# Incompressible 2D Navier-Stokes equation

I am trying to solve for and simulate the vorticity numerically (finite difference method), however there's one part I was hoping to get some help with.

I need to find the fluid velocity $\mathbf{u}$ from the vorticity $\omega$. I can write

$$\mathbf{u} = (\nabla \phi) \times \mathbf{\hat{z}} + \mathbf{u_0} ,$$

where $\mathbf{u_0}$ is known and $\phi$ is the fluid field potential and to find $\phi$ we solve

$$\mathbf{\hat{z }} \cdot (\nabla \times \mathbf{u}) \ = \nabla^2 \phi \ = \ \omega.$$

This is a problem with periodic boundary conditions and I know that the velocity won't change if $\phi$ is changed by a constant so I could choose a point in the plane such that $\phi = 0$. And this is where I don't know how to proceed. I'd really appreciate some help.

How could I go about choosing this point?

• A good way to deal with this problem is to apply Fourier transform to the whole Poisson equation in the periodic coordinate (or in both coordinates, if it is double-periodic). Feb 11 '18 at 3:12

If you want to solve for $\phi$, you could add in a condition, $$\int_\Omega \phi\,{\rm d}V =0,$$ which would make the solution unique. To do this numerically, you would probably need to use a Lagrange multiplier.