Pressure-Poisson formulation and projection methods

by Yi Zhang

Take divergence of the momentum equation, one will finally have

\nabla^2p=-\rho \nabla\cdot (u\cdot\nabla u-\nu\nabla^2 u)

This is the pressure-Poisson formulation, taking divergence-free condition of velocity into consideration. This equation presents the explicit form of pressure, instead of implicit in “standard” momentum formulations. Pressure-Poisson equation is widely used in projection methods, in which the velocity is decomposed into solenoidal part and irroational part by Hodge or Helmholtz decomposition. Chorin & Marsden’s this book contains extensive introduction for the latter, together with many other insightful comments.

Basically, in projection the velocity if splitted as


Or one can say that the velocity space is the direct product of two subspaces. The solenoidal nature of a flow field is projected to one of the subspaces: \nabla\cdot u^d=0. This projection, denoted by operator \mathcal{P}, function as

u^d=\mathcal{P}(u)=u-\nabla \phi=u-\nabla [\Delta^{-1}(\nabla\cdot u)]\Longrightarrow \mathcal{P}=I-\nabla\Delta^{-1}\text{div}

The stability of projection operator follows the fact that its norm is less or equal than one, which can be intuitively seen from the action of “projection”.

Generally, in a typical predictor-corrector method using pressure to correct the velocity pseudo-step, the velocity in predictor sub-step u^{n+1/2} is obtained using information from time step n, i.e. u^n, and p^n (explicit methods) or together with information at time step n+1 (implicit methods). Since u^{n+1/2} does not satisfy divergence-free condition, divergence operator is applied to the equation connecting u^{n+1/2} and p^{n+1}, in order to obtain the pressue p^{n+1} required to satisfy solenoidal u^{n+1} and u^n,  and this is the pressue-corrector sub-step. Usually, the second sub-step would give a Poisson equation of pressue, as the example by Idelsohn et al I cited in this post.