Efficient Pseudospectral Flow Simulations in Moderately Complex Geometries

Abstract

A computationally efficient pseudospectral method is developed for incompressible flow simulations in two-dimensional geometries involving periodicity in one direction and significant surface deformations. A pseudoconformal mapping is used to map the flow domain into a rectangle, thereby establishing an orthogonal curvilinear coordinate system within which the governing equations are formulated. The time integration of the spectrally discretized, two-dimensional momentum equations is performed by a second-order mixed explicit/implicit time integration scheme. The satisfaction of the continuity equation is obtained through the solution of a Poisson equation for the pressure and the use of the influence matrix technique. A highly efficient iterative solver has been developed for the solution of a generalized Stokes problem at each time step based on a spectrally preconditioned biconjugate gradient algorithm, which exhibits almost linear scalability, requiring an orderNlog2Nnumber of operations, whereNis the number of unknowns. Numerical results are presented for two-dimensional steady, oscillatory, and peristaltic flows within an undulating channel, which agree well with previous results that have appeared in the literature.