# Numerical Simulation of the Kelvin Wake Patterns
## 6 Conclusions  
In order to elucidate the effect of ship size on the Kelvin ship waves, this paper implements the numerical scheme, which combines the boundary integral method with the Jacobian-free Newton-Krylov method, for the simulation of nonlinear ship waves on an infinite fluid. Though imitating plenty of different Kelvin wave patterns, the mechanism of ship wave transformation is proposed, and then it is verified by comparing simulation results with real ship waves. Finally, the wake angle is discussed with the effects of Froude number, source strength and source type and the following conclusions can be drawn. 1. The wake angle tends to decrease with Froude number and the downtrend has two stages. The wake angle is inversely proportional to Froude number and decreases dramatically after Froude number reaches a threshold, which is around 2.0 for Rankine source and is around 2.8 for Kelvin source. 2. The wake angle tends to increase with source strength, meaning that ship size can affect the ship waves; the larger the ship size, the larger the wake angle generated will be. 3. Because the gradient of source strength is lower than the gradient of Froude number, the wake angle change caused by ship size is not as visually obvious as the ship speed. Meanwhile, it is hard for source strength to increase the wake angle when the Froude number is large. 4. With either the Kelvin source or Rankine source as a disturbance, the variation trends of wake angle are approximately identical with the effects of Froude number and source strength. However, a more accurate solution for nonlinear ship waves can be solved when the free surface flow passes Rankine source. This paper suggests that there will be a sudden change in the downtrend of wake angle when the ship speed reaches a threshold. The next stage in research will pay attention to exploring the cause of the sudden change and investigating the influencing factors of threshold, e.g., water depth, ship shape, etc. The theoretical conclusion about threshold value will be verified by model test. This study will provide reference for informing environmentally friendly shipping policies, e.g., speed limits, etc. Figure 12: Measured apparent wake angles \(\theta\) in degrees plotted against the Froude number for the flow past a Kelvin source with \(e^{\prime}=1\) (**a**) and a Rankine source with \(n^{\prime}=0.5\) (**b**). **Author Contributions:** Conceptualization, X.S. and M.C. ; methodology, X.S. and M.C. ; software, M.C. and J.W. ; validation, X.S. and C.L. ; formal analysis, X.S., J.W. and C.L. ; writing--original draft preparation, X.S. and M.C. ; writing--review and editing, X.S., M.C., J.W. and C.L. ; supervision, X.S. All authors have read and agreed to the published version of the manuscript. **Funding:** The work was supported by the National Key Research and Development Program of China (No. 2019 YFE0111600), National Natural Science Foundation of China (No. 52071049, 61971083 and 51939001), Liao Ning Revitalization Talents Program (No. XLYC2002078) and Dalian Science and Technology Innovation Fund (No. 2019J11 CY015). The authors would like to express sincere thanks for their support. **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable. **Data Availability Statement:** Not applicable. **Conflicts of Interest:** The authors declare no conflict of interest. **Nomenclature:**