## Appendix A  
### Rotation Framework for the Immersed Boundary  
The immersed boundary method (IBM) developed by [PERSON] et al. [20] is suitable for treating the moving problem in the current simulation. According to [PERSON] et al. [20], because the Courant-Friedrichs-Lewy (CFL) number--expressed as  
\[CFL=\frac{u_{conv}\times\Delta t}{\Delta x} \tag{1}\]  
where \(u_{conv}\) is the speed of the moving object--cannot be larger than 0.5 in order to avoid the fresh cell problem, the time step is restricted by \(u_{conv}\).  
This also implies that if objects have a high rotational speed, resulting in a high angular velocity, the time step must be very small. As a result, a significant consumption of computational resources can be anticipated. To address this issue, the multiple reference frame (MRF) approach is adopted to handle rotating objects. Equation (A2) shows the momentum equation in a rotating frame:  
\[\frac{\partial\rho u_{1}}{\partial t}+\
abla\cdot(\rho V_{r}u_{1})+\rho( \omega\times u_{1})=-\
abla p+\
abla\cdot\tau, \tag{2}\]  
where \(V_{r}=u_{1}-\omega\times r\) represents the relative velocity, which refers to the velocity observed from the rotating frame, and \(\rho(\omega\times u_{1})\) represents the term caused by the Coriolis and centripetal acceleration. However, a disadvantage of the MRF method is the challenging treatment of the interface between the stationary and rotating zones. Here, a method combining IBM and MRF is proposed as an alternative approach to handle rotating objects.  
Equation (A3) represents the momentum equation with the immersed boundary as a source term \(\rho F_{B}\):  
\[\frac{\partial\rho u_{1}}{\partial t}+\
abla\cdot(\rho u_{1}u_{1})=-\
abla p +\
abla\cdot\tau+\rho F_{B}, \tag{3}\]  
where \(F_{B}\) is the body force term, such that the velocity distribution on an arbitrary surface is equal to the desired velocity \(V_{s}\).  
\[u_{i}^{n+1}=u_{i}^{n}+\Delta t(RHS+\rho F_{B})=V_{s} \tag{4}\]  
If the velocity of the next time step \(u_{i}^{n+1}\) is equal to the desired value \(V_{s}\) on the immersed boundary, the source term \(F_{B}\) can be expressed as  
\[F_{B}=\frac{1}{\rho}(\frac{V_{s}-u^{n}}{\Delta t}-RHS) \tag{5}\]  
A comparison between Equations (A2) and (A3) reveals that if the relative velocity \(V_{r}\) is zero, which also implies that the angular velocity is a constant, Equation (A6) can be obtained.  
\[\rho(\omega\times u_{1})=\
abla\cdot(\rho u_{1}u_{1})-\rho F_{B} \tag{6}\]From an implementation perspective, the impact of rotation on the flow field due to the Coriolis and centripetal acceleration can be directly calculated using Equation (A6). As a result, employing the IBM can address the interface treatment for the rotating zone. However, the application of the MRF method in transient state problems remains controversial owing to the fresh cell problem. Therefore, in the current study, Equation (A6) is utilized solely to expedite the attainment of a quasi-steady state. Once the quasi-steady state is achieved, the time step is gradually reduced to the desired value. The CFL number, which is based on the maximum moving speed of the object, must be kept below 0.5 to avoid the fresh cell problem.