Measurements have been made of concentration fluctuations and turbulent fluxes for two passive plumes from an elevated and a ground-level source in a turbulent boundary layer. For the concentration fluctuations, results are presented for the variance, the intermittency, peak values of concentration, probability-density functions and spectra. The balance of terms in the variance transport equation is examined, as is the overall level of fluctuations along the plume. It is shown that most of the production of fluctuations occurs very near the source. Then, the level of fluctuation decays, roughly in accordance with a balance between advection and dissipation. For the turbulent fluxes of concentration, results are presented for the vertical and lateral fluxes, with the associated behaviour of the vertical and lateral eddy diffusivities. The balance of terms in the transport equations for the fluxes is examined. The essential differences between vertical diffusion from ground-level and elevated sources and between near-field and far-field behaviour are shown to be due to the relative importance of the advection and diffusion terms in these equations.

The process of levitation melting of metals is examined analytically and numerically for the case of axisymmetric toroidal high-frequency currents. The governing equations for the mean-velocity field and associated free-surface shape are derived under the assumption of low magnetic Reynolds number and the neglect of thermal effects. The form of the solution for high Reynolds number is discussed in general, and particularized to the case of high surface tension, in which limit a perturbation analysis about a spherical shape is presented. Finite-difference techniques are used to solve the Navier–Stokes equations in the sphere, and the surface perturbation is calculated. The asymptotic behaviour of the potential vorticity is illustrated by the numerical experiments.

When a piece of metal is placed above a coil carrying a high frequency current, the induced surface currents in the metal can provide a Lorentz force which can support it against gravity; at the same time the heat produced by Joule dissipation can melt the metal. This is the process of ‘levitation melting’, which is a well-established technique in fundamental work in physical and chemical metallurgy. Most theoretical studies of magnetic levitation have dealt only with solid conductors and so have a voided the interesting questions of interaction between the free surface, the magnetic field and the internal flow. These fluid dynamical aspects of the process are studied in this paper.

A particular configuration that is studied in detail is a cylinder levitated by two equal parallel currents in phase; this is conceived as part of a toroidal configuration which avoids a difficulty of conventional configurations, viz the leakage of fluid through the ‘magnetic hole’ at a point on the metal surface where the surface tangential magnetic field vanishes. The equilibrium and stability of the solid circular cylinder is first considered; then the dynamics of the surface film when melting begins; then the equilibrium shape of the fully melted body (analysed by means of a general variational principle proved in § 5); and finally the dynamics of the interior flow, which, as argued in § 2, is likely to be turbulent when the levitated mass is of the order of a few grams or greater.

Numerical solutions for the impulsively started spin-up of a thermally stratified fluid in a cylinder with an insulating side wall are presented. Previous experimental and numerical work on stratified spin-up had not provided a comprehensive and accurate set of flow-field data. Further, comparisons of this work with theory showed, in general, a substantial discrepancy. The theory was scaled using the homogeneous meridional-flow spin-up time scale and thus viscous-diffusion effects were excluded from the interior. It was anticipated that these effects could only be significant on the larger viscous-diffusion time scale. However, the comparisons with theory showed a faster rate of decay for the measurements even over the shorter meridional-flow spin-up time scale. Previous workers had suggested a number of explanations but the cause of the discrepancy was still unresolved. To provide data to extend the previous work, a numerical model was used. The model was first checked against accurate experimental measurements of stratified spin-up made using a laser-Doppler velocimeter. New accurate results which cover ranges of Ekman number (5·92 × 10−4 ≤ E ≤ 7·24 × 10−4), Rossby number (0·019 ≤ ε ≤ 0·220), stratification parameter (0·0 ≤ Sa−1 ≤ 1·03), and Prandtl number (5·68 ≤ σ ≤ 7·10) are presented. These results show the radial and vertical structure of the decaying azimuthal and meridional flows. The inertial–internal gravity oscillations excited by the impulsive spin-up are clearly seen. By making use of conclusions from the previous work and the results presented in this paper, it is established that viscous diffusion in the interior is the cause of the discrepancy with theory. Stratification causes the meridional spin-up flow to be confined closer to the boundary disks. This results in non-uniform spin-up of the interior and hence flow gradients in the interior. These gradients introduce viscous diffusion into the interior sooner than anticipated by the theory. A previous suggestion that the faster decay rate is due to angular momentum being injected into the interior from an oscillation of the meridional corner-jet flow is shown to be untenable.

