When two cylinders are counter-rotated at low Reynolds number about parallel horizontal axes below the free surface of a viscous fluid, the rotation being such as to induce convergence of the flow on the free surface, then above a certain critical angular velocity Ωc, the free surface dips downwards and a cusp forms. This paper provides an analysis of the flow in the neighbourhood of the cusp, via an idealized problem which is solved completely: the cylinders are represented by a vortex dipole and the solution is obtained by complex variable techniques. Surface tension effects are included, but gravity is neglected. The solution is analytic for finite capillary number [Cscr ], but the radius of curvature on the line of symmetry on the free surface is proportional to exp (−32π[Cscr ]) and is extremely small for [Cscr ] [gsim ] 0.25, implying (in a real fluid) the formation of a cusp. The equation of the free surface is cubic in (x, y) with coefficients depending on [Cscr ], and with a cusp singularity when [Cscr ] = ∞.

The influence of gravity is considered through a stability analysis of the free surface subjected to converging uniform strain, and a necessary condition for the development of a finite-amplitude disturbance of the free surface is obtained.

An experiment was carried out using the counter-rotating cylinders as described above, over a range of capillary numbers from zero to 60; the resulting photographs of a cross-section of the free surface are shown in figure 13. For Ω < Ωc, a rounded crest forms in the neighbourhood of the central line of symmetry; for Ω > Ωc, the downward-pointing cusp forms, and its structure shows good agreement with the foregoing theory.

We consider the general problem of geostrophic adjustment in a channel in the weakly nonlinear and dispersive (non-hydrostatic) limit. Governing equations of Boussinesq-type are derived, based on the assumption of weak nonlinear, dispersive and rotational effects, both for surface waves on a homogeneous fluid and internal waves in a two-layer system. Numerical solutions of the Boussinesq equations are presented, giving examples of the geostrophic adjustment in a channel for two different kinds of initial disturbances, both with non-zero perturbation potential vorticity. The timescales of rotational separation (that is, the separation of the Kelvin and Poincaré waves due to their dispersive properties) and that of nonlinear evolution are considered, with particular concern for the resonant interactions of nonlinear Kelvin waves and linear Poincaré waves described by Melville, Tomasson & Renouard (1989). A parameter measuring the ratio of the two timescales is used to predict when the free and forced Poincaré waves may be separated in the solution. It also distinguishes the cases in which the linear solutions are valid for the rotational separation from those requiring the full Boussinesq equations. Finally, solutions for the evolution of nonlinear internal waves in a sea strait are presented, and the effects of friction on the wavefront curvature of the nonlinear Kelvin waves are briefly considered.

In many geophysical situations, fluid is contained in long narrow fractures embedded within an impermeable medium of different thermal conductivity: and there may be a uniform vertical temperature gradient imposed upon the system. We show that whenever the slot is tiled to the vertical, convection develops in the fluid, even if the background temperature increases with height. We then investigate the transport of passive material governed by this flow. The dispersion coefficient of a passive contaminant transported by this flow DT = f[R sin2ϕ] k2 cot2 ϕ/D, where k and D are the thermal and compositional diffusivities, ϕ is the angle of tilt and R is a Rayleigh number for the slot.

Using typical values for the physical properties of a water-filled fracture, we show that the Earth's geothermal gradient produces a convective flow in a fracture through the mechanism above; this has an associated dispersion coefficient DT ∼ 102-103D in fractures about a centimetre wide. We show that this shear dispersion could transport radioactive material, of half-life 104 years, tens of metres along the fracture within one half-life; without this dispersion, the material would only diffuse a few metres along the fracture within one half-life.

If there is a background salinity gradient along the slot in addition to the thermal gradient, analogous steady flow solutions exist; if the flow is stable, the salinity may either enhance or reduce the steady flow and associated dispersion of passive tracer.

