To conclude the first step, i.e. expressing D[u, ω] as a function of the derivatives of S(r), one uses a Taylor expansion for S(r) when r → 0, remembering that S(0) = 0, D[u, ω] = -32 r S (0) -35S (iv) (0) + O(r). (D.47) Since D[u, ω] is finite, because H is, one has S (0) = 0. In the end, one recovers (8.65), where only S(0) = S (0) = 0 was used. The general expression of ∂ 3 npq φ (uuω) ijlis now derived to obtain the explicit expression of S (iv) (0): this is a lengthy calculation. Nevertheless, since we search for derivatives of S(r), only the corresponding part is considered in (D.38): the part with derivatives of k(r) vanishes with the appropriate indices contractions. in r j r l r p r q + δ jn . . . + δ ln . . . + δ pn . . . + δ qn . . . + δ ip . . . + δ iq . . . + δ jp . . . + δ jq . . . + δ ij . . . + δ pl . . . + δ ql . . . + δ pq . . .

  ijpq = δ ij δ pq + δ ip δ jq + δ iq δ jp . With this equation, one can obviously recover (D.46). Even though this would be tedious, this equation (D.48), combined with (D.44), can determine each non-zero component of ∂ 3 npq φ (uuω) ijlas a function of derivatives of S(r). This expression is of course an important result and could be used for further theoretical developments. One gets in particular

  j (-k, r)δ(pk)dr dp.Using the fact that the Dirac function is even, one obtains∂E T (k) ∂t + ... = -k j S T j (-k, p)dp -k j S T * j (-k, p)dp = -2k j S T j (-k, p)dp .This equation implies that the scalar correlation E T is real, as the Reynolds stress tensor (without helicity), and thus follows the property E T (k) = E T (-k), which leads to ∂E T (k) ∂t + ... = 2k j S T j (k, p)dp .

  k 2 + p 2 ) + νq 2 θ(k) θ(p)û j (q) + ... = -i k l k=r+s θ(r) θ(p)û l (s)û j (q)d 3 r + p l p=r+s θ(r) θ(k)û l (s)û j (q)d 3 r+ P jmn (q) q=r+s θ(k) θ(p)û m (r)û n (s)d 3 r ,which becomes, after ensemble average and convolution rules∂ ∂t + a(k 2 + p 2 ) + νq 2 S T j (k, p, t)δ(k + p + q) + ... = k l k=r+s T T jl (r, p)δ(k + p + q)d 3 r + p l p=r+s T T jl (k, r)δ(k + p + q)d 3 r+ + P jmn (q)q=r+s T T mn (k, p)δ(k + p + q)d 3 r.

The last term gives q=r+s F

 q=r+s m (k)F n (p)δ(r + k)δ(s + p)d 3 r + q=r+s E T (k) Rmn (r)δ(r + s)δ(k + p)d 3 r + q=r+s F m (p)F n (k)δ(r + p)δ(k + s)d 3 r = 2P jmn (q)δ(k + p + q)F m (p)F n (k).

H

  π 2 k 2 pq(x + yz) z(ky -px) E F , E F ij -py E F , E F ij -k(2x + 3yz) E F , π 2 k 2 p 2 q(xy + z)E 0 2(1 + 3xyz)E T 0 H (dir) , H (T ) ij + 2(3y 2 -1)E T 0 H (dir) , H (T ) ij -(1 -xyz -2z 2 )E T 0

  k j Rij = 0, one recovers the scalar flux Craya equation (4.10). Additional details on how T F,NL i is obtained are now provided. Before spherical-averaging, one has∂ û * i (p) θ(k) ∂t + ... = -ik j k=r+s θ(r)û j (s)û * i (p)d 3 r + iP imn (p) p=r+s θ(k)û * m (r)û * n (s)d 3 r, = -ik j θ(r)û j (s)û * i (p)δ(k -rs)d 3 rd 3 s + iP imn (p) θ(k)û * m (r)û * n (s)δ(p -rs)d 3 rd 3 s, = -ik j θ * (r)û * j (s)û * i (p)δ(k + r + s)d 3 rd 3 s + iP imn (p) θ(k)û m (r)û n (s)δ(p + r + s)d 3 rd 3 s.Ensemble average further gives∂F i (k, t)δ(k -p) ∂t + ... = k j S F * ji (r, p)δ(p + r + s)δ(k + r + s)d 3 rd 3 s + P imn (p) S F nm (k, r)δ(p + r + s)δ(k + r + s)d 3 rd 3 s.Integration over p simplify the first Dirac function, and integration over s in the rhs term of the equation erase the second one∂F i (k, t) ∂t + ... = k j S F * ji (r, -r -s)δ(k + r + s)d 3 rd 3 s + P imn (-r -s) S F nm (k, r)δ(k + r + s)d 3 rd 3 s,∂F i (k, t) ∂t + ... = k j S F * ji (r, k)d 3 r + P imn (k) S F nm (k, r)d 3 r.

