Coincident simultaneous measurements of the surface displacement and the horizontal velocity at the surface of steep and breaking waves are presented. The measurements involve a novel use of laser anemometry at the fluctuating air-water interface and clearly show the limitations of surface displacement measurements in characterizing steep and breaking wave fields. The measurements are used to examine the evolution of the surface drift velocity, spectra, wave envelopes, and forced long waves in unstable deep-water waves. Preliminary results of this work were reported by Melville & Rapp (1983).

We present a new theoretical model of ‘late-time’ phenomena related to the interaction of a planar shock with a local, discrete inhomogeneity in an ambient gas. The term ‘late-time’ applies to the evolution of the inhomogeneity and the flow field after interaction with the incident shock has ceased. Observations of a shock propagating through a bubble or a spherical flame have exhibited or implied the formation of vortex structures and have showed continual distortion of the bubble or flame. Our theory shows that this is due to the generation of long-lived vorticity at the edge of the discrete inhomogeneity. The vorticity interacts with itself through the medium of the fluid, and, depending on the geometry of the discrete inhomogeneity, can roll up into vortex filaments or vortex rings. To verify and amplify this theoretical description, we use numerical solutions of the fluid equations for conservation of mass, momentum, and energy to study the interaction of a weak shock with a cylindrical or spherical bubble. The simulated bubble has either a higher or lower density than the ambient gas. In this way, the calculations provide insights into the effects of both geometry and distortion of the local sound speed. The Mach number of the shock is 1.2, the ambient gas is air, and the pressure is 1 atmosphere. Because of the simple geometry of each bubble, the vorticity generated at the boundary rolls up into a vortex filament pair (cylindrical bubble) or a vortex ring (spherical bubble). The structural features and timescales of the phenomena observed in the calculations agree closely with recent experiments of Haas & Sturtevant, in which helium and Freon bubbles were used to provide the local departures from ambient density. The discussion of results includes a survey of alternative numerical methods, sources of uncertainty in velocities of interfaces or structures, as derived from the laboratory and numerical experiments, and the relationship of our analysis to other theories.

The three-dimensional development of a plane free shear layer subjected to small sinusoidal perturbations periodically placed along the span is experimentally studied. Both laser induced fluorescence and direct interface visualization are used to monitor the interface between the two fluids. The development of the different flow stabilities is obtained through analysis of the temporal and spatial evolution of the interface separating the two streams. It is shown that the characteristic time of growth of the two-dimensional shear instability is much shorter than that of the three-dimensional instability. The primary Kelvin-Helmholtz instability develops first, leading to the formation of an almost two-dimensional array of spanwise vortex tubes. Under the effect of the strain field created by the evolving spanwise vortices, the perturbed vorticity existing on the braids undergoes axial stretching, resulting in the formation of vortex tubes whose axes are aligned with the principal direction of the positive strain field. During the formation of these streamwise vortex tubes, the spanwise vortices maintain, to a great extent, their two-dimensionality, suggesting an almost uncoupled development of both instabilities. The vortex tubes formed through the three-dimensional instability of the braids further undergo nonlinear interactions with the spanwise vortices inducing on their cores a wavy undulation of the same wavelength, but 180° phase shifted with respect to the perturbation. In addition, it is shown that owing to the nature of the three-dimensional instability, the effect of vertical and axial perturbations are coupled. Finally, the influence of the amplitude and wavelength of the perturbation on the development of the two- and three-dimensional instabilities is described.

The evolution of the two- and three-dimensional structures in a temporally growing plane shear layer is numerically simulated with the discrete vortex dynamics method. We include two signs of vorticity and thus account for the effect of the weaker boundary layer leaving the splitter plate which is used to create a spatially developing mixing layer. The interaction between the two layers changes the symmetry properties seen in a single vorticity-layer calculation and results in closer agreement with experimental observations of the interface between the two streams. Our calculations show the formation of concentrated streamwise vortices in the braid region between the spanwise rollers, whereas the spanwise core instability is observed to grow only initially. Comparison with flow visualization experiments is given, and we find that the processes dominating the early stages of the mixing-layer development can be understood in terms of essentially inviscid vortex dynamics.

The stability properties of an inviscid, parallel, incompressible, free shear flow are studied. The shear profile is that of an unbounded, plane Couette flow containing a defect, or transition zone, whose magnitude ε is assumed to be small. The linearized eigenvalue problem is solved first for discretized models. When the defect has a finite thickness, the instability is confined to longitudinal wavenumbers, k [les ] 0(ε), in contrast to the more common 0(1) bandwidth, in units of inverse shear length. This observation motivates the application of a long-wave expansion to a smooth defect profile. A double expansion in both k and ε captures the whole waveband of the instability, and yields convergent expansions for the unstable eigenfunctions and for the dispersion relation describing their growth rate. The fastest growing modes are determined, and their back-reaction on the basic shear is calculated.

