Comparison of three simple models of Kelvin’s ship wake

Abstract

A theoretical explanation of observations of high-Froude-number ship wakes that are narrower than the classical Kelvin 39∘ angle was recently offered by Rabaud and Moisy. The explanation relies on the assumption that a ship hull does not create waves longer than its length. A validation of this theoretical model has also been given. The validation is based on the approximation of the flow created by a ship hull by means of a Gaussian distribution of pressure at the free surface. These two flow models predict a wake angle ψmax that decreases like 1/F as the Froude number F increases beyond F≈0.5. A third theoretical explanation was recently proposed by the authors. This theoretical explanation assumes that the wave pattern of a ship mostly consists of dominant waves that are created by the ship bow and stern, and is mostly determined by interference effects between these dominant waves. The analysis of interference effects on the Kelvin wake of a ship predicts a wake angle ψmax≈0.14/F2 for a monohull ship, or ψmax≈0.2b/F for a catamaran with beam/length ratio b. The ‘flow models’ underlying these three alternative theoretical explanations of narrow ship wakes are examined, and the corresponding theoretical predictions are compared to the 37 observations of ship wakes reported by Rabaud and Moisy for Froude numbers F within the wide range 0.1<F<1.7. The wake observations are found to be consistent with the predictions given by an analysis of interference between the bow and stern waves of a monohull ship, or a catamaran with beam/length ratio b within the range 0.4≤b≤0.8. Indeed, agreement is consistently strong for the 35 wake observations within the range 0.1<F<1.4. This range of Froude numbers includes the range F<0.6, where interference between transverse bow and stern waves is important, and corresponds to the vast majority of ships. The predictions given by the Rabaud–Moisy ‘cutoff-wavelength model’ and the ‘Gaussian pressure distribution model’ are in close agreement with two wake observations for 1.6<F<1.7 and may also be consistent with several wake observations for 0.6<F<1.4, but are not consistent with most observations. This finding and a critical examination of the assumptions underlying the Rabaud–Moisy model and the Gaussian pressure distribution model suggest that these theoretical models may not be realistic for most ships. This conclusion is further validated by numerical computations of wave patterns for F=1. The computed waves are largest along a ray angle that agrees with the prediction of the bow and stern waves interference model, but is noticeably smaller than predicted by the Gaussian pressure distribution model.