# How to assign initial velocity field and handle pressure-velocity coupling in FVM?

I am trying to solve the 2D incompressible Navier-Stokes equations for laminar flow over a backward facing step using the finite volume method.

This is the plot that I generated of a generic mesh showing u1, u2, and p nodes to help myself derive the discrete integrals in the conservation equations: The grid is staggered to avoid pressure checkerboard-ing. I am doing explicit time integration and I am using a uniform grid. Except near the boundaries, I apply central differencing scheme for derivatives, and I perform second order linear interpolations to obtain values at the control volume faces.

I am following the procedure from Ferziger and Peric in this picture: Currently my solution is blowing up, and I have a few reasons for why I suspect this to be the case, but I have some general questions that may help me narrow down why this is happening.

1. When I assign an inflow boundary condition of u1, how do I ensure that the initial velocity field is divergence-free? For example, if I assign a u1 inflow condition and the rest of the u1 nodes are 0, it seems like the velocity field will of course not be divergence free.

2. I am handling the pressure velocity coupling by taking the divergence of the momentum equation which leads to the equation shown in the picture below from Ferziger and Peric. I then apply Gauss's theorem to both sides of the equation to turn this into a surface integral equation that I can apply the finite volume method to.
Is this typically how the velocity-pressure coupling is handled for explicit time integration?
In all my reading and searching, I have not seen it done this way. I have read about pressure-correction methods like SIMPLE, but based on the procedure in the image above, it seems that I do not need to apply a pressure correction method.
Does it mean that if the velocity field at time n is divergence free, I can just solve the pressure Poisson equation in integral form and then use this pressure field to obtain a divergence free velocity field at time n+1? Or is this not an adequate way to handle the pressure-velocity coupling? Any help would be greatly appreciated.

• Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking.
– Community Bot
Dec 13, 2021 at 14:56