[maybee]

Hi,

I have read now a few introductions into LES, but still don`t understand exactly how the filtering works.

As far as I ve understood the filtering should be done by the eddie sizes, e.g. all eddies with a size below "2" will be filtered out and modelled.

One definition of the filtering is given here (I ve also read introductions in a lot of books, but can t link them here)

http://en.wikipedia.org/wiki/Filter_...ddy_simulation)

or here in chapter 2.2 (german)

http://www.ttm.tugraz.at/rene/Publications/LES.pdf

However, how does the filtering, especially the spacial filtering, actually work?
The starting position are a specific field like velocity u(x',t) which is multiplied by a filter function (weighting function, convolution kernel), e.g. see equation (2.9) in the second link where only one space coordinate is observed:

[IMG][/IMG]


I assume here that the field, e.g. u, is described in dependancy of space x' and time t. The filter function G is defined by

G(x-x',t)

where the filter can be a top hat filter as shown in the image above.
If we look at the filter function knowing x' is a length (part of space coordinate) and within the function the difference x-x' is used, x also has to be a length.
The result is that all velocities at the distance x from x' (x-x') bigger than "delta/2" are filtered out (see top hat filter).

My question is consequently: How does this filter the small eddies out of the velocity field?
If my assumptions are right it seems there are velocities filtered out in specific distances of x'.

Can someone explain to me whats going on here?

greetings
maybee

[FMDenaro]

Hello,
searching in this forum you can find similar discussions with many responses

[maybee]

Can you give me a link? I can't find any explanation for my question.

[FMDenaro]

Originally Posted by maybee

Can you give me a link? I can't find any explanation for my question.

http://www.cfd-online.com/Forums/mai...tion-help.html

[maybee]

hi again,

I have read now a little about the LES filtering and as far as I ve understood there is a great difference in the (A) theoretic mathematical description [describes an explicite filtering] of filtering :

Divide the physical dimension into a unmodelled part for the high energy eddies of the turbulence (later solved by DNS) and into a modelled part for the low energy eddies of the turbulence (e.g. usage of Smagarovski SGS model). The terms "high" and "low" are decided by a specified filter width.
The unmodelled part is then received by multiplying the physical dimension with a filter function over the whole space of the process and additionally over the time.


and


(B) the filtering procedure as it is actually generally used in LES [implicitely].
The filtering that takes place is

1. the filtering because of the mesh size (Quote:"induces a cut-off at Nyquist frequency") [this filtering happens in all simulations ?]

and

2. the filtering because of the discretisations used for the terms of the governing equations:

How does it actually take place? How do the discretisations filter parts of the physical dimension in such way that one part of turbulence is solved DNS like and one part is solved modelled?

-------------------------------------------------------------------------------------

With respect to the filtering mentioned in 2.:

This type of filtering which is done (B) practically implicitely by the discretisations is described in the (A) theoretical mathematical models as explicitely, right?

Quote of http://www.cfd-online.com/Forums/mai...ion-help.html:
"Many often the authors say simply "top-hat" when they use second order FD/FV and spectral cut-off when using spectral methods"

If the filtering is done by the choice of discretisation scheme like FD or FV of first,second... Order, how does LES use this filtering for splitting the physical dimension into two parts as described above? and respectively what is actually calculated in the subgrid modells?


I still have to read your paper though - I had a lot of work to do with some programming issues...

greetings
maybee

[FMDenaro]

The Nyquist frequency exists always, no matter which one among FD/FV/FEM/SM are used. It depends on the sampling on a finite computational grid as same as a measurement devise has a resolution limit.

The filter implictly introduced by a discretization depends on the local truncation error that you can analyse in wavenumber space.
Consider the exact derivative of the function exp(ikx), it is simply ik times the function.
Consider now some formula, for example the FD second order formula (f(i+1)-f(i-1))/2h. You can see that it produces ik_eff times the function, k_eff being the modified wavenumber. The difference (k-k_eff)/k represented in wavenumber space shows that k_eff smoothes the resolved component before Nyquist, implictly inducing a smooth filter. This happes for higher order FD and FV. Conversely Spectral Methods provided k_eff=k up to Nyquist, therefore they implicitly induce a cut-off filter

[maybee]

Thank you very much professor Denaro. I have some question about your last answer:

You can see that it produces ik_eff times the function, k_eff being the modified wavenumber. The difference (k-k_eff)/k represented in wavenumber space shows that k_eff smoothes the resolved component before Nyquist, implictly inducing a smooth filter. This happes for higher order FD and FV.

I have some basic knowledge about the FD Discretisation and therefore I know that the change in one physical dimension between two mesh points is described by the basic linear derivation (difference quotient), second derivates can be derived from the first derivates.
Starting with the first line of the quote I can t follow you exactly .
Few quesitons:
1.If I am right the wavenumber is defined by

wavenumber k = 2pi/eddiesize

Is the effective wavenumber defined by

wavenumber k_eff = 2pi/eddiesize_eff

where "eddiesize_eff" defines the limit of size of the eddies which is solved for DNS like?

2. The i in the function exp(ikx) isn t the same as the i in the second order derivation f(i+1)-f(i-1))/2h , right? Or is the function defined in dependancy of the mesh points?

3.

You can see that it produces ik_eff times the function, k_eff being the modified wavenumber. The difference (k-k_eff)/k represented in wavenumber space shows that k_eff smoothes the resolved component before Nyquist

I don t understand this.

Sorry to bother you again. I need to write a really basic introduction in how the LES works and unfortunately I am not able to do so yet.

[FMDenaro]

1) A periodic function can be expressed by means of Fourier series

f(x) = Sum [ F(k) exp (ikx)]

where F(k) is the Fourier coefficients and i is the imaginary unit (question n.2): sorry, is similar to the index of the grid node ). You can see that k is dimensional and is the inverse of a lenght. That is an equivalent of a space-frequancy. Given the lenght of periodicity L, one has k = n Pi/L (n=0,1,…nmax)
An exact derivative of f(x) is f’(x) = ik f(x).
A second order FD derivative is

D f(x) = F(k)[exp (ik(x+h))-exp(ik(x-h)]/2h = f(x)[exp (ikh)-exp(ik(-h)]/2h= i [sin(kh)/h] f(x)

2) You see that such formula produces k_eff = sin(kh)/h, which must be compared to k. Therefore, the transfer function k_eff/k is close to the unity only at low wavenumbers k but decreases going to k=pi/h. That implies that flow eddies too close to the grid size h are smoothed in their energy content