LES - spatial, time and spectral filtering
 

Hello,

I am reading Pierre Sagaut's book about LES for incompressible flow. I understood that in LES we solve the big turbulence scales and modelise the small's. The difference between resolved and modelised scales is fixed thanks to a filter. This filter can be spatial, spectral or time relative.
My questions are the following :

- What is the difference between filtering equation with a time filter and applying a time step to solve the equations ? Do we have to apply both ? Using time step is not it a way of time filtering ?

- Can we superpose two filters (i.e. spatial and spectral or spectral and time) ?

- I was thinking that by filtering space we implicitly filter time. Is that true ?

Thank you in advance for your answers,
Sincerely

A lot of well-assessed questions on issues that are often not clear in LES


- the time step defines only the cut-off filter implied by the Nyquist frequency pi/dt. This is an implicit filtering appearing when we discretize the time axis, no matter about the practical numerical method is used. Formally, the filter in time can be applied to the equations independently from the time step, for example if you want to resolve only large time-scales (see the analogy to the URANS formulation) the time-filter width can be larger than the time step.



- the spectral filter is also a spatial filter. You can filter in the spatial domain or in the spectral domain but you have always a correspondence between the two spaces. And yes, you can apply a filter any times you want on a variable. Just be careful that the spectral filter is idempotent, therefore you get the same original filtered field, at least if the same cut-off is used. This does not happen for filters such as the top-hat.



- Not exactly, you must think that a practical numerical discretization in time and space introduces implicitly a time-space filtering. That is differently induced by the chosen method of discretization. On the other hand, if you see the continuous equations, filtering in space does not correspond to a filter in time.

I suggest to try develop some simple example using the 1D Burgers equation.


Have a good reading

Forget to say that some of your questions are explained in my lecture notes on LES here https://www.researchgate.net/publica...CMJ6ACE7W9UnDA

But, suppressing some spatial scales wouldn't, in theory, suppress their associated time scales as well? I think Sagaut states this as well somewhere in his book...

Paolo,

that could be a good argument to discuss. Here is just my opinion.


Forget for now we are focusing on simulation of turbulence and let us start a general discussion to help also understanding of other people. If you discretize both space and time you introduce the cut-off frequencies pi/h and pi/dt. This is the Nyquist filter acting on any type of numerical solution (filtered or unfiltered governing equations). The relation between these two frequencies is governed by the CFL number, involving a characteristic local velocity. When CFL=O(1) it is clear the relation between the two cut-off frequencies.


Now let us filter the equation using a spatial filter of width Delta. As you know, in explicit filtering Delta can be chosen independently from the computational spatial step, that is Delta=N*h. That defines the new spatial frequency pi/Delta<pi/h. The key of the discussion is if such new frequency induces a new filter in time such that pi/Delta_time<pi/dt. That would imply that a new CFL value exists to link them. Is the Delta_time = Delta/U somewhere introduced in a practical numerical LES solution?

Considering now the turbulence field, this is somehow relating to the existence of a frozen velocity in the Taylor assumption. But we know that such velocity can be assumed only in specific cases and has a physical meaning. While Delta is a parameter we fix arbitrarily.
Should we invoke the ergodicity hypothesis ? I am not sure about what Sagaut exactly considered in his book.


Have you any idea?

Admittedly, the demonstration on Sagaut book is not among those strongly convincing for me. Yet, I haven't the book with me right now and I can't add any detail.

But, if I picture a DNS snapshot of a velocity field and the same snapshot for the filtered DNS velocity field and a probe point where I register the time evolution, the only way the filtered field can give an unaltered history in the probe point, with respect to DNS, is by altering the bulk velocity in a very peculiar way...

My opinion is that the time filtering effect is a practical consequence of the discrete time integration that induces a smooth transfer function even when we assume that no time-filtering is introduced.

I think there is an error in your explanation.

If you apply a spatial filter to different snapshots of DNS solutions, there is no reason why the history of a probe point should be the same as for the DNS.
Nevertheless you can state that the "change" of the history of a probe point is identical.

This should hold for the smooth and discrete case.

Any discrete time integration produces that... Just as example, if you integrate exactly in time an equation like df/dt + dF(f)/dx=0 you get


f(x,t)-f(x,t0)= - Int [t0,t] dF(f)/dx dt' = -(t-t0) <dF(f)/dx>


where <*> is a time-filtering of width (t-t0). But when you discretize it to solve numerically, a smoothing in the transfer function is introduced implicitly.


Again, that does not require to think about turbulence or any other physics one want to solve.

Dear FMDenaro,
i did not mention a time or spacial integration scheme. I only wanted to point out:
No matter what integration scheme you are using, if you have the exact space-time DNS solution and apply a spatial filter to different snapshots, i think you should compare the relative change at a given probe point and not the state.
Your last explanations sound reasonable.

Regards

Thank you for the answer. FMDenaro your publication really helped me !

Sincerely,

Martin

Good, I suggest not to stop to the reading of my notes, follow the references to the publications of different authors, give yourself enough time to search and understand different ideas on the topic and, finally get your idea. There is no one exact idea and the LES field is still reach of several formulations and interpretation about the role of filtering. And if you see conflicting ideas about the role of the spatial filtering, while this is quite normal in the LES field, adding the time-filtering issue is stille somehow an obscure topic.
Have a good study in LES!
Regards
Filippo

The post-filtering of DNS fields is always a very good experience to do. Unfortunately, this operation get into theoretical troubles about its meaning when comparing to a real LES simulation. Are we able to filter DNS with a filter mimicking at the best the filter in the real LES? Are fields obtained by applying the static filtering on the DNS at each time correlated as happens in LES? The effect of the SGS model is dynamically different.


But I always strongly suggest to make own experience in doing such exercise!