Measurements of heavy particle dispersion have been made using direct numerical simulations of isotropic turbulence. The parameters affecting the dispersion of solid particles, namely particle inertia and drift due to body forces were investigated separately. In agreement with the theoretical studies of Reeks, and Pismen & Nir, the effect of particle inertia is to increase the eddy diffusivity over that of the fluid (in the absence of particle drift). The increase in the eddy diffusivity of particles over that of the fluid was between 2 and 16%, in reasonable agreement with the increases reported in Reeks, and Pismen & Nir. The effect of a deterministic particle drift is shown to decrease unequally the dispersion in directions normal and parallel to the particle drift direction. Eddy diffusivities normal and parallel to particle drift are shown to be in good agreement with the predictions of Csanady and the experimental measurements of Wells & Stock.

The response of a floating ice plate to a moving load is given in terms of a pair of Green's functions. General expressions for these Green's functions are derived for the case of an infinite isotropic plate of uniform thickness supported on a fluid base of uniform depth. The distributions of stress and strain in the vicinity of a concentrated load receive significant contributions from waves of length comparable with the plate thickness and their description necessitates an exact description of thickness effects. Circumstances in which the classical thin-plate theory can be recovered are discussed. The steady-state response to a uniformly moving load displays a so-called ‘critical’ behaviour for load velocities in the neighbourhood of a threshold value at which radiation commences. At the critical speed the amplitude is limited by dissipative forces in the ice plate. To describe this a simple viscoelastic term is included in our model. Calculations indicate that thin-plate theory is accurate to within 5% for distances greater than twenty times the ice thickness.

A linear analysis of the free oscillations of captive drops or bubbles is discussed. The drop is surrounded by an immiscible liquid or gas and undergoes rotation as a rigid body in the presence of gravity. Using spectral analytical methods, we provide a general formulation for both elliptic and hyperbolic oscillation regimes of the frequency spectrum, for any combination of the Weber and Bond numbers. The method uses a Green function to reduce the inviscid Navier–Stokes equations and boundary conditions to an eigenvalue problem. Both the Green function and normal velocities at the interface are expanded in the orthogonal functional space generated by the Sturm–Liouville problem associated to the interface equation. The effect on the vibration modes of the density and geometrical parameters of the captive drop and surrounding medium is analysed. We present a complete analysis of the low-frequency spectra in the elliptic regime of a set of floating liquid zones and captive drops for a continuous range of Weber and Bond numbers. It is shown that, depending on the geometrical parameters of the system, the elliptic vibration spectrum presents a sui generis modal interaction for low wavenumbers and certain ranges of Weber number.

This paper presents and discusses the results of an extensive experimental investigation of a flat-plate turbulent boundary subjected to an adverse pressure gradient sufficiently strong to lead to the formation of a large separated region. The pressure gradient was produced by applying strong suction through a porous cylinder fitted with a rear flap and mounted above the boundary layer and with its axis in the spanwise direction. Attention is concentrated on the structure of the turbulent flow within the separated region and it is shown that many features are similar to those that occur in separated regions produced in a very dissimilar manner. These include the fact that structure parameters, like Reynolds stress ratios, respond markedly to the re-entrainment of turbulent fluid transported upstream from the reattachment region, the absence of any logarithmic region in the thin wall boundary layer beneath the recirculation zone and the lack of any effective viscous scaling in this wall region, and the presence of a significant low-frequency motion having timescales much longer than those of the large-eddy structures around reattachment.

Similarities with boundary layers separating under the action of much weaker pressure gradients are also found, despite the fact that the nature of the flow around separation is quite different. These similarities and also some noticeable differences are discussed in the paper, which concludes with some inferences concerning the application of turbulence models to separated flows.

