Local discontinuous Galerkin numerical solutions of non-Newtonian incompressible flows modeled by p-Navier-Stokes equations

We present a Local Discontinuous Galerkin (LDG) method for solving non-Newtonian incompressible flow problems. The problems are modeled by p-type Navier-Stokes equations, the extra stress tensor follows a p-power rule (p-NS). The aim of this paper is to present an efficient way for discretizing the governing equations by an LDG method, choosing both equal and mixed order local polynomial space for velocity and pressure. The velocity gradient is introduced as an auxiliary variable and the p-NS system is decomposed into a first order system including the projection of the nonlinear stress tensor components to the local discontinuous space. Every equation resulting from the splitting of the extra stress tensor is discretized under the DG element by element technique. In the divergence constraint equation, an artificial compressibility term (time derivative of the pressure with a small parameter) is added and the first order terms are expressed in a divergence form, and are discretized by utilizing the Lax-Friedrichs numerical fluxes. An upwind wave analysis is applied on the outflow boundary for constructing artificial outflow boundary conditions. For the time discretization, s-stage Diagonally Implicit Runge-Kutta schemes are applied. Numerical experiments on problems with known exact solutions are performed for verifying the expected convergence rates of the method. Benchmark problems are considered in order to check the performance of the method for solving flow problems described by p-NS systems.