Cette Thèse

Khaled, Memdouh, Saïd, Zakaria, Lamia, Idris, Arnab, Subbu Micheline Lahcène

Keywords: Large-eddy simulation, Supersonic boundary layer, SGS modeling, Cold-wall boundary layer, heat transfer, Strong Reynolds Analogy, Compressibility effects. viii Couche limite supersonique, Transferts pariétaux, Paroi refroidie, Analogie forte de Reynolds, Effets de compressibilité, Modélisation sous maille

Cette thèse a bénéficié d'une bourse

A mes parents,

A ma soeur, A mes deux frères.

Context of the study

The use of advanced space propulsion devices, such as optimized rocket nozzles, has been the subject of several conceptual design studies. Indeed, rocket engine nozzles are one of the most important components of the propulsion system which need a careful design in order to fully expand and accelerate the hot gases emitted by the combustion process into exhaust gases. Since most of the energy released in the combustion chamber should be converted into kinetic energy, the performance of the nozzle, in terms of thrust and specific impulse, should be well optimized for any flight regime. Indeed, the first-stage rocket engine should operate, not only at high altitude regime, but also at ground level especially during start-up and shutdown phases. Furthermore, due to the minimal duration of the atmospheric flight compared to the total duration of the launch, the first stage engines are optimized for low external pressures. However, when operating at ground level (ignition and first lift-off phase), the rocket nozzle encounters strong instabilities due to sideloads which are generated by boundary layer separation. This phenomena is prejudicial for the mechanical behavior of the structure and can cause damage or complete loss of the engine.

In addition to the aerodynamics loads, there exists a significant difference in temperature between the hot combustion gases and the cooled walls of the engine. This inevitably leads to strong wall-heat fluxes and extreme temperature gradients through the combustion chamber and subsequently the nozzle walls. To remedy this problem, various cooling techniques have been developed and used in the past, such as ablative, radiation, gas-turbine dump and film cooling [START_REF] Suslov | Convective and film cooled nozzle extension for a high pressure rocket subscale combustion chamber[END_REF] [START_REF] Suslov | Convective and film cooled nozzle extension for a high pressure rocket subscale combustion chamber[END_REF]. In the dump cooling, for instance, the liquid Hydrogen LH 2 at the exit of the turbine passes through cooling channels within the coating material, and is dumped at the end of the skirt of the nozzle through supersonic outlets. However, in the film cooling, part of the Liquid Hydrogen (LH 2 ) and the Turbine Exhaust Gas (TEG) is injected into the nozzle, at a specific cross section. This method can either be used alone for small engine nozzles with low heat fluxes, or simultaneously with cooling channels, to the lower section of the nozzle, as the Vulcain II engine for instance. The TEG film-injection at a relatively low temperature reduces the convective heat transfer between the hot gas and the inner surface of the nozzle. Hydrogen dump cooling, delivering a mass flow of 3 kg/s. The coolant flows through rectangular helical channels. At a nozzle expansion ratio of Σ = A e /A th = 32 (where A e is the exit section and A th is the throat area), Hydrogen-film is injected into the nozzle at supersonic speed in the flow direction. Downstream of the injection point, Hydrogen develops a cooling film that protects the lower nozzle section from heat impact of the accelerated hot combustion gases. In addition to Hydrogen, the turbine exhaust gases (TEG) are also injected with a mass flow rate of about 10 kg/s at almost the same position. To provide equal circumferential coolant distribution, the TEG is guided by two pipes from the upper part of the engine along side the upper nozzle part to a manifold ring arranged before the coolant injection [START_REF] Suslov | Convective and film cooled nozzle extension for a high pressure rocket subscale combustion chamber[END_REF] [START_REF] Suslov | Convective and film cooled nozzle extension for a high pressure rocket subscale combustion chamber[END_REF].

Heat transfer in supersonic nozzles

The convective heat transfer in supersonic nozzles has not been frequently investigated through experimental studies. The available experiments mainly investigated heat transfer in conical convergent-divergent type nozzles, focusing on comparison between experimental data and available theoretical correlations. The focus was in the validity of Nusselt-type correlations, used for predicting heat transfer coefficients in canonical flows (such as pipe flows) which have been discussed and used for predicting heat transfer in the convergent as well as in the supersonic part of the nozzle.

Among others, experimental results of convective heat transfer, subjected to large variations in pressure gradients in conical nozzles, are presented by [START_REF] Back | Convective heat transfer in convergent-divergent nozzle[END_REF] [4]. The 2.5 Mach convergent-divergent nozzle has a half-angle of 30 • for the convergent and a half-angle of 15 • for divergent. At the inlet, the imposed stagnation pressure ranged from 5.5 to 17.2 bar, the total temperature from 570 to 1110 K, and the thicknesses of the considered dynamic boundary layers at the inlet between 5% and 25% of the radius of the nozzle inlet. The used gas is a compressed air, heated by internal combustion of methanol. The mass flow ratio (mixture of methanol/air) is taken sufficiently small (even for the largest total temperature case) so that the combustion products can be assumed to air. The results show that the heat transfer coefficient reaches its maximum just upstream of the throat (region where the mass flow rate reaches its maximum value) and that a reduction of this coefficient occurs downstream of the separation zone, which appears in the case of a low total pressure. It was also observed a reduction of 10% of the heat transfer coefficient between the lowest and highest studied boundary layer thicknesses. This result was also observed by [START_REF] Bartz | Turbulent boundary layer heat transfer from rapidly accelerating flow of rocket combustion gazes and of heated air[END_REF] [7] under different boundary layer thickness and wall conditions, who presented a theoretical analysis of a convergent-divergent flow using the available skin-friction and heat-transfer coefficient calculation theories.

Interesting works on the convective heat transfer in supersonic nozzles were also conducted by Back et al. (1966 [3], 1967 [START_REF] Back | Flow phenomena and convective heat transfer in a conical supersonic nozzle[END_REF], 1968 [START_REF] Back | Heat-transfer measurements in the shock-induced flow separation region in a supersonic nozzle[END_REF]). Among them, measurements of heat transfer from a hot air flow in a conical supersonic water cooled convergent-divergent type nozzle, were carried out [START_REF] Back | Flow phenomena and convective heat transfer in a conical supersonic nozzle[END_REF] [START_REF] Back | Flow phenomena and convective heat transfer in a conical supersonic nozzle[END_REF]: the nozzle has a convergent and divergent half-angle of 45 • and 15 • respectively, an area ratio Σ equal to 6.6 and a ratio of radius of the throat to radius of throatcurvature equal to 0.625. The ratio of the gas wall-temperature to the total temperature T w /T r (where T r is the recovery temperature, nearly equal to the adiabatic wall-temperature) ranges from 0.3 to 0.7. Different turbulent boundary layer thicknesses δ in were imposed at the inlet of the nozzle (where H = δ * in /θ in is the shape factor) is slightly less than 0.5. At the inlet, the dynamic boundary layer has a thickness of 45% of the radius of the nozzle inlet, and is slightly thinner than the thermal boundary layer. Reynolds numbers based on the momentum thickness at the nozzle inlet θ in vary between 3000 and 18000. The results showed that the heat transfer coefficient along the nozzle depends on the structure of the boundary layer at the inlet, and on the way the boundary layer is affected by the acceleration of the flow. It was also reported that the heat transfer coefficient increases through the convergent, reaching its maximum downstream of the nozzle throat, to decrease again in the divergent. At low Reynolds number, the thinnest boundary layer is laminar along the nozzle, and the heat-transfer coefficient in the region near the throat seems unaffected by the the boundary layer thickness.

Boldmann et al. (1966) [START_REF] Boldman | Turbulence heat-transfer and boundary layer measurements in a canical nozzle with a controlled inlet velocity profile[END_REF] presented experimental velocity and temperature profiles of the boundary layers, heat-transfer, and turbulence measurements for a conical nozzle configuration with a cylindrical approach section operating with air at a nominal total temperature and pressure of 540 K and 20 bar, respectively. Experimental heat-transfer coefficients were compared to values determined by Nusselt-type pipe flow correlation. It was concluded that the experimental heattransfer coefficients could not be correlated by the Nusselt-type pipe flow correlation, where the predicted values of the heat-transfer coefficient were significantly higher by about 70% than the experimental values in the throat region of the nozzle. It was also noted that the heat-transfer coefficients were essentially unaltered even with the higher levels of turbulence intensity, and the maximum heat-transfer coefficient occurred slightly upstream of the geometric throat.

An experimental study of the effect of wall-cooling on the mean structure of a turbulent boundary layer in low-speed gas flow were presented by [START_REF] Back | Effect of wall cooling on the mean structure of a turbulent boundary layer in low-speed gas flow[END_REF] [2]. The study is interesting in a way to highlight the single effect of wall-cooling on the dynamics of the considered flow. The flow evolved at a Mach number of 0.06 and the Reynolds numbers Re θ varied from 1500 to 36000.

Wall-temperature to freestream temperature ratio T w /T ∞ spanned a range extending from 0.4 to 1. Semi-empirical analysis, based on supersonic-flow measurements by [START_REF] Coles | The turbulent boundary layer in a compressible fluid[END_REF] [START_REF] Coles | The turbulent boundary layer in a compressible fluid[END_REF] and [START_REF] Spalding | The drag of a compressible turbulent boundary layer on a smooth flat-plate with and without heat transfer[END_REF] [START_REF] Spalding | The drag of a compressible turbulent boundary layer on a smooth flat-plate with and without heat transfer[END_REF] were used to compare the obtained data. Experimental results were obtained at two different inlet conditions, based on the transition to a turbulent state of the upstream boundary layer: natural or forced transition. In this investigation, the ratio of the thermal to the velocity boundary layer thickness δ t /δ was assumed to be the same as the ratio of energy thickness to momentum thickness φ/θ. It was concluded that under forced transition, the ratio of displacement thickness to momentum thickness H = δ * /θ (H is the shape factor) was reduced.

The effect of wall-cooling was to increase the friction velocity above that of the adiabatic case (20% at T w /T ∞ = 0.5). It was also reported that the measured velocity and temperature profiles were in fair agreement with Coles' (1961) transformation theory [START_REF] Coles | The turbulent boundary layer in a compressible fluid[END_REF], and indicated a wakelike behavior in the outer part of the layer.

Supersonic turbulent boundary layers

For high-speed flows, the kinetic energy of the motion is a considerable fraction of the total energy of the fluid (which is not necessary true for low-speed flows) which yields to an important amount of viscous dissipation in the mean energy balance within the boundary layer. [START_REF] Morkovin | Effect of compressibility on turbulent flows, Mécanique de la Turbulence[END_REF] [START_REF] Morkovin | Effect of compressibility on turbulent flows, Mécanique de la Turbulence[END_REF] showed that 'in non-hypersonic boundary layers [...] the entropy (total-temperature) mode is very small for conventional rates of heat transfer' [START_REF] Bradshaw | Compressible turbulent shear layers[END_REF] [START_REF] Bradshaw | Compressible turbulent shear layers[END_REF], and announced that at moderate Reynolds numbers 'the essential dynamics of these shear flows will follow the incompressible pattern' [START_REF] Smits | Turbulent shear layers in supersonic flow[END_REF] [START_REF] Smits | Turbulent shear layers in supersonic flow[END_REF]. In fact, for an adiabatic compressible turbulent boundary layer, and due to viscous heating, compressibility effects arise mainly from the large change in the fluid properties :'a density gradient caused by the dissipative heating near the no-slip wall is the primary effect of increasing the mean flow Mach number' [START_REF] Lele | Compressibility effects on turbulence[END_REF] [START_REF] Lele | Compressibility effects on turbulence[END_REF]. It is then commonly concluded that adiabatic supersonic turbulent boundary layers at moderate Mach numbers (typically M ≤ 5) can be studied using the same models as low-speed flows, as long as the variations in the mean flow properties are accounted for. Based on this, it follows some interesting results, including mean velocity and fluctuations scalings, and a specific formulation of the Reynolds analogy, namely the SRA (Strong Reynolds Analogy).

Adiabatic case

Adiabatic supersonic turbulent boundary layers were first investigated through experiments (for a data compilation, see Fernholz & Finley (1977) [START_REF] Fernholtz | A critical compilation of compressible turbulent boundary layer data[END_REF]) mainly in order to validate the Morkovin's hypothesis. One of the main conclusions that follow the hypothesis is that the total temperature fluctuations are very weak within the boundary layer. Hence, velocity and temperature fluctuations are perfectly anti-correlated, and the Strong Reynolds Analogy linking the velocity and temperature fields is nearly equal to 1.

Recently, highly resolved direct numerical simulations (DNS) and large-eddy simulations (LES) helped getting more insight into the topic. [START_REF] Guarini | Direct numerical simulation of a supersonic turbulent boundary layer at Mach 2.5[END_REF] [START_REF] Guarini | Direct numerical simulation of a supersonic turbulent boundary layer at Mach 2.5[END_REF] studied an adiabatic compressible boundary layer at Mach number M = 2.5 and Reynolds number based on the momentum thickness Re θ = 1577, using DNS. In their simulation, it was assumed that the boundary layer grows slowly in the streamwise direction such as the turbulence can be treated as approximately homogeneous in this direction. The flow was then considered homogeneous in both streamwise and spanwise directions. It was concluded that the total temperature fluctuations were of the same order as the temperature fluctuations. The under-estimation of the R uT correlation compared to the experimental values seemed to be due to a difference of about a factor of 2 in the magnitude of the total temperature fluctuations. The modified SRA relation derived by [START_REF] Huang | Compressible turbulent channel flows: DNS results and modeling[END_REF] [43] was found to give a better agreement with the numerical data, compared to [START_REF] Gaviglio | Reynolds analogies and experimental study of heat transfer in the supersonic boundary layer[END_REF] relation [START_REF] Gaviglio | Reynolds analogies and experimental study of heat transfer in the supersonic boundary layer[END_REF]. Maeder et al. (2000) [START_REF] Maeder | Direct simulation of turbulent supersonic boundary layers by an extended temporal apporach[END_REF] made DNS of zero-pressure gradient adiabatic supersonic turbulent boundary layers at Mach numbers 3, 4.5 and 6 and a Reynolds number Re θ ≈ 3000. An extended temporal approach, where the streamwise direction is supposed periodic, was used. It was concluded that the compressibility effects on the turbulence statistics are small up to a Mach number of about 5, and that the total temperature fluctuations cannot be neglected, even for a Mach number of 3. The dilatational dissipation correlations were also found to be small throughout the considered Mach number range. Verifying the Morkovin's hypothesis, the structure parameter had approximately the same value for compressible and incompressible boundary layers. [START_REF] Pirozzoli | Direct numerical simulation and analysis of a spatially evolving supersonic turbulet boundary layer at M=2.25[END_REF] [START_REF] Pirozzoli | Direct numerical simulation and analysis of a spatially evolving supersonic turbulet boundary layer at M=2.25[END_REF] performed DNS of a supersonic turbulent boundary layer over an adiabatic wall at Mach number 2.25 and Re θ ≈ 4000. For the turbulent inflow condition, a generic laminar boundary layer profile is prescribed at the inlet, and a blowing/suction method forced (Schlichting, 1979 [82]).

the flow to transit to a fully turbulent state. It was reported that the total temperature was not uniform throughout the boundary layer, and that the total temperature fluctuations were nonnegligible across the boundary layer. Linking the velocity and the temperature fluctuations, it was shown that u ′ and T ′ are not perfectly anti-correlated, reaching a maximum of 0.85. The turbulent Prandtl number was not equal to unity, varying between 0.7 and 0.8 in the outer part of the boundary layer. [START_REF] Lagha | Near-wall dynamics of compressible boundary layers[END_REF] [START_REF] Lagha | Near-wall dynamics of compressible boundary layers[END_REF] studied compressible turbulent boundary layers with freestream Mach numbers ranging from 2.5 to 20 using DNS. A modified recycling/rescaling procedure was used to prescribe a fully turbulent inflow condition. It was shown that the zero-pressure gradient supersonic turbulent boundary layers exhibit close similarities to incompressible boundary layers, and that the incompressible results can correctly describe the main compressible turbulence statistics, as long as the density variation is taken into account. [START_REF] Duan | Direct numerical simulation of hypersonic turbulent boundary layers. Part 3. Effect of Mach number[END_REF] [START_REF] Duan | Direct numerical simulation of hypersonic turbulent boundary layers. Part 3. Effect of Mach number[END_REF] simulated turbulent boundary layers at Mach numbers ranging from 3 to 12, with Re θ ranging from 3000 up to 11350. With regards to the Morkovin's scaling, the DNS study showed that the van Driest transformation for the mean streamwise velocity well collapse different results with the incompressible references.

More recently, Pirozzoli & Bernardini (2011) [74] investigated supersonic turbulent boundary layers at Mach number M = 2 and moderate Reynolds numbers up to Re τ ≈ 1120, using highly resolved DNS. An extended domain in the streamwise direction, up to L x = 100δ in , was used in order to achieve a fully turbulent state of the flow, in order to ensure a better prediction of the turbulent statistics. Comparison of the velocity statistics up to the fourth-order showed nearly exact agreement with incompressible data, provided the momentum thickness Reynolds number matched, and provided the mean velocity and the velocity fluctuations were scaled according to the mean density variation. It was also found that an energy peak emerges in the logarithmic region of the boundary layer, associated with the appearance of large and streaky structures, referred to as superstructures. As expected for the range of the considered Mach number, compressibility effects were found to be weak. Again, the negligible total temperature fluctuations assumption is found to be violated, and the -R uT correlation reached its maximum in the buffer layer.

Isothermal case

The previous conclusions are 'certainly reasonable for supersonic flows without mass and heat sources.' [START_REF] Smits | Turbulent shear layers in supersonic flow[END_REF] [START_REF] Smits | Turbulent shear layers in supersonic flow[END_REF]. However, for an isothermal-wall boundary layer, the applicability of the Morkovin's hypothesis and the following assumptions are, at first sight, debatable. The number of studies investigating supersonic turbulent boundary layers over heated/cooled walls are limited. Early investigations dealt with theoretical calculations of flow properties, such as the skin-friction C f and the heat transfer C h coefficients, and theoretical calculations of the flow properties.

Hopkins & Inouye (1971) [START_REF] Hopkins | An evaluation of theories for predicting turbulent skin friction and heat transfer on flat plates at supersonic and hypersonic Mach numbers[END_REF] reviewed available theories, namely Sommer & Sort (1956) [START_REF] Sommer | Free-flight measurements of turbulent boundary-layer skinfriction in the presence of severe aerodynamic heating at Mach numbers from 2.8 to 7.0[END_REF], [START_REF] Spalding | The drag of a compressible turbulent boundary layer on a smooth flat-plate with and without heat transfer[END_REF] [START_REF] Spalding | The drag of a compressible turbulent boundary layer on a smooth flat-plate with and without heat transfer[END_REF], Van-Driest II (1956) [START_REF] Van Driest | Problem of Aerodynamic Heating[END_REF] and [START_REF] Coles | The turbulent boundary layer in a compressible fluid[END_REF] [START_REF] Coles | The turbulent boundary layer in a compressible fluid[END_REF] theories, for predicting turbulent skin-friction and heat-transfer coefficients on flat plates at supersonic and hypersonic Mach numbers. Considered flows evolved over adiabatic as well as non-adiabatic cold walls (0.3 ≤ T w /T r ≤ 1) and covered a wide range of supersonic and hypersonic Mach numbers (1.5 ≤ M ≤ 5.8). They suggested that the van-Driest II theory should be used to predict the best approximations of these quantities. [START_REF] Cook | Heat transfer for highly cooled supersonic turbulent boundary layers[END_REF] [START_REF] Cook | Heat transfer for highly cooled supersonic turbulent boundary layers[END_REF] extended the Hopkins & Inouye's (1971) [START_REF] Hopkins | An evaluation of theories for predicting turbulent skin friction and heat transfer on flat plates at supersonic and hypersonic Mach numbers[END_REF] study with highly cold-walls (T w /T r ≤ 0.3) at lower supersonic Mach numbers (1.5 ≤ M ≤ 1.8), and found that van-Driest II theory [START_REF] Van Driest | Problem of Aerodynamic Heating[END_REF] predicts with better certainty heat transfer, but was found to be less confident for predicting skin friction for the range of the considered variables. [START_REF] Huang | Skin friction and velocity profile family for compressible turbulent boundary layers[END_REF] [START_REF] Huang | Skin friction and velocity profile family for compressible turbulent boundary layers[END_REF] presented a theoretical method for calculating skin-friction coeffi-cients and mean velocity profiles for compressible turbulent boundary layers with isothermal and adiabatic walls. For the logarithmic region of the mean velocity profiles, the incompressible lawof-the-wall, transformed using the van-Driest theory [START_REF] Van Driest | Turbulent boundary layer in compressible fluids[END_REF], is adopted. The transformation reads

u c = √ B arcsin A+u D -arcsin A D with A = q w /τ w , B = 2C p∞ T w /Pr t and D = √ A 2 + B.
This equation has been extended to the outer and the viscous layers. Results reported that skin friction seems to be predicted nearly as well as using the van-Driest II theory for the range of the considered Mach numbers, while it shows lower results compared to the same theory for strongly cooled walls.

Experimental investigations of supersonic turbulent boundary layers over heated/cooled walls are limited to measurements of the basic quantities, such as mean velocity and temperature profiles, temperature fluctuations, shear stress profiles, etc.

Among others, [START_REF] Gran | The effect of wall cooling on a compressible turbulent boundary layer[END_REF] [START_REF] Gran | The effect of wall cooling on a compressible turbulent boundary layer[END_REF] experimentally studied Mach 4 supersonic turbulent boundary layers subjected to adverse and favorable pressure gradients, and with constant temperature wall T w /T r = 0.5. They found that the van-Driest transformed velocity profiles showed good agreement with low-speed results. They also studied the response of zero pressure gradient highspeed boundary layers to a near step change in the wall temperature. Two cases were analyzed: a step-down case which consists of an initially adiabatic turbulent boundary layer flowing onto a cooled wall, and a step-up case that consists of an initially cooled boundary layer flowing onto a wall with nearly the recovery temperature. They reported that the thermal boundary layer thickness, δ T , varies considering the direction of the wall temperature change (δ T ∼ x for the step-up case, and δ T ∼ x 1/2 for the step-down one). [START_REF] Laderman | Effect of wall temperature on a supersonic turbulent boundary layer[END_REF] [START_REF] Laderman | Effect of wall temperature on a supersonic turbulent boundary layer[END_REF] reported experimental results concerning mean flow properties of a Mach 3 turbulent boundary-layer with negligible pressure gradient, and with T w /T r ratio ranging from 0.54 to 0.94. He reported that the law-of-the-wake expression correctly correlates the measured mean velocity profiles, and that the turbulent transport properties (turbulent shear stress, mixing length and eddy viscosity) are found to be in good agreement with previous compressible adiabatic flows. These properties were indirectly extracted from the time-averaged conservation equations using the measured mean flow profiles. With decreasing heat transfer, it was observed that the experimental total temperature-velocity profiles did not agree well with the linear Crocco's [START_REF] White | Viscous fluid flow[END_REF] relation.

Furthermore, [START_REF] Laderman | Turbulent shear stresses in compressible boundary layers[END_REF] [START_REF] Laderman | Turbulent shear stresses in compressible boundary layers[END_REF] extended the study to account for the turbulent transport properties using hot-wire anemometry measurements of the shear stresses of a Mach 3 boundary layer at Re θ ≃ 3500, with T w /T r ratios of 0.94 and 0.71. The normalized shear stress distributions, τ /τ w , is found to be independent of the Mach number and the wall-temperature, for the considered range of wall-temperature (0.4 ≤ T w /T r ≤ 1.0). [START_REF] Gaviglio | Reynolds analogies and experimental study of heat transfer in the supersonic boundary layer[END_REF] [START_REF] Gaviglio | Reynolds analogies and experimental study of heat transfer in the supersonic boundary layer[END_REF] and [START_REF] Rubesin | Extra compressibilty terms for Favre-averaged two-equation models of inhomogeneous turbulent flows[END_REF] [START_REF] Rubesin | Extra compressibilty terms for Favre-averaged two-equation models of inhomogeneous turbulent flows[END_REF] proposed two different modifications of the SRA, called respectively GSRA and RSRA, to better account for the wall-heat transfer. [START_REF] Dupont | Etude expérimentale des champs turbulents dans une couche limite supersonique fortement chauffée[END_REF] [START_REF] Dupont | Etude expérimentale des champs turbulents dans une couche limite supersonique fortement chauffée[END_REF] studied experimentally the effect of strong wall-heating on the fluctuating field of a flat plate supersonic turbulent boundary layer. It was concluded that two combined but distinct mechanisms influence the total temperature production: the first is related to compressibility effects, and the second is due to the large temperature gradient originated from the heating. The Reynolds shear stress was found to remain unaffected by the wall-heating, and the velocity-temperature correlation was found to be constant, and close to -0.8.

Audiffren (1993) [1] studied the influence of wall-heating on the characteristics of a turbulent boundary layer at Mach 2.25 and Re θ ≈ 5000. It was reported that, except the inner region (y/δ < 0.2), there is no significant increase in the fluctuating Mach number. It was also concluded that the wall-heating was governed by some specific parameters, like the density ρ and the friction velocity, u τ , for the dynamic field, and the friction temperature, T τ , for the thermal field. It was found that the wall-heating has the effect of increasing the boundary layer thickness, and that the intensity of static and total temperatures increase as well. Finally, it was reported that the Reynolds shear stress u ′ v ′ did not undergo notable changes with wall-heating, and that the skewness and the flatness factors were also not influenced by the wall-heating. [START_REF] Huang | Compressible turbulent channel flows: DNS results and modeling[END_REF] improved collapsing the results, even if this improvement is partial. The outer scaling, using ρ R ij /τ w (where R ij is the Reynolds stress tensor) versus y/H (H being the height of the channel) seems collapsing all studied cases in a region sufficiently far from the wall.

