Divergence-Free Pressure Poisson Equation for Viscous Incompressible Flows with Solid Wall Boundaries

Properly setting the incompressibility constraint for incompressible Navier-Stokes equations has been an area of intense research since the pioneering MAC scheme was proposed in 1965. Nevertheless, the appropriate solution with the pressure Poisson equation in the presence of solid walls has remained an unsolved problem and a controversial topic. This is primarily because the conventional setups often fail to generate a truly solenoidal velocity field (i.e., a velocity field with zero divergence). In the present study, we clarify the longtime misunderstanding of the existing approaches and propose an answer to this problem. A reformulated pressure Poisson equation and its associated Neumann boundary condition are presented. They are derived in detail, analyzed for the pressure compatibility and numerical stability, and then validated successfully with test problems. The new formulations can produce strictly divergence-free velocity solutions to machine accuracy and enhance solution accuracy in general application scenarios. On this basis, we believe the proposed formulations constitute the proper pressure Poisson equation and Neumann boundary condition for solving viscous incompressible flows.