Mme Jean-Paul Bonnet

Bérengère Podvin

M Bertrand Aupoix

M Sagaut

Thomas Gomez

Gabin

Fabio

Carole Linda

I don't know anything, but I do know that everything is interesting if you go into it deeply enough.

Richard Feynman

Ma reconnaissance s'adresse d'abord à M. Michel Stanislas pour avoir accepté de diriger ma thèse. Ses conseils m'ont été d'une grande utilité tout au long de celle ci, sa rigueur dans le travail, sa démarche scientifique, son esprit de synthèse et ses conseils pour la revue bibliographique m'ont beaucoup aidé à réaliser ce rapport. Il aura pris toutes les dispositions pour que cette thèse se déroule dans les meilleures conditions et pour cela je lui dit merci infiniment. Ensuite, j'aimerais remercier M. Jean-Marc Foucaut pour l'encadrement de ma thèse. Les nombreux échanges que j'ai eus avec lui, m'ont beaucoup servi. Sa formule, toujours se ramener à la signification physique des résultats, explorer le fond plutôt que la forme me servira certainement bien au delà de la thèse. De plus son enthousiasme et sa bonne humeur contagieuse ont fait de ces années de thèse des moments inoubliables.

Ma gratitude s'adresse aussi à M.

• Le chapitre 3 décrit la procédure de reconstruction d'un champ de vitesse résolu en espace et en temps à partir des mesures aux fils chauds résolus en temps et des mesures PIV résolus en espace. Une revue de la littérature sur l'Estimation Stochastique Linéaire est proposé en début de chapitre, puis les différents algorithmes de reconstruction (inversion matricielle, décomposition en valeur singulière et régularisation Tikhonov) sont présentés. La majeure partie du chapitre est consacrée à la validation de ces algorithmes. Une base de données issue de la simulation numérique directe d'un écoulement de canal turbulent est utilisée à cet effet. Le champ de vitesse reconstruit est comparé à un champ de référence extrait de la DNS avec la visualisation des contours du champ de vitesse, l'analyse spectrale, les corrélations, etc. En fin de chapitre, la nécessité d'étendre l'estimation stochastique à des ordres supérieurs à 2 est discutée.

ix

• Le chapitre 4 s'intéresse à la qualité des données reconstruites. Une analyse statistique est réalisée sur le champ de PIV d'origine, les données issues des fils chauds et les données reconstruites. Les résultats obtenues sont comparés avec ceux de Carlier and Stanislas (2005) qui servent de référence. Les statistiques inclus les spectres, moyennes, écart types, fonctions de densité de probabilité et les coefficients d'asymétrie et d'applatissement.

• Le chapitre 5 est consacré aux méthodes de détection et de caractérisation des structures et aux résultats qui en découlent. Comme toutes ces méthodes font intervenir des opérations de morphologie mathématique telles que l'érosion et la dilatation, une revue de la littérature sur les principes de la morphologie mathématique est donnée en début de chapitre. Ensuite, les algorithmes de détection des régions de vitesse uniforme et de mouvements tourbillonnaires sont présentés. A l'inverse des régions de vitesse uniforme où un unique algorithme de détection est utilisé, pour les mouvements tourbillonnaires, les critères Q, λ, Γ 1 et Γ 2 de Graftieaux et al. (2001) et l'algorithme de détection de tourbillon de Nencioli et al. (2010) sont validés avec un modèle de tourbillon de type Lamb-Oseen avant d'être appliqués sur la base de donnée reconstruite. Les régions de vitesse uniforme sont caractérisées avec leur diamètre hydraulique moyen (dans le plan Y Z perpendiculaire à l'écoulemenent), leur durée de vie et leur contribution aux tenseurs de Reynolds. Pour les tourbillons, on leur associe un rayon, une circulation et une vorticité en plus de la durée de vie et la distance relative à la paroi. Les résultats obtenus sont ensuite discutés en détail, l'accent est porté sur l'adimensionnement des caractéristiques de ces structures cohérentes. En fin de chapitre, l'organisation spatiale des structures est étudiée avec des corrélations de leurs fonctions indicatrices. Le modèle simplifié à grandes échelles qui en découle est comparé avec ceux qui existent dans la littérature. [START_REF] Blackwelder | Streamwise vortices associated with the bursting phenomenon[END_REF]). . . . . . . . . . . 1.3 (a,b) Lift-up by isotropic wall-normal motions, (c) effect of mean shear and (d) diffusion [START_REF] Chernyshenko | The mechanism of streak formation in near-wall turbulence[END_REF]. . . . . .

Latin Symbols

1.4 Schematic of hairpin structure attached to the wall and the induced motion [START_REF] Adrian | Vortex organization in the outer region of the turbulent boundary layer[END_REF]). . . . . . . . . . . . . . . . . 1.5 Hairpin packet model [START_REF] Adrian | Vortex organization in the outer region of the turbulent boundary layer[END_REF]). . . . . . . . . . . .

2.2

Basic angular configuration for Stereoscopic PIV, with enforcement of the Scheimpflug condition. "u1" is the in-plane projection of the velocity measured by camera 1; "u2" is the in-plane projection of the velocity measured by camera 2; the true velocity vector (of in-plane component "u" and out-of plane component "w") is reconstructed through a triangulation of "u1" and "u2". . . . .

(a)

RMS wall normal velocity profiles obtained in the present study

by using single hot-wires probes l + w = 11.8 at Re θ = 9830 compared with [START_REF] Carlier | Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry[END_REF] at Re θ = 10000. . . . . . . . . 4.9 RMS wall normal velocity profiles obtained in the present study by using single hot-wires probes l + w = 21.96 at Re θ = 19660 compared with [START_REF] Carlier | Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry[END_REF] at Re θ = 20000. . . . . . . . . 4.10 RMS spanwise velocity profiles obtained in the present study by using single hot-wires probes l + w = 11.8 at Re θ = 9830 compared with [START_REF] Carlier | Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry[END_REF] at Re θ = 10000. . . . . . . . . 4.11 RMS spanwise velocity profiles obtained in the present study by using single hot-wires probes l + w = 21.96 at Re θ = 19660 compared with [START_REF] Carlier | Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry[END_REF] at Re θ = 20000. . . . . . . . . 4.12 PDF of normalised streamwise velocity fluctuations from the reconstructed field at 2 KHz (black line), the PIV at 4 Hz (green line), the HWA at 30 KHz (blue diamond) all at y + = 237 and Re θ = 9830 and from [START_REF] Carlier | Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry[END_REF] Turbulence occurs daily in various flow configurations (pipelines, jets, airplane wakes, etc.). It therefore covers a wide range of domains (for e.g in aeronautics, hydraulics, medical and chemical industry) and is of great importance to many engineering applications. According to [START_REF] Hinze | TurbulenceMcGraw-Hill[END_REF], the study of turbulence can be sorted in two distinct categories: a) wall turbulence and b) free turbulence. The first, which is the subject of the present study, includes all the flows where turbulence is generated and continously affected by fixed walls. In contrast, the free turbulence denotes turbulence generated in the abscence of any wall. Wall bounded flows are distinguishable from the others by a thick boundary layer attached to the wall within which high velocity gradients occur. To understand and model this boundary layer is of prime interest in the turbulence research field as it can help to improve the design of transport vehicles in order to reduce fuel consumption and green house gas emissions. Since 1904 when Prandtl simplified the general equations of fluid motion for the boundary layer near the wall, tremendous efforts have been put on the study of near wall turbulence. Despite the fact that turbulence is random in space and time, statistically distinct average values can be discerned [START_REF] Hinze | TurbulenceMcGraw-Hill[END_REF]). Coherent structures (vortices, streaks, ejections and sweeps), with characteristic length, velocity and time scales populate the boundary layer. They are involved in mass and momentum transport and play a key role in Reynolds stress production near the wall. For decades, they have been investigated by many authors who have contributed significantly to the understanding of their origin, the main characteristics (angle, size, energetic contribution ...etc) and the interaction process. Although the boundary layer turbulence is well detailed today, many questions remain unanswered. The present research focuses particularly on very large scale structures in high Reynolds turbulent boundary layer (TBL) which span several boundary layer thicknesses [START_REF] Abe | Very large-scale structures and their effects on the wall shear-stress fluctuations in a turbulent channel flow up to Reτ= 640[END_REF], Ganapathisubramani et al., 2006aHutchins and Marusic, 2007a[START_REF] Lee | Very-large-scale motions in a turbulent boundary layer[END_REF]. They play a crucial role in the turbulence production across the boundary layer [START_REF] Ganapathisubramani | Characteristics of vortex packets in turbulent boundary layers[END_REF], Ganapathisubramani et al., 2005aand Ganapathisubramani et al., 2006b and modulate small scales near the wall (Hutchins and Marusic, 2007b). [START_REF] Adrian | Vortex organization in the outer region of the turbulent boundary layer[END_REF] have proposed a model at very large scale which suggests that hairpin type vortices are bounding regions of low speed fluids with ejections between their legs and sweeps outside. But at high Reynolds numbers, the main characteristics of these structures (size, intensity and life time) and the way they interact with the near wall structures is not fully understood and remains of interest. At these Reynolds numbers, large scale structures are three dimensional complex features which wander in and out of the measurement domain and their main characteristics are known to vary with the wall normal position, being wider and less intense as they move away from the wall. Extracting them, therefore, is not an easy task both for experiments and simulation. Classical flow measurement techniques such as Hot Wires Anemometry (HWA) and Particle Image Velocimetry (PIV) are limited. HWA provides time resolved measurement but usually limited to one velocity component which does not account for the 3D nature of the flow. In PIV, the spatial resolution of the data and the interrogation window size set the range of scales that can be resolved. The first parameter defines the size of the smallest eddies detected and the last one the maximum size of the eddies. Optimizing both of them is difficult and many studies focus on a particular range of scales. Thus, new simulations and PIV experiments with enough spatial and temporal resolution and with a large field of view need to be performed at high Reynolds to complete the existing model. It is for this purpose that an experimental database at high Reynolds numbers (Re θ = 9830 and Re θ = 19660) was built in the frame of the WALLTURB project. Measurements were made in a Zero Pressure Gradient (ZPG) TBL over a flat plate using Stereo PIV at 4 Hz for the spatial resolution and HWA at 30 kHz for the temporal one. Based on such data, an interesting approach is to use Linear Stochastic Estimation (LSE) in order to reconstruct a fully time-resolved field with three velocity components and a good spatial resolution. Then large scales structures, low and high momentum regions as well as vortices are extracted from the reconstructed field and carefully examined in order to complete the understanding of the turbulent boundary at high Reynold numbers and improve existing models. An emphasis is put on the structures main characteristics (size, life time, energetic contribution and wall normal distribution) and their scaling.

The thesis is organized in five chapters and ends with a general conclusion which also includes perspectives. Its content is summmarized as follows:

• Chapter 1 is the literature review of near wall turbulence. The first part reviews the boundary layer theory, the main equations of the boundary layer and its organization are visited. The second part is dedicated to coherent structures. 60 years of investigation on near wall coherent structures and large scale coherent motions are summarized, their formation mechanisms and main characteristics are discussed.

• Chapter 2 describes the experimental set up from which the data used in the present study come from. This set up combines hot wire measurements at 30 KHz and stereo PIV measurements at 4 Hz. The working principles of these two measurements techniques are given at the beginning of the chapter.

• Chapter 3 describes the procedure used to reconstruct a space-time resolved three components velocity field from time resolved hot wire measurements at 30 KHz and space resolved PIV measurements at 4 Hz. A brief review of the Linear Stochastic Estimation is given at the beginning of the chapter, then LSE reconstruction algorithms (matrix inversion, Singular Values Decomposition and Tikhonov Regularization) are presented. Most of the chapter is devoted to the validation of the algorithms. A Direct Numerical Simulation (DNS) database of a turbulent wall bounded flow is used in this purpose. The reconstructed velocity field is compared to a reference field extracted directly from this DNS through velocity contours visualizations, spectral analysis, autocorrelations, etc. In the last section, the necessity to include high order terms in the estimation is discussed.

• Chapter 4 addresses the quality of the reconstructed data. A statistical analysis is performed on the original PIV, HWR data and the reconstructed data. These are compared to results obtained by [START_REF] Carlier | Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry[END_REF] which are used here as reference. The statistics consist of spectra, mean and root mean square profiles, probability density functions, skewness and flatness factors.

• Chapter 5 describes the methods used to detect and characterize coherent structures from the reconstructed field and presents the results obtained. As all the methods use morphological operations such as erosion and dilatation, a literature review of the principles of mathematical morphology is given at the beginning of the chapter. In the following section, the algorithm of detection of uniform momentum region (UMR) and vortex motion centers are described. Contrary to UMR, where a single algorithm is used, for vortices, the Q criterion, λ criterion, Γ 1 and Γ 2 criteria from [START_REF] Graftieaux | Combining PIV, POD and vortex identification algorithms for the study of unsteady turbulent swirling flows[END_REF] and the eddy detection algorithm from [START_REF] Nencioli | A vector geometry-based eddy detection algorithm and its application to a high-resolution numerical model product and high-frequency radar surface velocities in the Southern California Bight[END_REF] were validated on a Lamb Oseen vortex and then applied to the vortical motions in the database. After the detection, the structures are characterized. The mean hydraulic diameter of UMR in a YZ plane normal to the flow direction, their life time and contribution to Reynolds stresses are investigated. Similarly, the vortical motions radius, circulation, vorticity, life time and wall normal evolution are also investigated. The results obtained are discussed with emphasis on the scaling of the structures. At the end of the chapter, the global organization of the coherent structures is investigated through spanwise correlation of their respective binary indicative functions.

• The general conclusion summarizes the study, a more detailed discussion of the results obtained is provided. Our efforts are re-assessed and constructive suggestions are provided. The last part of the conclusion provides some perspectives to the present study. In addition to these characteristics, we can add the non-predictible behaviour of turbulent flows meaning that the evolution of the velocity with time at a given point is not known. Also, turbulence has the property of increasing the mixing. To characterize a turbulent motion, the notion of turbulent scale was introduced. Therefore the length, velocity and time scales refer respectively to the characteristic size, velocity and period of the turbulent motion under investigation.

Reynolds Averaged Navier Stokes equations

The evolution of an incompressible isothermal flow is governed by the following form of the Navier-Stokes equations:

               ∂u i ∂x i = 0 ρ( ∂u i ∂t + u j ∂u i ∂x j ) = - ∂p ∂x i + µ ∂ 2 u i ∂x 2 j (1.1)
Where u(i = 1, 3) is the instantaneous velocity, p is the pressure field, ρ is the density and µ is the dynamic viscosity. Through a Reynolds decomposition the instantaneous field can be expressed as the sum of mean (U i = ūi , P = p) and fluctuating (u i , p ) parts:

       u i = U i + u i p = P + p (1.
2)

The Reynolds decomposition can be introduced in 1.1 to derive the Reynolds Averaged Navier-Stokes equations:

             ∂U i ∂x i = 0 ∂U i ∂t + U j ∂U i ∂x j = - -1 ρ ∂P ∂x i + ∂ ∂x j (2νS ij -u i u j ) (1.3) Where S ij = 1 2 ( ∂U i ∂x j + ∂U j ∂x i
) is the strain tensor and ρu i u j is the Reynolds stress tensor. This last term expresses the effect of turbulence on the mean flow, it is a symetric tensor. The trace of u i u j is equal to twice the turbulent kinetic energy:

k = 1 2 u i u i (1.4)
To solve 1.3 we need to find the six unknows u i u j . This can be achieved based on semi empirical hypothesis linking the Reynolds stress to the mean flow. A standard model is the turbulent-viscosity introduced by Boussinesq in 1877. It is written as:

u i u j = ν t ∂U i ∂x j + ∂U j ∂x i (1.5)
ν t is the turbulent viscosity, it varies with the flow characteristics and can be modeled by solving additional transport equations (see for example [START_REF] Pope | Turbulent flows[END_REF]).

Turbulent boundary layer equation

Near wall turbulent flows at high Reynolds number have a boundary layer character, ie the flow consist of a thin near wall region called boundary layer where strong velocity gradients occur and a region outside the boundary layer where the flow is nearly inviscid. For such flows, the basic equations (1.1) and (1.3) can be considerably simplified. Considering a stationary boundary layer flow in a 2D plane XY , it is a matter of observation that gradients of mean values in the mean flow direction x which is loosely parallel to the wall, are less important than in the transverse direction y ( ∂ ∂x ∂ ∂y ) and the length scales L and l associated with these respective directions are such that l << L. Besides assuming that turbulent diffusion and convection time are equals (see for example "Turbulence et couche limite. 1989"), equation (1.3)

for (u 1 , u 2 ) = (u , v ) and (U 1 , U 2 ) = (U , V ) becomes: ∂U ∂x + ∂V ∂y = 0              U ∂U ∂x + V ∂U ∂y = - 1 ρ ∂P ∂x + ∂ ∂y (ν ∂U ∂y -u v ) 0 = - 1 ρ ∂P ∂y (1.6)
Equations for a Zero Pressure Gradient (ZPG) turbulent boundary layer can be obtained by cancelling the pressure gradient term in the second equation of (1.6).

Turbulent kinetic energy transport equation

The turbulent kinetic energy evolution in space and time is governed by:

∂k ∂t + U j ∂k ∂x j =                                            +(-u i u j ) ∂U i ∂x j (1) -2ν ∂u i ∂x j s ij (2) - 1 ρ ∂ ∂x j p u j (3) + ∂ ∂x j (-k u j ) (4) +2ν ∂ ∂x j u i s ij (5) (1.7)
Term ( 1) is the production term, usually positive, it takes the energy from the mean flow to feed turbulence. Term (2) is a viscous terms, always negative it dissipates the energy into heat essentially at small scale. The last three terms (3), ( 4) and ( 5) are quantities under derivatives and therefore can be viewed as turbulent kinetic energy fluxes. They transport the turbulent kinetic energy from one point to another in a process comparable to diffusion.

For the sake of clarity, in the rest of our study we will keep the notations u , v , w and U ,V , W for turbulent fluctuations and mean velocites in the streamwise, wall normal and spanwise directions respectively.

Boundary layer organization

In a boundary layer, the velocity profile is not driven by the same physical phenomena according to the region under investigation. Therefore, the boundary layer is splitted into two distinct regions. The inner region which is viscous close to the wall and the outer region or wake region which is fully turbulent. This description can be explained with the behaviour of the shear stress τ uv = -ρu v + µ ∂U ∂y which is dominated by -ρu v almost everywhere in the boundary layer except near the wall where the viscous term is high.

Inner region

This region covers about 10% of the boundary layer and extends up to y ≈ 0.1δ. Near the wall, convection terms are negligible compared to viscous and turbulent ones. In the case of a ZPG the streamwise momentum equation (1.6) is rewritten as:

∂τ uv ∂y = 0 (1.8)
Where τ uv = -ρu v + µ ∂U ∂y . By integration, we have τ uv = τ w where τ w is the wall shear stress. u τ and ν u τ are respectively the characteristic velocity and length scales in this region. Through dimensional analysis, the universal form of the profile can be deduced as:

U + = f (y + ) (1.9)
With U + = U u τ and y + = yu τ ν .

Viscous sublayer

In the near wall part of the inner region (y + < 5) viscosity is dominant so that:

τ w = µ ∂U ∂y (1.10)
Taking into account the no slip condition this equation can be integrated as:

U + = y + (1.11)

Outer region

This region covers 90% of the boundary layer from y ≈ 0.1δ to the outer edge of the boundary layer. In this region turbulence effects are dominant, u τ and δ are characteristic velocity and length scale respectively. Through a dimensional analysis we have:

U u τ = f ( y δ ) (1.12)
Generally the deficit law which includes the external velocity is preferred:

U ∞ -U u τ = f ( y δ ) (1.13)

Logarithmic region

At high Reynolds number there is a common region between the inner and the outer region in which the viscous terms are negligible. Asymptotic methods can be applied to the velocity profile of the inner and outer regions with y δ → 0 and y + → ∞ respectively to obtain a logarithmic velocity profile:

U + = 1 κ ln(y + ) + C (1.14)
Where κ (the Von Karman constant) and C are determined experimentally (see for e.g [START_REF] Zanoun | Evaluating the law of the wall in two-dimensional fully developed turbulent channel flows[END_REF]). Standard values are κ = 0.41 and C = 5.0 respectively. Note that there is also a common region called buffer layer between the logarithmic region and the viscous sublayer, for 5 < y + < 50.

Coherent structure investigation in a turbulent boundary layer

The boundary layer is populated by coherent structures (UMR, vortices, ejections and sweeps). These are involved in mass and momentum transport and play a key role in the energy production across the boundary layer. To characterize them and to understand their spatial organization is of prime interest in the field as it can help to improve current boundary layer models. In turn, a better prediction of the wall turbulence yields many engineering applications. For example, in the transport sector, the vehicle design can help to overcome the drag and to reduce the fuel consumption.

In the following, a literature review of CS is presented in two separate frames. The near wall coherent structures in the inner region and the large scale ones which appear at high Reynolds numbers. In both frames, the review focuses on the fundamental questions: how are the CS formed, which role do they play in the layer and what are their main characteristics (size, life time, intensity, energetic contribution, etc).

Near wall coherent structures

Streaks

General introduction: Streaks are streamwise elongated low or high momentum regions which populate the inner part of a turbulent boundary layer. They were first observed by [START_REF] Hama | Boundary-layer characteristics for smooth and rough surfaces[END_REF]; and [START_REF] Kline | Coherent structure of turbulent boundary layers[END_REF] suggested that they constitute a universal feature of bounded shear flows. In those years, their presence was highlighted by means of flow visualization techniques such as dye injection (see [START_REF] Oldaker | Spatial structure of the viscous sublayer in drag-reducing channel flows[END_REF]) and the hydrogen bubble technique (see an illustration from [START_REF] Kline | The structure of turbulent boundary layers[END_REF] depicted in Figure 1.1). Due to the improvement of numerical simulations and measurements techniques such as PIV, we now have access to the full velocity field associated with a given flow. Therefore, most of the studies interested in educing streaky structures take the information from this quantity. Either by thresholding the streamwise velocity component at a certain level (see [START_REF] Wallace | The wall region in turbulent shear flow[END_REF]) or from unconditional statistics such as autocorrelation built upon the same velocity field. While the first appproach requires choosing an adequate threshold to highlight the coherent structures, the second one requires enough samples to converge the statistics. As suggested by [START_REF] Landahl | On sublayer streaks[END_REF], streaky structures contribute to Reynolds stress production near the wall. They interact with vortices and play a key role in the near wall flow dynamics. For this reason, they have been investigated in various flow configurations. We can cite ZPG turbulent boundary layers [START_REF] Kline | The structure of turbulent boundary layers[END_REF], [START_REF] Runstadler | An experimental investigation of the flow structure of the turbulent boundary layer[END_REF] and [START_REF] Robinson | A review of quasicoherent structures in a numerically simulated turbulent boundary layer[END_REF]) and channel flows [START_REF] Oldaker | Spatial structure of the viscous sublayer in drag-reducing channel flows[END_REF]), [START_REF] Blackwelder | Streamwise vortices associated with the bursting phenomenon[END_REF], [START_REF] Smith | The characteristics of low-speed streaks in the near-wall region of a turbulent boundary layer[END_REF] and Acarlar and Smith (1987a)). Once extracted, most of the studies investigate their relevant characteristics which should provide information on their spatio-temporal organization as well as their impact on the turbulent boundary layer dynamics. In the literature, we have found investigations on the streaks length, width, spanwise angle, spanwise spacing, frequency of appearance and energetic contribution to the flow. Besides studies on their merging, bursting and meandering processes can be added. Some other studies investigate the origin of these streaky structure. This point is controversial and is detailed in the next section.

Origin of the near wall streaks: The origin of the streaks is still a subject of controversy. From the first study by [START_REF] Stuart | Hydrodynamic stability[END_REF] until now, two major theories that could explain the streaks origin have been proposed. The first one suggests Figure 1.2 -Schematic illustrating how counter-rotating vortex pairs generate streaks [START_REF] Blackwelder | Streamwise vortices associated with the bursting phenomenon[END_REF]).

that streaks originate from wall normal motions close to the wall. However, the authors do not agree on the nature of these motions, primarily it was claimed that low and high speed streaks are created by counter-rotating vortex pairs that pump low speed fluid away from the wall and high speed fluids towards the wall respectively (see an illustration from [START_REF] Blackwelder | Streamwise vortices associated with the bursting phenomenon[END_REF] in Figure 1.2 and related studies from [START_REF] Stuart | Hydrodynamic stability[END_REF] and [START_REF] Ersoy | Viscous flow induced by counter-rotating vortices[END_REF]) . Later, studies by [START_REF] Smith | On the dynamics of near-wall turbulence[END_REF], Acarlar and Smith (1987a) and [START_REF] Haidari | The generation and regeneration of single hairpin vortices[END_REF] suggested that instead of simple streamwise vortices, the vortices that give birth to streaks are part of bigger structures termed hairpins. It was claimed that counter-rotating streamwise legs of hairpins lift up the near wall fluid between them. This results in streamwise elongated low momentum regions with characteristics and features similar to boundary layer streaks. The second theory suggests that streaks are created by the combined action of wall normal motions, mean shear and diffusion. Figure 1.3 summarizes results from [START_REF] Chernyshenko | The mechanism of streak formation in near-wall turbulence[END_REF] who have investigated the theories on the origin of streaks trough numerical tests and compared their results with numerical experiments. In this figure, the surface denotes the boundary of a region of high concentration of a passive scalar (smoke) released from the wall. Initially, in Figure1.3 (a) the boundary is assumed to be flat but the lift up by wall normal motions will deform it ( Figure 1.3 (b)). Now, the distorted patterns in Figure 1.3 (b) are visualized in a plane parallel to the wall and just above the initial flat surface, one can observe the elliptical shape patterns. This indicates that the wall normal motions alone cannot generate streaks. In Figure 1.3 (c) The mean shear stretches and tilts downstream the lifted volumes but still no streaky patterns can be observed in the vizualisation plane. However, the tilting and the stretching increase the wall normal gradient leading to wall normal diffusion. Finally, a streaky pattern appears in the visualisation plane (see Figure 1.3 (d)).

In the same study, it was shown that removing the patterns of wall normal motions will not take away streaks. Therefore, these results suggest that contrary to the first theory which attributes the streaks origin to vortices and hairpins, the second theory is more likely to explain the mechanism of streaks formation. In the literature there are fewer references in favour of this theory compared to the fisrt one. However, since the effect of lift up, shear and diffusion are governed by linearized Navier-stokes equations and they are sufficient to explain the origin of streaks in the second theory, we can cite [START_REF] Landahl | On sublayer streaks[END_REF] who has investigated streaks origin with a theoretical model in a viscous sublayer. In his model, the fluctuating parts of the flow were following the linear interaction with the mean flow whereas the non-linearities were intermittent and locally distributed in space and time. With contours of u plotted in XZ plane they evidence streamwise elongated low and high speed streaks within the viscous sublayer. In addition, [START_REF] Butler | Optimal perturbations and streak spacing in wall-bounded turbulent shear flow[END_REF] have investigated the streaks within a channel flow with linear three dimensional optimal pertubation. Their results suggest that streaks formation is a linear process which can be predicted using optimal pertubations. By limiting the life time of the pertubation to the eddy turn-over time they obtain streaks spacing comparable to the experiment. Overall the two concepts are not exclusive, streaks characteristics investigated in real turbulent flows suggest that they are determined by both wall normal motions and the combined action of lift-up, shear and diffusion. However, choosing among the two concepts the one more likely to explain the streaks origin is still under investigation and beyond the scope of this thesis.

Main characteristics of streaks Low speed streaks are always associated with high speed ones. In the boundary layer, they appear respectively as alternated low and high momentum regions. These regions were found to oscillate in span (see [START_REF] Kline | The structure of turbulent boundary layers[END_REF] and [START_REF] Oldaker | Spatial structure of the viscous sublayer in drag-reducing channel flows[END_REF] ), this phenomenon is known as meandering. In the study by [START_REF] Kline | The structure of turbulent boundary layers[END_REF] low speed streaks and intermittent ejections were investigated in water channel flows. By means of hydrogen bubble technique oscillating low speed streaks were observed at y + = 2.7 and y + = 4.5. These authors found streaks shorterned and more wavy in the case of an adverse pressure gradient whereas they were large and quiescent in the case of a favourable one. When the latter was strong enough the bursting of streaks was cancelled and the flow was relaminarized. In the same study, [START_REF] Kline | The structure of turbulent boundary layers[END_REF] found that streaks reduce in number when moving away from the wall and they break up between y = 10 + and y = 30 + . Within the sublayer, low speed streaks tend to be longer than high speed ones and this difference is more pronounced at high Reynolds numbers (Robinson (1991)). As reported in [START_REF] Carlier | Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry[END_REF], except from their length which varies widely, there is an agreement on streaks characteristics in the literature. Low speed streaks appear under 10 + where they are quiet. They are between 500 + and 2000 + long, between 20 + and 40 + wide and between 5 + and 10 + high. The spanwise spacing varies widely between 50 + and 300 + . Particularly, [START_REF] Kline | The structure of turbulent boundary layers[END_REF] found an average spanwise spacing of sublayer streaks between 100 + and 150 + . Similarly, [START_REF] Oldaker | Spatial structure of the viscous sublayer in drag-reducing channel flows[END_REF] found an average spacing of 100 ± 15 + in water channel flows and 93 + in dilute polymer flows without drag reduction. [START_REF] Smith | The characteristics of low-speed streaks in the near-wall region of a turbulent boundary layer[END_REF] have investigated low speed streaks characteristics within a water channel turbulent boundary layer. High speed video cameras and hydrogen bubble wire were used for visualisation. For 740 ≤ Re θ ≤ 5830 they found that the average spanwise spacing of low speed streaks is λ s = 100 + independently of the Reynolds, and for 1 < y + < 30 the histogram of this spacing collapse on a log normal law. The evolution with wall distances up to y + = 30 was investigated at Re θ = 2030. The streaks spacing appeared to increase as y + increases due to the merging behaviour and intermittency of streaks. The merging behaviour started at y + ≥ 5 and was most pronounced for 10 ≤ y + ≤ 30, its effect was to fortify existing streaks and to reduce the number of low speed regions within the flow. The intermittency, manifested as appearance and disappearance for low speed streaks, was observed for y + ≥ 20. [START_REF] Lin | Quantitative characterization of coherent structures in the buffer layer of near-wall turbulence. Part 1: streaks[END_REF] performed stereo PIV experiment over a flat plate turbulent boundary layer at Re θ = 7800. From their database, streaks were retrieved by thresholding normalized streamwise velocity fluctuations with a value of 0.6 × σ max u /σ n u . Where σ n u and σ max u were respectively the standard deviation of the streamwise velocity component at a position n and the maximum standard deviation in the flow. Then, streaks characteristics namely frequency of appearance, spanwise angle, width and spanwise distance were investigated for various wall normal distances in the range 14.5 < y + < 48. For y + ≤ 30, the frequency of appearance N f of low speed streaks and high speed streaks evolved differently. For high speed streaks N f decreased fast with wall normal distance whereas for low speed streaks it was almost constant. For y + > 30 the frequency of both type of streaks decreased with wall normal distance. Most of the streaks were found to be aligned with the streamwise direction, many of them move with a spanwise angle up to 15°and values above 25°were attributed to the meandering. High speed streaks were found to be wider than low speed ones. Their width in wall units ranged from 42 to 47 whereas for low speed streaks it was around 31 for 14.5 < y + < 33 and up to 39 at y + = 48. The spanwise distance between the center of two nearby streaks was found to decrease with the wall distance. For both types of streaks this distance ranged between 114 + and 195 + .

Vortices

General introduction As defined by [START_REF] Robinson | A review of quasicoherent structures in a numerically simulated turbulent boundary layer[END_REF], a vortex is a coherent structure which exhibits circular or spiral instantaneous streamlines in a plane normal to its core when viewed in a reference frame moving with the center of the vortex core. With streaks, vortices are the most studied coherent structures in wall turbulence. They correspond to low pressure zones which perturb the mean pressure field and are involved in mass and momentum transport across the boundary layer. In addition, preventing vortex generation can substantially reduce the drag and heat transfer in wall bounded flows (see [START_REF] Schoppa | Coherent structure generation in near-wall turbulence[END_REF]). In literature, there are different methods to identify vortical structures. Some are based on the velocity gradient tensor: λ criterion (or swirling strength) and Q criterion [START_REF] Jcr | Eddies, stream, and convergence zones in turbulent flows[END_REF], [START_REF] Soria | Identification and classification of topological structures in free shear flows[END_REF] andKolář ( 2007))), isosurfaces of vorticity and vorticity lines [START_REF] Matheron | The structure of the vorticity field in turbulent channel flow. Part 1. Analysis of instantaneous fields and statistical correlations[END_REF]). Others use pressure minima [START_REF] Robinson | Statistical analysis of nearwall structures in turbulent channel flow[END_REF]) or closed or spiral streamlines [START_REF] Lugt | The dilemma of defining a vortex[END_REF] and [START_REF] Nencioli | A vector geometry-based eddy detection algorithm and its application to a high-resolution numerical model product and high-frequency radar surface velocities in the Southern California Bight[END_REF]). A detailed review of the vortex identification methods can be found in [START_REF] Zhou | Mechanisms for generating coherent packets of hairpin vortices in channel flow[END_REF] and [START_REF] Jiang | Detection and Visualization of[END_REF]. In near wall turbulence, 3D vortices are named according to their shape. Thus, horseshoe or loop vortices will refer to horseshoe type vortices with aspect ratio close to one. Moving away from the wall and when the Reynolds number increases, these structures are stretched into hairpins. Hairpins vortices are 3D vortices which exhibit two elongated trailing legs almost aligned with the mean flow, their tip is usually referred to as hairpin head and is connected to the legs via necks inclined at 45°. It is common to observe asymmetric hairpins (or cane vortices) with a head and a single leg. Hairpins can evolve into Ω shape vortices by self induction [START_REF] Moin | Evolution of a curved vortex filament into a vortex ring[END_REF] and [START_REF] Hon | Evolution of hairpin vortices in a shear flow[END_REF]). Λ shape vortices are idealized forms of hairpin introduced by [START_REF] Perry | On the mechanism of wall turbulence[END_REF] in order to model the turbulent boundary layer statistics. Quasi-streamwise vortices are the dominant features in the buffer layer while arch like vortices are more prominent in the wake region. In the log region they both exist often as part of the same vortical structure [START_REF] Robinson | A review of vortex structures and associated coherent motions in turbulent boundary layers[END_REF]). The three dimensional vortices can be mapped into planes [START_REF] Carlier | Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry[END_REF], [START_REF] Herpin | The organization of near-wall turbulence: a comparison between boundary layer SPIV data and channel flow DNS data[END_REF] ...etc): the XY plane (spanwise or transverse vortices), YZ plane (streamwise or longitudinal vortices) and planes inclined at 45°or 135°to the wall. Spanwise vortices include prograde (ω < 0) and retrograde (ω > 0) vortices. The prograde vortices rotate in the same sense as the mean shear and correspond to hairpin heads (Wu and Christensen ( 2006)), retrograde vortices are either signature of the lower part of the neck of an Ω vortex or are independent structures which rotate against the mean shear (Ganapathisubramani et al. (2006b) and [START_REF] Natrajan | Spatial signatures of retrograde spanwise vortices in wall turbulence[END_REF]). Streamwise vortices correspond to streamwise aligned vortex which are sometimes the legs of hairpins. In planes inclined at 45°(respectively 135°) to the wall, vortices detected are often part of 3D hairpin type vortices tilted upstream (respectively downstream) at 45°to the wall, these are referred to as backward (respectively forward) leaning hairpins. In the following lines, we will present different mechanisms proposed to explain the formation of near wall vortices and their main characteristic (size, intensity, inclination angle, convection velocity and organization). The next section focuses on hairpins vortices and the further one on streamwise and spanwise ones.

Hairpins [START_REF] Theodorsen | Mechanism of turbulence[END_REF] was the first to propose a hairpin model to describe the near wall dynamics. His model was derived from the vorticity transport equation and was used to explain the production and dissipation near the wall. 3D hairpin type vortices straddling low speed fluid were found with, their heads mainly inclined at 45°to the wall. This dominant inclination angle correspond to the angle where the vortex stretching is maximum [START_REF] Head | New aspects of turbulent boundarylayer structure[END_REF]) and it results from a competition between hairpin self backward induced velocities and shear. In fact, stretched vortices have smaller spanwise separation and thus higher backward induced velocity which rotates the hairpin up away from the wall. This action is counter-balanced by the mean shear. However, even if 45°is dominant near the wall, the angle distribution is wide owing to viscous dissipation which attenuates the backward induced velocity. This competition between lift up and shear is sometimes used to explain the origin of hairpin type vortices. For example, [START_REF] Zhou | Mechanisms for generating coherent packets of hairpin vortices in channel flow[END_REF] performed DNS of a low Reynolds number channel flow to study the generation mechanism of hairpin vortices. From a velocity field estimated via LSE with Q2 events as condition, the vortical structures were extracted using the imaginary part of the complex eigen value of the velocity gradient tensor. This primary structure, consisted of a pair of counter-rotating streamwise vortices with a narrow spanwise separation downstream. Owing to self induction, the vortex were found to lift up more rapidly at their downstream end. Further, with the opposing action of shear induced stretching the vortex pairs roll up into hairpins. [START_REF] Willmarth | Structure of turbulence in the boundary layer near the wall[END_REF] described a similar mechanism of hairpin vortex generation, but the primary structure consists of spanwise vorticity. Experimentally, hairpins were generated by injecting a low speed fluid into a laminar boundary layer [START_REF] Haidari | The generation and regeneration of single hairpin vortices[END_REF] and Acarlar and Smith (1987b)) and by introducing 3D distubances in a flat plate boundary layer using vibrating ribbon technique [START_REF] Klebanoff | The three-dimensional nature of boundary-layer instability[END_REF]) or an hemispherical bump (Acarlar and Smith (1987b)). For low speed fluid injections, the interaction with outerflow high speed fluid creates a shear layer which is stretched and deformed into a hairpin vortex. Secondary hairpins are formed in the hairpin legs plane of symmetry by the interaction between the low speed fluid lifted up in between the hairpins legs and the outer flow high speed fluid. For the vibrating ribbon technique , the hairpins are associated with high frequencies fluctuations observed during the breakdown of non linear 3D wave motions. This idea is also supported by [START_REF] Black | An analytical study of the measured wall pressure field under supersonic turbulent boundary layers[END_REF] who reported a periodic breakdown in the viscous sublayer.

