Dynamic SGS model procedure in Large eddy simulation
 

Dear guys
i'm working on kiva-3v and make smagorinsky and k-delta one-eq SGS models. the standard forms have been done and now i'm going to make sgs eddy viscosity in the dynamic form for both models.

i have one question. as you know dynamic procedure needs at least one level explicit filter which i use top hat for nodes. first i have all nodes velocity fields and i choose one direction to use a filter (actually it is not true but lets think just x direction in xyz system). i make a averaged value of all quantity i need to use in dynamic eq and use top hat three grid for three side by side cells in the x direction. you know kiva choose node by ifirst to ncells and it can call the front and back nodes by some commands so i use this procedure in kiva to make my dynamic form.

my question is this, is it a true way to make a dynamic filtering procedure? should i make filtering in all three direction and choose the averaged value? i just read some papers that recommend to use dynamic filtering in streamline.

what do you suggest me?

In principle, the type of test filtering for the dynamic procedure has not to be equal to the main one. That means you can use a 2D test filtering, this happens commonly for plane channel flow.
Note that for Cartesian grids, three subsequent 1D filter along x,y and z corresponds to a 3D filter over a volume.
A more specific answer depends on the problem you are solving.

Thank you.
I think in some way i get your meaning. but let me give some more explanation to get more to my problem
http://http.developer.nvidia.com/GPU...ks/22fig03.jpg
just consider one cells plane, then consider one row of that plane, is it true to act dynamic procedure by three side cells (top hat) and after finishing the row, just change and go to next row and go on,till finish the plane and go to next plane and go on...? (sth like we do in TDMA algorithm for 3d grid)?

1) do you want a 2d or a 3d test filtering?
2) what about your test-to-grid filter ratio, you wanto to fix the value =2?

1)i dont get your meaning by 2d or 3d test filter. what i have done, is sth like i explaned about the picture above in 3d mesh grid. i want to know if it is right or wrong and for your explanation about 2d or 3d test filter, i think it should be sth 1d test filter!

2) about my test-to-grid ration, i use grid filter as single cell volume^(0.33) and use three side by side cells volumes^(0.33) as test filter length. so in different position it gives me different test-to-grid ration between almost 1.2 to 2.7, again is this procedure true?

1) the type of test filtering depends also on the flow problem, if your problem is a plane channel flow,e.g. homogeneous in the x and z directions (stream-wise and spanwise), you can do a 2d filter, (an area average) in x,z for example using a 3x3 stencil. If you are solving homogeneous isotropic turbulence, a 3d test filtering (that is a volume average) should be considered, for example over a 3x3x3 stencil.

2) the test-to-grid ratio is an input parameter in the dynamic procedure, do you think to use a variable value depending on the local position? That implies some difference in the procedure

my grid is a engine cylinder with intake-exhaust port and combustion chamber geometry, which dose not allow me to use a uniform mesh. so i think i have to use a dynamic procedure with test-to-grid ratio by position and can't to use a constant ratio. consider a cylinder with a axis in z direction. i use a plane to do my dynamic procedure. for example between z=10mm and z=11 mm, all cells cubic cells between these planes are considered. i averaged of all node value firstly to make a averaged value for 8 nodes of each cells, then use three side by side cells and make a top hat 3-node filter. use this procedure for all cells of the plane then shift to other plane and use the same procedure.

you have also a piston moving in it?
I think you have to build, for each computational node, a local volume surrounding it in such a way that a 3D volume average can be computed. If you are using a structured grid a 3x3x3 local grid stencil can be used.
The numerical implementation can be done following your idea, fix a z=constant plane compute and store the area average values and repeat for each z plane. Then, compute the 1D average along z by using the 2D averaged field.
But if you are using an unstructured grid the number of volumes depends on the topology and the procedure can be more complex.

The final filtered field can be used in the dynamic procedure. However, the filter-to-grid ratio value must be also evaluated correctly.

please consider this image
http://i43.tinypic.com/2cqo1nq.jpg
for each cell, i have 4 node. so i compute the averaged value of velocity component (u,v,w) for each cell by its 8 nodes. then consider cell1,2 and 3 in row 1 @ z=cte, i use top hat test filter by these three cell, in the direction of row 1 axis, then i use this method to finish all cells in row 1 and go to next row (row2 and ...)
after finishing all the row of z=cte, i goto next plane in another z position
how does that procedure sound?

