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January 18, 2021, 16:12 
Questions about LES filtering

#1 
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Hello,
I'm confused about filtering in LES. I have some questions about LES. 1 How can we say that large scales are resolved like DNS in LES while by using filters, we are averaging over neighbor cells. For example filtered u velocity is equal to the average of neighbor cells velocity using box filter. But in DNS the velocity in each cell is not averaged. 2 Suppose that we have cubic cells all over the domain which they have equal length, width and height and our solver is collocated. So Delta, which is the filter width(using box filter) becomes equal to the width of the cell as it is equal to the cubic root of the volume of the cell. Then if we use box filter, the G kernel becomes zero because we don't have any cells nearby that it's distance with the center of a specific cell is less than Delta/2 (because the minimum distance with a neighbor cell's center is Delta). So the filtered velocity becomes zero all over the domain! 3 How does the dynamic subgridscale model detect that the cell is in laminar region(to set the turb viscosity equal to zero) or small scale region(to determine the turb viscosity)? 4 In this topic(LES Filtering: how are the small and large scales equations solved?) it is mentioned that in laminar region the wave lengths are small and the value of filter kernel is near 1. But how does a cfd code detect that the wave number is small in a cell? Please do not refer me to this topic: LES Filtering: how are the small and large scales equations solved? I have read that. Regards 

January 18, 2021, 16:38 

#2 
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Filippo Maria Denaro
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January 19, 2021, 00:34 

#3  
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Quote:
I used your replies in that topic and they were great. But two of my questions remained unanswered: Quote:
2 In this topic(https://www.cfdonline.com/Forums/ma...nssolved.html) it is mentioned that in laminar region the wave lengths are small and the value of filter kernel is near 1. But how does a cfd code detect that the wave number is small in laminar region while it is using same procedure of filtering on both laminar and turbulent regions? 

January 19, 2021, 04:08 

#4  
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Filippo Maria Denaro
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 A LES filter should be analysed by looking also at the transfer function G(k). You will see that the effect of the filter vanishes at low wavenumbers (large wavelenght) and the variable is practically unfiltered at that level. This justifies the fact the LES resolves in a DNSlike the large characteristic scales.  The CFD code often adopts the implicit filtering, that is the filter is never really applied but it acts only by measn of the numerical formulation  The dynamic procedure produces a testfiltering that in laminar region is practically equal to the gridfiltering. This way the eddy viscosity vanishes. You can read also these notes and the cited textbooks https://www.researchgate.net/publica..._Part_I_and_II 

January 19, 2021, 04:15 

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Thanks a lot, Prof.
What I noticed from this topic is that Filter definition has nothing to do with solving velocity field in LES and filter definition is just a formal definition. (what do we want to filter while the smallest scales can only be in the size of the grid?!) The velocity field is solved just like DNS but with additional terms related to eddy viscosity. Then, if we use dynamic subgridscale models, the filter now has a meaning because we use a test filter that filters the scales bigger than a size that is bigger than grid size. (test filter usually equals 2times of the grid width). Am I right? 

January 19, 2021, 04:28 

#6  
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Filippo Maria Denaro
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A LES code is often like a DNS code with the adding of the evaluation of the SGS model terms. The filter acts implicitly when the computational grid is coarser than the required for a DNS solution. But if you run a LES code on a DNSgrid the result is a DNS solution. However, the filter size induced implicitly by the discretization is larger than the grid size. Thus, the test filter has to take into account this issue. 

