Compressible Homogeneous Shear Flow Research Articles

A compressible turbulence model for the dissipation rate

In this work, the ability of a Reynolds stress model to compute turbulent homogeneous shear flow with significant compressibility effects is discussed. Several studies of compressible turbulent flows carried out in the past years have shown that the pressure strain correlation is mainly responsible for the strong changes in the magnitude of the Reynolds stress anisotropies. Two recent compressible models of this term are considered in conjunction with the standard model of the dissipation rate of the turbulent kinetic energy to predict compressible homogeneous flow highly sheared are tested. It is found that deficiencies appear in the calculations even if the pressure strain model is improved by compressibility corrections. Consistent with earlier studies, this deficiency is attributed to the use of the incompressible model for turbulent dissipation. However, a compressibility correction of this equation model uncovers the main focus of the present study. This correction makes the standard coefficients of this equation depend on the turbulent gradient Mach and Mach numbers. The proposed model is tested for low and strong compressibility cases from the DNS results of Sarkar. A comparison of the proposed model predictions with the DNS results shows good qualitative agreement. Therefore, compressibility correction for the incompressible model of the turbulent dissipation rate is found to be an important issue for the compressible homogeneous turbulent shear flow.

A Comparative Study for Compressible Turbulence Models

The incompressible models for the pressure-strain correlation are unable to correctly predict the turbulence flows evolving with significant compressibility. Huang and Fu use a damping function of the turbulent Mach number to modify two numerical coefficients of the incompressible model for the pressure strain developed by Launder, Reece and Rodi. This model predicts the spreading rate and the shear stress behavior in compressible turbulent mixing well. However, the model does not show the well-known compressibility effects on the compressible homogenous shear flow. In the present work, the model of Huang-Fu is revised, all resulting model coefficients become dependent on the turbulent Mach number, the gradient Mach number and the convective Mach number. The proposed model is tested in different compressible turbulent homogeneous shear flow and mixing layers cases. In general, the predicted results from the proposed model are in an acceptable agreement with DNS and experiment data.

A compressibility corrections of the pressure strain linear part models

The main focus of this paper is the analysis of the compressibility effects and the validation of some recent Reynolds stress models for computing compressible turbulent flows. The pressure strain correlation is one of the several terms appearing in the Reynolds stress equation which directly reflect the compressibility effects on the turbulence. For this reason, a special attention is paid to the modeling of this term in order to account for compressibility effects at high-speed. The models developed by Speziale Sarkar and Gatski (SSG) and Fu, Launder and Tselepidakis (FLT) for the pressure strain correlation are examined to be extended to compressible turbulent flows. A compressibility corrections of these models using the turbulent Mach number are proposed. The calculations have been performed for the compressible homogeneous shear flow and the turbulent plate mixing-layers. The comparison of the proposed compressibility modifications of the SSG and FLT models with its universal version shows some important ameliorations in results for the majority characteristic parameter of the structural compressibility effects. It?s found that the predicted results from the modified SSG and FLT models are in reasonable agreement with the accepted data.

Evaluation Study of Pressure-Strain Correlation Models in Compressible Flow

This paper is devoted to the second-order closure for compressible turbulent flows with special attention paid to modeling the pressure-strain correlation appearing in the Reynolds stress equation. This term appears as the main one responsible for the changes of the turbulence structures that arise from structural compressibility effects. The structure of the gradient Mach number is similar to that of turbulence, therefore this parameter may be appropriate to study the changes in turbulence structures that arise from structural compressibility effects. Thus, the incompressible model (LRR) of the pressure-strain correlation and its corrected form by using the turbulent Mach number, fail to correctly evaluate the compressibility effects at high shear flow. An extension of the widely used incompressible model (LRR) on compressible homogeneous shear flow is the major aim of the present work. From this extension the standard coefficients Ci became a function of the compressibility parameters (the turbulent Mach number and the gradient Mach number). Application of the model on compressible homogeneous shear flow by considering various initial conditions shows reasonable agreement with the DNS results of Sarkar. The ability of the models to predict the equilibrium states for the flow in cases A1 and A4 from DNS results of Sarkar is examined, the results appear to be very encouraging. Thus, both parameters Mt and Mg should be used to model significant structural compressibility effects at high-speed shear flow.

