# Numerical Analysis on Self-Propulsion of a Waterjet-Propelled Ship with Different Propulsion Models
## 4 Conclusions  
In the current research, numerical simulations and analysis of self-propulsion for a waterjet-propelled ship model are conducted. The grid independence study is performed for the bare hull to define the computational mesh in hull region. For the self-propulsion of the waterjet-propelled ship model, the computational cost, external flow filed, and internal ingested flow field of the control volume with the MRF model and body-force model are compared. The transient sliding mesh model is triggered when the net force of waterjet-propelled ship converges to zero (resistance/thrust balance) with the steady MRF model simulation at \(Fr=0.542\) (design speed). The main conclusions are presented below:  
1. Compared with the MRF model, the body-force model can reduce the number of grids by about 30%, and save about 28% in the computational time. In addition, the wave pattern and self-propelled hull resistance obtained by the body-force model agree well with that of MRF model, indicating that the body-force model is economical and efficient when the primary concerns are related to the external flow field.
2. The velocity profiles of the boundary layer at the waterjet centerline of the capture area with the MRF model and body-force model show little difference, and the non  
Figure 20: Non-dimensional axial velocity distribution at station 3 and 6 with three different models.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **MRF** & **BFM** & **SMM** \\ \hline \(NVR\) & 1.4973 & 1.4863 & 1.5234 \\ \(Q_{I}\) (m\({}^{3}\)/s) & 0.0589 & 0.0587 & 0.0599 \\ \(\beta_{M6}\) & 1.0116 & 1.0083 & 1.0094 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Results of \(NVR\), \(Q_{I}\) and \(\beta_{M6}\) at station 6 with the MRF model, body-force model (BFM) and sliding mesh model (SMM).  
uniformity of the capture area with the body-force model is slightly smaller than that with the MRF model.
3. As for the nozzle section, the momentum non-uniformity factor with the body-force model is slightly smaller compared with the MRF model. At the self-propulsion point, the nozzle velocity ratio (\(NVR\) ) and flow rate of waterjet with the body-force model are also slightly smaller.
4. As the speed grows, the resistance increment fraction \(t_{r}\) decreases, while the jet system thrust deduction fraction \(t_{j}\) increases. Additionally, the relative errors of \(1-t_{r}\), \(1-t_{j}\) and \(1-t\) between the MRF model and body-force model are less than 1.9%.
5. The volume flow rate and the net force of waterjet-propelled ship with the transient sliding mesh model fluctuate in a small range, while the values with the steady MRF and body-force models remain constant at self-propulsion point. Additionally, the axial velocity distribution in the upper half disk of station 3 with the sliding mesh model is more uneven than that with the MRF and body-force models. The nozzle velocity ratio, flow rate of waterjet and momentum non-uniformity factor at station 6 with the sliding mesh model are greater, compared with the body-force model.  
Conceptualization, Y.Z.; methodology, Y.Z.; software, Y.Z. and Z.L.; validation, Y.Z. and Z.L.; formal analysis, Y.Z. and Z.L.; investigation, Y.Z.; resources, A.Y.; data curation, Y.Z. and Z.L.; writing--original draft preparation, Y.Z.; writing--review and editing, Y.Z. and A.Y.; visualization, Y.Z. and Z.L.; supervision, A.Y.; project administration, A.Y.; funding acquisition, A.Y. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the innovation team project of the advanced ship propulsion technology (No. K90015-5), Marine Design and Research Institute of China.  
Not applicable.  
The hull and experimental data are provided by Science and Technology on Water Jet Propulsion Laboratory, Marine Design and Research Institute of China. We would like to express our deep appreciation to Science and Technology on Water Jet Propulsion Laboratory.  
The authors declare no conflict of interest.