A high-order discontinuous Galerkin method for the unsteady incompressible Navier-Stokes equations

We present a high-order discontinuous Galerkin discretization of the unsteady incompressible Navier-Stokes equations in convection-dominated flows using triangular and tetrahedral meshes. The scheme is based on a semi-explicit temporal discretization with explicit treatment of the nonlinear term and implicit treatment of the Stokes operator. The nonlinear term is discretized in divergence form by using the local Lax-Friedrichs fluxes; thus, local conservativity is inherent. Spatial discretization of the Stokes operator has employed both equal-order (P"k-P"k) and mixed-order (P"k-P"k"-"1) velocity and pressure approximations. A second-order approximate algebraic splitting is used to decouple the velocity and pressure calculations leading to an algebraic Helmholtz equation for each component of the velocity and a consistent Poisson equation for the pressure. The consistent Poisson operator is replaced by an equivalent (in stability and convergence) operator, namely that arising from the interior penalty discretization of the standard Poisson operator with appropriate boundary conditions. This yields a simpler and more efficient method, characterized by a compact stencil size. We show the temporal and spatial behavior of the method by solving some popular benchmarking tests. For an unsteady Stokes problem, second-order temporal convergence is obtained, while for the Taylor vortex test problem on both semi-structured and fully unstructured triangular meshes, spectral convergence with respect to the polynomial degree k is obtained. By studying the Orr-Sommerfeld stability problem, we demonstrate that the P"k-P"k method yields a stable solution, while the P"k-P"k"-"1 formulation leads to unphysical instability. The good performance of the method is further shown by simulating vortex shedding in flow past a square cylinder. We conclude that the proposed discontinuous Galerkin method with the P"k-P"k formulation is an efficient scheme for accurate and stable solution of the unsteady Navier-Stokes equations in convection-dominated flows.

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