Damped artificial compressibility iteration scheme for implicit calculations of unsteady incompressible flow

Abstract

AbstractPeyret (J. Fluid Mech., 78, 49–63 (1976)) and others have described artificial compressibility iteration schemes for solving implicit time discretizations of the unsteady incompressible Navier‐Stokes equations. Such schemes solve the implicit equations by introduing derivatives with respect to a pseudo‐time variable τ and marching out to a steady state in τ. The pseudo‐time evolution equation for the pressure p takes the form ∂p/∂ = −a2∂∇.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 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‐linear equations resulting from a fully implicit time differencing scheme 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. Ramshaw 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 pressence 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 δt,. Inded, if on sets a ˜ |u| as usual, the artificial sound waves cease to exist when δt is small and this adversely affects the iteration convergence rate. Approximate analytical expressions for a and b are proposed and the benefits of their use relative to the conventional values a ∼ |u| and b = 0 are illustrated in simple test calculations.