## 2 Numerical Simulation
### Experimental Validation  
As shown in Figure 5, referring to the level 1 accuracy class in standard ISO 9906-2012 [34], the external characteristic curves of the centrifugal pump are tested on a closed-loop test rig. The test system mainly consists of two parts: a closed-loop fluid circuit composed of a centrifugal pump, a water tank, inlet and outlet valves, and several pipes; and a signal measurement and acquisition facility composed of pressure gauges, electromagnetic flowmeter, and data acquisition card, etc. During the experiment, the water temperature is maintained at 20 \(\pm\) 1 \({}^{\circ}\)C, and the rotational speed of the centrifugal pump is kept at 2900 \(\pm\) 5 r/min using a frequency converter. The inlet valve is always fully open, and the outlet valve opening is adjusted to ensure the stable operation of the device under different flow conditions. An electromagnetic flowmeter (Model: CKLDG-D50-X, accuracy: \(\pm\)0.2%) is used for flow measurement, and the pressure gauge for pressure measurement has an accuracy of \(\pm\)0.1%. The head under each flow condition is calculated from the readings of two pressure gauges located at the inlet and outlet of the centrifugal pump:  
\[H=\frac{P_{2}-P_{1}}{\rho g}+\frac{{V_{2}}^{2}-{V_{1}}^{2}}{2g}+Z \tag{4}\]where \(P_{2}\) and \(P_{1}\) are the pressure at the outlet and inlet of the pump, respectively; \(V_{2}\) and \(V_{1}\) are the velocity at the outlet and inlet of the pump, respectively; \(\rho\) is density of water; \(g\) is gravitational acceleration; and \(Z\) is the height difference between the pressure gauge installation positions at the inlet and outlet pipe sections. The values of \(V_{out}\) and \(V_{in}\) are obtained by dividing the measured flow rate by the cross-sectional area of the outlet and inlet, respectively. Furthermore, the efficiency under each flow condition is calculated based on the above calculated head and shaft power:  
\[\eta=\frac{\rho gQ_{v}H}{P_{S}} \tag{5}\]  
where \(H\), \(Q_{v}\), and \(P_{s}\) are the head, flowrate, and shaft power of the pump, respectively. The shaft power is measured by a torque meter with a measurement accuracy of \(\pm\)0.1% and a frequency converter. The systematic error of the experiment is controlled within 0.15%. Through an uncertainty analysis, the uncertainties of the head and efficiency are determined to be 0.21% and 0.38%, respectively. Finally, the numerical results and experimental measurement results under each flow condition are compared, as shown in Figure 6. In Figure 6, the flow rate and head values are dimensionless, converted into the flow coefficient \(\varphi\) and the head coefficient \(\Psi\), respectively. The experimental measurements plotted with error bars in Figure 6 represent the total pump efficiency obtained through physical testing. It should be noted that while the CFD simulations specifically predict hydraulic efficiency \(\eta_{h}\), the experimentally measured total efficiency \(\eta\) inherently combines three components: \(\eta=\eta_{h}\times\eta_{v}\times\eta_{m}\), where \(\eta_{v}\) denotes volumetric efficiency, and \(\eta_{m}\) mechanical efficiency. To enable direct comparisons between simulation predictions and experimental data, the estimated hydraulic efficiency values are derived from measurements by applying the following relationship: \(\eta_{he}=\eta/(\eta_{v}\times\eta_{m})\). This combined volumetric-mechanical efficiency factor (\(\eta_{v}\times\eta_{m}\)) is established at 80% through a systematic evaluation of the manufacturer-provided engineering datasets, which integrate decades of operational performance records from equivalent pump configurations across diverse industrial applications. The value reflects validated correlations derived from accumulated field test data rather than theoretical assumptions, adhering to industry-standard efficiency characterization protocols that account for typical wear patterns and manufacturing tolerances observed in production hardware. Figure 5: Test rig for hydraulic performance curves.