A general solution of the three-dimensional Stokes equations is developed for the viscous flow past a square array of circular cylindrical fibres confined between two parallel walls. This doubly periodic solution, which is an extension of the theory developed by Lee & Fung (1969) for flow around a single fibre, successfully describes the transition in behaviour from the Hele-Shaw potential flow limit (aspect ratio B [Lt ] 1) to the viscous two-dimensional limiting case (B [Gt ] 1, Sangani & Acrivos 1982) for the hydrodynamic interaction between the fibres. These results are also compared with the solution of the Brinkman equation for the flow through a porous medium in a channel. This comparison shows that the Brinkman approximation is very good when B > 5, but breaks down when B [les ] O(1). A new interpolation formula is proposed for this last regime. Numerical results for the detailed velocity profiles, the drag coefficient f, and the Darcy permeability Kp are presented. It is shown that the velocity component perpendicular to the parallel walls is only significant within the viscous layers surrounding the fibres, whose thickness is of the order of half the channel height B′. One finds that when the aspect ratio B > 5, the neglect of the vertical velocity component vz can lead to large errors in the satisfaction of the no-slip boundary conditions on the surfaces of the fibres and large deviations from the approximate solution in Lee (1969), in which vz and the normal pressure field are neglected. The numerical results show that the drag coefficient of the fibrous bed increases dramatically when the open gap between adjacent fibres Δ′ becomes smaller than B′. The predictions of the new theory are used to examine the possibility that a cross-bridging slender fibre matrix can exist in the intercellular cleft of capillary endothelium as proposed by Curry & Michel (1980).

In this paper we examine the linear stability of an annular film surrounding a dielectric-fluid core in a tube in the presence of double layers of charges at the film core and at the film–tube interfaces, when the fluid-fluid interface is of low tension. In the absence of electrostatic forces, the surface tension force arising from the circumferential curvature destabilizes, and that from the axial curvature stabilizes the system. The competition is such that waves larger than the unperturbed interface circumference are unstable and those shorter are stable. For charged layers in the film, two cases are examined: (i) double-layer repulsion where the volume charge density is everywhere of the same sign and (ii) double-layer attraction where the diffusive layers next to the film interfaces are of opposite signs. In the first case, double-layer repulsion and surface tension lowering stabilize the destabilizing action of the circumferential component of the surface tension force, and a window of stability can exist. In the case of double layers of opposite signs, double-layer attraction destabilizes the system, and growth rates larger than those caused by pure capillarity can arise. Finally, for the case of a core bounded by an infinite electrolyte, surface tension lowering stabilizes the destabilizing action of the circumferential component of the surface tension force and destabilizes the longitudinal one, although the magnitudes of these effects may differ. As a result the thread can become unstable to waves shorter than the interface circumference.

The method of matched asymptotic expansions is applied to the flow analysis of a three-dimensional thin wing, moving uniformly in very close proximity to a curved ground surface. Four flow regions, i.e. exterior, bow, gap, and wake, are analysed and matched in an appropriate sequence. The solutions in expansions up to third order are given both in nonlinear and linear cases. It is shown that the flow above the wing is reduced to a direct problem, and the flow beneath it appears to be a two-dimensional channel flow. The wake assumes a vortex-sheet structure close to the curved ground, undulating with the amplitude of the ground curvature, and the flow beneath it is also two-dimensional channel flow. As a consequence, an equivalence is found between the extreme curved-ground effect and the corresponding flat-ground effect, which can be treated by the image method.

A numerical method is described, based on the hodograph formulation, for analysing in viscid, free-surface flows over a periodic wall. An efficient implementation of the wall boundary condition results in a straightforward method, accurate for a wide range of bottom undulation heights and flow parameters. It is demonstrated that a series of resonances is possible between the bottom undulations and the free surface. The steady, free-surface profiles are accurately calculated for a wide range of current velocities and are shown to be significantly dimpled by higher harmonics. A study of the flow field indicates that the free-surface shape strongly affects the velocities close to the wall, leading to distributions which change dramatically with current velocity. Some implications of the new results on the phenomena of wall dissolution or material deposition, Bragg scattering of surface waves and sediment transport in rivers, are discussed.

