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

Peyret (J. Fluid Mech., 78, 49-63 (1976)) and others have described ar tificial compressibility iteration schemes for solving implicit time d iscretizations of the unsteady incompressible Navier-Stokes equations. Such schemes solve the implicit equations by introducing derivatives with respect to a pseudo-time variable tau and marching out to a stead y state in tau. The pseudo-time evolution equation for the pressure p takes the form partial derivative p/partial derivative tau = -a(2) del . u, where a is an artificial compressibility parameter and u is the fluid velocity vector. We present a new scheme of this type in which c onvergence is accelerated by a new procedure for setting a and by intr oducing an artificial bulk viscosity b into the momentum equation. Thi s scheme is used to solve the non-linear equations resulting from a fu lly implicit time differencing scheme for unsteady incompressible flow . We find that the best values of a and b are generally quite differen t from those in the analogous scheme for steady flow (J. D. Ramshaw an d V. A. Mousseau, Comput. Fluids, 18, 361-367 (1990)), owing to the pr eviously unrecognized fact that the character of the system is profoun dly altered by the presence of the physical time derivative terms. In particular, a Fourier dispersion analysis shows that a no longer has t he significance of a wave speed for finite values of the physical time step Delta t. Indeed, if one sets a similar to \u\ as usual, the arti ficial sound waves cease to exist when Delta t is small and this adver sely affects the iteration convergence rate. Approximate analytical ex pressions for a and b are proposed and the benefits of their use relat ive to the conventional values a similar to \u\ and b = 0 are illustra ted in simple test calculations.