α

  i α j dλ = -2πzα i α j , 2π 0 α i α j dλ = -2πyα i α j , 2π 0 α i α j dλ = π[α i α j (x + 3yz) -δ ij (x + yz)],2π 0

τ

  i N * i = kE 0 e -iλ k -(1 -z 2 )(Y + + Y -) + Y * + (xz + 2yz 2 -y) -(1 -z 2 )Y * -+ 2py(x + yz)Y + , τ i N i = kE 0 e iλ k -(1 -z 2 )(Y + -Y -) + Y * + (xz + 2yz 2 -y) + (1 -z 2 )Y * -+ 2py(x + yz)Y + .

pe

  n k j Rni F j N * i = kp(x + yz)E 0 e -iλ Y + , p n k j Rni F j N i = kp(x + yz)E 0 e iλ Y + , 2F * m k j Rni P jmn N * i = kpE 0 e -iλ Y * + (x 2 -y 2 ), 2F * m k j Rni P jmn N i = kpE 0 e iλ Y * + (x 2 -y 2 ), 2F * m k j R nj P imn N * i = k 2 E 0 e -iλ (xy -z + 2y 2 z)Y * + + (y 2 -1)Y * -, 2F * m k j R nj P imn N i = k 2 E 0 e iλ (xy -z + 2y 2 z)Y * + + (1 -y 2 )Y * -.Henceτ * i N * i = k 2 E 0 e -iλ (xy + 2zy 2 -z)Y * + -(1 -y 2 )(Y * -+ Y + + Y -) + kpE 0 e -iλ (x + yz)Y + + (x 2 -y 2 )Y * + , (E.11) τ * i N i = k 2 E 0 e iλ (xy + 2zy 2 -z)Y * + + (1 -y 2 )(Y * --Y + + Y -) + kpE 0 e iλ (x + yz)Y + + (x 2 -y 2 )Y * + .(E.12) λ-integration: useful results are the following ones2π 0 e -iλ (Y + + Y -)dλ = 2πE F i N * i , 2π 0 e iλ (Y + + Y -)dλ = 2πE F i N i , iλ Y * -dλ = πE F i N i .This yields the following equations which have to be multiplied by pq/k:2π 0 (τ j + τ * j )N * j dλ = πkE 0 N * j k 2(1 -z 2 )E F j + (1 + y 2 -z 2 -xyz -2y 2 z 2 )E F j + 2py(x + yz)E F j -πkN * j kE 0 (xyz + 2y 2 z 2 -z 2 )E F j + (1 -y 2 )(2E F j -E F j ) + pE 0 y(x + yz)E F j + (z -z 3 + 2xyz + 2y 2 z 2 )E F j + τ * j )N j dλ = πkE 0 N j k 2(1 -z 2 )E F j + (1 + y 2 -z 2 -xyz -2y 2 z 2 )E F j + 2py(x + yz)E F j -πkN j kE 0 (xyz + 2y 2 z 2 -z 2 )E F j + (1 -y 2 )(2E F j -E F j ) + pE 0 y(x + yz)E F j + (z -z 3 + 2xyz + 2y 2 z 2 )E E.13) is contracted with N i and (E.14) with N * i . This yields the spherically averaged non-linear transfer of the scalar flux, and one can note that the () + and () -components are equal S

  Figure E.1: Effects of quadratic contributions of anisotropy in the non-linear transfers. (a) Quadratic anisotropic fluxes along with the integral and Kolmogorov wavenumbers k L and k η . Grey curves correspond to the cospectrum. (b) Time evolution of F r, Λ and b 33 without (black) and with (grey) the quadratic anisotropic contributions.

  . . . 2.2.1 Craya equation for Rij . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spherically-averaged equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Spherically-averaged descriptors . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Spherically-averaged final Lin equations . . . . . . . . . . . . . . . . . . . 2.4.3 Return to isotropy -Spectral tensor . . . . . . . . . . . . . . . . . . . . . 3 Dynamics of the Velocity Field in Shear-driven Turbulence 3.1 Homogeneous Shear-Released Turbulence (HSRT) . . . . . . . . . . . . . . . . . 3.1.1 Validation of HSRT with Rapid Distortion Theory . . . . . . . . . . . . . 3.1.2 Kinetic energy spectrum E(k, t) and spectral tensor φ ij (k, t) . . . . . . . . 3.1.3 Anisotropy descriptors b ij (t) and H () ij (k, t) . . . . . . . . . . . . . . . . . . 3.1.4 Modelling of the pressure-strain tensor Π ij . . . . . . . . . . . . . . . . . 3.1.5 Additional remarks on HSRT . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Decay of K(t) and R 13 (t) in Saffman and Batchelor HSRT . . . . . . . . . . . . . 3.3 Homogeneous Shear Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Exponential growth of the kinetic energy K(t) . . . . . . . . . . . . . . . 3.3.2 Non-linear transfers and the shear wavenumber . . . . . . . . . . . . . . . 3.3.3 Discussion on the scattering of integrated quantities in HST . . . . . . . . 3.4 Conclusion and perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

	(s)

2.2.2 Craya-Herring frame -E -Z decomposition . . . . . . . . . . . . . . . . . 2.2.3 Generalized Lin equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The closure problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 The EDQNM approximation . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Directional and Polarization transfers T E and T Z . . . . . . . . . . . . . . v Contents vi 2.4 3.4.1 Conclusions on HST and HSRT . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Exponential growth rate γ . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Table 1 .