Despite the 3D nature of hairpins, most of the studies which have investigated their characteristic (size, life time, intensity ...etc) in the wall region performed measurements within planes. In addition, reporting their characteristic can be confusing as the dominant structure in the wall region are quasi streamwise vortices which are sometimes (but not always) associated with the 3D hairpins. Therefore, the following characteristics of hairpin type structure is limited to 3D loop vortices, horseshoes, hairpins, canes and Ω shaped vortices in the wall region (buffer layer and a small portion of the log region) and at moderate Reynolds number where large and very large scale structures are not yet visible. Characteristics of hairpins educed in planes will be reported in the next section together with streamwise, spanwise and 135°and 45°inclined plane vortices.

The shape and the size of hairpins near the wall strongly depend on the Reynolds number, [START_REF] Head | New aspects of turbulent boundarylayer structure[END_REF] proposed ν/u τ as length scale for hairpins and ν/u 2 τ as mean life time of vortex pairs. These hairpins are found to be aligned in the streamwise direction in groups of two or three (hairpin packets) and induce a low momentum region (LMR) in between their legs. In a laminar boundary layer and for y+ < 200, [START_REF] Smith | A synthesized model of the near-wall behavior in turbulent boundary layers[END_REF] found an average streamwise spacing of 200 + between successive hairpins. [START_REF] Zhou | Formation of coherent hairpin packets in wall turbulence[END_REF] and [START_REF] Zhou | Mechanisms for generating coherent packets of hairpin vortices in channel flow[END_REF] with low Reynolds (Re τ = 180) channel flow DNS found an average streamwise spacing between asymetric hairpins heads in the range 165 + -220 + , [START_REF] Adrian | Vortex organization in the outer region of the turbulent boundary layer[END_REF] obtained 104, 139 and 144 for a turbulent boundary layer at Re θ equals to 930, 2370 and 6845 respectively. The hairpins are characterized by a dominant inclination angle of 45°with the wall, 15°-30°c orrespond to the range of angle made by a single line connecting packets hairpins heads to the wall [START_REF] Head | New aspects of turbulent boundarylayer structure[END_REF] and [START_REF] Smith | A synthesized model of the near-wall behavior in turbulent boundary layers[END_REF]).

Streamwise and spanwise vortices A complete literature on near wall streamwise vortex fomation can be found in [START_REF] Schoppa | Coherent structure generation in near-wall turbulence[END_REF], the concepts from them are summarized in this section.

The mechanisms explaining the formation of near wall streamwise vortices can be divided into two categories. The first category is known as offspring scenario which requires pre-existing vortices to trigger the formation of new ones, the second one is based on mean flow perturbation by instabilities. In the instability based mechanism, the newly generated vortex is sometimes observed to create a new instability in such way to close the regeneration cycle. Pre-existing vortices therefore act indirectly in the generation process similar to the offspring scenario. Instability based mechanisms include Taylor-Gortler instability [START_REF] Gortler | Uber eine dreidimensionale Instabilitat laminarer Grenzschichten an konkaven Wanden[END_REF]), Craik Leibovich type 2 instability [START_REF] Phillips | On the instability of wave-catalysed longitudinal vortices in strong shear[END_REF]), oblique mode interaction and streaks instability. In Gortler mechanism, local wall curvatures create instabilities which in turn form streamwise vortices. Craik Leibovich type 2 instability at the origin of streamwise vortices are created by the bounding surface periodicity. [START_REF] Benney | A non-linear theory for oscillations in a parallel flow[END_REF] has shown that the growth of secondary flow pertubations associated with the non linear interaction of primary oblique modes gives rise to longitudinal vortices in a mixing layer. Similarly, [START_REF] Waleffe | On the origin of streaks in turbulent shear flows[END_REF] used direct resonance technique (Benney and Gus-tavsson (1981)) to study the relationship between wall normal velocity oblique mode and streamwise vortex formation. This technique consists in applying a resonant forcing by the velocity on vertical vorticity, and only oblique modes are considered, as non linear effects are minor with these modes. [START_REF] Waleffe | On the origin of streaks in turbulent shear flows[END_REF] found that direct resonance trigger oblique wall normal velocity mode growth and the non linear interaction of these modes further create streamwise vortices. In the framework of streak instablity mechanism, Robinson (1991) suggested that local shear instablities consisting of inflectional velocity profile located at the streak crest catalyse the streamwise vortex formation. [START_REF] Hamilton | Regeneration mechanisms of near-wall turbulence structures[END_REF] studied streaks breakdown in the near wall region of a minimal Couette flow. The breakdown appears as a consequence of sinuous instability and complex non linear interactions which occured after this breakdown generated streamwise vortices. [START_REF] Schoppa | Coherent structure generation in near-wall turbulence[END_REF] made a review of all the aformentioned streamwise vortices generation mechanism and found that none of these can fully explain both the vortex generation and regeneration. Alternatively, they proposed a streak transient growth mechanism in which a pertubation of streamwise vorticity with a moderate amplitude ( ω 2 x = 0.5) will grow into an ω x sheet which is further stretched by ∂u ∂x into streamwise vortices. The stretching of ω x sheet by ∂u ∂x is associated with streak weaviness and no parent vortices is required to initiate the process. As mentioned in section 1.2.1, spanwise vortices include prograde (ω < 0 and retrograde (ω > 0) vortices. The prograde vortices correpond to the head of hairpins whose formation mechanism were described above. The retrograde vortices are either signature of the lower part of the neck of an Ω vortex or are independent structures which rotate against the mean shear. These vortices are less numerous than the the prograde ones within the near wall region, therefore, they are less documented. However, [START_REF] Kim | The production of turbulence near a smooth wall in a turbulent boundary layer[END_REF] suggested that Helmholtz type instability can explain the formation of both prograde and retrograde vortices, unstable high/low speed fluid interface can roll up into transverse vortices.

Wall region dominant structures are streamwise aligned counter-rotating vortex pairs, which are associated with both ejections and sweeps and are often observed to lift up a low momentum region. These vortices are more numerous and smaller than spanwise vortices essentially located in the outer region of the boundary layer. [START_REF] Herpin | Influence of the Reynolds number on the vortical structures in the logarithmic region of turbulent boundary layers[END_REF] have investigated streamwise and spanwise vortices within a flat plate turbulent boundary layer, and reported that y + = 90 corresponds to the maximum density of vortices near the wall. Their radius increases with the wall normal distance [START_REF] Tomkins | Spanwise structure and scale growth in turbulent boundary layers[END_REF] and [START_REF] Gao | Analysis of vortex populations in turbulent wall-bounded flows[END_REF]), while, their circulation and vorticity are found to decrease [START_REF] Gao | Analysis of vortex populations in turbulent wall-bounded flows[END_REF] and [START_REF] Herpin | Influence of the Reynolds number on the vortical structures in the logarithmic region of turbulent boundary layers[END_REF]). The mean radius evolution with the wall normal distance is independent on the Reynolds number when scaled in wall units [START_REF] Herpin | Influence of the Reynolds number on the vortical structures in the logarithmic region of turbulent boundary layers[END_REF]). Through 3D swirling strength [START_REF] Gao | Analysis of vortex populations in turbulent wall-bounded flows[END_REF] identified vortices in channel flow DNS datasets at Re τ = 937, mean radius of 13.7 + and 14.3 + and mean circulation of 175 + and 130 + were found at y + = 47 and y + = 110 respectively. In addition, the average spanwise distance between near wall counter-rotating streamwise vortices was 100 + [START_REF] Gupta | Spatial structure in the viscous sublayer[END_REF] and [START_REF] Lee | Identification of turbulent wall eddies through the phase relation of the components of the fluctuating velocity gradient[END_REF]). Near the wall, prograde vortices are convected with the mean streamwise velocity and retrograde vortices ones are moving sligthly faster than the mean in the range 40 < y + < 100 [START_REF] Herpin | Influence of the Reynolds number on the vortical structures in the logarithmic region of turbulent boundary layers[END_REF]). In addition, distribution of convection velocities in [START_REF] Gao | Analysis of vortex populations in turbulent wall-bounded flows[END_REF] suggest that the eddies are convected at 96% -98% of the local mean.

Ejections and sweeps

In order to investigate turbulent production and transport processes in the near wall region, [START_REF] Corino | ER & Brodkey, RS[END_REF] suspended colloidal size solid particles in a liquid (trichloroethylene) and tracked their motions with a high speed camera. As results, they observed abrupt motions of low speed fluid directed outward and originating from the wall (ejections) and inward motions of high speed fluid directed towards the wall (sweeps). These two were found to be part of a deterministic sequence of event which occurs randomly in space and time. Firstly, they noticed the appearance of a decelerated region with an axial velocity equal to 50% the local mean velocity, in the next step an accelerated mass of fluid from the upstream side moving wallward at angle 0 -15°gradually penetrates the retarded flow at y + ≈ 15. While this flow is accelerated, an ejection of low speed fluid away from the wall is observed. Finally, a sweep events comes to close the cycle. The interface between the mean flow and the retarted one was a zone of high shear. Ejected fluid elements interacted with this shear region and created chaotic motions. Therefore, in the boundary layer ejections are the main transport agent of turbulent energy outward from the wall whereas sweeps transfer it towards the wall [START_REF] Nychas | A visual study of turbulent shear flow[END_REF]).

Ejections originate from the wall between 5 < y + < 15 and interacts between 7 < y + < 30 [START_REF] Corino | ER & Brodkey, RS[END_REF]). Their inclination angle to the wall is small and weakly depends on the Reynolds number, [START_REF] Runstadler | An experimental investigation of the flow structure of the turbulent boundary layer[END_REF] proposed 10 -13°whereas [START_REF] Corino | ER & Brodkey, RS[END_REF] found 8.5°in average. The ejections frequency of occurence scales with outer flow units [START_REF] Rao | The 'bursting'phenomenon in a turbulent boundary layer[END_REF]) and increases with the Reynolds. Together with sweeps, they play a key role in the turbulent production and their interaction dissipates energy at small scales. Appart from [START_REF] Lu | Measurements of the structure of the Reynolds stress in a turbulent boundary layer[END_REF], who found that ejections contribute to about 70% and sweeps to 55% of the Reynolds shear stress -u v within the sublayer, the studies agrees on the predominance of sweeps in direct vicinity of the wall (for e.g. [START_REF] Raupach | Conditional statistics of Reynolds stress in rough-wall and smooth-wall turbulent boundary layers[END_REF] and [START_REF] Grass | Structural features of turbulent flow over smooth and rough boundaries[END_REF]). Moving away from the wall the sweeps contribution is diminished and the ejections ones increases [START_REF] Dennis | Coherent structures in wall-bounded turbulence[END_REF]). [START_REF] Corino | ER & Brodkey, RS[END_REF] estimated roughly that ejections contribute to 70% of -u v stress and sweeps to 30% by difference. At y + = 15 [START_REF] Wallace | The wall region in turbulent shear flow[END_REF] reported 70% from ejections and sweeps separetaly and -40% from their interaction.

Many theories has been proposed to explain the origin of ejections and sweeps. Ejections are known to be the results of streaks lift up described by [START_REF] Kline | The structure of turbulent boundary layers[END_REF] which are observed to oscillate in the buffer layer and break up in 10 ≤ y + ≤ 30 [START_REF] Dennis | Coherent structures in wall-bounded turbulence[END_REF]). They were also associated with outer region tranverse vortices which by modifying the pressure field induces condition for their generation [START_REF] Nychas | A visual study of turbulent shear flow[END_REF] and [START_REF] Praturi | A stereoscopic visual study of coherent structures in turbulent shear flow[END_REF]). Finally, [START_REF] Thomas | On the role of wall-pressure fluctuations in deterministic motions in the turbulent boundary layer[END_REF] found a close relationship between high pressure fluctuations during a bursting (ejection and sweep) cycle and inclined shear layers occuring on the upstream part of the bursting.

Quadrant analysis [START_REF] Wallace | The wall region in turbulent shear flow[END_REF]) and VITA (variable time averaging) technique are commonly used to evidence sweeps and ejections . Quadrant analysis, consists of splitting u v signal into four parts, Quadrant two (u < 0, v > 0) and quadrant four (u > 0, v < 0) correspond to ejections and sweeps events (also called Q2 and Q4 events respectively). Quadrant one (u < 0, v < 0) and quadrant three (u > 0, v > 0) correspond to the interaction between them (also called Q1 and Q3 events respectively). VITA technique compares the short term RMS of veocity fluctuation signal with the long term one, an adequately chosen threshold should provided ejections. For a complete review of ejections and sweeps detection methods the reader is refered to [START_REF] Yuan | A comparison study of conditional-sampling methods used to detect coherent structures in turbulent boundary layers[END_REF] and [START_REF] Wark | Experimental investigation of coherent structures in turbulent boundary layers[END_REF]).

Large and Very large scale coherent structures

In this section, the origin of Large and Very Large Scale motions which appear at relatively high Reynolds numbers is discussed as well as their statistical characteristics and related structures (hairpins, sweeps and ejections). Outer log region and wake region coherent structures characteristics are presented at both low and high Reynolds numbers to complete the above boundary layer description which was limited to near wall coherent structures at low Reynolds number.

Recent experiments and numerical studies have revealed the existence of very long meandering structures consisting of alternance of low and high speed fluid within the logarithmic and wake regions [START_REF] Adrian | Vortex organization in the outer region of the turbulent boundary layer[END_REF], [START_REF] Abe | Very large-scale structures and their effects on the wall shear-stress fluctuations in a turbulent channel flow up to Reτ= 640[END_REF], Ganapathisubramani et al. (2006a), Hutchins and Marusic (2007a) and [START_REF] Lee | Very-large-scale motions in a turbulent boundary layer[END_REF]). The terms Large Scale Motion (LSM) and Very Large Scale Motion (VLSM) refer respectively to structures with a streamwise extent of 1 -3δ and greater than 3δ. VLSMs result from the concatenation of streamwise aligned LSMs moving at different speeds. As suggested by channel flow DNS of [START_REF] Lee | Spatial organization of large-and very-large-scale motions in a turbulent channel flow[END_REF], streamwise convection velocities of large scale uniform momentum zones are proportional to both the velocity fluctuations sign and the magnitude, therefore, when two LSMs yielding small velocity differences are aligned along the streamwise direction, they are often observed to merge in a structure of larger streamwise extent (VLSM). However, [START_REF] Smits | High-Reynolds number wall turbulence[END_REF] have sorted LSMs into two categories: wall attached and detached LSMs, and suggested that wall attached LSMs with small spanwise extent are not involved in the merging scenario leading to VLSMs. In turn, [START_REF] Lee | Spatial organization of large-and very-large-scale motions in a turbulent channel flow[END_REF] futher suggest that LSMs result from the growth and merging of streamwise aligned structures of smaller scales. Because they play an important role in the turbulence production across the boundary layer [START_REF] Ganapathisubramani | Characteristics of vortex packets in turbulent boundary layers[END_REF], Ganapathisubramani et al. (2005a) and Ganapathisubramani et al. (2006b)) and are thought to be responsible for small scales amplitude modulation near the wall (Hutchins and Marusic (2007b)) many authors have investigated both of them in turbulent wall layers for various flow configurations (zero pressure gradient turbulent boundary layer, [START_REF] Ganapathisubramani | Characteristics of vortex packets in turbulent boundary layers[END_REF], [START_REF] Hambleton | Simultaneous orthogonalplane particle image velocimetry measurements in a turbulent boundary layer[END_REF], [START_REF] Hutchins | Three-dimensional conditional structure of a high-Reynolds-number turbulent boundary layer[END_REF] and [START_REF] Lee | Very-large-scale motions in a turbulent boundary layer[END_REF], pipe flows, [START_REF] Guala | Large-scale and very-largescale motions in turbulent pipe flow[END_REF], channel flows, [START_REF] Monty | A comparison of turbulent pipe, channel and boundary layer flows[END_REF], [START_REF] Christensen | Statistical evidence of hairpin vortex packets in wall turbulence[END_REF]). Some general conclusions can be drawn from these studies.

First, both low and high speed regions share similarities in averaged size. [START_REF] Sillero | Two-point statistics for turbulent boundary layers and channels at Reynolds numbers up to δ+ ≈ 2000[END_REF] have suggested that this is true only for pipe and turbulent boundary layer flows whereas there is a discrepancy between the two structures in channel flows. Conversely, Dennis and Nickels (2011b) found that low speed regions within a zero pressure turbulent boundary layer are slightly longer than high speed ones and their energetic contribution to the Reynolds shear stress u v is more significant. Secondly, the streamwise length of these structures scales with δ, increases with the wall normal distance in the log region and decreases beyond [START_REF] Ganapathisubramani | Characteristics of vortex packets in turbulent boundary layers[END_REF] and Ganapathisubramani et al. (2006a)). Their spanwise width increases monotonically with the wall normal distance [START_REF] Tomkins | Spanwise structure and scale growth in turbulent boundary layers[END_REF], Hutchins and Marusic (2007a) andLee andSung (2011)). [START_REF] Ganapathisubramani | Characteristics of vortex packets in turbulent boundary layers[END_REF] and [START_REF] Hambleton | Simultaneous orthogonalplane particle image velocimetry measurements in a turbulent boundary layer[END_REF] used stereo PIV to investigate LSM in the log region of a zero pressure gradient turbulent boundary layer at Re θ ≈ 2500 and found structures with average streamwise extent equal to 2δ. Through DNS, [START_REF] Lee | Very-large-scale motions in a turbulent boundary layer[END_REF] found 6δ for Re θ = 570 -2500 and further studies at Re θ = 35000 (PIV of a supersonic TBL by Ganapathisubramani et al. (2006a)) and Re τ = 6.6 × 10 5 (HWR measurements of an atmospheric surface layer by Hutchins and Marusic (2007a)) revealed structures whose length can go up to 8δ and 20δ respectively. The structure size greatly depends on the threshold applied either on the streamwise velocity fluctuations or on the streamwise velocity autocorrelations but, the literature agrees on the increase of the streamwise extent with the Reynolds number. Additionnally, [START_REF] Lee | Spatial organization of large-and very-large-scale motions in a turbulent channel flow[END_REF] found that the distribution of LSMs decreases as their length increases, [START_REF] Lee | Very-large-scale motions in a turbulent boundary layer[END_REF] and [START_REF] Ganapathisubramani | Characteristics of vortex packets in turbulent boundary layers[END_REF] noticed a peak at 150 + (0.25δ) and 100 + respectively. [START_REF] Tomkins | Spanwise structure and scale growth in turbulent boundary layers[END_REF] used PIV to investigate large scale structure in a flat plate turbulent boundary at high Reynolds Re θ = 7705 and suggested that the dominant large scale structure have a spanwise extent in the range 0.1δ -0.4δ. The LSM width distributions reported in [START_REF] Ganapathisubramani | Characteristics of vortex packets in turbulent boundary layers[END_REF] and [START_REF] Lee | Very-large-scale motions in a turbulent boundary layer[END_REF] at various wall normal positions yield peaks between 50 + -80 + and 60 + -115 + respectively. The LSM have an average inclination angle of 10°to the wall [START_REF] Lee | Very-large-scale motions in a turbulent boundary layer[END_REF]), the most probable angle increases with y faster near the wall than in the log region [START_REF] Lee | Spatial organization of large-and very-large-scale motions in a turbulent channel flow[END_REF]).

Together with a detailed analysis of large scale vortices, a large scale motion model was provided in [START_REF] Adrian | Vortex organization in the outer region of the turbulent boundary layer[END_REF]. This model illustrated in Figures 1.4 and 1.5 suggests that hairpin-type vortices are bounding the regions of low speed fluid, with ejections between their legs and sweeps outside [START_REF] Ganapathisubramani | Characteristics of vortex packets in turbulent boundary layers[END_REF], Ganapathisubramani et al. (2005b), Ganapathisubramani et al. (2006b)). Following Dennis and Nickels (2011a) the hairpins are mostly inclined at 35 -40°with the streamwise direction, Ganapathisubramani et al. (2005b), Ganapathisubramani et al. (2006b) suggest 45°. On average, they are aligned along the streamwise direction to form hairpin packets which move downstream with approximately the same convection velocity. The packets increase in spanwise scale as they evolve downstream. Shorter younger packets located close to the wall induce higher backward velocities and are less accelerated by the background flow than larger older ones far from it, therefore they are convected dowstream slower. [START_REF] Adrian | Vortex organization in the outer region of the turbulent boundary layer[END_REF] and [START_REF] Christensen | Statistical evidence of hairpin vortex packets in wall turbulence[END_REF] reported a mean inclination of the packets to the wall of 12°in the outer region of ZPG boundary layer and channel flow respectively. [START_REF] Tomkins | Spanwise structure and scale growth in turbulent boundary layers[END_REF] suggested that the streamwise separation between packets increases with the wall normal direction and ranges in 200 + -250 + at Re θ = 7705 and the spanwise spacing of counter rotating hairpin legs ranges in 100 + -120 + at y + = 100 and y + = 220. Far from the wall, for y + > 150 the number of vortices decreases with increasing wall normal distance except the spanwise prograde vortices which number is nearly constant in the log region before decreasing slowly in the outer region [START_REF] Herpin | Influence of the Reynolds number on the vortical structures in the logarithmic region of turbulent boundary layers[END_REF]. A more detailed report on the wall normal evolution of spanwise vortex population outside the buffer layer can be found in Wu and Christensen (2006). For y + > 100 the radius (and the vorticity) scaled in wall units keeps increasing (decreasing) with the wall normal distance but the rate is slow compare to the near wall region [START_REF] Herpin | Influence of the Reynolds number on the vortical structures in the logarithmic region of turbulent boundary layers[END_REF]) and the circulation is nearly constant [START_REF] Carlier | Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry[END_REF]). When scaled in Kolmogorov units η and τ k mean radius and mean vorticity are found to be independent of both the wall normal position and the Reynolds and corresponding probability density function (PDF) yield a log normal behaviour [START_REF] Herpin | Influence of the Reynolds number on the vortical structures in the logarithmic region of turbulent boundary layers[END_REF]). Precisely, in the log layer for 1300 < Re θ < 18950 [START_REF] Herpin | Influence of the Reynolds number on the vortical structures in the logarithmic region of turbulent boundary layers[END_REF] found a constant mean radius of ≈ 10η for spanwise vortices at y + ≤ 150 and of ≈ 8η above, the last value correponds to the mean radius of streamwise vortices in the whole layer. The mean vorticity associated with these eddies was found to be 1.5/τ k . [START_REF] Gao | Analysis of vortex populations in turbulent wall-bounded flows[END_REF] reported a mean vortex radius in the range 5 -6η, their detection method based on the swirling strength λ 3D and real eigen value of the velocity gradient tensor was fully 3D and accounted for vortices with any possible orientation. At

y + = 198 they reported a circulation of Γ + = 115.
The growth of the pre-multiplied energy spectra in high Reynolds numbers turbulent wall flows is attributed to LSM and VLSM which populate the outer part of the log region and the wake region and are associated with large value of Reynolds stresses [START_REF] Lee | Spatial organization of large-and very-large-scale motions in a turbulent channel flow[END_REF]). [START_REF] Guala | Large-scale and very-largescale motions in turbulent pipe flow[END_REF] found that VLSM in the outer region of a turbulent pipe for 3815 ≤ Re τ ≤ 7959 contribute to half the turbulent kinetic energy and that their cumulative Reynolds shear stress contribution is between 50% -60%. [START_REF] Balakumar | Large-and very-large-scale motions in channel and boundary-layer flows[END_REF] extend the investigation to channel (531 ≤ Re τ ≤ 1584) and ZPG boundary layer (1476 ≤ Re τ ≤ 2395) flows and found that VLSMs carry at least 45% of the turbulent kinetic energy and at least 40% of the Reynolds shear stress. Gana-pathisubramani et al. (2003) have investigated the energetic contribution of low momentum regions enveloped by cores of vorticity of opposite sign to the total shear stress at Re τ = 1060. These regions labelled as 'hairpin packets' were found to contribute to more than 25% of the total stress -u v even though they occupy less than 4% of the total streamwise spanwise area examined in the logarithmic region. [START_REF] Lee | Very-large-scale motions in a turbulent boundary layer[END_REF] found that the VLSMs separately contribute approximately to more than 45% of the total Reynolds shear stress included in all patches. the energetic contribution of vortices associated with large scale is less documented, their contribution is indirect in the sense that they induce uniform momentum regions which are characterized by strong Reynolds stresses. Wu and Christensen (2006) have investigated spanwise vortices contribution to the mean shear, prograde vortices where found to contribute between 8% -9% of the mean shear in a channel flow at Re τ = 570 and between 3% -4% in a ZPG boundary layer at Re τ = 3450, the contribution of retrograde vortices was insignificant. They also observed that the contributions decrease with increasing Re τ and that taking into account motions induced by the vortices increases these contributions by a factor of 4-5. Induced motions were identified with threshold of swirling strength parameter (Λ ci ≥ 1.5, see Wu and Christensen (2006) and [START_REF] Tomkins | Spanwise structure and scale growth in turbulent boundary layers[END_REF]) computed in a box center around the vortex center with sides two grid points larger than its core.

Chapter 2 Experimental set-up

The experimental database was obtained during the WALLTURB test campaign in the Laboratoire de Mécanique de Lille wind tunnel with stereo PIV and HWA. The test section of the wind tunnel is 20 m long with a cross section of 2 × 1 m 2 . The free stream velocity is 10 m/s and the turbulence level is 0.3%. A full description of this wind tunned can be found in [START_REF] Carlier | Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry[END_REF] and the WALLTURB program is described in Stanislas et al., 2012. Before describing the experimental set-up itself, a literature review of the HWA and PIV measurement techniques adapted to the present study is presented.

Hot wire anemometry

Hot wire anemometers are measurement tools which were widely developped in the early 60's thanks to progress in integrated electronics. They belong to the family of thermal anemometers which are based on elements whose resistances depend on their temperatures. The element, being inserted in a fluid flow, is cooled down by convection. The flow speed can then be deduced from the voltage necessary to keep the probe at a constant temperature. To keep the temperature and the resistance constant, the hot wire is set in a Wheastone's bridge. The Wheastone's bridge is balanced by a control loop. The voltage needed for this equilibrium depends on the flow velocity. The hot wire calibration procedure consists in finding this functional relation, the reference velocity for the calibration can be measured with an independent device (e.g. a Pitot tube). Through King's law (see 2.1) one can express the voltage E as function of the velocity u and fit calibration results to obtain the calibration coefficient b and n. In this study, instead of expressing the voltage (which is a known quantity) as a function of the velocity (which is unknown), we chose a direct approach described in section 2.3 where the velocity and its central moments are functions of voltages through a second order polynomial (see equation 2.2) similar to what was done by [START_REF] George | Polynomial calibrations for hot wires in thermally varying flows[END_REF].

E 2 = E 2 0 + b.u n (2.1) u = a 0 + a 1 e + a 2 e 2 (2.2)
It is important to mention that in addition to a good calibration which should be performed in the same conditions (temperature and velocity range) as the measurements, many other parameters can improve the accuracy of the results. The size of the probes should be small but the probes should be strong enough not to break up during the experiment. For a best performances, the probes diameters usually ranges between 0.5 and 5µm. It is recommended by [START_REF] Ligrani | Spatial resolution and measurement of turbulence in the viscous sublayer using subminiature hot-wire probes[END_REF] to keep the probes length as small as possible and its ratio to the probe diameter between 200 and 500. For gaz flow measurements, metallic hot wires made of platinum, rhodium/platinum and tungsten are often used. Such recommandations were followed during the experiments whose setup is descibed in section 2.3, platinum coated tungsten wires 0.5 mm long and 2.5µm in diameter were used, yielding a ratio of 200. To compare the Hot Wire Anemometry with the Particle Image Velocimetry (PIV) that will be describe in the next section, we can say that, despite the weakness of the probes which should be handled carefully, hot wire anemometers are much easier to use and cost less. Even though single probe measurements are limited to the streamwise component of velocity, X probes can be used to access the transverse velocities. Hot wires provide a high frequency velocity signal and are suited well to studies where a temporal resolution is needed. However, the spatial resolution of hot wire measurement is still low compared to PIV.

Particle Image Velocimetry (PIV)

Principle of PIV

PIV is a method allowing to access the velocity of a flow in a plane or in a volume. The principle of PIV is sketched in Figure 2.1. First the flow is seeded with small particles that follow its motion. Then the particles are illuminated twice at a short time interval ∆ t with a double pulsed laser and the light diffused by the particles is recorded via high quality lens on one or several cameras. The recorded images at time t and t are cut into small areas called interrogation windows and the average displacement is deduced using statistical methods such as cross-correlation between the two images. The velocity field in the plane of the light sheet is then deduced from this displacement.

Various PIV evaluation techniques are available in the literature, the original one is called "2D-2C" PIV, it gives access to only two components of the velocity field in the measurement plane. But flows are usually tri-dimensional and "2D-2C" PIV does not give access to all the physical phenomenon encountered in such flows. It has also been shown that with conventional "2D-2C" PIV, an out-of-plane motion leads to bias in the measurements (see [START_REF] Foucaut | Characterization of different PIV algorithms using the EUROPIV synthetic image generator and real images from a turbulent boundary layer[END_REF]). The stereo PIV or "2D-3C" PIV [START_REF] Prasad | Stereoscopic particle image velocimetry applied to liquid flows[END_REF]) was introduced to solve these problems. In this technique, the particles are illuminated and viewed from two different directions with two separate cameras. In order to measure the out-of-plane component, the two cameras are shifted from each other. Dual plane PIV and Holographic PIV are other techniques that can be used to access the out-of-plane component. Instead of a plane, the particles can be illuminated inside a volume and be viewed with several separate cameras. This method is called Tomographic PIV (or "3D-3C" PIV, [START_REF] Elsinga | Tomographic particle image velocimetry[END_REF]). One can also follow the flow evolution in time with high speed cameras, this method is known as time resolved PIV.

PIV accuracy

From the two light exposures at time t and t to the entire reconstruction of the flow velocity field there are several parameters that affect the PIV accuracy. For instance, large enough particle images are required to have good illumination and for an optimal evaluation, an homogeneous distribution of medium density of these particles is required. In addition, the spatial resolution of PIV is constrained by the window size which should be small enough to avoid the effect of the velocity gradient. Thus, the particle image diameter, the image density and the velocity gradient are important parameters on which depends the PIV accuracy as studied in [START_REF] Keane | Optimization of particle image velocimeters. I. Double pulsed systems[END_REF][START_REF] Keane | Optimization of particle image velocimeters. I. Double pulsed systems[END_REF][START_REF] Soria | An investigation of the near wake of a circular cylinder using a video-based digital cross-correlation particle image velocimetry technique[END_REF]. Besides we can add the out-of-plane component, the background noise and CCD fill ratio as studied in [START_REF] Foucaut | Characterization of different PIV algorithms using the EUROPIV synthetic image generator and real images from a turbulent boundary layer[END_REF]. Optimal parameters for PIV measurements are listed as follow:

• The particle image diameter should range between 2 and 3 pixels • The particle concentration should be around 0.04 particle per pixel • The fill ratio should be above 75 %.

• The percentage of saturated pixel should be less than 0.035 % • The effect of the velocity gradient and out-of-plane component are to be minimized. [START_REF] Keane | Optimization of particle image velocimeters. I. Double pulsed systems[END_REF] suggest that the image displacent displacement difference Du between the top and the bottom of the interrogation window should be less than the particle image radius.

In the experiment under study, polyethylene glycol micron particles where used with a image diameter of 2 pixels and a concentration close to the optimal value of 0.04 particle per pixel. Background removal by image division was used to cancel the global illumination and small reflections from the wall.

In addition to the optimization of each parameter, the way to retrieve the particle image displacement by correlation between the two windows can be optimized. Windows shifting and interpolation in the correlation domain are commonly used to achieve this (e.g. [START_REF] Westerweel | Fundamentals of digital particle image velocimetry[END_REF], [START_REF] Lecordier | Iterative sub-pixel algorithm for the cross-correlation PIV measurements[END_REF] and [START_REF] Scarano | Advances in iterative multigrid PIV image processing[END_REF]). The multipass algorithm, the multigrid algorithm and the image deformation scheme are standard avanced PIV evaluation algorithm that exist in the literature. Introduced by [START_REF] Westerweel | Fundamentals of digital particle image velocimetry[END_REF], the multipass algorithm is an iterative process in which the windows offset at each step is given by the displacement estimated from the previous interrogation. It significantly increases the signal to noise ratio leading to a better accuracy. The multigrid algorithm (see [START_REF] Soria | An investigation of the near wake of a circular cylinder using a video-based digital cross-correlation particle image velocimetry technique[END_REF]) is an improved version of the multipass, it follows the same principle except that the window size is reduced and the evaluation grid is refined at each step. Finally, deformation schemes introduced by [START_REF] Scarano | Iterative image deformation methods in PIV[END_REF] are suited to flows where there is a significant velocity gradient across the interrogation window.

In the experiment under study, a standard multigrid algorithm with discrete window offset was used to process the images from both cameras. The analysis was made with classical FFT-based cross-correlation method with integer shift on both windows. The displacement was determined with subpixel accuracy through a 1D gaussian peak fitting algorithm. The final interrogation window size was 30*42 pixels in the YZ plane with 70 % of overlap which is far from the 20% recommended in [START_REF] Foucaut | Characterization of different PIV algorithms using the EUROPIV synthetic image generator and real images from a turbulent boundary layer[END_REF] . To compensate for the stretching effect of the stereo PIV setup, a non square image interrogation window was used. The size of this window in the object space is about 7 × 7mm 2 .

Stereoscopic PIV

As mentioned earlier, the stereo PIV was introduced to measure the outof-plane velocity component which leads to unbiased data compared to the 2D-2C PIV. It follows the principle described in 2.2.1 except that it uses two cameras. Both cameras are set either in a translation configuration (the cameras are parallel to each other and orthogonal to the plane of the light) or in an angular configuration. The last configuration is commonly used, but in order to have focussed particle images in the whole field of view, the Scheimpflug conditions are used (see [START_REF] Prasad | Scheimpflug stereocamera for particle image velocimetry in liquid flows[END_REF] and [START_REF] Scheimpflug | Improved method and apparatus for the systematic alteration or distortion of plane pictures and images by means of lenses and mirrors for photography and for other purposes[END_REF]): the image plane of each of the two cameras must be titled with respect to the lens plane so that the intersect of both planes and the object plane constitute a common line (see Figure 2.2). Finally, for optimum measurement accuracy the enclosed angle between the camera viewing axis should be close to 90°(see [START_REF] Raffel | Particle image velocimetry: a practical guide[END_REF].

Stereo PIV Analysis

References on techniques to find and optimize the displacement were given in 2.2.2. In this part, we focus exclusively on the projection and the reconstruction. According to the projection methods used, stereo PIV algorithms can be sorted Figure 2.2 -Basic angular configuration for Stereoscopic PIV, with enforcement of the Scheimpflug condition. "u1" is the in-plane projection of the velocity measured by camera 1; "u2" is the in-plane projection of the velocity measured by camera 2; the true velocity vector (of in-plane component "u" and out-of plane component "w") is reconstructed through a triangulation of "u1" and "u2". into two main categories:

1. Methods based on vector projection called Vector Warping

Methods based on image projection called Image Mapping

Vector Warping methods project the velocity field obtained from the PIV analysis performed in the image plane. Image Mapping Methods project the image of the particle and the PIV analysis is performed in the object plane. A final geometric reconstruction is performed in order to obtain the three components of the velocity field (see [START_REF] Westerweel | Fundamentals of digital particle image velocimetry[END_REF]).

Mapping Functions: Mapping functions are mathematical transformations that allow to switch from the object to the image plane of each camera (projection) and vice-versa (back-projection). They can be expressed either with polynoms or with ratio of polynoms.

Lens non linearities and optical distortions result in a non uniform magnification which is not taken into account in the geometric reconstruction. To tackle this, [START_REF] Soloff | Distortion compensation for generalized stereoscopic particle image velocimetry[END_REF] proposed to combine the projection and the reconstruction in a single step. While in-plane velocity components are expressed through a third order polynom, the out of plane component is given by a second order polynom. In addition, the PIV analysis is performed in the image plane. This technique was used in the present study and more details on the reconstruction procedure can be found in [START_REF] Coudert | Double large field stereoscopic PIV in a high Reynolds number turbulent boundary layer[END_REF].

The pinhole model can also account for optical distorsions (see [START_REF] Calluaud | Stereoscopic particle image velocimetry measurements of the flow around a surface-mounted block[END_REF] and [START_REF] Wieneke | Stereo-PIV using self-calibration on particle images[END_REF] ). In this model, it is assumed that camera lenses act as a pinhole, all the rays originating from the object space and reaching the image space go through this pinhole.

Calibration Procedure

For calibration, a target is placed at the laser sheet position in the object plane. Markers are distributed on a precise grid on this target and target images are recorded on each camera. Marker positions in the image plane of each camera can easily be found with basic image processing tools. This provides enough correspondence between the image space and the object space to optimize the mapping functions parameters. For 2D-2D mapping, only a planar target is required but for 2D-3C mapping [START_REF] Soloff | Distortion compensation for generalized stereoscopic particle image velocimetry[END_REF] have suggested to translate the 2D target in order to recover depth information. A multilevel target with reference markers at different heights is another possibility. An alternative is a calibration based on the imaging geometry of the setup (viewing angles, focal distance, etc.) but this second method is less accurate than the first one where no a priori information on the setup is necessary.

Misalignment correction

Most of the time the light sheet and calibration plane are not perfectly aligned. in the framework of 2D and 3D Warping techniques [START_REF] Coudert | Back-projection algorithm with misalignment corrections for 2D3C stereoscopic PIV[END_REF] proposed a method to correct the misalignment by minimizing the reconstruction residual to align the views. First, the images recorded from each camera are back-projected in the object space. Then the images from the two different fields of view are cross correlated in the object space to retrieve the local offset between the two cameras. Each node of the common cartesian grid in the object space is shifted by half the offset in the opposite direction for each camera stereo pair. Finally, the two shifted grids are projected in the image plane of each camera and the 2D-2C PIV interrogation is performed on the two resulting mesh.

