Nyquist wavenumber
 

Hi all,
I am looking at this Energy spectrum in the wavenumber domain. The dashed line is the Nyquist wavenumber. My question is: how is it possible that results are shown after the Nyquist wavenumber? Shouldn't that be the limit after that there are no value?

Right, the extension of the spectra from a numerical simulation cannot be extended for k>Kc.

You should check for more details, is that from a published paper?

Thanks for your answer. It is the verification guide of FDS that you can find here https://pages.nist.gov/fds-smv/manuals.html. There is a chapter about turbulence from page 45 to 53. I cannot understand how can they have value after kc. Maybe they apply Taylor hypothesis?

From what I understand, the numerical simulation has greater resolution (64^3) than the CBS (32^3) and they adopted an explicit filter. Therefore, the extension of the spectra is reported until the numerical Nyquist frequency of the finer grid.

The filter in FDS is implicit, as attached. The CBS is the experiment of Comte Bellot and Corrsin, the two top figure are related to the 32^3 the bottom two 62^3. I really cannot understand

Actually, what I really do not understand from the figures is the number of wavenumber components reported. In the top figure, the grid is 32^3, that is the spectra should be reported for 16 wavenumbers. On the bottom (64^3), the number of components should be 32. But if you count the circle you see that they are doubled, creating a fictious doubled Nyquist frequency.My opinion is that either they wrongly performed the FFT or the number of grid in each direction has to be doubled. You should ask to the author of the verification.

Ok. Thank you :D I will do that. I wanted simply to be sure that my doubts are lecit. I have also an other question about the subgrid model, but I will open an other thread

I think the authors use the spherical wave number defined in 3D as



Here the maximum values are:

Mesh with 32x32x32:



Mesh with 64x64x64:



This fits well with the presented results.

Regards

From the figure, the Nyquist line seems at different wavenumbers. For example, in the 32^3 grid it appears exactly at the wavenumber 16

Yes, your right. With 32x32x32 the Nyquist wavenumber remains at the maximum value of 16. However, it is possible to make an analysis with .

Whether that makes sense is another question. I also did this in some analyses.

If the spectra is along r you should consider a sphere in the box, if the radius is greater than L you have spurious wavenumbers.

However, the figures in the FDS file have no clear explanation.

It could be a 3D spectrum, also because if it is not, they should have used subscript such as 11 or 22. Furthermore it is a isotropic turbulence, where a 3D spectrum can be calculated, and maybe deduced from E_11.

I didn't understand a thing. Also if we make a 3D spectrum, how is it possible to see the wavenumberS after the nyquist wavenumber? shouldn't it be the limit? Does this change in a 3D contest?

In isotropic homogeneous turbulence, the statistics are the same under rotation and traslation, this way the spectrum E11,E22,E33 has the same behavior. And the spectrum computed for a sphere of radius r=L should be also the same.

If you want to assume that dr=sqrt(dx^2+dy^2+dz^2) then you have the Nyquist frequency pi/dr but this is not what I see in the figure

Suppose your only have two modes in each direction:

x-Direction



y-Direction



z-Direction



Spherical



Now you may accumulate your data in Fourier space according these modes.

Not sure to understand you example, it is not for isotropic case, right?

I only wanted to show, what the authors may have done numerically. Perhaps the explanation is not quite well.

The FDS guide says that they consider as ,which they use as Filter, the geometric mean .

Actually this leads to a discrepancy in the 3d spectra the Nyquist wavenumber is not the filter limit. Could be?

Is this the theoretical Nyquist number with 2 points, or is it the numerical one for the discretization scheme? Not familiar with FDS, but if it uses an an implicit filter plus finite volume scheme, then the spectrum can contain infinite frequencies. The cut off wavenumber reported can thus be the theoretical or numerical, but depending on how the FFT is done on the finite volume solution, there can be very high frequencies present. The spectra shown here are nothing unusual for an implicit filter.

The Nyquist frequency is identified by the grid size, no matter about which numerical method is used. The filter width identifies the Nyquist frequency only for spectral methods, otherwise pi/Delta is not equal to pi/h.

I agree. But the question is different: is the line in the plot: Nyquist number (theoretical) or cut off number of the FDS scheme? Also, second question, how is the finite volume solution interpolated to do FFT?

In my opinion there is no doubt, on a 32^3 and 64^3 the Nyquist cut-off is at 16 and 32 wavenumbers, respectively.


Then, there is no need to interpolate, the FFT is taken from the node solution.