An extension of the study by [START_REF] Morinishi | Direct numerical simulation of compressible turbulent channel flow between adiabatic and isothermal walls[END_REF] [START_REF] Morinishi | Direct numerical simulation of compressible turbulent channel flow between adiabatic and isothermal walls[END_REF] was made by [START_REF] Tamano | Effect of different thermal wall boundary conditions on compressible turbulent channel flow at M=1.5[END_REF] [START_REF] Tamano | Effect of different thermal wall boundary conditions on compressible turbulent channel flow at M=1.5[END_REF] of compressible turbulent channel flows between adiabatic and/or isothermal walls at Mach numbers of 1.5. By analyzing the temperature variance transport equation, they found that the compressibility effects were not negligible near the low-temperature wall. The skewness and flatness factors, S(u ′ ) and F (u ′ ), were found to agree well with incompressible data, which did not depend on the thermal boundary conditions. The Morkovin's hypothesis seemed to be correct for higher-order turbulence statistics of u ′ . However, S(v ′ ) and F (v ′ ) did not agree with incompressible data, due to the compressibility effects.

Additional studies were reported by [START_REF] Liu | Compressibility and variable density effects in turbulent boundary layers[END_REF] [START_REF] Liu | Compressibility and variable density effects in turbulent boundary layers[END_REF] on compressible turbulent boundary layers using DNS. Four cases were considered : two of them were related to turbulent boundary layers with Reynolds number Re θ ≈ 2000 at Mach number M ≈ 0.06 with a heated wall at T w /T ∞ = 1.58, and the two others concerned adiabatic supersonic turbulent boundary layers with Re θ ≈ 2000 and at M = 1.8, where the wall-temperature to freestream temperature ratio T w /T ∞ = 1.58. It was concluded that the similarity laws could be approximately satisfied if the density-weighted transformation was applied. Even with the weak wall-temperature, T w , the friction velocity, u τ , of the heated case seemed to be larger than that of the supersonic case. Using u * rms , of the heated-wall case appeared to nearly coincide with the distribution of an isothermal incompressible turbulent boundary layer. The Reynolds shear stress, -u ′ v ′ , plotted against y + using local properties, showed good agreement with the corresponding incompressible results. The subsonic flow with heat transfer was found to have a larger density compared to the supersonic flow throughout the boundary layer. The turbulent Mach number, M t , had a very small value for the M = 0.06 subsonic case (M t ≤ 0.02 for the inner part of the boundary layer), while it showed a larger value, which reached a peak of ≈ 0.25 for the adiabatic supersonic case. Gosh et al. (2008) [START_REF] Ghosh | Direct and large-eddy simulation of supersonic turbulent flow in pipes, nozzles and diffusers[END_REF] made DNS and LES to investigate compressibility effects of a supersonic pipe flow at Mach number M ≈ 1.5 (based on the ratio of the bulk velocity to the speed of sound at wall-temperature) and with an isothermal wall boundary condition. Although studying supersonic pipe flows is different from supersonic plane channel or boundary layer flows, due to the effects of wall curvature and the non-zero pressure-gradient, interesting conclusions can be learned from such studies. It was observed that, near the wall, the density and temperature showed strong negative correlation, while in the core region, there were no mean density and temperature gradients. This was also confirmed by a strong pressure-density correlation in the core region, and a nearly zerovalue near the wall. In particular, it was also concluded that the Morkovin's hypothesis held for most statistics of the flow, and that the van-Driest transformation could be used for the mean velocity profile. A scaling, using local mean density and viscosity (i.e. semi-local scaling) was found to nearly collapse the peak locations of the velocity fluctuations, but not the peak magnitudes. As it is known, the mean density decreases with increasing Mach number, which leads to the decrease of the pressure-strain correlations, and then increase the Reynolds stress anisotropy. [START_REF] Duan | Direct numerical simulation of hypersonic turbulent boundary layers. Part 2. Effect of wall temperature[END_REF] [START_REF] Duan | Direct numerical simulation of hypersonic turbulent boundary layers. Part 2. Effect of wall temperature[END_REF] performed DNS of supersonic turbulent boundary layers at Mach number 5 and with T w /T r ranging from 0.18 to 1. One of the main conclusions of the study was that decreasing the wall-temperature increased the compressibility effects. The streamwise velocity distributions normalized using the van-Driest transformation collapsed with the incompressible results, and the r.m.s of the velocity fluctuations scaled using the Morkovin's scaling held also for the adiabatic profiles. The Walz's equation [START_REF] White | Viscous fluid flow[END_REF] [START_REF] White | Viscous fluid flow[END_REF] did not hold for the wall temperature varying cases, and the mismatch increased as the temperature decreased. The R uT correlation remained about the same, except near the wall. The modified SRA relation given by [START_REF] Huang | Compressible turbulent channel flows: DNS results and modeling[END_REF] [START_REF] Huang | Compressible turbulent channel flows: DNS results and modeling[END_REF] gave better results compared to the standard SRA and to other modified relations. It was also shown that the Prandtl number was nearly constant and insensitive to the wall cooling. For the budget of the turbulent kinetic energy, the semi-local scaling correctly collapsed profiles. It was reported also that the wall cooling has the effect of increasing the length and the coherence of the turbulent structures. [START_REF] Lagha | Near-wall dynamics of compressible boundary layers[END_REF] [START_REF] Lagha | Near-wall dynamics of compressible boundary layers[END_REF] showed that the streak length in terms of x/δ decreased when Re τ increased. It was also concluded that in an isothermal simulation with cold wall, the wall-parallel plane streaks were longer, whereas for a hot wall, they were shorter (in comparison with the adiabatic case).

In terms of large-eddy simulation, few works relying on either explicit or implicit LES of wallbounded flows were reported. Among others, [START_REF] Spyropoulos | Large-eddy simulation of a spatially evolving supersonic turbulent boundary layer flow[END_REF] [START_REF] Spyropoulos | Large-eddy simulation of a spatially evolving supersonic turbulent boundary layer flow[END_REF] reported LES of spatially evolving supersonic turbulent boundary layer at Mach number M = 2.25. A secondas well as a fourth-order accurate upwind biased finite differences schemes were used for the convective fluxes. In terms of wall-units, all considered grids had resolutions ranging between 59 ≤ ∆x + ≤ 88 for the streamwise direction, 0.77 ≤ ∆y + min ≤ 0.97 for the wall-normal direction and 11.4 ≤ ∆z + ≤ 42.1 for the spanwise direction. It was concluded that a decrease of about 20% of the computed skin-friction is found when lower order schemes are employed, mainly a third order upwind scheme for the convective terms and second order for the viscous terms. Because of the low considered Mach number, the modeling of the isotropic part of the shear stresses was not found to have a considerable effect on the skin-friction coefficient, C f . The insufficient amount of turbulent transport was attributed to the use of the dynamic Smagorinsky model, in which the eddy viscosity is computed using the smallest resolved scales. Supersonic flat-plate boundary layers have been investigated by [START_REF] Yan | Large-eddy simulation of supersonic flat-plate boundary layers using the monotonically integrated large-eddy simulation MILES technique[END_REF] [START_REF] Yan | Large-eddy simulation of supersonic flat-plate boundary layers using the monotonically integrated large-eddy simulation MILES technique[END_REF] using monotonically integrated large-eddy simulation (MILES) approach. In this simulation, the numerical dissipation induced by the scheme substitutes to the SGS eddy viscosity, mimicking from an energetic view-point the action of SGS terms on the flow dynamics. The simulated flows evolved at freestream Mach numbers of 2.88 and 4. An adiabatic as well as an isothermal case with T w /T r = 1.1 were performed. In terms of wall-units, grid resolutions were ∆x + = 18, ∆y + min = 1.5 and ∆z + = 6.5. It was reported that the mean streamwise velocity profiles using the van-Driest transformation were in good agreement with the viscous sublayer linear approximation and law-ofthe-wall (u + vd = 2.5 log y + + 5.7). The distributions of the streamwise Reynolds stresses scaled by ρ and τ w , were found to be very similar except close to the wall, and showed good agreement in the outer region of the boundary layer (y/δ > 0.2). The peak magnitude in the near-wall region (y/δ < 0.2) was supported by experimental and DNS results, although its location was not consistent with the reference data. The Reynolds shear stresses of both cases showed good agreement with the reference solutions. Finally, the turbulent Prandtl number Pr t was found to be in good agreement with the experimental value of 0.89. [START_REF] Brun | Large-eddy simulation of compressible channel flow[END_REF] [START_REF] Brun | Large-eddy simulation of compressible channel flow[END_REF] performed LES of fully turbulent compressible isothermal channel flows at Re θ ≈ 3000 and Re θ ≈ 4880, with Mach numbers varying in the range 0.3 ≤ M ≤ 5. Mesh refinement was used in the near-wall region so that the flow was solved in a DNS sense, while the SGS model acted mainly in the core region of the channel. A new integral scaling for wall-bounded flows (denoted y c+ = y + 0 µ w / µ dy + ) was presented and used to improve both the wall-unit definition and the van-Driest transformation. Both viscosity/conductivity and density effects were accounted for in the proposed scaling. Based on the present integral scaling, it was shown that a relative universality of the wall-layer property exists. A new constant c = 1/Pr m (Pr m is the mixing Prandtl number) for the modified Strong Reynolds Analogy, as proposed by Huang et al. (1995) [START_REF] Huang | Compressible turbulent channel flows: DNS results and modeling[END_REF], was also proposed. It was shown that accounting for the Pr m in the modified SRA yielded a good fit of the results. It was finally concluded that it is possible to deduce the turbulent properties of non-hypersonic boundary layers from the incompressible law-of-the-wall with equivalent integral Re c τ numbers.

Summary

With regards to heat and fluid flows in supersonic nozzles, it has been shown in the reviewed experimental studies, that the intensity of heat transfer in such systems depends on many factors such as gas flow rate, compressibility, nozzle dimensions and chemical reactions.

Nowadays, with the progress achieved in computer sciences and numerical flow simulations, space system designers, especially those in charge of rocket nozzles, are eager to introduce highfidelity numerical simulations in their design chains. More fundamental research towards basic studies such as near-wall turbulence is therefore needed.

The objective of this work is then to understand, through well-resolved numerical simulations, the governing flow parameters and the near-wall turbulence behavior of wall-bounded flows, such as those occurring in supersonic aerodynamics. The influence of the wall-heat transfer on the dynamics of such flows is of primordial interest.

From the previous review, it can be retained that both experimental and numerical investigations do agree with the fact that, for relatively low heated/cooled walls at moderate Mach numbers, the mean velocity profiles hold for the incompressible data as long as the van-Driest transformation is applied. However, the velocity fluctuations profiles does not universally scale with the incompressible counterparts when the standard Morkovin scaling is used. Consequently alternative scaling laws must be applied, whether near-wall region or far-from-wall region is studied (Morinishi et al., 2004 [66], Foysi et al., 2004 [31]). Also, the original SRA seems to fail linking velocity and temperature fluctuations, and the modified SRA relations, better accounting for the isothermal condition, are prefered (Cebeci & Smith, 1978 [13]; Gaviglio, 1987 [33]; Rubesin, 1990 [79]; Huang et al., 1995 [43]). In terms of flow organization, near-wall steaks are found to be more coherent when decreasing the wall temperature (Coleman et al., 1995 [15]

ϕ(x) = G ⋆ ϕ = D G(x -ξ, ∆)ϕ(ξ) dξ (2.1)
où ⋆ représente le produit de convolution à travers le domaine D et x = (x 1 , x 2 , x 3 ) = (x, y, z)

représente les coordonnées cartésiennes d'un point.

Le filtre est construit de sorte à respecter la propriété suivante:

D G(x -ξ, ∆) dξ = 1 (2.
ϕ -ϕ = ϕ ′′ = - ρ ′ ϕ ′ ρ = - ρ ′ ϕ ′′ ρ (2.
3) avec ϕ = U ou T , avec U = (u 1 , u 2 , u 3 ) = (u, v, w) est le vecteur vitesse et T est la température.

Système d'équations filtrées

Le système d'équations de Navier-Stokes compressibles filtrées s'écrit sous une forme conservative:

∂ρ ∂t + ∂ρ u i ∂x i = 0 ∂ρ u i ∂t + ∂ρ u i u j ∂x j + ∂p ∂x i - ∂ σij ∂x j ≈ - ∂τ ij ∂x j ∂ρ Ě ∂t + ∂ ρ Ě + p u j ∂x j - ∂ u i σij ∂x j + ∂ qj ∂x j ≈ - 1 γ -1 ∂ (pu j -p u j ) ∂x j -u j ∂τ ij ∂x j                  (2.4)
où ρ, p et u i représentent respectivement les champ résolus de densité, de pression et de vitesse.

L'équation d'état, l'énergie totale, le cisaillement visqueux et le flux de chaleur résolus s'écrivent:

p = ρr T ρ Ě = p γ -1 + 1 2 ρ u i u i σij = µ ∂ u j ∂x i + ∂ u i ∂x j - 2 3 µ ∂ u k ∂x k δ ij qj = - µC p P r ∂ T ∂x j                        (2.

Modélisation de sous-maille

Concept de la modélisation fonctionnelle

Le concept de base de l'approche LES repose sur le fait que l'action des échelles de sous-maille (isotropes et présentant un comportement universel) sur les échelles résolues (anisotropes avec un comportement moins déterministe) est principalement une action énergétique dissipative. De ce fait, un modèle de sous-maille agit essentiellement afin de dissiper l'énergie résolue par les grandes structures de l'écoulement. Dans notre étude, la contribution des échelles de sous-maille est considérée en utilisant une approche fonctionnelle explicite, en introduisant une viscosité turbulente reproduisant les effets des échelles de sous-maille sur la turbulence.

Un spectre typique d'une turbulence homogène, tel que représenté sur la Fig. 2.1, comprend une zone de production, une zone inertielle dans laquelle l'énergie est transférée depuis les grandes structures vers les plus petites (Outscatter), et une zone de dissipation d'énergie.

En effet, les grandes structures d'un écoulement turbulent (associées aux faibles nombres d'ondes) extraient l'énergie cinétique depuis le champ moyen. Ces structures, qui sont initialement relativement épaisses, vont s'aplatir créant de fines structures tourbillonnaires instables qui, en s'enroulant elles-mêmes, génèrent de petites structures (associées aux grands nombres d'ondes) sous formes de filaments. D'un point de vue énergétique, cela résulte en un transfert direct de l'énergie depuis les grandes structures vers les plus petites, jusqu'à ce que la viscosité moléculaire dissipe toute l'énergie cinétique en énergie interne [START_REF] Garnier | Large eddy simulation for compressible flows[END_REF] [START_REF] Garnier | Large eddy simulation for compressible flows[END_REF].

La modélisation des termes de sous-maille consiste donc à modifier le système d'équations régissant l'écoulement d'une manière à y intégrer les effets désirés de dissipation ou de production d'énergie.

Dans la présente étude, plusieurs modèles de sous-maille sont utilisés pour modéliser l'action des petites structures sur la turbulence : le modèle de Smagorinsky utilisant la procédure dynamique (noté DSM) proposé par [START_REF] Germano | A dynamic subgrid-scale eddy viscosity model[END_REF] [START_REF] Germano | A dynamic subgrid-scale eddy viscosity model[END_REF] et Lilly (1991) [START_REF] Lilly | A proposed modification of the Germano subgrid-scale closure method[END_REF] et étendu au cas compressible par [START_REF] Moin | A dynamic subgrid-scale model for compressible turbulence and scalar transport[END_REF] [START_REF] Moin | A dynamic subgrid-scale model for compressible turbulence and scalar transport[END_REF], le modèle de structures cohérentes (Coherent Structures, noté CSM) introduit par [START_REF] Kobayashi | The subgrid-scale models based on coherent structures for rotating homogeneous turbulence and turbulent channel flow[END_REF] [START_REF] Kobayashi | The subgrid-scale models based on coherent structures for rotating homogeneous turbulence and turbulent channel flow[END_REF], et le modèle WALE (Wall-Adapting Local Eddy-viscosity) proposé par [START_REF] Nicoud | Subgrid-scale stress modelling based on the square of the velocity gradient tensor[END_REF] [START_REF] Nicoud | Subgrid-scale stress modelling based on the square of the velocity gradient tensor[END_REF]. Afin de mieux quantifier la contribution des modèles de sous-maille, une approche implicite (Implicit Large-eddy simulation, notée ILES) est aussi utilisée, dans laquelle la dissipation induite par les schémas numériques et le maillage utilisé imite, d'un point de vue énergétique, l'action du modèle de sous-maille sur les grandes structures de la turbulence.

Modélisation du tenseur des contraintes de sous-maille

Le tenseur des contraintes de sous-maille, τ ij , dans l'équation (2.4) est défini comme:

τ ij = ρ ( u i u j -u i u j ) (2.6)
Il est modélisé en utilisant une formulation fonctionnelle, moyennant l'introduction d'une viscosité turbulente. Cela se traduit par: Pour les écoulements compressibles, [START_REF] Yoshizawa | Statistical theory for compressible turbulent shear flows with the application to subgrid modeling[END_REF] [START_REF] Yoshizawa | Statistical theory for compressible turbulent shear flows with the application to subgrid modeling[END_REF] propose un modèle de fermeture pour la partie isotrope du tenseur de sous-maille, τ kk , défini par: Par analogie à τ ij , le filtre test, appliqué au tenseur des contraintes de sous-maille, s'écrit:

τ ij - 1 3 τ kk δ ij = -2µ sgs S ij - 1 3 S kk δ ij (2.
τ kk = 2ρC I ∆ 2 |
T ij = ρu i u j - ρu i ρu j ρ ( 2
L ij = ρ u i u j - ρ u i ρ u j ρ (2.11)
Dans ce qui suit, on note:

S * ij = S ij - 1 3 S kk δ ij L Cs ij = L ij - 1 3 L kk δ ij (2.12)
Par analogie au modèle de fermeture du tenseur des contraintes de sous-maille, τ ij , il est possible d'écrire:

T ij - 1 3 T kk δ ij = -2 ρC s ∆ 2 | S| S * ij T kk = 2C I ∆ 2 ρ| S| 2      (2.13)
Utilisant l'expression du tenseur de Léonard (2.11) et le système d'équations précédent (2.13), cela donne:

L Cs ij = C s -2 ∆ 2 ρ| S| S * ij + 2∆ 2 ρ| S| S * ij M Cs ij = C s M Cs ij L kk = C I -2 ∆ 2 ρ| S| 2 + 2∆ 2 ρ| S| 2 M C I ij = C I M C I ij (2.
C s = L Cs ij M Cs ij M Cs ij M Cs ij C I = L kk M C I kk (2.15)
Les constantes retrouvées possèdent, néanmoins, deux propriétés mathématiques contraignantes:

• Elles peuvent prendre des valeurs négatives, ce qui implique que le modèle possède un caractère anti-dissipatif. Cette caractéristique est souvent interprétée comme la manifestation d'un transfert d'énergie depuis les petites structures vers les grandes (Backscatter).

• Elles ne sont pas bornées, et peuvent prendre des valeurs très larges, voire infinies.

Pour éviter toute instabilité numérique résultant de la procédure mathématique utilisée, C s et C I sont moyennées dans la direction homogène de l'écoulement:

C s = L Cs ij M Cs ij H M Cs ij M Cs ij H C I = L kk H M C I kk H (2.
F cs = Q E (2.18)
Q and E sont respectivement le tenseur du second invariant résolu et la norme d'un tenseur de gradient de vitesses résolues, donnés par:

Q = 1 2 W ij W ij -S ij S ij E = 1 2 W ij W ij + S ij S ij (2.19)
avec S ij et W ij sont le tenseur du taux de déformation des échelles résolues et le tenseur de vorticité des échelles résolues, respectivement. Il s'en suit que:

Q = - 1 2 ∂ u j ∂x i ∂ u i ∂x j E = 1 2 ∂ u j ∂x i ∂ u j ∂x i (2.
µ sgs = ρ∆ 2 C 2 w S * ij S * ij 3/2 S ij S ij 5/2 + S * ij S * ij 5/4
(2.21) avec

S * ij = 1 2 g 2 ij + g 2 ji - 1 3 g 2 kk δ ij g 2 ij = g ik g kj (2.22)
C w est une constante du modèle, prise égale à 0.5 (Nicoud et Ducros, 1999 [68]) et g ij = ∂ u i /∂x j .

Tout comme le modèle précédent, le modèle WALE présente l'avantage de reproduire le bon comportement asymptotique en y + 3 de la viscosité de sous-maille µ sgs à la paroi (Garnier et al., 2009 [32]).

Modélisation du flux de chaleur de sous-maille

Pour la modélisation du flux de chaleur, il est communément supposé que le transfert d'énergie depuis les échelles résolues vers les échelles de sous-maille est proportionnel au gradient de température résolue (Eidon, 1985 [START_REF] Eidson | Numerical simulation of the turbulent Rayleigh-Bénard problem using subgrid modeling[END_REF]; Garnier et al., 2009 [32]). Le flux de chaleur de sousmaille est donc modélisé, par analogie au tenseur des contraintes de sous-maille, en utilisant une formulation de type viscosité turbulente. Il s'écrit:

1 γ -1 ∂ (pu j -p u j ) ∂x j = - µ sgs C p Pr sgs ∂ T ∂x j (2.23)
Dans cette étude, le nombre de Prandtl turbulent, Pr sgs , est pris constant et égal à 0.9.

Simulation aux grandes échelles Implicite

En plus de l'opération explicite de filtrage, deux autres formes de filtres -implicites -sont associées aux simulations LES:

• En utilisant un maillage ne permettant pas de résoudre les plus fines échelles de l'écoulement étudié, une opération de filtrage, appelé filtre-maillage, est implicitement appliquée au système d'équations. En effet, du fait de la distribution spatiale des noeuds du maillage, une séparation entre les petites et les grandes structures de l'écoulement est implicitement générée, puisque des informations, qui sont plus petites qu'une certaine échelle associée à la résolution du maillage, ne peuvent être calculées.

• Etant donné que les schémas numériques utilisés induisent une erreur qui est dépendante de l'échelle calculée, ils introduisent donc une distinction entre les échelles suffisamment résolues et celles qui le sont moins. Une deuxième opération de filtrage est ainsi implicitement appliquée par le biais du schéma numérique utilisé, où l'erreur commise par l'approximation des opérateurs de dérivés tend à modifier la solution calculée (Garnier et al., 2009 [32]). Cette erreur se voit cependant minimiser avec l'utilisation de schémas numériques d'ordre élevés suffisamment précis.

En plus, à cause de la présence de forts gradients dans les écoulements supersoniques, des schémas numériques, robustes mais quelque peu dissipatifs -localement du moins -sont utilisés.

Cette dissipation numérique intrinsèque, inhérente à ce type de schémas et qui imite d'un point de vue énergétique l'action des échelles de sous-maille sur la turbulence, peut être considérée comme une viscosité turbulente implicitement introduite par un modèle de sous-maille. Ce concept est appelé ILES, ou Implicit large-eddy simulation. Cette approche de modélisation des écoulements turbulents est intéressante dans la mesure où elle permet de quantifier la réelle contribution du modèle de sous-maille sur la turbulence en comparant les résultats ILES aux calculs LES explicites.

Il est à noter que cette manière d'utiliser la dissipation numérique comme un modèle SGS repose sur l'hypothèse qui stipule que l'action des échelles de sous-maille sur les échelles résolues est strictement une action dissipative.

Le gros inconvénient de cette méthode c'est qu'elle ne permet pas un contrôle directe de la quantité d'énergie de sous-maille dissipée, puisque celle-ci dépend fortement du schéma de discrétisation spatiale utilisée.

Schémas numériques

Dans ce travail, les équations de Navier Stokes compressibles 3D sont résolues à l'aide du code CHOC-WAVES1 . Il s'agit d'un solveur compressible parallèlle DNS/LES mono-et multi-espèces, utilisant une forme de résolution type différences finies. Pour des raisons de simplicité, la version mono-espèce du code, utilisant l'air, supposé un gaz parfait, est utilisée par la suite.

Traitement des flux convectifs

Dans les écoulements supersoniques en présence de forts gradients et/ou de discontinuités de contact, l'utilisation de schémas numériques suffisamment robustes dans les régions de forts gradients ou proche d'une discontinuité (telle qu'une onde de choc) mais précis dans les régions de turbulence, est primordiale. Pour la description de la méthode de calcul du schéma WENO, une équation de conservation scalaire unidimensionnelle est utilisée:

∂u ∂t + ∂f (u) ∂x = 0 (2.24)
Le flux caractéristique f (u) est décomposé en une partie positive et une partie négative:

f (u) = f + (u) + f -(u) (2.25)
Les deux flux présentant des vitesses de propagation non-négative et non-positive, respectivement:

df + (u) du ≥ 0 df -(u) du ≤ 0 (2.26)
Cette décomposition s'effectue selon le schéma de Lax-Friedrichs, qui s'écrit:

f ± (u) = 1 2 (f (u) ± αu) (2.27) où α = max u | f ′ (u)
| est la plus grande valeur sur une ligne de maillage concernée. Le flux numérique à l'interface f i+1/2 est également décomposé:

f i+1/2 = f + i+1/2 + f - i+1/2 (2.28) Ainsi, comme indiqué sur la Fig. (2.3), le flux f + i+1/2 (respectivement f - i+1/2 ) est reconstruit en utilisant une interpolation des flux f + i (respectivment f - i ) sur les trois stencils S = {S 1 , ..., S 3 }.
Le schéma WENO utilise une combinaison convexe des trois flux suivants afin d'obtenir la précision maximale dans les régions à faibles gradients: Enfin, le flux numérique à l'interface à l'ordre 5 est calculé comme suit:

             f + (1) i+1/2 = 2 6 f + i-2 - 7 6 f + i-1 + 11 6 f + i f + (2) i+1/2 = - 1 6 f + i-1 + 5 6 f + i + 2 6 f + i+1 f + (3) i+1/2 = 2 6 f + i + 5 6 f + i+1 - 11 6 f + i+2 (2.29) S 1 S 2 S 3 i + 1/2 i -2 i -1 i i + 1 i + 2 i + 3 S
f i+1/2 = r l=1 w + l f + (l) i+1/2 (2.30)
Les coefficients de pondération non-linéaire sont définis par:

w + l = α + l r l=1 α + l α + l = d + l ǫ + β + l 2
(2.31)

d + l = d - l = d l
sont les poids qui permettent d'obtenir une précision optimale du schéma WENO (5 dans notre cas):

d 1 = 1/10 d 2 = 6/10 d 3 = 3/10 (2.32)
ǫ est un nombre très faible qui permet d'éviter la division par zéro (ǫ = 10 -6 ; Jiang & Shu (1996) [START_REF] Jiang | Efficient implementation of weighted ENO schemes[END_REF]), et β + l sont les indicateurs qui permettent de diminuer les poids des stencils contenant la discontinuité. Ils sont définis comme la somme des normes de toutes les dérivées des polynômes

S 1 S 2 S 3 S 4 i -2 i -1 i i + 1 i + 2 i + 3 S i + 1/2
             β + 1 = 13 12 f + i-2 -2f + i-1 + f + i 2 + 1 4 f + i-2 -4f + i-1 + f + i 2 β + 2 = 13 12 f + i-1 -2f + i + f + i+1 2 + 1 4 f + i-1 -4f + i + f + i+1 2 β + 3 = 13 12 f + i -2f + i+1 + f + i+2 2 + 1 4 f + i -4f + i+1 + f + i+2 2 (2.33)
La partie négative du flux numérique WENO, f - i+1/2 , est calculée en suivant exactement la même procédure, remplaçant f + par f -.

