[cfdmms]

Hi all!

I would like to know the difference between the Constant smagorinsky and Dynamic model Smagorinsky model.
So far I know it does not help me to clear my concept about this two models. For the constant smagorinsky the local value of Cs is not changing with the different types of flows and Mu_turbulent not goes to zero during laminar flows.

1. Is there any other factors for which the dynamic smagorinsky model was introduced?
2. And what are the other limitations of constant smagorinsky model over dynamic smagorinsky model?
3. Among them which model is able to give better solution and why?

It would be great help if someone help me regarding this. Thanks.

[sbaffini]

There are no obvious answers for your questions besides what you already mentioned: the dynamic constant vanishes in laminar flows and has the correct limiting behavior near the walls.

Indeed, while the constant is computed in such a way to force the model to satisfy an exact algebraic relation (Germano Identity), the final value usually used can only force this in a least squares sense and not exactly. Also, the basic dynamic formulation also suffers of additional assumptions not usually verified (a commuting filter, both basic and test; a constant smooth enough to be extraxcted from the test filter; ect.). Finally, the limitations of the underlying model are not avoided in any case.

Thus, is the dynamic procedure worth the effort? It depends. Still, the Germano identities are exact relations which can also be derived without all the usual strong assumptions and a model not satisfying them certainly has some flaws.

For the Smagorinsky model, the dynamic procedure certainly produced some advantages over its static version. However, there are models for which such advantages are null or negative.

I don't think, however, that there is a unique (short) answer to your question. I've seen the same Dynamic Smagorinsky model producing a wide variety of results on the same case only depending on the code numerics. Maybe, the next step in LES is to show code independence for explicitly filtered LES with a fixed model. Then, probably, something more accurate can be said on this matter.

[FMDenaro]

just to add some words....

- the "dynamic model" is rather a procedure than a model ... you can use it by introducing an eddy viscosity model but also by different modelling such as a scale similar.
- the Germano identity could be developed in such a way to take into account for the real discretization so that the computed dynamic functions have someway a "physical and numerical" relevance

[cfdmms]

Originally Posted by sbaffini

There are no obvious answers for your questions besides what you already mentioned: the dynamic constant vanishes in laminar flows and has the correct limiting behavior near the walls.

Indeed, while the constant is computed in such a way to force the model to satisfy an exact algebraic relation (Germano Identity), the final value usually used can only force this in a least squares sense and not exactly. Also, the basic dynamic formulation also suffers of additional assumptions not usually verified (a commuting filter, both basic and test; a constant smooth enough to be extraxcted from the test filter; ect.). Finally, the limitations of the underlying model are not avoided in any case.

Thus, is the dynamic procedure worth the effort? It depends. Still, the Germano identities are exact relations which can also be derived without all the usual strong assumptions and a model not satisfying them certainly has some flaws.

For the Smagorinsky model, the dynamic procedure certainly produced some advantages over its static version. However, there are models for which such advantages are null or negative.

I don't think, however, that there is a unique (short) answer to your question. I've seen the same Dynamic Smagorinsky model producing a wide variety of results on the same case only depending on the code numerics. Maybe, the next step in LES is to show code independence for explicitly filtered LES with a fixed model. Then, probably, something more accurate can be said on this matter.

Another question is, in the case of explicit filtering in Standard (Paper from Tellervo Brandt) and Dynamic Smagorinsky (Gullbrand paper from CTR), they have used test filters which are greater than grid filter. And for the test filter which contains more grid cells in it and does some numerical integration within the test filter using Trapizoidal or Simpson filter to control the numerical errors to obtain reduce grid sensitive results.

My question
1. How these integration methods (Trapizoidal, Simpson, 3rd order, 5th order filters etc) help to control the numerical errors in the case of explicit filtering?
2. And following this way, how explicit filters give better solution compared to implicit filter?

[FMDenaro]

The key is that the grid filter of measure h cut frequencies higher than Pi/h (Nyquist frewquency).
The implicit filter has a shape that depends on the discretization, FV and FD methods have generally an induced smooth filter that decreases the wavenumber content for resolved frequencies close to Nyquist.
Using an explicit filtering over a clustered cells, for example h_tf=3h, you can introduce a new Nyquist frequency that cut at pi/(3h) the wavenumber affected by the smoothing.
Note that in dynamic models, you have the test filtering to be used that is formally different from an explicit main filtering.
If you want, many details can be found here
http://adsabs.harvard.edu/abs/2011JCoPh.230.3849D

PS: you can find a complete paper of Gullbrand on jfm

[cfdmms]

Originally Posted by FMDenaro

The key is that the grid filter of measure h cut frequencies higher than Pi/h (Nyquist frewquency).
The implicit filter has a shape that depends on the discretization, FV and FD methods have generally an induced smooth filter that decreases the wavenumber content for resolved frequencies close to Nyquist.
Using an explicit filtering over a clustered cells, for example h_tf=3h, you can introduce a new Nyquist frequency that cut at pi/(3h) the wavenumber affected by the smoothing.
Note that in dynamic models, you have the test filtering to be used that is formally different from an explicit main filtering.
If you want, many details can be found here
http://adsabs.harvard.edu/abs/2011JCoPh.230.3849D

PS: you can find a complete paper of Gullbrand on jfm

Thanks FMDenaro for the paper. It seems very interesting.

If i am not wrong, it means, in the case of explicit filtering, high frequency motions are damped down and that decreases the numerical error which ultimately give better solution compared to implicit filter.

If you explain a bit, it will be a help.

Another question is why averaging is needed for the clustered cells within the explicit filter? How it improves the accuracy?

[FMDenaro]

yes, high (resolved) frequency are affected by the discretization and explicit filtering eliminates such frequencies so that the SGS model works on a filtered field that should be cleared by numerical errors. However, this requires a well refined computational grid.

Note that the "volume averaging" can be used an explicit filtering

[juliom]

Dear Community; I have read your comments but I still have a couple of questions, related to the implementation of the dynamic procedure.
What I understand from this post and Pope's book, is that the test filter is applied upon the grid filter (or primary filter). Thus M_ij and L_ij are known in term of U (overbar)(x,t) (the filtered velocity field from the computation).
Now, my questions is how to implement the test filter for M_ij and L_ij in FVM. Also, I am having problem to visualize the difference between
the two tensor operation with different test filters operation on them (M_ij).
Thank in advance
Respectfuly
Julio

[FMDenaro]

Different type of test filtering (assuming the same width lenght) would change the shape of the spectral content of the wavenumbers.
Just as an example, the spectral filter is idempotent, so you can perform the test filtering by simply setting the cut-off frequency as the spectral content is the same.
Conversely, if you apply the top-hat test filtering you get a smoothing of the spectral content