# Numerical Analysis on Self-Propulsion of a Waterjet-Propelled Ship with Different Propulsion Models
## 3 Results and Discussions  
### Resistance Calculations of Bare Hull  
In this section, the bare hull without waterjet is simulated, in which the inlet and outlet ducts are covered, and the bare hull resistance in calm water are calculated. Four levels of grid resolution (a coarse grid of 2.40 million points, a medium grid of 4.83 million points, a  
Figure 6: Geometry used with body-force model.  
fine grid of 7.22 million points, and the finest grid of 9.62 million points) are used for the grid independence study at \(Fr=0.3\).  
According to the least squares root method proposed by [PERSON] and [PERSON] (2016), a mesh independence study for the non-uniform refinement ratio is conducted. The non-uniform refinement ratio is defined as follows:  
\[\frac{h_{i}}{h_{1}}=\frac{\sqrt[3]{N_{1}}}{\sqrt[3]{N_{i}}} \tag{7}\]  
where \(N_{i}\) and \(h_{i}\) are the total grid number and the typical mesh size for mesh index \(i\), respectively. Additionally, \(N_{1}\) and \(h_{1}\) correspond to the finest mesh. The convergence fit curve of bare hull resistance \(R_{m}\) is shown in Figure 7, and the results of grid uncertainty analysis are listed in Table 2. The convergence ratio of the numerical can be obtained as follows:  
\[R_{G}=\frac{\varepsilon_{21}}{\varepsilon_{32}}=\frac{S_{2}-S_{1}}{S_{3}-S_{2}} \tag{8}\]  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Case** & **Grid Number** & **Calculated Result S** & **Experimental Data D** & **Relative Error E** & \(R_{G}\) \\ \hline \(S_{1}\) & 9.62 M & 98.54 & 95.05 & 3.67\% & \(-0.26\) (OC) \\ \hline \(S_{2}\) & 7.22 M & 98.48 & & 3.61\% & \\ \hline \(S_{3}\) & 4.83 M & 98.71 & & 3.85\% & \\ \hline \(S_{4}\) & 2.40 M & 99.76 & & 4.96\% & \\ \hline \end{tabular}
\end{table}
Table 2: Results of grid uncertainty analysis.  
Figure 7: Convergence of the total resistance.  
As the convergence ratio \(R_{G}\) is \(-0.26\), showing oscillatory convergence (OC), the grid uncertainty is calculated by using the maximum value \(S_{U}\) and the minimum value \(S_{L}\) of the oscillation based on the numerical calculation result:  
\[U_{G}=\left|\frac{1}{2}(S_{U}-S_{L})\right| \tag{9}\]  
The grid uncertainty is calculated as \(0.12\%D\) by taking the fine grid result 98.48 N as the minimum value and the medium grid result 98.71 N as the maximum. Considering the computational accuracy and cost, the fine grid of 7.22 million points is selected for the simulations.  
The bare hull resistance is calculated in a speed range of \(0.167\leq Fr\leq 0.542\). The model towing tests for calm water resistance were conducted in towing tank of Science and Technology on Water Jet Propulsion Laboratory, Marine Design and Research Institute of China, whose tank dimensions is 280 m \(\times\) 10 m \(\times\) 5 m and maximum carriage speed is 9 m/s. The results of numerical simulation and model test of bare hull are shown in Figure 8 and Table 3. It can be seen that the relative error between numerical and experimental results of bare hull resistance becomes larger with the increase in the ship's speed, which reaches 7.66% at \(Fr=0.542\) (design speed). [PERSON] and [PERSON] (2018) found that with the speed growth, the spray resistance became larger, its proportion to total resistance was also growing, and there was obvious water spray around the hull at high speeds in the model test. However, the VOF method cannot accurately capture the fluid spray in the numerical simulation, leading to relatively large differences between the numerical and experimental resistances.