This paper describes an experimental investigation of the conditions for which the asymptotic description of longitudinal dispersion given by Taylor (1954) would apply. At non-dimensional times following the release of a dye pulse that are significantly larger than those previously investigated, the integrated concentration curves were observed to be skewed. At relatively short times from release the concentration curves appear to be well described by the models presented by Sullivan (1971) and by Chatwin (1973). Some features of the asymptotic behaviour, namely the translation of the modal value of the integrated concentration curve at the discharge velocity and the constant temporal growth rate of the variance, are observed at the longest times following release. On the basis of these observations it is estimated that a non-dimensional time interval of tu*/d = O(105/R*), where R* = u*d/v, u* is the friction velocity, v the kinematic viscosity and d the tube diameter, is required for the Taylor result to become applicable. Thus application of Taylor's theory is significantly restricted in turbulent flows, especially those with irregular boundaries and those that are not stationary. There the variations in the flow must be small with respect to an equivalent ‘development time’ if a value of the ‘local’ longitudinal diffusion coefficient is to have meaning.

The breakdown of Rossby waves in a bounded system is studied for the case in which the wave amplitude is small. In a very long, laterally bounded, channel all waves are unstable via second-order resonant interactions except those of wavenumber π/L in the cross-channel direction (where L is the channel width), which are stable if their longitudinal wavenumber is greater than 0·681π/L. These waves are, however, unstable to weaker side-band interactions, so that all waves with non-zero longitudinal wavenumber are unstable. The transition from sideband to triad instability occurs where the group velocity of the basic wave is equal to the velocity of long waves.

In this paper we intend to predict the magnitude of the contribution to the Reynolds stress of bursting events: ‘ejections’, ‘sweeps’, ‘inward interactions’ and ‘outward interactions’. We shall do this by making use of the conditional probability distribution of the Reynolds stress − uv, which can be derived by applying the cumulant-discard method to the Gram-Charlier probability distribution of the two variables u and v. The Reynolds-stress fluctuations in openchannel flows over smooth and rough beds are measured by dual-sensor hot-film anemometers, whose signals are conditionally sampled and sorted into the four quadrants of the u, v plane by using a high-speed digital data processing system.

We shall verify that even the third-order conditional probability distribution of the Reynolds stress shows fairly good agreement with the experimental results and that the sequence of events in the bursting process, i.e. ejections, sweeps and interactions, is directly related to the turbulent energy budget in the form of turbulent diffusion. Also, we shall show that the roughness effect is marked in the area from the wall to the middle of the equilibrium region, and that sweeps appear to be more important than ejections as the roughness increases and as the distance from the wall decreases.

Numerical solutions are presented for the detailed characteristics of the drift current and Langmuir circulation system produced according to a theory described in part 1 of this paper. The motions that develop are traced from an initially quiescent state, and the results are compared with field observations of currents in Langmuir circulations. Qualitative features of the phenomenon appear to be reproduced by the theory and, with the appropriate choice of an empirical parameter, the solutions seem to be quantitatively consistent with field data.

This paper investigates the flow near the summit of steep, progressive gravity wave when the crest is still rounded but the flow is approaching Stokes's corner flow. The natural length scale in the neighbourhood of the summit is seen to be l =q2/2g, where g denotes gravity and q is the particle speed at the crest in a reference frame moving with the wave speed. We show that a class of self-similar smooth local flows exists which satisfy the free-surface condition and which tend to Stokes's corner flow when the radial distance r becomes large compared withl. The behaviour of the solution at large values of r/l is shown to depend on the roots of the transcendental equation \[ K \tan h K = \pi/2\surd{3}. \] The two real roots correspond to a damped oscillation of the free surface decaying like (l/r)½. The positive imaginary roots correspond to perturbations vanishing like higher negative powers of r.

The complete flow is calculated by transforming the domain onto the interior of a circle in the complex plane and expanding the potential at the surface in a Fourier series. The computation is checked by an independent method, based on approximating the flow by a sequence of dipoles. The profile of the surface is found to intersect its asymptote at large values of r/l. This implies that the maximum slope slightly exceeds 30°. The computed value 30·37° is in close agreement with that obtained by extrapolating the maximum slopes of steep gravity waves, as calculated by previous authors. The vertical acceleration of a particle at the crest is 0·388g. In the far field, however, the acceleration tends to the value ½g corresponding to the Stokes corner flow.

