James Lighthill died on 17 July 1998, at the end of a ten-hour swim round the Channel Island of Sark. He had earlier, at age 49, been the first person ever to do this, and he was carrying out the swim for the seventh time when the exertion revealed a mitral valve weakness which had never been diagnosed, and which led to his sudden death in the water. The swim was one of many long ‘adventure swims’ which Lighthill liked to take, all characterized by strong tidal currents and often heavy seas. And Lighthill took much pleasure through exercising his comprehensive understanding of fluid mechanics first in preparing for them through study of local conditions and then in adapting his performance when, as often, he found that in practice the currents were not as charted and, in fact, often more treacherous.

Many obituary notices have already appeared in the national press in the UK and USA, and now in the newsletters and journals of learned societies; and extensive conspectuses of Lighthill's contributions to fluid mechanics and applied mathematics, and to science generally and to the administration of science, will be published in Annual Review of Fluid Mechanics (2000), and in Biographical Memoirs of Fellows of the Royal Society (2000). The reader will learn, from those accounts, of the unique range and depth of Lighthill's contributions; and virtually all readers should expect to be surprised and impressed to read of facets of Lighthill's work of which they were previously totally unaware.

The uniqueness theorem of Simon & Ursell (1984), concerning the linearized two-dimensional water-wave problem in a fluid of infinite depth, is extended in two directions. First, we consider a two-dimensional geometry involving two submerged symmetric bodies placed sufficiently far apart that they are not confined in the vertical right angle having its vertex on the free surface as the theorem of Simon & Ursell requires. A condition is obtained guaranteeing the uniqueness outside a finite number of bounded frequency intervals. Secondly, the method of Simon & Ursell is generalized to prove uniqueness in the axisymmetric problem for bodies violating John's condition provided the free surface is a connected plane region.

It is demonstrated, by using a simple model, that bedforms in a short tidal embayment can develop due to a positive feedback between tidal currents, sediment transport and bedforms. The water motion is modelled by the depth integrated shallow water equations. The system is forced by a prescribed free-surface elevation at the entrance of the embayment. For the sediment dynamics a diffusively dominated suspended load transport model is considered. Tidal averaging is used to obtain the bottom profiles at the long morphological time scale.

The stability of a constantly sloping equilibrium bottom profile is studied for various combinations of the model parameters. It turns out that without a mechanism that generates vorticity this equilibrium profile is stable. In that case small-scale perturbations can at most become marginally stable if no bedload term in the bottom evolution equation is incorporated. If vorticity is generated, in our model by bottom friction torques, the basic state is unstable. The spatial patterns of the unstable modes and their growth rates depend, among other things, on the strength of the bottom friction, the width of the embayment and the grain size: if the sediment under consideration consists of large particles, the equilibrium will be more stable than when smaller particles are considered. Without a diffusive term in the bed evolution equation, small-scale perturbations become unstable. To avoid this physically unrealistic behaviour bedload terms are included in the sediment transport. Furthermore, it is shown that using an asymptotic expansion for the concentration as given in earlier literature is only valid for small or moderate mode numbers and the technique is extended to large mode numbers. A physical interpretation of the results is also given.

In this paper we study a dynamical system consisting of a rigid body and an inviscid incompressible fluid. Two general configurations of the system are considered: (a) a rigid body with a cavity completely filled with a fluid and (b) a rigid body surrounded by a fluid. In the first case the fluid is confined to an interior (for the body) domain and in the second case it occupies an exterior domain, which may, in turn, be bounded by some fixed rigid boundary or may extend to infinity. The aim of the paper is twofold: (i) to develop Arnold's technique for the system ‘body + fluid’ and (ii) to obtain sufficient conditions for the stability of steady states of the system. We first establish an energy-type variational principle for an arbitrary steady state of the system. Then we generalize this principle for states that are steady either in translationally moving in some fixed direction or rotating around some fixed axis coordinate system. The second variations of the corresponding functionals are calculated. The general results are applied to a number of particular stability problems. The first is the stability of a steady translational motion of a two-dimensional body in an irrotational flow. Here we have found that (for a quite wide class of bodies) the presence of non-zero circulation about the body does not affect its stability – a result that seems to be new. The second problem concerns the stability of a steady rotation of a force-free rigid body with a cavity containing an ideal fluid. Here we rediscover the stability criterion of Rumyantsev (see Moiseev & Rumyantsev 1965). The complementary problem – when a body is surrounded by a fluid and both body and fluid rotate with constant angular velocity around a fixed axis passing through the centre of mass of the body – is also considered and the corresponding sufficient conditions for stability are obtained.

