# Numerical Analysis on Self-Propulsion of a Waterjet-Propelled Ship with Different Propulsion Models
## 3 Results and Discussions
### Computations of Self-Propulsion
#### 3.2.3 Internal Ingested Flow of the Control Volume  
Figure 13 displays streamlines tracked upstream from the waterjet pump, which intersects with the incoming boundary layer at station 1a. After that, a program is developed using the MATLAB language to obtain the shape and flow variables of the capture area. Figure 14 exhibits the velocity distributions in the capture area with the MRF model and body-force model at \(Fr=0.542\), showing the accordance of the axial velocity distribution and the geometry of the capture area obtained by the two models. Figure 15 shows the velocity profiles of the boundary layer at waterjet centerline of the bare hull and the waterjet self-propelled ship using the MRF model and body-force model. The axial velocity at the waterjet centerline of the self-propelled ship is larger than that of the bare hull, due to the suction of the waterjet system. The velocity profiles of the boundary layer at the waterjet centerline of the self-propelled ship with the MRF model and body-force model exhibit little difference. Figure 12: Jet stream with the MRF model and body-force model (BFM). Figure 13: Waterjet streamlines calculated by RANS intersecting station 1a boundary layer. Similar to the effective wake field on a propeller, the effective wake fraction \(w\) is calculated to study the effect of the waterjet suction on the hull distorted flow at station 1a:  
\[1-w=\frac{\iint\limits_{A_{1}}U_{x}dA}{\iint\limits_{A_{1}}U_{0}dA} \tag{10}\]  
where \(U_{0}\) is the ship velocity, \(A_{1}\) is the capture area, and \(U_{x}\) is the axial velocity of \(A_{1}\). The momentum velocity coefficient \(C_{m}\) is defined to quantitatively describe the influence of the hull on the ingested flow as follows:  
\[C_{m}=\frac{1}{U_{0}}\frac{\iint\limits_{A_{1}}U_{x}^{2}dA}{\iint\limits_{A_{1 }}U_{x}dA} \tag{11}\]  
where \(A_{1}\) is the capture area, \(U_{x}\) is the axial velocity of \(A_{1}\), and \(U_{0}\) is the ship velocity. The effective wake fraction \(w\) and momentum velocity coefficient \(C_{m}\) of the capture area with the MRF model and body-force model are summarized in Table 5. It shows that the non-uniformity of the capture area with the body-force model is slightly smaller than that with the MRF model. With the increase in speed, the result error between the two models becomes larger, but the relative errors are less than 0.6%. Figure 14: Velocity distributions in the capture area with the MRF model and body-force model (BFM). Figure 15: Velocity profiles of boundary layer at waterjet centerline of the bare hull (BH), MRF model and body-force model (BFM). \(\zeta\) is the vertical distance from ship bottom plating. Figure 16 compares the non-dimensional axial velocity distribution with the MRF model and body-force model at station 3. In general, the velocity distributions with the two models on the inflow plane are similar. The high-speed flow is distributed near the bottom dead center, and the velocity in the lower half disk is higher than that in the upper half disk. Figure 17 exhibits the non-dimensional axial velocity distribution on the nozzle section with the MRF model and body-force model. The value \(\beta_{M6}\) is a momentum non-uniformity factor [13], related to the experimental procedure for determining momentum at station 6:  
\[\beta_{M6}=\frac{1}{A_{6}\overline{U}_{6}^{2}}\int UU_{x}dA_{6} \tag{12}\]  
where \(\overline{U}_{6}\) is the average velocity of the nozzle section. \(\beta_{M6}\) is unity for uniform flow, and it deviates from unity as the flow field varies. Table 6 lists the results of nozzle velocity ratio \(NVR\) (\(NVR=\overline{U}_{6}/U_{0}\)), flow rate of waterjet \(Q_{f}\) and momentum non-uniformity factor \(\beta_{M6}\) at station 6 with the MRF model and body-force model. Compared with the MRF model, the momentum non-uniformity factor at station 6 with the body-force model is slightly smaller, due to the simplification of the impeller and guide vanes.