Explicit filtering in LES
Hi,
I have been doing some LES simulations using the FiniteVolume formulation as an implicit tophat filter. I would now like to implement explicit filtering into my code. However I have so far only succeded in confusing myself.... If anyone can answer the following questions I would be most gratefull. In the momentum equation the velocities become filtered velocities and the pressures become filtered pressures. Do I need to determine these from the actual velocities using the filter?? can I do the opposite and determine the actual velocity (and therefore the fluctuation) by reversing the filter and applying it to the filtered variables???? Additionally I am unsure about whether to represent the subgrid dissipation as a source/sink term in the equation or as an additional viscosity  what do you recommend?? Thanks 
Re: Explicit filtering in LES
TJ,
One thing to note before getting to your questions. Use of a finitevolume formulation does not necessarily imply tophat filter. There are approximate ways of estimating what the implicitfilter looks like and it's width (if needed). But in general the implicit filter depends on your discretization schemes (numerical dispersion and dissipation terms). If you had actual velocities and knew what your filter is, then you can compute filtered velocities. However, in LES, you have the problems in reverse. You have filtered velocities. Even if you knew what the implicit filter was, you can not back out the actual velocity field except to within an approximation. This is because a lot of different actual velocity fields can result in the same filtered velocity fields (upon filtering). i.e., Filter(F) = <F> For a given field "F", <F> is unique, but for a given <F>, "F" is not. Often when you reverse a filter operation, you might end up with unbounded operators or operators with broad support which make the reverse filtering cumbersome. There are however models based on approximate inverse operators to filters that are used in LES modeling. Here is a good website that discuss such models. http://www.ifd.mavt.ethz.ch/cfd/resles.htm Explicit filtering : When people talk about explicit filtering what that often means is you take the LES generated field and filter it using an explicit filter. The LES generated field being implicitly filtered already, explicit filtering involves filtering of a field that is already implicitly filtered. (Hope I haven't confused anyone here). EF : explicit filter IF : implicit filter actual variable q produces an LES variable IF(q). Upon explicit filtering what you end up with is EF[IF(q)]. If the characteristic width of the explicit filter is much larger than that of the implicit filter, you can safely assume that the resulting field is an actual velocity field filtered using an explicit userspecified filter. i.e., EF[IF(q)] ~ EF(q) Thus the implicit filter, which is often the unknown, becomes inconsequential. Since the implicit filter also depends on the mesh spacing, any dependence on the mesh is also removed. Sometimes, the numerics create heuristic high wavenumber behavior that needs to kept in check (eg. dealiasing errors), explicit filtering removes all those heuristic scales. That's why explicit filtering is highly recommended for unstructured meshes. 
Re: Explicit filtering in LES
Kaylan,
Thanks for your reply  It answers most of my questions. can I just confim a couple of points... If I explicitly filter an implicitly filtered field I need to make the explicit filter width much greater than that of the implicit filter. My implicit filter width is equal to the cube root of the cell volume  how much bigger does the explicit filter have to be ? It seems I could sensibly do 2 or 3 times the implicit filter length  is this enough  if I did this would the resolution of the simulation be effected ? If I used a germano SGS model would my test filter be 4 or 6 * grid scale ?? If I feed the explicitly filtered velocities into the momentum eqns and Explicitly filtered pressures into my SIMPLEC algorythm do I get actual or filtered pressures & velocities out at the end of the time step ?? Thanks ever so much TJ 
Re: Explicit filtering in LES
Typically, it has been assumed that filter width is about twice the cube root of cell volume. However, it is true for nearisotropic grids. For highly anisotropic grid, characteristic filter width is hard to define since the implicit filter may itself be highly anisotropic. This however seems to have minimal impact of dynamic LES since the test filter also is typically anisotropic (i.e., the anisotropy of the implicit filter and the test filter may be nearly analagous).
By doing explicit filtering, you lose some small scale information. This is OK since smallest scales in almost all numerical methods are affected the most by the numerical discretization errors and are thus not usually fully representative of the fluid physics at that scale. You are right that the test filter needs to be fairly wide when using explicit filtering. You have to balance your accuracy requirements with your computational expense. Regarding your last question, I can only give my opinion. If you feed explicitly filtered fields (v,p) into a numuerical solver with an LES model (corresponding to the explicit filter), you should end up with variables filtered at the explicit filter level. But in reality, you may end up with a explicitly filtered variable + some small scale field. I am reluctant to give this field a name but I guess the term "numerical noise" is probably fits it well. Upon filtering this field, the "numerical noise" is removed and you get the explicitly filtered variable. SIMPLE and it variants (SIMPLER, SIMPLEC) use implicit timestepping which may be problematic in terms of convergence if you explicitly filter your velocity field after each implicit iteration. Typically most LES is done using small time steps (CFL < 1) using predictorcorrector type schsmes and there is no need to iterate on the velocity. 
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Re: Explicit filtering in LES
Thanks Kalyan sir for the post. The information regarding explicit filtering technique is very helpful.
But I have a question to Kalyan sir that whether this mentioned concept EF[IF(q)] ~ EF(q) is only applicable to Dynamic Smagorinsky Model (DSM) or it can be implemented on the Constant Smagorinsky Model (CSM) as well?? Quote:

