# Numerical Simulation of the Kelvin Wake Patterns
## 5 Discussions of Kelvin Wake Angle
### The Effect of Froude Number on Wake Angle  
To demonstrate the effect of Froude number on apparent wake angle, Figure 6 shows three free surface patterns for flow past the Rankine source, computed for \(F^{\prime}=0.7,\,2.5,\,5.0\) and \(\mu^{\prime}=0.4\) on a \(181\times 61\) mesh with \(\Delta x^{\prime}=0.3,\Delta y^{\prime}=0.3\). The wake angle decreases as Froude number increases. The reason is that the highest peaks do not locate in the outermost divergent under enough large Froude numbers, and then the wake angle gets smaller apparently. In Figure 6a, the Froude number is a moderately small value and the wake angle is almost the same as Kelvin angle. However, in Figure 6b,c, the wake angle is much smaller than the Kelvin angle, ending up with the wake angle and Froude approximately inversely proportional. Figure 5: Comparisons between the three types of real ship waves and the corresponding simulation results. The pictures of real ship wave patterns come from internet: (**a**) speed-boat wake, from [[https://commons.wikimedia.org/wiki/File:Fjordn_surface_wave_boat.jpg](https://commons.wikimedia.org/wiki/File:Fjordn_surface_wave_boat.jpg)]([https://commons.wikimedia.org/wiki/File:Fjordn_surface_wave_boat.jpg](https://commons.wikimedia.org/wiki/File:Fjordn_surface_wave_boat.jpg)) accessed on 5 January 2022; (**c**) pilot boat wake, from [[https://www.shutterstock.com/zh/image-photo/small-ferry-boat-route-mainland-during-1706813791](https://www.shutterstock.com/zh/image-photo/small-ferry-boat-route-mainland-during-1706813791)]([https://www.shutterstock.com/zh/image-photo/small-ferry-boat-route-mainland-during-1706813791](https://www.shutterstock.com/zh/image-photo/small-ferry-boat-route-mainland-during-1706813791)) accessed on 1 May 2022; (**e**) large vessel wake, from [[https://www.istockphoto.com/photo/aerial-top-view-oil-ship-tanker-full-speed-transportation-oil-from-refinery-on-the-gm1159334567-316959177](https://www.istockphoto.com/photo/aerial-top-view-oil-ship-tanker-full-speed-transportation-oil-from-refinery-on-the-gm1159334567-316959177)]([https://www.istockphoto.com/photo/aerial-top-view-oil-ship-tanker-full-speed-transportation-oil-from-refinery-on-the-gm1159334567-316959177](https://www.istockphoto.com/photo/aerial-top-view-oil-ship-tanker-full-speed-transportation-oil-from-refinery-on-the-gm1159334567-316959177)) accessed on 5 January 2022. Therefore, we should use specific data to obtain an accurate conclusion. The wake angle \(\theta\) is measured in the contour plot, whose highest peaks are easily found. As shown in Figure 7, the angle between the line connecting these peaks and the ship heading is the wake angle \(\theta\). To clearly illustrate the effect of Froude number \(F^{\prime}\), this paper keeps the source strength and source type invariant, obtains a series of data of wake angles with changing Froude number and sets two groups of data for comparing. The Cartesian coordinates of \(F^{\prime}\) and \(\theta\). are established, as shown in Figure 8. The Froude number \(F^{\prime}\) ranges from 0.8 to 5.0, the trend of the wake angle \(\theta\) obviously tends to decrease with Froude number. However, gradients are obviously different between the range of Froude number from 0.8 to 2.0 and the range of Froude number from 3.0 to 5.0. In the first interval, the wake angle decreases slowly with Froude number and the value of the wake angle fluctuates up and down; in the second interval, wake angle decreases dramatically with Froude number. In sum, the downtrend of the wake angle with Froude number has two stages; it is only when the Froude number reaches a threshold that the wake angle is inversely proportional to the Froude number; the threshold is independent of the source strength, around 2.0 when the Rankine source is used as a disturbance. This finding can be reflected in real situations; only when the ship speed is high enough will the transformation of wake angle be noticeable.