I am thinking about how to formulate 3-D navier-stokes equations by using stream function and vorticity. I remember some people have done this before but I don't know where to get it. I know this method is not quite suitable for 3-D problems. However, from my previous experience in 2-D calculation, under the curvilar coordinates, it takes much longer time to get a converged result by using SIMPLE method than that obtained by using stream function and vorticity method. So far, I haven't figured out what's the main reason. When I used the SIMPLE method to calculate the flow field over a sphere(Re=100), the pressure correction equation always gave me the troubles. I took nearly 4 hours to get a converged result. The worse thing is that I could only use the uniform grid. If I use the nonuniform grids, it just gave an oscillating result or blowed up eventually. So this is the reason why I am trying to use stream function and vorticity method. I need some experts' help. Any recommendation on this issue will be highly appreciated.

It is probably difficult to extend 2-D stream function and vorticity formulation directly to 3-D. ( unless you can derive such set of governing equations ). But, it is possible to use velocity-vorticity formulation. It is also possible to use velocity-potential-vorticity formulation. The starting point is to derive the proper set of governing equations along with the necessary boundary conditions. You should be able to find the equations in most fluid mechanics books. As for the convergence problem, I'll repeat the example I used here before. For incompressible flow, velocity field and pressure field are decoupled, and they are identical twins. The stream function-vorticity approach solve for the velocity field first through stream function-vorticity. Once the converged solution is obtained,one can easily find the other identical one (the pressure field) by integration of velocity field. On the other hand, the pressure-based approach try to solve the identical twins problem at the same time, iteratively or coupled. The effort is not just doubled because of the direct coupling of the velocity field and the pressure field. As for the speed-up methods, a hardware approach is usually more practical than the software approach. I don't understand why the code you use is good for uniform mesh only.

Thank you so much for your valuable suggestions. When using the SIMPLE approach, I had to use staggered grids to prevent from getting the false pressure field. I set the pressure nodes at the centers of the cells. The velocity nodes lie on the interfaces between the adjacent cells. When I use uniform grids, the velocities are exactly on the middle points between adjacent cells. So it is natural to take the gradient of pressure as the driving forces for the velocitis. But in the nonuniform grid system, if we still set the pressure nodes at the centers of the cells, the velocity nodes don't lie in the middle places between the pressure nodes any more. So when discritizing the momentum equations, we might bring in some errors due to the spatial displacement. I guess this might be one of the reasons why nonuniform grids don't work very well. You mentioned velocity-vorticity approach. Since the pressure terms in the momentum equations have been dropped out, one doesn't have to use staggered grids, am I right? Actually, I have derived a set of 3-D Navier-stokes equations by using stream function and vorticity approach. I am not quite sure whether there are mistakes during my derivation because the final equations look nasty. Moreover, I am not quite clear how to dertermine the boundary conditions for stream functions because there are three components.

I think Batchelor talks about the 3D vectorial counterpart of the "usual" streamfunction. You can also check the book by Chorin & Marsden which basically introduces 2D vortex element methods, but has a section devoted to 3D vectorial streamfunctions.

On staggered vs. non-staggered gridding: The absence of the pressure term does not automatically imply/justify using a non-staggered gridding system. You have to make sure that your discretized equations are _consistent_ with the continuous form. Here lies the answer to your choice of gridding systems. Depending on what type of discretization scheme you implement you may have to use either staggered or non-staggered grids to get a consistent solution.

Adrin Gharakhani