Then, if only input is chosen as nodes, then 64 points -> no energy beyond 32 wavenumber, because FFT only has 32 modes. This is in conflict with plot, so it can not be the case. From experience: finite volume solution could be reconstructred to give linear or higher polynomials. Supersampling these and then input into FFT gives more modes. Finite volume solution space is not just cell averages or node values. It is also inter-cell jumps in solution. FFT over jump gives high frequencies.


My guess that explains plot in first post perfectly: finite volume was done on 64 grid with constant cell values or node values. Then, local linear or high reconstruction (WENO things) are done to smoothing solution. Then FFT is done of this.

FFT then gives of course energy over mode 32. But author of plot realize that 32 is theoretical Nyquist of computational grid of 64. So introduce line in plot at 32, recognizing theoretical results. This fully explains plot and resolves all issues.

Well, in LES the meaning of the local averaging is not in the sense you described that is typical in Euler-based method. The volume (filtered) averaged variable is a continuous function by definition, in the sense described by the historical Schumann’s paper.

maybe I confuse things. If the solution to the LES equation is computed from finite volume, then the averaging operator is not a continuous filtering in space. It is a filtering of the solution within each element. So inhomogeneous filtering.



If I compute LES with fluent or starccm and look at solution, I see: one cell, one constant solution value u. Next cell, next solution value, different level. I see jumps between each element, looks like staircase. Then how is the volume filtered solution can be said to be continuous, if finite volume does not produce continuous solution?

No, you miss the theoretical basic about the local filtering. The resulting filtered function is continuous according to (let me use 1D notation)



f_fil(x) = 1/Delta Int[x-Delta/2; x+Delta/2] f(x')dx'



or in a more general convolution definition





f_fil(x) = Int[-Inf; +Inf] G(x-x';Delta) f(x')dx'






That means that at each position x of the space, you have a volume where you do the average.



Do not see the Fluent plot as it says nothing about LES theory but only it own assumption in the graphics.

ah, but this is critical point here. maybe we come from different background in finite volume. if this statement above is true, then yes, filtered field can be continuous. But from my learning at university, statement above is not correct. Why: averaging procedure you describe is done by finite volume discretization operator. It is not second operator, but in the finite volume scheme inside. Finite volume scheme comes from projection of PDE onto mean inside each control volume. Projection looks like:

int{control volume} div F dx. using green's theorem, we can rewrite volume integration to surface integration over normal face fluxes:

int{surface} Fn dS. Then finite volume people need to find way to compute unique fluxes at cell interfaces, for example with JST flux. This tells us that this integral depends on where it is in space. So where are the element boundaries to compute surface flux. if element boundaries change (grid change), then surface flux changes. So projection of PDE onto mean depends on definition of integration regions: discrete points in space, NOT full space.



If finite volume operator is continuous like you say, then it would be indepedennt of position of grid cells. and shape of grid cells. then all that would could would be number of grid cells in a region, but not shape of grid cells or position of grid interfaces.



Maybe here is my biggest problem: If you say that finite volume scheme gives continuous solution in space, but we only have limited information (solution at grid nodes), then what is the correct solution between grid nodes? Then there must be a unique, well defined solution at every point in space (on nodes, and away from nodes). However this can not be with only limited number of information points, non? What I learned is: finite volume solution is a constant in each control volume, and jumps in between.



Is my university professor wrong, or is this a different way of the same truth?





To go back to quote at top: no, I learned that the average is only done at discrete spatial steps: the grid cells.

Unfortunately, you learned at your university a correct but only a specific framework of the FVM. In particular, your background is typical of people using FVM in the framework of compressible inviscid flows and analysed by a discrete point of view.
Conversely, FVM stems from the fundamental Reynolds theorem governing conservation laws. Such equations are the way to see the evolution of material volume. You can express such equation in the Eulerian representation and they remains continuous, not discrete.
Only the type of numerical flux reconstruction will define the specific FV scheme but the resolved variable is a volume averaging. The assumption you did about a piecewise constant averaged value is a low order assumption. For example, polynomial reconstruction of averaged values are high-order but this is a different numerical topic (eg., see Barth).


In the LES framework you have to think in a different way. If you want, have a reading to one of my paper and its reference (I suggest starting from Schumann's paper)

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

tahnk you very much for taking the time. I do not understand much, but I will start reading the shumann paper. thank very much.

If you have the paper of Schumann is good to start with it.

