Surface water waves on depth dependent flows

Abstract

The present thesis provides essential insights into surface water waves propagating atop a horizontal current whose magnitude and direction may vary arbitrarily with water depth. A comprehensive theory in this regard is developed in the framework of linear wave theory in three dimensions, being readily applied to a wide range of realistic circumstances. General theoretical solutions to different boundary value problems are presented. In particular, explicit expressions with regard to the surface elevation and the vertical velocity are derived. The boundary value problems include the Cauchy-Poisson problem, surface disturbances generated by an initial impulsive and a time-dependent pressure, and a steady pressure that normally works as the model of moving vessels and oscillating travelling sources. Efforts focus especially on the dispersion relation and the effects of a subsurface shear current on surface waves. A subsurface shear current is most often found to have significant effects on surface waves. In particular, the presence of a current of uniform vorticity is analysed in detail for the problems of ship waves, an oscillating advancing source and wave interferences. A theory is especially presented to calculate waves from a general, time-dependent applied surface pressure acting on the surface of a horizontally directed shear current which may vary arbitrarily with depth in both direction and magnitude. It is based on deriving the response function in the context of waves generated by an impulsive applied pressure. Effective approaches to calculate wave resistance without undue difficulty are presented. Strikingly, a lateral radiation force – that is defined towards the starboard (right) – is firstly found apart from the well-known wave resistance along the stern-wise direction due to the presence of a shear current when a ship is making an oblique angle with the shear current. The lateral radiation force may amount to 20 percent of the normal wave resistance in some specific situations. As for waves on a current in arbitrary variation of water depth, an implicit dispersion relation is derived, which poses potential challenges in obtaining analytical solutions. Several semi-analytical approaches to solve/approximate the dispersion relation are hence derived. In particular, a direct integration approach – that solves the linearised Rayleigh equation and implicit dispersion relation in a coupled way – and approximations based on a perturbation method are presented. The proper criteria under which different perturbed approximate dispersion relations are applicable are determined. Furthermore, the analytical solutions of the dispersion relation under limited circumstances are derived, e.g. for a shear current of uniform vorticity and stationary waves for a specific class of shear profiles of non-zero curvature. Despite the fact that linear waves in the presence of a linear shear current have been extensively analysed in two dimensions, studies in three dimensions are scarce. This means that realistic three dimensional effects may be in some cases overlooked or yet discovered. The present thesis hence attempts to fill this gap based on theoretical as well as numerical analysis. Effects of a uniform vorticity are specially analysed in the context of ship waves, waves generated by an oscillating travelling source and interferences of waves generated by a two-point wavemaker of monohull ships. Fascinating and novel features are found due to the uniform vorticity S that is known either as the ’intrinsic shear Froude number’ Frsb = S_L/g or the ’shear Froude number’ Frs = |V|S/g (L: the reference length; |V|: moving speed of a wavemaker; g: the gravitational acceleration). In particular, asymmetrical ship wave patterns, the critical shear velocity above which the transverse ship waves vanish, the transitions between the sub-critical and supercritical situations due to the complex interplays of the shear current and seabed, non-constant Kelvin angles, and a somewhat similar effect as a finite water depth on wave interferences are shown for the problem of ship wakes. All of those novel features would not have been found if theoretical studies are constrained to 2D. Furthermore, the classical Doppler resonance occurs when the non-dimensional frequency τ = |V|ω/g (ω : the oscillating frequency) is equal to 1/4 in the absence of a shear current, while there may be multiple Doppler resonances – as many as 4 – for Frs > 1/3 in deep water due to the presence of a linear shear current. It is also indicated that the Doppler resonance may be profoundly modified even for a linear current of weak vorticity.