In this paper we investigate the temporal behaviour of a solute cloud in a heterogeneous porous medium using a stochastic modelling approach. The behaviour of the plume evolving from a point-like instantaneous injection is characterized by the velocity of its centre-of-mass and by its dispersion as a function of time. In a stochastic approach, these quantities are expressed as appropriate averages over the ensemble of all possible realizations of the medium. We develop a general perturbation approach which allows one to calculate the various quantities in a systematic and unified way. We demonstrate this approach on a simplified aquifer model where only the retardation factor R(x) due to linear instantaneous chemical adsorption varies stochastically in space. We analyse the resulting centre-of-mass velocity and two conceptually different definitions for the dispersion coefficient: the ‘effective’ dispersion coefficient which is derived from the average over the centred second moments of the spatial concentration distributions in every realization, and the ‘ensemble’ dispersion coefficient which follows from the second moment of the averaged concentration distribution. The first quantity characterizes the dispersion in a typical realization of the medium as a function of time, whereas the second one describes the (formal) dispersion properties of the ensemble as a whole. We show that for finite times the two quantities are not equivalent whereas they become identical for t→∞ and spatial dimensions d[ges ]2. The ensemble dispersion coefficient which is usually evaluated in the literature considerably overestimates the dispersion typically found in one given realization of the medium. We derive for the first time explicit analytical expressions for both quantities as functions of time. From these, we identify two relevant time scales separating regimes of qualitatively and quantitatively different temporal behaviour: the shorter of the two scales is set by the advective transport of the solute cloud over one disorder correlation length, whereas the second, much larger one, is related to the dispersive spreading over the same distance. Only for times much larger than this second scale, and spatial dimensions d[ges ]2, do the effective and the ensemble dispersion coefficients become equivalent due to mixing caused by the local transversal dispersion. Finally, the formalism is generalized to an extended source. With growing source size the convergence of the effective dispersion coefficient to the ensemble dispersion coefficient happens faster as the extended source already represents an ensemble of point sources. In the limit of a very large source size, convergence occurs on the time scale of advective transport over one disorder length. We derive explicit results for the temporal behaviour in the different time regimes for both point and extended sources.

The destabilization of the stationary basic flow occurring between two disks enclosed by a cylinder is studied experimentally when the radius of the disks is large compared to the spacing. In the explored range of the cell aspect ratio, when one disk only is rotating, circular vortices propagating to the centre are observed above a critical angular velocity. These structures occur naturally but can also be forced by small modulations of the angular velocity of the disk. For each rotation rate the dispersion relation of the instability is experimentally reconstructed from visualizations and it is shown that this dispersion relation can be scaled by the boundary layer thickness measured over the disk at rest. The bifurcation is found to be of supercritical nature. The effect of the forcing amplitude is in favour of a linear convective nature of this instability of the non-parallel inward flow existing above the stationary disk. The most unstable temporal frequency is found to be about four times the frequency of the rotating disk. The evolution of the threshold of this primary instability is described for different aspect ratios of the cell. Finally, two sets of experiments made under transient conditions are presented: one in order to investigate further a possible convective/absolute transition for the instability, and the other to compare with the impulsive spin-down-to-rest experiments of Savas (1983).

Bitumen froth is produced from the oil sands of Athabasca using the Clark's Hot Water Extraction process. When transported in a pipeline, water present in the froth is released in regions of high shear, namely at the pipe wall. This results in a lubricating layer of water that allows bitumen froth pumping at greatly reduced pressures and hence the potential for savings in pumping energy consumption. Experiments establishing the features of the self-lubrication phenomenon were carried out in a 25 mm diameter pipeloop at the University of Minnesota, and in a 0.6 m diameter pilot pipeline at Syncrude, Canada. The pressure gradient of lubricated flows in 25 mm, 50 mm and 0.6 m diameter pipes closely follow the empirical law of Blasius for turbulent pipe flow; the pressure gradient is proportional to the ratio of the 7/4th power of the velocity to the 5/4th power of the pipe diameter, but the constant of proportionality is about 10 to 20 times larger than that for water alone. We used Reichardt's model for turbulent Couette flow with a friction velocity based on the shear stress acting on the pipe wall due to the imposed pressure gradient to predict the effective thickness of the lubricating layer of water. The agreement with direct measurements is satisfactory. Mechanisms for self-lubrication are also considered.

A method to locate periodic structures in general three-dimensional Stokes flows with time-periodic boundary conditions is presented and applied to mixing cavity flows. Numerically obtained velocity fields and particle tracking schemes are used to provide displacement and stretching fields. From these the location and identification of periodic points can be derived. The presence or absence of these periodic points allows a judgement on the quality of the mixing process. The technique is general and efficient, and applicable to mixing flows for which no analytical velocity field is available (the case for all three-dimensional flows considered in this paper). Results are presented for three different mixing protocols in a three-dimensional time-periodic cavity flow, serving as an accessible test case for the methods developed. A major result is that periodic lines are obtained for these three-dimensional flows. These lines can be complex in geometry and their nature can change along a line from hyperbolic to elliptic. They can serve as practical criteria in the optimization of three-dimensional mixing processes.

