## 3 Numerical Modeling and Setup  
### Turbulence and Acoustic Modeling  
In this case, the Reynolds-Averaged Navier-Stokes (RANS) equations were used for turbulence flow. The zonal-mixed turbulence model SST (Shear Stress Transport) \(k\)-\(\omega\) model [37] was applied. Its \(k\) equation and \(\omega\) equation can be expressed as follows:  
\[\frac{\partial(\rho k)}{\partial t}+\frac{\partial(\rho u_{i}k)}{\partial x_{i} }=P-\frac{\rho k^{3/2}}{l_{k\omega}}+\frac{\partial}{\partial x_{i}}\Bigg{[}( \mu+\sigma_{k}\mu_{t})\frac{\partial k}{\partial x_{i}}\Bigg{]} \tag{1}\]  
\[\frac{\partial(\rho\omega)}{\partial t}+\frac{\partial(\rho u_{i}\omega)}{ \partial x_{i}}=C_{\omega}P-\beta\rho\omega^{2}+\frac{\partial}{\partial x_{i} }\Bigg{[}(\mu_{t}+\sigma_{\omega}\mu_{t})\frac{\partial\omega}{\partial x_{i} }\Bigg{]}+2(1-F_{1})\frac{\rho\sigma_{\omega 2}}{\omega}\frac{\partial k}{ \partial x_{i}}\frac{\partial\omega}{\partial x_{i}} \tag{2}\]  
where \(t\) denotes the time, \(\rho\) the density, \(u\) the velocity, and \(x\) the coordinate. \(P\) is the production term, \(\mu\) is the dynamic viscosity, \(\mu_{t}\) is the turbulent eddy viscosity, \(\sigma\) is the model constants, \(C_{\omega}\) is the coefficient of the production term, and \(l_{k\omega}\) is the turbulence scale. \(F_{1}\) is the zonal mixture function, which is used to combine the standard \(k\)-\(\varepsilon\) model and [PERSON]'s \(k\)-\(\omega\) model.  
To improve the sensitivity of streamline curvature and system rotation, the Spalart-Shur correction [38] was also applied, accompanied by the SST \(k\)-\(\omega\) turbulence model. It adds the correction term \(f_{r1}\) as a multiplier on the production term \(P_{k}\). Moreover, \(f_{r1}\) has its specific limiters:  
\[f_{r1}=\max[0,1+C_{s}(f_{r}^{*})] \tag{3}\]  
\[f_{r}^{*}=\max[\min(f_{rot},1.25),0] \tag{4}\]  
\[f_{rot}=(1+C_{r1})\frac{2r^{*}}{1+r^{*}}\Big{[}1-C_{r3}\tan^{-1}(C_{r2}\hat{r} )\Big{]}-C_{r1} \tag{5}\]  
where \(C_{s}\) is the correction scaling factor. The constants \(C_{r1}\), \(C_{r2}\), and \(C_{r3}\) are equal to 1.0, 2.0, and 1.0. The remaining functions are defined as follows:  
\[r^{*}=\frac{S}{\Omega} \tag{6}\]  
\[\Omega=\sqrt{2\Omega_{ij}\Omega_{ij}} \tag{7}\]  
\[\hat{r}=\frac{2\Omega_{ik}S_{jk}}{\Omega B^{3}}\Bigg{[}\frac{DS_{ij}}{Dt}+ \Big{(}\varepsilon_{imm}S_{jn}+\varepsilon_{jmn}S_{in}\Big{)}\Omega_{m}^{rot} \Bigg{]} \tag{8}\]  
\[B^{2}=\max\Big{(}S^{2},0.09\omega^{2}\Big{)} \tag{9}\]  
where \(\Omega^{rot}\) is the rotation rate of the reference frame, and \(DS_{ij}/D_{t}\) is the Lagrangian derivative of the strain rate tensor.  
In this study, the Lighthill acoustic analogy method was used to analyze the near-field of noise in the pump impellers. [PERSON] [39] introduced a sound power calculation method for the flow-induced noise:  
\[W_{A}=\alpha_{\varepsilon}\rho\varepsilon M_{t}^{5} \tag{10}\]  
where \(\alpha_{\varepsilon}\) is a constant which can be valued as 0.1, \(\rho\) is the density, \(\varepsilon\) is the eddy dissipation rate, and \(M_{t}\) can be expressed as follows:  
\[M_{t}=\frac{\sqrt{2k_{2D}}}{\varepsilon} \tag{11}\]where \(c\) is the sound speed, which is 1500 m/s in water. Thus, the sound power level, \(L_{sp}\), can be calculated as follows:  
\[L_{sp}=10\,\log_{10}\left(\frac{W_{A}}{W_{ref}}\right) \tag{12}\]  
where \(W_{ref}\) is the reference sound power in water, which is about \(6.7\times 10^{-19}\) W/m\({}^{3}\).