## 3 Local Heat Flux Estimation Procedure
By employing a sensitivity analysis on Equation (9), the \(s\left(\rho c_{p}\frac{\partial T}{\partial t}-k\frac{\partial^{2}T}{\partial y ^{2}}-k\frac{\partial^{2}T}{\partial z^{2}}\right)\) term was assessed to be three orders of magnitude higher than the \(h_{env}(T-T_{env})\) term, especially during the device operation. In fact, the onset of fluid motion guarantees a lower wall temperature of the whole adiabatic section, coupled with higher time-space temperature variations, drastically diminishing the relative weight of outer free convection in terms of \(q\) estimation. Equation (9) can be numerically solved by approximating derivatives of temperature with finite differences. However, since noise is intrinsically present in raw experimental temperatures, Equation (9) gives unstable results, especially due to the presence of derivative operators [30]. A convenient solution to such an issue is represented by the filtering technique, a so-called regularization method employed for noise removal in inverse problems inputs [31; 32; 33; 34]. In particular, such a regularization technique relies on the application of a Gaussian filter for high-frequency component suppression in raw data, thus fulfilling the same function of the regularization term in the Tikhonov method [35]. Nevertheless, when compared to other approaches, the filtering technique represents an effective and computationally efficient regularization method, especially when the studied physics is described by a large domain with a high number of unknown terms [36; 37; 38; 39]. For wall temperature samples discretized into M spatial steps \(\Delta y\), N spatial steps \(\Delta z\) and O time increments \(\Delta t\), the discrete Fourier transform of the measured M \(\times\) N \(\times\) O temperature distribution T results in:  
\[\mathcal{F}(T)=\widetilde{T}(u,v,w)=\sum_{g=0}^{M-1}\sum_{h=0}^{N-1}\sum_{i=0 }^{O-1}T(g,h,i)e^{-j2\pi\frac{\Delta t}{\Delta t}}se^{  
where \(u_{c}\) and \(u_{w}\) are the cut-off frequencies referred to space and time, respectively. In particular, two different cut-off frequencies were introduced since the physics of the problem could significantly vary from space to time in terms of frequency response. Moreover, a single cut-off frequency in space was considered since both spatial coordinates were believed to present similar signal-to-noise ratio due to the behavior of the IR camera sensor in space. Last, the Fourier image is converted back to the time-space domain:  
\[\mathcal{F}^{-1}\Big{(}\widetilde{T_{f}}\Big{)}=T_{f}(m,n,o)=\frac{1}{M\cdot N \cdot O}\sum_{k=0}^{M-1}\sum_{l=0}^{N-1}\sum_{p=0}^{O-1}\widetilde{T_{f}}(k,l, p)e^{j2\pi\frac{k}{M}m}e^{j2\pi\frac{l}{M}n}e^{j2\pi}\bar{b}^{o} \tag{13}\]  
However, the optimal cut-off frequencies to be given in Equation (12) for a good trade-off between smoothness of the filtered data and loose of meaningful frequencies are not known a priori in real cases, leading to the need of selecting a criterion for the choice of such regularization parameters. In the present analysis, the discrepancy principle [35] was implemented. According to this principle, the inverse problem solution can be considered satisfactorily correct if the deviation between the measured and filtered temperatures is equal to the standard deviation of the raw measurements [39; 40]. In other terms, the condition to be satisfied is the following:  
\[\frac{\parallel T_{f}-T\parallel_{2}^{2}}{N\cdot M\cdot O}\cong\sigma_{T} \tag{14}\]  
where \(\parallel\parallel_{2}\) stands for the 2-norm, \(N\cdot M\cdot O\) is the overall size of \(T\) and \(\sigma_{T}\) is the time-space standard deviation of the raw data, estimated by measuring the wall temperature distribution in isothermal conditions. Specifically, Equation (14) is represented by the following minimization problem:  
\[ \begin{array}{c}\underset{u_{c}\,\in\,\,R^{+}}{\text{min}}\,fun(u_{c},w_{c}) ;\,fun(u_{c},w_{c})=\frac{\parallel T_{fu_{c},w_{c}}-T\parallel_{2}^{2}}{N \cdot M\cdot O}-\sigma_{T}\\ w_{c}\,\in\,\,R^{+}\end{array}