Directional artificial bulk viscosity for shock capturing on high aspect ratio grids

Abstract

An improved shock-capturing method for high-order finite-difference schemes, which alleviates numerical stiffness on anisotropic grids, is proposed. This method is an extension of the artificial bulk viscosity (ABV) scheme, developed to capture shock waves with a minimal dissipation of the physical oscillations present in the flow, such as those associated with turbulence. This modified scheme generalizes the ABV treatment to situations where there are strong spatial inhomogeneities that often require anisotropic grids. To accomplish this, ABV is independently computed and applied along each computational grid direction, therefore taking a multi-valued, directional form rather than a scalar form. Scalar dissipation on high aspect ratio (AR) grids with explicit time-stepping schemes can cause the stable time step to be reduced by a factor of 1/AR. The proposed method removes this constraint and can allow for substantial speedups on high AR grids with shock waves. A two-dimensional test case illustrates the added numerical stability of the method and its ability to capture shock waves more sharply. Brief results from a large-eddy simulation of an over-expanded planar nozzle are given which demonstrate the method's robustness in practical applications.