Schéma WENO-BWO (Large Bande Optimisé)

Un des inconvénients du schéma WENO classique est sa dissipation numérique relativement importante, surtout dans les régimes de forte turbulence, où les indicateurs β l ne sont pas strictement nuls. Pour remédier, en partie, à ce problème, un schéma à large bande optimisé a été proposé par [START_REF] Martin | A bandwidth-optimized WENO scheme for the effective direct numerical simulation of compressible turbulence[END_REF] [START_REF] Martin | A bandwidth-optimized WENO scheme for the effective direct numerical simulation of compressible turbulence[END_REF]. Ce schéma comporte quatre stencils S = {S 1 , ..., S 4 } introduits de manière à limiter le décentrement, en rendant le plus symétrique possible le calcul du flux numérique à l'interface i + 1/2 (voir Fig. 2

.4).

Les flux numériques à l'interface s'écrivent donc:

                   f + (1) i+1/2 = 2 6 f + i-2 - 7 6 f + i-1 + 11 6 f + i f + (2) i+1/2 = - 1 6 f + i-1 + 5 6 f + i + 2 6 f + i+1 f + (3) i+1/2 = 2 6 f + i + 5 6 f + i+1 - 1 6 f + i+2 f + (3) i+1/2 = 11 6 f + i - 7 6 f + i+1 + 2 6 f + i+2 (2.34)
Le flux numérique à l'interface est calculé (n = 4):

f i+1/2 = n l=1 w + l f + (l) i+1/2 (2.35)
Les coefficients de pondération non-linéaire sont définis par:

w + l = α + l n l=1 α + l α + l = d + l ǫ + β + l 2 avec α + 4 = min | 1≤l≤n (α + l ) (2.36)
d + l sont les poids optimaux proposés par [START_REF] Martin | A bandwidth-optimized WENO scheme for the effective direct numerical simulation of compressible turbulence[END_REF] [START_REF] Martin | A bandwidth-optimized WENO scheme for the effective direct numerical simulation of compressible turbulence[END_REF]:

d + 1 = 0.094647545896 d + 2 = 0.428074212384 d + 3 = 0.408289331408 d + 4 = 0.068988910311 (2.37)
et les indicateurs de décentrement s'écrivent donc:

                   β + 1 = 13 12 f + i-2 -2f + i-1 + f + i 2 + 1 4 f + i-2 -4f + i-1 + f + i 2 β + 2 = 13 12 f + i-1 -2f + i + f + i+1 2 + 1 4 f + i-1 -4f + i + f + i+1 2 β + 3 = 13 12 f + i -2f + i+1 + f + i+2 2 + 1 4 f + i -4f + i+1 + f + i+2 2 β + 4 = 13 12 f + i+1 -2f + i+2 + f + i+3 2 + 1 4 f + i+1 -4f + i+2 + f + i+3 2 (2.38)
Une dernière procédure d'optimisation est aussi implémentée, en appliquant un limiteur aux coefficients de pondérations non-linéaires w + l qui, selon la nature de l'écoulement (présence de choc ou de forte turbulence), permettant d'ajuster localement la précision du schéma. Le limiteur agit ainsi essentiellement dans les régions loin des instabilités, afin de limiter un tant soi peu la dissipation numérique dans ces endroits [START_REF] Chaudhuri | Numerical study of compressible mixing layers using high-order WENO schemes[END_REF] [START_REF] Chaudhuri | Numerical study of compressible mixing layers using high-order WENO schemes[END_REF].

Schéma centré conservatif split-centered (Skew-Symmetric)

Il est bien connu que, même en absence d'ondes de choc, l'utilisation d'approximations centrées en différences finies, présentent des instabilités lorsqu'elles sont utilisées à des viscosités très faibles, voir nulles, ce qui est le cas des équations hyperboliques. Pour les écoulements sans discontinuités, plusieurs méthodes existantes permettent d'assurer la stabilité des schémas numériques. Pour une revue détaillée des différentes méthodes existantes, le lecteur peut se référer à l'article de [START_REF] Pirozzoli | Generalized conservative approximations of split convective derivative operators[END_REF] [START_REF] Pirozzoli | Generalized conservative approximations of split convective derivative operators[END_REF].

Une des méthodes connues, est l'utilisation de la forme fractionnée (splitted) des dérivées dans les termes convectifs des équations de Navier-Stokes. Cette méthode se base sur l'idée d'écrire les termes convectifs sous une forme du type 'skew-symmetric'. Cependant, les approximations obtenues ne peuvent être écrites dans une forme localement conservative, c'est-à-dire sous la forme d'une divergence telle qu'elles sont présentées dans l'équation (2.40). Partant de ces constats, [START_REF] Pirozzoli | Generalized conservative approximations of split convective derivative operators[END_REF] [START_REF] Pirozzoli | Generalized conservative approximations of split convective derivative operators[END_REF] propose une approximation localement conservative des termes convectifs des équations de Navier Stokes compressibles, s'écrivant sous la forme:

∂ρu k ϕ ∂x k (2.39)
où ϕ est un scalaire générique, réduit à l'unité pour l'équation de continuité, au vecteur vitesse u i (i = 1, 2, 3) pour l'équation de quantité de mouvement, et à l'enthalpie H = γ γ-1 p ρ + u2 2 pour l'équation de l'énergie totale.

Principe de base de la formulation

On cherche ici une approximation conservative de type différences finies des termes convectifs qui s'écrit sous la forme 2 :

∂ρu k ϕ ∂x k x=x j ≈ 1 h fi+1/2 -fi-1/2 (2.
∂f g ∂x x=x j ≈ D s (f g) j ≡ 1 2 D (f g) j + 1 2 f j Dg j + 1 2 g j Df j (2.43)
où D s représente l'approximation discrète de la forme fractionnée du terme convectif, Df j représente l'approximation discrète de la dérivée première de f au noeud x j , et f = ρu, g = ϕ ou f = ρϕ, g = u dans le cas de la forme fractionnée (2.41) ou (2.42), respectivement.

Une approximation centrée standard de la dérivée première s'écrit:

Df j = L l=1 a l D l f j (2.44)
tel que

D l f j = 1 h (f j+l -f j-l ) (2.45)
Dans le cas simple d'une approximation au second ordre, ce qui correspond à L = 1, on obtient:

fi+1/2 = 1 4 (f j + f j+l ) (g j + g j+l ) (2.46)
Alors qu'une approximation conservative de (2.40) s'écrit:

fi+1/2 = 1 2 (f j g j + f j+l g j+l ) (2.47)
Si on considère un seul terme (D l ) de (2.44), remplacé dans (2.43), cela donne:

D l s (f g) j = 1 2 D l (f g) j + 1 2 (f ) j D l (g) j + 1 2 (g) j D l (f ) j (2.48)
ou encore:

D l s (f g) j = 2 h ( f, g) j,l -( f, g) j-l,l (2.49)
tel que:

( f, g) j,l = 1 4 (f j + f j+l ) (g j + g j+l ) (2.50)
est un opérateur de moyenne discret à deux variables.

Il est à noter que l'équation (2.48) n'est pas écrite sous la forme conservative (2.40). Moyennant un petit développement, elle peut être reécrite telle que:

D l s (f g) j = 1 4 f l i+1/2 -f l i-1/2
(2.51) avec:

f l i+1/2 = 2 l-1 m=0 ( f, g) j-m,l (2.52)
Le flux total est ensuite obtenu en assemblant les flux partiels (2.52) à partir de l'expression (2.44):

fi+1/2 = L l=1 a l f l i+1/2 (2.53)
On obtient ainsi:

fi+1/2 = 2 L l=1 a l l-1 m=0 ( ρu, ϕ) j-m,l (2

Ecriture généralisée

Pour les écoulements présentant de fortes variations de densité, et en développant le produit triple dans le terme à droite de l'équation (2.40), il est possible d'obtenir une forme généralisée du flux, se présentant sous la forme:

∂ρuϕ ∂x = α ∂ρuϕ ∂x + β u ∂ρϕ ∂x + ρ ∂uϕ ∂x + ϕ ∂ρu ∂x + (1 -α -2β) ρu ∂ϕ ∂x + ρϕ ∂u ∂x + uϕ ∂ρ ∂x (2.56)
Dans le cas d'une écriture sous la forme (2.56), une approximation conservative semi-discrète ne peut être obtenue qu'à des conditions spécifiques. En particulier lorsque α = β = 1/4 (Pirozzoli, 2010) [START_REF] Pirozzoli | Generalized conservative approximations of split convective derivative operators[END_REF]. En suivant la même approche décrite précédemment, il vient que:

fi+1/2 = 2 L l=1 a l l-1 m=0 ( ρ, u, ϕ) j-m,l (2.57) tel que ( f, g, h) j,l = 1 8 (f j + f j+l ) (g j + g j+l ) (h j + h j+l ) (2.58)
avec les constantes a l sont données dans les tableau 2.1.

Ordre i = 1 i = 2 i = 3 i = 4 L=1 a 1 = 1/2 L=2 a 1 = 8/12 a 2 = -1/12 L=3 a 1 = 45/60 a 2 = -9/60 a 3 = 1/60 L=4 a 1 = 672/840 a 2 = -168/840 a 3 = -/840 a 4 = 3/840

Traitement des termes visqueux

Dans le code CHOC-WAVES, les flux visqueux sont approximés à l'aide d'un schéma centré d'ordre 4 à 5 stencils, dans lequel la dérivée (∂F v /∂x) i,j,k s'écrit:

∂F v ∂x i,j,k = F v i-2,j,k -8F v i-1,j,k + 8F v i+1,j,k -F v i+2,j,k 12∆x + O ∆x 4 (2.
∂u ∂x i-2,j,k = -25u i-2,j,k + 48u i-1,j,k -36u i,j,k + 16u i+1,j,k -3u i+2,j,k 12∆x + O ∆x 4 ∂u ∂x i-1,j,k = -3u i-2,j,k -10u i-1,j,k + 18u i,j,k -6u i+1,j,k + u i+2,j,k 12∆x + O ∆x 4 ∂u ∂x i+1,j,k = -u i-2,j,k + 6u i-1,j,k -18u i,j,k + 10u i+1,j,k + 3u i+2,j,k 12∆x + O ∆x 4 ∂u ∂x i+2,j,k = 3u i-2,j,k -16u i-1,j,k + 36u i,j,k -48u i+1,j,k + 25u i+2,j,k 12∆x + O ∆x 4
(2.60)

Il est à noter que les flux de sous-maille sont également évalués en utilisant le même schéma de discétisation.

Avancement en temps

Les schémas d'intégration temporelle implémentés dans le code sont principalement des algorithmes explicites du type TVD Runge-Kutta. Ces schémas obeïssent à une restriction sur l'incrément temporel qui doit être suffisamment faible, compte tenu du caractère instationnaire des écoulements étudiés. Dans cette étude, un schéma Runge-Kutta d'ordre 3 est choisi. La procédure d'intégration se fait en 3 étapes comme suit:

             U (1) = U n + ∆t L(U n ) U (2) = 1 4 3U n + U (1) + ∆t L(U (1) ) U n+1 = 1 3 U n + 2U (2) + 2∆t L(U (2) ) (2.61)
où ∆t est le pas de temps, U n est la valeur de la variable U à l'instant n, et U (k) sont les valeurs intermédiaires de U (k = 1, 2). Le critère de stabilité sur le pas de temps s'écrit:

∆t = CFL . min (∆t x , ∆t y , ∆t z ) (2.62) avec: ∆t x = min ∆x |u| + c , 1 2
∆x 2 γ (µ/P r + µ sgs /Pr sgs ) (2.

Conditions aux limites

Les conditions aux limites sont satisfaites en imposant des valeurs finies sur les cellules fantômes (2004) [START_REF] Klein | A digital filter based generation of inflow data for spatially developing direct numerical or large eddy simulation[END_REF]. Ceci-dit, des tests on montré que ces méthodes de filtre digital requièrent une longueur minimale de 14δ in dans la direction longitudinale du domaine pour permettre à la turbulence de se développer proprement et pleinement.

Pour une revue détaillée de la procédure du filtre digital et son extension au cas d'étude de couche limites supersoniques, le lecteur peut se référer aux travaux de thèse de Touber (2010) [START_REF] Touber | Unsteadiness in shock-wave/boundary-layer interactions[END_REF] sur l'interaction onde de choc/couche limite supersonique. Les profils d'entrée de vitesses, de densité et de pression s'écrivent: Le profil de vitesse moyenne longitudinale est défini en utilisant la vitesse moyenne transformée au sens de van-Driest (Guarini et al., 2000 [38]; Bernardini, 2009 [8]):

                         u (0, y, z, t) = u ′ (0, y, z, t) + u(y) v (0, y, z, t) = v ′ (0, y, z, t) + v(y) w (0, y, z, t) = w ′ (0, y, z, t) ρ (0, y, z, t) = ρ ′ (0, y, z, t) + ρ(y) p (0, y, z, t) = p ∞ (2.
u + vd =      1 κ log(1 + κy + ) + C 1 1 -e -y + η 1 - y + η 1 e -by + + Φ(y/δ) y < δ u vd∞ /u τ y ≥ δ (2.65) où Φ(y/δ) = 1 κ y δ 2 - y δ 3 + 6Π y δ 2 -4Π y δ et κ = 0.41, C = 5.1, π = 0.20, C 1 = C -log 1/κ, η 1 = 11 et b = 0.33.
La pression est supposée uniforme p ∞ à l'entrée du domaine, et les profils de densité et de température sont définis en utilisant la formule de Crocco-Busemann (White, 1991 [105]):

ρ w ρ = T T w = 1 + T aw T w -1 u u ∞ -r γ -1 2 M 2 ∞ T ∞ T w u u ∞ 2 (2.66)
où r est le facteur de recouvrement (pris égal à 0.89), et T aw est la température adiabatique de la paroi (supposée égale de la température de recouvrement, T r ), définie par:

T aw ≃ T r = T ∞ 1 + r γ -1 2 M 2 ∞ (2.67)
Dans la première partie de l'étude de la couche limite supersonique adiabatique, les profils moyens utilisés, à savoir u , v et ρ , correspondent aux données issus du calcul DNS de Pirozzoli & Bernardini (2011) [74].

La renormalisation des fluctuations est réalisée en divisant la couche limite en une région interne (désignée par la coordonnée y + ) et une région externe (désignée par la variable η = y/δ), et est définie comme suit:

                   u ′ i | inn (0, y + , z, t) = ψu ′ i (x rec , y + , z + L z /2, t) u ′ i | out (0, η, z, t) = ψu ′ i (x rec , η, z + L z /2, t) ρ ′ | inn (0, y + , z, t) = ρ ′ (x resc , y + , z + L z /2, t) ρ ′ | out (0, η, z, t) = ρ ′ (x resc , η, z + L z /2, t) (2.68)
Le paramètre de renormalisation ψ, prenant en compte l'évolution de la densité à la paroi entre l'entrée et la station de recyclage, est défini par (Pirozzoli et al., 2010) [START_REF] Pirozzoli | Direct numerical simulation of transonic shock/boundary layer interaction under conditions of incipient separation[END_REF]: A l'extérieur de la couche limite (η > 1), un bruit blanc est superposé aux fluctuations de vitesses, avec:

ψ = u τ, in u τ, rec ρ w, in ρ w, rec (2
u ′ i η>1 (0, η, z, t) = 0.1 u ∞ e -2(η-1) (2.70)
Une moyenne pondérée spatialement est définie, en utilisant les deux profils issus des régions interne et externe de la couche limite, afin de garantir une transition douce entre ces deux régions. Ainsi:

ϕ = ϕ inn [1 -W (η)] + ϕ out W (η) (2.71)
avec W est la fonction de pondération:

W (η) = 1 2   1 + tanh α(η-b) η(1-2b)+b tanh (α)   (2.72) où α = 4 et b = 0.2.

Initialisation des calculs

Conventions

Dans ce qui suit, et sauf mention contraire, seule la partie résolue d'une quantité ϕ donnée est prise en compte dans la discussion des résultats. Ce qui implique que

ϕ i ≡ ϕ i et ϕ ′ i ≡ ϕ ′ i .

Notations

Dans ce qui suit, chaque quantité moyenne présentée est obtenue suite à une moyenne temporelle sur la période d'échantillonnage (T s ) et une intégration spatiale dans la direction homogène de l'écoulement (z). Elle est notée ϕ , et est définie comme: (2011) [74], as well as to experimental results available in the literature. The issue of the sub-grid scale modeling through the SGS models' dissipation has also been addressed in order to get more insight into the modeling behavior.

ϕ(x, y) = 1 L z 1 T s Lz Ts ϕ(x,

Flow conditions and simulation parameters

The incoming boundary layer is spatially evolving at a Mach number M = 2 and Reynolds numbers Re τ in = ρ w u τ in δ in /µ w ≈ 245 (where u τ in is the friction velocity, δ in is the inflow boundary layer thickness) and Re Pirrozoli & Bernardini, 2011 [74]) mainly due to computational cost issues.

θ in = ρ ∞ u ∞ θ in /µ ∞ ≈ 1150 (θ in
As shown in table 3.1, different grid resolutions are used with uniformly spaced grid in the streamwise and spanwise directions, and clustered grid in the wall-normal direction according to

x y L x L y M ∞ L z z δ in x rec
x res y j = d y sinh(η j ) (Touber, 2010) [START_REF] Touber | Unsteadiness in shock-wave/boundary-layer interactions[END_REF], where d y = L y / sinh(β y ), L y is the total height of the domain, β y is the grid stretching parameter, and η j = β y (j -1)/(N y -1). The flowfield is initialized using a digital filter procedure based on Klein's method (Klein et al., 2004) [START_REF] Klein | A digital filter based generation of inflow data for spatially developing direct numerical or large eddy simulation[END_REF] where the r.m.s.

velocity profiles are extracted from the DNS of Bernardini and Pirozzoli (2011) [START_REF] Bernardini | Wall pressure fluctuations beneath supersonic turbulent boundary layers[END_REF]. A series of 3.4: Lines and symbols used in graphes' legends.

≈ 140 characteristic times, τ c = δ in /u ∞ is
= ρ w u τ δ/µ w ; Re θ = ρ ∞ u ∞ θ/µ ∞ ; C f = 2τ w /ρ ∞ u 2 ∞ ; H = δ * /θ; M τ = u τ /(γRT w )
❍ DSM - ■ CSM -- ▲ WALE • • • ◆ ILES • -• ▼ LES-L1 - ■ LES-L2 -- ◆ LES-N2 • • • ▲ LES-P2 • -• ▲ LES-Q2 • • - ▼ LES-R2 • -- ▼ LES-S2 -- • Table
In the following, the legends are summarized in table 3.4.

Flow Organization

For subsonic, and subsequently supersonic flows, it is known that the very near-wall region is occupied by alternating streaks of high-and low-speed fluid. These streaks are presumed to derive from elongated, counter-rotating streamwise vortices near the wall. For the near-wall region y + < 100, those streaks are found to significantly contribute to the turbulence production, which occurs during the bursting process i.e. low-speed streaks would gradually lift up from the wall, oscillate, and then break up violently, ejecting fluid away from the wall and into the outer layer (Smits & Dussauge, 2006) [START_REF] Smits | Turbulent shear layers in supersonic flow[END_REF].

In order to qualitatively assess the turbulent nature of the flow in the inner layer, wall-parallel slices of velocity and temperature fluctuations are plotted in Fig. shows this turbulent structures, with a highly intermittent character, and inclined to the wall in the downstream direction.

Turbulence Statistics

Auto-Correlation Coefficients

To ensure that the computational domain is sufficiently wide in the spanwise direction, we analyzed the two-point correlations functions, defined by:

R ϕ ′ ϕ ′ (r z ) = N z/2 k ϕ k ϕ k+kr H k r = 0, 1..., N z /2 (3.1)
where r z = k r ∆ z , ϕ ′ represents the fluctuations of flow variables, and the angled brackets . H represent an averaging operation over homogeneous directions.

As shown in Fig. (3.4), the auto-correlations distributions decrease to zero, which means that two points of the domain separated by L z /2 are completely independent, and thus the domain is wide enough so that the principal turbulence mechanisms are not inhibited. This confirms the previous observation made on the streaks development in a x-z wall-parallel plane.

Wall properties

The boundary layer produces a drag on the plate due to the viscous stresses which are developed at the wall. To get an estimate of the velocity gradient near the wall, one can consider the skin-friction coefficient, which is defined as

C f = 2τ w /ρ ∞ u 2 ∞
, where τ w is the local wall-shear stress and u ∞ is the free-stream velocity. The skin-friction coefficient, C f inc , and the Reynolds number, Re θ inc , are calculated using the van-Driest II theory for an adiabatic wall [START_REF] Pirozzoli | Direct numerical simulation and analysis of a spatially evolving supersonic turbulet boundary layer at M=2.25[END_REF]Pirozzoli & Bernardini, 2011), which is found to be a reliable transformation for collapsing data at different Mach regimes:

R u ′ u ′ (r z )/R u ′ u ′ (0) r z /L z (a) 0 0.1 0.2 0.3 0.4 0.5 -0.2 0 0.2 0.4 0.6 0.8 1 R ρ ′ ρ ′ (rz) /R ρ ′ ρ ′ (0) r z /L z (b) 0 0.1 0.2 0.3 0.4 0.5 -0.2 0 0.2 0.4 0.6 0.8 1 R T ′ T ′ (r z )/R T ′ T ′ (0) r z /L z (c) 0 0.1 0.2 0.3 0.4 0.5 -0.2 0 0.2 0.4 0.6 0.8 1 (d) R u ′ u ′ (r z )/R u ′ u ′ (0) r z /L z 0 0.1 0.2 0.3 0.4 0.5 -0.2 0 0.2 0.4 0.6 0.8 1 (e) R ρ ′ ρ ′ (rz) /R ρ ′ ρ ′ (0) r z /L z 0 0.1 0.2 0.3 0.4 0.5 -0.2 0 0.2 0.4 0.6 0.8 1 (f) R T ′ T ′ (r z )/R T ′ T ′ (0) r z /L z
C f inc = F c C f and Re θ inc = F θ Re θ (3.2)
where the transformation functions F c and F θ are:

F c = T w /T ∞ -1 arcsin 2 α and F θ = µ ∞ µ w (3.3)
where

α = T w /T ∞ -1 T w /T ∞ (T w /T ∞ -1) (3.4)
The incompressible skin-friction coefficient correlations used in Fig. (3.5) are given by:

C f inc, B = 0.026Re -1/4 θ inc C f inc, S = 0.024Re -1/4 θ inc C f inc, KS = 1 17
.08 (log 10 Re θ inc ) 2 + 25.11 (log 10 Re θ inc ) + 6.012 (3.5) where the subscripts S, B and KS refer to the prediction laws of [START_REF] Smits | Low-Reynolds number turbulent boundary layers in zero and favorable pressure gradients[END_REF] [START_REF] Smits | Low-Reynolds number turbulent boundary layers in zero and favorable pressure gradients[END_REF], Blasius and Kármán-Schoenherr [START_REF] Hopkins | An evaluation of theories for predicting turbulent skin friction and heat transfer on flat plates at supersonic and hypersonic Mach numbers[END_REF] [START_REF] Hopkins | An evaluation of theories for predicting turbulent skin friction and heat transfer on flat plates at supersonic and hypersonic Mach numbers[END_REF], respectively. Fig. (3.5) shows that the skin-friction coefficient, C f inc , is overall well reproduced by the present simulations. For the DSM model, C f inc lies within the analytical Blasius' and Kármán Schoenherr's curves, while it is over-estimated by a maximum of 7% by the CSM model. The same trend is also observed for the different grid resolutions, where only the LES-L1 grid predicted a value of C f inc that lies within the analytical curves, and all other cases over-predicted the skin-friction coefficient, the difference reaches a maximum of 18% for LES-R2 case. Note that the latter case contains two times less points in y and z directions than the reference case LES-P2. The distributions show over-prediction of p rmsw /τ w compared to the formula by Farabee & Casarella (1991) [START_REF] Farabee | Spectral features of wall pressure fluctuations beneath turbulent boundary layers[END_REF], where it is estimated by 6% for the DSM model, 13% for the WALE model, 22% for the CSM and 26% for the Implicit LES.

First-and second-order statistics

In supersonic turbulent flows, compressibility effects are assumed to be weak for values of turbulent Mach number, M t , that does not exceed 0.3 [START_REF] Smits | Turbulent shear layers in supersonic flow[END_REF] [START_REF] Smits | Turbulent shear layers in supersonic flow[END_REF]. We also note that coarsening the grid seems to weakly affect the quantities (see Fig. The mean total temperature T t is defined as:

T t = T + 1 2 γ -1 Rγ u i 2 + u ′ i u ′ i (3.8)
where R is the specific gas constant, taken equal to 287.031 J.Kg -1 K -1 .

