In a recent paper (Jansons & Johnson 1988) the authors discuss topographic Rossby waves over a random array of seamounts. It is noted that resonance is possible between a hill and an equal and opposite dale but such resonances are mentioned only briefly due to the small likelihood of correctly matched topography in the ocean. The present paper considers the resonances in detail showing how the normal modes formed by frequency splitting at resonance can be combined to give modes that slowly transfer energy from one region supporting topographic waves, across a region where such weaves are evanescent, to another region supporting waves. In addition to the simplest case of a hill—dale pair for which an exact energy-transferring mode is obtained, transferring modes are given for a three-hill system, for two hills near a coastal boundary, and for two-basin lakes. The analysis is simplified and the results generalized by extensive use of the invariance of the governing equation under conformal mappings. A transferring mode is given for a dale in a random array of seamounts showing energy leaking slowly from the dale to large distances and returning, with the rate of leakage depending on the area fraction of seamounts. It is concluded that although resonances and transferring modes are not likely to be important in random arrays on infinite planes, they are relevant to numerical models, which are necessarily restricted to finite domains, to coastal seamount chains, and to multi-basin lakes.

The Budgets For The Reynolds Stresses And For The Dissipation Rate Of The Turbulence Kinetic Energy Are Computed Using Direct Simulation Data Of A Turbulent Channel Flow. The Budget Data Reveal That All The Terms In The Budget Become Important Close To The Wall. For Inhomogeneous Pressure Boundary Conditions, The Pressure—Strain Term Is Split Into A Return Term, A Rapid Term And A Stokes Term. The Stokes Term Is Important Close To The Wall. The Rapid And Return Terms Play Different Roles Depending On The Component Of The Term. A Split Of The Velocity Pressure-Gradient Term Into A Redistributive Term And A Diffusion Term Is Proposed, Which Should Be Simpler To Model. The Budget Data Are Used To Test Existing Closure Models For The Pressure—Strain Term, The Dissipation Rate, And The Transport Rate. In General, Further Work Is Needed To Improve the models.

Impulsively started, viscous, incompressible flows around a circular cylinder are simulated by a Lagrangian vortex solution of the vorticity equation using random walks for diffusion and the vortex-in-cell method for convection in a fractional-step scheme. Vortices are introduced around the surface at each timestep to satisfy the zero-slip condition. In the range of Reynolds numbers 2.5 × 102 to 105, comparisons with two analytical solutions, valid for small times (t < 1), show reasonable agreement. For somewhat longer times (t < 5), for a similar range of Reynolds numbers, comparisons are made with accurate Eulerian numerical solutions and with careful flow-visualization experiments. Agreement is good provided a sufficiently large number of vortices is introduced per timestep. The number required increases as Reynolds number increases. If too few are introduced, the vorticity in the wake tends to roll up too tightly. The vortex method remains stable, whereas Eulerian schemes have been reported to become eventually unstable unless upwind differencing is used, reducing accuracy.

Low-Rossby-number flow past a circular cylinder in a rapidly rotating frame is studied when 1 < N < 2, where N is equal to E½/Ro in terms of the Ekman number E and Rossby number Ro. For this parameter range the E¼ boundary layer contains a singularity at the rear stagantion point. The asymptotic structure of this singularity is shown to consist of three distinct asymptotic regions, one of which is viscous while the others are inviscid. New accurate numerical solutions of the boundary-layer equation confirm this singularity structure. The use of Von Mises coordinates both simplifies the analysis, and enables numerical solutions to be found closer to the critical value N = 1, beneath which the flow separates upstream of the rear stagnation point.

