[el_mojito]

Hello everyone,

I am trying to implement an Approximate Deconvolution Model (ADM) for LES into a finite-difference code, and I have two rather basic question concerning the deconvolution operation. More specifically, I am trying to implement the ADM-RT model as discussed in the dissertation of Schlatter (https://www.research-collection.ethz...00.11850/38300), i.e. only a relaxation term derived from the deconvolved field.


The deconvolution is usually given like this:
where is the approximate inverse of the primary filter :
and the overline denotes a filtered value.


For example for N=2 this would give

Now here's my first question: are the first two the not-explicity filtered (or implicitly grid-filtered; current state) field, or is it a filtered field. So do I need to filter the field twice or three times for N=2? Every paper I look at uses a slightly different notation and not always fully consistent, so I'm kinda lost here as I do not completely understand the maths behind this inverse filtering step.

Once we have the deconvolved field, is the relaxation term then , where is for example the current density and is the deconvolved density? Or do I need to apply the explicit filter to these terms?
For example in Stolz 2001 (DOI 10.1063/1.1397277) the relaxation term is given as for the continuity equation. Given the notation in this paper this could either mean:

• current field - deconvolved field
• filtered current field - deconvolved filtered field
• current field - devoncolved field, then filtered

I am lost in translation... Any help is greatly appreciated!

[sbaffini]

Typically, but this might vary and depends from the specific reference, the ADM equations are written in such a way that:

1) The variable you solve for is already in filtered form, i.e.,

2) There is an explicit filtering operation performed elsewhere, e.g., the RHS residual, that actually ensures you 1

And this must be, to the best of my understanding, always the case, because there is no way to do ADM without an explicit filter or an explicit assumption about an implicit filter.

So, it is all just about the specific reference you are using and how it explains things.

Of course this won't help you at all and I can only say that I feel your pain.

Yet, let me give you a hint. The equation you are looking for is 2.38 on page 40 of the Schlatter thesis. It tells you 3 things:

1) In that equation he is solving for deconvolved u_i
2) In that equation the nonlinear product of deconvolved variables u_iu_j requires the application of both G (first ) and Qn (second).
3) The SGS term is -chi(u_i-Qn*G*u_i) where u_i is again the deconvolved one

From this equation you should be able to tell any other form in the thesis. For example, in the equation 2.39, only the second bar is ever applied, the first one, despite formally explicit, is the result of the formulation itself and must not be applied.

Welcome to the LES literature

EDIT: Let me add that the choice of which term actually needs explicit filtering is, indeed, very formulation dependent, and inconsistencies abund. For example, not filtering the linear diffusive term, or the pressure term, as implied by the equations used by Schlatter, requires commutation of the explicit filter with the derivatives. Yet, ADM is completely possible without this assumption. The book of Garnier and Sagaut has the equations in the correct form for this, if I remember correctly.

[FMDenaro]

Originally Posted by el_mojito

Hello everyone,

I am trying to implement an Approximate Deconvolution Model (ADM) for LES into a finite-difference code, and I have two rather basic question concerning the deconvolution operation. More specifically, I am trying to implement the ADM-RT model as discussed in the dissertation of Schlatter (https://www.research-collection.ethz...00.11850/38300), i.e. only a relaxation term derived from the deconvolved field.


The deconvolution is usually given like this:
where is the approximate inverse of the primary filter :
and the overline denotes a filtered value.


For example for N=2 this would give

Now here's my first question: are the first two the not-explicity filtered (or implicitly grid-filtered; current state) field, or is it a filtered field. So do I need to filter the field twice or three times for N=2? Every paper I look at uses a slightly different notation and not always fully consistent, so I'm kinda lost here as I do not completely understand the maths behind this inverse filtering step.

Once we have the deconvolved field, is the relaxation term then , where is for example the current density and is the deconvolved density? Or do I need to apply the explicit filter to these terms?
For example in Stolz 2001 (DOI 10.1063/1.1397277) the relaxation term is given as for the continuity equation. Given the notation in this paper this could either mean:

• current field - deconvolved field
• filtered current field - deconvolved filtered field
• current field - devoncolved field, then filtered

I am lost in translation... Any help is greatly appreciated!

