## 2 Materials and Methods
### Numerical Model
\begin{array}{l}\frac{\partial\rho k}{\partial t}+\frac{\partial\rho uk}{ \partial x}+\frac{\partial\rho vk}{\partial y}+\frac{\partial\rho uk}{ \partial z}=\frac{\partial}{\partial x}\left[\left(\mu+\frac{\mu_{t}}{\sigma_ {k}}\right)\frac{\partial k}{\partial x}\right]+\frac{\partial}{\partial y} \left[\left(\mu+\frac{\mu_{t}}{\sigma_{k}}\right)\frac{\partial k}{\partial y }\right]+\\ \frac{\partial}{\partial z}\left[\left(\mu+\frac{\mu_{t}}{\sigma_{k}}\right) \frac{\partial k}{\partial z}\right]+\mu_{t}-C_{\mu}\rho k\omega\\ \frac{\partial\omega\omega}{\partial t}+\frac{\partial\mu u\omega}{\partial x }+\frac{\partial\rho v\omega}{\partial y}+\frac{\partial\rho u\omega}{ \partial z}=\frac{\partial}{\partial x}\left[\left(\mu+\frac{\mu_{t}}{\sigma_ {k}}\right)\frac{\partial k}{\partial x}\right]+\frac{\partial}{\partial y} \left[\left(\mu+\frac{\mu_{t}}{\sigma_{k}}\right)\frac{\partial k}{\partial y }\right]+\\ \frac{\partial}{\partial x}\left[\left(\mu+\frac{\mu_{t}}{\sigma_{k}}\right) \frac{\partial\omega}{\partial z}\right]+\gamma\rho\Phi-\beta^{\prime}\rho \omega^{2}\end{array}  \tag{4}\]  
where \(\mu_{t}\) represents the turbulent viscosity coefficient, \(k\) represents turbulent kinetic energy, \(\omega\) represents specific dissipation rate, \(\mu\) represents the dynamic viscosity coefficient, \(\rho\) represents fluid density, \(\sigma_{k}=2.0,\)\(\sigma_{\omega}=2.0,\)\(\gamma=0.556,\)\(\beta^{\prime}=0.075\)[27]. Turbulent flow, characterized by stochastic perturbations and pulsating characteristics, exhibits complex irregularities distinct from the ordered motion of laminar flow. This inherent complexity precludes rigorous mathematical derivation of turbulent phenomena. In engineering practice, fluid motion is therefore represented by time-averaged mean values over infinitesimal intervals. Steady-state simulations consequently employ Reynolds-averaged treatment--assuming fully developed flow and statistically steady conditions--which simplifies governing equations while aligning with the research objective of analyzing fixed operating points in engineering optimization. For centrifugal pump steady-state simulations, rotor-stator interface selection is critical due to its direct impact on computational accuracy and efficiency. This study adopts the Multiple Reference Frame (MRF) method (also known as the frozen rotor approach) to model the interface between impeller and solute core regions. Compared to alternative methods, this approach offers significantly reduced computational expense. All pump walls were assigned no-slip and no-penetration boundary conditions, ensuring zero mass, momentum, and energy fluxes across the solid boundaries. The axis of rotation, which corresponds to the spin axis of the impeller fluid domain, is aligned with the impeller centerline; consequently, it exhibits zero relative velocity within the rotating reference frame. Pressure-velocity coupling was resolved using the coupled pressure-based SIMPLE algorithm [28]. For spatial discretization, convective terms employed the first-order upwind scheme to ensure stability under high-Reynolds-number flow conditions,while viscous terms utilized the second-order central differencing scheme, the standard approach for diffusion terms. Temporal discretization employed a steady-state implicit formulation, thereby omitting transient terms. To address the complex curved geometry of the impeller, an unstructured tetrahedral mesh was employed for discretization using Ansys Meshing. Local grid refinement was implemented in the blade region to accurately capture the effects of geometric discontinuities on flow characteristics. The resulting mesh configuration is illustrated in Figure 3.  
Figure 4 displays the monitored head and efficiency results under the design operating condition (flow rate: 100 m\({}^{3}\)/h) with varying mesh densities. The results demonstrate that upon reaching 1.6 million total elements, variations in both head and efficiency diminish significantly and stabilize.