Calculating energy/power spectra.

rr3245 · July 4, 2017, 12:55

Hi all,

I am trying to validate my OpenFOAM CFD code for flow past a circular cylinder. This includes calculating the power spectra of velocity at fixed points in the cylinder wake, and hopefully identifying Kolmogorov's -5/3 slope. I have read up on the technique required using Pope's book- Turbulent Flows Ch.6 (2000), however I am still unsure of how I go about implementation. Pope describes a two-point autocorrelation, but my sample points are fixed in position behind the cylinder, so I therefore don't understand how this might apply.

From my understanding there are two possible routes-
(1) Calculate the autocorrelation function $R_{ij}$ , and then take the Fourier transform $E_{ij}(\kappa) \equiv \frac{1}{\pi} \int^\infty_{-\infty} R_{ij}(\textbf{e}_1r_1) e^{-i\kappa_{1} r_{1}} dr_1$ . I have a fixed sampling point, so I am not sure how I would use this, unless I utilise Taylor's frozen hypothesis??
(2) Take the FFT of the individual velocity components (I am interested in the streamwise direction), then use $(1/2) 4\pi k^2|u(k)|^2$ .

Method (2) seems more straightforward, but I have only seen this in a forum post, not in a textbook. Does anyone have a reference for this in order to get a better understanding? Apologies for the scattered post, I am getting very confused by the whole thing. Regards, RH.

FMDenaro · July 4, 2017, 17:59

I am not sure to understand your point, you have some probes and you sample in time the values of the velocity? Or do you want to analyze the spectral content along the spanwise direction (that I suppose is periodic)?

rr3245 · July 5, 2017, 05:28

Quote:

Originally Posted by FMDenaro

I am not sure to understand your point, you have some probes and you sample in time the values of the velocity? Or do you want to analyze the spectral content along the spanwise direction (that I suppose is periodic)?

Thanks for your reply FMDenaro. Apologies if my question is scattered , I'll try to clarify.

In the literature, probes are placed across the spanwise direction (in the

-direction). These are then assumed to be independent, and averaged. Lets assume I only place one probe in the wake and I record all three components of velocity. I want to analyse the velocity time series and calculate the power spectra of the streamwise (or normal) velocity. This is to confirm my plots display the -5/3 law, and that there are spikes where the shedding frequency is present.

FMDenaro · July 5, 2017, 07:06

Well, I suggest to do the FFT of the three velocity components at the chosed position. However, you need a correct windowing technique as your signal is not necessarily periodic in time. Then, consider that depending on the position far from the cylinder along the wake, you can or not recover the theoretical inertial slope. Furthermore, the theoretical slope should be the same, interchanging in space and time, requiring further assumption about ergodicity.
Finally, I would suggest to do the spectral analysis also in space, you can do the FFT along z and for several positions along the streamwise direction

rr3245 · July 5, 2017, 09:42

Quote:

Originally Posted by FMDenaro

Well, I suggest to do the FFT of the three velocity components at the chosed position. However, you need a correct windowing technique as your signal is not necessarily periodic in time. Then, consider that depending on the position far from the cylinder along the wake, you can or not recover the theoretical inertial slope. Furthermore, the theoretical slope should be the same, interchanging in space and time, requiring further assumption about ergodicity.
Finally, I would suggest to do the spectral analysis also in space, you can do the FFT along z and for several positions along the streamwise direction

This is what I've decided to try (from Chapter 3.6 of Pope (2000)):

I have the 3 component velocities time series at one point, which I use to calculate the fluctuating velocity

. The aurocovariance can be found $R(s) \equiv \langle u(t)u(t+s) \rangle$ , which will require varying

. It is then possible to calculate the frequency spectrum $E(\omega)$

$E(\omega) = \frac{1}{\pi} \int_{-\infty}^{\infty} R(s) e^{-i\omega s} ds.$

i.e. $E(\omega) = 2 \mathcal{F}\{R(s)\}$ , where $\mathcal{F}\{\}$ is the Fourier transform.

Does this look OK?

ari003 · September 29, 2020, 12:53

Quote:

Originally Posted by rr3245

This is what I've decided to try (from Chapter 3.6 of Pope (2000)):

I have the 3 component velocities time series at one point, which I use to calculate the fluctuating velocity

. The aurocovariance can be found $R(s) \equiv \langle u(t)u(t+s) \rangle$ , which will require varying

. It is then possible to calculate the frequency spectrum $E(\omega)$

$E(\omega) = \frac{1}{\pi} \int_{-\infty}^{\infty} R(s) e^{-i\omega s} ds.$

i.e. $E(\omega) = 2 \mathcal{F}\{R(s)\}$ , where $\mathcal{F}\{\}$ is the Fourier transform.

Does this look OK?

Hello Sir, I m trying to get the autocorrelation from a series of time-velocity at a fixed point using python. I have searched a lot but the solutions were not satisfactory. Could you please share the procedure how did you find the auto-correlation and plotted it?

July 4, 2017, 12:55	Calculating energy/power spectra.	#1
rr3245 New Member anonymous Join Date: Nov 2016 Posts: 29 Rep Power: 9	Hi all, I am trying to validate my OpenFOAM CFD code for flow past a circular cylinder. This includes calculating the power spectra of velocity at fixed points in the cylinder wake, and hopefully identifying Kolmogorov's -5/3 slope. I have read up on the technique required using Pope's book- Turbulent Flows Ch.6 (2000), however I am still unsure of how I go about implementation. Pope describes a two-point autocorrelation, but my sample points are fixed in position behind the cylinder, so I therefore don't understand how this might apply. From my understanding there are two possible routes- (1) Calculate the autocorrelation function $R_{ij}$ , and then take the Fourier transform $E_{ij}(\kappa) \equiv \frac{1}{\pi} \int^\infty_{-\infty} R_{ij}(\textbf{e}_1r_1) e^{-i\kappa_{1} r_{1}} dr_1$ . I have a fixed sampling point, so I am not sure how I would use this, unless I utilise Taylor's frozen hypothesis?? (2) Take the FFT of the individual velocity components (I am interested in the streamwise direction), then use $(1/2) 4\pi k^2\|u(k)\|^2$ . Method (2) seems more straightforward, but I have only seen this in a forum post, not in a textbook. Does anyone have a reference for this in order to get a better understanding? Apologies for the scattered post, I am getting very confused by the whole thing. Regards, RH.

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July 4, 2017, 17:59		#2
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,775 Rep Power: 71	I am not sure to understand your point, you have some probes and you sample in time the values of the velocity? Or do you want to analyze the spectral content along the spanwise direction (that I suppose is periodic)?

July 5, 2017, 07:06		#4
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,775 Rep Power: 71	Well, I suggest to do the FFT of the three velocity components at the chosed position. However, you need a correct windowing technique as your signal is not necessarily periodic in time. Then, consider that depending on the position far from the cylinder along the wake, you can or not recover the theoretical inertial slope. Furthermore, the theoretical slope should be the same, interchanging in space and time, requiring further assumption about ergodicity. Finally, I would suggest to do the spectral analysis also in space, you can do the FFT along z and for several positions along the streamwise direction