The laminar—turbulent transition of the Taylor—Görtler (TG) vortex flow in the clearance between two concentric spheres with only the inner sphere rotating (spherical Couette flow) is investigated by velocity measurement and simultaneous spectral and flow-visualization measurements by measuring the intensity of light scattered by the aluminium flakes used in flow visualization in the case of a relatively small ratio of the clearance to inner-sphere radius (clearance ratio β = 0.14). An azimuthal velocity component has been measured by a constant-temperature hotwire anemometer at two different colatitudes (meridian angles) θ; θ = 80° and 90° (the equator). A critical Reynolds number, some transition Reynolds numbers, flow regimes and flow states are obtained by the simultaneous spectral and flow-visualization measurements. The flow state is expressed by the number of toroidal TG vortex cells N, that of spiral TG vortex pairs Sp, the wavenumber of the travelling azimuthal waves on the toroidal TG vortices m and the wavenumber of shear waves SH. The mean velocity distribution and the characteristic values of the fluctuating velocity, such as autocorrelation coefficient, power spectrum and turbulence intensity (r.m.s. value), are considered over a great range of Reynolds number Re. Three kinds of fundamental frequencies of the velocity fluctuation are discovered and their characteristics are clarified by means of the velocity measurement and the simultaneous spectral and flow-visualization measurements. The three kinds of fundamental frequencies expressed by fS, fW and fH correspond to the spiral TG vortices, the travelling azimuthal waves and the shear waves, respectively. These fundamental frequencies are independent of both θ and wall distances from the inner sphere, but depend strongly on Re. Although the rotation frequency of the travelling azimuthal waves (or wave speed) in the circular Couette flow decreases monotonically with increasing Reynolds number until it reaches a plateau, the values of the rotation frequencies of the spiral TG vortices, the travelling azimuthal waves and the shear waves in the spherical Couette flow, fS/SP, fW/m and fH/SH, are nearly constant as the Reynolds number is increased, and differ slightly from one another.

The velocity fields beneath an air—water interface have been determined in a laboratory facility for the cases of wind-generated waves, with wind speeds ranging from 1.5 to 13.1 m/s, and of wind-ruffled mechanically generated waves of about 22 mm amplitude and 1 Hz frequency, with wind speeds ranging from 1.7 to 6.2 m/s. The velocity was measured in a fixed frame of reference with a two-component, laser-Doppler anemometer. It was possible to determine the lengthscales and evaluate the behaviour of the mean, wave-related and turbulent components of the flows. The waves affect the mean flows, even though the profiles remain essentially logarithmic and the wave field conforms generally with the results of linear theory. In the wind-wave cases the turbulent quantities behave similarly to those in flows over flat plates. They have different trends in the mechanical-wave cases, suggesting a coupling between waves and turbulence. Finally, measured values of the mean wave-induced shear stress were negative, leading to an energy transfer from the waves to the mean flow.

The preferred mode of instability was investigated in an axisymmetric air jet of moderate Reynolds number. Natural instabilities are shown to scale with local shear-layer thickness and the preferred mode is shown to be a shear-layer instability. The spatial evolution of the preferred mode was examined by exciting the flow acoustically and then mapping the phase-locked velocity fluctuations. Throughout the potential core region the phase-locked profiles are shown to agree with the eigensolutions of the Orr—Sommerfeld stability equations provided the calculations are based on measured, mean velocity profiles. The excitation intensity was varied from low levels, where the flow was merely tagged, to high levels where the mean flow was substantially distorted, and over that range of excitation there was no apparent deterioration in the agreement with stability predictions.

An experimental investigation of double-diffusive convection in a two-layer, salt-stratified solution destabilized by lateral heating and cooling has been performed. Initially, diffusive regime phenomena are observed as the two uniform salinity layers are thermally driven and behave somewhat independently. As salt is transferred across the interface separating the layers, salinity stabilization decreases and complicated flow structure is observed at the interface. In the final stages before mixing, the stabilizing salinity gradient becomes small, the thermal/hydrodynamic boundary layers on the heated and cooled sidewalls penetrate the salinity interface and mixing, in the finger regime, occurs. The dimensionless mixing time is described with parameters associated with thermal and salinity buoyancy forces and the enclosure aspect ratio. Careful selection of the experimental conditions allows dimensionless interfacial salinity transport rates to be correlated with appropriate dimensionless parameters.

In a previous paper (Churilov & Shukhman 1987a) we investigated the nonlinear development of disturbances to a weakly supercritical, stratified shear flow; we now report a continuation of that study. The degree of supercriticality of the flow is assumed not too small so that — unlike Paper 1 — the critical layer that appears in the region of resonance of the wave with the flow is an unsteady rather than viscous one. The evolution equation with cubic and quintic nonlinearity has been derived. The nonlinear term is non-local in time, i.e. depends on the entire preceding development of the disturbance. This equation has been used in the analysis of the evolution of an initially small disturbance. It is shown that where wave amplitude A is small enough (A [Lt ] ν½, ν is the inverse of the Reynolds number), cubic nonlinearity dominates. In this case, as in Paper 1, the character of the evolution essentially depends on the sign of the quantity (η − 1), where η is the Prandtl number. However, independently of this sign the disturbance reaches — as it increases — the level A ∼ O(ν½) and then quintic nonlinearity becomes dominant. At this stage an ‘explosive’ regime occurs and amplitude grows as $A \sim (t_0 - t)^{-\frac{7}{4}}$. The results obtained, together with the findings of Paper 1, provide a full description of the development of small disturbances at a large (but finite) Reynolds number in different regimes which are determined by the degree of flow's supercriticality.

