Computation of flows with shocks using the Spectral Difference method with artificial viscosity, I: Basic formulation and application

The present work combines the Spectral Difference method with an artificial viscosity based approach to enable high-order computation of compressible fluid flows with discontinuities. The study uses an artificial viscosity approach similar to the high-wavenumber biased artificial viscosity approach (Cook and Cabot, 2005, 2004; Kawai and Lele, 2008) [1–3], extended to an unstructured grid setup. The model employs a bulk viscosity for treating shocks, a shear viscosity for treating turbulence, and an artificial conductivity to handle contact discontinuities. The high-wavenumber biased viscosity is found to stabilize numerical calculations and reduce oscillations near discontinuities. Promising results are demonstrated for 1D and 2D test problems. Until recently, compressible flow computations on unstructured meshes have generally been dominated by schemes restricted to second order accuracy. However, the need for highly accurate methods in applications such as large eddy simulation, direct numerical simulation and computational aeroacoustics, has seen the development of higher order schemes for unstructured meshes. In particular, there has been a rise in the popularity and application of locally discontinuous formulations. Methods such as Discontinu-ous Galerkin (DG) method [4,5], Spectral Volume (SV) method [6,7] and Spectral Difference (SD) method [8,9], Lifting Collocation Penalty (LCP) approach [10], etc. fall under this category. The SD method is a high-order approach based on the differential form of the conservative equations. This method combines elements from Finite-Volume and Finite-Difference techniques and is particularly attractive because it is conservative, has a simple formulation and straightforward implementation. The absence of volume or surface integrals also makes this method efficient. The origins of the SD method can be traced back to 1996, when Kopriva and Kolias [11] and Kopriva [12] introduced their formulation for the solution of the 2D compressible Euler equations on unstruc-tured quadrilateral meshes, which they called the 'Conservative Staggered-Grid Chebyshev Multi-Domain method'. Liu et al. [8] developed a general formulation of this approach on simplex cells and applied it to wave equations on triangular grids. Wang et al. [9] extended it to 2D Euler equations on triangular grids. It was further extended to the 2D N–S equations by May and Jameson [13], and Wang et al. [14]. Sun et al. [15] further developed it for three-dimensional Navier–Stokes equations on hexahedral unstructured meshes. Recently, Jameson [16] obtained a theoretical proof that the SD method is stable for all orders of accuracy in a Sobolev norm provided that the interior flux points are located at the zeros of …