 1 4: Values of the ratio β L = k T /k L and relative errors given for several simulations with σ = σ T and σ = σ T : the values are almost the same for all P r in [10 -5 ; 10 5 ].

	Cases	β th	β num Error (%)	Cases	β th	β num Error (%)
	σ = σ T = 1	2/3	0.6684	0.26	σ = 2, σ T = 3 0.9104 0.8744	3.95
	σ = σ T = 2	2/3	0.6684	0.26	σ = 2, σ T = 4	1.07	1	6.64
	σ = σ T = 3	2/3	0.6684	0.26	σ = 3, σ T = 2 0.4882 0.5110	4.46
	σ = σ T = 4	0.7165 0.7645	6.2	σ = 4, σ T = 1	0.31	0.299	3.5
	σ = 1, σ T = 4 1.418	1.496	5.21	σ = 4, σ T = 2 0.430 0.4468	3.76

Table 5 .

 5 1: Decay and growth laws of kinetic and scalar integrated quantities in HSRT and HST. Note that the time exponents for K, , L, K T , T and L T of HSRT are also valid in HIT.

  . Let's call E max the maximum of the kinetic energy spectrum. Dimensional arguments yield E max (t) ∼ L(t)K(t). Using the self-similar exponential growth of integrated quantities gives E max (t) ∼ exp(3βN t/2), where β is the exponential growth rate of the kinetic energy. Then, at high Reynolds numbers, one can assume that k max (t) ∼ k L (t) so that the time t can be expressed as follows: t ∼ -2 ln(k L )/(βN ). Injecting this formula into E

		max finally
	provides
	E max (t) ∼ k -3 L ,	(7.18)
	in agreement with numerical simulations.

  Table7.1: Comparison between the theoretical prediction β th for the kinetic energy exponential growth rate, and the numerical result β obtained with our anisotropic EDQNM modelling.

	th	0.894 0.716 0.596 0.511
	Present EDQNM : β	0.893 0.715 0.596 0.540
	Burlot's prediction : β Burlot	1	4/5	2/3	4/7

Table 7 .

 7 , but nevertheless in agreement with our new theoretical prediction, and (ii) smaller values for b 33 as well. Asymptotic values of the 2: Comparisons of the asymptotic values at large Re λ of one-point statistics, obtained with the present anisotropic EDQNM modelling, and with the axisymmetric EDQNM. For the present modelling, values at left correspond to the setting of eddy-damping constants [EDC1], and at right to [EDC2] (see section 7.3.7 for details).

		Froude number F r	Mixing intensity Λ	Growth rate β	Global anisotropy b 33
	σ	Present	Burlot	Present	Burlot Present Burlot	Present	Burlot
	1 0.443-0.265 0.306 1.607-1.580	1.56	0.893	1	0.265-0.287	0.410
	2 0.551-0.337 0.385 1.508-1.466	1.45	0.715	4/5	0.239-0.260	0.375
	3 0.624-0.387 0.435 1.440-1.387	1.37	0.596	2/3	0.224-0.242	0.346
	4 0.659-0.412 0.460 1.407-1.347	1.31	0.540	4/7	0.217-0.234	0.323

Table 8 .

 8 1: Summary of the different eddy-viscosities ν t , ν H t , and ν ω t , that intervene in the non-local expansions k p ∼ q of the non-linear transfers S NL E and S NL H in HHT.

Table 8 .

 8 2: Comparison of the different decay exponents α H obtained by EDQNM with (IC2) and by Levshin & Chkhetiani (2013) (LC13).

  a 1 δ ip δ jq δ kr + a 2 δ ip δ jk δ qr + δ ik δ jq δ pr + δ ij δ kr δ pq + a 3 δ ip δ jr δ qk + δ ir δ jq δ pk + δ iq δ pj δ kr + a 4 δ iq δ jr δ pk + δ ir δ jp δ kq + a 5 δ ij δ kp δ qr + δ ij δ kq δ pr + δ ik δ jp δ rq + δ ik δ jr δ qp + δ iq δ jk δ pr + δ ir δ jk δ pq

  All the ingredients to obtain the evolution equation of u 2 f , or equivalently D LL , have been presented. Starting from (2.1), without mean-velocity gradients, and multiplying by u k , and then combining it with (2.1) written for u k multiplied by u i yields