There are many issues in the deconvolution-based apporaches and, in my opinion, there is also a wrong way of approaching the method.
I worked on the deconvolution approach since 1995, I then discovered in 1997 at the VKI that the approach was developed also by Ferziger and one student. Since then, a lot of methodologies based on the deconvolution were proposed in the literature.
In general a deconvolution operation Q is "theoretically" the inverse of the top-hat filtering G, that is Q*G=I. An exact inversion is not possible and an approximate deconvolution is introduced such that u_bar=G*u -> u_dec=Qn*u_bar is a new high order filtered variable. For n->Inf, Qn*u_bar tends to recover the spectral resolution (sharp cut-off filter).
The SGS model that is hystorically used for inserting the deconvolution is the scale similar model.

Here is the joke, if you use a spectral method, the scale similar term vanishes by definition (the spectral filter is idempotent) and no deconvolution makes sense. The "idea" is to apply explicitly a top-hat filter G on the spectral variable and introducing the scale similar term that is then modelled by the deconvolution method.


You could find useful Sec.4 in



https://www.researchgate.net/publica...-uniform_grids

[el_mojito]

Originally Posted by FMDenaro

In general a deconvolution operation Q is "theoretically" the inverse of the top-hat filtering G, that is Q*G=I. An exact inversion is not possible and an approximate deconvolution is introduced such that u_bar=G*u -> u_dec=Qn*u_bar is a new high order filtered variable. For n->Inf, Qn*u_bar tends to recover the spectral resolution (sharp cut-off filter).

I understand what the deconvolution does, just not the how, but thanks for clearing it up anyway. That is why I am not sure how to correctly apply the formula in the beginning, before even thinking about any LES model (i.e. how many filtering steps are necessary for a given N: is it N or N-1).

I have found the Dissertation of Jörg Ziefle (ETH No. 17846) where he explicitly states "The zeroth deconvolution is handled as a special case, since the deconvolution step involves no filtering", which strengthens the point of sbaffini, that "only the second bar is ever applied". So my deconvolution step is now as follows (not optimized, I know):

Qvec_deconv % rho = 0.0_rk
Qvec_filter % rho = Qvec % rho
do nu = 0, NDCONV
   call adm_deconvolve_loop(Qvec_deconv%rho, Qvec_filter%rho, grid_data, nu)
   call adm_filter_x(Qvec_filter % rho, grid_data)
   call adm_filter_y(Qvec_filter % rho, grid_data)
   call adm_filter_z(Qvec_filter % rho, grid_data)
end do

where adm_deconvolve_loop performs the summation of Qvec_deconv and the filtered data with the correct coefficient for nu. So e.g. for N=1 the devonvolved field is two times the original field minus one time a once-filtered field.

[el_mojito]

Originally Posted by sbaffini

Yet, let me give you a hint. The equation you are looking for is 2.38 on page 40 of the Schlatter thesis. It tells you 3 things:

1) In that equation he is solving for deconvolved u_i
2) In that equation the nonlinear product of deconvolved variables u_iu_j requires the application of both G (first ) and Qn (second).
3) The SGS term is -chi(u_i-Qn*G*u_i) where u_i is again the deconvolved one

Regarding point 2):
Don't you mean Qn first, G second? You need the deconvolved variables first, then filter the product? Anyway, I am first implementing the ADM-RT, where only the relaxation term on the right-hand side is used, so everything else is basically gonna stay in "DNS mode".

[sbaffini]

Originally Posted by el_mojito

Regarding point 2):
Don't you mean Qn first, G second? You need the deconvolved variables first, then filter the product? Anyway, I am first implementing the ADM-RT, where only the relaxation term on the right-hand side is used, so everything else is basically gonna stay in "DNS mode".

No, if you look at equation 2.38 it should be pretty clear: Qn follows the second bar (or G), the one really applied.

Note that this is different from eq. 2.35, which is explicitly stated to be the one actually used in implementation. 2.35 describes the evolution of a filtered variable, which is obtained by explicitly filtering the nonlinear product of the deconvolved variables (i.e., the convective term). So, 2.38 is 2.35 under a further Qn application. I used 2.38 as example because it makes more clear which bars are actually applied and which ones aren't.

The fact that the first bar is not really applied can be also seen in relation to equation 2.49, where the ADM-RT version is introduced.

The appalling thing is that it is not just Schlatter, or Stolz & co. that are little clear on this, it's the whole LES world at large. You get this by experience before you ever manage to recognize this thing in written words.

[FMDenaro]

The implementation has an issue depending on the numerical scheme of your LES code.
If you are using a spectral method the deconvolution has theoretically no meaning! To make that a practical method, you should consider that G*u is the spectral representation implicitly built-in by your numerical method. Then you need to introduce a smooth filter Gs that is really applied on the spectral variable. Only after that a deconvolution makes sense.
I personally don’t like this approach as it is just a back and forth way of doing.

