## 2 Methodology
### CFD Modeling
#### 2.3.2 Mathematical Model
The equation for the turbulence kinetic energy is  
\[\frac{\partial k}{\partial t}+U_{j}\frac{\partial k}{\partial x_{j}}=P_{k}- \beta^{*}k\omega+\frac{\partial}{\partial x_{j}}\left[(\
u+\sigma_{\omega}v_{ T})\frac{\partial k}{\partial x_{j}}\right]. \tag{33}\]  
The specific dissipation rate is  
\[\frac{\partial\omega}{\partial t}+U_{j}\frac{\partial\omega}{\partial x_{j}}= \alpha S^{2}-\beta\omega^{2}+\frac{\partial}{\partial x_{j}}\left[(\
u+\sigma _{\omega}v_{T})\frac{\partial\omega}{\partial x_{j}}\right]+2(1-F_{1})\sigma _{\omega 2}\frac{1}{\omega}\frac{\partial k}{\partial x_{i}}\frac{\partial \omega}{\partial x_{i}}. \tag{34}\]  
The closure coefficients and auxiliary relations [31; 32] are  
\[F_{2}=\tan\h\left[\left[\max\left(\frac{2\sqrt{k}}{\beta^{*}\omega y^{*}}, \frac{500\
u}{y^{2}\omega}\right)\right]^{2}\right] \tag{35}\]  
\[P_{k}=\min\left(\tau_{ij}\frac{\partial U_{i}}{\partial x_{j}},10\beta^{*}k \omega\right) \tag{36}\]  
\[F_{2}=\tan\h\left\{\left\{\min\left[\max\left(\frac{\sqrt{k}}{\beta^{*}\omega y ^{*}},\frac{500\
u}{y^{2}\omega}\right),\frac{4\sigma_{\omega 2}k}{CD_{k \omega}y^{2}}\right]\right\}^{4}\right\} \tag{37}\]  
\[\frac{\partial\rho U_{i}}{\partial t}+\frac{\partial}{\partial x_{j}}\big{(} \rho U_{i}U_{j}\big{)}=-\frac{\partial p^{\prime}}{\partial x_{i}}+\frac{ \partial}{\partial x_{j}}\Bigg{[}\mu_{eff}\Bigg{(}\frac{\partial U_{i}}{ \partial x_{j}}+\frac{\partial U_{j}}{\partial x_{i}}\Bigg{)}\Bigg{]}+S_{M}, \tag{29}\]0  
\[\frac{\partial\rho U_{i}}{\partial t}+\frac{\partial}{\partial x_{j}}\big{(} \rho U_{i}U_{j}\big{)}=-\frac{\partial p^{\prime}}{\partial x_{i}}+\frac{ \partial}{\partial x_{j}}\Bigg{[}\mu_{eff}\Bigg{(}\frac{\partial U_{i}}{ \partial x_{j}}+\frac{\partial U_{j}}{\partial x_{i}}\Bigg{)}\Bigg{]}+S_{M}, \tag{29}\]1  
The constants are  
\[ \begin{array}{c}\alpha_{1}=\frac{5}{9},\ \alpha_{2}=0.44,\ \beta_{1}=\frac{3}{40},\ \beta_{2}=0.0828,\ \beta^{*}=\frac{9}{100}\\ \sigma_{k1}=0.85,\ \sigma_{k2}=1,\ \sigma_{w1}=0.5,\ \sigma_{w2}=0.856.\end{array}  \tag{40}\]  
In the context of the shear stress transport (SST) model, the \(\alpha_{1}\), \(\beta_{1}\), \(\sigma_{k1}\), and \(\sigma_{w1}\) constants are specifically tailored for the \(k\)-\(\omega\) model, which is effective near the walls. These constants play a key role in determining the turbulent viscosity and the overall behavior of the turbulent flow, especially in regions close to the wall where the effects of turbulence are most pronounced. On the other hand, the \(\alpha_{2}\), \(\beta_{2}\), \(\sigma_{k2}\), and \(\sigma_{w2}\) constants are used for the \(k\)-\(\varepsilon\) model, which is more suitable for modeling the bulk of the flow away from the boundaries. These parameters help to accurately represent the turbulent kinetic energy and its dissipation, which contributes to a more complete understanding of the flow behavior under various conditions. The SST model effectively combines the strengths of both the \(k\)-\(\omega\) and \(k\)-\(\varepsilon\) models by using the mentioned constants to provide a more accurate representation of the flow, especially in predicting flow separation and other critical phenomena in centrifugal pumps. The following simplifying assumptions were made when modeling the flow in a centrifugal pump using ANSYS CFX [33]:Stationary flow: the fluid flow is considered stationary, i.e., the time variations in the velocity and pressure are not taken into account, which simplifies the solution of the problem and reduces the computational costs. * Constant fluid properties: the parameters such as viscosity and density are assumed to be constant throughout the flow, eliminating complex thermodynamic processes and simplifying the calculations. * Incompressible fluid: the fluid is assumed to be incompressible, which eliminates the effect of density changes on the flow and simplifies the solution of the Navier-Stokes equations.