January 19, 2021, 04:38 

#7  
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Instead, one can show (or, at least, I think I am able to show) that the original box filtered field on the grid is, indeed, the result of the product of two filters, in this very precise sequence: 1. A sampling filter: the original continuous field is sampled at the cell center locations and everything in between is interpolated. Doesn't really matter here what interpolation is used (but keep in mind that different numerical methods imply different interpolations). 2. A box filter with given width Now, subtlety comes in from the fact that, if you apply a third filter, still a box one with the same width as in 2, it leaves untouched the cell center values of the field you got in 2, but it alters their interpolation between the cell centers. So, at the discrete level, you will always see the same cell center values but, formally, the underlying interpolation between them is changed. Took me several years to figure this out. Quote:
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January 19, 2021, 07:05 

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Just a new question: As I understood, in implicit filtering for solving NS equations, actually, we have no filtering because the smallest scales are equal to the width of the grid. (We don't need to filter our captured small scales!) The question is: Why should someone use explicit filtering to solve NS equations? I mean in every LES we want to capture the scales as small as possible according to the size of the cell, then we model the scales that are smaller than them. As you know, filters remove the small scales and keep the large scales. Then why should we filter(and remove some of) our captured small scales in the domain using explicit filtering? I'm not talking about using the explicit test filter in the SGS model. I'm talking about solving NS equations. Best Regards 

January 19, 2021, 07:29 

#9 
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This is both obvious and not so much at the same time, because there is some contrasting evidence.
On purely theoretical grounds, explicitly filtering has the main advantage of reducing the effect of the numerical error (the filter will act on the very scales where the error is more pronounced). The additional benefit is that you also have the chance of achieving a grid converged LES, with no error at all. It also appears that you can reliably model the scales lost in the explicit filtering process because, except for cutoff filters, they do not actually disappear, but are just lowered in their energetic component. Still, and this is my own personal bias I guess, I still don't get, as of today, how comes the same fate is not reserved to the error (that is, it shouldn't disappear as well, just lower its energy content as well). But there is evidence that this actually works. The main problem with this approach is that it puts a higher burden on the model. As stated above, this can be compensated, but there is some evidence that, in some cases, such compensation is not better than not filtering at all and just give a shot with the given grid as it is. The problem in deciding effectively which route is better is that, unfortunately, it is very dependent on the code and implementation, but the LES community doesn't even 100% agree on which equations they are actually solving, imagine picking up a code and implementation. From a practical perspective, there is practically no one doing explicit filtering in applications (with few exceptions, of course) 

January 19, 2021, 07:58 

#10 
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Thanks a lot, I really appreciate your help.
Just the last question: In all books talking about box filter, they also plot the filter in spectral space as you can see below: In a CFD book by Kajishima, he warns the reader that filter function damps high wavenumber components but in an oscillatory manner. It should be noted with caution that this filter provides negative values for certain wavenumber. I didn't get this! It's OK that the filter removes the scales bigger than filter size , but: Why should it be important to have the filter function in spectral space? How does the filter remove small scales in physical space, but does not remove them in the spectral space for the corresponding wavenumber of the scale? Does this affect the accuracy of results?(Especially when used in test filter) Kind Regards 

January 19, 2021, 11:13 

#11 
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Filippo Maria Denaro
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I suppose you are a newcomer in the LES field, so I will try to explain what is misleading in that figures. When you read a theoretical introduction in LES, some filter functions are always introduced and plotted. Be aware about the fact that the resulting filtered function is a continuous function since the convolution with the filter kernel is illustrated as an analytical process. Actually, this analytical convolution is never really applied but the whole LES procedure is on a numerical ground where you work with discrete values and discrete filters. What is more, neither a discrete filter is actually applied so the real shape of the filter depends on the type of discretization. The exact (continuous) tophat filter will be only approximatively obtained (implicitly) by using a Finite Volume numerical formulation. "In a CFD book by Kajishima, he warns the reader that filter function damps high wavenumber components but in an oscillatory manner. It should be noted with caution that this filter provides negative values for certain wavenumber." This is only in theory but never happens in practice since the computational grid introduces a cutoff (Nyquist theorem) that cut away the negative/positive oscillations after the first zero. Note that we don't use the spectral space in a practical way but we are only analysing the filter shape, comparing it in physical and in wavenumbers space. 

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