Vortex methods with variable viscosity?
Hi,
Someone posed a question to me recently : how could one use vortex methods to compute flows with variable viscosity? I have experience with vortex methods, but am quite perplexed at this. See, in the momentum equation in convective form, one has the term (div tau), where tau is the viscous stress tensor. If no compressive viscosity is present, then tau= mu def(u). Here, mu is shear viscosity and def(u) is the rate of deformation tensor. OK, take curl of (div tau) for the vorticity equation. You get some nasty terms combining velocity gradients and viscosity gradients. These terms look very difficult to tackle in the vortex method formulation. Anyone has ideas? Lorena. 
Re: Vortex methods with variable viscosity?
It's usually written
tau = 2*mu*D where D = [ Grad(u) + Grad(u)^{T}]/2 I've never dealt with vortex methods but in some variable viscosity codes they simply use the chain rule. Div[ 2*mu*D ] = 2*Grad(mu)dot D + 2*mu*Div[ D ] If you are not dealing with gas mixtures and the fluid is in the gas phase then Grad(mu) = d(mu)/dT*Grad(T) where mu = mu(T) and T = temperature. For gas mixtures, mu = mu(T,Y_{i}). Also, look through these http://scholar.google.com/scholar?q=%22vortex+methods%22+variable+viscosity&ie=UTF8&oe=UTF8&hl=en http://scholar.google.com/scholar?hl=en&lr=&q=%22vortex+methods%22+compressible 
Re: Vortex methods with variable viscosity?
Thanks, Runge.
It's incompressible flow, not a gas. In vortex methods, we use the vorticity equation, i.e., the curl of momentum eq. When taking the curl of Div[ 2*mu*D ] is when the ugly terms appear. All sorts of mixed derivatives. Lorena. 
Re: Vortex methods with variable viscosity?
Gases may be treated as incompressible.
My impression is that vortex methods became popular long ago when full NavierStokes simulations were impractical. It may be that nonisothermal vortex methods are too much of an aggrevation to be worth it. Be careful not to consider flows that violate the underlying assumptions of the procedure. Strictly incompressible flows are isothermal. Incompressibility is derived by considering that the NavierStokes equations (NSE) support two distinct times scales; one fast (acoustic) and one slow (convective). Kreiss has written about how one "projects" the NSE onto the "slow manifold" to remove the fast acoustic scale (to some order in Ma^2). For isothermal flows, the acoustic component is projected out if the velocity vector evolves on the manifold (nabla dot u) = 0. When one allows nonisothermal flows, the equation of the manifold changes also. Many people solve the "lowMach number" equations but fail to determine what the correct manifold is. You have presumed Div(u)=0 but now suppose that there is temperature variation. The manifold for variable temperature, lowMach number flows is no longer Div(u)=0. http://scitation.aip.org/getabs/serv...cvips&gifs=yes 
Re: Vortex methods with variable viscosity?
That's an interesting discussion. I will think about it.
Unfortunately, I don't know the application in this case, because the query comes from someone who wrote to me in the knowledge that I had experience with vortex methods. By the way, "supporters" of the vortex method approach today are mainly interested in the meshless nature of the method, and the lack of numerical diffusion due to the Lagrangian treatment of convection. It still has interersting areas of application, notwithstanding the progress in full NS simulation. 
Re: Vortex methods with variable viscosity?
The most straightforward approach would be to treat the additional terms as source terms in your vorticity equation and see what happens.

Re: Vortex methods with variable viscosity?
Hi ag,
Yes, indeed. The problem is that those terms contain velocity derivatives and viscosity derivatives in various products. In the context of vortex methods, it is not straightforward to obtain spatial derivatives from the scattered particle information. Schemes to provide viscous effects (as far as I know) all assume constant viscosity. For example, the method known as Particle Strength Exchange (PSE) uses an integral operator to approximate the Laplacian in the diffusion term (constant nu) and then discretizes the integral using all the particles as quadrature points. I think it is a difficult problem. 
Re: Vortex methods with variable viscosity?
Lorena,
There is nothing to be perplexed about; it is what it is :)) You are correct; if you assume spatially varying viscosity then you end up with terms involving viscosity gradients and velocity gradients, etc. This becomes computationally expensive and as such, it loses some of the appeal of vortex methods. However, evaluating velocity gradients  at least in 3D, which is my main interest  is not an additional burden since we already need it for evaluating vorticity stretch (IF we use vortex particles and not filaments, but even with filaments we can approximate the vel grad tensor easily). The evaluation of viscosity gradients is not that difficult either, fundamentally. Here you have a whole host of options; such as moving least squares to arbitrary order of accuracy, and even particle methods. For the latter, just as you use particles to discretize vorticity you can use it to discretize viscosity. Again, the problem is with cost  nothing more, in my opinion  and that's one of the reasons (there are others) vortex methods have not been too successful in dealing with compressible flow. I am aware of the stateoftheart with compressible vortex methods, but it has a very very long way to go before it can become useful. Now, going back to the question of variable viscosity NS equations. The main question is whether the viscosity variation represents the physical problem, or whether it results due to filtering of the NS equations for LES purposes. In the case of LES, you don't have the complications with viscosity gradients, etc.; you just have the grad(viscosity(grad(vorticity)), which is easily doable either with PSE or VRM (vorticity redistribution method). In the above, I've ignored the term associated with the transpose grad of vorticity for simplicity. NOTE: both PSE and VRM can solve the variable viscosity diffusion problem (with or without source terms) If the interest is in physical variation of viscosity, such as would be in combustion, you can check the IOP paper by Winckelmans (vortex workshop in santa barbara). Hope this help a bit. Adrin Gharakhani 
Re: Vortex methods with variable viscosity?