Since the mass flux and the total enthalpy are functions of y/δ only, integrating the total enthalpy equation across the boundary layer is independent of the streamwise location for adiabatic flows, which yields to:

∞ 0 ρu ρ ∞ u ∞ 1 - H t H t∞ dy = 0 (3.9)
Since the mass flux is always positive and H t /H t∞ < 1 near the wall, there must be a region where

H t /H t∞ > 1,
that is, there must be an overshoot in the total temperature distribution. In adiabatic flows, this overshoot is small so that it is difficult to measure it experimentally (less than 1% in supersonic turbulent boundary layer at M = 2.3 and Re θ = 5500 (Smits & Dussauge, 1996) [START_REF] Smits | Turbulent shear layers in supersonic flow[END_REF]. overshoot is expected to increase with positive heat flux, namely heated wall [START_REF] Smits | Turbulent shear layers in supersonic flow[END_REF] [START_REF] Smits | Turbulent shear layers in supersonic flow[END_REF].

The van-Driest transformed mean streamwise velocity u + vd , which accounts for the variation of the mean flow properties aiming to collapse a compressible velocity profile with its incompressible counterpart, is defined as:

u + vd = u + 0 ρ ρ w d u + (3.10)
The distribution of the van-Driest transformed mean streamwise velocity u + vd as a function of y + is reported in Fig. (3.10). Except a very good estimation of u + vd by the DSM model, the plot shows a slight under-estimation of u + vd in the Log-region compared to the DNS data, for the remaining SGS models (Fig. 3. 10-a). In the viscous sub-layer and up to the buffer region (y + < 30), all models collapse well with the DNS. As expected for y + < 5, the velocity evolves linearly with regards to y + . As said before, a weak under-estimation of u + vd by the WALE and the CSM models (5% and 10%, respectively) in the logarithmic region (C + 1/κ log y + , with C = 5.2 and κ = 0.41) is observed in the region 30 < y + < 100, even if the slope is well reproduced, and all models exhibit an under-estimation of u + vd in the wake region and in the outer part of the boundary layer. In Fig. (3.10-b), u + vd shows sensitivity to grid resolution, and the noted under-estimation decreases as the grid is refined. The spanwise resolution is found to influence more this under-estimation than the streamwise and the wall-normal resolutions. in the region 50 < y + < 300, are compared to:

-The curve fitting of DNS data [START_REF] Pirozzoli | Direct numerical simulation and analysis of a spatially evolving supersonic turbulet boundary layer at M=2.25[END_REF] [76] at Re δ 2 ≈ 2400, written:

ρ ρ w u ′2 u 2 τ ≈ 1.500 -1.086 log(y/δ) -v(y + ) ρ ρ w v ′2 u 2 τ ≈ 1.526 -v(y + ) ρ ρ w w ′2 u 2 τ ≈ 1.243 -0.510 log(y/δ) -v(y + ) (3.11)
-The subsonic experimental results by Perry & Li (1990) [START_REF] Perry | Experimental support for the attached-eddy hypothesis in zeropressure gradient turbulent boundary layers[END_REF] at Re δ 2 = 5000:

u ′ 2 i u 2 τ = B i -A i log(y/δ) -v(y + ) (3.12)
where B 1 = 2.39, A 1 = 1.03, B 2 = 1.6, A 2 = 0, B 3 = 1.20, A 3 = 0.475. In both Eq. (3.11) and

(3.12), v(y + ) represents a correction of deviation from the logarithmic profile due to viscous action in the turbulent wall-region, that increases with y + , which is given by: v(y + ) = 5.58y + -1/2 -22.4y + -1 + 22.0y + -5/4 -5.62y

+ -2 + 1.27y + -11/4 (3.13)
It can be seen from Fig. (3.12-e; 3.12-f) that even if the streamwise and the spanwise components show an overall weak under-estimation in the considered region (50 < y + < 300), the slope is well reproduced for almost all considered cases. The wall-normal component collapses well with the curve-fitting, with regards to the slope as well as the magnitude. When refining the mesh, u + 2 rms ρ /ρ w shows a small shift towards the reference, while the two other components remain unaffected by the grid refinement. It is also found that the results are more compatible with the DNS data rather than to the subsonic experimental results, which can be due to the Reynolds number effect.

The resolved turbulent kinetic energy, -c). It is found that the peak position of K is overall well retrieved by all SGS models, and is located around y + ≃ 13 for the DSM model and y + ≃ 12 for the remaining models. This location is found to be weakly sensitive to the grid resolution, where it is y + ≈ 14 for the LES-L1 grid, which constitutes a classical result.

K = u ′ i u ′ i /
For incompressible flows, the structure parameter is found to be constant, and lies between 0.14 and 0.17 [START_REF] Smits | Turbulent shear layers in supersonic flow[END_REF] [START_REF] Smits | Turbulent shear layers in supersonic flow[END_REF]. The structure parameter shows constant value ≈ 0.15 by all SGS models at 0.2 < y/δ < 0.8, and seems to be weakly sensitive to the grid resolution since all tested meshes exhibit values that lie between 0.14 and 0.17 (Fig. 3.14). [START_REF] Duan | Direct numerical simulation of hypersonic turbulent boundary layers. Part 3. Effect of Mach number[END_REF] [START_REF] Duan | Direct numerical simulation of hypersonic turbulent boundary layers. Part 3. Effect of Mach number[END_REF] also showed in a DNS study that this parameter lies between 0.14 and 0.16 at 0.1 < y/δ < 0.9 when varying the Mach number from 3 to 5.

The distributions of the r.m.s. vorticity components as a function of y + are plotted in Fig. (3.15), where the i th component of the r.m.s of vorticity is defined as:

Ω i = ω ′2 i 1/2 (3.14)
Both the CSM and the WALE models are seen to give better results compared to the DSM model, while in the near-wall region, the r.m.s. vorticity components are overall well estimated and fit well with the references (y + ≤ 10), they are under-estimated away from the wall, especially for the streamwise component. As expected, the vorticity fluctuations are not isotropic for y + ≥ 30, and only ω ′ 2 ≈ ω ′ 3 away from the wall. the DNS data. However, the relative difference is much lower compared to the WENO scheme's result.

Subgrid scale analysis -On evaluating grid resolution

In LES, the accuracy of the resolved scales highly rely on the mesh size. A locally refined grid usually lead to more resolved turbulent energy but will definitely be more costly in terms of CPU time and memory requirements. The strategy in LES is then to make the best compromise between accuracy and computational cost. The SGS dissipation of a given SGS model may have originated, in different proportions, either from the resolved velocity fluctuations or from the mean-averaged velocity gradients. Before going onto the details of the flow physics, the objective here is first to assess the evaluation of the LES resolution by investigating the contribution of the SGS model to the turbulent energy dissipation.

Near-wall asymptotic behavior of the SGS viscosity

When modeling the subgrid terms, some constraints, physical as well as numerical, have to be set. Of primordial interest is the physical constraint, in a way that the modeled SGS terms must region, while it is well estimated by the CSM and the WALE models. Also, the DSM model seems not vanishing at the wall, although the flow is well resolved at this region (∆y + min ≃ 1). However, the CSM model has a lower amount of µ sgs , that monotonically increases when the grid is coarsen (Fig. 3.18-b). The CSM model is found to be less dissipative than the DSM model, but do preserve the good asymptotic behavior of the flow at the wall. For the DSM model, the combined effect of a weakly high SGS viscosity amount and a wrong asymptotic behavior near the wall can be a reason to its under-estimation of the temperature distribution, especially at the wall.

Ratio of turbulent kinetic energy

The parameter Γ, initially introduced by Pope (2004) [START_REF] Pope | Ten questions concerning the large-eddy simulation of turbulent flows[END_REF] and used by [START_REF] Davidson | Large eddy simulations: How to evaluate resolution[END_REF] [START_REF] Davidson | Large eddy simulations: How to evaluate resolution[END_REF], is defined as:

Γ = K k sgs + K (3.15)
where K is the resolved turbulent kinetic energy, k sgs = τ ii /(2 ρ ) is the modeled part and u ′ i is the i th component of the resolved velocity fluctuation. It represents the ratio of the resolved turbulent kinetic energy K = u ′ i u ′ i /2 to the total kinetic energy k sgs + K . The isotropic part of the stress tensor is modeled according to [START_REF] Yoshizawa | Statistical theory for compressible turbulent shear flows with the application to subgrid modeling[END_REF] [START_REF] Yoshizawa | Statistical theory for compressible turbulent shear flows with the application to subgrid modeling[END_REF], where C I is either retrieved using the dynamic procedure in the DSM model, or set constant for the CSM and the WALE models.

Based on the parameter given by Eq. (3.15), [START_REF] Pope | Ten questions concerning the large-eddy simulation of turbulent flows[END_REF] [START_REF] Pope | Ten questions concerning the large-eddy simulation of turbulent flows[END_REF] suggests that when 80% of the kinetic energy is accounted for, the LES can be considered as well-resolved.

In order to check this criterion, the parameter Γ is plotted as a function of y + , varying the SGS models (Fig. 3.19-a) and the grids resolution (Fig. 3.19-b). At the wall, the DSM model shows higher value of Γ than the CSM model, with 55% of the kinetic energy being resolved for the DSM model against 45% for the CSM model. However, one should remember that in the vicinity of the wall, the viscous effects are dominant, and the SGS model does not have significant effect on the turbulence structure. At the frontier of the viscous sublayer, this tendency is inverted, and up to y + ≃ 100, the CSM model better predicts Γ, until it reaches 1 for both SGS models. Fig. (3.19-b) shows that Γ is sensitive to the grid resolution at the wall, showing a monotone decrease while coarsening the grid, until reaching ≃ 1 at y + ≃ 20 for all grids. Apart from a small difference in a region very near from the wall, Γ reaches 1 quickly. However, because the SGS model is not supposed to act in the viscous sublayer where the grid is refined enough, and based on Pope's assumption, the parameter Γ seems not to be adapted to conclude for the present LES mesh quality.

Subgrid scale dissipation

In order to quantify the contribution of the SGS model to the production/destruction of turbulence, it is interesting to quantify the contribution of the SGS terms to turbulence, such as ε ′ sgs and ε sgs , which are the production/destruction term of the resolved turbulent kinetic energy, K , The total SGS dissipation is then defined by:

ε sgs = ε ′ sgs + ε sgs (3.16)
Following the developments by [START_REF] Davidson | Transport equations in incompressible URANS and LES[END_REF] [START_REF] Davidson | Transport equations in incompressible URANS and LES[END_REF] for an incompressible flow, these terms can be explicitly formulated for the compressible case.

k sgs k res K ε ′ s g s ε s g s 1 2 u i u i 1 2 u ′ i u ′ i 1 2 τ ii = τ ′ * i j ∂ u ′ i ∂ x j = τ * i j ∂ u i ∂ x j

Subgrid scale dissipation due to resolved fluctuations

The term of production/destruction of the resolved turbulent kinetic energy, K , by SGS turbulence, which can also be interpreted as the SGS dissipation term due to the resolved fluctuations, was defined by [START_REF] Davidson | Transport equations in incompressible URANS and LES[END_REF] [START_REF] Davidson | Transport equations in incompressible URANS and LES[END_REF] as:

ε ′ sgs = τ * ij -τ * ij ∂u ′ i ∂x j (3.17)
where τ * ij = τ ij -τ kk δ ij /3 is the modeled SGS stress tensor, and u ′ i is the resolved fluctuations of the i th component of the filtered velocity field. Simplifying the terms between the square-brackets, yields:

τ * ij -τ * ij = τ ′ * ij = -2 µ sgs s * ij -µ sgs s * ij ≃ -2µ sgs s ′ * ij (3.18)
where s * ij = s ij -s kk δ ij /3 is the deviatoric part of the strain-rate tensor computed from the filtered velocity field, and s ′ * ij is the fluctuating tensor of s * ij :

s ij = 1 2 ∂ u i ∂x j + ∂ u j ∂x i s ′ ij = 1 2 ∂u ′ i ∂x j + ∂u ′ j ∂x i s ′ * ij = s ′ ij - 1 3 s ′ kk δ ij (3.19)
Then, it follows:

ε ′ sgs = τ ′ * ij ∂u ′ i ∂x j ≃ -2 µ sgs s ′ * ij s ′ ij + ω ′ ij ≃ -2 µ sgs s ′ * ij s ′ ij (3.20)
where s ′ * ij ω ′ ij is equal to zero, since it is the product of a symmetric and anti-symmetric tensors,

s ′ ij ω ′ ij : ω ij = 1 2 ∂ u i ∂x j - ∂ u j ∂x i s ′ * ij ω ′ ij = s ′ ij ω ′ ij - 1 3 s ′ kk δ ij ω ′ ij = s ′ ij ω ′ ij - 1 3 s ′ kk ω ′ kk (3.21)
It is interesting to quantify the ratio of the SGS dissipation term due to resolved velocity fluctuations in a wall-parallel plane ε ′ sgs x+z , since the streamwise and spanwise components of the SGS dissipation term make an important contribution to the total SGS dissipation [START_REF] Davidson | Large eddy simulations: How to evaluate resolution[END_REF] [START_REF] Davidson | Large eddy simulations: How to evaluate resolution[END_REF]. It is defined as:

ε ′ sgs x+z = -2 µ sgs s ′ * ij s ′ ij x+z = -2 µ sgs s ′ * 11 s ′ 11 + s ′ * 33 s ′ 33 + 2s ′ * 13 s ′ 13 (3.22)

Subgrid scale dissipation due to time-averaged field

The production/destruction term, ε sgs , in the modeled turbulent kinetic energy k sgs , also interpreted as the SGS dissipation due to the time-averaged velocity field, is [START_REF] Davidson | Transport equations in incompressible URANS and LES[END_REF] [START_REF] Davidson | Transport equations in incompressible URANS and LES[END_REF]:

ε sgs = τ * ij ∂ u i ∂x j = -2 µ sgs s * ij ( s ij + ω ij ) ≃ -2 µ sgs s * ij s ij (3.23)
where s * ij ω ij is equal to zero, u i is the time-averaged velocity field of the i th component, and: (2009) [START_REF] Davidson | Large eddy simulations: How to evaluate resolution[END_REF] showed that the higher resolution is, the higher is the ε ′ sgs /ε sgs ratio. that, for the same given grid, the DSM model gives weakly better estimation of the ratio (barely 5%). ε ′ sgs /ε sgs is very weak up to y + ≤ 10 and increases steeply to reach its maximum at y + ≃ 50 for both models. However, it is shown in Fig. (3.21-d) that this ratio tends to monotonically increase when the grid is coarsen. For all the tested grids, the ratio varies between 0.85 and 0.95.

s ij = 1 2 ∂ u i ∂x j + ∂ u j ∂x i ω ij = 1 2 ∂ u i ∂x j - ∂ u j ∂x i s * ij = s ij - 1 3 s kk δ ij (3.
In contract to ε ′ sgs /ε sgs , ε sgs /ε sgs increases when coarsening the grid (Fig. 3.21-e). for the WALE model, in the outer region of the boundary layer. This contribution is also found to be barely affected by the grid resolution. ε ′ sgs x+z /ε ′ sgs seems monotonically decreasing when the grid is coarsen, and the refinement in the spanwise direction is found to be more important.

Subgrid scale activity

The total SGS dissipation of a LES model, ε sgs , is defined as the sum of both the SGS dissipation due to the fluctuating flowfield, ε ′ sgs , and the one due to the mean-averaged flowfield, ε sgs .

The amount of turbulent dissipation is the central quantity used to assess the importance of the SGS model, i.e., to quantify the amount of modeling in a LES compared to a DNS (Geurts & Fröhlich, 2002) [START_REF] Geurts | A framework for predicting acuracy limitations in large-eddy simulation[END_REF]. The SGS activity parameter is defined:

ς = ε sgs ε sgs + ε (3.25)
where ε sgs is the total SGS dissipation, defined as the sum of ε ′ sgs and ε sgs , and ε is the molecular dissipation. It comes that 0 ≤ ς < 1 by definition, with ς = 0 corresponding to DNS and ς = 1 to LES at infinite Reynolds number (Geurts & Fröhlich, 2002) [START_REF] Geurts | A framework for predicting acuracy limitations in large-eddy simulation[END_REF]. -f). In the viscous sublayer and up to y + ≃ 5, SGS activity parameter ς is about 0.8 for all SGS models, and seems slightly over-predicted by the DSM model, compared to CSM and WALE models, reaching a maximum of 0.84 at the wall. In this region, ς seems weakly affected by the grid resolution, and all studied grids predict a value of 0.8 at the wall. In the transition region and up to y + ≃ 300, ς decreases otherwise for different SGS models, to reach a minimum of 0.5 for DSM for y + ≃ 200, 0.4 for the WALE model and 0.4 for the CSM model at y + ≃ 300. Same behavior is also observed for the different grids, where ς seems unaffected by the grid resolution. ς increases again in the outer region of the boundary layer, reaching a maximum of about 0.6 for the DSM model, 0.6 for the WALE model and 0.4 for the CSM model, to decrease again at the edge of the boundary layer. In this region, ς shows a monotone increase when coarsening the grid, with a maximum of 0.2 for LES-L1 grid to 0.5 for LES-S2 using the CSM model. (d-e-f) Grid sensitivity study using CSM model. For legend, see table 3 by: A direct comparison of µ sgs and µ can also be interesting. [START_REF] Davidson | Transport equations in incompressible URANS and LES[END_REF] [START_REF] Davidson | Transport equations in incompressible URANS and LES[END_REF] suggested that, if µ sgs ≫ µ, the SGS dissipation is much larger than the viscous one, and if this is not the case, the grid is more likely to be a DNS grid. At y + > 10 (Fig. 3.23), the ratio of ε sgs /ε ≤ 1 in a wide range of the boundary layer, and thus µ sgs ≈ µ, which means that the considered grids are suitable for LES study. where it reaches a maximum of 9% of T ∞ and decreases then up to a level of 2% of T ∞ outside the boundary layer. The same trend is also observed for the r.m.s of density fluctuations, ρ rms , which exhibits a peak of ≈ 4.5% of ρ ∞ in the same region, but continues to increase reaching a level of ≈ 5.5% at y/δ ≃ 0.8, to finally decrease to 2% of ρ ∞ outside the layer. At the wall, the r.m.s of the pressure fluctuations, p rms , reaches a maximum of 3.75% of p ∞ for the Implicit LES, and decreases within the layer reaching 0.5% near the edge of the boundary layer. For the ILES, a bump of T rms and ρ rms is present in the outer region of the boundary layer y/δ > 0.2, probably due to a lack of SGS energy dissipation in this region. All r.m.s quantities show a monotone increase when coarsening the grid. The observed bump was also found to be sensitive to the grid resolution, since it is also observed for the less y-direction refined grid (namely the LES-S2 grid), which can confirm an accumulation of non-dissipated energy in this region of the layer.

ς sgs = ε sgs ε sgs + ε ς ′ sgs = ε ′ sgs ε ′ sgs + ε (3.

Thermodynamic properties and Strong Reynolds Analogy

For an adiabatic supersonic turbulent boundary layer, it is commonly known that u ′ and T ′ are supposed to be perfectly anti-correlated and that the Strong Reynolds Analogy relation, linking the r.m.s of temperature and velocity fluctuations, is nearly equal to 1. By definition, the r.m.s of temperature fluctuations is defined by:

T ′ T ′ = γ -1 Rγ 2 u 2 u ′ u ′ + 2 T ′ T ′ t -T ′ t T ′ t (3.27)
where T ′ t is the total temperature fluctuations.

If it is assumed that the following condition holds (Guarini et al., 2000 [38]; Pirozzoli et al., 2004 [76]): Eq. (3.27) is then written:

T ′ T ′ T 2 ≫ T ′ t T ′ t -2 T ′ T ′ t T 2 (3.
T ′ T ′ 1/2 ≈ γ -1 Rγ u u ′ u ′ 1/2 (3.29)
and the R uT and R uv correlations are written as:

R uT = -1 + T ′ t T ′ t 2 T ′ T ′ (3.30) R uv = -R vT 1 - v ′ T ′ t v ′ T ′ (3.31)
Finally, if the total temperature is supposed uniform and the total temperature fluctuations are neglected (which is not necessary the case as shown in fig. 3.9), the SRA and the velocity-temperature correlation R uT are then written:

SRA = T ′ T ′ / T (γ -1) M 2 ∞ u ′ u ′ / u ≈ 1 R uT = u ′ T ′ u ′ u ′ T ′ T ′ ≈ -1 (3.32)
where M = u / c is the local Mach number.

The SRA is found to be sensitive to the SGS model: for instance, the DSM and the WALE models under-predict the SRA (≈ 0.9, while the Implicit LES over-predict it (≈ 1.1). The CSM model however estimates a value of SRA ≈ 1 in almost all the boundary layer. The SRA is also found to be weakly sensitive to the grid resolution, with a slight increase of the value towards the coarser grid.

Previous reported computations predicted lower values of -R uT , ranging between ≈ 0.55 and 0.8 (Pirozzoli et al., 2004 [76]; Duan et al., 2010 [23]; Pirozzoli & Bernardini, 2011 [74]). In the present simulations, -R uT lays between 0.5 and 0.6 at the outer region of the boundary layer 0.2 < y/δ < 0.8, and rises to 0.8 at y/δ ≃ 0.02, where the models predict almost the same value.

The velocity-temperature correlation -R uT is found to be slightly sensitive to the grid resolution, decreasing monotonically when coarsening the grid. At the vicinity of the boundary layer, -R uT drops rapidly, a feature that was not obvious in previous DNS studies, except in Guarini et al.

(2000) [START_REF] Guarini | Direct numerical simulation of a supersonic turbulent boundary layer at Mach 2.5[END_REF] and Pirozzoli & Bernardini (2011) [74]. This weak anti-correlation between u ′ and T ′ 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 can be due to the non-negligible total-temperature fluctuations within the boundary layer (Pirozzoli et al., 2004 [76]; Pirozzoli & Bernardini, 2011 [74]).

For isothermal supersonic turbulent boundary layers, an extended form of the SRA, which accounts for the heat flux at the wall, is available. The extended form of the SRA, proposed by [START_REF] Cebeci | Analysis of turbulent boundary layers[END_REF] [START_REF] Cebeci | Analysis of turbulent boundary layers[END_REF], is written as:

ESRA = T ′ T ′ / T (γ -1) M 2 ∞ u ′ u ′ / u ≈ -1 + T w -T t∞ T u u ∞ (3.33)
Other forms of SRA, proposed by [START_REF] Gaviglio | Reynolds analogies and experimental study of heat transfer in the supersonic boundary layer[END_REF] [START_REF] Gaviglio | Reynolds analogies and experimental study of heat transfer in the supersonic boundary layer[END_REF] and [START_REF] Huang | Compressible turbulent channel flows: DNS results and modeling[END_REF] [START_REF] Huang | Compressible turbulent channel flows: DNS results and modeling[END_REF], are also available, respectively named GSRA and HSRA. They write:

T ′ T ′ / T (γ -1) M 2 ∞ u ′ u ′ / u ≈ 1 β 1 - ∂ T t ∂ T -1 (3.34)
where β = 1 and β = Pr t , respectively. One should note that Pr t = 0.9 is used to plot this formula.

It is found that the classical SRA correlation better predicts the anti-correlation between u Grid sensitivity study using CSM model. For legend, see table 3.4. the relationship between the density and the temperature fluctuations are as follows:

R ρT = ρ ′ T ′ ρ ′ ρ ′ T ′ T ′ ≈ ρ ′ ρ ′ T ′ T ′ T ρ = -1 (3.35)
In a wide region of the boundary layer (see Fig. 3.27), ρ ′ and T ′ are anti-correlated, and -R ρT weakly exceeds unity (≈ 1.1). The correlation -R ρT is also found to be insensitive to the SGS models as well as to grid resolutions. The -R uv correlation's behavior (Fig. 3.27) is also confident with theoretical observations: constant in the region 0.1 ≤ y/δ ≤ 0.8 and then decrease above this region [START_REF] Spina | The physics of supersonic turbulent boundary layers[END_REF] [START_REF] Spina | The physics of supersonic turbulent boundary layers[END_REF]. At 0.2 < y/δ < 0.8, all models show trends, regardless of the grids (0.45 < -R uv < 0.5). As it has been observed by [START_REF] Pirozzoli | Direct numerical simulation and analysis of a spatially evolving supersonic turbulet boundary layer at M=2.25[END_REF] [START_REF] Pirozzoli | Direct numerical simulation and analysis of a spatially evolving supersonic turbulet boundary layer at M=2.25[END_REF], -R uv and R vT are found to be fairly correlated, and nearly equal to ≈ 0.5 in the outer-region of the boundary layer (0.2 < y/δ < 0.8). This results are also in good agreement with the subsonic experimental data of Klebanoff (-R uv ≈ 0.5). The influence of the mesh and the SGS model on R vT in this region is weak.

The resolved turbulent Prandtl number Pr t is defined as:

Pr t = ρu ′ v ′ ∂ T /∂y ρv ′ T ′ ∂ u /∂y = 1 - ρv ′ T ′ t ρv ′ T ′ 1 - ∂ T t ∂ T -1 (3.36)
Assuming a uniform total temperature in Eq. (3.36) yields to Pr t = 1. Fig. (3.28) shows that this assumption is not satisfied, where in a wide region of the boundary layer y/δ > 0.2, Pr t < 0.8.

Pirozzoli & Bernardini (2011) [74] also found values of Pr t < 0.8 at y/δ > 0.4. This tendency is accentuated when coarsening the grid, where Pr t shows a monotone decrease. write Pr t as:

Pr t ≈ 1 - ∂ T t ∂ T -1 (3.38)
Eq. (3.38) is known as Huang's SRA relation [START_REF] Huang | Compressible turbulent channel flows: DNS results and modeling[END_REF] [START_REF] Huang | Compressible turbulent channel flows: DNS results and modeling[END_REF].

Fig. (3.29) shows that is assumption is also not verified, except for the DSM model at y/δ < 0.3 and the WALE model at y/δ < 0.2, but that the maximum error does not exceed 15% for the CSM model at y/δ < 0.4.

Instantaneous scatter plots

Scatter plots of fluctuating variables display fluctuations extracted each time-step at a given streamwise and wall-parallel position (1 × 1 × 64) of the computational domain. In the following, the analysis is made using the CSM-P2 test-case (see table 3 the following relations:

T ′ T ′ / T (γ -1)M 2 ∞ u ′ u ′ / u ≃ 1, -R uT ≃ 1 (3.39)
Note that Spina et al. (1994) [START_REF] Spina | The physics of supersonic turbulent boundary layers[END_REF] reported that the correlation coefficient -R uT is rather close to 0.8 or 0.9.

