DAMPED ARTIFICIAL COMPRESSIBILITY ITERATION SCHEME FOR IMPLICIT CALCULATIONS OF UNSTEADY INCOMPRESSIBLE FLOW

摘要

Peyret (J. Fluid Meek, 78,49-63 (1976)) and others have described artificial compressibility iteration schemes for solving implicit time discretizations of the unsteady incompressible Navier-Slokes equations. Such schemes solve the implicit equations by introducing derivatives with respect to a pseudo-time variable x and marching out to a steady state in t. The pseudo-time evolution equation for the pressure of takes the form ap/at=a~2▽.u, where a is an artificial compressibility parameter and u is die fluid velocity vector. We present a new scheme of this type in which convergence is accelerated by a new procedure for setting a and by introducing an artificial bulk viscosity b into the momentum equation. This scheme is used to solve the non-Iinoar equations resulting from a fully implicit time differencing schema for unsteady incompressible flow. We find that the best values of a and b are generally quite different from those in the analogous scheme for steady flow (J. D. Rainsbaw and V. A. Mousseau, Comput; Fluids, 18, 361-367 (1990)), owing to the previously unrecognized fact that the character of the system is profoundly altered by the presence of the physical time derivative terms. In particular, a Fourier dispersion analysis shows that a no longer has the significance of a wave speed for finite values of the physical time step A. Indeed, if one sets as usual, the artificial sound waves cease to exist when A/is small and this adversely affects the iteration convergence rale. Approximate analytical expressions fi? a and b arc proposed and the benefits of their use relative to the conventional values a ~ |u| and b=0 are illustrated in simple test calculations.