Oliver Soulard

Jerome Griffond

Philippe Arnault

Jean Clérouin

Nicolas Desbiens

Tristan Ulysse

Loïc

Alexandre Luis

Robinson

Jean-Baptiste Augustin

Estelle Raphael Paul

Gerome Nicolas

Guillaume

Richard

Jean-Cedrik Lucas

Mathieu En

Nils, Yoannis Thibaud Baptiste

Carmine E Donato

Marco

Di Lione

Richard Et Mayeul

Kn Knudsen

Viscous stress tensor h Perturbation amplitude λ Perturbation wavelength g Acceleration A t Atwood number δ Mixing zone size α Rayleigh-Taylor mixing zone growth rate ∆u Shock impulsion

Keywords: Mixing, turbulence, variable density flow, non-uniform transport coefficients, compression, plasma, inertial confinement fusion, DNS, EDQNM Projection . . . . . . .

This thesis is dedicated to the study of turbulent mixing in flows with variable density and non-uniform transport coefficients. We use a new direct numerical simulation (DNS) code based on a two-dimensional domain decomposition, capable of taking into account variable density and diffusive contributions. At first, we consider the case of turbulence in weakly-coupled plasmas under isotropic compression, which can experience a sudden dissipation of kinetic energy due to the growth of the viscosity coefficient due to temperature increase. In this case, in addition to DNS we use a spectral model based on the Eddy-Damped Quasi-Normal Markovian closure. We evidence the sensitivity of the flow dynamics to initial conditions for homogeneous isotropic turbulence and an inhomogeneous spherical turbulent layer. In the latter case, we find, also, the first hint of a sudden diffusion effect. The importance of initial conditions is also shown in the study of the variable density unstably stratified homogeneous turbulence. If the initial density contrasts are sufficiently strong, the large scales of the flow are modified with the consequent modification of the self-similar scaling laws. Finally, we consider an idealized configuration of inertial confinement fusion implosion, with both variable density and transport coefficients effects. During the compression, we evidence the competition between the plasma molecular diffusion, which is enhanced by the temperature increase, and the turbulent diffusion, which on the contrary decreases due to the increased viscous dissipation. In the last phase of the implosion, we highlight a sudden diffusion process, where compressed spherical mixing layers are quickly diffused.

Résumé : Cette thèse est consacrée à l'étude du mélange turbulent dans des écoulements à densité variable et à coefficients de transport non uniformes. Nous utilisons un nouveau code de simulation numérique directe (DNS) basé sur une décomposition bidimensionnelle du domaine, capable de prendre en compte la densité variable et les contributions diffusives. Dans un premier temps, nous considérons le cas de la turbulence dans les plasmas faiblement couplés en compression isotrope, qui peuvent subir une dissipation soudaine d'énergie cinétique due à la croissance du coefficient de viscosité par une augmentation de la température. Dans ce cas, en plus des DNS, nous utilisons un modèle spectral basé sur la fermeture Eddy-Damped Quasi-Normal Markovian. On démontre la sensibilité de l'écoulement aux conditions initiales pour une turbulence homogène isotrope et une couche sphérique turbulente. Dans ce dernier cas, nous trouvons aussi la première indication d'un effet de diffusion soudain. L'importance des conditions initiales est également mise en évidence par l'étude de la turbulence homogène stratifiée instable à densité variable. Si les contrastes de densité initiale sont suffisamment forts, les grandes échelles de l'écoulement sont modifiées avec pour conséquence la modification des lois d'échelle des états autosemblables. Enfin, nous considérons une configuration idéalisée d'implosion de fusion par confinement inertiel, avec des effets de densité variable et des coefficients de transport. Pendant la compression, nous mettons en évidence la compétition entre la diffusion moléculaire du plasma, qui est renforcée par l'augmentation de la température, et la diffusion turbulente, qui au contraire diminue en raison de la croissance de la dissipation visqueuse. Dans la dernière phase de l'implosion, nous mettons en évidence un processus de diffusion soudaine, où les couches de mélange sphériques en compression sont rapidement diffusées. Radial position where averaged value of the CH mass fraction reaches a value of 0.01 r 50

Mots-clés :

Radial position where averaged value of the CH mass fraction reaches a value of 0.5

Chapitre 1

Introduction

Turbulent mixing

Mixing is an irreversible process that increases entropy, decreases Gibbs free energy and exergy [START_REF] Tailleux | On the energetics of stratified turbulent mixing, irreversible thermodynamics, boussinesq models and the ocean heat engine controversy[END_REF]], and brings homogenization to the molecular level. According to [START_REF] Eckart | An analysis of the stirring and mixing processes in compressible fluids[END_REF], mixing in fluids can be thought of as a process in three steps :

1. An initial or entrainment phase, where large volumes of the mixed materials are visible, steep gradients are found at the interface between these regions, which are otherwise smooth. This phase is dominated by the large scales coherent structure of the flow.

2. An intermediate or stirring phase, when distortion caused by advection, provokes a rapid increase in the extension of interfacial surfaces, with the final results to increase the mean value of the initial gradients.

3. A final or diffusion phase where the gradients disappear, and the fluid becomes homogeneous.

Mixing can be laminar or turbulent, with noticeable differences in the stirring phase.

The main features that distinguish turbulent from the laminar mixing are the large range of flow scales and the associated greater interfacial surface, which permits to the molecular diffusive effect to proceed more effectively.

Although laminar mixing present interesting application [START_REF] Erwin | Theory of laminar mixing[END_REF]], especially in the case where high viscosity prevents the onset of turbulence [START_REF] Ottino | Laminar mixing of polymeric liquids ; a brief review and recent theoretical developments[END_REF], the most common mixing processes are driven by turbulence, which is present in many natural and engineering phenomena, with length scales and Reynolds numbers spanning 20 orders of magnitude [START_REF] Ottino | Mixing, Chaotic Advection, and Turbulence[END_REF]].

Many natural phenomena present turbulent mixing as a crucial factor, from the atmosphere dynamics [START_REF] Monin | Basic laws of turbulent mixing in the surface layer of the atmosphere[END_REF], where it is involved in the transport of heat from the subtropical latitudes to the polar regions, to the oceans circulation [START_REF] Polzin | Spatial variability of turbulent mixing in the abyssal ocean[END_REF][START_REF] Mashayek | Efficiency of turbulent mixing in the abyssal ocean circulation[END_REF] where it helps sustain the deep global ocean overturning circulation. Also, turbulent mixing has a significant impact on astrophysical processes, from stars [START_REF] Baglin | Rotation and mixing in the outer layers of a stars. turbulent mixing due to the meridional circulation velocity field[END_REF]] to interstellar medium of galaxies [START_REF] Slavin | Turbulent mixing layers in the interstellar medium of galaxies[END_REF] where turbulent mixing layer are substantial contributors to interstellar radiations. Moreover, applications involving combustion [START_REF] Pitz | Combustion in a turbulent mixing layer formed at a rearward-facing step[END_REF]Daily 1983, Reitz 2002], chemical industry reactors [START_REF] Nienow | Mixing in the process industries[END_REF]], hypersonic propulsion systems [START_REF] Parent | Numerical investigation of the turbulent mixing performance of a cantilevered ramp injector[END_REF] and inertial confinement fusion [Haines et al. 2014a] are all concerned in some measure by the understanding of this fundamental problem. Dimotakis (2005), in his review, distinguishes between three levels of possible turbulent mixing. Level-1 is the simplest and one of the widely studied ; it involves the mixing of a passive scalar, which does not act back on the flow dynamics. In Level-2, the mixing is coupled to the dynamics of the flow, such as in cases driven 1.1. Turbulent mixing 3 by buoyancy forces. The Level-3 mixing produces changes to the fluid, e.g., chemical composition change due to combustion, that in turn modify the flow dynamics.

Buoyancy driven mixing

A classic example of level-2 mixing, which is of interest in this thesis, is caused by buoyancy-driven flows. A case occurring when a fluid with non-uniform density distribution is subjected to pressure gradients or external accelerations, in both stably or unstably stratified configuration, i.e. the density of the fluid respectively increases or decreases in the direction of the force.

If one takes the vorticity, ω, equation for flow with non uniform density and constant body forces [START_REF] Tennekes | A first course in turbulence[END_REF],

∂ t ω + u • ∇ω = 1 ρ 2 ∇ρ × ∇p Baroclinic term + ω • ∇u -ω∇ • u Vorticity stretching + ∇ × ∇ • τ ρ Vorticity diffusion (1.1)
where u is the velocity field, ρ the density, p the pressure and τ the viscous stress tensor. The main difference with non-stratified flows is the generation of vorticity from the baroclinic effects, resulting from misalignment between pressure and density gradients.

The stability criterion for non-uniform density fluid can be a tricky question, given the vast difference in possible configurations. For instance, if one considers incompressible flows with a constant pressure gradient, imposed by a constant acceleration, and the density as the only non-uniform quantity, the stability can be determined by the direction of pressure and density gradients.

Otherwise, if the acceleration varies in time, even if it is always in the opposite direction of the density gradient, the configuration can be destabilized by the Faraday instability [Faraday 1831, Gréa and[START_REF] Gréa | What is the final size of turbulent mixing zones driven by the faraday instability[END_REF].

Another example is the double diffusion process, due to the simultaneous presence of different scalar species [START_REF] Huppert | Double-diffusive convection[END_REF], i.e. concentration and temperature. Finally, discarding the incompressible constraint, one has to take into account the entropy [START_REF] Lighthill | Waves in fluids[END_REF]].

Stably stratified mixing Within the framework of buoyancy-driven mixing, a great deal of study is devoted to turbulence in stably stratified medium [START_REF] Fernando | Turbulent mixing in stratified fluids[END_REF]]. This field is of particular importance because oceans, lakes, and atmospheres and many astrophysical bodies are stably stratified. This stable stratification would imply in principle that vertical motions are inhibited, imposing a limit to the vertical exchange of matter. Nevertheless, these exchanges happen thanks to the help of turbulent mixing. In stably stratified fluid, the transition to a turbulent state can be caused, for example, by shear flows generating Kelvin-Helmholtz or Homlboe waves, or the breaking of internal waves generated by gravitational restoring forces acting on vertically displaced fluid [START_REF] Lamb | Internal wave breaking and dissipation mechanisms on the continental slope/shelf[END_REF]].

Unstably stratified mixing

The simplest case of unstable stratification is obtained when density gradients and the acceleration have the opposite direction. In this configuration, the flow evolution and the related mixing mechanism are determined by the pressure gradient imposed by the acceleration profile. If the acceleration is constant, the flow is subjected to the Rayleigh Taylor instability, while if the acceleration is an impulsion, such as a shock wave, the Richtmyer-Meshkov instability takes place.

Rayleigh-Taylor

The Rayleigh-Taylor (RT) instability has been individually investigated by [START_REF] Rayleigh | Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density[END_REF] and [START_REF] Taylor | The instability of liquid surfaces when accelerated in a direction perpendicular to their planes[END_REF], who studied the interface of a heavy fluid of density ρ 2 , on top of a light fluid of density ρ 1 . It is one of the most common instability mechanism. For example, it is the reason why when one turns a glass over, the water inside falls, even if the atmospheric pressure should be able to keep the water inside the glass. This instability has been the object of extensive study, and different review articles are available [START_REF] Sharp | Overview of Rayleigh-Taylor instability[END_REF][START_REF] Kull | Theory of the Rayleigh-Taylor instability[END_REF][START_REF] Boffetta | Incompressible Rayleigh-Taylor turbulence[END_REF], Zhou 2017a]. Studies on this problem have identified three main phases in the evolution of the flow : linear, nonlinear, turbulent.

Linear The theory describing the first phase of the instability was proposed by [START_REF] Rayleigh | Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density[END_REF] and [START_REF] Taylor | The instability of liquid surfaces when accelerated in a direction perpendicular to their planes[END_REF] who studied the case of incompressible flows. They assumed that the amplitude, h, of the perturbations was small compared to their wavelength, λ. This hypothesis allows to linearize the equations and neglecting the effects of viscosity or surface tension they obtained :

h = h 0 exp   t g 2π λ A t   (1.2)
where g is the acceleration and A t = ρ 2 -ρ 1 ρ 2 +ρ 1 . So that the dispersion relation can be written as

ζ 2 = g 2π λ A t , (1.3)
which allows us to observe that when acceleration and Atwood number are positive, the flow is unstable, ω 2 > 0, for all perturbation wavelength, and the growth rate increase unbounded when λ → 0. When viscosity, surface tension, and compressibility are considered, the dispersion relation is modified. [START_REF] Chandrasekhar | Hydrodynamic and Hydromagnetic Stability[END_REF] shows that viscosity cannot stabilize the flow, it dampens the small wavelengths growth rate, which nevertheless stays always positive, imposing a maximum to the growth rate for a wavelength λ max > 0. On the contrary, the same author demonstrates how surface tension is capable of stabilizing an unstable configuration for sufficiently small wavelengths, which nonetheless remains unstable if perturbations have sufficiently long wavelengths. Regarding compressibility, [START_REF] Livescu | Compressibility effects on the Rayleigh-Taylor instability growth between immiscible fluids[END_REF] proves that it can have both stabilizing and destabilizing effects. In the linear phase, the initial perturbations develop without interacting with each other. However, at some point at the end of this first phase, they reach an amplitude comparable with their wavelength, and nonlinear interactions begin to be relevant, so that linear arguments are no longer valid. Non-linear When the flow transition in the nonlinear phase, we observe a first qualitative difference depending on the Atwood number. For small density contrasts, A t 1, the evolution of the light and heavy fluid are similar, and we observe an interpenetrating bubble. When A t ∼ 1, the evolution is asymmetric, the lighter fluid moves into the heavy fluid in the form of bubbles while the heavy fluid in the form of spikes. In this regime [START_REF] Youngs | Numerical simulation of turbulent mixing by Rayleigh-Taylor instability[END_REF] found that the mixing zone size δ = 2h has a quadratic dependence on time

δ = α i A t gt 2 .
Where α is the mixing zone growth rate. Discussion around this value has gone for 30 years, due to contrasting results obtained in experiments and numerical simulations. A combined effort resulted in the α-group collaboration [START_REF] Dimonte | A comparative study of the turbulent rayleigh-taylor instability using high-resolution three-dimensional numerical simulations : the alpha-group collaboration[END_REF]], which proved that numerical simulation and experiment converged to two different results, with the second value twice as much as the first. This incoherence was connected to the large scale perturbations and confinement effects in the experimental set-up that were not taken into account into numerical computations.

Nowadays discussions around the parameter alpha are still ongoing, for instance Zhou (2017a) in his review after collecting the value of α published after the α-group paper, found a spread of its value depending on Atwood number and initial conditions wavelengths (lower values for short initial wavelength, higher values for large initial wavelength ).

Turbulence At the end of the nonlinear phase, the bubbles and spikes begin to interact with each other creating a turbulent mixing regime. This phase addressed by [START_REF] Boffetta | Incompressible Rayleigh-Taylor turbulence[END_REF] in a recent review paper, present all the characteristic of turbulent mixing from multi-scale properties to the development of the Kolgomorov cascade [START_REF] Cabot | Reynolds number effects on Rayleigh-Taylor instability with possible implications for type Ia supernovae[END_REF].

Richtmyer-Meshkov

The work on the Richtmyer-Meshkov (RM) instability begins with the theoretical analysis of [START_REF] Richtmyer | Taylor instability in shock acceleration of compressible fluids[END_REF] and the experimental confirmation of the results by [START_REF] Meshkov | Instability of the interface of two gases accelerated by a shock wave[END_REF]. In principle, this instability can be considered as a particular case of RT instability with an impulsion acceleration. As for the Rayleigh-Taylor, this instability has been the object of extensive study and reviews [START_REF] Brouillette | The Richtmyer-Meshkov instability[END_REF], Zhou 2017a].

As in the case of RT, also for Richtmyer-Meshkov, we can divide the evolution into three phases : linear, nonlinear, turbulent, which can be observed in Figure 1.3.

Linear

The theory for the linear phase of the instability was derived by [START_REF] Richtmyer | Taylor instability in shock acceleration of compressible fluids[END_REF], who assumed that the amplitude, h, of the perturbations was small compared to their wavelength,λ, without the effects of viscosity or surface tension. With these hypotheses we obtain :

dh dt = h 0 ∆u 2π λ A t (1.4)
where ∆u is the shock impulsion, h 0 , and A t = ρ 2 -ρ 1 ρ 2 +ρ 1 are the initial amplitude and the Atwood number just after the shock has impacted the interface. The numerical solution of (1.4) shows that in this phase, the growth of the perturbation is linear in time, unlike the exponential growth of the RT. Moreover, the interface is unstable for every value of the Atwood number, even the negative [START_REF] Brouillette | The Richtmyer-Meshkov instability[END_REF]]. The effects of viscosity and surface tension were considered by [START_REF] Mikaelian | Effect of viscosity on Rayleigh-Taylor and Richtmyer-Meshkov instabilities[END_REF], who shows that viscosity induces damping of the perturbation while surface tension oscillations. Short wavelength perturbations oscillate faster but are more damped ; longer-wavelength perturbations oscillate slower and are less damped.

As for the Rayleigh-Taylor, in the linear phase, the initial perturbations develop without interaction with each other, but when their amplitude has increased sufficiently, the nonlinear phase takes place.

Non-linear

We observe the asymmetry between the interpenetration between the two fluids. The heavy fluid tends to the formation of spikes, whereas the light fluid organizes as bubbles. Some analytical tools allow studying this phase for single-mode perturbation [START_REF] Brouillette | The Richtmyer-Meshkov instability[END_REF]]. However, usually, the wavenumbers of the initial perturbation span many orders of magnitude. Also, as the RM flow develops, the Kelvin-Helmholtz instability develops, which further complicates the computational task. Ultimately, a three-dimensional turbulent mixing zone develops on the interface.

At the end of the non-linear stage, the time evolution of the overall mixing zone size h follows power-law ∼ t θ with values of θ ranging from 0.3 to 1, (see Zhou (2017a) for a collection of published values). As for the α coefficient in the self-similar phase of a Rayleigh-Taylor mixing zone, there was a debate on the value of the exponent θ, addressed by the θ-group work [START_REF] Thornber | Late-time growth rate, mixing, and anisotropy in the multimode narrowband richtmyer-meshkov instability : The θ-group collaboration[END_REF]. They determined that the growth rate exponent is θ = 0.292 ± 0.009, in good agreement with prior studies ; however, the exponent is decaying slowly in time.

Turbulence At the end of the nonlinear phase, the bubble and spikes begin to interact with each other creating a turbulent regime. The mixing between the two fluid is intensified, and the interface is no longer visible. Also, the mixing zone size continues to follow the evolution law ∼ t θ .

Open problems

From the short review on Rayleigh-Taylor and Richtmyer-Meshkov instabilities, we observe similarities in the development and evolution of the flow together with differences. The instability mechanisms are the same, and for both cases, the discussion on the dependence of the long-time flow evolution on certain factors is still ongoing. Here we present some open questions : What is the effect of initial conditions ? How well can these instabilities mix ? Is the anisotropy imposed by the acceleration maintained in the smallest scale of turbulence ? What happens if the acceleration is inverted during the development of the instability ? What is the effect of the Atwood (and Mach) number on the flow ? Initial condition (IC) influence One of the questions that arise from the study of RT and RM turbulence is about the role of IC, and this question seems to be still open to debate.

In the Rayleigh-Taylor context, this dependence has been used to justify the difference between experimental and numerical studies. For instance, [START_REF] Ramaprabhu | A numerical study of the influence of initial perturbations on the turbulent Rayleigh-Taylor instability[END_REF] show that initial perturbation wavelength may influence the late time self-similar evolution, depending on the wavelength. With short wavelengths [START_REF] Ramaprabhu | A numerical study of the influence of initial perturbations on the turbulent Rayleigh-Taylor instability[END_REF][START_REF] Youngs | The density ratio dependence of self-similar Rayleigh-Taylor mixing[END_REF]], it seems reasonable to assume a loss of the initial perturbation proprieties, whereas long-wavelengths impose a dependence on the initial conditions. [START_REF] Banerjee | 3d simulations to investigate initial condition effects on the growth of Rayleigh-Taylor mixing[END_REF] using Direct Numerical Simulation (DNS) concluded that RT mixing has a strong dependence on IC. Whereas the results of [START_REF] Soulard | Large-scale analysis of unconfined selfsimilar Rayleigh-Taylor turbulence[END_REF] prove that there is no permanence of large eddies for RT turbulence, which at late times is dominated by nonlinear interaction with loss of information of IC.

For Richtmyer-Meshkov, the influence of initial conditions is likely to be significant in most applications, in particular for the nonlinear and turbulent phases. For example, [START_REF] Budzinski | Influence of initial conditions on the flow patterns of a shock-accelerated thin fluid layer[END_REF] prove that changes in the initial interface shape give different patterns during the evolution of the flow. [START_REF] Thornber | The influence of initial conditions on turbulent mixing due to Richtmyer-Meshkov instability[END_REF]] found a dependence on initial conditions for the self-similar exponent θ, which seems to depend, similarly to the RT case, to the range of scale used for the initialization.

Moreover, from a theoretical point of view [START_REF] Soulard | Permanence of large eddies in Richtmyer-Meshkov turbulence with a small Atwood number[END_REF], evidence that the initial condition can influence late time flow evolution. They prove the permanence of large eddies, in case of low Atwood number, relating the self-similar growth rate of the turbulent mixing zone to the infrared slope of the velocity spectrum.

Mixing rate

The discussion about how the quantification of mixing is another subject of debate, with different quantity proposed (see Zhou (2017a), for instance). One that is widely used is molecular mix fraction Θ [START_REF] Youngs | Numerical simulation of turbulent mixing by Rayleigh-Taylor instability[END_REF]], which can be linked to α [START_REF] Gréa | The rapid acceleration model and the growth rate of a turbulent mixing zone induced by Rayleigh-Taylor instability[END_REF]].

The value reported for RT turbulence [START_REF] Ramaprabhu | A numerical study of the influence of initial perturbations on the turbulent Rayleigh-Taylor instability[END_REF], Youngs 2003; 2013] varies from 0.68 to 0.8. The reasons for the discrepancy are to be found in the dependence on the initial perturbation, with long initial wavelengths that result in a low value of the mixing parameter.

The same quantity can be measured in the RM mixing layer. The range of Θ from numerical simulation goes from 0.3 -0.8 [START_REF] Thornber | The influence of initial conditions on turbulent mixing due to Richtmyer-Meshkov instability[END_REF][START_REF] Oggian | Computing multimode shock-induced compressible turbulent mixing at late times[END_REF]. This difference can be attributed to differences in initial interface thickness and shape, as well as the characteristics of the initial perturbations, in particular, the range of scale used.

Anisotropy Both RT and RM have a preferential direction imposed by constant or impulsive acceleration so that the question about the anisotropy of the flow has to be addressed.

In general, for RT flows, the large turbulent scales are anisotropic due to the action of gravity, but the flow becomes isotropic at small scales in the core of the mixing layer. However, there are still discussions open, for instance, [START_REF] Livescu | High-reynolds number Rayleigh-Taylor turbulence[END_REF] found anisotropy for the largest and smallest scales where the buoyancy effects continue to be significant due to the cancellation between nonlinear transfer and viscous dissipation. Furthermore, Gréa et al. (2016b) show that the level of anisotropy is not very sensitive to initial energy and buoyancy spectra.

For the RM case, [START_REF] Soulard | Permanence of large eddies in Richtmyer-Meshkov turbulence with a small Atwood number[END_REF] prove large scales keep their initial anisotropy during the flow evolution, implying that the return to isotropy of the turbulent mixing zone is only partial. These results confirm the observation of [START_REF] Thornber | The influence of initial conditions on turbulent mixing due to Richtmyer-Meshkov instability[END_REF], [START_REF] Lombardini | Transition to turbulence in shockdriven mixing : a mach number study[END_REF]. This anisotropy, is, of course, more significant in the direction of the shock propagation, with the ratio of the component turbulent kinetic energy (TKE) in the parallel and perpendicular directions that tends towards a constant value.

Acceleration inversion and demixing for RT

The assumptions made about the direction of the acceleration with respect to the density gradient are not always valid in real applications, for instance, in astrophysical contexts like the Crab nebula [START_REF] Ebisuzaki | Rayleigh-Taylor instability and mixing in sn 1987a[END_REF]]. In this case, one has to reconsider the theoretical results under the light of the variable acceleration, which changes direction during the flow evolution. One of the consequences, for example, is that the expression of the mixing zone width self-similar evolution is no longer valid. When the sign of the acceleration is reversed, there is a partial demixing for miscible fluids and complete demixing for immiscible fluids.

Numerical simulations by [START_REF] Ramaprabhu | The rayleigh-taylor instability driven by an accel-decel-accel profile[END_REF] show that a deceleration between two acceleration phases destroys the spikes bubbles structures changing the flow topology. It increases the molecular mixing and the rate of return to isotropy within the mixing layer, retarding the following growth towards self-similarity. However, [START_REF] Burlot | Etude et modélisation de la turbulence homogène stratifiée instable[END_REF] shows how the final state of the mixing zone depends on the time between the two acceleration phases.

The role of the Atwood number For RT, as the Atwood number increases, the self-similar constant α increases. On the contrary, the late times' value of the mixing parameters shows no influence on the At. While the small-scale anisotropy may be more persistent for higher Atwood number, which can be explained by the fact that the effects of gravity that cause the anisotropy increase with the Atwood number (for a complete discussion see the corresponding section in Zhou (2017a)).

In the RM case, the self-similar exponent θ seems to have a little dependence on the At, while an increase in Atwood number increases the time needed to reach the self-similar state, and it has the opposite effect of the molecular mixing [START_REF] Thornber | The influence of initial conditions on turbulent mixing due to Richtmyer-Meshkov instability[END_REF].

Mach number

For the RM, the dependence on Mach number has been studied. The experimental results of Weber et al. (2014b) show the relative insensitivity of Θ to the Mach number, which has been confirmed by the numerical simulation of [START_REF] Lombardini | Transition to turbulence in shockdriven mixing : a mach number study[END_REF]. [START_REF] Scase | Rotating rayleigh-taylor instability[END_REF], chemical reactions [START_REF] Chertkov | Reactive Rayleigh-Taylor turbulence[END_REF], and magnetic field [START_REF] Chandrasekhar | Hydrodynamic and Hydromagnetic Stability[END_REF]] are among the effects that are sometimes present in applications where they can influence the evolution of the buoyancydriven instability.

Others effects Rotation

Converging Geometry

Rayleigh-Taylor or Richtmyer-Meshkov instability have been mostly studied in a planar geometry, a simplified setting useful to understand the fundamental driving processes. However, in some applications, for instance, inertial confinement fusion compression, the interface geometry can be spherical or cylindrical, also called converging geometry.

These configurations were investigated by [START_REF] Bell | Taylor instability on cylinders and spheres in the small amplitude approximation[END_REF] and [START_REF] Plesset | On the stability of fluid flows with spherical symmetry[END_REF] for inviscid potential flow and by [START_REF] Chandrasekhar | The character of the equilibrium of an incompressible fluid sphere of variable density and viscosity subject to radial acceleration[END_REF] who considered the effect of viscosity. They found that, in this situation, the perturbation growth is modified by pure geometrical effects, independent of the density of the fluid.

Modelling

All the fundamental studies using experiment or high fidelity numerical simulation are usually performed to understand the phenomena involved in RT and RM. The objective of the studies is to find the main ingredients of these buoyancy-driven flows and create meaningful models for real-life problems, where the range of spatial scales exceeds the current numerical capabilities. Multiple models with different levels of complexity have been proposed during the years. The first and most simple is the buoyancy-drag model [Zhou 2017a], which computes the evolution of the amplitudes of the mixing region. It is essentially an equation of motion that balances the inertia, buoyancy, and drag forces.

The Reynolds-Averaged Navier-Stokes (RANS) models are the second step in complexity. They are a class of method which attempt to analyze and characterize flows at high Reynolds number, with well-developed turbulence, which constitutes one of the shortcomings of RANS modeling for mix calculations in RT and RM. They are not designed to compute the production of turbulence from instabilities (Zhou (2017b) in his review discusses different RANS models with corresponding merits and shortcomings). Multiple models have been proposed during the years, K -, Besnard-Harlow-Rauenzahn (BHR), Reynolds stresses, multifluid, spectral with an increasing capability to reproduce the complex phenomena of the mixing.

All the proposed models have some free parameters that have to be fixed. The problem that arises at this point is that usually, they are calibrated to reproduce selfsimilar states. However, nothing ensures that they can reproduce the transient phase, which is more important from an engineering viewpoint since the self-similar state is never reached in real applications. For instance, Gréa et al. (2016a) demonstrated for different initial conditions, how different mixing models are capable of getting the self-similar state but behave very badly and differently in the transient period.

Why is buoyancy-driven mixing important ?

Concerning the hydrodynamic instabilities that we have discussed, the main reason that has driven the efforts over the years has been the quest to understand the role of turbulent mixing in Inertial Confinement Fusion (ICF) implosions.

For many years this has been identified as having a detrimental effect on inertial confinement fusion target. The inward acceleration, coupled with the fact that the density increases with the radius, is the basic context for the onset of baroclinic instability. Moreover, these instabilities are only a part of the phenomena involved in an ICF implosion, where plasma, radiation, and kinetic effects play significant roles. Furthermore, the fact that this is a context of high energy density (HED) physics makes experimental studies to understand the driving mechanisms even more complicated. Besides, numerical simulations used in the context of ICF are costly in terms of computational resources and usually take into account all physical phenomena that occur during an implosion. This makes it difficult to understand the role of hydrodynamics in this context. That is why, in this thesis, we propose to study mixing in a simplified framework, where we consider the effect of transport coefficients variation in an idealized ICF target and the effect of variable density in the homogeneous context.

The remainder of this chapter is organized as follows : After an introduction to the inertial confinement fusion in section 1.2, we discuss the radiation hydrodynamics simulation used to investigate ICF implosion and the assumption made to arrive at the idealized cases studied in this thesis. After this discussion, the following two sections are dedicated to the review of variable viscosity and variable density turbulence, the two main effects that are conserved in the idealized case. Finally, we discuss the plan of the thesis.

Inertial confinement fusion : an introduction

The idea behind ICF [Atzeni and Meyer-ter Vehn 2004b] is to compress a fuel usually composed by a mixture of hydrogen isotopes deuterium, D, and tritium, T, to incredible pressures, around ∼ 100 Gbar, densities ∼ 50 gcm -1 , and temperature ∼ 10 keV . Then the objective is to burn the fuel in a short time, ∼ 50 ps, in which inertia keeps the fuel confined. High power radiation, usually provided via multiple high-intensity lasers, is the external source that provides the energy that drives the 1.2. Inertial confinement fusion : an introduction 13 compression.

In the "direct drive" approach [START_REF] Betti | Inertial-confinement fusion with lasers[END_REF], the lasers are focused directly on the spherical target. The rapid heating causes vaporization of the shell materials and a consequent inward implosion, compressing the fuel. Moreover, it creates a shock wave, which further increases the fuel temperature and pressure and results in a self-sustaining burn.

Another approach, currently employed at the National Ignition Facility (NIF) and Laser Megajoule (LMJ), is the "indirect drive" method [START_REF] Lindl | Development of the indirect-drive approach to inertial confinement fusion and the target physics basis for ignition and gain[END_REF], in which the lasers heat the inner walls of a gold cavity containing the target, which is uniformly irradiated by X-rays. This radiation heats the outer surface of the spherical shell, causing a high-speed ablation and consequent implosion of the capsule in the same way as in the "direct drive." These two approaches are schematically represented in Figure 1.4. Following Atzeni and Meyer-ter Vehn (2004b) we can schematically divide the implosion in three phases :

1. Deposition of the energy by laser beams and ablation of the outer materials.

Implosion of the capsule that reaches the extreme thermodynamic condition

required for the fusion.

3. Ignition of the fuel and hot spot generation.

The numbers correspond to those of the images at the bottom of Figure 1.4.

Once the laser has deposited the energy via direct or indirect drive, the outer surface of the ablator starts to vaporize and expand outwards, forcing the inner part of the capsule shell to move inwards, starting the compression. The pressure that is acting on the ablator/fuel interface at this stage is around 100 Mbar, with an implosion velocity of 300 km/s. At the same time, as the inwards implosion of the ablator, a strong shock intensified by the spherical convergence propagates towards the center of the capsule. When it reaches the center, the shock is reflected and starts to travel outwards, when at some point encounters the inwards moving shell, the shell is decelerated. It is at this point that a significant part of the kinetic energy is transformed in internal energy, and the density increases considerably. At the same time, the sequence of shock and the compression heat the gas to a very high temperature, that is, a hot spot has formed. The hot-spot temperature is around 10 keV with a pressure of the order of 100 Gbar and density around 50 g/cm 3 . At this point, one expects a sudden, around 50 ps, rise in temperature up to 100 keV that would signal the ignition.

During the compression, perturbation to this idealized configuration may alter the dynamics of the implosion, modifying the desired energy output or preventing the fusion at all. These perturbations, which may develop at the ablator/fuel interface, see 1.4(c), are amplified by the Rayleigh-Taylor instability and may lead to the mixing of cold, dense material in the hot-spot plasma, de facto decreasing the possible yield of the fusion.

ICF hydrodynamic simulations

ICF implosions are usually modeled using radiation hydrodynamics codes, that are capable of accounting for effects coming from all the sources involved in the compression processes from the laser deposition to radiation effects [START_REF] Marinak | Three-dimensional hydra simulations of national ignition facility targets[END_REF]]. From the hydrodynamic point of view, since they deal with highly compressible materials, these codes usually employ Arbitrary Lagrangian Eulerian (ALE) scheme with artificial viscosity to stabilize shock, or other compressible hydrodynamic formulations [START_REF] Cabot | Reynolds number effects on Rayleigh-Taylor instability with possible implications for type Ia supernovae[END_REF], and they can solve 1D [START_REF] Vold | Plasma viscosity with mass transport in spherical inertial confinement fusion implosion simulations[END_REF], 2D [START_REF] Gittings | The RAGE radiation-hydrodynamic code[END_REF] or 3D [START_REF] Marinak | Advances in hydra and its applications to simulations of inertial confinement fusion targets[END_REF] problems. In the case where the fuel reaches temperatures such as the mean free path of the ions become comparable to the size of the hot-spot, a kinetic description has to be used [START_REF] Hoffman | Approximate models for the ion-kinetic regime in inertial-confinement-fusion capsule implosions[END_REF].

