Order of FFT and using Taylor's Frozen Flow Hypothesis for Grid Turbulence Matter?

loesung · March 27, 2022, 19:01

I am studying grid turbulence. I have a question about the following procedure (if the steps be modified):

1. I am given streamwise velocity field,
2. take it's FFT,
3. And then take the ensemble (temporal) average of the cross-spectra
4. Apply Taylor's frozen flow hypothesis (TFFH) after that.
5. Finally I do spectral analysis on data produced by (4)

My understanding is that ensemble averaging $\langle \cdot \rangle$ is usually averaging different *time* samples. However, can we invoke TFFH at step 2 - take FFT in *spatially* streamwise direction and then average in the spatial (streamwise) direction (by TFFH)? And would that be equivalent to the original procedure (above)?

From what I've seen, usually 1-5 is done as presented above; I accept that TFFH is valid for homogeneous isotropic flows, but I feel a bit nervous in envoking TFFH so early by spatially taking the fft and spatially averaging.

What's the 'best practice' here? Thanks in advance

LuckyTran · March 31, 2022, 08:56

Frozen flux hypothesis allows you to convert a temporal energy spectrum into a spatial energy spectrum and vice versa. You apply the frozen flux hypothesis after you have obtained one of the energy spectrum.

The most common use case is when you have a single point measurement and have temporal data at this location and no other locations. Calculate the temporal energy spectrum from the velocity measurements there and then use the frozen field hypothesis to relate it to the spatial energy spectrum.

The other invokation of going from spatial to temporal correlation is rarely performed. Spatially resolved measurements don't have the spatial resolution that is comparable to the temporal resolution you can achieve and anybody that is doing a spatially resolved measurement (i.e. PIV) will spend the effort to take enough to get statistics. It can be done, but I can't remember the last time I've seen it.

If you had data in space and time, you wouldn't need Taylor's hypothesis. You could and should calculate the spatial and temporal energy spectra directly.

March 27, 2022, 19:01	Order of FFT and using Taylor's Frozen Flow Hypothesis for Grid Turbulence Matter?	#1
loesung New Member Join Date: Mar 2022 Posts: 23 Rep Power: 4	I am studying grid turbulence. I have a question about the following procedure (if the steps be modified): 1. I am given streamwise velocity field, 2. take it's FFT, 3. And then take the ensemble (temporal) average of the cross-spectra 4. Apply Taylor's frozen flow hypothesis (TFFH) after that. 5. Finally I do spectral analysis on data produced by (4) My understanding is that ensemble averaging $\langle \cdot \rangle$ is usually averaging different time samples. However, can we invoke TFFH at step 2 - take FFT in spatially streamwise direction and then average in the spatial (streamwise) direction (by TFFH)? And would that be equivalent to the original procedure (above)? From what I've seen, usually 1-5 is done as presented above; I accept that TFFH is valid for homogeneous isotropic flows, but I feel a bit nervous in envoking TFFH so early by spatially taking the fft and spatially averaging. What's the 'best practice' here? Thanks in advance

March 31, 2022, 08:56		#2
LuckyTran Senior Member Lucky Join Date: Apr 2011 Location: Orlando, FL USA Posts: 5,728 Rep Power: 66	Frozen flux hypothesis allows you to convert a temporal energy spectrum into a spatial energy spectrum and vice versa. You apply the frozen flux hypothesis after you have obtained one of the energy spectrum. The most common use case is when you have a single point measurement and have temporal data at this location and no other locations. Calculate the temporal energy spectrum from the velocity measurements there and then use the frozen field hypothesis to relate it to the spatial energy spectrum. The other invokation of going from spatial to temporal correlation is rarely performed. Spatially resolved measurements don't have the spatial resolution that is comparable to the temporal resolution you can achieve and anybody that is doing a spatially resolved measurement (i.e. PIV) will spend the effort to take enough to get statistics. It can be done, but I can't remember the last time I've seen it. If you had data in space and time, you wouldn't need Taylor's hypothesis. You could and should calculate the spatial and temporal energy spectra directly.