# Flow Characteristics and Loss Models of the Transonic Radial Turbine
## 4 Loss Models
### Shock Wave-Induced Loss  
The shock wave-induced loss is caused by the shock wave near the trailing edge in transonic conditions, which contains the loss caused by the shock wave itself and the loss caused by the interaction of shock wave and the boundary layer, the leakage flow and the others. The physical mechanism for this is very uncertain. Equation (7) describes the loss caused by the normal shock wave itself.  
\[L_{shock}=T\Delta s=-TRln(\frac{p_{2}^{*}}{p_{1}^{*}})=\frac{TR}{k-1}\bigg{[} kn\bigg{(}\frac{2}{(k+1)Ma^{2}}+\frac{k-1}{k+1}\bigg{)}+ln\bigg{(}\frac{2k}{k+1}Ma^{2}- \frac{k-1}{k+1}\bigg{)}\bigg{]}(Ma>1) \tag{7}\]Figure 9: The distribution of efficiency and the percentage of four traditional losses. (a) 70,000 rpm. (b) 80,000 rpm. (c) 90,000 rpm. (d) 100,000 rpm.  
Considering the loss caused by the interaction between shock waves and other flows, a loss coefficient \(c\) needs to be added to the shock wave-induced model as Equation (8).  
\[L_{shock}=cT\Delta s=-cTRln(\frac{p_{2}^{*}}{p_{1}^{*}})=\frac{cTR}{k-1}\Big{[}kln \bigg{(}\frac{2}{(k+1)Ma^{2}}+\frac{k-1}{k+1}\bigg{)}+ln\bigg{(}\frac{2k}{k+1}Ma^ {2}-\frac{k-1}{k+1}\bigg{)}\bigg{]}(Ma>1) \tag{8}\]  
The loss models and the total to total efficiency should satisfy Equation (9), so the shock wave-induced loss coefficient \(c\) can be solved.  
\[\frac{L_{i}}{\Delta H_{0}}+\frac{L_{p}}{\Delta H_{0}}+\frac{L_{ctr}}{\Delta H_ {0}}+\frac{L_{trl}}{\Delta H_{0}}+\frac{L_{shock}}{\Delta H_{0}}+\eta_{tt}=1 \tag{9}\]  
There are four different rotation speed lines of the turbine as described in Figure 9; the data of rotation speed 70,000 rpm and 100,000 rpm were used to calculate the shock wave-induced loss coefficient \(c\) and the other data were used to verify whether \(c\) is correct. SSE means the sum of squares due to error between the left and the right of Equation (9). SSE changes with different shock wave-induced loss coefficients \(c\) as shown in Figure 10. Figure 10 shows that when \(c\) is 1.95, the SSE is the smallest. So, the shock wave-induced loss coefficient \(c\) is determined as 1.95.  
The total-to-total efficiency can be predicted by the loss models and Equation (9). Figure 11 shows the error of the predicted efficiency and the CFD efficiency. The solid line represents the error without the shock wave-induced loss model, and the dotted line represents the error with the shock wave-induced loss model. Figure 11 shows that when the expansion ratio is low, the turbine works in subsonic conditions without shock wave loss, so the two kinds of lines are coincident. The two types of lines begin to separate at high expansion ratios, as the shock wave starts to cause losses. The error of the predicted efficiency and the CFD efficiency in high expansion ratio conditions is larger than that in low expansion ratio conditions. The largest error is about 3.5% with the shock wave-induced loss model, much smaller than the 10% without the shock wave-induced loss model. It means that the shock wave-induced loss model can make the prediction of the efficiency more accurate.  
Figure 12 shows the distribution of different losses as the fraction of energy when the turbine operates at different expansion ratios and different rotation speeds, which contains the shock wave-induced loss model. The passage loss is the greatest loss of the rotor, which ranges from 8 to 10 percent when the expansion ratio increases and ranges little with the increase of the rotation speed. Tip clearance loss decreases with the increase in the expansion ratio. With the increase in the expansion ratio, the pressure difference between the inlet and the outlet increases, causing the increase in the radial component of velocity \(C_{m4}\) at the inlet and \(C_{m6}\) at the outlet. So, \(C_{x}\) and \(C_{r}\) will decrease with the increase in the expansion ratio, leading to the decrease in the tip clearance loss. Incidence loss is near zero when the turbine works at the design rotation speed of 80,000 rpm. The farther  
Figure 10: The SSE vs. c.