I have to distinguish between the test filtering procedure and its numerical implementation ...
If you want to use a top-hat filter, in a 2d plane say (x,y), you have to chose its width. Therefore, you write the continuous expression:

f_test(x,y) = (1/Deltax/Deltay) * Int [y-Deltay/2,y+Deltay/2] dy' Int[x-Deltax/2,x+Deltax/2] f(x',y') dx'

For example, you can choose Deltax=2*Dx and Deltay=2*Dy.

Now you have to numerically compute the above integrals by using a proper discretization. For example, you can use a 2D Simpson rule. Therefore, centred at i,j you need of a stencil of 9 nodes going from i-1,j-1 up to i+1,j+1. The weights of the integration formula can be found in many textbooks. You can do that for any i,j nodes in the x,y plane. Then repeat for all the z planes.
Finally, if you want to make the test filter three-dimensional, you have to average again the 2d filtered field using the 1d formula for the z direction.

Professor Denaro;
Could you please provide me with technical information, papers or book about the 3x3 stencil or 3x3x3 stenticl you mentioned for 2D and 3D explicit filtering respectively?
I am trying to upgrade the LES code I have from Standard Smagorinsky to the dynamic procedure. However, I am very confused. Also, you mentioned that for channelf flows (my case) an area average is enough (over the homogeneous directions). Nonetheless, the test filter is applied over the Strain Tensor and Strain norm wich are tensors. Thus, I cannot visualize the procedure clearly. Definitely, because the explicit procedure for the test filtering is not clear enough. I read your paper: What does Finite Volume - Based implicit filtering really resolve in LES and in the section 4.1 describes the discretization of the test-filter. However, it is not clear (at least for me) the expression you came up with.

Very respectfully
Julio

Hello Julio,
be careful, the test-filtering is applied on the main filtered velocity field, not on the stress tensor (derivative could not commute with filtering)...

the test filtering can be practically implemented as a simple discretization of the integral over a wider stencil, is a numerical issue.
You can find some further details here:

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

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

Thank you very much professor for sharing this paper with me. I am reading the paper and I hope to answer the questions regarding the test filtering. Also, thank you for clarifying that the test filter acts on the velocity field and it will yield to the filtered strain tensor.

Respectfully
Julio Mendez

good, if you have further questions do not hesitate to contact me

Dear Professor; I read the paper already, as well as other references from the same paper. Although it helped me a lot (now I understand why you said that the filtering is a numerical issue) I still have few questions.
Equation (15) from page 37; you wrote the expression for a 2D local average (box filter) which will be applied over the homogeneous directions. The computational stencil of 2*deltax*2*deltay. I am assuming that delta is the grid size and the test filter is 2*delta.
You mentioned the you used the fourth order Simpson rule. However, I was not able to derive the expression and obtain the same factor (1/36, 16 and 4 ) from your papers. Are these values unique for your gird size, and therefore for the width of the test filter?.
What paper or textbook do you recommend me to read to obtain my own coefficients? in case those coefficients are not universial for the Box (top-hat) and for a thest filter with a width 2 times greater than the grid size (delta)
Thanks
Julio Mendez

the formula is generally valid for uniform meshes, you can find the same formula at page 75 of the book of Ferziger & Peric

have also a look at section 5.1 here:

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

Thank you very much professor for your kindness.
What is surprising for me, is that the filter is a volume average (top-hat).
Finally, I am thinking about the influence of the test filter width. Does it only influence the location at which V_(i+1,j,k) and so forth are evaluated, while the expression remain unchanged?
I mean, if the test filter is 2*delta, the V_(i+1,j,k) is the very next volume, whereas if the test filter is 3*delta is the second volume next to V(i,j,k) ?