We describe observations of slowly decelerating stratified flow over undular bottom topography in an estuary. The flow, which initially was supercritical with respect to the first internal wave mode, approached a resonance after it had become subcritical. A series of acoustic images showed large-amplitude first-mode trapped waves during this phase of the tide. We derive a criterion for quasi-steady response, and present an extension of Yih's class II linear finite-amplitude solutions that accounts for the waves observed.

A Fizeau optical interferometer has been used to visualize the wake behind a circular cylinder at low Reynolds numbers Re. As well as showing the vortex shedding mechanism and development of the far wake in a new light, the shed vortex strength and age were derived from the results. The vortex velocity distributions, at downstream distances of 5 to 16 diameters, were found to be those of convected Oseen vortices. These measurements confirmed the existence of a transition at Re ≈ 100, which reflects the emergence of convection as the dominant process in the near wake.

A thermally stratified internal hydraulic jump in a fresh water ambient, which has a temperature equal to or greater than 4 °C, is investigated theoretically and experimentally. The thermal jump solutions obtained using a nonlinear buoyancy function are compared with the density jump solutions obtained using a linear buoyancy function. The study reveals considerable difference between thermal and density jump behaviour in a range of temperature above 4 °C. The error of treating a thermal jump as a density jump is found to increase with Froude number and temperature difference and decrease with ambient temperature. Thermal jump-experiments are conducted at ambient water temperatures of 4 °C and 16 °C. The two sets of experiments have identical Froude number, Reynolds number and temperature difference. Experimental observations compare favourably with the thermal jump theory. The analysis of temperature fluctuations suggests that ambient temperature is also modifying the internal mixing characteristics of thermal jumps.

In the present paper we determine the oscillatory flow generated by surface gravity waves near a sea bottom covered with large-amplitude ripples. The vorticity equation and Poisson equation for the stream function are solved by means of a numerical approach based on spectral methods and finite-difference approximations. In order to test the numerical algorithm and in particular the numerical scheme used to generate vorticity along the ripple profile, we also perform an asymptotic analysis, which holds as the time t tends to zero. The main features of the time development of vorticity are analysed and particular attention is paid to the dynamics of the large vortices generated by flow separation at the ripple crests and along the ripple profile. Some of the results obtained by Longuet-Higgins (1981) are recovered; in particular, the present results show a vortex pair shed from the ripple crest every half-cycle. The determination of flow separation along the ripple profile induced by the pressure gradient and the inclusion of viscous effects allows us to obtain accurate quantitative results and detect some important phenomena never observed before.

In particular it is shown that: (i) Whenever a vortex structure moves towards the bottom, a secondary vortex is generated near the ripple profile, which interacts with the primary vortex and causes it to move away from the bottom, (ii) Depending on the values of the parameters, the time development of the free shear layer shed from the ripple crest may produce two or even more vortex structures, (iii) Occasionally vortices generated previously may coalesce with the free shear layer shed from the ripple crest, generating a unique vortex structure.

This paper presents the results of an analytical investigation of the steady translation of a vertical surface-piercing plate at a small angle of attack. This problem is the antisymmetric equivalent of the symmetric thin-ship problem solved by Michell. The linearized boundary-value problem is transformed into an integral equation of the first kind by the method of Green functions. The Kelvin–Havelock Green function is used to satisfy the linearized free-surface boundary condition and radiation condition. A pressure Kutta condition is imposed at the trailing edge. Effective algorithms are developed to evaluate the hypersingular kernel without recourse to numerical integration. The resulting integral equation is solved by a collocation method with a refined scheme of discretization. After establishing the convergence of the present algorithm, computations are carried out for a surface-piercing rectangular plate of aspect ratio 0.5. The integrated lateral-force and yaw-moment coefficients show good agreement with experimental data. Other parameters of the flow such as pressure distributions, drag, strength of leading-edge singularity and free-surface profiles on the plate are also presented. The incompatibility between the pressure Kutta condition and the linearized free-surface condition does not affect the global solution.

