## 5 Conclusions  
In this paper, the skin-friction coefficient model of windage loss in a rotor-stator gap is verified and the influence patterns of gap size, Reynolds number and medium type on the skin-friction coefficient model are clarified. A method for predicting the optimal gap behind the radial impeller is proposed. The main conclusions are as follows:  
* The effects of the relative gap on the two turbulent state skin-friction coefficient models are studied. The results show that the relative deviation between numerical calculation and model prediction results decreases with the increase of relative gap and the maximum deviation is \(-12.4\%\) when the relative gap is \(0.0127\). The area occupied by the core rotating fluid outside the boundary layers increases which increases the skin-friction coefficient of the rotor surface. The model of Region III assumes boundary layer merging. In fact, there are two states of boundary layer merging and separation in the flow of Region III, so the model accuracy of Region III is lower than that of Region IV. * As the Reynolds number increases, the skin-friction coefficient decreases gradually. Since the windage loss is positively correlated with the third power of the rotational  
Figure 18: Contour of velocity distribution of the gaps behind the turbine and compressor. speed, windage loss actually increases. In addition, at higher Reynolds numbers, the prediction results of the boundary layer separation state are more accurate and the maximum deviation is only 6.1%. The boundary layers of the rotor and stator wall transfer from merging to separation resulting in a larger velocity gradient and higher skin-friction coefficient near the wall. The model of the boundary layer merging state still has the state of separation. Since the Reynolds number is higher than \(10^{7}\) and has not been experimentally corrected, the accuracy of the Region IV model decreases at higher Reynolds numbers. * The accuracy of the two turbulent state models under three mediums of air, water-liquid and critical CO\({}_{2}\) was studied. Due to the difference in physical properties, the vorticity distribution of air and critical CO\({}_{2}\) is diverse in the gap. The results show that when the physical properties of the medium change, the accuracy of the model of boundary layer separation state at low Reynolds number decreases, but all show a trend of decrease in the process of transition from merging to separation. The model accuracy of the remaining states is not sensitive to the change in the physical properties of the working medium. * According to the two turbulent state models, a method for predicting the optimal gap behind the radial impeller is proposed which is verified on the sCO\({}_{2}\) principle prototype of the Sandia National Laboratory in the United States. The theoretical optimal gap size is obtained and verified by numerical calculation. The results show that the theoretical model is still reliable when the working fluid becomes sCO\({}_{2}\). However, since the method is aimed at the optimal gap under a fixed Reynolds number, the windage loss obtained by this method under the design condition is the smallest. When the structural design of the turbine is completed, the optimal gap under variable conditions is different from the design condition. In addition, the difference between actual windage loss and theoretical loss is analyzed and four aspects that need to be paid attention to in the further development of high-precision windage loss models are pointed out. Conceptualization, Z.Z. and Q.D. ; software, Z.Z. ; investigation, Z.Z. ; data curation, L.H. ; writing--original draft preparation, Z.Z. ; writing--review and editing, J.L. and Z.F. ; visualization, Z.Z. and L.H. ; supervision, Q.D. All authors have read and agreed to the published version of the manuscript. This research was funded by Joint Funds of the National Natural Science Foundation of China: U20A20303 and National Key R&D Program of China: 2017 YFB0601804. Not applicable.