In this part we will analysis the Strouhal number and the kelvin Helmholtz instability by the frequency. We have defined 8 monitor points in order to calculate the evolution of the physical parameters on the defined points ( especially the pressure).

Here we have chosen a horizontal line, since the wake moves periodically and alternatively, so the horizontal line is the average position of the wake.

*Figure 3.4 – Position of the monitor points*

In the TABLE 3.1. are the coordinates of the eight monitor points. The block and the coordinate (k, i and j) are defined in the mesh reference of the NSMB.

monitor point | block | k | i | j |

1 | 13 | 1 | 83 | 12 |

2 | 13 | 1 | 64 | 26 |

3 | 13 | 1 | 55 | 30 |

4 | 13 | 1 | 45 | 36 |

5 | 13 | 1 | 39 | 41 |

6 | 5 | 1 | 6 | 22 |

7 | 5 | 1 | 11 | 24 |

8 | 5 | 1 | 20 | 28 |

*Table 3.1 – Coordinates of the monitor points*

Then we will use the FFT approach to observe the variation of the PSD as a function of the St ( dimensionless value of the frequency ).

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.

In mathematics, a continuous wavelet transform (CWT) is used to divide a continuous-time function into wavelets. Unlike Fourier transform, the continuous wavelet transform possesses the ability to construct a time-frequency representation of a signal that offers very good time and frequency localization.

*Figure 3.7 – Wavelet transform of the 4th monitor point on the 1st cylinder on 2D*

In this picture, in order to observe the Kármán vortex frequency, which corresponds to a $St=0.23$ ( 514 points in a period ), we have use a defined scales from 450 to 600. We have found exactly the same value of the $St_K=0.23$.

Compared to the FFT approach, AR estimate has a better stability for short segments of signal, a better spectral resolution and a better resolution as function of time. We will present the results on the 4th monitor point in the following part.

*Figure 3.8 – AR transform of the 4th monitor point on the 1st cylinder on 2D*

Here we have observed the $St_K=0.23$ which corresponds well to our previous study. But in this case, we can't observe the phenomena of shear layer vortex. So in the following part, we will at first filter the low frequency signal ( for example, $Fc=2.2$ for the strouhal number ), then we excuter a AR transform.

*Figure 3.9 – High-pass filtered signal of the variation pressure*

Here we define the $Fc=2.20$ in order to observe the Strouhal number corresponding to the Kelvin Helhomltz instability.

*Figure 3.10 – AR transform of the 4th monitor point on the 1st cylinder on 2D with a high-pass filter Fc=2.2*

With a high-pass filter, we can observe the Kelvin Helhomltz instability phenomena which corresponds to a $St=2.3$.

The Proper Orthogonal Decomposition (POD) is a post-processing technique. It is a multi-variate statistical method that aims at obtaining a compact representation of the data. This method may serve two purposes, namely order reduction by projecting high-dimensional data into a lower-dimensional space and feature extraction by revealing relevant, but unexpected, structure hidden in the data.

It takes a given set of data and extracts basis functions, that contain as much “energy” as possible. By taking first modes of POD, we can modelize almost all the problem because they have all the energy.

Let $ q_{k}(x), k = 1 ... N_t $ be a set of observations (called snapshots) at point x of the domain that could be obtained by a PIV experimental measurements or by numerical simulation. The goal of POD is to find functions $\Phi$ such that :

$$ \frac{<\mid(q,\Phi)\mid^{2}>}{\mid\mid\Phi\mid\mid^{2}} $$

is maximized. $<.>$ is an average, $\mid\mid.\mid\mid$ the induced norm. Solving the optimization problem leads to an eigenvalue problem, where the functions $\Phi$ are the eigenfunctions.

In our problem, the resolution gives us the eigenfunctions and by taking first modes, we can modelize the velocity :

*Figure 3.11 - First mode for velocity corresponds to the mean (numerical results)*

*Figure 3.12 – POD first modes*

*Figure 3.13 – POD reconstruction*