The bursting mechanism in two different high-Reynolds-number boundary layers has been analysed by means of conditional sampling. One boundary layer develops on a smooth, flat plate in zero pressure gradient; the other, also in zero pressure gradient, is perturbed by a rough-to-smooth change in surface roughness and the new internal layer has not yet recovered to the local equilibrium condition at the measurement station. Sampling on the instantaneous uv signal in the logarithmic region confirms the presence of two related structures, ‘ejections’ and ‘sweeps’ which, in the smooth-wall layer, appear to be responsible for most of the turbulent energy production, and to effect virtually all that part of the spectral energy transfer that is universal. Ejections show features similar to those of Falco's ‘typical eddies’ while sweeps appear to be inverted ejections moving down towards the wall. The inertial structures associated with ejections show attributes of the true universal motion (Townsend's ‘attached’ eddies) of the inner layer and these are therefore identified as ‘bursts’. In the outer layer, these become ‘detached’ from the wall. The large-scale structures associated with sweeps also appear to be ‘detached’ eddies (‘splats’), but these induce low-wave-number inactive motion near the wall and this is not universal even though the sweep itself is. Neither ejections nor sweeps detected in the rough-to-smooth layer are near a condition of energy equilibrium. The relation of ejections and sweeps to the law of the wall and other accepted laws is discussed.

The paper presents a linear stability analysis of cohesionless granular materials during rapid shear flow. The analysis is based on the governing equations developed in the kinetic theory of Lun et al. (1984) for granular flows of smooth, nearly elastic, uniform spherical particles. The primary flow is taken to be a uniform, simple shear flow and the effects of small perturbations in velocity components, granular temperature and solids fraction are considered. The inelasticity of the particles is characterized by a constant coefficient of restitution which is assumed to be close to unity. Some permissible solutions are sinusoidal plane waves in which the wavenumber vector is continuously turned by the mean shear flow and its magnitude varied as time proceeds. The initial growth (or decay) rates for these perturbations are sought. The resulting linearized equations for the flow perturbations turn out to be exceedingly long and complex; they are determined by the use of computer algebra. It is found that, in general, long wavelengths are the most unstable and that short wavelengths are dampened by ‘viscous action’. ‘Instability’ increases with decreasing coefficient of restitution. Numerical results for initial growth rates were obtained for several values of mean solids fraction and particle coefficient of restitution. Flows tend to be more stable at both high and very low concentrations than at moderate concentrations. These results appear to be consistent in the main with recent computer simulations of granular flows of disk-like particles by Hopkins & Louge (1991).

Equations of motion and boundary conditions for a flowing granular material, developed in earlier publications, are here extended to allow for drag forces resulting from relative motion of the material and interstitial air. These are solved for fully developed flow down an inclined plane, through which a constant flow of air passes upward. The results are compared with measurements from an experimental aerated chute, in which the inclination of the chute, the flow rate of the granular material, and the flow of air are all varied. Using parameter values from independent measurements, as far as possible, the theory is found to give a good qualitative account of the observed behaviour. With a reasonable assigned value for the one parameter that cannot be determined independently the quantitative agreement is also satisfactory.

In the past few years kinetic theory has been used to derive equations of motion for rapidly shearing granular materials, and there have been empirical extensions of these to take into account stress transmitted by sustained sliding and rolling contacts between particles. The equations are complicated and solutions have been generated only for very simple flows. In this paper three forms for the equations of motion are considered; one representing interaction by collisions only, one which is a high-density asymptotic form of this, and a third which includes terms representing the ‘frictional’ stresses associated with the sustained contacts referred to above. Solutions are found for fully developed flow under gravity down an inclined plane, and it is shown that the relation between the flow rate and the depth of the flowing layer predicted by the first two sets of equations is not in accord with observations. The third form appears to eliminate much of the discrepancy, but its predictions have not been explored over the whole parameter space. It is emphasized that the form of the solutions should be studied over a wide range of operating conditions in order to assess the usefulness of proposed equations.

