The linear stability of a uniformly internally heated, self-gravitating, rapidly rotating fluid sphere is investigated in the presence of an azimuthal magnetic field B0(r, θ)ϕ and azimuthal shear flow U0(r, θ)ϕ (where (r, θ, ϕ) are spherical polar coordinates). Solutions are calculated numerically for magnetic field strengths that produce a Lorentz force comparable in magnitude to that of the Coriolis force. The critical Rayleigh number Rc is found to reach a minimum here and the qualitative behaviour of the thermally driven instabilities in the absence of a shear flow (U0 = 0) is similar to that found by earlier workers (e.g. Fearn 1979b) for the simpler basic state B0 = r sin θ. The effect of a shear flow is followed as its strength (measured by the magnetic Reynolds number Rm) is increased from zero. In the case where the ratio q of thermal to magnetic diffusivities is small (q [Lt ] 1) the effect of the flow becomes significant when Rm = O(q). For Rm > q three features are evident as Rm is increased: the perturbation in the temperature field (but not the other variables when Rm < O(1)) becomes increasingly localized at some point (rL, θL); the phase speed of the instability tends towards the fluid velocity at that point; and Rc increases with Rm with the suggestion that Rc ∝ Rm/q for Rm [Gt ] q although the numerical resolution is insufficient to verify this. Greater resolution is achieved for a simpler problem which retains the essential physics and is described in the accompanying paper (Fearn & Proctor 1983). The possible significance of these results to the geomagnetic secular variation is discussed.

The role that differential rotation plays in the hydromagnetic stability of rapidly rotating fluids has recently been investigated by Fearn & Proctor (1983) (hereinafter referred to as I) as part of a wider study related to the geodynamo problem. Starting with a uniformly rotating fluid sphere, the strength of the differential rotation was gradually increased from zero and several interesting features were observed. These included the development of a critical region whose size decreased as the strength of the shear increased. The resolution of the two-dimensional numerical scheme used in I is limited, and consequently it was only possible to consider small shear strengths. This is unfortunate because differential rotation is probably an important effect in the Earth's core and a more detailed study at higher shear strengths is desirable. Here we are able to achieve this by studying a rapidly rotating Bénard layer with imposed magnetic field B0 = BMsϕ and shear U0 = UMsΩ(z)ϕ, where (s, ϕ, z) are cylindrical polar coordinates. In the limit where the ratio q of the thermal to magnetic diffusivities vanishes (q = 0), the governing equations are separable in two space dimensions and the problem reduces to a one-dimensional boundary-value problem. This can be solved numerically with greater accuracy than was possible in the spherical geometry of I. The strength of the shear is measured by a modified Reynolds number Rt = UMd/k, where d is the depth of the layer and κ is the thermal diffusivity, and the shear becomes important when Rt [ges ] O(1). It is possible to compute solutions well into the asymptotic regime Rt [Gt ] 1, and details of the behaviour observed are dependent on the nature of Ω(z). Specifically, two cases were considered: (a) Ω(z) has no turning point in 0 < z < 1, and (b) Ω(z) has a turning point at z = zT, 0 < zT < 1 (Ω′(zT = 0, Ω″(zT) ≠ 0). In both cases, as Rt increases a critical layer centred at z = zL develops, with width proportional to (a) Rt−1/3, (b) Rt−¼. In the case where Ω(z) has a turning point, the critical layer is located at the turning point (zL = zT). The critical Rayleigh number Rc increases with (a) Rc ∝ Rt, (b) Rc ∝ RRt−¼, and the instability is carried around with the fluid velocity at the critical layer. The relevance of these results to the geomagnetic secular variation is discussed.

The no-slip condition on rigid boundaries necessitates a correction to the critical value of the Rayleigh–Darcy number for the onset of convection in a horizontal layer of a saturated porous medium uniformly heated from below. It is shown that the use of the Brinkman equation to obtain this correction is not justified, because of the limitations of that equation. These limitations are discussed in detail. An alternative procedure, based on a model in which the porous medium is sandwiched between two fluid layers, and the Beavers–Joseph boundary condition is applied at the interfaces, is described, and an expression for the correction is obtained. It is found that the correction can be of either sign, depending on the relative magnitudes of the parameters involved.