However , it is interesting to test the validity of the strict SRA assumption for the instantaneous quantities. For instance, the relation between the instantaneous temperature and velocity fluctuations is given by:

C p T ′ + u u ′ = 0 (3.40)
This can be rewritten as:

T ′ T = -(γ -1) M 2 u ′ u (3.41)
where M = u / c is the local Mach number.

p ′ p = n ρ ′ ρ = n n -1 ρT ′ ρ T (3.42)
It follows that (n -1)ρ ′ / ρ ≈ T ′ / T and that R ρT = -1, where n = 0 is found to be an excellent thermodynamic fluctuations, it comes that:

s ′ C v = -(γ -1) ρ ′ ρ + T ′ T = -γ ρ ′ ρ + p ′ p = γ T ′ T -(γ -1) p ′ p (3.43)
If we neglect the pressure fluctuations in the near-wall region, we can write:

s ′ C v = -γ ρ ′ ρ = γ T ′ T (3.

Turbulence behavior

Anisotropy invariants map

The behavior of turbulent wall-bounded flows can be analyzed by examining the evolution of anisotropy through the turbulent stresses, u ′ i u ′ j , which can be qualified using the anisotropy tensor, defined as

a ij = u ′ i u ′ j -2Kδ ij /3
, where δ ij is the Kronecker tensor. The normalized anisotropy tensor, b ij = a ij /2K, is then simply defined:

b ij = u ′ i u ′ j 2K - 1 3 δ ij (3.45)
The anisotropy tensor has three invariants, the first being simply the trace of the tensor and is zero by definition. Therefore, any turbulent state can be fully characterized by the second and the third invariants, given by:

II = b ij b ji = 1 2 b 2 ii III = b ij b jk b ki = 1 3 b 3 ii (3.46)
In the map shown in According to [START_REF] Lumley | Computational modeling of turbulent flows[END_REF] [START_REF] Lumley | Computational modeling of turbulent flows[END_REF], any realizable quantity associated with the fluctuating field must be within the anisotropy map or on its boundaries

(C 1 C 2 , C 1 C 3 and C 2 C 3
). An analysis of the variation of these points and curves in the anisotropy-invariants map can help to highlight the change of the turbulence state.

As it can be seen from

Skewness and flatness factors

Higher-order moments such as the skewness and the flatness factors of the velocity fluctuations can be analyzed for better understanding the turbulence nature from statistics view-point.

The skewness and flatness coefficients of a given velocity fluctuation are defined as:

S(u ′ i ) = u ′ i 3 u ′ i 2 3/2 F (u ′ i ) = u ′ i 4 u ′ i 2 2 (3.47)
It is worth recalling that the skewness coefficient is a dimensionless quantity that measures the

Case

u

+ rms v + rms w + rms -u ′ v ′ + K + ε + -v ′ T ′ T rms ρ rms p rms Compressible 1 1 1 2 2 0 1 0 0 0 Incompressible 1 2 1 3 2 0 2 0 - 0
Table 3.5: Power indices n of near-adiabatic-wall asymptotic behaviors.

Near-wall asymptotic behavior

By means of the continuity equation and the non-slip wall boundary conditions, Tamano (2002) [START_REF] Tamano | Direct Numerical Simulation of Wall-Bounded Compressible Turbulent Flow[END_REF] and [START_REF] Morinishi | Direct numerical simulation of compressible turbulent channel flow between adiabatic and isothermal walls[END_REF] [66]proposed a comparison of analytical adiabatic and isothermal nearwall asymptotic behaviors for compressible and incompressible turbulent channel flows, expressed as a power of y + .

In the following, we examine the near-adiabatic-wall asymptotic behaviors of the different turbulent quantities, namely the velocity fluctuations, the turbulent kinetic energy, the molecular dissipation and the fluctuations of thermodynamic quantities. The turbulent fluctuations φ ′ of a given quantity φ are expanded in terms of Taylor series of y + as follows:

φ ′ = ξ 1,φ (x, z, t) + ξ 2,φ (x, z, t)y + + ξ 3,φ (x, z, t)y +2 + O(y +3 ) (3.48)
We can observe that the no-slip condition at the wall implies ξ 1 for all velocity fluctuating components are zero. For an incompressible flow, satisfying the continuity equation at the wall additionally yields ξ 2 = 0 for v ′ as ∂v ′ /∂y | w = 0 and thus v rms ∝ y +2 . On the other hand, u rms ∝ y + and w rms ∝ y + . It follows that

K + ∝ y +2 and u ′ v ′ ∝ y +3 .
The temperature and pressure distributions have a non-zero value at the wall, which implies that T rms ∝ y + 0 and p rms ∝ y

+ 0 . Consequently, it yields v ′ T ′ ∝ y + 2 .
For a compressible turbulent flow, the main difference with an incompressible one comes from the density variation with ∂ρ/∂t | w = 0 and ρ rms presents a non-zero value asymptotic behavior, which yields ∂u i /∂x i | w = 0. According to Eq. (3.48), ∂v ′ /∂y | w = 0. It yields u rms , v rms and w rms present asymptotes ∝ y. It follows that K ∝ y 2 , u ′ v ′ ∝ y 2 , and then v ′ T ′ ∝ y 1 .

Table 3.5 summarizes the different power indices n (∝ y + n ) of near-adiabatic-wall asymptotic behavior of the different quantities. Figs.

(3.39) to (3.45) plot the near-wall asymptotic behavior of the velocity fluctuations u + rms , v + rms , w + rms according to the Morkovin's scaling, as well as the Reynolds shear stress u ′ v ′ + , the normalized kinetic energy K + , the turbulent heat flux -v ′ T ′ , the temperature fluctuations T rms and the density fluctuations ρ rms as a function of y + in log-log coordinates. Fig. (3.39; 3.41) show that, at the wall and up to the frontier of the viscous sublayer, u + rms and w + rms vary linearly with decreasing y + . All SGS models exhibit acceptable behavior for the different quantities, and all grid-resolution cases almost fairly compare with the near-wall required asymptotes, except at the wall region (∆y + min ≈ 1), due to the wall-grid LES requirement, where ∆y + min 1.

The difference in indices between the compressible and incompressible flows was mainly observed

for v + rms , -u ′ v ′ + and -v ′ T ′ .
Those asymptotes are plotted in Fig. (3.40;3.42;3.44), and show that, up to the considered wall-region (∆y + 1), the near-wall asymptotic behavior of v + rms , -u ′ v ′ + and v ′ T ′ + = R vT are better estimated using the incompressible indices (∝ y + 2 , y + 3 and y + 2 , respectively), even if the Morkovin's scaling is not used for the Reynolds shear stress. In fact, according to [START_REF] Tamano | Direct Numerical Simulation of Wall-Bounded Compressible Turbulent Flow[END_REF] [START_REF] Tamano | Direct Numerical Simulation of Wall-Bounded Compressible Turbulent Flow[END_REF], the theoretical compressible asymptotes of these quantities hold for the very near-wall region, i.e. at 0.1 ≤ y + ≤ 1. However, since LES modeling requires a minimal wall-resolution ∆y + min 1, this region is not covered by this near-wall asymptotic behavior. Considering the turbulent kinetic energy component K + , it varies linearly with decreasing y + 2 , while the molecular dissipation ε exhibits a non-zero constant behavior near the wall (Fig. 3.43), tendency that was also confirmed by [START_REF] Morinishi | Direct numerical simulation of compressible turbulent channel flow between adiabatic and isothermal walls[END_REF] [START_REF] Morinishi | Direct numerical simulation of compressible turbulent channel flow between adiabatic and isothermal walls[END_REF]. This behavior is found to be unaffected by varying the SGS model or the grid resolution. [START_REF] Morinishi | Direct numerical simulation of compressible turbulent channel flow between adiabatic and isothermal walls[END_REF] [START_REF] Morinishi | Direct numerical simulation of compressible turbulent channel flow between adiabatic and isothermal walls[END_REF] also reported that the near-adiabatic-wall behavior of the thermodynamic quantities T rms , ρ rms and p rms for a compressible flow, have a constant non-zero value asymptote (y + 0 ) with decreasing y + . As it is shown in Fig. (3.45), T rms /T ∞ , ρ rms /ρ ∞ and p rms /p ∞ exhibit an asymptote ∝ y + 0 when decreasing y + . However, for a compressible flow near an isothermal wall, T rms should vary linearly with decreasing y + , while ρ rms and p rms do conserve a constant non-zero value asymptote with the same boundary condition.

Hence, for a near-adiabatic-wall region (1 ≤ y + ≤ 6), all statistics showed overall good asymptotic behavior when compared to their incompressible flow counterparts discussed by [START_REF] Morinishi | Direct numerical simulation of compressible turbulent channel flow between adiabatic and isothermal walls[END_REF] [START_REF] Morinishi | Direct numerical simulation of compressible turbulent channel flow between adiabatic and isothermal walls[END_REF]. u ′ v ′ + and v ′ T ′ + also showed acceptable behaviors compared to the incompressible asymptotes even if the variation in the mean density profile is not taken into account.

Turbulent energy dissipation rate

In homogeneous compressible turbulence with constant viscosity, the turbulent energy dissipation is commonly written as the sum of two components, namely the solenoidal dissipation ε s and the dilatational dissipation ε d . However, in inhomogeneous turbulent flows, an inhomogeneous component of the dissipation, ε I , is also expressed.

By definition, the turbulent energy dissipation ε is defined as:

ε ≡ τ ′ ik ∂u ′ i ∂x k (3.49)
where τ ′ ik is defined as [START_REF] Huang | Compressible turbulent channel flows: DNS results and modeling[END_REF] [START_REF] Huang | Compressible turbulent channel flows: DNS results and modeling[END_REF]:

τ ′ ik = µ ′ ∂u ′ i ∂x k + ∂u ′ k ∂x i - 2 3 µ ′ ∂u ′ l ∂x l δ ik -µ ′ ∂u ′ i ∂x k + ∂u ′ k ∂x i - 2 3 µ ′ ∂u ′ l ∂x l δ ik + µ ∂u ′ i ∂x k + ∂u ′ k ∂x i - 2 3 µ ∂u ′ l ∂x l δ ik + µ ′ ∂ u i ∂x k + ∂ u k ∂x i - 2 3 µ ′ ∂ u l ∂x l δ ik (3.50)
The total energy dissipation can be cast as the sum of three parts ε = ε 1 + ε 2 + ε 3 , where:

ε 1 = µ ∂u ′ i ∂x k ∂u ′ i ∂x k + ∂u ′ k ∂x i - 2 3 µ ∂u ′ i ∂x k ∂u ′ l ∂x l δ ik ε 2 = µ ′ ∂u ′ i ∂x k ∂u ′ i ∂x k + ∂u ′ k ∂x i - 2 3 µ ′ ∂u ′ i ∂x k ∂u ′ l ∂x l δ ik ε 3 = µ ′ ∂ u i ∂x k ∂ u i ∂x k + ∂ u k ∂x i - 2 3 µ ′ ∂u ′ i ∂x k ∂ u l ∂x l (3.51)
The quantity ε 1 = ε s + ε d + ε I is also expressed as the sum of three contributions, namely, the solenoidal dissipation, ε s , the dilatational dissipation , ε d , and the inhomogeneous dissipation, ε I , written:

ε s = 2 µ ω ′ ij ω ′ ij ε d = 4 3 µ ∂u ′ l ∂x l ∂u ′ k ∂x k ε I = 2 µ ∂ 2 u ′ i u ′ j ∂x i ∂x j -2 ∂ ∂x i u ′ i ∂u ′ j ∂x j (3.52)
Note that in our case, ε s is simply deduced from

ε s = ε 1 -ε d -ε I . 0 0.2 0.4 0.6 0.8 1 1.2 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 y/δ ε 1 , ε 2 , ε 3 ε 1 /ε ε 2 /ε ε 3 /ε (a) 0 0.2 0.4 0.6 0.8 1 1.2 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 y/δ ε s , ε d , ε I ε s /ε ε d /ε ε I /ε (b)
Figure 3.46: Ratios of the turbulent energy dissipation rate terms as a function of y/δ. Although, this result is in good concordance with the classical data of homogeneous shear flows [START_REF] Sarkar | Compressible homogeneous shear: simulation and modeling[END_REF] [81], a difference exists compared to the DNS data of [START_REF] Huang | Compressible turbulent channel flows: DNS results and modeling[END_REF] [START_REF] Huang | Compressible turbulent channel flows: DNS results and modeling[END_REF],

which shows a level of 3% in a turbulent channel flow. Fig. (3.47-b) shows the ratio ε d /ε s as a function of the turbulent Mach number. As found by [START_REF] Huang | Compressible turbulent channel flows: DNS results and modeling[END_REF] [START_REF] Huang | Compressible turbulent channel flows: DNS results and modeling[END_REF], the relationship between ε d /ε s and M t is not linear. Furthermore, this result shows that the Sarkar's dilatational dissipation model formulated for the problem of compressible shear layers growth rate is note applicable in our case.

Conclusion

In this chapter, large-eddy simulations of an adiabatic supersonic turbulent boundary layer at Mach experimental data.

For the three considered SGS models, the main statistical quantities show an overall acceptable agreement. As a first observation, the CSM and the WALE models show better results compared to the DSM. In fact, both models give the correct rate of decay as the wall is approached, which ensure that the SGS viscosity vanishes properly within the viscous sublayer. The results show that the DSM predicts the main statistical quantities with better accuracy, especially when estimating the peaks of the velocity fluctuations and the r.m.s. of the wall-pressure distribution. However, it shows less confidence in predicting the mean temperature profile especially at the wall, where the WALE and the CSM models perform better. The temperature and the velocity fluctuations are not perfectly anti-correlated, where R uT is in a good agreement with the DNS, and attains 0.55 in a wide range of the boundary layer. Furthermore, it is found that the correlations linking the thermodynamic quantities derived from the SRA, such as R ρT and R uv , are in acceptable agreements compared to the theoretical references and DNS results. The use of an Implicit LES shows tendencies that do not vary much compared to the explicit LES models, but it seems that the lack of SGS dissipation affected some thermodynamic quantities, such as T rms and ρ rms , where a bump in the outer region of the boundary layer is observed. This bump can express an accumulation of non-dissipated energy that moved away from the wall.

By discussing the different components of the subgrid scale dissipation, an attempt to set a criterion to evaluate the quality of LES grid resolutions have been made. By analyzing the different components of the turbulent energy dissipation rate, the present LES shows confidence to correctly predict the dissipation rate. In fact, it is found that the dissipation is mainly solenoidal throughout the boundary layer, which is a classical finding for the considered case. As expected for adiabatic flows, the inhomogeneous part is negligible, due to the weak value of µ ′ . Also, the dilatational dissipation ε d does not exceed 10% of the solenoidal component ε s .

Finally, it is worth mentioning that the CSM and the WALE models show similar computational performance in terms of CPU resources, using almost 1/4 less time than the DSM model.

Even though the WALE and CSM models give very similar results for most of the discussed quantities, it is considered that the CSM model will be used for the study of the isothermal supersonic turbulent boundary layer, due to its better performance for predicting the wall-temperature.

In following, only the CSM model will be used for the simulation of a cold-wall supersonic turbulent boundary layer.

Case

Re

Re τ = ρ w u τ δ/µ w ; Re θ = ρ ∞ u ∞ θ/µ ∞ ; C f = 2τ w /ρ ∞ u 2 ∞ ; H = δ * /θ; M τ = u τ /(γRT w ) 1/2 .

Wall properties

Table 4.1 compares the boundary layer properties with the DNS data.

Since T w is imposed, the computed ratio T w /T ∞ is found to be high compared to the adiabatic case and the DNS data (where a nearly-adiabatic wall condition is also used). This increase in T w /T ∞ is due to a decrease in the freestream temperature T ∞ at the given station. If normalized with the inlet freestream temperature, T w /T ∞ in ≈ 1.712, which is in very good agreement with the reference data. p rmsw /τ w is slightly over-predicted, it seems that the value moves towards the analytical and the DNS references data, probably due to an increase in the estimation of the friction velocity.

Near-wall asymptotic behavior

According to [START_REF] Tamano | Direct Numerical Simulation of Wall-Bounded Compressible Turbulent Flow[END_REF] [START_REF] Tamano | Direct Numerical Simulation of Wall-Bounded Compressible Turbulent Flow[END_REF] and [START_REF] Morinishi | Direct numerical simulation of compressible turbulent channel flow between adiabatic and isothermal walls[END_REF] [START_REF] Morinishi | Direct numerical simulation of compressible turbulent channel flow between adiabatic and isothermal walls[END_REF], the main difference between the near-wall asymptote of an adiabatic flow and that of an isothermal flow, is the T rms asymptote, which is ∝ y + for the isothermal case. As a consequence, the index n of the near-wall heat flux asymptote R vT is supposed to increase. Since the near-adiabatic-wall asymptote of R v ′ T ′ ∝ y + 2 , then the isothermal-case asymptote should be ∝ y + 3 . Table 4.2 summarizes the different power indices n (∝ y + n ) of useful quantities.

Case

u shows the near-wall asymptotic behaviors of K + and ε + , and shows that both of them exhibit the correct near-wall asymptotes, with ∝ y + 2 and ∝ y + 0 , respectively. It can be concluded that the nearly-adiabatic boundary layer shows overall similar results compared to the adiabatic one. Because of the very weak increase of the wall-temperature due to the wall condition, the Reynolds number is found to weakly decrease, and thus the wall-shear, τ w , to slightly increase. As suspected, the near-wall asymptotic behavior of T rms and R vT is modified compared to the adiabatic case. A small increase in the r.m.s of the temperature as well as the density fluctuations is also observed, while p rms conserves its adiabatic low fluctuating level.

+ rms v + rms w + rms -u ′ v ′ + K + ε + -v ′ T ′ T rms ρ rms p rms Compressible 1 1 1 2 2 0 2 1 0 0 Incompressible 1 2 1 3 2 0 3 1 - 0
Finally, the near-adiabatic wall condition is found to weakly improve the SRA relation.

LES of cold-wall supersonic turbulent boundary layer

In this section, the cold-wall supersonic turbulent boundary layer case is simulated. The considered domain has the same dimensions as the adiabatic case, i.e., L x ×L y ×L z = 106δ in ×9.13δ in ×3.18δ in in the streamwise x, wall-normal y and spanwise z directions. Here δ in , denotes the boundary layer thickness at the inflow plane, and is taken equal to its adiabatic counterpart. To discretize the physical domain, N x × N y × N z = 1536 × 180 × 120 grid points are used. The corresponding mesh is uniformly spaced in the streamwise and spanwise directions, while it is clustered in the wallnormal direction (β = 6.50 ∆y + min ≃ 0.49 and ∆z + = 13.1, respectively in the streamwise, the wall-normal and the spanwise directions.

At the inlet, the Reynolds numbers characterizing the flow are Re τ ≃ 498 and Re θ ≃ 1275.

The imposed wall-temperature is T w /T aw = 0.5, which gives T w /T ∞ ≈ 0.85. To prescribe fully turbulent inflow boundary condition, the recycling/rescaling procedure is used, where the velocity profile is defined using Eq. (2.65), while the temperature and the density profiles are defined by Eq. (2.66), according to the new wall-condition. As for the previous computations, the flowfield is initialized using the digital filter procedure using the isothermal mean velocity, density and temperature profiles. After sweeping the initial transient flow (≈ 100τ c ), statistics are sampled each time step for a period covering only 30τ c , where τ c = δ in /u ∞ . The simulation was performed over about 288 hours using 256 processors, for a total of 73720 CPU hours.

Flow properties

In order to study the unique effect of the wall-temperature on the flow dynamics, it is important to match a given Reynolds number for the different cases, namely the adiabatic and the coldwall boundary layer cases. Based on the Fernholz and Finley (1980) [START_REF] Fernholtz | A critical commentary on mean flow data for two-dimensional compressible boundary layers[END_REF] classification criterion, [START_REF] Duan | Direct numerical simulation of hypersonic turbulent boundary layers. Part 2. Effect of wall temperature[END_REF] [START_REF] Duan | Direct numerical simulation of hypersonic turbulent boundary layers. Part 2. Effect of wall temperature[END_REF] suggested to match a given Reynolds number, Re δ 2 , while varying the walltemperature (Re δ 2 = Re θ µ ∞ /µ w ). They also reported that the same conclusions are drawn by matching Re τ or Re θ . Since the adiabatic boundary layer analysis was based on matching Re τ for comparing the DNS and the LES data, it was agreed in the present study that the comparison of the adiabatic case with the present cold-wall case will be made by matching Re τ . However, in the second-half of the domain, the Reynolds number based on the friction velocity, Re τ , ranges from

≈
β q = q w /C p ρ w u τ T w ; T τ = q w /ρ w C p u τ .
Failing to match Re τ or Re δ 2 in the considered part of the domain, the following results are compared to the reference adiabatic case (CSM-P2) with matching Re θ , taken equal to ≈ 2366. where

F c = ϑ M (arcsin α + arcsin β) 2 and F θ = µ ∞ µ w ( 4
ϑ M = 0.2rM 2 ∞ α = 2A 2 -B (4A 2 + B 2 ) 1/2 and β = B (4A 2 + B 2 ) 1/2 A = ϑ M T w /T ∞ 1/2 and B = 1 + ϑ M T w /T ∞ -1 (4.3)
where r is the recovery factor, taken equal to 0.89. Note that according to this definition, [START_REF] Duan | Direct numerical simulation of hypersonic turbulent boundary layers. Part 2. Effect of wall temperature[END_REF]. The same streaky pattern is also visible in the instantaneous temperature fluctuations field (Fig. 4.12-b). However, the correspondence between high-speed streaks and low-value temperature fluctuations is not evident. Instead, a close structural matching between high-speed velocity fluctuations and high-value temperature fluctuations is formed in this region, which can be a footprint of a strong correlation between u ′ and T ′ in the very near-wall region.

Re θ inc = Re δ 2 .

Flow organization

Noting the number of the resolved streaks in the spanwise direction, it can be concluded that the adopted length, L z ≈ 3δ in , is suitable to fully resolve the most energetic turbulent scales.

Near-wall asymptotic behavior

The near-wall asymptotic behavior of the cold-wall case will follow the same asymptotes as the nearly-adiabatic-wall boundary layer, as reported in table 4.2. Fig. (4.13) shows the near-cold-wall asymptotic behaviors of T rms /T ∞ and R vT compared to the adiabatic case. It is found that the asymptote of T rms /T ∞ weakly agree with the theoretical tendency ∝ y + , while R vT agree well with y + 3 decay.

Turbulence statistics

The anisotropy invariants map of the cold-wall boundary layer, presented in Fig. decrease compared to the adiabatic case, with a constant value of about 0.14 in a wide region of the layer (0.1 ≤ y/δ ≤ 0.8).

It is known that the anisotropy coefficient v 2 rms /u 2 rms decreases when moving towards the wall. A decrease in the ratio means that the wall seems to force the turbulence in favor of the streamwise direction compared to the wall-normal direction [START_REF] Deleuze | Structure d'une couche limite turbulente soumise à une onde de choc incidente[END_REF] [START_REF] Deleuze | Structure d'une couche limite turbulente soumise à une onde de choc incidente[END_REF]. Figs. (4.15-a; 4.15-b) show respectively the anisotropy ratios v 2 rms /u 2 rms and w 2 rms /u 2 rms as a function of y + . Both quantities are found to decrease throughout the layer when the wall is cooled, where in the very near-wall region, the relative difference is found to be more significant for w 2 rms /u 2 rms . This indicates a decrease of the spanwise and the wall-normal intensity fluctuations compared to the streamwise one when cooling the wall. The same anisotropic behavior was also observed by [START_REF] Duan | Direct numerical simulation of hypersonic turbulent boundary layers. Part 2. Effect of wall temperature[END_REF] [START_REF] Duan | Direct numerical simulation of hypersonic turbulent boundary layers. Part 2. Effect of wall temperature[END_REF], who concluded that, as a result to this decrease, a less mixing in the wall-normal and spanwise directions occurs, which yields a more elongated streaks compared to the adiabatic case, as observed in Fig. The turbulent Mach number for the isothermal boundary layer is shown in Fig. (4.16). M t reaches a peak of 0.372 at y/δ ≈ 0.026, and is slightly higher than 0.3 up to y/δ ≈ 0.38. However, even if M t > 0.3 in the inner region of the layer, the compressibility effects are suspected to be small, and can be taken into account via the variation of the mean density. [START_REF] Walz | Boundary layers of flow and temperature[END_REF] [START_REF] Walz | Boundary layers of flow and temperature[END_REF] equation is defined: with the adiabatic supersonic flow data, which where found to be in a good agreement with the incompressible data of Klebanoff (M ≃ 0). The weak over-prediction by LES of the wall-normal component is known to be due to an under-estimation of this value by experiments (Pirozzoli & Bernardini, 2011) [74].

T T ∞ = T w T ∞ + T r -T w T ∞ u u ∞ + T ∞ -T r T ∞ u 2 u 2 ∞ ( 4
The van-Driest transformed mean streamwise velocity u + vd is shown as a function of y + (Fig. 4. 19-a). The cold-wall case shows a very good agreement throughout the layer, where it fits properly with the incompressible logarithmic law.

For isothermal boundary layers, it is possible to express the temperature profile T /T w in a logarithmic form using a turbulent total temperature T i , as defined by [START_REF] Debiève | Supersonic turbulent boundary layer subjected to step changes in wall temperature[END_REF] [START_REF] Debiève | Supersonic turbulent boundary layer subjected to step changes in wall temperature[END_REF]: where Pr sgs = 0.9. This leads to:

T i = T + Pr m 2C p u 2 (
T i T w = 1 - Pr m T τ T w u + (4.7)
where T τ is the friction temperature defined using the wall-heat flux q w , respectively written:

T τ = q w /ρ w C p u τ , q w = -λ w ∂ T /∂y| w (4.8)
Finally, integrating through the layer, the transformed total temperature distribution gives:

T t + = T + i 0 T w T d T i + = C T + 1 κ log y + (4.9)
where T i + = -T i /(T τ Pr m ). ). In the inner layer, T trms reaches 6% of T t∞ for the wall-cooling case, while it is less that 4% for the adiabatic case. In the outer region, however, both cases exhibit nearly equal levels. When normalizing by T t , the peak of T trms is found to increase and reach ≈ 9% of T t .

