# 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
At the self-propulsion point, the nozzle velocity ratio and flow rate of waterjet with the body-force model are also  
Figure 16: Comparison of non-dimensional axial velocity distribution of the MRF model and body-force model (BFM) at station 3. (**a**) \(Fr=0.3\); (**b**) \(Fr=0.4\); (**c**) \(Fr=0.5\); (**d**) \(Fr=0.542\). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\(\mathbf{Fr}\)} & \multirow{2}{*}{\(\mathbf{MRF}\)} & \multirow{2}{*}{\(\mathbf{BFM}\)} & \multirow{2}{*}{\(\mathbf{(1-w)}\) Error} & \multirow{2}{*}{\(\mathbf{MRF}\)} & \multirow{2}{*}{\(\mathbf{BFM}\)} & \multirow{2}{*}{\(\mathbf{C_{m}\mathbf{B}}\)} & \multirow{2}{*}{\(\mathbf{C_{m}\mathbf{ Error(=\frac{C_{mB}-C_{mM}}{C_{mM}})}}\)} \\  & & & & & & \\ \hline
0.300 & 0.8467 & 0.8465 & \(-\)0.02\% & 0.8711 & 0.8710 & \(-\)0.01\% \\ \hline
0.400 & 0.8521 & 0.8511 & \(-\)0.12\% & 0.8768 & 0.8763 & \(-\)0.06\% \\ \hline
0.500 & 0.8569 & 0.8543 & \(-\)0.30\% & 0.8817 & 0.8794 & \(-\)0.26\% \\ \hline
0.542 & 0.8555 & 0.8512 & \(-\)0.50\% & 0.8808 & 0.8762 & \(-\)0.52\% \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Calculation \((1-w)\) and \(C_{m}\) of the capture area with the MRF model and body-force model. slightly smaller. Additionally, the relative errors of the three variables with the two models are less than 1.2%. Figure 18 presents the detailed flow characteristics at longitudinal section inside the waterjet with the MRF model and body-force model under \(Fr=0.542\) (design speed). The axial velocity distribution ahead the pump inflow plane and behind the nozzle section with the two models are similar. However, there is a significant difference in the axial velocity distribution between station 3 and station 6, because the impeller and guide vanes are removed in the body-force model. Based on the recommended procedures and guidelines for propulsive performance predictions [4], a control volume is applied to analyze the performance of the waterjet system using the momentum flux method, as shown in Figure 1. Surface 1 is the imaginary capture area. Surface 2 is the imaginary boundary of inlet stream tube, which separates the internal ingested flow of the control volume from the external flow field. Surface 3 is the physical boundary of the waterjet system. Surface 4 is the surface between the stagnation lines and merge lines of the hull and duct. Surface 6 is the nozzle section at station 6. The horizontal component of the gross thrust vector can be obtained by writing the horizontal momentum flux balance over the control volume:  
\[T_{\text{\small gross}}=-\iint\limits_{A_{1}+A_{6}}\rho u_{x}(u_{k}n_{k})dA \tag{13}\]  
where \(\rho\) is the density of the fluid, \(u\) is the velocity vector and \(n\) is the unit vector normal to the control surface pointing outward of the control volume [11]. The net thrust is defined as the force vector acting upon the internal material boundary of waterjet system, which acts directly on the hull to balance the resistance. The horizontal component of the net thrust is abbreviated as \(T_{\text{\small net}}\) and defined as follows:  
\[T_{\text{\small net}}=-\iint\limits_{A_{3}+A_{4}}\sigma_{x}dA-\iint\limits_{ V_{3-5}}\rho F_{px}dV \tag{14}\]  
where \(\sigma_{x}\) is the horizontal component of the surface stresses and \(F_{px}\) is the horizontal component of the pump force per unit mass. For a waterjet-driven hull, the thrust deduction fraction \(t\) is not the same as the conventional thrust deduction fraction employed in propeller/hull interaction theory.