A pseudospectral matrix element method for solution of three-dimensional incompressible flows and its parallel implementation

A new pseudospectral matrix element (PSME) method employing the primitive variable formulation of the Navier-Stokes equations was used to simulate 3-dimensional time-dependent driven cavity flow at a Reynolds number of 3200 with an aspect ratio of 3 in the spanwise direction as well as 3-dimensional flow over a backward step. The new method only requires functions which are c0 continuous across the interface between two adjacent elements. It also ensures that the continuity equation is satisfied everywhere, in the interior (including the inter-element points) and on the boundary. The resulting complex geometry for flow over a backward step can be divided into a number of overlapping subdomains by a domain decomposition, of simpler geometry with patched grid points, in which the solution is more easily obtained. With an iterative procedure between subdomains, the complete solution is found by the Schwarz alternating procedure (SAP). With an eigenfunction expansion for the pressure, storage requirements for the 3D inversion step, O(N6), are reduced to O(N3) if the inverse of pressure equation is not stored. The parallel implementation of the three most time-consuming procedures: (i) computing the partial derivatives of scalar fields in terms of dotproduct; (ii) transforming between eigenfunction space and physical space for the pressure in terms of matrix multiplications; and (iii) performing the forward and backward sweeps of an LU decomposition to solve for the pressure have been efficiently performed on a parallel computer. The numerical results have reproduced dynamic longitudinal Taylor-Görtler-like (TGL) vortices in qualitative agreement with the experimental results of Koseff et al. and also indicated other 3-dimensional effects on the flow development. Computational results for both 2-and 3-dimensional flow over a backward-facing step at different Reynolds numbers are also presented in this paper. No pronounced 3-dimensional effects are observed for Reynolds numbers up to 450 except in the boundary layer along the spanwise direction.