I'm trying to develop a code to solve the stream function-vorticity equations using the Finite Element Method in order to simulate a 2D incompressible flow problem. I was wondering what the pros and cons are, whether coupling a turbulence model is possible, whether formulating the boundary conditions may face difficulty and whether the evaluation of the pressure field is flawed possibly due to decoupling of the pressure variable from the governing equations. Note an accurate evaluation of the pressure field is particularly important for my case of study.
I very much appreciate helping me out.
Using the vorticity-stream function formulation in 2D is a good way of reducing the computational cost if compared to a primitive variable formulation. The main issue should be addressed on the BCs for the vorticity, just think about the fact that a wall produces a vorticity that is flow dependent.
Quite more complex is the extension to 3D problem, often the velocity-vorticity is preferred.
Turbulence model for the vorticity-stream function formulation are almost unexplored.
Hi, Armin Hajighasem Kashani .
If I can add my two cents to the excellent answers above:
I used this formulation for computations of the continuous-phase field in my four-way coupled particulate flow code. Some notes that might be relevant to you based on my humble experience:
1. It can be quite difficult to extend this formulation to 3D. So If you plan to look at 3D phenomena at a later stage, I would advise against using this formulation.
2. If you are planning on handling flow in irregular geometries, you can face some serious difficulties formulating physically meaningful and numerically accurate boundary conditions for vorticity. The solution can be highly sensitive to the slightest of changes in those.
3. Pressure field computations should be straightforward and they can be easily done post solution. But if you are interested in looking at the temporal evolution of your pressure field, I am not entirely sure this formulation would be the best option.
4. If you use appropriate discretization schemes, the vorticity-streamfunction formulation can handle multiple, 'highly' non-linear source terms.
Hello, I agree with your comments, there is a general observation to understand the quality in the vorticity-velocity formulation: the time-space local truncation error is associated to the discretization of the vorticity, that is a derivative of the velocity. Generally that means you get more accuracy in the velocity field, associated to the fact that the divergence-free constraint is implicitly satisfied while the primitive forrmulation has some finite error producing a spurious effect of kinetic energy production.
However, I would highlight my opinion about the adoption of this formulation in model-based approaches for turbulence, such as in LES. I think that, after the filtering of the vorticity equation, the decomposition and the modelling of the non-linear part is not very theoretically explored. The concept of an eddy viscosity for the vorticity is somehow is far from be flawless.
Dear Tapan K. Sengupta
that is quite intriguing, I admit I never approached a development of LES in the framework of the vorticity equation. My main concern is about the concept of the eddy viscosity concept for the vorticity field. Then, the filtering of the stretching term would produce something new for me.
Approaching with ILES could work but does not give an idea of what the unresolved terms should do. Maybe a coupling with the filtered enstrophy equation would appear... but that would produce a new unresolved term...
Dear Tapan K. Sengupta
it is curious that in 1950, von Neumann published on Tellus the paper "Numerical Integration of the Barotropic Vorticity Equation", maybe the first numerical forecast of that time.Despite some interesting studies, the vorticity equation has not gained appeal in the present years. Maybe the field of weather forecast is more suitable for this formulation owing to the quasi 2D turbulence??
Dear Tapan K. Sengupta
The vorticity equation is one of the fundamental equations used in meteorology and the Ertel’s vorticity theorem remains until today one of the most powerful tools in the theoretical studies.
As far as the numerical aspects are concerned the studies of energy transfer are also based on the vorticity equation, see for example the following paper based on the ECMWF system
https://journals.ametsoc.org/jas/article/70/4/1233/27256/Coarse-Graining-the-Vorticity-Equation-in-the
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