The Fable of the Wall-Normal Pressure-Gradient Boundary       Condition -- Or: Why Muenchhausen was a Liar

Abstract

We revisit the issue of finding proper boundary conditions for the field equations describing incompressible flow problems, for quantities like pressure or vorticity, which do not have immediately obvious ``physical'' boundary conditions. It is shown that a number of very popular choices for boundary conditions as implemented in numerical simulations of incompressible flow lead to mathematically ill-posed problems. In particular, schemes that make use of such boundary conditions will, in general, not converge towards solutions to the incompressible Navier-Stokes equations. We will also discuss some little-known (ir)regularity properties of Navier-Stokes solutions.