An experimental investigation of various aspects of buoyancy transfer across a diffusive density interface that separates stably stratified, turbulently convecting layers of relatively fresh cold water overlying hot salty water is described. It is argued that the interfacial layer should possess a double boundary-layer structure, in which the thicknesses of the salt and heat interfacial layers are determined by a balance between the opposing effects of diffusion and entrainment. Based on this argument, a simple theory, that predicts the interfacial-layer thickness, the diffusive heat and salt fluxes across the density interface, and the time variation of the temperature and salt concentrations in the convecting layers, is proposed for the case in which the convection is driven by a constant heat flux supplied to the lower layer. During a certain time interval, the theory and experiment agree well, but thereafter distinct differences can be seen. Measurements suggest that these differences may be due to the distortion of the density interface at low interfacial stabilities by turbulent eddies, which leads to a change in the buoyancy transfer mechanism. When the Richardson number falls below a critical value Riv, the interface was found to migrate slowly upwards and the mechanism of entrainment was the detachment of thin sheets of fluid by eddies scouring the interface.

In this paper we consider the transient evolution of a swirling, recirculating flow in a truncated cylinder. In particular, we consider an initial time period during which the evolution of the flow is controlled by inertia. Such flows exhibit a mutual interaction between the swirl and the poloidal recirculation, whereby any axial gradient in swirl alters the recirculation, which, in turn, redistributes the swirl. This interaction may be visualized as a flexing of the poloidal vortex lines, the best known example of which is the inertial wave. Physical arguments and numerical experiments suggest that, typically, a strong, oscillatory recirculation will develop. We examine the exchange of energy between the swirl and recirculation, and show that the direction of transfer depends on the relative signs of ψ and ηuθ/ηz. In addition, there is a limit to the amount of energy that may be exchanged, since conservation of angular momentum imposes a lower bound on the kinetic energy of the swirl. The characteristic reversal time for the recirculation is estimated by considering the history of fluid particles on the endwalls. Its magnitude depends on the relative strengths of the swirl and recirculation. When the recirculation is large, the reversal time exceeds the turn-over time for a poloidal eddy and, consequently, the vortex lines accumulate at the stagnation points on the endwalls. This leads to accelerated local diffusion on the axis. An elementary one-parameter model is proposed for these nonlinear oscillations. In the limit of very weak recirculation, this model is consistent with the exact solution for inertial waves, while for strong recirculation, it confirms that the reversal time is greater than the turn-over time, and that the vortex lines accumulate on the axis.

Unsteady free-surface flow at the bow of a steadily moving, two-dimensional body is solved using a modified Eulerian-Lagrangian technique. Lagrangian marker particles are distributed on both the free surface and the far-field boundary. The flow field corresponding to an inviscid, double-body solution is used for the initial condition. Solutions are obtained over a range of Froude numbers for bodies of three different shapes: a vertical step, a faired profile, and a bulbous bow. A transition Froude number exists at which the bow wave begins to overturn and break. The value of the transition Froude number depends on the bow shape. A stagnation point is observed to be present below the free surface during the initial stage of the wave formation. For flows occurring above the transition Froude number, the stagnation point remains trapped below the free surface as the wave overturns. Below the transition Froude number, the stagnation point rises to the surface as the crest of the transient bow wave moves upstream and away from the body.

The shape of a moving inviscid axisymmetric drop is considered as a function of surface tension and of the intensity of the internal circulation. In a frame of reference moving with the drop, the drop is modelled as a region of diffused vorticity which is bounded by a vortex sheet, and is imbedded in streaming flow. First, an asymptotic analysis is performed for a slightly non-spherical drop whose circulation is very close to that required for the spherical shape. The results indicate that steady drop shapes may exist at all but a number of distinct values of the Weber number, the lowest two of which are 4.41 and 6.15. For highly deformed drops, the problem is formulated as an integral equation for the shape of the drop, and for the strength of the bounding vortex sheet. A numerical procedure is developed for solving this equation, and numerical calculations are performed for Weber numbers between 0 and 4.41. Limiting members in the computed family of solutions contain spherical drops, and inviscid bubbles with vanishing circulation. Computed new shapes include saucer-like shapes with a rounded main body and an elongated tip. The relationship between inviscid drops and drops moving at large Reynolds numbers is discussed.