The velocity field and turbulence structure in the near wake of a thick flat plate with a tapered trailing-edge geometry are examined using both hydrogen-bubble flow visualization and hot-film anemometry measurements. Tests were conducted for Re1 = 8.5 × 105 in the region 0 < x+ < 6400 behind the trailing edge. The probe and visualization results indicate a similarity between both (i) velocity and turbulence structure variations wih x+ in the near wake, and (ii) the corresponding changes in similar flow characteristics with y+ within a turbulent boundary layer. In particular, visualization data in the vicinity of the wake centreline reveal the existence of strong streamwise flow structures in the region close (x+ < 270) to the trailing edge. The streamwise orientation of the observed structures diminishes as x+ increases. From hot-film measurements, two separate regions along the wake centreline can be distinguished: (i) a linear growth region which extends over 0 < x+ < 100, wherein the centreline velocity varies linearly with x+; and (ii) a logarithmic growth region for x+ > 270, wherein the centreline velocity varies as log x+. The similarity in behaviour between these regions and the comparable wall region of a turbulent boundary layer suggests the existence of a common functionality. This similarity is demonstrated by a simple linear relationship of the form y+ = Kx+, which is shown to approximately collapse the velocity behaviour both across a turbulent boundary layer and along the wake centreline to a unified set of empirical relationships.

Immiscible displacements in porous media with both capillary and viscous effects can be characterized by two dimensionless numbers, the capillary number C, which is the ratio of viscous forces to capillary forces, and the ratio M of the two viscosities. For certain values of these numbers, either viscous or capillary forces dominate and displacement takes one of the basic forms: (a) viscous fingering, (b) capillary fingering or (c) stable displacement. We present a study in the simple case of injection of a non-wetting fluid into a two-dimensional porous medium made of interconnected capillaries. The first part of this paper presents the results of network simulators (100 × 100 and 25 × 25 pores) based on the physical rules of the displacement at the pore scale. The second part describes a series of experiments performed in transparent etched networks. Both the computer simulations and the experiments cover a range of several decades in C and M. They clearly show the existence of the three basic domains (capillary fingering, viscous fingering and stable displacement) within which the patterns remain unchanged. The domains of validity of the three different basic mechanisms are mapped onto the plane with axes C and M, and this mapping represents the ‘phase-diagram’ for drainage. In the final section we present three statistical models (percolation, diffusion-limited aggregation (DLA) and anti-DLA) which can be used for describing the three ‘basic’ domains of the phase-diagram.

The effects of a sharp density interface and a rigid flat plate on oscillating-grid induced shear-free turbulence were investigated experimentally. A two-component laser-Doppler velocimeter was used to measure turbulence intensities in and above the density interface (with matched refractive indices) and near the rigid flat plate. Energy spectra, velocity correlations, and kinetic energy fluxes were also measured. Amplification of the horizontal turbulent velocity, coupled with a sharp reduction in the vertical turbulent velocity, was observed near both the density interface and the flat plate. These findings are in agreement with some previous results pertaining to shear-free turbulence near rigid walls (Hunt & Graham 1978) and near density interfaces (Long 1978). The results imply that, near the density interface, the turbulent kinetic energy in the vertical velocity component is only a small fraction of the total turbulent kinetic energy and indicate that the effects of the anisotropy created by the density interface or the flat plate are confined to the large turbulence scales.

The interaction of a sharp density interface with oscillating-grid-induced shear-free turbulence was experimentally investigated. A linear photodiode array was used in conjunction with laser-induced fluorescence to measure the concentration of dye that was initially only in the less dense layer. A laser-Doppler velocimeter was used to measure the vertical velocity in and above the density interface at a point where the dye concentration was also measured. Potential refractive-index-fluctuation problems were avoided using solutes that provided a homogeneous optical environment across the density interface. Internal wave spectra, amplitudes and velocities, as well as the vertical mass flux were measured. The results indicate that mixing occurs in intermittent bursts and that the gradient (local) Richardson number remains constant for a certain range of the overall Richardson number Rj, defined in terms of an integral lengthscale, buoyancy jump and turbulence intensity. The spectra of the internal waves decay as f−3 at frequencies below the maximum Brunt-Väisälä frequency. These findings give support to a model for oceanic mixing proposed by Phillips (1977) in which the internal waves are limited in their spectral density by sporadic local instabilities and breakdown to turbulence. The results also indicate that, for a certain Rj range, the thickness of the interfacial layer (normalized by the integral lengthscale of the turbulence) is a decreasing function of Rj. At sufficiently high Rj the interfacial thickness becomes limited by diffusive effects. Finally, we discuss a simple model for entrainment at a density interface in the presence of shear-free turbulence.

