[kinematic_presser]

Hello guys,
I am still somewhat new to CFD (just starting my masters degree) so please bear with me if my question is dumb.

I am simulating fluid flow in a channel for an assignment. The aim of the assignment is to simulate 2 dimensional incompressible Poiseulle flow, moving from left to right. It is a very simple simulation. There are no obstacles and it is a simple rectangular domain. For the velocity, I enforce a velocity inlet condition at the inlet and a homogeneous Neumann condition at the outlet. I enforce a homogeneous Dirichlet condition (no-slip) at the top and bottom wall. This all makes sense to me as the INSE are second order in u .

But, when I was looking over some example tutorial cases in OpenFOAM, I noticed that their simulations were enforcing a boundary condition for the pressure on both the inlet AND the outlet, and correspondingly, on both the bottom AND the top wall. They typically used a homogeneous Neumann condition on the top and the bottom wall, a homogeneous Neumann condition on the inlet, and a Dirichlet condition on the outlet.

This does not make sense to me from a mathematical point of view, because the INSE are first order in p in both the x and y coordinate, so specifying boundary conditions at only, for example, the bottom and left wall should already be enough to fully specify the solution.

I know that the INSE are invariant under constant pressure shifts p --> p+p0 which gives the system a little extra freedom. I also know that typically the pressure is simulated using the pressure Poisson equation, which is of course second order. But this is a consequence of the INSE, not a governing equation in its own right.

I would appreciate some elucidation on this topic. Thanks.