By carefully controlled phase-coupled input of simultaneous two- and three-dimensional disturbances, the nonlinear evolution and breakdown of the laminar flow in a boundary layer was examined. This involved the generation of plane Tollmien–Schlichting waves and pairs of oblique waves so as to promote nearresonance conditions which have been theoretically shown to lead to the rapid development of three-dimensionality in unstble boundary layers. Special emphasis is placed on the two prominent mechanisms, namely resonant-triads of Orr–Sommerfeld modes and the secondary instability of the streamwise periodic flow to spanwise periodic three-dimensional disturbances. The sensitivity of these mechanisms on the amplitudes and wavenumbers of the input disturbances was of special focus.

The simultaneous two- and three-dimensional wave generation was accomplished using a spanwise array of line heaters suspended just above the wall at the approximate height of the critical layer in the laminar boundary layer. These were operated to produce, through local heating, time-periodic spanwise-phase-varying velocity perturbations. Of primary emphasis in this paper are conditions obtained by the combined forcing of fundamental plane waves with wavenumbers (α, 0) and pairs of subharmonic oblique waves (½α, ± β). The reslults document resonant growth of energy in the subharmonic modes, the formation of staggered lambda vortex patterns with a cross-stream scale commensurate with the seeded ± β condition, and their subsequet transition to turbulence. Complete documentation of the flow field at these various stages is presented using smoke-wire flow visualization and through phase-conditioned hot-wire surveys measuring all three velocity components in three space dimensions.

The flow structure and entrainment mechanisms in the far field of a round vertical buoyant jet have been studied experimentally by use of an optical technique based on laser-induced fluorescence (LIF). A large number of essentially instantaneous tracer concentration profiles were recorded for each experimental run by combining LIF with linear photodiode array imaging and high-speed digital data acquisition. Analysis of the resulting high-resolution flow images indicates that the far-field region is dominated by the periodic passage of structures spanning the entire radial flow extent. Ambient fluid is entrained by vortical motions and is transported to regions deep into the flow interior. Correlation analysis discloses that the passage frequency of the structures scales with the local mean velocity and flow width. Conditional averaging of the data indicates that the downstream frontal region of the structure is well mixed and at higher concentration level than the back and side regions where ambient fluid is intermittently present. This results in an axial concentration gradient within the structure, analogous to the ramp-like pattern previously observed in heated air jets. In comparison to the momentum-driven flow the ambient fluid presence in the flow interior is greatly increased when body forces are the driving mechanism. This appears to result from the influence of buoyancy forces in the production of turbulent vortices at the integral scale. An important feature of both the momentum-driven and buoyancy-driven flows investigated is the strongly intermittent character of the concentration field. This raises the issue of the appropriateness of gradient-diffusion theories for the description of such flows.

Our main objective is to identify a boundary-value problem capable of describing the dynamics of fluids having moving contact lines. A number of models have been developed over the past decade and a half for describing the dynamics of just such fluid systems. We begin by discussing the deficiencies of the methods used in some of these investigations to evaluate the parameters introduced by their models. In this study we are concerned exclusively with the formulation of a boundary-value problem which can describe the dynamics of the fluids excluding that lying instantaneously in the immediate vicinity of the moving contact line. From this perspective, many of the approaches referred to above are equivalent, that is to say they give rise to velocity fields with the same asymptotic structure near the moving contact line. Part of our objecive is to show that this asymptotic structure has only one parameter. A substantial portion of our investigation is devoted to determining whether or not the velocity field in a particular experiment has this asymptotic structure, and to measuring the value of the parameter.

More specifically, we use the shape of the fluid interface in the vicinity of the moving contact line to identify the asymptotic structure of the dynamics of the fluid. Experiments are performed in which silicone oil displaces air through a gap formed between two parallel narrowly-spaced glass microscope slides sealed along two opposing sides. Since we were unable to make direct measurements of the shape of the fluid interface close to the moving contact line, an indirect procedure has been devised for determining its shape from measurements of the apex height of the meniscus. We find that the deduced fluid interface shape compares well with the asymptotic form identified in the studies referred to above; however, systematic deviations do arise. The origin of these deviations is unclear. They could be attributed to systematic experimental error, or, to the fact that our analysis (valid only for small values of the capillary number) is inadequate at the conditions of our experiments.

The effects of the surface density field on wind-driven subtropical gyre circulation are examined within the content of a continuously stratified, potential vorticity conserving model. It is found that the ventilated fluid downwelled from the mixed layer is confined to a relatively thin layer in subtropical regions; the maximum depths of the ventilated region occur in the eastern and southern regions of the gyre. As a consequence of the relative thinness of the ventilated region in the wind-driven gyre, the transport associated with the surface density field is a small part of the total transport in subtropical regions. Thus the surface density field and the potential vorticity associated with ventilated fluid play a minor role in subtropical gyre dynamics and the potential vorticity of unventilated recirculating fluid is found to play the major role in subtropical gyre dynamics. A method of calculating the flow in the relatively large recirculating region is developed and the results of a specific example are discussed.