A number of processes in which air is entrained in a flow appear to involve the formation of a thin air film between a relatively fast liquid stream and a region of slow recirculation. Eventually, the film breaks into bubbles. This study addresses a possible mechanism causing this process. The linear stability of a vertical film of a viscous gas bounded by liquid in uniform motion on one side, and by liquid at rest on the other side, is studied. Instabilities are found that, depending on the parameter values of the undisturbed flow, are controlled by two basic mechanisms. One is due to the velocity jump across the film and can be related to the usual Kelvin–Helmholtz instability. The second one is controlled by the viscosity jump across the air liquid interfaces. The relation between the remainder of the discrete spectrum and the spectrum of other parallel shear flows bounded by solid or free surfaces is also discussed.

When a suspension of small particles is overlain by a clear fluid whose density is greater than that of the interstitial fluid, but less than that of the bulk suspension, the settling of the dense suspended particles can lead to vigorous convection in the overlying fluid. This novel situation is investigated experimentally and theoretically. A sharp interface is observed between the convecting upper region and a stagnant lower region in which there is unimpeded sedimentation at low Reynolds number. There is no transport of fluid from the upper region into the lower, though there is mixing of both buoyant fluid and entrained particles from the lower region into the upper. The interface between the two regions is found to descend at a constant velocity. Systematic laboratory measurements have determined how this velocity depends on the densities of the layers and the distributions of settling velocities of the particles. A theoretical description is developed which calculates the evolution of the density of the lower region due to differential sedimentation of polydisperse particles. Buoyancy arguments based on the calculated density profile are used to place upper and lower bounds on the amount of particle entrainment into the upper layer and on the rate of fall of the interface between the convecting and sedimenting regions. The theoretical predictions are in good agreement with the experimental observations. The analysis of the interaction between convection and sedimentation in the system considered here may be particularly relevant to the description of evolving crystal-rich layers in magma chambers and of silt-laden outflow from rivers, and has a wide range of other industrial, environmental and geological applications.

We investigate the onset of Rayleigh–Bénard convection in a horizontal porous layer with anisotropic permeability. The permeability is transversely isotropic, whereas the orientation of the longitudinal principal axes is arbitrary. This is sufficient to achieve qualitatively new flow patterns with a tilted plane of motion or tilted lateral cell walls. The critical Rayleigh number and wavenumber at marginal stability are calculated. There are two different types of convection cells (rolls): (i) the plane of motion is tilted, whereas the lateral cell walls are vertical; (ii) the plane of motion is vertical, whereas the lateral cell walls are tilted as well as curved. It turns out that type (i) occurs when the transverse permeability is larger than the longitudinal permeability, and for the converse case type (ii) is preferred.

An exact equation governing the transport of energy associated with disturbances in an arbitrary steady flow is derived. The result is a generalization of the familiar concept of acoustic energy and is suggested by a perturbation expansion of the general energy equation of fluid mechanics. A disturbance energy density and flux are defined and identified as exact fluid dynamic quantities whose leading-order regular perturbation representations reduce in various special cases to previously known results. The exact equation on disturbance energy is applied to a simple example of nonlinear wave propagation as an illustration of its general utility in situations where a linear description of the disturbance is inadequate.

