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} $,$$\mathbf{u} = (\nabla \phi) \times \mathbf{\hat{z}} + \mathbf{u_0} ,$$
where $\mathbf{u_0} $ is known and $ \phi $ is the fluid field potential and to find $ \phi $ we solve
$ \mathbf{\hat{z \ . }} \ (\nabla \times \mathbf{u}) \ = \nabla^2 \phi \ = \ \omega$ .$$ \mathbf{\hat{z }} \cdot (\nabla \times \mathbf{u}) \ = \nabla^2 \phi \ = \ \omega.$$
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?
