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Toward second-moment closure modelling of compressible shear flows

Published online by Cambridge University Press:  23 September 2013

Carlos A. Gomez  and
Sharath S. Girimaji
Carlos A. Gomez
Affiliation:
Department of Aerospace Engineering, Texas A&M University, College Station, TX 77843-3141, USA
Sharath S. Girimaji*
Affiliation:
Department of Aerospace Engineering, Texas A&M University, College Station, TX 77843-3141, USA
*
†Email address for correspondence: [email protected]
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Abstract

Compressibility 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.

JFM classification

Turbulent Flows: Compressible turbulence Turbulent Flows: Shear layer turbulence Turbulent Flows: Turbulence modelling
Type
Papers
Information
Journal of Fluid Mechanics , Volume 733 , 25 October 2013 , pp. 325 - 369
Copyright
©2013 Cambridge University Press 

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References

Adumitroaie, V., Ristorcelli, J. R. & Taulbee, D. B. 1999 Progress in Favré–Reynolds stress closures for compressible flows. Phys. Fluids 11 (9), 2696–2719.Google Scholar
ANSYS, 2010 ANSYS® FLUENT Theory Guide, ANSYS®, Inc, Southpointe 275 Technology Drive Canonsburg, PA 15317, Release 13.0.Google Scholar
Aupoix, B. 2004 Modelling of compressibility effects in mixing layers. J. Turbul. 5, N7.Google Scholar
Bertsch, R. 2010 Rapidly sheared compressible turbulence: characterization of different pressure regimes and effect of thermodynamic fluctuations. Master’s thesis, Texas A&M University.Google Scholar
Bertsch, R. L., Suman, S. & Girimaji, S. S. 2012 Rapid distortion analysis of high Mach number homogeneous shear flows: characterization of flow-thermodynamics interaction regimes. Phys. Fluids 24 (12), 125106.CrossRefGoogle Scholar
Bowersox, R. D. W. 2009 Extension of equilibrium turbulent heat flux models to high-speed shear flows. J. Fluid Mech. 633, 61–70.Google Scholar
Cambon, C., Coleman, G. N. & Mansour, N. N. 1993 Rapid distortion analysis and direct simulation of compressible homogeneous turbulence at finite mach number. J. Fluid Mech. 257, 641–665.CrossRefGoogle Scholar
Chaouat, B. & Schiestel, R. 2005 A new partially integrated transport model for subgrid-scale stresses and dissipation rate for turbulent developing flows. Phys. Fluids 17 (6), 065106.Google Scholar
Chinzei, N., Masuya, G., Komuro, T., Murakami, A. & Kudou, K. 1986 Spreading of two-stream supersonic turbulent mixing layers. Phys. Fluids 29 (5), 1345–1347.Google Scholar
Clemens, N. T. & Mungal, M. G. 1995 Large-scale structure and entrainment in the supersonic mixing layer. J. Fluid Mech. 284, 171–216.Google Scholar
Crow, S. C. 1968 Viscoelastic properties of fine-grained incompressible turbulence. J. Fluid Mech. 33, 1–20.Google Scholar
Duan, L., Beekman, I. & Martín, M. P. 2011 Direct numerical simulation of hypersonic turbulent boundary layers. Part 3. Effect of Mach number. J. Fluid Mech. 672, 245–267.Google Scholar
Durbin, P. A. & Speziale, C. G. 1994 Realizability of second-moment closure via stochastic analysis. J. Fluid Mech. 280, 395–407.Google Scholar
Durbin, P. A. & Zeman, O. 1992 Rapid distortion theory for homogeneous compressed turbulence with application to modelling. J. Fluid Mech. 242, 349–370.CrossRefGoogle Scholar
Eliasson, P. 2005 EDGE, a Navier–Stokes solver for unstructured grids. Tech. Rep. FOI-R-0298-SE. FOI.Google Scholar
Eliasson, P. & Peng, S.-H. 2008 Drag prediction for the DLR-F6 wing-body configuration using the EDGE solver. J. Aircraft 45 (3), 837–847.CrossRefGoogle Scholar