The collapse of a spherical vapour cavity in the vicinity of a compliant boundary is examined numerically. The fluid is treated as a potential flow and a boundary-element method is used to solve Laplace's equation for the velocity potential. Full nonlinear boundary conditions are applied on the surface of the cavity. The compliant wall is modelled as a membrane with a spring foundation. At the interface between the fluid and the membrane, the pressure and vertical velocity in the flow are matched to the pressure and vertical velocity of the membrane using linearized conditions. The results of calculations are presented which show the effect of the parameters describing the flow (the initial cavity size and position, the fluid density and the pressure driving the collapse) and the parameters describing the compliant wall (the mass per unit area, membrane tension, spring constant and coating radius) on the interaction between the two. When the wall is rigid, the collapse of the cavity is characterized by the formation of a re-entrant jet that is directed toward the wall. However, if the properties of the compliant wall are chosen properly, the collapse can be made to occur spherically, as if the cavity were in an infinite fluid, or with the reentrant jet directed away from the wall, as if the cavity were adjacent to a free surface. This behaviour is in qualitative agreement with the experiments of Gibson & Blake (1982) and Shima, et al. (1989). Calculations of the transfer of energy between the flow and the coating are also presented.

This paper reports a temporal instability analysis of a moving thin viscous liquid sheet in an inviscid gas medium. The results show that surface tension always opposes, while surrounding gas and relative velocity between the sheet and gas favour, the onset and development of instability. It is found that there exist two modes of instability for viscous liquid sheets – aerodynamic and viscosity-enhanced instability – in contrast to inviscid liquid sheets for which the only mode of instability is aerodynamic. It is also found that axisymmetrical disturbances control the instability process for small Weber numbers, while antisymmetrical disturbances dominate for large Weber numbers. For antisymmetrical disturbances, liquid viscosity, through the Ohnesorge number, enhances instability at small Weber numbers, while liquid viscosity reduces the growth rate and the dominant wavenumber at large Weber numbers. At the intermediate Weber-number range, Liquid viscosity has complicated effects due to the interaction of viscosity-enhanced and aerodynamic instabilities. In this range, the growth rate curve exhibits two local maxima, one corresponding to aerodynamic instability, for which liquid viscosity has a negligible effect, and the other due to viscosity-enhanced instability, which is influenced by the presence and variation of liquid viscosity. For axisymmetrical disturbances, liquid viscosity always reduces the growth rate and the dominant wavenumber, aerodynamic instability always prevails, and although the regime of viscosity-enhanced instability is always present, its growth rate curve does not possess a local maximum.

A numerical model enabling the prediction of the axial and secondary velocity fields in three-dimensional configurations at moderate Reynolds numbers and Womersley parameters is presented. Steady and unsteady entrance flows in a 90° curve tube (δ = ⅙) under various flow conditions are analysed. The good quality agreement between axial and secondary velocities for a sinusoidally varying flow rate at a Womersley parameter of α = 7.8, obtained from a finite-element calculation, and those obtained from laser-Doppler measurements justify the use of the numerical model.

Halfway into the deceleration phase for a sinusoidally varying flow rate (200 < Re < 800, α = 7.8) a strong resemblance is found to the steady flow case (Re = 700). In contrast with steady flow, near the inner wall reversed axial flow regions are found halfway into and at the end of the deceleration phase. Throughout the flow cycle the Dean-type secondary flow field highly influences axial flow resulting in a shift of the maximal axial velocity towards the outer wall, C-shaped axial isovelocity lines and an axial velocity plateau near the inner wall. Further downstream in the curved tube the Dean-type secondary vortex near the plane of symmetry is deflected towards the sidewall (’tail’–formation), as is also found for steady flow. An increase of the Womersley parameter (α = 24.7) results in a constant secondary flow field which is probably mainly determined by the steady component of the flow rate. A study on the flow phenomena occurring for a physiologically varying flow rate suggests that the diastolic phase is only of minor importance for the flow phenomena occurring in the systolic phase. Elimination of the steady flow component (− 300 < Re < 300) results in a pure Dean-type secondary flow field (no ‘tail’-formation) for α = 7.8 and in a Lyne-type secondary flow field for α = 24.7. The magnitude of the secondary velocities for α = 24.7 are of O(10−2) as compared to the secondary velocities for α = 7.8.