A high-order semi-explicit discontinuous Galerkin solver for 3D incompressible flow with application to DNS and LES of turbulent channel flow

摘要

We present an efficient discontinuous Galerkin scheme for simulation of theincompressible Navier-Stokes equations including laminar and turbulent flow. Weconsider a semi-explicit high-order velocity-correction method for timeintegration as well as nodal equal-order discretizations for velocity andpressure. The non-linear convective term is treated explicitly while a linearsystem is solved for the pressure Poisson equation and the viscous term. Thekey feature of our solver is a consistent penalty term reducing the localdivergence error in order to overcome recently reported instabilities inspatially under-resolved high-Reynolds-number flows as well as small timesteps. This penalty method is similar to the grad-div stabilization widely usedin continuous finite elements. We further review and compare our method toseveral other techniques recently proposed in literature to stabilize themethod for such flow configurations. The solver is specifically designed forlarge-scale computations through matrix-free linear solvers including efficientpreconditioning strategies and tensor-product elements, which have allowed usto scale this code up to 34.4 billion degrees of freedom and 147,456 CPU cores.We validate our code and demonstrate optimal convergence rates with laminarflows present in a vortex problem and flow past a cylinder and showapplicability of our solver to direct numerical simulation as well as implicitlarge-eddy simulation of turbulent channel flow at $Re_{\tau}=180$ as well as$590$.