help Energy spectrum 2.0
 

Hi all,

this post can be placed here and in the the lounge as well.

It is a really frustrating period for me, my advisor thinks that the Energy spectrum is the Fourier Transform of the time history of the specific Turbulent kinetic energy.

I demonstrated that the mean value of turbulent kinetic energy is equal to the integral of the energy spectrum over the frequency, which could be a definition of the energy spectrum, as you can see at page 53 of the book "Turbulence" of Uriel Frisch, but maybe because I am young and just arrived he does not believe in what I am saying.

I also tried to do the Fourier Transform of the time history of the specific Turbulent kinetic energy and I showed him that it is not demonstrated that the mean value of turbulent kinetic energy is equal to the integral of the energy spectrum over the frequency. (He thinks that I am wrong in some scaling factor after doing the fft, which it is impossible since, using the proper definition of the Energy spectrum, this demonstration checks)

I think that that the name of this spectrum confused him, I have seen some confusion here as well https://www.cfd-online.com/Forums/ma...-spectrum.html .

The problem here is scientific and psichological:(. I would like to do a demonstration that the Fourier Transform of the time history of the specific Turbulent kinetic energy is not the Energy spectrum, but it seems difficult to me prove this idea. I know that in this group there are expert scientists and I am searching for advices, I hope you can help me.

Have a read here, the concepts are described
https://www.io-warnemuende.de/tl_fil...Chap4_WS08.pdf

I am not searching for more references about that. I already showed him how the things must be done. I just would like to show him why his theory is wrong

First of all, the position you have towards your advisor is delicate... you should work in such a way that you show the theory on well known literature.

Without being present to your discussion is not possible to say that he is totally wrong. Just consider the kinetic energy function in a position as k(t)=0.5*(u(t)^2+v(t)^2+w(t)^2). Now, just forget what the physical meaning of k but ask yourself what happens if you perform the FT of this function. How do you would name the coefficients? Just remember that we can analyse a function either in the physical domain or in the frequency domain but that is always the same function, differently represented. The function k remains the kinetic energy represented in the Fourier space. The coefficients represent the power density of the function. I think (but I may be worng) that your discussion should be focus on the meaning of "energy spectrum" and "spectrum of the kinetic energy".

You are right, it is a very delicate position.

He just think that I should do:
Tinc is the time step
delta_f is the frequency step
k(t)=0.5*(u(t)^2+v(t)^2+w(t)^2)
E(f)=abs(fft(k(t)))*Tinc

but if I do

mean(k(t))

and

sum(E(f))*delta_f

they are different and this does not respect the definition of E(f)

When you say the coefficients represent te power density of the function, are you talking about the energy spectrum or spectrum of the kinetic energy.

You are right, but I don't know where to start. Do you have any suggestion about that?

I also thought that maybe I can replicate the result of some important publication, but it is difficult to find the raw data

I think that a possible way is to start from a general framework, not necessarily the turbulence field.
For example, start from the various denominations explained here https://en.wikipedia.org/wiki/Spectral_density

Thanks, I have already seen it, but I don't know how can this help me. Could you be more specific, which definition would you use?

See the first part
"The power spectrum https://wikimedia.org/api/rest_v1/me...e9f81152fa470e of a time series https://wikimedia.org/api/rest_v1/me...b0818107fa5f5c describes the distribution of power into frequency components composing that signal.[1] According to Fourier analysis, any physical signal can be decomposed into a number of discrete frequencies, or a spectrum of frequencies over a continuous range. The statistical average of a certain signal or sort of signal (including noise) as analyzed in terms of its frequency content, is called its spectrum. "


This way you can present the power spectrum of the function k(t), that is the kinetic energy in one spatial point (or a volume average k_av(t) if you do that).

The difference is that one can define the spectral density of turbulent kinetic energy, E(|f|) as the total contribution from E3D(f) over a sphere of radius |f| (ie. all contributions with wavelength L = 2p /|f|). The integral over all frequencies is the total kinetic energy (averaging is considered).


Note that the FT of the correlations produces a tensor, each component being Eij(f). You would discuss only of the main diagonal entries.


I cannot say more because you have an advisor and you need to work with him to assess the issues, I am not there to discuss with you...