The radial distribution of the volumetric vapour (or void) fraction in steady, fully developed turbulent two-phase flow is described for vertical low-quality bubbly flows in a circular pipe. The analysis is based on the phasic equations of conservation of momentum in the axial and radial directions. Mixing-length theory is used to model the turbulent stresses in the continuous phase. The predicted flow structure shows three distinct regions. The ‘outer’ region, that is, the region away from the wall and the centre-line, has a uniform void distribution. For upflow, a bubble layer is predicted near the wall, while for downflow, vapour coring is predicted, with a peak in void fraction at the centre-line. These predictions are in agreement with observed void profiles.

When a particle is placed in a fluid in which there is a non-uniform concentration of solute, it will move toward higher or lower concentration depending on whether the solute is attracted to or repelled from the particle surface. A quantitative understanding of this phenomenon requires that the equations representing conservation of mass and momentum within the fluid in the vicinity of the particle are solved. This is accomplished using a method of matched asymptotic expansions in a small parameter L/a, where a is the particle radius and L is the length scale characteristic of the physical interaction between solute and particle surface. This analysis yields an expression for particle velocity, valid in the limit L/a → 0, that agrees with the expression obtained by previous researchers. The result is cast into a more useful algebraic form by relating various integrals involving the solute/particle interaction energy to a measurable thermodynamic property, the Gibbs surface excess of solute Γ. An important result is that the correction for finite L/a is actually O(Γ/C∞a), where C∞ is the bulk concentration of solute, and could be O(1) even when L/a is orders of magnitude smaller.

We have used flow-visualization and spectral techniques to study the spatial and temporal properties of the flow that precedes the onset of weak turbulence in a fluid contained between concentric cylinders with the inner cylinder rotating (the circular Couette system). The first three flow regimes encountered as the Reynolds number is increased from zero are well-known – Couette flow, Taylor-vortex flow, and wavyvortex flow. The present study concerns the doubly periodic regime that follows the (singly periodic) wavy-vortex-flow regime. Wavy-vortex flow is characterized by a single frequency f1, which is the frequency of travelling azimuthal waves passing a point of observation in the laboratory. The doubly periodic regime was discovered in studies of power spectra several years ago, but the fluid motion corresponding to the second frequency f2 was not identified. We have found that f2 corresponds to a modulation of the azimuthal waves; the modulation can be observed visually as a periodic flattening of the wavy-vortex outflow boundaries. Moreover, in addition to the previously observed doubly periodic flow state, we have discovered 11 more doubly periodic flow states. Each state can be labelled with two integers m and k, which are simply related to physical characteristics of the flow: m is the number of azimuthal waves, and k is related to the phase angle between the modulation of successive azimuthal waves by Δϕ = 2πk/m. This expression for the phase angle was first conjectured from the flow-visualization measurements and then tested to an accuracy of 0·01π in spectral measurements. Recently Rand (1981) has used dynamical-systems concepts and symmetry considerations to derive predictions about the space–time symmetry of doubly periodic flows in circularly symmetric systems. He predicted that only flows with certain space–time symmetries should be allowed. The observed flow states are in agreement with this theory.

This paper presents the first infinite-series solutions to the creeping-flow equations for the axisymmetric motion of a sphere of arbitrary size towards an orifice whose diameter is either larger or smaller than the sphere. To obtain the solution the flow field is partitioned in the plane of the opening, and for the flow to the left and right of the fluid interface separate solutions are developed that satisfy the viscous-flow boundary conditions in each region and unknown functions for the axial and radial velocity components in the plane of the opening. The continuity of the fluid stress tensor at the matching interface leads to a set of dual integral equations which are solved analytically to determine the unknown functions for the velocity components in the matching plane. A boundary collocation technique is used to satisfy the no-slip boundary conditions on the surface of the sphere.