The parameter that determines if one can use the continuum mechanics approach to investigate a particular fluid problem is the Knudsen number, Kn, the ratio bet-

ICF hydrodynamic simulations

ween the mean free path of the fluid constituents and the characteristic size of the problem. Usually, large imploding targets as the one used at the NIF, kinetic effect seems to be negligible. For instance, this can be verified using the expression given by [START_REF] Molvig | Knudsen layer reduction of fusion reactivity[END_REF], that gives Knudsen numbers of ∼ 10 -2 at bang time, so that the continuum assumption is wholly justified. This assumption would not be verified in other ICF implosions, for instance, is smaller capsules such as the ones used in OMEGA experiments [Rinderknecht et al. 2014b].

Another question that has to be addressed is related to the Magnetohydrodynamics (MHD) effects. There exists the possibility of selfgenerating magnetic field during the compression due to the asymmetry of the implosion [START_REF] Igumenshchev | Self-generated magnetic fields in direct-drive implosion experiments[END_REF]. These fields can have intensity up to 10 4 Tesla, in the last phases of the implosion [START_REF] Walsh | Self-generated magnetic fields in the stagnation phase of indirect-drive implosions on the national ignition facility[END_REF]]. The magnetic energy density is

E B = B 2 µ 0 ∼ 10 14 kg m s 2 .
This value has to be compared with typical kinetic energy density for an ICF implosion that is :

E kin = 1 2 ρv 2 ∼ 10 17 kg m s 2 .
The ratio between magnetic and kinetic energy density is ∼ 10 -3 so that MHD effect can be neglected.

ICF simulations are performed mainly to reproduce implosion experiments in order to explain different than expected behaviors and also to suggest possible improvements in the quest to obtain nuclear fusion with desired yields. In recent years, computations mostly concentrated on understanding the impact of various perturbation sources on the evolution of the spherical compressed flow. The investigation considered both low-foot and high-foot implosions, which are differentiated by the shape of the driving laser pulses. In particular low foot implosion have low compression velocities (320 -330 kms -1 ) but higher convergence ratios, the ratio between the initial and final diameter of the target, (40 -45), compared to the high foot case where compression velocity can reach 380 kms -1 but with low convergence ratios of ∼ 35. In Figure 1.5, the effects of the various perturbations sources are apparent. From the tent perturbation at the capsule poles to fill tube defects and the roughness of the ablation front that grows during compression. While the results for the two cases are different, it is evident the presence of denser colder matter that reaches the center at the bang time in the two simulations. This is mainly caused by the tent perturbation growth and the hydrodynamic instabilities due to shocks and the strong inward acceleration.

From ICF radiative hydrodynamics simulation to idealized simulation framework

At the moment, many results of ICF implosions simulation exist [START_REF] Clark | Radiation hydrodynamics modeling of the highest compression inertial confinement fusion ignition experiment from the national ignition campaign[END_REF]2016, Weber et al. 2014a;;2017]. They have been used to study multiple aspects of actual ICF experiments. These simulations take into account all the physical phenomena that are important in an ICF context (radiation, alpha heating, alpha burn). However, usually, they did not take into account transport coefficients, although recently Weber et al. (2014a) have shown how important it is to take into consideration viscosity to have a realistic representation of the small scale behavior of the flow in the hot-spot. [START_REF] Clark | Three-dimensional simulations of low foot and high foot implosion experiments on the national ignition facility[END_REF] in the simulations presented in section 1.3 take into consideration the viscosity, but they do not consider the effect of molecular 1.4. From ICF radiative hydrodynamics simulation to idealized simulation framework 17

diffusivity.

The different physical mechanisms in action during an ICF compression are challenging to disentangle if one uses the full simulations just described. A solution is to employ idealized theoretical frameworks allowing to isolate one phenomenon from the others, and in that way, have a deeper comprehension of the involved physics. As an example, in Figure 1.6, we show different levels of idealization for an inertial confinement fusion implosion. For instance, one can discard all the nuclear physics effects and radiative effects, leaving a problem of a spherical mixing zone between a heavy and light material under compression. Furthermore, one can make the hypotheses that locally, the spherical mixing zone could be approximated by a plane mixing layer under the effect of vertical acceleration, leaving with a Rayleigh-Taylor problem. The last step would be to consider the center of the planar mixing zone where inhomogeneous effects are negligible so that one can study homogeneous problems.

In this thesis, we have chosen to study two of the three idealized configurations depicted in Figure 1.6. In particular, we choose to study the compression of spherical turbulent mixing zones composed by Deuterium-Tritium (DT) and Carbon-Hydrogen (CH) in the plasma state, taking into consideration plasma transport coefficients. At the same time, we have used homogeneous turbulence simulations whereby we investigated compressed turbulent plasma and the unstably stratified homogeneous turbulence using the variable density approximation.

The next two sections are dedicated to introducing the two main subjects of this thesis, namely : variable viscosity effects in turbulent flows and variable density turbulence.

Variable viscosity flows

Variable viscosity flows, are not specific of the ICF context, but on the contrary represent an important part of real-life flow. From geophysics [Turner andCampbell 1986, Christensen and[START_REF] Christensen | 3-d convection with variable viscosity[END_REF] to Magnetohydrodynamics (MHD) [START_REF] Pantokratoras | Study of mhd boundary layer flow over a heated stretching sheet with variable viscosity : a numerical reinvestigation[END_REF][START_REF] Makinde | Mhd variable viscosity reacting flow over a convectively heated plate in a porous medium with thermophoresis and radiative heat transfer[END_REF], from re-entry vehicle aerodynamics [K. [START_REF] Reddy | Hypersonic turbulent flow simulation of fire ii reentry vehicle afterbody[END_REF] to blood vessel simulation [START_REF] Layek | Unsteady viscous flow with variable viscosity in a vascular tube with an overlapping constriction[END_REF], from oil industry applications [START_REF] Joseph | Core-annular flows[END_REF]] to rising plumes [START_REF] Carey | Variable viscosity effects in several natural convection flows[END_REF].

Viscosity variations, in many real cases due to viscosity stratification, play an essential role in the stability of parallel flows. For instance, [START_REF] Govindarajan | Instabilities in viscosity-stratified flow[END_REF] have reviewed these effects. They highlight the importance of viscosity variations even at high Reynolds numbers, which can alter the stability of the flow.

Most of the cited works are either theoretical or numerical. One of the first experimental investigations on the nature of flows with different viscosities is due to [START_REF] Campbell | Turbulent mixing between fluids with different viscosities[END_REF]. They studied the injection of a fluid of low viscosity ν 1 in a host fluid with a higher viscosity ν 2 and higher density. When the ratio ν 2 /ν 1 is close to one, classical turbulent mixing phenomenology is found, while when this viscosity ratio increases, at fixed Reynolds number, the mixing is less and less effective until a critical viscosity ratio when it is inhibited. [START_REF] Talbot | Mélange et dynamique de la turbulence en écoulements libres à viscosité variable[END_REF] conducted an experimental study using a propane jet issuing into an air/neon mix. This setting was chosen because of the lower viscosity of propane with respect to the air/neon mix. Moreover, it allows the author to investigate the 1.5. Variable viscosity flows effects of viscosity variations in fluid with almost similar densities. These results were then confronted with ones obtained for a case where the jet and the host fluid have identical viscosity, air into air. [START_REF] Talbot | Mélange et dynamique de la turbulence en écoulements libres à viscosité variable[END_REF] proves that the efficacy of the mixing, is higher for the jet with variable viscosity with respect to the constant viscosity case, with the same initial momentum. Moreover, he shows how viscosity variations influence velocity signals, shown in Figure 1.7 for the two jets, at a position with the same local Reynolds number. In particular, from Figure 1.7, we can see that the turbulence developed by the variable viscosity jet has higher frequency fluctuations, suggesting the presence of finer structure with respect to the constant viscosity jet. From [START_REF] Talbot | Mélange et dynamique de la turbulence en écoulements libres à viscosité variable[END_REF] Concerning more fundamental aspects of variable viscosity turbulence, most of the investigations were concentrated on the Homogeneous Isotropic Turbulence (HIT) decay. One of the first works by [START_REF] Lee | Validity of Taylor' s dissipation-viscosity independence postulate in variable-viscosity turbulent fluid mixtures[END_REF] used direct numerical simulation of mixing of two fluids with different viscosities showing that the Taylor postulate, independence of dissipation from viscosity, is still valid in this type of fluid mixtures. [START_REF] Gréa | The effects of variable viscosity on the decay of homogeneous isotropic turbulence[END_REF] using DNSs and the Eddy Damped Quasi-Normal Markovian (EDQNM) spectral model show that in the case where viscosity depends linearly on a scalar field, the effective viscosity is proportional to the viscosity fluctuations variance and it is lower than the mean viscosity. A recent investigation by [START_REF] Gauding | One-point and two-point statistics of homogeneous isotropic decaying turbulence with variable viscosity[END_REF] confirmed the results of [START_REF] Lee | Validity of Taylor' s dissipation-viscosity independence postulate in variable-viscosity turbulent fluid mixtures[END_REF], and they show how in variable viscosity HIT there is an increased level of small scales intermittency due to the presence of smaller scales in the low viscosity region of the flow. Moreover, they demonstrate the presence of an inverse energy cascade contribution, from small to large scales, due to the viscosity gradients. The possible effects of viscosity on the large scales of turbulence is evidenced by [START_REF] Voivenel | Variable viscosity jets : Entrainment and mixing process[END_REF], who investigated the entrainment of a jet issuing in a more viscous ambient fluid, found that variable viscosity affects the dynamics of the flow at all scales. Moreover, [START_REF] Danaila | Self-similarity criteria in anisotropic flows with viscosity stratification[END_REF] using the same experience as [START_REF] Voivenel | Variable viscosity jets : Entrainment and mixing process[END_REF] showed how the similarity assumption for the jet is valid in regions where the viscosity is uniform but is not valid anymore in regions where the flow has intense viscosity gradients. Unlike in the works mentioned above, viscosity in ICF increases during the compression due to the growth of temperature, and its effect become more and more important as compression progresses. Weber et al. (2014a) show that thanks to the viscosity growth into the capsule hot-spot, small scale fluctuations are entirely suppressed. On the same subject, Haines et al. (2014b) provided examples of plasma viscous effects on instability growth. The authors performed two-dimensional hydrodynamic simulations of Rayleigh-Taylor and Kelvin-Helmholtz instabilities with plasma transport coefficients, at a fixed temperature of 1 keV . They found that if the characteristic length of the instability is small enough, around ∼ 100µm, viscosity effects become noticeable, and they are dominant at smaller scales, around ∼ 1µm.

Variable density turbulence

The usual approximation in the fundamentals studies on buoyancy-driven mixing is the Boussinesq approximation [START_REF] Boffetta | Incompressible Rayleigh-Taylor turbulence[END_REF], which assumes incompressible flows and small variation in the density. However, in most of the applications, these assumptions may become limiting. Density variation may become important in certain types of flows, and consequently, their effects have to be carefully taken into account. Moreover, the Boussinesq approximation assumes symmetry in the mixing ; it does not differentiate the mixing of a heavy in a light material or the inverse. A famous example was given by [START_REF] Miller | Which way is up ? A fluid dynamics riddle[END_REF], and it is shown in Figure 1.9. 2005) [START_REF] Sandoval | The dynamics of variable-density turbulence[END_REF], in his thesis, introduced the theoretical framework of the variable density turbulence. His work was based on the evidence, provided by [START_REF] Joseph | Fluid dynamics of two miscible liquids with diffusion and gradient stresses[END_REF], that the mix of two incompressible flows, i.e. flows in which the velocity field, u, is solenoidal ∇ • u = 0, is no longer incompressible and the divergence of the velocity field depends on the density, ρ, and the diffusion coefficient, D, variations i.e.

∇ • u = -∇ • D ρ ∇ρ .
Within this approximation, acoustic phenomena are decoupled from the problem, and the density is not dependent on pressure, meaning that with this assumption, only low Mach number (the ratio between the flow velocity and the speed of sound) flows can be studied. Sandoval used this approximation to investigate the turbulent mixing generated by buoyancy effects, with the idea to improve one and two-point turbulence models. He used direct numerical simulations of the Navier-Stokes equations to study the decay of homogeneous isotropic and buoyancy generated turbulence with variable density effects. In both cases, Boussinesq approximation was not applicable, because the ratio between the initial rms value of the density fluctuations and the initial mean density was higher than 0.1. [START_REF] Livescu | Buoyancy-driven variable-density turbulence[END_REF]2008) have performed direct numerical simulations of buoyancy-driven turbulence, with different initial conditions. They evidenced that the pure light fluid mixes more rapidly than the pure dense fluid and that when the density contrast grows, one of the limiting factors from a numerical resolution point of view is the sharp gradients in the density field. [START_REF] Rao | Nonlinear effects in buoyancy-driven variable-density turbulence[END_REF] studied the L 2 -spatial average of the density gradient, evidencing the intense mixing of density field at small scales in buoyancy-driven turbulence and, also, the possible blow-up of density gradient in a finite time.

From an experimental point of view [START_REF] Prestridge | Experimental adventures in variable-density mixing[END_REF] provided a review of the experimental efforts in the investigation of variable density mixing. For instance, the author evidenced that, there are peculiarities in variable density mixing related only to density effect, but independent on the way mixing has started. On the same note, [START_REF] Gerashchenko | Density and velocity statistics in variable density turbulent mixing[END_REF] evidenced how the same non-Boussinesq mechanism, that modifies the mixing in a high-density jet issuing in a low-density ambient fluid is the same that was identified in homogeneous buoyancy-driven turbulence by [START_REF] Livescu | Buoyancy-driven variable-density turbulence[END_REF] and in Rayleigh-Taylor mixing layer by [START_REF] Livescu | New phenomena in variable-density rayleigh-taylor turbulence[END_REF]. In particular, they found that buoyancy has a considerable impact on turbulent quantities such as velocity fluctuations, turbulent kinetic energy, and Reynolds stresses and that the higher is the Atwood number of the jet, the slower is the mixing. In a following investigation on the same experimental set-up [START_REF] Charonko | Variable-density mixing in turbulent jets with coflow[END_REF] found a negative turbulent kinetic energy production near the center of the dense jet, i.e. the mean flow receive energy from the fluctuations. To better understand this phenomenon, [START_REF] Lai | A Kármán-Howarth-Monin equation for variable-density turbulence[END_REF] derived a Karman-Howarth-Monin equation for variable density turbulence to investigate the energy scale-by-scale budget in the jet. They identified the total inter-scale energy transfer rate across a scale r, as the sum of a linear Π U and nonlinear contribution Π. The linear contributions take into account vortex stretching and other effects from the mean flow gradient, while the nonlinear term is the classical energy cascade. These transfers are depicted in Figure 1.10. The overall net inter-scale transfer is similar for the two jets, but in the Boussinesq jet, the two terms Π, and Π U have the same sign, and both contribute to the forward cascade from large to smaller scales. In the non-Boussinesq case, the linear term has a positive sign, indicating an inverse cascade from small to larger scales.

As an example of the difference between variable density and Boussinesq approximation, we show the results of a spherical mixing zone compression. In this simulation, the mixing zone is composed of a heavy material that surrounds a light material, with Atwood number equal to 0.7. The compression forces accelerate the mixing layer towards the center of the spherical shell, leading to a Rayleigh-Taylor flow in spherical geometry. In Figure 1.11 are plotted the contours of θ, a density related quantity (the definition of θ, is given in chapter 2). These results are computed using the codes described in chapter 2 and 3. From the same initial condition in (I), the two simulations at the instant (II) show a different flow topology. In the Boussinesq computation, the light/heavy and the heavy/light sides of the mixing zone have the same flow structure, as already pointed out in Figure 1.9. On the other hand, if we observe the variable density results, we find a difference between the flow that is developing in the light material and the one in the heavy material. At the center of the capsule, we observe spikes of heavy materials penetrating, without the typical mushroom topology, that, on the contrary, we observe in the Boussinesq case. This is due to the higher inertia of the heavy material that avoids the development of the Kelvin-Helmholtz instability that causes the mushroom shape. The higher inertia is also the cause of the slow mixing of the heavy fluid with respect to the lighter fluid, as observed by [START_REF] Livescu | Buoyancy-driven variable-density turbulence[END_REF]2008).

Plan of the thesis

The plan of the thesis is the following :

In chapter 2, we present the theoretical developments that allow us to investigate the different topics covered in this thesis, the compressed turbulence, the spherical turbulent mixing zone, and the unstably stratified homogeneous turbulence. In chapter 3, we give an overview of the numerical methods used to solve the equation derived in 2 together with the validation steps for the new code developed.

In chapter 4, we consider a weakly coupled plasma of hydrogen under compression and we investigate the possible influence of initial condition on the sudden dissipation effect, knowing that classically, turbulence has a dependence on initial conditions. To this end, we use the direct numerical simulations of the equation derived in chapter 2 together with a spectral model, knowing that the DNS code cannot explore high Reynolds number flows. We use the eddy damped quasi normal markovian ( EDQNM) closure, knowing that the classical version is capable of simulating the decay of homogeneous isotropic turbulence for high Reynolds number. With this model, we perform a parametric study of the compression as a function of the initial Rey-1.7. Plan of the thesis 25 nolds number and compression number to understand the sensitivity of the sudden dissipation effect to the initial conditions. We establish, theoretically, the connection between initial condition and the statistical proprieties of the flow during compression, to have analytical relations between initial and final states. Furthermore, we extend our analysis to an inhomogeneous spherical configuration.

However, the analysis of chapter 4 does not take into consideration the density field, that is why we introduce the Unstably Stratified Homogeneous Turbulence (USHT) approximation in chapter 5. In this chapter, with constant viscosity and diffusivity, with the help of direct numerical simulations, we investigate the effects of the intensity of the initial density field variance on the self-similar states of USHT, confronting the results of simulations with both variable density and Boussinesq approximations. Moreover, using Implicit Large Eddy Simulations (ILES), we investigate how different initial conditions affect the self-similar behavior of the flow.

Finally, in chapter 6, we analyze the influence of the molecular and turbulent diffusivity on the evolution of the fuel/ablator mixing zone and the hot spot contamination. To pursue this analysis, we have derived the equation for a spherical mixing zone under compression with the variable density approximations. In order to be able to simulate the DT/CH mixtures, we have integrated into the direct numerical simulation code, a plasma physics model capable of predicting the transport coefficient values in a plasma mixture for different thermodynamic conditions. In this chapter, we introduce the different theoretical frameworks that have been used through this work. In section 2.1, we introduce the subject of compressed turbulence and the change of reference frame, a useful tool in this study. The problem of mixing in variable density unsteady stratified homogeneous turbulence is discussed in section 2.2. Section 2.3 is focused on spherical compression, with a summary of the literature findings and the theoretical framework that we have developed for this work. Finally, section 3.4 is devoted to the discussion on how to generate initial conditions for the different cases discussed.

Compressed turbulence

The problem of compressed turbulence has drawn much attention for several years. From reciprocating engines to astrophysics and inertial fusion applications, the interaction between turbulence and the mean compressing field is a fundamental aspect to understand. The relative importance between the effect of the mean field and the nonlinear interactions can be quantified by the time scale associated with the mean field t M and to the turbulence t F [START_REF] Hamlington | Modeling of non-equilibrium homogeneous turbulence in rapidly compressed flows[END_REF], and their ratio

C M = t F t M When C M
1, the turbulence dynamic is mainly influenced by the mean flow, and the nonlinear transfer and viscous dissipation can be neglected. In this regime, numerous authors have used the Rapid Distortion Theory (RDT) [Hunt and Carruthers Chapitre 2. Theory 1990] to obtain theoretical results and improve existing turbulence models. The RDT demonstrates, even if in a limit case, how the compression affects turbulence and provides theoretical solutions against which turbulence models can be tested [START_REF] Cambon | On the application of time dependent scaling to the modelling of turbulence undergoing compression[END_REF], Coleman and Mansour 1993[START_REF] Blaisdell | Rapid distortion theory for compressible homogeneous turbulence under isotropic mean strain[END_REF][START_REF] Durbin | Rapid distortion theory for homogeneous compressed turbulence with application to modelling[END_REF][START_REF] Hamlington | Modeling of non-equilibrium homogeneous turbulence in rapidly compressed flows[END_REF].

When compression and turbulence have similar time scales, i.e. C M ∼ 1, RDT hypotheses are no longer valid and Direct Numerical Simulation (DNS) of the Navier-Stokes equations become useful to investigate the physics of this problem. [START_REF] Rogallo | Numerical experiments in homogeneous turbulence[END_REF] and [START_REF] Wu | Simulation and modeling of homogeneous, compressed turbulence[END_REF] performed DNS of compressed turbulence and applied their results to improve turbulence models. In particular, they have studied a case where compressibility does not affect the fluctuating field, i.e. the turbulent Mach number M T = urms c is close to zero (here u rms is the rms velocity of the turbulent fluctuations and c the speed of sound). In his paper, [START_REF] Rogallo | Numerical experiments in homogeneous turbulence[END_REF] proposed a mean velocity field together with a coordinate system that moves with the mean flow, that have been employed in most of the works on isotropic compression [START_REF] Cambon | On the application of time dependent scaling to the modelling of turbulence undergoing compression[END_REF], Coleman and Mansour 1993[START_REF] Blaisdell | Numerical simulation of compressible homogeneous turbulence[END_REF], Davidovits and Fisch 2016a]. The extension to compressible turbulence has been done by [START_REF] Blaisdell | Numerical simulation of compressible homogeneous turbulence[END_REF], who removed the low Mach number hypothesis and performed a direct numerical simulation.

In this work, we use the same method as in [START_REF] Rogallo | Numerical experiments in homogeneous turbulence[END_REF]. We start from Navier-Stokes and mass conservation equations with constant dynamic viscosity µ

∂ t (ρU i ) + ∂ j (ρU j U j ) = -∂ i P + µ∂ 2 jj U i (2.1) ∂ t ρ + ∂ j (ρU j ) = 0 (2.2)
in a stationary Cartesian reference frame with space coordinates x i and time t. The density ρ, pressure P and velocity U i are decomposed into base and fluctuation :

ρ = ρ B + ρ (2.
3)

P = P B + p (2.4) U i = U B i + u i (2.5)
and density fluctuations, ρ , are neglected using the low Mach number hypothesis. The assumption ρ ρ holds during an isotropic compression while it is not necessarily true for non-isotropic compression, as shown by [START_REF] Coleman | Simulation and modeling of homogeneous compressible turbulence under isotropic mean compression[END_REF] and reference therein. If we consider homogeneous velocity fluctuations u i , the base flow velocity in Eq. (2.5) must have uniform spatial gradient [START_REF] Sagaut | Homogeneous Turbulence Dynamics[END_REF]. This requires that the base velocity U B (x, t) be a linear function of position x i and time t U B i (x, t) = -S ij x j , (2.6)

Compressed turbulence 29

where the tensor deformation S ij depends only on time ; in isotropic compressions its expression simplifies as S ij = S(t)δ ij , allowing us to define the compression rate S(t).

The assumption in this type of compression is to consider uniform base density,

ρ B = ρ B (t).
Together with the base velocity expression (2.6) and the continuity equation (2.2), this gives the time evolution of the base density :

ρ B = ρ B 0 exp 3 t 0 S(s)ds , (2.7)
where ρ B 0 is the base density at initial time t = 0. If we consider a cubic domain with edge length R(t) we can define the compression parameter Λ as the normalized size of the domain, Λ = R(t) R 0 , and we observe that both R(t) and Λ decrease in time.

The volume of the cubic domain is V (t) = R(t) 3 , which implies that the base density can be written as a function of the compression parameter ρ B (t) = ρ 0 Λ(t) -3 .

Using Eq. (2.7) we find the relation between the compression time rate and the compression parameter

Λ = exp - t 0 S(s)ds or S(t) = - 1 Λ dΛ dt (2.8)
and we rewrite the time evolution of the base density as

ρ B = ρ B 0 Λ -3 .
(2.9)

Injecting the base/fluctuations decomposition from Eqs. (2.3), (2.4), (2.5) for U i , P , ρ in Eq. (2.1), and applying the volume average operator * , we obtain an equation for the base flow

∂ t U B i + U B j ∂ j U B i = - ∂ i P B ρ B -∂ j u i u j , (2.10)
and ∂ j u i u j is zero for spatial homogeneity. If needed we can use (2.10) to obtain the base pressure P B , from the knowledge of the base velocity.

The equation for the perturbation

∂ t u i + U B j ∂ j u i + u j ∂ j U B i + u j ∂ j u i = - ∂ i p ρ B + µ ρ B ∂ 2 jj u i (2.11)
using the expression for the base velocity becomes

∂ t u i + u j ∂ j u i -S(t)x j ∂ j u i -S(t)u j = - ∂ i p ρ B + µ ρ B ∂ 2 jj u i .
(2.12)

Chapitre 2. Theory

The inhomogeneous term can be eliminated using a change of reference frame [START_REF] Rogallo | Numerical experiments in homogeneous turbulence[END_REF]] :

xi = Λ -1 (t)x j (2.13)
The detailed computation that shows how to choose the particular transformation is developed in appendix A. Using (2.13) and (2.9) in the perturbation momentum equation (2.12) we obtain

∂ t u i + 1 Λ u j ∂ j u i -S(t)u i = -Λ 2 ∂ i p + Λ µ ρ B 0 ∂ 2 jj u i . (2.14)
At this point we have two possibilities to further simplify this equation :

-We can eliminate the forcing term -S(t)u j .

-We can eliminate the time dependence due to Λ in front of the viscous term. In the first case the rescalings of the variables are [START_REF] Cambon | On the application of time dependent scaling to the modelling of turbulence undergoing compression[END_REF][START_REF] Viciconte | Self-similar regimes of turbulence in weakly coupled plasmas under compression[END_REF]

] ũi (x, t) = u i (x, t)Λ(t), t = t 0 Λ -2 (s)ds, p(x, t) = p(x, t)Λ 5 (t) (2.15)
and the corresponding equation is

∂ t ũi + ũj ∂ j ũi = -∂ i p + Λ 3 (t) µ ρ B 0 ∂ 2 jj ũi (2.16)
In the second case [Davidovits and Fisch 2016a] ũi (x, t) = u i (x, t)Λ -2 (t), t = t 0 Λ(s)ds, p(x, t) = p(x, t)Λ -1 (t) (2.17)

and the corresponding equation becomes

∂ t ũi + ũj ∂ j ũi + 3 S(t) Λ u i = -∂ i p + µ ρ B 0 ∂ 2 jj ũi . (2.18)
In equations (2.14), (2.16) and (2.18) the spatial operators are to be considered in the moving frame, depending on xi .

Unstably stratified homogeneous turbulence

Turbulent mixing driven by buoyancy forces is an important phenomenon common to different fields, from astrophysics to inertial confinement fusion. In these contexts, to investigate the fundamental properties of the turbulence generated by baroclinic instabilities, the homogeneous approximation is an alternative to the full inhomogenous simulation of mixing zones. This approximation, called Unstably Stratified Homogeneous Turbulence (USHT) [START_REF] Griffond | Unstably stratified homogeneous turbulence as a tool for turbulent mixing modeling[END_REF]], applies at the center of a fully developed mixing zone, as depicted in Figure2.1, where the separation between the mean density gradient and the integral scale of the turbulence allows the decoupling between the mean inhomogenous flow and the homogeneous fluctuations. With respect to the complete configuration of full Rayleigh-Taylor mixing, the USHT approximation removes inhomogeneity and transport mechanisms but conserves the similar properties regarding buoyancy production, nonlinear transfers, and dissipation.

Most investigations on USHT [Burlot et al. 2015a, Gréa et al. 2016b] have used the Boussinesq approximation, which is only adapted when the density fluctuations are small compared to the mean value. To extend the homogeneous approximation to more general cases, we derive the unstably stratified homogeneous turbulence equations using the variable density approximation. As explained in section 1.6, this approximation was introduced by [START_REF] Sandoval | The dynamics of variable-density turbulence[END_REF] to study the decay homogeneous turbulence with buoyancy effects, generated by the mixing between light and dense fluids.

In this section, we will reformulate the theoretical framework for variable density buoyancy-driven turbulence decay to obtain a non-decaying flow sustained by buoyancy forces i.e. USHT flow.

We consider the mixing of two materials with different densities but with the same dynamic viscosity µ and diffusivity D. We start from the conservation equations for momentum ρU i , mass ρ, species ρY α , where U i are the velocity components and Y α Chapitre 2. Theory is the mass fraction of the material α = 1, 2 :

∂ t ρU i + ∂ j ρU i U j = -∂ i P + ∂ j τ ij + ρgδ i3
(2.19)

∂ t ρ + ∂ j ρU j = 0 (2.20) ∂ t ρY α + ∂ j ρU j Y α = D∂ j (ρ∂ j Y α ) (2.21)
where g is the gravity acceleration and τ ij is the viscous tensor defined by

τ ij = µ ∂ j U i + ∂ i U j - 2 3 ∂ k U k δ ij .
We add to the system a mixing law :

1 ρ = Y 1 ρ 1 + Y 2 ρ 2 (2.22)
where ρ 1 and ρ 2 are the local microscopic densities of each species, assumed constants and not equal to one another. Using the same approach as [START_REF] Sandoval | The dynamics of variable-density turbulence[END_REF] and [START_REF] Livescu | Buoyancy-driven variable-density turbulence[END_REF], we start from equation (2.21) for the conservation of species, that, using the conservation of mass (2.20), reduces to

∂ t Y 1 + U j ∂ j Y 1 = 1 ρ D∂ j (ρ∂ j Y 1 ), (2.23)
where without loss of generality we have chosen α = 1. The mixing law (2.22) is rearranged to express Y 1 as a function of the density ρ

Y 1 = A ρ -B with A = ρ 1 ρ 2 ρ 2 -ρ 1 and B = ρ 1 ρ 2 -ρ 1 .
(2.24)

Injecting (2.24) in equation (2.23) we have

∂ t ρ + U j ∂ j ρ = ρD∂ j ∂ j ρ ρ .
(2.25)

The comparison of (2.25) with the equation for conservation of mass leads to the following

∂ j U j = -D∂ j ∂ j ρ ρ , (2.26)
that is, a relation expressing the dependence of the velocity field compressibility on the mixture density. We use Eqs (2.25) and (2.26) together with the conservation of momentum (2.19) to have a closed system of equation in the variable density approximation

∂ t U i + U j ∂ j U i = -∂ i Π -Π∂ i Θ + gδ i3 + ν∂ j [(∂ j U i + ∂ i U j )] + ν (∂ j U i + ∂ i U j ) ∂ j Θ,
(2.27)

∂ t Θ + U j ∂ j Θ = D∂ 2 jj Θ,
(2.28)

∂ j U j = -D∂ 2 jj Θ.
(2.29) Π is the reduced pressure P ρ and Θ is a new scalar value defined as a Θ = log ρ ρ 0 , where ρ 0 is a reference density. At this point we use the Reynolds decomposition :

U i = U i + u i (2.30) Θ = Θ + θ (2.31)
and we assume homogeneity, in order to obtain the equations for the fluctuating quantities, i.e. their statistical properties do not vary with position in space. One of the hypotheses of the USHT approximation is that U 3 and ∂ 3 Θ are uniform, together with U 1 = 0 and U 2 = 0. Using Eq. (2.29), with the previous assumptions, we obtain for the mean and fluctuations :

∂ 3 U 3 = -D∂ 2 33 Θ ⇒ U 3 = -D∂ 3 Θ + f (t) ∂ j u j = -D∂ 2 jj θ.
Furthermore, we impose the function f (t), such that U 3 = 0.

Equations for the scalar Θ

Starting from Eq. (2.28) and decompositions (2.30) and (2.31), we obtain the following equation for the mean Θ

∂ t Θ + u j ∂ j θ = D∂ 2 jj Θ (2.32)
and the fluctuation θ

∂ t θ + u j ∂ j θ -u j ∂ j θ + u j ∂ j Θ = D∂ 2 jj θ but ∂ j Θ = ∂ 3 Θ so that ∂ t θ + u j ∂ j θ -u j ∂ j θ + u 3 ∂ 3 Θ = D∂ 2 jj θ.
(2.33)

Equations for u i

The same reasoning can be applied to the momentum equation (2.27). With the decomposition of the reduced pressure Π = Π + π, the mean momentum equation Chapitre 2. Theory is

∂ t U i + U j ∂ j U i + u j ∂ j u i + Π ∂ i Θ -π∂ i θ = -∂ i Π + gδ i3 + ν∂ 2 jj U i + + ν (∂ j U i + ∂ i U j ) ∂ j Θ + ν∂ i (∂ j U j )
and knowing that U i = 0 we have

u j ∂ j u i + Π ∂ i Θ -π∂ i θ = -∂ i Π + gδ i3 .
(2.34)

The equation for the fluctuating part is

∂ t u i + u j ∂ j u i -u j ∂ j u i + π∂ i θ -π∂ i θ = -∂ i π -Π ∂ i θ -π∂ 3 Θ δ i3 + ν∂ 2 jj u i + ν∂ i (∂ j u j ) + ν (∂ j u i + ∂ i u j ) ∂ j θ + ν (∂ 3 u i + ∂ i u 3 ) ∂ 3 Θ .
(2.35)

The homogeneity required for the fluctuations imposes that Π does not depend on space. The same condition holds for ∂ 3 Θ . From the mean velocity equations (2.34), that assuming turbulent quantities are small enough, there is a relation between two mean quantities Π and Θ : in fact we have Π ∂ 3 Θ = g. The gradient of Θ is related to the mixing zone size L via

∂ 3 Θ = 1 L (2.36)
so that Eq. (2.33) and (2.35) become

∂ t u i + u j ∂ j u i + π∂ i θ = -∂ i π -gL∂ i θ - 1 L πδ i3 + ν∂ 2 jj u i + ν∂ i (∂ j u j ) + ν (∂ j u i + ∂ i u j ) ∂ j θ + ν L (∂ 3 u i + ∂ i u 3 ) (2.37) ∂ t θ + u j ∂ j θ = D∂ 2 jj θ - 1 L u 3 (2.38) ∂ j u j = -D∂ 2 jj θ.
(2.39)

Assuming small fluctuations for θ, we can write the equations for the Boussinesq case :

∂ t u i + u j ∂ j u i = -∂ i π + gθδ i3 + ν∂ 2 jj u i (2.40) ∂ t θ + u j ∂ j θ = D∂ 2 jj θ - 1 L u 3 (2.41) ∂ j u j = 0.
(2.42)

For both variable density and Boussinesq approximations, we introduce the buoyancy frequency

N = g L .
(2.43)

Spherical compressions 35

Spherical compressions

In section 2.1, we have discussed the isotropic compression of homogeneous isotropic turbulence. In this section, we discuss the same isotropic compression, but in the case of a spherical mixing zone. A sketch representing the two configurations is shown in Figure 2.2. We make the hypothesis that the mixing layer is generated by the implosion of a capsule of plastic containing a mix of deuterium and tritium, a configuration of interest for inertial confinement fusion application. The investigations of mixing in converging spherical geometry have been concentrated mostly on the curvature effect on the baroclinic instability that may develop at the interface.

