ABSTRACT
This chapter reviews the properties of global spectral methods and suggests ways of their efficient parallel implementation when applied to the Navier-Stokes equations governing incompressible fluid flows. It formulates the governing equations for incompressible fluid flows, and presents an efficient high-order accuracy time-stepping scheme. The chapter presents examples of simulations of turbulence in simple geometries and of unsteady laminar flows in complex geometries obtained using conventional supercomputers and hypercube computers. It reviews the spectral element discretization appropriate for the formulation. The spectral accuracy is recovered if an elemental interface passes exactly through the sharp corner. A row of three-dimensional spectral elements is considered as computational domain with periodic boundary conditions imposed on all sides. Similar Fourier pseudospectral simulations of a Beltrami flow performed on a 128-node and on a 1024-node Ncube are reported in Pelz. Spectral element simulations are capable of resolving precisely the Hopf bifurcation that the flow undergoes as the Reynolds number increases.