Turbulent kinetic energy production and flow structures in compressible homogeneous shear flow

The production of turbulent kinetic energy (TKE) and flow structures in compressible homogeneous turbulent shear flow (HTSF) are investigated by using direct numerical simulation. A theoretical analysis suggests that the vertical turbulent transport process should be responsible for the production of TKE in HTSF. It is manifested based on a conditional average method that the pure TKE production becomes increasingly larger in strain regions than in vortex regions of the flow. The velocity-derivative correlation in the shear plane is employed to identify the streaky structures in HTSF, which also tend to occur predominantly in strain regions of turbulence. Localized analyses of conditional averages reveal that the streaky structures in compressible HTSF are closely related to the negative productions of TKE. A comparative study implies that flow compressibility has a considerable effect on the spatial distributions and patterns of the strain- and vortex-dominated fields, which in turn cause the discrepancies in distribution of the TKE production and streaky structures between incompressible and compressible HTSFs.

Toward second-moment closure modelling of compressible shear flows

AbstractCompressibility profoundly affects many aspects of turbulence in high-speed flows, most notably stability characteristics, anisotropy, kinetic–potential energy interchange and spectral cascade rate. We develop a unified framework for modelling pressure-related compressibility effects by characterizing the role and action of pressure in different speed regimes. Rapid distortion theory is used to examine the physical connection between the various compressibility effects leading to model form suggestions for pressure–strain correlation, pressure–dilatation and dissipation evolution equations. The closure coefficients are established using fixed-point analysis by requiring consistency between model and DNS asymptotic behaviour in compressible homogeneous shear flow. The closure models are employed to compute high-speed mixing layers and boundary layers. The self-similar mixing-layer profile, increased Reynolds stress anisotropy and diminished mixing-layer growth rates with increasing Mach number are all well captured. High-speed boundary-layer results are also adequately replicated even without the use of advanced thermal-flux models or low-Reynolds-number corrections.

An Extension of the SSG Model on Compressible Turbulent Flow

This work focuses on the performance and validation of some recent Reynolds stress models in compressible homogeneous shear flow. The SSG model developed by Speziale Sarkar and Gatski has shown a great success in simulating a variety of incompressible complex turbulent flows. On the other hand, it has not predicted correctly the compressible turbulence at high speed shear flow. Thus, a compressibility correction for this model is the major aim of this study. In the present work, two recent compressible models for the pressure strain-strain correlation have been used to modify the linear term of the SSG model. These modifications make the linear term dependent on a turbulent Mach number. In addition, compressibility correction model for the slow part of the pressure strain is proposed. The obtained results are compared with DNS results of Sarkar. The results show that important parameters characteristic of compressibility in homogeneous turbulent shear flow are well captured by the extended SSG model.

A priori evaluation of the Pantano and Sarkar model in compressible homogeneous shear flows

In this study, a Reynolds stress closure, including the Pantano and Sarkar model of the mean part of the pressure–strain correlation is used for the computation of compressible homogeneous at high-speed shear flow. Several studies concerning the compressible homogeneous shear flow show that the changes of the turbulence structures are principally due to the structural compressibility effects which significantly affect the pressure field and then the pressure–strain correlation. Eventually, this term appears as the main term responsible for the changes in the magnitude of the Reynolds stress anisotropies. The structure of the gradient Mach number is similar to that of turbulence, therefore this parameter may be appropriate to study the changes in turbulence structures that arise from structural compressibility effects. Thus, the incompressible model of the pressure strain correlation and its corrected form by using the turbulent Mach turbulent only, fail to correctly evaluate the compressibility effects at high shear flow. An extension of the widely used incompressible Launder, Reece and Rodi model on compressible homogeneous shear flow is the major aim of the present work. From this extension, the standard coefficients C i become a function of the extra compressibility parameters (the turbulent Mach number M t and the gradient Mach number M g ) through the Pantano and Sarkar model. Application of the model on compressible homogeneous shear flow by considering various initial conditions shows reasonable agreement with the DNS results of Simone et al. and Sarkar. The observed trend of the dramatic increase in the normal Reynolds stress anisotropies, the significant decrease in the Reynolds shear stress anisotropy and the increase of the turbulent kinetic energy amplification rate with increasing the gradient Mach number are well predicted by the model. The ability of the model to predict the equilibrium states for the flow in cases A 1 to A 4 from DNS results of Sarkar is examined, the results appear to be very encouraging. Thus, both parameters M t and M g should be used to model significant structural compressibility effects at high-speed shear flow.