The correlation between the velocity and the temperature fluctuations, -R uT , is shown in Fig.

v ′ T ′ /u ∞ T ∞ . v ′ T ′ /u ∞ T ∞ is
u ′ v ′ /u 2
∞ as a function of y/δ. Except a weak decrease in a near-wall region, it is found that the cooling does not have a significant effect on both quantities throughout the layer, where -R uv exhibits its typical adiabatic-wall value (≃ 0.45). [START_REF] Audiffren | Turbulence d'une couche limite soumise à une variation de densité due à une onde de choc ou un chauffage pariétal[END_REF] [1] showed that the wall heating has a weak influence on u ′ v ′ , which seems also to be applicable in the case of wall cooling.

Counter to the adiabatic case, R vT and -R uv are not correlated, at least in the inner region of the boundary layer, which was expected since the total temperature fluctuations are not negligible.

Scatter plots

Scatter plots of the fluctuating variables display fluctuations extracted each time-step at a given streamwise and wall-parallel position (1 × 1 × 120) of the computational domain. where c = 1 by Gaviglio (1987) [START_REF] Gaviglio | Reynolds analogies and experimental study of heat transfer in the supersonic boundary layer[END_REF], c = 1.34 by [START_REF] Rubesin | Extra compressibilty terms for Favre-averaged two-equation models of inhomogeneous turbulent flows[END_REF] [START_REF] Rubesin | Extra compressibilty terms for Favre-averaged two-equation models of inhomogeneous turbulent flows[END_REF] and c = Pr t by Huang et al.

(1995) [START_REF] Huang | Compressible turbulent channel flows: DNS results and modeling[END_REF]. For c = 1, the equation gives ∂ T t /∂ T ≃ 1.95 in a very near-wall region (y + ≃ 9). The total temperature fluctuations is defined as [START_REF] Lechner | Turbulent supersonic channel flow[END_REF] [57]:

T ′ t ≈ T ′ + 1 C p u u ′ + u ′ i u ′ i -u ′ i u ′ i 2 (4.11)
which can be expanded to: To a first order, T ′ t / T t can be written:

T ′ t T t = T T t T ′ T + (γ -1) M 2 u ′ u + u ′ 2 i -u ′ 2
T ′ t T t ≈ (γ -1) M 2 T T t u ′ u (4.13)
Fig. (4.30) confirms that approximating T ′ t / T t as written in Eq. (4.13) does not hold for the near-wall region, and can only be adopted in a region sufficiently far from the wall (y + ≥ 100).

Conclusion

In this chapter, basic properties and flow characteristics of isothermal supersonic turbulent boundary layers were investigated using LES. The nearly-adiabatic boundary layer statistics show overall comparable results with the adiabatic LES data. The only difference concerns the near-wall asymptotic behavior, where T rms and v ′ T ′ admit respectively y + and y + 3 as near-wall asymptotes. For the cold-wall supersonic turbulent boundary layer, first-as well as second-order statistics are analyzed. It is found that the compressibility effects are not enhanced due to the cooling, where the turbulent Mach number does not exceed 0.4 throughout the boundary layer. The velocity fluctuations scaled according to the Morkovin's hypothesis exhibit acceptable agreement with experimental data of an adiabatic supersonic turbulent boundary layer results. As expected, the total temperature fluctuations are found to be higher than that of the adiabatic case. As a consequence, the velocity and the temperature fluctuations are not anti-correlated, as concluded for adiabatic flows.

Instead, a strong relationship between u ′ and T ′ exists in the near-wall region. These tendencies were also verified through instantaneous scatter plots.

Chapter 5 Conclusions & perspectives

This study, conducted within the ATAC 1 research program under the auspices of the French Space Agency 2 , is a part of a large activity dedicated to experimental and numerical analysis of side-loads in rocket nozzles. The focus here is to study the dynamics of a supersonic boundary layer and its effect on heat transfer at solid walls. This configuration represents a basic element towards full characterization of complex unsteady turbulent flows in rocket nozzles.

The current investigation deals with high-fidelity numerical simulations of wall heat transfers in three-dimensional supersonic turbulent boundary layers (TBL). The use of three-dimensional direct-and large-eddy simulations of TBL was shown to well determine the physical mechanisms behind the turbulent transport/diffusion formation and the general properties of the boundary layer, which depends critically on the development of large-scale turbulent structures.

In this study, the equations governing the considered flows are solved using an in-house solver, named CHOC-WAVES, which is a three-dimensional multi-species solver with variable thermodynamic properties running on massively parallel computers with an Immersed Boundary Method, in order to handle complex geometries. The code is based on low-dissipation high-order accurate methods in finite difference formulation, typically fourth-or sixth-order in space and third order in time.

Our choice has been directed towards the use of a skew-symmetric split-centered conservative scheme for the convective fluxes. In terms of turbulence modeling, several subgrid scale models have been used to model the action of the small scale structures on the turbulence: the dynamic 1 Aérodynamique des Tuyères et Arrière-Corps. 2 Centre National d' Études Spatiales, CNES Smagorinsky model (DSM), the coherent structures model (CSM) and the WALE model, as well as an Implicit approach, in order to assess the contribution of the subgrid scale terms. For the considered subgrid scale models, the main statistical quantities showed an overall acceptable agreement compared to results available in the literature. Both the CSM and the WALE models showed good near-wall scaling and better prediction of the wall-properties. The use of an Implicit LES highlighted a lack of subgrid scale dissipation that affected some thermodynamic quantities, where a bump in r.m.s of the temperature fluctuations in the outer region of the boundary layer was observed, which could be due to the manifestation of an accumulation of non-dissipated energy in this region. Based on the different subgrid scale dissipation terms, an attempt was made to set a criterion for evaluating the quality of LES grid resolutions.

Based on this analysis, the CSM model was selected for the study of an isothermal supersonic turbulent boundary layer. The obtained results, corroborated by the state-of-the-art literature data, showed that the main conclusions made for wall-bounded turbulent shear flows incompressible adiabatic flows can be adopted for the case of adiabatic as well as isothermal supersonic flows, as long as the variations in the mean flow properties are taken into account, which is the basis of the well-known Morkovin's hypothesis. Strong compressibility effects were found to be negligible, and could be taken into account by density scaling. The temperature as well as the velocity fluctuating fields were found to be weakly correlated, and were linked through the Strong Reynolds Analogy. The second part of the study was dedicated to the analysis of basic properties and flow characteristics of LES of cold-wall supersonic turbulent boundary layer. The velocity fluctuations, which were scaled according to the Morkovin's hypothesis, showed acceptable agreement with the experimental results. As expected, the total temperature fluctuations were found to be higher when compared to the adiabatic case. Furthermore, the velocity and temperature fluctuations were not anti-correlated, since a strong correlation linked u ′ and T ′ in the near-wall region (y + ≃ 9). These tendencies were also verified through instantaneous scatter plots of the fluctuating quantities at different distances from the wall.

Based on the present work, and taking into account the current progress in numerical methods and high-performance computing, the large-eddy simulation is found to be a promising tool for studying such complex phenomena.

As an outlook, to further understand the near-wall dynamics in supersonic bounded flows, some future works can be addressed:

• Comparative study of strongly cooled/heated boundary layers at fixed wall-heat flux. This point will help to clarify the change in the large-scale structures dynamics, and will bring more light into the mechanisms of energy transfer between the different turbulent fluctuating components in case where energy is added/extracted from the fluid. Another important point is to highlight the stabilizing/destabilizing effects of turbulence under strong thermal constraints.

• Study of the dynamics and wall heat-transfer in a wall-jet blowing parallele along a wall in order to be closer to the case of film cooling in supersonic nozzles.

1 Introduction 1 . 1

 111 Context of the study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Heat transfer in supersonic nozzles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Supersonic turbulent boundary layers . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Adiabatic case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Isothermal case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Équations Mathématiques et Schémas Numériques 2.1 Équations mathématiques et modèles de sous-maille . . . . . . . . . . . . . . . . . . 2.1.1 Système d'équations filtrées . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Modélisation de sous-maille . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2.1 Concept de la modélisation fonctionnelle . . . . . . . . . . . . . . . 2.1.2.2 Modélisation du tenseur des contraintes de sous-maille . . . . . . . . 2.1.2.3 Modélisation du flux de chaleur de sous-maille . . . . . . . . . . . . 2.1.2.4 Simulation aux grandes échelles Implicite . . . . . . . . . . . . . . . 2.2 Schémas numériques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Traitement des flux convectifs . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1.1 Schéma WENO (Weighted Essentially Non-Oscillatory) . . . 2.2.1.2 Schéma WENO-BWO (Large Bande Optimisé) . . . . . . . . . .

Figure 1 . 1 :

 11 Figure 1.1: Vulcain II engine, at the P5 test bench in DLR Lampoldshausen, Germany (left); Nozzle geometry with the dump cooling design (right). (Suslov et al., 2010) [96].

Fig. 1 .

 1 Fig. 1.1 shows an example of the Vulcain II engine during a hot gas run on the P5 test bench at the DLR center in Lampoldshausen, Germany. The Vulcain II is a first-stage liquid propellant LOX/H 2 gas generator engine. Its vacuum thrust is about 1350 kN, with a mass flow of 320 kg/s and a specific vaccum impulse ISP of 433 s. The upper part of the structure features Liquid

Figure 1 . 2 :

 12 Figure 1.2: General diagram of a turbulent boundary layer on a flat plate(Schlichting, 1979 [82]).

Debiève

  et al. (1997) [START_REF] Debiève | Supersonic turbulent boundary layer subjected to step changes in wall temperature[END_REF] investigated the effect of a step change in the wall-temperature on a turbulent boundary layer in a supersonic flow at Mach number of 2.3 with a wall-to-recovery temperature ratio T w /T r taken as 1.0, 1.5 and 2.0. The aim of their study was to describe the development of the boundary-layer flow to the new condition at the wall. A deviation from Crocco's law was observed in the outer part of the boundary layer, and the temperature field seems to have had a little effect on the velocity field in the same region. It was also reported that the correlation between the velocity and the temperature fluctuations appears unaffected by the wall-heating.On the other hand, numerical studies dealing with supersonic wall-bounded flows involving isothermal walls mainly focused on studying canonical problems, such as turbulent channel flows and flat-plate boundary layers.[START_REF] Coleman | A numerical study of turbulent supersonic isothermal wall channel flow[END_REF] [START_REF] Coleman | A numerical study of turbulent supersonic isothermal wall channel flow[END_REF] simulated a compressible turbulent flow, at Mach numbers of 1.5 and 3 and Reynolds number Re θ ≈ 3000, in a plane channel with cold walls using DNS. Results are found to agree well with the incompressible data, when the scaling accounts for the mean property variations. It was also concluded that, even if the assumption, which states that the instantaneous total temperature remains constant, is approximately satisfied, the van-Driest transformation is found to be overall very successful. The isothermal-wall flow is strongly influenced by sharp gradients of the mean density and temperature, to the point that most of ρ ′ and T ′ are the result of solenoidal mixing of density and temperature by the turbulence.[START_REF] Huang | Compressible turbulent channel flows: DNS results and modeling[END_REF] [START_REF] Huang | Compressible turbulent channel flows: DNS results and modeling[END_REF] exploited DNS results of two supersonic fully developed channel flows between very cold isothermal walls developed by[START_REF] Coleman | A numerical study of turbulent supersonic isothermal wall channel flow[END_REF] [START_REF] Coleman | A numerical study of turbulent supersonic isothermal wall channel flow[END_REF]. They concluded that the compressibility effects due to turbulent fluctuations might not be as strong as at a given M t in boundary layers, and that when turbulent heat fluxes are scaled by the mean density variation ρ /ρ w , the profiles collapse onto the corresponding incompressible curves. A new formulation of the SRA, named HSRA, was also proposed and it seemed better accounting for the effect of the isothermal condition on the turbulence. Lechner et al. (2001) [57] studied a Mach 1.5 supersonic turbulent channel flow at Re θ ≈ 3000, where the results were compared to Coleman et al. (1995)[START_REF] Coleman | A numerical study of turbulent supersonic isothermal wall channel flow[END_REF]. It was reported that the density fluctuations were produced by the mean density gradient in the wall-layer and peak there. In the wall-layer, the temperature fluctuations were observed to be perfectly correlated with the density fluctuations. In this specific case of cooling, and by analyzing the scatter plots of the different thermodynamic fluctuations, it was concluded that the temperature fluctuations were conditioned by sweeps and ejections. In fact, sweeps (respectively ejections) carry positive (respectively negative) temperature fluctuations. The total temperature fluctuations reached maximum values of about 10% of the mean total temperature in the wall layer, and were due to temperature, velocity and kinetic energy fluctuations. In the core region, they were reduced by a factor of 3.[START_REF] Morinishi | Direct numerical simulation of compressible turbulent channel flow between adiabatic and isothermal walls[END_REF] [START_REF] Morinishi | Direct numerical simulation of compressible turbulent channel flow between adiabatic and isothermal walls[END_REF] compared two compressible turbulent channel flows at Mach 3 and Re θ ≈ 3000, with cold/cold walls and cold/adiabatic walls. The main objective of the study was to reveal the effects of the cold-wall boundary conditions on the dynamic of the flow. They reported that the semi-local scaling provided a universal profile for the Reynolds shear stress, while no universal scaling was found for the turbulent heat flux. It was reported that the Morkovin's hypothesis was not applicable to the asymptotic behavior in a region very near from the wall. Foysi et al. (2004) [31] studied compressibility effects and turbulence scaling in a supersonic channel flow with cold isothermal walls using DNS, at a Mach number up to M ≈ 3.5 and Re τ ≈ 1030. When comparing their results to incompressible data, it was observed that the inner scaling with y + = yu τ /ν w fails. The use of the semi-local scaling y * = y/δ * ν (with δ * ν = ν /u * τ and u * τ = τ w / ρ , τ w is the wall-friction) used by Huang et al. (1995)

  the local scaling u local τ = u τ ρ /ρ w and y + = yu local τ / ν (where ρ and ν are the local properties), it was observed that the magnitude and the location of the maximum u * rms = u rms /u local τ agreed well with the incompressible data. The streamwise component of the velocity fluctuations,

Figure 2 . 1 :

 21 Figure 2.1: Spectre d'énergie typique dans un écoulement turbulent homogène.

Figure 2 . 2 :

 22 Figure 2.2: Spectre d'énergie de l'approche Dynamique du modèle de Smagorinsky.

2. 2 . 1 . 1

 211 Schéma WENO (Weighted Essentially Non-Oscillatory) Les schémas ENO (Essentially Non-Oscillatory) sont basés sur l'idée de déterminer le flux numérique à partir d'une reconstruction d'ordre élevée sur un seul stencil adaptatif sélectionné pour éviter au maximum l'interpolation à travers les discontinuités. Ces schémas souffrent, néanmoins, de problème de convergence vers une solution stationnaire, ainsi que d'une perte de précision (Pirozzoli, 2010) [72]. Les schémas WENO (Weighted Essentially Non-Oscillatory), qui viennent remédier aux limites des schémas ENO, sont basés sur l'idée d'une construction d'un flux numérique d'ordre élevé à partir d'une combinaison linéaire convexe d'une reconstruction pondérée de polynômes d'ordre plus faible (stencils). La pondération des stencils a pour but de maximiser la précision du schéma dans les régions dites lisses, tout en annulant l'effet du stencil adaptatif près de la discontinuité. Dans le code CHOC-WAVES, plusieurs variantes du schéma WENO sont disponibles, tels que le schéma WENO classique de Jiang & Shu (1996) [45], le schéma WENO à large bande de Martin et al. (2006) [64] ou encore le schéma Mapped-WENO de Henrick et al. (2005) [40]. Ces schémas peuvent être utilisés soit à l'ordre 3 (r = 2) ou à l'ordre 5 (r = 3), la précision du schéma étant en 2r -1.

Figure 2 . 3 :

 23 Figure 2.3: Choix des stencils pour la reconstruction des flux dans le cas du schéma WENO du 5 ème ordre.

Figure 2 . 4 :

 24 Figure 2.4: Choix des stencils pour la reconstruction des flux dans le cas du schéma WENO du 5 ème ordre à Large Bande optimisé.

(

  ρϕ, u) j-m,l (2.55) avec (2.54) qui correspond à la forme (2.41), l'expression (2.55) à celle (2.42), et a l = a 1 = 1/2.

59 )

 59 Les composantes du flux F v contiennent des dérivés du type ∂()/∂x, ∂()/∂y et ∂()/∂z. Dans les directions normale (y) et d'envergure (z) de l'écoulement, le calcul des dérivées est approximé par une formulation centrée. Dans la direction longitudinale (x), le calcul des dérivées se fait à l'aide de différentes formulations d'ordre 4 sur un stencil réduit S = {i -2, i -1, i, i + 1, i + 2}, ce qui donne:

63 )Figure 2 . 5 :

 6325 Figure 2.5: Domaine de calcul et conditions aux limites de la configuration étudiée.

  Méthode de recyclage/renormalisationDans notre étude, la condition d'entrée turbulente est préscrite en utilisant la méthode de recyclage/renormalisation décrite par Pirozzoli et al. (2010)[START_REF] Pirozzoli | Direct numerical simulation of transonic shock/boundary layer interaction under conditions of incipient separation[END_REF], originalement proposée par Lund et al. (1998)[START_REF] Lund | Generation of turbulent inflow data for spatiallydevelopping boundary layer simulations[END_REF]. La méthode consiste à recycler les fluctuations de vitesses et de densité à partir d'une station dite de recyclage notée x rec , et à les introduire à l'entrée du domaine, après une procédure de renormalisation des quantités, afin de prendre en compte l'épaississement de la couche limite.

64 )

 64 Les profils moyens génériques de vitesses et de densité sont introduits à l'entrée du domaine de calcul, et seules les fluctuations sont recyclés à partir de la station x rec . Le fait de figer les profils moyens et de ne recycler que les fluctuations permet un meilleur contrôle des grandeurs caractéristiques de la couche limite à l'entrée du domaine, notamment l'épaisseur de la couche limite δ in et la vitesse de frottement u τ in . Ceci conduit à une meilleure stabilité des calculs et à la réduction du temps de convergence statistique, en comparaison avec la méthode originale de Lund et al. (1998)[START_REF] Lund | Generation of turbulent inflow data for spatiallydevelopping boundary layer simulations[END_REF].

. 69 )

 69 Dans la direction d'envergure, les fluctuations sont recyclées à partir d'une position z + L z /2 (voir Eq. 2.68) afin d'éviter l'introduction de toute information faussée (périodicité du champ injecté) à l'entrée du domaine, pouvant résulter d'une condition de quasi-périodicité (Pirozzoli et al., 2010) [75].

  Afin de minimiser le temps d'établissement d'un régime pleinement turbulent où la couche limite atteint un état d'équilibre, le calcul est initialisé, comme l'a fait auparavant Chaudhuri et al. (2010)[START_REF] Chaudhuri | Numerical study of compressible mixing layers using high-order WENO schemes[END_REF], en utilisant la méthode du filtre digital basée sur la procédure deKlein et al. (2004) [START_REF] Klein | A digital filter based generation of inflow data for spatially developing direct numerical or large eddy simulation[END_REF]. La méthode consiste à imposer, dans tout le domaine, des profils moyens de vitesses, de température et de pression, en y superposant des corrélations statistiques à partir de perturbations réalistes en utilisant les profils de fluctuations de vitesses obtenus via une DNS. Ici, les données DNS deBernardini & Pirozzoli (2011) [START_REF] Bernardini | Wall pressure fluctuations beneath supersonic turbulent boundary layers[END_REF] sont utilisées. Cette méthode présente l'avantage de définir, avant le lancement du calcul, des champs moyen et fluctuant réalistes, qui sont déjà plus ou moins corrélées entre-eux.

Chapter 3 Adiabatic

 3 Supersonic Turbulent Boundary layer This chapter concerns large-eddy simulations of a supersonic turbulent boundary layer evolving over an adiabatic flat-plate. The study focuses mainly on the mean flow as well as statistical characteristics of the boundary layer. First, an assessment of the LES results is presented in order to check the proper convergence of the statistics and the conformity of the structure of the flow as a canonical zero-pressure gradient turbulent boundary layer. The obtained results in terms of mean and fluctuating quantities are compared to the recent DNS data of Pirozzoli & Bernardini

  is the momentum thickness at the inlet). The computational domain has a size of L x × L y × L z = 106 δ in × 9.13 δ in × 3.18 δ in in the streamwise (x), wall-normal (y) and spanwise (z) directions, respectively. One should note that the domain was reduced by a factor of 3 compared to the DNS reference(Bernardini & Pirozzoli, 2011 [9];

Figure 3 . 1 :

 31 Figure 3.1: Computational domain size.

3 . 3

 33 Fig.(3.3-a) shows typical alternating high-and low-speed streaks, which correspond to positive and negative velocity fluctuations, respectively. This alternative behavior is interpreted as the

Figure 3 . 2 :

 32 Figure 3.2: Instantaneous flow field for the LES-L1 grid with the CSM model: (center) Q-criterion isosurface colored with the temperature field [25]; (top) Instantaneous temperature field in a x-y plane; (bottom) Temperature fluctuations field in the x-z plane at y + ≃ 9.

Figure 3 . 3 :

 33 Figure 3.3: Instantaneous (a) velocity and (b) temperature fluctuation fields in the x-z plane at y + ≃ 9, for LES-L1 using the CSM model. Contour levels are shown for (a) -0.2 ≤ u ′ /u ∞ ≤ 0.2 and (b) -0.2 ≤ T ′ /T ∞ ≤ 0.2, from dark to light shades.

Fig. ( 3 . 5 )

 35 Fig. (3.5) shows the incompressible skin-friction distribution, C f inc , as a function of the Reynolds number based on the incompressible momentum thickness, Re θ inc , and is compared to formulas by Smits et al. (1983)[START_REF] Smits | Low-Reynolds number turbulent boundary layers in zero and favorable pressure gradients[END_REF], Blasius and Kármán-Schoenherr[START_REF] Hopkins | An evaluation of theories for predicting turbulent skin friction and heat transfer on flat plates at supersonic and hypersonic Mach numbers[END_REF], as well as to DNS data

Figure 3 . 4 :

 34 Figure 3.4: Instantaneous auto-correlation coefficients at y + ≃ 9. (a-c) SGS models study using LES-P2 grid; (d-f) Grid sensitivity study using CSM model. For legend, see table3.4.

Fig. ( 3 Figure 3 . 5 :

 335 Fig. (3.6) depicts the variation of the normalized r.m.s. wall-pressure fluctuations as a function

Fig. ( 3 Figure 3 . 6 :

 336 Figure 3.6: Normalized r.m.s. wall-pressure distribution p rms w /τ w as a function of Reynolds number Re τ . (--) Farabee & Casarella (1991) [28]; (• -•) Bernardini & Pirozzoli (2011) [9]. (a) SGS models study using LES-P2 grid; (b) Grid sensitivity study using CSM model. For legend, see table 3.4.

  .4.

  Fig. (3.7) shows M t as a function of y/δ. It can be seen that M t doesn't vary much neither with SGS models nor with grid resolutions, and the peak value reaches M max t ≈ 0.325 at y/δ ≃ 0.04, except for LES-R2 grid where the peak of M t attains almost 0.35. The compressibility effects are then supposed to be weak for the considered flow, and only variations of the thermodynamic quantities are supposed to vary across the boundary layer. Distributions of the mean-averaged quantities as a function of y/δ are plotted in Fig. (3.8). The profiles collapse overall well with the DNS data, and the CSM model seems best estimating the averaged quantities. The normalized wall-temperature, T w /T ∞ , is estimated by 1.717 for the DNS, 1.666 for the CSM model, 1.664 for the WALE model and 1.62 for the DSM model. ρ u /ρ ∞ u ∞ is best estimated by the CSM model. The under-estimation of the wall-temperature by the DSM model could originates mainly from a too high damping of the turbulent fluctuations in this region, caused by a wrong prediction of the near wall asymptotic behavior of the SGS eddy viscosity µ sgs .

Figure 3 . 7 :

 37 Figure 3.7: Turbulent Mach number, M t , as a function of y/δ. (a) SGS models study using LES-P2 grid;(b) Grid sensitivity study using CSM model. For legend, see table3.4.

Fig. ( 3 . 9 )Figure 3 . 8 :

 3938 Fig.(3.9) shows the normalized mean total temperature T t /T t∞ . As it can be seen, the total temperature is not constant throughout the layer. The overshoot of the total temperature is retrieved by all models and for different grid resolutions, and does not exceed 2% for all cases. This

Figure 3 . 9 :

 39 Figure 3.9: Distribution of the normalized time-averaged total temperature, T t /T t∞ , as a function of y/δ. • DNS Pirozzoli et al. (2004) [76] at Re θ = 4260. (a) SGS models study using LES-P2 grid; (b) Grid sensitivity study using CSM model. For legend, see table 3.4.

  .

  4.

Figure 3 . 10 :

 310 Figure 3.10: Distribution of the van-Driest transformed mean streamwise velocity u + vd as a function of y + . C = 5.2; κ = 0.41. (a) SGS models study using LES-P2 grid; (b) Grid sensitivity study using CSM model. For legend, see table 3.4.

