## 2 Materials and Methods  
### Mathematical Model  
The mathematical algorithms that form the theoretical background of Computational Fluid Dynamics are typically based on the finite volume method, with ANSYS Fluent relying on this approach. The background, continuity, and momentum equations are present. Equation (1) provides the continuity equation, and Equation (2) presents the momentum equations. In the continuity equations, \(\mathrm{S_{m}}\) represents the source term, \(\rho\) denotes density, t stands for time, and \(\overset{\rightarrow}{\mathrm{v}}\) represents the velocity vector. In the momentum equation, \(\rho\) denotes pressure, \(\overset{\rightarrow}{\mathrm{\overline{r}}}\) represents the stress tensor, \(\overset{\rightarrow}{\mathrm{\overline{g}}}\) is the gravitational acceleration, and \(\overset{\rightarrow}{\mathrm{F}}\) represents the body forces. \[\frac{\partial\rho}{\partial\mathrm{t}}+\
abla.\left(\rho\overset{ \rightarrow}{\mathrm{v}}\right)=\mathrm{S_{m}} \tag{1}\]  
\[\frac{\partial}{\partial\mathrm{t}}\left(\rho\overset{\rightarrow}{\mathrm{v }}\right)+\
abla.\left(\rho\overset{\rightarrow}{\mathrm{v}}\overset{ \rightarrow}{\mathrm{v}}\right)=\
abla p+\
abla.\left(\overset{\rightarrow}{ \mathrm{\overline{r}}}\right)+\rho\overset{\rightarrow}{\mathrm{\overline{ g}}}+\overset{\rightarrow}{\mathrm{\overline{F}}} \tag{2}\]  
The moving reference frame (MRF) method was employed in the analyses, which can be expressed as the defining movement relative to a reference point. This methodology comprises two types of functions: absolute and relative. Absolute function definitions were employed for the analyses. \[\overset{\rightarrow}{\mathrm{\overline{\omega}}}=\omega\dot{\mathrm{a}} \tag{3}\]  
\[\overset{\rightarrow}{\mathrm{v_{r}}}=\overset{\rightarrow}{\mathrm{v}}- \overset{\rightarrow}{\mathrm{\overline{u_{r}}}} \tag{4}\]  
\[\overset{\rightarrow}{\mathrm{\overline{u_{r}}}}=\overset{\rightarrow}{ \mathrm{\overline{v_{t}}}}+\overset{\rightarrow}{\mathrm{\overline{\omega}}} \times\overset{\rightarrow}{\mathrm{\overline{r}}} \tag{5}\]  
Here, \(\overset{\rightarrow}{\mathrm{\overline{\omega}}}\) represents angular velocity (3), \(\overset{\rightarrow}{\mathrm{\overline{v_{r}}}}\) denotes relative velocity (4), \(\overset{\rightarrow}{\mathrm{\overline{v_{r}}}}\) refers to absolute velocity, \(\overset{\rightarrow}{\mathrm{\overline{u_{r}}}}\) stands for the velocity of the moving frame relative to the inertial reference frame (5), and \(\overset{\rightarrow}{\mathrm{\overline{v_{t}}}}\) signifies the translational frame velocity. Using these expressions, continuity and momentum equations can be rewritten as Equations (6) and (7). \[\frac{\partial\rho}{\partial\mathrm{t}}+\
abla.\rho\overset{\rightarrow}{ \mathrm{\overline{v_{r}}}}=0 \tag{6}\]  
\[\frac{\partial}{\partial\mathrm{t}}\rho\overset{\rightarrow}{\mathrm{ \overline{v_{t}}}}+\
abla.\left(\rho\overset{\rightarrow}{\mathrm{\overline{v _{r}}}}\overset{\rightarrow}{\mathrm{\overline{v_{t}}}}\right)+\rho\left[ \overset{\rightarrow}{\mathrm{\overline{\omega}}}\times\left(\overset{ \rightarrow}{\mathrm{\overline{v}}}  
Equations for the Realizable k-\(\varepsilon\) turbulence model are given in (8) and (9). \[ \begin{array}{ll}\frac{\partial}{\partial t}(\rho\mathrm{k})+\frac{\partial}{ \partial\chi_{\mathrm{j}}}\big{(}\rho\mathrm{ku}_{\mathrm{j}}\big{)}=&\frac{ \partial}{\partial\chi_{\mathrm{j}}}\Big{[}\Big{(}\mu+\frac{\mu_{\mathrm{k}}}{ \sigma_{\mathrm{k}}}\Big{)}\frac{\partial\varepsilon}{\partial\chi_{\mathrm{j} }}\Big{]}\\ &+\mathrm{G}_{\mathrm{k}}+\mathrm{G}_{\mathrm{b}}-\rho\varepsilon-\mathrm{Y}_{ \mathrm{M}}+\mathrm{S}_{\mathrm{k}}\\ \end{array}  \tag{8}\]  
\[