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Dynamic Smagorinsky model- Filtering Concept

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Old   July 31, 2013, 11:55
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Filippo Maria Denaro
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Originally Posted by CFDJosh View Post
Yes, the LES equation is independent of filtering, but interpretation of what filtering actually means is absolutely dependent on what filter you actually use. If you want to talk about scales, e.g. large scale, small scale, SGS scale, scale similarity, etc, you're implicitly assuming that your filter extracts some portion of the spectrum, more or less, and more less leaves the rest alone.

The problem is "scale" isn't a mathematically precise term. It only becomes precise when you express your solution field in terms of a Fourier series (or any kind of series with basis functions defined in terms of length, actually).

So if your turbulence model is built on a scale-similarity assumption and comparing scales (like Germano), you should use a filter that does a good job of extracting chunks of the spectrum. Top hat doesn't, and in general, all filters of the form (u_i-1 + B * u_i + u_i+1)/(2+B) do a pretty bad job of separating scales. Hence they're not going to do a particularly good job of doing what you want in that sort of turbulence model.

Ok, you were talking about the correct sense of "scale separation", I agree that only the spectral cut-off defines univocally a separation of scales. However, there is no real reason to prefere the Fourier filter in LES... the separation acts only formally but you are never able to separate the evolution of the turbulent spectrum spectrum since resolved and unresolved scales remain coupled each other in their dynamics. Furthermore, using the Fourier cut-off (idempotent filter) the scale-similar part is identically zero (apart from the de-aliasing)... The real key between sharp cut-off and smooth transfer function is in identifying the meaning and beahviour of the filtered field in order to be used correctly in the SGS model. Of course also the filter width in smooth filter must be properly defined.
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Old   July 31, 2013, 13:30
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You shouldn't be reusing your grid filter as a test filter if what you want to do is Germano, regardless of what kind of filter you're using.

The dynamic model is built on the assumption that the residual turbulent stress tensor is related to the test-filtered strain rate in the same manner as the subgrid scale stress tensor is related to the resolved strain rate (this is separate from the validity of the Boussinesq hypothesis on which it's all built to begin with), thus allowing you to get your dynamic parameter. This assumption is not justifiable if the test filter you're using does not result in a scale separation structurally similar to that imposed by your grid filter.

The fact that the entire spectrum is coupled does not make the choice of filter irrelevant; it makes it critical.
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Old   July 31, 2013, 13:43
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Originally Posted by CFDJosh View Post
You shouldn't be reusing your grid filter as a test filter if what you want to do is Germano, regardless of what kind of filter you're using.

The dynamic model is built on the assumption that the residual turbulent stress tensor is related to the test-filtered strain rate in the same manner as the subgrid scale stress tensor is related to the resolved strain rate (this is separate from the validity of the Boussinesq hypothesis on which it's all built to begin with), thus allowing you to get your dynamic parameter. This assumption is not justifiable if the test filter you're using does not result in a scale separation structurally similar to that imposed by your grid filter.

The fact that the entire spectrum is coupled does not make the choice of filter irrelevant; it makes it critical.
Actually, the Germano identity is exact, despite the type of primary and test filter you adopt (you can also use a mix of filter types) ... it is the Smagorinsky approximation next introduced, which is based on the use of a unique constant model, to be critical in the sense you said.
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Old   August 1, 2013, 07:12
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When we are talking about the grid independent LES solution using implicit filter, in which extent it is true? Because implicit filter directly depends on the grid size, which means when we are changing the grid size solution is changing. Ok when we lower the grid resolution fine enough then solution seems grid independent, but rather than that what are the other criteria when we can call it grid independent? In most of the cases we see that when the solution is not changing with the change of grid size (should be fine enough)we call it grid independent though computationally it is very expensive. What is the contribution of explicit filtering (another alternative way to obtained grid independence) in that case?

Another question, what will happen when the computational domain is very large to take the turbulence length scale up to Kolmogorov's scale or inertial sub-range? In that case, how grid independence can be obtained?
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Old   August 1, 2013, 07:38
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When we are talking about the grid independent LES solution using implicit filter, in which extent it is true? Because implicit filter directly depends on the grid size, which means when we are changing the grid size solution is changing. Ok when we lower the grid resolution fine enough then solution seems grid independent, but rather than that what are the other criteria when we can call it grid independent? In most of the cases we see that when the solution is not changing with the change of grid size (should be fine enough)we call it grid independent though computationally it is very expensive. What is the contribution of explicit filtering (another alternative way to obtained grid independence) in that case?

