Penalty and characteristic-based operator splitting with multistep scheme finite element method for unsteady incompressible viscous flows

Abstract

The penalty and characteristic-based operator splitting with multistep scheme (penalty-MCBOS) has been developed for the unsteady incompressible Navier-Stokes N-S equations. At each time step, the N-S equations are split into the diffusive part, the convective part by adopting the operator splitting method and the pressure part by applying the penalty method with low penalty parameters. For the diffusive part, the temporal discretisation is based on backward difference method and is solved by preconditioned conjugate gradient (PCG) method. For the convective part, the temporal discretisation is performed by characteristic-Galerkin method. It is solved explicitly and the multistep technique is used. The pressure can be calculated from the pressure part without necessitating the solution of the pressure Poisson equation. The plane Poisseuille flow, the lid-driven cavity flow, the lid-driven triangular cavity flow and the backward-facing step flow are investigated to validate the present model. It is shown that the numerical results are in good agreement with the previously published data, and the present model has high efficiency. For the backward-facing step problem at Re = 3,000, the flow becomes periodical in time and dynamic evolution process of vortex can be simulated.