A fundamental understanding of nonlinear oscillations of a viscous liquid drop is needed in diverse areas of science and technology. In this paper, the moderate- to large-amplitude axisymmetric oscillations of a viscous liquid drop, which is immersed in dynamically inactive surroundings, are analysed by solving the free boundary problem comprised of the Navier–Stokes system and appropriate interfacial conditions at the drop–ambient fluid interface. The means are the Galerkin/finite-element technique, an implicit predictor-corrector method, and Newton's method for solving the resulting system of nonlinear algebraic equations. Attention is focused here on oscillations of drops that are released from an initial static deformation. Two dimensionless groups govern such nonlinear oscillations: a Reynolds number, Re, and some measure of the initial drop deformation. Accuracy is attested by demonstrating that (i) the drop volume remains virtually constant, (ii) dynamic response to small-and moderate-amplitude disturbances agrees with linear and perturbation theories, and (iii) large-amplitude oscillations compare well with the few published predictions made with the marker-and-cell method and experiments. The new results show that viscous drops that are released from an initially two-lobed configuration spend less time in prolate form than inviscid drops, in agreement with experiments. Moreover, the frequency of oscillation of viscous drops released from such initially two-lobed configurations decreases with the square of the initial amplitude of deformation as Re gets large for moderate-amplitude oscillations, but the change becomes less dramatic as Re falls and/or the initial amplitude of deformation rises. The rate at which these oscillations are damped during the first period rises as initial drop deformation increases; thereafter the damping rate is lower but remains virtually time-independent regardless of Re or the initial amplitude of deformation. The new results also show that finite viscosity has a much bigger effect on mode coupling phenomena and, in particular, on resonant mode interactions than might be anticipated based on results of computations incorporating only an infinitesimal amount of viscosity.

The evolution of passive magnetic fields is considered in random flows made up of single helical waves. In the absence of molecular diffusion the growth rates of all moments of a magnetic field are calculated analytically, and it is found that the field becomes increasingly intermittent with time. The evolution of normal modes of the ensemble-averaged field is determined; it is shown that the flows considered give fast dynamo action, and magnetic field modes with either sign of magnetic helicity may grow.

The electromagnetic forces in a ferrofluid depend on the domain occupied by the fluid. We study here the equilibrium positions of a ferrofluid drop with a boundary which is partially or totally free. The method used is based on the minimization of the energy with respect to the shape of the drop. We show bifurcations of the solutions and hysteresis phenomena when the parameters vary.

The statistical evolution of a two-dimensional polygonal, or ‘dry’, foam during diffusion of gas between bubbles lends itself to a very simple mathematical description by combining physical principles discovered by Young. Laplace, Plateau, and von Neumann over a period of a century and a half. Following a brief review of this ‘canonical’ theory, we report results of the largest numerical simulations of this system undertaken to date. In particular, we discuss the existence and properties of a scaling regime, conjectured on the basis of laboratory experiments on larger systems than ours by Glazier and coworkers, and corroborated in computations on smaller systems by Weaire and collaborators. While we find qualitative agreement with these earlier investigations, our results differ on important, quantitative details, and we find that the evolution of the foam, and the emergence of scaling, is very sensitive to correlations in the initial data. The largest computations we have performed follow the relaxation of a system with 1024 bubbles to one with O(10), and took about 30 hours of CPU time on a Cray-YMP supercomputer. The code used has been thoroughly tested, both by comparison with a set of essentially analytic results on the rheology of a monodisperse-hexagonal foam due to Kraynik & Hansen, and by verification of certain analytical solutions to the evolution equations that we found for a family of ‘fractal foams’.