Sorry, I am not sure about your question ..If you would define the filter width equal to 3*delta then just define the integral over [x-3/2*delta, x+3/2*delta] and use a proper discretization involving a wider stencil.
This can be done for all i,j,k nodes over overlapping areas

Thank you very much professor; I am still struggling with this but I expect to fully understand with the material you gave me.
Professor, I read the other paper (Direct and Large-Eddy Simulation VIII Volume 15 of the series ERCOFTAC Series pp 27-32).
There, you used two different test width: 3*deltaX*3*deltaZ and 5*deltaX*3deltaZ. Also, in the section 5.1 from: A new development of the dynamic procedure
in large-eddy simulation based on a Finite Volume integral approach. Application to
stratified turbulence you specified a stencil of 3x3 for alpha = 2 and 5x5 for alpha = 4.
Having said that: the width of the test filter what basically means is the amount of volumes I use to do the filtering or (volume average) rather than antoher mesh with different delta that overlap the main mesh?.
On the other hand, the velocity field is the one I apply the test filter on. Hence, I have to interpolate the velocity from the cell (staggered grid) to the center of the node to have these velocity ready for the filtering procedure?
Finally, professor do you have the expression (as you presented for 2*deltaX*2deltaY in On the relevance of the type of contraction of the Germano identity in the new integralbased dynamic Smagorinsky model) for the other test width you presented on the other two works I referred above?

Thanks in advances
Very Respectfully
Julio Mendez

While using the top-hat as test filtering the filter width is not strictly the size of the volume. This aspect is illustrated in the LES literature and you can find some details in my JCP paper you read. The best way is define the filter width from the transfer function.

Furthermore, since you filter separately the velocity components u,v,w you can apply the test filtering on each component without interpolation on centred node

Thank you very much professor. I will read thoroughly the paper. I need to really understand the implementation.

Thank you very much for you time.

Respectfully
Julio Mendez

Dear Professor;
I read the papers and I think that now the physics and the numerical representation is clear. Thank you very much for your kindness answering my questions and sharing your papers; very illustrative your papers. Nonetheless; I have a couple of questions regarding the notation you used and other questions related to the numerical implementation.
The paper from JCP in the section 4.1 (equations 59), you introduced the variable m. that basically defines the width of the test filter. Am I right?
Then I read equations (60 and 61) for m = 2 and m = 4 respectively. In other words the width of the test filter is twice the computational grid for equation (60) and 4 times the computational grid for equation (61). Is it right?
Since deltaX and delta Y are constant in these two equations, you applied a wider stencil for equation (61) compared to equation (60). Is this also right?
However, in point 4.3 you introduce the different cases and and you mentioned that case 1 (equation 60 for m = 2) you ued alpha = 2, 3 and 4. Alpha = (delta_exe)/(delta_eff). I am confused with this, because once you define equation (60 for m=2) the value for alpha is directly constrained; is it not? If no, what is the difference between alpha and m?
Professor; I am using the staggered velocity (MAC). Without any deep analysis I think that I can apply equations (60) and (61) over the velocity cell on the staggered configuration.is it always true?
Also, I would like to know your recommendation to treat the nodes near the boundaries.
Finally; Professor you claimed that the best results were obtained for m = 4 which is the opposite to what Lund and Germano concluded. Is your conclusion based on the integral formulation only?

Thanks in advance for your time.
Very respectfully
Julio Mendez

The Eq.(59) introduces m just as a multiplying factor for the size of the computational grid. But that does not define rigorously the filter width as 2 and 4 the mesh size. According to the transfer functions In fig.6, you can see some estimations giving greater values, for example see Eq(64).

As a consequence, if you read Sec.4.1, fixing the alpha value does not say what discrete test filtering is used. Alpha and m are really different things...

Eq.(60) and (61) can be used for the scalar component u,v,w separately even on staggered grids.

What do you need to treat near the boundary? the 2D test filtering has only the need of periodic links between the values.

Of course, all the conclusions in my paper are well suited for the integral-based formulation. Lund and Germano used always the differential formulation.

Thank you very much professor;
Professor; If I assumed that the discretization process does not introduce any deviation due to:truncation, round-off, numerical scheme and so forth. Hence, delta_eff is strictly function of h and delta_exx is strictly function of the width of the test filter. Can I assume that m represents the filter width? In several papers the authors use equation (59) with m = 2 to define the width of the test filter as twice the computational grid. That is why I am confused.
Finally, why are your conclusions different from Germano and Lund (besides that are based on different approaches), even though both are based on conservation laws. From your experience what are causing such deviation from the other authors ?

I am very thankful for your time and patience. This discussion has been very helpful for me

Very Respectfully
Julio Mendez

of course, it is not possible to eliminate the truncation error, but if it were disregardeble then the filter width would be defined by the extension of the integrals in continuous form...