Freund, J. B., Lele, S. K. & Moin, P. 2000 Compressibility effects in a turbulent annular mixing layer. Part 1. Turbulence and growth rate. J. Fluid Mech. 421, 229–267.Google Scholar
Fujiwara, H., Matsuo, Y. & Chuichi, A. 2000 A turbulence model for the pressure–strain correlation term accounting for compressibility effects. Intl J. Heat Fluid Flow 21 (3), 354–358.Google Scholar
Gatski, T. B. & Jongen, T. 2000 Nonlinear eddy viscosity and algebraic stress models for solving complex turbulent flows. Prog. Aeronaut. Sci. 36, 655–682.CrossRefGoogle Scholar
Gatski, T. B. & Speziale, C. G. 1993 On explicit algebraic stress models for complex turbulent flows. J. Fluid Mech. 254, 59–78.Google Scholar
Germano, M. 1992 Turbulence: the filtering approach. J. Fluid Mech. 238, 325–336.CrossRefGoogle Scholar
Girimaji, S. S. 1996 Fully explicit and self-consistent algebraic Reynolds stress model. Theor. Comput. Fluid Dyn. 8, 387–402.Google Scholar
Girimaji, S. S. 1997 A Galilean invariant explicit algebraic Reynolds stress model for turbulent curved flows. Phys. Fluids 9 (4), 1067–1077.Google Scholar
Girimaji, S. S. 2000 Pressure-strain correlation modelling of complex turbulent flows. J. Fluid Mech. 422, 91–123.Google Scholar
Girimaji, S. S. 2004 A new perspective on realizability of turbulence models. J. Fluid Mech. 512, 191–210.Google Scholar
Girimaji, S. S. 2006 Partially-averaged Navier–Stokes model for turbulence: a Reynolds-averaged Navier–Stokes to direct numerical simulation bridging method. Trans. ASME: J. Appl. Mech. 73 (3), 413–421.CrossRefGoogle Scholar
Girimaji, S. S., Jeong, E. & Srinivasan, R. 2006 Partially-averaged Navier–Stokes method for turbulence: fixed point analysis and comparison with unsteady partially averaged Navier–Stokes. Trans. ASME: J. Appl. Mech. 73 (3), 422–429.Google Scholar
Goebel, S. G. & Dutton, J. C. 1991 Experimental study of compressible turbulent mixing layers. AIAA J. 29 (4), 538–546.Google Scholar
Hall, J. L., Dimotakis, P. E. & Rosemann, H. 1993 Experiments in nonreacting compressible shear layers. AIAA J. 31 (12), 2247–2254.Google Scholar
Hellsten, A. 2005 New advanced $k$ – $\omega $ turbulence model for high-lift aerodynamics. AIAA J. 43 (9), 1857–1869.Google Scholar
Huang, S. & Fu, S. 2008 Modelling of pressure–strain correlation in compressible turbulent flow. Acta Mechanica Sin. 24 (1), 37–43.Google Scholar
Johansson, A. V. & Hallbäck, M. 1994 Modelling of rapid pressure–strain in Reynolds-stress closures. J. Fluid Mech. 269, 143–168.Google Scholar
Jones, W. P. & Musonge, P. 1988 Closure of the Reynolds stress and scalar flux equations. Phys. Fluids 31 (12), 3589–3604.Google Scholar
Khlifi, H., Abdallah, J., Aïcha, H. & Taïeb, L. 2011 A priori evaluation of the Pantano and Sarkar model in compressible homogeneous shear flows. C. R. Mec. 339, 27–34.Google Scholar
Kim, J. & Park, S. O. 2010 New compressible turbulence model for free and wall-bounded shear layers. J. Turbul. 11 (10), 1–20.Google Scholar
Kline, S. J., Cantwell, B. J. & Lilley, G. M. 1982 Proc. 1980-81 AFOSR-HTTM-Stanford Conf. on Complex Turbulent Flows, Vol. 1, Stanford University.Google Scholar
Launder, B. E., Reece, G. J. & Rodi, W. 1975 Progress in the development of a Reynolds-stress turbulence closure. J. Fluid Mech. 68, 537–566.Google Scholar
Lavin, T. A., Girimaji, S. S., Suman, S. & Yu, H. 2012 Flow-thermodynamics interactions in rapidly-sheared compressible turbulence. Theor. Comput. Fluid Dyn. 26 (6), 501–522.Google Scholar
