## 2 Analytical Model  
### Fluctuating Reynolds Stress as Point Source  
First, let us define the force \(\mathbf{f}\) generated by gas leakage. When a leak occurs through a small hole, turbulence is generated as the fluid upstream of the pinhole accelerates toward the hole and can be continuously affected by real physical boundaries such as solid walls. In fact, turbulent fluctuations, known as FRS, contribute to the mean motion of the fluid, meaning that FRS is actually an additional momentum flux Landau (1940). If the momentum in the \(x_{i}\) direction crosses a unit surface area in the \(x_{j}\) direction, the net flux of \(x_{i}\)-momentum with a negative sign can be expressed as Landau (1940)  
\[T^{\prime}_{ij}=-\rho v^{\prime}_{i}v^{\prime}_{j}, \tag{1}\]  
where the primes represent fluctuating states and \(v\) is the velocity along a given direction. In fact, the FRS in Equation (1) is for instantaneous fluctuations. Therefore, it must be averaged over a period of time, as follows  
\[\overline{T^{\prime}_{ij}}=-\rho\overline{v^{\prime}_{i}v^{\prime}_{j}}. \tag{2}\]The FRS on the hole wall is the most effective source of AE generated by leakage. For turbulent flow with a radial velocity of \(v_{r}\) through a circular hole, FRS occurs along the vertical axes (axial and tangential axes), as shown in Figure 1a. The CF generated by the gas leaking through the pinhole can be written as follows  
\[P_{z}=-\rho\left.\frac{\partial\left(\overline{v_{r}v_{z}^{\prime}}\right)}{ \partial x_{z}}\right|_{\text{wall}}\text{ and }P_{\theta}=-\rho\left.\frac{\partial\left( \overline{v_{r}v_{\theta}^{\prime}}\right)}{\partial x_{\theta}}\right|_{ \text{wall}}, \tag{3}\]  
where \(P_{z}=P_{\theta}\equiv P_{\text{w}}\). In this study, we determined the FRS from the reported date [20]. Figure 1 (in Appendix B) shows the plot of the Reynolds stress quantity (RSQ) vs. the wall-distance divided by layer distance (\(\delta\)). In the figure, RSQ is defined as [20]  
\[RSQ=-20\frac{\overline{u^{\prime}v^{\prime}}}{U^{2}}. \tag{4}\]  
where \(\overline{u^{\prime}v^{\prime}}\) is assumed to be the mean value of the product of \(v_{r}^{\prime}v_{w}^{\prime}\), and \(U\) is the mean velocity of \(\dot{v}_{r}^{\prime}\). The experimental data is well fitted by a fifth-order polynomial  
\[RSQ=0.028-0.013\left(\frac{y}{\delta}\right)+0.013\left(\frac{y}{\delta} \right)^{2}-0.155\left(\frac{y}{\delta}\right)^{3}+0.193\left(\frac{y}{\delta} \right)^{4}-0.065\left(\frac{y}{\delta}\right)^{5}\]  
Substituting RSQ into Equation (3), we obtained  
\[P_{w}=6.62\times 10^{-4}\rho U^{2}. \tag{5}\]  
According to the mathematical model of hole leakage [21; 22; 23], the characteristics of gas flow, such as mass flow rate (\(Q\)) and mean flow velocity in the leak hole, can be divided into sonic flow and subsonic flow according to the critical pressure ratio (CPR)  
\[CPR=\frac{p_{a}}{p_{cr}}=\left(\frac{2}{\gamma+1}\right)^{\frac{\gamma}{\gamma -1}}, \tag{6}\]  
where \(p_{a}\) is the atmosphere pressure of the surrounding environment, \(p_{cr}\) is the critical pressure when the gas in the leakage section changes from subsonic to sonic flow, and \(\gamma\) is  
Figure 1: (**a**) Two forms of the PS vector along the axial and tangential directions caused by a pinhole leakage, and (**b**) a radial cross-section containing the PS and a given point S, in which the red line represents the arc connecting PS and S.  
the isentropic index. When the gas pressure in the gas container (\(p_{0}\)) is lower than \(p_{cr}\), the gas flow is in subsonic flow state, and the mass flow rate is given by  
\[Q=C_{D}A_{lh}P_{0}\sqrt{\frac{2\gamma}{\gamma-1}\frac{M}{ZRT_{2}}\left[\left( \frac{P_{a}}{P_{0}}\right)^{\frac{2}{\gamma}}-\left(\frac{P_{a}}{P_{0}}\right)^ {\frac{\gamma+1}{\gamma}}\right]}, \tag{7}\]  
where \(C_{D}\) is the flow correction factor of the leakage hole (0.6\(\sim\)1.0), \(A_{lh}\) is the cross-sectional area of the hole, and \(Z\) is the compressibility factor of the gas. When \(p_{0}\geq p_{cr}\), the gas flow is in a sonic flow state.