## 2 Materials and Methods
### Numerical Model  
The geometric model of the wide-range centrifugal pump impeller used in this study is illustrated in Figure 1. Key design parameters include a single-stage head \(\geq\)4.5 m, a flow rate range of 0-200 m\({}^{3}\)/d, a rated rotational speed of 3000 r/min, an impeller outer diameter of 72.85 mm (matching a guide vane outer diameter of 88.9 mm), a shaft diameter of 17.4 mm, an outlet width of 6.29 mm, 6 blades, and a minimum blade thickness of 3 mm at the outlet. The computational domain, constructed from the 3D solid model, was divided into rotating impeller and stationary guide vane subdomains (Figure 2), coupled through interface boundaries for fluid-field interaction. The numerical simulation was conducted using the commercial CFD software ANSYS Fluent 2022 R1. The governing equations encompass mass conservation, momentum conservation, and energy conservation. Given that this study focuses on the hydraulic performance of ESP with stable medium temperature, thermal diffusion and conduction effects were neglected during computation. The fluid model was simplified to an adiabatic, incompressible, viscous Newtonian fluid. Continuity Equation (Mass Conservation Equation):  
\[\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}+\frac{\partial w}{ \partial z}=0 \tag{1}\]  
Figure 1: Wide-range electric submersible centrifugal pump: **(a)** impeller; **(b)** guide vane. Figure 2: Computational domain: **(a)** global domain; **(b)** impeller subdomain; **(c)** guide vane subdomain. where \(u,v,\) and w represent the velocity components in the \(x,y,\) and \(z\) directions, respectively. Momentum Equations (Navier-Stokes Equations):  
\[ \begin{array}{l}\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v \frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}=F_{x}-\frac{1}{ \rho}\frac{\partial\eta}{\partial x}+\frac{\mu}{\rho}\
abla^{2}u\\ \frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v }{\partial y}+w\frac{\partial v}{\partial z}=F_{y}-\frac{1}{\rho}\frac{ \partial\eta}{\partial y}+\frac{\mu}{\rho}\
abla^{2}v\\ \frac{\partial w}{\partial t}+u\frac{\partial w}{\partial x}+v\frac{\partial v }{\partial y}+w\frac{\partial v}{\partial z}=F_{z}-\frac{1}{\rho}\frac{ \partial\eta}{\partial z}+\frac{1}{\rho}\
abla^{2}w\end{array}  \tag{2}\]  
where \(\rho\) denotes the density, \(p\) denotes the pressure on the fluid element, \(\mu\) represents the dynamic viscosity, \(F_{x},F_{y},\) and \(F_{z}\) represent the body forces acting on the element. When gravity is the sole body force acting numerically downward along the \(z\)-axis, \(F_{x}=0,F_{y}=0,\)\(F_{z}=\rho g\). Given the focus on near-wall flow characteristics affecting frictional resistance, the standard k-\(\omega\) turbulence model was selected for numerical simulation. This model demonstrates high accuracy under complex pressure gradients and effectively resolves near-wall flow features and boundary layer separation. Turbulent viscosity was calculated using the correlation equations for turbulent kinetic energy k and specific dissipation rate \(\omega\) (Equation (3)), where the transport equations for k (Equation (4)) and \(\omega\) (Equation (5)) characterize turbulent energy distribution and dissipation, respectively:  
\[\mu_{t}=\rho\frac{k}{\omega} \tag{3}\]  
\[