Cette étude porte sur le calcul numérique du champ acoustique rayonné par des écoulements subsoniques turbulents présentant des inhomogénéités de température. Des méthodes hybrides sont développées grâce à un développement de Janzen-Rayleigh des équations de Navier-Stokes. L'écoulement est résolu par un calcul quasi incompressible puis les perturbations acoustiques sont propagées selon deux méthodes : les équations d'Euler linéarisées (EEL) et l'approximation à faible nombre de Mach perturbée(PLMNA). Les méthodes sont validées sur des cas simples puis appliquées à une couche de mélange isotherme et anisotherme en développement spatial. / This study focuses on the numerical calculation of the acoustic field radiated by subsonic turbulent flows with temperature inhomogeneities. Hybrid methods are developed through a Rayleigh-Janzen expansion of the Navier-Stokes equations. The flow is solved in a quasi-incompressible way then the acoustic disturbances are propagated by two methods : the linearized Euler's equations (EEL) and the perturbed low Mach number approximation (PLMNA). The methods are validated on simple cases and then applied to an isothermal and non isothermal spatially evolving mixing layer.

This thesis is concerned with the study of parallel mixing of two dissimilar gases under compressible conditions in the confined environment. A number of numerical studies are reported in the literature for the compressible mixing of two streams of gases where (1) both the streams are of similar gases at the same temperatures, (2) both the streams are at different temperatures with similar gases, and (3) dissimilar gases are with nearly equal temperatures. The combination of dissimilar gases at large temperature difference, mixing under compressible conditions, as in the case of scramjet propulsion, has not been adequately addressed numerically. Also many of the earlier studies have used two dimensional numerical simulation and showed good match with the experimental results on mixing layers that are inherently three dimensional in nature. In the present study, both two-dimensional (2-d) and three dimensional (3-d) studies are reported and in particular the effect of side wall on the three dimensionality of the flow field is analyzed, and the reasons of the good match of two dimensional simulations with experimental results have been discussed. Both two dimensional and three dimensional model free simulations have been conducted for a flow configuration on which experimental results are available. In this flow configuration, the mixing duct has a rectangular cross section with height to width ratio of 0.5. In the upper part of the duct hydrogen gas at a temperature of 103 K is injected through a single manifold of two Ludweig tubes and in the lower part of the duct nitrogen gas at a temperature of 2436 K is supplied through an expansion tube, both the gases are at Mach numbers of 3.1 and 4.0 respectively. Measurements in the experiment are limited to wall pressures and heat flux. The choice of this experimental condition gives an opportunity to study the effect of large temperature difference on the mixing of two dissimilar gases with large molecular weights under compressible conditions. Both two dimensional and three dimensional model free simulations are carried out using higher order numerical scheme (4th order spatial and 2nd order temporal) to understand the structure and evolution of supersonic confined mixing layer of similar and dissimilar gases. Two dimensional simulations are carried out by both SPARK (finite difference method) and OpenFOAM (finite volume method based open source software that was specially picked out and put together), while 3D model free simulations are carried out by OpenFOAM. A fine grid structure with higher grid resolution near the walls and shear layer is chosen. The effect of forcing of fluctuations on the inlet velocity shows no appreciable change in the fully developed turbulent region of the flow. The flow variables are averaged after the attainment of statistical steady state established through monitoring the concentration of inert species introduced in the initial guess. The effect of side wall on the flow structure on the mixing layer is studied by comparing the simulation results with and without side wall. Two dimensional simulations show a good match for the growth rate of shear layer and experimental wall pressures. Three dimensional simulations without side wall shows 14% higher growth rate of shear layer than that of two dimensional simulations. The wall pressures predicted by these three dimensional simulations are also lower than that predicted using two dimensional simulations (6%) and experimental (9%) results in the downstream direction of the mixing duct. Three dimensionality of the flow is thought of as a cause for these differences. Simulations with the presence of side wall show that there is no remarkable difference of three dimensionality of the flow in terms of the variables and turbulence statistics compared to the case without side walls. However, the growth rate of shear layer and wall surface pressures matches well with that predicted using two dimensional simulations. It has been argued that this good match in shear layer growth rate occurs due to formation of oblique disturbances in presence of side walls that are considered responsible for the decrease in growth rate in 3-d mixing layers. The wall pressure match is argued to be good because of hindrance from side wall in the distribution of momentum in third direction results in higher wall pressure. The effect of dissimilar gases at large temperature difference on the growth rate reduction in compressible conditions is studied. Taking experimental conditions as baseline case, simulations are carried out for a range of convective Mach numbers. Simulations are also carried out for the same range of convective Mach numbers considering the mixing of similar gases at the same temperature. The normalized growth rates with incompressible counterpart for both the cases show that the dissimilar gas combination with large temperature difference shows higher growth rate. This result confirms earlier stability analysis that predicts increased growth rate for such cases. The growth rate reduction of a compressible mixing layer is argued to occur due to reduced pressure strain term in the Reynolds stress equation. This reduction also requires the pressure and density fluctuation correlation to be very near to unity. This holds good for a mixing layer formed between two similar gases at same temperature. For dissimilar gases at different temperatures this assumption does not hold well, and pressure-density correlation coefficient shows departure from unity. Further analysis of temperature density correlation factor, and temperature fluctuations shows that the changes in density occur predominantly due to temperature effects, than due to pressure effects. The mechanism of density variations is found to be different for