Different is the case if G*u is the primary filtered implicitly implemented by some FV method. In this case the deconvolution is effective.
Have you started from the implementation of the standard ADM method?

The implementation can be easy starting from the differential form of the deconvolution operator.

[sbaffini]

Another way of looking at it, but still excluding the case of spectral cut off G, is to actually consider Qn*G as the actual explicit LES filter, which is one of the reasons I mentioned eq. 2.38. The fact that Schlatter mentions differences with respect to the original formulation is just because of the way commutation error is treated. Indeed, a correct implementation would actually lead to the Mathew formulation, e.g., https://doi.org/10.1016/j.ijheatfluidflow.2006.02.007

[sbaffini]

Originally Posted by sbaffini

Another way of looking at it, but still excluding the case of spectral cut off G, is to actually consider Qn*G as the actual explicit LES filter, which is one of the reasons I mentioned eq. 2.38. The fact that Schlatter mentions differences with respect to the original formulation is just because of the way commutation error is treated. Indeed, a correct implementation would actually lead to the Mathew formulation, e.g., https://doi.org/10.1016/j.ijheatfluidflow.2006.02.007

Which now, I realize, is kind of what you, Filippo, actually do with your FV approach using the implicit FV filter as G. Am I correct?

[FMDenaro]

Originally Posted by sbaffini

Which now, I realize, is kind of what you, Filippo, actually do with your FV approach using the implicit FV filter as G. Am I correct?

Paolo, unfortunately I have no access to this article. From the abstract I can suppose that an explicit filtering is applied on the non linear term, thus if they do not commute with the divergence the approach should result in a FV framework.

[sbaffini]

Originally Posted by FMDenaro

Paolo, unfortunately I have no access to this article. From the abstract I can suppose that an explicit filtering is applied on the non linear term, thus if they do not commute with the divergence the approach should result in a FV framework.

Pages 220-222 from Sagaut

[FMDenaro]

Originally Posted by sbaffini

Pages 220-222 from Sagaut

Not sure from those pages if they really solve the form (7.42)

[el_mojito]

Originally Posted by sbaffini

No, if you look at equation 2.38 it should be pretty clear: Qn follows the second bar (or G), the one really applied.

Note that this is different from eq. 2.35, which is explicitly stated to be the one actually used in implementation. 2.35 describes the evolution of a filtered variable, which is obtained by explicitly filtering the nonlinear product of the deconvolved variables (i.e., the convective term). So, 2.38 is 2.35 under a further Qn application. I used 2.38 as example because it makes more clear which bars are actually applied and which ones aren't.

The fact that the first bar is not really applied can be also seen in relation to equation 2.49, where the ADM-RT version is introduced.

The appalling thing is that it is not just Schlatter, or Stolz & co. that are little clear on this, it's the whole LES world at large. You get this by experience before you ever manage to recognize this thing in written words.

Wait, now I'm kinda lost. The nonlinear term in equation 2.38 reads

which second bar do you mean? I understand it such that you take eqn 2.35, where you deconvolve and then explicity filter the product (as you wrote) and then convolute it with Q_n to get eqn. 2.38, so you still need to apply Q_n first, then G to get the term in parantheses, no? (If you were to actually evolve this equation)

[el_mojito]

Originally Posted by FMDenaro

The implementation has an issue depending on the numerical scheme of your LES code.
If you are using a spectral method the deconvolution has theoretically no meaning! To make that a practical method, you should consider that G*u is the spectral representation implicitly built-in by your numerical method. Then you need to introduce a smooth filter Gs that is really applied on the spectral variable. Only after that a deconvolution makes sense.
I personally don’t like this approach as it is just a back and forth way of doing.

Different is the case if G*u is the primary filtered implicitly implemented by some FV method. In this case the deconvolution is effective.

It's a finite difference scheme with either compact sixth-order or regular eighth-order differences, not a spectral scheme.

Originally Posted by FMDenaro

Have you started from the implementation of the standard ADM method?

The implementation can be easy starting from the differential form of the deconvolution operator.

It's not a LES code, it's a DNS code, so there's no LES implementation to start from.
I wanted to start with the ADM-RT model because the relaxation term can simply be added in the same way other forcing terms such as sponge forces are added without touching any other parts of the code.

