Higher entropy conservation and numerical stability of compressible turbulence simulations Albert E. Honein * , Parviz Moin Center for Turbulence Research, Stanford University, Building 500, Stanford, CA 94305, USA Received 16 February 2004; received in revised form 14 June 2004; accepted 14 June 2004 Available online 15 July 2004 Abstract We present a numerical formulation for the treatment of nonlinear instabilities in shock-free compressible turbu- lence simulations. The formulation is high order and contains no artificial dissipation. Numerical stability is enhanced through semi-discrete satisfaction of global conservation properties stemming from the second law of thermodynamics and the entropy equation. The numerical implementation is achieved using a conservative skew-symmetric splitting of the nonlinear terms. The robustness of the method is demonstrated by performing unresolved numerical simulations and large eddy simulations of compressible isotropic turbulence at a very high Reynolds number. Results show the scheme is capable of capturing the statistical equilibrium of low Mach number compressible turbulent fluctuations at infinite Reynolds number. Comparisons with the entropy splitting technique [J. Comput. Phys. 162 (2000) 33; J. Comput. Phys. 178 (2002) 307], staggered method [J. Comput. Phys. 191(2) (2003) 392], and skew-symmetric like schemes [J. Comput. Phys. 161 (2000) 114] confirm the superiority of the current approach. We also discuss a flaw in the skew-symmetric splitting implemented in the literature. Very good results are obtained based on the proper splitting. Ó 2004 Elsevier Inc. All rights reserved. Keywords: Nonlinear numerical stability; Conservation properties; Skew-symmetric form; Compressible turbulence simulations; High order central schemes 1. Introduction Nonlinear instabilities have been a major hurdle in turbulence simulations [5,6]. They become more pronounced when high order non-dissipative methods are used in under-resolved simulations, where ali- asing errors significantly increase. This is the typical situation in large eddy simulations (LES), where high order schemes are preferred in order to keep truncation errors smaller than subgrid scale terms [7]. In incompressible simulations, instabilities were successfully suppressed without artificial dissipation by * Corresponding author. E-mail addresses: honein@stanfordalumni.org (A.E. Honein), moin@stanford.edu (P. Moin). 0021-9991/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.jcp.2004.06.006 Journal of Computational Physics 201 (2004) 531–545 www.elsevier.com/locate/jcp