Potential Flow Solver
Is it correct to say there are basically three types of flow solvers? As I understand it these can be classified as potential flow solvers, density based solvers and pressure based solvers. Turbulence models are just turbulence models   they fall outside any of these categories.

I would say the three main categories of solvers based on assumptions of the physics are potential, Euler, and Navier Stokes. Of course there are all sorts of sub categories, for example steady, unsteady, compressible (subsonic, transonic, supersonic, hypersonic), incompressible, laminar, RANS, URANS, LES, DNS, etc.
On the other hand, density based solvers and pressure based solvers says something about how a solution is obtained (along with giving an idea of the type of flow being solved for). 
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Here is my real question: Seems that if one wants to solve for, say, the flow over an airfoil at a high angle of attack, a nonlinear potential flow solver coupled with some kind of a turbulent wake model would be preferable to either a densitybased or a pressurebased model, simply because (at least in the outer region) you are solving a single PDE instead of three or four. Is this too naive? 
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If you mean by fundamental a solver that solves just the navier stokes equations, they are everywhere  i would wager that all meaningful commercial solvers do that, and almost each grad student in cfd probably has written his own ... They are all as general as the navier stokes equations themselves. I would guess that about 20% of the world's supercomputers do nothing but solving the ns equations at a given time :) Quote:
I am not aware of people computing turbulence in wakes, jets or around bodies ever doing anything else but Navier Stokes, but i have to admit i m not familiar with the nonlinear potential flow. A quick google search gave me very few pappers on this from the 80s, i guess its not considered anymore. 
Thanks for this. It is quite illuminating.

The Navier Stokes solvers do solve the NS equations. It's the Reynolds Averaged Navier Stokes (RANS) solvers which solve (or try to solve) for the mean flow by inserting eddy viscosity.
Potential codes have been hooked up to boundary layer codes to calculate the lift and drag on an airfoil. But, in general, that is for flow without separation (and therefore not high angle of attack) and it is not for 3D viscous flow. That's not to say some haven't pushed this class of code somewhat into these regions. In general, the performance of potential codes should be compared to the performance of 2D incompressible/subsonic (uncoupled pressure based) solvers. Sure they are faster, but, for a 2D geometry, I'm not sure many would really care due to the high performance of computers. Maybe if someone is doing optimization or a trajectory, but even then I doubt it would really matter much anymore. In general, a coupled Euler density solver (as apposed to uncoupled pressure based) is meant for transonic and above (i.e. greater than supercritical). In this region the accuracy of potential solvers degrades due to the assumptions for potential flow. Therefore, in general, full potential flow and coupled Euler density solvers target two different flow regions. Of course one could use an Euler density solver to solve for low subsonic speeds (the realm of potential and Euler uncoupled pressure based solvers), but the convergence rate will suffer and at some point one will need to go to Mach number preconditioning. Also, in general, flux splitting schemes (which have not been adapted to potential as far as I know) capture and track vorticity better. 
Oh, just want to be clear.
Full potential was used in the past, and is still used, for optimizing wings at transonic speeds. However, as far as I understand, it is challenging for them, due to the lack of a good boundary layer model and the assumption of isentropic flow, to capture the position of the shock. Therefore, the wave drag is off. 
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