## 3 Results and Discussion  
### Relationship between Wind Resistance and Experimental Capsule Radius  
This study mainly focused on the wind resistance optimization of the CHIEF high-speed machine with a maximum acceleration of 1500 g. Therefore, the effect of the radius was investigated under the operating condition of an angular velocity of 57 rad/s, corresponding to 1500 g. This study considered a radius range of 4350-4650 mm. We simulated the torque resistance of an armature with various radii. Figure 7 presents the relationship between the wind resistance torque, average sectional speed within the MRF motion reference system, and average effective viscosity across sections as they vary with the radius. A schematic of the section location is shown in the upper-left corner of Figure 7. Figure 7 shows that as the radius increases, the wind resistance exhibits an upward trend, with a maximum increase of approximately 19%. This finding agrees with the results obtained by [PERSON] (2016). However, notably, [PERSON] attributed the increase in wind resistance to the growing mass of entrained air and the reduction in entrainment speed of the air as the radius expands, which leads to an increased relative velocity between the entrained air and the armature, thereby enhancing the wind resistance. Nevertheless, within the radius range of 4350-4650 mm, the simulation results did not lead to a reduction in the entrainment speed. We used the average sectional speed to visually represent the variation in the air entrainment speed, as indicated by the solid blue circles in Figure 7. The average sectional speed remained at approximately 140 m/s, without any noticeable decline. To analyze the reasons for the increase in the simulated wind resistance, the expression for windage power in Table 1 was examined, revealing that only the dynamic viscosity appears in the formula. However, during high-speed rotational motion, turbulent shear stress cannot be neglected. The turbulent shear stress can be represented by Equation (14) ([PERSON], 2016), as follows:  
\[\tau_{t}=\rho\varepsilon_{M}\frac{\partial\overline{u}}{\partial y}, \tag{14}\]  
Figure 7: Wind resistance torque, average sectional velocity, and average effective viscosity versus radius change. where \(\varepsilon_{M}\) represents the eddy diffusivity coefficient and \(\overline{u}\) denotes the turbulent mean velocity. Combining the turbulent shear stress with the laminar viscous shear stress results in an effective shear stress, as expressed in Equation (15), as follows:  
\[\tau=\rho(v+\varepsilon_{M})\frac{\partial\overline{u}}{\partial y}=\rho v_{eff} \frac{\partial\overline{u}}{\partial y}. \tag{15}\]  
In the equation, \(v_{eff}\) represents the effective dynamic viscosity, whereas \(v\) denotes the dynamic viscosity. For high-Reynolds-number flows, in the viscous sublayer of the boundary layer, \(v\) is significantly larger than \(\varepsilon_{M}\), where laminar viscosity losses dominate. In contrast, outside the boundary layer in the turbulent region, \(\varepsilon_{M}\) is much greater than \(v\), where the viscous losses caused by turbulent shear stresses become dominant. Notably, \(\varepsilon_{M}\) is not a constant physical property but depends on spatial coordinates and is determined by the conditions of the flow field. The wind resistance power was analyzed based on the Reynolds transport theorem. By employing tensor notations in Cartesian coordinates, the Reynolds transport equation can be given as follows [22]:  
\[\frac{\partial}{\partial t}(\rho E)+\frac{\partial}{\partial x_{i}}\left[u_{i }(\rho E+p)\right]=\frac{\partial}{\partial x_{i}}\left[k_{eff}\frac{\partial T }{\partial x_{i}}+u_{j}(\tau_{ij})_{eff}\right]+S_{h}, \tag{16}\]  
where \(E\) represents the total energy, \(\rho\) denotes the density, \(k_{eff}\) represents the effective thermal conductivity, \(S_{h}\) is a constant, and \(\left(\tau_{ij}\right)_{eff}\) signifies viscous heating.