Lee, K. & Girimaji, S. S. 2013 Flow-thermodynamics interactions in decaying anisotropic compressible turbulence with imposed temperature fluctuations. Theor. Comput. Fluid Dyn. 27 (1–2), 115–131.Google Scholar
Lee, K., Girimaji, S. S. & Kerimo, J. 2008 Validity of Taylor’s dissipation-viscosity independence postulate in variable-viscosity turbulent fluid mixtures. Phys. Rev. Lett. 101 (7), 074501.Google Scholar
Lien, F. S. & Leschziner, M. A. 1994 Assessment of turbulent-transport models including nonlinear RNG eddy-viscosity formulation and second-moment closure for flow over a backward-facing step. Comput. Fluids 23 (8), 983–1004.Google Scholar
Livescu, D., Jaberi, F. A. & Madnia, C. K. 2002 The effects of heat release on the energy exchange in reacting turbulent shear flow. J. Fluid Mech. 450, 35–66.Google Scholar
Livescu, D. & Madnia, C. K. 2004 Small scale structure of homogeneous turbulent shear flow. Phys. Fluids 16 (8), 2864–2876.Google Scholar
Livescu, D. & Ristorcelli, J. R. 2007 Buoyancy-driven variable-density turbulence. J. Fluid Mech. 591, 43–71.Google Scholar
Livescu, D. & Ristorcelli, J. R. 2008 Variable-density mixing in bouyancy-driven turbulence. J. Fluid Mech. 605, 145–180.Google Scholar
Lumley, J. L. 1978 Computational modeling of turbulent flows. Adv. Appl. Mech. 18, 123–176.Google Scholar
Marzougui, H., Khlifi, H. & Lili, T. 2005 Extension of the Launder, Reece and Rodi model on compressible homogeneous shear flow. Eur. Phys. J. B 45, 147–154.Google Scholar
Menter, F. R. 1994 Two-equation eddy-viscosity turbulence models for engineering applications. AIAA J. 32 (8), 1598–1605.Google Scholar
Mishra, A. A. & Girimaji, S. S. 2010 Pressure-strain correlation modeling: towards achieving consistency with rapid distortion theory. Flow Turbul. Combust. 85, 593–619.Google Scholar
Owen, F. K. & Horstman, C. C. 1972 On the structure of hypersonic turbulent boundary layers. J. Fluid Mech. 53, 611–636.Google Scholar
Pantano, C. & Sarkar, S. 2002 A study of compressibility effects in the high-speed turbulent shear layer using direct simulation. J. Fluid Mech. 451, 329–371.Google Scholar
Papamoschou, D. & Roshko, A. 1988 The compressible turbulent shear layer: an experimental study. J. Fluid Mech. 197, 453–477.Google Scholar
Park, C. H. & Park, S. O. 2005 A compressible turbulence model for the pressure–strain correlation. J. Turbul. 6 (2), 1–25.Google Scholar
Pennisi, S. & Trovato, M. 1987 On the irreducibility of Professor G. F. Smith’s representations for isotropic functions. Intl J. Engng Sci. 25, 1059–1065.Google Scholar
Pope, S. B. 1975 A more general effective-viscosity hypothesis. J. Fluid Mech. 72 (2), 331–340.Google Scholar
Pope, S. B. 1994 Lagrangian PDF methods for turbulent flows. Annu. Rev. Fluid Mech. 26, 23–63.Google Scholar
Pope, S. B. 2000 Turbulent Flows, 1st edn. Cambridge University Press.Google Scholar
Reynolds, W. C. 1976 Computation of turbulent flows. Annu. Rev. Fluid Mech. 8, 183–208.Google Scholar
Ristorcelli, J. R. 1993 A representation for the turbulent mass flux contribution to Reynolds-stress and two-equation closure for compressible turbulence. ICASE Rep. 93-88. NASA Langley Research Center, Hampton, VA.Google Scholar
Ristorcelli, J. R., Lumley, J. L. & Abid, R. 1995 A rapid-pressure covariance representation consistent with the Taylor–Proudman theorem materially frame indifferent in the two-dimensional limit. J. Fluid Mech. 292, 111–152.Google Scholar
Rodi, W. 1976 A new algebraic relation for calculating the Reynolds stresses. Z. Angew. Math. Mech. 56, T219–T221.Google Scholar
