# Sample Time Length

Up to Velocimeters - single pointI think this is a relatively simple question that could get very complicated.

What are some good methods for determing the time to take a representative sample in a turbulent flow? Convergence of averages, is this alright to use by itself? Any way of using the power spectrum? Can any conclusions be made with respect to having an adequate sample time if found to follow the 5/3 rule (i.e. captures all time scales for the inertial sub-range)?

Thanks in advance.

Cheers,

Matt.

It depends on your situation. If you have a flow that is steady in the mean, then you want to take averages that are long compared to the longest time scales of your flow. So if you have a boundary layer, the largest scales will be of the order of the boundary layer height, h, and you can use a characteristic velocity, U, for this calculation as the free stream velocity. From those you can get a characteristic time scale by T = L/U. Then take averages over t >> T. The longer the better. Note that T is a characteristic time scale of your flow and you want stationary statistics that are a sample of the larger physical process that you are describing. Thus you want to capture as many of the long time scale events as possible and practical for the experiment time constraints and data analysis.

For wind tunnel boundary layer studies, we took something like 2 minute averages for LDV data with a free stream velocity of 27.5 m/s and a bondary layer thickness of about 0.10 m at the test section. Trowbridge (1998) took 18 min averages with a LDV and in Trowbridge (1999) they took 3.28 min averages.

For more info, look in a turbulence text for stationarity. Most will have it in about chapter 3 where they discuss the averaging of time series. Or find another refernece that has a similar experiment as yours and use that as a benchmark.

If your measurements are not Eulerian (at a fixed location) or in an unsteady flow at scales larger than your process of interest... then I hope that someone else offers insight as I am not familiar yet with the appropriate selection of averaging time for these periods.

I have not looked at this with regards to the power spectra. The -5/3 slope is an indication that the energy in the turbulence is being cascaded from the larger scales to smaller scales. This is the inertial subrange. I am 90% certain (without checking) that in this region the production of turbulent energy = the dissipation of turbulent kinetic energy. The other terms in the governing equations are just there to redistribute the vorticity to smaller scales in the other directions. So you should see the -5/3 slope when you take data in this physical scenario. (See Mike Gregg's paper from 1987 from JGR I think, a review with a nice plot of the spectra that comes through the scales of internal waves, to anisotropic turbulence, to inertial subrange, to isotropic turbulence, to dissipation of the turbulent energy.) I think that if you take longer averages and uyou are measuring scales of turbulence that follow this physical scenario, then you are only reducing the noise (scatter) and increasing the frequency resolution in your spectral estimate by taking longer time averages. Off the top of my head, I would guess that the stationarity of the data should be checked in the time domain. If you want to check, take data at some length. Then take a few more time series of data. If the means and standard deviations are similar and the histograms look similar, then you are probably taking long enough time series. Longer time series will reduce the standard deviation due to the white noise... bringing the noise towards the noise level seen at the high frequencies in your spectra. So the variance left in your signal becomes more representative of the turbulent fluctuations.

Yes, it could get long. The short answer is that the texts says to take samples over times long compares to the longest time scale of interest, but not so long as to make the data unstationary compared to larger variations in the flow.

Hope that helps some, sorry it si so long.

Dave

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I forgot to attach some papers. Here.

I am also interested to hear what others in the community have to say on the topic of averaging. It is rather important when you can only get short samples.

(guess I can only attach one paper) The other is Trowbridge et. al. Near bottom turbulence... J. Phys. Ocean. 1999 vol 29 p 3056.

Dave

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