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Velocity / Energy Spectra for planar homogeneous turbulence 

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June 3, 2022, 14:58 
Velocity / Energy Spectra for planar homogeneous turbulence

#1 
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Jeffrey Johnston
Join Date: Oct 2020
Location: Belfast, Northern Ireland
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Hello,
I realise there are a lot of posts about turbulence energy spectra on this forum. I've had a look through and have picked up some helpful info, but I'm still stuck. I also posted my own thread months ago, but I know a lot more about what I'm doing now than then, so thought I should post again. I'm trying to figure out where specifically I am going wrong. It may be in my understanding of how to calculate the turbulent kinetic energy spectrum, or it may be an issue with how I've implemented it in my python / numpy script. I am conducting LES simulations of an atmospheric boundary layers using a horizontally periodic domain. This is very similar to an old paper (here) that includes an energy spectrum for validating that the inertial range (5/3 law) is being partially resolved. Here's my method/code as it stands. Most important questions are in bold u, v and w are 2D arrays of resolved velocities at the cell centres for a horizontal plane through my domain at a given time (we assume the turbulence to be statistically homogeneous in the plane). dx = dy =10 is my cell spacing, and nx = ny = 300 is the number of cells in each direction.
I've attached an image of the plot I get form the above method. I am comparing this against figure 3 in the linked paper, which I've also attached here. As mentioned, I think the units I have for E are different from the paper. The shape is similar to an extent, but the yaxis scale is very different, making me think I've incorrectly calculated E from my FFT arrays. Also, theres a weird peak in the high wavenumbers. It's possible this is genuinely something going on in my simulation data, but I am also thinking it could be a flaw in the energy spectrum code above. Thank you for taking the time to look at this and I appreciate your input. Also, feel free to use any or all of my code above if you are in a similar situation. 

June 3, 2022, 15:44 

#2 
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Filippo Maria Denaro
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Before discussing any of your doubts about the simulation and the computed spectra, I have a simple question:
Have you tested your code by generating a known periodical signal such as sin(kx) for various k and checking that it is correctly captured by the FFT and the plot? 

June 3, 2022, 19:44 

#3  
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Jeffrey Johnston
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Quote:
However, I'm not entirely sure how to extend the sin test to a 2D case for the code I'm using here. Should the signal be something like sin(kx + ky)? And I'm not sure what results I should expect. Regards, 

June 4, 2022, 05:25 

#4  
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Filippo Maria Denaro
Join Date: Jul 2010
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Quote:
The plot of you computed spectrum is really different from those in the papers. I cannot say if the problem is in the LES code or in your spectrum. There are fundamental textbooks about discrete Fourier transform, you can find all the basic theory. Just as an example of how I used the multidimensional discrete Fourier serie, have a look at Sec. 3.2 here https://www.researchgate.net/publica...dy_Simulations 

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energy spectrum, les, turbulence 
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