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When using the pressure projection method to solve the incompressible Navier-Stokes equations do we apply Neumann boundary conditions for pressure only where there are associated no-slip velocity boundary conditions? For example suppose we were trying to solve the vortex shedding problem. The velocity boundary conditions are shown in the figure velocity boundary conditions

Then would I use a Neumann condition on pressure for the top, bottom, and circle? If so what about the left and right sides where the velocity is not no-slip? Would I have to specify a Dirichlet pressure here and if so then what would that be? I know that in order to solve the Poisson type problem for pressure that at least one node (I am using finite elements) must be specified in order for the system to have a unique solution, but should we be specifying pressure at all nodes where there is not a no-slip boundary condition on velocity? The related question Flow past a cylinder - Projection Method - Boundary Conditions seemed to indicate that Neumann conditions on pressure are only used when that boundary also has a no-slip velocity boundary condition. Is this correct?

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For the incompressible Navier-Stokes equations, in practice, where ever there is a Dirichlet boundary condition on the velocity, Neumann boundary condition is applied for the pressure. It does not matter if it no-slip or "velocity-inlet" or "velocity-outlet". You should not specify the pressure where the velocity is specified. Your figure shows velocity specified at all the boundaries, so you would need to apply Neumann BCs for pressure at all the boundaries including the left and right faces of the domain.

The problem with a "velocity-outlet" is that this velocity is not known apriori. and if it chosen poorly it will affect the solution upstream. The other alternative is to use a "do nothing" boundary condition at the outflow, for example by setting derivative of the velocity to be zero at the outflow and specfying the pressure.

For concrete example using finite difference you could look at:

Van Kan, "A second-order accurate pressure-correction scheme for viscous incompressible flow." SIAM Journal on Scientific and Statistical Computing 7.3 (1986): 870-891. http://epubs.siam.org/doi/abs/10.1137/0907059

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