Results of an experimental and numerical study of parametrically excited nonlinear cross-waves in the vicinity of the cut-off frequency, are reported. Experiments are performed at three cross-wave modes and in the whole range of existence of cross-waves. Numerical studies are based on the solution of the nonlinear Schrödinger equation with a boundary condition at the wavemaker which corresponds to parametric excitation. The validity of the scaling procedure adopted in the model is verified experimentally. Dissipation is incorporated in the model equation and in the wavemaker boundary condition. The influence of the wave breaking on the range of existence of cross-waves is discussed and the relation between the maximum possible steepness of cross-waves and the limits of their existence is obtained.

Surface wave motions in a container with a square base, which is subject to a vertical oscillation, are considered when the amplitude of the oscillation is small and the frequency of the oscillation is close to twice the natural frequency of the system. Subcritical wave motions are found for single modes as well as mixed modes. Here, single modes are described by either one of the two horizontal coordinates whereas mixed modes depend on both coordinates. It is found that in some subcritical region a stable single mode and a stable mixed mode coexist, producing complex basins of attraction.

A numerical study of the generation of Tollmien-Schlichting (T–S) waves due to the interaction between a small free-stream disturbance and a small localized variation of the surface geometry has been carried out using both finite–difference and spectral methods. The nonlinear steady flow is of the viscous–inviscid interactive type while the unsteady disturbed flow is assumed to be governed by the Navier–Stokes equations linearized about this flow. Numerical solutions illustrate the growth or decay of the T–S waves generated by the interaction between the free-stream disturbance and the surface distortion, depending on the value of the scaled Strouhal number. An important result of this receptivity problem is the numerical determination of the amplitude of the T–S waves.

By means of an original optical technique we have studied the spatio-temporal behaviour in a Rayleigh–Bénard convection experiment of small rectangular geometry. The experimental technique allows complete reconstruction of the temperature field integrated along the roll axis. Two main spatiotemporal regimes have been found, corresponding to localized oscillations and travelling waves respectively. Several parameters are proposed for the quantitative characterization of this complex behaviour.

The appearance of chaotic particle trajectories in steady, laminar, incompressible flow through a twisted pipe of circular cross-section is demonstrated using standard dynamical systems diagnostics and a model flow based on Dean's perturbation solutions. A study is performed to determine the parameters that control fluid stirring in this mixing device that has no moving parts. Insight into the chaotic dynamics are provided by a simple one-dimensional map of the pipe boundary onto itself. The results of numerical experiments illustrating the stretching of material lines, stirring of blobs of material, and the three-dimensional trajectories of fluid particles are presented. Finally, enhanced longitudinal particle dispersal due to the coupling between chaos in the transverse direction and the non-uniform longitudinal transport of particles is shown.

We study the instability of finite-amplitude waves on uniform vortex layers of finite thickness bounded by a plane rigid surface. A weakly nonlinear analysis of vorticity interface perturbations, and spectral stability calculations using the full equations of motion, together show that steady progressive waves are unstable to general subharmonic pertubations in the range 0.094 < d/λ < 1.7, where d is the mean layer thickness and λ is the primary wavelength. The relevance of this instability to ultimate interface filamentation is tested by performing several numerical contourdynamical simulations of the nonlinear interface evolution for initial disturbances consisting of the finite amplitude wave plus eigenfunctions obtained from the spectral calculations. The results indicate that within the band of unstable wavelengths, small perturbations to the steady non-uniform flow given by the finite amplitude wave motion (vortex equilibrium) are able to grow in magnitude, until at a time tf, the wave extremum encounters a hyperbolic critical point of the velocity field after which filamentation occurs. Arguments are put forward based on the unsteady simulations with the purpose of identifying the preferred frame of reference for viewing the kinematical events controlling the filamentation process. An estimate for tf is then made, and the mechanism of filamentation found is discussed in relation to the recently proposed nonlinear-cascade mechanism of Dritschel (1988a).

For small-amplitude vortical and entropic unsteady disturbances of potential flows, Goldstein proposed a partial splitting of the velocity field into a vortical part u(I) that is a known function of the imposed upstream disturbance and a potential part ∇ϕ satisfying a linear inhomogeneous wave equation with a dipole-type source term. The present paper deals with flows around bodies with a stagnation point. It is shown that for such flows u(I) becomes singular along the entire body surface and its wake and as a result ∇ϕ will also be singular along the entire body surface. The paper proposes a modified splitting of the velocity field into a vortical part u(R) that has zero streamwise and normal components along the body surface, an entropy-dependent part and a regular part ∇ϕ* that satisfies a linear inhomogeneous wave equation with a modified source term.

