hi, I am working in 2D, trying to solve depth-integrated N-S in vorticity/streamfunction formulation. One of the main difficulty it is how to set up good boundary conditions for vorticity and stremfunction. I solve the problem for enclosed flow like Lid-driven cavity but I don't know what to do with open boundaries. Can you send me any advice or book I should read about this topic ? Thanks

the easiest way to do this is to transform the NS equations to the streamfunction-velocity formulation. Getting the BCs for the vorticity is tough, but the streamfunction and velocity BCs are much easier.

Hi Andrew, thanks for fast replying, ok I already transformed NS eq. to streamfunction-vorticity form. I also derived both streamfunction and vorticity no-slip boundary conditions. I begin to have problems in open flows, for example : when you are in a cavity with moving walls, or no, you can set streamfunction constant on all walls ; there is no problem because the averaged flow rate inside the cavity is zero. Once I try to simulate Poiseuille or Couette flow I experience lot of problems: the solution is conserving flow rate introducing non physical circulation. Now my domain has one no slip wall and three open boundaries : How should I write th boundary condition for streamfunction and vorticity in this case ? May it be normal gradient equal to zero for both variables? Do you have any references or article to suggest ?

I tried to do the exact same problem, and that is why I moved from the streamfunction-vorticity equations to the streamfunction-velocity. It was too tough to get the BC's for the vorticity.

I used a paper by Gupta to solve my equations. It is a good paper.

I believe the following paper (and references in it) will help you:

T. Lundgren and P. Koumoutsakos, "On the generation of vorticity at a free surface," JFM, vol. 382, pp. 351-366, 1999

Adrin Gharakhani

THANKS !!!