## 5 Results and Discussion
### Quantitative Description of the Local Heat Flux Variations  
To achieve a quantitative and repeatable description of channel-wise heat flux variations, two coefficients of variation, namely \(cvt\) (over time) and \(cvs\) (along space), were introduced and evaluated by means of Equations (25) and (26). Due to the high number of available data provided by the method of Section 3, such quantities were adopted to statistically reduce the wall-to-fluid heat flux distributions, as suggested in [27] and [46] for different types of PHPs. In Equations (25) and (26), the \(std\) and \(mean\) operators are the standard deviation and the arithmetic mean, respectively, \(N\) is the total number of considered axial coordinates (140), \(M\) are the considered time instants (1000) and \(q_{n}\) is the wall-to-fluid heat flux distribution related to the \(n\)-th channel.  
\[cvt\ n=\frac{\sum_{i=1}^{N}\left[\frac{std(|q_{n}(i+1,\ldots,M)|)}{mean(|q_{n }(i+1,\ldots,M)|)}\right]}{N} \tag{25}\]  
\[cvs\ n=\frac{\sum_{t=1}^{M}\left[\frac{std(|q_{n}(i=1,\ldots,M)|)}{mean(|q_{n }(i=1,\ldots,M)|)}\right]}{M} \tag{26}\]  
The use of the average in Equations (25) and (26) is due to the fact that, from a statistical point of view, a specific time-space coordinate, i.e., a single pair of \(i\) and \(t\), is not representative of the actual variation of the time-space heat flux distributions. In particular, the statistical coefficient \(cvt_{n}\) describes the irregularity of heat flux variations in each FPPHP channel over time: for elevated \(cvt_{n}\) values, the fluid oscillations are regarded as strongly irregular, denoting the presence of stopovers during the observation window. On the contrary, a low \(cvt_{n}\) denotes fluid oscillations which are substantially regular and persistent in terms of amplitude over time. To provide an order of magnitude for the  
Figure 11: Wall-to-fluid heat flux averaged over time, as a function of the axial coordinate for three meaningful channels (horizontal orientation, \(Q=100\) W).  
regularity of thermofluidic interactions quantified by \(cvt_{n}\), sine-shaped functions (highest regularity) present coefficient of variation equal to about 50%, whereas constant functions are characterized by coefficient of variation of 0%. Other complementary remarks can be obtained by \(cvs_{n}\): high values of such a quantity denote great variations of the heat flux from end to end of the channel, while low values suggest that the entire channel is simultaneously interested by the same heat transfer phenomenon. To give a better thermal description of the overall device, the average and the standard deviation of \(cvt_{n}\) and \(cvs_{n}\), denoted by the subscript \(av\) and \(std\), respectively, were computed by considering all the analyzed FPPHP channels \(num\) as follows:  
\[cv_{av}=\frac{\sum_{n=1}^{num}cv_{n}}{num} \tag{27}\]  
\[cv_{std}=\sqrt{\frac{\sum_{n=1}^{num}(cv_{av}-cv_{n})^{2}}{num}} \tag{28}\]  
where \(cv\) is the generic coefficient of variation (either over time or along space). The coefficient of Equation (27) describes the average heat flux variation either over time (\(cvt_{av}\)) or along space (\(cvs_{n}\)) in the entire adiabatic section of the device. The coefficient of Equation (28) quantifies instead time (\(cvt_{std}\)) and space (\(cvs_{std}\)) variations of the estimated heat flux from channel to channel. It has to be stressed that these parameters offer a critical evaluation of the goodness of numerical codes, since they represent the real statistical behavior of the FPPHP.