# Wind Reversal in Bubbly Natural Convection
## 3 Results
### Wind Reversal: Detection and Analysis  
The bulk dynamics is characterized by the angular orientation \(\theta_{c}\) of the large-scale circulation (LSC) plane and by the amplitude of the temperature fluctuation \(\delta\) in the azimuthal direction. Typically, \(\delta\) is a measure of the magnitude of the LSC. Taking advantage of the data accessibility provided by the direct numerical simulation, the flow dynamics were explored. Several numerical probes are placed within the fluid volume, allowing point-wise velocity and temperature time series to be extracted. The eight probes (Figure 3) are located along the azimuthal direction every \(\pi/4\) rad starting from \(\theta_{0}=0\) rad on the horizontal plane midway between the two plates (cold and hot) at a radial location \(r/H=0.125\) (where, \(r\) the radial coordinate and \(H\) the height of the cylindrical domain). Figure 3: Probe location in the horizontal plane midway between the lower and upper plates. Figure 2: Bubbles location in the domain for three time instants: (**a**) time = 500—phenomenon beginning; (**b**) time = 520—before the reversal; (**c**) time = 530—after the reversal. The bubbles size is considered at each time. According to [PERSON] et al. [30], we fitted the adimensional probe temperature \(\psi_{i}\) i = 0,...7, separately at each time, using the following function:  
\[\psi_{i}^{*}=\psi_{m}+\delta cos(i\pi/4-\theta_{c}) \tag{9}\]  
where \(\psi_{m}=1/8\sum_{i=0}^{7}\psi_{i}\) is the average temperature of the eight probes. The \(\delta\) and \(\theta_{c}\) are estimated by enforcing the minimum of the square error function \(Er_{\psi}=\sum_{i=0}^{7}(\psi_{i}^{*}-\psi_{i})^{2}\). In order to avoid the non-linearity of the error function due to the cosine of \(\theta_{c}\) (coming from Equation (9)), that equation is rewritten as follows. \[\psi_{i}^{*}=\psi_{m}+A\cdot sin(i\pi/4)+B\cdot cos(i\pi/4);\hskip 14.226378 pt \delta=\sqrt{A^{2}+B^{2}},\hskip 14.226378 pt\theta_{c}=arctg\left(\frac{A}{B}\right) \tag{10}\]  
\[\left\{ \begin{array}{l}Er_{\psi}=\sum_{i=0}^{7}(\psi_{i}^{*}-\psi_{i})^{2} \\ \\ \frac{\partial Er_{\psi}}{\partial A}=0\\ \\ \frac{\partial Er_{\psi}}{\partial B}=0\end{array} \right.\hskip 14.226378 pt \Longrightarrow\hskip 14.226378 pt\left\{ \begin{array}{l}A=\frac{1}{4}\sum_{i=0}^{7} \psi_{i}sin\left(i\frac{\pi\pi}{4}\right)\\ \\ \\ B=\frac{1}{4}\sum_{i=0}^{7}\psi_{i}cos\left(i\frac{\pi}{4}\right)\end{array} \right. \tag{11}\]  
Figure 4 shows the \(\psi_{i}\) and the corresponding fit \(\psi_{i}^{*}\) at three time instants with Figure 4a corresponding to the beginning of the phenomenon. The parameters \(\psi_{m}\), \(\delta\) and \(\theta_{c}\) have been listed in Table 1.  
The fitted temperature \(\psi_{i}^{*}\) shows a good agreement with the sampled temperature \(\psi_{i}\) sampled; moreover, we observe that \(\delta\) and \(\theta_{c}\) are strongly dependent on time. The reorientation of the LSC was observed in turbulent convection but its dynamics is still only partially understood. An interesting aspect of the LSC observed in literature is its spontaneous changing \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Dimensionless Time** & \(\psi_{m}\) & \(\delta\) & \(\theta_{c}\) \\ \hline
500.0 & \(5.019\times 10^{-1}\) & \(1.037\times 10^{-1}\) & 6.141 \\ \hline
521.8 & \(4.547\times 10^{-1}\) & \(5.404\times 10^{-2}\) & 2.942 \\ \hline
530.0 & \(4.960\times 10^{-1}\) & \(6.910\times 10^{-3}\) & 4.124 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters values of the fit function \(\psi_{i}^{*}\) at three time instants. The time (T) is adimensionalized by means of the free fall velocity and the height of the cylindrical domain. Figure 4: Dots indicates the mean temperature of eight probes, \(\psi\), solid line refers to temperature fitted by Equation (9).