The stability of quasi-geostrophic three-layer stratified flow in a channel is examined. The mean zonal velocity $\overline{U}_i$ is uniform within each layer (i = 1, 2, 3). Thus, as in the two-layer model of Phillips (1954), the only source of energy for growing disturbances is the potential energy stored in the sloping interfaces. Attention is focused upon the case in which ε = Δρ2/Δρ1 [Lt ] 1 (Δρ1, Δρ2 are the changes in density across the upper and the lower interfaces). Two scales of instability are possible: long waves (wavenumber O(1)) associated with the upper interface and short waves (wave-number O(ε−½)) associated with the lower interface. It is found that short waves are unstable only when S (the ratio of the slope of the lower interface to that of the upper interface) is greater than one or less than zero, i.e. when the gradients of potential vorticity in the two lower layers have opposite signs. The short waves have the largest growth rates when S2ε (the ratio of the potential energy stored in the lower interface to that stored in the upper interface) [gsim ] 1. The results of this analysis are used in an accompanying paper to interpret some experiments with three-layer eddies.

The stability of eddies with three-layer stratification is examined experimentally. When the difference in density between the upper two layers is much greater (or less) than that between the lower two layers baroclinic instability on two different lengthscales (the Rossby radii associated with the upper and the lower interfaces) is possible. The vortices are created using modifications of two techniques described by Griffiths & Linden (1981) in their study of two-layer eddies.

‘Constant-flux’ eddies are generated by the release of a constant flux of buoyant fluid from a small source positioned at the surface of a two-layer fluid. In a second variation of this experiment, the source is positioned at the interface between two layers and fluid of intermediate density is injected. As the horizontal lengthscale increases, the vortices evolve from a stable to an unstable state. It is showns that the size at which the vortices become unstable may be significantly altered by the presence of a second interface. The results agree qualitatively with the conclusions of a linear stability analysis of quasi-geostrophic three-layer flow in a channel (Smeed 1988), but it is necessary to examine the effects of horizontal shear and Ekman dissipation to explain the experimental results.

‘Constant-volume’ eddies are produced by the release of a volume of buoyant fluid, initially contained within a cylindrical barrier, at the surface of a two-layer fluid. After the barrier is removed, the buoyant fluid spreads a distance of the order of the Rossby radius. Similarly, vortices are created by releasing a volume of fluid of density intermediate between the initial two layers. Within a few rotation periods the vortices become unstable to disturbances similar to those observed in two-layer experiments. Qualitative agreement is found between the observed wavelength and the fastest growing mode predicted by the linear stability theory (Smeed 1988). When the disturbances reach large amplitude a change in lengthscale is often observed.

Mechanisms of turbulent mixing are explored by numerical simulations of one-dimensional and two-dimensional mixing with Pr < 1. The simulations suggest that the local rate of strain γ mixes the scalar field by at least two interacting mechanisms: the mechanism of generation, pinching and splitting of extrema proposed by Gibson (1968a) which acts along lines where the scalar-gradient magnitude is small; and a new mechanism of alignment, pinching and amplification of the gradients which acts along lines where the scalar-gradient magnitude is large. After extrema are generated, they split to form new extrema of the same sign, and saddle points. These zero-gradient points are connected by minimal-scalar-gradient lines which continuously stretch at rates of order γ, becoming longer than the viscous scale LK. For Pr < 1, this extends the influence of the local rate of strain to lengths of at least the order of the inertial-diffusive scale LC > LK; that is, larger than the maximum assumed possible by Batchelor, Howells & Townsend (1959). Roughly orthogonal maximal-scalar-gradient lines are also embedded in the fluid, and compressive mixing along these lines also reflects the magnitude and direction of the local rate of strain over distances larger than LK. Because the two rate-of-strain mixing mechanisms act along lines, they can be modelled by one-dimensional numerical simulation. Both are Prandtl-number independent and together they provide a plausible physical basis for the universal scalar similarity hypothesis of Gibson (1968b) that turbulent mixing depends on γ for all Pr.

