Dear All,

Could anyone suggest if there is commutation error in LES when finite volume scheme is applied as implicit filtering? If so, how to minimize this error? Many thanks!

Ray

Remedy : use equal size of cells. Cells should be small enough to capture small scale structures of the flow. (in the case of top-hat filter)

The above description can be proven using simple examples. Small cells means the flow inside the cell should be expressed simple equation (one eddy in a cell is simpler to model than two or more eddies in a cell). So small size cells and changing continuously the cell sizes with small increasing factor can reduce the commutation error.

References could be found as follows;

1. The basic equations for the large eddy simulation of turbulent flows in complex geometry, S. Ghosal & P. Moin

2. Construction of commutative filters for LES on unstructured meshes, Alison L. Marsden, Oleg V. Vasilyev, Parviz Moin

Jongdae Kim.

[sbaffini]

I dislike hijacking such an old post but, as this seems to be among the first results when googling for "les commutation error", it is worthwile having a correct, specific information.

The correct answer to the original question:

Is there any commutation error in LES when finite volume scheme is applied as implicit filtering?

is ABSOLUTELY NOT. Not per se, at least (more on this later).

It actually requires being really into LES and finite volumes to actually understand why there is no commutation error in this case but, as a useful simplification, you can get it by considering that, in FV, you don't suddenly take terms out of the integrals (which are your filter) as you do with spatial derivatives and spatial filters in general.

To be more specific, when you invoke commutation in classical LES explanations you are actually taking a spatial derivative out of a filter. In FV you don't take spatial derivatives out of volume integrals, you change them both in surface integrals by Green-Gauss theorem. So no commutation is ever involved in implicit LES performed with the FV approach.

Indeed, if you think more about it, it would be non sense in FV to invoke commutation between volume integrals and spatial derivatives, as this would change your integral form of the equation back into the differential one.

There is, however, a caveat here. When this lack of commutation error for implicit LES with FV is properly taken into account, it turns out that the form of the SGS stress tensor requiring modeling is different from what you are typically told, and it actually involves linear terms as well (i.e., pressure and viscous ones).

It would require too much space to delve into the topic here, so here are some useful references:

1) U. Schumann, Subgrid scale model for finite difference simulations of turbulent flows in plane channels and annuli. J. Comp. Phys., 18:376–404, 1975 (available here https://elib.dlr.de/53585/1/frt-75-jcomph-376-kl.pdf)

The work predated by most of the following ideas... it is very much cited also outside the commutation error fv community, yet without actually being read at all.

2) A.W. Vreman and B.J. Geurts, A new treatment of commutation errors in large-eddy simulation. In I.P. Castro, P.E. Hancock and T.G. Thomas, Advances in Turbulence IX, pages 199–202, CIMNE, Barcelona, Spain, 2002 (available here http://www.vremanresearch.nl/etc9.pdf)

The first ever work dealing with the topic. It is just cool that it was published in the same year of this post.

3) G. de With and A.E. Holdø, The use of turbulent inflow conditions for the modelling of a high aspect ratio jet. Fluid Dyn. Res., 37:443–461, 2005

The first to recognize the Vreman work above. Who says that people using Fluent don't understand things?

4) F.M. Denaro, G. De Stefano, D. Iudicone and V. Botte, A finite volume dynamic large eddy simulation method for buoyancy driven turbulent geophysical flows. Ocean Modelling, 17:199–218, 2007 (available here https://www.researchgate.net/publica...physical_flows)

Denaro and coworkers worked extensively on the topic. This is just one of a series of works starting more than a decade before and still going on.

5) P. Lampitella, Large Eddy Simulation For Complex Industrial Flows, Ph.D. Thesis (available here https://www.politesi.polimi.it/bitst...Lampitella.pdf)

Of course, I did my Ph.D. on this topic

Roughly speaking, the point is that you don't have commutation errors anymore, but you have additional SGS terms that might be worth considering in your SGS model. Spoiler: a simple scale similar model would do the job.