A complete description of the PIV set-up is available in [START_REF] Coudert | Double large field stereoscopic PIV in a high Reynolds number turbulent boundary layer[END_REF].

Facility U ∞ (m/s) δ(m) u τ (m/

Flat plate set-up

The present experiment was carried out with a free stream velocity U ∞ equals to 5 and 10 m/s. The Reynolds number based on momentum thickness Re θ were respectively equal to 9830 and 19660. A Clauser chart fit was used to estimate the friction velocities u τ . Table 2.1 summarizes the main characteristics of the boundary layer for the two free stream velocities. The Hot Wire Rake (HWR) displayed in Figure 2.3 (a) was positioned streamwise at x = 18 m from the boundary layer starting point. This hot wire rake is made of 143 single hot wire probes grouped in 13 vertical combs along the spanwise direction z with 11 probes on each of them. The probes are logarithmically distributed as shown in Figure 2.3 (b). The vertical combs were distributed symmetrically around the center comb located in the plane of symmetry of the wind tunnel, corresponding to z=0. The symmetric pairs of vertical combs were positionned at ±4 mm, ±12 mm, ± 28mm, ±60 mm, ±100 mm and ±140 mm. The logarithmic spacing of each probe on the vertical combs from the wall to the free stream was 0.3 mm, 0.9 mm, 2.1 mm, 4.5 mm, 9.3 mm, 18.9 mm, 38.1 mm, 76.5 mm, 153.3 mm, 230.1 mm and 306.9 mm. The first two rows are below the PIV measurement plane and were not used in the present study. The sensing wires are 0.5 mm long and 2.5 µm in diameter (respectively l + w = 6.1 and d + w = 0.003 at Re θ = 9830 and l + w = 11.8 and d + w = 0.006 at Re θ = 19660). The acquisition time of the hot wire signal was 6 s, the sampling frequency was 30 KHz and measurements were repeated over 534 and 570 blocks for Re θ = 9830 and Re θ = 19660 respectively, to ensure convergence. Because hot wire measurements are limited to one dimension, a stereo-PIV system described in [START_REF] Delville | The WALLTURB joined experiment to assess the large scale structures in a high Reynolds number turbulent boundary layer[END_REF] was used simultaneously, allowing measurements at 4 Hz. This stereo PIV system is highlighted in green in Figure 2.3 and detailed in Figures 2.4 and 2.5. Four cameras capture the flow in a plane parallel to the HWR (Y Z plane), the laser sheet positionned 1 cm upstream of the rake (see Figure 2.4 ) covers the entire boundary layer over an area of 30 × 30 cm 2 . The four cameras can be viewed as two stereo PIV subsystems on top of each other (see Figure 2.5). Each system is located in a 45°configuration with parameters H x = 1.10 m and L x = 2.20 m (see Figure 2.4). The field of view of each system is around 28 × 17 cm 2 . Through a small overlap region between both systems field, the final field of view was set to 28 × 32 cm 2 . This value is close to the boundary layer thickness and to the HWR field for comparison. The resulting velocity field has 3 components with a spatial resolution of 2 mm in the spanwise and wall normal directions.

Each camera lens was equiped with a polarization filter in order to separate the light from the two PIV measurement planes. In a first approach, high order statictics from these PIV measurements were used as reference data to calibrate hot wire probes (see [START_REF] Tutkun | In situ calibration of hot wire probes in turbulent flows[END_REF]). From equation 2.2 a Reynolds decomposition as: u = U + u and e = E + e , where E and e are respectively the ensemble average and the fluctuating part of instantaneaous voltage e, allows to express the mean velocity and central moments as combination of voltage moments φ i (i = 0 -9) up to 6th order:

U = a 0 φ 0 + a 1 φ 1 + a 2 φ 2 (2.3) u 2 = a 2 1 φ 3 + 2a 1 a 2 φ 4 + a 2 2 φ 5 (2.4) u 3 = a 3 1 φ 6 + 3a 2 1 a 2 φ 7 + 3a 1 a 2 2 φ 8 + a 3 2 φ 9 (2.5)
The detailed expression of voltage moments φ i (i = 0-9) can be found in Tutkun (2009). The calibration coefficients a j (j = 0 -8) are retrieved by solving this non linear system with a least square fit. However, a stastitical analysis performed on the streamwise velocity field revealed that the PIV measurements are somewhat filtered compared to results obtained by [START_REF] Carlier | Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry[END_REF] (see the root mean square comparison of both studies in Figure 4.6). This is why, following the same calibration procedure, single hot wire measurements from [START_REF] Carlier | Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry[END_REF] were used as referenced data for the second and third order moments instead of PIV measurements. More details on the anemometry system used in this experiment can be found in [START_REF] Delville | The WALLTURB joined experiment to assess the large scale structures in a high Reynolds number turbulent boundary layer[END_REF].

Chapter 3

Linear stochastic estimation

General introduction

Stochastic estimation was first introduced in turbulence by [START_REF] Adrian | On the role of conditional averages in turbulence theory[END_REF] to characterize conditionally structures in homogeneous turbulent shear flows. It is based on the assessement of a conditionally averaged field g given an event E. This conditional average is estimated through a Taylor series expansion of the event truncated at a chosen level. This formulates as follows:

< g i |E >= L ij E j + M ijk E j E k + ... (3.1)
Where the operator <; |; > stands for the conditional averaging process. The unknown coefficients L, M, ... are found by minimizing the mean square error between the estimate (right hand side of equation (3.1)) and the conditional average (left hand side of equation (3.1)). This formulation includes normally quadratic and higher order terms but, it has been shown that if the Joint Probability Density Function (JPDF) between the variable being estimated and each of the conditional data is a gaussian, a linear stochastic estimation is sufficient (see [START_REF] Naguib | Stochastic estimation and flow sources associated with surface pressure events in a turbulent boundary layer[END_REF] and [START_REF] Druault | Quadratic stochastic estimation of farfield acoustic pressure with coherent structure events in a 2D compressible plane mixing layer[END_REF]). This point will be discussed in detail in section 3.5. In this case, (3.1) simplifies as:

< g i |E >= L ij E j (3.2)
To illustrate the method, let us consider the example proposed in [START_REF] Guezennec | Stochastic estimation of coherent structures in turbulent boundary layers[END_REF], where a conditionally averaged velocity vector u at a point x + r is estimated linearly given a condition on u at a point x as:

ûi (x; r) = L ij (r)u j (x) (3.3) with û(x; r) < u(x + r)|u(x) >.
Then, from a set of simultaneous measurements of u(x) and u(x + r), an error e i can be defined between the effective velocity at point x + r and the estimate and minimized in a least square sense with respect to the unknown coefficients L ij . So that we have:

e i = [ ûi (x; r) -u i (x + r)] 2 (3.4)
and:

∂e i ∂L ij (r) = 0 (3.5)
Leading to the following system of equations:

u j (x)u k (x)L ik (r) = u j (x)u i (x + r) (3.6)
The resolution of this system provides the unknown coefficients L ik allowing then to estimate ûi through equation (3.3) for any given u(x). Note that the estimates of the conditional velocity field are obtained from unconditional statistics such as correlations which appear in equation (3.6). In pratice, we might be interested in the evolution of the velocity field at a point x + r and a time t + τ given an event at x and t. Then the unknown coefficients are not only function of the spatial separation r but also of the time lag τ. Equation (3.3) can be rewritten as:

ûi (t, x; τ, r) = L ij (τ, r)u j (t, x) (3.7)
This is known as a single-point single-time linear stochastic estimation (see [START_REF] Guezennec | Stochastic estimation of coherent structures in turbulent boundary layers[END_REF] and [START_REF] Durgesh | Multi-time-delay LSE-POD complementary approach applied to unsteady high-Reynolds-number near wake flow[END_REF]). If the event used for the estimation is defined at several points, the formulation is called multi-point linear stochastic estimation, and formulates for two points as follows:

ûi (x 1 , x 2 , r) = L ij (r)u j (x 1 ) + N ij (r)u j (x 2 ) (3.8)
Similarly, we can use the information from multiple times to estimate a conditional event, this is known as multi-time formulation. [START_REF] Lasagna | Multi-time delay, multi-point linear stochastic estimation of a cavity shear layer velocity from wall-pressure measurements[END_REF] have investigated the shear layer flow inside a cavity using simultaneous mea-surements of a single hot wire probe, traversed on a fine grid in the shear layer and multiple wall mounted condenser microphones. Their results show that the multi-time formulation can improve the accuracy of classical linear and non linear stochastic estimations and that in cases where the number of samples taken to compute the correlations is large enough, multi-point and single-point multi-time stochastic estimation are equivalent. Besides, in the literature, many studies on different types of turbulent fields have shown that linear estimates are suprisingly accurate and relatively simple to form [START_REF] Guezennec | Stochastic estimation of coherent structures in turbulent boundary layers[END_REF], [START_REF] Adrian | Stochastic estimation of conditional eddies in turbulent channel flow[END_REF]). We can also mention the work of [START_REF] Christensen | Statistical evidence of hairpin vortex packets in wall turbulence[END_REF] and [START_REF] Lee | Very-large-scale motions in a turbulent boundary layer[END_REF] who have used LSE to estimate a conditionally averaged 2D velocity field associated with a swirling motion using correlations between PIV streamwise and wall normal velocity fluctuations and the swirling strength.

In the following section, we will describe our formulation of LSE, present the different algorithms of reconstruction that we have tested and discuss the limitation of LSE. Algorithms using a matrix inversion and Singular Value Decomposition (SVD) were implemented and validated on a DNS database. DNS has the advantage over experimental data to provide enough information (u, v, w) to test the reconstruction algorithm. Similarly, a Tikhonov regularization was tested on the same database. This last algoritm was provided by Nguyen van Linh (see [START_REF] Linh | Reconstruction of finely resolved velocity vector fields in turbulent flows from low resolution measurements[END_REF]).

Linear stochastic estimation

In the present study, LSE is used to reconstruct a fully time-resolved 3 component velocity field from a set of simultaneous hot wire and PIV measurements with the same spatial resolution as the original PIV data. Given a set of observables located in space at x and in time at t , the LSE allows the linear approximation of the conditional estimate of some quantity at a position x and time t. In our case, the conditional variables to reconstruct at high frequency are the 3 components of the velocity u (t, x) = (u , v , w )(t, x) in the PIV Y Z plane, x = (x 1 , ..., x N p ) with N p the number of points. The set of observables includes the streamwise velocity u (t , x ) measured at the N h hot-wires probes on the two dimensional rake, whose coordinates are given by x = (x 1 , ..., x N h ). A single-time formulation for the linear approximation of the velocity component ûi (t , x) is implemented as:

ûi (t , x) = N h k=1 u (t + τ(x k ), x k ).a k,i (x) i = 1, 2, 3 (3.9)
Where a k,i (x) are coefficients relating the conditional field to the observers, and, for each observer x k , k=1-N h on the rake, τ(x k ) is the time delay which maximize the streamwise velocity fluctuation correlation with the corresponding point in the PIV plane.

Linear Regression

Least square estimation

As mentioned in section 3.1, in order to find the best coefficients a k,i (x) to estimate ûi (t , x) using the hot wire measurements u (t , x ), we have to minimize an error function. In a mean square sense, this function corresponds to the residual sum of squared errors at the instant t where the PIV is known:

RSS = N p j=1        u i ( t, x j ) - N h k=1 u ( t + τ(x k ), x k )a k,i (x j )        2 i = 1, 2, 3 (3.10)
In a matrix form, the error function can be rewritten as:

RSS = ||XA -Y|| 2 2 (3.11)
Where A is a matrix of size R N p ×N h which contains all the coefficients a k,i (x),

X and Y correspond respectively to measured hot wire u ( t + τ(x k ), x ) and PIV u i ( t, x) velocity fields and ||.|| 2 2 is the square of the L2 norm ||.|| 2 . Minimizing RSS with respect to A is equivalent to set its derivative to 0, that is:

(X) T (XA -Y) = 0
(3.12) hence:

A LSE = (X) T X -1 (X) T Y (3.13)
The Matrices R = (X) T X and S = (X) T Y of size N h × N h and N h × N p are the hot-wire hot-wire and hot-wire PIV correlations respectively. For each observer

x k , k=1-N h on the rake, the time delays τ(x k ) are the ones which maximize the streamwise velocity fluctuation correlation with the corresponding point in the PIV plane. Note that in our case, the same time delays were used for the streamwise, spanwise and wall normal velocity components. The inverse of R = (X) T X was found in two different ways. The first approach using LU is already implemented in CImg library with the function "invert" and the matrix

A is deduced as:

A = R -1 S (3.14)
The second way is to use the SVD of R as described in [START_REF] Cordier | Calibration of POD reduced-order models using Tikhonov regularization[END_REF],with this approach R is written as:

R = UΣV T = N h i=1 u i σ i v T i (3.15)
Where U and V are orthogonal matrices (UU T = VV T = I), Σ is the diagonal matrix which contains the singular value σ i of R arranged in non-increasing order from the top to the bottom of its diagonal (σ 1 ≥ ... ≥ σ N h ≥ 0). Then:

A = N h i=1 h i 1 σ i u T i Sv i (3.16)
The h i coefficients are used to filter small singular values with threshold equals to σ lim :

h i =        1 if σ i > σ lim 0 if σ i ≤ σ lim (3.17)

Tikhonov Regularisation

The idea of the Tikhonov regularization is to impose a penalty term on the size of the typical LSE coefficients:

A = argmin A ||XA -Y|| 2 2 + α||A|| 2 2 (3.18)
Where the function argmin A is the mathematical function called argument of minimum, it corresponds to the set of coefficients A which minimizes ||XA -Y|| 2 2 + α||A|| 2 2 . α is the parameter that regularizes how much the coefficients given by LSE are shrunk towards 0. Then, the derivative of the residual function with respect to A is set to zero:

(X T X + αI)A -X T Y = 0 (3.19)
Where I is the identity matrix. A is deduced as follows:

A = (X T X + αI) -1 X T Y (3.20)

Validation of the Algorithms

A DNS database of a turbulent wall-bounded flow was used to test the different reconstruction algorithms (Matrix inversion, SVD and Tikhonov regularization). This simulation uses the numerical procedure described in [START_REF] Marquillie | Direct numerical simulation of a separated channel flow with a smooth profile[END_REF]. The flow is at a Reynolds number Re τ = 550 based on the friction velocity and Re h = 11180 based on the maximum velocity U max and half the channel height h. Cartesian coordinates in space are (x, y, z) for streamwise, vertical and spanwise directions respectively. The domain size L x × L y × L z normalized by half the channel height is 2π × 2 × π and the number of grid points in the corresponding directions are (576,257,288). Fluctuating streamwise velocities in a plane normal to the flow direction resolved in space and time, were extracted from this DNS database. These include 10000 snapshots with (N y , N z ) = (257, 288) grids points at a sampling frequency of 40 Hz. Then, these data were subsampled in space to simulate hot wire measurements with 26 × 29 grid points in the wall normal and spanwise direction respectively, while keeping the same frequency of 40 Hz. "Hot wire probes" depicted in Figure 3.1 are logarithmically distributed in y and evenly in z. "PIV" data were simulated by subsampling at the same plane 4 Hz, resulting in N t = 1000 snapshots with the original spatial resolution of 257 × 288. In this section, the term, "Hot Wire" refers to the first set sampled at 40 Hz and the term "PIV" to the second set. The implementation of the inverse method is straigthforward, we need to compute the S correlations between PIV and Hot Wire data and R between Hot Wire and Hot Wire at the PIV time steps and then get the coefficients matrix through equation (3.14). The LSE solution is then obtained from equation (3.9). For the SVD and Tikhonov regularization, there are two crucial parameters to optimize σ lim and α respectively. To do so, for a wide range of parameters tested, we have kept only the one which minimizes the root mean square (RMS) difference χ between the velocity field ûi reconstructed midway between two original PIV time steps and the reference field u i extracted directly from the DNS data. This difference is:

χ( ûi , u i ) = ( ûi -u i ) 2 u 2 i i = 1, 2, 3 (3.21)
A tenth-fold cross validation (see A.1) procedure was used to find optimal values of σ lim and α.

Figures 3.2, 3.4 and 3.6 provide a comparison of the three components of the velocity from the original field extracted from DNS database (a) and reconstructed with matrix inversion (b), SVD (c) and Tikhonov regularization (d). Data from the direct matrix inversion show high frequency noise. Velocities reconstructed with Tikhonov and SVD are subject to a filtering effect, probably due to the regularization imposed through cut off by σ lim and α. For comparison, all fields are displayed with the same velocity range. Consequently, the velocity map from the original field and the one recontructed with matrix inversion are saturated. For the streamwise component, among the three methods tested, Tikhonov regularisation ( 3.2 (d)) gives the best match with the orignal field plotted in (3.2 (a)). For the wall normal and the spanwise velocity component, the filtering effect of SVD and Tikhonov is higher, there is nevertheless some correlation between the largest scale in the original field and the reconstructed one. Figures 3.3 (black dots) extracted from the DNS database with the one reconstructed by matrix inversion (black line), SVD (blue line) and Tikhonov regulization (red line). These spectra are plotted against the spanwise wavenumber k z multiplied by half the channel size at four different wall normal positions y = 0.030h, y = 0.090h, y = 0.189h and y = 0.674h depicted in Figure 3.1. From the three figures, we can notice that LSE filtering effect is higher for the wall normal and spanwise velocity spectra than for the streamwise one. This was predictable as the hot wire data provide only the streamwise velocity component u which is used to reconstruct all three velocity components. The correlation S between PIV and hot wire data that is used in the reconstruction is diplayed in Figure 3.8. It is computed for a probe located at y = 0.2041h and z = 0 and normalised with the corresponding RMS. The streamwise/streamwise correlation (3.8 (a)) peaks at 1.0 while the streamwise/wall normal (3.8 (b)) and the streamwise/spanwise (3.8 (c)) have lower amplitude, respectively 0.41 and 0.24. This is the reason why they are displayed in the same range between -0.2 and 0.2.

In addition, similarly to the velocity fields, spectra from Tikhonov regularization and SVD are more filtered than the ones from matrix inversion, essentially near the wall at y = 0.0027h were the spanwise length scales are relatively small. Far from it, the filtering effect is reduced and for E 11 the energy in the largest spanwise scales remains comparable. Globally the spectra are relatively similar, with an overall loss of energy sometimes more marked at small scale, but the overall spectral content seems to be preserved by the reconstruction and notably the large scales of interest here.

R ûi u i (y) = 1 N z N t N t t i =1 N z z k =1 ûi (t i , y, z k )u i (t i , y, z k ) σ ûi (y)σ u i (y) i = 1, 2, 3 (3.22)
The results of figures 3.2-3.7 are confirmed by the plots in Figure 3.9, which presents the cross correlations between the reconstructed field ûi and the original field u i (equation (3.22)), normalised by the R.M.S. These correlations are higher for the Tikhonov regularization than for the other methods for the three velocity components. We can notice that in the optimal case, the streamwise correlation with a value close to 0.9 is higher than the wall normal and spanwise ones (around 0.45). In addition the streamwise correlations display wavy pattern. Each crest corresponds to a hot wire probe position highlighted by the crosses at the top part of Figure 3.9 (c). Because the LSE is based on spatio-temporal correlations between the hot wire signal and the original PIV field, points in the PIV domain which are between two hot wire probes are less correlated 1998)). Near the border ( y = 0 and y = 2h see Figure 3.1), even if the number of probes is more important, the streamwise correlation is lower than in the center. This can be explained by the fact that large scales associated with the streamwise velocity component are mostly located in the center (e.g Figure 3.2). As evidence on the E 11 spectra, these scales are less filtered than the smaller ones near the wall.

χ( û, u ) χ( v, v ) χ( ŵ,
There is, therefore, a better match between the original streamwise velocity field and the reconstructed one in the center of the channel than near the border. However, for the SVD, there is a drop of the streamwise correlation in the center of the channel, this can be explained by the fact that the RMS difference χ( ûi , u i ) varies significanty with a small variation of σ lim . This point is discussed further in A.2. The wall normal and spanwise reconstructed velocity fields are more filtered than the streamwise one. Notice that near the border (below y + = 44), these correlations are found to inscrease sligthly toward the wall. This can be explained by the correlation of w with the near wall streaks waviness and the one of v with near wall ejections and sweeps which are also indirectly linked to the streamwise aligned streaky structure near the wall. Additionally, the large scales associated with v and w are relatively smaller than the ones associated with u . This is why the correlation in the center of the channel is lower. Notice that the wavy patterns observed on streamwise correlations are almost cancelled on the spanwise and wall normal ones. Probably due to a low correlation between spanwise and wall normal velocity fluctuations with the streamwise one (see Figure 3.8).

Table 3.1 displays the RMS difference χ( ûi , u i ) from equation (3.21) for each of the three methods tested and for each of the three velocity components. The Tikhonov regularization gives the best reconstruction for all three components. The RMS difference on v and w (≈ 0.9) are higher than the one on u (0.45 previous analysis show that the reconstructed three component velocity field and the reference one are fairly correlated. In addition, corresponding spectra yield the same trend even though there is a pronouced attenuation observed with v and w . Hence, instead of giving a RMS difference χ( ûi , u i ), a RMS attenuation A rms ( ûi , u i ) appears more representative. This attenuation is defined as:

A rms ( ûi , u i ) = û2 i u 2 i i = 1, 2, 3 (3.23)
The values obtained A rms ( ûi , u i ) are given Table in 3.2. The optimal parameters α and σ lim for each velocity components are also displayed. The optimal values of σ lim are 0.012 for v and 0.011 for w , which is higher than the optimal value of 0.002 for u . This reflects a bad conditioning of the linear system with many values of the coefficient matrix shrunk towards 0 during the resolution leading to a more important error. The same comparison can be done for α.

The streamwise correlation drop observed in the center of the channel on SVD results in Figure A.3 (c) has compelled us to perfom a reconstruction with parameters σ lim and α optimized for each wall normal position. The results of this investigation are summarized in section A.2. After the analysis of the different reconstruction algorithms on the DNS data, we have seen that among all the methods tested, Tikhonov regularization is the one that provides the best reconstruction. Even if the error for the regularization with α varying with y is smaller than the one where α is kept constant, the difference between the two errors of order 0.01 is small and we could not guarantee that it will lead to a better reconstruction for the real database. For this reason, in the remaining part of our study, we have decided to keep only the Tikhonov regularization with a constant α as the reconstruction method for the real database.

Limitations of LSE

As mentioned in section 3.1 to find out wether a linear stochastic estimation is sufficient, we can study the behaviour of the JPDF between the variable being estimated and the conditional data. When this JPDF is close to gaussian, quadratic terms and high order ones can be neglected (see [START_REF] Naguib | Stochastic estimation and flow sources associated with surface pressure events in a turbulent boundary layer[END_REF] and [START_REF] Druault | Quadratic stochastic estimation of farfield acoustic pressure with coherent structure events in a 2D compressible plane mixing layer[END_REF]). Given equation 3.9, we have built the JPDF between the three components of the original PIV field u , v , w (t, x) at 4 Hz and the Hot Wire streamwise velocity field used for the estimation u (t + τ(x ), x ) at Re θ = 9830 and Re θ = 19660. For each observer x k , k=1-N h on the rake, the hot wire time steps t + τ(x ) were chosen to maximize the streamwise velocity fluctuation correlation with the corresponding point in the PIV plane. As the resolution of the PIV and Hot Wire velocity fields are not the same , the JPDFs were computed only from PIV points close to hot wire sensors and were averaged over 534 blocks for Re θ = 9830 and over 570 blocks for Re θ = 19660. Figure 3.10 and 3.11 give the JPDFs obtained at Re θ = 9830 and Re θ = 19660 respectively and compared with gaussian distributions given by equation 3.24 and obtained through a least square fit on the corresponding JPDFs. These JPDFs were built from a single probe located at y = 5.3 mm and z = 0. JPDFs from the other probes follow the same trend and are not displayed.

g 1 (y, z) = H 0 + A 0 .exp - ((y -y 0 )cos(θ 0 ) -(z -z 0 )sin(θ 0 )) 2 2σ 2 y - ((y -y 0 )sin(θ 0 ) + (z -z 0 )cos(θ 0 )) 2 2σ 2 z (3.24)
The optimal parameters H 0 ,A 0 , y 0 , z 0 , σ y , σ z and θ 0 are given in tables 3.3 and 3.4. As can be seen from figures 3.10 and 3.11, the gaussian distribution on the right matches well with the JPDFs on the left. The streamwise/wall normal JPDF (Figure 3.10 and 3.11 (c)) is the less gaussian of the three, but the global shape still matches quite well the gaussian distribution. These results suggest that a linear stochastic estimation should be sufficient to perfom the reconstruction following equation 3.9. [START_REF] Druault | Quadratic stochastic estimation of farfield acoustic pressure with coherent structure events in a 2D compressible plane mixing layer[END_REF] pushed further the investigations by comparing the contribution of additional quadratic terms in the stochastic estimation with linear ones. Their results suggest that if the contribution from these terms are small enough, a Quadratic Stochastic Estimation (QSE) will not improve the estimation. Because these quadratic terms involves triple correlations of type u 2 1 .u between the original PIV and Hot wire velocity fields which are not converged given our number samples (534 for Re θ = 9830 and 570 for Re θ = 19660) we could not follow such investigations. Also, in the case of a QSE a least square minimization to find the estimation coefficients is not recommended (see [START_REF] Brereton | Stochastic estimation as a statistical tool for approximating turbulent conditional averages[END_REF]. New methods have to be tested (see for e.g [START_REF] Naguib | Stochastic estimation and flow sources associated with surface pressure events in a turbulent boundary layer[END_REF] [START_REF] Druault | Quadratic stochastic estimation of farfield acoustic pressure with coherent structure events in a 2D compressible plane mixing layer[END_REF]) and will certainly require more time. Therefore, we have limited our reconstruction to LSE and we keep QSE as a perspective of the present study.

H 0 A 0 y 0 z 0 σ y σ z θ 0 (°) P (u , u

Conclusion

With a Linear Stochastic Estimation procedure based on correlations computation, a 3 component field is reconstructed at high frequency from stereo-PIV at 4 Hz and hot wire data at 30 kHz. The reconstructed field has the same spatial resolution (2 × 2 mm 2 in YZ plane) as the PIV at 4 Hz. Estimation coefficients are obtained by minimizing a least square error between both fields at the instant where the PIV is known. To do so, various algorithms (matrix inversion, Singular Value Decomposition and Tikhonov Regularization) were tested and a DNS database of turbulent wall bounded flow was used for validation. This simulation uses the numerical procedure described in [START_REF] Marquillie | Direct numerical simulation of a separated channel flow with a smooth profile[END_REF].

The flow is at a Reynolds number Re τ = 550 based on the friction velocity and

Re h = 11180 based on the maximum velocity U max and half the channel height h. Fluctuating streamwise velocities in a plane normal to the flow direction resolved in space and time, were extracted from this DNS database. These include 10000 snapshots with (N y , N z ) = (257, 288) grids points at a sampling frequency of 40 Hz. Then, these data were subsampled in space to simulate hot wire measurements with 26 × 29 grid points in the wall normal and spanwise direction respectively, while keeping the same frequency of 40 Hz. "Hot wire probes" depicted in Figure 3.1 are logarithmically distributed in y and evenly in z. "PIV" data were simulated by subsampling at the same plane 4 Hz, resulting in N t = 1000 snapshots with the original spatial resolution of 257 × 288.

To evaluate the quality of the reconstruction, velocity fluctuations reconstructed midway between the the original PIV time steps by using each of the three methods (matrix inversion, SVD and Tikhonov regularization) are compared to a refence field extracted directly from the DNS data. 2D velocity visualizations in the YZ plane, the spectra, autocorrelation and RMS difference between both fields were investigated. Among the three methods tested, Tikhonov regularization gives the best results. For the streamwise velocity component a good match is observed between the reference field and the reconstructed one. For the spanwise and wall normal velocities, there is little direct correlation between the largest scale in the reference field and the one reconstructed with Tikhonov and SVD, but the reconstructed large scales are comparable in size and shape to the original one, but with lower intensity. In the optimal case (Tikhonov regularization), the streamwise correlation between the reconstructed field and the reference one present a wavy patterns which is explained by the loss of correlation between the hot wire probes positions. This correlation (≈ 0.9) is higher than the wall normal and spanwise one (≈ 0.45). This was predictable as only the streamwise velocity component is used as observable to reconstruct all the three components. In addition, the RMS attenuation (see equation (3.23)) of the reconstructed field compared to the reference one is higher for v and w (≈ 0.35) than for u (≈ 0.8). Continuer ici cf notes The spectra plotted against the spanwise wavenumber at different wall normal positions depicted in Figure 3.1 were investigated. The streamwise spectra from Tikhonov reconstruction are comparable to the original one with an overall loss of energy marked at small scales, the large scales of interest in the present study are well preserved. For the wall normal and spanwise spectra, all wavenumbers are attenuated by a factor of order 10 but the global spectral content is conserved. LSE reconstruction appears here as a broadband filter attenuating all wavelengths more or less uniformly but keeping a good representativity of the spectral content of the original signal.

Based on the previous results, the algorithm using Tikhonov regularization was retained as reconstruction algorithm in the present study. Additionnally, the necessity of including higher order terms in the estimation was adressed directly with the WALLTURB data. The Joint Probability Density Functions between the three velocity components of the original PIV at 4 Hz and the hot wire streamwise velocity are close to gaussian, indicating that quadratic terms can be neglected in the stochastic estimation.

Chapter 4

Data reconstruction

In order to characterize the quality of the reconstructed data, a statistical analysis was performed on the original PIV and HWR data as well as on the reconstructed data to compare them to those of [START_REF] Carlier | Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry[END_REF]. First, energy spectra from the different data sets are compared and the filtering of small scales by LSE is discussed. Then the same comparison is made with the mean velocity, and the turbulence intensity profiles. Finally, the probabity density functions are compared together with the corresponding skewness and flatness factors. The conclusions that arise from these comparisons are discussed at each step. As in [START_REF] Kunkel | Study of the near-wall-turbulent region of the high-Reynolds-number boundary layer using an atmospheric flow[END_REF] and [START_REF] Perry | An experimental study of the turbulence structure in smooth-and rough-wall boundary layers[END_REF], a Taylor hypothesis is used with the local mean velocity to switch from frequency to wavenumber (k x = 2πf U ). Despite small discrepancies at the highest wall normal locations, the match is quite good between the two datasets. The -5/3 and -1 slopes are given as reference.

Spectra

In Figures 4.3 and 4.4, the premultiplied streamwise frequency spectra from the LSE reconstruction (crosses) are plotted together with the HWA spectra (lines) at different wall normal positions and at Re θ = 9830 and Re θ = 19660 respectively. The symbol * is used for an outer-scaling based on U ∞ and δ. Very close to the wall, at y/δ ≈ 0.065, the two spectra match at small frequencies (or large scales). Only the high frequencies (or small scales) are affected by the LSE filtering. At the corresponding height, the hot wire spectrum yields two prominent peaks which are more evident in the log region at y + = 237 (4.3). They correspond to the streamwise u 2 intensity profile peaks observed by [START_REF] Marusic | High Reynolds number effects in wall turbulence[END_REF]. They confirmed that, the small scale peak or inner peak is located near the wall at y + = 15 and is independent of the Reynolds number while they found that the large scale outer one is the signature of superstructures and varies in position and shape with Reynolds throughout the logarithmic region. Therefore y + = 237 correspond to an intermediate location where the contribution from both peaks is still discernable. Notice also that, with LSE spectra the near wall peak is suppressed due to the filtering of small scales.

Moving away from the wall, the LSE filtering affects both small and high frequencies. However, large scales spectra from LSE are attenuated uniformly with respect to the HWA ones. Contrary to small scales, their filtering is attributed to the influence of the lack of correlation between the probes on the reconstruction. The different behaviour of the filter observed in Figure 4.3 when we move from the near wall region to the top part of the boundary layer can also be explained by the fact that due to the wall constraint, structures near the wall are more elongated in the streamwise direction than the ones located far from it, and as the reconstruction is mostly based on the streamwise velocity component, it will provide a better result close to the wall. Additionally, the influence of the probes distribution in span on the LSE reconstructed spectra is discussed in appendix A.3.

Overall, the pre-multiplied spectra obtained here are in good agreement with the spanwise spectra (Figures 3.3) investigated in section 3. The spectra are attenuated uniformly with respect to the original ones and the eneregy loss is pronounced only at high frequencies. The present results support the idea that even though the global energy is attenuated by LSE reconstruction, the spatial organization of the coherent structure might be preserved.

Mean and root mean square profiles

In a previous analysis of the present PIV and HWR data, [START_REF] Tutkun | In situ calibration of hot wire probes in turbulent flows[END_REF] did show that the HWR generates an obstruction which is maximum in the plane of symmetry and affects mostly the mean velocity field. For this reason, the side rows of the HWR are used here for comparison of the mean velocity to minimize the influence of the obstruction (for RMS, all valid probes data are used after substraction of the local mean U (y, z)). Figures 4.6 and 4.7 display profiles of mean U + and root mean square (RMS) streamwise velocities u + at Re θ = 9830 and Re θ = 19660 respectively. A good match is observed between the results from side rows of the HWR, the PIV, the results of [START_REF] Carlier | Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry[END_REF] and the logarithmic law with κ = 0.41 and C = 4.9. The central row is also plotted to illustrate the blockage effect. To get around this problem, for the HWA only, each time the streamwise velocity fluctuations are computed we use a local mean U (y, z) rather than U (y). Concerning the RMS, a good match is observed between all rows of the HWR and the PIV data (side rows only are plotted for clarity). Slight underestimations are visible compared to the data of [START_REF] Carlier | Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry[END_REF]. These remain in the range of the corresponding measurement errors of ±0.28 wall units at both Reynolds numbers. Measurement errors are highlighted on the plots with the error bars in black lines and were obtained by considering a PIV measurement error of 0.1 pixel (see [START_REF] Foucaut | Characterization of different PIV algorithms using the EUROPIV synthetic image generator and real images from a turbulent boundary layer[END_REF]) and maximum displacements of 10 pixel at 5 m/s and 10 m/s. The RMS reconstructed by LSE at 2.0 KHz display an underestimation and a wavy pattern. Similar patterns were observed in section 3.4 on the streamwise correlations (Figure 3.9 (a)) between a field reconstructed from DNS data and a reference field. The RMS of the reconstructed fields follow the same trend as the one obtained by [START_REF] Carlier | Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry[END_REF] but their amplitude are lower denoting a loss of energy. Most of it is due to the filtering of the small scales by LSE reconstruction discussed in the previous section. Hence, it is more pronounced near the wall in agreement with the premutiplied spectra in Figures 4.3 and 4.4. Besides, the PIV plane and the HWA are separated streamwise by ∆x 0 = 1 cm (∆x + 0 = 125.33 at Re θ = 9830 and ∆x + 0 = 233.33 at Re θ = 19660), therefore the correlation between PIV and HWA streamwise velocity fluctuations reaches values close to 0.8 and not 1.0 as might be expected. For clarity, this correlation is displayed later (Figures 4.18 and 4.19), together with the streamwise/wall normal and the streamwise/spanwise ones to explain the shape of the PDFs. [START_REF] Carlier | Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry[END_REF], but they still remain in the range of the corresponding RMS measurement errors of ±0.28 wall units at both Reynolds numbers. A small discontinuity is observed at mid height of the boundary layer (y/δ ≈ 0.5). It corresponds to the connection of the two SPIV subsystems, each made of two cameras and mounted on top of one another as described in section 2.3. The RMS of the reconstructed field has a lower amplitude than the one from HWR and [START_REF] Carlier | Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry[END_REF], denoting a loss of energy. As for u + , this loss of energy is attributed to the filtering of small scales by LSE reconstruction and to the streamwise separation between the PIV plane and HWA. It could also be explained by the fact that the correlation between streamwise velocity fluctuations from HWA and wall normal velocity fluctuations from PIV at 4 Hz used to reconstruct v is weaker than the streamwise velocity correlation used to reconstruct u as discussed in section 3.4. For comparison, the RMS attenuation of the reconstructed field compared to the PIV at 4 Hz is 0.28 and 0.30 for v + at Re θ = 9830 and Re θ = 19660 respectively and 0.65 and 0.68 for u + at same Reynolds numbers. These values are comparable to the ones obtained with DNS data of 0.34 and 0.79 (see table 3.2), despite the streamwise separation between the PIV plane and the hot wire rake. Figures 4.10 and 4.11 display RMS spanwise velocities w + at Re θ = 9830 and Re θ = 19660 respectively. As for u + and v + , RMS from PIV measurements at 4 Hz are underestimated compared to the data of [START_REF] Carlier | Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry[END_REF], except for the region y/δ < 0.3 at Re θ = 19660 the RMS remain in the range of the corresponding measurement errors. The energy loss observed on w + is still attributed to the filtering of small scales by LSE reconstruction, to streamwise separation between the PIV plane and the HWA and also to the low correlation between streamwise velocity fluctuations from hot wire and spanwise velocity fluctuations from PIV at 4 Hz. The attenuation of the reconstructed field compared to the the PIV at 4 Hz are 0.24 and 0.27 for w + at Re θ = 9830 and Re θ = 19660 respectively which is comparable to 0.33 obtained with the DNS database. [START_REF] Carlier | Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry[END_REF]. PDFs from PIV at 4 Hz display some slight oscillations. This is probably due to a peak locking effect which describe a displacement bias error of a periodic pattern on pixel intervals (see [START_REF] Raffel | Particle image velocimetry: a practical guide[END_REF][START_REF] Westerweel | Fundamentals of digital particle image velocimetry[END_REF]). This effect is smoothed out by LSE probably because of the use of averaged quantities (such as correlations) in the reconstruction. Figure 4.14 and 4.15 are the same plots as in figures 4.12 and 4.13 but for wall normal velocity fluctuations. Again PDFs from the reconstructed field at 2 KHz match well with the ones from [START_REF] Carlier | Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry[END_REF]. PDFs from PIV at 4 Hz are overstimated close to zero but present no oscillations. This can still be the effect of peak locking on PIV measurements combined with noise caused by strong velocity gradients near the wall. Figure 4.16 and 4.17 are the same plots as in figures 4.12 and 4.13 but for spanwise velocity fluctuations. PDFs from the reconstructed field at 2.0 KHz is not matching properly with the one obtained by [START_REF] Carlier | Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry[END_REF] and the values are more spread around zero. This behaviour was not observed on the streamwise and the wall normal PDFs, it can be explained by the shape of the correlation S between the PIV and hot wire data (Figures 4.18 and 4.19) that is used for reconstruction. With the streamwise and wall normal component, the correlations (S x and S y respectively) yield a strong peak at the probe location whereas with the spanwise one the peaks are located away from the probe. Given the logarithmic distribution of the probes (Figure 4.5), the later shape is expected to affect the quality of the spanwise velocity component reconstruction. Again PDFs of the normalised spanwise velocity from the PIV at 4 Hz display some oscillations which can be attributed to peak locking.