The ingestion of convected vorticity by a high-solidity rotating blade row is a potent noise source in modern aeroengines, due largely to the high level of mutual aerodynamic interactions between adjacent blades. In order to model this process we solve the problem of determining the unsteady lift on an infinite cascade of finite-chord flat plates due to an incident vorticity wave. The method of solution is the Wiener–Hopf technique, and we consider the case of the reduced frequency, Ω, being large, allowing application of asymptotic analysis in the formal limit Ω → ∞. This approach yields considerable simplification, both in allowing the truncation of an infinite reflection series to just two terms, and in allowing algebraic expressions for the Wiener–Hopf split functions to be found. The unsteady lift distribution is derived in closed form, and the accuracy of the asymptotic Wiener–Hopf factorization demonstrated for even modest values of Ω by comparison with exact (but less tractable) methods. Our formulae can easily be incorporated into existing noise prediction codes: the advantage of our scheme is that it handles a regime in which conventional numerical approaches become unwieldy, as well as providing significant physical insight into the underlying mechanisms.

As a model of the thermohaline circulation of the ocean we study the two-dimensional Boussinesq equations forced by prescribing the surface temperature and the surface salinity flux. We simplify the equations of motion using an expansion based on the small aspect ratio of the domain. The result is an amplitude equation governing the evolution of the depth averaged salinity field. This amplitude equation has multiple, linearly stable equilibria. The simplified dynamics has a Lyapunov functional and this variational structure permits a simple characterization of the relative stability of the alternative steady solutions.

Even when the thermal and salinity surface forcing functions are symmetric about the equator there are asymmetric solutions, representing pole to pole circulations. These asymmetric solutions are stable to small perturbations and are always found in conjunction with symmetric solutions, also stable to small perturbations. Recent numerical solutions of the full two-dimensional equations have shown very similar flow patterns.

We analyse the linear water-wave equations for shallow channels with arbitrary rapidly varying bottoms. We develop a theory for reflected waves based on an asymptotic analysis for stochastic differential equations when both the horizontal and vertical scales of the bottom variations are comparable to the depth but small compared to a typical wavelength so the shallow water equations cannot be used. We use the full, linear potential theory and study the reflection–transmission problem for time-harmonic (monochromatic) and pulse-shaped disturbances. For the monochromatic waves we give a formula for the expected value of the transmission coefficient which depends on depth and on the spectral density of the O(1) random depth perturbations. For the pulse problem we give an explicit formula for the correlation function of the reflection process. We compare our theory with numerical results produced using the boundary-element method. We consider several realizations of the bottom profile, let a Gaussian-shaped disturbance propagate over each topography sampled and record the reflected signal for each realization. Our numerical experiments produced reflected waves whose statistics are in good agreement with the theory.

Numerical work of many people on the bifurcations of uniformly travelling water waves (two-dimensional irrotational gravity waves on inviscid fluid of infinite depth) suggests that uniformly travelling water waves have a reversible Hamiltonian formulation, where the role of time is played by horizontal position in the wave frame. In this paper such a formulation is presented. Based on this viewpoint, some insights are given into bifurcations from Stokes’ family of periodic waves. It is demonstrated numerically that there is a ‘fold point’ at amplitude A0 ≈ 0.40222. Assuming non-degeneracy of the fold and existence of an associated centre manifold, this explains why a sequence of p/q-bifurcations occurs on one side of A0, with 0 < p/q [les ] ½, in the order of the rationals. Secondly, it explains why no symmetry-breaking bifurcation is observed at A0, contrary to the expectations of some. Thirdly, it explains why the bifurcation tree for periodic uniformly travelling waves looks so much like that for the area-preserving Hénon map. Fourthly, it leads to predictions of a rich variety of spatially quasi-periodic, heteroclinic and chaotic waves.

Closed-form analytical jump conditions across normal shock waves in pure vapour–droplet flows have been derived for different boundary conditions. They are equally applicable to partly and fully dispersed shock waves. Collectively they may be called the generalized Rankine–Hugoniot relations for wet vapour. A phase diagram is constructed which specifies the type of shock structure obtained in vapour–droplet flow given some overall parameters. It is shown that in addition to the partly and fully dispersed shock waves that are possible in any relaxing medium, there also exists a class of shock waves in wet vapour in which the two-phase relaxing medium reverts to a single-phase non-relaxing one. An analytical expression for the limiting upstream wetness fraction below which complete evaporation will take place inside a shock of specified strength has been deduced. A new theory has been formulated which shows that, depending on the upstream wetness fraction, a continuous transition exists for the shock velocity between its frozen and fully equilibrium values. The mechanisms of entropy production inside a shock are also discussed.

