# Numerical Simulation of the Kelvin Wake Patterns
## 4 Simulation Results  
According to the solution of the system of nonlinear equations, a perspective view of the nonlinear Kelvin wake pattern is given in Figure 4, for the flow past the Ranking source with \(F^{\prime}=0.7\) and \(\mu^{\prime}=0.3\), computed on a \(151\times 51\) mesh with \(\Delta x^{\prime}=0.3\), \(\Delta y^{\prime}=0.3\). As shown in this free surface pattern, there are two different types of waves. One type of wave is the transverse wave, whose direction of propagation is consistent with the ship heading and the crests are slightly lower, being the main part of the whole pattern. The other waves are divergent waves, distributed symmetrically along the ship heading, whose crests appear to form ridges pointing diagonally away from the source, with the large steepness. The amplitude of the ship waves decays in proportion to the square root of distance along the \(x-\)axis, forming the well \(-\)known V\(-\)shaped Kelvin ship wave.  
In the numerical scheme proposed in this paper, there are three parameters to regulate the simulation results, with the polymorphic ship wave patterns appearing. The parameters include Froude number, source strength and source type. By adjusting these parameters, the wake angle, wave amplitude, prominent waves and wavelength can be changed. After plenty of experiments on the wake waveform, this paper proposed the following mechanism of ship wave transformation. The wake angle tends to decrease with Froude number but increase with the strength of the disturbance; the divergent waves will be prominent with the sufficiently large Froude number or the Rankine source; the wavelength dramatically increases with Froude number. It is suitable to choose the large Froude number, low source strength and Rankine source to simulate the ship wave with prominent divergent waves, small wake angle and long wavelength. In the opposite case, the ship wave with prominent transverse waves, large wake angle and short wavelength will be simulated. Here, three kinds of real ship wave patterns are used as examples, shown in Figure 5.  
1. The ship waves have small wake angle, prominent divergent waves and long wavelength. In this case, the vessel's overall length is short but its speed is high. In Figure 5a, there is a wake wave from the speedboat. Therefore, the Rankine source is utilized to make divergent waves conspicuous, then continuously increasing the Froude number until the wake angle is suitable. The simulation result is consistent with the real ship wave pattern.
2. The wake angles are the same as Kelvin angle, with prominent transverse waves and either the value of overall length or speed is not large, like the pilot boat in Figure 5c (note that: due to the far perspective, the entire wake pattern looks slightly smaller. However, the primary waveform is also consistent with Figure 5d). It is better to choose the Kelvin source, low source strength and the moderately small value of the Froude number. The perspective view of the pattern for \(F^{\prime}=0.7\) and \(\epsilon^{\prime}=0.5\) is presented in Figure 5d.
3. The ship waves have large wage angle, prominent divergent waves and short wavelength. Generally, only the large vessels can profile these wakes (Figure 5e). It is right to enhance the source strength to increase the wake angle, and then utilize the Rankine source to make the divergent waves prominent. The perspective view of the pattern for \(F^{\prime}=0.7\) and \(\mu^{\prime}=0.5\) is presented in Figure 5f.  
In conclusion, the simulation method adopted in this paper can simulate the polymorphic ship waves, and the simulation results are consistent with real ship waves. Therefore, it is feasible to discuss the wake angle on that basis.  
Figure 4: Perspective view of the free surface using Ranking source, for the case \(F^{\prime}=0.7\) and \(\mu^{\prime}=0.3\), computed on a \(151\times 51\) mesh with \(\Delta x^{\prime}~{}=~{}0.3,~{}\Delta y^{\prime}~{}=~{}0.3\).