PDF

Skewness and flatness

To study the Gaussianity of the PDFs described in the previous section, the skewness S G and the flatness F G can be computed. Based respectively on the third and fourth order moments of the random variable G, they are defined as:

S G = G 3 G 2 3/2 (4.1) F G = G 4 G 2 2 (4.2)
The skewness S G = 0 if the PDF of G is symmetric and the flatness F G = 3 if it is Gaussian. Figures 4.24 and 4.25 display, as a function of wall normal distance, the skewnesses of streamwise velocity fluctuations from the reconstructed field at 2 KHz (black lines), the PIV at 4 Hz (green lines), the HWA at 30 KHz (blue Re θ = 20000. Despite, small discrepancies at y ≈ 0.27δ on HWA data and the amplitude difference, the flatness follows the same trend as Carlier's ones. The discrepancies here are probably due to a lack of convergence that appears on high order moments. Flatness from LSE also matches fairly well with Carlier's ones despite the amplitude difference. v and w component flatnesses displayed (Figures 4.32,4.33,4.34 and 4.35) are also in good agreement with Carlier's ones at both Reynolds numbers.

Conclusion

In an attempt to characterize the quality of the reconstructed data, a statistical analysis was performed on HWA, PIV and LSE velocity fields. The spectra, mean, RMS, PDFs, skewnesses and flanesses from the different sets of data were computed and compared to the one obtained by [START_REF] Carlier | Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry[END_REF] for validation. As a result, pre-multiplied spectra from LSE are attenuated compared to HWA ones. Close to the wall y < 0.065δ, only the small scales are filtered while the large ones remain intact. Moving away from it, all the frequencies are affected by the LSE filtering but the spectra from LSE at large scales are attenuated uniformly with respect to the HWA ones. Their filtering is attributed to the influence of the lack of correlation between the probes on the reconstruction.

Mean velocity profiles from PIV and HWA side rows data collapse well on [START_REF] Carlier | Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry[END_REF] profile which matches the log law. The mean profile from HWA central rows is underestimated. In the study, the blockage effect responsible of this underestimation is cancelled by using the local mean velocity Chapter 5

Characterization of coherent structures

Robinson (1991) defined a coherent structure (CS) as "a three dimensionnal region of the flow over wich at least one fundamental flow variable (velocity component, density, temperature, etc) exhibits significant correlation with itself or another variable over a range of space and/or time that is significantly larger than the smallest local scale of the flow". As explained in the literature review on coherent structures in section 1.2, there are three main types of CS encountered in high Reynolds wall bounded flows: uniform momentum regions (including streaks), vortices and ejections and sweeps. they are involved in mass and momentum transfer across the boundary layer and play a key role in turbulence production. Thus, a complete modelization of the boundary layer should incorporate a detailed study of its CS. The present chapter fits into this context. Various methods of detecting and measuring (size, intensity, energy, etc.) large scale 3D coherent structures are presented. Due to the fact that the v , w velocity component are more filtered than the u one (see section 3.4), ejections and sweeps detection which implies taking information from the three veolcity components into account (see for e.g Lin et al. ( 2008)) is not performed here. Even if our investigation is limited to uniform momentum regions and vortices, ejections and sweeps are sometimes used to explain the results obtained. As a summary, in section 5.1 the mathematical formalism behind mathematical morphology and classical morphogical operations is presented. Most of the definitions in that section are from [START_REF] Serra | Image analysis and mathematical morphology[END_REF]. In section 5.2, detection methods are presented. Uniform momentum regions (UMR) are extracted by thresholding streamwise velocity fluctuations, after what a binary image is generated to separate the structures from the rest of the field. Vortex centers are detected from v , w velocity component only through classical operations (Q criterion, λ criterion, vortex center and vortex core identification algorithms respectively Γ 1 and Γ 2 from [START_REF] Graftieaux | Combining PIV, POD and vortex identification algorithms for the study of unsteady turbulent swirling flows[END_REF] and the eddy detection algorithm from [START_REF] Nencioli | A vector geometry-based eddy detection algorithm and its application to a high-resolution numerical model product and high-frequency radar surface velocities in the Southern California Bight[END_REF]) found in the literature. As with UMR, a binary function is also generated to distinguish the vortex center patches from the rest of the field. Morphological operations (such as erosions and dilatations) combined with various filtering techniques were applied to the binary images (including UMR and vortices patches) in order to smooth the noise and enhance their visiblity. In section 5.3, measurements methods are presented. UMR are characterized by their mean hydraulic diameter in the Y Z plane, their life time and their contributions to Reynolds stresses. Vortices are characterized by their radius, their circulation and vorticity in the Y Z plane, their life time and frequency are investigated as well. In section 5.4, the optimal characterization method is applied to the Wallturb database, and results obtained are discussed with emphasis put on the structure scaling and their global organization. Finally, a summary of the chapter is given in section 5.5.

Mathematical morphology

Introduction:

Morphological operations belong to the wide family of image processing techniques which aim at improving the visibility of various parts of an image and detecting individual patterns from these parts. The processing tasks include: removing various types of noise from the image; smoothing the image patterns and enhancing the contrast among adjacent regions. They can be sorted in four main categories: operations based on image histogram, on simple mathematics and on convolution (among which mathematical morphology). A detailed review of these operations can be found in [START_REF] Young | Fundamentals of image processing[END_REF]. Among the convolution methods, Morphological algorithms are easy to implement benefit from a complete theretical background [START_REF] Serra | Image analysis and mathematical morphology[END_REF])and are suited to problems where a quantitative description of the geometrical structure of the image object is needed. They first appeared in late 60's to process binary images form biomedical and geological data [START_REF] Matheron | The structure of the vorticity field in turbulent channel flow. Part 1. Analysis of instantaneous fields and statistical correlations[END_REF]). Then from 70's to 90's they were extended successively to gray level images [START_REF] Serra | Image analysis and mathematical morphology[END_REF]), image/signal processing [START_REF] Maragos | Morphological systems for multidimensional signal processing[END_REF]) and generalised to arbirtrary lattice [START_REF] Heijmans | Morphological image operators[END_REF]). These pionnering years have paved the way to the field of morphological imaging processing which incorporate nonlinear filters, design methodologies, applications systems and theoretical concepts [START_REF] Maragos | Morphological filtering for image enhancement and feature detection[END_REF]). These later are described in the next section.

Principles:

Image analysis is based on the spatial aspect of an object in an image and aims at measuring its feature (size, orientation...). Let X be the set of values representing an image. A morphological operation is divided into two steps: First, a transfomation φ(X) of the original set of values into a new one and second, a measure m(X) (e.g weigth, volume, surface area...) performed on transformed set. A morphological operator must satisfy four principles:

1. Invariance by translation: Applying the transformation φ on X before a translation gives the same result as if it was applied after translation.

φ(X h ) = (φ(X)) h t (5.1)
With h t the size of the translation vector and X h t the set of value after translation.

2. Invariance under change of scale: The transformation φ(X) of set X does not depend on the image magnification β.

φ(β.X) = β.φ(X) (5.2)
With β > 0.

3. Local knowledge: For a bound set Z in which we want to know φ(X) there must be a bound set Z in which X is known.

[

φ(X ∩ Z)] ∩ Z = φ(X) ∩ Z (5.3)
4. Semi-continuity: For every increasing tranformation φ and for any decreasing sequence of closed sets tending towards a limit δ 0 , there must be a sequence of transformed sets tending towards a limit φ(δ 0 ).

Where an increasing transformation φ is defined as a transformation where

φ(A) is always included in φ(B) if A is included in B.
Note that restriction on increasing transformations and decreasing sequences in principle 4 can be removed (see [START_REF] Matheron | The structure of the vorticity field in turbulent channel flow. Part 1. Analysis of instantaneous fields and statistical correlations[END_REF]).

In addition to the four principles, there are two additional principles that a mathematical morphology operator should respect. The first one, known as criteria principle, consists of a set of constraints that are specific to the case under study (see for eg [START_REF] Serra | Image analysis and mathematical morphology[END_REF]) and the second one suggests that to include an algorithm that gives instructions for numerical data manipulation, it should satisfy mathematical operations closely related this case. A good morphological operation should take into account both the structural description of the object and the physics of the problem which is described. As the criteria principle handles the first part, models should be built for the second part. A model can be defined as a set of hypotheses which can be formalized mathematically and which have an impact on the final result. They are often used in mathematical morphology to test some hypothesis. From a probabilistic approach there are three types of models, those which include randomness to have access to a given result, those which must include randomness as a constraint (for example a model to describe a genetic random process can't ignore randomness) and finally those without randomness. If objects we are working on are completely known the model is deterministic ( or purely geometric) this type of modelling is often used in pattern recognition.

Erosion and dilatation:

An erosion is a special case of hit or miss tranformation. Hit or Miss transformation (Serra 1964(Serra , 1969) ) of a given set X by a structuring element B composed of two sets B 1 and B 2 and centered at a point x 0 denoted by (B

x 0 1 , B x 0 2 ) is the set of x 0 where B x 0 1 is included in X and B x 0 2 is included in the complement X c of X: X B = x 0 : B x 0 1 ⊂ X; B x 0 2 ⊂ X c
(5.4)

We have an erosion when the set B x 0 2 is empty. An eroded set Y of a set X is the set of center x 0 of the structuring element B x 0 included in X. Y = x 0 : B x 0 ⊂ X (5.5)

From equation (5.5) the erosion of a set X by a structuring element B could be redefined as the Minkowski substraction of X by the transposed set B of B (see appendix B.1 for the demonstration):

Y = X B (5.6) With B = ∪ y 0 ∈B {-y 0 }.
Dilatation is also a special case of hit or Miss transformation.

It is the dual of erosion (see Appendix B.1.2 for the definition of a dual transformation). In fact each erosion of a set X by a structuring element B leads to an enlargement of the pores in the background image thus a dilatation of the complement X c of X. This dilatation can be viewed as a Minkowski addition of X c by B:

X c ⊕ B = (X B) c (5.7)
Following a classical Minkowski addition the dilatation is rewritten as:

X ⊕ B = ∪ x 0 ∈X y 0 ∈ B{x 0 + y 0 } (5.8)
From Morgan theorem we can infer a symmetrical expression of the dilatation (see appendix B.1 for the statement of the theorem and the demonstration)

X ⊕ B = ∪ x 0 ∈X Bx 0 (5.9)
As a summary in the present study we hold these two definitions for dilatation and erosion. The dilatation of a set X by a structuring element B is the Minkowski addition of X and the transposed set B of B. the resulting set δ B (X) is given by:

δ B (X) = X ⊕ B = ∪ x 0 ∈X y 0 ∈ B {x 0 + y 0 } (5.10)
And the erosion of a set X by a structuring element B is the Minkowski substraction of B from X. The resulting set B (X) is given by:

B (X) = X B = ∪ x 0 -y 0 ∈X y 0 ∈ B {x 0 }
(5.11)

Algebraic properties:

1. Dilatation and erosion are invariant under translation. It means that any point inside or outside the structuring element could be taken as origin (ie the center) and the result will be the same modulo a shift.

2. Dilatation is distributive, for a set X and structuring elements B and B we have:

X ⊕ (B ∪ B ) = ∪ x 0 ∈X (B x 0 ∪ B x 0 ) = ( ∪ x 0 ∈X B x 0 ) ∪ ( ∪ x 0 ∈X B x 0 ) (5.12)
As a consequence of this we can dilate or erode an image by a structuring element piece by piece and recombine intermediate results.

3. Dilatation and erosion are iterative operations. It means that a complex structuring element B can be decomposed in a Minkowski sum of several ones much simpler we have:

(X B) B = X (B ⊕ B ) (5.13) (X ⊕ B) ⊕ B = X ⊕ (B ⊕ B ) (5.14)
4. Increasing transformations: Erosion and dilatation are increasing transformations. We have:

X ⊂ X ⇒        X B ⊂ X B X ⊕ B ⊂ X ⊕ B ∀ B (5.15)
and, by duality:

B ⊂ B ⇒ X B ⊃ X B ∀ X (5.16)

Opening and Closing:

Opening and closing were introduced by Matheron in 1967. An opening operation of a set X by a structuring element B is an erosion followed by a dilatation, the resulting set X B is written as:

X B = (X B) ⊕ B
(5.17)

A closing operation of a set X by B is a dilatation followed by an erosion, the resulting set X B is written as:

X B = (X ⊕ B) B
(5.18)

Opening and closing are dual regarding complement, meaning that the opening of X c with B is the complement of the closing of X with B. While the opening eliminates the small island, sharp peaks or capes and cuts the narrow Closing Opening isthmus, the closing operation merges narrow channels, small lakes and the long thin gulfs as illustrated in Figure 5.1. Here are listed some algebric properties of these two transformations:

1. The opening of X by B is anti-extensive (X B ⊂ X), increasing (X ⊂ X ⇒ X B ⊂ X B ) and idempotent ((X B ) B ⊂ X B ).

By duality the closing of

X by B is extensive (X B ⊃ X), increasing (X ⊂ X ⇒ X B ⊂ X B ) and idempotent ((X B ) B ⊂ X B ).
3. Opening and closing are translation invariant.

Detection methods

Uniform momentum regions

Algorithm of detection

The algorithm of detection of uniform momentum regions has four main steps listed as follows:

1. Distribution function definition 2. Thresholding 3. Mathematical morphology processing

Cleaning

In the following lines, each step is detailed with emphasis on the choice made to obtain an optimal result. Distribution function It is known that low (respectively high) momentum regions, defined as regions of low (respectively high) velocity fluctuations, play a major role in generating turbulence near the wall. Thus many investigations on these uniform momentum regions have been performed over decades (see the literature review in section 1.2.1 for examples). All these studies use the same principle: a suitably chosen threshold is applied directy on the fluctuating part of the velocity field. In order to ensure a normalisation of the fluctuations, the fluctuating part was divided here by its standard deviation averaged in time and spanwise directions (stationnarity and homogeneity). The distribution function is then:

Fd u = u (t, y, z) σ u (y) (5.19)
For the purpose of clarity, in the following a quantity g (t, y, z) is written as g and σ u (y) is written as σ u .

Thresholding One of the most important problem to solve in uniform momentum regions identification is to find the adequate threshold for the streaky structures. The final result depends greatly on this parameter and any mistake in the choice of the threshold could lead to a misdetection of coherent structures in the boundary layer. for Re θ = 9830 and Re θ = 19660 respectively. The two symetrical vertical dashed bars at ± -1 represent the threshold chosen for low and high momentum regions identification. These are standard threshold values in the literature and were applied in previous studies of the group [START_REF] Carlier | Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry[END_REF]. In addition, the threshold is chosen to keep as many energetic structures as possible and to limit the noise. Below a threshold value C tl = -1 for low momentum regions (respectively C th = 1 for high momentum regions), the cleaning procedure described in the next paragraph gives poor results. Cleaned structures are still noisy and cannot be used for characterization in section 5.3.1. The thresholding step can be summarized as:

F i =        1 if Fd u < C tl 0 if Fd u ≥ C tl (5.20)
and

F i =        1 if Fd u > C th 0 if Fd u ≤ C th (5.21)
for low and high momentum regions respectively. The indicative function F i is a binary function obtained after the thresholding operation at C t .

Mathematical morphogy At this stage, morphological operations erosion and dilatation previously described are used to remove the noise from the thresholded streaky structures. To do so, a suitably chosen 3D structuring element is applied to the binary field generated from equations 5.20 and 5.21. This step is Cleaning The cleaning procedure is generally used to remove small object left by morphological operations. In the present study we used a volume-size based cleaning procedure. Given a threshold C v and a 3D mask ( values also given in appendix B.2), the volume around a pixel of the indicative function F i is computed and the pixel is deleted if this volume is not big enough:

F i =        1 if volume > C v 0 if volume ≤ C v (5.22)

Vortices

The aim of this part is to describe the algorithm used to detect vortices within the flow and the results obtained from it. The detection is carried out in a 2D plane (z, y) for each time t and the vortex position is referenced by its center coordinates (z, y) in the flow field at different time steps.

Detection algorithms

Reference to different methods commonly used to detect vortices were given in the literature review in section 1.2.1. As mentioned there, a global review of the vortex detection methods can be found in [START_REF] Holmén | Methods for Vortex Identification[END_REF]. Only five of them are used here (Q criterion, λ criterion, vortex center and vortex core identification algorithms respectively Γ 1 and Γ 2 from [START_REF] Graftieaux | Combining PIV, POD and vortex identification algorithms for the study of unsteady turbulent swirling flows[END_REF] and eddy detection algorithm from [START_REF] Nencioli | A vector geometry-based eddy detection algorithm and its application to a high-resolution numerical model product and high-frequency radar surface velocities in the Southern California Bight[END_REF]). Before applying these algorithms to the real database, we have tested and validated them on a Lamb-Oseen axisymmetrical vortex centered at (z 0 , y 0 ) = (5, 5) and defined at each point (z,y) in a field of size (10, 10) by the 2D velocity field (w, v):

w(z, y) = - Γ ∞ sin(θ 0 ) cos(θ 0 ) 2π(z -z 0 ) 1 -exp - (z -z 0 ) 2 + (y -y 0 ) 2 r 2 (5.23) v(z, y) = Γ ∞ cos 2 (θ 0 ) 2π(z -z 0 ) 1 -exp - (z -z 0 ) 2 + (y -y 0 ) 2 r 2 (5.24)
Where r = 2 and Γ ∞ = 2 in our case. The λ criterion computation is described in [START_REF] Zhou | Mechanisms for generating coherent packets of hairpin vortices in channel flow[END_REF]. Here the velocity gradient tensor D = ∇u is 2D:

D =        ∂w ∂z ∂w ∂y ∂v ∂z ∂v ∂y        (5.25)
To compute the derivatives, we used centered finite differences at all points except boundaries:

∂u i ∂x j = u i,j+1 -u i,j-1 2∆x j (5.26)
At the boundary we used forward and backward finite differences for lower and upper bounds respectively:

∂u i ∂x j low = u i,j+1 -u i,j
∆x j (5.27)

∂u i ∂x j up = u i,j -u i,j-1 ∆x j (5.28)
The λ coefficients are the imaginary parts of the complex eigenvalues of the 2D velocity gradient tensor obtained from the characteristic equation:

det(D -λI) = 0 (5.29)
where I is the 2D identity matrix. The Q criterion is the second invariant of the same tensor:

Q = 1 2 ((tr(D)) 2 -tr(DD)) (5.30)
For 3D incompressible flows we have tr(D) = ∇.u = 0 which gives:

Q = -tr(DD) 2 (5.31)
When large scale vortices are superimposed on small scale ones, methods based on the vorticity or the velocity gradient tensor computation show their limits. Generally they are not able to detect the vortex position due to the high frequency background. Γ 1 and Γ 2 allow then to overcome this problem. They are defined as follows:

Γ 1 (O) = 1 N s (OP ∧ u P ) • n OP • u P (5.32) Γ 2 (O) = 1 N s (OP ∧ ( u P -ũO )) • n OP • u P (5.33)
Where O is the central point at which we want to compute Γ 1 and Γ 2 , P is a point in the rectangular domain s centered at O, N is the number of points in s, n is a unit vector normal to the plane in which vortices are detected, as vortices are detected in a 2D plane in this study, n is colinear to OP ∧ u P and plays no role in Γ 1 and Γ 2 computation. u P is the instantaneous velocity field at each point P. Note that ũO = 1 N s u and u P = U P + u P . Where U P is the mean velocity and u P is the fluctuating part. In the test case the grid spacing to compute velocity gradients is 2.10 -3 m in each direction, the velocity field used for Γ 1 and Γ 2 computation is given by the equations (5.23) and (5.24). For the experimental database, the grid spacing is the same in the wall normal direction y and in the spanwise direction z, instead of using instantaneaous velocity, we computed Γ 1 and Γ 2 from fluctuations. This will be discussed again in the next paragraph. Finally, the domain s for Γ 1 and Γ 2 computation is a window of size 5x5 grid points in both cases. Additionally the vortex detection algorithm from Nencioli(see [START_REF] Nencioli | A vector geometry-based eddy detection algorithm and its application to a high-resolution numerical model product and high-frequency radar surface velocities in the Southern California Bight[END_REF]) was tested. This algorithm is based on the geometry of the veloity vector around an eddy center, and, four constraint were proposed to identify it. For a 2D eddy in the spanwise/wall normal plane, they are listed as follows:

1. The sign of the wall normal velocity should change as we cross the eddy center along a spanwise section and its magnitude should increase away from it.

2. The sign of the spanwise velocity should change as we cross the eddy center along a wall normal section, its magnitude should also increase away from it and the rotation sense should be the same as for v.

3. The velocity should be minimum at the eddy center.

4. The velocity vector directions around the eddy center should change with a constant sense of rotation and the directions of two neighboring vectors should lay within the same or two adjacents quadrants.

More details regarding the validation procedure are given in section B.3. For the Lamb oseen vortex, contours of λ, Q, Γ 1 and Γ 2 are found to peak at the vortex center and Nencioli's detection algorithm allows us to obtain its exact position. For the experimental database, Γ 1 and Γ 2 implementation were slightly modified. There is a blockage effect (discussed in [START_REF] Stanislas | Progress in Wall Turbulence: Understanding and Modeling[END_REF]) due to the hot wire rake which affects SPIV measurements and which increases V and W values. Knowing from the turbulent boundary layer hypothesis that, w = 0 and v is negligeable compared to ū and compared to its respective fluctuations v and w so that instantaneous velocity components v and w are almost equal to the fluctuations v and w . We have chosen to compute Γ 1 and Γ 2 from equations (5.32) and ( 5 Among all the algorithms tested in the database, only contours of Γ 1 ( |Γ 1 | > 0.7) were able to find the position of the vortex center (see section B.3). Even if [START_REF] Graftieaux | Combining PIV, POD and vortex identification algorithms for the study of unsteady turbulent swirling flows[END_REF], have suggested the maximum of |Γ 1 |, between 0.9 -1.0 as detection criterion, our threshold is reasonable given that a window with fixed size is used on eddies with a wide range of scales and shapes either circular or elliptical. Therefore in the rest of the study, they will be named vortical motions instead of vortex. Additionnally, it was observed that applying a high threshold to |Γ 1 | may hide its temporal evolution. To track the same vortical motions center at consecutive time steps with a threshold of 0.7 is not always possible. Consequently, a low threshold, |Γ 1 | > 0.35 was used. The resulting indicative function was further processed in order to keep only a centroid coordinates (t,y,z) which correspond to the vortical motions center. More details regarding this processing and the threshold optimization can be found in section B.3.

Measurement methods

Uniform momentum regions

Once the low and high momentum regions are extracted, it is possible to characterize them with their mean hydraulic diameter d h in the YZ plane, their life time t l , and the u u and u v Reynolds stresses computed inside them compared with the total stresses in the flow. After a labeling of uniform momentum regions, the hydraulic diameter is defined at each time t for the structure labelled n as: 5.34) where P e and A r are respectively the perimeter and the cross section of the uniform momentum region labeled n in Y Z at the time step t. The life time t l is the difference between the highest and the smallest time index encountered in a label.

d h (t, n) = 4 * A r P e (

Vortical motions

The vortical motions core identification algorithm follows the one used in [START_REF] Nencioli | A vector geometry-based eddy detection algorithm and its application to a high-resolution numerical model product and high-frequency radar surface velocities in the Southern California Bight[END_REF] with slight modifications. This core region is defined for each vortical motion center detected from |Γ 1 | > 0.35 contours (see section 5.2.2) and is deduced from contour lines of the streamfunction. The eddy radius r corresponds to the average distance between the points lying on the core boundary and the vortical motion center. Following [START_REF] Vollmers | Detection of vortices and quantitative evaluation of their main parameters from experimental velocity data[END_REF], the circulation Γ = v. dl is computed on the same boundary by adding the contributions from all the points. Since the circulation Γ is linked to the vorticity ω through a surface S (Γ = s w. dS). The vorticity magnitude along the x axis ω 0 of a vortical motion in the YZ plane is deducted from its circulation Γ and its radius r as follows:

ω 0 = Γ πr 2 (5.35)
Detailed validation of these algorithms can be found in appendix C.1. In addition, vortical motions Reynolds stresses (u u and u v ) contributions to the flow are computed inside the core region detected. The life time of the vortical motions are obtained during the centroid detection procedure as described in section B.3.

Results

Vortical motions reconstruction representativity

As was discussed in section 3.4, the LSE reconstruction attenuations obtained on v (≈ 0.34) and w (≈ 0.33) from which 2D vortical motions are detected are more important than the one obtained on u (≈ 0.79) that is thresholded to identify uniform momentum regions. Vortical motions statistics that will be detailed in section (5.4.3) may, therefore, be biased, compared to the uniform momentum ones given in section 5.4.2. Therefore, it is necessary to discuss the representativity of LSE results. The original PIV at 4 Hz being the only exact information available, we have chosen to compare the vortical motions statistics (radius and circulation) obtained from these fields with the one obtained from a velocity field reconstructed at the same frequency in between the original PIV time steps. The statistics obtained from LSE at 2 KHz are also given.

For a relevant comparison, the same detection procedure where the 2D vortical motions positions are provided by the contours of |Γ 1 | thresholded at 0.35, cleaned through 3D morphological operations and filtered in time was performed on the PIV data at 4 Hz and, the radius and circulation are computed in the same manner as with the 2 KHz LSE reconstructed data. Notice that such a comparison cannot be made with the uniform momentum regions, as low time resolved 4 Hz PIV data does not account for the 3D uniform momentum regions detected in this study. Given that the attenuation on u is less important, it is considered not necessary to discuss the representativity of LSE detected uniform momentum regions. It is also important to mention that the aim here, is not to discuss vortical motions statistics obtained from the original PIV at 4 Hz, but to try to find out via a comparison with these statistics what is kept in the reconstruction with LSE at 2 KHz.

Number of vortical motions: Figures 5.4 displays the number N vy of vorti-

cal motions detected at a given wall normal location range, normalised with the total number of vortical motions detected N v at Re θ = 9830 (see Table 5.1). To build these histograms, the domain was split into 23 intervals of equal size in the wall normal direction. For a purpose of clarity, the histograms intervals are represented with half their size, the left side of the interval is used for PIV data (green) and the right one for LSE data (yellow). As it is known that there is a loss of energy between the hot wire probes positions, these are displayed with crosses to see whether the number of vortical motions detected is affected. The evolution of the number of vortical motions detected from both fields is comparable. For the first interval, this number is low for both original and reconstructed data compared to nearby regions, this can be justified by the fact that the PIV fields start at 0.4 mm from the wall. Small vortical motions whose centers are located on the border (or very close to it) could not be processed and are not taken into account in the distributions. Close to the wall (y ≤ 0.25δ), the number of vortical motions is found to decrease rapidly with y. Further away, PIV data yield an almost constant value till the outer edge of the boundary layer whereas LSE ones is affected by the filtering due to the logarithmic distribution of the probes. Close to the probes, at y/δ in [0.25, 0.30], [0.50, 0.55] and [0.75, 0.80] the detected vortical motions are more numerous than in between the probes; where a loss of energy is observed due to LSE filtering (for example at y/δ in [0.35, 0.45] and y/δ in [0.60, 0.70]). Thus, in these regions the number of detected vortical motions should be readjusted towards higher values and a smoother trend with wall distance would result. Except for the region y ≤ 0.2δ, where the number of vortical motions detected from the LSE at 4 Hz is less important than the one detected from the PIV, the figures shows that the LSE reconstruction creates more vortical motions. In fact, conditional averaging techniques such as LSE tend to enforce symmetry on the velocity field (Robinson (1991)). This is linked with the symmetry of the averaged correlations that is used in LSE reconstruction (see 4.18) thanks to the homogeneity of the flow in the spanwise direction. 3D vortical structures with opposite sense of rotation, separated by several time scales and long enough to appear in the averaged quantities might be reconstructed by LSE as a counter-rotating vortex pair. Therefore, for y > 0.2δ, taking into account the non uniform probe distribution, it is not suprising to find with LSE that the number of vortical motions is more important than the one found in the original PIV. Long asymmetric vortical motions are probably duplicated in the averaging process and the shortest ones remain intact. Near the wall (y ≤ 0.2δ), the reversed tendency may indicate a different spatial repartition of the vortical motions. But it is likely that the 4 Hz data are not fully converged, making the interpretation a bit difficult. As will be seen in the section 5.4.3, with 2 KHz LSE histograms (Figure 5.12), this number of vortices scales with 1/y and decreases faster near the wall than what is observed at 2 KHz. Even if not displayed here, the evolution of the number of vortical motions at Re θ = 19660 follows the same trend.

Radius: Figure 5.5 (a) displays the histogram of the radius at Re θ = 9830 scaled with the boundary layer thickness. A comparison of the histogram from original PIV (green line), LSE at 4 Hz (yellow line) and LSE at 2 KHz shows that the LSE tends to overestimate the number of vortical motions (at large radius) in agreement with the wall normal evolution of the number of vortical motions detected (5.4). The same histogram is plotted for vortical motions detected below and above y = 0.2δ (5.5 (b) and (c) respectively). Near the wall (y < 0.2δ), the vortical motions are small but intense, they are therefore fairly well reconstructed with a mean radius of 4 × 10 -2 δ (LSE) compared to 3 × 10 -2 δ (PIV). Above 0.2δ, the vortical motions are larger with low intensity, the smallest ones are filtered by LSE and only the largest one are fairly well reconstructed but they are more numerous due to the symmetrization of the flow. At this location, the mean radius from PIV around 5 × 10 -2 δ is less important than the one from the LSE at 4 Hz (1.2 × 10 -1 δ). LSE tends to overestimate the vortical motions radius far from the wall this is discussed in section 5. 4.3. In Figures 5.5 (d) and (e), the histograms at Re θ = 19660 (plotted with dot) are added, the smallest radius detected near the wall (y < 0.2δ) are found to be more numerous with LSE at 2 KHz than with the PIV (contrary to 5.5 (b)). The origin of this difference is not well understood, but, as mentioned earlier the 4 Hz data might not be fully converged as testified by the dicrepancies between the LSE at 4 Hz and 2 KHz. Overall, the evolution of the radius histograms near the wall matches well with the one from PIV, although more vortical motions are detected in the LSE. This tendency is further pronounced above 0.2δ and vortical motions which radius are small are filtered.

Circulation: Figure 5.6 (a) displays the histogram of the circulation at Re θ = 9830 scaled with the free stream velocity U ∞ and the boundary layer thickness δ. The whole field is investigated. Contrary to the radius, a very good agreement is observed between PIV at 4 Hz and LSE at 4 Hz and 2 KHz. This might be explained by the fact that the circulation is computed as Γ = v. dl, the attenuation of the velocity magnitude due to LSE is compensated by an overestimation of the eddy radius. Notice that in 5.5 (b) where the histograms are plotted at both Reynolds, the effect of this parameter is consistent with the one observed on original PIV data from LSE at 2 Khz. Summary From this analysis, due probably to the symmetric properties of the correlations, LSE tends to create more vortical motions. The circulation is preserved by the reconstruction. Considering the radius, two regions can be distinguished: -The first one at y < 0.2δ, where the vortical motions are fairly well reconstructed with radius comparable to the PIV at 4 Hz. -The second one above 0.2δ, where LSE filters the smallest vortical motions, resulting in an overestimated mean radius.

Uniform momentum regions

Figure 5.7 displays the histograms of the mean hydraulic diameter d h /δ of low (LMR) and high (HMR) momentum regions and the histogram of their life time t * l scaled with outer scales U ∞ and δ at both Reynolds in dashed and plain lines respectively. The histograms were normalised with the number of samples N s used to build them. For the diameter, it is obvious that the two types of structures follow the same trend with a common peak at dh = 0.03δ and mean values of dh ≈ 0.1δ (vertical lines). This is true for both Reynolds number, showing an invariance with this parameter. Based on PIV measurements at Re θ = 7705, [START_REF] Tomkins | Spanwise structure and scale growth in turbulent boundary layers[END_REF] suggest that 0.1δ -0.4δ is the spanwise range of the dominant large scale structures. This is partly supported by the histograms in Figure 5.7 (a) and (c) which fall at zero at dh = 0.4δ.

The life time histograms are similar but the one for LMR are slightly shifted toward longer times with a lower peak at Re θ = 9830. Studies in the literature do not agree on this point: [START_REF] Sillero | Two-point statistics for turbulent boundary layers and channels at Reynolds numbers up to δ+ ≈ 2000[END_REF] have suggested that the two structures share similarities in average size in pipe and turbulent boundary layer flows whereas there is a discrepancy between them in channel flows. In opposite, Dennis and Nickels, 2011b found that within a zero pressure gradient turbulent boundary layer low speed regions are slightly longer than high speed ones and argue that the hairpins heads increase the length of low speed streaks. Although the scale range is different from the one in the present study, the observations of Dennis and Nickels, 2011b match well. The mean life time t * l is 2.5 and 3.6 at Re θ = 9830 and 2.6 and 3.2 at Re θ = 19660 for HMR and LMR respectively denoting a sligthtly decreasing trend for LMR. Given that most of the studies reports a streamwise length instead of the life time, it is difficult to make relevant comparisons. For example, Dennis and Nickels, 2011b used a Taylor hypothesis based on the local mean velocity. They reported a length ratio between LMR and HMR of 1.15 which is lower than the values of 1.4 and 1.2 obtained here at Re θ = 9830 and Re θ = 19660 respectively. Besides, considering from the literature that the streamwise extent of the structures scaled with δ increases with the Reynolds number, it is a matter of concern that t * follows the opposite trend for LMR. Given that, the Reynolds number goes from Re θ = 9830 to Re θ = 19660 we can assume that the life time of LMR which goes from 3.6 to 3.2 is nearly constant as the one of HMR. As mentioned in the literature review on the CS, the UMR streamwise extent is highly dependent on the detection procedure, particularly on the selected threshold value.

Figures 5.8 displays the streamwise and shear Reynolds stresses contribution from LMR and HMR (dashed and plain lines respectively) separately, compared to the total stresses (empty circles) at both Reynolds numbers. The joined contribution is also plotted (filled circle). The contributions from both regions move away from each other with the distance from the wall. Even if not shown here, the stresses at both Reynolds almost collapse on each other when scaled in wall units. The percentage contribution against the wall normal distance are depicted in Figure 5.9. The contribution from LMR (42% on average) is almost constant below 0.5δ and increases further away from the wall. The one from HMR keeps decreasing across the boundary layer. These tendencies will be discussed later.

The fact that the energetic contribution from low momentum region is more important than the high momentum ones is not new and was reported in Dennis and Nickels, 2011b for the shear stress only. The 42% contribution to the shear stress obtained here is above the value of 25% obtained by [START_REF] Ganapathisubramani | Characteristics of vortex packets in turbulent boundary layers[END_REF] for low momentum regions enveloped by cores of vorticity of opposite sign. Their feature extraction algorithm looks for zones of strong shear stress thresholded at 2σ + u v (where σ + u v is the RMS of -u v ) and they argued that low thresholds (0.5σ + u v ) can lead to contribution as high as 55%. The uniform momentum regions extraction procedure used here as well as the range of scale investigated being different, probably explain these differences.

Q2 and Q3 (resp. Q1 and Q4 ) contribution to the streamwise and shear Reynolds stresses inside LMR (resp. HMR) are displayed in 5.10 and 5.11. It can be seen that inside LMR and HMR, the major contributions to Reynolds stresses arise from outward ejections (or Q2) and wallward sweeps (or Q4) respectively. The contributions from Q1 and Q3 peak near the wall but remains insignificant all across the boundary layer. Bearing this in mind, we can now explain the contribution in percent displayed in Figure 5.9. From the literature, it is known that sweeps contribution to u v is predominant near the wall (for e.g. [START_REF] Raupach | Conditional statistics of Reynolds stress in rough-wall and smooth-wall turbulent boundary layers[END_REF] and [START_REF] Grass | Structural features of turbulent flow over smooth and rough boundaries[END_REF]), with an energetic contribution which diminishes away from it, while the ejections ones increases [START_REF] Dennis | Coherent structures in wall-bounded turbulence[END_REF]). LMR and HMR contribution in percent displayed in Figure 5.9 matches well with such a trend.

Vortical motions

Once extracted, the vortical motions are characterized by their position, radius, circulation and vorticity. The position corresponds to the centroid provided by the contours of |Γ 1 | thresholded at 0.35, cleaned through morphological operations and filtered in time, the radius is the average distance between the vortical motion center and the points on the contour which defines the vortical motion core, the circulation is derived from the same contour following equation (C.3) and the vorticity is estimated from the circulation and the radius following equation (C.6). In addition, the evolution with the wall normal position of the Number of vortical motions: Figures 5.12 displays the number of vortical motions N vy detected in a given wall normal location range, normalised with the total number of vortical motions detected N v = 14.03 × 10 6 at Re θ = 9830 and N v = 9.91 × 10 6 at Re θ = 19660 respectively. A peak is observed for y/δ in [0.05, 0.10] ( 175 < y + < 350 at Re θ = 9830 and 340 < y + < 675 at Re θ = 19660). Globally the number of vortical motions detected decreases as we move from the peak away from the wall as suggested by [START_REF] Herpin | Influence of the Reynolds number on the vortical structures in the logarithmic region of turbulent boundary layers[END_REF]. However, the peak that they evidence at y + ≈ 90 is not in the range observed here, as vortical motions close to the wall are not taken into account. Additionnally, the attached edddy hypothesis [START_REF] Townsend | The structure of turbulent shear flow[END_REF]) suggest that the number of vortices scale with 1/y. Taking into account the logarithmic distribution of the probes (plotted for Re θ = 9830) which affects the number of vortices detected (see section 5.4.1), the evolution collapse well on a 0.02/y curve. Separate wall normal evolution of the number of positive and negative vortical motions are given in D.1 for Re θ = 9830 only (Figures D.1,D.2). A good homogeneity is observed.