A laser light-scattering technique has been used to study the relative particle concentration field in a round turbulent air jet. Measurements were made in the far field of a smoke-marked turbulent jet exhausting into a secondary air stream. Radial distributions of mean particle concentration, concentration fluctuation intensity and intermittency were measured at several streamwise locations. Concentration fluctuation power spectra and the micro- and integral scales of the concentration fluctuations were measured on the jet axis. The effects of ambiguity noise and noise due to optical path attenuation on the performance of the laser light-scattering system are discussed.

This paper is concerned with a general study of the modal structure for stratified viscous plane Couette flow with a constant buoyancy frequency. When the overall Richardson number Ri is zero, the velocity and temperature modes are distinct but as Ri is increased there is an intricate interaction between them. Some simple analytical results are obtained for large and small values of the Reynolds number and more detailed results are given for $Ri = 0, \frac{1}{8}, \frac{1}{4}$ and ½. The present theory would appear to be reasonably complete for 0 [les ] Ri [les ] ¼; for Ri > ¼, however, an important open question concerns the relationship between the limiting form of the viscous modes as the Reynolds number tends to infinity and the spectrum of internal gravity waves.

High frequency surface waves are generated by the forced heaving of either two half-immersed spheres in infinite water or by a half-immersed sphere in a hemispherical lake. The virtual-mass coefficients can be found, to leading order, in terms of wave-free limit potentials.

Large-scale structures in the form of instability waves are an inherent part of a shearlayer mixing process. Such structures are shown to be present in an acoustically and aerodynamically well behaved jet even at high Mach numbers. They do not directly radiate significant acoustic power in a subsonic jet, but do govern the production of the turbulent fluctuations which radiate broad-band jet noise. Over the whole subsonic Mach number range, a significant increase in jet noise can be produced by exciting the shear layer with a fluctuating pressure at the nozzle of only 0·08 % of the jet dynamic head but with the correct Strouhal number. Such excitation by internal acoustic, aerodynamic or thermal fluctuations could explain the variability of jet noise measurements between different rigs and could also be responsible for some components of ‘excess’ noise.

A statically stable stratification with buoyancy frequency N2(z) = z2 is found to cause large changes in the modal structure for viscous plane Couette flow (as compared with the case when N2(z) = 1) and it also has a strongly destabilizing effect on the flow. On minimizing with respect to both wavenumber and Richardson number, it is found that the flow is unstable if the Reynolds number is greater than about 183. A study of the Reynolds stress and vertical buoyancy flux shows that there is a large transfer of energy from the basic flow to the velocity disturbances and this is consistent with such a surprisingly low value of the minimum critical Reynolds number.

Johnson's (1973) description of a solitary wave in water of slowly varying depth is extended to a channel of slowly varying breadth and depth b and d on the assumption that the scale for the variation of b and d is large compared with d5/2a3/2. It is inferred from conservation of energy that the amplitude of the wave is proportional to $b^{-\frac{2}{3}}d^{-1}$ (cf. Green's law $a\propto b^{-\frac{1}{2}}d^{-\frac{1}{4}}$ for long waves of small amplitude). Comparison with experiment (Perroud 1957) yields fairly satisfactory agreement for a linearly converging channel of constant depth. The agreement for a linearly diverging channel is not satisfactory, but the experimental data are inadequate to support any firm conclusion.

The flow of viscous lubricant in narrow gaps is considered for those geometries in which cavitation arises. A detailed review is presented of those boundary conditions which have been proposed for terminating the lubrication regime (i.e. those valid where the cavity forms). Finally it is shown that a uniform cavity-fluid interface remains stable to small disturbances provided that \[ \frac{d}{dx}\left(P+\frac{T}{r}\right) < 0, \] in which T and r represent the surface tension of the fluid and the radius of curvature of the interface respectively whilst dP/dx is the gradient of fluid pressure immediately upstream of the interface.

The structure of the turbulent shear flow over a propagating wave of fixed frequency is examined. The vertical and horizontal velocities were measured in a wind-wave facility at Stanford University. The structure of the wave-perturbed turbulence was found to depend significantly on the ratio of the local mean velocity to the wave speed. The results support the idea of ‘cat's-eye’ type flow about the mean critical height.