Thanks ever so much for your comments, Adrin.
Now, this was a query of someone who wrote to me in the knowledge that I had experience with vortex methods, so it's not really a problem I'm working on. For this reason, I don't know the application. I've written him to ask. The following thing pops in my head after reading your reply: if one were to use particles to discretize the viscosity, just as they are used to discretize the vorticity, then ... wouldn't that mean that no longer are the particles confined to the vorticity support? Is this what you mean by the problem of cost? Because as far as I can remember, this was the main drawback of the compressible method of Jeff Eldredge: he needed particles everywhere to track the dilatational. I've found the combustion paper of Thirifay and Winckelmas.. will have a look. Quite an interesting discussion, thanks again. Lorena. 
Re: Vortex methods with variable viscosity?
Lorena,
"if one were to use particles to discretize the viscosity, just as they are used to discretize the vorticity, then ... wouldn't that mean that no longer are the particles confined to the vorticity support?" No. There is no problem there, whatsoever. Remember, you're solving for the vorticity field, so your support is the vorticity, and as a result you _only_ need viscosity values at points you have vorticity. You don't care how the viscosity changes in areas where vorticity itself is zero (negligible). That is, you can't diffuse a zerovalued vorticity any further :) The problem of cost comes from the additional cost of having to discretize the viscosity field and evaluating its gradients. And in 2D, you have the additional cost of evaluating the velocity gradients (by direct differentiation of the BiotSavart integrals). The cost associated with the compressible flow computation  low Mach compressibility is not that expensive, actually, with vortex methods  is that, as you said, the computational domain is no longer compact and you need particles everywhere. On top of that you have additional variables and derivatives, etc. that you have to evaluate. In this case, I believe it is better to use the primitive variable formulation in the SPH or other meshless method context. Adrin Gharakhani 
Re: Vortex methods with variable viscosity?
Yes, of course! No need to track viscosity everywhere ... (duh!) :) Thanks for the clarification.
I did for a moment think about the SPH method for _this_ problem (variable viscosity). Who knows what the application is, the guy in India who asked the question in the first place has not replied to this. But one could probably use SPH derivatives for the viscosity, as one could do PSE derivatives. In either case, one would have to do the remeshing thing. Going a bit off topic : are you familiar with Mark Stock's calculations of the Rayleigh Taylor instability with vortex methods? This is one problem I would like to get into. I suppose that one needs a similar approach as we were discussing for the barotropic generation of vorticity. Best, Lorena. 
Re: Vortex methods with variable viscosity?
> are you familiar with Mark Stock's calculations of the Rayleigh Taylor instability with vortex methods?
Yes, I'm familiar. His work is vortexincell based not "particle" based  nothing wrong with VIC; indeed, it's probably the happy medium between gridbased and purely meshless based methods (in terms of cost reduction while still maintaining very low levels of numerical diffusion) Adrin Gharakhani 
Re: Vortex methods with variable viscosity?
hmm, VIC ... a lot of people switch to VIC. I think Winckelmans switched, or at least was doing _some_ VIC calculations.
I haven't studied the details of VIC, but does this make it easier to calculate grad(p), because you have the grid to do it? My interest would be to do barotropic generation of vorticity in a particle method (no grid!). Maybe the SPH would be a better way than vortex method? I will send you an offprint of my paper on the vortex method with an alternative to remeshing. The address would be at Applied Scientific? I've looked it up on the web.... Thanks for your conversation, Lorena. 
Re: Vortex methods with variable viscosity?
I have your thesis and recent IJNMF/AIAA papers; if there is something new/unpublished then I'd appreciate a copy
Thanks Adrin Gharakhani 
Re: Vortex methods with variable viscosity?
In the low Mach number limit, you don't need any info on grad(p) for vorticity generation (you can use the Lagrangian fluid acceleration instead). This has been done already for quite a long time by Ghoniem's group at MIT. Check out JCP, Vol 184, 2003, for example, for the purely gridfree version and no remesh using VRM
Adrin Gharakhani 
Re: Vortex methods with variable viscosity?
This is most useful. Thanks a lot.
Also, nice to "meet you" in this forum. Best wishes, Lorena 
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