	Interestingly, the velocity derivative skewness can be expressed as
						S(t) = lim r→0	D LLL LL D 3/2	.	(A.73)
	Finally, D LLL (r, t) can be computed from the non-linear kinetic energy transfers T (k, t), and
	details are provided in Bos et al. (2012)
	D LLL (r, t) = 12r	0	∞	T (k, t)	3(sin(kr) -(kr) cos(kr)) -(kr) 2 sin(kr) (kr) 5	dk.	(A.74)
	A.4.3 Towards the Kármán-Howarth equation
				∂R ik ∂t	=	∂ ∂r j	(T ijk + T kji ) + 2ν	∂ 2 R ik ∂r j ∂r j	.	(A.75)
								).	(A.72)

  It is revealed inFig. A.4 that -D LTT /( T r) closely approaches 2/3 at very large Reynolds numbers (the constant is 2/3 in the present simulations because T

	k	t)	sin(kr) (kr) 2 -	cos(kr) kr	dk.	(A.101)

  The previous equation for Rij is finally integrated over the whole domain (which simplifies the Dirac function δ(k -p)) to obtain the Craya equation. The total derivative reads d t

	,	(C.4)
	Rotation effect
	where α i = k i /k. Writing (C.3) for ûj (k), multiplying it by û * i (p), summing it to (C.3) written for
	û * i (p) and multiplied by ûj (k), and finally taking the ensemble average, one obtains the equation of the second-order spectral tensor Rij , which is, in Fourier space
	Rij (k, t)δ(k -p) =< û * i (p, t)û j (k, t) > .	(C.5)

  ijpq given in (2.47) and (2.48) yields the fourth order-expansions (2.44) and (2.45). Finally, the expansion of the spectral Reynolds tensor in terms of the deviatoric tensors H

	The tensors U	()4 ijpq have the same properties as H ijpq . Combining these two equations with the definitions ()
	of H ijpq and H (dir)	(pol)
				(dir) ij	, H ij (pol)	,
	H ijpq and H (dir)	(pol)

  49)

	Contributions of H	(dir) ijpq and H	(pol) ijpq in S ij L(dir)	L(pol) ij and S

  53)

	Contributions of H ijpq and H (dir)	(pol) ijpq in S ijpq L(dir)	and S ijpq : lengthy calculations provide the following L(pol)
	compact formula
	S k	E 0 A + ln H	()				16 3465	E A + li H jlpq + A + () lj H ilpq + A + () lp H ijlq + A + () lq H ijlp ()
	-	2 7	A + ln δ ij H	() lnpq + δ pq H	() lnij + δ ip H	() lnjq + δ iq H lnjp + δ jp H () lniq + δ jq H ()	() lnip	=	16 3465	H	(4)

abrs α a α b α l α n α r α s P ijpq d 2 k = ijpq [EH () ],

  From this equation and (C.3), the helical Craya equation (8.5) is obtained (without mean velocity gradients). Regarding the evolution equation of K H =< u i ω i > /2 in physical space, the solenoidal property of ω i provides < ω i ∂ i p >= 0 in the homogeneous framework.D.5.1 Non-linear helical transfer T HThe non-linear helical transfer is computed using (8.6). This expression can be simplified considering the real and imaginary contributions of τ ij intoT H (k, t) = -i ijl k l τ ij (k, t).(D.29)Using the quasi-normal closure already detailed in Chapter 2 and Appendix C, there are three terms to compute ijl k l τ ij = 2k l k n ijl P ipq R pj R qn + P jpq R pn R qi + P npq R pi R qj .(D.30)

						One has
		∂ω i ∂t	+ u l	∂ω i ∂x l	+ U l	∂ω i ∂x l	= ω l	∂u i ∂x l	+ A il ω l + ν	∂ 2 ω i ∂x l ∂x l	,	(D.27)
	and the spectral counterpart is thus
	∂ ∂t	-A lj k l	∂ ∂k

j + νk 2 ωi (k) = A il ωl (k) + ik l u i ω l -u l ω i (k).

(D.28)

  Simplification of H : the explicit derivation of R H ij gives∂ 2 R H ij ∂r p ∂r q = (δ ip δ jq + δ iq δ jp ) r j δ pq + r i r p δ jq + r i r q δ jp + r j r p δ iq + r j r q δ ip r 2 + ijl r l δ pq + r p δ lq + r q δ lp rwhere h and h can be expressed as specific components of the velocity and vorticity fields thanks to various relations coming from (D.33): one gets

					h (0) =<	∂u 2 ∂x 1	∂ω 2 ∂x 1	> -	1 2	<	∂u 1 ∂x 1	∂ω 1 ∂x 1	>,	(D.35)
					h (0) = lim r→0	∂ 3 ∂r 3 < u 2 u 3 >=< u 2	∂ 3 u 3 1 ∂x 3	>,	(D.36)
			u l	∂u i ∂x j	∂ 2 ω i ∂x j ∂x l	> + < u l	∂ω i ∂x j	∂ 2 u i ∂x j ∂x l	>= 0,	< ω l	∂u i ∂x j	∂ 2 u i ∂x j ∂x l	>= 0,	(D.32)
	and yields (8.55).
							h r 2 -	h r 3 -δ ij δ pq	h r	+	h r 2 -	h r 3 +
	+	r h r	-3	h r 2 + 3	h r 3
	-	r p r q δ ij r 2	h -3	h r 2 + 3	h r 3 +	r i r j r p r q r 4	h -6	h r	+ 15	h r	-15	h r
							f 2		+	f r	-	f r 2 +	r l r p r q 2r 3	rf (iv) + f -6	f r	+ 6	f r 2	,	(D.33)
	where the prime denotes the spatial derivative ∂/∂r. From this general equation, one notably obtains
	the quantity of interest here
							<		∂u i ∂x j	∂ω i ∂x j	>= 2 h + 4	h r	,	(D.34)

i

  This expression of ∆φ is essential since it links φ (ωuu) ijl to S(r). Now, let's express D[u, ω] as a function of the derivatives of φ