Hopefully this will help shed some light on this (apparently) obscure topic

[FMDenaro]

Very old topic, having so few answers that is quite strange google highlights that


Paolo has written the fundamental keys about that, I would only add that the misleading seems due to the hystorical confusion about the generic FV formulation, out of the LES field.

Indeed, you can often find in the CFD literature the discretization of the divergence form addressed as "FV method", as it appears also in the paper of Schumann. People used to translate that also in the LES field, calling implicit filtering the use of the divergence form wherein filter is commuted.
Actually, as Paolo wrote, the integral form is what should be denoted as the basic equation for the FV formulation, as clearly addressed in the textbook of Peric & Ferziger.


Paolo reported also a reference to one of my papers, dated 2007. Actually, the original formulation, wherein the commutation is not introduced, dates back to the years before, the topic being reported in the textbook of Sagaut at Sec. 7.9.1, pag 276. Therein you can find the original references to our papers.

[sbaffini]

Originally Posted by FMDenaro

Paolo reported also a reference to one of my papers, dated 2007. Actually, the original formulation, wherein the commutation is not introduced, dates back to the years before, the topic being reported in the textbook of Sagaut at Sec. 7.9.1, pag 276. Therein you can find the original references to our papers.

Just to mention that I actually knew that (as you know). Just that it would have been misleading to cite those early works in the present informal context. I think I gave proper context in my thesis.

[FMDenaro]

Originally Posted by sbaffini

Just to mention that I actually knew that (as you know). Just that it would have been misleading to cite those early works in the present informal context. I think I gave proper context in my thesis.

Yes, however, if I am remembering the correct paper, Vreman et al. showed the filter applied explicitly on the divergence of the flux but they did not change it in a surface integral by means of Gauss. In other words, that was not an integral-based FV formulation. Or I am wrong?

[sbaffini]

There are really few details in the paper, but if you look at equations 3 and 4 it seems that they, indeed, invoke the divergence theorem. But, as you mentioned, on a larger stencil, in order to have an explicit filter

[FMDenaro]

Originally Posted by sbaffini

There are really few details in the paper, but if you look at equations 3 and 4 it seems that they, indeed, invoke the divergence theorem. But, as you mentioned, on a larger stencil, in order to have an explicit filter

yes Paolo, is not clearly addressed the full procedure in that paper but it seems that the surface integral of the fluxes is applied to define the explicitly filtering procedure on a larger stencil. And I don't know what and how the computation is performed on the grid, maybe using overlapping volume...

[eric_r]

Dear all, new member here so I cannot open my own post, but my question fits here I hope.


I am a researcher on cfd methods, and I am trying to learn about LES. I understand that in finite volume schemes, people often claim that things like: "A finite-volume discretization of eq. (1) corresponds to a convolution with
a top-hat filter returning the filtered solution" (from Complex Effects
in Large Eddy Simulations). I have trouble understanding this fully. What I can understand is that given any turbulent field, a finite volume discretization would - as a first step! - project the field onto the FV basis, that is compute the averages in the elements or - which is the same - apply a top hat filter in physical space in each element. So far, so good. However, this is only the initial step of the FV volume. Once the FV operator is computed, and the solution is advanced in time, the non-linearities of the FV operator (through e.g. flux functions, upwinding etc) will make the solution deviate from the filtered DNS solution. Is this correct?


Or to put it in another way:
Given a DNS field at times t0 and t1, start by discretizing the solution with a top hat filter to get the field DNS_f_t0. Use this as an initial condition for a FV scheme, and compute the solution using FV until t1. Will the solution of the filtered DNS at time t1 DNS_f_t1 be the same as the solution of the finite volume scheme FV_t1? if so, why?



I guess my main confusion is how a linear operation like the filtering can account for the non-linear actions of the FV scheme....


