On pressure boundary conditions for the incompressible Navier‐Stokes equations

TLDR

In incompressible flows, pressure acts as a Lagrange multiplier enforcing divergence‑free velocity, propagates instantaneously, and while well‑defined by conservation laws, its Poisson‑form representation requires careful specification of boundary conditions. The study demonstrates that mass and momentum conservation, together with a continuity argument, yield the correct pressure Poisson equation boundary conditions. These conditions are a Neumann boundary condition derived by applying the normal component of the momentum equation at the boundary.

Abstract

Abstract The pressure is a somewhat mysterious quantity in incompressible flows. It is not a thermodynamic variable as there is no ‘equation of state’ for an incompressible fluid. It is in one sense a mathematical artefact—a Lagrange multiplier that constrains the velocity field to remain divergence‐free; i.e., incompressible—yet its gradient is a relevant physical quantity: a force per unit volume. It propagates at infinite speed in order to keep the flow always and everywhere incompressible; i.e., it is always in equilibrium with a time‐varying divergence‐free velocity field. It is also often difficult and/or expensive to compute. While the pressure is perfectly well‐defined (at least up to an arbitrary additive constant) by the governing equations describing the conservation of mass and momentum, it is (ironically) less so when more directly expressed in terms of a Poisson equation that is both derivable from the original conservation equations and used (or misused) to replace the mass conservation equation. This is because in this latter form it is also necessary to address directly the subject of pressure boundary conditions, whose proper specification is crucial (in many ways) and forms the basis of this work. Herein we show that the same principles of mass and momentum conservation, combined with a continuity argument, lead to the correct boundary conditions for the pressure Poisson equation: viz., a Neumann condition that is derived simply by applying the normal component of the momentum equation at the boundary. It usually follows, but is not so crucial, that the tangential momentum equation is also satisfied at the boundary.