## 2 Description of Experimental Setup
### Uncertainty Analysis  
The uncertainties \(q\) were analyzed by the method described in the Guide to the expression of Uncertainty (GUM) [33]. It can be classified into two groups. In Type \"\(A\)\", standard uncertainty is evaluated using statistical methods. In Type \"\(B\)\", standard uncertainty is evaluated by means other than statistical methods [33].  
The equation to estimate the experimental uncertainty (type \"\(A\)\" uncertainty) of a quantity \(X\), estimated from \(n\) independent observations \(X_{i}\), is represented by:  
\[q_{A}=\frac{1}{\sqrt{n}}\bigg{[}\frac{1}{n-1}\sum(X_{i}-\overline{X})^{2} \bigg{]}^{1/2}. \tag{6}\]  
When performing the cycle averaging of the data obtained over 45 successive cycles, the Type \"\(A\)\" uncertainty of the measured velocity is \(\pm 0.02\) ms\({}^{-1}\). The dynamic uncertainty measurement is \(\pm 1.85\times 10^{-3}\) bar.  
For the Type \"\(B\)\" evaluation of standard uncertainty, we used data provided in the calibration apparatus for the velocity (\(\pm 1\%\)) and the pressure (\(\pm 5\times 10^{-3}\) bar). Considering that these intervals represent a rectangular distribution between the values \([-y,+y]\), the Type \"\(B\)\" evaluation of standard uncertainty was calculated according the following equation:  
\[q_{B}=\frac{y}{\sqrt{3}} \tag{7}\]  
The combined standard uncertainty is:  
\[q_{C}=\sqrt{q_{A}^{2}+q_{B}^{2}}. \tag{8}\]  
The expanded uncertainty \(q\) is obtained by multiplying the combined standard uncertainty \(q_{c}\) by a coverage factor \(k=2\) corresponding to a 95% confidence interval:  
\[q=kq_{C}. \tag{9}\]  
Figure 6: Evolution of the fluid velocity during the compression phase (or hot blow when the hot gas enters the regenerator) and expansion phase (or cold blow when the cold gas enters the regenerator).  
The total relative percent uncertainty is computed using the following formula:  
\[\text{Total relative percent uncertainty}=\frac{\text{expended uncertainty }(q)}{\text{value of measurment}}100[\%]. \tag{10}\]  
The uncertainties of frequency \(Fr\) and porosity \(\varepsilon\) were estimated to be 0.64% and 1%, respectively. The thermocouples were calibrated under static and dynamic states to ensure the accuracy uncertainties of \(\pm 0.1\)\({}^{\circ}\)C (Lanzetta and [PERSON] [32]). The calculated total relative percent uncertainties for pressure and velocity are given in Table 1 for different structures of the regenerator with all experimental data.  
Considering that the errors are uncorrelated variables, the standard uncertainties can be calculated using Equations (11) and (12) according to the principle of uncertainty propagation.  
\[Z(r,\ldots,v,\ldots w), \tag{11}\]  
\[q_{Z}=\pm\sqrt{\left(\frac{\partial Z}{\partial r}\right)^{2}q_{r}^{2}+\ldots \left(\frac{\partial Z}{\partial v}\right)^{2}q_{v}^{2}+\ldots\left(\frac{ \partial Z}{\partial w}\right)^{2}q_{w}^{2}}, \tag{12}\]  
where \(r,\ldots,v,\ldots,w\) are measured quantities with uncertainties \(q_{r},\ldots q_{v},\ldots q_{w}\), and the measured values are used the compute the function \(Z(r,\ldots,v,\ldots w)\).  
The maximum uncertainties of the kinetic Reynolds number \(Re_{\omega}\) and the friction factor \(f\) are 1.55% and 8.9%, respectively, with a 95% confidence level.  
The relative uncertainties depend on the frequency \(Fr\) and vary from:  
1. 0.72% to 1.55% for the kinetic Reynolds number \(Re_{\omega}\);
2. 7.6% to 2.5% for the pressure drop \(\Delta P\);
3. 8.9% to 3.6% for the friction factor \(f\).