Spectral Element Methods for Transitional Flows

 

Abstract

We consider spectral element solutions of the unsteady incompressible Navier-Stokes equations in general two- and three-dimensional domains. The spectral element method is based upon rapidly convergent, high-order weighted residual techniques employing tensor-product polynomial bases of degree N in each of K deformed quadrilateral or hexahedral elements. Because of its low numerical dissipation and dispersion, the method is well suited to transitional Reynolds number applications where the physical viscosity is small. For high Reynolds numbers, the Galerkin formulation is stabilized using a recently developed filter that is local and preserves both interelement continuity and spectral accuracy. Time advancement is based on consistent, high-order, operator splittings that permit large time steps (typ. a convective CFL of 1--5) and yield independent convective, viscous, and pressure subproblems, the latter of which are solved using Jacobi and Schwarz-preconditioned CG, respectively. The code scales to 1000s of processors, achieving performance in excess of 400 GFLOPS on 4096 processors of the Intel ASCI Red machine at Sandia National Laboratory. We present several example applications that demonstrate current capabilities, including flow in an arteriovenous graft model at Reynolds number 2700, turbulent bouyancy driven convection at Rayleigh number 10^6, and hairpin vortex formation in the wake of a roughness element for a boundary layer at Reynolds numbers 500--3500.