The evolution of an internal wavebreaking event in a continuous stable stratification is examined using schlieren colour imagery combined with streak photography. For standing but strongly interacting wavemodes forced to breaking, the Richardson number defined by local gradients of density and velocity is critically low only in the immediate vicinity of the breaking region where convective overturning also occurs. Breaking and vertical mixing results from the rapid development of three-dimensional interleaving density microstructure within a confined volume which persists and gravitates to modify weakly the surrounding density distribution. Continual mixing through internal motion is seen as a widespread repetition of such events.

Based on the observations, a general description of the process is proposed and applied in two types of simple kinematical model to provide an estimate of ‘mixing efficiency’, the ratio of potential energy gained through stratification weakening to kinetic energy expended in motions on the scales of the mixing event. These models do not rely on an assumed similarity between momentum and buoyancy transfer. They yield, for complete homogenization of small discrete volumes, an efficiency of ¼ in a linear stratification. Mixing across a density interface is predicted to have a lower efficiency, and lower efficiency is expected where homogenization is incomplete.

Using experimental measurements, estimates are made of the efficiency of conversion of kinetic energy into potential energy through vertical mixing in a continuously stratified fluid. In the experiments kinetic energy was supplied continuously at a rate of ε to a fundamental internal wave mode in a rectangularly bounded and initially linear stable stratification. Mixing resulted from the instability of this wave and its consequent ‘breaking’. Potential energy was gained by the system at rate $\dot{p}$ through the gradual weakening of the stratification.

The instability is predictable using wave-interaction theory, affording a means of estimating the amount of kinetic energy lost (at rate εV) to laminar viscosity without first cascading to the fine scales characteristic of turbulent mixing.

With account taken of this viscous loss the mixing efficiency $\dot{p}/\epsilon_{\rm M}$, based upon the residual kinetic energy input εM (= ε − εV) was found to be approximately constant, and not significantly correlated with the rate at which energy was supplied, nor with the estimated instantaneous minimum gradient Richardson number. The average value for eight separate experimental runs using two different experimental configurations was 0·26 with a sample standard deviation of 0·06.

Measured density profiles also afforded an estimate of the effective vertical diffusivity κd of density as a result of mixing. Vertically averaged values of the product of κd and the squared local static buoyancy frequency N, $\overline{\kappa_{\rm d}N^2}$, were found to have an average for seven runs of 0·24εM, with a standard deviation for the coefficient of 0·1, and no significant correlation with energy supply rate.

These results, the first of their kind to correct for incidental losses, substantiate the values previously assumed in estimates of dissipation and vertical diffusion in the ocean and the atmosphere, and validate the assumption of similarity between buoyancy and mass transfer on which they are based. The efficiency value also agrees with the kinematic prediction for localized homogenization in small discrete volumes made in the companion paper (McEwan 1983). On the basis of that work it is inferred from the present results that the mixing efficiency is only weakly dependent upon Prandtl number provided that this is of order unity or greater.

The present investigation considers the vortex wakes behind finite-length yawed cylinders that are stationary or vibrating transversly in a uniform flow. In the Reynolds-number range 160–103 a number of three-dimensional and nominally two-dimensional wake flows are observed and attributed to the dominance of end conditions over rather large spans of the cylinder (aspect ratios up to 100).

The mechanism of peristaltic transport of an incompressible viscoelastic fluid by means of an infinite train of sinusoidal waves travelling along the wall of the duct is studied in the case of a plane flow. The main assumptions are that the relevant Reynolds number is small enough to neglect inertia forces, and that the ratio of the wavelength and the channel height is large, which implies that the pressure is constant over the cross-section. For sufficiently small values of the ratio of the wave amplitude and the mean height of the channel, details of the fluid motion are studied analytically within a second-order approximation with respect to the amplitude ratio. Under these conditions the integral constitutive equation of finite linear viscoelasticity is relevant. Particular attention is given to the pressure–discharge characteristics of the peristaltic pump and to the pumping efficiency. The results are influenced by specific values of the complex viscosity of the fluid, which can be determined using standard rheometers. In general, the rate of discharge turns out to be a non-monotonic function of the wave speed. This leads to an optimal wave speed, for which the memory of the fluid particles extends over several wave periods. From an energetic point of view, relatively small wave speeds are the best, where the fluid changes its state slowly such that the memory and with it the elasticity of the fluid do not influence the flow field at all. As the dimensionless memory parameter tends to zero, the analytical results reduce to the well-known case of a Newtonian fluid.

