## 5 Results and Discussion
### Time Evolution of Fluid Oscillations  
The FPPHP oscillatory behavior over time, which was qualitatively observed in scalograms of Section 5.2, was assessed by investigating some of its features, such as continuity and channel activity. Specifically, the continuity \(t_{c}\) is here defined as the persistency of fluid oscillations over time around the dominant frequency, i.e., in the range \(B_{f}\!=\!f_{d}\pm 15\%\,f_{d}\). It can be conceptually expressed, for a given device channel, as  
\[t_{c}=\frac{\text{Time during which fluid oscillates at a frequency close to the dominant one}}{\text{Total acquisition time}} \tag{5}\]  
Channels activity \(t_{a}\) is instead referred to presence of fluid oscillations:  
\[t_{a}=\frac{\text{Time during which fluid oscillations are perceeviable}}{\text{Total acquisition time}} \tag{6}\]  
Both properties are believed to complete the fluid oscillations description in terms of time evolution, providing a better quantification of the device heat transfer capabilities. The evaluation of continuity and activity is based on the time-wise determination of the main oscillation frequencies, i.e., referred to as the power peaks in scalograms. However, a reliable and unambiguous evaluation of scalogram peaks is greatly hampered by the \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(f_{d}\) [Hz] & \multicolumn{5}{c}{Q [W]} \\ \hline
**Channel** & _50_ & _100_ & _150_ & _200_ & **250** \\ \hline
1 & - & 0.66 & 0.87 & 1.07 & 1.32 \\
2 & - & 0.66 & 0.93 & 1.07 & 1.15 \\
3 & - & 0.62 & 1.87 & 1.07 & 1.32 \\
4 & - & 0.71 & 0.93 & 1 & 1.23 \\
5 & - & 0.71 & 0.93 & 1.07 & 1.32 \\
6 & - & 1.42 & 0.87 & 1 & 1.23 \\
7 & - & 1.42 & 0.87 & 1.07 & 1.23 \\
8 & - & 1.42 & 0.93 & 1 & 1.23 \\
9 & - & 0.87 & 0.81 & 1.07 & 1.23 \\
10 & - & 0.76 & 0.87 & 1.07 & 1.32 \\
11 & - & 0.66 & 0.87 & 1.07 & 1.23 \\
12 & - & 0.76 & 0.87 & 1 & 1.23 \\
13 & - & 0.71 & 1.74 & 1.07 & 1.23 \\
14 & - & 0.71 & 0.87 & 1.07 & 1.23 \\
15 & - & 0.66 & 0.87 & 1.07 & 1.32 \\
16 & - & 0.66 & 0.93 & 1.07 & 1.15 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Dominant oscillation frequencies evaluated in the horizontal orientation for every channel at varying heat load. Dashes stand for absence of dominant fluid oscillation. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multicolumn{2}{c}{_Orientation_} & \multicolumn{5}{c}{_Q [W]_} \\ \hline \multirow{2}{*}{_Vertical BHM_} & _50_ & _70_ & _90_ & _120_ & _150_ & _200_ & _250_ \\ \cline{2-8}  & \(1\) & \(1\) & \(F\) & \(F\) & \(F\) & \(F\) & \(F\) \\ \hline \multirow{2}{*}{_Horizontal_} & _50_ & _100_ & _150_ & _200_ & _250_ & & \\ \cline{2-8}  & \(1\) & \(F\) & \(F\) & \(F\) & \(F\) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Device working regimes identified in [29]. “I”: intermittent flow, “F”: full activation. presence of low-power peaks, which are not always representative of significant fluid oscillations. Hence, a procedure for scalogram denoising, aimed at discerning meaningful from neglectable power peaks, is required. In the present analysis, the \(K\)-means clustering was employed to highlight high-power regions in scalograms. In particular, such a method relies on an iterative data-partitioning algorithm that assigns \(N\) observations to \(K\) separate clusters defined by centroids [46]. \(K\) is preliminarily chosen, depending on the specific application. The observations assignment relies on within-cluster variance minimizations through the evaluation of squared Euclidean distances from centroids, progressively moved until the minimization condition is satisfied. Given a set of observations \(\mathbf{O}=\{\mathbf{O_{1}},\ \mathbf{O_{2}},\ \dots\ \mathbf{O_{n}}\}\), the partition into \(k\leq K\) subsets \(\mathbf{S}=\{\mathbf{S_{1}},\ \mathbf{S_{2}},\ \dots\ \mathbf{S_{k}}\}\) is performed according to the following minimization problem:  
\[\underset{\mathbf{S}}{argmin}\sum_{k=1}^{K}\sum_{s\in\mathbf{S_{k}}}\|s-\mu_{k}\| \tag{7}\]  
where \(\mu_{k}\) is the mean of the \(k\)-th subset.