Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-05T13:33:30.387Z Has data issue: false hasContentIssue false
  • English
  • Français

A Modified Gas-Kinetic Scheme for Turbulent Flow

Published online by Cambridge University Press:  03 June 2015

Marcello Righi
Marcello Righi*
Affiliation:
School of Engineering, Zurich University of Applied Sciences, Technikumstrasse 9, 8401 Winterthur, Switzerland
*
*Corresponding author.Email:[email protected]
Get access

Abstract

The implementation of a turbulent gas-kinetic scheme into a finite-volume RANS solver is put forward, with two turbulent quantities, kinetic energy and dissipation, supplied by an allied turbulence model. This paper shows a number of numerical simulations of flow cases including an interaction between a shock wave and a turbulent boundary layer, where the shock-turbulent boundary layer is captured in a much more convincing way than it normally is by conventional schemes based on the Navier-Stokes equations. In the gas-kinetic scheme, the modeling of turbulence is part of the numerical scheme, which adjusts as a function of the ratio of resolved to unresolved scales of motion. In so doing, the turbulent stress tensor is not constrained into a linear relation with the strain rate. Instead it is modeled on the basis of the analogy between particles and eddies, without any assumptions on the type of turbulence or flow class. Conventional schemes lack multiscale mechanisms: the ratio of unresolved to resolved scales – very much like a degree of rarefaction – is not taken into account even if it may grow to non-negligible values in flow regions such as shocklayers. It is precisely in these flow regions, that the turbulent gas-kinetic scheme seems to provide more accurate predictions than conventional schemes.

