Numerical aspects of compressible turbulence simulations

Nonlinear instabilities present a long standing hurdle for compact, high order, non dissipative, finite difference computation of compressible turbulence. The spectral-like accuracy of these schemes, while attractive, results in significant aliasing errors that corrupt the solution. As a result, successful simulations have been limited to moderate Reynolds numbers ( Re) and low-order or upwind schemes with inherent numerical dissipation. However, resorting to dissipative schemes in discretizing the nonlinear terms was shown to have a detrimental effect on turbulence. A recent LES approach is to abandon the subgrid model altogether and rely on the scheme dissipation to mimic the effect of small scales.; A dissipative monotone integrated LES (MILES) algorithm based on a multidimensional flux-corrected transport (FCT) algorithm has been developed and tested for decaying compressible isotropic turbulence. Agreement with the benchmark experiments of Comte-Bellot and Corrsin is very sensitive to the parameters involved in the FCT algorithm, while the evolution of thermodynamic fluctuations do not compare well with direct numerical simulations.; An under-resolved simulation of inviscid, compressible, isotropic turbulence at low Mach number is chosen as a severe benchmark to investigate the nonlinear stability properties of nondissipative schemes. The behavior of this benchmark is predicted by performing a fully de-aliased spectral simulation on a 32 3 grid with turbulent Mach number of Mto = 0.07. The kinetic energy and thermodynamic fluctuations are found to decay for finite Re, and remain constant at infinite Re for a long time before the occurrence of numerical shocks. Extending the proof of Kraichnan (Journal of the Acoustical Society of America, 27(3), 1955), this inviscid statistical equilibrium is demonstrated to be a consequence of the discrete equivalent of the Liouville theorem of classical statistical mechanics.; Several existing non-dissipative methods are evaluated and instabilities are found to occur at finite Re, the value varying from one scheme to another. These instabilities arise due to violation of two conservation properties stemming from the entropy equation. New non-dissipative formulations respecting these conservation properties are proposed for schemes of any order, finite difference or spectral, and valid for regular and staggered grids. The numerical implementation is simply achieved using a conservative skew-symmetric splitting of the nonlinear terms. Furthermore, robust versions of the schemes in Nagarajan et al. (Journal of Computational Physics , 191(2), 2003) and Ducros et al. (Journal of Computational Physics, 161(1), 2000) have been formulated and found to be effective at very high Re.