Parallel Nodal Interior-Penalty Discontinuous Galerkin Methods for the Subsonic Compressible Navier-Stokes Equations: Applications to Vortical Flows and VIV Problems

Abstract

We present a Discontinuous Galerkin (DG) solver for the compressible Navier-Stokes system, designed for applications of technological and industrial interest in the subsonic region. More precisely, this work aims to exploit the DG-discretised Navier-Stokes for two dimensional vortex-induced vibration (VIV) problems allowing for high-order of accuracy. The numerical discretisation comprises a nodal DG method on triangular grids, that includes two types of numerical fluxes: 1) the Roe approximate Riemann solver flux for non-linear advection terms, and 2) an Interior-Penalty numerical flux for non-linear diffusion terms. The nodal formulation permits the use of high order polynomial approximations without compromising computational robustness. The spatially-discrete form is integrated in time using a low-storage strong-stability preserving explicit Runge-Kutta scheme, and is coupled weakly with an implicit rigid body dynamics algorithm. The proposed algorithm successfully implements polynomial orders of $p\ge 4$ for the laminar compressible Navier-Stokes equations in massively parallel architectures. The resulting framework is firstly tested in terms of its convergence properties. Then, numerical solutions are validated with experimental and numerical data for the case of a circular cylinder at low Reynolds number, and lastly, the methodology is employed to simulate the problem of an elastically-mounted cylinder, a known configuration characterised by significant computational challenges. The above results showcase that the DG framework can be employed as an accurate and efficient, arbitrary order numerical methodology, for high fidelity fluid-structure-interaction (FSI) problems.