The first investigations are due to [START_REF] Bell | Taylor instability on cylinders and spheres in the small amplitude approximation[END_REF] and [START_REF] Plesset | On the stability of fluid flows with spherical symmetry[END_REF]. They used a potential flow approach to study the stability of an accelerated interface between two incompressible fluids. They found a geometric influence that has a major role in the development of instabilities, especially in case of compressions with a large ratio between the initial and final radii. Further investigation by [START_REF] Chandrasekhar | The character of the equilibrium of an incompressible fluid sphere of variable density and viscosity subject to radial acceleration[END_REF] showed the role of viscosity, in particular on the selection of the most unstable modes during the compression.

The following works by [START_REF] Mikaelian | Rayleigh-Taylor and Richtmyer-Meshkov instabilities and mixing in stratified spherical shells[END_REF] and [START_REF] Amendt | Bell-Plesset effects for an accelerating interface with contiguous density gradients[END_REF] were dedicated to extending the investigation on Rayleigh-Taylor instability in spherical geometry with the objective of developing a model for inertial confinement fusion application. Other works by [START_REF] Prosperetti | Viscous effects on perturbed spherical flows[END_REF] and [START_REF] Lin | Rayleigh-Taylor instability of violently collapsing bubbles[END_REF] concentrated on the application of bubbles collapse. [START_REF] Sakagami | Three-dimensional Rayleigh-Taylor instability of spherical systems[END_REF] presented the first direct numerical simulation of the three-dimensional problem. In this paper, they were able to investigate both linear and nonlinear regimes of the spherical perturbation growth, and they found a good agreement for the linear evolution, between theoretical prediction and simulation results. [START_REF] Youngs | Turbulent mixing in spherical implosions[END_REF] proposed an implosion problem and investigated the role of the mesh refinement and dissipation on turbulent statistics. Lombardini et al. (2014a) using a Large Eddy Simulations (LES) studied the Richtmyer-Meshkov driven mixing in low convergence ratio spherical implosion. They evidenced how Bell-Plesset effect has little impact on the dynamics of the imploding mixing layer. In particular, they show how the mean flow is dominated by baroclinic instabilities (Richtmyer-Meshkov, Rayleigh-Taylor) and the turbulent fluctuation inertial subrange approaches the same behavior that is obtained in planar geometry.

In the present study, we suppose that a shock wave has already deposited the vorticity at the interface between the external ablator and the fuel and that the spherical mixing zone at this instant has radius R.

Navier-Stokes for binary mixtures

We use the same conservation equations (2.19)-(2.21), where the variables depend on time t and position x relative to a cartesian stationary reference frame and we recall that Y α is the mass fraction of the material α = 1, 2. Moreover, we add an equation for the temperature T , as well the ideal gas law, which, in the regimes considered is good approximation for the equation of state (see for instance [START_REF] Vold | Plasma viscosity with mass transport in spherical inertial confinement fusion implosion simulations[END_REF]).

∂ t ρU i + ∂ j ρU i U j = -∂ i P + ∂ j τ ij ,
(2.44)

∂ t ρ + ∂ j ρU j = 0, (2.45) ∂ t ρY α + ∂ j ρU j Y α = ∂ j φ j,α , (2.46) nk B γ -1 (∂ t T + U j ∂ j T ) = -P ∂ j U j + ∂ j (κ∂ j T ) + ∂ i U j τ ij , (2.47) P = nk B T. (2.48) (2.49)
where P is the pressure, n is the number density of particles, γ = 5/3, k B is the Boltzmann constant, and κ the temperature diffusion coefficient. The viscous stress tensor is modelled with a Newtonian constitutive law and the diffusive flux with a Fickian law :

τ ij = µ(x, t) ∂ j U i + ∂ i U j - 2 3 ∂ k U k δ ij φ j,α = ρD(x, t)∂ j Y α ,
and we consider that transport coefficients ν and D are functions of space x i and time t. In order to close the system we provide the mixture law for a general twocomponent mixture with elements 1 and 2 :

1 ρ = Y 1 ρ 1 + Y 2 ρ 2 with ρ 1 = nM 1 1 + Z 1 and ρ 2 = nM 2 1 + Z 2 (2.50)
with Z α and M α the ionization number and the atomic mass of species α = 1, 2.

Base flow and perturbation

In the following we will consider the solution of the previous equations (2.44)-(2.48) as a superposition of a baseflow and a perturbation, that is, if F (x, t) is one of the unknowns of the problem, we can write it as

F (x, t) = F B (x, t) + f (x, t).
(2.51)

Base Flow

In this part we propose to characterize the base flow. In section 2.1 we have already chosen the base component of the velocity field in 2.6. Due to the spherical symmetry of the problem, we propose to derive the base components depending only on the radius r.

First the base velocity is rewritten as

U B i (r, t) = -S(t)x i . (2.52)
From the density equation in spherical coordinates

∂ t ρ B + 1 r 2 ∂ r r 2 ρ B U B = 0 (2.53)
and substituting the expression (2.52) for U B (r, t), we obtain an equation for the base density ρ B :

∂ t ρ B -rS(t)∂ r ρ B = 3S(t)ρ B (2.54)
that admits self-similar solution in the form

ρ B (r, t) = ρ 0 Λ(t) -3 G r Λ .
(2.55)

Chapitre 2. Theory

The base temperature equation, neglecting thermal diffusion, reduces to a form that resembles (2.54) :

∂ t T B -rS(t)∂ r T B = 3(γ -1)S(t)T B (2.56)
with a solution :

T B (r, t) = T 0 Λ(t) -3(γ-1) H r Λ .
(2.57)

To obtain the equation for the base number density of particle n B , we use the mixing law (2.50)

ρY 1 = An + Bρ (2.58) A = M 1 M 2 M 2 (1 + Z 1 ) -M 1 (1 + Z 2 ) (2.59) B = M 1 (1 + Z 2 ) M 2 (1 + Z 1 ) -M 1 (1 + Z 2 ) (2.60)
together with Eq. (2.46). Neglecting the diffusive term, we get

∂ t n B + 1 r 2 ∂ r r 2 n B U B j = 0 (2.61)
that has a self similar solution

n B (r, t) = n 0 Λ(t) -3 G r Λ .
(2.62)

From the equation of state we deduce that the base pressure

P B (r, t) = P 0 Λ(t) -3γ G r Λ H r Λ (2.63)
and the reduced base pressure

Π B (r, t) = P B ρ B = Π 0 Λ(t) -3(γ-1) H r Λ (2.64) We assume that G(0) = G 0 = H(0) = H 0 = 1.
This means that the subscript 0 corresponds to the initial value at the center of the capsule.

Let us consider the case where G r Λ = 1, corresponding to a uniform base density. Using the momentum equation, (2.44) for the base component and neglecting viscous effect we obtain

- dS(t) dt + S(t) 2 r = -∂ r Π B (r). (2.65)
This equation has a solution if Π B is a quadratic function of the radius r. This requirement determines the expression of the self similar function H r Λ :

H(r, t) = 1 - r 2 η 2 Λ(t) 2 and Π B (r, t) = Π 0 Λ(t) -3(γ-1) 1 - r 2 η 2 Λ(t) 2 (2.66)
where η is the characteristic scale of the temperature gradient. With this choice of Π B , Eq. ( 2.65) reduces to

- dS(t) dt + S(t) 2 = -2 Π 0 η 2 Λ(t) -3γ+1 (2.67)
that can be rewritten using the relation between the compression parameter and the compression rate (2.8) :

d 2 Λ(t) dt 2 + Ω 2 0 Λ(t) -3γ+2 = 0 with Ω 0 = 2 Π 0 η 2 . (2.68)
This equation is solved with initial conditions Λ(0) = 1 and dΛ dt (0) = S 0 and γ = 5/3 (monoatomic gas). Finally the evolution of Λ with time is

Λ(t) = 1 -2S 0 t + (S 2 0 + Ω 2 0 )t 2 (2.69)
where S 0 = S(0), i.e. the compression rate at the beginning of the simulation. The value of Ω 0 is specified by fixing S 0 and assigning the minimum value of Λ :

Λ min = Ω 0 Ω 2 0 + S 2 0 so that Ω 0 = S 2 0 Λ 2 min 1 -Λ 2 min .
(2.70)

The choice of Λ min , imposes the convergence ratio, that is defined as the ratio between the initial, R 0 , and final radius R end , C R = R 0 R end , so that is the inverse of Λ min . These choices are also important because Ω 0 is linked to the characteristic scale of the temperature gradient h, through the relation (2.68). Therefore it is worth exploring how the choices of S 0 and Λ min influence the time evolution of the compression parameter Λ and the radial evolution of the self-similar function H(r, t).

We plot Λ as function of time in Figure 2.3. In Figure2.3a we consider a constant S 0 but different values Λ min . If Λ min is sufficiently small, the compression parameter curves are similar, only with different values at t = 1 ns. The other effect illustrated in Figure 2.3 is a difference in the time of the minimum of Λ, in particular,

t min(Λ) = 1 -Λ 2 min S 0 (2.71)
if in this equation Λ min is sufficiently small, the compression time t min(Λ) depends only on S 0 . We explore the compression parameter dependence on the initial compression rate S 0 at fixed Λ min (or fixed convergence ratio) in Figure2.3b. We observe how the increase of S 0 leads to a decrease in the compression time, although with a similar evolution of Λ. For instance, Figure2.3b shows that when S 0 increases from 500µs -1 to 1000µs -1 , the minimum of Λ is reached in half the time, from 2 ns to 1 ns.

Concerning the self-similar function H in (2.66), once Π 0 is fixed, i.e. once the thermodynamic condition at the center of the capsule are defined, h depends only on S 0 and Λ min . An example of this dependence is plotted in Figure2.4. As for the compression parameter, we show in 2.4a how H(r) depends on the inverse of convergence ratio and in Figure2.4b its dependence on initial compression rate S 0 . At this point, we have all the elements to describe the base flow completely :

U B (x, t) = -x • Λ Λ(t) (2.72) ρ B (t) = ρ 0 Λ(t) -3 (2.73) Θ B (t) = -3 log Λ(t) (2.74) n B (t) = n 0 Λ(t) -3 (2.75) Y B = 1 (2.76) T B (r, t) = T 0 Λ(t) -3(γ-1) 1 - 1 η 2 r 2 Λ(t) 2
(2.77)

P B (r, t) = P 0 Λ(t) -3γ 1 - 1 η 2 r 2 Λ(t) 2 (2.78) Π B (r, t) = Π 0 Λ(t) -3(γ-1) 1 - r 2 η 2 Λ(t) 2
(2.79)

Perturbations

The perturbations equations follow from the system of equations (2.44) -(2.49) and the base flow expressions determined in section 2.3.2.1 :

∂ t u i + u j ∂ j u i -S(t)x j ∂ j u i -S(t)u i = -∂ i π -π∂ i θ -Π B (x, t)∂ i θ +∂ j ν ∂ j u i + ∂ i u j - 2 3 ∂ k u k δ ij +ν ∂ j u i + ∂ i u j - 2 3 ∂ k u k δ ij ∂ j θ (2.80) ∂ t θ + (u j -S(t)x j )∂ j θ = ∂ j u j (2.81) ∂ t n + ∂ j nu j + ∂ j (n B u j ) + ∂ j (nU B j ) = ∂ j (D∂ j n) -∂ j (n + n B )D∂ j θ (2.82)
Equations (2.80), (2.81), (2.82) can be simplified with the hypothesis that the fluctuations of the density number of particle are small with respect to its base value, n n B 1. With this assumption we get the following system of equations :

∂ t u i + u j ∂ j u i -S(t)x j ∂ j u i -S(t)u i = -∂ i π -π∂ i θ -Π B (x, t)∂ i θ +∂ j [ν(∂ j u i + ∂ i u j )] + ν(∂ j u i + ∂ i u j )∂ j θ
(2.83)

∂ t θ + (u j -S(t)x j )∂ j θ = ∂ j u j (2.84) ∂ j u j = -∂ j (D∂ j θ) (2.85)
The main difference with the classic incompressible approximation is that the divergence of the velocity field is no longer equal to zero but is connected to the scalar field θ. In this approximation we decouple the energy equation from the problem and compressibility effects are only a result of the mixing.

Change of reference frame and rescaling

Equations (2.83)-(2.84) have inhomogeneous and forcing terms representing the base flow effects on the velocity and scalar perturbations. To eliminate these terms we use the change of reference frame (2.13) and the rescaling (2.15) defined in section 2.1. We obtain the following equations for the perturbation in the non-inertial frame :

∂ t ũi + ũj ∂ j ũi = -∂ i π -π∂ i θ -Λ 2 Π B (x, t)∂ i θ + ∂ j [ν (∂ j ũi + ∂ i ũj )] + ν (∂ j ũi + ∂ i ũj ) ∂ j θ
(2.86)

∂ t θ + ũj ∂ j θ =∂ j D∂ j θ (2.87) ∂ j ũj = -∂ j D∂ j θ (2.88)
Chapitre 3 The objective of this chapter is to present shortly the numerical methods used in this thesis. At first, we introduce the Fourier pseudo-spectral method and the thirdorder Runge-Kutta Strong Stability Preserving (SSP) time advancement scheme . We focus on the solution of the pressure Poisson equation in the incompressible and variable density cases. In particular, for the latter case, we present the Generalized Minimal Residual (GMRES) iterative solver, employed to solve the non-local elliptic equation. In section 3.2.4, we discuss the case of variable transport coefficients and the implicit solution of the diffusive part of the equations.

Numerical methods

Contents

Finally, we discuss the convergence of the iterative GMRES scheme and the different steps used to validate the code.

Equations

The equations derived in chapter 2 are not in conservative form. So that during computation, conservation properties may not be assured [START_REF] Canuto | Spectral methods in fluid dynamics[END_REF]]. The use of the following vector calculus identity

(u∇)u = ω × u + 1 2 ∇u 2 where ω = ∇ × u, (3.1)
semi-conserves kinetic energy (at least for inviscid flow), and assures numerical stability, which is not granted if we use the standard formulation. Using this substitution, we redefine the pressure to include an additional kinetic energy contribution, into a new variable π * , as

π * = π + 1 2 u 2 . (3.2)
The resulting formulation of the equation is called rotation formulation.

Here we rewrite equations in a more compact form that is common to the two studies of this thesis, spherical compression, and USHT, with incompressible and variable density approximations.

-The incompressible approximation yields

∂ t u i + N ω 1 = -∂ i π * + ∂ j (ν∂ j u i ) + F θ 6 (3.3) ∂ t θ + N u (θ) 2 = +∂ j (D∂ j θ) + F L 7 (3.4) ∂ j u j = 0. (3.5)
-The variable density approximation yields

∂ t u i + N ω 1 +π * ∂ i θ -N θ (u i , θ) 3 -N ν (u i , θ) 4 = -∂ i π * + F(x i , θ) 5 +∂ j (ν∂ j u i ) + F π 8 (3.6) ∂ t θ + N u (θ) 2 -∂ j (D∂ j θ) = F L 7 (3.7) ∂ j u j = -∂ j (D∂ j θ) (3.8)
In the previous equations, we have used the following short notation :

1. N ω = ω × u
This term, also called the Lamb vector l, comes from the reformulation of the nonlinear term to have better energy conservation and stability properties.

N

u (θ) = u j ∂ j θ
This is the nonlinear advection term in the scalar θ equation.

3

.

N θ = u k u k 2 ∂ i θ
This terms appears when we use the identity (3.1) to introduce the rotation formulation, in order to redefine the pressure as in Eq. (3.2).

4.

N ν (u i , θ) = ∂ j (ν∂ i u j ) + ν (∂ j u i + ∂ i u j ) ∂ j θ
This term contains nonlinear viscous contributions. They are among the new nonlinearity introduced by the variable density approximation.

In addition to the above terms which are always present, the following ones may appear, depending on the problem which is specifically studied. In the spherical compression case 5.

F(x i , θ) = -Λ 2 Π B (x i , t)∂ i θ
This is a forcing term due to the base pressure gradient. In this case we do not have other forcing terms, so that 6.

F θ = 0 7. F L = 0 8. F π = 0
In the USHT study :

5. F(x i , θ) = F(θ) = -gL∂ i θ
This forcing term comes from the mean homogeneous pressure.

F θ = gθ

This term corresponds to the linear approximation of the forcing from the mean pressure.

7.

F L = - 1 L u 3
This forcing term comes from the mean θ gradient in the scalar equation.

8

.

F π = - 1 L π * δ i3
This forcing term comes from the mean θ gradient in the momentum equation.

Direct numerical simulations

Spectral methods are well-established tools in fundamental studies in fluid mechanics [START_REF] Canuto | Spectral methods in fluid dynamics[END_REF] because they are among the highest precision methods. When the solution is smooth, the decay of the error is exponential, as we increase the resolution. Since spectral methods are a particular case of methods of weighted residuals [START_REF] Canuto | Spectral methods in fluid dynamics[END_REF], the choice of test and trial functions characterizes different types of spectral methods. In particular, in this discretization, these functions have global support that coincides with the domain of the problem under consideration. Trial functions are used as basis functions for the series expansion of the solution, while the test functions ensure that the differential equation is satisfied as close as possible by the approximation series.

The choice of the trial function depends mostly on the boundary conditions of the problem. There are three classes of polynomials that are widely employed : Fourier, Chebyshev, and Legendre. For the test functions, there are usually three alternatives :

1. If one chooses to have the same as trial functions we have Galerkin spectral scheme ;

2. If one selects translated Dirac delta functions centered on the mesh points, the scheme is called collocation or pseudo-spectral ;

3. If one instead wants the same trial and test functions which do not satisfy boundary conditions, we have the tau spectral scheme ;

In this work, we use a Fourier pseudo-spectral (collocation) scheme as it has been already proven very effective in fluid dynamics problem [START_REF] Orszag | Numerical simulation of three-dimensional homogeneous isotropic turbulence[END_REF].

Pseudo-spectral computation using FFT

One of the advantages of using the Fourier spectral method is that there are libraries that implement the discrete Fourier transform algorithm very effectively. All the most famous implementations are based on the Fast Fourier Transform (FFT), an algorithm of [START_REF] Cooley | An algorithm for the machine calculation of complex Fourier series[END_REF]. One of the most famous is the Fastest Fourier Transform in the West (FFTW) library developed at MIT by [START_REF] Frigo | The design and implementation of fftw3[END_REF]. In pseudo-spectral methods, the derivatives are calculated in the Fourier space since they reduce to the multiplication of Fourier coefficients by the wavenumbers. All products are instead done in physical space. In this thesis, we have developed a DNS code based on the Parallel Three-Dimensional Fast Fourier Transforms (PDFFT++), a library for large-scale computer simulations on parallel platforms, developed by [START_REF] Pekurovsky | P3dfft : A framework for parallel computations of Fourier transforms in three dimensions[END_REF]. This library allows the two-dimensional or "pencil" decomposition of the computational domain depicted in Figure 3.1(b), whereas the old version of the DNS code used a one dimensional or "slab" decomposition 3.1(a). If one has a three-dimensional problem where the domain dimensions are N x × N y × N z , where x, y, and z are the three spacial dimensions, the 1D decomposition allows dividing the domain into planes which are distributed on the available processors N P . Once the direction of the decomposition is chosen, here we chose z, every processor receives a slab of dimension N x × N y × Nz N P . With this decomposition, one can, at most, use the number of processors equal to the size of the domain. For instance, in this case, N P = N z is the maximum possible number of processors.

The two-dimensional decomposition allows to have "pencils" or "columns" instead of planes. If we consider the same domain as before, the new sub-domains distributed on every processor have dimension N x × Ny N P 1 × Nz N P 2 . Now the number of the processors that can be potentially used for the domain decomposition is N P = N P 1 × N P 2 . If we consider the maximums of N P 1 and N P 2 , we have

N P = N y × N z .
If all the dimensions are equal, N x = N y = N z = N , 1D, and 2D decomposition allows using respectively at most N and N 2 processors, making it clear why it is interesting to develop a new code based on the 2D decomposition.

The scaling properties of the new direct numerical simulation code, whose numerical details are detailed in the following section, are presented in

Time advancement scheme

The time advancement is done in Fourier space so that Fourier coefficients are the dependent variables. The numerical scheme chosen is a third-order Runge-Kutta SSP [START_REF] Gottlieb | Strong stability-preserving high-order time discretization methods[END_REF]. Consider the following dynamical system

dF dt = L(F, t). (3.10)
The time advancement between t n and t n+1 is done using the following three steps

F 1 = F (t n ) + ∆tL(F (t n ), t), (3.11) F 2 = 3 4 F (t n ) + 1 4 F 1 + ∆tL(F 1 , t + ∆t) , (3.12) F (t n+1 ) = 1 3 F (t n ) + 2 3 F 2 + ∆tL F 2 , t + ∆t 2 . (3.13)
The sub-steps of the scheme are different if one considers either the incompressible approximations or the variable density one, as detailed hereafter.

Sub-steps for incompressible computation

One of the advantages of doing the time advancement in Fourier space, is that we can compute implicitly the viscous contribution. For that, one can define a new variable v as

v i = u i exp νk 2 (t -t 0 ) (3.14)
such that, in wave-number space, the equation for v i does not have any explicit viscous term. After this substitution, we use the classical projection method of [START_REF] Chorin | Numerical solution of the Navier-Stokes equations[END_REF].

Starting at time t n , -Compute an intermediate velocity v * i v * i = v n i -∆tF T (N ω ) exp νk 2 (t -t 0 ) . (3.15)
-Using the incompressibility condition we can derive a Poisson equation for π

* ∂ 2 ii π V = ∂ i v * i ∆t . (3.16)
solving this equation is straightforward in Fourier space. -We obtain the velocity at time t n+1 from

v n+1 i = v * i -∆t∂ i π * .
(3.17)

Here we have used the symbol π V instead of π. The use of the exponential factor to take into account the viscous effects modifies the pressure terms as :

π V = π exp νk 2 (t -t 0 ) .

Sub-steps for variable density computation

In this case, the time-advancement steps are a bit more complicated. At time t n -Compute intermediate variables u * i (without taking into account π * ) and θ * . In this step, we compute explicitly only a part of the viscous contribution :

u * i = u n i + ∆t[N ω (u n i ) + N θ (u n i , θ n ) + F(x i , θ n ) + N ν (u n i , θ n )],
(3.18)

θ n+1 = θ n -∆tN u (θ n ). (3.19)
-The updated velocity u n+1 is obtained by taking into account π

u n+1 i = u * i + ∆t(-∂ i π * -π * ∂ i θ n -F π * ). (3.20)
-Applying the divergence operator to Eq. (3.20) we obtain the Poisson equation

∂ 2 ii π * + π * ∂ 2 ii θ n + ∂ i π * ∂ i θ n + (F π ) 3 = 1 ∆t -∂ i u n+1 i + ∂ i u * i . (3.21)
At this point we use the expression for the divergence of u n+1 i given by the variable density approximation ∂ i u n+1 i = -∂ j (D∂ j θ n+1 ) and we obtain

∂ 2 ii π * + π * ∂ 2 ii θ n + ∂ i π * ∂ i θ n + (F π ) 3 = 1 ∆t ∂ j D∂ j θ n+1 + ∂ i u * i . (3.22)
This is a non local Poisson equation that has to be solved with an iterative method. Once π * is computed, we can finally use Eq. (3.20) to obtain the velocity at time t n+1 .

Solution of the Poisson equation

It is worth devoting here a detailed description to the solution method for the Poisson equation, since this task has proven to present difficulties, especially in the variable density case, and has required a dedicated implementation of the original method proposed by Di Pierro (2017). In the incompressible calculations, the Poisson equation in Fourier space has a straightforward solution. We start from the Fourier transform of equation (3.16)

k 2 π * = k i v * i ∆t ,
where k is the module of the wavevector, whose solution is straightforward :

π = k i v * i k 2 ∆t .
In the algorithm for the variable density solution, the equation is more complicated given the non-locality of π * in Fourier space :

k 2 π + F T π∂ 2 ii θ n + F T [∂ i π∂ i θ n ] = 1 ∆t F T ∂ j D∂ j θ n+1 + k i û * i . (3.23)
Di [START_REF] Pierro | On a preconditionment for the spectral solution of incompressible variable density flows[END_REF], using 1D differentiation matrices, showed that the operator on the left-hand side of the Eq. (3.23) is ill-conditioned. To improve the convergence, he proposes to use the constant density operator as a preconditioner and to introduce a new variable φ, such that π = e -θ n /2 φ.

The resulting equation for φ is written in physical space as

∂ 2 ii φ + φ 2 ∂ 2 ii θ n - φ 4 ||∂ i θ n || 2 = e θ n+1/2 ∆t ∂ j D∂ j θ n+1 + ∂ i u * i . (3.24)
The benefit of this choice is not only connected to the decrease of the condition number but it saves us the computation of the scalar product ∂ i π∂ i θ n . We can rewrite this equation in condensed form, calling L the operator acting on φ and b the righthand side of the equation :

Lφ = b. (3.25)
The simplest iterative method for solving Eq. (3.24) is the fixed point iteration method [Di [START_REF] Pierro | A projection method for the spectral solution of nonhomogeneous and incompressible navier-stokes equations[END_REF]. One of the main disadvantages is the slow convergence of the method when compared with other iterative solvers, and the dependence of the convergence on the mesh size. With this method, the convergence is obtained

if max (|∂ i θ|) < k (3.26)
where k is the biggest resolved wavenumber allowed by the mesh. More explicitly, characteristic scales of θ gradients must be of the same size as the grid discretization. See Di [START_REF] Pierro | A projection method for the spectral solution of nonhomogeneous and incompressible navier-stokes equations[END_REF], Di Pierro (2012) for details. In our work, we would like to achieve relative rapid convergence without such constraints on the grid-∂ i θ relation.

The quest to find the best all-purpose iterative method is a rather difficult one. In particular, it has been shown by [START_REF] Nachtigal | How fast are nonsymmetric matrix iterations[END_REF] that the best iterative solver in one case may perform poorly in other situations. In his work, Di Pierro (2017) tested four iterative schemes to solve a non-local Poisson equation. They tested conjugated gradient (CG), Richardson minimal residual (RMR), and the generalized minimal residual methods (GMRES). They find that the GMRES method is the best option, even if it has a higher cost per iteration, and it is more complex to implement. In light of these results, we have decided to implement this algorithm to solve Eq. (3.24). Details of the implementation are given in section 3.2.3.1.

Generalized Minimal Residual (GMRES) algorithm

The generalized minimum residual (GMRES) is an iterative method to solve linear systems proposed by [START_REF] Saad | Gmres : A generalized minimal residual algorithm for solving nonsymmetric linear systems[END_REF]. It is part of a class of iterative solvers based on Krylov spaces. The methods in this class are among the best iterative methods available to solve problems involving non-symmetric matrices and operators. Such methods are conjugated gradient, bi-conjugated gradient, and the GMRES. The idea behind this approach is to converge to the exact solution of the system (3.25), using the sequence of approximate solutions

φ n ∈ φ 0 + K n (r 0 , L) (3.27)
where K n (r 0 , L) is the Krylov subspace generated by the discretized operator L and the starting vector

r 0 = b -Lφ 0 K n (r 0 , L) = {r 0 , Lr 0 , L 2 r 0 ...L n-1 r 0 } (3.28)
where φ 0 is an initial choice of φ.

The main feature of the GMRES is that the sequence (3.27) is built to minimize the residuals r = b -Lφ 2 . It can be shown [START_REF] Stoer | Introduction to numerical analysis[END_REF] that the solution of the problem Lφ = b lies in the affine spaces created by φ n ∈ φ 0 +K n (r 0 , L), that is, the solution φ can be decomposed in this new basis φ n = K n g for some vector g. The problem with the definition (3.28) is that when n increases, the vectors in the base become more and more linearly dependent so that the method becomes unstable. Instead, the alternative is to construct a new orthonormal basis for K n using Arnoldi iterations [START_REF] Stoer | Introduction to numerical analysis[END_REF]. The corresponding algorithm is as follows :

-Choose first guess x 0 , and the size of the basis m.

-Compute r 0 = b -Lφ 0 , β = r 0 2 and q 1 = r 0 /β.

-To compute the orthonormal base we use the Gram-Schmidt process : For j = 1, .., m do -Compute w = Lq j . w is the vector that we would have in the Krylov base using the classic power iteration. -For w to be orthonormal to the other vectors in the basis we perform the following operation : For i = 1, .., j do h i,j = (w, q i ), where (, ) is the inner product (w, q i ) = w T q i w = w -h i,j q i -Compute h j+1,j = w 2 and q j+1 = w/h j+1,j -At the end we have the orthonormal basis Q m defined by Q m = [q 1 , ...q m ] In this new basis, φ n is decomposed as

φ n = Q n y n (3.29)
The previous Arnoldi process produces the coefficient h i,j that are the coefficients of an upper Hessenberg matrix [START_REF] Stoer | Introduction to numerical analysis[END_REF] H n that satisfies the following equality

LQ n = Q n+1 H n . (3.30)
Recall that the objective of the method is to minimize the norm of the residual r n 2 = b -Lφ 2 , knowing that the updated solution at each step is φ = φ 0 + φ n . Thus we can rewrite the minimization as

min r n 2 = min b -L(φ 0 + φ n ) 2 = min r 0 -Lφ n 2 . (3.31)
Using relations (3.29) and (3.30) we can rewrite the problem as

min Q T n+1 r 0 -H n y 2 . (3.32)
In this equation the matrix Q n+1 contains the orthonormal vector computed during the Arnoldi iterations. That is the matrix-vector product is zeros for all but one column and Eq. (3.32) simplifies to (3.33) where β = r 0 2 and e 1 is the n-dimensional vector e 1 = [1, 0..., 0]. Equation (3.33) is a least square problem, that we solve using Givens rotations [START_REF] Stoer | Introduction to numerical analysis[END_REF]. Once y is computed the solution is

min βe 1 -H n y 2 ,
φ = φ 0 + Q n y.
To sum up the steps for the GMRES algorithm, they are presented in a concise form in Algorithm 1 :

Algorithm 1 GMRES 1:
Choose the dimension of the Krylov subspaces m, the initial guess φ 0 . 2: Compute the initial residual r 0 = b -Lφ 0 , β = r 0 2 and q 1 = r 0 /β 3: for j = 1, ...m do 4:

Compute w = Lq j 5:

for j = 1, ...m do 6: h i,j = (w, q i ) 7: w = w -h i,j q i 8:
end for 9:

Compute h j+1,j = w 2 and q j+1 = w/h j+1,j 10: end for 11: Define the matrix Q m = [q 1 , ...q m ] 12: Compute φ = φ 0 + Q n y m where y m = min βe 1 -H n y m 2 13: If satisfied with the results stop, else set φ 0 = φ m and go to 3

Non uniform transport coefficients

In this work, we will consider the compression of weakly coupled plasma mixtures. In these computations, transport coefficients vary considerably in time and space, reaching considerably large values, so that an explicit treatment may require a very stringent restriction on the time step, to satisfy numerical stability constraints. That is why we use an implicit treatment of diffusive terms in the momentum and scalar equation. This method has already be employed in the case of variable viscosity by [START_REF] Gréa | The effects of variable viscosity on the decay of homogeneous isotropic turbulence[END_REF], using finite differences in physical space to solve the implicit problem. In the present work, we choose to use the GMRES solver already presented in the previous section.

The following sub-steps are performed before the projection algorithm presented the section 3.2.2.2 Chapitre 3. Numerical methods -The values of transport coefficients ν and D depend on mass fraction, temperature, and θ. Therefore the first step is to compute their values at time t n .

ν n = ν(Y n , T n , θ n ) and D n = D(Y n , T n , θ n ).
- [START_REF] Ferziger | Computational methods for fluid dynamics[END_REF] argue that the main diffusive contributions that need implicit treatment are the terms

∂ j (ν∂ j u i ) and ∂ j (D∂ j θ) .
While the other viscous nonlinear viscous contribution N ν (u i , θ) can be computed explicitly. Furthermore we make the hypothesis that D n+1/2 = D n and ν n+1/2 = ν n , so that

θ n+1/2 = θ n + ∆t∂ j (D n+1/2 ∂ j θ n+1/2 ), (3.34) u n+1/2 i = u n i + ∆t∂ j (ν n+1/2 ∂ j u n+1/2 i
).

(3.35)

-At this stage we modify the intermediate variable u * using u n+1/2 (and not u n ) adding the nonlinear effect of the viscous contribution computed explicitly :

u * i = u n+1/2 i + ∆t[N ω u n+1/2 i + N θ u n+1/2 i , θ n+1/2 (3.36) + F x i , θ n+1/2 + N ν u n+1/2 i , θ n+1/2 ].
(3.37)

Preconditioning and initial guesses

In this chapter we have presented three equations that have to be solved using iterative methods (3.34), (3.35), (3.24). In all three cases, we have decided to use preconditioning to improve the performance of the GMRES algorithm. In general, preconditioning is employed for ill-conditioned problems to lower the condition number of the operator and obtain better performances from the iterative method, i.e accelerate the rate of convergence.

