Mark J. Cooker

Pressure-impulse theory for liquid impact problems

A mathematical model is presented for the high pressures and sudden velocity changes which may occur in the impact between a region of incompressible liquid and either a solid surface or a second liquid region. The theory rests upon the well-known idea of pressure impulse, for the sudden initiation of fluid motion in incompressible fluids. We consider the impulsive pressure field which occurs when a moving fluid region collides with a fixed target, such as when an ocean wave strikes a sea wall. The boundary conditions are given for modelling liquid-solid and liquid-liquid impact problems. For a given fluid domain, and a given velocity field just before impact, the theory gives information on the peak pressure distribution, and the velocity after impact. Solutions for problems in simple domains are presented, which give insight into the peak pressures exerted by a wave breaking against a sea wall, and a wave impacting in a confined space. An example of liquid-liquid impact is also examined. Results of particular interest include a relative insensitivity to the shape of the incident wave, and an increased pressure impulse when impact occurs in a confined space. The theory predicts that energy is lost from the bulk fluid motion and we suggest that this energy can be transferred to a thin jet of liquid which is projected away from the impact region.

The interaction between a solitary wave and a submerged semicircular cylinder

Numerical solutions for fully nonlinear two-dimensional irrotational free-surface flows form the basis of this study. They are complemented and supported by a limited number of experimental measurements. A solitary wave propagates along a channel which has a bed containing a cylindrical bump of semicircular cross-section, placed parallel to the incident wave crest. The interaction between wave and cylinder takes a variety of forms, depending on the wave height and cylinder radius, measured relative to the depth. Almost all the resulting wave motions differ from the behaviour which was anticipated when the study began. In particular, in those cases where the wave breaks, the breaking occurs beyond the top of the cylinder. The same wave may break in two different directions: forwards as usual, and backwards towards the back of the cylinder. In addition small reflected waves come from the region of uniform depth beyond the cylinder. Experimental results are reported which confirm some of the predictions made. The results found for solitary waves are contrasted with the behaviour of a group of periodic waves.

Reflection of a high-amplitude solitary wave at a vertical wall

The collision of a solitary wave, travelling over a horizontal bed, with a vertical wall is investigated using a boundary-integral method to compute the potential fluid flow described by the Euler equations. We concentrate on reporting new results for that part of the motion when the wave is near the wall. The wall residence time, i.e. the time the wave crest remains attached to the wall, is introduced. It is shown that the wall residence time provides an unambiguous characterization of the phase shift incurred during reflection for waves of both small and large amplitude. Numerically computed attachment and detachment times and amplitudes are compared with asymptotic formulae developed using the perturbation results of Su & Mirie (1980). Other features of the flow, including the maximum run-up and the instantaneous wall force, are also presented. The numerically determined residence times are in good agreement with measurements taken from a cine film of solitary wave reflection experiments conducted by Maxworthy (1976).

Nonlinear three-dimensional gravity–capillary solitary waves

Steady three-dimensional fully nonlinear gravity–capillary solitary waves are calculated numerically in infinite depth. These waves have decaying oscillations in the direction of propagation and monotone decay perpendicular to the direction of propagation. They travel at a velocity

Wave impact pressure and its effect upon bodies lying on the sea bed

U

Water waves in a suspended container

smaller than the minimum velocity

Three-dimensional gravity-capillary solitary waves in water of finite depth and related problems

c_{min}

Characteristics of pressure pulses propagating through water-filled cracks

of linear gravity–capillary waves. It is shown that the structure of the solutions in three dimensions is similar to that found by Vanden-Broeck & Dias ( J. Fluid Mech. vol. 240, 1992, pp. 549–557) for the corresponding two-dimensional problem.

Nonlinear three-dimensional interfacial flows with a free surface

Abstract This paper is concerned with the very large, sudden pressures produced by water-wave impact on a vertical wall. Numerical and theoretical models are briefly described and we show that there are significant pressure gradients along an impermeable sea bed, away from the wall. This aspect of wave impact pressure has not previously been noted. The effects of such transient pressure gradients on bodies lying on the bed are examined in the light of pressure impulse theory. Our theoretical results indicate that bodies may be displaced by these pressure gradients when ordinary fluid drag is insufficient to move them. We discuss the impulse and the speed imparted to a rigid body lying within a region of pressure impulse which has constant horizontal gradient, and we show that the impulse is related to the bodys volume, and shape. Analytic results are given for a hemi-ellipsoidal “boulder”, which is free to translate in a direction perpendicular to the wall. The analysis for the hemi-ellipsoid encompasses the special cases of a semi-circular cylinder and a semi-circular disc. The case of a circular cylinder which touches the bed is also discussed, and it is shown that there is a significant vertical impulse on the body in addition to the horizontal component of the impulse.

Three-dimensional capillary-gravity waves generated by a moving disturbance

Abstract Analysis and experiments are carried out on a horizontal rectangular wave tank which swings at the lower end of a pendulum. The walls of the tank generate waves which affect the motion of the pendulum. For small displacements of the tank, linearised shallow water equations are used to model the motion, and there exist time-periodic solutions for the system whose periods are governed by a transcendental relation. Numerical and analytic solutions of this relation show that the fundamental period is greater than both the period of the empty tank (moving like a simple pendulum) and the fundamental period of the standing wave which occurs when the tank is removed from its supports and held fixed. For a rectangular tank the theory compares well with some experimental measurements. Qualitative observations are also made of the effect of breaking waves on the tank motion: for a tank which has a mass small compared with its load the energy dissipated by breaking waves can rapidly reduce the amplitude of swing of the tank. Potential flow theory is used with linearised free-surface boundary conditions to find time periodic motions for a tank with a hyperbolic cross section.