Higher entropy conservation and numerical stability of compressible turbulence simulations

We present a numerical formulation for the treatment of nonlinear instabilities in shock-free compressible turbulence 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.