I am looking to solve the incompressible Navier-Stokes equations in 3D, using an inflow boundary condition specifying a velocity:

$\mathbf{u} = \mathbf{g}_0 \,\, \forall \,\, \mathbf{x} \in \Gamma_u$

and I want to use a Neumann type boundary condition where I specify pressure at the outlet

$-p\mathbf{I} + \tau = p_0\mathbf{I} \,\, \forall \,\, \mathbf{x} \in \Gamma_h$

I am looking to use a projection method to solve for the combined velocity and pressure fields, however I am not certain as to how to incorporate the Neumann term above into such a scheme. Most of the literature talks about deriving a velocity based boundary condition for the discrete poisson equation - but that is all for problems where the Neumann boundary $\Gamma_h = \phi$.

I was wondering whether anyone here had any insights on this. I can give more details of the mathematics that I am looking at if needed. Any references, or web-pages, or some brief pointers on this will be very helpful.


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