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When reading about turbulent flows and turbulence spectrum I frequently come across the terms 'wavelength' 'wavenumber'. I understand that the energy is transferred from larger eddies to smaller ones and so there is a spectrum of different sized eddies spinning at different velocities. But what does the wavelength and frequency of turbulence mean? Is wavelength the diameter of the various sized eddies? Does the frequency of turbulence mean the angular velocity of vortices or the frequency of variation of different state variables?

I have a feeling that the wavelength is related to some characteristic length of the eddies but why term this 'length' the 'wavelength'? How is it related to waves?

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Wavelength is with reference to Kolmogorov's theory of turbulence. Kolmogorov postulated a theory for how turbulence is conveyed from large to small scale eddies, as you described. The wavelength and frequency origins are buried deep in the mathematics of the theory. Basically, the theory postulates an energy spectrum for turbulence, which divides the energy spectrum by eddy wavelengths. This originates from a Fourier series that is used to characterize the distribution of perturbations over different length and time scales, followed by more complicated mathematics to obtain the energy spectrum (https://atmos.washington.edu/~breth/classes/AS547/lect/lect2.pdf page 4). The result is an energy spectrum, which can be integrated to determine the turbulent kinetic energy composition of a particular wavelength. The author in the attachment describes the energy spectrum of a particular wavelength as the result of all eddies whose characteristic length l is 1/2 of the corresponding wavelength. I am not the best with turbulence, but I hope this provides some insight.

Other sources: https://clouds.eos.ubc.ca/~phil/courses/atsc500/pdf_files/09-kolm.pdf https://www.dartmouth.edu/~cushman/courses/engs250/Kolmogorov.pdf

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