	(uuω) ijl	and φ	(ωuu) ijl	. One has
		∂ 3 φ ∂r n ∂r p ∂r q (uuω) ijl	=<	∂u i ∂x q	∂u j ∂x p	∂ω l ∂x n	> + <	∂u i ∂x p	∂u j ∂x q	∂ω l ∂x n	>
								+ < u i	∂ω l ∂x n	∂ 2 u j ∂x p ∂x q	> + < u j	∂ω l ∂x n	∂ 2 u i ∂x p ∂x q	>,	(D.44)
		∂ 3 φ ∂r n ∂r p ∂r q (ωuu) ijl	=<	∂u l ∂x n	∂u j ∂x q	∂ω i ∂x p	> + <	∂u l ∂x n	∂u j ∂x p	∂ω i ∂x q	>
								+ < u j	∂u l ∂x n	∂ 2 ω i ∂x p ∂x q	> + < ω i	∂ω l ∂x n	∂ 2 u j ∂x p ∂x q	> .	(D.45)
	Using (D.32), one obtains (8.64), with φ	(ωuu) ili	-φ	(ωuu) lii	= 2r l ∆φ/r. The explicit calculation of the above
	expression yields
	∂ 3 ∂r j ∂r j ∂r l	φ	(uuω) ili		+ φ	(ωuu) ili	-φ	(ωuu) lii
	=	1 r 2	∂ ∂r	r 2 ∂ ∂r	-2 S + 5	S r	+ 3	S r 2
											(uuω) ∂r l φ ili

  , writing that at k = k H η there is a balance between convection and viscous dissipation yields νkH η 2 E(k H η ) ∼ θk H η H(k H η ) 2, where θ = θ kkk . With the relation (8.21), one further has E/H ∼ / H . Then, for the characteristic time, θ ∼ 1/(νk H η 2 ) is chosen, in agreement with dissipation being dynamically important at large wavenumbers in the definition (2.40) of θ kpq . Finally, the classical inertial scaling (8.20) is used for H, so that Here, the evolution equation of E has been used: the final result can also be obtained starting from the evolution equation of H(k), and writing νk H

	νk H η	2 E(k H η ) H(k H η )	∼	1 η νk H	2 k H η	H	-1/3 k H η	-5/3	⇔	k H η	14/3 ∼ 2 H	-4/3 ν -2 ,	(D.52)
	from which one recovers (8.22). η	2 H(k H η ) ∼ θk H η	3 E(k H η )H(k H η ).
	The wavenumber k H

  which yields (4.44). The linear spherically-averaged directional scalar transfer is defined by (4.45). Equation (4.46) is recovered usingA ln H (T ) pq S k α i α j α l α n α p α q d 2 k =

											8πk 2 105	2A + li H	(T ) jl + 2A + lj H il + A ln H (T ) ln δ ij , (T )
	S k	A ln k l	∂H pq α p α q α i α j (T ) ∂k n	d 2 k =	8πk 2 105	2A + li	k	(T ) lj ∂H ∂k	+ 3H	(T ) lj
	+ 2A + lj k	∂H li (T ) ∂k	+ 3H li (T )	+ A ln k	(T ) ln ∂H ∂k	+ 3H ln (T )	δ ij .
		S k	A ln k l	∂E T H ∂k n (T ) pq α p α q	d 2 k =	8πk 2 15	A ln k	∂H ln E T (T ) 0 ∂k	+ 3H ln E T (T ) 0	,

  α j + yα j ) + (y + xz) 2α i (α j + yα j ) -P ij . Computation of τ F * i (p, k): the starting point is τ F * i (p, k) = p n k j R nj F i + Rni F j + 2k j F * m P jmn Rni + P imn R nj . k j = -kp(xy + z) E + X , p n R ni = pE 0 (α i + zα i ), α n α n = -x. F j P ij , p n k j Rni F j = 3 2 kpE 0 (α i + zα i )E F j (α j + yα j ), 2k j F * m P jmn Rni = 3 2 kqE 0 (α i + yα i )E F jα j + zα j + 2y(α j + xα j ) ,

