## 2 Numerical Analysis Method
### Check-Valve Shape and Analysis Conditions
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Conditions** \\ \hline Turbulence model & Realizable \(\mathrm{k}-\epsilon\) \\ Real gas equation & SRK EOS \\ Inlet pressure (MPa) & 70 \\ Differential pressure (MPa) & 2, 4, 6, 8, 10 \\ \(\mathrm{H_{2}}\) temperature (K) & 233, 293, 363 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Analysis conditions. \[C_{P}=\frac{p-p_{\infty}}{\frac{T}{2}p_{\infty}M^{2}} \tag{4}\]  
In Equation (4), P is the static pressure at the computation site, P\({}_{\infty}\) is the pressure in the free flow, and M is the Mach number in the far field [19]. In order to analyze the changes in the flow and turbulence characteristics inside the check valve in response to changes in the inlet temperature and outlet pressure of hydrogen, a measurement position was selected in the X-axis direction of the check valve, as shown in Figure 1b. Measurement positions at 5 mm intervals were selected based on the core part, where the flow changes significantly due to the decrease in flow path area, and the results were analyzed from the point after x = 65 mm to the check-valve outlet, where flow characteristics change significantly due to the change in flow path area. In addition, the average pressure and velocity results were measured according to the X-axis direction measurement position to analyze the overall flow characteristic changes in the core part and the discharging part, where local flow changes occurred more and less. To analyze the flow-separation result in the L-shaped curved tube area, the line measurement position in the Y-axis direction was selected to compare the U-direction velocity result with the pressure coefficient result. If the mesh geometry in the numerical analysis is dense or rough, it will affect the reliability of the analysis results, so it is necessary to configure the appropriate mesh elements and geometry [20]. Therefore, in order to analyze the effect of mesh formation conditions on the results, a mesh dependency validation was performed, as shown in Figure 2a. The mesh dependency was performed at the check-valve hydrogen-inlet temperature (T\({}_{\text{in}}\)) of 233 K and outlet pressure (P\({}_{\text{out}}\)) of 60 MPa. The flow velocity results at the y = 6.8 mm position in the y-axis direction of the core part area, which is expected to be affected by the number of mesh elements due to the decrease in the flow area, are compared and analyzed. As shown in Figure 2a, the error of the obtained results tends to stabilize gradually as the number of mesh elements increases. As the number of mesh elements increases from about 9 \(\times\) 10\({}^{6}\) to 1.1 \(\times\) 10\({}^{7}\), the results affected by the number of mesh elements show a difference of up to about 5%, but when the mesh condition changes after 1.4 \(\times\) 10\({}^{7}\), the error rate of the results is within 1%. Therefore, a mesh element number of 1.4 \(\times\) 10\({}^{7}\) was selected and applied to the numerical analysis because the mesh element number and shape have a small effect on the results, satisfy the grid dependency, and have the advantage of analysis time. In addition, the mesh geometry applied in the numerical simulation consisted of a tetrahedral lattice system, 12 prism layers were created to simulate the flow at the check-valve wall, and mesh conditions with Y+ ranging from 30 to 300 were applied. The results are shown in Figure 2b. Figure 2: _Cont._