I can suggest also a very simple exercise, use a function f(x)= sin(k*x), apply the definition


f_bar(x;Delta)=1/Delta Integral [x-Delta/2; x+ Delta/2] sin(k*x') dx'


and see how the filtered function behaves depending on Delta.

Thank you, can you tell me if this the correct Schuman paper here:

https://www.google.com/url?sa=t&rct=...fn9aMHi2U6Txwk

Yes, note how he stated the similar approach of Deardoff about local filtering.

Be also aware that in 1975 the formalism of the LES theory was poor, you need to have a reading of further literature.



The key is that when you fix a set of finite volumes, you can assume that the value computed in the center node is the sample of the continuous locally averaged function.

is this only approximately so - for many finite volumes - or always?


I started reading your paper - I understand from the beginning two things:

a) the filter in finite volume is somewhat of a lucky accident? because the integrals to compute averages are the same formally as box filters. Is this correct interpretation?


b) the continuous form is nice because no need to commute filter and ddx operators - the integral is always the outer operation. Correct like so? If this is correct: but then how to compute the integral matters - right?


thanks you for pointing me to this point of view on finite volume. it is very new to myself!

a) not exactly a "lucky accident" but yes, the local volume averaging can be seen as the theoretical "top-hat" (or box) filter. One of the possible filter. Of course, you have to consider the real filter type if an implicit approach is used.

b) the surface integral of the flux can be computed in one of the numerical flux reconstruction method.

I do not understand this last point. Is it not always a box filter, because it is an average?



Aaaah! So this means that WENO2 and WENO4 give different *integrands* (under the integral), but the "outer" filter is always a box filter / finite volume filter?



So my point of view has changed now. I understand: in your paper it is shown that the Schumann approach needs not to compute commutation of filter and ddx operator. So finite volume schemes are always box-filtered LES forms. Different numerical fluxes (things like JST and MUSL and so) just change the integrand. This means that they provide better or worse things to integrate - closer to the true solution. But it always stays a nicely box-filtered LES. Is this summary correct?


I have to stop reading your paper now. It is very difficult for me and has a lot of detail. I will continue tomorrow and need more time to learn. Thank you!

You must first observe that the integral form of the LES equation has never an explicit averaging procedure. The LHS is the time derivative of a variable that results filtered implicitly by the numerical method adopted for computing the surface integral of the flux. Therefore, each numerical method produces an approximation of the theoretical box-filter.


Just to be clear, the implicit filtering is often assumed also in finite difference method (that is the discretiation of the differential equation) an you can find in literature that the box-filter is assumed implicitly from a second order central discretization.

Ok! So we can not say that finite volume method is box filter, but specific finite volume method approximates box filter. Is the same true for other filters? My prof works on discontunes galerkin methods - where the outer integral is an L2 projection. is this idea then valid still? I believe my prof does not know this!

The continous form address exactly the box filter of width corresponding to the box lenght. Depending on the accuracy of the surface integral of the fluxes, the FVM would produce implicitly an approximation of the box filter. In my paper you can see different numerical flux reconstruction and the consequent transfer functions induced by the schemes.



The integral form of the conservation law (the basis of FMV) is also a mathematical weak form (the basis of FEM), being the shape function specifically built. In other words, the FVM can be see as a specific FEM.

Interesting discussion, the provided paper on what implicit filtering in fv does provides some alternative point of view. still, the point of view given by the author that in integral methods the filter and derivative need not commute is true but only moves the problem somewhere else: how to approximate the fluxes. That remains some nonlinear task, which interacts with the nonlinear closures. So nothing is gained here, the problem is just looked at from a different perspective. Greens integral theorem tells us that it does not matter if we push the problem to a discrete ddx, or a discrete flux.

Clearly, avoiding to commute does not solve for the unresolved part of the non-linear flux and a closure model is still required.
The gain is in the fact you have a clear and recognized filter shape induced by the surface integral of the fluxes, whatever the accuracy order of flux reconstruction is. Conversely, if you commute you have a differential form and the shape of the filtering is not well recognized using some FD discretization. That means you can specifically use a SGS model for this type of filtering. Furthermore, in a further article, I showed that the unresolved terms is less relevant, see Sec.4 here
https://www.researchgate.net/publica...er_mixed_model




you will find that without commuting, a new expression of the Germano identity is obtained and no need to extract the model function from filtering is required.

Thanks for the paper, I will have to read it later. I understand that FD filters can be ill defined. What I wonder, though is: different reconstruction procedures under the integral will still give different coarse scale solution, and thus need different sgs models - is this correct?