Fig. ( 3 Figure 3 . 11 :

 3311 Fig. (3.11) depicts the van-Driest transformed mean streamwise velocity deficit as a function of y/δ. All SGS models are found to exhibit overall acceptable results. While the CSM model fits well with the DNS, both the DSM and the WALE models under-estimate the velocity deficit, while the Implicit LES over-estimate it. Figs. (3.12-a; 3.12-b) show the normalized velocity fluctuations as a function of y/δ and Figs.(3.12-c; 3.12-d) the velocity fluctuations in Morkovin's scaling as a function of y + . Except a good fitting in the near-wall region (y/δ < 0.05 and y + < 10), all models present under-estimated magnitudes compared to the DNS, especially for the u + rms component (5% for the DSM, 8% for the CSM and 10% for the WALE model). The location of the streamwise component peak is well estimated by all the models (0.02δ). It is also found that the wall-normal fluctuations, v + rms , as well as the spanwise fluctuations, w + rms , and the Reynolds shear stress u ′ v ′ + are better estimated by the WALE and the CSM model. In Fig. (3.12-b), normalized velocity fluctuations are found to be weakly sensitive to the grid resolution. Fig. (3.12-e; 3.12-f) shows the normalized velocity fluctuations in Morkovin's scaling as a function of y + in Log-scale, where results, at Re δ 2 ≈ 1800

2

 2 for the SGS model study and for the grid sensitivity study is shown in Figs. (3.13-a) and (3.13-b), respectively, as well as the modeled part of the turbulent kinetic energy using the DSM model, k sgs = τ kk /(2 ρ ), in Fig. (3.13

Fig. ( 3 Figure 3 .

 33 Fig.(3.16) shows the different normalized shear stresses as a function of y + . The CSM and the

  .

  4.

Figure 3 . 13 :Figure 3 . 14 :

 313314 Figure 3.13: Normalized turbulent kinetic energy K/u 2 τ as a function of y/δ. (a) SGS models study using LES-P2 grid; (b) Grid sensitivity study using CSM model; (c) Resolved turbulent kinetic energy K + and SGS turbulent kinetic energy k +sgs for the LES-P2 grid using the DSM model as a function of y + . For legend, see table3.4.

Figure 3 . 15 :

 315 Figure 3.15: Distribution of r.m.s. vorticity components as a function of y + . (a) SGS models study using LES-P2 grid; (b) Grid sensitivity study using CSM model. For legend, see table 3.4.

  .

  4.

Figure 3 . 16 :

 316 Figure 3.16: Turbulent shear stress -ρu ′ v ′ , mean viscous shear stress µ ∂ u /∂y and total shear stress -ρu ′ v ′ + µ ∂ u /∂y as a function of y + . (a) SGS models study using LES-P2 grid; (b) Grid sensitivity study using CSM model. For legend, see table 3.4.

Fig. ( 3 Figure 3 .

 33 Fig.(3.17) depicts a comparison between the WENO-BWO scheme, the 4 th order and the 6 th order split-centered schemes, through the van-Driest transformed mean streamwise velocity, u + vd , and the velocity fluctuations in Morkovin's scaling, as a function of y + . Except in the viscous sublayer, u + vd exhibits an over-estimation compared to the DNS data, which reaches a maximum of about 25% in the Log-region. The streamwise velocity fluctuations shows also the same tendency, where the peak is over-predicted by about 25% compared to the reference data. The wall-normal and the spanwise fluctuations components are however under-estimated. The observed trends are the footprint of an excessive numerical dissipation due to the WENO scheme. The 4 th and the 6 th order split-centered schemes show similar results, which are slightly under-estimated compared to

Figure 3 . 18 :

 318 Figure3.18: Normalized SGS viscosity, µ sgs /µ, as a function of y + . (a) SGS models study using LES-P2 grid; (b) Grid sensitivity study using CSM model. For legend, see table3.4.

Figure 3 . 19 :

 319 Figure 3.19: Ratio of the turbulent kinetic energy Γ as a function of y + . (a) SGS models study using LES-P2 grid; (b) Grid sensitivity study using CSM model. For legend, see table 3.4.

Figure 3 . 20 :

 320 Figure 3.20: Contribution of the different SGS dissipation terms to turbulence.

24 )Figs. ( 3 .

 243 Figs. (3.21-a) and (3.21-d) plot the ratio of the SGS dissipation due to the fluctuating flowfield to the total SGS dissipation, respectively varying the SGS model and the grid resolution. Davidson

Fig. ( 3 . 21 Figure 3 .

 3213 Figure 3.21: (a-d) Ratio of the SGS dissipation due to the fluctuating flowfield to the total SGS dissipation ε ′ sgs /ε sgs as a function of y + ; (b-e) Ratio of the SGS dissipation due to the mean-averaged flowfield to the total SGS dissipation ε sgs /ε sgs as a function of y + ; (c-f) Ratio of the SGS dissipation due to the fluctuating flowfield in a wall-parallel plane to the total SGS dissipation due to the fluctuating flowfield ε ′ sgs x+z /ε ′ sgs as a function of y + . (a-b-c) SGS models study using LES-P2 grid; (d-e-f) Grid sensitivity study using CSM model. For legend, see table 3.4.

  .4.

Figs. ( 3 .

 3 Figs. (3.21-c) and (3.21-f) shows the ratio of the SGS dissipation due to the fluctuating flowfield in a wall-parallel plane ε ′ sgs x+z to the total SGS dissipation due to the fluctuating flowfield ε ′ sgs , respectively varying the SGS model and the grid resolution. The contribution of the resolved fluctuations in a wall-parallel plane varies between 0.4 for the CSM and the DSM models and 0.5

Fig. ( 3

 3 Fig. (3.22) shows the SGS activity parameter ς as a function of y + , varying the SGS model (Figs. 3.22-a; 3.22-b; 3.22-c) and the grid resolution (Figs. 3.22-d; 3.22-e; 3.22-f). In the viscous sublayer

Fig. ( 3 Figure 3 . 22 :

 3322 Fig. (3.22-b; 3.22-e) show the SGS activity parameter, ς ′ sgs , due to the fluctuating flowfield and Figs. (3.22-c; 3.22-f) the SGS activity parameter ς sgs due to the mean-averaged flowfield, defined

  .4.

Figure 3 . 23 :

 323 Figure 3.23: Ratio of the SGS dissipation to the viscous dissipation as a function of y + . (a) SGS models study using LES-P2 grid; (b) Grid sensitivity study using CSM model. For legend, see table 3.4.

26 )From

 26 Figs. (3.22-b; 3.22-e) and Figs. (3.22-c; 3.22-f), it can be deduced that the SGS dissipation is mainly driven by viscous effects in a near-wall region (up to y + ≃ 10), while the fluctuating flowfield mainly acts in the transition region and up to the edge of the boundary layer (10 ≤ y + ≤ 1000). From Figs. (3.22-a; 3.22-b; 3.22-c), it can be concluded that the DSM model seems more dissipative than both the CSM and the WALE models, especially in the outer region of the boundary layer, and referring to the definition of the SGS activity parameter, ς, the DSM model behaves better than the CSM model, in a way that it is higher for the same grid resolution. For the WALE model, ς ′ sgs is very close to that of the DSM model throughout the layer.

Figs. ( 3 .

 3 Figs. (3.24-a; 3.24-b; 3.24-c) shows the cross-stream distribution of the normalized r.m.s of some thermodynamic quantities with varying SGS models and Figs. (3.24-d; 3.24-e; 3.24-f) with varying the grid resolutions. Overall, the results show similar levels of these quantities when varying the SGS models. The r.m.s of temperature fluctuations, T rms , exhibits a peak near the wall (y/δ ≃ 0.05)

28 ) 1 Figure 3 . 24 :

 281324 Figure 3.24: Normalized r.m.s. of the thermodynamic quantities as a function of y/δ. (a-b-c) SGS models study using LES-P2 grid;(d-e-f) Grid sensitivity study using CSM model. For legend, see table3.4.

  .

  4.

Figure 3 . 25 :

 325 Figure 3.25: SRA and -R uT correlations as a function of y/δ. Circle: Pirozzoli et al. (2004) [76]. (a) SGS models study using LES-P2 grid; (b) Grid sensitivity study using CSM model. For legend, see table 3.4.

Figure 3 . 26 :

 326 Figure 3.26: Modified SRA as a function of y/δ. (a, b, c) SGS models study using LES-P2 grid; (d, e, f)

Pirozzoli 2 ( 3 . 37 )Figure 3 . 27 :Figure 3 . 28 :

 2337327328 Figure 3.27:-R ρT , -R uv and R vT correlations as a function of y/δ. (a, c, e) SGS models study using LES-P2 grid; (b, d, f) Grid sensitivity study using CSM model. For legend, see table3.4.

  .4.

Fig. ( 3 Figure 3 . 30 :

 3330 Fig. (3.30) depicts scatter plots of the normalized temperature -velocity fluctuations. It shows

Figure 3 . 31 :

 331 Figure 3.31: Scatter plots of the strict SRA in a wall-normal plane for the CSM-P2 case. (a) y + ≃ 9; (b) y + ≃ 28; (c) y + ≃ 100.

Figure 3 . 32 :

 332 Figure 3.32: Scatter plots of the normalized temperature -density fluctuations in a wall-normal plane for CSM-P2 case. (a) y + ≃ 9. (b) y + ≃ 28. (c) y + ≃ 100.

Fig. ( 3 . 32 )

 332 Fig.(3.32) shows scatter plots of normalized temperature fluctuations T ′ / T against normalized density fluctuations. At y + ≃ 9, R ρT is found to be less than 1, and the relation (n -1)ρ ′ / ρ ≈ T ′ / T seems better verified as the distance from the wall increases, reaching 0.98 at y + ≃ 100.

Fig. ( 3 . 33 )Figure 3 . 33 :

 333333 Fig. (3.33) shows that the pressure fluctuations are weakly unimportant compared to the density fluctuations, which confirm the validity of the Eq. (3.42) in the current study. Using the entropy relation s = C v log(p/ρ γ ) and assuming a linear relationship between the

44 )

 44 Scatter plots of the entropy fluctuations as a function of the density fluctuations Figs. (3.34-a; 3.34-b; 3.34-c) and pressure fluctuations Figs. (3.34-a; 3.34-b; 3.34-c) in a wall-normal plane are presented. A linear fitting of the values shows that the slope of the (s ′ , ρ ′ ) scatter plot is -1.12 in a very near-wall region (y + ≃ 9), and it increases until reaching -1.25 at y + ≃ 100. The scatter plot relating the entropy fluctuations to the pressure fluctuations (3.34-d; 3.34-e; 3.34-f) shows also that the pressure fluctuations can weakly be neglected, which probably deviates the previous scatter plot from the theoretical slope (∼ -γ).Scatter plots of the total temperature fluctuations as a function of the temperature and velocity fluctuations are presented in Fig.(3.35). It shows that an instantaneous quadratic relationship exists between the total and the static temperature fluctuations. Similar relation is also found with the streamwise velocity component.

Figure 3 . 34 :

 334 Figure 3.34: Scatter plots of (a-c) the normalized entropy-density fluctuations and (d-f) the normalized entropy-pressure fluctuations in a wall-normal plane for CSM-P2 case. (a, d) y + ≃ 9; (b, e) y + ≃ 28; (c, f) y + ≃ 100.

Figure 3 . 35 :

 335 Figure 3.35: Scatter plots of (a-c) the normalized total-temperature -temperature fluctuations and (d-f) the normalized total temperature -velocity fluctuations in a wall-normal plane for the nearly-adiabatic CSM-P2 case. (a, d) y + ≃ 9; (b, e) y + ≃ 28; (c, f) y + ≃ 100.

Fig. ( 3

 3 .36), point C 1 represents perfectly isotropic turbulence, where II = III = 0. Point C 2 is two-components isotropic turbulence limit, where II = 1/12 and III = -1/108, and C 3 is one-component turbulence limit, where II = 1/3 and III = 2/27. The upper boundary C 2 C 3 represents two-dimensional turbulence, while the two lower bounds C 1 C 2 and C 1 C 3 correspond to axisymmetric compression and expansion turbulence, respectively.

Fig. ( 3 2 C 3 Figure 3 . 36 :

 323336 Figure 3.36: Anisotropy invariants map.

Figure 3 . 37 :Figure 3 .

 3373 Figure 3.37: Anisotropy invariants maps. (a) SGS models study using LES-P2 grid; (b) Grid sensitivity study using CSM model. For legend, see table 3.4.

  Figure 3.38: (a, b) Skewness and(c, d) Flatness factors as a function of y + . For legend, see table3.4.

  Figure 3.38: (a, b) Skewness and(c, d) Flatness factors as a function of y + . For legend, see table3.4.

Figure 3 . 39 :

 339 Figure 3.39: Near-wall asymptotic behavior of the streamwise velocity fluctuations u rms in Morkovin's scaling as a function of y + . (a) SGS models study using LES-P2 grid; (b) Grid sensitivity study using CSM model. For legend, see table 3.4.

  .

  4.

Figure 3 .

 3 Figure 3.40: Near-wall asymptotic behavior of the wall-normal velocity fluctuations v rms in Morkovin's scaling as a function of y + . (a) SGS models study using LES-P2 grid; (b) Grid sensitivity study using CSM model. For legend, see table 3.4.

  .

  4.

Figure 3 . 41 :

 341 Figure 3.41: Near-wall asymptotic behavior of the spanwise velocity fluctuations w rms in Morkovin's scaling as a function of y + . (a) SGS models study using LES-P2 grid; (b) Grid sensitivity study using CSM model. For legend, see table 3.4.

  .

  4.

Figure 3 .

 3 Figure 3.42: Near-wall asymptotic behavior of the normalized Reynolds shear stress u ′ v ′ + as a function of y + . (a) SGS models study using LES-P2 grid; (b) Grid sensitivity study using CSM model. For legend, see table 3.4.

  .4.

Figure 3 . 43 :

 343 Figure 3.43: Near-wall asymptotic behavior of (a-b) the normalized kinetic Energy and(c-d) the normalized molecular dissipation as a function of y + . For legend, see table3.4.

  .4.

Figure 3 . 44 :

 344 Figure 3.44: Near-wall asymptotic behavior of the normalized turbulent heat flux, R vT , as a function of y + . (a) SGS models study using LES-P2 grid; (b) Grid sensitivity study using CSM model. For legend, see table 3.4.

Figure 3 . 45 :

 345 Figure 3.45: Near-wall asymptotic behavior of (a) the temperature, (b) the density and (c) the pressure fluctuations as a function of y + . For legend, see table 3.4.

  .4.

Fig. ( 3

 3 Fig. (3.46-a) shows the ratios of ε 1 /ε, ε 2 /ε and ε 3 /ε as a function of y/δ. It is shown that ε 1 dominates the other contributions. This result is found to be a classical finding for an adiabatic turbulent boundary layer, since µ ′ is negligible for such a flow. Fig. (3.46-b) shows the ratios ε s /ε, ε d /ε and ε I /ε as a function of y/δ, and shows that the inhomogeneous part of the dissipation is also negligible.Fig. (3.47-a) shows the ratio of dilatational dissipation to solenoidal dissipation as a function of y/δ. The ratio is found to be constant throughout the boundary layer, reaching a level of 10%.

Figure 3 . 47 :

 347 Figure 3.47: Ratio of the dilatational to the solenoidal dissipation as a function of (a) y/δ and (b) turbulent Mach number M t .

Fig. ( 4

 4 Fig. (4.1-a) shows the skin-friction coefficient, C f inc , as a function of the Reynolds number, Re θ inc . It is found that C f inc gives similar results as the adiabatic case, where it is over-predicted by 6% compared to the Blasius curve, for both the adiabatic and the nearly-adiabatic cases. However, Re θ inc is found to decrease and moves towards the DNS data for the nearly-adiabatic boundary layer, due to the weak increase of the predicted wall-temperature. As seen in Fig (4.1-b), even if

Fig. ( 4 Figure 4 . 1 :

 441 Figure 4.1: (a) Incompressible skin-friction coefficient C finc as a function of Reynolds number based on incompressible momentum thickness Re θinc . (b) Normalized r.m.s. wall-pressure distribution p rms w /τ w as a function of the Reynolds number Re τ . ◆ Nearly-adiabatic condition; ■ Adiabatic condition. For legend, see caption in Fig. (3.5).

Fig. ( 4

 4 Fig. (4.3-a) shows a comparison of the asymptotic behavior of T rms for the adiabatic and nearly-adiabatic wall condition. As found by Morinishi et al. (2004)[START_REF] Morinishi | Direct numerical simulation of compressible turbulent channel flow between adiabatic and isothermal walls[END_REF], T rms follows y + for the nearly-adiabatic wall. The near-wall asymptotic behavior of R vT compared to y + 2 and y + 3 is shown in Fig.(4.3-b). It shows that, in the considered wall-region (y + ≃ 1), y + 2 is found to be a good asymptote of R vT , which is in contrast with the theoretical compressible near-wall asymptote (∝ y + 3 ).

Figure 4 . 2 :

 42 Figure 4.2: (a) Near-wall asymptotic behavior of the normalized turbulent kinetic energy and (b) the normalized dissipation rate as a function of y + . (-) Nearly-adiabatic condition; (--) Adiabatic condition.

Figure 4 . 3 :Figure 4 . 4 :

 4344 Figure 4.3: (a) Near-wall asymptotic behavior of the normalized r.m.s temperature fluctuations T rms /T ∞ and (b) Near-wall asymptotic behavior of the normalized heat flux R vT as a function of y + . (-) Nearlyadiabatic condition; (--) Adiabatic condition.

4. 1 . 4

 14 Turbulence statisticsThe time-averaged mean temperature distribution as well as the streamwise velocity component multiplied by the density ratio ρ /ρ ∞ are presented in Fig.(4.4-a). Although a small difference (≈ 2%) in the temperature profile in the inner region of the boundary layer (y/δ < 0.2), both quantities predict the same tendency. For the total temperature, however, a weak improvement of the profile is observed in the inner region of the boundary layer (Fig.4.4-b).

Fig. ( 4

 4 Fig. (4.5-a) shows the van-Driest transformed mean streamwise velocity u + vd as a function of y + , where both results are found to almost perfectly collapse, excepting in the wake region, due to the weak decrease of the Reynolds number with the wall condition. The velocity fluctuations remain also unaffected by the wall condition (Fig. 4.5-b). In Fig. (4.6) the normalized r.m.s. of the thermodynamic quantities as a function of y/δ are shown. Due to the imposed temperature condition, a small increase of the r.m.s. of the temperature fluctuations (Fig. 4.6-a), and subsequently the density fluctuations (Fig. 4.6-b), is observed in a wide region of the boundary layer (a maximum of 15% and 7% at peak values, respectively). However, the pressure fluctuations remains small (< 3% of p ∞ ) and unaffected by the wall-condition. Fig. (4.7-a) shows the SRA and the R uT correlation as a function of y/δ. It shows that u ′

Figure 4 . 5 :

 45 Figure 4.5: (a) Distribution of the van-Driest transformed mean streamwise velocity u + vd as a function of y + . (b) r.m.s velocity fluctuations in Morkovin's scaling as a function of y + . The two constants of the log-region profile are C = 5.2 and κ = 0.41. Circle: DNS Pirozzoli & Bernardini (2011) [74]. (-) Nearly-adiabatic condition; (--) Adiabatic condition.

Figure 4 . 6 :Figure 4 . 7 : 4 . 1 . 5 Figure 4 . 8 :

 464741548 Figure 4.6: Normalized r.m.s. of the thermodynamic quantities as a function of y/δ. (-) Nearly-adiabatic condition; (--) Adiabatic condition.

Figure 4 . 9 :

 49 Figure 4.9: Scatter plots of the strict SRA in a wall-normal plane for the nearly-adiabatic CSM-P2 case.(a) y + ≃ 9. (b) y + ≃ 28. (c) y + ≃ 100.

Figure 4 . 10 :

 410 Figure 4.10: Scatter plots of the normalized temperature -density fluctuations in a wall-normal plane for CSM-P2 case. (a) y + ≃ 9. (b) y + ≃ 28. (c) y + ≃ 100.

Figure 4 . 11 :

 411 Figure 4.11: Incompressible skin-friction coefficient, C finc , as a function of Reynolds number based on the incompressible momentum thickness, Re θinc . (• • • ) Smits et al. (1983) [90]; • Adiabatic DNS (M=2), Pirozzoli & Bernardini (2011)[74]; ■ Isothermal (T w /T ∞ ≈ 0.6, M=0.06),[START_REF] Back | Effect of wall cooling on the mean structure of a turbulent boundary layer in low-speed gas flow[END_REF] [2]; ▲ Adiabatic LES (M=2); ◆ Isothermal LES (T w /T ∞ ≈ 0.85, M=2).

Fig. ( 4 - 1 / 4 θ

 414 Fig. (4.11) shows C f inc as a function of Re θ inc compared to the analytical formula C f inc, S = 0.024Re -1/4 θ inc by Smits et al. (1983) [90] and to the experiment by Back et al. (1970) [2] of a turbulent boundary layer at M = 0.06 over a cooled-wall with T w /T ∞ ≈ 0.6. The present isothermal boundary layer shows good agreement compared to the analytical formula, where it is slightly under-predicted by 3% compared to the theoretical curve.

Fig. ( 4

 4 Fig.(4.12-a) highlights the instantaneous velocity fluctuations field in a wall-parallel plane at y + ≃ 9. As it is observed in the adiabatic case, it shows an alternating high-and low-speed streaks, respectively corresponding to positive and negative velocity fluctuations, typical of a fully turbulent wall-bounded flow. However, the observed streaks are found to be smaller and more elongated than their adiabatic counterparts. This behavior was also observed by[START_REF] Coleman | A numerical study of turbulent supersonic isothermal wall channel flow[END_REF] [START_REF] Coleman | A numerical study of turbulent supersonic isothermal wall channel flow[END_REF] and[START_REF] Duan | Direct numerical simulation of hypersonic turbulent boundary layers. Part 2. Effect of wall temperature[END_REF] [START_REF] Duan | Direct numerical simulation of hypersonic turbulent boundary layers. Part 2. Effect of wall temperature[END_REF]. The same streaky pattern is also visible in the instantaneous temperature

  (4.14-a), shows the same turbulence state observed in the adiabatic-wall boundary layer. First, a two-components state in the near-wall region with a maximum anisotropy reached at y + ≃ 8 is observed, and then gradually switched to a state of axisymmetric expansion up to the edge of the boundary layer, where an isotropic turbulence behavior is reached. Compared to the adiabatic case, a weak shift of the map is observed along the two-components boundary. This behavior discussed by[START_REF] Shahab | Numerical investigation of the influence of an impinging shock wave and heat transfer on a developing turbulent boundary layer[END_REF] [83] can be explained by a strengthening of the streamwise fluctuating velocity component relative to the wall-normal component. The structure parameter (Fig.4.14-b) is also found to weakly

Figure 4 .

 4 Figure 4.12: (a) Instantaneous velocity fluctuations field and (b) temperature fluctuations field in the x-z plane at y + ≃ 9. Contour levels are shown for -0.25 ≤ u ′ /u ∞ ≤ 0.25 and -0.12 ≤ T ′ /T ∞ ≤ 0.12, from dark to light shades.

Figure 4 .

 4 Figure 4.13: (a) Near-wall asymptotic behavior of the normalized r.m.s temperature fluctuations, T rms /T ∞ and (b) Near-wall asymptotic behavior of the normalized heat flux, R vT , as a function of y + . (-) Isothermal case; (--) Adiabatic case.

( 4 .

 4 [START_REF] Brun | Large-eddy simulation of compressible channel flow[END_REF].

Figure 4 .

 4 Figure 4.14: (a) Anisotropy invariants map. (b) Structure parameter as a function of y/δ. • DNS Pirozzoli & Bernardini (2011). (-) Isothermal case; (--) Adiabatic case.

Figure 4 . 15 :Figure 4 . 16 :

 415416 Figure 4.15: Anisotropy ratios as a function of y + . (a) v 2 rms /u 2 rms ; (b) v 2 rms /u 2 rms . (-) Isothermal case; (--) Adiabatic case.

  Fig.(4.17-b) shows T /T ∞ as a function of u /u ∞ . The computed T /T ∞ (lines) are found to well collapse with the Walz's (1969)[START_REF] Walz | Boundary layers of flow and temperature[END_REF] equation (symbols), which confirms the validity of this equation for predicting the temperature-velocity relationship for isothermal boundary layers, at least for the considered conditions. The[START_REF] Walz | Boundary layers of flow and temperature[END_REF] [START_REF] Walz | Boundary layers of flow and temperature[END_REF] equation is defined:

. 4 )≈ 1 T

 41 Figure 4.17: (a) Distributions of the time-averaged mean quantities as a function of y/δ. (b) Predicted Walz's equation. (-) Isothermal case; (--) Adiabatic case.

4 . 5 )Figure 4 .

 454 Figure 4.18: r.m.s velocity fluctuations in Morkovin's scaling as a function of y/δ. Symbols: Adiabatic (M = 2.28), Piponniau (2009) [71]; Diamond: PIV; Triangle: LDA. (-) Isothermal case; (--) Adiabatic case.

Fig. ( 4 Figure 4 .

 44 Fig. (4.19-b) shows the transformed total temperature, T t + , as a function of y + for the coldwall case. It is found that the distribution follows the logarithmic-law slope 1/κ, to a constantC T , which depends on the Prandtl number and the ratio of the wall-temperature to the recovery temperature T w /T r(Debiève et al., 1997 [21]). In the present LES simulation, the logarithmic-law

( 4 .

 4 23-a). As expected, u ′ and T ′ are not found to be anti-correlated as commonly assumed for adiabatic supersonic boundary layers, since the total temperature fluctuations are non-negligible.

Figure 4 . 20 :Figure 4 . 21 :Figure 4 . 22 :

 420421422 Figure 4.20: Reynolds stress -ρu ′ v ′ , mean viscous shear stress µ ∂ u /∂y and total shear stress -ρu ′ v ′ + µ ∂ u /∂y as a function of y + . (-) Isothermal case; (--) Adiabatic case.

Fig. ( 4 . 24 )

 424 Fig.(4.24) shows the heat flux v ′ T ′ when normalized by the fluctuating r.m.s quantities(a) and by the free stream quantities (b). A significant decrease of R vT and v ′ T ′ /u ∞ T ∞ is observed with wall-cooling. In a very-near wall region, R vT is negative: equal to -0.35 at the wall, it continues to decrease and attains a peak of -0.45 at y/δ ≃ 0.01, where it starts to increase up to y/δ ≃ 0.1. At y/δ ≃ 0.2, it reaches 0.36 and remains constant (≈ 0.4) up to the edge of the boundary layer. The same tendency is observed with the freestream normalization v ′ T ′ /u ∞ T ∞ .