Analyzing the influence of compressibility on the rapid pressure–strain rate correlation in turbulent shear flows

The influence of compressibility on the rapid pressure–strain rate tensor is investigated using the Green’s function for the wave equation governing pressure fluctuations in compressible homogeneous shear flow. The solution for the Green’s function is obtained as a combination of parabolic cylinder functions; it is oscillatory with monotonically increasing frequency and decreasing amplitude at large times, and anisotropic in wave-vector space. The Green’s function depends explicitly on the turbulent Mach number M t , given by the root mean square turbulent velocity fluctuations divided by the speed of sound, and the gradient Mach number M g , which is the mean shear rate times the transverse integral scale of the turbulence divided by the speed of sound. Assuming a form for the temporal decorrelation of velocity fluctuations brought about by the turbulence, the rapid pressure–strain rate tensor is expressed exactly in terms of the energy (or Reynolds stress) spectrum tensor and the time integral of the Green’s function times a decaying exponential. A model for the energy spectrum tensor linear in Reynolds stress anisotropies and in mean shear is assumed for closure. The expression for the rapid pressure–strain correlation is evaluated using parameters applicable to a mixing layer and a boundary layer. It is found that for the same range of M t there is a large reduction of the pressure–strain correlation in the mixing layer but not in the boundary layer. Implications for compressible turbulence modeling are also explored.

Extension of the Launder, Reece and Rodi model on compressible homogeneous shear flow

This article describes the second order closure progress that was made to calculate compressible homogeneous shear flow with significant compressibility. Several DNS results show that compressibility has an important effect on the pressure-strain correlation. The term recognized as the principal responsible for the change in the magnitude of Reynolds-stress anisotropies. Thus, the pressure-strain incompressible models do not correctly predict compressible turbulence at high-speed shear flow. A method of including compressibility effects in the pressure-strain correlation is the subject of the present study. The concept of the growth rate of turbulent kinetic energy can be used to construct a compressible correction to the Launder, Reece and Rodi model for the pressure-strain correlation. This correction concerns essentially the C 1,C 3 and C 4 coefficients which become in a compressible turbulence situation a function of the turbulent Mach number. The application of the new model shows good agreement with DNS results of Sarkar for cases A 1, A 2 and A 3. These cases correspond to a moderate mean shear rate, so that nonlinear effects are important.

A compressible turbulence model for the pressure–strain correlation

Recent DNS (Direct Numerical Simulation) results indicate that the main compressibility effect comes from the reduced pressure–strain term due to reduced pressure fluctuations. Using the concept of moving equilibrium in compressible homogeneous shear flow, we develop a new compressible pressure–strain model by modifying the incompressible linear pressure–strain model, which is characterized by a decrease in the primary redistribution term but an increase in the secondary redistribution term. The model was applied to a compressible mixing layer. The predicted results show that the growth rate is reduced with increasing convective Mach number and the Reynolds stresses are reduced. All the components of the pressure–strain term and the shear stress anisotropy are reduced with the increase of the compressibility, whereas the normal stress anisotropy is increased—then results are in accord with experimental and DNS results.

The effects of heat release on the energy exchange in reacting turbulent shear flow

The energy exchange between the kinetic and internal energies in non-premixed reacting compressible homogeneous turbulent shear flow is studied via data generated by direct numerical simulations (DNS). The chemical reaction is modelled by a one- step exothermic irreversible reaction with Arrhenius-type reaction rate. The results show that the heat release has a damping effect on the turbulent kinetic energy for the cases with variable transport properties. The growth rate of the turbulent kinetic energy is primarily in uenced by the reaction through temperature-induced changes in the solenoidal dissipation and modifications in the explicit dilatational terms (pressure–dilatation and dilatational dissipation). The production term in the scaled kinetic energy equation, which is proportional to the Reynolds shear stress anisotropy, is less affected by the heat release. However, the dilatational part of the production term increases during the time when the reaction is important. Additionally, the pressure–dilatation correlation, unlike the non-reacting case, transfers energy in the reacting cases, on the average, from the internal to the kinetic energy. Consequently, the dilatational part of the kinetic energy is enhanced by the reaction. On the contrary, the solenoidal part of the kinetic energy decreases in the reacting cases mainly due to an enhanced viscous dissipation. Similarly to the non-reacting case, it is found that the direct coupling between the solenoidal and dilatational parts of the kinetic energy is small. The structure of the flow with regard to the normal Reynolds stresses is affected by the heat of reaction. Compared to the non-reacting case, the kinetic energy in the direction of the mean velocity decreases during the time when the reaction is important, while it increases in the direction of the shear. This increase is due to the amplification of the dilatational kinetic energy in the x2-direction by the reaction. Moreover, the dilatational effects occur primarily in the direction of the shear. These effects are amplified if the heat release is increased or the reaction occurs at later times. The non-reacting models tested for the explicit dilatational terms are not supported by the DNS data for the reacting cases, although it appears that some of the assumptions employed in these models hold also in the presence of heat of reaction.