The accuracy and convergence of the present solution is tested by detailed numerical comparison with the exact bipolar co-ordinate solutions of Brenner (1961) for the drag on a sphere translating perpendicular to an infinite plane wall up to a distance of 0·1 sphere radii and is found to be in agreement to five significant digits. The converged-series collocation solutions are presented for the sphere in motion in quiescent fluid or for flow past a rigidly held sphere positioned axisymmetrically near a fixed orifice. Solutions are also presented for the zero-drag velocity of a neutrally buoyant sphere in a flow through an orifice, and the pressure–volume flow relation for a ball-valve geometry.

The stability of the finite-area Kármán ‘vortex street’ to two-dimensional disturbances is determined. It is shown that for vortices of finite size there exists a finite range of spacing ratio κ for which the array is stable to infinitesimal disturbances. As the vortex size approaches zero, the range narrows to zero width about the classical von Kármán value of 0·281.

The Miles–Phillips model of the linear coupling between waves on the ocean surface and a fluctuating wind field is generalized to include the average effect of the nonlinear water-wave interactions in the dynamic equations for gravity–capillary waves. A statistical-linearization procedure is applied to the general problem and yields the optimum linear description of the nonlinear terms by linear terms. The linearized dynamic equations are stochastic with solutions that have stable moments, i.e. the average nonlinear interactions quench the linear instability generated by the coupling to the mean wind field. In particular, an asymptotic steady-state power-spectral density for the water-wave field is calculated exactly in the context of the model for various wind speeds.

An analytical theory is developed to describe how negative pressure, (or ‘mud suction’, as it is sometimes referred to) develops underneath a body as it detaches itself from the ocean bottom. Biot's quasistatic equations of poro-elasticity are used to model the ocean bottom, and a general three-dimensional time-dependent analysis of the problem is worked out first using the boundary-layer approximation recently proposed by Mei and Foda. Then, explicit leading-order analytical solutions are presented for the problems of extrication of slender bodies as well as axisymmetric bodies from the ocean bottom.

Weakly nonlinear theory and finite-difference calculations are used to describe steadystate and oscillatory convective heat transport in water-saturated porous media. Two-dimensional rolls in a rectangular region are considered when the imposed temperature difference between the horizontal boundaries is as large as 200 K, corresponding to a viscosity ratio of about 6·5. The lowest-order weakly nonlinear results indicate that the variation of the Nusselt number with the ratio of the actual Rayleigh number to the corresponding critical value R/Rc, is independent of the temperature difference for the range considered. Results for the Nusselt number obtained from finite-difference solutions contain a weak dependence on temperature difference which increases with the magnitude of R/Rc. When R/Rc = 8 the constantviscosity convection pattern is steady, while those with temperature differences of 100 and 200 K are found to oscillate.

The motion of a sphere along the axis of rotation of an incompressible viscous fluid that is rotating as a solid mass is investigated by means of numerical methods for small values of the Reynolds and Taylor numbers. The Navier–Stokes equations governing the steady axisymmetric flow can be written as three coupled, nonlinear, elliptic partial differential equations for the stream function, vorticity and rotational velocity component. Two numerical methods are employed to solve these equations. The first is the method of series truncation in which the dependent variables are expressed as series of orthogonal Gegenbauer functions and the equations of motion are then reduced to three coupled sets of ordinary differential equations, which are integrated numerically subject to their boundary conditions. In the second method, specialized finite–difference techniques of solution are applied to the two-dimensional partial differential equations. These techniques employ finite-difference equations with coefficients that depend upon the exponential function; a particularly suitable form of approximation for use in calculating numerical solutions is obtained by expanding the exponential coefficients in powers of their exponents.

Calculated results obtained by the two methods are in good agreement with each other. The calculations have been carried out according to theoretical assumptions that simulate the experiments of Maxworthy (1965) in which the sphere experiences no resultant torque exerted by the surrounding fluid and is free to rotate with constant angular velocity. Numerical estimates of this angular velocity and of the drag exerted by the fluid on the sphere are found to agree well with the experimental results for Reynolds and Taylor numbers in the range from zero to unity. The results for small values of the Reynolds number are also consistent with theoretical work of Childress (1963, 1964) which is valid as the Reynolds number tends to zero.

