Stability of Pressure Boundary Conditions for Stokes and Navier–Stokes Equations

The stability of a finite difference discretization of the time-dependent incompressible Navier–Stokes equations in velocity-pressure formulation is studied. In paticular, we compare the stability for different pressure boundary conditions in a semiimplicit time-integration scheme. where only the viscous term is treated implicitly. The stability is studied in three different ways: by a normal-mode analysis, by numerical computation of the amplification factors, and by direct numerical simulation of the governing equations. All three approaches identify the same pressure boundary condition as the best alternative. This condition implicitly enforces the normal derivative of the divergence to be zero on the boundary by coupling the normal derivative of the pressure to the normal component of the curl of the vorticity. Using this boundary condition, we demonstrate that the time-step is determined only by the convective term.