## 4 Results and Discussion  
### Analysis under Different Relative Gaps and Reynolds Number  
In this section, the numerical calculation under different Reynolds numbers is carried out for the models of Region III and Region IV with water-liquid as the medium. The differences between the prediction results and the theoretical model are comprehensively compared to explore the influence of the gap size on the accuracy of the skin-friction model. For Geometry 1, all of the \(Re\) fall into Region III. From Figure 7, it can be seen that the numerical calculation and model prediction results show a negative deviation when the \(Re\) is small and the maximum relative deviation is \(-12.4\%\) (\(Re=3.77\times 10^{6}\)). With the increase of \(Re\), the deviations become positive and the deviation is 3.6% when \(Re=3.93\times 10^{7}\). The \(Re\) calculated by Geometry 3 falls into Region IV, almost all of which are negative deviations with a maximum value of \(-5.2\%\) and no regular change. It should be noted that only  
Figure 6: Grid-independence verification. (**a**) Mesh of the computational domain. (**b**) Circumferential velocity distribution (\(R/R_{0}=0.5\), \(Re=2.91\times 10^{7}\)). (**c**) 3.27 million grid solution with discretization error (\(R/R_{0}=0.5\), \(Re=2.91\times 10^{7}\)). Geometry 2 (\(S/R_{0}=0.0255\)) crosses two flow regions among six different \(Re\). When the flow is in Region III, the maximum deviation is 9.8% (\(Re=4.06\times 10^{5}\)) and less than 8% in Region IV. On the whole, the deviation is larger when the flow is in Region III. The following analyzes the causes of the deviation from the perspective of the flow in the gap. Figure 8 shows the velocity distribution of the flow in the gap of the cross-section (section of the fluid domain along the radial direction). It can be seen that due to the small \(S/R_{0}\) of Geometry 1, when \(Re=4.06\times 10^{5}\), there is a velocity gradient between the boundary layer and the core rotating fluid of the rotor and stator. The boundary between the two regions is not obvious. The flow in the gap is approximately a Couette flow and the velocity is linearly distributed. With the increase of \(Re\), it can be seen from Figure 8b,c that as the centrifugal force of the fluid near the wall increases, the viscous force decreases which leads to the increase of velocity gradient and the decrease of boundary layer thickness. According to the non-slip boundary condition, the fluid velocity near the rotor and stator wall is consistent with its corresponding velocity. At the position near the shaft (\(Re=3.77\times 10^{6}\), \(R/R_{0}=0.2\)), the velocity gradient difference between the boundary layers and the core rotating fluid at two Reynolds numbers is not alike. The boundary layers of the two walls convert from merging to separation. It can be predicted that with the increase of \(Re\), the boundary layer separation phenomenon first appears in the low-velocity zone near the shaft and gradually expands along the radius direction until it covers the entire rotor surface, thus establishing the flow state of Region IV. What needs illustration is that for a sufficiently small size gap, there will be no boundary layer separation phenomenon. Figure 9 shows the circumferential velocity distribution of three radius ratio positions in the gap of Geometry 1 when the Reynolds number is 3.77 \(\times\) 10\({}^{6}\). It can be seen that the velocity distribution in the gap is basically the same at different radius ratios. In the boundary layer zones, the velocity gradient gradually increases along the radial direction, which is due to the wall boundary layer thinning. The velocity increases gradually along the radial direction, the centrifugal force of the fluid near the wall increases and the effect of the viscous force decreases. These make the boundary layer thickness near the wall decrease and the velocity gradient expand.