The modifications of algorithm 1 are described in algorithm 2, where the operator M is the preconditioner.

The next question is the choice of M . Di [START_REF] Pierro | On a preconditionment for the spectral solution of incompressible variable density flows[END_REF] demonstrates that the constant density operator is very effective as a preconditioner for the solution of the Poisson equation. Using the same argument, we propose the implicit constant viscous operator for the variable viscosity and diffusivity computations. In particular the constant preconditionings are

Convergence and validation 55

Algorithm 2 GMRES with preconditioning 1: Choose the dimension of the Krylov subspaces m, the initial guess φ 0 . 2: Compute the initial residual r 0 = b -Lφ 0 , β = r 0 2 and q 1 = r 0 /β 3: for j = 1, ...m do 4:

Compute z j = M -1 q j 5:

Compute w = Lz j 6:

for j = 1, ...m do 7: h i,j = (w, q i ) 8: w = w -h i,j q i 9:
end for 10:

Compute h j+1,j = w 2 and q j+1 = w/h j+1,j 11: end for 12: Define the matrix

Q m = [q 1 , ...q m ]. 13: Compute φ = φ 0 + M -1 Q n y m where y m = min βe 1 -H n y m 2 14:
If satisfied with the results stop, else set φ 0 = φ m and go to 3

1. M ν = 1 1+ν∆tk 2 2. M D = 1 1+D∆tk 2 3. M π = -1 k 2
where M ν and M D are the preconditioner for the variable viscosity and variable diffusivity substeps (3.34), (3.35) and M π is the preconditioning matrix for the Poisson equation (3.24).

The last choice to make is to decide the initial guesses to start the algorithm. In our computation they are

1. u n+1/2 0 = u n 1+ν∆tk 2 2. θ n+1/2 0 = θ n 1+D∆tk 2 3. φ 0 = k i û * i k 2 ∆t e θ/2
where u n+1/2 0 and θ n+1/2 0 are the initial guesses for the variable transport coefficients substeps. They are also the solution that we would have in case of constant viscosity and diffusivity. φ 0 is the initial guess for the GMRES algorithm applied to the Poisson equation and it corresponds to the φ that we obtain in the solution of the incompressible case.

Convergence and validation

In this section, we describe the convergence criteria for the GMRES iterative schemes together with the validation steps for the new direct numerical simulation code. We start with the incompressible decay of homogeneous isotropic turbulence (HIT), to validate the computations of the nonlinear convective terms. We move on to the decay of HIT with variable viscosity to validate the GMRES algorithm for the computation of the diffusive terms. The first step toward the validation of the GMRES algorithm for the pressure Poisson equation concerns a case of unstably stratified homogeneous turbulence with low non-Boussinesq effects, such that variable density and Boussinesq approximations should have the same results. When one deals with an iterative algorithm to solve linear systems, it is useful to have an idea of what are the conditions under which the iterative scheme can be considered converged. In particular, the GMRES algorithm has a free parameter, the size of the Krylov subspace m, which should be carefully tuned. The more m grows, the more efficient the GMRES method becomes, but at the same time, it became computationally more expensive. Hence, for our computations, a trade-off has to be decided between the increased precision and the increased costs.

Convergence of the iterative GMRES algorithm

Numerical experiments show that a good compromise for the most challenging cases, with mesh sizes of 1024 3 and increasing density contrast, a Krylov base of m = 20 have good conservation and convergence properties.

In Figure 3.3, we show an example of the effects of m on the convergence of the GMRES for a 256 3 USHT computation. The decrease of the residuals is plotted in Figure 3.3a, while the consequences of the converged/un-converged iterative algorithm on the kinetic energy spectrum are shown in 3.3b. This convergence ana-lysis, as a function of m, considers only one time step. To further validate the DNS code, we evaluated the conservation of mass of the numerical method, during the computations. which, if we separate the solenoidal u S j , ∂ j u S j = 0, and dilatational u D j , ∂ j u D j = 0, velocity contributions, becomes

N t

∂ j u D j = -D∂ 2 jj θ. (3.39)
Therefore, to check the conservativity, during the simulation we compute the quantity

V D = ||∂ j u D j + D∂ 2 jj θ||. (3.40)
An example of the values of V D during an USHT simulation is given in Figure 3.4, together with the norm of the GMRES residuals at the same time.

Validation : Decay of incompressible homogeneous isotropic turbulence

The first step in validating the new direct numerical simulation code is to compute the time evolution of the isotropic homogeneous turbulence (HIT) decay. The results are compared with the previous version of the DNS code parallelized in slab, described in [START_REF] Griffond | Unstably stratified homogeneous turbulence as a tool for turbulent mixing modeling[END_REF]. We consider the case of a HIT with a resolution of 512 3 with an initial Reynolds number of Re 0 = K(0) 2 ν (0) = 30, where is the turbulence dissipation, K the integrated turbulent kinetic energy and ν the kinematic viscosity.

The results of the simulations are shown in Figures 3.5 and 3.6. The evolution of the kinetic energy is plotted in 3.5a, as a function of the dimensionless time τ , defined as the time of the simulation divided by the eddy turnover time t E = 1 K 1/2(0) kp , where k p is the peak of the energy spectrum at t = 0. We observe the excellent agreement between the two simulations, on the one-point statistics. Subsequently, the spectra,

Convergence and validation 59

in Figure 3.5b, show how also the spatial dependencies are similar between the two simulations. In Figure 3.5b, the two results are superimposed, so that we cannot evaluate the differences between the two simulations. That is why in Figure 3.5c we show the relative error between the two spectra, computed as follow

Er p (k) = E N (k) -E O (k) E O (k) (3.41)
where E N (k) and E O (k) are respectively the resulting spectra of new and old code.

Lastly, the qualitative comparison between the speed fields in Figure 3.6 confirms the similarity between the two results.

Validation : Decay of incompressible homogeneous isotropic turbulence with variable viscosity

To validate the GMRES algorithm for the variable transport coefficients, we compute a case of homogeneous isotropic turbulence with variable viscosity. [START_REF] Gréa | The effects of variable viscosity on the decay of homogeneous isotropic turbulence[END_REF] has investigated this problem using spectral direct numerical simulations where viscous effects are accounted for using an implicit finite difference scheme for equation 3.35. The viscosity has a linear relation with the scalar, θ

ν = ν A + Vθ (3.42)
In this set up θ ∈ [-1, 1] and its averaged value on the volume is zero θ = 0, so that the mean value of viscosity is ν = ν A .

We compute the decay of homogeneous isotropic turbulence with a resolution of 256 3 with an average viscosity of ν A = 0.5. It has a maximum equal to 0.99 and a minimum of 0.01. The diffusivity is constant and equal to the minimum of viscosity, D = ν A , such that the Schmidt number based on the averaged values is Sc = ν A D = 50. In Figure 3.7a, we present the comparison of the turbulent kinetic energy as a function of the dimensionless time τ defined in section 3.3.2. The evolution of the integrated kinetic energies shows good agreement between the two simulations, and the two curves are superimposed. On the contrary, we observe some differences in the kinetic energy spectra. While at large scales, the two simulations have a good agreement, they show some differences when we look at smaller scales, with the finite difference computation that has a higher dissipation with a lower value of the spectrum at higher k. The difference is caused by the two numerical methods used to solve the variable viscosity step of the computation. The finite difference schemes have a higher numerical dissipation, confronted with the iterative spectral GMRES.

Validation : Variable density results with small non-Boussinesq effects.

To validate the GMRES algorithm for the Poisson equation, we use unstably stratified homogeneous turbulence (USHT) results with little effects coming from the density field, that is, a case where the small fluctuations approximation is valid.

In this configuration, we confront USHT results from computations using the Boussinesq and the variable density approximations. The 1024 3 simulations, are initialized with velocity fluctuations but without scalar fluctuations. The evolution of kinetic energy K, and the scalar variance θθ are reported in Figure 3.8.

The variance, in Figure 3.8b, shows a very good agreement between the two simulations, while for the kinetic energy, in Figure 3.8a, a little difference is observed during the decay phase, which is due to the different development of the turbulent energy cascade. The two integral quantities are plotted as a function of the dimensionless quantity N t, where t is the simulation time and N is the stratification frequency defined, for the USHT computations, as N = g/L, with g and L introduced in section 2.2.

The analogies and differences between the two simulations can be further evidenced by the kinetic energy, Figure 3.9a, and variance spectra, Figure 3.9b.

From the same initial condition at N t = 0, the two simulations continue to have similar results on both spectra at N t = 2.5, after the minimum of the decay. Eventually, at N t = 5, the non-Boussinesq effects, explained in chapter 6, start to affect both large and small scales beginning to drive the two results apart. The similarity between the two approximations is confirmed qualitatively by the contours of the scalar θ in Figure 3.10. We observe the formation of the characteristic structures of USHT turbulence, with the light fluid moving upwards and the heavy downwards.

Initial conditions

In this section, we describe the method used to generate the initial conditions for the resolution of the dynamical equations. The initializing procedure for the homogeneous case is presented in section 3.4.1, while the initialization steps for the spherical compression case are resumed in section 3.4.2.

Homogeneous isotropic turbulence

Initial conditions for homogeneous isotropic turbulence are created in Fourier space, using the following kinetic energy spectrum

E(k) = A k k l s exp   - s 2 k k l 2   (3.43)
and random phases for the Fourier components. In Eq. (3.43), k l is the peak wavenumber, s is the infrared slope, and A is a constant parameter used to adjust the amplitude of the spectrum such that it matches a specified initial total kinetic energy K(t = 0), via the relation

K(t = 0) = ∞ 0 E(k)dk (3.44)
For the initialisation of the scalar field we use the same expression as (3.43) and a relation similar to (3.44) to achieve the desired initial scalar variance. Once the initial velocity and scalar fields are generated, there is a difference between the incompressible and variable density initializations.

For the incompressible case, we have to enforce the solenoidal constraint on the velocity field u i . This requirement reduces in Fourier space to impose the velocity perpendicular to the wavevector :

u i k i = 0.
The initial condition for variable density computations requires an additional step. In fact for this case we know that velocity and the scalar θ have to satisfy the relation :

∂ j u jV D = -D∂ 2 jj θ. (3.45)
In order to obtain consistent initial conditions we follow the same steps as [START_REF] Sandoval | The dynamics of variable-density turbulence[END_REF]. Using the Helmholtz theorem we can decompose the initial velocity field u into sum of solenoidal (∇ × H) and irrotational (∇ψ) parts :

u V D = -∇ψ + ∇ × H. (3.46)
We replace the divergence-free vector ∇ × H with the incompressible initialization u i :

u iV D = -∂ i ψ + u i . (3.47)
When we compute the divergence of (3.47), we have

∂ i u iV D = -∂ 2 ii ψ. (3.48)
If we compare this relation with (3.45), the scalar field ψ is determined by ψ = Dθ. So that the initial velocity fields for the variable density simulations are computed by

u iV D = -D∂ i θ + u i (3.49)

Spherical compressions

In section 2.3, we have described how, for the spherical compressions computations, we decompose the velocity and scalar fields as a superposition of base flow and perturbations, and we have discussed how the former is defined and imposed via analytical relations. In this part, we describe the initialization of the perturbations field. First, we use the Reynolds decomposition to separate the mean part of the velocity and the scalar fields from the fluctuations :

θ = θ + θ (3.50) u i = u i + u i .
(3.51) For this problem the mean is defined as a tangential average on the spherical angular variables ψ and ζ :

Q(r) = 2π 0 π 0 Q(t, r, ζ, ψ)r 2 sin(ζ)dψdζ
where Q could be u i or θ.

Initialization of velocity

In all the cases considered, the initial value of mean velocity field u i is always zero. Thus we have to provide only the fluctuating velocity in the spherical layer. We proceed as follows :

1. An homogeneous isotropic velocity field is initialized as described in subsection 3.4.1, with prescribed kinetic energy and peak wavenumber.

2. We compute the vorticity ω = ∇ × u in physical space to which we apply a filter function F (x), which detail are given in the coming subsection 3.4.3, to obtain the filtered vorticity ω F .

3. Once the filtered vorticity is computed, it is transformed back to Fourier space (the • indicates variables in Fourier space). At this point we use the vector calculus identity

ik × (ik × û) = ik(ik.û) -û(ik.ik)
that can be rewritten as

ûF = ik × ωF -ik(ik.û F ) k 2 2 .
(3.52)

From (3.52) we see that there is the second term -ik(ik.û F ), i.e. the gradient of the divergence of the filtered velocity field, that depends on the type of computation that we want to perform. In fact :

-For incompressible initial condition this term is zero, and the filtered velocity is The initial condition for the scalar θ is defined through the mean profile, with no fluctuations. In particular we use an analytical function of the radius r.

ûF = ik × ωF k 2 2 . ( 3
θ M (r) = 1 2 log 1 -At 1 + At 2 -tanh 1 d r -2R 0 R 0 + tanh 1 d r + 2R 0 R 0 (3.55)
where At = ρmax-ρ min ρmax+ρ min = exp(θmax)-1 exp(θmax)+1 is the Atwood number. d is a parameter that is defined as :

d = 2 R 0 θ max ∞ 0 θ M (1 -θ M )dr (3.56)
The integral in (3.56) is usually employed in the definition of the a mixing zone width [Andrews andSpalding 1990, Gréa 2013]. So that the specification of d directly imposes the initial width of the spherical mixing layer. The influence of the Atwood number and d on the radial profile of scalar is plotted in Figure 3.13. We observe on both figures 3.13a and 3.13b that the scalar gradient is steeper the smaller the d the larger the At. An example of the three dimensional contours for the scalar θ is shown

Filter

In the procedure of section 3.4.2.1, for the initial velocity field, we use a filter function to limit the fluctuation inside a defined spherical shell. The filter F s (x), is a smoothed square wave, initialized with the characteristic of the spherical layer, such as inner (r m ) and outer (r M ) radii :

F s (x) = 1 for r m ≤ |x| ≤ r M ,
and F s (x) = 0 elsewhere.

(3.57)

The two values r m and r M depend on the mean radial profile θ M defined in the previous section 3.4.2.2. We define r m and r M , respectively, such as the radial positions for which θ M = 0.01 and θ M = 0.99θ max The definition of the filter function, as a discontinuous function in Eq. (3.57), can create numerical oscillations due to the Gibbs phenomenon. One way to avoid this numerical problem is to smooth out high wavenumbers oscillation using a convolution with a smoothing function,

S(k) = exp C s k 2 1 + k 2 2 + k 2 3 N 2
in Fourier space. In the previous equation, N is the grid resolution and C s = 128 is a parameter used to determine the size of the transition zone (see Jause-Labert (2012) for details). Once the function F (k) = Fs S(k) is transformed back to physical space to obtain F (x), the step 2 of the section 3.4.2.1 procedure can continue. In Figure 3.14, we plot the filter function after the smoothing and the corresponding profile of θ M . In inertial confinement fusion application, the extreme temperatures involved imply that the fusion plasmas are in a weakly coupled regime. In this regime, the kinetic effects are dominant with respect to potential interaction, and the plasma coupling parameter is small compared to 1. In a weakly-coupled plasma, the viscosity µ has a strong dependence on temperature, in particular for light elements it has been shown [Braginskii 1995

] that µ ∼ T 5 2 . (4.1)
For heavier material the dependence is different and less intense. As an example [START_REF] Ticknor | Transport properties of an asymmetric mixture in the dense plasma regime[END_REF] showed that silver (Ag) dynamic viscosity evolves following

µ ∼ T 3 2 . (4.2)
The theoretical framework for compressed turbulence, introduced in section 2.1, has been applied to plasma compression by Davidovits and Fisch (2016a). They studied an adiabatic compression of Homogeneous Isotropic Turbulence (HIT) in the low Mach number limit, taking into consideration viscosity variations due to the mean temperature increase, as in Eq. (4.1). Besides, the turbulence production is impulsed solely by the compression terms coming from the mean velocity field, while other production sources such as shear or density gradient are neglected. Despite being crude, this framework has already proven an interesting starting point to explore important mechanisms at work during the compression, namely turbulence production and dissipation, but also nonlinear transfer. In particular, they evidenced how, during the compression, there is a moment when viscosity effects begin to dominate the dynamics of the flow until the turbulent kinetic energy is dissipated. In this context, the dissipation is rapid with respect to the compression duration, so they called this phenomenon sudden dissipation. In the following paper, Davidovits and Fisch (2016b) take into account the ionization state of the compressed plasma as a phenomenon that modifies the viscosity dependence to the temperature. In particular they consider µ ∼ T β where the exponent β is ionization dependent. They find that there is no sudden dissipation for β < 1.

In a recent paper, [START_REF] Campos | Self-consistent feedback mechanism for the sudden viscous dissipation of finite-Mach-number compressing turbulence[END_REF] have investigated the same phenomenon without the low Mach number approximation, allowing the direct study of the turbulent kinetic energy dissipation on the temperature of the system. These simulations show how the sudden dissipation effect acts on the solenoidal as well as the dilatational part of the kinetic energy. Furthermore, their examination of the internal energy source terms shows that the adiabatic compression term dominates the viscous dissipation. They conclude that, at least for the cases studied, the proposed self-consistent mechanism to increase the local temperature in compressed plasma has a minimal impact ; in particular, the adiabatic contribution is two or three orders of magnitude higher than the frictional viscous heating.

Meanwhile, the initial conditions and particularly the large scale asymmetries of the capsule, due for instance to the support tent or the fill tube [START_REF] Clark | Radiation hydrodynamics modeling of the highest compression inertial confinement fusion ignition experiment from the national ignition campaign[END_REF], play a determining role by having a strong imprint on the turbulent mixing. Still, how initial conditions influence the balance between the stirring processes and the microscopic dissipation remains unclear in ICF imploding capsules.

Concerning turbulence, it is known since the work of Batchelor [START_REF] Batchelor | The role of big eddies in homogeneous turbulence[END_REF][START_REF] Batchelor | The Theory of Homogeneous Turbulence[END_REF] followed by [START_REF] George | The decay of homogeneous isotropic turbulence[END_REF] that the large-scale structures or big eddies play an essential part in the dynamics of turbulent flows. This role has been identified, for instance, through different self-similar solutions, corresponding to the turbulent and final decays of HIT, showing a dependence on the initial distribution of energy at large scales. These solutions have been investigated in particular by spectral models based on Eddy Damped Quasi-Normal Markovian (EDQNM) closures [START_REF] Orszag | Lectures on the statistical theory of turbulence[END_REF][START_REF] Lesieur | Turbulence in Fluids[END_REF]] allowing a systematic exploration of the influence of initial conditions at very high Reynolds number [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][START_REF] Mons | Is isotropic turbulence decay governed by asymptotic behavior of large scales ? An eddy-damped quasi-normal Markovian-based data assimilation study[END_REF]]. Note that this method has been similarly generalized to other problems such as Rayleigh-Taylor and unstably stratified turbulence [START_REF] Soulard | Large-scale analysis of self-similar unstably stratified homogeneous turbulence[END_REF], Burlot et al. 2015b, Soulard et al. 2015].

In this chapter, we identify different regimes of turbulence in plasma under compression by exploiting the similarity properties of the system and the possibility of finding self-similar solutions in the moving frame corresponding to the coordinate transformation. Back in the laboratory frame, this gives information about the importance of initial conditions. To achieve this goal, we use DNS and spectral models based on classical EDQNM closures. The two methods are complementary : DNS provides the details of the turbulent fields in space, and their time evolution ; while, by construction, the EDQNM model directly predicts the time-evolving turbulent spectra of two-point correlations, and allows exhaustive parametric studies as well as the exploration of high Reynolds number regimes, due to its low computational cost. To extend our analysis to a less academic geometry and an inhomogeneous flow, we also consider a spherical turbulent layer with parameters relevant to ICF, as a paradigm of the hot spot contamination by turbulence.

Equation for plasma compression

In chapter 2 in section 2.1 we have derived the equation describing the dynamics of the velocity field for a neutral gas under compression. We recall that for a velocity field ũi in the moving frame, moving with the mean compressing flow, we have two possibilities :

∂ t ũi + ũj ∂ j ũi = -∂ i p + Λ 3 (t) µ ρ B 0 ∂ 2 jj ũi (4.3)
and

∂ t ũi + ũj ∂ j ũi + 3 S(t) Λ ũi = -∂ i p + µ ρ B 0 ∂ 2 jj ũi (4.4)
depending if one wants to take into account compression effects by having a time dependent viscosity or a forcing term. In this chapter we consider the following time dependence for the compression parameter Λ :

Λ = 1 -S 0 t (4.5)
The dynamic viscosity in plasmas does not behave as in neutral gases. As already pointed out in the introduction of this chapter, it grows with temperature. In this work we will consider an adiabatic compression and with this hypothesis the mean temperature T has the following dependence on the compression parameter Λ :

T (t) = T 0 Λ(t) -2 (4.6)
where T 0 is the initial temperature. Using relation (4.6) together with the Braginskii (1995) law in (4.1) we deduce that the dynamic viscosity for a weakly coupled plasma under adiabatic compression is

µ = µ 0 Λ(t) -5 . (4.7)
If we inject this expression of the dynamic viscosity µ in (4.3) and (4.4) we have

∂ t ũi + ũj ∂ j ũi = -∂ i p + ν 0 Λ -2 ∂ 2 jj ũi (4.8)
and

∂ t ũi + ũj ∂ j ũi + 3 S(t) Λ ũi = -∂ i p + ν 0 Λ -5 ∂ 2 jj ũi , (4.9)
where ν 0 = µ 0 /ρ B 0 is the kinematic viscosity. In Eq. (4.9) there is a new time dependence in front of the viscous term, so that the previous proposed rescaling (2.17) has to be modified [Davidovits and Fisch 2016a

] : ũi (x, t) = u i (x, t)Λ 3 (t), t = t 0 Λ -4 (s)ds, p(x, t) = p(x, t)Λ 9 (t) (4.10)
and consequently Eq. (2.18) is modified as

∂ t ũi + u j ∂ j ũi + 2SΛ 4 ũi = -∂ i p + ν 0 ∂ 2 jj ũi . (4.11)
In the moving frame, an initially homogeneous turbulence remains homogeneous during the compression. So that in this frame we are left with the problem of the decay of HIT with time-varying viscosity or with a new forcing term. These two different choices provide two different equations that model the same phenomenon.

In section 4.2, we derive the EDQNM model for both cases, while the direct numerical simulations are performed only for Eq. (4.8).

To fully characterize the flow regime we use two nondimensional numbers : the Reynolds number Re and the compression number Cp, defined at the initial time as Re = u 0 0 ν 0 , and Cp = u 0 0 S 0 , (4.12)

where u 0 and 0 are characteristic velocity and length scales of the initial turbulent flow. Re measures the ratio between turbulent and physical viscosity, and Cp the ratio between the turbulent frequency to the compression rate. They give information about the initial conditions of the system and help to determine which physical phenomenon can become dominant during different phases of compression.

Modification to the time integration scheme for the DNS with time-varying viscosity

The DNSs are performed using the methods described in chapter 3 in section 3.2.2.1 for incompressible computations, with the only modification due to timedependent viscosity. In particular for the step that concerns the implicit treatment of viscosity. In section 3.2.2.1 for the constant viscosity case, after transforming to Fourier space, we use the change of variable

v i = u i exp νk 2 (t -t 0 ) . (4.13)
When the viscosity is a function of time this substitution is no longer adequate. As an alternative we propose to use

v i = u i exp ν 0 k 2 N ( t -t0 )) (4.14)
where N ( t) is

N ( t) = Λ( t) -2 d t = Λ 3 3S 0 (4.15)
where the compression parameter dependence on the rescaled time t, defined in Eq.(2.15), is Λ( t) = 1/(1 + S 0 t). This step is particularly crucial in these DNSs because the viscosity increases considerably during the simulation, causing numerical instability if it is treated explicitly.

Statistical approach : EDQNM Model

The Eddy Damped Quasi-Normal Markovian (EDQNM) closure has proven an efficient closure for different types of turbulent flows. The first EDQNM model concerned isotropic turbulence, in which a damping timescale (Θ kpq ) -1 was used to adjust non linear decorrelation of the third-order cumulants generating the exchange of energy between three wavenumbers k, p, q (see e.g. [START_REF] Orszag | Lectures on the statistical theory of turbulence[END_REF], [START_REF] Lesieur | 3D isotropic turbulence at very high Reynolds numbers : EDQNM study[END_REF]). More or less sophisticated extensions of the EDQNM model were proposed to account for additional distortions to turbulence : mean velocity gradients, buoyancy force for stratified flows, Lorentz-Laplace force in magnetohydrodynamics, Coriolis force for rotating flows [START_REF] Sagaut | Homogeneous Turbulence Dynamics[END_REF]. In short, when accounting for such additional forcing in the Navier-Stokes equation, one can account for the corresponding linear term at different levels of the closure (namely, choosing to retain its influence on increasingly higher-order statistics, e.g. second-or third-order statistical moments). Since we deal with a rather strong effect of the additional linear terms arising in equations (4.11) and (4.8), we anticipate that its direct impact on non linear transfers will be second-order in amplitude with respect to its direct effect on energy spectra, and we retain a simple version of the EDQNM closure, in which spectral energy transfers T (k) will be closed with the same rationale as for isotropic turbulence. We shall see in section 4.2.3 that the resulting model compares very well with DNS.

Therefore, in the following, we derive the EDQNM closure for the compressed turbulence with plasma effect, for each of the two options mentioned above for behavior equations.

First case : Time-dependent viscosity

In the moving frame the Navier-Stokes equation with time-dependent viscosity is

∂ t ũi + ũj ∂ j ũi = -∂ i p + ν 0 Λ -2 ∂ 2 jj ũi (4.16)
It is then clear that, when t goes to infinity, the dissipation term in Eq. (4.16) tends to infinity as well, thus causing the sudden dissipation of kinetic energy. As done for the classical Navier-Stokes equation, one can derive the two-point velocity correlation evolution equation from Eq. (4.16), and derive the evolution equation for the two-point velocity spectrum and then obtain the Lin equation for the kinetic energy spectrum E(k). In the present case, the corresponding dynamical equation for the kinetic energy spectrum is

∂ tE(k, t) + 2ν 0 (1 + S 0 t) 2 k 2 E(k, t) = T (k, t) (4.17)
where the energy transfer term T (k, t) is closed using the classical EDQNM closure

T (k, t) = ∆ k Θ kpq k pq E(q, t)b(k, p, q) k 2 E(p, t) -p 2 E(k, t) dpdq . (4.18)
b(k, p, q) is the classical geometrical coefficient related to the geometry of the triad [START_REF] Sagaut | Homogeneous Turbulence Dynamics[END_REF]] and Θ kpq is the characteristic time appearing during the combined markovianization and eddy-damping process. Its expression is thus provided by Θ kpq ( t) = 1 -e µ kpq t+(1+2S 0 t) 2 ν 0 (k 2 +p 2 +q 2 ) t

µ kpq + (1 + 2S 0 t) 2 ν 0 (k 2 + p 2 + q 2 ) (4.19)
with µ kpq = ν k + ν p + ν q and ν k = 0.36

k 0 p 2 E(p)dp 1/2 .

Second case : Forcing term

If one wants to use the compression equation with the forcing term

∂ t ũi + u j ∂ j ũi + 2SΛ 4 ũi = -∂ i p + ν 0 ∂ 2 jj ũi (4.20)
with Λ( t) = 1/ 3 1 + 3S 0 t. The corresponding Lin equation for this case reads

∂ t + 2ν 0 k 2 E(k, t) = T (k, t) -D( t)E(k, t) (4.21)
where D( t) = 2SΛ 4 = 2S 0 /(1 + S 0 t). The energy transfer term T (k, t) is the same as in the previous case, with the only difference in the detailed expression of Θ kpq :

Θ kpq ( t) = 1 -e µ kpq t+ν 0 (k 2 +p 2 +q 2 ) t µ kpq + ν 0 (k 2 + p 2 + q 2 ) . (4.22)
where µ kpq has the same expression as in (4.19).

Statistical approach : EDQNM Model 75

Validation

We now present comparisons between EDQNM and DNS results. We use DNS data from Davidovits and Fisch (2016a) from simulations using 128 3 grid points, and from our simulations using 256 3 grid points. A close comparison permits to validate the closure and confirms that the statistical approach can be used for an extensive parametric study. Besides, comparing the different DNS data allows evaluating the influence of a modification of numerical parameters on DNS statistics.

The numerical integration of EDQNM model Equations (4.17) or (4.21) is done using a simple trapezoidal integration quadrature as regards the integral appearing in the transfer T (k). Time-marching is done using the third-order strong stability preserving Runge-Kutta method presented in section 3.2.2, a treatment that we believe is original for solving these Lin-type equations. This scheme improve the numerical stability of the time integration process, permitting the use of a larger time step with respect to the first-order Euler time-integration method used in previous works.

Whereas statistics obtained in DNS is but a by-product of the velocity field for a particular realization, obtained by averaging over the computational domain, the resolution of EDQNM equations directly provides the two-point statistics that can be integrated to obtain one-point statistics such as total kinetic energy. These statistics stand for the ensemble averaging that should be done if many DNS realizations could be afforded. Therefore, when comparing DNS and EDQNM, one needs to bear in mind these differences between the two approaches. Also, initial conditions in DNS use fields that are δ-correlated, thus with zero third-order correlations. This is not the case in EDQNM in which energy transfers are immediately triggered at the beginning of the computation, and therefore have non zero values from the start.

Comparison with DNS of Davidovits et al.

The time evolution of kinetic energy is shown in Figure 4.1 as a function of Λ, at different compression rates S 0 . The figure presents results from the EDQNM model and from DNS by Davidovits and Fisch (2016a). The initial kinetic energy is set to unity, and the time evolution goes from right to left, starting at the domain size Λ = 1.

The figure shows that kinetic energy evolution initially increases for compression rate S 0 larger than 1, and decreases for S 0 < 1. It then undergoes a sudden drop, triggered at a value of Λ increasing with S 0 . This drop is not observed for S 0 = 0.1 since kinetic energy decays too fast from the beginning.

Figure 4.1 also shows that EDQNM results agree very well with DNS data for S 0 > 1. When S 0 ≤ 1, the curves depart slightly, but the overall trend is rather good. Such differences can be due to various factors :

1. slight differences in initial conditions (that are not fully detailed in Davidovits and Fisch (2016a)) ; 2. the effect of confinement at low-wave number, when the computational domain is too small in DNS, so that DNS simulation over a 128 3 grid is under-resolved initially, although this resolution rapidly becomes adequate when viscosity increases ;

3. there are differences between the two formulations of the model, based either on the time-dependent viscosity closure or on the forcing term closure. Although the two formulations should be formally equivalent, their numerical implementation can induce departures, as observed on the figure.

Comparison with new DNS data

To have better control over the parameters of the simulation, we perform new Direct Numerical Simulations with known initial conditions and an increased resolution of 256 3 . This permits further validation and comparison of the EDQNM model results with those of our DNS data, shown in Figure 4.2. The initial conditions are generated using the method described in section 3.4.1, using the spectrum (3.43) with s = 4.

For the large compression rates S 0 ≥ 10, Figure 4.2 shows an almost perfect agreement between EDQNM and DNS kinetic energy evolution, better than in Figure 4.1. Also, the agreement is much improved for the lesser compression rates S 0 . This is the result of the increase of resolution in the DNS, and better controlled initial conditions as well.

Direct Numerical Simulations and EDQNM results with increased resolution

To quantify and analyze the different phenomena involved in the flow dynamics, and in particular, to illustrate the competition between turbulence production and viscous dissipation, we present results from DNS with increased resolution along with predictions by the spectral EDQNM model.

We illustrate this in Figures 4.3 an 4.4, with results obtained using an initial kinetic energy spectrum of the Batchelor form E 0 (k) ∼ k 4 exp (-2(k/k peak ) 2 ), where k peak corresponds to the maximum of E 0 . The initial Reynolds number is Re 0 = 250 and the compression number Cp 0 = 0.1, based on 0 = 1/k peak and u 0 = K 1/2 . Turbulence is therefore relatively weak in this example, while the compression is rapid compared to the turbulent timescale.

In Figure 4.3, we show the evolution of turbulent kinetic energy K = +∞ 0 E(k)dk as a function of the compression parameter Λ(t). We observe a very good agreement between simulations of 512 3 grid-point DNS and the EDQNM model during all the phases of the kinetic energy evolution. Moreover, the spectral distribution of energy at four instants plotted in Figure 4.4 (bottom row), show an excellent agreement between DNS and EDQNM at all scales. This supports the relevance of the closure as a model for compressed turbulence.

The case presented in Figures 4.3 and 4.4 is typical of the regime extensively discussed in Davidovits and Fisch (2016a). At the beginning (Λ(t) from 1 to O(10 -1 )), the dynamic of the flow is dominated by the compression effects leading to the increase of kinetic energy. Progressively, viscosity grows and dissipates energy in the small scales and eventually at larger scales, as indicated by the spectra. This counterbalances the turbulence production mechanisms and finally triggers the sudden viscous dissipation effect. These mechanisms are qualitatively observed in Figure 4.4 (top row), which shows the spatial distribution of kinetic energy in the DNS domain at the same four instants as the presented spectra. One particularly observes the intensification of kinetic energy, especially at the instant (III) corresponding to the peak of kinetic energy in Figure 4.3, and the substantial reduction of energy levels at (IV), along with a smoothing of the field due to the damping of small-scale fluctuations.

The critical value Λ M of the compression parameter corresponds to the maximum of kinetic energy in Figure 4.3. It indicates how much the turbulence can be compressed before the appearance of the sudden viscous dissipation effect. We have therefore computed the evolution of Λ M with the two relevant nondimensional pa- rameters, Reynolds and compression numbers, the initial spectrum remaining of the Batchelor form. The corresponding map of Λ M in the (Re, Cp) coordinates is shown in Figure 4.5. This parametric study was permitted by the EDQNM model, which allows exploring six decades of initial Reynolds numbers Re 0 and three decades of initial compression number Cp 0 . It required as many as 10000 EDQNM simulations, which would not be possible using DNS due to its high computational cost.