Life time of vortical motions: Figures 5.13 displays the histogram H of the life time of the vortical motion at Re = 9830 normalised with the number of samples N s = 2.12 × 10 6 used to build the histograms. The histogram was obtained after the time-based filter and before the centroid coordinates are computed (see Figure B.15). The change of slope at 0.05 appears because the vortical motions with a life time t * l below 0.05 were removed from the field in order to clean it (see Figure B.21). Globally, we can notice that the number of vortical motions decreases as their life time increases with an average life time of t * l = 0.11. The same histogram at Re θ = 19660 yield a similar evolution and is displayed in Figure 5.14 on top of the previous one. The average life time found at this Reynolds is t * l = 0.19. Therefore, the vortical motions mean life time obtained here increases with the Reynolds number. As far as we know, the life time of large scale vortical motions is not documented in the literature, generally, time is converted into space through a Taylor hypothesis of frozen turbulence (see [START_REF] Dennis | On the limitations of Taylor's hypothesis in constructing long structures in a turbulent boundary layer[END_REF]) and instead of vortical motions the streamwise length of "induced" low momentum regions is reported. Hence, at this stage, a relevant comparison cannot be made. Finally, in Figure 5.13, we can notice that positive ( red dashed line) and negative (blue dashed line) life times have similar evolution.

Radius: Figure 5.15 displays the histogram H of the radius at Re = 9830 scaled in Kolmogorov units normalised with the number of vortical motions N v = 14.03 * 10 6 used to build it. Due to the spatial resolution of the experiment, the smallest vortical motions mostly located close to the wall (below 20η) could not be processed and are not taken into account in the distributions. The mean value of 50η obtained here is far above that of 5 -6η and 8η reported by [START_REF] Gao | Analysis of vortex populations in turbulent wall-bounded flows[END_REF] and [START_REF] Herpin | Influence of the Reynolds number on the vortical structures in the logarithmic region of turbulent boundary layers[END_REF] respectively. The present results evidence large vortical motions within the turbulent boundary. In PIV, the size of the interrogation window and the field of view set the range of scales that can be resolved. The first parameter defines the size of the smallest eddies resolved and the second one the maximum size of the detected eddies. Due to the limited number of pixels of digital cameras, optimizing both of them is difficult and many studies focus on a particular range of scale [START_REF] Herpin | Influence of the Reynolds number on the vortical structures in the logarithmic region of turbulent boundary layers[END_REF][START_REF] Gao | Analysis of vortex populations in turbulent wall-bounded flows[END_REF] looked particularly at the small scales). [START_REF] Herpin | Influence of the Reynolds number on the vortical structures in the logarithmic region of turbulent boundary layers[END_REF] used windows of size 11.2 + and 20.6 + with a resolution of 6 + and A 0

x 0 σ 0 0.059 3.1 0.82 Table 5.2 -Optimal parameter values to fit a log normal distribution described by equation (5.36) on the histogram of the vortex radius plotted in Figure 5.15.

3.2 + at Re θ = 10140 and Re θ = 18950 respectively. In the present study, 88 + and 163 + sized window with a spatial resolution of 25 + and 47 + were used at Re θ = 9830 and Re θ = 19660 respectively. Besides, assuming that Adrian's model persists at the scales investigated here, our vortical motion detection being performed in a 2D plane Y Z allows us to track only the hairpins legs which are almost aligned in the streamwise direction. Moreover, as these structures are sligthly inclined toward the wall, their size projected in the Y Z plane is probably slightly overestimated. Additionally, LSE reconstruction being based on averaged quantities (correlations), it certainly returns a mean view of what is in flow. Given the fact that the spanwise spacing of the structures varies and that they are inclined with respect to the wall, it is probable that events occuring from time to time at different spanwise locations but often enough to appear in averaged quantities will be reconstructed as a single event of larger size which is, in fact, a cluster of different events of small size. [START_REF] Herpin | Influence of the Reynolds number on the vortical structures in the logarithmic region of turbulent boundary layers[END_REF] found that PDF of vortical motion radius scaled in Kolmogorov units are universal with Reynolds number with some scatter at large radius. As shown by Figure 5.16 (a), the histograms obtained here do not collapse. The superposition is fairly good in the range investigated by [START_REF] Herpin | Influence of the Reynolds number on the vortical structures in the logarithmic region of turbulent boundary layers[END_REF] but a significant difference appears at large radius. A log normal distribution (green dashed lines), described by equation (5.36) with the fitting parameters given in table 5.3 collapse well on the histogram of Figure 5.15. This agrees well with Herpin et al. ( 2013)) results who also noticed a log normal behaviour on the PDF of vortical motions radius. Nevertheless, the value of the standard deviation σ 0 is fairly different between the two studies: 0.33 for [START_REF] Herpin | Influence of the Reynolds number on the vortical structures in the logarithmic region of turbulent boundary layers[END_REF] c) and (d) display the histograms of the radius at both Reynolds scaled respectively with the boundary layer thicknesses δ, the displacement thickness δ * and the momentum thickness θ (all given in table 5.3). The integral scale δ * and θ were computed following equations (5.37) and (5.38). The two histograms show similar evolution but do not collapse. The best overlap is obtained with δ * and θ. The mean values are 0.073δ and 0.056δ with a peak observed around 0.025δ for Re θ = 9830 and Re θ = 19660 respectively. Therefore, the mean radius decreases when the Reynolds numbers increases.

δ * = δ 0 1 - U U ∞ dy (5.37) θ = δ 0       1 - U U ∞ 2       dy (5.38)
Figures 5.17 and 5.18 display 2D histograms of the vortical motion radius scaled with δ and plotted against the wall normal position for Re θ = 9830 and Re θ = 19660 respectively. The probe position (highlighted with crosses) is observed to affect the histogram amplitude. In between the hot wire probes, due to the LSE filtering, the number of vortical motions detected is low compared to the others locations. In contrast, close to the probe positions, the velocity field is less filtered and the number of vortical motions detected is more important. Above y ≈ 0.9δ, the 2D histogram shows that the number of vortical motions detected is very low. a) display at both Reynolds and against the wall normal position the evolution of the mean radius of vortical motions scaled in Kolmogorov units. The evolutions are comparable at both Reynolds, but they do not collapse. For 0 < y < 0.2δ the mean radius increase with the wall normal distance. For 0.2δ < y < 0.9δ the values oscillate in the range 58η -84η for Re θ = 9830 and 70η -110η Positive vortical motion (red dashed line), negative vortical motion (blue dashed line), both positive and negative vortical motions (grey plain line) and mean radius (grey vertical dashed line). A log normal distribution described by equation (5.36) and fitted on the histogram is added in green dashed lines. Fitting parameters are given in table 5.2.

for Re θ = 19660, and, for y > 0.9δ the radius decrease with the wall normal distance. The peaks of r observed in the oscillating range (0.2δ < y < 0.9δ) and the values obtained at y > 0.9δ are located in between the hot wire probes inside the regions where the number of vortical motions detected is low compared to the others (see Figures 5.17 and 5.18). Therefore, these values should be taken with caution. In contrast, the through are close to the probes positions inside regions where the number of vortical motions detected is important. These values are likely to occur in the real flow. [START_REF] Herpin | Influence of the Reynolds number on the vortical structures in the logarithmic region of turbulent boundary layers[END_REF] reported that the mean radius scaled in Kolmogorov units is independent on both the Reynolds and wall normal position with constant value at ≈ 8η. Taking into account the influence of the probe distribution, we may obtained constant radius only in the range 0.2δ < y < 0.9δ but with greater magnitude from ≈ 75η at Re θ = 9830 up to ≈ 90η at Re θ = 19660). As mentioned earlier, the difference observed with the present results, can be explained by the target radius range due to PIV limitation, the detection method and the LSE averaging.

In Figures 5.19 (b), (c) and (d) the mean radius scaled respectively with the boundary layer, the displacement and the momentum thicknesses are plotted at both Reynolds against the wall normal position. Contrary to 5.19 (a), the mean radius is found to increase in the range 0.2δ < y < 0.9δ. Besides, the evolutions almost collapse outside this range at y < 0.2δ and y > 0.9δ. Given the fact that the same reconstruction procedure was used at both Reynolds, and that large scales are less affected by the LSE filtering, the discrepancy observed in the 0.2δ < y < 0.9δ could not originate from this filtering. It suggests that in the range 0.2δ < y < 0.9δ the eddy mean radius decreases when the Reynolds number increases in consistency with the histograms (see Figure 5.16). The radius is found to scale better with δ * and θ.

Circulation: Figures 5.20 momentum θ thicknesses. The histograms suggest that the number of vortical motions decreases as their circulation increases. Apart from (a), the mean circulation is found to decrease slightly with increasing Reynolds number. Mean values scaled in outer units are found at 5 × 10 -3 and 3.7 × 10 -3 for Re θ = 9830 and Re θ = 19660 respectively. In the four scaling cases, the evolutions are comparable to each other and no universality in Reynolds is observed. As for the radius, a better scaling is obtained with δ * and θ.

Figures 5.21 and 5.22 display 2D histogram of the circulation scaled in outer units and plotted at both Reynolds against the wall normal position. The histograms are comparable to the radius ones (see Figures 5.17 with the free stream velocity and respectively (b) the boundary layer δ, (c) the displacement δ * and (d) the momentum θ thicknesses plotted at both Reynolds against the wall normal position. The curves profiles are comparable to the radius ones (see Figures 5.19 ) with an oscillating range in 0.2δ < y < 0.9δ and a relatively good collapse is observed near the wall and the border of the boundary layer with δ, δ * and θ. In the central part the collapse is worse than with the radius for the four scaling.

Vorticity: Figures 5.24 display at both Reynolds the histograms of the vorticity scaled in (a) Kolmogorov units, then with the free stream velocity U ∞ and respectively (b) the boundary layer δ, (c) the displacement δ * and (d) the momentum θ thicknesses. The four histograms yield similar trends. The number of vortical motions is found to decrease as the vorticity increases but below 0.13U ∞ /δ for Re θ = 9830 and 0.21U ∞ /δ for Re θ = 19660 the tendency is reversed. In outer scales, the histograms yield mean values of 0.35U ∞ /δ and 0.4U ∞ /δ respectively denoting a slight increasing trend with the Reynolds number. [START_REF] Herpin | Influence of the Reynolds number on the vortical structures in the logarithmic region of turbulent boundary layers[END_REF] found that the PDF of vortical motion vorticity scaled in Kolmogorov units yields a log normal behaviour with a mean values of 1.5/τ k and a Reynolds number universality. As with the radius, log normal distributions (not displayed here) match fairly well with the present results. The standard deviation σ 0 in equation (5.36) is in this case closer to the data of [START_REF] Herpin | Influence of the Reynolds number on the vortical structures in the logarithmic region of turbulent boundary layers[END_REF] (0.6 here compared to 0.5). The histograms collapse only at the large vorticities and a mean value of 2.4 × 10 -2 /τ k is obtained at both Reynolds numbers. As mentioned before, the difference in the target radius range, the detection procedure and the avering effect of LSE don't allow us to make relevant comparisons. Figures 5.27 display the evolution of the mean vorticity scaled in (a) Kolmogorov units, then with the free stream velocity and respectively (b) the boundary layer δ, (c) the displacement δ * and (d) the momentum θ thickesses plotted at both Reynolds against the wall normal position. From the regions very close to the wall where the mean vorticity peaks at ≈ 0.82U ∞ /δ, a monotonous decrease is observed up to δ. We should recall that the vorticity is derived from the vortical motion circulation and vortical motion radius through equation (C.6) and the previous analysis did show that in the region 0.2δ < y < 0.9δ the mean radius and the circulation are affected by LSE filtering and display oscillation with mean values overestimated at the peak located close to the probes. Due to the autosimilarity between the circulation and radius profile, these oscillations are still visible but are attenuated. The tendency observed on the mean radius and mean circulation is reversed, the peaks of vorticity are close to the probe positions where an important number of vortical motions are detected (see for example Figures 5.25 and 5.26)). In contrast, the through in Figures 5.27 are located in between probes, in regions where few vortical motions are detected, and therefore should be taken with caution. [START_REF] Herpin | Influence of the Reynolds number on the vortical structures in the logarithmic region of turbulent boundary layers[END_REF] also found that the vorticity decreases with the wall normal distance. The present results suggest that such a trend persists for vortices of larger size. A good collapse is observed with the four scaling but particularly with δ.

The previous analysis shows that the vortical motions radius, circulation and vorticity scale better with U ∞ , δ * and θ, these two are retained as characteristic length scales of the vortical motions. However, in the range 0.2δ < y < 0.9δ, the evolution suggests than the vortical motion mean radius might be proportionnal to the Kolmogorov length at each Reynolds number taken separately. Therefore, the large scales vortical motions obtained here can be cluster of small vortical motions that scale with η as suggested by [START_REF] Herpin | Influence of the Reynolds number on the vortical structures in the logarithmic region of turbulent boundary layers[END_REF].

Global organization

Coherent structures organization

The aim of this part is to make a general appraisal of the coherent structures organization inside the turbulent boundary layer. To do so, 1D correlations in the z direction were used to estimate the spanwise spacing between the coherent structures. The evolution of this spacing with the wall normal distance was investigated and the results obtained are discussed.

Given two 3D binary indicative functions F il and F ih corresponding to the low and high momentum regions respectively, the spanwise correlation between the two functions R hl at y and for a spacing ∆ z is written as follows: R hl (y, ∆ z ) = 1 N (t 0 , z 0 ) t 0 ,z 0 F ih (t 0 , y, z 0 ).F il (t 0 , y, z 0 + ∆ z ) (5.39)

Where N (t 0 , z 0 ) is the total number of points used to average the correlation, t 0 and z 0 grid points cover the computation domain as far as a point at z 0 + ∆ z is inside this domain. The correlations are symmetric around ∆ z = 0, with pronounced peaks in the outer region (y = 0.27δ and y = 0.55δ). The plateau observed near the wall at y = 0.037δ and y = 0.14δ is probably due to a lack of convergence (less visible at Re θ = 19660 where the number of samples is more important). The average spacing between the low and high momentum regions is found to increase slowly from ∆ z = ±0.34δ up to ∆ z = ±0.46δ for both Reynolds number. A mean value is ∆ z = ±0.4δ which means an average spacing of order of δ for structures of the same type.

The literature review (see 1.2.1) on turbulent boundary layer coherent structures show that the streaks spacing has been studied mostly in the near wall region at moderate Reynolds number. For 740 ≤ Re θ ≤ 5830, [START_REF] Smith | The characteristics of low-speed streaks in the near-wall region of a turbulent boundary layer[END_REF] reported an average spacing of λ + s = 100 between low speed streaks indepently of the Reynolds which is well accepted value in the literature. At Re θ = 2030, the same authors did also investigated the evolution of this spacing with y up to y + = 30. In analogy with the present study, this spacing increases with y. They attribute, this behaviour to the merging and the intermittency of streaks. Ganapathisubramani et al. (2005a) did not investigated directly this spacing but reported two point spatial correlation of all three velocity fluctua- Similarly, the spanwise correlation between the positive and negative vortical motions was computed at different wall normal positions, and its evolution is displayed in Figures 5.29 Tomkins and [START_REF] Tomkins | Spanwise structure and scale growth in turbulent boundary layers[END_REF] have suggested a spanwise spacing between counter rotating hairpin legs of 100 + -120 + (0.05δ) between y + = 100 (0.045δ) and y + = 220 (0.1δ) for a flat plate turbulent boundary layer at Re θ = 7705. The value of the order 0.1δ and 0.12δ obtained at y = 0.037δ for Re θ = 9830 and Re θ = 19660 respectively are about twice in external scaling and three times in wall scaling. This may be explained by the fact that the vortices are not detected in the same plane. Additionally, the correlations in Figures 5.29 (c) and (d) are assymetric, it suggests that counter rotating vortical motions are more frequently associated with low speed regions than with high speed ones.

Figures 5.31 display the correlation between the positive vortical motions and low and high momentum regions respectively R v p l and R v p h at different wall normal positions depicted in Figure 5.28. The spacing between a positive vortical motion and a low (resp. high) momentum region is ∆ z = -0.072δ (resp. ∆ z = 0.05δ) for Re θ = 9830 and ∆ z = -0.074δ (resp. ∆ z = 0.056δ) for Re θ = 19660 at y = 0.037δ. This spacing is found to increase with the wall normal distance with a mean value of 0.14δ for both UMR and HMR. The correlation between negative vortical motions and UMR (not shown here) gives similar results with HMR located at ≈ 0.072δ on the left and LMR at ≈ 0.053δ on the right at y = 0.037δ. These correlations are in agreement with the 3D views of the coherent structures depicted in Figure 5.32 where the positive vortical motions (highlighted in red) are bounded on the left side by low momentum regions ( highlighted in green) and on the right side by high momentum regions (highlighted in yellow).

The large scale coherent structure organization obtained in the present study (see 5.32) agrees well with the literature (Dennis and Nickels, 2011b etc.) but also, suprisingly, with the one reported by [START_REF] Blackwelder | Streamwise vortices associated with the bursting phenomenon[END_REF] for the inner layer structures (5.33). Figure 5.32 shows positive (red) and negative (blue) streamwise vortical motions in between low (green) and high (yellow) momentum regions. The vortical motions have the adequate sign to weak lift up in low momentum region and sweeping in the high ones. Recall that, the spacing between these UMR was found to increase with y similarly to [START_REF] Smith | The characteristics of low-speed streaks in the near-wall region of a turbulent boundary layer[END_REF] investigation near the wall (y + < 30). The analogy with Figure 5.33 from Blackwelder and Eckelmann, 1979 is striking, it seems that at this Reynolds numbers the flow is rebuilding an equivalent to the near wall cycle, but at a larger scale and above it. A model of the coherent structures obtained in the present study at Re θ = 9830 is displayed in Figure 5.34. It is derived from the CS average size and spacing and can be compared with the one obtained by [START_REF] Lin | Etude détaillée des structures Cohérentes de la zone tampona l'aide de données de PIV stéréoscopique[END_REF] near the wall (y + < 50) for Re θ = 7800 displayed in Figure 5.35. Even though the scales investigated are different, there is an analogy between the two models. The spatial organization of the structure yield a low momentum region closely border by a negative vortical motion on its left side and a positive [START_REF] Blackwelder | Streamwise vortices associated with the bursting phenomenon[END_REF].

one on its right. Ejections and sweeps type events mostly occur within sweeps and were found to contribute to about 96% of the corresponding UMR Reynolds stresses in the present study. Due to a limited field of view, [START_REF] Lin | Etude détaillée des structures Cohérentes de la zone tampona l'aide de données de PIV stéréoscopique[END_REF] could not measure the streamwise extent of the UMR but his model shows vortices relatively shorter than UMR in agreement with the present result. The ratio obtained between the vortical motions and LMR width is 0.75 compared to 0.67 in the present study. UMR spanwise spacing-width ratio is 1.66 compared to 4. This difference can be attributed to the selected thresholds. Given the same detection pprocedure, they used a threshold of ±0.65 which lower than ±1 used in the present study and thus provides larger structures. Besides,they found HMR to be wider than LMR whereas their hydraulic diameter is constant here.

To the model depicted in Figure 5.34, we could add separetely a high momentum region (HMR) border on the left by a positive vortical motion and on the right by a negative one. Given that the average spanwise spacing between LMR and HMR is 0.4δ and the one between a single UMR and a vortical motion is 0.14δ, it is not possible to superimposed HMR and LMR models together. Probably, the two models often occurs separetely in the boundary layer.

Conclusion

The aim of this chapter was to extract and characterize uniform momentum regions in order to provide a large scale model of the turbulent boundary layer. Before analysing the results, the reconstruction representativity of the vortical motions by LSE was investigated via a comparison of vortical motion statistics (number, radius and circulation histograms) obtained from the orignal PIV field at 4 Hz and an LSE field reconstructed in between the original PIV time steps at the same frequency. The evolution of the number of vortical motions from both fields plotted against the wall normal direction were comparable. For y > 0.2δ, LSE data was affected by the logarithmic distribution of hot wire probes and LSE reconstruction tends to create more vortices than in the PIV. Near the wall, this tendency is reversed. The probable lack of convergence did not allow to make a full interpretation. Near the wall at y < 0.2δ, the small vortical motions are intense and fairly well reconstructed by LSE with mean radius comparable to the PIV data at 4 Hz. Above this range, the vortical motions are less intense, the smallest ones are filtered and the large ones are more numerous than in the PIV. The provided LSE model tends to overestimate the vortical motion mean radius, while the mean circulation remains the same.

Based on the analysis of an LSE reconstruction at 2KHz, low momentum regions are found to be slightly more energetic and longer than the high momentum ones but on average, their cross sections are nearly equal. Their mean hydraulic diameter is 0.1δ at Re θ = 9830 and Re θ = 19660 and intantaneous values can go up to 0.4δ in agreement with [START_REF] Tomkins | Spanwise structure and scale growth in turbulent boundary layers[END_REF]. The mean life time of LMR scaled in outer units decreases with Reynolds number, from 3.6 at Re θ = 9830 to 3.2 at Re θ = 19660. the one of HMR is nearly constant (around 2.6). It was found that UMR contribute to more than 50% of the total stresses in the boundary layer. More than 96% of the LMR (resp. HMR) contribution to the total stresses comes from Q2 (resp. Q4) motions all across the boundary layer.

Q4 contribution is known to be more important near the wall (for e.g. [START_REF] Raupach | Conditional statistics of Reynolds stress in rough-wall and smooth-wall turbulent boundary layers[END_REF] and [START_REF] Grass | Structural features of turbulent flow over smooth and rough boundaries[END_REF]), whereas Q2 one increases away from it [START_REF] Dennis | Coherent structures in wall-bounded turbulence[END_REF]). The present results match well, HMR contribution decreases fast from the wall to the top of the boundary layer whereas LMR contribution increases slightly with an average contribution of ≈ 42%. Besides, UMR contributions to Reynolds stresses scaled in wall units collapse almost at both Reynolds.

Concerning the vortical motions, their number was found to decrease as their life time increases. The mean life time scaled in outer units is 0.11 at Re θ = 9830 and 0.19 at Re θ = 19660. Against the distance from the wall, the number of vortical motions collapses on a 0.02/y curve in agreement with the attached eddy hypothesis [START_REF] Townsend | The structure of turbulent shear flow[END_REF]). The mean radius of 50η obtained in the present study for Re θ = 9830 is far above the one reported in the literature (5 -6η [START_REF] Gao | Analysis of vortex populations in turbulent wall-bounded flows[END_REF]), 8η Herpin et al. (2013)). This is probably mostly due to the fact that the small vortical motions are not resolved by the PIV and that the LSE reconstruction based on correlations, returns an average view of the flow, where vortical motions radius is overestimated.

Scaled in external units, the vortical motion mean radius and mean circulation decrease with increasing Reynolds number. The mean radius goes from 0.073 to 0.056 and the mean circulation from 5 × 10 -2 to 3.7 × 10 -2 . Below 0.2δ, both mean radius and mean circulation increase with y. Beyond 0.9δ, they decrease. In between a plateau with some waviness due to the HWA probes spacing is observed. Integral scales δ * and θ provided a better scaling of the radius than η and the boundary layer thickness δ. However, the mean radius scaled in η is relatively constant in the range 0.2δ < y < 0.9. It is thought that, the large vortical motions obtained in this range can be cluster of small vortices effectively scaling with η. No proper scaling of the circulation was obtained but, still δ * and θ give the best results.

The vorticity histograms collapse well only at large radius with the four scaling. Mean vorticity values of 0.35U ∞ /δ and 0.4U ∞ /δ were found at Re θ = 9830 and Re θ = 19660 respectively. The mean vorticity is found to decrease with the wall normal distance. Finally, δ * or θ and U ∞ are retained as scaling quantities for the vortical motions as they provided the best results.

The coherent structures organization was investigated at four wall normal locations displayed in Figure 5.28. Respective indicative functions were correlated in the spanwise direction. The organization obtained matches well with the literature on large scale coherent structures (Dennis and Nickels, 2011b etc.) but also, suprisingly, with the one reported by [START_REF] Blackwelder | Streamwise vortices associated with the bursting phenomenon[END_REF][START_REF] Blackwelder | Streamwise vortices associated with the bursting phenomenon[END_REF][START_REF] Lin | Etude détaillée des structures Cohérentes de la zone tampona l'aide de données de PIV stéréoscopique[END_REF] for the inner layer structures. The vortical motions have the adequate sign to create lift up in low momentum region and sweeping in the high ones. Counter rotating vortex pairs with short life times (t * l ≈ 0.15δ) are found to be more frequently associated with LMR than with HMR and this occur separetely. However, the paires of vortices might be due to tendency of LSE to enforce symmetry on the velocity field. The mean spanwise spacing between LMR and HMR increases with the wall normal distance with a mean value of ≈ 0.4δ at both Reynolds number. The spacing between positive and negative vortical motions follows the same wall normal trend with a mean value of 0.28δ which is two times larger than the one between a UMR and a single vortical motion.

Conclusion and perspectives

This work lies in the context of large scale coherent structures investigations in a near wall turbulent layer at high Reynolds numbers. Due to their 3D feature and meandering, these structures are not easy to extract and characterize with standard algorithms. For that reason, in the frame of the WALLTURB project, an ambitious experiment at high Reynolds numbers (Re θ = 9830 and Re θ = 19660) was carried out. Hot wire measurements at 30 KHz gave the streamwise velocity component but with a low resolution in space. These were combined with Stereo PIV at 4 Hz to have a spatially resolved three components velocity field in a YZ plane normal to the flow direction. To take advantage of these two sets of data, a LSE reconstruction procedure was implemented here in order to reconstruct a three component velocity field with enough resolution in space and time to assess the large scale structures. As the experiment was conducted in 2006, a primary objective here was to validate the LSE reconstruction procedure conceive during the design of the set-up. Then, it was to extract and characterize large scale coherent structures from the reconstructed field. Reliable algorithms needed therefore to be implemented. Finally, results obtained from this processing had to be examined to extract a physical knowlegde in order to improve the modelling of the flat plate turbulent boundary layer.

The present conclusion is divided into two parts. The first part summarizes our study and the second part points out the limitations and proposes some perspectives.

Chapter 1 was dedicated to the literature review, it provides the theoretical background necessary for the present study. Chapter 2 describes briefly the experimental set up and flow configuration used in the present study.

In chapter 3 the LSE reconstruction procedure was detailed. Three different reconstruction algorithms (matrix inversion, Singular Value Decomposition and Tikhonov regularization) were tested. A DNS database (from [START_REF] Marquillie | Direct numerical simulation of a separated channel flow with a smooth profile[END_REF]) of a turbulent channel at a Reynolds number Re h = 11180 was used for validation. The DNS data were subsampled in space and time in order to simulate HWA and PIV measurements. The accuracy of the reconstruction was adressed by comparing velocity fluctuations reconstructed midway between the original PIV time steps and a reference field extracted from the DNS data.

The analysis shows that the LSE reconstruction implemented with Tikhonov regularization provides the best results. In this case, the streamwise correlation between the reconstructed field and the reference one (≈ 0.9) is higher than the wall normal and spanwise ones (≈ 0.45). Besides, the RMS attenuation (see equation (3.23)) is higher with v and w (≈ 0.35) than with u (≈ 0.80). In fact, the streamwise velocity component from HWA being used as observable to reconstruct all the three velocity components, a better reconstruction of u is obtained.

A comparison of the streamwise spectra from both fields shows that LSE reconstruction filters mostly the small scales and keeps the large ones of interest in the present study. Wall normal and spanwise spectra are more attenuated than the streamwise one (by a factor ≈ 10). Nevertheless, their shape is well preserved by the reconstruction so that they always remains parallel to the reference ones. LSE reconstruction can therefore be viewed as a broadband filter which attenuates all the wavenumbers more or less uniformly but keeps a good representativity of the spectral content of the original signal. Additionally, the JPDFs between the three velocity components of the original PIV at 4 Hz and the hot wire streamwise velocity are close to gaussian indicating that high order terms (≥ 2) can be neglected in the Stochastic Estimation.

In chapter 4, despite the Reynolds number difference, the LSE reconstruction at 2 KHz applied to the WALLTURB data is in good agreement with the one validated on the DNS data (spectra, RMS attenuation, etc.). The statistical analysis performed on HWA, PIV and LSE velocity fields was compared to the ones of [START_REF] Carlier | Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry[END_REF]. The mean velocity profile from HWA is understimated in the center of the rake due to the high concentration of probes. The influence of this blockage on the fluctuations is minimized by using a local mean U(y, z) rather than U(y).

Inside the range of the RMS measurement error (±0.28 + ), RMS profiles from PIV and HWA data collapse with the one from [START_REF] Carlier | Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry[END_REF]. The ones from LSE follow the same trend as the PIV one but are attenuated. Part is due to the filtering of small scales by the LSE. Also, the streamwise separation between the HWR and the PIV laser sheet (∆x 0 = 1 cm) provides a lower correlation peak between the PIV and HWA velocity fluctuations (≈ 0.8 for u ).

PDFs from PIV, HWA and LSE data match well with the ones of [START_REF] Carlier | Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry[END_REF]. PDFs of w from LSE are flattenned compared to the others. This can be explained by the shape of the streamwise spanwise correlation.

Chapter 5 describes the methods used to detect and characterize coherent structures and provides a discussion of the results obtained. UMR detection consists of thresholding the streamwise velocity fluctuations normalised by the standard deviation. The standard threshold value of ±1 for HMR and LMR respectively, was selected in order to keep the most energetic structures and to limit the noise. Further, classical morphological operations (erosion and dilatation) were combined with a volume size based cleaning in order to enhance UMR visibility and smooth the remaining noise.

Contours of |Γ 1 | (see [START_REF] Graftieaux | Combining PIV, POD and vortex identification algorithms for the study of unsteady turbulent swirling flows[END_REF]) thresholded at 0.35 were used to isolate the regions in which the vortical motions centers lays in. Morphological operations combined with a life time based filter were applied to these regions again to smooth the noise and enhance vortical motion centers visibility. The vortical motion core was deduced from contours lines of the streamfunction through a modified version of the [START_REF] Nencioli | A vector geometry-based eddy detection algorithm and its application to a high-resolution numerical model product and high-frequency radar surface velocities in the Southern California Bight[END_REF] algorithm. The vortical motion radius r corresponds to the average distance between points lying on the core boundary and the vortical motion center. The circulation Γ = v. dl is computed on the same boundary by adding the contributions from all points. The vorticity magnitude ω 0 along the x axis of a vortical motion in the Y Z plane is deduced from its circulation and its radius as ω 0 = Γ πr 2 . Vortical motions reconstruction representativity was investigated via a comparison of their statistics (number, radius and circulation histograms) obtained from the original PIV field at 4 Hz and an LSE field reconstructed in between the original PIV time steps at the same frequency. A comparison of the evolution of the number of vortical motions from both fields showed that LSE globally tends to create more structures than in the original PIV. This is probably linked with the symmetry of the correlations that are used in the reconstruction process. Besides, the reconstructed vortical motions yield larger radius. Two distinct regions were found: -The one below 0.2δ where the eddy radius are comparable in both fields. -The one above 0.2δ where the small vortices are filtered by LSE, the large ones are still present but they are more numerous than in the PIV. For the circulation, a good match was observed between both fields.

The characterization of coherent structures has allowed to built a model of the TBL at large scale. In this model, counter rotating vortical motions are frequently associated with LMR and HMR separately. Despite the difference in the targeted range of scales, the large scale organization obtained in this study is in good agreement with the one obtained by Dennis and Nickels, 2011b but also suprisingly with the classical very near wall picture. Particularly, the vortical motions have the adequate sign to lift up a LMR and sweep down a HMR in agreement with [START_REF] Blackwelder | Streamwise vortices associated with the bursting phenomenon[END_REF] and [START_REF] Lin | Etude détaillée des structures Cohérentes de la zone tampona l'aide de données de PIV stéréoscopique[END_REF] conlusions for the inner layer. We found that on average, UMR and vortical motions are respectively 0.1δ and 0.15δ wide, the corresponding width ratio (0.67) is comparable to the one obtained by [START_REF] Lin | Etude détaillée des structures Cohérentes de la zone tampona l'aide de données de PIV stéréoscopique[END_REF] (0.75) at y + < 50. The UMR average spacing is 0.4δ, the spacing increases with y similarly to [START_REF] Smith | The characteristics of low-speed streaks in the near-wall region of a turbulent boundary layer[END_REF] investigation at y + < 30. The counter rotating vortical motion spacing also increases with y with an average of 0.28δ. Besides, we found that UMR live longer (t * l ≈ 3.4 for LMR) than vortices (t * l ≈ 0.15). The number of vortical motions plotted against the distance from the wall, fit on an 0.02/y profile in agreement with the attached eddy hypothesis [START_REF] Townsend | The structure of turbulent shear flow[END_REF]). The PDF of their radius and vorticity matches well with log normal distributions similarly to [START_REF] Herpin | Influence of the Reynolds number on the vortical structures in the logarithmic region of turbulent boundary layers[END_REF] results.

The effect of the Reynolds number was also investigated. The mean radius plotted against y was found to scale better with δ * or θ. No proper scaling was found for the mean circulation but the mean vorticity scales rather well with δ and U ∞ . Figure 5.36 gives the circulation histogram and mean wall normal evolution scaled properly with δ * , δ and U ∞ , showing a reasonable overlap. Concerning the life time, the one of UMR appears nearly constant whereas the one of vortical motions increases with the Reynolds number. Finally, at both Reynolds numbers, UMR contribute to more than 50% of the total Reynolds stresses and 96% of this contribution arise from Q2 and Q4 events which are associated with LMR and HMR respectively.

Overall, this study presents a unique method to extract and characterize large scale coherent structures in the boundary layer. The developed algorithms can be useful for further studies where a characterization of large scale coherent structures is needed.

Despite substantial efforts, the lack of references on studies with similar flow configuration and at comparative Reynolds does not allow to validate entirely these results. In particular, UMR are described with a characteristic life time rather than a streamwise length makes the comparison difficult. It was envisaged to use a Taylor hypothesis to switch from time to space but with such long times it is delicate to assume that the turbulence is frozen. The difference obtained on the vortical motions statistics with respect to the ones taken from the literature is usually attributed to either the difference in the target range of scales investigated, the LSE reconstruction and the detection methods. It therefore does not stand as a full validation.

In terms of perspectives, the LSE reconstruction method is easy to implement and can be extended further to other Reynolds numbers and flow configuration (e.g adverse pressure gradient). This would allow a deeper insight in the present large scale model. Besides, the DNS database can be used to optimize the probe distribution on the hot wire rake with regard to the reconstruction accuracy. As seen previously, one difficulty in this study comes from the v and w component which are attenuated in the LSE reconstruction. Combining single and cross hot wire probes could be explored to tackle this. A multi-time formulation of the LSE can also be tested. The experiment could be done over a much longer time to ensure the convergence of the statistics at 4 Hz which are useful to evaluate the vortical motion reconstruction representativity. 3D vortex detection techniques can be investigated to avoid the projection in the Y Z plane which also overestimates the eddy radius. Vortical motion reconstruction representativity can be studied with time resolved DNS data at moderate Reynolds numbers. It should answer such questions as: -how does the LSE enforce the symmetry on the velocity field, -which type of vortical motions (size and life time) are duplicated in the reconstruction process, -how does the LSE and detection techniques influence the vortical motion characteristics at moderate Reynolds numbers ... wall normal position contrary to α u , α w and at some point α v evolutions which are smoother. The peaks obtained for σ lim correspond to locations where we could not reach the algorithm stopping criteria, that is to have a constant error χ( û1 , u 1 ) after 100 iterations maximum where the value of σ lim is refined at each step. Therefore the linear system was badly conditioned at these positions. The reconstruction error was found to vary rapidly with small variations of σ lim . To illustrate this, in Figure A.3, streamwise, wall normal and spanwise autocorrelations from SVD and Tikhonov regularization are compared with the one obtained with the same methods but varying the optimal parameters α and σ lim . We can notice that, for the three velocity components the autocorrelations from Tikhonov regularization with α varying with y matches well with the one with a constant value of α with slight improvement. The same autocorrelations for SVD drops at locations where the algorithm did not converge. The streamwise autocorrelation (obtained for a constant σ u = 0.002) is improved in the centre of the channel with values of σ u (obtained for each y position and display in Figure A.2 a) which are close to the unique choice of 0.002. Therefore, it confirms that the reconstruction error can vary much more with small variations of σ lim than with the ones of α. This different behaviour of the error probably comes from the different regularization approaches. With SVD, diagonal elements of the hot-wire hot-wire correlation matrix R which are smaller are filtered (see equation 3.17), while with Tikhonov regularization larger values are added to diagonal elements of R (via the term αI) instead of filtering the smallest ones (see equation 3.20). More investigations can be done to study this in detail but it is beyond the scope of this thesis. Notice that, on averaged the wall normal and spanwise autocorrelations (Figures A.3 b) and c)) obtained with σ lim varying with the wall normal direction are lower than the case were σ lim is kept constant. The hot wire-PIV correlation S being lower in the wall normal and spanwise directions than in the streamwise one, it is therefore more difficult to reach the convergence and the sensitivity of the error to small variation of σ lim is increased. Its also explain why the peaks are more pronounced on σ v and σ w than for σ u (Figure A.2). In figure A.1, we can notice that the curve of α u displays a wavy pattern and peak close to the bounds, it suggests that the system conditioning required higher values of α u in between the probe positions (highlighted with crosses) were the LSE filtering is higher and close to the border than in the central region where scales are larger and less filtered. The deeps observed on the curve of α v and α w near the border suggest that the corresponding linear systems are better conditionned at the correponding wall normal locations. F i is the indicative binary function and W i, He and De correspond respectively to the width, height and depth of the structuring element. As was mentionned before, one important problem to solve during uniform momentum regions identification is to find the adequate threshold. To this we should add the choice of the size of the structuring element, if the same structuring elements is used for the closing and the opening or even more if different values of width, height and depth are chosen in each step of the morphological operations. As the final results will depend on the choice of the threshold and lengths of structuring elements used in mathematical morphology the problem is complex and has to be approached tactfully.