Effects of pressure fluctuations on turbulence growth in compressible homogeneous shear flow

Direct numerical simulation results are used to study compressibility effects on turbulent energy growth in homogeneous shear flow. Normalized amplitude of the pressure fluctuations is shown to decrease as the turbulent Mach number increases. This decrease causes the reduction in the pressure–strain term, changes the anisotropy of the Reynolds stress, and reduces the growth rate of turbulent kinetic energy as is the case of mixing layers. The transport equation for the pressure variance is examined in detail. It is shown that when the turbulent Mach number is high the pressure-variance dissipation term is not negligible and contributes to the reduction in the pressure fluctuations. The pressure-variance dissipation is closely related to the high-wavenumber part of the pressure-variance spectrum. Profile of the spectrum is observed to depend on the turbulent Mach number. A statistical theory called the two-scale direct-interaction approximation is applied to obtain model expressions for pressure-related terms. The pressure–dilatation correlation is a dominant term in the pressure-variance equation; its model expression as derived contains the material derivative of turbulent kinetic energy. This quantity represents nonequilibrium properties of turbulent field and can explain the very small value of the pressure dilatation in boundary layers and channel flows. The model expression for the pressure–strain correlation is also derived; the normalized pressure variance can be a parameter that describes compressibility effect on the pressure strain.

The stabilizing effect of compressibility in turbulent shear flow

Direct numerical simulation of turbulent homogeneous shear flow is performed in order to clarify compressibility effects on the turbulence growth in the flow. The two Mach numbers relevant to homogeneous shear flow are the turbulent Mach number Mt and the gradient Mach number Mg. Two series of simulations are performed where the initial values of Mg and Mt are increased separately. The growth rate of turbulent kinetic energy is observed to decrease in both series of simulations. This ‘stabilizing’ effect of compressibility on the turbulent energy growth rate is observed to be substantially larger in the DNS series where the initial value of Mg is changed. A systematic comparison of the different DNS cases shows that the compressibility effect of reduced turbulent energy growth rate is primarily due to the reduced level of turbulence production and not due to explicit dilatational effects. The reduced turbulence production is not a mean density effect since the mean density remains constant in compressible homogeneous shear flow. The stabilizing effect of compressibility on the turbulence growth is observed to increase with the gradient Mach number Mg in the homogeneous shear flow DNS. Estimates of Mg for the mixing layer and the boundary layer are obtained. These estimates show that the parameter Mg becomes much larger in the high-speed mixing layer relative to the high-speed boundary layer even though the mean flow Mach numbers are the same in the two flows. Therefore, the inhibition of turbulent energy production and consequent ‘stabilizing’ effect of compressibility on the turbulence (over and above that due to any mean density variation) is expected to be larger in the mixing layer relative to the boundary layer, in agreement with experimental observations.

Statistical analysis of the rate of strain tensor in compressible homogeneous turbulence

Recent analysis of direct numerical simulations of compressible homogeneous shear flow turbulence has unraveled some of the energy transfer mechanisms responsible for the decrease of kinetic energy growth when the flow becomes more compressible. In this complementary study, attention is focused on the rate of strain tensor. A Helmholtz decomposition of the velocity field leads to a consideration of a solenoidal and an irrotational rate of strain tensor. Their eigenvalue distributions, eigenvector orientations, and the relative alignment between the eigenvectors and the vorticity and pressure gradient vectors are examined with the use of probability density functions. The irrotational rate of strain tensor is found to have a preferred structure in regions of strong dilatation. This structure depends on the mean shear, and is quite different from that of the solenoidal rate of strain tensor. Compressibility strongly affects the orientation properties of the pressure gradient vector.

Compressibility effects on the growth and structure of homogeneous turbulent shear flow

Compressibility effects within decaying isotropic turbulence and homogeneous turbulent shear flow have been studied using direct numerical simulation. The objective of this work is to increase our understanding of compressible turbulence and to aid the development of turbulence models for compressible flows. The numerical simulations of compressible isotropic turbulence show that compressibility effects are highly dependent on the initial conditions. The shear flow simulations, on the other hand, show that measures of compressibility evolve to become independent of their initial values and are parameterized by the root mean square Mach number. The growth rate of the turbulence in compressible homogeneous shear flow is reduced compared to that in the incompressible case. The reduced growth rate is the result of an increase in the dissipation rate and energy transfer to internal energy by the pressure–dilatation correlation. Examination of the structure of compressible homogeneous shear flow reveals the presence of eddy shocklets, which are important for the increased dissipation rate of compressible turbulence.