Vector penalty-projection method for incompressible fluid flows with open boundary conditions

Abstract

A new family of methods, the so-called vector penalty-projection methods (V PPr;"), were introduced recently by Angot et al. [1, 2] to compute the solution of unsteady incompressible fluid flows and to overcome most of the drawbacks of the usual incremental projection methods. In this work, we deal with the time-dependent incompressible Stokes equations with outflow boundary conditions using the present method. The spatial discretization is based on the finite volume scheme on a MAC staggered grid and the time discretization is based on the backward difference formula of second-order BDF2 (namely also Gear's scheme). We show that the (V PPr;") method provides a second-order convergence for both velocity and pressure in space and time even in the presence of open boundary conditions with small values of the augmentation parameter r; typically 0 · r · 1 and a penalty parameter " small enough; typically " = 10¡10. The resulting constraint on the discrete divergence of velocity is not exactly equal to zero but is satis¯ed approximately as O(" ±t) with a penalty parameter " as small as desired. Finally, the e±ciency and the second-order accuracy of our method are illustrated by several numerical cases.