Pressure-Poisson formulation and projection methods

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.