For periodic disturbances, explicit expressions for u(R) are given for three-dimensional flows past a single obstacle and for two-dimensional mean flows past a linear cascade. For weakly sheared flows, it is shown that if the mean flow has only a finite number of isolated stagnation points, u(R) will be finite along the body surface. On the other hand, if the mean flow has a stagnation line along the body surface such as in two-dimensional flows then the component of u(R) in this direction will have a logarithmic singularity.

For incompressible flows, the boundary-value problem for ϕ* is formulated in terms of an integral equation of the Fredholm type. The theory is applied to a typical bluff body. Detailed calculations are carried out to show the velocity and pressure fields in response to incident harmonic disturbances.

The time-space evolution of a steep water wave reaching the breaking stage is observed by means of a visualization technique. In particular, the asymmetry of the wave profile in the near-breaking region is displayed. Measurements at breaking onset on a sample of breaking waves show a relation between the rate of asymmetry growth and the breaker type. The shape evolution of a plunging crest after breaking has started, and the related splash-up phenomenon and its part in the air-entrainment process are also observed.

A solid long slender body with curved centreline is held at rest in a fluid undergoing a uniform flow. Assuming that the Reynolds number Re based on body length is fixed, the force per unit length on the body is obtained as an asymptotic expansion in terms of the ratio κ of the cross-sectional radius to body length. In the limit of large Re, this result is no longer valid and an asymptotic expansion in κRe is necessary. A uniformly valid solution is obtained from these two expansions. The total force and torque acting on a body with a straight centreline are explicitly determined. The limiting cases of small and large Re are studied in detail.

Chaotic mixing of fluids in slow flows is ubiquitous but incompletely understood. However, relatively simple experiments provide a wealth of information regarding mixing mechanisms and indicate the need for complementary theoretical developments in dynamical systems. In this work we presnt a versatile cavity flow apparatus, capable of producing a variety of two-dimensional velocity fields, and use it to conduct a detailed experimental study of mixing in low-Reynolds-number flows. Since the goal is detailed understanding, only two time-periodic co-rotating flows induced by wall motions are considered: one continuous and the other discontinuous. Both types of flows produce exponential growth of intermaterial area, as expected from chaotic flows, and a mixture of islands and chaotic regions. A procedure for identifying periodic points and determining their movements is presented as well as how to make meaningful comparisons between periodic flows. We observe that periodic points move very much as a planetary system; planets (hyperbolic points) have moons (elliptic points) with twice the period of the planets; furthermore the spatial arrangement of periodic points becomes symmetric at regular time intervals. Detailed analyses reveal complex behaviour: birth, bifurcation, and collapse of islands; formation and periodic motion of coherent structures, such as islands and large-scale folds. However, the richness and complexity of the results obtained indicate that these two-dimensional time-periodic systems are far from completely understood and that other wall motions might deserve a similar level of scrutiny.

A theoretical description of the low-Reynolds-number collision and rebound of two rigid or elastic spheres separated by a thin layer of viscous fluid with pressure-dependent physical properties is presented. It has previously been shown by Davis et al. (1986) that the hydrodynamic pressure which builds up in the thin fluid layer must become large enough to elastically deform the spheres near the axis of symmetry, if they are to rebound subsequent to colliding. Under these extreme pressures, however, it is expected that the fluid may also compress and that its viscosity may increase by several orders of magnitude. It is shown that these pressure-dependent effects may significantly alter the minimum separation reached during approach of the spheres, as well as the maximum separation and relative velocity attained during rebound of the spheres. In particular, the pressure buildup during the collision process is predicted to become sufficiently large under some conditions so that the corresponding viscosity increase causes the fluid in the gap between the colliding spheres to behave nearly as a solid and to limit the close approach of the opposing surfaces. Also, the storage of energy via the compression of the fluid in the gap allows rigid spheres to bounce as this energy is released subsequent to their collision. However, it is found that this rebound is very weak relative to that which is predicted for elastic spheres.

It is shown that hydrodynamic interactions between non-Brownian, non-spherical, sedimenting particles give rise to an increase in the number of neighbouring particles in the vicinity of any given particle. This result suggests that the suspension is unstable to particle density fluctuations even in the absence of inertia; a linear stability analysis confirms this inference. It is argued that the instability will lead to convection on a lengthscale (nl)−½, where l is a characteristic particle length and n is the particle number density. Sedimenting suspensions of spherical particles are shown to be stable in the absence of inertial effects.