The differential and integral forms produce many differences...just as example, see the different Germano identities

Thanks professor;
Assuming that all those error are negligible. The width of the test filter is defined by m in the integrals?. Also, alpha and m are directly related by the value of the width of the test filter "m" ?
Thanks
Julio Mendez

The answer is no ... consider the 1D case and the continuous integral between [-h,+h]. If you see the trasfer function, it has an infinite number of zeros along the wavenumber axis. That does not define a specific cut in the frequencies according to a value of m.
In literature, a way to define the filter width can be to evaluate the wavenumber for which the transfer function is equal to 0.5. But, while this is effective in 1D, when you have a multidimensional case also the transfer function is multidimensional and this estimation is more complex.

Only the discretization of the domain introduces a grid-filtering (projective cut-off)

Thank you professor. This is a very complex study domain. I hope to master this (or at least understand) details in the next years with more study and hands-on experience.
This discussion has been very rich for me. Now, I need to move to temporal filtering.
My advisor wants me to implement the temporal filtering rather than a spatial filtering...

Thank you very much professor.

Very Respectfully
Julio Mendez

be careful, the time filtering does not exclude the co-existance of a spatial filtering in LES

I imagined it professor. Do you recommend something specific to read?

there are several papers of Pruett on temporal filtering, but you can find also different authors working on that...
In addition, I suggest the state-of-the art books about LES (Sagaut, Layton et al., etc.)

Dear colleagues;
Is the dynamic procedure a model based on scale similarities? I am very confused because in Germano's papers he never defined it explicitly as a scale similarity model. Instead, he used the central moment to obtain obtain the value of the constant. My issue is also to visualize the role of the central moment because what I end up always seeing is that he introduced the Germano identity and this allowed him to obtain the Cs. So, where does the central moment come into play?? Because L_ij is based on the resolved scales and some how it uses the scale similarity framework.

Thanks in advance!!

The Germano identity is exact, just it follows as a consequence of a filter hierarchy...
Then you decide the type of approximation by introducing the SGS model, it can be an eddy viscosity, a scale similar or a mixed one (1 or 2 coefficients).

Thank you very much dear professor; but where does the averaging come into play, because he claimed that this procedure is invariant to the particular averaging operation.
Finally professor; are you referring to the type of the approximation of the SGS model for the test level ? Because, what we are focus on is the value of the C_s.

Thanks

Consider the filtered momentum equation without any model, where you have the exact unresolved term. Now, apply any other filter further to this filtered equation and you get the Germano identity without any approximation due to the model...

Then, you can introduce an SGS model both a test and primary level...

for example see;
http://ntrs.nasa.gov/archive/nasa/ca...0000039436.pdf

Thank you very much professor; very interesting your explanation and the paper you suggested.

good, just remember that the width of the test filter must be greater than that of the primary

Dear prof. Denaro,

I have faced a problem and hope you can provide me some insight:)

I have tried to simulate channel395 in OpenFOAM through both dynamic Smagorinsky (local) and an explicit SGS stress tensor adding to momentum equations.

The first approach is a well-known eddy-viscosity method which calculates nuSgs and adds it to nu to constitute nuEff and finally calculates divDevReff from something like:

fvm::laplacian(nuEff, U)

The second approach does not calculate nuSgs directly, but instead calculates SGS stress tensor B from the following relation:

B = -2 * nu_t * S_ij

where nu_t is equal to nuSgs and S_ij is the resolved strain rate tensor. The only difference from the first approach is that divDevReff is now an explicit source term which is added to momentum equation as the following term:

fvc:: div(B) - fvm::laplacian(nu, U)

Well, I would expect to get the same results in term of viscous stress (u_tau) in the case of channel395 with default grid spacing, but it is not as thought. The first approach (currently used in OF) yields a reasonable Re_tau=370 but the second approach strongly over-predicts viscous stress, yielding a Re_tau=500!!:confused:

I would like to ask if such a big difference should be expected from the second approach.


Sincerely,
Syavash

To tell my opinion, I would consider fvm::laplacian(nuEff, U) a wrong approach. The second one appers correct as it retains the SGS viscosity under the divergence operator. Remember that in the dynamic procedure it is a point-wise and time-dependent function.