The procedure used here is to remove the noise from the original image without altering its form, which somehow, contains the physical information. Erosions, dilatations and the volume based cleaning procedure where combined all together and tested step by step in an interactive way with caution on the size of the structuring elements and the threshold used for the cleaning. Therefore the method presented here is certainly not universal but follows the previous constraints in such way to stay close to the raw field. In the present study, two different structuring elements given in grid points were used respectively s 1 = (2, 2, 2) and s 2 = (3, 2, 2). The cleaning was perfomed twice with a different mask in time than the one used in the spanwise and wall normal directions, for the first cleaning step we have mask t 1 = 3 and mask yz 1 = 1 with volume threshold field. A comparison between the cleaned field and the raw one is displayed in B.3, small objects were eliminated and we have kept all the main features from the raw field. Even if not illustrated here, the same cleaning procedure is applied to the high momentum regions at both Reynolds Re θ = 9830 and Re θ = 19660.

B.3 Vortices

The aim of this part is to detail the validation procedure of algorithm of detection of vortices. λ, Q, Γ 1 and Γ 2 criteria presented in section 5.2.2 with Nencioli's new detection algorithm [START_REF] Nencioli | A vector geometry-based eddy detection algorithm and its application to a high-resolution numerical model product and high-frequency radar surface velocities in the Southern California Bight[END_REF]) are firstly tested on a Lamb-Oseen vortex described by the equations (5.23) and (5.24) and are then applied to the experimental database. Nencioli (see [START_REF] Nencioli | A vector geometry-based eddy detection algorithm and its application to a high-resolution numerical model product and high-frequency radar surface velocities in the Southern California Bight[END_REF]) has implemented a new 2D detection algorithm. Based on a defintion of a vortex and its main characteristics, he proposed four constraints to identify a vortex in a spanwise wall normal plane:

B.3.1 Validation

1. The sign of the wall normal velocity should change as we cross the eddy center along a spanwise section and its magnitude should increase away from it.

2. The sign of the spanwise velocity should change as we cross the eddy center along a wall normal section, its magnitude should also increase away from it and the rotation sense should be the same as for v.

3. The velocity should be minimum at the eddy center.

4. The velocity vector directions around the eddy center should change with a constant sense of rotation and the directions of two neighboring vectors should lay within the same or two adjacents quadrants.

Two additional parameters should be added to the four constraints. The first one, a, defines how many grid points away from the eddy center the increase in magnitude of the wall normal velocity along a spanwise section and of the spanwise velocity along a wall normal section is checked, and a also sets the points around the vortex center at which the change of velocity vectors direction is inspected in the fourth constraint. The second parameter, b, defines the domain in which the minimum of the velocity field should be found. To illustrate the four constraints, Figure B.5 displays from left to right and from top to bottom the results of the four constraints in increasing order applied to the velocity field of the Lamb-Oseen vortex given by the equations (5.23) and (5.24) with paramaters a = 2 and b = 1. For the pairs of points with opposite sign of wall normal velocity like the one denoted by the filled red circles in Figure B.5 (a) (pairs of points with only one point which has its wall normal velocity equal to zero were considered), only the pairs for which the points in the same spanwise section located ±a grids points away (blue cross) where the wall normal velocity keeps an opposite sign as the one in the pair (highlighted with filled red circles) and increases in magnitude fulfill the first condition. Such points are highlighted with filled and empty red circles. From the first condition, the sense of rotation of the vortex is known, if the wall normal velocity switches from positive to negative values when the center of the eddy is crossed from right to left, then the vortex rotates counterclockwise and if it switches from negative to positive values when the center of the eddy is crossed from right to left then the vortex rotates clockwise. This method used to detect the sense of rotation of a vortex might be wrong for turbulent flows where large and small scales are superimposed and will be discussed further. Similarly, the second constraint is applied to all the pairs of points which fulfill the first constraint, the points located in the same wall normal plane as the pair (highlighted with filled red circles) and ±a grid points away are checked and only the ones where the sign of the streamwise velocity is still of opposite sign as in the pair and where its magnitude is still increasing are kept for the next step. The third constraint is then applied to the pairs of points which fulfill both the first and second constraints, a local minimum of the velocity magnitude is sought for the points located ±b grid points around each point of the valid pairs, in Figure B.5 (c) the filled red circle illustrates a valid point, the research domain is highlighted by the blue dashed box and the red cross in the center indicates that a velocity minimum was found within this domain. A second search around this minimum in a domain with the same size (highlighted with green dashed box in Figure B.5 (c)) should be performed to be certain that the minimum is local and if not the point is deleted. The fourth constraint is there to differentiate 'true vortex' from the meandering and shear zones that could pass through the three previous constraints, for more details, this constraint is well explained in [START_REF] Nencioli | A vector geometry-based eddy detection algorithm and its application to a high-resolution numerical model product and high-frequency radar surface velocities in the Southern California Bight[END_REF]. The velocity vector direction change is checked at the points located ±(a -1) grid points away from the position of the local minimum of the velocity, these points are on the green box in

B.3.2 Application to the present study

After the validation of our algorithms, we started the detection procedure by applying the Q criterion to the experimental database. Figure B.6 shows contours of Q and λ criterions plotted on velocity fluctuations. The top figures correspond to a time t = 2.5 s and the bottom ones correpond to t = 1.65 s. As suggested by the theory, the two different methods provide the same result. Contours ofQ ≈ 80 and λ ≈ 10 allow to detect the center of vortical motions near the wall. The same observation is valid for t = 1.65 s plotted at the bottom. Even if a lot of noise still remains, we can see that Q and λ criterions are able to identify some small vortical motions near the wall but fail to identify the bigger ones far away from the wall. This result was expected as Q and λ criteria are based on the velocity gradient tensor D, near the wall the velocity gradient is higher, vortical motions constrained by the wall are smaller and rotate rapidly. They are easily identified by both Q and λ criterions. Away from the wall, the velocity gradient is lower, vortical motions are free, bigger and less intense. This is why contours of Q and λ in these vortical motions are lower and can easily get blurred by noise.

To improve the results, we have applied a gaussian filter to the velocity fluctuations before computing the criteria. The gaussian distribution g 2 (z, y) has the following form:

g 2 (z, y) = 1 2πσ 2 exp - z 2 + y 2 2σ 2 (B.9)
With a standard deviation σ = 2.

The gaussian filter is then convolved with the velocity field (w, v) in a 5 * 5 box, this step is written as follows:

w(z, y) = j=2 k=2 j=-2 k=-2 w(z + k, y + j).g 2 (k, j) j=2 k=2 j=-2 k=-2 g 2 (k, j) (B.10) v(z, y) = j=2 k=2 j=-2 k=-2 v(z + k, y + j).g 2 (k, j) j=2 k=2 j=-2 k=-2 g 2 (k, j) (B.11)
The filtered velocity field is compared with the original one in figure B.7, we can see that the filtered vectors (b), (d) still have the same shape as the unfiltered ones (a), (c), the effect of the gaussian filter is to smooth the original data. [START_REF] Graftieaux | Combining PIV, POD and vortex identification algorithms for the study of unsteady turbulent swirling flows[END_REF], have suggested the maximum of |Γ 1 |, between 0.9 -1.0 as detection criterion, our threshold is reasonable given that the window size s is fixed while a wide range of scales are encountered in our data. In fact in their study, the influence of the ratio between the window size and the eddy radius on Γ 1 is investigated on a Lamb-Oseen vortex type only. It is found that reducing this ratio tends to sharpen Γ 1 distribution around the eddy, the influence on its magnitude is not clearly mentioned. Another explanation, may come directly from the shape of the vortical motions in our database, which are sometimes observed to have an elliptical shape motions rather than a circular one. In contrast, values of 2 π < |Γ 2 | < 1 suggested in [START_REF] Graftieaux | Combining PIV, POD and vortex identification algorithms for the study of unsteady turbulent swirling flows[END_REF] fail to identify vortical motions in the boundary layer. The major difference between Γ 1 and Γ 2 is in the term ũO (see equations (5.32) and ( 5 The top figures correspond to t = 2.5 s and the bottom ones to t = 75 ms. We can see that around each contour of Γ 1 there is a vortical motion. This can give an idea about the relevance of Γ 1 which is able to track large and small scales vortical motions at the same time. However, as the flow is turbulent, the same vortical motion may loose its shape at a certain time step and recover it later. For example, in to values bigger than 0.5, 0.4 and 0.3. From these plots, we can notice that we gain much coherence in time for the vortical motion labels when threshold values are low. Similar visualizations were made with threshold values between 0.3 and 0.4 to choose only one value that represents the vortical motion (see Figure B.17 for the threshold of 0.36, 0.35 and 0.34). Our final choice of 0.35 was dictated by a compromise between having a long temporal coherence in the vortical motion labels and limiting the high frequency noise that comes out when the threshold is low. After the thresholding step, this noise is eliminated with morphological operations as it was done for UMR in section 5.2.1. In the present study, the mathematical procedure includes two dilations with structuring elements of size s 1 = (1, 2, 2) and s 2 = (2, 1, 1) and one erosion with a structuring element s 2 = (2, 1, 1). A volume-size-based cleaning algorithm described in 5.22 was performed four times with two different parameters, for the first one, 3D mask parameters to compute the volume around each pixel were mask t 1 = 1 in time and mask yz 1 = 1 in the spanwise wall normal plane with volume threshold C 1 and C 2 of 4, in the second one, we have set the same parameters at mask t 2 = 6 in time and mask yz 2 = 1 in the spanwise wall normal plane with C 1 and C 2 equal to 6. The whole cleaning procedure is displayed in Figures B.18,B.19 and B.20 where the vortical motion center indicative function deducted from isosurfaces of Γ 1 is highlighted in red.

As described in Figure B.14, after the morphological operations the indicative function of vortical motion centers is labelled in 3D, the life time of each label is computed and the field is filtered in order to eliminate the labels whose life time is short. The value of t * > 0.05 was chosen according to the comparison with the field obtained after morphological operations (see Figure B.21). In this figure one can see that thresholding at this level allows us to filter the short time vortical motion labels found in the domain while keeping the longer one of (2010), it is recommended that some sensitivity test be made to choose the best values for a and b but in our experimental database, a unique choice of the couple (a, b) allows only detection of a few numbers of vortices filtering the others. For all the couples (a, b) tested there was no evident correlation between the input value of (a, b) and the vortex characteristics (size, position). For this reasons, we discuss only results obtained with a looped algorithm where We can see that most of the main vortical motions (the ones distinguishable at sight) are identified as with Γ 1 but the sense of rotation of some of them is wrong. Recall that the sense of rotation is deduced from the first constraint and it is the sense of rotation in the pair where a vortical motion center is expected. As mentionned before, this method is restrictive in turbulent flows where large and small scales interact with each other, we can find small scale vortices superimposed on larger ones and in case they rotate with an opposite sign, the first constraint will fail to give the sense of rotation of the vortex. As in previous cases, a gaussian filter was applied to the velocity field before applying the algorithm but it automatically eliminates the pairs in the center of the vortical motion and thus eliminates the vortical motion itself. As this algorithm is mostly based on small scales, it is not adapted to turbulent flows where both large and small scales should be taken into account. In the following parts, the method with Γ 1 summarized in Figures B.13, B.14 B.15 is the only criterion used to obtain vortical motion centers. Since the circulation Γ is linked to the vorticity ω through a surface dS (Γ = s w. dS). The vorticity magnitude along the x axis ω 0 of a vortex in the (z, y) plane is deduced from its circulation Γ and its radius r as follows:

ω 0 = Γ πr 2 (C.6)
In order to provide a clear vision of the vortical motion characterization algorithm, all the steps as described in [START_REF] Nencioli | A vector geometry-based eddy detection algorithm and its application to a high-resolution numerical model product and high-frequency radar surface velocities in the Southern California Bight[END_REF] [2,31] and Γ ∞ equal to 1 and 10. Assuming that the Lamb Oseen velocity is defined on cartesian grid of 2 mm spacing as in the LSE data, these values of r and Γ ∞ will match well the radius and circulation ranges expected in the the database. A radius range of [4,62] mm and infinite circulations of 2.10 -3 m 2 /s and 2.10 -2 m 2 /s were chosen such as to reproduce the values obtained with the same algorithm applied on the database and to have a similar evolution of the streamfunction ψ that is |∆ψ| ≈ 0.01 from one streamfunction contour line to the closest one. An illustration of the result obtained with r = 6 and Γ ∞ = 1 is depicted in Figure C.2 where contours of ψ are plotted over the Lamb-Oseen velocity field centered at (z 0 , y 0 ) = (10, 10). The domain of size ±2a where ψ is computed is highlighted with the red box and Figures C. Γ a∞ = 0.93 which is sligthly smaller than the original value Γ ∞ = 1. Note that the step on streamfunction contour line should be small enough in order to get the exact value of the radius but not too small because the number of contour lines will increase drastically with a and slow down the code. From all the tests performed on the LSE data a step of ∆ ψ = 0.01 showed a good compromise between resolution and computational time.

We have also investigated the effect of the number of contour points used to compute Γ on the circulation Γ a∞ . An example, where Γ was computed from contour points located 4 grid points away from the center of a Lamb Oseen vortex defined by r = 6 and Γ ∞ = -1 is displayed in Figure C.3. In Table C.1, we can notice that the value provided for Γ a∞ varies less with the number of points used to compute Γ , with 10 points we have Γ a∞ = -0.93 which is not very far from the input value of -1 obtained with 50 points. As on average, with LSE data the number of points on the contour lines retained as eddy core was always above 45, we expect this parameter to have no influence on the circulation obtained with the algorithm.

We have investigated the results obtained with Nencioli's algorithm when small and large scale vortices are superimposed together. For this purpose, we built a first Lamb Oseen vortex from equations 5. In addition the step where the distance between the outermost closed contour line and the boundary edge was checked (d 0 > 1.5 see Figure C.1) is cancelled in the simplified version. These two modifications can be explained by the fact that during the processing the spacing ∆ ψ = 0.01 on the streamfunction contour line is set constant and if we consider a contour line that is not too far from the border (d 0 ≤ 1.5) it will be farther in a window of larger size. In tables C.2 and C.3, the results obtained with the simplified version of the algorithm are given with the corresponding input values of r in the range [2,31] and Γ ∞ set to 1 and 10. Even if there is an underestimation of the radius at small radius and an overestimation at the large ones, in general, the values of r a and Γ a ∞ from the algorithm match well with the input values of r and Γ ∞ and present similar evolution. Finally, the simplified version of the algorithm was used as vortical motion core detection criterion on LSE data. On these data radius ranged in [4,62] mm and circulations ranged in [1.10 -3 , 4.10 -2 ] m 2 /s corresponding to r in the range [2, 31] and Γ ∞ varying from 0.5 to 40 for a Lamb Oseen vortex described by equations (5.23) and (5.24) on a cartesian grid of 2mm spacing. From all the tests that we have done with r in the range [2,31] and Γ ∞ equal to 1 and 10 we expect our algorithm to provide results with enough accuracy.

C.1.2 Application to the present study

The simplifed version of the algorithm of vortex core detection summarized in To evaluate the quality of the method, reconstructed velocity fluctuations are compared to refence ones extracted directly from the DNS database. Furthermore, a statistical analysis is performed on HWA, PIV and LSE velocity fields. The results obtained are compared to those from [START_REF] Carlier | Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry[END_REF]. Algorithms were developed to extract coherent structures from the reconstructed field. Uniform momentum regions are characterized with their mean hydraulic diameter in the YZ plane, their life time and their contribution to Reynolds stresses. The vortical motions are characterized by their position, radius, circulation and vorticity in addition to their life time and their number computed at a fixed position from the wall. The spatial organization of the structures was investigated through a correlation of their respective indicative functions in the spanwise direction. The simplified large scale model that arised is compared to the ones in the literature.

Keywords: high Reynolds numbers turbulent boundary layer, coherent structures, Linear Stochastic Estimation, detection methods

2. 1

 1 Experimental arrangement for Particle Image velocimetry taken from Raffel et al., 1998. . . . . . . . . . . . . . . . . . . . . . . .

  View of the 143 probes HWR built by Institut PPRIME installed in the LML wind tunnel. (b) Position of the rake relative to the stereo PIV measurement plane, probes on the rake are logarithmically distributed in wall-normal direction; . . . . . . 2.4 Top view of the stereo PIV setup . . . . . . . . . . . . . . . . . . 2.5 Front view of the stereo PIV setup . . . . . . . . . . . . . . . . . 3.1 Location of "Hot Wire" probes (black dots) in the DNS field and wall normal positions where the frequency spectra in Figures 3.3, 3.5, and 3.7 were averaged. . . . . . . . . . . . . . . . . . . . . . xxvii 3.2 Comparison between the original streamwise velocity field u extracted from DNS database and the reconstructed one û. a) Original streamwise velocity, b) streamwise velocity reconstructed through matrix inversion, c) same as b) for the SVD and d) same as b) for the Tikhonov regularization. . . . . . . . . . . . . . . 3.3 Streamwise velocity spectra from the original DNS field (•) compared to the one reconstructed with matrix inversion ( ), SVD ( ) and Tikhonov regularization ( ). At different wall normal locations (Figure 3.1). a) y = 0.030h, b) y = 0.090h, c) y = 0.189h and d) y = 0.674h. . . . . . . . . . . . . . . . . . . . . . . 3.4 Comparison between the the original wall normal velocity field v extracted from DNS database and the reconstructed one v. a) Original wall normal velocity, b) wall normal velocity reconstructed through matrix inversion c), same as b) for the SVD and d) same as b) for the Tikhonov regularization. . . . . . . . . . . . . . . 3.5 Wall normal velocity spectra from the original DNS field (•) compared to the one reconstructed with matrix inversion ( ), SVD ( ) and Tikhonov regularization ( ). At different wall normal locations (Figure 3.1). a) y = 0.030h, b) y = 0.090h, c) y = 0.189h and d) y = 0.674h. . . . . . . . . . . . . . . . . . . . . 3.6 Comparison between the original spanwise velocity field w extracted from DNS database and the reconstructed one ŵ. a) Original spanwise velocity, b) spanwise velocity reconstructed through matrix inversion, c) same as b) for the SVD and d) same as b) for the Tikhonov regularization. . . . . . . . . . . . . . . . . . . . . 3.7 Spanwise velocity spectra from the original DNS field (•) compared to the one reconstructed with matrix inversion ( ), SVD ( ) and Tikhonov regularization ( ). At different wall normal locations depicted on the probes distribution in (Figure 3.1). a) y = 0.030h, b) y = 0.090h, c) y = 0.189h and d) y = 0.674h. . . 3.8 (a) Streamwise/streamwise S x , (b) streamwise/wall normal S y and (c) streamwise/spanwise correlation between the PIV and hot wire data used for reconstruction plotted for a single probe located at y = 0.2041h and z = 0 which is highlighted with the black dot. . 3.9 Evolution with wall normal direction of the R ûi u i correlations for the three different methods tested (Inversion , SVD and Tikhonov regularization ). (a) Streamwise correlation, (b) wall normal correlation and (c) spanwise correlation as given in equation 3.22. The probes positions are highlighted in a) with crosses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 Comparison of the JPDFs between the original PIV field (u , v , w ) and the Hot Wire velocity field u 1 at Re θ = 9830 with gaussian distributions fitted on these JPDFs. (a) JPDF of u and u 1 ; (b) fitted gaussian (equation 3.24); (c) JPDF of v and u 1 ; (d) fitted gaussian distribution; (e) JPDF of w and u 1 ; (d) fitted gaussian distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11 Comparison of the JPDFs between the original PIV field (u , v , w ) and the Hot Wire velocity field u 1 at Re θ = 19660 with gaussian distributions fitted on these JPDFs. (a) JPDF of u and u 1 ; (b) fitted gaussian (equation 3.24); (c) JPDF of v and u 1 ; (d) fitted gaussian distribution; (e) JPDF of w and u 1 ; (d) fitted gaussian distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Streamwise spectrum from HWA (lines) at Re θ = 9830 and at different wall normal positions compared with Carlier and Stanislas (2005) (crosses) at Re θ = 10000 plotted. . . . . . . . . . . . . . . 4.2 Streamwise spectrum from HWA (lines) at Re θ = 19660 and at different wall normal positions compared with Carlier and Stanislas (2005) (crosses) at Re θ = 20000 plotted. . . . . . . . . . . . . . . 4.3 Streamwise premultiplied frequency spectra from LSE reconstruction (crosses) compared with HWA (solid lines) at Re θ = 9830. . 4.4 Streamwise premultiplied frequency spectra from LSE reconstruction (crosses) compared with HWA (solid lines) at Re θ = 19660. 4.5 Location of Hot Wire probes (black dots) and wall normal positions where the spectra in Figures 4.3 and 4.4 were averaged. Defect probes where the velocity magnitude was abberant were not used for reconstruction. . . . . . . . . . . . . . . . . . . . . 4.6 Mean & RMS streamwise velocity profiles obtained in the present study by using single hot-wires probes l + w = 11.8 at Re θ = 9830 compared with Carlier and Stanislas (2005) at Re θ = 10000. The vertical dashed lines indicate the region of the boundary layer where LSE reconstruction is performed, the (×) denote the locations of hot-wires probes. . . . . . . . . . . . . . . . . . . . . . . 4.7 Mean & RMS streamwise velocity profiles obtained in the present study by using single hot-wires probes l + w = 21.96 at Re θ = 19660 compared with Carlier and Stanislas (2005) Re θ = 20000. The vertical dashed lines indicate the region of the boundary layer where LSE reconstruction is performed, the (×) denote the locations of hot-wires probes. . . . . . . . . . . . . . . . . . . . . . . . . . .

  (red dots) at y + = 259 and Re θ = 10000. . . . . . . . . . . . . . . . . . . . . . 4.13 PDF of normalised streamwise velocity fluctuations from the reconstructed field at 2 KHz (black line), the PIV at 4 Hz (green line), the HWA at 30 KHz (blue diamond) all at y + = 442 and Re θ = 19660 and from Carlier and Stanislas (2005) (red dots) at y + = 485 and Re θ = 20000. . . . . . . . . . . . . . . . . . . . . . 4.14 PDF of normalised wall normal velocity fluctuations from the reconstructed field at 2 KHz (black line), the PIV at 4 Hz (green line) all at y + = 237 and Re θ = 9830 and the one from Carlier and Stanislas (2005) (red dots) at y + = 259 and Re θ = 10000. . . . . 4.15 PDF of normalised wall normal velocity fluctuations from the reconstructed field at 2 KHz (black line), the PIV at 4 Hz (green line) all at y + = 442 and Re θ = 19660 and the one from Carlier and Stanislas (2005) (red dots) at y + = 485 and Re θ = 20000. . . 4.16 PDF of spanwise velocity fluctuations from the reconstructed field at 2 KHz (black line), the PIV at 4 Hz (green line) all at y + = 237 at Re θ = 9830 and from Carlier and Stanislas (2005) (red dots) at y + = 259 and Re θ = 10000. . . . . . . . . . . . . . . . . . . . . . 4.17 PDF of spanwise velocity fluctuations from the reconstructed field at 2 KHz (black line), the PIV at 4 Hz (green line) all at y + = 442 at Re θ = 19660 and from Carlier and Stanislas (2005) (red dots) at y + = 485 and Re θ = 20000. . . . . . . . . . . . . . . . . . . . . . 4.18 (a) Streamwise streamwise S x , (b) streamwise/wall normal S y and (c) streamwise/spanwise correlation between the PIV and hot wire data used for reconstruction at Re θ = 9830 and plotted for a single probe located at y = 1916 + and z = 0 which is highlighted with the black dot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.19 (a) Streamwise streamwise S x , (b) streamwise/wall normal S y and (c) streamwise/spanwise correlation between the PIV and hot wire data used for reconstruction at Re θ = 19660 and plotted for a single probe located at y = 3567 + and z = 0 which is highlighted with the black dot. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.20 PDF of normalised streamwise velocity fluctuations from HWA at 30 KHz and Re θ = 9830 plotted at different wall normal positions. 4.21 PDF of normalised streamwise velocity fluctuations from HWA at 30 KHz and Re θ = 19660 plotted at different wall normal positions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.22 PDF of normalised streamwise velocity fluctuations from the reconstructed field at 2 KHz and Re θ = 9830 plotted at different wall normal positions. . . . . . . . . . . . . . . . . . . . . . . . . 4.23 PDF of normalised streamwise velocity fluctuations from the reconstructed field at 2 KHz and Re θ = 19660 plotted at different wall normal positions. . . . . . . . . . . . . . . . . . . . . . . . . 4.24 Skewness of streamwise velocity fluctuations from the reconstructed field at 2 KHz (black line), the PIV at 4 Hz (green line) and the HWA at 30 KHz (blue line) all at Re θ = 9830 with the one from Carlier and Stanislas (2005) (red dots) at Re θ = 10000 all against the distance from the wall. . . . . . . . . . . . . . . . . 4.25 Skewness of streamwise velocity fluctuations from the reconstructed field at 2 KHz (black line), the PIV at 4 Hz (green line) and the HWA at 30 KHz (blue line) all at Re θ = 19660 and the one from Carlier and Stanislas (2005) (red dots) at Re θ = 20000 all against the distance from the wall. . . . . . . . . . . . . . . . . 4.26 Skewness of wall normal velocity fluctuations from the reconstructed field at 2 KHz (black line) and the PIV at 4 Hz (green line) both at Re θ = 9830 and the one from Carlier and Stanislas (2005) (red dots) at Re θ = 10000 all against the distance from the wall. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.27 Skewness of wall normal velocity fluctuations from the reconstructed field at 2 KHz (black line) and the PIV at 4 Hz (green line) both at Re θ = 19660 and the one from Carlier and Stanislas (2005) (red dots) at Re θ = 20000 all against the distance from the wall. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.28 Skewness of spanwise velocity fluctuations from the reconstructed field at 2 KHz (black line) and the PIV at 4 Hz (green line) both at Re θ = 9830 and the one from Carlier and Stanislas (2005) (red dots) at Re θ = 10000 all against the distance from the wall. . . 4.29 Skewness of spanwise velocity fluctuations from the reconstructed field at 2 KHz (black line) and the PIV at 4 Hz (green line) both at Re θ = 19660 and the ones from Carlier and Stanislas (2005) (red dots) at Re θ = 20000 all against the distance from the wall. . . 4.30 Flatness of streamwise velocity fluctuations from the reconstructed field at 2 KHz (black line), the PIV at 4 Hz (green line) and the HWA at 30 KHz (blue line) all at Re θ = 9830 with the one from Carlier and Stanislas (2005) (red dots) at Re θ = 10000 all against the distance from the wall. . . . . . . . . . . . . . . . . . . . . . 4.31 Flatness of wall normal velocity fluctuations from the reconstructed field at 2 KHz (black line), the PIV at 4 Hz (green line) and the HWA at 30 KHz (blue line) all at Re θ = 19660 and the one from Carlier and Stanislas (2005) (red dots) at Re θ = 20000 all against the distance from the wall. . . . . . . . . . . . . . . . . 4.32 Flatness of wall normal velocity fluctuations from the reconstructed field at 2 KHz (black line) and the PIV at 4 Hz (green line) both at Re θ = 9830 and the one from Carlier and Stanislas (2005) (red dots) at Re θ = 10000 all against the distance from the wall. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.33 Flatness of wall normal velocity fluctuations from the reconstructed field at 2 KHz (black line) and the PIV at 4 Hz (green line) both at Re θ = 19660 and the one from Carlier and Stanislas (2005) (red dots) at Re θ = 20000 all against the distance from the wall. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.34 Flatness of spanwise velocity fluctuations from the reconstructed field at 2 KHz (black line) and the PIV at 4 Hz (green line) both at Re θ = 9830 and the one from Carlier and Stanislas (2005) (red dots) at Re θ = 10000 all against the distance from the wall. . .4.35 Flatness of spanwise velocity fluctuations from the reconstructed field at 2 KHz (black line) and the PIV at 4 Hz (green line) both at Re θ = 19660 and the one from Carlier and Stanislas (2005) (red dots) at Re θ = 20000 all against the distance from the wall. . . 5.1 Illustration of opening and closing. . . . . . . . . . . . . . . . . 5.2 Energetic contribution of u σ u at various wall normal positions at Re θ = 9830. The dashed lines are the threshold chosen for low and high momentum regions identification. . . . . . . . . . . . 5.3 Energetic contribution of u σ u at various wall normal positions at Re θ = 19660. The dashed lines are the threshold chosen for low and high momentum regions identification. . . . . . . . . . . . 5.4 Wall normal evolution of the number of vortical motions N vy detected at fixed position normalised with the total number of vortical motions N v detected from the original PIV (green) and the LSE (yellow) at Re θ = 9830. The hot wire probes positions are highlighted with crosses. . . . . . . . . . . . . . . . . . . . . . . 5.5 Histograms H of the vortical motion radius scaled with the boundary layer thickness and normalised with the number of vortical motions N v detected from: the original PIV at 4 Hz (green), the LSE reconstruction at 4 Hz (yellow) and the same reconstruction at 2 KHz (grey). In (a) the whole field, (b & d) y/δ ≤ 0.2 and (c & e) y/δ > 0.2 are investigated. The plain lines corresponds to Re θ = 9830 and the circles to Re θ = 19660. Corresponding mean radius and mean circulation are highlighted in the dashed lines. 5.6 Histograms H of the vortical motion circulation scaled with the boundary layer thickness and the free stream velocity, and normalised with the number of vortical motions N v detected from: the original PIV at 4 Hz (green), the LSE reconstruction at 4 Hz (yellow) and the same reconstruction at 2 KHz (grey). The plain lines corresponds to Re θ = 9830 and the circles to Re θ = 19660. Corresponding mean radius and mean circulation are highlighted in the dashed lines. . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Histograms of mean hydraulic diameter ( (a) & (c)) and life time ( (b) & (d)) of Low and high momentum regions for Re θ = 9830 ((a) & (b)) and Re θ = 19660 ((c) & (d)). . . . . . . . . . . . . . . . . . 5.8 Wall normal evolution of LMR and HMR contribution to Reynolds stresses. ( (a) & (b)) streamwise stress ( (c) & (d))shear stress for Re θ = 9830 ((a) & (c)) and Re θ = 19660 ((b) & (d)). . . . . . . . . 5.9 Contribution in percent of LMR and HMR to the Reynolds stresses. ((a) & (b)) streamwise stress ((c) & (d)) shear stress for Re θ = 9830 ((a) & (c)) and Re θ = 19660 ((b) & (d)). . . . . . . . . . . . . . . 5.10 Wall normal evolution of Q2 and Q3 (resp. Q1 and Q4 ) contribution to the streamwise stress inside LMR (resp. HMR) for Re θ = 9830 ((a) & (c)) and Re θ = 19660 ((b) & (d)) . . . . . . . . 5.11 Wall normal evolution of Q2 and Q3 (resp. Q1 and Q4 ) contribution to the shear stress inside LMR (resp. HMR) for Re θ = 9830 ((a) & (c)) and Re θ = 19660 ((b) & (d)) . . . . . . . . . . . . . . . 5.12 Wall normal evolution of the number of vortical motions N vy detected at fixed position normalised with the total number of vortical motions N v detected at Re θ = 9830 (blue) and Re θ = 19660 (red). The curve a/y is plotted with a = 0.02. The hot wire probes positions are highlighted with crosses. . . . . . . . . . . . . . . 5.13 Histogram of the life time of the vortical motions at Re θ = 9830. Positive vortical motion (red dashed line), negative vortical motion (blue dashed line), both positive and negative vortical motions (grey plain line) and the mean life time (grey dashed line). N s is the total number of labelled samples. . . . . . . . . . . . . 5.14 Histogram of the life time of the VPN vortical motions at Re θ = 9830 (grey line) and Re θ = 19660 (black line) normalised with the number of labelled samples N s used to build the histogram. The corresponding mean life time are highlighted by the dashed line. 5.15 Histogram of the vortical motion radius at Re θ = 9830 scaled in Kolmogorov units and normalised with the number of vortical motions N v . Positive vortical motion (red dashed line), negative vortical motion (blue dashed line), both positive and negative vortical motions (grey plain line) and mean radius (grey vertical dashed line). A log normal distribution described by equation (5.36) and fitted on the histogram is added in green dashed lines. Fitting parameters are given in table 5.2. . . . . . . . . . . . . . 5.16 Histogram of the vortical motion radius at Re θ = 9830 (grey line) and at Re θ = 19660 (black lines) scaled with: (a) the Komogorov length η; (b) boundary layer δ, (c) displacement δ * and (d) momentum θ thickness. All normalised with the number of vortical motions N v . The corresponding mean radius are highlighted by the vertical dashed lines. . . . . . . . . . . . . . . . . . . . . . . 5.17 Histogram of the radius at Re θ = 9830 scaled with the boundary layer thickness, normalised with the number of vortical motions N v and plotted against the wall normal position. The hot wire probes positions are highlighted with crosses. . . . . . . . . . . 5.18 Histogram of the radius at Re θ = 19660 scaled with the boundary layer thickness, normalised with the number of vortical motions N v and plotted against the wall normal position. The hot wire probes positions are highlighted with crosses. . . . . . . . . . . 5.19 Wall normal evolution of the mean radius at Re θ = 9830 (grey line) and Re θ = 19660 (black line) scaled with: (a) the Kolmogorov length η; (b) boundary layer δ, (c) displacement δ * and (d) momentum θ thickness. The hot wire probes positions are plotted with crosses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.20 Histograms of the vortical motion circulation at Re θ = 9830 (grey line) and Re θ = 19660 (black line) scaled in (a) Kolomorov units, then with the free stream velocity U ∞ and (b) the boundary layer, (c) the displacement and (d) the momentum thicknesses. All normalised with the number of vortical motions N v . The corresponding mean circulations are highlighted in the dashed lines. 5.21 Histogram of the circulation at Re θ = 9830 scaled with the free stream velocity and the boundary layer thickness, normalised with the number of vortical motions N v and plotted against the wall normal position. The hot wire probes positions are highlighted with crosses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.22 Histogram of the circulation at Re θ = 19660 scaled with the free stream velocity and the boundary layer thickness, normalised with the number of vortical motions N v and plotted against the wall normal position. The hot wire probes positions are highlighted with crosses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.23 Wall normal evolution of the mean circulation at Re θ = 9830 (grey line) and Re θ = 19660 (black line) scaled in (a) Kolomorov units, then with the free stream velocity U ∞ and (b) the boundary layer δ; (c) the displacement δ * and (d) the momentum θ thickness. The hot wire probes positions are plotted with crosses. . . . . . . . 5.24 Histograms of the vortical motion vorticity at Re θ = 9830 (grey line) and Re θ = 19660 (black line) scaled in (a) Kolomorov units, then with the free stream velocity U ∞ and (b) the boundary layer, (c) the displacement and (d) the momentum thicknesses. All normalised with the number of vortical motions N v . The corresponding mean circulations are highlighted in the dashed lines. 5.25 Histogram of the vorticity at Re θ = 9830 scaled with the free stream velocity and the boundary layer thickness, normalised with the number of vortical motions N v and plotted against the wall normal position. The hot wire probes positions are highlighted with crosses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.26 Histogram of the vorticity at Re θ = 19660 scaled with the free stream velocity and the boundary layer thickness, normalised with the number of vortical motions N v and plotted against the wall normal position. The hot wire probes positions are highlighted with crosses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.27 Wall normal evolution of the mean vorticity at Re θ = 9830 (grey line) and Re θ = 19660 (black line) scaled in (a) Kolomorov units, then with the free stream velocity U ∞ and (b) the boundary layer; (c) the displacement and (d) the momentum thicknesses. The hot wire probes positions are plotted with crosses. . . . . . . . . . . 5.28 Location of Hot Wire probes (black dots) and wall normal positions where the correlation R hl from equation (5.39) are computed at Re θ = 9830. Defect probes were the velocity magnitude was abberant were not used for reconstruction. . . . . . . . . . . . . 5.29 Spanwise correlation of high and low momentum regions (R hl ); positive and negative vortical motions (R pn ) plotted at different wall normal positions depicted in 5.28. (a) and (c) correspond to Re θ = 9830; (b) and (d) correspond to Re θ = 19660. . . . . . . . 5.30 (a) R uu correlation at y + = 92, (b) R uu correlation at y = 0.5δ obtained by Ganapathisubramani et al. (2005a). The contour levels for R uu range from -0.1 to 1.0 with spacing of 0.1. Zero contours are not shown. . . . . . . . . . . . . . . . . . . . . . . . 5.31 Spanwise correlation of the positive vortical motions and the low momentum regions (Rv p l), the high momentum regions (Rv p h) plotted at different wall normal positions depicted in 5.28. (a) and (c) correspond to Re θ = 9830; (b) and (d) correspond to Re θ = 19660. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.32 3D views of coherent structures. The low and the high momentum regions are displayed in green and yellow respectively, the positive and the negative vortical motions are displayed in red and blue respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.33 Counter rotating streamwise vortical motions lifting up a low momentum region (Blackwelder and Eckelmann, 1979). . . . . 5.34 Model of the coherent structures organization built with their "average" size and spacing at Re θ = 9830. The low regions is displayed in green, the positive and negative vortical motions are displayed in red and blue respectively. . . . . . . . . . . . . . . 5.35 Coherent structure organisation below y + = 50 for Re θ = 7800 (Reference Lin (2006)). Private communication from Michel Stanislas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.36 (a) Histograms and (b) wall normal evolution of the vortical motion circulation at Re θ = 9830 (grey line) and Re θ = 19660 (black line) scaled with the free stream velocity U ∞ the boundary laye and displacement and thicknesses. The histograms are normalised with the number of vortical motions N v . The corresponding mean circulations are highlighted in the dashed lines. The probe positions is highlighted with crosses. . . . . . . . . . . . . . . . . . A.1 Evolution of Tikhonov regularization parameter with the wall normal position. a) α u used for the streamwise velocity, b) α v used the wall normal velocity and c) evolution of α w used for the spanwise velocity. The probes positions are highlighted in a) with crosses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Evolution of SVD optimal parameter with the wall normal position. a) σ u used for the streamwise velocity, b) evolution of σ v used for the wall normal velocity and c) evolution of σ w used for the spanwise velocity. . . . . . . . . . . . . . . . . . . . . . . . . A.3 Evolution with wall normal direction of R ûi u i correlations for the SVD and Tikhonov regularization with optimal parameters varying with the wall normal position (SVDy ; Tikhy ) and the same parameters kept constant (SVD ; Tikh ). a) Streamwise velocity correlation, b) wall normal velocity correlation and c) spanwise velocity correlation as in equation 3.22. . . . . . . . A.4 Comparison between the streamwise velocity field obtained with SVD and Tikhonov regularization performed on DNS database when the optimal parameters are kept constant and the same field when the optimal parameters depend on y. a) Streamwise velocity from SVD with constant σ lim , b) same as a) with σ lim varying with y, c) streamwise velocity from Tikhonov regularization with constant α and d) same as c) with α varying on y. . . . . . . . . A.5 Comparison between the wall normal velocity field obtained with SVD and Tikhonov regularization performed on DNS database when the optimal parameters are kept constant and the same field when the optimal parameters depend on y. a) Wall normal velocity from SVD with constant σ lim , b) same as a) with σ lim varying with y, c) wall normal velocity from Tikhonov regularization with constant α and d) same as c) with α varying on y. . . . . . . . . A.6 Comparison between the spanwise velocity field obtained with SVD and Tikhonov regularization performed on DNS database when the optimal parameters are kept constant and the same field when the optimal parameters depend on y. a) Spanwise velocity from SVD with constant σ lim , b) same as a) with σ lim varying with y, c) spanwise velocity from Tikhonov regularization with constant α and d) same as c) with α varying on y. . . . . . . . . A.7 Comparison between the premultiplied streamwise frequency spectrum obtained from HWA (black lines), averaged in z at y/δ = 0.511 and the same spectrum obtained from LSE reconstruction averaged in z (black crosses), at z = 0 (blue crosses) and at z/δ = 0.45 (red crosses) all at y/δ = 0.506 and at Re θ = 9830 . . . . . . A.8 Comparison between the premultiplied streamwise frequency spectrum obtained from HWA (black lines), averaged in z at y/δ = 0.511 and the same spectrum obtained from LSE reconstruction averaged in z (black crosses), at z = 0 (blue crosses) and at z/δ = 0.45 (red crosses) all at y/δ = 0.506 and at Re θ = 19660 . . . . . A.9 Comparison between the premultiplied streamwise frequency spectrum obtained from HWA (black lines), averaged in z at y/δ = 0.511 and the same spectrum obtained from LSE reconstruction averaged in z (black crosses) and averaged in z only at PIV point close to the probe positions (green crosses) all at y/δ = 0.506 and at Re θ = 9830. . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.10 Comparison between the premultiplied streamwise frequency spectrum obtained from HWA (black lines), averaged in z at y/δ = 0.511 and the same spectrum obtained from LSE reconstruction averaged in z (black crosses) and averaged in z only at PIV point close to the probe positions (green crosses) all at y/δ = 0.506 and at Re θ = 19660. . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1 Indicative function F i of high momentum regions from equation (5.21) at different steps of the cleaning procedure (first figure). B.2 Indicative function F i of high momentum regions from equation (5.21) at different steps of the cleaning procedure (last figure). B.3 Illustration of cleaning procedure. (top) indicative function F i of high momentum regions from equation (5.21) and (bottom) same function after cleaning. . . . . . . . . . . . . . . . . . . . . . . . B.4 (a) contours of λ ; (b) contours of Q; (c) contours of Γ 1 and (d) contours of Γ 2 all plotted on the velocity field of the Lamb-Oseen vortex given by equations (5.23) and (5.24). . . . . . . . . . . . B.5 Illustration of vortex center identification algorithm as described in Nencioli et al. (2010). (a) First ; (b) Second ; (c) third and (d) fourth steps of the algorithm applied on a velocity field of the Lamb-Oseen vortex given by equations (5.23) and (5.24). The blue crosses indicate points around the pair (highlighted with filled red circle) at which the increase in magnitude and the change of sign of the velocity is checked. The blue and the green dashed box highlight the domains in which a minimum of the velocity is searched around a point in a pair. The green box in plain lines highights points at which the velocity vector direction change is checked in the last step. The filled and empty red circles indicates points in pairs which satistfies the constraint at the corresponding steps, the red cross indicate same type of points which are also local velocity minimum. . . . . . . . . . . . . . . . . . . . . . . B.6 Comparison of vortical motion identification with Q criterion and λ criterion. (a) Contours of Q plotted on the velocity fluctuations at t = 2.5 s; (b) contours of λ plotted on the same velocity field as (a); (c) contours of Q plotted on the velocity fluctuations at t = 1.65 s and (d) contours of λ plotted on the same velocity field as (c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.7 Comparison of velocity fluctuations before and after a gaussian filtering. (a) and (c) Original velocity fluctuations; (b) and (d) velocity fluctuations after the gaussian filtering. . . . . . . . . . B.8 Comparison of results obtained with and without the gaussian filter described in equations B.9, B.10 and B.11. (a) Contours of Q plotted on the velocity fluctuations at t = 2.5 s; (b) contours of Q plotted on the velocity fluctuations at t = 2.5 s after the gaussian filter; (c) contours of Q plotted on the velocity fluctuations at t = 1.65 s and (d) contours of Q plotted on the velocity fluctuations at t = 1.65 s after the gaussian filter. . . . . . . . . . . . . . . . . B.9 Results obtained with the vortical motion center and core identification criterions respectively Γ 1 and Γ 2 proposed by Graftieaux et al., 2001. (a) Contours of Γ 1 plotted on the velocity fluctuations at t = 2.5 s ; (b) contours of Γ 2 plotted on the same velocity field as (a); (c) contours of Γ 1 plotted on the velocity fluctuations at t = 1.65 s and (d) contours of Γ 1 plotted on the same velocity field as (c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.10 Comparison of results obtained with and without the gaussian filter described in equations B.9, B.10 and B.11. (a) Contours of Γ 2 plotted on the velocity fluctuations at t = 2.5 s ; (b) contours of Γ 2 plotted on the velocity fluctuations at 2.5 s after the gaussian filter; (c) contours of Γ 2 plotted on the velocity fluctuations at t = 1.65 s and (d) contours of Γ 2 plotted on the velocity fluctuations at 1.65 s after the gaussian filter. . . . . . . . . . . . . . . . . . . . . . . B.11 Investigation of results obtained with Γ 1 at small scales. (a) contours of Γ 1 plotted on the velocity fluctuations at t = 2.5 s the green box denotes the small scales investigated; (b) contours of Γ 1 plotted on the velocity fluctuations at t = 2.5 s the domain is reduced to the green box in (a) ; (c) contours of Γ 1 plotted on the velocity fluctuations at t = 75 ms and (d) contours of Γ 1 plotted on the velocity fluctuations at t = 75 ms the domain the is reduced to the green box in (c). . . . . . . . . . . . . . . . . . . . . . . . . . B.12 Study of the evolution of contours of Γ 1 ( |Γ 1 | > 0.7) indicating the vortical motion centers with consecutive time steps. (a) Contours of Γ 1 plotted in the velocity fluctuations at t = 1.640 s ; (b) contours of Γ 1 plotted on the velocity fluctuations at 1.641 s ; (c) contours of Γ 1 plotted on the velocity fluctuations at t = 1.643 s and (d) contours of Γ 1 plotted on the velocity fluctuations at 1.644 s. . . B.13 First part of the centroid detection algorithm . . . . . . . . . . B.14 Second part of the centroid detection algorithm . . . . . . . . . B.15 Last part of the centroid detection algorithm . . . . . . . . . . . B.16 From the top to the bottom isosurfaces of |Γ 1 | bigger than 0.5, 0.4 and 0.3 respectively. . . . . . . . . . . . . . . . . . . . . . . . . .