Longuet-Higgins’ exact expression for the increase in the Lagrangian-mean elevation of the free surface due to the presence of periodic, irrotational surface gravity waves is rederived from generalized Lagrangian-mean theory. The raising of the Lagrangian-mean surface as wave amplitude builds up illustrates the non-zero divergence of the Lagrangian-mean velocity field in an incompressible fluid.

Hydrodynamic damping of floating bodies is due mainly to wave radiation and viscous damping. The latter is particularly important in controlling those responses of the body for which the wave damping is small. The roll response of ship hulls near resonance in beam seas is an example of this. The present paper applies a discrete vortex method as a local solution to model vortex shedding from the bilges of a barge hull of rectangular cross-section and hence provides an analytic method for predicting its coupled motions in three degrees of freedom, including the effects of the main component of viscous damping. The method provides a frequency-domain solution satisfying the full linearized boundary conditions on the free surface.

We have carried out high-precision measurements of the heat transport in intermediate-size rectangular layers of convecting normal liquid 4He with Prandtl numbers of 0.52 and 0.70. The containers used for these experiments had horizontal dimensions, in units of the height d, of 13.4 × 5.95 (cell I) and 18.2 × 8.12 (cell II). The slopes N1 of the Nusselt curves were 0.56 and 0.70 respectively for cell I and cell II. These values are significantly lower than predictions for N1 for horizontally unbound layers, but comparable with results obtained in cylindrical containers of liquid helium with roughly the same number of convection rolls. For the two containers, the onset of the first instability after the onset of convection occurred at Rayleigh numbers R1 that were in reasonable quantitative agreement with the predictions of Busse and Clever for the skewed-varicose instability. For both containers, the transition at R1 was characterized by long transients ranging from ∼ 102 to ∼ 103 vertical-thermal-diffusion times. A decrease in the Nusselt number was also observed. As the Rayleigh number was increased above R1, a new steady state evolved and then additional transitions were observed. These transitions occurred at Rayleigh numbers labelled R2, R3,…, with a total of five transitions seen in cell I and a total of three transitions seen for cell II. The transition for each cell at R2 can be related quantitatively to the skewed-varicose instability, and the transition at R3 is associated with an oscillatory instability. For cell II, the time-dependence beginning at R3 persisted to the highest Rayleigh number studied, R = 11.7Rc. However, for container I, two more regimes of time-independent flow were observed; the last of these was at an unexpectedly high Rayleigh number of 6.7Rc. This work extends to lower Prandtl number recent studies made on moderate-size rectangular layers of convecting water and alcohol.

The joint effects of weak nonlinearity and weak linear damping on the dominant, antisymmetric gravity wave (excited by torsional oscillations of a plane wavemaker about a vertical axis) near its cutoff frequency in a rectangular channel are investigated, following Barnard, Mahony & Pritchard (1977). The evolution equations for the envelope of this mode are derived from the variational formulation previously developed for the parametrically excited cross-wave problem (Miles & Becker 1988). They are equivalent to those of Barnard et al., after correcting their damping and self-interaction terms, and, after appropriate normalization, differ from the cross-wave evolution equations only in the boundary condition at the wavemaker. Analytical approximations and the results of numerical integration for stationary envelopes (as observed in the experiments of Barnard et al.) are presented. The present results are somewhat closer to the observations of Barnard et al. than are their calculations, but the differences between the two calculations are not qualitatively significant.

This paper describes the induced pressure distribution, free-surface waves, vortical flow and wave drag of an exact solution of low-aspect-ratio flat-ship theory. An energy balance is derived which relates the spray drag, the energy carried away by the far-field waves and the vortical flow to the total wave drag.

