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

u=u^d+\nabla\phi

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.

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