I hope this is not a trivial or stupid question!
Thank you


Eric

[FMDenaro]

Originally Posted by eric_r

Dear all, new member here so I cannot open my own post, but my question fits here I hope.


I am a researcher on cfd methods, and I am trying to learn about LES. I understand that in finite volume schemes, people often claim that things like: "A finite-volume discretization of eq. (1) corresponds to a convolution with
a top-hat filter returning the filtered solution" (from Complex Effects
in Large Eddy Simulations). I have trouble understanding this fully. What I can understand is that given any turbulent field, a finite volume discretization would - as a first step! - project the field onto the FV basis, that is compute the averages in the elements or - which is the same - apply a top hat filter in physical space in each element. So far, so good. However, this is only the initial step of the FV volume. Once the FV operator is computed, and the solution is advanced in time, the non-linearities of the FV operator (through e.g. flux functions, upwinding etc) will make the solution deviate from the filtered DNS solution. Is this correct?


Or to put it in another way:
Given a DNS field at times t0 and t1, start by discretizing the solution with a top hat filter to get the field DNS_f_t0. Use this as an initial condition for a FV scheme, and compute the solution using FV until t1. Will the solution of the filtered DNS at time t1 DNS_f_t1 be the same as the solution of the finite volume scheme FV_t1? if so, why?



I guess my main confusion is how a linear operation like the filtering can account for the non-linear actions of the FV scheme....


I hope this is not a trivial or stupid question!
Thank you


Eric

The key is that you write the full FV equation, that is you apply the volume averaging on each term of the differential NSE equations (written as in a DNS formulation). This way, you get an equation for the time evolution of the filtered (top-hat in physical space) variable, you are not filtering the DNS solution. Of course, the non linear term are responsible of the closure problem and an SGS model is required (or, in a different formulation, you set it to zero in a so-called LES no-model formulation).
Note that the filtering operation is implicitly acted on by the FV method, is not directly performed.

[sbaffini]

Dear eric,

I'm at the cell phone, so I can't go into details but, you need to notice that:

1) The first zero of the implicit FV operator is actually at twice the grid resolution, so its effect is not as pronounced as one would expect. It is a second order effect, that indeed needs to be taken into account for high order fv schemes.

2) As such, large part of the filtering you have for a fv computation is still due to the grid, which is of a projective, i.e., lossy, nature

3) On top of these, the numerical schemes, especially the nonlinear ones, introduce further filtering, which is not always well characterized

4) Finally, when there is a projective filter, you have a one to many relationship between the filtered field and the unfiltered ones that could have produced it

5) Besides this, as I mentioned in my first post above, the implicit fv filtering requires closing even the linear terms in addition to the non linear ones

If you put all of this together, the answer is NO, the computed FV solution would not agree with the DNS one.

[vesp]

I think the confusion comes from the often imprecise wording: the FV discretization (meaning the volume integration / projection on the mean) is a top hat filter, but the FV scheme is not. Often one sees that people so the analysis for FV and just consider the first part, so they describe the effects of FV in implicit filtering LES as a top hat, which is incomplete - this might be the case in the book you cited

[FMDenaro]

Originally Posted by vesp

I think the confusion comes from the often imprecise wording: the FV discretization (meaning the volume integration / projection on the mean) is a top hat filter, but the FV scheme is not. Often one sees that people so the analysis for FV and just consider the first part, so they describe the effects of FV in implicit filtering LES as a top hat, which is incomplete - this might be the case in the book you cited

Actually the key is that the FV scheme must be congruent to governing of the time evolution of the local mean. Whatever flux reconstruction you use in a FV method, the global update of the scheme must be in terms of the discrete volume integration /projection. If the scheme would change the type of implicit filtering that is no longer congruent to the volume integral operation.

[sbaffini]

Another key aspect, without even getting to the discretization, is that any fv scheme requires the unfiltered, fully resolved variables at the volume faces while, by definition, you actually know only the fv filtered ones projected on a coarse grid. That is, the fv equations are inherently unclosed even for linear equations.