# FFT approach estimate

As there are several methods of FFT approach method, we will at first identify which method works better for our study. Here we will talk about two methods, Periodogram PSD estimate and Welch PSD estimate. With several calculations, we have found that, periodogram PSD estimate works well for the low frequency signal; Welch PSD estimate has a good estimation for the high frequency signal. So in our study, we will fix a critical valueof the frequency $f_{cri}$ ( $St_{cri}$ ), if $f<f_{cri}$ ( $St<St_{cri}$ ), we will use the periodogram PSD estimate; if $f>f_{cri}$ ( $St>St_{cri}$ ), we will use the Welch PSD estimate. In the following charts, we can clearly identify the Von Kármán vortex shedding which corresponds to a maximum value of the PSD ( the first peak in the diagram ). Based on the study of M.Elhimer (2014), the experimental Kelvin Helhomltz instability frequency value is around 1800 Hz, on which we will have a discuss in detail in the following part.

Figure 3.5 – Variation of PSD as a function of St of monitor ponit 1 on the first cylinder on 2D

In the chart, we find that the Strouhal number equals to 0.23 which matches well our previous calculate ( $St_K=0.23$ ).

Figure 3.6 – Variation of PSD as a function of frequency of monitor point 1 on the first cylinder on 2D

In this chart we find that the Von Kármán vortex frequency $f_K=174.22 Hz$ and we have as well marked the experimental Kelvin Helhomltz instability frequency value which is $f_{SL}=1800 Hz$.

In these two pictures, the black lines represent the final PSD estimate : equals to green line when $f<f_{cri}$ for the low frequency signal; equals to red lines when $f>f_{cri}$ for the high frequency signal. In both pictures, the results of the Von Kármán shedding matches well with our previous study, we have marked the experimental values for the Kelvin Helhomltz instability  with $f=1800 Hz$ and $St=2.3$. We have also performed this study for the other monitor points and also for the 2nd cylinder, the results are almost the same.