  (a) vortical motion positions provided by the algorithm for a = 2 and b = 1 plotted on velocity fluctuations at t = 2.5s; (b) vortical motion positions provided by the algorithm for a = 3 and b = 2 plotted on the same velocity field as (a); (c) vortical motion positions provided by the algorithm for a = 4 and b = 3 plotted on the same velocity field as (a) and (d) vortical motion positions provided by the algorithm for a = 5 and b = 4 plotted on the same velocity field as (a). . . . . . . . . . . . . . B.23 Comparison of results obtained with the vortex center detection algorithm of Nencioli et al. (2010) and the one obtained with contours of Γ 1 ( |Γ 1 | > 0.7) as described in Graftieaux et al. (2001); (a) vortical motion positions provided by Nencioli's algorithm plotted on velocity fluctuations at t = 2.5s for parameters a = 3 to a = 7 and b = 2 to b = 6; (b) vortical motion centers provided by contours of Γ 1 ( |Γ 1 | > 0.7) plotted on the same velocity field as (a); (c) vortical motion positions provided by the Nencioli's algorithm plotted on velocity fluctuations at t = 1.65s for parameters a = 3 to a = 7 and b = 2 to b = 6 and (d) vortical motion positions provided by contours of Γ 1 ( |Γ 1 | > 0.7) plotted on the same velocity field as (c). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.1 Algorithm of detection as described in the refence article (Nencioli et al. (2010)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.2 Illustration of vortex core detection as described in Nencioli et al. (2010), the red box highlights the domain of size ±2a where the streamfunction ψ is computed. (a) Contours of ψ with parameter a = 2; (b) same with a = 3; (c) same with a = 4 and (d) contour which defines the eddy radius obtained at a = 4 all plotted on the Lamb-Oseen velocity field provided by equations (5.23) and (5.24) with Γ ∞ = 1 and r = 6. . . . . . . . . . . . . . . . . . . . . C.3 Lamb oseen velocity field from (5.23) and (5.24) with r = 6 and Γ ∞ = -1, the circulation Γ was integrated with points (red dots) located 4 grid points away from the center and lying on the green contours. (a) 10 points are used for the integration of Γ and Γ ∞ ; (b) same with 30 points; (c) same with 50 points and (d) same with 100 points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.4 Illustration of vortex core detection as described in Nencioli et al. (2010), the red box highlights the domain of size ±2a where the streamfunction is computed. (a) Superimposed Lamb oseen velocity fields from the equations (5.23) and (5.24) with r = 3 and Γ ∞ = 1 for the first center (red dot) at (z 01 , y 01 ) = (11, 11) and r = 6 and Γ ∞ = 1 for the second center (green dot) at (z 02 , y 02 ) = (10, 10); (b) contours of ψ with a = 2 plotted on the velocity field of (a); (c) same as (b) with a = 3 and (d) contour which defines the eddy radius obtained at a = 3 and plotted on the Lamb oseen velocity field of (a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.5 Illustration of core detection with a simplified version of the algorithm described in Nencioli et al. (2010) and summarized in Figure C.6. (a) Contours of ψ with parameter a = 5 and (b) contour which defines the eddy radius obtained at a = 5 and all plotted on the superimposed Lamb oseen velocity field from (5.23) and (5.24) with r = 3 and Γ ∞ = 1 for the first center (red dot) at (z 01 , y 01 ) = (11, 11) and r = 6 and Γ ∞ = 1 for the second center (green dot) at (z 02 , y 02 ) = (10, 10). . . . . . . . . . . . . . . . . . . C.6 Simplified version of the algorithm of detection described in Nencioli et al. (2010) used in the present study. . . . . . . . . . . . C.7 Investigation of results obtained with the simplified version of the vortex core detection algorithm described in C.6. The blue contours correspond to negative vortical motions, the red contours correspond to the positive ones and the green dots denote vortical motion centroids (a) contours which define the eddies core plotted on 2D velocity fluctuations (v , w ) at t = 0.01s; (b) the same as (a) at t = 0.075s; (c) the same as (a) at t = 0.12s and (d) the same as (a) at t = 0.23s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.8 Investigation of results obtained with the simplified version of the vortex core detection algorithm described in C.6. The blue contours correspond to negative vortical motions, the red contours correspond to the positive ones and the green dots denote vortical motion centroids (a) contours which define the eddies core plotted on 2D velocity fluctuations (v , w ) at t = 0.80s; (b) the same as (a) at t = 2.23s; (c) the same as (a) at t = 3.54s and (d) the same as (a) at t = 4.6s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.9 3D views of the contours which define the eddy core obtained with the simplified version of the algorithm described in C.6 at different time intervals. . . . . . . . . . . . . . . . . . . . . . . . C.10 3D views of the contours which define the eddy core obtained with the simplified version of the algorithm described in C.6 at different time intervals. . . . . . . . . . . . . . . . . . . . . . . . D.1 Number of positive vortical motions detected N vp normalised with the number of measurement blocks N B = 534 and plotted against the wall normal position for Re θ = 9830. . . . . . . . . . D.2 Number of negative vortical motions detected N vn normalised with the number of measurement blocks N B = 534 and plotted against the wall normal position for Re θ = 9830. . . . . . . . . . D.3 Positive (VP) and negative (VN) vortical motions streamwise ((a) & (b)) shear ((c) & (d)) stresses contribution at Re θ = 9830 ((a) & (c)) and Re θ = 19660 ((b) & (d)). . . . . . . . . . . . . . . . . . . D.4 Percentage of streamwise ((a) & (b)) shear ((c) & (d)) stresses contribution from Positive (VP) and negative (VN) vortical motions to the total stresses (ALL) at Re θ = 9830 ((a) & (c)) and Re θ = 19660 ((b) & (d)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  was the first in 1883 to evidence the turbulent flow in rectilinear pipe with circular cross section. A laminar flow is a stationary flow with parallel streamlines far from the inlet. This flow becomes unstable when velocity increases, leading the Reynolds number to reach a critical value. The flow then becomes turbulent. Turbulent flows are distinguishable by their main characteristics:• High Reynolds number (non-dimensional number defined as Re = LU ν given a length scale L, a velocity scale U and a viscosity ν),• Random spatio-temporal evolution of the flow variables, • three-dimensional and rotational velocity fields, • Non-linearity (which involves many length and time scales and thus complexifies the flow physics) • Dissipation

Figure 1 . 1 -

 11 Figure 1.1 -Structure of a flat plate TBL visualised with hydrogen bubbles at y + = 9.6 (Kline et al. (1967)). The flow direction is from left to right.

Figure 1 .

 1 Figure 1.3 -(a,b) Lift-up by isotropic wall-normal motions, (c) effect of mean shear and (d) diffusion Chernyshenko and Baig (2005).

Figure 1 .

 1 Figure 1.4 -Schematic of hairpin structure attached to the wall and the induced motion (Adrian et al. (2000)).

Figure 1 .

 1 Figure1.5 -Hairpin packet model[START_REF] Adrian | Vortex organization in the outer region of the turbulent boundary layer[END_REF]).

Figure 2 . 1 -

 21 Figure 2.1 -Experimental arrangement for Particle Image velocimetry taken from Raffel et al., 1998.

  Figure 2.3 -(a) View of the 143 probes HWR built by Institut PPRIME installed in the LML wind tunnel. (b) Position of the rake relative to the stereo PIV measurement plane, probes on the rake are logarithmically distributed in wallnormal direction;

Figure 2

 2 Figure 2.4 -Top view of the stereo PIV setup

Figure 3 . 1 -

 31 Figure 3.1 -Location of "Hot Wire" probes (black dots) in the DNS field and wall normal positions where the frequency spectra in Figures 3.3, 3.5, and 3.7 were averaged.

  Figures 3.2, 3.4 and 3.6 provide a comparison of the three components of the velocity from the original field extracted from DNS database (a) and reconstructed with matrix inversion (b), SVD (c) and Tikhonov regularization (d). Data from the direct matrix inversion show high frequency noise. Velocities reconstructed with Tikhonov and SVD are subject to a filtering effect, probably due to the regularization imposed through cut off by σ lim and α. For comparison, all fields are displayed with the same velocity range. Consequently, the velocity map from the original field and the one recontructed with matrix inversion are saturated. For the streamwise component, among the three methods tested, Tikhonov regularisation ( 3.2 (d)) gives the best match with the orignal field plotted in (3.2 (a)). For the wall normal and the spanwise velocity component, the filtering effect of SVD and Tikhonov is higher, there is nevertheless some correlation between the largest scale in the original field and the reconstructed one. Figures 3.3, 3.5 and 3.7 display respectively the comparison of streamwise E 11 , wall normal E 22 and spanwise E 33 energy spectra from the original field

  Figure 3.8 -(a) Streamwise/streamwise S x , (b) streamwise/wall normal S y and (c) streamwise/spanwise correlation between the PIV and hot wire data used for reconstruction plotted for a single probe located at y = 0.2041h and z = 0 which is highlighted with the black dot.

  Figure 3.10 -Comparison of the JPDFs between the original PIV field (u , v , w ) and the Hot Wire velocity field u 1 at Re θ = 9830 with gaussian distributions fitted on these JPDFs. (a) JPDF of u and u 1 ; (b) fitted gaussian (equation 3.24); (c) JPDF of v and u 1 ; (d) fitted gaussian distribution; (e) JPDF of w and u 1 ; (d) fitted gaussian distribution.

Figures 4 .

 4 Figures 4.1 and 4.2 display the comparison of streamwise spectra E 11 from HWA (lines) at Re θ = 9830 and Re θ = 19660 respectively with the ones obtained by Carlier and Stanislas (2005) (crosses) at Re θ = 10000 and Re θ = 20000 at various wall normal distance. These E 11 spectra are averaged in the spanwise direction and over 534 blocks at Re θ = 9830 and 570 blocks at Re θ = 19660.As in[START_REF] Kunkel | Study of the near-wall-turbulent region of the high-Reynolds-number boundary layer using an atmospheric flow[END_REF] and[START_REF] Perry | An experimental study of the turbulence structure in smooth-and rough-wall boundary layers[END_REF], a Taylor hypothesis is used with the local mean velocity to switch from frequency to wavenumber (k x =

Figure 4 . 1 -

 41 Figure 4.1 -Streamwise spectrum from HWA (lines) at Re θ = 9830 and at different wall normal positions compared with Carlier and Stanislas (2005) (crosses) at Re θ = 10000 plotted.

Figure 4

 4 Figure 4.2 -Streamwise spectrum from HWA (lines) at Re θ = 19660 and at different wall normal positions compared with Carlier and Stanislas (2005) (crosses) at Re θ = 20000 plotted.

Figure 4

 4 Figure 4.3 -Streamwise premultiplied frequency spectra from LSE reconstruction (crosses) compared with HWA (solid lines) at Re θ = 9830.

Figure 4

 4 Figure 4.4 -Streamwise premultiplied frequency spectra from LSE reconstruction (crosses) compared with HWA (solid lines) at Re θ = 19660.

Figure 4

 4 Figure 4.5 -Location of Hot Wire probes (black dots) and wall normal positions where the spectra in Figures 4.3 and 4.4 were averaged. Defect probes where the velocity magnitude was abberant were not used for reconstruction.

Figure 4

 4 Figure 4.6 -Mean & RMS streamwise velocity profiles obtained in the present study by using single hot-wires probes l + w = 11.8 at Re θ = 9830 compared with Carlier and Stanislas (2005) at Re θ = 10000. The vertical dashed lines indicate the region of the boundary layer where LSE reconstruction is performed, the (×) denote the locations of hot-wires probes.

Figure 4

 4 Figure 4.7 -Mean & RMS streamwise velocity profiles obtained in the present study by using single hot-wires probes l + w = 21.96 at Re θ = 19660 compared with Carlier and Stanislas (2005) Re θ = 20000. The vertical dashed lines indicate the region of the boundary layer where LSE reconstruction is performed, the (×) denote the locations of hot-wires probes.

Figures 4 .

 4 Figures 4.8 and 4.9 display RMS wall normal velocities v + at Re θ = 9830 and Re θ = 19660. As for u + , RMS from PIV measurements at 4 Hz are underestimated compared to the data of[START_REF] Carlier | Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry[END_REF], but they still remain in the range of the corresponding RMS measurement errors of ±0.28 wall units at both Reynolds numbers. A small discontinuity is observed at mid height of the boundary layer (y/δ ≈ 0.5). It corresponds to the connection of the two SPIV subsystems, each made of two cameras and mounted on top of one

Figure 4

 4 Figure 4.8 -RMS wall normal velocity profiles obtained in the present study by using single hot-wires probes l + w = 11.8 at Re θ = 9830 compared with Carlier and Stanislas (2005) at Re θ = 10000.

Figure 4

 4 Figure 4.9 -RMS wall normal velocity profiles obtained in the present study by using single hot-wires probes l + w = 21.96 at Re θ = 19660 compared with Carlier and Stanislas (2005) at Re θ = 20000.

Figure 4 .

 4 Figure 4.10 -RMS spanwise velocity profiles obtained in the present study by using single hot-wires probes l + w = 11.8 at Re θ = 9830 compared with Carlier and Stanislas (2005) at Re θ = 10000.

Figure 4 .

 4 Figure 4.11 -RMS spanwise velocity profiles obtained in the present study by using single hot-wires probes l + w = 21.96 at Re θ = 19660 compared with Carlier and Stanislas (2005) at Re θ = 20000.

Figures 4 .

 4 Figures 4.12 and4.13 display PDFs of normalised streamwise velocity fluctuations from the reconstructed PIV at 2 KHz (black lines), the PIV at 4 Hz (green lines), the HWA at 30 KHz (blue diamonds) at Re θ = 9830 and Re θ = 19660 respectively, together with the ones from[START_REF] Carlier | Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry[END_REF] (red dots) at Re θ = 10000 and Re θ = 20000. PDFs from HWA and the reconstructed field at 2.0 KHz match well with the ones from[START_REF] Carlier | Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry[END_REF]. PDFs from PIV at 4 Hz display some slight oscillations. This is probably due to a peak locking effect which describe a displacement bias error of a periodic pattern on pixel intervals (see[START_REF] Raffel | Particle image velocimetry: a practical guide[END_REF][START_REF] Westerweel | Fundamentals of digital particle image velocimetry[END_REF]). This effect is

Figure 4 .

 4 Figure 4.12 -PDF of normalised streamwise velocity fluctuations from the reconstructed field at 2 KHz (black line), the PIV at 4 Hz (green line), the HWA at 30 KHz (blue diamond) all at y + = 237 and Re θ = 9830 and from Carlier and Stanislas (2005) (red dots) at y + = 259 and Re θ = 10000.

Figures

  Figures 4.20, 4.21, 4.22 and 4.23 display PDFs of normalised streamwise

  Figures 4.20, 4.21, 4.22 and 4.23 display PDFs of normalised streamwise

Figure 4 .

 4 Figure 4.13 -PDF of normalised streamwise velocity fluctuations from the reconstructed field at 2 KHz (black line), the PIV at 4 Hz (green line), the HWA at 30 KHz (blue diamond) all at y + = 442 and Re θ = 19660 and from Carlier and Stanislas (2005) (red dots) at y + = 485 and Re θ = 20000.

Figure 4 .

 4 Figure 4.14 -PDF of normalised wall normal velocity fluctuations from the reconstructed field at 2 KHz (black line), the PIV at 4 Hz (green line) all at y + = 237 and Re θ = 9830 and the one from Carlier and Stanislas (2005) (red dots) at y + = 259 and Re θ = 10000.

Figure 4 .

 4 Figure 4.15 -PDF of normalised wall normal velocity fluctuations from the reconstructed field at 2 KHz (black line), the PIV at 4 Hz (green line) all at y + = 442 and Re θ = 19660 and the one from Carlier and Stanislas (2005) (red dots) at y + = 485 and Re θ = 20000.

Figure 4 .

 4 Figure 4.16 -PDF of spanwise velocity fluctuations from the reconstructed field at 2 KHz (black line), the PIV at 4 Hz (green line) all at y + = 237 at Re θ = 9830 and from Carlier and Stanislas (2005) (red dots) at y + = 259 and Re θ = 10000.

Figure 4 .

 4 Figure 4.17 -PDF of spanwise velocity fluctuations from the reconstructed field at 2 KHz (black line), the PIV at 4 Hz (green line) all at y + = 442 at Re θ = 19660 and from Carlier and Stanislas (2005) (red dots) at y + = 485 and Re θ = 20000.

  Figure 4.18 -(a) Streamwise streamwise S x , (b) streamwise/wall normal S y and (c) streamwise/spanwise correlation between the PIV and hot wire data used for reconstruction at Re θ = 9830 and plotted for a single probe located at y = 1916 + and z = 0 which is highlighted with the black dot.

Figure 4 .

 4 Figure 4.20 -PDF of normalised streamwise velocity fluctuations from HWA at 30 KHz and Re θ = 9830 plotted at different wall normal positions.

Figure 4 .

 4 Figure 4.21 -PDF of normalised streamwise velocity fluctuations from HWA at 30 KHz and Re θ = 19660 plotted at different wall normal positions.

Figure 4 .

 4 Figure 4.22 -PDF of normalised streamwise velocity fluctuations from the reconstructed field at 2 KHz and Re θ = 9830 plotted at different wall normal positions.

Figure 4 .

 4 Figure 4.23 -PDF of normalised streamwise velocity fluctuations from the reconstructed field at 2 KHz and Re θ = 19660 plotted at different wall normal positions.

Figure 4 .

 4 Figure 4.24 -Skewness of streamwise velocity fluctuations from the reconstructed field at 2 KHz (black line), the PIV at 4 Hz (green line) and the HWA at 30 KHz (blue line) all at Re θ = 9830 with the one from Carlier and Stanislas (2005) (red dots) at Re θ = 10000 all against the distance from the wall.

Figures 4 .

 4 Figures 4.26 and 4.27 display the skewnesses of wall normal velocity fluctuations from the reconstructed field at 2 KHz (black lines) at Re θ = 9830 and Re θ = 19660 respectively compared to Carlier and Stanislas (2005) at Re θ = 10000 and Re θ = 20000 respectively. Here again the agreement is quite good. Figures 4.28 and 4.29 give finally the same results as figures 4.26 and 4.27 but for the spanwise velocity fluctuations. They confirm the good agreement obtained for the two other velocity components.

Figures 4 .

 4 Figures 4.30 and 4.31 display the flatnesses of streamwise velocity fluctuations from the reconstructed field at 2 KHz (black lines), the PIV at 4 Hz (green lines), the HWA at 30 KHz (blue lines) all at Re θ = 9830 and Re θ = 19660 respectively compared to Carlier and Stanislas, 2005 (red dots) at Re θ = 10000 and

Figure 4 .

 4 Figure 4.25 -Skewness of streamwise velocity fluctuations from the reconstructed field at 2 KHz (black line), the PIV at 4 Hz (green line) and the HWA at 30 KHz (blue line) all at Re θ = 19660 and the one from Carlier and Stanislas (2005) (red dots) at Re θ = 20000 all against the distance from the wall.

Figure 4 .

 4 Figure 4.26 -Skewness of wall normal velocity fluctuations from the reconstructed field at 2 KHz (black line) and the PIV at 4 Hz (green line) both at Re θ = 9830 and the one from Carlier and Stanislas (2005) (red dots) at Re θ = 10000 all against the distance from the wall.

Figure 4 .

 4 Figure 4.27 -Skewness of wall normal velocity fluctuations from the reconstructed field at 2 KHz (black line) and the PIV at 4 Hz (green line) both at Re θ = 19660 and the one from Carlier and Stanislas (2005) (red dots) at Re θ = 20000 all against the distance from the wall.

Figure 4 .

 4 Figure 4.28 -Skewness of spanwise velocity fluctuations from the reconstructed field at 2 KHz (black line) and the PIV at 4 Hz (green line) both at Re θ = 9830 and the one from Carlier and Stanislas (2005) (red dots) at Re θ = 10000 all against the distance from the wall.

Figure 4 .

 4 Figure 4.29 -Skewness of spanwise velocity fluctuations from the reconstructed field at 2 KHz (black line) and the PIV at 4 Hz (green line) both at Re θ = 19660 and the ones from Carlier and Stanislas (2005) (red dots) at Re θ = 20000 all against the distance from the wall.

Figure 4 .

 4 Figure 4.30 -Flatness of streamwise velocity fluctuations from the reconstructed field at 2 KHz (black line), the PIV at 4 Hz (green line) and the HWA at 30 KHz (blue line) all at Re θ = 9830 with the one from Carlier and Stanislas (2005) (red dots) at Re θ = 10000 all against the distance from the wall.

Figure 4 .

 4 Figure 4.31 -Flatness of wall normal velocity fluctuations from the reconstructed field at 2 KHz (black line), the PIV at 4 Hz (green line) and the HWA at 30 KHz (blue line) all at Re θ = 19660 and the one from Carlier and Stanislas (2005) (red dots) at Re θ = 20000 all against the distance from the wall.

Figure 4 .

 4 Figure 4.32 -Flatness of wall normal velocity fluctuations from the reconstructed field at 2 KHz (black line) and the PIV at 4 Hz (green line) both at Re θ = 9830 and the one from Carlier and Stanislas (2005) (red dots) at Re θ = 10000 all against the distance from the wall.

Figure 4 .

 4 Figure 4.33 -Flatness of wall normal velocity fluctuations from the reconstructed field at 2 KHz (black line) and the PIV at 4 Hz (green line) both at Re θ = 19660 and the one from Carlier and Stanislas (2005) (red dots) at Re θ = 20000 all against the distance from the wall.

Figure 4 .

 4 Figure 4.34 -Flatness of spanwise velocity fluctuations from the reconstructed field at 2 KHz (black line) and the PIV at 4 Hz (green line) both at Re θ = 9830 and the one from Carlier and Stanislas (2005) (red dots) at Re θ = 10000 all against the distance from the wall.

Figure 4 .

 4 Figure 4.35 -Flatness of spanwise velocity fluctuations from the reconstructed field at 2 KHz (black line) and the PIV at 4 Hz (green line) both at Re θ = 19660 and the one from Carlier and Stanislas (2005) (red dots) at Re θ = 20000 all against the distance from the wall.

Figure 5 . 1 -

 51 Figure 5.1 -Illustration of opening and closing.

  Figure 5.2 -Energetic contribution of u σ u at various wall normal positions at Re θ = 9830. The dashed lines are the threshold chosen for low and high momentum regions identification.

Figure 5 . 3 -

 53 Figure 5.3 -Energetic contribution of u σ u at various wall normal positions at Re θ = 19660. The dashed lines are the threshold chosen for low and high momentum regions identification.

  .33) directly with velocity fluctuations ũ O = 1 N s u and u P instead of instantaneous velocities ũO = 1 N s u and u P .

Figure 5

 5 Figure 5.4 -Wall normal evolution of the number of vortical motions N vy detected at fixed position normalised with the total number of vortical motions N v detected from the original PIV (green) and the LSE (yellow) at Re θ = 9830. The hot wire probes positions are highlighted with crosses.

1 -

 1 Number of vortical motions detected from the orginal PIV and the reconstructed LSE both at 4 Hz for Re θ = 9830 and Re θ = 19660.

  Figure 5.5 -Histograms H of the vortical motion radius scaled with the boundary layer thickness and normalised with the number of vortical motions N v detected from: the original PIV at 4 Hz (green), the LSE reconstruction at 4 Hz (yellow) and the same reconstruction at 2 KHz (grey). In (a) the whole field, (b & d) y/δ ≤ 0.2 and (c & e) y/δ > 0.2 are investigated. The plain lines corresponds to Re θ = 9830 and the circles to Re θ = 19660. Corresponding mean radius and mean circulation are highlighted in the dashed lines.

  Figure 5.7 -Histograms of mean hydraulic diameter ( (a) & (c)) and life time ( (b) & (d)) of Low and high momentum regions for Re θ = 9830 ((a) & (b)) and Re θ = 19660 ((c) & (d)).

  Figure 5.8 -Wall normal evolution of LMR and HMR contribution to Reynolds stresses. ( (a) & (b)) streamwise stress ( (c) & (d))shear stress for Re θ = 9830 ((a) & (c)) and Re θ = 19660 ((b) & (d)).

  Figure 5.10 -Wall normal evolution of Q2 and Q3 (resp. Q1 and Q4 ) contribution to the streamwise stress inside LMR (resp. HMR) for Re θ = 9830 ((a) & (c)) and Re θ = 19660 ((b) & (d))

Figure 5 .

 5 Figure 5.13 -Histogram of the life time of the vortical motions at Re θ = 9830. Positive vortical motion (red dashed line), negative vortical motion (blue dashed line), both positive and negative vortical motions (grey plain line) and the mean life time (grey dashed line). N s is the total number of labelled samples.

Figure 5 .

 5 Figure 5.14 -Histogram of the life time of the VPN vortical motions at Re θ = 9830 (grey line) and Re θ = 19660 (black line) normalised with the number of labelled samples N s used to build the histogram. The corresponding mean life time are highlighted by the dashed line.

Figures

  Figures5.19 (a) display at both Reynolds and against the wall normal position the evolution of the mean radius of vortical motions scaled in Kolmogorov units. The evolutions are comparable at both Reynolds, but they do not collapse. For 0 < y < 0.2δ the mean radius increase with the wall normal distance. For 0.2δ <

  Figures5.19 (a) display at both Reynolds and against the wall normal position the evolution of the mean radius of vortical motions scaled in Kolmogorov units. The evolutions are comparable at both Reynolds, but they do not collapse. For 0 < y < 0.2δ the mean radius increase with the wall normal distance. For 0.2δ <

Figure 5 .

 5 Figure 5.15 -Histogram of the vortical motion radius at Re θ = 9830 scaled in Kolmogorov units and normalised with the number of vortical motions N v . Positive vortical motion (red dashed line), negative vortical motion (blue dashed line), both positive and negative vortical motions (grey plain line) and mean radius (grey vertical dashed line). A log normal distribution described by equation (5.36) and fitted on the histogram is added in green dashed lines. Fitting parameters are given in table 5.2.

  Figure 5.16 -Histogram of the vortical motion radius at Re θ = 9830 (grey line) and at Re θ = 19660 (black lines) scaled with: (a) the Komogorov length η; (b) boundary layer δ, (c) displacement δ * and (d) momentum θ thickness. All normalised with the number of vortical motions N v . The corresponding mean radius are highlighted by the vertical dashed lines.

Figure 5 .

 5 Figure 5.17 -Histogram of the radius at Re θ = 9830 scaled with the boundary layer thickness, normalised with the number of vortical motions N v and plotted against the wall normal position. The hot wire probes positions are highlighted with crosses.

Figure 5 .

 5 Figure 5.18 -Histogram of the radius at Re θ = 19660 scaled with the boundary layer thickness, normalised with the number of vortical motions N v and plotted against the wall normal position. The hot wire probes positions are highlighted with crosses.

  Figure 5.20 -Histograms of the vortical motion circulation at Re θ = 9830 (grey line) and Re θ = 19660 (black line) scaled in (a) Kolomorov units, then with the free stream velocity U ∞ and (b) the boundary layer, (c) the displacement and (d) the momentum thicknesses. All normalised with the number of vortical motions N v . The corresponding mean circulations are highlighted in the dashed lines.

  Figures 5.21 and 5.22 display 2D histogram of the circulation scaled in outer units and plotted at both Reynolds against the wall normal position. The histograms are comparable to the radius ones (see Figures 5.17and 5.18).

Figure 5 .

 5 Figure 5.23 display the mean circulation scaled in (a) Kolmogorov units, then

Figure 5 .

 5 Figure 5.22 -Histogram of the circulation at Re θ = 19660 scaled with the free stream velocity and the boundary layer thickness, normalised with the number of vortical motions N v and plotted against the wall normal position. The hot wire probes positions are highlighted with crosses.

Figures 5 .

 5 Figures 5.25 and 5.26 display 2D histogram of the vortical motion vorticity scaled in outer units and plotted at both Reynolds against the wall normal position. As with the radius (see Figures 5.17and 5.18), the logarithmic distribution of the probes affects the histograms distribution.

  Figures 5.25 and 5.26 display 2D histogram of the vortical motion vorticity scaled in outer units and plotted at both Reynolds against the wall normal position. As with the radius (see Figures 5.17and 5.18), the logarithmic distribution of the probes affects the histograms distribution.

  Figure 5.23 -Wall normal evolution of the mean circulation at Re θ = 9830 (grey line) and Re θ = 19660 (black line) scaled in (a) Kolomorov units, then with the free stream velocity U ∞ and (b) the boundary layer δ; (c) the displacement δ * and (d) the momentum θ thickness. The hot wire probes positions are plotted with crosses.

Figure 5 .

 5 Figure 5.25 -Histogram of the vorticity at Re θ = 9830 scaled with the free stream velocity and the boundary layer thickness, normalised with the number of vortical motions N v and plotted against the wall normal position. The hot wire probes positions are highlighted with crosses.

Figure 5 .

 5 Figure 5.26 -Histogram of the vorticity at Re θ = 19660 scaled with the free stream velocity and the boundary layer thickness, normalised with the number of vortical motions N v and plotted against the wall normal position. The hot wire probes positions are highlighted with crosses.

  Figure 5.27 -Wall normal evolution of the mean vorticity at Re θ = 9830 (grey line) and Re θ = 19660 (black line) scaled in (a) Kolomorov units, then with the free stream velocity U ∞ and (b) the boundary layer; (c) the displacement and (d) the momentum thicknesses. The hot wire probes positions are plotted with crosses.

Figure 5 .

 5 Figure 5.28 -Location of Hot Wire probes (black dots) and wall normal positions where the correlation R hl from equation (5.39) are computed at Re θ = 9830. Defect probes were the velocity magnitude was abberant were not used for reconstruction.

  Figures 5.29(a) and (b) display the evolution of the correlation R hl obtained from equation (5.39) with the spacing ∆ z plotted at four different wall normal positions displayed in Figure 5.28 for Re θ = 9830 and Re θ = 19960 respectively. The correlations are symmetric around ∆ z = 0, with pronounced peaks in the outer region (y = 0.27δ and y = 0.55δ). The plateau observed near the wall at

  Figure 5.29 -Spanwise correlation of high and low momentum regions (R hl ); positive and negative vortical motions (R pn ) plotted at different wall normal positions depicted in 5.28. (a) and (c) correspond to Re θ = 9830; (b) and (d) correspond to Re θ = 19660.

Figure 5 .

 5 Figure 5.30 -(a) R uu correlation at y + = 92, (b) R uu correlation at y = 0.5δ obtained by Ganapathisubramani et al. (2005a). The contour levels for R uu range from -0.1 to 1.0 with spacing of 0.1. Zero contours are not shown. .