Another question, what will happen when the computational domain is very large to take the turbulence length scale up to Kolmogorov's scale or inertial sub-range? In that case, how grid independence can be obtained?
well, the first answer is that while using implicit filtering no grid-independent LES solution exists. You simply tends towards the DNS solution ...

As the second question is concerned, I honestly have not well understood the meaning of what you asked for...
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Old   August 1, 2013, 07:54
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Dear CFDJosh,

i genuinely don't understand some of your points. More specifically:

- you started by "you want a filter with a monotonically decreasing transfer function with F(k_max) = 0" (i guess the 9 was a misprint)

- after that, and i quote again "...if your turbulence model is built on a scale-similarity assumption and comparing scales (like Germano), you should use a filter that does a good job of extracting chunks of the spectrum. Top hat doesn't...";

- then you wrote "You shouldn't be reusing your grid filter as a test filter if what you want to do is Germano, regardless of what kind of filter you're using";

- finally "...This assumption is not justifiable if the test filter you're using does not result in a scale separation structurally similar to that imposed by your grid filter".

So, i completely miss your final point. What i know is that:

- scale similarity is not part of the Germano identity; it only comes in when using the same model and constants at the two filtering levels.

- actually, when a proper analysis is performed:

https://tcg.mae.cornell.edu/pubs/Pope_NJP_04.pdf (final point)

scale-similarity turns out to be required only when the same constant is used at the two filtering levels. Indeed, it is not even a requirement, it's just that the constant so computed is that relative to the test filter level, hence you might want that to be really close to the one relative to the basic filter level.

- Carati and Vanden Ejinden have shown that scale-similarity is, in practice, embedded in the dynamic procedure independently from the test filter (top-hat included), in the sense that it is the test filter itself that determines the resolved field. The overall approach might be debatable but, still, scale-similarity can be proven for top-hat filters.

- Liu et al. (JFM 94, but i guess there are also more recent works on this) have shown that the subgrid energy flux (t_ij S_ij) is mostly independent from the basic filter employed and, more importantly, for a top-hat test filter there is substantial correlation between the flux at the test and basic filter level. The same authors also provide the current, most general, definition of scale-similarity (a la Bardina) by adopting a top-hat filter. What is important to note is that scale-similarity between L_ij (Leonard tensor) and t_ij (subgrid stress tensor) is lost for a cut-off filter.

- the dynamic procedure is currently not that nice mathematical approach that one would expect (actually is quite the contrary in my opinion)

- actually, there is some evidence corroborating the fact that the Dynamic Smagorinsky model actually works because of the higher order dependence on S_ij rather than anything else (Jimenez, CTR... 1993 if i remember correctly). Hence the Whole filter stuff might just be a huge waste of time.

- the textbook spectral energy distribution with a clear cut-off and a nice inertial range down to the cut is quite difficult to achieve besides the HIT case, where static Smagorinsky already works good. This is true, in my experience, even for spectral codes

Could you further elaborate on this?
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Old   August 1, 2013, 08:30
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Originally Posted by sbaffini View Post
Dear CFDJosh,

- actually, there is some evidence corroborating the fact that the Dynamic Smagorinsky model actually works because of the higher order dependence on S_ij rather than anything else (Jimenez, CTR... 1993 if i remember correctly). Hence the Whole filter stuff might just be a huge waste of time.

Could you further elaborate on this?
Dear Paolo,
Thats really interesting! If I am not wrong, you are mentioning here to filter the starin rate S_ij only for Dynamic Smagorinsky.

What do you mean by the Whole filter stuff here?
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Old   August 1, 2013, 10:05
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Well,

i am referring to the use of an "enhanced" velocity field in S and S_ij. More specifically, there is a huge amount of literature referring to the so called Multilevel LES approach where, different combinations of small-large scale velocity contributions are used in the construction of a modified form of Smagorinsky model.

You can find this approach also under the nomenclature: "Accentuation Techniques", "High Pass Filtered Eddy Viscosity", etc.

Some non linear models can also obtain the same effect.