similar and dissimilar gases, while for similar gases the density variations are due to pressure variations. For dissimilar gases density variation is also affected by temperature variations in addition to pressure variations. It has been observed that the traditional k-ε turbulence model within the RANS (Reynolds Averaged Navier Stokes) framework fails to capture the growth rate reduction for compressible shear layers. The performance of k-ε turbulence model is tested for the mixing of dissimilar gases at large temperature difference. For the experimental test case the shear layer growth rate and wall pressures show good match with other model free simulations. Simulations are further carried out for a range of convective Mach numbers keeping the mixing gases and their temperatures same. It has been observed that a drop in the growth rate is well predicted by RANS simulations. Further, the compressibility option has been removed and it has been observed that for the density and temperature difference, even for incompressible case, the drop in growth rate exists. This behaviour shows that the decrease in growth rate is mainly due to the interaction of temperature and species mass fraction on density. Also it can be inferred that RANS with k-ε turbulence model is able to capture the compressible shear layer growth rate for dissimilar gases at high temperature difference. The mixing of heat and species is governed by the values of turbulent Prandtl and Schmidt numbers respectively. These numbers have been observed to vary for different flow conditions, while affecting the flow field considerable in the form of temperature and species distribution. Model free simulations are carried out on an incompressible convective Mach number mixing layer, and the results are compared with that of a compressible mixing layer to study the effect of compressibility on the values of turbulent Prandtl / Schmidt numbers. It has been observed that both turbulent Prandtl and Schmidt numbers show an almost constant value in the mixing layer region for incompressible case. While, for a compressible case, both turbulent Prandtl and Schmidt numbers show a continuous variation within the mixing layer. However, the turbulent Lewis number is observed to be near unity for both incompressible and compressible cases. The thesis is composed of 8 chapters. An introduction of the subject with critical and relevant literature survey is presented in chapter 1. Chapter 2 describes the mathematical formulation and assumptions along with solution methodology needed for the simulations. Chapter 3 deals with the two and three dimensional model free simulations of the non reacting mixing layer. The effect of the presence of side wall is studied in chapter 4. Chapter 5 deals with the effect of compressibility on the mixing of two dissimilar gases at largely different temperatures. The performance of k-ε turbulence model is checked for dissimilar gases in Chapter 6. Chapter 7 is concerned with the effect of compressibility on turbulent Prandtl and Schmidt numbers. Finally concluding remarks are presented in chapter 8. The main aim of this thesis is the exploration of parallel mixing of dissimilar gases under compressible conditions for both two and three dimensional cases. The outcome of the thesis is (a) a finding that the presence of sidewall in a mixing duct does not make flow field two dimensional, instead it causes the formation of oblique disturbances and the shear layer growth rate is reduced, (b) that it has been shown that the growth rates of dissimilar gases are affected far more by large temperature difference than by compressibility as in case of similar gases, (c) that the growth rates of compressible shear layers formed between dissimilar gases are better predicted using k-εturbulence model and (d) that for compressible mixing conditions the turbulent Prandtl and Schmidt numbers vary continuously in the mixing layer region necessitating the use of some kind of model instead of assuming constant values.

L’approximation dite « anélastique » permet de filtrer les ondes acoustiques grâce à un développement asymptotique deséquations de Navier-Stokes, réduisant ainsi le pas en temps moyen, lors de la simulation numérique du développement d’instabilités hydrodynamiques. Ainsi, les équations anélastiques sont établies pour un mélange de deux fluides pour l’instabilité de Rayleigh-Taylor. La stabilité linéaire de l’écoulement est étudiée pour la première fois pour des fluides parfaits, par la méthode des modes normaux, dans le cadre de l’approximation anélastique. Le problème de Stokes issu des équations de Navier-Stokes sans les termes non linéaires (une partie de la poussée d’Archiméde est prise en compte) est défini ; l’éllipticité est démontrée, l’étude des modes propres et l’invariance liée à la pression sont détaillés. La méthode d’Uzawa est étendue à l’anélastique en mettant en évidence le découplage des vitesses en 3D, le cas particulier k = 0 et les modes parasites de pression. Le passage au multidomaine a permis d’établir les conditions de raccord (raccord Co de la pression sans condition aux limites physiques). Les algorithmes et l’implantation dans le code AMENOPHIS sont validés par les comparaisons de l’opérateur d’Uzawa développé en Fortran et à l’aide de Mathematica. De plus des résultats numériques ont été comparés à une expérience avec des fluides incompressibles. Finalement, une étude des solutions numériques obtenues avec les options anélastique et compressible a été menée. L’étude de l’influence de la stratification initiale des deux fluides sur le développement de l’instabilité de Rayleigh-Taylor est amorcée. / The « anelastic » approximation allows us to filter the acoustic waves thanks to an asymptotic development of the Navier-Stokes equations, so increasing the averaged time step, during the numerical simulation of hydrodynamic instabilitiesdevelopment. So, the anelastic equations for a two fluid mixture in case of Rayleigh-Taylor instability are established.The linear stability of Rayleigh-Taylor flow is studied, for the first time, for perfect fluids in the anelastic approximation.We define the Stokes problem resulting from Navier-Stokes equations without the non linear terms (a part of the buoyancyis considered) ; the ellipticity is demonstrated, the eigenmodes and the invariance related to the pressure are detailed.The Uzawa’s method is extended to the anelastic approximation and shows the decoupling speeds in 3D, the particular casek = 0 and the spurius modes of pressure. Passing to multidomain allowed to establish the transmission conditions.The algorithms and the implementation in the existing program are validated by comparing the Uzawa’s operator inFortran and Mathematica langages, to an experiment with incompressible fluids and results from anelastic and compressiblenumerical simulations. The study of the influence of the initial stratification of both fluids on the development of the Rayleigh-Taylor instability is initiated.