vorticity boundary condition
 

I'm solving a 2D forced convection problem by using finite difference method. It involves solving the vorticity-stream function equations. I met some problem when I'm trying to solve the vorticity equation, which can be stated as follows:

How to deal with the vorticity boundary condition? after some searching in google, I realized that vorticity boundary condition is actually changing as time goes forward! In other word, vorticity boundary condition has to upgraded at each time step. But since the vorticity boundary condition is not constant, what values should we adopt? Say we're calculating the n+1 step, should we use the vorticity boundary condition at n+1 step or at n step? One other thing, because vorticity boundary condition is actually calculated from stream function distribution, what choice we make about vorticity boundary condition involves the distribution of stream function, at least at the boundary and at the layer that is closest to the boundary(suppose we use Thom method to calculate vorticity boundary condition). In this sense, it seems that we have to use the previous time step stream function values, because solving the vorticity-stream function equation series requires vorticity to be first solved, then the stream function. Does that mean we use the n time step stream function to get the vorticity boundary condition for n+1 time step vorticity calculation?

Thank you for reading my post, any suggestion or advice will be greatly appreciated :)