## THERMAL SCIENCE

International Scientific Journal

### A COMPRESSIBLE TURBULENCE MODEL FOR THE DISSIPATION RATE

**ABSTRACT**

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 incomp- ressible model of the turbulent dissipation rate is found to be an important issue for the compressible homogeneous turbulent shear flow.

**KEYWORDS**

PAPER SUBMITTED: 2022-04-08

PAPER REVISED: 2022-07-15

PAPER ACCEPTED: 2022-07-18

PUBLISHED ONLINE: 2022-12-17

**THERMAL SCIENCE** YEAR

**2023**, VOLUME

**27**, ISSUE

**Issue 2**, PAGES [1611 - 1626]

- Launder, B., et al., Progress in the Development of a Reynolds-Stress Turbulence Closure, Journal of Fluid Mech., 68 (1975), 3, pp. 537-566
- Speziale, C. G., et al., Modelling the Pressure Strain Correlation of Turbulence an Invariant Dynamical Systems Approach, Journal of Fluid Mech., 227 (1991), Apr., pp. 245-272
- Fu, S., et al., Accommodating the Effects of High Strain Rates in Modelling the Pressure Strain Correlation, UMIST Technical Report: TFD/5, Manchester, UK, 1987
- Speziale, C. G., Sarkar, S., Second Order Closure Models for Supersonic Turbulent Flows, NASA Langley Research Center, ICASE Report: 91-9, Hampton, Va., USA, 1991, pp. 1-22
- Speziale, C. G., et al., Evaluation of Reynolds Stress Turbulence Closures in Compressible Homogeneous Shear Flow, Journal of Applied Math. Physics, 46 (1994), Special Issue, pp. 5717-5736
- Blaisdell, G. A., et al., Compressibility Effects on the Growth and Structure of Homogeneous Turbulent Shear Flow, Journal of Fluid Mech., 256 (1993), 6S, pp. 443-485
- Sarkar, S., The Stabilizing Effects of Compressibility in Turbulent Shear Flows, Journal of Fluid Mech., 282 (1995), Apr., pp. 163-186
- Simone, S., et al., The effect of Compressibility On Turbulent Shear Flow: A Rapid Distortion-Theory and Direct Numerical Simulation Study, Journal of Fluid Mech., 330 (1997), 1, pp. 307-338
- Fujihiro, H., Effects of Pressure Fluctuations on Turbulence Growth Compressible Homogeneous Shear Flow, Journal of Physics of Fluids, 11 (1999), 6, pp. 1625-1635
- Adumitroiae, V., et al.,Progress in Favre Reynolds Stress Closures for Compressible Flows, Journal of Physics of Fluids, 11 (1999), 9, pp. 2696-2719
- Hung, S., Song Fu., Modelling of Pressure Strain Correlation Compressible Turbulent Flow, Acta Mech., Sin., 24 (2008), 1, pp. 37-43
- Park, C. H., Park, S. O., Compressible Turbulence Model for The Pressure-Strain Correlation, Journal of Turbulence, 6 (2005), 2, pp. 1-24
- Vreman, A. W., et al., Compressible Mixing Layer Growth Rate And Turbulence Characteristics, Journal of Fluid Mech., 330 (1996), Apr., pp. 235-258
- Hechmi, K., Taieb, L., On the Compressibility Effects in Mixing Layers, Thermal Science, 20 (2016), 5, pp. 1473-1484
- Hechmi, K., Taieb, L., An Extension of the SSG Model on Compressible Turbulent Flow, Journal of Applied Fluid Mechanics, 5 (2012), 4, pp. 101-111
- Hechmi, K., et al., A Priori Evaluation of The Pantano and Sarkar Model in Compressible Homogeneous Shear Flow, Comptes Rendus Mécanique, 339 (2011), 1, pp. 27-34
- Hechmi, K., A Compressibility Corrections of the Pressure Strain Linear Part Model, Thermal Science, 22 (2018), 1B, pp. 453-466
- Hechmi, K.,Taieb, L., A compressibility Correction of the Pressure Strain Correlation Model in Turbulent Flow, Comptes Rendus Mécanique, 341 (2013), 7, pp. 567-581
- Hechmi, K., Adnen, B., A Compressible Model for Pressure Strain, Fluids, 34 (2022), 7, pp. 1-20
- Zeman, O., Dilatational Dissipation, the Concept and Application in Modelling Compressible Mixing Layers, Physics of Fluids, A2 (1990), 2, pp. 178-188
- Sarkar, S., et al., The Analysis and Modelling of Dilatational terms in Compressible Turbulence, Journal of Fluid Mechanics, 227 (1991), Apr., pp. 473-493
- Taulbee, D., Van Osdol, J., Modelling Turbulent Compressible Flows, the Mass Fluctuating Velocity and Squared Density, Proceedings, 29th Aerospace Sciences Meeting, Reno, Nev., USA, 1991
- Fujiwra, H., et al., A Turbulence Model for the Pressure Strain Correlation term Accounting for Compressibility Effects, International Journal of Heat and Flows, 21 (2000), 3, pp. 354-358
- Fujiwara, H., Arakawa, C., Modelling of Turbulent Flows with Emphasis on Pressure Dilatation Correlation, Engineering Turbulence Modelling and Experiments, 3 (1996), pp. 151-160
- Tennekes, H., Lumley, J. L., A First Course in Turbulence, MIT Press, Cambridge, Mass., USA, Paper 91-0524, 1972
- Pantano, C., Sarkar, S., A Study of Compressibility Effects in the High-Speed Turbulent Shear Layer Using Direct Simulation, Journal of Fluid Mech., 451 (2002), Feb., pp. 329-371
- Stefan, H., A Model for the Reduction of the Turbulent Energy Redistribution by Compressibility, Journal of Physics of Fluids, 15 (2003), 11, pp. 3580-3583
- Aicha, H., et al., Evaluation Study of Pressure-Strain Correlation Models Compressible Turbulent Flow, Journal of Applied Fluid Mechanics, 9 (2016), 6, pp. 2685-2693