	2k j F * m P imn R nj =	3 2	k 2 E 0 E F j
	Useful relations are
	p n R

nj

The terms to compute are

p n k j R nj F i = -3 2 kp(xy + z)E 0 E

  + (k, t) = φ + (k, t)e iλ + φ -(k, t)e -iλ , Y -(k, t) = φ + (k, t)e iλ -φ -(k, t)e -iλ .(E.8)The pressure part of the non-linear transfer has no contribution with the present helical decomposition (E.6) because α i N i = 0. The non-linear transfers associated with φ + and φ -are consequently(k, t) = P ij (k, t)T F,NL j (k, t) = T F + (k, t)N i (k, t) + T F -(k, t)N * i (k, t). (E.10)Computation of τ i (k, p): in this case, a symmetric expression of the closure (2.36) is used, more convenient for calculations

	so that
	T F,NL i
	T F,QN ij
				t)N * j (k, t),	φ -(k, t) =	1 2	E F j (k, t)N j (k, t).	(E.7)
	Convenient notations are used for computation
	Y T F + (k, t) =	1 2	T F,NL i	(k, t)N * i (k, t),	T F -(k, t) =	1 2	T F,NL	(E.9)

i (k, t)N i (k, t),

  α i α p α l H () np dλ = α i α l α p α q H () pq -5yz 2 + y -2xz + α i α p H

		35 4	yz 3 -	15 4	yz +	15 4	xz 2 -	3 4	x
	+ α i α p H	() pl -5yz 3 + 3yz -3xz 2 + x +	1 2	H	()
									() pl -y + xz + 2yz 2 ,
	2π
	0

il xz 2 + yz 3 -x -yz , 2π 0 α n

It is worth noting at this point the recent DNS by[START_REF] Ishihara | Energy spectrum in high-resolution direct numerical simulations of turbulence[END_REF] where Re λ

is reached with resolution 12288 3 .

For this section only, we choose KT =< θ

> /2 instead of KT =< θ 2 > for consistency with some of the reference papers

kpE 0 E Q i (-xy -z -zx 2 + z

) + α i E Q l (α l (xy + 2z)

Remerciements

Abbreviations

CBC

Comte-Bellot and Corrsin

Statistics and Structure Functions

The first objective of this appendix is to gather equivalences between physical and spectral formulations, in order to compute third-order statistical quantities with EDQNM, such as derivative skewnesses, in homogeneous turbulence. Equivalences in homogeneous isotropic turbulence (HIT) are not that straightforward, and thus deserve some details, since errors are found in reference papers [START_REF] Pope | Turbulent Flows[END_REF][START_REF] Antonia | Similarity of decaying isotropic turbulence with a passive scalar[END_REF][START_REF] Ristorcelli | Passive scalar mixing: Analytic study of time scale ratio, variance, and mix rate[END_REF]. In the following sections, as many details as possible are given, that could be used for other purposes as well.

Extensions to homogeneous turbulence without any particular symmetries are proposed. The second objective is to recall some basic results about structure functions.

A.1 Evolution equations and definitions

The fluctuating vorticity ω i is divergence-free and the vorticity tensor is defined as

The enstrophy W ii =< ω 2 > is linked to the kinetic energy dissipation rate through

This is always valid in homogeneous turbulence, as shown hereafter. The evolution equations of the fluctuating velocity and vorticity are given in (2.1) and (D.27). Let's define as well the two useful symmetric and antisymmetric tensors

For the scalar field, the evolution equation of the fluctuating part is given in (4.2). Let's define as well the scalar covariance tensor < ∂ i θ ∂ j θ >=< ξ i ξ j > and the derivative scalar variance

The evolution equations of the enstrophy < ω 2 > and < ξ 2 > are detailed for homogeneous flows in section A.2, and then for HIT in section A.3.

A.3.2 Second and third-order statistics

In what follows, second-order statistics, such as dissipation rates, and third-order statistics, such as skewnesses, are computed and simplified within the HIT framework. First, the velocity derivative skewness and kinetic palinstrophy are defined as

(A.39)

The aim of the previous calculations is to express B ijkpqr and C ijkpqr , which appear in the evolution equation of the enstrophy, as a function of B 111111 and C 111111 only. One can proceed similarly for the passive scalar field. The mixed-derivative skewness and scalar palinstrophy are defined as

Dissipation rate and enstrophy < ω 2 >: the fourth order tensor

is used to simplify the expression of = ν < ω 2 >. B ijkl can be expanded as

Then, the incompressibility B iikl = 0 gives 3a 1 + a 2 + a 3 = 0. In addition, < ∂ j (u i ∂ i u j ) >= 0 and < ∂ 2 ij (u i u j ) >= 0 yield B ijji = 0 which provides a 1 + a 2 + 3a 3 . Hence

Using the results of section A.3, < (∂ 2 11 u 1 ) 2 >= 2P/35, where P = k 4 E(k)dk, so that u 2 f (iv) (0) = 2P/35. Hence, the longitudinal correlation can be expanded in HIT as