In recent work on shear-flow instability, the tacit assumption has been made that the two-dimensional stability of finite-amplitudes waves in plane Poiseuille flow follows a simple and well-understood pattern, namely one with a stability transition at the limit point in Reynolds number. Using numerical stability calculations we show that the application of heuristic arguments in support of this assumption has been in error, and that a much richer picture of bifurcations to quasi-periodic flows can arise from considering the two-dimensional superharmonic stability of such a shear flow.

When coils carrying high-frequency currents are placed in the neighbourhood of a stream of liquid metal (or other electrically conducting fluid), the magnetic pressure on the liquid surface causes a deflection of the stream. This effect is studied for a two-dimensional stream on the assumptions that the width of the stream is small compared with the scale characterizing the applied magnetic pressure distribution, and that the effect of gravity may be neglected over this scale. The relationship between the angle of deflection of the stream and the power supplied to the perturbing currents is determined. More complex deformations associated with distributed current sources are considered. Experiments are performed in which a thin sheet of mercury is deflected by two antiparallel line currents. The agreement between theory and experiment is reasonable, despite a tendency towards three-dimensionality in the latter. A second configuration is considered in which a thin current-carrying circular jet is deflected by a vertical line current. The path of the deflected jet is calculated. The limitations of the analysis are briefly discussed.

Direct numerical simulations of decaying high-Reynolds-number turbulence are presented at resolutions up to 8002 for general periodic flows and 20482 for periodic flows with large-scale symmetries. For turbulence initially excited at large scales, we characterize a transition of the inertial energy-spectrum exponent from n ≈ − 4 at early times to n ≈ − 3 when the turbulence becomes more mature. In physical space, the first regime is associated with isolated vorticity-gradient sheets, as predicted by Saffman (1971). The second regime, which is essentially statistical, corresponds to an enstrophy cascade (Kraichnan 1967; Batchelor 1969) and reflects the formation of layers resulting from the packing of vorticity-gradient sheets. In addition to these small-scale structures, the simulation displays vorticity macro-eddies which will survive long after the vorticity-gradient layers have been dissipated (McWilliams 1984). We validate the linear description of two-dimensional turbulence suggested by Weiss (1981), which predicts that coherent vortices will survive in regions where vorticity dominates strain, while vorticity-gradient sheets will be formed in regions where strain dominates. We show that this analysis remains valid even after vorticity-gradient sheets have been formed.

The form of steep capillary waves is of interest as a possible initial condition for the formation of air bubbles at a free surface. In this paper the limiting forms of pure capillary waves and of quasi-capillary waves are studied analytically. Crapper's finite-amplitude solution is expressed in a simple form, and is shown to be one of several exact elementary solutions to the pure-capillary free-surface condition. Among others are the solution z = w+sinh w, where w is the velocity potential, and also z = w3. The latter solution, though it represents a self-intersecting flow, can be used as the first in a sequence of approximations to the form of the steepest wave. Hence it is shown that the influence of gravity on the shape of the limiting ‘bubble’ is very small. The result is confirmed by an examination of Hogan's numerical calculations of limiting capillary-gravity waves.

In the crest of a limiting wave the particle velocity is almost constant and equal to the phase speed. This property makes it possible to apply a quasi-static approximation so as to determine the form of the crest, and hence to find an expression for the complete profile of a capillary-gravity wave of limiting steepness. It appears that there exists a solitary wave of capillary-gravity type on deep water.

An exact analytical study is presented for the electrophoretic motion of a dielectric sphere in the proximity of a large non-conducting plane. The applied electric field is parallel to the plane and uniform over distances comparable with the particle radius. The particle and plane surfaces are assumed uniformly charged and the thin-double-layer assumption is employed. The presence of the wall causes three basic effects on the electrophoretic velocity: first, an electro-osmotic flow of the suspending fluid exists owing to the interaction between the electric field and the charged wall; secondly, the electrical field lines around the particle are squeezed by the wall, thereby speeding up the particle; and thirdly, the wall enhances viscous retardation of the moving particle. In the analysis, corrections to Smoluchowski's classic equation for the electrophoretic velocity in an unbounded fluid are presented for various separation distances between the particle and the wall. Of particular interest is the electrophoresis for small gap widths, in which case the net effect of the plane wall is to enhance the particle velocity. The particle mobility can be increased by as much as 23% when the surface-to-surface spacing is about 0.5% of the sphere radius. For the case of moderate to large separations, the electrophoretic velocity of the particle is reduced by the wall, but this effect is much weaker than for sedimentation. In addition to the translational migration, the electrophoretic sphere rotates at the same time in the direction opposite to that which would occur if the sphere sedimented parallel to a plane wall. The ratio of rotational-to-translational speeds of the sphere is in general larger for electrophoresis than for sedimentation.

