Vortex methods for airfoil analysys
 

Hello,

I have recently learned about vortex (random walk, diffusion etc) mesh-free methods and wondering if they could be efficient in an analysys of incompressible flows past airfoils. As I have learned from the literature, people usually use them for analysys of flows past bluff bodies at low reynolds numbers.

I would like to adopt the algorithm describet in the thesis of Shankar Subramaniam:

http://www.eng.fsu.edu/~dommelen/pap..._a/node78.html

by modifying it: I wouls like to increase diffusion radius and decrease the number of vortices as vortices move downstreams in the far wake region and eventually remote vortices, taking their influence into account using asymptotics, so that the number of vortices will not increase indefinitely with time.

If it is worth of doing, is this method will work accurately enough for airfoils at Re 10^4-10^6?

There is also concern about computation time, the fast multipole methods allow to reduce computation time of the vortex velocity field, making it O(N) rather tah O(N^2), but will it be fast enoug to run a computation in a reasonable time on a PC?

There is fundamentally no reason not to use vortex method for your problem; in fact, it is even a better method. Are you thinking of running a 2D problem? In either case, you can probably still use diffusion for the case of Re=10^4 but for Re=10^6 it will make little sense to include diffusion; for all practical purposes the latter is an inviscid problem. If you insist on accounting for diffusion then you'd essentially be solving a DNS problem, which means that your inter-particle spacing would have to be in the order of the diffusion length scale. For the vorticity redistribution method of Shankar and van Dommelen you are talking about sqrt(8*dt/Re) as the optimal case. You can work the numbers with respect to the size of the geometry and the wake to see that you'll end up with an unreasonably large number of particles. Even for Re=10^4 you'd already be pushing the limit.

As for the suggestion to slowly increase the interparticle spacing to reduce the number of particles, this has been demonstrated recently (see Lakkis & Ghoniem, J. Comp. Phys, I think 2010). This is not a very straight-forward matter. Keep in mind that you can't just increase the interparticle size arbitrarily without losing (quite a bit of) accuracy.

As for fmm; 2D fmm is very robust/fast and you should get fast turn around times on a multicore cpu. 3D is more complicated.

Adrin

Thanks Adrin,

I am thinking about 2D problem at Re about 10^5. At this reynolds number there is almost always flow separation on airfoils of a reasonable thikness, so that inviscid methods should be modified to take separation/transitional bubbles into account. But even this is not a big problem, as numerous inviscid-viscous interaction methods + turbulence models could, in principle, handle marginal separation (as e.g. it is done in x-foil).

My problem is that I would like to compare performance of special stepped or vortex trapping airfoils with conventional ones and usual inviscid-viscous methods will not work for them, as there will be recirculation regions, vortex shedding and circular boundary/mixing layers. I also doubt that RANS calculations will reliably do.

So i hoped that the vortex diffusion methods with average distange, as you said sqrt(8*dt/Re), in regions of high velocity gradient (i.e. in the boundary layer and small separation/recirculation regions) will handle the problem if flow of vortices transforms into a dilute "vortex gas" in the regions where flow is virtually inviscid. Indeed, a fine resolution of wake is not so critical for airfoils as for say a cylider. Might this fact handle the problem with much smaller amount of vortices and much smaller than in DNS total amount of calculation at this reynolds number?

Could it happen that for steamlined bodies and virtually inviscid flows, errors related to adaptive diffusion radius could be negligible in comparison with flows past bluff bodies, such as cylinder?

OK, for 2D you have quite a bit of room to play. Basically, I believe, you'd need to do DNS near the boundary layer (and in the separation region) and beyond the immediate neighborhood of the object an inviscid flow will do just fine. You have to realize that as you increase the interparticle spacing from sqrt(8*dt/Re) to sqrt(R*dt/Re), where R > 8, the amount of circulation that is given away by each vortex particle to its neighbors becomes smaller and smaller, and for practical interparticle spacings the neighboring particles practically receive no circulation (which is the equivalent of an inviscid flow assumption).

There are other issues that you need to consider very carefully. Shankar's thesis used flow over a circular cylinder for benchmarking, so he could use the method of images to impose the no-flux boundary condition and obtain the wall-slip vortex sheet strength (the amount of vorticity generated at the wall). That is no longer the case for your more complex problem. So, you will have to implement a 2D panel method as well.

I don't know what your time frame is and whether this is a MS/PhD thesis, or industry, etc, but there is quite a bit of implementation work involved than initially meets the eye. We can discuss this in a private setting if you wish

Adrin

Thanks Adrin,

This is not my main work, I got interested myself in this kind of airfoilos and potential application of vortex methods for airfoil analysys at low reynolds numbers. I am folowing this topic "for fun" when have spare time, so I am not restricted by a deadline, but do not have much time for it.

On the first glance the vortex methods seem to be easy to implement, so they attracted my attention, as they do not require to solve PDEs and to rely on uncontrolled and doubtful turbulence modelling. Such small things as 2D panel code is not a problem, but the rest seems to take much time to implement (fmm, pse through diffusion, adaptive radius/ swich off diffusion, paricle creation and elimination, asymptotics for far wake etc).

It would be interesting to discuss this topic with you.