The definition of [START_REF] Bos | Reynolds number effect on the velocity increment skewness in isotropic turbulence[END_REF] is used for the two-point second-order longitudinal structure function

where δu L is the longitudinal velocity increment. It is obviously possible to link D LL and f (r, t), and this deserves some details. Considering x i and x i as independent variables, using ∂r/∂r i = r i /r, and expending the incompressibility condition ∂R ji /∂r i = 0, one gets [START_REF] Von Karman | On the statistical theory of isotropic turbulence[END_REF] 2f (r, t) -2g(r, t) = -r ∂f ∂r ,

(A.62) so that the Reynolds stress tensor can be expressed only as a function of f :

Half the trace, R(r) = R ii (r)/2, is important since it was used by [START_REF] Saffman | The large-scale structure of homogeneous turbulence[END_REF] to demonstrate some invariant properties which will be detailed hereafter. In agreement with [START_REF] Davidson | On freely decaying, anisotropic, axisymmetric saffman turbulence[END_REF], one can write

Injecting (A.63) into (A.61) gives

in agreement with [START_REF] Saffman | Calculation of velocity structure functions for vortex models of isotropic turbulence[END_REF]. Finally, following Kolmogorov (1941b), the secondorder longitudinal structure function scales in the inertial range as ( r) 2/3 . Furthermore, since f (r → ∞) = 0, one has also, at large scales, D LL ∼ 2u 2 . And at small r, typically near the Kolmogorov scale η, δu L ∼ r∂ 1 u 1 , so that D LL ∼ r 2 /(15ν). This yields

Finally, it is possible to compute f (and thus D LL ) from the kinetic energy spectrum. Details are given for D LL in [START_REF] Bos | Reynolds number effect on the velocity increment skewness in isotropic turbulence[END_REF], and one has

(A.67) consistent with another expression in [START_REF] Saffman | Calculation of velocity structure functions for vortex models of isotropic turbulence[END_REF]. This relation allows to determine the structure functions from the high Reynolds numbers kinetic energy spectra obtained with EDQNM, as revealed in Fig. A.1a: the r 2 and r 2/3 scalings for the inertial range are recovered, Appendix B

Non-local Expansions of the Non-Linear Transfers

In this part, details on the calculation of non-local expansions of the non-linear transfers are given. These non-local expansions are crucial for the scalar field when the Prandtl number is such that P r 1. These expansions where extensively discussed for the velocity field by [START_REF] Lesieur | Amortissement autosimilaire d'une turbulence á grand nombre de reynolds[END_REF]; [START_REF] Métais | Statistical predictability of decaying turbulence[END_REF]. The main elements of the computation are gathered here. The need to evaluate these non-local contributions arises from the logarithmic discretization of the wavenumber space that cannot take into account the elongated triads. The criterion to quantify the non-local transfer is inf(k, p, q) sup(k, p, q) ≤ a,

Details on the Spherically-Averaged Lin Equations

Here, all the calculations yielding to the spherically-averaged Lin equations of the velocity field of Chapter 2 are fully detailed. It includes notably the computation of the non-linear and linear transfers, and the complete spherical and λ integrations. Moreover, additional theoretical considerations are developed: (i) quadratic contributions of anisotropy in the non-linear transfers, (ii) the modelling of anisotropy resulting from a truncation at the fourth-order of the expansion into spherical harmonics of the spectral Reynolds tensor.

C.1 Spectral evolution equations C.1.1 Craya equation

The evolution equation of the fluctuating field in the presence of both mean velocity gradients and rotation is

The counterpart of (2.1) in Fourier space, using

where ûi is the Fourier transform of u i , A ij = d j U i is a space uniform gradient matrix, k is the wavenumber modulus, and u m u n (k) is the convolution product

Thanks to the incompressibility condition ûi k i = 0 in Fourier space, the pressure term can be erased by projecting (C.2) on the plane perpendicular to k i p. Using the operator

Each of the three rhs terms can be written

The quasi-normal approximation consists into neglecting the fourth order cumulants, which gives

and simplifies into

The quasi-normal term T QN ijn is finally recovered since

C.2 Calculations of T E and T Z

C.2.1 Relations between frameworks

In Fig. 2.2, a new frame (β,γ,α) has been presented, attached to the plane of the triad. The new frame (β,γ,α) is obtained from the Craya frame (e (1) , e (2) , e (3) = α) by rotations of angles λ, λ and λ around k, p and q. Hence, one has

, (C.17)

where γ is normal to the plane of the triad

The vectors β, β and β are perpendicular to k, p and q but still in the plane of the triad

With these definitions, one gets

with x = cos a, y = cos b, and z = cos c. To perform the following calculations, one has to keep in mind fundamental geometrical relations valid in any triangles

Finally, since p and q have symmetric behaviours, one has both

Starting from (2.39), the main term to compute is k l T QN ijl , divided into two parts

Computation of τ - ij : starting from the expression of τ - ij given in (C.23) and using the fact that p m R ml = -k m R ml thanks to the incompressibility condition, and that k m Rmj = 0, one finds

Using (2.30), geometrical relations and equalities such as W i β i = 1 and β i γ i = 0, one has

where X = Ze 2iλ , X = Z e 2iλ and X = Z e 2iλ . Similarly,

Then, using

→ Now, the contribution of τ - ij to the polarization transfer is evaluated. Using q sin b sin c = p(1 -z 2 ), this yields

All other useful quantities have already been detailed. Thus, under a symmetric form

Using previous geometric relations, the total contribution of τ + ij to T E becomes

With the definitions of τ - ij and τ + ij given in (C.23), and using the fact that τ ij is real (because there is no helicity), one finds

Consequently, (2.41) and (2.42) are obtained.