The main theme here is that you can achieve a "dynamic-procedure like" effect by just acting on the order of the dissipation (the powers of S really involved in the SGS model). Evidence indicates that more often than not higher is better.

A somehow similar approach is that based on hyperviscosity or, to some extent, ADM, as in both cases you have an energy dumping which is high order and tied only to the smallest resolved scales (in contrast to a classical Smagorinsky, which can affect a certain amount of the resolved spectrum).

The work of Jimenez i was citing before:

http://ctr.stanford.edu/ResBriefs95/jimenez

together with the work of Pope i also cited, both provide some analysis on the dynamic procedure which shed a different light on the dynamic procedure. As you can see, both of them don't explicitly cite any specific filter as it is not the most relevant factor in the analysis. Of course, i'm not neglecting the adequacy of certain class of SGS model with respect to the basic filter, but that is another story. Also differences are still expected as in any numerical procedure acting on a delicate part of the spectrum as the near-cut-off one.

In conclusion, what i want to say is that the dynamic procedure is far from being mathematically exact or even robust and similar effects can be achieved with different SGS approaches without too much concern to the filter employed. As a consequence, i think it is not correct to ascribe sepcific deficiencies of the dynamic procedure to the test filter, especially if there is no theoretical flaw in their use.

Nonetheless, i think that using a test filter which can more closely replicate the basic filter is the starting point for any dynamic procedure. Then you use additional numerical tricks to let it work better. Consider that just the averaging and clipping of the constant can be done in so much different ways that you can achieve order 200% variation in your results just by that.
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Old   August 2, 2013, 01:30
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Another question, what will happen when the computational domain is very large to take the turbulence length scale up to Kolmogorov's scale or inertial sub-range? In that case, how grid independence can be obtained?
Dear Filippo, I have quoted my second question here and sorry to confuse you.
Actually I would like to know, if the computational domain ranging (say) in kilometers (km) (example, let say fire in the landscapes), in that case what would be the best affordable grid resolutions, that means in the other sense what would be the turbulence length scale for the simulation or how this can be measured? Along with this, how grid independence can be obtained for the mentioned case?
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Old   August 2, 2013, 04:32
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If your domain has an integral lenght of order of Km your case is not quite affordable with LES ... I mean that I suppose that the dissipative scales of your problem are at least 3-4 order of magnitude lower. Your LES grid, provided that the filter lenght lie in the inertial range (order of meters?), should be as small as to have O(10^9 - 10^10) unknowns.
My experience of LES in geophysical flows is limited to domain with integral scale of order of 10^2 m where I used a grid size of order of meter or few less.
In any case I suggest you to have an estimation of the Taylor and Kolmogorov scales of your problem, take also in to account that if your case is not homogeneous the resolution required can be more critical.

Finally, using implicit filtering you can not search for a real grid independent LES solution... I can suggest considering a couple of different grids based on the requirements above described and check the statistics of the two solutions.
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Old   August 14, 2013, 11:21
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Hi all!

In the case of explicit filtering, do I need to filter the energy equation as well along with the momentum equation in LES? If only the momentum equation is explicitly filtered but energy equation is not, then I would like to know whether we will get any inconsistency?
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Old   August 14, 2013, 11:39
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Hi all!

In the case of explicit filtering, do I need to filter the energy equation as well along with the momentum equation in LES? If only the momentum equation is explicitly filtered but energy equation is not, then I would like to know whether we will get any inconsistency?

if the momentum equation is coupled with energy and contains some energy variable that is explicitly filtered, then you must have the same type of filtered variable in the energy equation.
Maybe some trick can be introduced but I don't see any advantage in using a mixed explicit/implicit filtering...
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Old   August 16, 2013, 01:21
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Maybe some trick can be introduced but I don't see any advantage in using a mixed explicit/implicit filtering...
Hi Filippo! Thanks for your reply.
Please explain what sort of tricks you are talking about? And what type of inconsistency will be observed if mixed explicit/implicit filtering used?
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Old   August 16, 2013, 03:48
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Hi Filippo! Thanks for your reply.
Please explain what sort of tricks you are talking about? And what type of inconsistency will be observed if mixed explicit/implicit filtering used?
I wrote that I see no advantage but, provided that the explicit filtered velocity and the implicit filtered temperature are correctly used in the coupled equations, no inconsistence would exist.
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