Divergence-Free Pressure Boundary Condition for Solid Walls in Viscous Incompressible Flows

The solution of the incompressible Navier-Stokes equations with pressure Poisson equation has been an area of intensive research and extensive applications. Nevertheless, the proper formulation of pressure boundary conditions for solid walls has remained theoretically unestablished, since the existing specifications are unable to generate a truly solenoidal velocity field (i.e., a velocity field with zero divergence). Some prior researches have claimed divergence-free boundary conditions but lack rigorous validity. In this study, we clarify a longtime misconception and point out that the regular pressure boundary condition for solid walls will make the system ill-posed and non-divergence-free, because the system doesn’t need any pressure boundary condition in theory. However, solving the pressure Poisson equation still requires a boundary closure. To resolve this contradiction, we propose a new pressure Poisson equation and associated Neumann boundary condition based on the corrective pressure. They are derived in detail, examined for the pressure compatibility, analyzed for numerical stability with the normal mode method, and validated successfully with test problems. The new formulation produces strictly divergence-free velocity solutions with enhanced accuracy and stability due to its theoretical soundness.