Keywords

47.27.em47.27.nb47.32.Ff47.40.Hg47.40.Nm47.45.Ab47.85.Gj47.85.ldGas kinetic theoryturbulence modelingcompressible flowshock boundary layer interaction.
Type
Research Article
Information
Communications in Computational Physics , Volume 16 , Issue 1 , July 2014 , pp. 239 - 263
Copyright
Copyright © Global Science Press Limited 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Babinsky, H. and Harvey, J. K.Shock Wave-Boundary-Layer Interactions. Cambridge University Press, New York, 2011.Google Scholar
[2]Bhatnagar, P.L., Gross, E.P., and Krook, M.A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev., 94(3): 511–525, 1954.CrossRefGoogle Scholar
[3]Bookey, P., Wyckham, C., and Smits, A.Experimental investigations of Mach 3 shock-wave turbulent boundary layer interactions. AlAA Paper No. 2005–4899, 2005.Google Scholar
[4]Cercignani, C.The Boltzmann Equation and its Applications. Springer, New York, 1988.CrossRefGoogle Scholar
[5]Chen, H., Kandasamy, S., Orszag, S., Shock, R., Succi, S., and Yakhot, V.Extended Boltzmann kinetic equation for turbulent flows. Science, 301(5633): 633–636, 2003.Google Scholar
[6]Chen, H., Orszag, S.A., Staroselsky, I., and Succi, S.Expanded analogy between Boltzmann kinetic theory of fluids and turbulence. J. Fluid Mech., 519(1): 301–314, 2004.CrossRefGoogle Scholar
[7]Chou, S.Y. and Baganoff, D.Kinetic flux-vector splitting for the Navier-Stokes equations. J. Comput. Phys., 130(2): 217–230, 1997.Google Scholar
[8]Delery, J.Experimental investigation of turbulence properties in transonic shock/boundary-layer interactions. AIAA J., 21: 180–185, 1983.Google Scholar
[9]Delery, J., Marvin, J.G., and Reshotko, E.Shock-wave boundary layer interactions. AGAR-Dograph, 280, 1986.Google Scholar
[10]Dolling, D. S. and Erengil, M. E.Unsteady wave structure near separation in a Mach 5 compression rampinteraction. AIAA J, 29(5): 728–735, 1991.Google Scholar
[11]Dupont, P., Haddad, C., and Debieve, J.F.Space and time organization in a shock-induced separated boundary layer. J. Fluid Mech., 559: 255–278, 2006.Google Scholar
[12]Edwards, J. R.Numerical simulations of shock/boundary layer interactions using time-dependent modeling techniques: A survey of recent results. Prog. Aerosp. Sci., 44(6): 447–465, 2008.Google Scholar
[13]Garnier, E., Adams, N., and Sagaut, P.Large Eddy Simulation for Compressible Flows. Springer, 2009.CrossRefGoogle Scholar
[14]Goldberg, U., Peroomian, O., and Chakravarthy, S.Application of the k-e-R turbulence model to wall-bounded compressive flows. AIAA Paper No. 1998-0323, 1998.Google Scholar
[15]Jameson, A.Solution of the Euler equations for two dimensional transonic flow by a multigrid method. Appl. Math. Comput., 13(3-4): 327–356, 1983.Google Scholar
[16]Kogan, M. N.Rarefied gas dynamics. Plenum Press, New York, 1969.Google Scholar
[17]Li, Q., Xu, K., and Fu, S.A high-order gas-kinetic Navier-Stokes flow solver. J. Comput. Phys., 229(19): 6715–6731, 2010.CrossRefGoogle Scholar
[18]Liao, W., Luo, L., and Xu, K.Gas-kinetic scheme for continuum and near-continuum hypersonic flows. J. Spacecraft Rockets, 44(6): 1232–1240, 2007.Google Scholar
[19]Mandal, J.C. and Deshpande, S.M.Kinetic flux vector splitting for Euler equations. Comput. Fluids, 23(2): 447–478, 1994.Google Scholar
[20]May, G., Srinivasan, B., and Jameson, A.An improved gas-kinetic BGK finite-volume method for three-dimensional transonic flow. J. Comput. Phys., 220(2): 856–878, 2007.Google Scholar
[21]Pirozzoli, S., Bernardini, M., and Grasso, F.Direct numerical simulation of transonic shock/boundary layer interaction under conditions of incipient separation. J. Fluid Mech., 657: 361–393, 2010.Google Scholar
[22]Pope, S.B.Turbulent Flows. Cambridge University Press, Cambridge, 2000.Google Scholar
[23]Righi, M.A gas-kinetic scheme for the simulation of turbulent flows. In Mareschal, M. and Santos, A. (Eds.), Proceeding of the 28th Internaltional Symposium on Rarefied Gas Dynamics, Zaragoza, pages 481–488. American Institute of Physics, 2012.Google Scholar
[24]Righi, M.A finite-volume gas-kinetic method for the solution of the Navier-Stokes equations. Royal Aeronautical Society, Aeronaut. J., (117)(1192), 2013.Google Scholar
[25]Sartor, F., Losfeld, G., and Bur, R.Piv study on a shock-induced separation in a transonic flow. Exp. Fluids, 53(3): 815–827, 2012.Google Scholar
[26]Settles, G.S., Fitzpatrick, T.J., and Bogdonoff, S.M.Detailed study of attached and separated compression corner flowfields in high Reynolds number supersonic flow. AIAA J., 17(6): 579–585, 1979.Google Scholar
[27]Settles, G.S., Vas, I.E., and Bogdonoff, S.M.Details of a shock-separated turbulent boundary layer at a compression corner. AIAA J., 14(12): 1709–1715, 1976.Google Scholar
[28]Succi, S., Filippova, O., Chen, H., and Orszag, S.Towards a renormalized lattice Boltzmann equation for fluid turbulence. J. Stat. Phys., 107(1-2): 261–278, 2002.CrossRefGoogle Scholar
[29]Wallin, S. and Johansson, A.V.An explicit algebraic Reynolds stress model for incompressible and compressible turbulent flows. J. Fluid Mech., 403: 89–132, 2000.Google Scholar
[30]Wilcox, D. C.Turbulence Modeling for CFD, 3rd edition. DCW Industries, Inc., La Canada CA, 2006.Google Scholar
[31]Xu, K.Gas-kinetic schemes for unsteady compressible flow simulations. Von Karman Institute, Computational Fluid Dynamics, Annual Lecture Series, 29th, Rhode-Saint-Genese, Belgium, 1998.Google Scholar
[32]Xu, K.A gas-kinetic BGK scheme for the Navier-Stokes equations and its connection with artificial dissipation and Godunov method. J. Comput. Phys., 171(1): 289–335, 2001.Google Scholar
[33]Xu, K. and Huang, J.A unified gas-kinetic scheme for continuum and rarefied flows. J. Comput. Phys., 229(20): 7747–7764, 2010.Google Scholar
[34]Xu, K., Mao, M., and Tang, L.A multidimensional gas-kinetic BGK scheme for hypersonic viscous flow. J. Comput. Phys., 203(2): 405–421, 2005.Google Scholar
[35]Xu, K. and Prendergast, K.H.Numerical Navier-Stokes solutions from gas kinetic theory. J. Comput. Phys., 114(1): 9–17, 1994.Google Scholar
[36]Xuan, L. and Xu, K.A new gas-kinetic scheme based on analytical solutions of the BGK equation. J. Comput. Phys., 2012.Google Scholar
[37]Yoon, S. and Jameson, A.Lower-upper symmetric-Gauss-Seidel method for the Euler and Navier-Stokes equations. AIAA J., 26(9): 1025–1026, 1988.Google Scholar