## 2 Numerical Setup
### Numerical Methods  
The incompressible flow of the impeller is governed by equations of conservation of mass and momentum:  
\[\frac{\partial u_{i}}{\partial x_{i}}=0 \tag{1}\]  
\[\frac{\partial u_{i}}{\partial t}+u_{j}\frac{\partial u_{i}}{\partial x_{j}}= f_{i}-\frac{1}{\rho}\frac{\partial P^{*}}{\partial x_{i}}+\
u\frac{\partial^{2}u_{i}} {\partial x_{j}\partial x_{j}} \tag{2}\]  
where \(x_{i}\) is the direction component of the Cartesian coordinate system; \(u_{i}\) is the velocity component; \(P^{*}\) is the pressure considering the conversion of turbulent kinetic energy and centrifugal force; \(f_{i}\) is the component of the volume force including the centrifugal force and Coriolis force; and \(\
u\) is the kinematic viscosity coefficient.  
The standard \(k\)-\(\varepsilon\) turbulence model is employed to simulate the turbulent flow in the centrifugal impeller. The transport equations of the turbulent kinetic energy \(k\) and turbulent dissipation rate \(\varepsilon\) are as follows:  
Figure 3: Enlarged view of the structured mesh.  
Figure 2: Geometry of the computation domain.  
\[\frac{\partial k}{\partial t}+\frac{\partial}{\partial x_{i}}(ku_{i})=\frac{1}{\rho} \frac{\partial}{\partial x_{j}}\Bigg{[}(\mu+\frac{\mu_{t}}{\sigma_{k}})\frac{ \partial k}{\partial x_{j}}\Bigg{]}+G_{k}+G_{b}-\rho\varepsilon-Y_{M}+S_{k} \tag{3}\]  
\[\frac{\partial\varepsilon}{\partial t}+\frac{\partial}{\partial x_{i}}( \varepsilon u_{i})=\frac{1}{\rho}\frac{\partial}{\partial x_{j}}\Bigg{[}(\mu+ \frac{\mu_{t}}{\sigma_{\varepsilon}})\frac{\partial\varepsilon}{\partial x_{j }}\Bigg{]}+C_{1\varepsilon}\frac{\varepsilon}{k}(G_{k}+C_{3\delta}G_{b})-C_{2 \varepsilon}\rho\frac{\varepsilon^{2}}{k}+S_{\varepsilon} \tag{4}\]  
in which \(G_{k}\) represents the turbulent kinetic energy term caused by the laminar velocity gradient; \(G_{b}\) is the turbulent kinetic energy term produced by buoyancy; \(Y_{M}\) represents the contribution term of the turbulent pulsation expansion in the compressible flow to the global flow to the dissipation rate; \(C_{1}\), \(C_{2}\), and \(C_{3}\) are constants; \(\partial_{k}\) and \(\sigma_{\varepsilon}\) and are the turbulent Prandtl numbers of the \(k\)-equation and \(\varepsilon\)-equation; and \(S_{k}\) and \(S_{\varepsilon}\) are user-defined turbulent kinetic energy terms and turbulent dissipation source terms. According to the recommended values [25], the values of \(C_{1\varepsilon}\), \(C_{2\varepsilon}\), \(C_{\mu}\), \(\sigma_{k}\), and \(\sigma_{\varepsilon}\) are fixed at 1.44, 1.92, 0.09, 1.0, and 1.3, respectively. Since the flow is incompressible, the source terms \(G_{b}\), \(Y_{M}\), \(S_{k}\), and \(S_{\varepsilon}\) are set as zero.  
The centrifugal impeller works at an off-designed flow rate \(Q\) = 0.76L/s, which is set as the mass flow rate condition at the inflow boundary. The zero-gauge pressure condition is set at the outflow boundary. The no-slip condition is set for velocity on all solid walls.  
The numerical simulation was performed using ANSYS-Fluent 16.0. The governing equations are discretized in space by second-order central difference scheme and discretized in time by a second-order implicit scheme. The velocity and pressure decoupling use the SIMPLEC algorithm. The convergence residual is set to \(10^{-4}\), and the maximum iteration is 15 times in each time step. In order to lower computational time and resource request, a steady-state simulation was first carried out using the same boundary conditions and computational parameters, and the converged results with residual smaller than \(10^{-4}\) were used to initialize the unsteady simulation. In the unsteady simulation, 360 physical time steps were set as the impeller rotates for one revolution; thus, the physical time step size is \(2.3\times 10^{-4}\) seconds. We first run the unsteady simulation for 1.656 seconds, corresponding to 20 revolutions of the impeller, and then continue for one more revolution, which produces data for the following analysis and discussion.