## 4 Conclusions  
The comprehensive cooling properties of variable cross-section serpentine channels with different thickness turning vane under both stationary and rotating conditions are explored is this research. According to the results, specific conclusions are:  
1. When stationary, after arranging the turning vane, the wall \(Nu\) of each region increases in different degree with the arguments of the turning vane thickness, especially for the turning zones and the tip wall. The turning vane can ameliorate the flow separation phenomenon at inlets of pass2 and pass3. Moreover, affected by the rectification of the turning vane, the uniformity is improved in pass2 and pass3, which reduces the disturbance and weakens the heat transfer there.
2. In the range of \(Re\) = 10,000-50,000, as the turning vane thickness increases, the \(Nu\) of the turning area and the tip wall increases. Compared with case1, the \(Nu_{\text{up}}/Nu_{0}\) is increased by 56.5% at the maximum (case6, \(Re\) = 50,000), \(fff_{0}\) is reduced by 14.2% at the maximum (case2, \(Re\) = 10,000), and the \(TP\) is increased by 4.5% at the maximum (case2, \(Re\) = 10,000). When \(Re\) is low (\(Re\) \(\leq\) 30,000), the turning vane with smaller thickness should be selected, and when \(Re\) is high (\(Re\) > 30,000), the turning vane with larger thickness is better.
3. When rotating, the wall \(Nu\) of pass1 and pass3 has a relative relationship of \(\text{{}^{\prime}RS}_{\text{TR}}\) > \(\text{{}^{\prime}RS}_{\text{LE}}\)', while the \(Nu\) presents a state of \(\text{{}^{\prime}RS}_{\text{LE}}\) > \(\text{{}^{\prime}RS}_{\text{TR}}\)' in pass2 owing to centrifugal force and Coriolis force. The \(Nu\) of turning zones and the tip wall in channels with  
Figure 19: The variation of \(Nu_{\text{all}}/Nu_{0}\) with \(Ro\) for different channels.  
turning vane also improves remarkably. The improvement degree increases with the increase of the turning vane. Under the influence of turning vane, the velocity of fluid in pass2 and pass3 is lower, while the \(Nu\) of the RS\({}_{\rm LE}\) in pass2 and the RS\({}_{\rm TR}\) in pass3 are significantly reduced.
4. In the range of \(Ro\) = 0-0.5 under rotating condition, compared with case1, the \(Nu_{\rm up}\)/\(Nu_{0}\) is increased by 33.0% at the maximum (case6, \(Ro\) = 0.1), and the \(Nu_{\rm all}\)/\(Nu_{0}\) is increased by 4.0% at the maximum (case6, \(Ro\) = 0.1).  
To sum up, the comprehensive cooling properties of the variable cross-section serpentine channel under stationary and rotating conditions can be significantly improved by arranging turning vane of suitable thickness.  
Conceptualization and investigation, T.X. and D.S.; Data curation, T.X.; Writing--original draft preparation, T.X.; Writing--review and editing, T.X., D.S., D.Z. and Y.X. All authors have read and agreed to the published version of the manuscript.  
This research was funded by National Science and Technology Major Project: J2019-IV-0022-0090 and Fundamental Research Funds for the Central Universities: xjlgf2018009.  
Not applicable.  
All authors in the paper express great gratitude for the financial support by National Science and Technology Major Project: J2019-IV-0022-0090 and Fundamental Research Funds for the Central Universities: xjlgf2018009.  
The authors declare no conflict of interest.