The flow between two finite rotating disks enclosed by a cylinder is investigated both numerically and experimentally. For this finite geometry the full stationary Navier–Stokes equations are solved numerically without similarity assumptions. Experimental results are obtained by means of stereophotography of small tracer particles. The results are in good agreement with the numerical solution. Owing to the presence of the cylinder sidewall, the solution is found to be unique for all values of the parameters considered. When the disks rotate in opposite senses with counter-rotation above 15%, a stagnation point appears at the slower-rotating disk. This stagnation point is associated with a two-cell structure in the meridional plane and is experimentally observed as a ring of particles at the slower-rotating disk. Near the axis of rotation the solution is found to satisfy similarity demands; for weak counter-rotation the solution is of Batchelor type near the axis of rotation, but for strong counter-rotation a Stewartson profile is found to be more adequate for the description of the tangential velocity near the axis.

The diffraction of a gravity wave of length λ that is obliquely incident upon, and the radiation from, a periodic row of vertical circular ducts of radii a and horizontal spacing b with mouths at a distance h below the free surface of a deep ocean are determined through integral-equation and variational formulations. Numerical results for the reflection coefficient, the pressure-amplification factor (the ratio of the complex amplitude of the wave-induced pressure in the depths of the duct to that of the incident wave at the level of the mouths), and the radiation impedance (the real and imaginary parts of which are measures of the radiation damping and the virtual mass or stiffness of the fluid external to the ducts) are presented as functions of a/λ with a/b and a/h as parameters for the special case of normal incidence with λ > b (which implies that the crests of the scattered waves are parallel to the midplane of the ducts). These results, which complement those of Simon (1981) for a single duct, are of practical interest for wave-power absorption.

The Lighthill theory has been extended so that it may be used to determine the flow noise induced by a turbulent boundary layer over a plane homogeneous flexible surface. The influence of the surface properties and the mean flow on the sound generated is brought out explicitly through the use of a Green function. It is found that there is an analogy between the sound generated by turbulence and equivalent sources placed between a surface with the same compliance as the physical surface and a hypothetical vortex sheet positioned at the outer edge of the boundary layer. This analogy is used to determine the spectrum of the surface-pressure fluctuations under statistically stationary turbulence. The form of this wall-pressure spectrum is investigated in detail for three particular types of surface: rigid surfaces, bending plates and sound-absorbent liners.

The algebraic stress model developed by Launder & Ying for the secondary flow of the second kind was employed with the κ and ε model for the prediction of fully developed turbulent flows in square, rectangular and trapezoidal ducts using the numerical procedure designed for ducts of arbitrary cross-sectional shape. Results of the calculation are compared extensively with available experimental data, with strong emphasis on the local structures of turbulence to reveal full features of this particular stress model.

When a film of liquid runs down a vertical wall and enters a large pool a number of stationary horizontal ripples are observed on the film just above the point of entry. These ripples are explained and analysed by the method of matched expansions and the results have a bearing on more-complicated problems. Next the case when the wall itself is moving vertically downwards is analysed. This is of interest in roll-coating technology, where it is normally desired to achieve smooth entry of the descending film, without air entrainment. The paper gives sufficient conditions for smooth entry, and these are consistent with known experimental results.

The interference of flow around two circular cylinders forming a cross can be considered as an element of a gauze screen or an offshore structure. The most important feature of the interference was found to be a significant increase of local drag coefficient due to the symmetric formation of secondary flow patterns. The pressure distribution measured around both cylinders at various spanwise stations was distinctly different for the two cylinders in the region of interference. A strong secondary flow was found behind the upstream cylinder and in front of the downstream cylinder. The secondary flow distorted and displaced the separation streamlines on the upstream and downstream cylinders. This has been verified by oil-film flow visualization.

