Scaling of far-field wake angle of nonaxisymmetric pressure disturbance.

Abstract

It has been recently emphasized that the angle of maximum wave amplitude α in the wake of a disturbance of finite size can be significantly narrower than the maximum value α_{K}=sin^{-1}(1/3)≃19.47^{∘} predicted by the classical analysis of Kelvin. For axisymmetric disturbance, a simple argument based on the Cauchy-Poisson initial-value problem suggests that the wake angle decreases following a Mach-like law at large velocity, α≃Fr_{L}^{-1}, where Fr_{L}=U/sqrt[gL] is the Froude number based on the disturbance velocity U, its size L, and gravity g. In this paper we extend this analysis to the case of nonaxisymmetric disturbances, relevant to real ships. We find that, for intermediate Froude numbers, the wake angle follows an intermediate scaling law α≃Fr_{L}^{-2}, in agreement with the recent prediction of Noblesse et al. [Eur. J. Mech. B/Fluids 46, 164 (2014)]. We show that beyond a critical Froude number, which scales as A^{1/2} (where A is the length-to-width aspect ratio of the disturbance), the asymptotic scaling α≃Fr_{B}^{-1} holds, where now Fr_{B}=A^{1/2}Fr_{L} is the Froude number based on the disturbance width. We propose a simple model for this transition, and provide a regime diagram of the scaling of the wake angle as a function of parameters (A,Fr_{L}).