Measurements of both the velocity and the temperature field have been made in the thermal layer that grows inside a turbulent boundary layer which is subjected to a small step change in surface heat flux. Upstream of the step, the wall heat flux is zero and the velocity boundary layer is nearly self-preserving. The thermal-layer measurements are discussed in the context of a self-preserving analysis for the temperature disturbance which grows underneath a thick external turbulent boundary layer. A logarithmic mean temperature profile is established downstream of the step but the budget for the mean-square temperature fluctuations shows that, in the inner region of the thermal layer, the production and dissipation of temperature fluctuations are not quite equal at the furthest downstream measurement station. The measurements for both the mean and the fluctuating temperature field indicate that the relaxation distance for the thermal layer is quite large, of the order of 1000θ0, where θ0 is the momentum thickness of the boundary layer at the step. Statistics of the thermal-layer interface and conditionally sampled measurements with respect to this interface are presented. Measurements of the temperature intermittency factor indicate that the interface is normally distributed with respect to its mean position. Near the step, the passive heat contaminant acts as an effective marker of the organized turbulence structure that has been observed in the wall region of a boundary layer. Accordingly, conditional averages of Reynolds stresses and heat fluxes measured in the heated part of the flow are considerably larger than the conventional averages when the temperature intermittency factor is small.

The steady and unifrom flow of a viscous fluid past a unifrom cavity in a gemoetry with small, yet arbitrary, film thickness is considered. A mathematical model for describing steady perturbations to such a flow is presented in which the perturbation to the cavity-fluid interface is represented by a small amplitude harmonic wave of wavenumber n. A linearized perturbation analysis then permits the formulation of a boundary-value problem involving the homogeneous Reynolds equation, the solution to which determines both n and the perturbed pressure field.

Numerical and approximate analytic solutions are determined for the cylinderplane geometry in which fluid flows between a rotating cylinder and a Perspex block. Whilst these compare well with experimental data over the whole range \[ 0.1 < \eta U/T < 3, \] closest agreement between theory and experiment is attained for small values of both ηU/T and n.

The problem of the collapse of a vapour/gas bubble attached to a solid wall and initially perturbed from a hemispherical shape is solved numerically by the variational method, in which the bubble's viscosity and compressibility in liquid are neglected. The effects of surface tension on the collapsing bubble are taken into account. The rebounding processes of a non-hemispherical gas bubble are simulated: the gas inside the bubble undergoes an adiabatic process. The results of numerical calculations are given for two initial shapes: one is close to a prolate spheroid, the other is close to an oblate spheroid. The governing equations for the motion of a bubble can be written in matrix form, which is simpler than that derived from perturbation theory. This analysis using the variational method may be applied to more complicated problems.

Singular perturbation techniques are used to calculate the migration velocity of a spherical particle sedimenting, at low Reynolds numbers, in a stagnant viscous fluid bounded by one or two infinite vertical plane walls. The method is then used to study the migration of a pair of spherical particles sedimenting either in unbounded fluid or in fluid bounded by a plane vertical wall. The migration phenomenon is studied experimentally by recording the trajectory of a spherical particle settling through a viscous fluid bounded by parallel vertical plane walls. Duct- to particle-diameter ratios in the range of 27 to 48 were used with the Reynolds number based on the particle radius being between 0·03 and 0·136.

In all cases the particle is observed to migrate away from the walls until it reaches an equilibrium position at the axis of the duct. The experimentally determined migration velocities agree well with those predicted by the present theory.

Large (gigajoule) amounts of energy can in principle be stored as kinetic energy in liquid metal circulating round a torus and can be extracted at the gigawatt level by Alfvén waves propagating along an imposed axial field. A major limitation on the energy that may be so stored is the disruption of these primary Alfvén waves by secondary flows in meridional planes, associated with out-of-balance centrifugal forces ahead of and behind the waves and non-uniform magnetic pressures at the wave fronts. Vorticity, created at the wave, itself propagates in secondary Alfvén waves.

This paper gives a linearized treatment of these secondary motions and the associated perturbations of the imposed axial field and compares the resulting disruption of the primary wave mode with crude estimates made in an earlier paper. The main case treated is the discharge of the stored energy into a matched resistor by an Alfvén step wave but the secondary consequences of standing primary waves are also explored. The nature of the solutions depends on the electromagnetic characteristics of the walls normal to the imposed field. The problem is mathematically interesting because it involves the joint solving of elliptic and hyperbolic equations that are coupled by the boundary conditions at these walls.

An extension is made of the α-effect model of the earth's dynamo into the nonlinear regime following the prescription of Malkus & Proctor (1975). In this model, the effects of small-scale dynamics on the α-effect are suppressed, and the global effects of induced velocity fields examined in isolation. The equations are solved numerically using finite-difference methods, and it is shown that viscous and inertial forces are unimportant in the final equilibration, as suggested in the above paper.