As expected, Figure 4.5 shows two general trends. First, when the initial Reynolds number increases, Λ M decreases since the viscosity coefficient needs to grow enough to impact the main energetic scales. Second, for decreasing values of the compression number Cp 0 , the critical compression parameter Λ M also becomes smaller since turbulent production is stronger and needs more time before being balanced by dissipation.

Upon closer inspection, the isolines of Λ M in Figure 4.5 permit to identify two additional kinds of dynamics different from that presented in Figure 4.3, wherein turbulent production dominance is followed by that of viscous dissipation as Λ decreases. The first additional regime occurs at small initial Re 0 and large Cp 0 . It is also observed in Davidovits and Fisch (2016a) : dissipation immediately prevails from the very beginning of the compression phase, and yields a decay of turbulent kinetic energy, and a narrow energy spectrum limited to large scales, as in Figure 4.6, case (A). The second additional regime appears at large values of Re 0 , and Cp 0 . It corresponds to the presence of important nonlinear energy transfers. In this new regime, two values of the compression parameter, corresponding to two kinetic energy maxima, may exist. This generates a discontinuity observed in Figure 4.5 (top-right corner, at large Re and Cp, where the label (C) appears), which can be explained by considering the following concurrent phenomena. First, kinetic energy grows as Λ -2 , which means it remains constant in the moving frame. In the meantime, energy is transferred to small scales by nonlinear effects. It is then suddenly dissipated when reaching the viscous scales, in the classical mechanism of enstrophy blow up well known in HIT decay. This sudden energy loss can counterbalance the growth of kinetic energy if the Reynolds number is not large enough, leading to the first maximum of K(Λ). Meanwhile, large scales keep gaining energy due to the compression until dissipated by the sudden growth of the viscosity coefficient, resulting in the second maximum shown in the spectrum in Figure 4.6, case (C).

In summary, we have explored the influence of initial nondimensional parameters on the dynamics of turbulent plasma under compression by combining the results of DNS and EDQNM simulations. This has permitted to identify the importance of the various physical mechanisms involved during the compression. In particular, we have shown that nonlinear energy transfer can have a decisive role at high Reynolds numbers. In the following section, we propose theoretical arguments in support of these conclusions.

Self-similar solutions in plasmas under compression

The main goal of this part is to derive the different self-similar solutions for turbulent plasma under compression. These solutions are first expressed in the moving frame and then transformed back in the laboratory frame using Eq. (2.15). As already stated, we stress that this derivation is limited to a restricted scenario, where a part of the phenomena typically involved are not considered (e.g., shear and buoyancy effects). But again, this kind of analysis may be useful, although not conclusive, to understand the different mechanisms involved in the evolution of turbulence in plasma under compression. Section 4.4.1 is dedicated to the theoretical analysis, and in Section 4.4.2 we show the actual existence of these regimes using DNS and EDQNM simulations.

Self-similar scaling

Three self-similar solutions are found, corresponding to the rapid compression regime, the cascade regime, and the viscous regime. They are listed hereafter, and for convenience, the corresponding scaling laws are gathered in Table 4.1

Rapid compression regime

The rapid compression (RC) regime, historically referred to as Rapid Distorsion Theory -called RDT in [START_REF] Cambon | Rapid distortion analysis and direct simulation of compressible homogeneous turbulence at finite Mach number[END_REF], although unlike in sheared turbulence, wavenumbers are not distorted-, is the simplest self-similar solution which can be obtained. Assuming that the turbulent timescale is much larger than the compression time, such that Cp 1, turbulence can be considered frozen in the moving frame, leading to the following energy spectrum :

Ẽ( k, t) = E 0 ( k), or E(k, t) = E 0 (Λ(t)k) × Λ(t) -1 , (4.23)
whence the evolution of kinetic energy and of the integral length scale I :

K(t) = +∞ 0 E(k, t)dk ∼ Λ -2 (t), I (t) = 3π 4 +∞ 0 k -1 E(k, t)dk +∞ 0 E(k, t)dk ∼ Λ(t).
This self-similarity of spectrum Ẽ( k, t) is of course only strictly valid for wavenumbers unaffected by viscosity. It is clear that the RC regime can only last for a limited duration, due to the Λ -2 growth of the viscosity and to the non-linear transfer in the moving frame. Note also that K ∼ Λ -2 does not necessary imply a RC phase.

For instance, redistribution of energy by non linear transfers can modify the shape of spectra and the value of integral length scale ˜ I , while still conserving the total kinetic energy.

Cascade regime

The decay of HIT is extensively discussed in most turbulence books, as for instance in [START_REF] Batchelor | The Theory of Homogeneous Turbulence[END_REF][START_REF] Lesieur | Turbulence in Fluids[END_REF][START_REF] Sagaut | Homogeneous Turbulence Dynamics[END_REF]. According to these monographs, the decay rate of energy depends on the cascading process governed by large scales of turbulence, but not on the value of the viscosity coefficient. For compressed turbulence, although the viscosity coefficient varies in the moving frame, one can expect similarly that the dynamics at large Reynolds number is driven only by the turbulent energy flux from large to small scales. Considering a self-similar evolution of kinetic energy as K ∼ t-nc , the integral length scale evolves as ˜ I ∼ t1-nc/2 . Assuming the permanence of big eddies [Batchelor 1949, Lesieur and[START_REF] Lesieur | 3D isotropic turbulence at very high Reynolds numbers : EDQNM study[END_REF], which can be expressed by lim k→0 Ẽ( k, t) = ks (s being the infra-red spectral slope), we deduce from self-similarity :

n c = 2(s + 1) s -3 . (4.25)
This provides the classical decay exponent generally encountered in turbulence : n c = 10/7 1.43 for Batchelor spectrum (s = 4) and n c = 6/5 = 1.2 for Saffman spectrum (s = 2). Note that the permanence of big eddies is relatively well observed for s ≤ 3 while not entirely true for s = 4 due to backscatter effects transferring energy from small to large scales.

Returning to the laboratory frame using Eq. (2.15) and the expression for Λ(t), this leads to the kinetic energy evolution as Λ → 0 for the cascade regime :

K ∼ Λ nc-2 and I (t) ∼ Λ nc/2 , (4.26)
leading in particular to a growth of kinetic energy and decrease of integral scale as K ∼ Λ -4/5 , I ∼ Λ for Saffman turbulence and as K ∼ Λ -4/7 , I ∼ Λ 2 for Batchelor turbulence.

Viscous regime

The last regime is closely related to the sudden viscous dissipation effect occurring in a turbulent plasma under compression, discussed in Davidovits and Fisch (2016a). When the Reynolds number becomes small enough, the decay of HIT enters a final regime, which is driven by dissipation and in which the nonlinear turbulent transfer is negligible. This phase exhibits a self-similar solution known as the final decay regime of HIT when the viscosity coefficient is constant [START_REF] Batchelor | The Theory of Homogeneous Turbulence[END_REF]]. We propose here a generalization for the time-varying viscosity case corresponding to a weakly-coupled plasma.

As for the turbulent decay presented in Section 4.4.1.2, we consider a self-similar evolution in the moving frame of the form K ∼ t-nv . The scaling for the integral scale can be obtained from the self-similar evolution of the turbulent time ∼ t and the viscosity coefficient ν( t), yielding ˜ I ∼ ν( t) t 1/2 . Assuming again lim k→0 Ẽ( k, t) = ks as large scales remain unaffected by viscosity, we obtain, for Λ → 0 :

n v = 3 2 (s + 1). (4.27)
Similarly to the cascade phase, we deduce the scaling laws for the kinetic energy and integral scale in the laboratory frame for the viscous phase,

K ∼ Λ nv-2 and I (t) ∼ Λ -1/2 , (4.28)
leading in particular to a sudden viscous dissipation of the form K ∼ Λ 5/2 in Saffman turbulence and K ∼ Λ 11/2 for Batchelor turbulence. Therefore, the viscous regime corresponding to the sudden dissipation effect clearly induces an important sensitivity to initial conditions, contrary to the cascade regime, for instance, where the variations of the growth exponent are smaller. Notably in the viscous regime, the integral scale of turbulence grows despite the compression and is not dependent on initial conditions.

RC regime Cascade regime Viscous regime

K ∼ Λ -2 ∼ Λ nc-2 ∼ Λ nv-2 I ∼ Λ ∼ Λ nc/2 ∼ Λ -1/2 Re ∼ Λ 2 ∼ Λ nc+1 ∼ Λ (nv+1)/2 Cp ∼ Λ -1 ∼ Λ 0 ∼ Λ (nv+1)/2

Simulations of self-similar regimes

We now confirm the actual appearance, in simulations, of the different self-similar solutions analytically found in the theoretical study of Section 4.4.1. We consider two types of initial conditions, as representative of many encountered situations : Saffman turbulence (s = 2) corresponding to equipartition of energy at large scale among the different wave-vectors, and Batchelor turbulence (s = 4) generated by backscatter effects from small scale perturbations. In Section 4.4.2.1, we consider the case of high Reynolds number turbulence, and for this, we use the EDQNM model. In Section 4.4.2.2, we model the compression of a spherical turbulent layer using both DNS and EDQNM.

High Reynolds number compressed turbulence with EDQNM simulations

The Reynolds number Re = u /ν for a weakly-coupled turbulent plasma under compression is expected to decrease, as assessed by the scaling laws for u = K 1/2 and = I in either the rapid compression, the cascade or the viscous regime. Thus, we use the EDQNM model to reach a very high initial Reynolds number, beyond the possibilities of DNS, to obtain the different self-similar solutions derived in Section 4.4.1.

We choose initial spectra in the von Karman type E 0 (k) ∼ k s exp [-s/2(k/k peak ) 2 ], both for the Saffman (s = 2) or the Batchelor (s = 4) case. We set the Reynolds number at Re 0 = 10 7 and the compression number Cp 0 = 0.47. Accordingly, this corresponds to a flow with high intensity turbulent fluctuations and fast compression. In Figure 4.7, we observe, for the turbulent kinetic energy evolution, a greater sensitivity to initial conditions in the viscous regime. Still, the integral length scale increase in this regime is similar for both Saffman and Batchelor cases. We conclude that differences in the integral scale I dynamics are only due to the cascade phase, thus demonstrating the imprint of nonlinear mechanisms.

The Reynolds number decrease observed in Figure 4.8-top is mainly due to the growth of viscosity, while initial conditions have relatively low influence except during the viscous phase. The compression number Cp (Figure 4.8-bottom) increases from its initial low value of Cp = 0.47 during the RC phase and reaches a plateau at about ten times this value during the cascade phase.

The fact that simulations agree well with the scaling laws proposed in Sec. 4.4.1 is a consequence of the fact that the distribution of energy at large scales remains constant during the compression. This constancy is verified in Figure 4.9, for both Saffman and Batchelor turbulences, in which the infrared slopes of turbulent kinetic spectra are maintained over three decades of k. In particular, energy backscatter usually alters the slope in Batchelor turbulence, but this seems of little influence here. Besides, the presence of nonlinear transfer over a few decades of scales during the cascade regime is attested by the observed Kolmogorov scaling k -5/3 in spectra E(k) at Λ = 10 -2 in Figure 4.9. On the contrary, the absence of inertial subrange shows that, during the sudden viscous dissipation phase, energy is systematically dissipated from small to large scales.

In summary, we have evidenced in this section the existence of the self-similar regimes using EDQNM simulations at high Reynolds number with long evolution times, thus reaching very small compression parameter Λ. This demonstrates the influence of the initial distribution of energy at large scales, in particular during the viscous dissipation phase. In the following, we show that these results still apply in configurations more relevant to ICF, i.e when the Reynolds number is smaller and in the presence of inhomogeneities.

Spherical turbulent layer

An essential question in ICF is how the turbulent ablator/fuel interface eventually contaminates the hot spot at the center of the spherical capsule. We, therefore, consider here a spherical turbulent layer configuration, more relevant to the ICF problem than homogeneous turbulence. Of course, this case is still simplified, for it discards important physical phenomena that are present in ICF, for instance, density effects and the stagnation shock in the fuel, which has to be taken into account before the bang time. Nonetheless, it is progress towards more realistic predictions. Therefore, in this section, we investigate the self-similar solutions and the influence of large scale perturbations in the case of a spherical turbulent layer under compression using both DNS and EDQNM.

Although some estimates of physical parameters of experimental ICF are very difficult to make, we choose initial conditions that can be relevant to actual flow situations. We evaluate the Reynolds and compression numbers based on the ICF simulations corresponding to the NIF shot N120205 presented in Weber et al. (2014a). During the phase considered just before the bang time, the ablator/fuel interface passes from a radius of 340 µm to 54 µm in 1 ns giving S 0 ∼ 10 9 s -1 . The initial fuel viscosity is around ν ∼ 10 cm 2 s -1 for ρ ∼ 10 g cm -3 and T ∼ 500 eV corresponding to a plasma coupling parameter Γ ∼ 0.1 for the deuterium-tritium fuel. The integral scale and fluctuating velocity are respectively 0 ∼ 10 -2 cm and u 0 ∼ 10 6 cm s -1 , leading to Re 0 ∼ 10 3 and Cp 0 ∼ 10 -1 . In our simulations, we therefore choose Re 0 = 450 and Cp 0 = 0.1, figures that are relevant to ICF configurations.

The initial spectra are of von Karman type already introduced in Section 4.4.2, E 0 (k) ∼ k s exp [-s/2(k/k peak ) 2 ]. These conditions are sufficient to initialize the EDQNM model, which assumes homogeneity and isotropy of turbulence. Its predictions will be used to compare to the DNS results. The initialization of the DNS fluctuating velocity field is performed using the procedure described in section 3.4. for the incompressible velocity field in a spherical layer. The DNSs are done in a three-dimensional 2π-periodic domain so that it is important to ensure minimal interaction between adjacent spherical turbulent layers. Therefore, we choose an outer radius r M small enough compared to the size of the domain H = 2π, so that the velocity decreases sufficiently between the layers. We have experienced that r M /H = 1/π is adequate for that, and we also choose an inner radius such that (r M -r m )/H = 1/(2π). The thickness of the turbulent spherical layer MZ is determined on the spherically integrated kinetic energy profiles and corresponds to the region where K ≥ 0.1 max r K(r). The integral length scale I has to be smaller than MZ to avoid a fast turbulent diffusion of the layer during the compression. In consequence, we choose I0 / MZ0 ∼ 0.07. This scale separation, together with the constraint of resolving all the scales of turbulence, calls for 1024 3 DNS grid points.

Figure 4.10 shows the evolution of kinetic energy from DNS and EDQNM, and The kinetic energy evolution in Figure 4.10 exhibits two self-similar phases corresponding to rapid compression followed by the viscous dissipation regimes. As ex- As for the simulations presented previously, DNS and EDQNM agree very well if one considers the non dimensional parameters (Figure 4.12) and the one-point statistics K, I (Figure 4.13), but it is also true for the kinetic energy spectra (Figure 4.14). This may appear surprising in the sense that EDQNM is a model for homogeneous isotropic turbulence, whereas the spherical turbulent layer is spatially inhomogeneous. This can be explained by the fact that turbulence lengthscales remain smaller than the size of the turbulent layer throughout the compression, as shown in Figure 4.13. Thus the evolution of turbulent structures is not influenced by the large-scale inhomogeneity, and the flow remains quasi-homogeneous as regards its statistics.

The different scaling laws derived in Sec. 4.4.1 are also recovered in the spherical turbulent layer simulations confirming in particular that viscous dissipation is more important in the Batchelor case compared to the Saffman case, as shown in Figure 4.11. One, however, remarks in Figure 4.13 That the integral length scale I is larger in Saffman turbulence than in the Batchelor case. This can be explained by non-negligible nonlinear turbulent transfer around Λ M , since, during the compression phase, I always evolves as Λ 2 in the rapid compression regime, while it scales as Λ -1 in the viscous regime. and at the different kinetic energy profiles in Figure 4.15 which, in the ICF context would correspond to studying the contamination of the hot spot by the turbulence.

It is interesting to look at the MZ evolution during the compression in

In the beginning, for Λ ≤ Λ M , the evolution of the turbulent fronts of the turbulent layer is slow and MZ ∼ Λ 2 . However, the dynamics of MZ changes when entering the viscous phase, which corresponds to a diffusive increase of turbulent kinetic energy towards the center, as seen in the radial profiles in Figure 4.15. This phase also is sensitive to initial conditions since kinetic energy in the center in the Saffman case is higher than in the Batchelor case. In summary, our simulations show that the sudden viscous phase not only dissipates the turbulent kinetic energy but also enhances its transport, possibly leading in the context of ICF to the contamination of the hot spot. Accordingly, it seems particularly important to predict the time at which the viscous phase occurs, especially if it is before the bang time. From the data of the simulations of Weber et al. (2014a), one can obtain a quantitative estimate of the order of the corresponding timescales. Although some physical phenomena are overlooked and there remain uncertainties concerning their relative magnitude, it seems that the bang time appears before the viscous phase, hence possibly explaining why the contamination of the hot spot is not very important (by bang time the fuel-ablator interface has converged from 340µm to 54µm giving Λ bang ∼ 0.16 > Λ M ). However, it seems possible to consider different initial perturbations in which the viscous phase may appear earlier than the bang time. This would clearly result in important modifications of the turbulent mixing in the fuel.

Conclusion

In this chapter, we have investigated compressed turbulence in a weakly-coupled plasma by identifying the different regimes resulting from the competition between turbulence production, nonlinear energy transfer, and viscous dissipation. Depending on the relative importance of these terms, three self-similar regimes can appear, namely rapid compression, cascade and viscous phases. All three are clearly observed in our DNS and EDQNM simulations. While the rapid compression and viscous phases have been discussed in Davidovits and Fisch (2016a), the cascade phase is precisely characterized for the first time thanks to the EDQNM model that permits to explore very high Reynolds number configurations. Besides, this new model has proven its efficiency at lower Reynolds numbers, by matching DNS results regarding one-point statistics and two-point correlation spectra.

For sufficiently large initial Reynolds numbers and small initial compression numbers, weakly-coupled plasmas under compression experience growth of kinetic energy due to a rapid compression or cascade phase. However, the viscous phase always prevails at the end of the compression leading to the sudden dissipation phenome-non. By performing a parametric study with the EDQNM model, we have been able to explore the phase space of nondimensional numbers, Re and Cp, and to locate the values of the critical compression parameter at the maximum of kinetic energy corresponding to the beginning of the viscous phase. This also reveals the complex interplay between turbulence production, transfer, and dissipation, leading to some configurations to two successive growth/decay phases of kinetic energy in place of one during the compression.

The scaling laws for the different self-similar regimes have been derived from a theoretical analysis and recovered in DNS and EDQNM simulations. As in other classical turbulence problems, it demonstrates the dependence of the flow dynamics on the initial distribution of energy at large scales. This appears particularly crucial during the viscous phase since it can change the decay exponents of kinetic energy drastically. Consequently, a much higher amount of turbulent fluctuations remain at the end of compression when initial kinetic energy is concentrated at large scales.

Thanks to these results about the refined stages of the flow dynamics and the conditions of their appearance, we were able to extend our study to the case of an inhomogeneous spherical turbulent layer under compression, using parameters representative of ICF capsules. While recovering the different phases already identified in the homogeneous configurations, the simulations permit us to observe the diffusion of the layer towards the center, mimicking the contamination of the hot spot by turbulence. The results show an enhanced enlargement of the layer during the viscous phase, along with great sensitivity to initial conditions. Therefore, it raises the question of whether the sudden viscous phenomenon is favorable to achieve ignition in ICF. On the one hand, viscous dissipation participates in a global temperature increase by converting kinetic energy into internal energy, increasing the fusion reactions. On the other hand, turbulent mixing can be transported into the core, cooling the hot spot. In all cases, the appearance of the viscous phase indicates an important change in the plasma dynamics, and it seems important to figure out when this happens, before or after the bang time for some configurations. In the introduction, we have presented the problems that are still open in the context of the baroclinic instability. The complexity of this type of flows makes fundamental studies difficult because of the different interacting mechanisms, such as inhomogeneity, anisotropy, and compressibility [Zhou 2017a]. This is the reason why, during the years, some simplified frameworks have been introduced to study the buoyancy generated turbulence.

The first simplified set-up was proposed by [START_REF] Batchelor | Homogeneous buoyancygenerated turbulence[END_REF], who investigated the decay of homogeneous turbulence, with passive scalar and buoyancy effects. They limited their analysis to Boussinesq approximations, that is, cases with small scalar fluctuations that correspond to low Atwood number Rayleigh-Taylor turbulence. They found, using linear analysis of the Navier-Stokes equations together with 96 Chapitre 5. Variable density effects in unstably stratified turbulence DNS, that the history of the motion depends on the large scales of the flow. They also show that similar behavior is still present when the nonlinear terms are taken into account, that is, the self-similar state of the flow at large times depends on the behavior of the spectra near k = 0.

Subsequently, studies on a similar problem were carried out by [START_REF] Sandoval | The dynamics of variable-density turbulence[END_REF] and [START_REF] Livescu | Buoyancy-driven variable-density turbulence[END_REF]2008), who eliminated the small fluctuation hypothesis using the variable density approximations.

Although they consider buoyancy effects in homogeneous configurations, they limit their investigations to a case of turbulence decay, which is an essential difference with the Rayleigh-Taylor turbulence. One of the features of the latter is the continuous conversion of potential energy into kinetic energy, which increases turbulence intensity.

The closer simplified configuration, which retains most of the Rayleigh-Taylor features together with the idealization of homogeneous turbulence, is the Unstably Stratified Homogeneous Turbulence (USHT).

The USHT was investigated experimentally by [START_REF] Thoroddsen | Experiments on homogeneous turbulence in an unstably stratified fluid[END_REF], who studied the effects of an unstable linear density profile on the evolution of homogeneous grid-generated turbulence, in a thermally stratified wind tunnel. They found that, contrary to the stably stratified case, where vertical velocity fluctuations are inhibited, the buoyancy forces feed energy into the vertical motions.

The derivation of the USHT equations and the first numerical simulations were performed by [START_REF] Griffond | Unstably stratified homogeneous turbulence as a tool for turbulent mixing modeling[END_REF], [START_REF] Soulard | Large-scale analysis of self-similar unstably stratified homogeneous turbulence[END_REF], who identified this idealized framework configuration as a tool to obtain a deeper understanding of buoyancydriven mixing. The underlying hypothesis of the unstably stratified homogeneous turbulence is the homogeneity. This is justified if the integral scale of turbulence is small compared to the mixing zone size L. The ratio between these two quantities has been measured by [START_REF] Vladimirova | Self-similarity and universality in rayleigh-taylor, boussinesq turbulence[END_REF] in incompressible Rayleigh-Taylor mixing layer, where they found L ≈ 0.2.

The direct numerical simulation of the USHT equations presents, as usual, for DNS, limitation on the Reynolds number of the flow. That is why Burlot et al. (2015a), proposed a spectral model based on the EDQNM closure to investigate USHT dynamics at large Reynolds number.

They found that for several initial conditions, the late time study of the self-similar states is limited due to confinement effect and that large numerical simulations for both DNS and EDQNM are required to reach these states.

The Rayleigh-Taylor turbulent mixing zone, at late times, presents a self-similar evolution, as discussed in the introduction of this thesis. The unstably stratified homogeneous turbulence maintains the same characteristic.

It has been shown [START_REF] Griffond | Unstably stratified homogeneous turbulence as a tool for turbulent mixing modeling[END_REF][START_REF] Soulard | Large-scale analysis of self-similar unstably stratified homogeneous turbulence[END_REF]] that, for the case of constant buoyancy frequency N , the second-order moments grow exponentially with 5.1. Velocity field induced by an isolated eddy in a variable density field 97 time, for example, the turbulent kinetic energy

u i u i = exp (βN t).
(5.1)

Here the parameter β plays the same role as the α for the Rayleigh-Taylor turbulence.

The value of β can be directly connected to the large scale of the flow and in particular, to the infrared slope of the energy spectrum [START_REF] Poujade | Growth rate of Rayleigh-Taylor turbulent mixing layers with the foliation approach[END_REF]. In fact, [START_REF] Soulard | Large-scale analysis of self-similar unstably stratified homogeneous turbulence[END_REF] show that

β = 4 3 + s , for s ≤ 4. (5.2)
So that the large scales of the flows are the determinant factor to determine the growth rate of an USHT field second order moments.

Velocity field induced by an isolated eddy in a variable density field

Here we consider a blob of vorticity located at x = 0 within a domain V [START_REF] Davidson | Turbulence : an introduction for scientists and engineers[END_REF]] with characteristic length B . Inside this domain, the density is nonuniform, while outside of V , it is constant. The following developments are based on the work of [START_REF] Soulard | Permanence of large eddies in variable-density homogeneous turbulence[END_REF]. If we use the approximation that the velocity is still incompressible we can relate it where the integral is all over the space. If we consider value of x B , we can find value of A using the Taylor expansion of the |xx |

1 |x -x | = 1 r -∂ i 1 r x i + 1 2 ∂ 2 ij 1 r x i x j + • • • , (5.5)
where r = |x|. If one substitutes this expression in Eq (5.4) and after some simplifications one obtains

4πA(x) = ∂ i 1 r × x × ω 2 dx + 1 2 ∂ 2 ij 1 r x i x j ω dx + • • • . (5.6)
The first integral is a measure of the linear momentum introduced into the fluid by the blob of vorticity and we shall call it linear impulse L. To finally obtain the expression for the velocity, We take the curl of (5.6), to finally obtain the expression for the velocity

4πu(x) = (L • ∇)∇ 1 r + ∇ ∂ 2 ij 1 r × x i x j ω dx + • • • . (5.7)
This expression evidences how, if L = 0 the velocity in the far field goes like u(x) ∼ r -3 while in the case of L = 0, we have u(x) ∼ r -4 . In case of constant density, it has been shown [START_REF] Davidson | Turbulence : an introduction for scientists and engineers[END_REF]] that L is an integral invariant for the isolated blob of vorticity, so that the far-field decay of the velocity is not modified during the blob evolution.

Taking the divergence of the incompressible Navier-Stokes equation provides a relation between the velocity and the pressure. Which allows expressing the pressure p as a function of the velocity u using the Biot-Savart law

4π p ρ = [∇ • (u • ∇u)] dx |x -x | .
(5.8)

Again we can expand the integral using the Taylor series (5.5) which after some simplification leads to

4π p ρ = ∂ 2 ij 1 r u i u j dx + • • • . (5.9)
which shows how the pressure field decays as p ∼ r -3 . However, when the density is not uniform, equation (5.8) is no longer valid. In this case, we start from the incompressible Navier-Stokes equation (5.10) where the specific volume τ = 1/ρ is non uniform. Taking the divergence of (5.10) we obtain (5.11) This equation cannot be simply inverted to obtain an equation similar to (5.8), however an implicit solution can still be found

∂ t u i + u j ∂ j u i = -τ ∂ i p + ν∂ 2 jj u i ,
∂ 2 ii (τ p) -∂ j (p∂ j τ ) = -∂ 2 ij (u i u j ).
4πτ p = ∂ j [∂ i (u i u j ) -p∂ j τ )] dx |x -x | (5.12)
which can be expanded, as in the constant density case using the Taylor series

4πτ p = -∂ i 1 r p∂ j τ dx + ∂ 2 ij 1 r u i u j dx - 1 2 p(x i ∂ j τ + x j ∂ i τ )dx + • • • (5.13)
From equations (5.9) and ( 5.13), we observe that in the two cases, the pressure scaling is not the same, due to the additional terms coming from the τ gradients. [START_REF] Soulard | Permanence of large eddies in variable-density homogeneous turbulence[END_REF] show that injecting the far-field expansion of the velocity u, Eq (5.7), into the evolution Eq (5.10) and using the scaling obtained from the pressure, one obtains an evolution equation for the linear impulse L :

∂ t L i = p∂ i τ dx.
(5.14)

In particular, this result shows that linear impulse can be created in the vorticity blob by the correlation of pressure and density gradient, so that in this case the velocity in the far-field u(x) ∼ r -3 .

(5.15)

In the same paper, the author analyzes the case of homogeneous turbulence with variable density. They show that the permanence of large eddy is not achieved in the case when the infrared slope of the kinetic energy spectra s is greater than 2. This conclusion is, however, only verified if the density fluctuations are sufficiently large ; if it is not the case, a kinetic energy spectrum with s = 4 will maintain this value during the flow evolution.

Unstably stratified homogeneous turbulence equations

The equations for the unstably stratified homogeneous turbulence derived in section 2.2 are recalled for both approximations, variable density, and Boussinesq.

These equations are derived for a turbulent mixing zone between two fluids with different densities, in a uniform gravity field. In the following u i is the velocity, π the reduced pressure, θ = log(ρ/ρ 0 ) is a scalar value connected to the density, g is the acceleration and L is the mixing zone width. ν and D are the molecular viscosity and 100 Chapitre 5. Variable density effects in unstably stratified turbulence diffusivity that in this chapter are considered constant.

∂ t u i + u j ∂ j u i + π∂ i θ = -∂ i π -gL∂ i θ - 1 L πδ i3 + ν∂ 2 jj u i + ν∂ i (∂ j u j ) + ν (∂ j u i + ∂ i u j ) ∂ j θ + ν L (∂ 3 u i + ∂ i u 3 ) (5.16) ∂ t θ + u j ∂ j θ = D∂ 2 jj θ - 1 L u 3
(5.17)

∂ j u j = -D∂ 2 jj θ.
(5.18)

The variable density assumption introduces new nonlinear terms due to the scalar gradients, together with the relation (5.18) connecting the divergence of the velocity field to the Laplacian of the scalar.

If one assumes small fluctuations for θ, equations (5.16), (5.17), (5.18) reduce to the Boussinesq case : where u is the characteristic velocity and is the integral scale, already introduced in chapter 4, defined as

∂ t u i + u j ∂ j u i = -∂ i π + gθδ i3 + ν∂ 2 jj u i (5.19) ∂ t θ + u j ∂ j θ = D∂ 2 jj θ - 1 L u 3 (5.
= 3π 4 +∞ 0 k -1 E(k, t)dk +∞ 0 E(k, t)dk (5.24)
where E(k, t) is the kinetic energy spectrum. The Reynolds number helps to understand the relative importance of inertial over viscous terms, while the Froude number indicates the ratio between inertial and forcing terms.

Numerical methods

We solve equations (5.16)-(5.18) and (5.19)-(5.21) using the pseudo-spectral method and the third-order Runge-Kutta SSP scheme described in chapter 3. We recall that in all the simulations, the velocity and scalar mean values are imposed to be 0.

The solution of both variable density and Boussinesq systems of equations faces, in part, the same numerical challenges due to the properties of the USH turbulence. The growth of the integral scale, together with the decrease of the Kolgomorov length scale, η = ( /ν 3 ) -1/4 , requires a careful choice of the initial conditions and mesh sizes.

On the one hand, one should have a higher k peak (the wavenumber of the energy spectrum peak) possible, knowing that it decreases during the simulation. [START_REF] Pope | Turbulent Flows[END_REF] estimates that for a DNS free of confinement k peak /k min ≥ 10. In our cases, we stop our simulations when k peak /k min ∼ 8, which is a compromise between the minimization of confinement effects and the concurrent need to have long enough simulations to observe the establishment of self-similar states.

On the other hand, the k peak is limited by the mesh size, which imposes the smallest resolved scale. Furthermore, during the simulations, due to the turbulence kinetic energy increase, the Kolgomorov length scale decreases. Consequently, the computations resolution deteriorates. [START_REF] Pope | Turbulent Flows[END_REF] proposes the limit k max × η = 1.5, where k max is the maximum resolved wavenumber, as a lower limit to satisfy in order to have resolved simulations. For the Boussinesq case at the end of the DNS simulation, we have k max × η ∼ 4, which satisfies the proposed criterion.

Moreover, we recall that, in addition to the issues mentioned above, for variabledensity approximation, the non-locality, in spectral space, of the Poisson equation requires an iterative method to invert the linear system and compute the pressure contribution. We use the generalized minimal residual (GMRES) algorithm described in section 3.2.3.1, which ensures convergence even in cases with significant density fluctuations. Nevertheless, the convergence of this method is one of the variable density USHT critical features. As shown by Di [START_REF] Pierro | On a preconditionment for the spectral solution of incompressible variable density flows[END_REF], in the Poisson equation, the pressure operator conditioning number depends on the density contrast in the computational domain. That is why, in USHT computations, where the scalar variance grows with time, the number of GMRES iteration required to achieve the desired convergences increases as the simulation advances.

The increasing computational cost of the DNSs would not be manageable for a parametric study on the initial conditions, for instance, the direct numerical simulation Ud3, presented in section 5.4, demanded 1000000 CPU hours dispatched on 4196 cores.

For this reason, to explore the influence of initial conditions, we propose to use the implicit large eddy simulation method of [START_REF] Mathew | A new approach to LES based on explicit filtering[END_REF] and referred to as ADM-LES.

Chapitre 5. Variable density effects in unstably stratified turbulence

The ILES allows us to study the same initial conditions that one would have for the DNS with coarser meshes and reduced computational time.

Validation ILES for variable density

The effectiveness of the ADM-LES for USHT computation in Boussinesq approximation has already be proven by Griffond et al. (2015b). Here we show how this simple large eddy simulation method can be useful in variable density computation too. The ADM-LES is based on a low pass filter and an approximate deconvolution [START_REF] Grinstein | Implicit large eddy simulation : computing turbulent fluid dynamics[END_REF]. In this method, as explained in [START_REF] Mathew | A new approach to LES based on explicit filtering[END_REF], the filtering and the deconvolution can be combined into a single step, which reduces to simply filtering the variable at each time step. A comparison between DNS and ILES data for the same case with a domain respectively of 1024 3 and 512 3 mesh points is plotted in Figure 5.2. These cases correspond to the simulation Ud3 V D and Ul3 V D , which details are summarised in table 5.1. From the comparison of the one point integrated kinetic energy in Figure 5.2b, we observe a very good agreement between the two simulations with the ILES results having the same temporal dependence as the DNS.