Figure 4 .Figure 4 .Figure 4 . 2 ∞

 4442 Figure 4.23: (a) -R uT correlation and (b) u ′ T ′ /u ∞ T ∞ heat flux as a function of y/δ. (-) Isothermal case; (--) Adiabatic case.

Figs. ( 4 .

 4 Figs. (4.25-a) and (4.25-b) plot respectively -R uv correlation and the normalized shear stress

Fig. ( 4 Figure 4 . 26 :

 4426 Fig. (4.26) plot the scattered temperature fluctuations as a function of the streamwise velocity fluctuations (top). As expected, it is shown that the present results does not verify the Morkovin's hypothesis, where the correlation linking u ′ and T ′ is found to have a positive slope in the near-wall region, and tend to zero when moving away from the wall.The modified SRA relations linking u ′ and T ′ are written:

Lechner

  et al. (2001) [START_REF] Lechner | Turbulent supersonic channel flow[END_REF] found similar results in a DNS of supersonic isothermal channel flow, where ∂ T t /∂ T = 2.08 gave a value of c = 1.05.Assuming a polytropic behavior of the thermodynamic quantities, the fluctuations ρ ′ and T ′ are assumed to be perfectly correlated. This result is not verified in the present cold-wall boundary layer, where a linear-fitting slope of 0.65 is found in the near-wall region (Fig.4.27).As written for the adiabatic case, when neglecting the pressure fluctuations, the normalized entropy fluctuations, s ′ /C v , should have a slope proportional to -γ when shown against ρ ′ / ρ . In a very near-wall region, this slope is found to be proportional to -0.91, which can be explained by a non-negligible level of pressure fluctuations (Fig.4.28).

Figure 4 . 27 :

 427 Figure 4.27: Scatter plots of the normalized temperature -density fluctuations in a wall-normal plane for the cold-wall boundary layer. (a) y + ≃ 9; (b) y + ≃ 28; (c) y + ≃ 100.

Figure 4 . 28 :

 428 Figure 4.28: Scatter plots of (a-c) the normalized density -pressure fluctuations and (d-f) the normalized entropy -density fluctuations in a wall-normal plane for the cold-wall boundary layer. (a, d) y + ≃ 9; (b, e) y + ≃ 28; (c, f) y + ≃ 100.

i 2 u 2 ( 4 . 12 )Fig. ( 4

 24124 Fig. (4.29) shows the scattered total temperature fluctuations as a function of the temperature fluctuations, velocity fluctuations and kinetic energy fluctuations. As it was concluded by Lechner et al. (2001)[START_REF] Lechner | Turbulent supersonic channel flow[END_REF], in the near-wall region, the contribution of the temperature, velocity and kinetic energy fluctuations is of the same order, and none of those quantities can be neglected ((γ -1) M 2 ≃ 0.095 and T / T t ≃ 0.96).

Figure 4 . 29 :

 429 Figure 4.29: Scatter plots of (a-c) the normalized total-temperature -temperature fluctuations, (d-f) the normalized total-temperature -velocity fluctuations and (g-i) the normalized total-temperature kinetic energy fluctuations in a wall-normal plane for the cold-wall boundary layer. (a, d, g) y + ≃ 9; (b, e, h) y + ≃ 28; (c, f, i) y + ≃ 100.

Figure 4 . 30 :

 430 Figure 4.30: Scatter plots of the total-temperature -velocity fluctuations in a wall-normal plane for the cold-wall boundary layer. (a) y + ≃ 9. (b) y + ≃ 28. (c) y + ≃ 100.

3. 5

 5 Incompressible skin-friction coefficient C finc as a function of Reynolds number based on incompressible momentum thickness Re θinc . (--) Blasius; (• -•) Kármán Schoenherr; (• • • ) Smits et al. (1983); ■ Pirozzoli et al. (M=2.25); ◆ Guarini et al. (M=2.5); ▲ Maeder et al. (M=3); • Pirozzoli & Bernardini (M=2). (a) SGS models study using LES-P2 grid; (b) Grid sensitivity study using CSM model. For legend, see table 3.4. . . . . . . . . . . . . . . . . 3.6 Normalized r.m.s. wall-pressure distribution p rmsw /τ w as a function of Reynolds number Re τ . (--) Farabee & Casarella (1991) [28]; (• -•) Bernardini & Pirozzoli (2011) [9]. (a) SGS models study using LES-P2 grid; (b) Grid sensitivity study using CSM model. For legend, see table 3.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Turbulent Mach number, M t , as a function of y/δ. (a) SGS models study using LES-P2 grid; (b) Grid sensitivity study using CSM model. For legend, see table 3.4. . . . . . . . . . . . 3.8 Distributions of the time-averaged mean quantities as a function of y/δ. (a) SGS models study using LES-P2 grid; (b) Grid sensitivity study using CSM model. For legend, see table

3

 3

  .4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Distribution of the normalized time-averaged total temperature, T t /T t∞ , as a function of y/δ. • DNS Pirozzoli et al. (2004) [76] at Re θ = 4260. (a) SGS models study using LES-P2 grid; (b) Grid sensitivity study using CSM model. For legend, see table 3.4. . . . . . . . . . 3.10 Distribution of the van-Driest transformed mean streamwise velocity u + vd as a function of y + . C = 5.2; κ = 0.41. (a) SGS models study using LES-P2 grid; (b) Grid sensitivity study using CSM model. For legend, see table 3.4. . . . . . . . . . . . . . . . . . . . . . . . . . 3.11 Normalized mean velocity deficit u vd ∞ -u vd /u τ as a function of y/δ. (a) SGS models study using LES-P2 grid; (b) Grid sensitivity study using CSM model. For legend, see table 3.4. . 3.12 (a-b) r.m.s of velocity fluctuations components u i, rms /u τ as a function of y/δ; (c-d) r.m.s velocity fluctuations in Morkovin's scaling as a function of y + ; (e-f) r.m.s velocity fluctuations in Morkovin's scaling as a function of y + in Log-scale. (• -•) Perry & Li (1990) [70] in Eq.

3. 12 .

 12 (--)[START_REF] Pirozzoli | Direct numerical simulation and analysis of a spatially evolving supersonic turbulet boundary layer at M=2.25[END_REF] [START_REF] Pirozzoli | Direct numerical simulation and analysis of a spatially evolving supersonic turbulet boundary layer at M=2.25[END_REF] in Eq. 3.11. For legend, see table3

  .

2 τ

 2 4. . . . . . . . . 3.13 Normalized turbulent kinetic energy K/u as a function of y/δ. (a) SGS models study using LES-P2 grid; (b) Grid sensitivity study using CSM model; (c) Resolved turbulent kinetic energy K + and SGS turbulent kinetic energy k + sgs for the LES-P2 grid using the DSM model as a function of y + . For legend, see

  (a) Distribution of the van-Driest transformed mean streamwise velocity u + vd as a function of y + ; (b) r.m.s velocity fluctuations in Morkovin's scaling as a function of y + . (• -•) WENO-BWO scheme; (• • •) 4 th split-centered; (--) 6 th split-centered. . . . . . . . . . . . . 3.18 Normalized SGS viscosity, µ sgs /µ, as a function of y + . (a) SGS models study using LES-P2 grid; (b) Grid sensitivity study using CSM model. For legend, see table 3.4. . . . . . . . . . 3.19 Ratio of the turbulent kinetic energy Γ as a function of y + . (a) SGS models study using LES-P2 grid; (b) Grid sensitivity study using CSM model. For legend, see table 3.4. . . . . 3.20 Contribution of the different SGS dissipation terms to turbulence. . . . . . . . . . . 3.21 (a-d) Ratio of the SGS dissipation due to the fluctuating flowfield to the total SGS dissipation ε ′ sgs /ε sgs as a function of y + ; (b-e) Ratio of the SGS dissipation due to the mean-averaged flowfield to the total SGS dissipation ε sgs /ε sgs as a function of y + ; (c-f) Ratio of the SGS dissipation due to the fluctuating flowfield in a wall-parallel plane to the total SGS dissipation due to the fluctuating flowfield ε ′ sgs x+z /ε ′ sgs as a function of y + . (a-b-c) SGS models study using LES-P2 grid; (d-e-f) Grid sensitivity study using CSM model. For legend, see table 3.4. 3.22 SGS Activity parameter ς as a function of y + . (a-b-c) SGS models study using LES-P2 grid; (d-e-f) Grid sensitivity study using CSM model. For legend, see table 3.4. . . . . . . . . . . 3.23 Ratio of the SGS dissipation to the viscous dissipation as a function of y + . (a) SGS models study using LES-P2 grid; (b) Grid sensitivity study using CSM model. For legend, see table 3.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.24 Normalized r.m.s. of the thermodynamic quantities as a function of y/δ. (a-b-c) SGS models study using LES-P2 grid; (d-e-f) Grid sensitivity study using CSM model. For legend, see

3. 30

 30 Scatter plots of (a-c) the normalized temperature -streamwise velocity fluctuations and (d-f) the normalized temperature -wall-normal velocity fluctuations in a wall-normal plane for the CSM-P2 case. (a, d) y + ≃ 9; (b, e) y + ≃ 28; (c, f) y + ≃ 100. . . . . . . . . . . . . . . 3.31 Scatter plots of the strict SRA in a wall-normal plane for the CSM-P2 case. (a) y + ≃ 9; (b) y + ≃ 28; (c) y + ≃ 100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.32 Scatter plots of the normalized temperature -density fluctuations in a wall-normal plane for CSM-P2 case. (a) y + ≃ 9. (b) y + ≃ 28. (c) y + ≃ 100. . . . . . . . . . . . . . . . . . . . . 3.33 Scatter plots of the normalized density -pressure fluctuations in a wall-normal plane for CSM-P2 case. (a) y + ≃ 9; (b) y + ≃ 28; (c) y + ≃ 100. . . . . . . . . . . . . . . . . . . . . 3.34 Scatter plots of (a-c) the normalized entropy-density fluctuations and (d-f) the normalized entropy-pressure fluctuations in a wall-normal plane for CSM-P2 case. (a, d) y + ≃ 9; (b, e) y + ≃ 28; (c, f) y + ≃ 100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.35 Scatter plots of (a-c) the normalized total-temperature -temperature fluctuations and (df) the normalized total temperature -velocity fluctuations in a wall-normal plane for the nearly-adiabatic CSM-P2 case. (a, d) y + ≃ 9; (b, e) y + ≃ 28; (c, f) y + ≃ 100. . . . . . . . . 3.36 Anisotropy invariants map. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.37 Anisotropy invariants maps. (a) SGS models study using LES-P2 grid; (b) Grid sensitivity study using CSM model. For legend, see table 3.4. . . . . . . . . . . . . . . . . . . . . . .

  S| = 2 S ij S ij est le second invariant du tenseur de déformation, et C s est la constante du modèle, souvent déterminée dynamiquement.

	où |
		7)
	où µ sgs est la viscosité turbulente de sous-maille, et S ij = 1/2 (∂ u i /∂x j + ∂ u j /∂x i ) est le tenseur
	du taux de déformation des échelles résolues.
	La viscosité turbulente, µ sgs , est généralement modélisée sous la forme:
	µ sgs = ρC s ∆ 2 | S|	(2.8)

Table 2 .

 2

1: Coefficients des schémas centrés du 2 ème au 8 ème ordre. En comparaison avec les formes (2.54) et (2.55) des flux numériques, il est à noter que dans la forme (2.58), l'effet de la variation de densité est bien pris en compte, et ceci indépendamment des deux autres variables u et ϕ. Dans la suite du travail, et sauf mention contraire, le schéma centré split-centered du 6 ème ordre est utilisé.

  et une valeur de température T w est fixée à la paroi à travers l'équation d'énergie.A la sortie du domaine, une condition de sortie supersonique est définie, alors qu'une condition de bord libre est appliquée en haut du domaine.La génération de conditions d'entrée, turbulente et instationnaire, demeure une des principales difficultés lors de la réalisation d'une DNS ou d'une LES. Ces conditions doivent être dépendantes du temps, ainsi que suffisamment précises pour décrire l'état turbulent de l'écoulement. Il est ainsi inévitable de spécifier des fluctuations réalistes, qui soient en cohérence avec l'écoulement Elle consiste à imposer à l'entrée du domaine un champ moyen de vitesse et d'y superposer des fluctuations aléatoires filtrées, qui sont corrélées avec des propriétés statistiques d'ordre élevées données, connues au préalable à partir de données expérimentales ou de simulations DNS(Jarrin et al., 2008 [44]).Une méthode alternative, bien adaptée à ce type d'écoulement, est proposée par Lund et al.(1998) pour la simulation LES de couches limites turbulentes, dans laquelle l'écoulement produit ses propres conditions d'entrée, à partir d'un processus de renormalisation du profil de vitesse, situé à une station suffisamment éloignée de la sortie du domaine de calcul. Cette technique présente

	2.2.4.2 Conditions d'entrée turbulente
	2.2.4.1 Conditions de parois, de borne supérieure et de sortie
	La condition de paroi adiabatique (Adiabatic no-slip boundary condition) est satisfaite en imposant
	une condition de gradient nul pour la température ainsi que toutes les composantes de la vitesse.

(Ghost cells) se trouvant en dehors du domaine de calcul. En fonction de la condition physique à satisfaire, les variables conservatives du système d'équations régissant l'écoulement sont définies dans ces mailles. L'ensemble des conditions imposées dans cette étude est présenté sur la Fig. (3.1). Pour la condition de paroi isotherme, la même condition est imposée pour les composantes du vecteur vitesse, moyen. Cette étape s'avère délicate à traiter, vu la sensibilité des équations de Navier-Stokes aux conditions aux limites. Dans cette étude, nous nous intéressons à une turbulence pleinement développée, dont les grandeurs moyennes sont supposées être moins sensibles aux variations des conditions d'entrée. Dans le cadre spécifique des simulations DNS ou LES spatio-temporelles de couches limites turbulentes, plusieurs méthodes sont proposées dans la littérature : • La solution la plus simple consiste à imposer un profil laminaire à l'entrée et d'y superposer des fluctuations aléatoires afin que s'effectue une transition naturelle vers la turbulence. Cette approche, utilisée entre autre par Pirozzoli et al. (2004) [76] dans le cas d'une DNS de couche limite supersonique, a donné des résultats probants. Son coût reste prohibitif puisqu'une grande partie du domaine de calcul est consacrée à la transition de la couche limite vers un état pleinement turbulent.

• Il est également possible d'imposer, en entrée du domaine, des profils de quantités moyennes, auxquelles on superpose un bruit blanc. Cette méthode pose le problème de générer des fluctuations qui ne vérifient que statistiquement les propriétés turbulentes réelles de l'écoulement, ce qui conduit à une atténuation inévitable des fluctuations introduites à l'entrée. Elle nécessite également l'emploi de grands domaines de calculs afin que ces fluctuations se réorganisent en grandeurs réalistes, ce qui induit un coût de calcul important.

• Par ailleurs, ils existent des techniques alternatives à celles déjà citées, utilisant une approche de génération de la turbulence dite synthétique. Cette approche est intéressante à condition de disposer de quelques données statistiques expérimentales ou issues de calculs DNS. l'avantage de réduire le domaine de calcul et, de ce fait, le coût de la simulation. Une extension de cette méthode au cas compressible à aussi été proposée par Urban & Knight (2001) [100], Stolz & Adams (2003) [95], Sagaut et al. (2004) [80] et récemment par Pirozzoli et al. (2010) [75].

Méthode du filtre digital

Récemment proposée par

Klein et al. (2004)

[START_REF] Klein | A digital filter based generation of inflow data for spatially developing direct numerical or large eddy simulation[END_REF]

, cette méthode a néanmoins montré certaines limites, puisqu'elle est significativement lourde en terme de temps de calcul. En effet, à chaque pas de temps, la méthode requiert de corréler un ensemble de plans de données aléatoires de taille

(N x × N y × N z ).

Une deuxième méthode, toujours utilisant l'approche du filtre digital, proposée par

[START_REF] Xie | Efficient generation of inflow conditions for large-eddy simulation of street-scale flows[END_REF]

[START_REF] Xie | Efficient generation of inflow conditions for large-eddy simulation of street-scale flows[END_REF]

, reprend les mêmes bases que la méthode de

Klein et al. (2004)

[START_REF] Klein | A digital filter based generation of inflow data for spatially developing direct numerical or large eddy simulation[END_REF]

, sauf qu'au lieu de filtrer et de corréler un ensemble tridimensionnel de données, la méthode génère et filtre seulement un ensemble bidimensionnel de plans de données aléatoires, corrélé ensuite avec l'ensemble des plans de données du dernier pas de temps. Cette méthode présente l'avantage d'être performante et moins gourmande en termes de ressources informatique que la méthode originale de Klein et al.

Table 3 .

 3 The First-half of the domain is used for the recycling/rescaling procedure, and only the second-half of the domain is fully dedicated to extract and analyze the results. In the latter part of the domain, Reynolds number Re τ is ranging from 510 to 640 (which corresponds to Re θ from 2090 to 2900). 1: Grid resolution sensitivity study using the CSM model. Subscript (+) denotes the normalization by the friction velocity, with y + = yρ w u τ /µ w .

	Tables 3.2 and 3.3 reports statistical properties of the considered test-cases at a given down-

achieved to sweep the initial transient flowfield. Then, turbulence statistics are sampled and extracted each time step from time series covering τ ≈ 300τ c . By plotting the time evolution of the main boundary layer statistics, such as the boundary layer thickness and the friction velocity, this sampling time is judged to be sufficient to reach a statistical convergence of the considered quantities.

A reference simulation (e.g. LES-P2 case) is performed over about 40 hours using 64 processors, for a total of about 2560 CPU hours. stream station x res ≃ 84.8δ in , which corresponds to Re τ ≈ 580.

Table 3 .

 3

2: Boundary layer properties using LES-P2 grid for different subgrid models. Re τ

Table 3 .

 3 3: Boundary layer properties for different grids using the CSM model.

	Case	Re τ	Re θ	10 3 C f δ * /δ 10 2 θ/δ	H	T w /T ∞	M τ	p ′ 2 w /τ w
	DNS	583.9 2865.9	2.53	0.259	8.50	2.96	-	0.0712	-
	LES-L1 541.0 2582.0	2.73	0.294	9.38	3.14	1.675	0.0740	3.01
	LES-L2 599.4 2670.4	2.93	0.289	9.17	3.16	1.659	0.0767	3.46
	LES-N2 599.2 2691.8	2.99	0.291	9.28	3.14	1.669	0.0773	3.43
	LES-P2 590.7 2597.5	2.98	0.282	9.08	3.11	1.666	0.0773	3.34
	LES-Q2 581.7 2565.5	2.89	0.279	8.99	3.11	1.666	0.0762	3.49
	LES-R2 651.0 2669.6	3.19	0.266	8.79	3.02	1.661	0.0800	3.66
	LES-S2 597.7 2388.7	2.99	0.251	8.26	3.04	1.666	0.0776	3.30

1/2

; θ refers to the momentum thickness.

  .1).The objective behind the analysis of the Strong Reynolds Analogy is to verify the applicability of the Morkovin's hypothesis. Reynolds Analogies in supersonic flows imply that the total temperature fluctuations are negligible and the turbulent Prandtl number is one. In particular, this yields to Resolved turbulent Prandtl number Pr t and Huang's relation (Eq. 3.36) as a function of y/δ.

	2				2
	1.6	1 -	∂ T t ∂ T	-1	1.6	1 -	∂ T ∂ T t	-1
	1.2				1.2
	t			t
	Pr			Pr
	0.8				0.8
	0.4				0.4
	0 0.2 0.4 0.6 0.8 1 1.2 0	0 0.2 0.4 0.6 0.8 1 1.2 0
	(a)		y/δ	(b)		y/δ
	Figure 3.29:

(a) SGS models study using LES-P2 grid; (b) Grid sensitivity study using CSM model. For legend, see table

3.4.

Table 4 .

 4 1: Boundary layer properties of the wall boundary condition study using LES-P2 grid.

		τ	Re θ	10 3 C f δ * /δ 10 2 θ/δ	H	T w /T ∞	M τ
	DNS	583.9 2865.9	2.53	0.259	8.50	2.96	1.717	0.0712
	Adiabatic	590.7 2597.5	2.98	0.282	9.08	3.11	1.666	0.0773
	Nearly-adiabatic 572.1 2560.8	2.94	0.297	9.03	3.29	1.75	0.0786

Table 4 .

 4 2: Power indices n of near-adiabatic-wall asymptotic behaviors.

  ). The grid spacing, expressed in term of wall-units, gives ∆x + = 34.0,

	Case	Re τ	Re θ	Re δ 2	∆x +	∆y +	∆z +
	Adiabatic	505-633	2038-2828 1408-1962 34.2-34.4 1.18-1.19 12.3-12.4
	Cool	1004-1376 2348-3330 2649-3751 34.0-38.6 0.49-0.55 13.1-14.8
	Table 4.3: Reynolds numbers ranges for the adiabatic and isothermal-wall boundary layers and wall-units
	grid spacing.

Table 4 .

 4 1000 to ≈ 1380 (see table 4.4). Re θ, inc 10 3 C f δ * /δ 10 2 θ/δ H T w /T ∞ 10 2 M τ 4: Boundary layer properties of the isothermal boundary layers. C finc and Re θinc are calculated using van-Direst II theory (Eq. 4.2).CaseT w /T r T w /T ∞ 10 2 β q 10 -2 T τ

	Case	Re τ	Re θ
	Adiabatic 557.2 2368.2 1640.0	3.05	0.280	8.90	3.14	1.66	7.818
	Cool	1040.3 2366.9 2670.1	3.09	0.196	10.9	1.79	0.85	7.864
			Nearly-adiabatic	1	1.71	0.0	0.0
			Cool	0.5	0.85	-5.55	0.141

Table 4 .

 4 5: Boundary layer thermal properties of the isothermal boundary layers. q w = -λ w ∂ T /∂y| w ;

  The presented results are extracted at a station x res ≃ 53.8δ in , which gives a Reynolds number Re τ ≈ 1040.3.The basic flow properties are reported in table 4.3, and the thermal flow properties of the cold-wall and the nearly-adiabatic-wall boundary layers are given in table 4.5.For a cold-wall boundary layer,[START_REF] Hopkins | An evaluation of theories for predicting turbulent skin friction and heat transfer on flat plates at supersonic and hypersonic Mach numbers[END_REF] [START_REF] Hopkins | An evaluation of theories for predicting turbulent skin friction and heat transfer on flat plates at supersonic and hypersonic Mach numbers[END_REF] and[START_REF] Cook | Heat transfer for highly cooled supersonic turbulent boundary layers[END_REF] [START_REF] Cook | Heat transfer for highly cooled supersonic turbulent boundary layers[END_REF] recommended the use of the van-Driest II theory transformation to predict C f inc . The transformed incompressible skin-friction coefficient, C f inc , and the incompressible Reynolds number based on momentum thickness, Re θ inc , are transformed according to C f inc = F c C f and Re θ inc = F θ Re θ , respectively, where the transformation functions F c and F θ are defined by:

  table 3.4. . . . . . . . . . . . . . . . . . . . . . . . . 3.14 Structure parameter -u ′ v ′ /2K as a function of y/δ. (a) SGS models study using LES-P2 grid; (b) Grid sensitivity study using CSM model. For legend, see table 3.4. . . . . . . . . . 3.15 Distribution of r.m.s. vorticity components as a function of y + . (a) SGS models study using LES-P2 grid; (b) Grid sensitivity study using CSM model. For legend, see table 3.4. . . . . 3.16 Turbulent shear stress -ρu ′ v ′ , mean viscous shear stress µ ∂ u /∂y and total shear stress -ρu ′ v ′ + µ ∂ u /∂y as a function of y + . (a) SGS models study using LES-P2 grid; (b) Grid sensitivity study using CSM model. For legend, see table 3.4. . . . . . . . . . . . . .

	3.17

table 3 .

 3 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.25 SRA and -R uT correlations as a function of y/δ. Circle: Pirozzoli et al. (2004) [76]. (a) SGS models study using LES-P2 grid; (b) Grid sensitivity study using CSM model. For legend, see table 3.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.26 Modified SRA as a function of y/δ. (a, b, c) SGS models study using LES-P2 grid; (d, e, f) Grid sensitivity study using CSM model. For legend, see table 3.4. . . . . . . . . . . . . . 3.27 -R ρT , -R uv and R vT correlations as a function of y/δ. (a, c, e) SGS models study using LES-P2 grid; (b, d, f) Grid sensitivity study using CSM model. For legend, see table 3.4. . . 3.28 Resolved turbulent Prandtl number as a function of y/δ. (--) DNS curve-fitting in Eq. (3.37). (a) SGS models study using LES-P2 grid; (b) Grid sensitivity study using CSM model. For legend, see table 3.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.29 Resolved turbulent Prandtl number Pr t and Huang's relation (Eq. 3.36) as a function of y/δ. (a) SGS models study using LES-P2 grid; (b) Grid sensitivity study using CSM model. For legend, see table 3.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Compressible High-Order Code using Weno AdaptiVE Stencils, c Abdellah Hadjadj & Collaborateurs, CORIA

Le développement numérique qui suit est repris de l'article de Pirozzoli (2010)[START_REF] Pirozzoli | Generalized conservative approximations of split convective derivative operators[END_REF].

Chapter 4 Isothermal Supersonic Turbulent Boundary Layer

This chapter is divided into two parts. First, a comparison is made between an isothermal wall with the theoretical adiabatic temperature (refereed to as nearly-adiabatic) and an adiabatic-wall (q w = 0) boundary layers. The objective of this study is to highlight the main differences resulting in varying the wall condition. The second part of the chapter is dedicated to the study of a cold-wall boundary layer, and the results are analyzed and compared to the adiabatic case.

Nearly-adiabatic wall Conditions

Flow conditions and simulation parameters

The isothermal nearly-adiabatic wall condition is achieved by imposing a fixed-wall temperature, given by the recovery temperature from the relation:

where r = 0.89 is the recovery factor.

The test case is performed using the LES-P2 grid (see table 3.1). The turbulent inflow conditions as well as the mean velocity and the temperature profiles are the same as used in the adiabatic

List of Figures