High-Order Spectral Volume Method for the Navier-Stokes Equations on Unstructured Grids

In this paper, the spectral volume (SV) method is extended to solve the Navier-Stokes equations by treating the viscous terms with a mixed formulation named local Discontinuous Galerkin approach. The SV method combines two key ideas, which are the basis of the finite volume and the finite element methods, i.e., the physics of wave propagation accounted for by the use of a Riemann solver and high-order accuracy achieved through high-order polynomial reconstructions within spectral volumes. The formulation of the SV method for the two-dimensional compressible Navier-Stokes equations is described. Accuracy studies are performed on the scalar advection diffusion and the Navier-Stokes equations using problems with analytical solutions. It is shown that the designed order of accuracy is achieved for all orders of polynomial reconstructions. The solver is then used to solve other viscous laminar flow problems to shown its potential.