How to Handle Changing Vorticity Boundary Conditions in 2D Forced Convection?

[bearcharge]

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 :)

 

[Achy47]

Hello,

Thank you for reaching out about your question on vorticity boundary conditions in your 2D forced convection problem. I understand that this can be a challenging aspect to deal with in your finite difference method solution.

First, let me confirm that your understanding is correct - the vorticity boundary condition does indeed change as time progresses. This is because the vorticity at the boundary is affected by the flow and other boundary conditions at each time step.

In terms of what values to use for the vorticity boundary condition, it is best to use the values at the previous time step (n) to calculate the vorticity at the current time step (n+1). This is because, as you mentioned, the vorticity boundary condition is calculated from the stream function distribution, which is solved at the previous time step. Using the current time step values could lead to inconsistencies and errors in your solution.

Regarding your concern about using the previous time step stream function values for the vorticity boundary condition, this is a common approach and is acceptable. However, if you are using a more advanced method such as the Thom method, it may be necessary to use a more refined approach to calculate the vorticity boundary condition. This could involve using multiple previous time step stream function values, or other techniques to ensure accuracy.

I hope this helps and provides some guidance for your problem. If you have any further questions, please don't hesitate to ask. Good luck with your solution!