## 2 Numerical Simulation and Experimental Validation
### Numerical Simulation  
The computational domain is divided into the impeller rotating domain, the volute domain, the inlet domain, and the outlet domain, as shown in Figure 3a. The interaction between the rotating impeller and the stationary guide structures is modeled using the Multi-Reference Frame (MRF) method, with the rotating and stationary domains connected via an interface. Different flow conditions are simulated by controlling the impeller rotational speed, with the inlet and outlet boundaries set as pressure-inlet and pressure-outlet, respectively. The turbulence intensity at the inlet boundary is 5%, with a hydraulic diameter of 0.281 m, corresponding to a turbulence viscosity of 10. The steady-state flow field results after 5000 steps are used as initial conditions for the transient simulations. The steady-state solution is obtained using the SIMPLE algorithm, while the transient solution is obtained using the PISO algorithm. The time step for transient numerical simulations is selected based on the Courant-Friedrichs-Lewy (CFL) condition, ensuring that CFL \(<\) 1 within the computational domain. The time step is set to correspond to the time required for the impeller to rotate by 1\({}^{\circ}\), and the residual convergence criterion is set to 1.0 \(\times\) 10\({}^{-5}\). The pressure term is discretized using a second-order scheme, while the momentum, turbulence kinetic energy, and turbulence dissipation rate are discretized using a second-order upwind scheme. The impeller is meshed with structured grids, while other components are meshed using unstructured grids with ICEM (a commercial grid generation software, version number 2022R1), as shown in Figure 3b. Local mesh refinement is applied to complex flow passage components, such as the volute. One disadvantage of the eddy viscosity model is its insensitivity to the curvature of streamlines and system rotation [31]. However, rotational turbulence is inevitable in large, high-speed, highly-curved centrifugal impellers. The presence of rotational turbulence not only diminishes the accuracy of predictions made by two-equation models, but also tends to prematurely estimate the location of flow separation. Considering the limited capability of traditional eddy viscosity models in predicting curvature effects, this study enhances numerical accuracy by modifying the eddy viscosity coefficient in the SST k-\(\omega\) model to  
Figure 3: Schematic of the fluid domain and impeller mesh structure: (**a**) computational domain schematic, (**b**) structured mesh of the impeller. account for curvature effects in near-wall turbulence, thereby improving the accuracy of vortex structure identification. The modified eddy viscosity coefficient \(v_{t}\) is expressed as [32]:  
\[v_{t}=\frac{C_{\mu}^{*}}{C_{\mu}}\frac{k}{\omega} \tag{1}\]  
\[C_{\mu}^{*}=C_{\mu}\bigg{(}\alpha_{1}(|\eta_{3}|-\eta_{3})+\sqrt{1-\min(\alpha_ {2}\eta_{3},0.99)}\bigg{)}^{-1} \tag{2}\]  
In the expression, \(\eta_{1}=(Sk/\epsilon)^{2}/2\), \(\eta_{2}=(\Omega k/\epsilon)^{2}/2\), \(\eta_{3}=\eta_{1}-\eta_{2}\), \(\alpha_{1}=0.055\), \(\alpha_{2}=0.5\), \(C_{\mu}\) is the model constant, \(\Omega\) is the vorticity, \(\epsilon\) is the dissipation rate, \(k\) is the turbulent kinetic energy, and \(\omega\) is the specific dissipation rate. The modified turbulence model is validated by selecting a 180\({}^{\circ}\) U-shaped channel flow to assess its ability to capture the flow characteristics over curved walls and its applicability. The computational domain is shown in Figure 4, which includes a 10H-long inflow section, a 180\({}^{\circ}\) bend, and a 12H-long outflow section. The grid is arranged with 151 \(\times\) 211 nodes. For computational convenience, a layer of mesh is stretched normal to the surface. Both the inner and outer solid walls are treated as no-slip boundaries. The inlet is specified with a uniform inflow (Normal Velocity, 10.07 m/s), while the outlet is set with a pressure outlet (Relative Pressure, 0 Pa).