The theory of lateral waves is reviewed in the context of long surface gravity waves and is used to interpret the results of experiments performed in a wave tank to simulate tsunami propagation following a disturbance on a continental shelf. It is suggested that since the lateral wave precedes the direct wave at coastal stations its detection might prove useful in tsunami warning systems.

It is shown that the well-known Jeffery–Hamel solution of the Navier–Stokes equations admits generalization to the case in which the viscosity μ and density ρ are arbitrary functions of the angular co-ordinate θ. When |Rα| [Lt ] 1, where R is the Reynolds number and 2α the angle of divergence of the planes, lubrication theory is applicable; this limit is first treated in the context of flow in a channel of slowly varying width. The Jeffery–Hamel problem proper is treated in §§ 3–6, and the effect of varying the viscosity ratio λ in a two-fluid situation is studied. In § 5, results already familiar in the single-fluid context are recapitulated and reformulated in a manner that admits immediate adaptation to the two-fluid situation, and in § 6 it is shown that the singlefluid limit (λ → 1) is in a certain sense degenerate. The necessarily discontinuous behaviour of the velocity profile as the Reynolds number (based on volume flux) increases is elucidated. Finally, in § 7, some comments are made about the realizability of these flows and about instabilities to which they may be subject.

A lifting surface of small aspect ratio is analysed for motion with constant forward velocity, parallel and in close proximity to a rigid plane surface of infinite extent. The gap flow beneath the lifting surface is represented by a simple nonlinear solution in the cross-flow plane, and appropriate conditions are imposed at leading and trailing edges. The transition between these two conditions depends on the kinematics of the gap flow as well as the planform geometry. For steady-state motion of a delta wing with sufficiently large angle of attack, the transition point is upstream of the tail. For oscillatory heaving motion of a delta wing the transition point is cyclic if the heave velocity is sufficiently large. Illustrative computations of the lift force are presented.

Dynamical equations for the interaction of high-wavenumber, high-frequency internal waves with a prescribed, linear, large-scale internal-wave field are obtained from the Boussinesq–Euler equations. The relationship of these ‘induced-diffusion’ interactions to the Taylor–Goldstein equation is discussed. Exact equations are derived in the induced-diffusion limit of McComas & Bretherton (1977) for the evolution of the first and second moments of the small-scale flow when the large-scale flow is assumed random. Estimates of corrections to the induced-diffusion approximation for the Garrett–Munk internal-wave model indicate the domain of applicability of these equations. Computations of the autocorrelation function and action transport in wavenumber and physical space are presented. Severe limitations are found on the applicability of two-time perturbation theory and the resonant-interaction approximation. The high transfer rates found by McComas & Bretherton in the induceddiffusion regime are reduced significantly in the present calculations.

We investigate the stability of gravity currents, in a rotating system, that are infinitely long and uniform in the direction of flow and for which the current depth vanishes on both sides of the flow. Thus, owing to the role of the Earth's rotation in restraining horizontal motions, the currents are bounded on both sides by free streamlines, or sharp density fronts. A model is used in which only one layer of fluid is dynamically important, with a second layer being infinitely deep and passive. The analysis includes the influence of vanishing layer depth and large inertial effects near the edges of the current, and shows that such currents are always unstable to linearized perturbations (except possibly in very special cases), even when there is no extremum (or gradient) in the potential vorticity profile. Hence the established Rayleigh condition for instability in quasi-geostrophic models, where inertial effects are assumed to be vanishingly small relative to Coriolis effects, does not apply. The instability does not depend upon the vorticity profile but instead relies upon a coupling of the two free streamlines. The waves permit the release of both kinetic and potential energy from the mean flow. They can have rapid growth rates, the e-folding time for waves on a current with zero potential vorticity, for example, being close to one-half of a rotation period. Though they are not discussed here, there are other unstable solutions to this same model when the potential vorticity varies monotonically across the stream, verifying that flows involving a sharp density front are much more likely to be unstable than flows with a small ratio of inertial to Coriolis forces.

