A fast algorithm for simulation of a spatially-evolving, two-dimensional planar mixing layer

Abstract

An algorithm of exceptional computational efficiency is developed for numerical simulation of spatially-evolving instability waves in an incompressible fluid. The efficiency of the algorithm is demonstrated in a simulation of forced instability waves in a two-dimensional planar mixing layer, computationally analogous to the forced-flow experiment of Miksad. The Navier-Stokes equations in vorticity-streamfunction form are solved on a rectangular domain with inflow-outflow boundary conditions. The finite difference algorithm exhibits (1) an asymptotic count of 2MNlog 2 M operations per discrete time-step on an M × N mesh, comparable in computational efficiency to pseudospectral methods used to model temporally-evolving instabilities, (2) global 2nd-order accuracy in time and space and (3) structure which is vectorizable for implementation on array processors. Execution rate on a Cray X-MP is 0.056 seconds per time-step for a computational mesh of 256 × 144 resolution. Among the recent advances in numerical techniques which contribute to this efficiency are semi-implicit temporal discretization, approximate factorization of the matrix representation of the implicit diffusion operator, and fast direct solution of Poisson's equations. Moreover, the algorithm is shown to extend naturally and efficiently to three spatial dimensions.