# Plot Turbulence Spectrum Using 2D Spatial FFT

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October 21, 2021, 08:31
Plot Turbulence Spectrum Using 2D Spatial FFT
#1
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Jeffrey Johnston
Join Date: Oct 2020
Location: Belfast, Northern Ireland
Posts: 21
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Hello!

I am trying to replicate the results of a study (here) that performs LES of an atmospheric boundary layer. One of the results produced is a turbulence spectrum. I want to recreate this, but I'm a bit lost.

Here's the description in the paper:

Quote:
 Figure 3 shows velocity spectra taken at 90 m (the turbine's hub height) above the surface. The spectra were obtained by taking two-dimensional fast Fourier transforms in space in the horizontal plane at 90 m. We then summed the energy in wavevectors of equal magnitude (i.e. we integrated the energy between concentric rings in the two-dimensional wavespace) to produce the spectra shown in Figure 3. Spectra are shown for both the vertical component and the magnitude of the horizontal components of the velocity vector. A line of −5/3 slope is shown for reference to judge LES's ability to reproduce the cascade of energy from larger to smaller scales.
So here is my first point of confusion. I thought the turbulence energy spectrum had to be based on kinetic energy. That is, I thought I had to calculate before performing the FFT. But in this paper, the (horizontal) velocity magnitude is calculated instead, . So, have I misunderstood the turbulence spectrum? Does the -5/3 slope region exist in a a spectral transform of the magnitude of the velocity? And will it exist for only the horizontal magnitude? Do we have to assume that the vertical component is negligible here (a reasonable assumption in this case)?

My second point of confusion is how to obtain a single 2-axis plot from a 2D fft. Performing a fft of the 2D array of velocity magnitudes produces a 2D array of velocity intensities. I assume that this resultant array is what the we call "wavespace"? But what is a wavevector? How do I find ones of equal magnitude and sum them? And what are the concentric rings all about?

Many thanks,

October 21, 2021, 13:23
#2
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Filippo Maria Denaro
Join Date: Jul 2010
Posts: 6,290
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Quote:
 Originally Posted by NotDrJeff Hello! I am trying to replicate the results of a study (here) that performs LES of an atmospheric boundary layer. One of the results produced is a turbulence spectrum. I want to recreate this, but I'm a bit lost. Here's the description in the paper: So here is my first point of confusion. I thought the turbulence energy spectrum had to be based on kinetic energy. That is, I thought I had to calculate before performing the FFT. But in this paper, the (horizontal) velocity magnitude is calculated instead, . So, have I misunderstood the turbulence spectrum? Does the -5/3 slope region exist in a a spectral transform of the magnitude of the velocity? And will it exist for only the horizontal magnitude? Do we have to assume that the vertical component is negligible here (a reasonable assumption in this case)? My second point of confusion is how to obtain a single 2-axis plot from a 2D fft. Performing a fft of the 2D array of velocity magnitudes produces a 2D array of velocity intensities. I assume that this resultant array is what the we call "wavespace"? But what is a wavevector? How do I find ones of equal magnitude and sum them? And what are the concentric rings all about? Many thanks,

Often one performs the FFT of a velocity component, the corresponding Fourier coefficient is "squared" and you have dimensionally one component of the kinetic energy. There are theoretical reasons to deduce that.

In the paper the 2D plot is reduced to a 1D plot along the unique k wavenumber computed as described, for different rings.

 Tags fft, les, spectral analysis, turbulence, waves