Return to isotropy: some details to obtain T (RTI) in equation (2.69) are given. Using previous relations, one gets

and, with p ↔ q symmetry

so that (2.69) is recovered.

Final quadratic anisotropic non-linear transfers: the quadratic anisotropic isotropic transfer term is conservative and reads

The quadratic anisotropic directional transfer term is

The quadratic anisotropic polarization transfer term is

, and similarly for the polarization. The direct retro-action of the fourth-order terms on the second-order ones is done through the additional linear transfers S

where

are the linear transfers computed with the second-order expansions.

Appendix D Additional Results for the Velocity Field in Homogeneous Turbulence

In this appendix, additional results, with respect to the ones presented in Chapter 3 which focused on shear flows, are proposed. Notably, (i) rapid distortion theory (RDT) is briefly addressed; (ii) some calculations about axisymmetric turbulence and plane distortion are presented; (iii) the pressure fluctuations are investigated to derive the equation for the pressure spectrum; and finally, (iv) some details about the helical field in skew-isotropic turbulence are given.

D.1 Rapid Distortion Theory

In this part, some details about the Rapid Distortion Theory (RDT) are given, in order to derive initial conditions for anisotropic flows and a solution for short times as well. This section is inspired by [START_REF] Townsend | The structure of turbulent shear flow[END_REF]. Firstly, the non-linear terms from Navier-Stokes equation (2.1) in physical space are discarded ∂ ∂t

In RDT, solutions valid for short times only are derived: indeed, non-linear terms are negligible only at the beginning of the simulation, when anisotropy grows thanks to production terms. In the early times of the flow, the Reynolds number is important, and thus viscous effects are small with respect to inertial ones. For this reason, ν∆u i is neglected as well. Then, the following Fourier decomposition for the fluctuating velocity and pressure fields is used

Then, with dk i /dt = -A ji k j , equation (D.1) becomes, after projecting on the plane perpendicular to

Thanks to the previous linearity assumption, it is possible to compute at each time the fluctuating velocity a i (t) = G ij (t, t 0 )a j (t 0 ), (D.4) where G ij is the Green's function and t 0 the initial time (chosen to be 0 in the simulations). The wavenumber at each time is given by

Appendix E

Details on Spherically-Averaged Scalar Lin Equations

In this appendix, all the calculations yielding to the scalar spherically-averaged Lin equations of the passive scalar field and scalar flux are fully detailed. Additional theoretical results are presented as well, such as the quadratic anisotropic contributions in the non-linear transfers with illustrations in USHT, and an alternative modelling for the scalar flux.

E.1 Scalar-scalar correlation

In this section, the spectral scalar-scalar correlation is first addressed: the quasi-normal procedure along with the calculations of the linear and non-linear transfers are detailed.

E.1.1 Scalar Craya equation

The scalar-scalar correlation E T is defined in (4.5). Its evolution equation is obtained by multiplying (4.3) by θ * (p) and summing it to the evolution equation of θ * (p) multiplied by θ(k). Ensemble average and integration over the whole domain gives the scalar Craya equation (4.6). The calculation of the scalar non-linear transfer T T,NL deserves some additional details. Then, ensemble average gives

and the return to isotropy part

The influence of the quadratic contributions of anisotropy in the non-linear transfers is illustrated for USHT. Their impact is comparable to the case of sustained shear flows, illustrated in Appendix C. The scalar quadratic contributions are numerically the most intense ones in USHT, compared to the scalar flux and kinetic ones. of the total cospectrum quadratic transfer Q F,NL

3

. One can remark that these anisotropic contributions mainly act at large scales. It is worth noting that the flux of the (first-order in anisotropy) isotropic scalar transfer S T,NL(iso) is more than ten times higher than Π T(iso) Q

. Secondly, the impact of the quadratic contributions on the one-point statistics is revealed in Fig. E.1b: they slightly increase the global level of anisotropy of the flow. The main difference with the case without these quadratic contributions (in grey) is observed for the Froude number. Nevertheless, this is has less impact than changing the eddydamping constants, as seen before in section 7.3.7. Finally, the inclusion in the simulations of these quadratic contributions does not change at all the growth rate β of the kinetic energy, nor the scalings of the spectra.

E.2.7 Scalar flux in HHTSG

In this section, the framework of homogeneous helical turbulence with a mean scalar gradient (HHTSG) is considered, and some details about how to obtain S F,NL(hel) i and S Q,NL i , given in (8.75) and (8.77), are provided.

Computation of τ F i (k, p): one can use p ↔ q symmetry here, and k(x + yz) = p(y + xz) = q(z + xy) as well, so that