## 2 Materials and Methods
### Theoretical Method  
The simulation results yielded a maximum Reynolds number scale of \(10^{9}\), corresponding to turbulent flow conditions. The k-\(\omega\) shear stress transport (SST) turbulence model equation is strongly applicable for numerically calculating turbulent flow situations under rotational conditions. For instance, both [PERSON] et al. (2016) and [PERSON] et al. (2017) independently applied the k-\(\omega\) SST turbulence model to simulate electric motors and a hydro turbine, respectively, and they substantiated these simulations using experimental data. Compared with the RNG k-\(\varepsilon\) turbulence model, the SST k-\(\omega\) turbulence model more accurately captures small eddies within the viscous sublayer ([PERSON], 2015). Therefore, we adopted the k-\(\omega\) SST computational equations. The following is the set of conservation equations along with the k-\(\omega\) SST turbulence model equations ([PERSON], 2015):  
Mass conservation equation, as follows:  
\[\frac{\partial\rho}{\partial t}+\
abla(\rho U)=0, \tag{1}\]  
Figure 2: Dimensions of the CHIEF high-speed machine and simplified model, where (**a**) represents the lateral dimensioned view, (**b**) is the top plan with dimensions, (**c**) is the simplified model side view, and (**d**) is the simplified model top view. where \(U\) is the velocity vector and \(\rho\) is the density. Momentum conservation equation, as follows:  
\[\frac{\partial(\rho U)}{\partial t}+\
abla\cdot(\rho UU)=-\
abla p+\mu_{eff} \
abla^{2}U+F, \tag{2}\]  
where \(F\) is a miscellaneous body force term, \(\mu\) is the dynamic viscosity, \(\mu_{t}\) is the eddy viscosity, and \(\mu_{eff}=\mu+\mu_{t}\) is the effective kinetic viscosity. Energy conservation equation, as follows:  
\[\frac{\partial}{\partial t}(\rho E)+\
abla\cdot[U(\rho E+p)]=\
abla\cdot\left[ k_{eff}\
abla T+U\cdot\tau_{eff}\right]+S, \tag{3}\]  
where \(p\) denotes static pressure; \(E\) represents total energy; \(k\) signifies the thermal conductivity; \(k_{t}\) denotes the turbulent thermal conductivity; \(k_{eff}=k+k_{t}\) represents the effective thermal conductivity; \(T\) is the temperature; \(S\) accounts for the heat source term that includes both heat from chemical reactions and volumetric heat sources; and \(\tau_{eff}\) signifies the viscous heating term, as presented in Equation (4). \[\tau_{eff}=\mu_{eff}\left(\frac{\partial u_{j}}{\partial x_{i}}+\frac{\partial u _{j}}{\partial x_{j}}\right)=\rho v_{eff}(\frac{\partial u_{j}}{\partial x_{ i}}+\frac{\partial u_{j}}{\partial x_{j}}), \tag{4}\]  
where \(v_{eff}=\mu_{eff}/\rho\) represents the effective kinematic viscosity. The variable \(\mu_{t}\) can be determined by solving the two k-\(\omega\) SST turbulence model equations, as shown in Equations (5) and (6). \[\frac{\partial(\rho k)}{\partial t}+\frac{\partial(\rho ku_{i})}{\partial x_{ i}}=\frac{\partial}{\partial x_{i}}\left(\Gamma_{k}\frac{\partial k}{ \partial x_{j}}\right)+G_{k}-Y_{k}+S_{k} \tag{5}\]  
and  
\[\frac{\partial(\rho\omega)}{\partial t}+\frac{\partial(\rho\omega u_{i})}{ \partial x_{i}}=\frac{\partial}{\partial x_{i}}\left(\Gamma_{\omega}\frac{ \partial\omega}{\partial x_{j}}\right)+G_{\omega}-Y_{\omega}+D_{\omega}+S_{ \omega}, \tag{6}\]  
where \(G_{\omega}\) denotes the generation of \(\omega\); \(G_{k}\) is the production of turbulent kinetic energy; \(\Gamma_{k}\) and \(\Gamma_{\omega}\) are the effective diffusivities of \(k\) and \(\omega\), respectively; \(Y_{k}\) and \(Y_{\omega}\) denote the dissipation of \(k\) and \(\omega\) due to turbulence, respectively; \(S_{k}\) and \(S_{\omega}\) are user-defined source terms; and \(D_{\omega}\) denotes the cross-diffusion term. \[\mu_{t}=\frac{\rho k}{\omega}\frac{1}{max}\frac{\left[\frac{1}{\alpha^{*}}, \frac{S\Gamma_{k}}{\alpha_{1}\omega}\right]}, \tag{7}\]  
\[\alpha^{*}=\alpha_{\infty}^{*}(\frac{\alpha_{0}^{*}+Re_{t}/R_{k}}{1+Re_{t}/R_ {k}}), \tag{8}\]  
and  
\[Re_{t}=\frac{\rho k}{\mu\omega}, \tag{9}\]  
where \(R_{k}=6\), \(\alpha_{0}^{*}=\beta_{i}/3\), and \(\beta_{i}=0.072\).