## 2 Materials and Methods
### Solution Strategy and Grid Independence  
In the present study, 2D axi-symmetric models of Venturi tubes were simulated using a pressure-based solver. The working fluids considered were water and vapor, and the surrounding environment was maintained at 25 \({}^{\circ}\)C. The specific boundary conditions used for calculations in this study are listed in Table 2. For pressure solution, the PRESTO scheme was selected, and the first-order upwind discretization scheme was employed to discretize turbulent kinetic energy and turbulent dissipation rates. The momentum and vapor were discretized using the second-order upwind discretization scheme and QUICK scheme, respectively. Pressure-velocity coupling was carried out using the SIMPLE algorithm. To ensure convergence, the consistency of mass flow rates at the inlet and outlet was checked, and a convergence criterion of residual of the continuity equation below \(10^{-5}\) was set. All CFD simulations were performed under steady-state conditions.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Cases**} & \multirow{2}{*}{
\begin{tabular}{c} **P\({}_{\text{in}}\)** \\ **(MPa)** \\ \end{tabular} } & \multirow{2}{*}{\(\rm\sigma\)} & \(\bf L_{C}\) & \(\bf D_{in}\) & \(\bf L_{C}/D_{in}\) & \(\bf L_{d}\) & \(\bf D_{out}\) & \(\bf L_{d}/D_{out}\) \\  & & & **(mm)** & **(mm)** & **(mm)** & **(mm)** & **(mm)** & **(mm)** \\ \hline
1 & & 0.2595 & 48 & 50.00 & 0.960 & 140 & 39.42 & 3.55 \\
2 & & 0.2603 & 48 & 50.  
Accurate simulations in computational fluid dynamics (CFDs) require careful consideration of the mesh size used. Therefore, conducting a sensitivity analysis is crucial to evaluate the impact of varying grid sizes on the accuracy and consistency of simulation results. As shown in Figure 2, ANSYS ICEM was employed to generate a structural mesh and apply local refinement to create half of two-dimensional Venturi geometric model, given its axial symmetry characteristics. Additionally, initial grid layers were positioned within the turbulent core region to ensure that most wall-adjacent grids had Y+ values exceeding 30. The heights of the first layer of grids were set to 0.1 mm, 0.08 mm, and 0.05 mm for model circumstances of 0.4 MPa, 0.8 MPa, and 1.2 MPa, respectively. The Y+ distribution of the Venturi tube is depicted in Figure 3 across three distinct pressure conditions. It is evident that the Y+ values for all pressure conditions predominantly reside within a reasonable range of 30 to 300 (suitable for standard wall functions). For the Venturi geometric model with a divergence segment length of 190 mm, four different grid sizes were generated with respective cell counts of 21,064, 41,584, 80,804, and 121,211. Figure 4 displays the absolute pressure profile along the Venturi wall for four distinct mesh sizes. It is evident that no significant deviation in the pressure distribution occurs upon further grid refinement beyond a certain point when using 80,804 grid cells. Therefore, it can be asserted that a grid containing 80,804 cells is sufficient to conduct the present simulation. The identical mesh generation approach was also utilized for both the model parameters of 140 mm and 240 mm divergence section lengths, resulting in mesh element counts of 74,632 and 87,484, respectively.  
\begin{table}
\begin{tabular}{c c c} \hline
**Section** & **Boundary Conditions** & **Remark** \\ \hline Working fluid & - & Primary phase: water \\  & & Second phase: vapor \\ Inlet & Pressure inlet & Table 1 (0.4–1.2 MPa) \\ Outlet & Pressure outlet & 0.1 MPa \\ Wall & No slip condition & The wall treatment method: standard \\  & & wall functions \\ Turbulence & Intensity and hydraulic & Intensity: 3\% \\  & diameter & Hydraulic diameter: 50 mm \\ \hline \end{tabular}
\end{table}
Table 2: Boundary conditions applied to the calculation.  
Figure 2: The refinement of the Venturi tube throat.