## 4 Results and Discussion
### Verification of Angular and Axial Dependences
For a more practical application, we consider the dependence of \(A_{\text{obs}}\) on Reynolds number, expressed as follows:  
\[Re=\frac{C_{D}}{\mu}DP_{0}\sqrt{\frac{2\gamma}{\gamma-1}\frac{M}{RT_{1}}\left[ \left(\frac{P_{a}}{P_{0}}\right)^{\frac{2}{\gamma}}-\left(\frac{P_{a}}{P_{0}} \right)^{\frac{\gamma+1}{\gamma}}\right]\left(\frac{2}{\gamma+1}\right)^{\frac {2+1}{\gamma-1}}}\left(P_{0}<P_{cr}\right), \tag{54}\]  
\[Re=\frac{C_{D}}{\mu}DP_{0}\sqrt{\frac{\gamma M}{RT_{1}}\left(\frac{2}{\gamma+1 }\right)^{\frac{\gamma+1}{\gamma-1}}}\left(P_{0}>P_{cr}\right). \tag{55}\]  
Figure 3f shows the plot of the \(A_{\text{o,ang}}\) values observed at 90\({}^{\circ}\) versus the Reynolds number (Re). With increasing Re up to 3 \(\times\) 10\({}^{4}\), the value of \(A_{\text{o,ang}}\) increases very gradually. Above Re = 3 \(\times\) 10\({}^{4}\), the value of \(A_{\text{o,ang}}\) increases rapidly with increasing Re. The observed \(A_{\text{o,ang}}\) values are fitted by an exponential growth \(A_{\text{o,ang}}(Re)=A_{\text{o}}^{Re}e^{Re/\tau}+A_{0}\). As shown  
Figure 3: (**a–e**) Observed mean maximum amplitudes (circles) and \(\sigma\)s (bars) obtained from five different hole diameters: (**a**) D0.2n10, (**b**) D0.3n10, (**c**) D0.5n10, (**d**) D0.8n10 and (**e**) D1.2n10 at \(P_{0}\) = 2 bar (black), 3 bar (red) and 4 bar (green), simulated amplitudes (lines) and (**f**) plot of mean amplitudes (circle) at \(\vartheta=90^{\circ}\) vs. Reynolds number and the fit (line) by one-phase exponential growth. in Figure 3f, the observed values are in good agreement with the values calculated by the equation with \(A_{\rm o}^{Re}=5.776\times 10^{-4}\), \(\tau=5277\) and \(A_{0}=0.1216\) (\(R^{2}=0.99553\)). We then expanded the angular dependence of \(A_{\rm obs}\) to four different \(\eta\) positions from \(0^{\circ}\) to \(180^{\circ}\) for D0.2. As can be seen in Figure 4, all \(A_{\rm obs}\) values agree well with their corresponding \(A_{\rm sim}\) values. Next, the axial dependence of the AE amplitude was studied as a function of \(\eta\) at \(\theta=180^{\circ}\) using five different LSs. Ten sensors were mounted on the cylinder surface at \(\theta=180^{\circ}\): eight sensors were mounted downwards, and two sensors were mounted upwards relative to the position of the LS. Figure 5 shows the \(A_{\rm obs}\) values observed at \(\theta=180^{\circ}\) from five different LSs with D0.2-D1.2 and three different internal pressures. Within the standard deviation, all results show that the \(A_{\rm obs}\) values decrease exponentially with increasing \(\eta\). As in the case of the angular dependence, simulations of \(A_{\rm obs}\) can be performed by normalizing the calculated displacement values and then matching them to the observed relative amplitude values in the region of \(-0.2\ \leq\eta/\rm{m}\leq 0.8\). The simulated angular dependence of the amplitude is formulated as follows:  
\[A_{\rm sim}(\eta)=u_{\rm N}(\eta)\times(A_{\rm obs,max}-A_{\rm o,axi})+A_{\rm o,axi}, \tag{56}\]  
where \(A_{\rm obs,max}\) and \(A_{\rm o,axi}\) are the estimated values of \(A_{\rm obs}\) from extrapolation to \(\eta=0\) and \(\eta/\rm{m}=0.8\), respectively. It should be noted that the normalized displacement, \(u_{\rm N}(\eta)=u_{\rm leak}(\eta)/u_{\rm leak}(0)\), is invariant to FRS and frequency. As shown in Figure 5a-e, except the case for D1.2P3 and D1.2P4, all experimental values are in good agreement with the simulated values within the standard deviation. For D1.2P3 and D1.2P4, there is a rather large deviation between \(A_{\rm sim}\) and \(A_{\rm obs}\). Figure 5f shows the plot of the \(A_{\rm 0,axi}\) values versus Re. Similar to the case of the angular dependence, the value of \(A_{\rm 0,axi}\) increases very gradually with increasing Re up to \(3\times 10^{4}\).