## 2 Materials and Methods
### Numerical Methods  
To explore the effect of the flow disturbance caused by the transducer recesses on the performance of the simultaneous verification of multi-UFMs arranged in tandem, we consider five cases with different numbers of tested UFMs ranging from 1 to 5 (\(1\leq n\leq 5\)), the geometries and configurations of \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Straight Length Upstream** & **Straight Length Downstream** \\ \hline
**Master Meter** & 25D & 5D \\
**Checking Meter** & 25D & 5D \\
**Test Meter** & 150D & 20D \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Straight pipe configuration in the working standard facility. Figure 3: Schematic of the working standard facility for verification of the tested UFMs. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{1}{c}{**DN250**} & **1-UFM1** & **2-UFM1** & **2-UFM2** & **3-UFM1** & **3-UFM2** & **3-UFM3** \\ \hline
**CH4 (wt.\%)** & 94.7567 & 94.7637 & 94.7546 & 94.7589 & 94.7546 & 94.7560 \\ N2 (wt.\%) & 1.2633 & 1.2604 & 1.2596 & 1.2600 & 1.2596 & 1.2609 \\ CO2 (wt.\%) & 0.7658 & 0.7493 & 0.7599 & 0.7533 & 0.7599 & 0.7644 \\ C2H6 (wt.\%) & 2.8918 & 2.9065 & 2.9039 & 2.9084 & 2.9039 & 2.8970 \\ C3H8 (wt.\%) & 0.1964 & 0.1949 & 0.1960 & 0.1943 & 0.1960 & 0.1963 \\ P (MPa) & 6.5 & 6.5 & 6.5 & 6.5 & 6.5 & 6.5 \\ T (K) & 293 & 293 & 293 & 293 & 293 & 293 \\ \(\rho\) (kg/m\({}^{3}\)) & 51.278 & 51.268 & 51.277 & 51.272 & 51.277 & 51.279 \\ \(\mu\) (Pa-s) & 1.2583 & 1.2582 & 1.2583 & 1.2583 & 1.2583 & 1.2583 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Physical and chemical properties of the pressurized natural gas under test. which were carefully measured and modeled according to the actual UFMs. In addition, in order to deepen the understanding of the influence of the change of flow field on SVUFMST, we also consider three cases with same type but different diameters (DN150). Figure 2 depicts a wetted UFM Daniel 3400 (DN250 and 150) studied in this paper, which has four paths and eight transducer recesses along the paths in different configurations. The simulations were carried out in natural gas using 50D and 10D straight pipes upstream and downstream of the tested UFMs, respectively. The natural gas is set as an incompressible mixed-flow medium according to the experimental conditions. The 50D straight pipe upstream was determined by conducting numerical verifications based on the previous work supposed by [PERSON] et al. [14] to achieve the fully developed flow at the 1 st tested UFM inlet. The constant axial flow velocity U\({}_{0}\) (for DN250) ranged from 0.88 m/s to 29.22 m/s and the fully developed outflow were used as the inlet and outlet boundary conditions, respectively, in our CFD simulations. In this study, the relatively high pressure in the pipe and flowrate led to a very large Re in the range of 106-107 and, hence, only turbulent flow needs to be considered. In consideration of the high accuracy of the tested UFMs (1.0% full scale), it is very important to obtain reasonable calculation accuracy for examining the SVUFMST's performance. Thus, we use 3D CAD software to build the model according to the actual geometric measurement results of the tested flowmeter. Then, we use advanced polyhedral mesh technology to render the mesh and refine it in both near-transducer-recesses regions and flowmeter bodies. Because considering the high Reynolds number environment in which the inertial force dominates the flow in the tube while the viscous force affects the near-wall and near-transducer-recesses flow, we also add boundary layer grids in these regions with a height of 0.0001D of the first layer of elements and an exponential growth row (growth factor: 1.2, rows: 10). Figure 4 depicts the grid of a single UFM and a detailed profile of the recesses. This mesh improves the prediction accuracy of the velocity profiles, especially the steep velocity gradients on the near-wall region and negative velocity near transducer-recesses region.