We have performed a series of experiments on two-dimensional gravity currents for which the inflow rate at the origin is a power-law function of time, tα−1. The theoretical results of Huppert (1982), for currents in which there is a balance between buoyancy and viscous forces, are found to be valid for a wide range of conditions. A large numbers of experiments at a critical value of α = 7/4 show very precise agreement with the theory, while values of a parameter that separates regions in which either viscous forces or inertial forces dominate are well within limits one would expect from the order-of-magnitude arguments used.

Gravity currents or intrusions for which the fluid within the current contains a substance that gives it a diffusivity different from that of its surroundings are very common both in natural and technological applications. The interface between the two fluids can become the site of either a ‘fingering’ or a ‘diffusive’ type of instability, and vigorous convection and material exchange occurs (Turner 1975). This transfer of material has several important effects upon the dynamics of the intrusion. Horizontal momentum can be transferred across the interface to create a stress which in many cases dominates the more conventional viscous stresses. Entrainment into the convective plume beneath the intrusion and, in a container of finite depth, the formation of a secondary, bottom-boundary current, creates a flow external to the main intrusion which modifies its behaviour even more. Two cases have been studied: the release of a fixed volume of fluid and the injection of fluid at a constant rate for both types of interface. Several experiments on the motion of an intrusion for which the interface is essentially non-diffusive are presented for comparison.

The wall-pressure fluctuations beneath a turbulent boundary layer have been conditionally sampled on a basis of the high-frequency activity of the pressure fluctuations themselves, the high-frequency activity of the streamwise velocity fluctuations in the vicinity of the wall, and the excursions in velocity in the vicinity of the wall. This has led to the identification of a characteristic wall-pressure fluctuation pattern which is associated with the burst–sweep cycle of events in the wall region. The pattern has the form of an overpressure over a streamwise extent of about 1·5−2·0δ*, with a region of underpressure and a pressure minimum to either side of it, the distance between pressure minima being about 3·0−3·5δ*. This pattern is convected at a velocity 0·67 times the freestream velocity. Its phase relationship with velocity fluctuations close to the wall and the wall shear-stress fluctuations during the burst–sweep cycle have been established. It appears to be produced by the inclined shear layer which forms the upstream surface of the large organized structures in the layer, and calculated pressure patterns support this conclusion.

The phase relationships indicate that fluid involved in the bursting process is subjected to a favourable streamwise pressure gradient by the characteristic wall-pressure pattern at the time that the lift-up of low-speed streaks in the wall region begins. In addition, order-of-magnitude estimates suggest that the adverse pressure gradients associated with the characteristic pressure pattern, even if their phasing with streak lift-up were appropriate, would be insufficient to initiate the lift-up. It is therefore concluded that the streamwise pressure gradients associated with the pressure patterns do not play an active role in the dynamics of the wall flow and are not the direct cause of the bursting process.

The evolution of temperature variance and heat flux in decaying grid turbulence with a linear cross-stream temperature gradient is studied by producing the temperature gradient by means of two different methods: (a) by placing a ‘mandoline’ (Warhaft & Lumley 1978) downstream from the grid but with its wires differentially heated for the present study, and (b) by differentially heating ribbons of nichrome (a ‘toaster’) placed in the plenum chamber of the wind tunnel. For the former method the initial thermal/mechanical lengthscale ratio Lθ/L was varied by changing the mandoline configuration. For this method it is shown that the gradient causes Lθ/L to equilibrate to a value of about 0·9 regardless of its initial value, and that when this value is achieved the temperature variance increases approximately linearly with time. The toaster was used to produce a temperature gradient without the associated initial temperature variance (and initial thermal lengthscale) that is necessarily produced by the mandoline wires; for the toaster the temperature variance was produced solely by the action of turbulence against the temperature gradient. For this experiment too, the thermal variance grew linearly with time, and Lθ/L was approximately the same as the equilibrium value for the mandoline experiments. The equilibrium value of the ratio of temperature-variance production to temperature-variance dissipation was approximately 1·5 for all of the experiments. The ratio of the mechanical-dissipation/thermal-dissipation timescales was also found to equilibrate, but there was considerably more scatter in the data for this parameter. The values of the equilibrium length- and timescale ratios were not affected by the magnitude of the temperature gradient, which was varied for both experiments. Good transverse homogeneity in the thermal field was achieved in all cases, in contrast with previous experiments (using heated grids).

