High Order Filter Methods for Wide Range of Compressible Flow Speeds

Abstract

This paper extends the accuracy of the high order nonlinear filter finite difference method of Yee and Sjögreen [Development of Low Dissipative High Order Filter Schemes for Multiscale Navier-Stokes/MHD Systems, J. Comput. Phys., 225 (2007) 910–934] and Sjögreen and Yee [Multiresolution Wavelet Based Adaptive Numerical Dissipation Control for Shock-Turbulence Computation, RIACS Technical Report TR01.01, NASA Ames research center (Oct 2000); Also J. Scient. Comput., 20 (2004) 211–255] for compressible turbulence with strong shocks to a wider range of flow speeds without having to tune the key filter parameter. Such a filter method consists of two steps: a full time step using a spatially high-order non-dissipative base scheme, followed by a post-processing filter step. The post-processing filter step consists of the products of wavelet-based flow sensors and nonlinear numerical dissipations. For low speed turbulent flows and long time integration of smooth flows, the existing flow sensor relies on tuning the amount of shock-dissipation in order to obtain highly accurate turbulent numerical solutions. The improvement proposed here is to solve the conservative skew-symmetric form of the governing equations in conjunction with an added flow speed and shock strength indicator to minimize the tuning of the key filter parameter. Test cases illustrate the improved accuracy by the proposed ideas without tuning the key filter parameter of the nonlinear filter step.