this post is very old and many new studies appeared in literature, I suggest a search to find the stateoftheart in the field of implicit/explicit filtering

Explicit filtering in LES
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However, as I am quite new in the field of LES, I have some questions regarding implicit/ explicit filtering techniques in LES. Say for an example: if any LES code (which is basically physics based model) is operating by deafault in implicit technique and some one would like to implement the explicit filtering in it (as from CTR briefs it assumes that explicit filter will be able to give better solution over implicitly filtered LES), what would be the best possible way to do it? 
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Conversely, the explicit filter applied to the updated resolved velocity is not consistent to the filtered equations. 
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I have basically two questions regarding this post. If we have to implement an specific filter (explicit filter) let's say that the kernel is the spectral cutoff which indeed reduces the degree of freedom of the computation. According to your previous statement, implementing an explicit is basically to filter the velocity field (u_i) from every time steps using the filter kernel decided? However, you mentioned the convective term; which is made by u_i*u_j. I lean towards the idea of applying the filter procedure on the u_i field at every time steps and then use that field for the next time step. The other questions is, how can I really end up with a grid independent solution if the solved velocity field (prior the filtering with the specific kernel) has an implicit filter (second order schemes Tophat ). Hence the filter operation will be on the solved velocity field which has an implicit filter on it. From different papers, it is clearly shown that the cutoff kernel is very effective imposing the cutoff wavelength. However, the usually se spectral methods. I was wondering if the same effect is achieved even if the derivatives are computed using finite differences. I know that this is very cumbersome, for someone who is just beginning in this field .!! thank you very much for your time. 
Many authors suggested the correct way to adopt explicit filtering as the application of a filtering (it can be a spectral cutoff but that is not mandatory) of width Delta > h (this latter being the computational size) on the nonlinear term.
It can be shown that filtering the velocity field at each time step is not consistent with the filtered equation. This way, a LES solution will be gridindependent when Delta is maintained fixed and h goes toward zero. That does not depends on the type of the filter that can be either projective or smooth. 

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correct ;) 
thank you professor; Now I see why it is computational expensive..!!!

Professor; In Pope's book section 13.5 (LES in wavenumber space) he considered sharp spectral filter, which is natural choice for LES in wavenumber. So, now I wondering if this type of filters are only possible for spectral or pseudo spectral methods. Nonetheless, I think that if a sharp cutoff is applied as an explicit filter in physical space, then it is possible to retain the advantages of the sharp cutoff no matter how I introduce the filtering.. Am I right?
Thanks before hand Respectfully JM 
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using spectral methods the cutoff is a natural way to get an explicit filtering, conversely, using FV or FD methods you have to perform a Fourier transform, apply the filter and come back to the physical space, a task quite cumbersome. The second part of your question is not very clear to me... 
Excuse me professor; I will make my statement clearer. Although it is a little disappointing that sharp cutoff are only possible in spectral methods. It means that any one using FD or FV can only apply the top hat (based on the numerical scheme and order)
What I wanted to say, is that since the kernel function of the convolution integral can take any form, i.e: the form of the top hat, Gaussian , spectral etc. I thought that it was a matter of changing the filter kernel of the convolution integral and use any integration method to apply the filter on the specific property. Hence, if I apply an explicit filter (once my velocity field is computed) using the kernel I wish, then the final solution will keep the advantages of the sharp cutoff.... Thanks Julio 
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theoretically yes ... but practically is very costexpensive working with FV/FD methods and perform at each time step FFT/IFFT only to apply the spectral cutoff. What is more, the case would be feasible ony for particolar geometry with periodical conditions. Conversely, the tophat filter would be useful for general flow conditions. 
Thank you professor; It seems that the only options for FD and FV are Gaussian and Tophat. My main concern is due to important facts that I have come across in the literature; that only cutoff reduces the degree of freedom of the problem. And, smooth filters (gaussian, top hat) are not effective to attenuate wavenumbers beyond the filter width.
The more I read, the least I understand :S I think this is not a good sign!!! Respectfully and thank you very much Julio Mendez 
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