When a temperature gradient is imposed on the free surface of a thin liquid layer, fluid motion can develop due to thermocapillarity. Previous work using linear theory has shown that the layer can become unstable to a pair of obliquely travelling hydrothermal waves. Here, we shall study the nonlinear behaviour of this system to determine possible equilibrium waveforms for the instability when the critical point from the linear theory is slightly exceeded. We find that for all Prandtl numbers and small Biot numbers, possible waveforms are composed of only one of the unstable linear waves. For small Prandtl number and larger Biot numbers, a combination of the two linear waves is a possible waveform. Further analysis of these equilibrium states shows that both exhibit the Eckhaus and Benjamin-Feir sideband instability and a corresponding phase instability. Thus, they become modulated on long length-and timescales as the system develops.

A theoretical and numerical investigation of streamwise-oriented Dean vortices in curved channel flow is presented. The principal results are obtained from three-dimensional pseudospectral simulations of the incompressible time-dependent Navier-Stokes equations. With increasing Reynolds number, a sequence of transitions similar to that observed in non-turbulent Taylor-Couette flow is found. The transition from laminar curved channel Poiseuille flow to axisymmetric Dean vortex flow is studied using linear and weakly nonlinear analyses; these results are compared to the full simulations. Using the code, two transitions that cause the axisymmetric vortices to develop waves travelling in the streamwise direction at higher Reynolds numbers are discovered. The linear stability of axisymmetric Dean vortex flow to non-axisymmetric perturbations is examined. Associated with the two transitions are two different non-axisymmetric flows: undulating and twisting Dean vortex flow. Undulating vortices are similar to wavy Taylor vortices. Twisting vortices, with a much shorter streamwise wavelength, are dissimilar; to our knowledge, they have no counterpart in the Taylor-Couette problem. At sufficiently high Reynolds numbers, linear growth rates associated with twisting vortices far exceed those associated with undulating vortices. For the channel curvatures studied, angular speeds of both kinds of travelling waves are only weakly dependent on Reynolds number and wavenumber. A bifurcation limits the vortex spacings that can be examined and suggests an Eckhaus stability boundary. The development of wavy vortex flows from small-amplitude disturbances shows that full development of undulating vortices may require a streamwise distance greater than one circumference, whereas for sufficiently large Reynolds numbers, twisting vortices reach equilibrium amplitude within half this distance and are therefore more likely to be observed experimentally. We suggest twisting vortices are due to a shear instability.

Experimental and theoretical studies of vibration-induced flow and mixing of dry granular materials are described. Tests were performed on rounded polystyrene beads contained in a rectangular box having transparent front and back walls and a flexible, nominally horizontal bottom which could be driven at various frequencies and amplitudes. The amplitude of the bottom vibrations was a maximum at the centre and decreased towards the vertical sidewalls. Slow recirculating flows were observed; they had the form of two vortices in which the velocity was upwards at the vertical centreline and downwards along the vertical sidewalls. The streaming velocities were measured as a function of bed vibration frequency and displacement amplitude. An explanation proposed for the recirculating flows is that the vibrating base sends ‘acoustic’ waves upwards through the bed. These waves ‘fluidize’ the granular material but are in turn attenuated because of the dissipative nature of the collisions between the ‘fluidized’ particles. Thus the slow recirculating flows in the granular material are analogous to the more familiar ‘acoustic streaming’ in air. An approximate analysis of these streaming motions is developed by making use of a modification of the constitutive theory of Jenkins & Savage (1983). A number of simplifying assumptions are introduced to make the analysis tractable. The general flow patterns of the streaming motions are predicted, but the velocities are overestimated as a result of the simplifying assumptions. The analysis is restricted to a rather narrow range of conditions.