An analysis is given of a secondary instability that obtains in a wide class of wall-bounded parallel shear flows, including plane Poiseuille flow, plane Couette flow, flat-plate boundary layers, and pipe Poiseuille flow. In these flows it is shown that two-dimensional finite-amplitude waves are (exponentially) unstable to infinitesimal three-dimensional disturbances. This secondary instability seems to be the prototype of transitional instability in these flows in that it has the characteristic (convective) timescales observed in the typical transitions. In the case of plane Poiseuille flow, two-dimensional nonlinear equilibria and quasi-equilibria exist, and the stability of the secondary flow is determined by a three-dimensional linear eigenvalue calculation. In flows without equilibria (e.g. pipe flow), a time-dependent stability analysis is performed by direct spectral numerical calculation of the incompressible three-dimensional Navier–Stokes equations.

The energetics and vorticity dynamics of the instability are discussed. It is shown that the two-dimensional wave mediates the transfer of energy from the mean flow to the three-dimensional perturbation but does not directly provide energy to the disturbance. The instability is of an inviscid character as it persists to high Reynolds numbers and grows on convective timescales. Maximum vorticity (inflexion-point) arguments predict some features of the instability like phase-locking of the two-dimensional and three-dimensional waves, but they do not explain its essential three-dimensionality. The inviscid vorticity dynamics of the instability shows that vortext-stretching and tilting effects are both required to explain the persistent exponential growth. The instability is not centrifugal in nature.

The three-dimensional instability requires that a threshold two-dimensional amplitude be achieved (about 1% of the centreline velocity in plane Poiseuille flow): the growth rates are relatively insensitive to amplitude for moderate two-dimensional amplitudes. With moderate two-dimensional amplitudes, the critical Reynolds numbers for substantial three-dimensional growth are about 1000 in plane Poiseuille and Couette flows and several thousand in pipe Poiseuille flow. The asymptotic (as R approaches infinity) growth rate in plane Poiseuille flow is approximately 0·15h/U0, where h is the half-channel width and U0 is the centreline velocity.

It is possible to make some progress identifying experimental features of transitional spot structure with aspects of the nonlinear two-dimensional/linear three-dimensional instability. The principal excitation of the eigenfunction of the three-dimensional (growing) disturbance is localized within a given periodicity length (in both the stream and cross-stream directions) near the vorticity maxima of the two-dimensional flow; its planwise structure corresponds to that of observed streaks in early transitional spots; its vortical structure resembles that of a streamwise vortex lifting off the wall. As the three-dimensional disturbance grows to finite amplitude, the flows become chaotic with statistical structure similar to that observed experimentally in moderate-Reynolds-number turbulent shear flows.

The wave forces exerted on a floating 3-dimensional body can be found by expressing the velocity potential of the surrounding fluid as the field of a distribution of point wave sources over the wetted part of the body surface. The problem then reduces to one of finding the solution to a 2-dimensional Fredholm integral equation of the second kind, to give the (unknown) surface source density. A simplification is possible for bodies that have a vertical axis of symmetry: for this type of body we can distribute ‘rings of sources’ over the body surface, and the problem then reduces to the solution of 1-dimensional Fredholm equations of the second kind. This approach has been adopted before, but earlier work has made use of expressions for the fundamental ring-source potentials which are not always suitable for numerical computation. It is possible to derive many alternative expressions for the ring-source potentials, but it appears that no single expression is computationally convenient in every situation; the present paper discusses the computational merits of three different types of expression, the aim being to provide a comprehensive scheme for the evaluation of the ring-source potentials. The ring-source/integral-equation method will be used to calculate the wave forces exerted on certain specific bodies of revolution and results are presented here. A brief discussion of the problem of ‘irregular values’ is also given: these only occur when the body intersects the free surface.

Detailed measurements of the time-dependent velocities induced inside and outside the opening of an acoustically excited, two-dimensional Helmholtz resonator imbedded in a grazing flow are presented. The remarkably clear structure of the perturbation field which evekes a pulsating source and a coherently pulsating vortex-image pair is described.