## 3 Results and Discussions  
### Radial Hydraulic Force on the Separate Turbine  
The radial hydraulic force is generated on the turbine because of inner unsteady flow characteristics. The numerical simulations of flow characteristics were carried out at three different rotation speeds values (\(0.8n_{\mathrm{T}}\), \(n_{\mathrm{T}}\) and \(1.2n_{\mathrm{T}}\)). At the mid-planes of turbine section, normal to the rotating axis, pressure distributions are presented in Figure 7. High pressures exist in the suction, and there is the obvious pressure difference between the suction and impeller channel. The turbine suction plays the role of the rectifier and bears the majority of the water impact. The static pressure distribution on the coupling surface between impeller and solute was obtained by numerical simulation. The force on each node of the decoupling surface was calculated when the static pressure is evenly distributed near each grid node of the coupling surface. Then, the forces on the \(x\)-axis and \(y\)-axis were calculated, respectively, by the quadrature method. Based on the decomposition and synthesis theorem of the force, the size and direction of the total force was obtained as shown in the Equations (1)-(6). \[F_{i}=P_{i}(\frac{2\pi R_{2}B_{2}}{N}) \tag{1}\]  
Figure 6: Comparison between experiment result and numerical simulation. (**a**) Turbine speed characteristic curves. (**b**) Pump performance characteristic curves. Figure 7: Pressure distributions in the turbine. \[F_{i}^{x}=-F_{i}(\frac{x_{i}}{R_{2}}) \tag{2}\]  
\[F_{i}^{y}=-F_{i}(\frac{y_{i}}{R_{2}}) \tag{3}\]  
\[F_{x}=\sum_{i=1}^{N}F_{i}^{x} \tag{4}\]  
\[F_{y}=\sum_{i=1}^{N}F_{i}^{y} \tag{5}\]  
\[F=\sqrt{F_{x}^{2}+F_{y}^{2}} \tag{6}\]  
where \(F\) is the radial hydraulic force (N), \(P_{i}\) is the pressure at the \(i\)th grid node, \(R_{2}\) is the impeller radius (m), \(b_{2}\) is the impeller inlet/outlet width in the decoupling surface (m), \(N\) is the coupling surface mesh node number, \(F_{i}^{x}\) is the component of pressure on the \(i\)th grid node in the \(x\)-axis direction, \(F_{i}^{y}\) is the component of pressure on the \(i\)th grid node in the \(y\)-axis direction, \(F_{x}\) is the component of radial force in \(x\)-axis direction and \(F_{y}\) is the component of radial force in \(y\)-axis direction. A radial hydraulic force coefficient \(K\) was defined the force variation, as follows:  
\[K=\frac{F}{\pi\rho R_{2}B_{2}\mu^{2}} \tag{7}\]  
where \(u\) is the circumferential velocity in the largest base circle of the impeller (m/s). Figure 8 shows the initial direction and magnitude of radial hydraulic force on the turbine section under three operation conditions. The value of the force on the normal impeller rotation is smaller than that on abnormal operation conditions. The direction of the force on the suction is in the second quadrant, while that on the impeller is in the first quadrant. Figure 9 presents the force variations in one full cycle of impeller rotation. The radial hydraulic force on the suction seems to vary randomly but has an elliptical trajectory like the force on the impeller. The suction is a rectifier to improve the unsteady flow characteristics of the inflow for increasing the water energy utilization efficiency of the impeller. These unsteady flows in the suction generate random variations of the force. Remarkably, the initial states and changing trends of the forces on the turbine are close for different conditions; the fire turbopump is typically used in normal conditions. Thus, the force characteristics on the normal impeller rotation were selected as research subjects for further study, as shown in Figures 10 and 11. \[F_{i}=P_{i}(\frac{2\pi R_{2}B_{2}}{N}) \tag{1}\]0 Initial direction and magnitude of radial hydraulic force: (**a**) On the volute; (**b**) On the impeller. \[F_{i}=P_{i}(\frac{2\pi R_{2}B_{2}}{N}) \tag{1}\]1 Time-domain results of radial hydraulic forces. (**a**) Turbine suction. (**b**) Turbine impeller.