Solution of Flow in Complex Geometries by the Pseudospectral Element Method

Abstract

A multi-grid domain decomposition approach with the Schwarz alternating procedure has been developed for the solution of flow in complex geometries by the pseudospectral element method in primitive variable form. The approach for flow problems is first to divide the computational domain into a number of simple blocks (fine-grid or coarse-grid subdomains) with the inter-overlapping zone, of which the overlapped grids may or may not be coincided with each other. An isoparametric mapping is next to map each block onto a simple square (or cube), where c 0 pseudospectral elements (quadrilateral or hexahedral), generated by linearly interpolating the shape function which defines the geomery of their parent elements (blocks), are further used to partition the mapped domain. Schwarz alternating procedure is used to exchange data among the blocks, where the multi-grid (two grid) technique is implemented to remove the high frequency error that occurs when the data interpolation from the fine-grid subdomain to the coarse-grid subdomain is conducted. The solution of the pressure equation in each block can be efficiently solved by the two-level preconditioned minimal residual method in terms of eigenfunction expansion technique, which greatly reduces the inverse of the preconditioned matrix to the simplest "algebraic" form with the least storage requirement, i.e., O( N 3) in 3D and O( N 2) in 2D. Numerical experiments have been performed for both the two-dimensional flow over a cylinder in a channel (Re = 100-1000) and three-dimensional bifurcation flow (Re = 500) to account for the versatility of the proposed technique. The shedding frequency behind a cylinder, Strouhal number increases with increasing Reynolds number, which is different from those found in a square cylinder. As for the bifurcating case, the streamwise velocity profiles of the two-dimensional flow model underestimates the three-dimensional results due to the negligence of boundary effect.