[Boone]

Hi everyone,

I study flows in biperiodic channel and I would like to plot turbulent energy spectra in 1D (streamwise direction).

I had the instantaneous fields of my 3 velocity components at a fixed distance from the channel wall (y+40).

I perform a Fourier transform thanks to Python as follows:
1) I store 1 line of my instantaneous field of U in the streamwise direction in a vector (same for V and W)
2) I perform the FFTs:

FFT_U=abs(np.fft.fft(U))**2

FFT_V=abs(np.fft.fft(V))**2

FFT_W=abs(np.fft.fft(W))**2

3) I plot

0.5(FFT_U+FFT_V+FFT_W)

as a function of the wave number in the streamwise direction.

I obtain a 1D spectrum that have a shape in agreement with the literature (at the same friction Reynolds number) but the energy levels are significantly different.

I am not sure about the formulae used to compute the kinetic energy spectra. Is the following expression right ?

abs(np.fft.fft(U))**2+abs(np.fft.fft(V))**2+abs(np.fft.fft(W))**2

Thanks !
Sincerely,
Boone

[FMDenaro]

Originally Posted by Boone

Hi everyone,

I study flows in biperiodic channel and I would like to plot turbulent energy spectra in 1D (streamwise direction).

I had the instantaneous fields of my 3 velocity components at a fixed distance from the channel wall (y+40).

I perform a Fourier transform thanks to Python as follows:
1) I store 1 line of my instantaneous field of U in the streamwise direction in a vector (same for V and W)
2) I perform the FFTs:

FFT_U=abs(np.fft.fft(U))**2

FFT_V=abs(np.fft.fft(V))**2

FFT_W=abs(np.fft.fft(W))**2

3) I plot

0.5(FFT_U+FFT_V+FFT_W)

as a function of the wave number in the streamwise direction.

I obtain a 1D spectrum that have a shape in agreement with the literature (at the same friction Reynolds number) but the energy levels are significantly different.

I am not sure about the formulae used to compute the kinetic energy spectra. Is the following expression right ?

abs(np.fft.fft(U))**2+abs(np.fft.fft(V))**2+abs(np.fft.fft(W))**2

Thanks !
Sincerely,
Boone

If you search in the forum, you will find several posts that answer you. I personally posted years ago a matlab script for doing that.

[Boone]

Dear FMDenaro, thank you for your answer.


I effectively found these posts that help me proposing the expression I wrote in my first message:
Plot Turbulence Spectrum Using 2D Spatial FFT
Energy spectrum-wrong spectrum shape
Energy Spectrum for a 3D turbulent Flow
Computing the energy spectrum of a turbulent channel flow

I plotted (abs(FFT(U)))^2+(abs(FFT(V)))^2+(abs(FFT(W)))^2 as function of the streamwise wave number. I am surprised by the energy levels which are very high when compared to the levels obtained by some author of the literature (see: fig 5. of Spectral analysis of turbulence based on the DNS of a channel flow Igor A. Bolotnov et al. 2010 Computers & Fluids)

[FMDenaro]

Originally Posted by Boone

Dear FMDenaro, thank you for your answer.


I effectively found these posts that help me proposing the expression I wrote in my first message:
Plot Turbulence Spectrum Using 2D Spatial FFT
Energy spectrum-wrong spectrum shape
Energy Spectrum for a 3D turbulent Flow
Computing the energy spectrum of a turbulent channel flow

I plotted (abs(FFT(U)))^2+(abs(FFT(V)))^2+(abs(FFT(W)))^2 as function of the streamwise wave number. I am surprised by the energy levels which are very high when compared to the levels obtained by some author of the literature (see: fig 5. of Spectral analysis of turbulence based on the DNS of a channel flow Igor A. Bolotnov et al. 2010 Computers & Fluids)

1) if you want to plot the 1D spectra, why do you sum the contribution of U,V and W?


2) If you are working with the non-dimensional equations, your velocity fields are already non-dimensionalized by u_tau.

[Boone]

1) I thought that is was possible to compute the total contribution of the 3 velocities along 1 direction (streamwise, x). My mesh is homogeneous in this direction and in the spanwise direction (z). Isn't it the case ?

2) I am working with dimensional equations. My fields of velocity are in m/s. That is why I normalized by nu*Utau the energy spectrum.

------

Do you have any other idea that can cause this energy difference ?

Is there a difference between the energy spectrum and the density energy spectrum ?

Thanks !

[FMDenaro]

Originally Posted by Boone

1) I thought that is was possible to compute the total contribution of the 3 velocities along 1 direction (streamwise, x). My mesh is homogeneous in this direction and in the spanwise direction (z). Isn't it the case ?

2) I am working with dimensional equations. My fields of velocity are in m/s. That is why I normalized by nu*Utau the energy spectrum.

------

Do you have any other idea that can cause this energy difference ?

Is there a difference between the energy spectrum and the density energy spectrum ?

Thanks !

The velocity is normalized by utau, therefore the first issue is in your computation of u_tau. Then, the kinetic energy is normalized by u_tau^2.

[Boone]

In the literature, see: fig 5. of Spectral analysis of turbulence based on the DNS of a channel flow Igor A. Bolotnov et al. 2010 Computers & Fluids, it seems that the energy spectrum is normalized by Utau*nu [m3/s2] which seems coherent to obtain a normalized energy spectrum.

I think that if a want to first normalize the velocity I have to do:
abs(np.fft.fft(U/Utau))**2 but the resulting energy spectrum will be homogeneous to meter.

[FMDenaro]

Yes. Spectral density has dimension m^3/s^2.
Have a check here to see if your different scaling is due to these issues

https://journals.ametsoc.org/view/jo...-17-0056.1.xml