  (c) and (d) for Re θ = 9830 and Re θ = 19660 respectively. The spanwise distance between counter rotating structures is ∆ z = ±0.1δ at y = 0.037δ (130 + ) (∆ z = ±0.12δ at Re θ = 19660). As for the uniform momentum regions, this spacing increases with y up to ∆ z = ±0.45δ at y = 0.55δ for both Reynolds numbers. A mean value is ∆ z = ±0.28δ

  Figure 5.31 -Spanwise correlation of the positive vortical motions and the low momentum regions (Rv p l), the high momentum regions (Rv p h) plotted at different wall normal positions depicted in 5.28. (a) and (c) correspond to Re θ = 9830; (b) and (d) correspond to Re θ = 19660.

Figure 5 .

 5 Figure 5.32 -3D views of coherent structures. The low and the high momentum regions are displayed in green and yellow respectively, the positive and the negative vortical motions are displayed in red and blue respectively.

Figure 5 .

 5 Figure 5.33 -Counter rotating streamwise vortical motions lifting up a low momentum region[START_REF] Blackwelder | Streamwise vortices associated with the bursting phenomenon[END_REF].

Figure 5

 5 Figure5.34 -Model of the coherent structures organization built with their "average" size and spacing at Re θ = 9830. The low regions is displayed in green, the positive and negative vortical motions are displayed in red and blue respectively.

Figure 5 .

 5 Figure 5.35 -Coherent structure organisation below y + = 50 for Re θ = 7800 (Reference Lin (2006)). Private communication from Michel Stanislas.

  Figure 5.36 -(a) Histograms and (b) wall normal evolution of the vortical motion circulation at Re θ = 9830 (grey line) and Re θ = 19660 (black line) scaled with the free stream velocity U ∞ the boundary laye and displacement and thicknesses. The histograms are normalised with the number of vortical motions N v . The corresponding mean circulations are highlighted in the dashed lines. The probe positions is highlighted with crosses.

Figure A. 1 -

 1 Figure A.1 -Evolution of Tikhonov regularization parameter with the wall normal position. a) α u used for the streamwise velocity, b) α v used the wall normal velocity and c) evolution of α w used for the spanwise velocity. The probes positions are highlighted in a) with crosses.

  Figure A.3 -Evolution with wall normal direction of R ûi u i correlations for the SVD and Tikhonov regularization with optimal parameters varying with the wall normal position (SVDy ; Tikhy ) and the same parameters kept constant (SVD ; Tikh ). a) Streamwise velocity correlation, b) wall normal velocity correlation and c) spanwise velocity correlation as in equation 3.22.

Figure A. 7 -

 7 Figure A.7 -Comparison between the premultiplied streamwise frequency spectrum obtained from HWA (black lines), averaged in z at y/δ = 0.511 and the same spectrum obtained from LSE reconstruction averaged in z (black crosses), at z = 0 (blue crosses) and at z/δ = 0.45 (red crosses) all at y/δ = 0.506 and at Re θ = 9830

Figure A. 10 -

 10 Figure A.10 -Comparison between the premultiplied streamwise frequency spectrum obtained from HWA (black lines), averaged in z at y/δ = 0.511 and the same spectrum obtained from LSE reconstruction averaged in z (black crosses) and averaged in z only at PIV point close to the probe positions (green crosses) all at y/δ = 0.506 and at Re θ = 19660.

  Figure B.2 -Indicative function F i of high momentum regions from equation (5.21) at different steps of the cleaning procedure (last figure).

Figure B. 3 -

 3 Figure B.3 -Illustration of cleaning procedure. (top) indicative function F i of high momentum regions from equation (5.21) and (bottom) same function after cleaning.

Figure B. 4

 4 Figure B.4 shows from left to right and from top to bottom positive contours of λ, Q, Γ 1 and Γ 2 plotted on the velocity field of the Lamb-Oseen vortex described by the equations (5.23) and (5.24). For the four criteria investigated contours are centered around the vortex position with maximum values close to the vortex center.

  Figure B.5 (d). To have a vortex, the change should remain constant as we cover all these points successively and the directions of two neighboring vectors should lay within the same or two adjacent quadrants. Finally, the four constraints implementation allows us to have the center of a Lamb-oseen vortex displayed in Figure B.5 (d) by the filled red circle.

  Figure B.5 -Illustration of vortex center identification algorithm as described in Nencioli et al. (2010). (a) First ; (b) Second ; (c) third and (d) fourth steps of the algorithm applied on a velocity field of the Lamb-Oseen vortex given by equations (5.23) and (5.24). The blue crosses indicate points around the pair (highlighted with filled red circle) at which the increase in magnitude and the change of sign of the velocity is checked. The blue and the green dashed box highlight the domains in which a minimum of the velocity is searched around a point in a pair. The green box in plain lines highights points at which the velocity vector direction change is checked in the last step. The filled and empty red circles indicates points in pairs which satistfies the constraint at the corresponding steps, the red cross indicate same type of points which are also local velocity minimum.

  Figure B.6 -Comparison of vortical motion identification with Q criterion and λ criterion. (a) Contours of Q plotted on the velocity fluctuations at t = 2.5 s; (b) contours of λ plotted on the same velocity field as (a); (c) contours of Q plotted on the velocity fluctuations at t = 1.65 s and (d) contours of λ plotted on the same velocity field as (c).

  .33)). Contours of Γ 2 were also plotted with filtered velocity fluctuations. The results show a slight improvement (see Figure B.10) but not enough to keep Γ 2 as a vortical motion core identification criterion. We have tried to investigate results obtained with Γ 1 at small scale. The green boxes in Figure B.11 (a) and (c) denote small scale vortical motions which are not identifiable at sight. Figure B.11 (b) and (d) correspond respectively to a zoom in the green boxes displayed in figure B.11 (a) and (c).

  Figure B.12 (a), (b), (c) and (d) contours of Γ 1 ( |Γ 1 | > 0.7) are plotted on the velocity fluctuation field in the Y Z plane at different time steps respectively 1.640, 1.641, 1.643 and 1.644 s. A large scale coherence in time is observed among the four planes but it is not tracked by |Γ 1 | contours which appear and dissapear with distortions of the vortical motion. Therefore, applying a high threshold on |Γ 1 | may hide the vortical motion temporal evolution, instead, a low threshold typically |Γ 1 | > 0.35 is applied and the resulting indicative function is processed in order to keep only a centroid coordinates (t,y,z) which correspond to the vortical motion position. Details of this processing are depicted in Figures B.13, B.14 and B.15.

Figure

  Figure B.16 displays from the top to the bottom isosurfaces of Γ 1 correponding

  Figure B.14 -Second part of the centroid detection algorithm

Figure B. 16 -

 16 Figure B.16 -From the top to the bottom isosurfaces of |Γ 1 | bigger than 0.5, 0.4 and 0.3 respectively.

Figure B. 17 -

 17 Figure B.17 -From the top to the bottom isosurfaces of |Γ 1 | bigger than 0.36, 0.35 and 0.34 respectively.

Figure B. 18 -

 18 Figure B.18 -First step of mathematical morphology applied to the vortical motion center indicative function obtained by thresholding Γ 1 . From the top to the bottom respectively isosurfaces of Γ 1 bigger than 0.35, isosurfaces obtained after application to the previous indicative function (top) two cleaning operations following the equations (5.22) with mask parameters to compute the volume around a pixel mask t and mask yz and volume threshold C 1 and C 2 set to [1, 1, 4, 4] and [2, 1, 6, 6] consecutively and isosurfaces obtained after application to the previous indicative function (middle) one dilatation with a structuring element of s 1 = (1, 2, 2).

Figure B. 19 -

 19 Figure B.19 -Second step of mathematical morphology applied to the vortical motion center indicative function obtained in the first step (bottom of Figure B.18 ), the original field is displayed at the top for comparison. From the top to the bottom respectively isosurfaces of Γ 1 bigger than 0.35, isosurfaces obtained after application to the indicative function obtained at the end of the first step (see Figure B.18) one erosion with a structuring element of s 1 = (2, 1, 1) and isosurfaces obtained after application to the previous indicative function (middle) two cleaning operations following equation (5.22) with mask parameters to compute the volume around a pixel mask t and mask yz and volume threshold C 1 and C 2 set to[1, 1, 4, 4] and [2, 1, 6, 6] consecutively.

Figure

  Figure B.20 -Last step of mathematical morphology applied to the vortical motion center indicative function obtained in the second step (bottom of Figure B.19 ), the original field is displayed at the top for comparison. From the top to the bottom respectively isosurfaces of Γ 1 bigger than 0.35 and isosurfaces obtained after application to the indicative function obtained at the end of the second step (see Figure B.19) one dilatation with a structuring element of s 1 = (2, 1, 1).

  z a )(y cy a ) w a (z bz i )(y cy i )+ w c (z bz i )(y iy a ) + w b (z iz a )(y cy i ) + w d (z iz a )(y iy a ) (C.5)

  are summarized in Figure C.1. validation To validate the detection algorithm summarized in Figure C.1, we have checked the results obtained with a Lamb-Oseen vortex for r in the range

  2

1 -

 1 (a) , (b) and (c) correspond to a = 2, a = 3 and a = 4 respectively. The contour line that defines the eddy radius was found at a = 4 and is plotted in blue in FigureC.2 (d). From the N points (z i , y i ) on this contour, we have computed a radius r a = 5.78 and an infinite circulation Γ a∞ = 0.93. This infinite circulation is derived from Γ of equation (C.3) as follows: Investigation of the effect of the number of contour points nb p used to integrate the circulation Γ a∞ on the values obtained. The input values used to build the Lamb oseen velocity field from (5.23) and (5.24) were r = 6 and Γ ∞ = -1 and the circulation was integrated at a radius equal to 4. -(z i -z 0 ) 2 +(y i -y 0 ) 2 r 2 a (C.7)r a = 5.71 is very close to the original value r = 6 but slightly smaller, this point was expected and is discussed in[START_REF] Nencioli | A vector geometry-based eddy detection algorithm and its application to a high-resolution numerical model product and high-frequency radar surface velocities in the Southern California Bight[END_REF] through comparison with other eddy core detection methods. The same remark is valid for the

  Figure C.3 -Lamb oseen velocity field from (5.23) and (5.24) with r = 6 and Γ ∞ = -1, the circulation Γ was integrated with points (red dots) located 4 grid points away from the center and lying on the green contours. (a) 10 points are used for the integration of Γ and Γ ∞ ; (b) same with 30 points; (c) same with 50 points and (d) same with 100 points.

  Figure C.4 (d) corresponds to a = 3 and displays in blue line only the contour retained by the algorithm as the eddy core. By averaging the distance between points lying on this contour and the center of the larger vortex located at (z 02 , y 02 ) = (10, 10), we found r a = 2.44 which is very far from the size of the largest vortex equal to 6. Large scale being of interest in the present study, we refined the algorithm in order to handle this problem. Instead of starting with low values of a at the first step and increasing it by one at each following step, we choose to start with the largest value possible for a and decrease it at each step. Far from the vortical motion center (far beyond the core region) unclosed contour lines are expected and they become circular or elliptical as we move close to the center (inside the core region). Therefore, this new approach will allow us to retain the core of the largest vortical motion which surrounds a center, enhancing the extraction of large scale vortical motions of interest in the present study. The choice of the largest value possible for a is a compromise between the boundaries of the domain and the size of the largest vortical motion expected. According to the tests perfomed on LSE data with the refined version of the algorithm, a value of a = 29 is sufficient to extract the largest vortical motion in the flow. Hence, at the beginning step of the algorithm, a is always set to 29 except when half the distance from the vortical motion center to the closest boundary of the domain is below 29, a is set to this distance. Each step follows the same procedure described in Figure C.1 except that a is decreased by one from one step to another. An illustration, of the refined version of the algorithm is depicted in Figure C.5. Figure C.5 (a) correspond to a = 5 (half the distance from the center of the largest vortical motion to the closest boundary) and displays contours of ψ computed in a domain of size ±2a.

  Figure C.5 (b) also corresponds to a = 5 and display in blue line only the contour retained by the algorithm as eddy core. The values derived from this contour were r a = 5.36 and Γ a∞ = 0.80, they are close to the input values of r = 6 and Γ ∞ = 1. The simplified version of the vortex core detection algorithm used on LSE data is described in Figure C.1 and summarized in Figure C.6. We have chosen to compute contour of the streamfunction ψ only one time with the largest value possible for a which remains constant for each vortical motion center processed.

  Figure C.4 -Illustration of vortex core detection as described in Nencioli et al. (2010), the red box highlights the domain of size ±2a where the streamfunction is computed. (a) Superimposed Lamb oseen velocity fields from the equations (5.23) and (5.24) with r = 3 and Γ ∞ = 1 for the first center (red dot) at (z 01 , y 01 ) = (11, 11) and r = 6 and Γ ∞ = 1 for the second center (green dot) at (z 02 , y 02 ) = (10, 10); (b) contours of ψ with a = 2 plotted on the velocity field of (a); (c) same as (b) with a = 3 and (d) contour which defines the eddy radius obtained at a = 3 and plotted on the Lamb oseen velocity field of (a).

  Figure C.7 -Investigation of results obtained with the simplified version of the vortex core detection algorithm described in C.6. The blue contours correspond to negative vortical motions, the red contours correspond to the positive ones and the green dots denote vortical motion centroids (a) contours which define the eddies core plotted on 2D velocity fluctuations (v , w ) at t = 0.01s; (b) the same as (a) at t = 0.075s; (c) the same as (a) at t = 0.12s and (d) the same as (a) at t = 0.23s

Figure D. 1 -

 1 Figure D.1 -Number of positive vortical motions detected N vp normalised with the number of measurement blocks N B = 534 and plotted against the wall normal position for Re θ = 9830.

Figure D. 2 -

 2 Figure D.2 -Number of negative vortical motions detected N vn normalised with the number of measurement blocks N B = 534 and plotted against the wall normal position for Re θ = 9830.

  Mots clés: couche limite turbulente à hauts nombres de Reynolds, structures cohérentes, Estimation Stochastique Linéaire, méthodes de détectionLarge scale organization of wall turbulenceThis study lies in the context of large scale coherent structures (uniform momentum regions and vortical motions) investigation in a near wall turbulent boundary layer at high Reynolds numbers (Re θ = 9830 and Re θ = 19660). With a Linear Stochastic Estimation procedure based on correlations computation, a full time-resolved 3 component field is reconstructed at high frequency from stereo-PIV at 4 Hz and hot wire data at 30 kHz . A DNS database of turbulent channel flow was used to validate the reconstruction method.

  Boundary layer theory . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.1 Definition of Turbulence . . . . . . . . . . . . . . . . . . 5 1.1.2 Reynolds Averaged Navier Stokes equations . . . . . . . 6 1.1.3 Turbulent boundary layer equation . . . . . . . . . . . . 7 1.1.4 Turbulent kinetic energy transport equation . . . . . . . 7 1.1.5 Boundary layer organization . . . . . . . . . . . . . . . . Structure of a flat plate TBL visualised with hydrogen bubbles at y + = 9.6 (Kline et al. (1967)). The flow direction is from left to right. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

	R pn	ν t	Spanwise correlation between positive and negative votices Turbulent viscosity
		h y	Channel half height in the DNS binary indicative functions Wall normal coordinate
	R	H ω 0	Histogram amplitude Hot wire hot wire correlation matrix Vorticity of vortical motion
	R ûu	z	Cross correlation between a reconstructed field û and a Spanwise coordinate
		k x φ	Streamwise wave number reference one u Set transformation operator in mathematical morphology
	k z Greek Symbols Spanwise wave number R v p h Spanwise correlation between positive vortices and high momentum region binary indicative functions φ i Voltage moment used to calibre the hot wires ψ Streamfuntion List of Figures
	R v p l	l w	Length of the hot wire probes Spanwise correlation between positive vortices and low
		σ lim	momentum region binary indicative functions Regularisation parameter for SVD
	A a j a k,i A rms C C t C tl , C th C t D d h d w E 11 ,E 22 , E 33 e E e f Fd u N v N vy N y N z p P Q1 Q2 Q Q3 Q4 r Re Re h Re τ Re θ R hl t l t U ∞ U max k u τ u, v, w u 1 , v 1 , w 1 U ,V , W U 1 ,V 1 , W 1 u , v , w u 1 , v 2 , w 3 x κ λ λ s µ 1.1 8 Matrix of LSE coefficients a k,i LSE Coefficient used to estimate a velocity field ûi given an observer x k structed field û and a reference one u Constant in the Logarithmic velocity profile Threshold for uniform momentum regions detection func-tion Fd u Threshold for low and high momentum regions detection functions Object volume based threshold Velocity gradient tensor matrix Hydraulic diameter of uniform momentum regions Diameter of hot wire probes Streamwise , wall normal and spanwise spectra nstantaneous voltage in the Hot Wire Anemometry control loop Mean part of the instantaneous voltage e in Hot Wire Anemometry Fluctuating part of the instantaneous voltage e in Hot Wire Anemometry Frequency Detection function of uniform momentum regions Number of vortical motions detected Number of vortical motions detected at a given wall nor-mal location range Number of grid points in y the DNS data Number of grid points in z the DNS data Pressure fluctuation Mean part of the pressure field Event with u > 0 and v > 0 Event with u < 0 and v > 0 Vortex detection criterion Q Event with u < 0 and v < 0 Event with u > 0 and v < 0 Radius of vortical motions Reynolds number Reynolds number based on the channel half height h Reynolds number based on the friction τ Reynolds number based on the momentum thickness θ Spanwise correlation between high and low momentum region binary indicative functions streamwise/spanwise correlations matrices between PIV and hot wire data Life time of coherent structures Time coordinate Free stream velocity Maximum velocity in the DNS Turbulent kinetic energy Friction velocity Instantaneous streamwise, wall normal and spanwise ve-locities Instantaneous streamwise, wall normal and spanwise ve-locities Mean streamwise, wall normal and spanwise velocities Streamwise, wall normal and spanwise velocity fluctua-Streamwise, wall normal and spanwise velocity fluctua-tions Streamwise coordinate Von Karman constant Swirling strength Streaks spanwise spacing Dynamic viscosity tions Γ 2 1 Literature review 5 Vortex core detection criterion from Graftieaux Mean streamwise, wall normal and spanwise velocities χ Root mean square difference between a velocity recon-structed via LSE and a reference one extracted from the DNS data δ Boundary layer thickness δ * Displacement thickness ∆x 0 Streamwise separation between the HWR and the PIV plane ∆ z Coherent structures spanwise spacing η Kolmogorov length scale Γ ∞ Infinite circulation of a Lamb-Oseen vortex Γ Circulation of vortical motions Γ 1 Vortex center detection criterion from Graftieaux τ uv 1 Shear stress tensor Introduction Root mean square attenuation coefficient between a recon-N s Number of samples N t Number of time steps in simulated PIV data subsampled from DNS S u ,S v , S w Streamwise , wall normal and spanwise velocity fluctua-tions skewnesses S x ,S y , S z Streamwise/streamwise, streamwise/wall normal and β τ k Kolmogorov time scale ogy sponding one in the PIV plane Image magnification coefficient in mathematical morphol-τ Time delay between a point on the HWR and the corre-Coefficients used to calibrate the hot wires N Total number of points N p Number of PIV points S Hot wire PIV correlation matrix α σ u Standard deviation of the streamwise velocity fluctuations Regularisation parameter for Tikhonov 1.1
	F i F u ,F v , F w ν 1.2 Coherent structure investigation in a turbulent boundary layer Indicative function for uniform momentum region Streamwise , wall normal and spanwise velocity fluctua-10 Viscosity 1.2.1 Near wall coherent structures . . . . . . . . . . . . . . . 11 tions flatnesses 1.2.2 Large and Very large scale coherent structures . . . . . 21
			xiii xvii xxi

1.2 Schematic illustrating how counter-rotating vortex pairs generate streaks

  B.17From the top to the bottom isosurfaces of |Γ 1 | bigger than 0.36, 0.35 and 0.34 respectively. . . . . . . . . . . . . . . . . . . . . . B.18 First step of mathematical morphology applied to the vortical motion center indicative function obtained by thresholding Γ 1 .From the top to the bottom respectively isosurfaces of Γ

	1 bigger
	than 0.35, isosurfaces obtained after application to the previous
	indicative function (top) two cleaning operations following the
	equations (5.22) with mask parameters to compute the volume
	around a pixel mask t and mask yz and volume threshold C 1 and
	C 2 set to [1, 1, 4, 4] and [2, 1, 6, 6] consecutively and isosurfaces
	obtained after application to the previous indicative function
	(middle) one dilatation with a structuring element of s 1 = (1, 2, 2).
	B.21 Comparison between the indicative function obtained at the end
	of morphological operation ( top figure) and the same function
	filtered to keep vortical motion labels with t

B.19 Second step of mathematical morphology applied to the vortical motion center indicative function obtained in the first step (bottom of Figure B.18 ), the original field is displayed at the top for comparison. From the top to the bottom respectively isosurfaces of Γ 1 bigger than 0.35, isosurfaces obtained after application to the indicative function obtained at the end of the first step (see Figure B.18) one erosion with a structuring element of s 1 = (2, 1, 1) and isosurfaces obtained after application to the previous indicative function (middle) two cleaning operations following equation (5.22) with mask parameters to compute the volume around a pixel mask t and mask yz and volume threshold C 1 and C 2 set to [1, 1, 4, 4] and [2, 1, 6, 6] consecutively. . . . . . . . . . . . . . . . B.20 Last step of mathematical morphology applied to the vortical motion center indicative function obtained in the second step (bottom of Figure B.19 ), the original field is displayed at the top for comparison. From the top to the bottom respectively isosurfaces of Γ 1 bigger than 0.35 and isosurfaces obtained after application to the indicative function obtained at the end of the second step (see Figure B.19) one dilatation with a structuring element of s 1 = (2, 1, 1). . . . . . . . . . . . . . . . . . . . . . . . * > 0.05 (bottom figure). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.22 Investigation of results obtained with the vortex center detection algorithm of Nencioli et al. (2010) at t = 2.5s while varying parameters a and b.

Table 2 .

 2 1 -Main flow properties with δ the boundary layer thickness, u τ the friction velocity and θ the momentum thickness. The Reynolds numbers are Re τ = u τ .δ ν and Re θ = U ∞ .θ ν .

	s)	Re τ	Re θ

Table 3 .

 3 2 -Comparison of the RMS attenuation obtained with the different methods (equation (3.23)) for the three velocity components. The optimal parameters for SVD and Tikhonov regularization are also given.

	). The

Table 3 .

 3 3 -Optimal parameter values to fit through a least square procedure JPDFs P (u , u 1 ), P (v , u 1 ) and P (w , u 1 ) at Re θ = 9830 with gaussian distributions described by equation 3.24.

	1 )	1.59.10 -3 1.72 128.49 129.51 11.64 35.01 45.40
	P (v , u 1 )	1.65.10 -2 3.24 122.53 129.88 26.62 15.75 192.22
	P (w , u 1 )	-3.49.10 -4 1.60 128.29 128.43 26.42 24.28 196.29
		H 0	A 0	y 0	z 0	σ y	σ z	θ 0 (°)
	P (u , u 1 )	1.91.10 -3 4.69	129	130	11.23 36.05 45.19
	P (v , u 1 )	4.05.10 -3 0.90 122.59 130.37 26.71 16.97 130.37
	P (w , u 1 )	-4.53.10 -4 0.42 128.03 128.58	27	24.50 15.93

Table 3

 3

.4 -Optimal parameter values to fit through a least square procedure JPDFs P (u , u 1 ), P (v , u 1 ) and P (w , u 1 ) at Re θ = 19660 with gaussian distributions described by equation 3.24.

Table 5 .

 5 3 -Boundary layer δ, displacement δ * and momentum θ thicknesses at Re θ = 9830 and Re θ = 19660. The large radii obtained here have compelled us to investigate other length scales. Figures 5.16 (b), (

	compared to 0.82 in the present

Table C .

 C 2 -Comparison between the input values r and Γ ∞ = 1 used to build a Lamb oseen velocity field from (5.23) and (5.24) and the values r a and Γ a∞ provided by the simplified version of the algorithm described in C.6.

	r	r a	Γ a∞	r	r a	Γ a∞	r	r a	Γ a∞
	2	0.99 2.20	12	11.74 9.73	22	23.60	10.81
	3	r 2 1.84 4.67 r a 0.85 0.16 Γ a∞ 13	r 12 13.66 10.57 23 r a Γ a∞ 10.92 0.88	r 22 24.70	r a 22.74 1.03 Γ a∞ 10.83
	4	3 3.79 9.25 1.68 0.39 14	13 14.76 10.61 24 13.16 1.01	23 25.57	23.81 1.04 10.73
	5	4 4.75 9.33 3.60 0.86 15	14 15.72 10.54 25 14.22 1.01	24 26.67	24.87 1.04 10.75
		5	4.65 0.90	15	15.28 1.02	25	25.94 1.04
	6	6 5.77 9.50 5.71 0.93 16	16 16.66 10.47 26 16.35 1.02	26 27.76	27 10.76	1.04
	7	7 6.77 9.59 6.77 0.95 17	17 17.74 10.49 27 17.41 1.02	27 28.59	28.06 1.04 10.66
		8	7.06 0.85	18	18.48 1.03	28	29.13 1.04
	8	9 7.76 9.62 8.03 0.86 18	19 18.65 10.41 28 19.54 1.03	29 29.68	30.19 1.04 10.67
	9	10 8.72 9.62 8.99 0.87 19	20 19.73 10.43 29 20.61 1.03	30 31.61 11 31.26 1.04
	10	11 9.77 9.71 9.96 0.88 20	21 20.61 10.34 30 21.67 1.03	31 32.73	32.32 1.04 11.01
	11	10.70 9.67	21	22.70 10.90 31	33.54	10.91

Table C .

 C 3 -Comparison between the input values r and Γ ∞ = 10 used to build a Lamb oseen velocity field from (5.23) and (5.24) and the values r a and Γ a∞ provided by the simplified version of the algorithm described in C.6.

  Organisation à grandes échelles de la turbulence de paroi Ce travail porte sur l'étude des structures cohérentes dans une couche limite de plaque plane à hauts nombres de Reynolds (Re θ = 9830 et Re θ = 19660). L'estimation Stochastique Linéaire est utilisée pour reconstruire un champ de vitesses résolu en espace et en temps à partir des mesures aux fils chauds à 30 KHz et des mesures PIV à 4 Hz. Une base de données DNS d'un écoulement de canal turbulent a été utilisée pour valider la procédure de reconstruction. Le champ de vitesse reconstruit est comparé à un champ de référence extrait de la DNS avec la visualisation des contours du champ de vitesse, l'analyse spectrale, les corrélations, etc. Par ailleurs, une analyse statistique est réalisée sur le champ de PIV originale, les données issues des fils chauds et celles reconstruites. Les résultats obtenus sont comparés à ceuxde Carlier et Stanislas (2005) qui servent de référence. Des algorithmes ont été développés pour extraire les structures cohérentes du champ reconstruit. Les régions de vitesse uniforme sont caractérisées avec leur diamètre hydraulique, leur durée de vie et leur contribution aux tenseurs de Reynolds. Pour les tourbillons, on leur associe un rayon, une circulation et une vorticité en plus de leur durée de vie et leur nombre calculé à une distance fixe de la paroi. L'organisation spatiale des structures est étudiée avec la corrélation de leurs fonctions indicatives. Le modèle simplifié à grandes échelles qui en découle est comparé à ceux qui existent dans la littérature.

Remerciements

U (y, z) rather than U (y). In the range of the RMS measurement error around ±0.28, RMS profiles from PIV and HWA data collapse on the one from [START_REF] Carlier | Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry[END_REF]. The RMS profile from LSE follows the same trend as the PIV one but its amplitude is low denoting a loss of energy. This loss is attributed to the filtering of small scales by LSE and to the streamwise separation between the HWR and the PIV laser sheet (∆x 0 = 1 cm). This separation provides a correlation peaks of 0.8 between the PIV and HWA streamwise velocity fluctuations rather than 1 as can be expected. Streamwise/wall normal and streamwise/spanwise correlation from which the v and w velocity fluctuations are reconstructed peak around ±0.2. Hence, the energy loss is amplified on the corresponding RMS profiles.

PDFs from PIV, HWA and LSE data match well with the one obtained by [START_REF] Carlier | Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry[END_REF]. The PDFs from PIV display some oscillations which are attributed to the peak locking effect. This effect is smoothed out by the LSE reconstruction. Additionally, PDFs of w from LSE are flattened compared to the others. To explain this, the shape of the streamwise/spanwise correlation that is used to reconstruct this component was considered. As displayed in Figures 4. 18 and 4.19, contrary to the streamwise/streamwise and the streamwise/wall normal correlation, the peak of this correlation is away from the probe location. Despite the strong attenuation, LSE skewnesses and flatnesses obtained match almost the one of [START_REF] Carlier | Experimental study of eddy structures in a turbulent boundary layer using particle image velocimetry[END_REF] and of the HWA. As expected, slight discrepancies appear with fourth order velocity moments which are attributed to a lack of convergence.

Appendix A LSE A.1 Cross validation

As in the present work, from PIV fields sampled at 4 Hz, a time resolved velocity field is reconstruted in between the original PIV time steps at a frequency of 2 KHz, a tenth-fold cross validation procedure was used to find optimal values of σ lim and α in equations (3.17) and (3.20) respectively. It consists in dividing the original set of 1000 PIV fields into two separate sets of 900 and 100 respectively. Then 90% of the original PIV data is used to find the matrix coefficients A while the error is estimated from the remaining set of 100. In order to have enough convergence, this operation is repeated 10 times with the two separate sets chosen randomly, and the reconstruction errors obtained are averaged. Optimal values of α and σ lim minimize this error.

A.2 Study of the influence of the wall normal position on LSE reconstruction

The streamwise velocity correlation drop in the center of the channel observed on SVD results in Figure A.3 (a) has compelled us to perfom a reconstruction with parameters σ lim and α optimized for each wall normal position. The wall normal direction being subject to a strong velocity gradient, taking it into account can lead to better results. Figures A.1 and A.2 display respectively the evolution of optimal values of α and σ lim with the wall normal position. These parameters were obtained by solving the linear system for each wall normal location. The evolution of σ u , σ v and σ w corresponding respectively to the streamwise, wall normal and spanwise velocities displays some peaks at random In table A.1, the attenuation factor of the SVD and Tikhonov with constant optimal parameter σ lim and α are compared to the ones obtained with the same parameters varying with the wall normal direction. For the SVD, the attenuation factor and reconstruction error ((3.21)) is comparable in both case for the three velocity component. However, considering σ lim varying with y, the reconstruction error was found to vary rapidly with small variations of σ lim , the solution of the linear system obtained for each wall normal locations is not always converged. With Tikhonov, the solution is converged, the reconstruction error is still comparable in both case but the attenuation factor differs significantly. The origin of this difference was not fully understood. Figures A.4,A.5 and A.6 display the velocity fields obtained from SVD and Tikhonov with optimal parameters α and σ lim varying with the wall normal direction and with the same parameters kept constant. The three velocity components from Tikhonov with α varying with y match well with the case where α is kept constant. The matching is also observed between the velocity fields obtained from SVD with σ lim varying with y and the case where it is kept constant. But the structure coherence is lost at some wall normal locations, this is more pronounced on the wall normal and spanwise velocities for which the linear system solution convergence is more difficult.

A.3 Influence of the probes distribution in span on the LSE reconstructed spectra

Knowing that the flow is homogeneous in the spanwise direction and that the probe distribution is not uniform in the same direction. We have tried to investigate the effect of the spanwise location of the PIV points used to compute the premutiplied frequency spectra on the results. Figures A.7 and A.8 display the spectra of the streamwise fluctuations from the HWA (black lines) averaged in z at y/δ = 0.511 and the same spectra from the LSE reconstruction at y/δ = 0.506 averaged at z = 0 (blue crosses) and at z/δ = 0.45 (red crosses) at Re θ = 9830 and Re θ = 19660 respectively. The spectra averaged for all z including the ones located in between the probes where the filtering is important are also plotted in black crosses at the same wall normal position for comparison. As expected, given the probes concentration in the center of the rake and due to the border effect which will cut the large scales, the probe located in the center of the rake is less affected by the LSE filtering than the one close to the border at small frequencies and at a significant proportion of high frequencies. At very high frequencies, f * > 3 this tendency is reversed, the results suggest that the smallest scales reconstructed near the border are more energetic than the same scales located in the center of the rake. These smallest scales are the most filtered by LSE, and, therefore results obtained in this range should be taken with caution. Similarly, the premultiplied frequency spectra of the streamwise fluctuations from the HWA (black lines) averaged in z at y/δ = 0.511 were compared with the spectra from the LSE reconstruction (green lines) averaged in z only at PIV points close to probe positions at the same wall normal position. Results are depicted in Figures A.9 and A.10 for Re θ = 9830 and Re θ = 19660 respectively. The spectra averaged for all z are also plotted in black crosses at the same wall normal position for comparison. As expected, the spectra derived only from single probe positions are slightly more energetic than the one averaged for all z, this is attributed to the energy loss at PIV points reconstructed in between the probe positions. (Hadwiger, 1957)) which is defined as the (possibly infinite) intersection:

When y 0 sweeps B the erode set Y could be defined as the set of center

x 0 of B such that x 0 + y 0 belongs to X. It means that x 0 belongs to the translate X -y 0 of X when y 0 sweeps B:

2. Dilatation: Following equation 5.7 we can write:

From the last equality in equation 1 we can write:

According to Morgan theorem for a set X and a set I which is finite or not integer we have:

So that:

Assuming that X is the union of all its points X = ∪ y 0 ∈X

x 0 and with a permution of union order in equation B.6 we have the symetrical expression:

Definition of a dual transformation

As a set X becomes φ(X) after a transformation by an operator φ its complement representing the background of the image becomes φ

B.2 Uniform momentum regions

Uniform momentum region detection algorithm described in section 5.2.1 was applied directly on the experimental database for validation. The following lines, gives more details on this procedure.

Dilatation and erosion 3D functions needed to smooth the noise from the thresholded streaky structures are available under the CImg library, respectively img.dilate() and img.erode() where img is the input image. The size of the structuring element should be specified according to the shape of the structure to process. As we are working on 3D uniform momentum regions we should specify three parameters: width, height and depth of the structuring element each corresponding here to the streamwise, wall normal and spanwise directions respectively. We can also expect a width greater than the other dimension as the uniform regions are elongated in the streamwise direction. A dilation and an erosion are written as: The vortical motion core identification algorithm follows the one used in [START_REF] Nencioli | A vector geometry-based eddy detection algorithm and its application to a high-resolution numerical model product and high-frequency radar surface velocities in the Southern California Bight[END_REF] to find the vortex core with slight modifications which are detailed below. It was applied to the centroids deduced from Γ 1 , the radius, the circulation and the vorticity of each vortical motion are computed.

As described in [START_REF] Nencioli | A vector geometry-based eddy detection algorithm and its application to a high-resolution numerical model product and high-frequency radar surface velocities in the Southern California Bight[END_REF], contours of the streamfunction are computed in a domain of size ±2a along y and z direction. With a the minimum size of detected eddies set to 2 at the beginning. Then an increase of the velocity magnitude in radial direction is checked on cardinal points of the outermost closed contour line around the center. That is |v| should increase as we move away from the outermost closed contour line around the center in z direction at y = 0 and |w| should increase as we move away from the outermost closed contour line around the center in y direction at z = 0. In the last step, the distance d 0 between the domain defined by a and the contour line that defines the boundary edge is computed, contour lines with values of d 0 above 1.5 grid points are kept as eddy boundaries. Note that the parameter a is increased by a grid points in each direction when the test fails and the algorithm restarts from the beginning. In order to have enough resolution, for the validation of the algorithm, we have chosen to increase a by one grid point in each step. The way to compute the streamfunction from the velocity field is also described in [START_REF] Nencioli | A vector geometry-based eddy detection algorithm and its application to a high-resolution numerical model product and high-frequency radar surface velocities in the Southern California Bight[END_REF]. If ψ(z 0 , y 0 ) = 0 denotes the streamfunction value at a centroid of coordinates (z 0 , y 0 ) in the (z,y) plane, its value at a point of coordinates (z k , y j ) = (z 0 + k.∆ z , y 0 + j.∆ y ) -2a≤k≤2a -2a≤j≤2a located around the centroid is written as follows:

Where

and sign is the function giving the sign of an integer i, sign(i) = i |i| and sign(0) = 0. The radius of the vortex in equations r in equations 5.24 and 5.23 is estimated by averaging the distance between the center of the vortex and the points on the contour that defines the vortex core. The circulation Γ = v. dl is computed on the same contour by adding the contributions from all the points on this contour. Following [START_REF] Vollmers | Detection of vortices and quantitative evaluation of their main parameters from experimental velocity data[END_REF] for a velocity field defined by equations 5.23 and 5.24 the circulation Γ is given as follows:

Where N is the total number of points on the countour line which defines the edge of the vortex and w i and v i are respectively the spanwise and wall normal velocity fluctuations at a point of coordinates (z i , y i ) on the same contour. Because countour points do not necessary follow a cartesian grid, a bilinar interpolation was used to access the velocity field at these points. For each point of coordinates (z i , y i ) at which we want to access the 2D velocity field (w i , v i ), we consider the four closest grid points (z a , y a ), (z b , y b ), (z c , y c ) and (z d , y d ) forming a rectangle around the contour point and their respective velocity fluctuations (w a , v a ), (w b , v b ), (w c , v c ) and (w d , v d ) so that we have: Appendix D

Results

D.1 Vortical motions

Energetic contribution of vortical motions: In order to complete the study of the energetic contribution of coherent structures which was started with uniform momentum regions in section 5.4.2, we have investigated the ones from positive and negative vortical motions separetely. Figure D.3 display the streamwise and shear Reynolds stresses contribution from positive and negative vortical motions at both Reynolds. Contary to uniform regions, where low speed regions are found to contribute more than the high speed ones, positive and negative vortical motions have the same energetic contribution to Reynolds stresses. However, as displayed in Figure D.4, the percentage of these contribution to the total stresses is insignificant. Positive and negative vortical motions contribute separately to 1.7% of the total stresses at Re θ = 9830 and to 0.68% at Re θ = 19660. These values are far below the ones of low and high momentum regions at corresponding Reynolds around 42% and 23% repectively. This low contribution from vortical motions and its decrease with Reynolds has been reported in Wu and Christensen (2006) but for spanwise vortices. Such low contributions, could also be explained by the fact that vortical motions are detected in 2D plane and the outcoming structure is not fully 3D. Therefore, it is not excluded that some inclined patches could be lost in the 2D planes resulting in a lower energetic contribution.