how to compute the tranfer spectrum of TKE
 

Hello and good day to everyone,
I'm new here on the forum and would like to say hello to everyone on board!
I have a question for the turbulence experts here:

I'm computing homogeneous isotropic turbulence (HIT) in a box with a code in physical space (FV method) and I am using an external FFT tool to compute the energy spectra from E(k) = sum over all shell contributions of u*conj(u).

The spectra and dissipation rate of the (HIT) comes out allright, but now I'm stuck at how to compute the transfer spectra T...
For HIT, we have dE/dt = T - 2nu k^2 E(k), where T is the transfer between the modes and the integral of T over all k is zero....

Now how do I compute the T starting from my solution in physical space?
Could I do the following:

1) my code is set up in conservation form, so U_t + F_x+G_y+H_z+viscousFluxes=0.

2) to get the equation for E, we multiply by u=(u,v,w) as a scalar product

3) that would give me u*F_x+u*G_y+u*H_z, where the first term would be something like u*F(2)_x+v*F(3)_x+w*F(4)_x (where 2:4 is the entry corresponding to the u,v,w momentum).

4) Then, I would take an FFT of this whole term, to get T(k1,k2,k3) and then sum up the T according to k=sqrt(k1^2 + k2^2 +k3^2)

5) then plot T(k) over k

This is what I have done so far, but something seems to be wrong... the T(k) integral does not sum up to zero, and I'm getting some strong positive values in the low wavenumbers....

is there anything wrong with my approach? Is there a better way to do it? How is this computation normally done?

I'm sorry if this might be a stupid question, but I'm new and not so experienced in turbulence!

thank you very much!
Kind regards!