analysing turbulence you always find energy spectra in wavenumber space, i.e. E(k) over k. My question is, how to get such a spectrum from a timeseries of let's say a velocity measurement. I assume there must be a way as in most laboratory experiments you measure timeseries at a fixed location rather than spatially resolved data. I appreciate any kind of hint.
Re: wavenumber spectrum
One of the ways to do it is to invoke Taylor's frozen field hypothesis, which states that the time scale of evolution of the turbulent field is much larger than the mean flow field. Hence, one may assume that the entire turbulent field is frozen while being advected by the mean flow.
Using this it is easy to convert a time varying signal into a spatially varying function. Simply multiply the time-sampled points by the mean flow velocity to get the spatial coordinates corresponding to those points. An FFT of this function of space will provide you a spectrum in wavenumber space. Note to get an enegry spectrum, you must transform the velocity squared, or use the Parseval's theorem. Also note that typically your spatial data will be unevenly spaced, so taking the transform is a bit complicated.
Of course, this does not work when the velocity associated with the turbulent fluctuations is comparable to the mean flow speed. It will also not work when the mean flow reverses direction (as in a periodic flow) because your spatial data then ceases to be monotonic.
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