The differences between the two methods are evidenced by the kinetic energy spectra of Figure 5.2a. The two spectra are in excellent agreement for all the wave numbers resolved by the ILES method until the sharp cut-off due to the low pass filtering around k 200.

The difference that is limited to the smallest scale of the simulation has to be considered when one discusses ILES results. All in all, the utility of this kind of under resolved simulation is undeniable, as it decreases the computational cost of the parametric study to 150000 CPU hours per simulation, retaining most of the physical properties of the direct numerical simulations. 106 Chapitre 5. Variable density effects in unstably stratified turbulence

Results

In this section we confront the results between the two direct numerical simulations Ud3 with Variable Density (VD) and Boussinesq (B) approximations. The effective Atwood number computed using the maximum and minimum value

Non dimensional numbers

Results

107

of θ, that is found in the computational domain at every time step,

A t = exp(θ max ) -exp(θ min ) exp(θ max ) + exp(θ min ) (5.25)
is plotted in 5.4c. In this figure, we can observe how, at the beginning, there is a phase where the A t decays, which corresponds to the decrease in the scalar variance, in Figure 5.5b, discussed in section 5.4.2. After the minimum, reached around N t 1.5 for the two simulations, the Atwood number increases during the evolution of the flow, reaching values close to 1 at the end. We remark how the values for the VD simulation are always higher than B simulation, suggesting the presence of larger fluctuations of density in the domain. The evolution of the one-point statistics is plotted in Figure 5.5. From the evolution of the turbulent kinetic energy in Figure 5.5a, we can observe how the two approximations have analogies and differences.

One-point statistics

At the beginning of the simulation, the kinetic energy in the VD approximations grows more rapidly with respect to the Boussinesq case. The different behavior during this first transient phase can be explained by the variable density effects, which are intense enough to modify the large scale of the flow.

During the following phase, from N t 1, when the nonlinear effects begin to dominate the dynamics, we observe a period where the two approximations have a good agreement. This period is, however, short, and from N t 3, the kinetic energy of the two simulations shows, again, different temporal dependencies. After this phase, where the nonlinearities become the principal mechanism, we observe the flow entering the self-similar state. As explained in the introduction, in this phase the second-order moments grow exponentially, with the following scaling

K ∼ exp(βN t) θθ ∼ exp(βN t) u 3 θ ∼ exp(βN t) (5.26)
where β depends on the infrared part of the spectra. It is in this phase that the large scale flow modifications due to variable-density effects become evident. In particular, we observe that the kinetic energy of the VD simulation has a growth rate of β 2 = 4/5 corresponding to s = 2, while for the B simulation β 4 = 4/7 corresponding to s = 4. Additionally, we also remark how the infrared slope alteration, not only modifies β, but also the time when the flow enters the self-similar regimes, anticipating it.

The scalar variance θθ time evolutions for both simulations, plotted in 5.5b, have a decay from N t = 0 to N t 1.5, followed by phase of growth reaching the self similar state at N t 4, where the modification of the growth rate is also observed.

As expected, we also find the same behavior in the self-similar phase for the vertical flux u 3 θ .

Two-point statistics

The kinetic energy E(k, t), the scalar variance E θ (k, t), and the vertical flux E u 3 θ are presented in Figure 5.6.

From the same initial condition, we observe how the infrared part of the spectra of the three quantities, in the variable density approximation, is not constant and on the contrary, is modified, from a slope of k 4 to k 2 . This modification induces the difference already observed on the one point quantities in section 5.4.2. For most of the temporal evolution, the main difference between the two approximations is due to this large scale effects.

However, we observe at N t = 3.7 and more clearly at N t = 4.73, that differences are also present in the nonlinear cascading processes, with small scales differences that are more and more evident, as the simulations proceed.

In the inertial zone of the VD simulation, we start to observe a scaling close to the k -5/3 , while for the Boussinesq, this dependence is less evident. At N t = 4.73, the confinement effects are evident for both cases. This is a known problem for the direct numerical simulations of USHT configuration, which prevents the study of large Reynolds number configurations.

The difference just discussed is observed on the spectra of the three quantities, with the large scale justifying the difference observed for most of the simulations, while towards the end, we can appreciate the increased influence of the nonlinearities and the onset of confinement effects.

Anisotropy

Anisotropy is one of the main characteristics of the unstably stratified homogeneous turbulence, and it is one of the main features of buoyancy-driven flows such as Rayleigh-Taylor. The acceleration direction imposes the anisotropy in USHT configuration, and as shown in section 5.3, it is present in both Boussinesq and variable density approximations.

As a further illustration in Figure 5.7 we show the contours of the vertical component of the velocity in a plane parallel 5.7a, 5.7b and perpendicular 5.7c, 5.7d to the vertical direction, for both simulations Ul3. The two figures show a marked anisotropy in the vertical direction, with the variable density case that has higher values.

A more quantitative way of looking at the anisotropy is using the anisotropy coefficient

b ij = u i u j u i u i -δ ij 1 3 (5.27)
for the case of i = j.

Results

111

This coefficient is a simple but effective way of looking at the anisotropic behavior of a flow. When this value is zero, it means that the kinetic energy is distributed equally among all the components. While if it is different means that the energy distribution is not isotropic, and we have preferential directions.

The evolution of the three anisotropy coefficients is plotted in Figure 5.8. As we already remarked qualitatively, the direction of anisotropy of the flow is the direction 3, following the acceleration. b 33 is positive with values around 0.4 for most of the simulations, with a slight decrease towards the end. The difference between variable density and Boussinesq results is small and seems to increase from N t = 2 to N t = 4, and then decreases until the end of the simulation.

The difference, observed in the phase where nonlinear effects are important, may be caused by the different redistribution of energy produced by the further nonlinearities of the variable density equation.

Since the primary sources of the anisotropy are the large scales, the confinement effects in the last steps of the simulation may play a role in the final decrease. In Figure 5.9 are reported the kinetic energy spectra of u 2 in 5.9a and u 3 in 5.9b. We observe how the difference between the two velocity components increases during the simulation for both variable density and Boussinesq approximations. The density influence on the infrared spectra is more evident on the vertical than on the horizontal component. The other interesting difference is observed at higher wavenumber, where the difference between the two approximations is higher in the vertical direction than the horizontal.

Probability density functions

In this section, we compute the probability density functions for the scalar θ and two components of its gradient ∂ 3 θ and ∂ 1 θ. The objective is to understand, in the physical space, how the fluctuations of θ are influenced by the variable density effects.

Probability density function of θ

The initial condition plotted in Figure 5.10a shows a symmetric distribution of θ around the mean value 0. The probability density functions stay symmetric for both simulations, but the values that θ can take decrease from N t = 0 to N t = 1.8, Figure 5.10b, which is implied by the narrowing of the pdf, which is a decrease of the variance of θ, already observed in section 5.4.2.

Until this point, the two simulations have similar results, with the VD, which already has a small asymmetry between positive and negative values of θ.

At N t = 3.7, while the Boussinesq result is still symmetric, the variable density has a clear asymmetry toward the positive value of θ, which becomes more evident, as the simulation progresses, until becoming striking at N t = 4.73. That is, during the variable density simulation, the heavy material mixes more slowly, so that the probability of finding positive fluctuations of θ of increasing values, grows during the simulations, an effect already evidenced by [START_REF] Livescu | Variable-density mixing in buoyancy-driven turbulence[END_REF].

Probability density function of ∂ 1 θ

As for θ, the initial condition of the gradient ∂ 1 θ is plotted in Figure 5.11a, and show symmetry of the values around the mean 0.

After the decrease that we observe until N t = 1.8, Figure 5.11c, we find that at N t = 3.8, the two simulations have still similar probability density functions.

The main difference is obtained towards the end of the simulation at N t = 4.73. In Figure 5.11d, we observe that the tails of the probability density function of the VD simulation are much wider than the Boussinesq. Confirming the observation of [START_REF] Rao | Nonlinear effects in buoyancy-driven variable-density turbulence[END_REF], that is, intense variable density effects may be the cause of the existence of sharp and intense gradients in the domain.

Initial condition variation

In this chapter, as already stated in section 5.2.2, the only difference in initial conditions is the initial ratio of the scalar variance and the kinetic energy. Since we want to keep the initial Reynolds number constant, this reduces to varying the scalar variance θθ . The characteristics of the different simulations are reported in table 5.1.

The modification of the initial scalar variance implies a variation of the initial Atwood number of the flow, defined by Eq. (5.25).

Initial condition variation

Non dimensional numbers

The evolution of the nondimensional numbers of the flow is plotted in Figure 5.12, for the 10 LES simulation.

The Reynolds numbers are shown in Figure 5.12a. We observe how, for both variable density and Boussinesq simulations, the strong Reynolds number dependence on the initial value of scalar variance. That is, the production of turbulence is connected to the scalar fluctuations, the stronger they are, the higher the Reynolds number is. On the same figure, we observe how the difference between VD and B simulation decreases when the initial scalar fluctuations are less intense. For instance, for the simulations Ul01 and Ul00, no noticeable difference seems to exist.

The Froude number in Figure 5.12b has a similar dependence on the initial conditions. The difference observed for Ul3 decreases for all the other cases becoming non-existent for the two cases Ul01 and Ul00. The notable difference with respect to the Reynolds number is the tendency for all the 10 simulations to converge to a value of the Froude number 2.

The Atwood number, as explained in the introduction of this section, decreases for all the simulations due to the scalar variance decrease. After the non-dimensional numbers, we investigate the influence of the initial conditions on the one-point statistics : integrated kinetic energy, integrated scalar variance, and vertical flux.

One point statistics

The kinetic energy evolution for the 10 LES simulation is plotted in Figure 5.13a. Here we have the same effect observed in the previous section, that is, the difference between variable density and Boussinesq simulation decreases with the decrease of the initial scalar variance. One of the main consequences of the reduction of variable density effects concerns the growth rate of the kinetic energy in the self-similar phase, which becomes the same for Ul01 and Ul00. That is, we expect no modification of the infrared part of the kinetic energy spectra. Similar behaviors are observed for the other two quantities in Figures 5.13b and 5.13c, where the differences that are evident for the case Ul3 are less and less so as the initial scalar variance decreases.

In particular, the disappearance of variable density effects, observed between Ul05 and Ul01, suggests that the non-Boussinesq limit for the USHT simulations is between these two initial conditions.

Two point statistics

The influence of the initial variance on the spectra is illustrated using as examples simulations Ul1 and Ul05. In Figure 5.14, we show the spectra for the four simulations at the same time instants during the evolution of the flow.

We observe how the effects of variable density, which act on both large and small scales, are more evident for the simulation Ul1, Figure 5.14a, where the scalar fluctuations at the beginning of the simulation are more intense, with respect to Ul05.

In the simulation Ul05, Figure 5.14b, the large scale effects are still visible, but they do not concern the energetic scales near the peak of the four spectra. Even if their influence is still present on the integrated quantity, as shown in Figure 5.13.

Conclusion

This chapter concerns the study of the variable density effects on the unstably stratified homogeneous turbulence (USHT). In particular, following the work of [START_REF] Soulard | Permanence of large eddies in variable-density homogeneous turbulence[END_REF] on the permanence of large eddies, we are interested in the possible modification of the the self-similar state growth rate β, which is related to the energy distribution at large scales. The investigation is carried out, confronting the results of variable density and Boussinesq direct numerical simulations. Moreover, we report 10 LES simulations with different initial conditions to study in what measure the variance of the initial scalar field θ influences the late time self-similar flow evolution.

The study of USHT configurations is accompanied by the numerical challenges associated with the growing spectral range, which requires adequate numerical discretization. On the one hand, one has to consider the rapid growth of the integral scale, which could cause the confinement of the flow and on the other, the development of the turbulence cascade, which involves more and more small scales.

Furthermore, the variable density approximation requires an iterative method to solve the Poisson equation, which, in this case, is not straightforward to solve. In this work, we propose the GMRES algorithm, which has been used in both direct numerical simulation and large eddy simulations. The latter has been introduced to allow the parametric study on initial conditions, that with the DNS alone, would not have been possible.

From the results of the numerical simulations, we find that in the variable density approximation, the permanence of the large eddies is not verified for initial conditions having an infrared spectrum with ∼ k 4 . In particular, the presence of a scalar field, with sufficiently high variance, causes the modification of the large scales of the flow, reflected by the change of the infrared part of the kinetic energy spectra, which was already reported by [START_REF] Soulard | Permanence of large eddies in variable-density homogeneous turbulence[END_REF]. In unstably stratified homogeneous turbulence, the infrared exponent has a direct influence on the growth rates β of the self-similar state. That is why we observe differences between the β found in Boussinesq and variable density results. These are evident in the time evolution of the integrated quantities : kinetic energy, scalar variance, and vertical flux.

From the analysis of the two-point statistics, in addition to the large scale differences, we observe that the two approximations lead to different nonlinear cascade processes, evidenced by the differences in the inertial and dissipative parts of the spectra. Moreover, the anisotropy of the flow seems to be enhanced in the variable density case, even if the difference with the Boussinesq result is small. Furthermore, we compute the Probability Density Function (PDF) of the scalar θ. Thanks to the PDF, we show that, during the simulation, in the variable density case, the scalar fluctuations tend to create a longer tail towards large positive values, while in the Boussinesq case, the PDF stays symmetric. That is, in the variable density case, after the initial transient, it is more probable to find heavy fluid particles than light ones. While in the Boussinesq case, the probability is the same during the computation. This result seems to confirm previous observations of [START_REF] Livescu | Buoyancy-driven variable-density turbulence[END_REF].

Likewise, the study of the density gradients shows that during the flow evolution, the variable density PDFs have more extended tails with respect to the Boussinesq's, which indicates an increased probability of sharper interfaces and the possible blowup of the density gradient proposed by [START_REF] Rao | Nonlinear effects in buoyancy-driven variable-density turbulence[END_REF].

Finally, we use the LES simulations to perform a parametric analysis of initial conditions. We chose to vary the initial scalar variance and to observe its effect on spectra and self-similar growth rates. As we expect for the Boussinesq simulation, no dependence on initial condition is observed, while in case of variable density, we find that the decrease of the scalar variance decreases the influence of the scalar field in the dynamics of the flows. In particular we find that the non-Boussinesq behaviour has its onset for a value of θθ K between 0.01 and 0.05. The results presented in this chapter are only the beginning of the analysis of the variable density unstably stratified turbulence. Future studies may go deeper into the investigation of its small scales properties. Or one could push further the studies on the initial conditions dependencies.

Chapitre 6 In chapter 4, we have discussed how the weakly coupled plasma produced in inertial confinement fusion compressions experiences a tremendous growth of the dynamic viscosity µ, as the temperature T increases. Considering the small dimensions of a hot spot in ICF devices, ∼ 10 -100[µm], and the temperatures achieved, above 10[keV ], hydrodynamic instabilities can be partially damped [Haines et al. 2014b] and small scale velocity structures can be significantly dissipated [Weber et al. 2014a]. We have investigated the sudden viscous dissipation effect proposed by Davidovits and Fisch (2016a), which by rapidly converting the kinetic energy of turbulent motions into internal energy, could, in principle, enhance the hot spot temperature, helping the fusion reactions to take place. The different regimes and scaling laws characteristics of compressed turbulent plasmas have been explored theoretically and with spectral EDQNM model, also evidencing an important sensitivity to initial conditions during the sudden viscous dissipation phase due to the distribution of energy fluctuations between scales.

Sudden diffusion of plasma turbulent mixing layers under compression

Besides, inhomogeneous simulations of turbulent kinetic energy layers under compression also investigated in chapter 4 have shown an increased transport of turbulence toward the center of the capsule during the implosion. These observations, although not accounting for plasma molecular diffusion, suggest that a new mechanism producing mixing of the heavy elements from the ablator into the DT fuel may take place in ICF. Can sudden viscous dissipation come together with sudden diffusion ? Mixing is indeed an important issue in ICF as having depleting cooling effects on the hot spot [Betti andHurricane 2016, Ma et al. 2017]. The asymmetry of the implosion [START_REF] Haines | Detailed high-resolution three-dimensional simulations of omega separated reactants inertial confinement fusion experiments[END_REF], the presence of defects on the capsule, like fill tube [START_REF] Hammel | High-mode Rayleigh-Taylor growth in nif ignition capsules[END_REF], MacPhee et al. 2017] or support tent [START_REF] Weber | Improving ICF implosion performance with alternative capsule supports[END_REF] are among the large scale mechanisms known as principally responsible for mixing in ICF targets. Other contributions may come from the fine scale perturbations amplified by hydrodynamic instability [START_REF] Hammel | High-mode Rayleigh-Taylor growth in nif ignition capsules[END_REF]. Plasma transport coefficients have been shown to play a role in the mixing zone dynamics (see [START_REF] Vold | Plasma viscosity with mass transport in spherical inertial confinement fusion implosion simulations[END_REF]). In their paper, Rinderknecht et al. (2014a) present an experiment from the OMEGA facility, having a more than expected ablator-fuel mix prior to the deceleration phase, a period in which hydrodynamic instability effects are expected to be negligible. Post-processing experimental data and using one-dimensional radiation-hydrodynamic calculations, they demonstrated that the main process driving mixing is the molecular diffusion. On the same experimental platform, [START_REF] Zylstra | Diffusion-dominated mixing in moderate convergence implosions[END_REF] provide evidence that implosion with a moderate temperature (< 6keV ) and moderate convergence are dominated by diffusive mix and that any turbulence-related effect is less significant.

Still, the question of how and when nonuniformities and strong temporal variations of plasma transport coefficients due to temperature and mass fraction mixture [START_REF] Ticknor | Transport properties of an asymmetric mixture in the dense plasma regime[END_REF]] act on the mixing zone dynamics is pending.

Here we study the idealized case of the DT/CH mixing layer in spherical compressions representative of ICF. Here the plasma transport coefficients are taken into account using the Pseudo-Ion-in-Jellium (PIJ) model, proposed by [START_REF] Arnault | Modeling viscosity and diffusion of plasma for pure elements and multicomponent mixtures from weakly to strongly coupled regimes[END_REF], and further validated in [START_REF] Ticknor | Transport properties of an asymmetric mixture in the dense plasma regime[END_REF]. These simulations will be compared to constant viscosity and diffusivity simulations to identify the importance of the transport coefficients on the evolution of the mixing zones.

Description of the pseudo-ions in jellium (PIJ) model 123

This chapter is organized as follows : At first, we present the pseudo-ions in jellium model and some results concerning the role of the coupling parameter in plasma mixtures. Then, after recalling the equations derived in chapter 2 for the spherical turbulent layer under compression, we detail the configurations for direct numerical simulations (DNS) of DT/CH mixing layer. We then identify the different regimes corresponding to the dominance of turbulent or molecular diffusion on the mixing from a parametric study on initial conditions.

Description of the pseudo-ions in jellium (PIJ) model

In this section, we recall the principle of the pseudo-ions in jellium model, whose details can be found in [START_REF] Arnault | Modeling viscosity and diffusion of plasma for pure elements and multicomponent mixtures from weakly to strongly coupled regimes[END_REF]. The most precise estimates, to compute transport coefficients in a plasma, come from the quantum molecular dynamics simulations. This method is very accurate, although computationally demanding and not of practical use in hydrodynamic simulations, which need the values of viscosity and diffusivity at every mesh point and for every time steps.

We consider a plasma as a gas of ions, atoms which, due to extreme heating, lose part of their electrons, and the free electrons. The ions are composed of the atom nucleus plus, if the elements are not fully ionized, bound electrons.

In the pseudo-ions in jellium model, the jellium is a background of uniform electric density representing the free electrons, in which we place the pseudo ions, i.e. a first-order approximation of the real ions accounting for the approximation on the electrons density assumed uniform. One of the most significant quantities in plasma physics, and especially for the PIJ model, is the coupling parameter Γ, the ratio of the Coulomb electrostatic potential energy of nearest neighbors, and the thermal energy. This parameter, in general, can vary from values less than 1 to more than 100 [START_REF] Piel | Plasma physics : an introduction to laboratory, space, and fusion plasmas[END_REF]], identifying plasma regime with different properties.

When Γ 1, the kinetic energy of the single particles of the plasma exceeds the potential energy of their interaction, this regime is called weakly-coupled plasma, and it is characterized by high temperature and low density. On the contrary, when Γ 1, plasma is in the strongly-coupled regime, which corresponds, usually, to colder and denser plasma with strong interaction between particles.

An example of the different plasma regimes that are present in nature or engineering application is given by [START_REF] Piel | Plasma physics : an introduction to laboratory, space, and fusion plasmas[END_REF] and reported in fig. 6.1.

The coupling parameter is defined as

Γ = Z * e 2 ak B T , (6.1)
where e is the electron charge, Z * is the ionization of the pseudo-ion, or pseudo- ionization, T is the temperature, k B is the Boltzmann constant and a is the Wigner-Seitz radius, the radius of the sphere whose volume is equal to the average volume occupied by one atom. The pseudo ionization Z * is computed using the scaling law from the Thomas-Fermi approximation, a description of the electronic structure of atoms using only information from electronic density function. Once the value of the coupling parameter Γ is determined, the model uses simple kinematic formulas with collision frequencies and scaling laws to compute viscosity and diffusivity in pure elements as well as in plasma bi-component mixtures.

One of the advantages of this model is that it can predict viscosity and diffusivity for different plasma regimes spanning from the high-temperature low-density weaklycoupled regime to the low-temperature high-density strongly-coupled regime. Figure 6.2 gives an example of the capability of the PIJ model in a plasma mixture of hydrogen and silver, validated against results obtained with a more accurate method [START_REF] Ticknor | Transport properties of an asymmetric mixture in the dense plasma regime[END_REF].

Two examples of the coupling parameter values in plasma mixture that are of interest for inertial confinement fusion are given in Fig. 6.3. We observe that the values of the coupling parameter are higher in the DT/gold mixture than in the DT/CH mixture, nevertheless for both cases, Γ decreases with temperature and the mass fraction of the heavier element. These demonstrate that plasma mixtures can be weakly or strongly coupled depending on temperature or composition, thus requiring the use of models, like the particle-ion in jellium, for case in which thermodynamic conditions vary significantly during the simulations.

How the PIJ results depend on mass fraction, density and temperature

We have observed how the PIJ model is able to compute the viscosity and diffusivity of plasma mixtures in different thermodynamic conditions. In this section, we show how the results of the model depend on the three quantities that the PIJ routine demands as inputs : mass fraction, density, and temperature. The mixture used as an example is the DT/CH. The results are plotted in Fig. 6.4.

In Fig. 6.4a viscosity and diffusivity show an inverse dependence on the CH mass fraction Y CH , the first decreases while the second increases when Y CH grows. On the contrary, in Fig. 6.4b, we observe that the two transport coefficients have the same dependence on the density : they decrease when the density increases. Finally, Fig. 6.4c shows the temperature dependence of viscosity and diffusivity, in this case, for the pure DT. We observe how the curves, for sufficiently high temperatures, follow the kinematic scaling ∼ T 5/2 [Braginskii 1995].

Theoretical framework

In this section, we recall the system of equations derived in chapter 2, that describe the evolution of a spherical mixing zone under compression. The flow is decomposed into a base component and a perturbation one. On the one hand, the base flow accounts for the radial velocity and the deuterium-tritium thermodynamics quantities during the adiabatic compression of an idealized ICF target. It can be derived analytically from conservation laws. On the other hand, the fluctuations account for the turbulence and the fuel/ablator mixing zone and are computed from direct numerical simulations.

Equations for DT/CH mixtures

We specialize the equation derived in section 2.3 for the DT fuel/CH ablator mixture. We recall that, in this study, radiative effects and combustion terms due to fusion reactions are not taken into account. In this specific case the mixture law (2.50), becomes for deuterium-tritium (DT) fuel and ablator (CH) :

1 ρ = 1 -Y ρ DT + Y ρ CH , ( 6.2)
where ρ DT and ρ CH are respectively the microscopic densities of the light and heavy materials. They are defined as

ρ DT = nM DT 1 + Z DT and ρ CH = nM CH 1 + Z CH (6.3)
The particularity of plasma is the strong dependence of the kinematic viscosity µ and the molecular diffusion D coefficients on the temperature T , density ρ, and mass fraction Y of CH.

At this stage, we recall that to solve the problem, it is convenient to decompose the flow quantities into a base flow (noted with suffix B ) and fluctuations (identified by small letters). For instance velocity components are written as

U i = U B i + u i .

Base flow

The objective pursued by selecting an arbitrary base flow is twofold. First, we wish to obtain simplified equations for the perturbations when expressed in a noninertial frame (see section 2.1). Also, the base thermodynamics conditions are selected to represent the hot spot characteristics in ICF capsules. We start by recalling the choice of the radial base velocity field, U B i , accounting for the main compression, (6.4) introducing the uniform compression time rate S(t), assumed positive for a compression. Any non-distorted length scale evolves proportionality to Λ during the compression. In particular, the radius R corresponding to the fuel/ablator interface, if unperturbed by instabilities, would decreases as R(t) = R 0 Λ(t). Thereafter, initial values at t = 0 are identified with the suffix 0 .

U B i (x, t) = -S(t)x i ,
While the expressions of the other base flow quantities are :

ρ B (t) = ρ 0 Λ(t) -3 , (6.5a) n B (t) = n 0 Λ(t) -3 , (6.5b) Y B (t) = Y 0 = 0. (6.5c)
Where the base density ρ B is taken uniform and the base mass fraction is set to one, Y B = 0, corresponding to pure DT fuel. The base temperature T B and pressure P B are derived assuming an adiabatic compression in which viscous and diffusion effect have no impact on the base flow. They are :

T B (r, t) = T 0 Λ(t) -3(γ-1) 1 - 1 h 2 0 r 2 Λ(t) 2 .
(6.6)

P B (r, t) = P 0 Λ(t) -3γ 1 - 1 h 2 0 r 2 Λ(t) 2 .
(6.7)

The length scale h 0 > 0 expresses the temperature gradient at the beginning of the compression.

The compression parameter Λ(t) expression is (6.8) with the characteristics frequencies S 0 = S(0), the initial compression rate, and

Λ(t) = 1 -2S 0 t + (S 2 0 + Ω 2 0 )t 2 ,
Ω 0 = 2P 0 ρ 0 h 2 0 1/2
expressing pressure gradient effects. The position of the fuel/ablator interface, R(t) = R 0 Λ(t), is principally driven by the initial impulsion S 0 at the beginning of the compression, and then decelerated by pressure gradient effects, Ω 0 , leading to a minimum convergence radius (see Figure 6.5).

The adiabatic compressions studied here differ qualitatively from realistic implosions of ICF capsules, which are driven by shocks. However, it is possible to choose parameters such that the duration and convergence ratio are representative of ICF implosions. Also, we can impose at the minimum convergence ratio the thermodynamics conditions relevant to hot spots in ICF capsules (see Atzeni and Meyer-ter Vehn (2004a)). These conditions are gathered in Tab. 6.1 and in figure 6.5.

We can remark that the initial parameters at t = 0 correspond to compressed and heated plasma state where DT and CH are already fully ionized with Z DT = 1 and Z CH = 3.5. The inner radius diameter is R 0 = 300 µm, the duration of the compression 1 ns and a convergence ratio of 20 is achieved. The base temperature at the center of the capsule varies from 75 eV to 30 keV, with the base pressure reaching 10 3 Gbar and fuel density 40 gcm 3 . In figure 6.5 is plotted the temperature and pressure profiles at different times, revealing the gradients responsible for the deceleration of the capsule radius. It also generates Rayleigh-Taylor instability at the fuel/ablator interface. However, due to the fact that Ω 0 = 50ns -1 S 0 , its effect remains very weak during the implosion.

Perturbation equations

The equations describing the dynamics of the perturbations around the mean flow are derived in section 2.3.2.2. We recall that we work in a non-inertial frame with the new time t and position variable x corresponding to the reference frame deforming with the base radial velocity :

t = t 0 Λ -2 (s)ds, x = x Λ(t)
. (6.9)

Moreover, we recall the rescaling that we use in order to eliminate inhomogeneous forcing terms due to base velocity in the velocity equation.

ũ(x, t) = u(x, t)Λ(t), π(x, t) = π(x, t)Λ 2 (t), θ(x, t) = θ(x, t). (6.10)

The equations for the perturbation in the non-inertial frame are : (6.11a) (6.11b)

∂ t ũi + ũj ∂ j ũi = -∂ i π -π∂ i θ -Λ 2 Π B ∂ i θ + ... ∂ j [ν (∂ j ũi + ∂ i ũj )] -ν (∂ j ũi + ∂ i ũj ) ∂ j θ,
∂ t θ + ũj ∂ j θ = ∂ j D∂ j θ ,
∂ j ũj = -∂ j D∂ j θ . (6.11c)
In Eq. (6.11a) for the velocity perturbation, one inhomogeneous term remains which is proportional to the base reduced pressure, Π B , and accounts for the buoyancy production. The quantity, π, expressed in the new coordinate system not only accounts for the reduced pressure fluctuations, but also for the diagonal components of the viscous stress tensor. We emphasize again that for the perturbation, the variable density condition (6.11c) holds. Therefore, compressibility is accounted only for the base flow in this work, while it is neglected for the perturbations.

Numerical methods

The perturbation equations in the non-inertial reference frame are solved using the pseudo-spectral method for variable density turbulence with nonuniform transport coefficients described in chapter 3. In particular, we recall that in the variable density approximation, the non-locality in spectral space requires an iterative method to invert the linear system. We use the generalized minimal residual (GMRES) algorithm described in section 3.2.3.1, which ensures convergence even if density fluctuations are important contrary to fixed point methods.

The strong dependence of the dynamic viscosity µ and the molecular diffusion D coefficients on the temperature T , density ρ, and mass fraction Y of CH are taken into account using the particle-ion-in-jellium (PIJ) model, described in section 6.1. Viscous and diffusive contributions are taken into account implicitly, also using a GMRES algorithm.

The typical cost of a simulation is roughly 200000 CPU hours dispatched over 2048 cores.

Initial conditions

To define more precisely the initial conditions in the simulations, we introduce the volume average of a quantity * as * . It is also convenient to use * as the tangential average of the same quantity (thus depending on r only). The fluctuations around these radial averaged profiles are noted as * . At this stage quantities are expressed by a triple decomposition with for instance the full velocity given by

U i = U B i + u i = U B i + u i + u i .
The initial conditions are generated using the methods described in section 3.4.2. The initial density profile is radial, such that θ = θ. It forms a diffuse spherical interface between fuel and ablator of width L 0 = 5µm, centered on radius R 0 = 300µm. Therefore, θ is zero at the center of the capsule, while in the ablator region of pure CH, θ = 0.16. This latter value is obtained from Eq. ( 6.3) as the density number of particles is constant, n = n 0 , to ensure the variable density approximation. The Atwood number between DT and CH is then given by At = ρ CH -ρ DT ρ CH +ρ DT = 0.08. Along with the small value of the mean base pressure gradient, this small Atwood number value explains why Rayleigh-Taylor instability remains limited.

Also, zero-mean random velocity perturbations with u i = 0 are added around the interface, using the filtering procedure described in section 3.4.2.1 and a classical von Karman spectra (3.43) with integral scale, 0 , and rms fluctuations, u 0 . Here, the spectra are of Batchelor type, i.e., the distribution of energy at small wavenumber k scales as k 4 .

Along with the initial size of the mixing layer L 0 , the simulations are characterized by an initial Reynolds number, Re 0 = u 0 0 ν , here defined with the DT viscosity at the center of the capsule (see Tab. 6.2). The values range from Re 0 = 0, for simulations without turbulence, to Re 0 = 217, reaching the limits of our available computational resources. In addition, by introducing the compression parameter, Cp 0 = u 0 0 S 0 , we observe that the compression is relatively rapid compared to turbulence in the simulations. This choice is motivated to avoid the contamination of the whole domain by mixing before the bang time.

At this stage, it is important to assess the validity of the variable density approximation from the initial conditions. For S1 simulations, which have the largest rms velocity u 0 = 1.73kms -1 , we find a Mach number value of M = u 0 /c = 0.02, using the sound speed provided by c = (γP 0 /ρ 0 ) 1/2 = 79kms -1 . This small value ensures the validity of the variable density approximation during the whole simulation. The advantage of using the variable density approximation in our code has to be stressed. Indeed, a fully compressible code would have been very demanding and would have failed to resolve the small scales of turbulence due to the shock-capturing schemes [START_REF] Thornber | Numerical dissipation of upwind schemes in low mach flow[END_REF]. By contrast, the initial base velocity at interface radius R 0 is 300kms -1 , which gives a Mach number of M = 3.8. This shows that energy contained in the perturbation is small compared to the base component, justifying our choice to neglect the feedback of the perturbations on the base flow.

Resolution

A total of 9 simulations are presented in this study, varying the initial Reynolds numbers and using either a plasma model for transport coefficients (type a) or constant ones (type b) to evidence its effects.

The simulations start from an initial computational cubic domain of 1200µm size. This choice results from a compromise to encompass the fuel/ablator interface to avoid confinement effects, and also to guarantee that the finest turbulent structures are correctly resolved.

Therefore, the mesh size depends on the initial Reynolds number and the simulation type a or b. The most demanding simulations, S1 b and S2 b , uses a 1024 3 and 512 3 mesh size respectively and correspond to a constant viscosity and molecular diffusion configuration. The small scales of turbulence are not fully resolved in these simulations, which enter the category of Implicit Large Eddy Simulations (ILES). By contrast, all other simulations can be considered as DNS . In particular, simulations with plasma coefficients (type a) are less demanding as the diffusion and viscosity growths rapidly smooth the small scales of turbulence.

As an illustration, we provide in Figure 6.6 the 3D representations of the density θ field at different instants for S1 a,b simulations. This evidences the strong imprint of transport coefficients, which dissipate turbulence during the implosion (S1 a ). We now quantify this phenomenon in the following section. 134 Chapitre 6. Sudden diffusion

S1 a S1 b 3 Q Q Q Q Q Q Q s t= 0

One-dimensional evaluation of the transport coefficients effects on the implosion 135

One-dimensional evaluation of the transport coefficients effects on the implosion

In section 6.2.5, we have illustrated the implosion characteristics that we use in this chapter and the thermodynamic properties of the base flow at the beginning and during the compression. Nonetheless, in section 6.1.1, we have shown how the transport coefficients have a strong dependence on temperature density and composition.