Experiments with a current of buoyant fluid at the free surface of a lower layer are described, and the observations are compared with the computed mode of maximum growth rate for a flow with a uniform potential vorticity. The current is observed to be always unstable, but, contrary to the predicted behaviour of the one-layer coupled mode, the dominant length scale of growing disturbances is independent of current width. On the other hand, the structure of the observed disturbances does vary: when the current is sufficiently narrow compared with the Rossby deformation radius (and the lower layer is deep) disturbances have the structure predicted by our one-layer model. The flow then breaks up into a chain of anticyclonic eddies. When the current is wide, unstable waves appear to grow independently on each edge of the current and, at large amplitude, form both anticyclonic and cyclonic eddies in the two-layer fluid. This behaviour is attributed to another unstable mode.

Computed numerical results are presented for the laminar buoyancy-induced flows driven by combined thermal and saline transport near a vertical melting ice surface in saline water. Results are presented which indicate that conventional boundary-layer flow occurs for some combinations of ambient salinity and temperature in the ranges 0 to 31‰ and −1 to 20°C, respectively. These conditions are very common in terrestrial waters. The analysis used herein is the first to model fully the many complicated aspects of these flows. The present analysis includes simultaneous transport of salt and thermal energy as well as the effect of interface motion. The formulation uses the most recently available transport properties and a very accurate equation of state for the density of pure and saline water. The interface temperature and salinity, which are not known a priori, are here determined jointly from the transport equations, and from species-conservation and phase-equilibrium relations at the ice surface. At low ambient temperatures, the flow is found to be dominated by the upward saline buoyancy, resulting in upward flow. However, at high temperatures and low salinities, the downward thermal buoyancy overcomes the upward saline buoyancy near the surface to cause downward flow. For choices of ambient conditions between these extremes, the opposing saline and thermal buoyancy are about equal in strength. The resulting tendency for bi-directional flow at these intermediate circumstances caused numerical stability problems which made it impossible to obtain convergent solutions for some cases. However, calculated solutions were obtained at ambient salinities below 5‰, for ambient temperatures between 8 and 20°C, and at temperatures below 4°C, for ambient salinities between 0 and 31‰. These solutions indicate the limits of the range of conditions for which laminar boundary layer flow occurs. They further suggest that outside these ranges, the flow may be laminar and bi-directional. The very strong buoyancy which characterizes some of these conditions suggests that they may become turbulent at short downstream distances. The computed results are seen to be in excellent agreement with the limited experimental data and observations of previous studies.

Time-exposure photographs of the buoyancy-driven flow adjacent to a submerged vertical ice surface melting in 10‰ saline water are presented for ambient water temperatures between 1 and 15°C. For ambient temperatures greater than 1·9°C, the thermal and saline components of the buoyancy force are at least partially opposed to each other. Since most past studies of ice melting in saline water have concentrated on oceanic salinities, little is known about the complicated flow behaviour which results from these opposed buoyancy effects at ambient water salinities between fresh water and oceanic salinities (35‰).

The results presented here provide new insight concerning the subtle mechanisms which arise in such flows. Photographs of the entire flow field document the many different and complicated flow configurations that arise. At 10‰, as the ambient temperature is increased from 1 to 15°C, regimes of upward, bi-directional and split flow are observed. In the latter circumstance, the flow is laminar and bi-directional over part of the ice surface and turbulent over the rest of the surface. Bi-directional flow results from reversal of part of the upward wake above the top of the ice surface. Split flow appears to be a consequence of the transition to turbulence of either the inner or outer portion of the laminar bi-directional flow. Measured velocity profiles, surface heat-transfer rates and interface temperatures agree well with the analytical results reported in a previous study for conditions that result in conventional boundary-layer flow. These experimental results, together with those of previous studies, indicate the approximate extent of the different flow regimes that arise for a vertical ice surface melting in cold, low-salinity water.