If granular materials comprising particles of identical material but different sizes are sheared in the presence of a gravitational field, the particles are segregated according to size. The small particles fall to the bottom and the larger ones drift to the top of the sheared layer. In an attempt to isolate and study some of the essential segregation mechanisms, the paper considers a simplified problem involving the steady two-dimensional flow of a binary mixture of small and large spherical particles flowing down a roughened inclined chute. The flow is assumed to take place in layers that are in motion relative to one another as a result of the mean shear. For relatively slow flows, it is proposed that there are two main mechanisms responsible for the transfer of particles between layers. The first mechanism, termed the ‘random fluctuating sieve’, is a gravity-induced, size-dependent, void-filling mechanism. The probability of capture of a particle in one layer by a randomly generated void space in the underlying layer is calculated as a function of the relative motion of the two layers. The second, termed the ‘squeeze expulsion’ mechanism, is due to imbalances in contact forces on an individual particle which squeeze it out of its own layer into an adjacent one. It is assumed that this mechanism is not size preferential and that there is no inherent preferential direction for the layer transfer. This second physical mechanism in particular was proposed on the basis of observations of video recordings that were played back at slow speed. Since the magnitude of its contribution is determined by the satisfaction of overall mass conservation, the exact physical nature of the mechanism is of less importance. By combining these two proposed mechanisms the net percolation velocity of each species is obtained. The mass conservation equation for fines is solved by the method of characteristics to obtain the development of concentration profiles with downstream distance. Although the theory involves a number of empirical constants, their magnitude can be estimated with a fair degree of accuracy. A solution for the limiting case of dilute concentration of fine particles and a more general solution for arbitrary concentrations are presented. The analyses are compared with experiments which measured the development of concentration profiles during the flow of a binary mixture of coarse and fine particles down a roughened inclined chute. Reasonable agreement is found between the measured and predicted concentration profiles and the distance required for the complete separation of fine from coarse particles.

The classical point-vortex model for a Kármán vortex street is linearly stable only for a single, isolated, marginally stable, case. This property has been shown numerically to hold for streets formed by symmetric rows of uniform vortices of equal area. That result is extended here to the case in which the areas of the vortices in the two rows are not necessarily equal. The method used is an analytic perturbation valid when the vortex areas are small, and applied using an automatic symbolic manipulator.

Measurements of the lateral components of the vorticity fluctuation have been made in the self-preserving turbulent wake of a circular cylinder. Each component was obtained separately using two X-wires separated in the appropriate lateral directions. The two velocity derivatives which make up the streamwise vorticity component were also determined but not simultaneously. An approximation to the streamwise vorticity was made from these measurements. Moments, probability density functions and spectra of the three vorticity components across the wake are presented and discussed. The high-wavenumber behaviour of the spectra is compared with calculations, based on local isotropy. Satisfactory agreement with the calculations is obtained for the lateral vorticity components over a significant high-wavenumber range. The approximated streamwise vorticity spectrum tends towards the isotropic calculation at very large wavenumbers.

In the present paper we examine low-Prandtl-number thermal convection using a highly truncated modal approach. For the horizontal structure we assume a hexagonal planform as in Toomre Gough & Spiegel (1977) but including a vertical vorticity mode. The system develops a non-zero vertical vorticity component through a finite-amplitude instability. Following this, the system displays a Hopf bifurcation giving rise to periodic oscillations. The mechanism for this instability is associated with the growth of swirl in the azimuthal direction. We have found three different types of periodic solutions, possibly associated with subharmonic bifurcations, and their structure has been examined.

A large part of the present work is devoted to exploring the cases of mercury and liquid helium - or air - as the best-known examples of low and intermediate-Prandtl-number fluids. Results for mercury are quite satisfactory as far as frequencies and fluxes are concerned and they show reasonable agreement with experimental measurements at mildly supercritical Rayleigh values. On the other hand, for liquid helium or air agreement is poor.

The small-amplitude free vibrations of a spherical drop immersed in an outer immiscible fluid and in partial contact with a solid support are considered when both fluids are assumed to be viscous and incompressible, while gravity effects are neglected. Using the normal-mode decomposition and the Green-function method, the solution of the linearized Navier-Stokes equations is reduced to the solution of an eigenvalue problem. The model includes as particular cases the viscous model for a free drop proposed by Prosperetti (1980) and the inviscid model for a supported drop previously proposed by the authors.

The influence of the viscosity and of the support size are analysed both for the bubble and for the drop. At large values of the viscosity, the free drop shows significant differences with respect to the unsupported drop and a singular behaviour of the eigenvalue problem as the support size tends to zero.

The comparison with the available experimental data shows a quite satisfactory agreement for both the vibration frequency and the damping constant, provided that the support angle is not too large.

The primary objective of this paper is to provide a detailed confirmation of the validity of potential-flow theory for describing steep gravity waves produced in an experimental tank. Very high-resolution computations are carried out which use a refined mixed Eulerian-Lagrangian solution scheme under the assumptions of potential flow. The numerical results for a plunging breaker produced by a programmed piston-type wavemaker are found to be in excellent agreement with tank measurements up to and including overturning. The calculated free-surface elevations are almost indistinguishable from measured profiles, even close to where the wave plunges. The horizontal and vertical water-particle velocities measured with a laser anemometer throughout the water depth at two longitudinal stations are also well predicted by the theory. In contrast to the fully nonlinear theory, predictions based on linearized theory become poorer as the wave packet moves down the tank. To allow other investigators to evaluate the computations and experiments, the Fourier amplitudes and phases which completely specify the time history of the wavemaker's velocity are given in Appendix B.