Hence, the question that arises is how much their effects depend on the characteristics of the implosion or the values of the thermodynamic quantities. A parametric study using DNS simulations is not feasible due to the high computational cost. An alternative is to use a one-dimensional model of the perturbation equations (6.11a)-(6.11c), in spherical coordinates such as

∂ t ũr + ũr ∂ r ũr = -∂ r π -π∂ r θ -Λ 2 Π B ∂ r θ + 1 r 2 ∂ r 2νr 2 ∂ r ũr -2ν∂ r ũr ∂ r θ, (6.12a) ∂ t θ + ũr ∂ r θ = 1 r 2 ∂ r Dr 2 ∂ r θ , (6.12b) ∂ r ũr = - 1 r 2 ∂ r Dr 2 ∂ r θ . (6.12c)
together with a one-dimensional initial condition for θ, without velocity perturbations, using the radial profile discussed in section 3.4.

The parameter that we use to investigate to what extent the implosion conditions impact the molecular diffusion in the mixing layer is the mixing layer width L, whose definition is given in section 6.4.2, and whose value is computed at the end of the implosions, L(Λ min ) = L E .

We perform two parametric studies obtaining the maps, shown in Figure 6.7. In one case, we set the thermodynamic conditions to the ones chosen in section 6.2.2 and vary the initial compression rate and the convergence ratio. In the other, the compression characteristics are constant, and we change the initial temperature and density of the DT at the center of the capsule.

On Figure 6.7a , we observe that the faster the compression, the lower the final value of the mixing zone size, since the diffusion has less time to act. This tendency is observed for every convergence ratio. For the same reason, as the minimum of the compression parameter decreases, i.e. the capsule is compressed to a smaller radius, the values of L E increase.

On the map in Figure 6.7b, the value of the final mixing zone width grows with the increase of the initial temperature, and it has the inverse dependence of the initial density. The first effect is explained by the fact that the higher is the temperature, the greater is the diffusion. The dependence on the density is similar to that observed in Figure 6.4b.

Results

In this section, we extract the different turbulent quantities from simulations to observe the effects of plasma transport coefficients on the dynamics of turbulent layers.

Global statistics

Figure 6.8 shows the variations of the turbulent kinetic energy, K = u i u i /2, and of the variance θ θ for high Reynolds number simulations S1 a,b . This comparison shows the role of plasma transport coefficients in the mixing evolution.

For both simulations, the dynamics of the flow is first driven by compression effects, leading to an increase of kinetic energy. This phase, identified as the rapid compression in chapter 4, leads to a Λ -2 growth of K . Simultaneously, the density fluctuations initially at rest and expressed by θ θ rise rapidly due to the advection term. After this initial phase, simulations S1 a and S1 b start to differ. On the one hand, for the S1 a simulation, the turbulent kinetic energy growth is progressively slowed down due to the viscosity increase. Then, the sudden viscous dissipation effect occurs [Davidovits and Fisch 2016a], since the turbulent production mainly due to compression cannot balance viscous terms. Note that a scaling K ∼ Λ 11/2 is expected for the self-similar decay of a Batchelor spectra with constant implosion rate (see chapter 4).

On the other hand, the S1 b simulation goes from rapid compression to a nearly cascade regime dominated by turbulent nonlinear transfers. This phase lasts until the end of the compression with scaling close to Λ -4/7 as expected for a Batchelor spectrum [START_REF] Viciconte | Self-similar regimes of turbulence in weakly coupled plasmas under compression[END_REF]. A sharp decrease is observed at the end of all the simulations, which can be attributed to the deceleration of the compressed matter.

The smaller values of θ θ in S1 a simulations compared to S1 b are the first indication of enhanced mixing due to plasma transport coefficients.

Mixing layer width

We now compare the time evolutions of the size of the mixing layers in Figure 6.9 for high Reynolds simulations S1 a,b , with or without plasma effects, and also for S5 to stress the role of plasma transport coefficients when turbulence is absent.

The mixing layer sizes L can be evaluated in simulations using the following integral function of the radial averaged mass fraction Y (r, t) [Andrews andSpalding 1990, Gréa 2013].

L(t) = 6 +∞ 0 Y (t, r)(1 -Y (t, r))dr (6.13)
During the implosion, the mixing layers in S1 a,b simulations first experience growth due to the rapid compression regime identified in section 6.4.1. This growth is slowed due to the dissipation of turbulent energy, either by the cascade process in S1 b or directly by transport coefficient in S1 a . This process is very similar to the growth of a mixing layer in the plane Richtmyer-Meshkov instability as the turbulence scales are small compared to the radius of the interface. Despite the active role of plasma viscosity in destroying turbulence and relaminarizing the flow, S1 a simulation has only a slightly lower growth rate as S1 b simulation until the time (III) t = 0.93ns. This can be explained as the dynamics of a layer is principally driven by the large energetic scales, which are not dissipated by viscous effects acting at smaller scales. The molecular diffusion increase can also compensate for the loss of turbulent diffusion in the S1 a simulation. Eventually, this process completely dominates the turbulence at the end of the simulation leading to a sudden diffusion of the layer also present in S5 simulation at Re = 0.

It is during this last sudden diffusion phase, that S1 a,b simulations become significantly different, with the mixing layer width in S1 a increasing rapidly much above the values obtained in S1 b simulation.

The effect of initial conditions, mainly with varied Reynolds number detailed in Table 6.2, is now investigated in Figure 6.10. For the range of Re investigated, the final sizes of mixing layers are weakly sensitive to the initial level of turbulence in simulations accounting for plasma transport coefficients (Figure 6.10a). The difference between simulations with or without varying plasma transport coefficients mainly occurs during the sudden diffusion phase, appearing earlier in low Reynolds number simulations. Besides, the relaminarization process due to viscosity increase has a low impact on the mixing layer dynamics. The simulations at higher Reynolds number would allow us to determine a critical Re where the sudden diffusion no longer occurs. Still, these simulations are presently out-of-reach, too computationally demanding.

Results

139

Λ 10 -1 10 0 L R 10 -2 10 -1 Re=217 Re=153 Re=108 Re=32 Re=0 (a) Λ 10 -1 10 0 L R 10 -2 10 -1 Re=217 Re=153 Re=108 Re=32 (b)

Radial profiles

In this section, we explore the mean radial structures of mixing zones and compare their evolutions when the relaminarization process occurs due to plasma transport coefficients.

We show in Figure 6.11 the mean mass fraction of ablator, Y , extracted from S1 a,b . The selected times correspond to the phases identified in Figure 6.9 in section 6.4.2. Whereas at early times there are small differences between simulations accounting or not for varying transport coefficients, at late times, the sudden diffusion phase in S1 a marks a strongly different behavior. This phenomenon is very effective at contaminating the capsule center with heavy ablator elements.

In parallel, we present the mean profiles of turbulent kinetic energy u i u i /2 and variance θ θ in Figure 6.12 also for S1 a,b simulations. These profiles are classically maximum at the center of the mixing layers. The relaminarization process due to viscous effects is marked by the dissipation of turbulent variances and occurs very soon, as shown by the differences between the simulations. Interestingly, relaminarization is close to symmetric between the fuel and ablator sides on θ θ radial profile, but, this is not the case on kinetic energy profiles where dissipation seems to occur firstly on the DT side. Finally, only molecular diffusion is responsible for the sudden diffusion of the mixing layer as turbulent quantities are completely quenched at late times in S1 a simulation.

Mixing parameter

We further shed light on mixing in S1 a,b simulations using the molecular mixing parameter Ξ(r, t) defined from the mass fraction of the ablator as (see [START_REF] Youngs | Three-dimensional numerical simulation of turbulent mixing by Rayleigh-Taylor instability[END_REF]

) : Ξ(r, t) = Y (1 -Y ) Y (1 -Y ) = 1 - Y Y Y (1 -Y ) . (6.14)
The mixing parameter value thus reaches 1 when mixing is completed. This quantity is important in ICF for expressing the ratio between the amount of fusion reactions in the mixing zone and the amount obtained without mass fraction fluctuations [START_REF] Youngs | Three-dimensional numerical simulation of turbulent mixing by Rayleigh-Taylor instability[END_REF]]. Therefore, molecular mixing reflects to what extent the local mass fraction departs from the mean but does not give the relative amount of the two species in the mix [START_REF] Danckwerts | The definition and measurement of some characteristics of mixtures[END_REF]].

We compare the temporal evolution of Ξ between S1 a and S1 b simulations in Figure 6.13. Here, the mixing layer width is shown by the specific radii r 01 , r 99 where Y = 0.01 reaches 0.01 and 0.99 respectively. From the time (I) to (II) corresponding to the rapid compression phase, the mixing parameter values are similar, since the process is mainly driven by turbulence.

From the time (II) to (III), although the sizes of the mixing layer are comparable between both simulations, the values of the molecular mixing parameter in S1 a become gradually larger than in S1 b , owing to the varying transport coefficients of the plasma. At time (III) and during the sudden diffusion phase, the values of Ξ in S1 a become very close to 1, indicating that the mixing is almost complete. By contrast, in S1 b simulation, the mixing parameter also grows but at a smaller pace, as the mixing is driven by the turbulent cascade.

Transport coefficient evolutions

In this section, we present the temporal and spatial evolutions of plasma transport coefficients evaluated from the PIJ model [START_REF] Arnault | Modeling viscosity and diffusion of plasma for pure elements and multicomponent mixtures from weakly to strongly coupled regimes[END_REF][START_REF] Ticknor | Transport properties of an asymmetric mixture in the dense plasma regime[END_REF][START_REF] White | Correlation and transport properties for mixtures at constant pressure and temperature[END_REF]]. This model is based on a formulation in terms of collision frequencies with scaling laws to cover the different regimes from weakly to strongly-coupled plasmas and to address binary mixtures with chemical elements of arbitrary atomic number Z. In that respect, it extends the theory limited to kinetic regime of [START_REF] Molvig | Classical transport equations for burning gas-metal plasmas[END_REF], [START_REF] Kagan | Thermo-diffusion in inertially confined plasmas[END_REF], and better represent the early time evolution of the simulations that exhibit moderate coupling (Γ = 0.3).

We show in Figure 6.14 the mean radial profiles of kinematic viscosity ν(r, t) and diffusion D(r, t) at different instants extracted from the S1 a simulation. The temporal evolutions of viscosity and diffusion are also shown for specific radii r 01 and r 99 . Several classical features of transport coefficients in plasma are thus recovered. Viscosity and molecular diffusion follow the kinetic scaling laws Braginskii (1995) and thus experience tremendous growth, up to two orders of magnitude, due to At the end of the simulation (t 1 ns), r 01 reaches zero as mixing is entirely spread inside the capsule. The mean viscosity and diffusion are then plotted at the center of the capsule, r = 0, and therefore decrease since Y becomes greater than 0.01 at r = 0 temperature increase. The mean radial profiles of ν exhibit a strong dependence on the mixture composition expressed by the mass fraction Y . The presence of heavy CH ions in the pure DT plasma indeed increases the effective plasma coupling parameter, leading to 1 -2 orders of magnitude lower viscosity on the ablator side. Here, the plasma coupling parameter indeed evolves from 3 × 10 -2 to 2 × 10 -3 in pure DT, and from 3 × 10 -1 to 10 -2 in pure CH. Conversely, the molecular diffusion is 2 -3 times higher on the ablator side. Indeed, the ion density number on the ablator side is lower than on the DT side, considering that the particle number distribution (ion and electron) n is constant in the mixing layer. However, the spatial variations of D are less spectacular than for viscosity. Consequently, the Schmidt number ν/D varies between 3 and 4 close to the DT side of the mixing layer, while it reaches values of around 0.05 on the CH side. The time evolution of the mean variances of viscosity ν ν and diffusion, D D , and their radial profiles, ν ν and D D , are plotted at different instants in Figure 6.15. These quantities reveal how fast viscosity and diffusion vary inside the mixing layer justifying the use of implicit iterative methods for numerical simulations. Transport coefficient variances increase during the compression until the decay of turbulence smooths the mass fraction fluctuations. Interestingly, while radial profiles of molecular diffusion variances remain maximum and well-centered around the mixing layer center, r/R = 1, viscosity variances shift toward the DT edge where even a small amount of CH drives huge variations of viscosity. This effect has also been observed on kinetic energy, and θ variance profiles in section 6.4.3.

Bi-dimensional Maps

In this section, we exhibit the instantaneous turbulent fields extracted from S1 a,b simulations to better understand how the relaminarization process and sudden diffusion effect induced by plasma transport coefficients operate.

We start by showing the two-dimensional contours of local kinetic energy K = Figure 6.16 clearly evidences the asymmetric relaminarization process due to the viscosity growth in S1 a simulation. The turbulent kinetic energy is firstly dissipated on the DT side of the mixing layer, following the spatial variations of viscosity, as detailed in section 6.4.5. Also, the fact that turbulent structures are larger in S1 a simulation compared to S1 b shows that dissipation acts at small scales before reaching larger ones during the final phase of the compression. On the contrary, the constant viscosity of S1 b simulation allows the development of small structures by classical nonlinear energy cascade.

We now compare the structure of the θ variance in Figure 6.17 at the center of the mixing zone, i.e. r = r 50 , with kinetic energy contours at the same position. Interestingly, the characteristic sizes of the scalar field θ θ are larger than for the kinetic energy contours, indicating that the dissipation is more effective for the variance of θ than for the turbulent kinetic energy, as already seen in Figure 6.8.

Spherical harmonics spectra

The two-dimensional maps offer only a qualitative appreciation of the transport coefficient effects on the turbulence within the mixing layer. To quantify the information given by the maps of section 6.4.6, we use the spectral analysis of fluctuating fields on spherical surfaces proposed by Lombardini et al. (2014b). The natural spectral basis to represent these fields are the spherical harmonics. For instance, a function f (r i , ψ, φ) defined on a sphere of radius r i can be decomposed using the real spherical harmonic basis as (6.15) where Y lm (ψ, φ) are the real spherical harmonics [Abramowitz andStegun 1965, Groemer 1996], and f lm are the expansion coefficients (see also Appendix C.2).

f (r i , ψ, φ) = ∞ l=0 l m=-l f lm Y lm (ψ, φ),
Within this basis, it can be shown [Lombardini et al. 2014b] that the angular power spectrum C l associated with the two-point correlation of the quantity f taken on the sphere of radius r i (and assumed statistically homogeneous and isotropic on the same sphere) can be computed from the expansion coefficient as

C l = 1 2l + 1 l m=-l f 2 lm . (6.16)
Considering high l spherical harmonics, corresponds to very small characteristic dimensions with respect to sphere curvature, there is a direct relationship between the

Results

147

angular power spectra C l and the local one-dimensional planar spectrum E(κ), where the "wavenumber" κ is defined as κ 2 = l(l + 1)/r 2 i . In the limit of l 1, κ l/R and it can be shown that lC l ∼ κ -α [Lombardini et al. 2014b].

We thus compute the angular power spectra, using the field interpolated on spheres with radii r 01 , r 50 , and r 99 . Because of the mixing, these radii vary during the compressions. Therefore, in order to compare the different spectra at different times and radii, we plot lC l as a function of the non-dimensional wavenumber q l , that is directly related to the spherical harmonics number l, by the relation q l = R r i l 2π . In Figure 6.18, we compare the angular power spectra of kinetic energy lC l,K and scalar field variance lC l,θ , between the simulations S1 a with varying plasma transport coefficients and S1 b with constant transport coefficients. At the time (I), the kinetic energy spectra of the two initial conditions are superimposed, while, as we explained in section 6.2.5, the initial conditions have no scalar fluctuations. At the time (II), as observed on the bi-dimensional maps of section 6.4.6, in the constant viscosity simulation, nonlinear phenomena produce a turbulent cascade. The inertial range, exhibiting a l -5/3 slope of the kinetic energy spectrum, extends up to the maximum resolved q l . On the contrary, the spectrum of the S1 a simulation shows less energetic scales, with no inertial range due to the increased value of viscosity. At small q l , which coincides with the most energetic spherical harmonics, the two spectra have a similar energy distribution. This implies that the mixing zone evolution, in both simulations, is still driven by large scale turbulent diffusion, confirming the results of sections 6.4.2 on the mixing zone width. The scalar spectra, lC l,θ , exhibit the same behavior, with the l -5/3 scaling recovered for the constant transport coefficient simulations, and the relaminarization effects in the S1 a case. When the simulations reach time (III), the S1 b kinetic energy spectrum maintains an inertial zone at intermediate wavenumber.

Still, the l -5/3 scaling does not extend to the q l,max , and we observe the beginning of a dissipative range. In contrast, for the S1 a case, the energy-containing harmonics, for both kinetic energy and scalar, are reduced to the small q l , suggesting that at this time, the dynamics of the mixing zone is dominated entirely by viscous and diffusive effects. At the time (IV), the S1 a spectra, for both quantities, show a very limited spherical harmonics range since the transport coefficients dissipate almost all the fluctuations. These results are consistent with the bi-dimensional maps of section 6.4.6, where at the time (IV), minimal fluctuations are visible. On the other hand, the constant coefficient spectra suggest that turbulence is still the primary driving phenomenon of the mixing zone evolution.

The temporal variation of the kinetic energy and scalar spectra of the S1 a simulation, in Figure 6.19, is a further indication of the relaminarization caused by the transport coefficients. This effect smooths velocity and density fluctuations equally, leaving only large scales fluctuations at the end of the implosion. q l q l q l q l q l q l q l q l t = 0 ns (I) q l q l q l q l q l q l r 99 r 50 r 01

Figure 6.19 -Spherical harmonics spectra computed at the beginning (I) and at the end (IV) of the simulation S1 a at three radial positions r 01 , r 50 , r 99 . Black solid lines correspond to kinetic energy spectra. Red dashed lines correspond to scalar spectra.

Conclusion

This chapter presents numerical simulations of turbulent plasma mixtures under compression using a fluid approach and the PIJ [START_REF] Arnault | Modeling viscosity and diffusion of plasma for pure elements and multicomponent mixtures from weakly to strongly coupled regimes[END_REF][START_REF] Ticknor | Transport properties of an asymmetric mixture in the dense plasma regime[END_REF][START_REF] White | Correlation and transport properties for mixtures at constant pressure and temperature[END_REF]] modeling of the transport coefficients, which vary with temperature, density, and composition. The parameters are chosen to be representative of adiabatic implosions of DT/CH layers, mimicking the characteristic times and hot spot thermodynamic conditions of an ICF capsule before the bang time. To achieve full resolution of low Mach number turbulent fluctuations, we use a Rogallo noninertial frame following the compression and the variable density approximation.

Simulations with or without varying plasma coefficients evidence the influence of plasma viscosity and molecular diffusion on the dynamics of mixing layers. This aspect is crucial in the context of ICF as the hot spot contamination by heavy materials has deleterious effects on the capsule yield. A complex relaminarization process has been observed, occurring first on the DT side where viscosity is higher, as shown by radial profiles and angular spectra. Although this phenomenon dissipates the small scales of turbulence and leads to a more homogeneous DT/CH mixing layer, it does not drastically reduce the dynamics of the mixing zone, mainly driven by larger scales. However, we show that during the late time evolution of the compression, the plasma molecular diffusion overcomes the turbulent one, leading to enhanced diffusion of the DT/CH layer anticipated in chapter 4. This sudden diffusion effect comes along with the sudden viscous dissipation of turbulence already observed by Davidovits and Fisch (2016a). This gives credit to the interpretation of mixing due to physical diffusion in recent experiments performed at the Omega facility by Rinderknecht et al. (2014a), [START_REF] Zylstra | Diffusion-dominated mixing in moderate convergence implosions[END_REF]. The objective of the works presented in this thesis is to study turbulent mixing in fluid with strong transport coefficients and density variation. We consider both canonical flows like the homogeneous isotropic turbulence and unstably stratified homogeneous turbulence and inhomogeneous configurations like spherical implosions.

The contributions presented in this thesis are, for the most part, based on the analysis of direct numerical simulations of the Navier-Stokes equations with the working hypothesis for every case considered.

In the first part of the thesis, we have presented -The derivation of model equations for spherical compression and USHT in the variable density approximation. -The spectral GMRES algorithm employed to solve the Poisson equation in the variable density simulations. -The development of a massively parallel spectral direct numerical simulation code based on the two-dimensional domain decomposition for both incompressible and variable density case. -The GMRES algorithm used to compute the viscous and diffusive contribution in turbulent plasma mixtures whose transport coefficients are computed using a plasma physics code interfaced with the DNS code. In the following chapters, we have used the codes developed to investigate three major topics :

-The sudden dissipation effect of turbulent kinetic energy in plasma under compression in the context of homogeneous isotropic turbulence with timevarying viscosity, for which we have developed a spectral model based on EDQNM closure.

-Then, still in a homogeneous approximation, we study the effect of variable density on the turbulence properties in an unstably stratified configuration. -Finally, we consider an inhomogeneous configuration of a spherical implosion, with transport coefficients and variable density effects.

Sudden dissipation effect : spectral modeling and influence of initial conditions

In the context of turbulent weakly-coupled plasma under compression, we have identified three different self-similar regimes resulting from the competition between turbulence production, non-linear energy transfer, and viscous dissipation, which are observed in our DNS and EDQNM simulations.

In the case where turbulence is sufficiently intense and compression sufficiently rapid, we observe that weakly-coupled plasma under compression experiences growth of kinetic energy. However, the viscous effects always prevail at the end of the compression leading to the sudden dissipation phenomenon.

Using the EDQNM model, we have explored the phase space of the nondimensional numbers, Re and Cp, initial values thanks to which we have identified the critical values of the compression parameter corresponding to the beginning of the viscous phase. This analysis has revealed a strong dependence of critical values on the initial conditions and evidenced the complex interplay between turbulence production, transfer, and dissipation, leading for some configurations to two successive growth/decay phases of kinetic energy instead of one during the compression.

We have identified three self-similar regimes during the isotropic compression of a weakly-coupled turbulent plasma. Using a theoretical analysis, we have derived the scaling laws which have been verified using DNS and EDQNM simulations.

The scaling demonstrates the dependence of flow dynamics on the initial distribution of energy at large scales, which becomes crucial during the viscous phase since it changes the decay exponents of kinetic energy drastically.

We further extended our study to the case of an inhomogeneous spherical turbulent layer under compression, using parameters representative of ICF capsules.

In this configuration, we were able to observe the different phases already identified in the homogeneous settings together with the enhanced enlargement of the layer during the viscous phase and high sensitivity to initial conditions.

Unstably stratified homogeneous turbulence

The other subject of the thesis concerns the study of the variable density effect in the unstably stratified homogeneous turbulence, a canonical flow that retains some of the features of the Rayleigh-Taylor mixing layer.

We consider two approximations, Boussinesq (B) and Variable Density (VD). The objective of the investigation is twofold : to understand in what measure the variance of the initial scalar field influences the late time self-similar flow evolutions, and the limitations of the Boussinesq approximation and the condition of the onset of non-Boussinesq effects.

The numerical solution of the USHT equation with the variable density approximation is considerably challenging in the cases considered due to the high-density contrasts that are already present at the beginning of the simulation. Together with the direct numerical simulation, we have introduced the implicit large-eddy simulations, which are less computationally demanding, thanks to which we can perform a parametric study of initial conditions.

We show that depending on the initial scalar field variance, in VD approximation, contrary to Boussinesq, the large scale of the flow can be modified. In particular, we confirm a recent observation of [START_REF] Soulard | Permanence of large eddies in variable-density homogeneous turbulence[END_REF] demonstrating how variable density effects modify the infrared slope of the kinetic energy spectrum if the intensity of the scalar fluctuations is sufficiently important.

In particular, we observe that, from an initial condition where the large scales of the flow have a dependence in k 4 , during the initial transient, the spectrum changes to k 2 , if the initial variance is sufficiently high. This large scale modification is extremely important because of the direct influence of the infrared exponent on the growth rate of the self-similar phase. Furthermore, using the probability density function, we demonstrate other differences between variable density and Boussinesq approximation. The pdfs of the scalar computed from Boussinesq results have a symmetric distribution around the mean value, while the VD cases show an asymmetrization of the probability density functions towards the positive values of θ.

Moreover, the investigation of the scalar gradients shows that in the variable density case, the tails of the probability density functions are much wider than the Boussinesq case implying the presence of sharper interfaces that can be observed qualitatively on the scalar contours too. Finally, performing the parametric study on the initial conditions, we find that the non-Boussinesq effects become important for a scalar variance to kinetic energy ratio between 0.01 and 0.05.

Sudden diffusion effect in spherical mixing zones of plasma under compression

The last chapter of the thesis presents numerical simulations of turbulent plasma mixtures under compression using a fluid approach and the uses of the PIJ model to compute the transport coefficients, which vary with temperature, density, and composition.

The theoretical framework derived allows for the study of variable-density flow with extreme variations of viscosity and diffusivity, both taken into account thanks to the GMRES iterative scheme.

The parameters of inhomogeneous spherical configuration, together with the thermodynamic variable of the hot spot, are chosen to be representative of adiabatic implosions of DT/CH layers, mimicking the characteristic of an ICF capsule before the bang time. To evidence the effects of plasma transport coefficients, we present two sets of simulations : In the first, viscosity and diffusivity are computed using the PIJ model ; in the second, both transport coefficients are maintained constant. In the two cases, different initial Reynolds numbers are considered.

During the compression, we can identify different phases based on the mixing zone width, which evolution evidence the influence of plasma viscosity and molecular diffusion on the dynamics of mixing layers.

In the first rapid compression phase, both simulations present similar results, and no effects of transport coefficients are observed. In the following period, turbulence is the primary factor influencing the evolution of the mixing zone, and the growth of viscosity comes into play. During this phase, we observe a complex relaminarization process, occurring first on the side closer to the pure DT , where viscosity is higher than the part closer to the ablator, and move outwards towards the CH as the temperature in the capsule increases. This non-homogenous behavior has been demonstrated with both radial profile and angular power spectra. Although this phenomenon dissipates the small scales of turbulence and leads to a more homogeneous DT/CH mixing layer, it does not drastically reduce the dynamics of the mixing zone, mainly driven by larger scales.

In the last phase of the compression, however, we show that plasma molecular diffusion overcomes the turbulent one, leading to enhanced diffusion of the DT/CH layer anticipated in chapter 4. This aspect is crucial in the context of ICF as the hot spot contamination by heavy materials has deleterious effects on the capsule yield. This sudden diffusion effect comes along with the sudden viscous dissipation of turbulence already observed in chapter 4 and can explain some recent experimental results on the Omega laser facility.

Annexe B

EDQNM models B.1 EDQNM Model for isotropic turbulence

In this first section we derive a spectral equation based on an EDQNM closure for a case of homogeneous isotropic turbulence.

We start from the Navier-Stokes equations for an incompressible flow

∂ t u i + ∂ j u i u j = -∂ i p + ν∂ jj u i , (B.1)
together with the continuity equation

∂ i u i = 0. (B.2)
We apply the Fourier transform to B.1 to obtain its spectral counterpart

B.1.1 Double correlation equation

The goal of a spectral model is to derive a closed equation for the two point correlation in spectral space. To this end we use equation (B.5) to derive an equation for the double correlation tensor R ij .

(∂ t + 2νk 2 )R ij (k) = - i 2 P iαβ (k) R 3
S jαβ (-k, p)dp

+ i 2 P jαβ (k)

B.1.2 Triple correlation equation

A similar method is used to derive an equation for the triple correlation tensor S iαβ . If one follows with the computation one finds an equation for S iαβ that depends on quadruple correlation. Giving an example of the closure problem of turbulence, caused by the intrinsic nonlinear nature of the process .

One solution to close the system is to use a quasi normal hypothesis on the fourth order moment (quadruple correlation), meaning that they can be expressed as a sum of the products between second order moments g(x 1 )g(x 2 )g(x 3 )g(x 4 ) = g(x 1 )g(x 2 ) g(x 3 )g(x 4 ) + g(x 1 )g(x 3 ) g(x 2 )g(x 4 ) + g(x 1 )g(x 4 ) g(x 2 )g(x 3 ) .

This assumption together with some computation allows us to obtain a closed equation for the triple correlation tensor

(∂ t + ν(k 2 + k 2 + (k + k ) 2 ))S ijh (k, k ) = -iP iαβ (k) (R jα (k )R hβ (-k -k )) -iP jαβ (k ) (R iα (k)R hβ (-k -k ) ) -iP hαβ (-k -k ) (R iα (k)R jβ (k )) . (B.10)
This equation has a solution, which expresses S ijh as a sum of second order tensors products : At this point we introduce the markovian hypotheses on the double correlation, we consider that R ij values do not depend on their past, so that they can be taken out of the integral sign

S ijh (k, k ) = -i t 0 e -ν(k 2 +k 2 +(k+k ) 2 )(t-s) ×    P iαβ (k)R jα (k )R hβ (-k -k ) +P jαβ (k )R iα (k)R hβ (-k -k ) +P hαβ (-k -k )R iα (k)R jβ (k )
S ijh (k, k ) = -i    P iαβ (k)R jα (k )R hβ (-k -k ) +P jαβ (k )R iα (k)R hβ (-k -k ) +P hαβ (-k -k )R iα (k)R jβ (k )    × t 0
e -ν(k 2 +k 2 +(k+k ) 2 )(t-s) ds, (B.12) defining the quantity Θ kpq = t 0 e -ν(k 2 +k 2 +(k+k ) 2 )(t-s) ds we can rewrite the expression of the third order correlation tensor

S ijh (k, k ) = -iΘ kpq    P iαβ (k)R jα (k )R hβ (-k -k ) +P jαβ (k )R iα (k)R hβ (-k -k ) +P hαβ (-k -k )R iα (k)R jβ (k )    .
(B.13)

B.1.3 Lin equation for E(k)

At this point we use the derived expression for S ijk in the double correlation equation (B.6), obtaining

(∂ t + 2νk 2 )R ii (k) = R 3 Θ kpq P iαβ (k)    P ilm (k)R lα (p)R mβ (q) +P αlm (p)R il (k)R mβ (q) +P βlm (q)R il (k)R mα (p)    dp (B.14)
where we have already imposed the case i = j, to consider only the trace of the tensor. However, [START_REF] Ogura | A consequence of the zero-fourth-cumulant approximation in the decay of isotropic turbulence[END_REF] shows that the quasi-normal approximation leads to nonphysical behaviour of the solution, in particular it can lead to negative energies. That is why [START_REF] Orszag | Lectures on the statistical theory of turbulence[END_REF] proposes to add to equation (B.10), an eddy-damping term proportional to the third-order correlation tensor µ kpq (t)S ijk = (µ k + µ p + µ q )S ijk (B.15) where the expression of µ k is given by [START_REF] Pouquet | Evolution of high Reynolds number two-dimensional turbulence[END_REF] µ It can be shown that the integral in the wavenumber space is limited to the triads k + p + q = 0. Here we introduce the isotropic spectrum E(k) connected to the trace of the double correlation tensor via

R ii = E(k) 2πk 2 (B.18)
After some algebra and geometric consideration on the triads k,p,q one finally obtains

(∂ t + 2νk 2 )E(k) = Θ kpq k pq E(q)b(k, p, q) k 2 E(

Spherical harmonics and Mollweide projection C.1 Mollweide Projection

The Mollweide projection is an equal-area, pseudo-cylindrical map projection [START_REF] Snyder | Map projections-A working manual[END_REF]. It has been previously used in other application for the projection of spherical results onto a plane such as the cosmic microwave background radiation [START_REF] Bennett | Nine-year wilkinson microwave anisotropy probe (wmap) observations : final maps and results[END_REF]. The first step is to interpolate the field of interest on a sphere of given radius R, to get data as function of the polar angular coordinates (ψ,φ). The Mollweide projection establish a relation between these variables and the map coordinates X and Y . They represent respectively the equator and the central meridian.

C.2 Spherical harmonics

We follow Lombardini et al. (2014b). Here we sum up the principal steps :

1. At time t we have a field in the Cartesian reference frame f (x, y, z). We interpolate it on a sphere of radius R obtaining a f (R, θ, φ). Where P m l are the associated Legendre polynomials. The continuous spherical harmonics transform is defined as

f lm = 1 4π Ω f (R, θ, φ)Y lm (θ, φ) dΩ (C.5)
3. The coefficient of the angular power spectrum can then be now computed

C l = 1 2l + 1 m=+l m=-l |f lm | 2 (C.6)

  driven mixing . . . . . . . . . . . . . . . . . . . . . . 3 1.1.2 Converging Geometry . . . . . . . . . . . . . . . . . . . . . . . 11 1.1.3 Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.1.4 Why is buoyancy-driven mixing important ? . . . . . . . . . . . 12 1.2 Inertial confinement fusion : an introduction . . . . . . . .

Figure 1

 1 Figure 1.1 -(a) Numerical simulation of turbulent mixing in the ocean (NASA/Chris Henze) (b) Numerical simulation of the turbulent mixing with chemical reaction in a burner. (Universität Duisburg-Essen)

Figure 1

 1 Figure 1.2 -(a) Sequence of images relative to the development of Rayleigh-Taylor instability in the linear and nonlinear phases. The Atwood number is 0.15. (b) Threedimensional visualization of the structure of the bubbles with the same initial condition of (a). From[START_REF] Wilkinson | Experimental study of the single-mode threedimensional Rayleigh-Taylor instability[END_REF]

Figure 1

 1 Figure 1.3 -Time evolution of Richtmyer-Meshkov instability with an Atwood number of 0.5 and incident shock Mack number of 1.21. The initial condition is on the top-right position. The time is noted near every image, together with the distance traveled in parentheses. These images display the complete evolution of a Richtmyer-Meshkov mixing zone from the initial development to the final turbulent state. Orlicz et al. (2013)

Figure 1

 1 Figure 1.4 -Schematic of the indirect drive (upper left) and direct drive (upper right) approaches in inertial confinement fusion experiments. The four images at the bottom describe schematically different stages in the implosion.In the last image on the right, the effects of hydrodynamics instability are evidenced. From[START_REF] Betti | Inertial-confinement fusion with lasers[END_REF]

Figure 1

 1 Figure 1.5 -Temperature and density contours of low-foot (left) and high-foot (right) implosions simulation of an indirect drive inertial confinement fusion target using the radiation hydrodynamic code HYDRA [Marinak et al. 2001]. The top images show the computational results for a radius of 200µm while the bottom images the contours of temperature and density at the bang time. Figure adapted from Clark et al. (2016).