[sbaffini]

Originally Posted by el_mojito

Wait, now I'm kinda lost. The nonlinear term in equation 2.38 reads

which second bar do you mean? I understand it such that you take eqn 2.35, where you deconvolve and then explicity filter the product (as you wrote) and then convolute it with Q_n to get eqn. 2.38, so you still need to apply Q_n first, then G to get the term in parantheses, no? (If you were to actually evolve this equation)

Sorry, bad wording, by second bar I was referring to the only one ever applied which, of course, is also the only one ever appearing in eq. 2.38.

However, the whole thing is that in 2.38 you are solving for deconvolved u_i, as you see the unsteady term is in this form. It is not always so, but in this case it means you are not applying Qn first ever. The form of the equations "guarantees" that you are already solving for deconvolved variables.

Starting from this, in 2.38, you filter the non linear product with G (this is where I used a bad wording, as it only ever applies once here in 2.38, and is explicit). After that, you just apply Qn as well to the nonlinear product.

In the end Qn*G is the only operator (but let's call it filter, that's what it is) you ever apply, and you do that in just two places:

1) The non linear product in the convective term

2) The relaxation term

So, again, the culprit is that, in some circumstances, even explicit filtering approaches actually have some "implicit part" in the way they are written. In this case, specifically, what you do on the nonlinear product of the convective term implies that you are solving for the deconvolved variables, so that you don't actually have to deconvolve first ever. It would be clear if you try to apply, say, a first order explicit euler discretization to the equation; you end up with an equation that indeed updates deconvolved u_i.

[el_mojito]

Originally Posted by sbaffini

However, the whole thing is that in 2.38 you are solving for deconvolved u_i, as you see the unsteady term is in this form. It is not always so, but in this case it means you are not applying Qn first ever. The form of the equations "guarantees" that you are already solving for deconvolved variables.

Ah, of course you are correct. I was forgetting that if you solve for a deconvolved u_i, you don't actually apply Qn to get it in the first place. Thank you!

[sbaffini]

Originally Posted by el_mojito

Ah, of course you are correct. I was forgetting that if you solve for a deconvolved u_i, you don't actually apply Qn to get it in the first place. Thank you!

To be honest, I am either too old to remember coorectly all the formulations I've seen or, indeed, this is not really always the case. That is, your doubt is legit because when this is implied it is not, typically, explicitly stated in words (pun intended... maybe). In practice I didn't notice uniformity in the way such things are presented. And unless you spend hours per day on just it, it is easy to loose focus on such details.

[FMDenaro]

Originally Posted by el_mojito

It's a finite difference scheme with either compact sixth-order or regular eighth-order differences, not a spectral scheme.





It's not a LES code, it's a DNS code, so there's no LES implementation to start from.
I wanted to start with the ADM-RT model because the relaxation term can simply be added in the same way other forcing terms such as sponge forces are added without touching any other parts of the code.

No matter about your DNS code, you will use it on unresolved condition, that is on a coarse grid. You will just add the SGS model.
From your words I assume you have the FD discretization of the convective terms in divergence form, this way the primary filter is implied implicitly by your discretization and is a smooth filter. That means you don't apply any explicit smooth filter like would be required by a spectral method.
I have some old notes that could be useful, I will post that but I first try to recast them because I don't work on the deconvolution since many years ago.

[FMDenaro]

Originally Posted by el_mojito

It's a finite difference scheme with either compact sixth-order or regular eighth-order differences, not a spectral scheme.





It's not a LES code, it's a DNS code, so there's no LES implementation to start from.
I wanted to start with the ADM-RT model because the relaxation term can simply be added in the same way other forcing terms such as sponge forces are added without touching any other parts of the code.

Have a look to this note, maybe could be of some help


https://www.researchgate.net/publica...ased_modelling

[sbaffini]

Originally Posted by el_mojito

It's a finite difference scheme with either compact sixth-order or regular eighth-order differences, not a spectral scheme.





It's not a LES code, it's a DNS code, so there's no LES implementation to start from.
I wanted to start with the ADM-RT model because the relaxation term can simply be added in the same way other forcing terms such as sponge forces are added without touching any other parts of the code.

Maybe you could give the Mathew approach a try, as it is really best suited for cases as yours. Basically, he shows that ADM is indeed just applying Qn*G to your solved variables at each time step.

He also recalls that, the RT can be interpreted as filtering again, ideally with something slightly more dissipative than Qn*G, every 1/(chi dt) time steps.

Finally, in his extension of the model, he also introduces scale similar like models: f(Qn*G*u_star)-f(u_star).

The Mathew implementation is simple, elegant, fits any LES implementation without approximation (not even commutation errors) and, obviously, is very well suited for cases where the underlying scheme is already high order (as for ADM in general).