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Dynamic -equation model for large-eddy simulation of compressible flows

  • Xiaochuan Chai (a1) and Krishnan Mahesh (a1)


This paper presents a dynamic one-equation eddy viscosity model for large-eddy simulation (LES) of compressible flows. The transport equation for subgrid-scale (SGS) kinetic energy is introduced to predict SGS kinetic energy. The exact SGS kinetic energy transport equation for compressible flows is derived formally. Each of the unclosed terms in the SGS kinetic energy equation is modelled separately and dynamically closed, instead of being grouped into production and dissipation terms, as in the Reynolds averaged Navier–Stokes equations. All of the SGS terms in the filtered total energy equation are found to reappear in the SGS kinetic energy equation. Therefore, these terms can be included in the total energy equation without adding extra computational cost. A priori tests using direct numerical simulation (DNS) of decaying isotropic turbulence show that, for a Smagorinsky-type eddy viscosity model, the correlation between the SGS stress and the model is comparable to that from the original model. Also, the suggested model for the pressure dilatation term in the SGS kinetic energy equation is found to have a high correlation with its actual value. In a posteriori tests, the proposed dynamic -equation model is applied to decaying isotropic turbulence and normal shock–isotropic turbulence interaction, and yields good agreement with available experimental and DNS data. Compared with the results of the dynamic Smagorinsky model (DSM), the -equation model predicts better energy spectra at high wavenumbers, similar kinetic energy decay and fluctuations of thermodynamic quantities for decaying isotropic turbulence. For shock–turbulence interaction, the -equation model and the DSM predict similar evolutions of turbulent intensities across shocks, owing to the dominant effect of linear interaction. The proposed -equation model is more robust in that local averaging over neighbouring control volumes is sufficient to regularize the dynamic procedure. The behaviour of pressure dilatation and dilatational dissipation is discussed through the budgets of the SGS kinetic energy equation, and the importance of the dilatational dissipation term is addressed.