Direct numerical simulation (DNS) is used to examine low Froude number free-surface turbulence (FST) over a two-dimensional mean shear flow. The Navier–Stokes equations are solved using a finite-difference scheme with a grid resolution of 1283. Twenty separate simulations are conducted to calculate the statistics of the flow. Based on the velocity deficit and the vertical extent of the shear of the mean flow, the Reynolds number is 1000 and the Froude number is 0.7. We identify conceptually and numerically the surface layer, which is a thin region adjacent to the free surface characterized by fast variations of the horizontal vorticity components. This surface layer is caused by the dynamic zero-stress boundary conditions at the free surface and lies inside a thicker blockage (or ‘source’) layer, which is due to the kinematic boundary condition at the free surface. The importance of the outer blockage layer is manifested mainly in the redistribution of the turbulence intensity, i.e. in the increase of the horizontal velocity fluctuations at the expense of the vertical velocity fluctuation. A prominent feature of FST is vortex connections to the free surface which occur inside the surface layer. It is found that as hairpin-shaped vortex structures approach the free surface, their ‘head’ part is dissipated quickly in the surface layer, while the two ‘legs’ connect almost perpendicularly to the free surface. Analysis of the evolution of surface-normal vorticity based on vortex surface-inclination angle shows that both dissipation and stretching decrease dramatically after connection. As a result, vortex structures connected to the free surface are persistent and decay slowly relative to non-connected vorticities. The effects of surface and blockage layers on the turbulence statistics of length scales, Reynolds-stress balance, and enstrophy dynamics are examined, which elucidate clearly the different turbulence mechanisms operating in the respective near-surface scales. Finally we investigate the effect of non-zero Froude number on the turbulence statistics. We show that the most significant effect of the presence of the free surface is a considerable reduction of the pressure–strain correlation at this surface, compared to that at a free-slip at plate. This reduction is finite even for very low values of the Froude number.

The effect of the downstream pressure (defined here as the wake pressure behind the tail of the reflecting wedge) on shock wave reflection in steady flows is investigated both numerically and analytically. The dependence of the shock wave configurations on the downstream pressure is studied. In addition to the incident-shock-wave-angle-induced hysteresis, which was discovered a few years ago, a new downstream- pressure-induced hysteresis has been found to exist. The numerical study reveals that when the downstream pressure is sufficiently high, an inverse-Mach reflection wave configuration, which has so far been observed only in unsteady flows, can be also established in steady flows. Very good agreement between the analytical predictions and the numerical results is found.

Rayleigh–Bloch surface waves are acoustic or electromagnetic waves which propagate parallel to a two-dimensional diffraction grating and which are exponentially damped with distance from the grating. In the water-wave context they describe a localized wave having dominant wavenumber β travelling along an infinite periodic array of identical bottom-mounted cylinders having uniform cross-section throughout the water depth. A numerical method is described which enables the frequencies of the Rayleigh–Bloch waves to be determined as a function of β for an arbitrary cylinder cross-section. For particular symmetric cylinders, it is shown how a special choice of β produces results for the trapped mode frequencies and mode shapes in the vicinity of any (finite) number of cylinders spanning a rectangular waveguide or channel. It is also shown how one particular choice of β gives rise to a new type of trapped mode near an unsymmetric cylinder contained within a parallel-sided waveguide with locally-distorted walls. The implications for large forces due to incident waves on a large but finite number of such cylinders in the ocean is discussed.

Trapped modes around a row of bottom-mounted vertical circular cylinders in a channel are examined. The cylinders are identical, and their axes equally spaced in a plane perpendicular to the channel walls. The analysis has been made by employing the multipole expansion method under the assumption of linear water wave theory. At least the same number of trapped modes is shown to exist as the number of cylinders for both Neumann and Dirichlet trapped modes, with the exception that for cylinders having large radius the mode corresponding to the Dirichlet trapped mode for one cylinder will disappear. Close similarities between the Dirichlet trapped modes around a row of cylinders in a channel and the near-resonant phenomenon in the wave diffraction around a long array of cylinders in the open sea are discussed. An analogy with a mass–spring oscillating system is also presented.

