## 2 Numerical Simulation
### Governing Equations and Turbulent Model  
The CFD research in this work mainly focuses on the three-dimensional incompressible unsteady flow problems within a centrifugal pump under different flow rate conditions and is solved by applying the Reynolds-Averaged Navier-Stokes (RANS) method. The solution process is based on the continuity equation and the Navier-Stokes equation:  
\[\frac{\partial\rho}{\partial t}+\
abla\cdot\left(\rho\vec{u}\right)=0 \tag{1}\]  
\[\frac{\partial\left(\rho\vec{u}\right)}{\partial t}+\
abla\cdot\left(\rho\vec {u}\vec{u}\right)=-\
abla P+\
abla\cdot\left[\mu\left(\
abla\vec{u}+\left( \
abla\vec{u}\right)^{T}\right)\right]+\overline{f} \tag{2}\]  
where \(t\) is time (s); \(\vec{u}\) and \(p\) represent the velocity vector (m/s) and static pressure (Pa), respectively; \(\rho\) and \(\mu\) stand for density (kg/m\({}^{3}\)) and effective viscosity (kg/(m\(\cdot\)s)), respectively; and \(\overline{f}\) is external body force term (N/m\({}^{3}\)). The effective viscosity \(\mu\) in  
Figure 2: Mid-plane view of the inner fluid region of the pump.  
Equation (2) combines the molecular viscosity \(\mu\)L and turbulent eddy viscosity \(\mu_{t}\), i.e., \(\mu=\mu_{t}+\mu_{t}\). For the SST \(k\)-\(\omega\) turbulence model used in this study, the turbulent eddy viscosity \(\mu_{t}\) is calculated based on the turbulent kinetic energy \(k\) and specific dissipation rate \(\omega\), as detailed in [31].  
To close the above two major governing equations and meet the solution requirement, a suitable turbulence model needs to be supplemented. Considering that the high-speed rotating impeller with complex twisted blades inside the centrifugal pump generates a strong fluid shear effect and has a high Reynolds number, this study selects the SST \(k\)-\(\omega\) turbulence model proposed by [PERSON] [31]. This turbulence model combines the advantages of both the \(k\)-\(\varepsilon\) model and the \(k\)-\(\omega\) model. In the calculation of turbulent viscosity, it takes into account the transport of turbulent shear stress, improving the prediction accuracy for separated flows and flows with adverse pressure gradients. Therefore, it has been widely applied in predicting the internal flow field of centrifugal pumps [28; 32; 33].  
The core idea of the SST \(k\)-\(\omega\) model lies in using a blending function to achieve the transition from the high Reynolds number turbulent core region far from the wall to the low Reynolds number region near the wall:  
\[\varphi= F_{1}\varphi_{1}+(1-F_{1})\varphi_{2} \tag{3}\]  
where \(F_{1}\) is the factor of function and \(\varphi\) is the coefficient of the SST \(k\)-\(\omega\) turbulence model, while \(\varphi_{1}\), and \(\varphi_{2}\) stand for coefficients of the \(k\)-\(\omega\) turbulence model and \(k\)-\(\varepsilon\) turbulence model, respectively.  
For the region far away from the wall, the flow turbulence is solved solely by the \(k\)-\(\varepsilon\) turbulence model, and the factor \(F_{1}\) is set to 0. Conversely, in the region close to the wall, the factor \(F_{1}\) is set as 1. This is to more accurately consider the influence of viscous forces on turbulence, thereby more precisely capturing the development and dissipation phenomena of turbulence near the boundary layer.