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1. Bedford, K. W. & Yeo, W. K. 1993 Conjunctive filtering procedures in surface water flow and transport. In Large Eddy Simulation of Complex Engineering and Geophysical Flows (ed. Galperin, B. & Orszag, S. A. ), pp. 513539. Cambridge University Press.
2. Chai, X. & Mahesh, K. 2011 Simulations of high speed jets in cross-flows. AIAA Paper 2011-650.
3. Chollet, J. P. & Lesieur, M. 1981 Parametrization of small scales of three-dimensional isotropic turbulence utilizing spectral closures. J. Atmos. Sci. 38, 27472757.
4. Comte-Bellot, G. & Corrsin, S. 1971 Simple Eulerian time correlation of full- and narrow-band velocity signals in grid-generated isotropic turbulence. J. Fluid Mech. 48, 273.
5. Deardorff, J. W. 1973 Three-dimensional numerical modeling of the planetary boundary layer. In Workshop on Micrometeorology (ed. Haugen, D. A. ), pp. 271311. American Meteorological Society.
6. Dubois, T., Domaradzki, J. A. & Honein, A. 2002 The subgrid-scale estimation model applied to large eddy simulations of compressible turbulence. Phys. Fluids 14 (5), 17811801.
7. Erlebacher, G., Hussaini, M. Y., Kreiss, H. O. & Sarkar, S. 1990 The analysis and simulation of compressible turbulence. Theor. Comput. Fluid Dyn. 2, 7395.
8. Garnier, E., Adams, N. & Sagaut, P. 2009 Large Eddy Simulation for Compressible Flows (Scientific Computation), 1st edn. Springer.
9. Génin, F. & Menon, S. 2010 Dynamics of sonic jet injection into supersonic cross-flow. J. Turbul. 11, 113.
10. Germano, M., Piomelli, U., Moin, P. & Cabot, M. 1991 A dynamic subgrid-scale eddy viscosity model. Phys. Fluids 3, 1760.
11. Ghosal, S., Lund, T. S., Moin, P. & Akselvoll, K. 1995 A dynamic localization model for large-eddy simulation of turbulent flows. J. Fluid Mech. 286, 229.
12. Ghosh, S. & Mahesh, K. 2008 Numerical simulation of the fluid dynamic effects of laser energy deposition in air. J. Fluid Mech. 605, 329354.
13. Horiuti, K. 1985 Large eddy simulation of turbulent channel flow by one-equation modeling. J. Phys. Soc. Japan 54, 28552865.
14. Kraichnan, R. H. 1964 Direct-interaction approximation for shear and thermally driven turbulence. Phys. Fluids 7 (7), 10481062.
15. Kraichnan, R. H. 1976 Eddy viscosity in two and three dimensions. J. Atmos. Sci. 33, 15211536.
16. Larsson, J. & Lele, S. K. 2009 Direct numerical simulation of canonical shock/turbulence interaction. Phys. Fluids 21, 126101.
17. Lee, S., Lele, S. K. & Moin, P. 1993 Direct numerical simulation of isotropic turbulence interacting with a weak shock wave. J. Fluid Mech. 251, 533562.
18. Lesieur, M. & Métais, O. 1996 New trends in large-eddy simulations of turbulence. Annu. Rev. Fluid Mech. 28, 4582.
19. Lilly, D. K. 1967 The representation of small-scale turbulence in numerical simulation experiments. In IBM Scientific Computing Symposium on Environmental Sciences, Yorktown Heights, pp. 195210. IBM.
20. Lilly, D. K. 1991 A proposed modification of the Germano subgrid-scale closure method. Phys. Fluids A 3, 1760.
21. Mahesh, K., Lele, S. K. & Moin, P. 1997 The influence of entropy fluctuations on the interaction of turbulence with a shock wave. J. Fluid Mech. 334, 353379.
22. Mason, P. J. 1994 Large-eddy simulation: a critical review of the technique. Q. J. R. Meteorol. Soc. 120 (515), 126.
23. Meneveau, C. 1994 Statistics of turbulence subgrid-scale stresses: necessary conditions and experimental tests. Phys. Fluids 6 (2), 815833.
24. Meneveau, C. & Katz, J. 2000 Scale-invariance and turbulence models for large-eddy simulation. Annu. Rev. Fluid Mech. 32 (1), 132.
25. Menon, S. & Kim, W. W. 1996 High Reynolds number flow simulations using the localized dynamic subgrid-scale model. In AIAA 34th Aerospace Sciences Meeting and Exhibit. AIAA Paper 1996-0425.
26. Métais, O. & Lesieur, M. 1992 Spectral large-eddy simulation of isotropic and stably stratified turbulence. J. Fluid Mech. 239, 157194.
27. Moeng, C.-H. 1984 A large-eddy-simulation model for the study of planetary boundary-layer turbulence. J. Atmos. Sci. 41, 20522062.
28. Moin, P., Squires, K., Cabot, W. & Lee, S. 1991 A dynamic subgrid-scale model for compressible turbulence and scalar transport. Phys. Fluids A 3 (11), 27462757.
29. Muppidi, S. & Mahesh, K. 2011 DNS of roughness-induced transition in supersonic boundary layers. J. Fluid Mech. 693, 2856.
30. Park, N. & Mahesh, K. 2007 Numerical and modeling issues in LES of compressible turbulent flows on unstructured grids. AIAA Paper 2007-0722.
31. Piomelli, U., Cabot, W. H., Moin, P. & Lee, S. 1991 Subgrid-scale backscatter in turbulent and transitional flows. Phys. Fluids A 3, 17661771.
32. Pomraning, E. & Rutland, C. J. 2002 Dynamic one-equation nonviscosity large-eddy simulation model. AIAA J. 40, 689701.
33. Ristorcelli, J. R. 1997 A pseudo-sound constitutive relationship for the dilatational covariances in compressible turbulence. J. Fluid Mech. 347, 3770.
34. Ristorcelli, J. R. & Blaisdell, G. A. 1997 Consistent initial conditions for the DNS of compressible turbulence. Phys. Fluids 9 (1), 46.
35. Sarkar, S., Erlebacher, G., Hussaini, M. Y. & Kreiss, H. O. 1991 The analysis and modeling of dilatational terms in compressible turbulence. J. Fluid Mech. 227, 473493.
36. Schumann, U. 1975 Subgrid scale model for finite difference simulations of turbulent flows in plane channels and annuli. J. Comput. Phys. 18 (4), 376404.
37. Shaw, R. H. & Schumann, U. 1992 Large-eddy simulation of turbulent flow above and within a forest. Boundary-Layer Meteorol. 61, 4764.
38. Smagorinsky, J. 1963 General circulation experiments with the primitive equations. I. The basic experiment. Mon. Weath. Rev. 91, 99165.
39. Smith, L. M. & Woodruff, S. L. 1998 Renormaliztion-group analysis of turbulence. Annu. Rev. Fluid Mech. 30, 275310.
40. Speziale, C. G. 1991 Analytic methods for the development of Reynolds-stress closures in turbulence. Annu. Rev. Fluid Mech. 23, 107157.
41. Speziale, C. G., Erlebacher, G., Zang, T. A. & Hussaini, M. Y. 1988 The subgrid-scale modeling of compressible turbulence. Phys. Fluids 31 (4), 940942.
42. Spyropoulos, E. T. & Blaisdell, G. A. 1996 Evaluation of the dynamic model for simulations of compressible decaying isotropic turbulence. AIAA J. 34 (5), 990998.
43. Vreman, B., Geurts, B. & Kuerten, H. 1995 Subgrid-modeling in LES of compressible flow. Appl. Sci. Res. 54, 191203.
44. Yakhot, A., Orszag, S. A. & Yakhot, Y. 1989 Renormalization-group formulation of large-eddy simulations. J. Sci. Comput. 4, 139.
45. Yee, H. C, Sandham, N. D & Djomehri, M. J 1999 Low-dissipative high-order shock-capturing methods using characteristic-based filters. J. Comput. Phys. 150 (1), 199238.
46. Yoshizawa, A. 1986 Statistical theory for compressible turbulent shear flows, with the application to subgrid modeling. Phys. Fluids 29, 2152.
47. Yoshizawa, A. & Horiuti, K. 1985 A statistically-derived subgrid-scale kinetic energy model for the large-eddy simulation of turbulent flows. J. Phys. Soc. Japan 54, 28342839.
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