Parasitic ripple generation on short gravity waves (4 cm to 10 cm wavelengths) is examined using fully nonlinear computations and laboratory experiments. Time-marching simulations show sensitivity of the ripple steepness to initial conditions, in particular to the crest asymmetry. Significant crest fore–aft asymmetry and its unsteadiness enhance ripple generation at moderate wave steepness, e.g. ka between 0.15 and 0.20, a mechanism not discussed in previous studies. The maximum ripple steepness (in time) is found to increase monotonically with the underlying (low-frequency bandpass) wave steepness in our simulations. This is different from the sub- or super-critical ripple generation predicted by Longuet-Higgins (1995). Unsteadiness in the underlying gravity–capillary waves is shown to cause ripple modulation and an interesting ‘crest-shifting’ phenomenon – the gravity–capillary wave crest and the first ripple on the forward slope merge to form a new crest. Including boundary layer efects in the free-surface conditions extends some of the simulations at large wave amplitudes. However, the essential process of parasitic ripple generation is nonlinear interaction in an inviscid flow. Mechanically generated gravity–capillary waves demonstrate similar characteristic features of ripple generation and a strong correlation between ripple steepness and crest asymmetry.

We investigate the scattering of a plane acoustic wave by an axisymmetric vortex in two dimensions. We consider vortices with localized vorticity, arbitrary circulation and small Mach number. The wavelength of the acoustic waves is assumed to be much longer than the scale of the vortex. This enables us to define two asymptotic regions: an inner, vortical region, and an outer, wave region. The solution is then developed in the two regions using matched asymptotic expansions, with the Mach number as the expansion parameter. The leading-order scattered wave field consists of two components. One component arises from the interaction in the vortical region, and takes the form of a dipolar wave. The other component arises from the interaction in the wave region. For an incident wave with wavenumber k propagating in the positive X-direction, a steepest descents analysis shows that, in the far-field limit, the leading-order scattered field takes the form i(π−θ)eikX+½cosθcot(½θ) (2π/kR)1/2ei(kR−π/4), where θ is the usual polar angle. This expression is not valid in a parabolic region centred on the positive X-axis, where kRθ2=O(1). A different asymptotic solution is appropriate in this region. The two solutions match onto each other to give a leading-order scattering amplitude that is finite and single-valued everywhere, and that vanishes along the X-axis. The next term in the expansion in Mach number has a non-zero far-field response along the X-axis.

The interaction of waves on deep water with spatially varying currents may be described by a ray theory, with the wave amplitudes determined by the principle of conservation of wave action (CWA). However, all previous deep water derivations of CWA are restricted to the case of an irrotational current. In this paper, both the ray theory and CWA are derived by a WKB method without the assumption of irrotationality. Also derived is a new equation for a spatially varying phase shift which is not predicted by the usual ray theory, and which, in general, displaces the positions of the wave crests by a distance on the order of a wavelength. This phase shift, which is caused by variations of the current velocity with depth, vanishes in the irrotational case, and so is in accord with the irrotational theory.

We divide the interaction between wind and ocean surface waves into three parameter regimes, namely slow, intermediate and fast waves, that are distinguished by the ratio c/u∗ (c is the wave phase speed and u∗ is the friction velocity in the wind). We develop here an analytical model for linear changes to the turbulent air flow caused by waves of small slope that is applicable to slow and to fast waves. The wave-induced turbulent shear stress is parameterized here with a damped mixing-length model, which tends to the mixing-length model in an inner region that lies close to the surface, and is then damped exponentially to zero in an outer region that lies above the inner region. An adjustable parameter in the damped mixing-length model controls the rate of decay of the wave-induced stress above the inner region, and shows how the results vary from a model with no damping, which corresponds to using the mixing-length model throughout the flow, to a model with full damping, which, following previous suggestions, correctly represents rapid distortion of the wave-induced turbulence in the outer region.

Solutions for air flow over fast waves are obtained by analysing the displacement of streamlines over the wave; they show that fast waves are damped, thereby giving their energy up to the wind. There is a contribution to this damping from a counterpart of the non-separated sheltering mechanism that gives rise to growth of slow waves (Belcher & Hunt 1993). This sheltering contribution is smaller than a contribution from the wave-induced surface stress working against the orbital motions in the water. Solutions from the analysis for both slow and fast waves are in excellent agreement with values computed by Mastenbroek (1996) from the nonlinear equations of motion with a full second-order closure model for the turbulent stresses. Comparisons with data for slow and intermediate waves show that the results agree well with laboratory measurements over wind-ruffled paddle-generated waves, but give results that are a factor of about two smaller than measurements of purely wind-generated waves. We know of no data for fast waves with which to compare the model. The damping rates we find for fast waves lead to e-folding times for the decay of the waves that are a day or longer. Although this wind-induced damping of fast waves is small, we suggest that it might control low-frequency waves in a fully-developed sea.

The weakly nonlinear amplitude growth of slightly oblique instability waves in an incompressible free shear layer is shown to be first influenced by three-dimensionality in a limiting case for large Reynolds number when a particular order relationship is chosen between the spanwise scale and the amplitude of the small disturbance. The formulation resembles that for purely two-dimensional motion but includes the effect of vortex stretching in the nonequilibrium, nonlinear, viscous critical layer.