Comparison of numerical methods for the solution of viscous incompressible and low Mach number compressible flow

Abstract

The novelty of the paper is the application of the discontinuous Galerkin method (DGM) to the solution of viscous compressible flow with low Mach number (M≈10−2−10−3). The compressible Navier–Stokes equations are written in the conservative form using conservative variables. They are discretized by the DGM in space. In time we use either the backward difference formula (BDF) or the discontinuous Galerkin method. The resulting technique is called the space-time discontinuous Galerkin method (STDGM).The goal of the presented analysis is to prove the ability of the developed DGM to solve viscous compressible low Mach number flows described by the compressible Navier–Stokes equations without any modification and to obtain results comparable with simulations of incompressible viscous flow described by the incompressible Navier–Stokes equations, solved by the finite element method (FEM). We present numerical results for a flow in a simplified channel of the human vocal tract, for a flow around the airfoil NACA0012 and moreover for flow-induced airfoil vibrations. The results prove the robustness of the discontinuous Galerkin method with respect to the low Mach number.