## 4 Discussion
### Internal Flow Field and Selection of AFTs  
For the AFT in this study, there was a viscous boundary layer on the blade surface and the internal pipe wall, which resulted in the presence of a thin reg-lv 46. The relatively large area of reg-lv at the leading edge of the blade suction surface is caused mainly by the direct impact of the flow on the blade (\(\theta\) > 0\({}^{\circ}\) in Figure 6a), resulting in flow separation and vortex formation. Along the flow direction, \(\theta\) gradually decreases because of the straightening effect of the impeller, which weakens the flow separation and vortex phenomena at the leading edge of the blade suction surface. As a result, the area of reg-lv in successive stages of the impeller gradually decreases in the direction of the flow (Figure 10).  
When \(H\) and \(n\) are constant, as M\({}_{1}\) and M\({}_{2}\) increase, the areas of reg-lv and reg-hv increase and then decrease, whereas \(v\) in the reg-hv area gradually decreases. For the same AFT, the areas of reg-lv and reg-hv, as well as the velocity in the reg-hv region, follow an increasing trend as \(H\) increases. These trends in the internal flow field of the AFT with respect to the structural parameters and operating conditions are consistent with the variation in \(P_{\rm max}\). For D33, when \(H\) and n are constant, both the area and \(v\) of reg-hv are maximized.  
\(BP\) is a quantitative indicator that characterizes the impact of water flow on the AFT. The variation in \(BP\) with \(x/L\) in existing studies is consistent with the results presented in this paper [48]. From the perspective of energy transfer, \(\overline{\Delta BP}\) reflects the work performed by the turbine and is positively correlated with \(P_{\rm max}\)[49]. In this study, the response of \(\overline{\Delta BP}\) to structural parameters and operating conditions is consistent with that of \(P_{\rm max}\). \(\overline{\Delta BP}\) for D33 is 26.5% greater than that for D34, D43, and D44.  
According to Figures 12-14, at the \(x/L\) position of 0\(-\)0.1 on the suction side, the \(BP\) of the blade is less than 0. The negative pressure at this location is caused by the presence of the reg-lv region, which generates many vortices and low-energy fluid pockets. At the \(x/L\) = 0.93 position, a sharp pressure change occurs due to flow separation at the trailing edge [49; 50]. Numerous studies have shown that when negative pressure zones occur on a blade, cavitation can occur, with the water flow in these regions being prone to cavitation and bubble formation. When the bubble clusters collapse after they leave the negative pressure region, they produce high-intensity shock waves traveling at speeds of 10-100 m/s, which can erode the blade and shorten its lifespan [51]. The larger the \(NP\) value is, the greater the likelihood that cavitation will occur on the blade [52]. Since the \(NP\) of D33 is on average 32.6% greater than those of D34, D43, and D44 are, it is more susceptible to cavitation, whereas D44 has the smallest \(NP\) value.  
When the maximum \(P_{\rm max}\) is reached, the optimal distribution characteristics of the internal flow field (the maximum area and \(v\) of reg-hv) and the maximum \(\overline{\Delta BP}\) are the optimal targets, and D33 is optimal. When the cavitation phenomenon is considered, compared with D33, for D34 and D44, \(P_{\rm max}\) is reduced by 10.7% and 15.1%, respectively; \(\overline{\Delta BP}\) is reduced by 23.7% and 27.9%, respectively; but \(NP\) is reduced by 30.5% and 37.9%, respectively.