# Landau global oscillator model

In this part, we will study the Landau global oscillator model with a Re =132812.

Once we have the data of the vibration position, we will use the Landau global oscillator model to analysis the physical parameters. The amplification envelopes of the signals have been treated by the Landau global oscillator model (Landau, 1944), Provansal and al (Provansal and al., 1987). Here is the Stuart-Landau equation:
\begin{equation}
\frac{dA}{dt}=\sigma A-\frac{1}{2} l |A|^2 A
\end{equation}

Where A is the complex amplitude of the transverse position in
the flow axis, which is the most significant quantity of the symmetry (varies on time especially for the transition regime); $\sigma$ is the relative growth rate; $l$ is the Landau constant.

Then we have the Landau equation :
\begin{equation}
\frac{d|A|}{dt}=\sigma_r |A|-\frac{1}{2} l_r |A|^2 A
\end{equation}
where $\sigma_r$ is the amplification rate in the linear regime and $l_r$ is the Landau constant in the non-linear regime.

# Critical reduced velocity $u^*_{critical}$

In this part, we are going to calculate the critical value of the reduced velocity. At first, we will have a diagram of $log(A)$ as a function of real time, A is the amplitude of the vibration of the cylinder. In our study, the position of the cylinder is dimensionless but the calculate time is real time. Figure 3.1 –Different regime of the a signal, example $u^*=4$

In the linear regime, the predominant term is $\sigma_r$; in the non-linear regime, the predominant term is the $l_r$. In other words, in the following study, in order to calculate the $\sigma_r$ we will analyze the signal in the linear regime; on the contrary, the signal in the non-linear regime is more interesting for us to study the Laudau constant $l_r$.

Once we have the signal data, we should at first identify the linear regime and then draw the diagram of $log(A)$ as a function of the period. We will do the same process to have the function of the fitting line for each case of reduced velocity, especially the slope of these fitting lines, amplification rate in the linear regime $\sigma_r$. In order to have the value of the critical reduced velocity, we should draw the $\sigma_r$ as a function of $u^*$, the intersection between the fitting line and $\sigma_r=0$ indicates the value of the critical reduced velocity $u^*$. Figure 3.2 – Critical reduced velocity $u^*$​

With this study, we have obtained $u^*_{critical}=3.55$ when $\sigma_r=0$

# Landau constant $l_r$

In order to calculate the Landau constant, we should at first identify the transition regime and the the saturated regime, where the non-linear term $\frac{1}{2} l_r |A|^2 A$ dominates in the Landau equation. With the diagram of $A$ as function of $T A^2$, we can have the slope of the fitting line of these points. Figure 3.3 – Landau constant $l_r$ for $u^*=3.3$

For the case $u^*=3.3$ we have obtained that the Landau constant $l_r=-5.2<0$, which indicates a sub-critical regime, since here we have $u^*=3.3<u^*_{critical}$ ($u^*_{critical}=3.55$).

# Spectral analysis

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 ).

# 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.

# CWT-Continuous wavelet transform

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 $St_K=0.23$.

# AR-Autoregressive power spectral density estimate

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 method of Proper Orthogonal Decomposition

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