Fractional step artificial compressibility schemes for the unsteady incompressible Navier–Stokes equations

Abstract

We propose a novel, time-accurate approach for solving the unsteady, three-dimensional, incompressible Navier–Stokes equations on non-staggered grids. The approach modifies the standard, dual-time stepping artificial-compressibility (AC) iteration scheme by incorporating ideas from pressure-based, fractional-step (FS) formulations. The resulting hybrid fractional-step/artificial-compressibility (FSAC) method is second-order accurate and advances the Navier–Stokes equations in time via a two-step procedure. In the first step, which is identical to the convection–diffusion step in pressure-based FS methods, a preliminary velocity field is calculated, which is not divergence-free. In the second step, however, instead of deriving a pressure-Poisson equation as in FS methods, the projection of the velocity field into the solenoidal vector space is implemented using a dual-time stepping AC formulation. Unlike the standard dual-time stepping AC formulations, where the dual-time iterations are carried out with the entire non-linear system, in the FSAC scheme the convective and viscous terms are computed only once or twice per physical time step. Numerical experiments show that the proposed method provides second-order accurate solutions and requires considerably less CPU time than the widely used standard AC formulation. To demonstrate its ability to compute complicate problems, the method is also applied to a flow past cylinder with endplates.