Rotta, J. C. 1951 Statistiche theorie nichthomogener turbulenz. Z. Phys. 129, 547–572.Google Scholar
Samimy, M. & Elliot, G. S. 1990 Effects of compressibility on the characteristics of free shear layers. AIAA J. 28 (3), 439–445.Google Scholar
Sarkar, S. 1992 The pressure-dilatation correlation in compressible flows. Phys. Fluids 4 (12), 2674–2682.Google Scholar
Sarkar, S. 1995 The stabilizing effect of compressibility in turbulent shear flow. J. Fluid Mech. 282, 163–186.Google Scholar
Sarkar, S., Erlebacher, G. & Hussaini, M. Y. 1991a The analysis and modelling of dilatational terms in compressible turbulence. J. Fluid Mech. 227, 473–493.Google Scholar
Sarkar, S., Erlebacher, G. & Hussaini, M. Y. 1991b Direct simulation of compressible turbulence in a shear flow. Theor. Comput. Fluid Dyn. 2, 291–305.Google Scholar
Schlichting, H. & Gersten, K. 2000 Boundary Layer Theory, 8th edn. Springer.Google Scholar
Simone, A., Coleman, G. N. & Cambon, C. 1997 The effect of compressibility on turbulent shear flow: a rapid-distortion-theory and direct-numerical-simulation study. J. Fluid Mech. 330, 307–338.Google Scholar
Sjögren, T. & Johansson, A. V. 2000 Development and calibration of algebraic nonlinear models for terms in the Reynolds stress transport equations. Phys. Fluids 12 (6), 1554–1572.Google Scholar
Smith, G. F. 1971 On isotropic funcions of symmetric tensors, skew-symmetric tensors and vectors. Intl J. Engng Sci. 9, 899–916.Google Scholar
Smits, A. J. & Dussauge, J. 2006 Turbulent Shear Layers in Supersonic Flow, 2nd edn. Springer Science+Business Media.Google Scholar
Speziale, C. G. 1991 Analytical methods for the development of Reynolds-stress closures in turbulence. Annu. Rev. Fluid Mech. 23, 107–157.Google Scholar
Speziale, C. G., Gatski, T. B. & Sarkar, S. 1992 On testing models for the pressure–strain correlation of turbulence using direct simulations. Phys. Fluids 4 (12), 2887–2899.Google Scholar
Speziale, C. G., Sarkar, S. & Gatski, T. B. 1991 Modelling the pressure–strain correlation of turbulence: an invariant dynamical systems approach. J. Fluid Mech. 227, 245–272.Google Scholar
Suman, S. & Girimaji, S. S. 2010 On the invariance of compressible Navier–Stokes and energy equations subject to density-weighted filtering. Flow Turbul. Combust. 85 (3), 383–396.CrossRefGoogle Scholar
Sutherland, W. 1893 The viscosity of gases and molecular force. Phil. Mag. Ser. 5 36 (223), 507–531.Google Scholar
Tavoularis, S. & Corrsin, S. 1981 Experiments in nearly homogeneous shear flow with a uniform mean temperature gradient. Part 1. J. Fluid Mech. 104, 311–347.Google Scholar
Thacker, W. D., Sarkar, S. & Gatski, T. B. 2007 Analyzing the influence of compressibility on the rapid pressure–strain rate correlation in turbulent shear flows. Theor. Comput. Fluid Dyn. 21 (3), 171–199.Google Scholar
Vreman, A. W., Sandham, N. D. & Luo, K. H. 1996 Compressible mixing layer growth rate and turbulence characteristics. J. Fluid Mech. 320, 235–258.Google Scholar
Wallin, S. & Johansson, A. V. 2000 An explicit algebraic Reynolds stress model for incompressible and compressible turbulent flows. J. Fluid Mech. 403, 89–132.Google Scholar
Walz, A. 1969 Boundary Layers of Flow and Temperature, 1st edn. MIT.Google Scholar
White, F. M. 1991 Viscous Fluid Flow, 2nd edn. McGraw-Hill.Google Scholar
Wilcox, D. C. 1988 Reassesment of the scale-determining equation for advanced turbulence models. AIAA J. 26 (11), 1299–1310.Google Scholar
Wilcox, D. C. 1993 Turbulence Modelling for CFD, 1st edn. DCW Industries.Google Scholar