  Figure

Figure 1

 1 Figure 1.7 -Instantaneous velocity signals on the axis of : (Left) a jet with the same viscosity of the issuing fluid, (right) jet with lower viscosity of the issuing fluid.From[START_REF] Talbot | Mélange et dynamique de la turbulence en écoulements libres à viscosité variable[END_REF]

Figure 1

 1 Figure 1.8 -Concentration plots of the two dimensional Rayleigh-Taylor computation, for different characteristic domain sizes. The leftmost results was obtained without plasma transport coefficients, the others corresponds to the spacial scales indicated just below every plot. From Haines et al. (2014b)

Figure 1

 1 Figure1.9 -Vertical slices of a Rayleigh-Taylor simulation at Atwood number of 0.5, one of which is upside down. From[START_REF] Miller | Which way is up ? A fluid dynamics riddle[END_REF]

Figure 1 .

 1 Figure 1.10 -Energy transfer mechanism in Boussinesq top and non-Boussinesq jets. From Prestridge (2018)

Figure 1

 1 Figure 1.11 -Contours of θ, a density related quantity, at three instants during the compression of a spherical mixing zone. (Top) Results obtained using the Boussinesq approximation. (Bottom) Results obtained with the variable density approximation

Figure 2

 2 Figure 2.1 -Unstably stratified homogeneous turbulence approximation with respect to a Rayleigh-Taylor simulation. (From Griffond et al. (2015a)).

Figure 2

 2 Figure 2.2 -Sketches of the isotropic compressions for the (a) homogeneous isotropic case of section 2.1 (b) spherical compression case of section 2.3. The dashed line represents the initial condition while the solid line represents a later time during the compression.

Figure 2

 2 Figure 2.3 -(a) Time evolution of the compression parameter Λ for different value of Λ min with S 0 = 1000. (b) Time evolution of the compression parameter Λ for different value of S 0 (S 0 = 500, 625, 750, 875, 1000 µs -1 , increases in the direction of the arrow), with Λ min = 0.05

Figure 2

 2 Figure 2.4 -Radial profile of the parabolic function H, in Eq. (2.66), at the beginning of the simulation (Λ = 1) with fixed value of Π 0 . (a) Plot for different values of Λ min , which increases in the direction of the arrow, with S 0 = 1000. (b) Plot for different value of S 0 (S 0 = 500, 625, 750, 875, 1000 µs -1 increases in the direction of the arrow), with Λ min = 0.05

  -spectral computation using FFT . . . . . . . . . . . . . 46 3.2.2 Time advancement scheme . . . . . . . . . . . . . . . . . . . . 48 3.2.3 Solution of the Poisson equation . . . . . . . . . . . . . . . . . 50 3.2.4 Non uniform transport coefficients . . . . . . . . . . . . . . . . 53 3.2.5 Preconditioning and initial guesses . . . . . . . . . . . . . . of the iterative GMRES algorithm . . . . . . . . . 56 3.3.2 Validation : Decay of incompressible homogeneous isotropic turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.3.3 Validation : Decay of incompressible homogeneous isotropic turbulence with variable viscosity . . . . . . . . . . . . . . . . 59 3.3.4 Validation : Variable density results with small non-Boussinesq effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.4 Initial conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.4.1 Homogeneous isotropic turbulence . . . . . . . . . . . . . . . . 63 3.4.2 Spherical compressions . . . . . . . . . . . . . . . . . . . . . . . 64 3.4.3 Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Figure 3

 3 Figure 3.1 -From the site http ://www.2decomp.org

Figure 3

 3 Figure 3.2 -Speed up of the code

Figure 3

 3 Figure 3.3 -(a) Evolution of GMRES algorithm residuals for the pressure Poisson equation as a function of the Krylov subspaces dimension m (b) Kinetic energy spectra of an unstably stratified homogeneous turbulence simulation, computed at the same time with different Krylov subspaces dimension m.

Figure 3

 3 Figure 3.4 -Evolution of the variable density condition (dashed line) and norms of the residual (solid line) during an 256 3 USHT computation with N = 1.4 and no initial scalar fluctuations.

Figure 3

 3 Figure 3.5 -(a) Evolution of the turbulent kinetic energy, normalized with its initial value, as a function of the non-dimensional time τ . (b) Kinetic energy spectra at three times during the decay. Solid lines indicate results using the new code while dashed line the ones from the previous version. (c) Relative error Er p between the old and new code.

Figure 3

 3 Figure 3.6 -Contours of turbulent kinetic energy at τ = 2 computed with (a) the previous version and with (b) the new version of the DNS code.

Figure 3

 3 Figure 3.7 -(a) Evolution of the turbulent kinetic energy, normalized with its initial value, as a function of the non-dimensional time τ (b) Kinetic energy spectra at three times during the decay. Solid lines indicate GMRES results while dashed line the finite differences ones.

Figure 3

 3 Figure 3.8 -(a) Evolution of the turbulent kinetic energy and (b) scalar variance, as a function of the non-dimensional time N t. Solid lines : Variable density computations. Solid lines with circles : Boussinesq results.

Figure 3

 3 Figure 3.9 -(a) Kinetic energy and (b) Scalar variance spectra at three times during the USH turbulence computations. Solid lines : Variable density computations. Dashed lines : Boussinesq results.

  Figure 3.10 -Contours of the scalar θ at N t = 2.5 computed with (a) the Boussinesq and (b) the new variable density approximations.

Figure 3 .

 3 Figure 3.11 -Example of initial condition for the component v 2 of the velocity field, with k l = 30 and K = 0.2.

Figure 3

 3 Figure 3.12 -Example of initial condition for the scalar field for the spherical compression computations. Here the Atwood number is0.07

Figure 3

 3 Figure 3.13 -Initial radial profile of the mean component of the scalar field θ. (a) Plot with constant Atwood number At = 0.2 and different values of the parameter d = 0.01, 0.05, 0.10, 0.15, 0.2, which increase in the direction of the arrow. (b) Plot with constant d = 0.05 and different values of the Atwood number,A t = 0.1, 0.15, 0.2, 0.25, 0.3 , which increase in the direction of the arrow.

Figure 3

 3 Figure 3.14 -Solid line : mean radial profile of the scalar θ for an Atwood number of 0.46. Dashed dotted line : the corresponding filter function F s after the application of the smoothing function.

Figure 4

 4 Figure 4.1 -Turbulent kinetic energy evolution as a function of the compression parameter Λ. Solid lines : DNS results from Davidovits and Fisch (2016a). Dotted lines : EDQNM forcing term closure. Dash-dotted lines : EDQNM time-dependent viscosity closure. Different colors for decreasing compression rates : red : S 0 = 100 ; green : S 0 = 10 ; black : S 0 = 1 ; blue : S 0 = 0.1.

Figure 4

 4 Figure 4.2 -Turbulent kinetic energy evolution as a function of the compression parameter Λ. Solid lines : our DNS results. Dash-dotted line : EDQNM time-dependent viscosity closure. Different colors for different compression rates : red : S 0 = 100 ; green : S 0 = 10 ; black : S 0 = 1 ; blue : S 0 = 0.1. Viscosity coefficient ν 0 = 5 × 10 -2 in all simulations.

Figure 4

 4 Figure 4.3 -Evolution of turbulent kinetic energy K, normalized by its value at t = 0, as a function of the compression parameter Λ for both DNS at resolution 512 3 and EDQNM simulations of a HIT compression case with Re 0 = 250 and Cp 0 = 0.1. The critical compression parameter Λ M corresponds to the kinetic energy maximum before the beginning of the viscous phase.

Figure 4

 4 Figure 4.4 -Top row : distribution of turbulent kinetic energy K in the DNS at four instants of the evolution, indicated in the kinetic energy curve of Figure 4.3. The scaling of the box corresponds to the moving frame of reference. Bottom row : Associated energy spectra E( k) for both DNS and EDQNM simulations, at the same instants.

Figure 4 Figure 4 . 6 -

 446 Figure 4.5 -Iso-contour map of the critical compression parameter Λ M corresponding to maximum of kinetic energy, as a function of the initial Reynolds number Re 0 and the compression number Cp 0 . Results from EDQNM simulations. The black circle corresponds to the parameters used in Figures 4.3 and 4.4, and the red triangle to that of Figure 4.10. The red line is the contour line at Λ M = 1.

Figure 4 .Figure 4 . 7 -Figure 4 Figure 4

 44744 Figure 4.7 -Evolution of (Top) the kinetic energy K and (Bottom) the integral scale I as a function of the compression parameter Λ using EDQNM simulations. Solid line : Batchelor initial condition (s = 4). Dashed line : Saffman initial condition (s = 2). The scaling laws corresponding to the self-similar solutions derived in Sec. 4.4.1 are also shown.

Figures 4 .

 4 Figures 4.7 and 4.8-bottom exhibit very clearly the three different regimes that can be identified by the evolution changes in K, I , and Cp when Λ decreases. The dynamical changes in the evolution of Re are less evident due to transitions between similar powerlaws, especially the RC to cascade one, but are still present. For all curves, the self-similar scaling laws proposed in Sec. 4.4.1 fit the results of simulations adequately. The succession of rapid compression, cascade, and viscous regimes is clear, and the values of Λ at which the RC-cascade transition occurs seem to be similar for Batchelor and Saffman turbulences.

Figure 4

 4 Figure 4.10 -Evolution of turbulent kinetic energy K as a function of the compression parameter Λ for both 1024 3 DNS and EDQNM simulations corresponding to the case of the spherical turbulent layer. Solid line : DNS ; Dashed line : EDQNM simulation. Saffman and Bachelor initial conditions with respectively (s = 2) and (s = 4) are used, as indicated in the legend.

Figure 4 .

 4 11 visualizations of the kinetic energy field extracted from DNS at different stages of the compression. The Reynolds and compression numbers Re and Cp, the integral scales I and the turbulent layer size MZ are plotted in Figures 4.12 and 4.13 respectively. Energy spectra at the same compression stages as the kinetic energy distribution in Figure 4.10 are plotted in Figure 4.14. The kinetic energy radial profiles averaged over spherical shells in physical space are shown in Figure 4.15.

Figure 4

 4 Figure 4.11 -Distribution in a plane cut of turbulent kinetic energy u 2 i 2 in the DNS at different instants indicated in Figure 4.10. Top row : the Batchelor case (s = 4) ; and bottom row : the Saffman case (s=2). The scaling corresponds to that of the moving frame.

FigureFigure 4

 4 Figure 4.12 -Evolution of (Top) the Reynolds number Re and (Bottom) the compression number Cp as a function of the compression parameter Λ for the spherical turbulent layer case extracted from DNS (solid line) and EDQNM (dashed line) simulations. Black lines : Batchelor initial condition (s = 4). Red lines : Saffman initial condition (s = 2).

Figure 4 .

 4 Figure 4.13 -Evolution of the integral scale I (DNS and EDQNM) and the turbulent layer size MZ (DNS) as a function of the compression parameter Λ corresponding to the case of the spherical turbulent layer with (red lines) Saffman and (black lines) Batchelor initial conditions.

Figure 4 Figure 4

 44 Figure 4.14 -Energy spectra as a function of wavenumber, E(k), corresponding to the spherical turbulent layer case both in DNS (Plain line) and EDQNM (Dashed line) and taken at different values of the compression parameter Λ. (Right) Batchelor initial condition (s = 4). (Left) Saffman initial condition (s = 2).

  field induced by an isolated eddy in a variable density field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.2 Unstably stratified homogeneous turbulence equations . . . 99 5.2.1 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.2.2 Initial conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.3 Flow Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.4.1 Non dimensional numbers . . . . . . . . . . . . . . . . . . . . . 106 5.4.2 One-point statistics . . . . . . . . . . . . . . . . . . . . . . . . 107 5.4.3 Two-point statistics . . . . . . . . . . . . . . . . . . . . . . . . 108 5.4.4 Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.4.5 Probability density functions . . . . . . . . . . . . . . . . . . . 112 5.5 Initial condition variation . . . . . . . . . . . . . . . . . . . . 114 5.5.1 Non dimensional numbers . . . . . . . . . . . . . . . . . . . . . 115 5.5.2 One point statistics . . . . . . . . . . . . . . . . . . . . . . . . 116 5.5.3 Two point statistics . . . . . . . . . . . . . . . . . . . . . . . . 117 5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

Figure 5

 5 Figure 5.1 -Isolated blob of vorticity in a domain with variable density located at x = 0. Figure modified from Soulard et al. (2019).

  constant in time. The different USHT flows considered in this work are characterized by two nondimensional numbers : the Reynolds and the Froude numbers which are defined as

Figure 5

 5 Figure 5.2 -Comparison of DNS (red) and ILES (black) results. (a) Kinetic energy spectra. (b) Total kinetic energy.

Figure 5

 5 Figure 5.3 -Contours of the scalar field θ of simulations Ud3 B (left) and Ud3 V D (right). At four different times during the flow evolution.

Figure 5

 5 Figure 5.4 -Temporal evolution of nondimensional quantities of the flow. (a) Reynolds number. (b) Froude number. (c) Atwood number. Red : Variable density. Black : Boussinesq.

Figure 5 .

 5 Figure 5.4 shows the evolution of the nondimensional number characteristic of the flow. The Reynolds number that, at the beginning, has a value of 5 grows considerably and reaches different values for the variable density and Boussinesq approximations. At the end of the simulations, we have Re 287 for U l3 V D and Re 220 for U l3 B , which is the first indication of different behaviors between the two simulations.The effective Atwood number computed using the maximum and minimum value

Figure 5 . 5 -

 55 Figure 5.5 -Temporal evolution of one-point statistics. (a) Turbulent kinetic energy. (b) Variance of the scalar. (c) Vertical flux. Red : Variable density. Black : Boussinesq

Figure 5

 5 Figure 5.6 -Temporal evolution at four instants, N t = 0, 1.8, 3.7, 5 of : (a) Turbulent kinetic energy spectra. (b) Scalar variance spectra. (c) Vertical flux spectra. The time increases in the direction of the arrows. Red : Variable density. Black : Boussinesq

Figure 5

 5 Figure 5.7 -Vertical component of the velocity at N t = 4.73. Top : Contours in a plane parallel to the acceleration. Bottom : Contours in a plane perpendicular to the acceleration. (a) and (c) Boussinesq results (b) and (d) Variable density

Figure 5 . 8 -

 58 Figure 5.8 -Evolution of the anisotropy coefficients.In red variable density and black Boussinesq results.

Figure 5

 5 Figure 5.9 -Spectral energy spectra at four instants during the flows evolution N t = 0, 1.8, 3.7, 4.73 of :(a) the horizontal component u 2 and (b) vertical component u 3 . In red variable density and black Boussinesq results

Figure 5

 5 Figure 5.10 -Probability density function of the scalar at four instants during the flows evolution : (a)N t = 0, (b)N t = 1.8, (c) N t = 3.7, (d) N t = 4.73. In red variable density and black Boussinesq results

Figure 5

 5 Figure 5.11 -Probability density function of one of the scalar gradient horizontal component at four instants during the flows evolution : (a)N t = 0, (b)N t = 1.8, (c) N t = 3.7, (d) N t = 4.73. In red variable density and black Boussinesq results

Figure 5

 5 Figure 5.12 -Temporal evolution of nondimensional number characteristic of the flows. (a) Reynolds number. (b) Froude number. (c) Atwood number. In red variable density and black Boussinesq results

Figure 5

 5 Figure 5.13 -Temporal evolution of one-point statistics. (a) Turbulent kinetic energy. (b) Variance of the scalar. (c) Vertical flux. In red variable density and black Boussinesq results. The different simulations are indicated in the legend of every figure.

Figure 5 Figure 5

 55 Figure 5.14 -Temporal evolution of kinetic energy spectra at four instants for : (a) Ul1 simulations. (b) Ul05 simulation. In red variable density and black Boussinesq results.

Contents 6 . 1

 61 Description of the pseudo-ions in jellium (PIJ) model . . . 123 6.1.1 How the PIJ results depend on mass fraction, density and temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 6.2 Theoretical framework . . . . . . . . . . . . . . . . . . . . . . 128 6.2.1 Equations for DT/CH mixtures . . . . . . . . . . . . . . . . . . 128 6.2.2 Base flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 6.2.3 Perturbation equations . . . . . . . . . . . . . . . . . . . . . . . 130 6.2.4 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . 131 6.2.5 Initial conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 132 6.2.6 Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 6.3 One-dimensional evaluation of the transport coefficients effects on the implosion . . . . . . . . . . . . . . . . . . . . . . 135 6.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 6.4.1 Global statistics . . . . . . . . . . . . . . . . . . . . . . . . . . 136 6.4.2 Mixing layer width . . . . . . . . . . . . . . . . . . . . . . . . . 137 6.4.3 Radial profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 6.4.4 Mixing parameter . . . . . . . . . . . . . . . . . . . . . . . . . 140 6.4.5 Transport coefficient evolutions . . . . . . . . . . . . . . . . . . 142 6.4.6 Bi-dimensional Maps . . . . . . . . . . . . . . . . . . . . . . . 144 6.4.7 Spherical harmonics spectra . . . . . . . . . . . . . . . . . . . . 146 6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

Figure 6 . 1 -

 61 Figure 6.1 -Different plasma regimes characterized by the temperature T , measured in Kelvin, and the electron density, i.e., the number of electron for unit volume. On this figure, the dotted lines represent the border between weakly and strongly coupled plasma, defined for a coupling parameter of Γ = 1 3 . From Piel (2017).

Figure 6

 6 Figure 6.2 -Viscosity of a hydrogen (H) silver (Ag) mixture. PIJ simulation ( continuous lines) are compared with results from orbital free molecular dynamics simulation (symbols). From Ticknor et al. (2016).

Figure 6 6Figure 6

 66 Figure 6.3 -Evolution of the coupling coefficient in plasma mixtures, as a function of the mass fraction of the heavier element, for different temperature and constant density. (Left) : Mixture DT/CH with density ρ = 5 10 -3 g/cm 3 . (Right) : Mixture DT/ Gold with density ρ = 20g/cm 3

Figure 6

 6 Figure 6.5 -Description of the implosion characteristics studied in the present work. (Left) : Time evolution of the temperature T B , the pressure P B and density ρ B at the center of the domain. The thick solid line represents the radius of the capsule. (Right) : Radial profiles of base temperature and pressure, at four instants during the compression as indicated in the left figure by the black triangles on the time axis.

Figure 6

 6 Figure 6.6 -Three dimensional contours of mass fraction of CH, Y , using volume rendering at different instants and for simulations with plasma, S1 a , and constant, S1 b , viscosity and diffusion coefficients. The different times correspond to Figure 6.5.

Figure 6

 6 Figure 6.7 -Dependence of the final size of the mixing zone, on initial compression velocity and final size of the domain (left) and on initial temperature and density of the capsule center (right)

Figure 6

 6 Figure6.8 -Evolution of the turbulent kinetic energy normalized by its initial value (black) and the scalar variance θ θ (red), as functions of the compression parameter Λ. The solid lines represent the results of plasma transport coefficients simulation S1 a , and the dashed-dotted lines are for S1 b results. Scaling laws corresponding to the self-similar regimes identified in[START_REF] Viciconte | Self-similar regimes of turbulence in weakly coupled plasmas under compression[END_REF] are also plotted.

Figure 6

 6 Figure6.9 -Evolution of the renormalized mixing layer width as a function of the compression parameter Λ. The solid black line for the case of plasma transport coefficients (S1 a ). The black dashed line is for the constant case (S1 b ). The solid green line is the reference case with Re = 0 (S5 a ) where the mixing layer is only enlarged by molecular diffusion.

Figure 6 .

 6 Figure 6.10 -Evolution of the mixing layer width as a function of the compression parameter Λ for the simulations of table 6.2. (a) Simulation with plasma transport coefficients S1 a -S5 a . (b) Simulations with constant transport coefficients S1 b -S4 b .

Figure 6

 6 Figure 6.11 -Ablator (CH) tangential averaged mass fraction Y at four times during compression. (a) Initial condition at t = 0 ns and Λ = 1 (b) (II) t = 0.5 ns and Λ = 0.5 (c) (III) t = 0.93 ns and Λ = 0.08 (d) (IV) t = 1 ns and Λ = 0.05. Solid line for the S1 a simulation, dashed-dotted line for the S1 b simulation. The lagrangian position of the unperturbed fuel/ablator interface is also indicated in the figure.

Figure 6

 6 Figure 6.12 -Radial profiles of the variance of θ (top) and kinetic energy K (bottom), at four times during compression. (a) Initial condition at t = 0 ns and Λ = 1 (b) (II) t = 0.5 ns and Λ = 0.5 (c) (III) t = 0.93 ns and Λ = 0.08 (d) (IV) t = 1 ns and Λ = 0.05. Solid line for the S1 a simulation, dashed-dotted line for the S1 b simulation.

Figure 6

 6 Figure 6.13 -Contour maps of the molecular mixing parameter Ξ as a function of the compression parameter Λ and normalized radial position r/R. (Top) S1 a simulation ; (Bottom) S1 b simulation. The red lines show the evolution of the mass fraction dependent radii r 01 /R and r 99 /R, corresponding respectively to Y = 0.01 and Y = 0.99.

Figure 6

 6 Figure 6.14 -Transport coefficient values extracted from S1 a simulation. (a) and (b) : mean radial profiles of viscosity and diffusion at four different instants. (c) :Time evolution of mean viscosity and diffusion at the edges of the mixing layer at r 01 (solid line) and r 99 (dashed lines) corresponding to Y = 0.01 and 0.99 respectively. At the end of the simulation (t 1 ns), r 01 reaches zero as mixing is entirely spread inside the capsule. The mean viscosity and diffusion are then plotted at the center of the capsule, r = 0, and therefore decrease since Y becomes greater than 0.01 at r = 0

Figure 6

 6 Figure 6.15 -Transport coefficient variances extracted from S1 a simulation.(Top) mean radial profiles of (a) viscosity, ν ν , and (b) diffusion, D D , at four different instants. (c) : Time evolution of mean variance of viscosity ν ν and diffusion D D .

Figure 6

 6 Figure 6.16 -Mollweide projection of the local turbulent kinetic energy in the mixing layer. The three rows correspond to the radial positions r 99 , r 50 , r 01 defined in S1 b simulation, at times I, II, III, IV. Top and bottom half contours corresponds respectively to S1 a and S1 b results.

Figure 6

 6 Figure 6.17 -Top : Mollweide projection of the kinetic energy. Bottom : Mollweide projection of the scalar variance. The radius of the spheres corresponds to the center of the mixing zone. Top and bottom half contours correspond respectively to S1 a and S1 b results.

Figure 6

 6 Figure 6.18 -Spherical harmonics spectra computed at the center of the mixing zone, r 50 , at the instants I, II, III, IV. Left : kinetic energy spectra ; right : scalar spectra. The solid black lines correspond to the simulation S1 a , the dashed lines to the simulation S1 b . The red solid line represents the l -5/3 power law.

  diffusion effect in spherical mixing zones of plasma under compression . . . . . . . . . . . . . . . . . . . . . . . . . 154 7.4 Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 7.5 Publications & Conferences . . . . . . . . . . . . . . . . . . . 155

∂

  t ûi + ik j R 3 ûi (p)û j (kp)dp = -ik i p -νk 2 ûi . (B.3)while the incompressibility condition is simply expressed byk i u i = 0. (B.4)The pressure can be eliminated from B.3, using the fact that the velocity field is solenoidal, resulting in∂ t ûi (k) + νk 2 ûi (k) = -i 2 P iαβ (k) R 3 ûα (p)û β (kp)dp (B.5)where P iαβ (k) = k α P iβ (k) + k β P iα (k) is the Kraichnan projector, andP ij (k) = δ ijk i k j k 2 .

R 3 S

 3 iαβ (k, p)dp.(B.6) Where we observe how the double correlation R ij , depends on the triple correlation S abc . T Equation (B.6) is simply obtained taking the product of (B.5) by ûj (k ) summed to the (B.5) rewritten for ûj (k ) and multiplied by ûi (k ), and using an the ensemble average operator * .The definition of the double R ij and triple S iαβ correlation tensors are the followingûi (k)û j (k ) = R ij δ(k + k ), (B.7) ûi (k)û α (p)û β (kp) = S iαβ (k, p)δ(k + k ), (B.8) ûj (k )û α (p)û β (kp) = S jαβ (k , p)δ(k + k ).(B.9)

3 ΘP

 3 is a model constant. When we add this new term to equation (B.10) the term Θ kpq is modified and becomesΘ kpq = t 0 e -(µ kpq +ν(k 2 +k 2 +(k+k ) 2 ))(t-s) ds So that finally (∂ t + 2νk 2 )R ii (k) = R kpq P iαβ (k) ilm (k)R lα (p)R mβ (q) +P αlm (p)R il (k)R mβ (q) +P βlm (q)R il (k)R mα (p)

  a parametric angle defined by 2λ + 2 sin λ = π sin φ (C.3) Equations (C.2), (C.3) have to be solved via an iterative Newton-Raphson method.

2.

  Using a discrete spherical harmonics transform we obtain the values f lm . The spherical harmonics are defined by Y lm (θ, φ) =    N (l,m) P m l (cos θ) cos(mθ), if m ≥ 0 N (l,|m|) P |m| l (cos θ) sin(|m|θ), if m < 0 (C.4)

  Mélange, écoulements à densité variable , coefficients de transport non-uniformes, compression, plasma, fusion par confinement inertiel, DNS, EDQNM

	vi	List of Symbols
	Θ, θ	Logarithm of the density divided by a reference
		density
	N	Buoyancy frequency
	*	Volume average of a quantity *
	*	Tangential average of a quantity * ( depending on
		r only
	L	Mixing zone width
	H, G	Self-similar functions
	T	Temperature
	r	Radial position
	T	Mass fraction
	n	Number density of particles
	k B	Boltzmann constant
	κ	Temperature diffusion coefficient
	φ	Diffusive flux
	M	Atomic mass
	Z	Ionization number
	F T	Fourier transform
	k	Wavenumber
	∆t	Time step
	K n	Krilov subspace of size n
	E(k)	Kinetic energy spectrum
	E θ (k)	Scalar variance energy spectrum
	E u 3 θ (k) Vertical flux spectrum
	K	Integrated kinetic energy
	Re	Reynolds number
	Cp	Compression number
	I	Integral length scale
	β	Growth rate of USHT one-point statistics
	F r	Froude number
	θθ	Volume averaged scalar variance
	θu 3	Volume averaged vertical flux
	b ij	Anisotropy coefficient
	Γ	Coupling parameter
	h 0	Characteristic length scale of temperature gradient
	c	Speed of sound
	Ξ	Molecular mixing parameter
	r 01

Chapitre 2 Theory Contents 2.1 Compressed turbulence . . . . . . . . . . . . . . . . . . . . . . 27 2.2 Unstably stratified homogeneous turbulence . . . . . . . . . 30 2

  .2.1 Equations for the scalar Θ . . . . . . . . . . . . . . . . . . . . .

33 2.3 Spherical compressions . . . . . . . . . . . . . . . . . . . . . . 35

	2.3.1 Navier-Stokes for binary mixtures . . . . . . . . . . . . . . . . 36
	2.3.2 Base flow and perturbation . . . . . . . . . . . . . . . . . . . . 37
	2.3.3 Change of reference frame and rescaling . . . . . . . . . . . . . 42

Table 4 .

 4 1 -Scaling laws corresponding to the different self-similar regimes for the different turbulent quantities and non dimensional numbers. The coefficients n c and n v are given by Eqs. (4.25) and (4.27) as a function of the infrared spectral slope s.

Table 6 .

 6 1 -Temperature T , pressure P and density ρ at the center of the capsule, for different instants. The last column indicates the radius of the unperturbed fuel/ablator interface R.

	t [ns]	T [eV ]	P [M bar]	ρ[gcm -3 ]	R[µm]
	0	75	2.87 × 10 -1	5 × 10 -3	300
	0.5	293	8.7	3.9 × 10 -2	150
	0.93	12.8 × 10 3	1.08 × 10 5	11.08	24
	1	30 × 10 3	9.14 × 10 5	40	15

  Table 6.2 -Simulation characteristics in terms of initial Reynolds number, compression number, integral length scale, and mixing layer size. Type a corresponds to simulations with plasma transport coefficients while for type b viscosity and molecular diffusion are kept constant during the computation.

	Name		Type	Mesh	Re 0	Cp 0	0 /R 0	L 0 /R 0
				size
	S1	a b	DNS ILES	512 3 1024 3	217 217	0.27 0.27	0.07 0.07	0.016 0.016
	S2	a b	DNS ILES	512 3 512 3	153 153	0.2 0.2	0.07 0.07	0.016 0.016
	S3	a b	DNS DNS	512 3 512 3	108 108	0.13 0.13	0.07 0.07	0.016 0.016
	S4	a b	DNS DNS	256 3 256 3	32 32	0.1 0.1	0.05 0.05	0.016 0.016
	S5	a	DNS	256 3	0	0	0	0.016

  p) -p 2 E(k) dpdq (B.19)where b is geometric factor depending on the angles of the triad.At this point if one defines the transfer term

T (k, t) = Θ kpq k pq E(q)b(k, p, q) k 2 E(p) -p 2 E(k) dpdq (B.20)

we obtain the following Lin equation

(∂ t + 2νk 2 )E(k) = T (k, t) (B.21)

Annexe C

Acknowledgments

Initial conditions

The initial conditions are generated using the methods described in section 3.4.1, for both variable density and Boussinesq approximations.

A total of 12 simulations are presented in this study, using either the Variable Density (VD) or Boussinesq (B) approximation to study non-Boussinesq effects. The 12 initial conditions have the same initial Reynolds number, Froude number, integral scale, and buoyancy frequency N but with a different ratio between scalar variance and kinetic energy.

Flow Topology

In this section, we discuss the differences in the flow topology during the time evolution of USHT direct numerical simulation Ud3. In Figure 5.3, we show the contours of the scalar fields resulting from Boussinesq and variable density approximations. We observe that from the same initial condition at N t = 0, the two flows have a similar evolution until N t = 1.8.

The first differences appear on the contours at N t = 3.7, when in the scalar field of the variable density case, we can differentiate between the more spike-like and bubble-like structures, while this differentiation is not evident in the Boussinesq 104 Chapitre 5. Variable density effects in unstably stratified turbulence case, which shows a more symmetric behavior.

The different behaviors of the scalar field become more evident as the simulations advance with the last two contours showing striking differences.

The variable density tendency to form spike-like structures reach some extreme, with the presence of thin layers of heavy fluid moving towards the lighter fluid.

Perspective

The results of this work, together with the code developed, allow different possible future researches. For the spherical implosion with plasma transport coefficients, the first improvement would be to explore other cases of implosion. In particular, one could use thermodynamic paths extracted from real Inertial Confinement Fusion (ICF) experiment or simulation. Moreover, the studies of these idealized implosions could help to include the transport coefficients effects in models used in ICF design.

In the case of variable density unstably stratified turbulence, one could investigate in-depth the small scale properties and their difference with the Boussinesq cases. Furthermore, the parametric study on initial conditions could be widened, including the variation of other quantities such as the Froude number or the buoyancy frequency. Moreover, the USHT problem could be explored in the case of variable acceleration.

Finally, the variable density code with little modification can be adapted to simulate Richtmyer-Meshkov, Rayleigh-Taylor, and Faraday turbulent mixing zones.

Publications & Conferences

Viciconte, G., Gréa, B. J., and Godeferd, F. S. (2018). Self-similar regimes of turbulence in weakly coupled plasmas under compression. Physical Review E, 97(2), 023201.

Viciconte, G., Gréa, B. J. and Godeferd, F. S., Arnault, P. and Clérouin, J (2019). Sudden diffusion of turbulent mixing layers in weakly-coupled plasmas under compression. Accepted for publication in Physical Review E.

Compression

A.1 Moving frame

Here we sum up the derivation of the relation between frames coordinates , that is found in [START_REF] Rogallo | Numerical experiments in homogeneous turbulence[END_REF]. The moving frame (x i ) has to be linearly related to the fixed frame

where the tensor B ij is derived imposing that the new coordinates moves with the mean flow

In this work we will only consider isotropic compression so that

With this simplification the previous equations reduce to

so that we have the following relation

Following other works on isotropic compression [START_REF] Wu | Simulation and modeling of homogeneous, compressed turbulence[END_REF][START_REF] Cambon | On the application of time dependent scaling to the modelling of turbulence undergoing compression[END_REF]], B(t) can be expressed as a function of a characteristic length R of the compressing domain :

where R 0 = R(t = 0), so that we have the following

.

Considering a particle moving with the base velocity U B , its distance to the center decreases by a factor Λ(t) = expt 0 S(s)ds , referred to as the compression parameter. So that, finally, we have the following relations xi = Λ(t) -1 x j (A.1)

for the change of reference frame.

Annexe A. Compression

A.2 Rescaling

Once the reference frame change derived in section A.1 is applied to the inhomogeneous dynamical equations derived in section 2.1, they reduce to an expression without explicit space dependence

At this point we have Navier-Stokes equation with a time dependent viscosity Λ µ ρ B 0 and a forcing term -S(t)u i . We can further simplify Eq. (A.2) choosing to eliminate the forcing term or to eliminate the time dependence before the viscous term. Since the space variable x i is rescaled using the compression parameter Λ, we look for the rescaling of velocity u i , time t and pressure p as a function of Λ :

When we use Eqs. (A.3), (A.4), (A.5) we obtain, for the time derivative in Eq. (A.2)

where the time derivatives relation for ũi is

The full expression of Eq. (A.2) using the rescalings for velocity, pressure and time is

If we group together the forcing terms we have

We want a Navier-Stokes-like equation so the time derivative and the convection term factors have to be equal, Λ a+e = Λ 2a-1 (A.10)

A.2. Rescaling 159

imposing the equality e = a -1. We can then divide all the terms in Eq. (A.9) by Λ 2a-1 obtaining :