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 $u^*$‚Äč

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 $u^*=3.3<u^*_{critical}$ ($u^*_{critical}=3.55$).