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LES: Difference between physical space and fourier

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Old   June 28, 2000, 10:03
Default LES: Difference between physical space and fourier
  #1
K.Senthil Kumar
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dear users

Can u please give the difference between physical space and fourier space with regard to LES or any reference material on that.

Sincerely ur's

Senthil.

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Old   June 29, 2000, 10:17
Default Re: LES: Difference between physical space and fou
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Patrick Godon
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The physical space is the everyday world in which you have your flow, water, air, etc.. with the 'primitive' variables being velocities and density (and pressure, temperature, etc.. if you use them).

When you carry out simulations of turbulence, it is very convenient to analyse the results in what we called the Fourier space. THis just means that some of the variables are expanded in Fourier series. Usually, the kinetic energy (square of the velocity) is expanded in Fourier series in one dimension or another. Then one can look at the amplitude of the coefficients of the Fourier expansion, as a function of the discrete index of the expansion. This is the Fourier space.

for example if you look at a function of x

f(x)=v(x)**2 (kinetic energy)

then the Fourier transform is just

f(x) = sum (over k) ak exp(-ikx)

where ak (a index k) are the coefficient of the expansion, and k is the discrete index.

A given k corresponds to a wavelength L of the order of D/k, where D is the size of the computational domain in the physical space. So k=1 corresponds to the large scale while a large k (e.g. k=100) corresponds to a small physical scale.

Now if you look a the amplitude of the Fourier coefficient (a index k) of the kinetic energy (f=v**2) as a function of k, you will have what we call a spectrum of the energy.

If you have large Eddies in your turbulence, then you can expect the coefficient a to be large for small k (for example in 2D turbulence, small vortices merge to form larger vortices, and eventually all the energy is contained in a few big vortices). And the oposite too. However, most turbulent flows look very chaotic and all the vortices are transient (then they are called Eddies) and it is very difficult to see anything by looking at the flow in the physical space. Then one looks at the flow in the Spectral space (Fourier Space is a Spectral Space). Then one can see some characteristic features that one cannot see otherwise. For example 3D homogeneous turbulence have an energy spectrum, where the coefficients 'a' are a decaying exponential function of the index 'k' (the Kolmogorov spectrum). And the exponential decay coefficient is the same for all 3D homogeneous turbulence, but it is different for 2D flow for example. So by looking at the slope of the decaying 'a' as a function of 'k' one can find out if the turbulence is 2D or 3D, and many other things...

I hope this helps.

Patrick
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Old   July 3, 2000, 01:59
Default Re: LES: Difference between physical space and fou
  #3
Senthil
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Dear Mr.Patrick

so can you please explain when to use gaussian filter and when to use tophat filter in LES. Thanks in advance senthil k.
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Old   July 3, 2000, 08:36
Default Re: LES: Difference between physical space and fou
  #4
Patrick Godon
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Hi again,

I am not doing LES, but rather DNS (direct numerical simulations, and not just Large Eddies Simulations). So I am not sure what is done in LES (I guess the small scales are negelcted). ALso I do not know Gaussian and Tphat filters. I guess you are talking about spectral filters, applied in the Fourier space (am I correct?). Gaussian filter has probably the form of a Guassian in the spectral space, again I guess.

The only filters that I know are the ones used in Spectral Methods, and are spectral filters. THere is an exponential cut-off filter that cut off the highest frequencies, and there are stronger filters that cut off also lower frequenceis (frequencies in the spectral-Fourier space). In the case of DNS the methods usually is to use a hyper-viscosity that mimics the effect of real viscosity in the flow in that sens that it damps the high frequencies without affecting the slope of the energy spectrum in the Fourier space. THe hyperviscosity is applied in the real space as an additional term in the velocities (or momenta) equations, with a high order derivatives term.

In any case, if you are simulating turbulence (DNS as well as LES, I guess) and you want to use a filter, you need to make sure that your energy spectrum is not affected directly by the filter. And if it is affected, then you need to know how it is affected and what does this imply for the meaning of your results.

For example, if you are interested in the large scale structure of the flow and don't really mind what's happening on the small scale (you just want the energy to be dissipated in the small scales, like in 3D homogeneous hydrodynamic turbulence, for example), then you can use a filter that cuts off (or reduces) the high frequencies. For each specific problem one needs to assess the effect of the filter by changing the parameters of the filter and rerun the same problem.

So, without knowing your flow problem and the methods you are talking about, I would just suggest that if you are simulating turbulence and you want to use a filter, you need use a filter that will affect only the highest frequencies, and that leaves the lower frequencies unchanged, so that the large scale structure of the flow is not affected by the filter. The best filter allows even to resolve the fine structure of the flow (ideal in DNS).

Patrick
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