Spectral element methods for turbulence

This chapter introduces spectral element methods (SEMs), which extend the efficiencies of high-order global tensor-product bases of spectral methods to unstructured multidomain configurations suited to the simulation of turbulence in complex geometries. Like its monodomain counterpart, the SEM exhibits rapid convergence with increasing polynomial order that yields minimal numerical dissipation and dispersion. Fast matrix-free operator evaluation facilitates the use of preconditioned iterative solvers that are necessary for unstructured discretizations. Temporal discretizations based on high-order operator splitting approaches are introduced to decouple the advancement of incompressible and low-Mach flow equations into distinct advection, diffusion, and pressure substeps and the splitting errors are explained and analyzed. A brief survey of viable multilevel iterative solvers for the pressure Poisson problem—the stiffest substep—is presented. The discussion closes with several SEM-based examples of turbulent flow in complex domains.