## 4 Numerical Results
### Outlet Flow of the Crescent Pump  
Figure 12 depicts the three types of outlet flow from one gear pair under 2000 rpm and 20 MPa: the kinematic outlet flow (\(Q_{\text{kin}}\)) defined by Equation (13), the with-trapped outlet flow (\(Q_{\text{wt}}\)) by taking the trapped flow into consideration defined by Equation (22), and the actual outlet flow (\(Q_{\text{out}}\)) evaluated by the proposed model defined by Equation (23).  
\[Q_{\text{wt}}=Q_{\text{kin}}+q_{\text{trap}} \tag{22}\]  
\[Q_{\text{out}}=Q_{\text{kin}}+q_{\text{trap}}-q_{\text{leak}}-q_{\text{tri}} \tag{23}\]  
Referring to Figure 12, it can be seen that the three types of flow were subject to ripples with the period of approximately 27.7\({}^{\circ}\), which was expected since the tooth number of the gear shaft (driving  
Figure 11: Frequency spectra of simulated and experimental outlet pressure ripples at 500 rpm, 20 MPa.  
Figure 10: Comparison of outlet pressure ripples regarding different operating conditions: (a) 500 rpm, 5 MPa; (b) 500 rpm, 20 MPa; and (c) 2000 rpm, 20 MPa.  
gear) was 13, as shown in Equation (24). The kinematic outlet flow (\(Q_{\rm kin}\)) yielded between 74.88 and 80.14 L/min; the with-trapped outlet flow (\(Q_{\rm wt}\)) yielded between 78 and 80.14 L/min; and the actual outlet flow (\(Q_{\rm out}\)) yielded between 75.25 and 78.33 L/min.  
\[\lambda=360^{\circ}/z_{1}=27.7^{\circ} \tag{24}\]  
Figure 13 depicts the outlet flow from two gear pairs with an index angle (\(\gamma\)) under 2000 rpm and 20 MPa (\(\zeta=1\)). It can be seen that within a period, the flow from one gear pair was characterized by an ascending stage followed by a descending stage, and one could observe a peak value and a valley value. The flow from the two gear pairs were in the same shape since the displacement was \(\zeta=1\). A phase difference was also observed between the flow from the two gear pairs due to the index angle.  
Figure 14 depicts the outlet flow from the tandem pump (\(\zeta=1\)) under 2000 rpm and 20 MPa with respect to different index angles (0, \(\lambda/4\), \(\lambda/2\)), noting that the index angle \(\gamma\) was bounded between 0 and \(\lambda\). It can be seen that the outlet flow of the tandem pump exhibited a periodic behavior with the same period of 27.7\({}^{\circ}\) and roughly the same mean outlet flowrate though the index angle  
Figure 12: Three types of outlet flow from one gear pair under 2000 rpm and 20 MPa.  
Figure 13: Outlet flow from two indexed gear pairs under 2000 rpm and 20 MPa (\(\zeta=1\)).  
varied. However, the outlet flow exhibited different shapes and different ripples as the index angle varied. The outlet flow yielded between 151.59 and 158.64 L/min when \(\gamma=0\), between 153.25 and 156.96 L/min when \(\gamma=\lambda/4\), and between 154.01 and 156.34 L/min when \(\gamma=\lambda/2\).  
Figure 15 depicts the outlet flow from the tandem pump (\(\gamma=\lambda/2\)) under 2000 rpm and 20 MPa with respect to different displacement ratios (0, 0.5, 1), noting that the displacement ratio \(\zeta\) was bounded between 0 and 1. Expectedly, the outlet flow exhibits a periodic behavior with the period of 27.7\({}^{\circ}\). It could also be seen that under the same operating conditions, the pump yielded different mean outlet flowrate and different flow ripples as the displacement ratio varied. The outlet flow yielded between 75.25 and 78.33 L/min when \(\zeta=0\), between 115.02 and 116.88 L/min when \(\zeta=0.5\), and between 153.91 and 156.34 L/min when \(\zeta=1\).  
Figure 14: Outlet flow from the tandem pump (\(\zeta=1\)) under 2000 rpm and 20 MPa regarding different index angles.  
Figure 15: Outlet flow from the tandem pump (\(\gamma=\lambda/2\)) under 2000 rpm and 20 MPa regarding different displacement ratios.