We present results from an experimental and numerical investigation into the structure of vortex patterns and the dynamics of their interactions for the incompressible flow in the near field of a round coaxial jet issuing into a quiescent ambient fluid. A two-colour planar laser-induced-fluorescence technique is used to document the flow field via still photographs and ciné sequences over a limited range of parameters. We examine the effects of varying the velocity ratio as well as the absolute velocities of the two coaxial streams for equal densities and for a single area ratio. Results show that a variety of widely differing near-field vortex patterns can arise, with very different interaction dynamics, which can depend both on the velocity ratio and on the absolute velocities of the two streams. The observed vortex structures and their dynamics are interpreted in terms of the instability of the initially cylindrical and concentric vorticity layers separating each of the fluid streams, and their subsequent rollup to form wake-like or shear-layer-like vortices. Our results show that in addition to the velocity jump across each of these vorticity layers, an accounting of the layer thicknesses and the wake defect within each layer can be essential to understanding the resulting near-field structure that occurs. Ensuing dynamical interactions between the vortices formed from each layer can produce a strong coupling between the development of the two layers. These resulting vortex structures and interaction dynamics are also seen to produce widely differing mixing patterns in the jet near field.

The stability of a pair of counter-rotating vortices to three-dimensional disturbances in the presence of a stretching flow is studied for vortices of small circular cross-section. The problem is reduced to a system of two first-order, linear ordinary differential equations, which can be integrated numerically to obtain the change in the perturbation of the vortex pair with time. The stability of the vortex pair depends upon four dimensionless constants, two of which characterize the stretching flow. Computations indicate that stretching usually exerts a stabilizing influence on the vortex pair, although in many cases the perturbation amplitude may initially increase and then decrease at some later time due to the effects of stretching. The results of the study are applied to investigate stability of hairpin vortices that are typically observed in turbulent shear flows. An estimate of the percentage increase in perturbation amplitude of a hairpin vortex in a homogeneous turbulent shear flow is given as a function of the stretch of the hairpin for different values of the dimensionless perturbation wavenumber and the microscale Reynolds number Reλ = λq/ν (based on the Taylor microscale λ and the turbulent kinetic energy ½q2). The maximum percentage growth of a perturbation of the legs of a hairpin vortex in a turbulent shear flow is found to decrease with increase in Reλ.

We consider the parallel flow of two immiscible fluids in a Hele-Shaw cell. The evolution of disturbances on the fluid interfaces is studied both theoretically and experimentally in the large-capillary-number limit. It is shown that such interfaces support wave motion, the amplitude of which for long waves is governed by a set of KdV and Airy equations. The waves are dispersive provided that the fluids have unequal viscosities and that the space occupied by the inner fluid does not pertain to the Saffman-Taylor conditions (symmetric interfaces with half-width spacing). Experiments conducted in a long and narrow Hele-Shaw cell appear to validate the theory in both the symmetric and the non-symmetric cases.

The basic model problem of separation as predicted by the time-mean boundary-layer equations is studied, with the Cebeci-Smith model for turbulent stresses. The changes between laminar and turbulent flow are investigated by means of a turbulence ‘factor’ which increases from zero for laminar flow to unity for the fully turbulent regime. With an attached-flow starting point, a small increase in the turbulence factor above zero is found to drive the separation singularity towards the trailing edge or rear stagnation point for flow past a circular cylinder, according to both computations and analysis. A separated-flow starting point is found to produce analogous behaviour for the separation point. These findings lead to the suggestion that large-scale separation need not occur at all in the fully turbulent regime at sufficiently high Reynolds number; instead, separation is of small scale, confined near the trailing edge. Comments on the generality of this suggestion are presented, along with some supporting evidence from other computations. Further, the small scale involved theoretically has values which seem reasonable in practical terms.