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Integral Length Scale - 2D

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Old   May 15, 2018, 07:05
Default Integral Length Scale - 2D
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Hello Everyone,

I want to calculate the integral length scale directly from the energy spectrum. According to Wiki (https://en.wikipedia.org/wiki/Integral_length_scale)
this can be done via


l=\frac{\int_0^\infty E(k)k^{-1}\,\text{d}k}{\int_0^\infty E(k)\,\text{d}k}.


But I am working on 2D-flows and I have reason to suspect, that this formula should include a factor of 2 in this case, though I couldn't find anything about this.
Does anyone know whether there is indeed a factor of 2 or if I am mistaken?


Thanks a lot and greetings
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Old   May 16, 2018, 03:09
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What is your specific reasoning for the factor 2 in 2D?

Actually, my opinion is that any evalution of the integral lenght is for the order of magnitude, so that a factor of 2 would not change so much.
See https://www.researchgate.net/publica...Numerical_Data
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Old   May 16, 2018, 03:48
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I used a gaussian Spectrum with

E(k)=\frac 18 \sigma^2 k^3 L^4 \text{exp}(-\frac{k^2L^2}{4}).

I calculated the longitudinal correlation function via

R_{11}(r \vec{e}_1)=\int_{R^2}e^{i \vec{k} \cdot  \vec{e}_1 r}\Phi_{11}(\vec{k}) \, \text{d}\vec{k}.


In this case I obtained

R_{11}(r \vec{e}_1)/v_{\text{rms}}^2= f(r) = e^{-\frac{r^2}{L^2}}.


The integral length scale l is the integral over the correlation function

l=\int_0^\infty f(r) \,\text{d}r = \sqrt{\pi}/2 L.


But if I calculate the same quantity via the method in my first post,
I get

l = \sqrt{\pi}/4 L.
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Old   May 17, 2018, 22:52
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These formulae look quite arbitrary. What is the result for 3D?
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Old   May 18, 2018, 10:46
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I'm sorry, but I do not understand. What do you think is arbitrary?
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Old   May 19, 2018, 16:56
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I think you could start by the general definition in terms of the integral of the autocorrelation, considering that spectra and correlation are related each other by direct/inverse Fourier transformation.
But in general the determination of the integral scale is somehow still debated, as you can see for example here
http://www.turbulence-online.com/Pub...apers/WG02.pdf
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