The morning glory is a meteorological phenomenon which occurs in northern Australia and takes the form of a series of roll clouds. The morning glory is generated by the interaction of nocturnal seabreezes over Cape York Peninsula and propagates in a south-westerly direction over the Gulf of Carpentaria. In the present work, it is shown that the morning glory can be modelled by the resonant flow of a two-layer fluid over topography, the topography being the mountains of Cape York Peninsula. In the limit of a deep upper layer, the equations of motion reduce to a forced Benjamin–Ono equation. In this context, resonant means that the underlying flow velocity of the seabreezes is near a linear long-wave velocity for one of the long-wave modes. The morning glory is then modelled by the undular bore (simple wave) solution of the modulation equations for the Benjamin–Ono equation. This modulation solution is compared with full numerical solutions of the forced Benjamin–Ono equation and good agreement is found when the wave amplitudes are not too large. The reason for the difference between the numerical and modulation solutions for large wave amplitude is also discussed. Finally, the predictions of the modulation solution are compared with observational data on the morning glory and good agreement is found for the pressure jump due to the lead wave of the morning glory, but not for the speed and half-width of this lead wave. The reasons for this are discussed.

Numerical computations of the steady, two-dimensional, incompressible, uniform velocity but stably stratified flow past a normal flat plate (of unit half-width) in a channel are presented. Attention is restricted to cases in which the stratification is weak enough to avoid occurrence of the gravity wave motions familiar in more strongly stratified flows over obstacles. The nature of the flow is explored for channel half-widths, H, in the range 5 [les ] H [les ] 100, for Reynolds numbers, Re, (based on body half-width and the upstream velocity, U) up to 600 and for stratification levels between zero (i.e. neutral flow) and the limit set by the first appearance of waves. The fourth parameter governing the flow is the Schmidt number, Sc, the ratio of the molecular diffusion of the agent providing the stratification to the molecular viscosity. For cases of very large (in the limit, infinite) Sc a novel technique is used, which avoids solving the density equation explicitly. Results are compared with the implications of the asymptotic theory of Chernyshenko & Castro (1996) and with earlier computations of neutral flows over both flat plates and circular cylinders. The qualitative behaviour in the various flow regimes identified by the theory is demonstrated, but it is also shown that in some cases a flow zone additional to those identified by the theory appears and that, in any case, precise agreement would, for most regimes, require very much higher Re and/or H. Some examples of multiple (i.e. non-unique) solutions are shown and we discuss the likelihood of these being genuine, rather than an artefact of the numerical scheme.

Regularized point-vortex simulations are presented for vortex sheet motion in planar and axisymmetric flow. The sheet forms a vortex pair in the planar case and a vortex ring in the axisymmetric case. Initially the sheet rolls up into a smooth spiral, but irregular small-scale features develop later in time: gaps and folds appear in the spiral core and a thin wake is shed behind the vortex ring. These features are due to the onset of chaos in the vortex sheet flow. Numerical evidence and qualitative theoretical arguments are presented to support this conclusion. Past the initial transient the flow enters a quasi-steady state in which the vortex core undergoes a small-amplitude oscillation about a steady mean. The oscillation is a time-dependent variation in the elliptic deformation of the core vorticity contours; it is nearly time-periodic, but over long times it exhibits period-doubling and transitions between rotation and nutation. A spectral analysis is performed to determine the fundamental oscillation frequency and this is used to construct a Poincaré section of the vortex sheet flow. The resulting section displays the generic features of a chaotic Hamiltonian system, resonance bands and a heteroclinic tangle, and these features are well-correlated with the irregular features in the shape of the vortex sheet. The Poincaré section also has KAM curves bounding regions of integrable dynamics in which the sheet rolls up smoothly. The chaos seen here is induced by a self-sustained oscillation in the vortex core rather than external forcing. Several well-known vortex models are cited to justify and interpret the results.

Initially turbulent bluff body wakes decay in the presence of a stable background density gradient to form chains of comparatively stable and long-lived vortex structures, most of the late-time properties of which have been shown to be independent of the initial generating Froude number (for a sphere of diameter, D, moving at speed, U, F = 2U/ND, where N is the buoyancy frequency). Results of experiments with vertical interrogation planes are described, where any anticipated F-dependence might be most evident, as the competing effects of horizontal inertial forcing and the restoring buoyancy force can be measured directly by simultaneous measurement of horizontal and vertical velocity components. Experiments were conducted at sufficiently large values of Re [ges ] 3 × 103 and F [ges ] 4 that turbulence can occur over many scales in the near wake, and the scaling properties might then extrapolate to ocean engineering applications.

When F [ges ] 4, the fluid motions in the intermediate, non-equilibrium régime always occur in coherent patches whose vertical extent is smaller than the total wake height. The patches of vorticity have longer horizontal than vertical coherence lengths, and may be termed layers, even though they are far from uniform in the horizontal. The degree to which the complex vertical structure is later dominated by the mean wake defect depends strongly on F.

The total wake height, LV, depends on the initial value of F so that LV/D ∼ F0.6. LV is established early and remains almost unchanged up to Nt ≈ 30. At later times, the non-equilibrium wake exchanges potential with kinetic energy and re-adjusts according to local dynamical constraints, so that, within each layer, the quasi-two-dimensional flow proceeds without any further dependence on, or memory of, the initial value of F. The flow is everywhere stable to overturning Kelvin–Helmholtz instabilities and local length and velocity scales evolve so that the local horizontal and vertical Froude numbers, FH, FV, are both of order 0.1.

Although Osmidov-length arguments for vertical scale selection appear to be physically appropriate, they do not correctly predict the measured F-dependence in either LV, or in the layer height, lV. Thus the physical mechanism responsible remains elusive, as the alternative laminar instability mechanisms are not presented with the appropriate, scale-free initial conditions over the parameter range in which they have been shown to operate.

Ultimately, the measurements support the application of low FH and FV scaling theories to the late wake flow. The preceding non-equilibrium stage, when the vertical structure of the late wake is determined, does not yield so readily to assumptions involving the smallness of the vertical velocity component.

Flow visualizations and phase-averaged particle image velocimetry (PIV) measurements of a jet in crossflow configuration at very low Reynolds numbers (Rej ≃ 100) are performed in a water tunnel for jet-to-cross-stream velocity ratios R ranging from 1.5 to 4.5. The PIV vector fields and flow visualizations, carried out by injecting methylene blue dye and by the laser induced fluorescence (LIF) technique, are analysed to characterize the effect of R on the formation and evolution of large-scale vortices. It is shown that two distinct flow regimes are established depending on R, with R ≃ 3 being a transitional value. At low R, the longitudinal vorticity dynamics is dominated by the so-called wake-like structures which are shown to be strictly connected to the streamwise counter-rotating vortices (CRVP) which drive the destabilization of the jet flow. On the other hand, at large R, vortices with positive and negative vorticity are coupled together. The establishment of these different behaviours is interpreted physically as an effect of the jet Reynolds number which plays an essential role on the destabilization mechanisms which lead to the formation of the jet shear-layer structures. In any case, the onset of instability is driven by mechanisms which are different from those characteristic of free jets.

We study the head-on collision between two weakly deformable droplets, each of radius a (in the range 10–150 μm), moving towards one another with characteristic impact speeds ±U′c. The liquid comprising the drop has density ρd and viscosity μd. The collision takes place in an incompressible continuum gas with ambient density ρg [Lt ] ρd, ambient pressure p′∞ and viscosity μg [Lt ] μd The gas–liquid interface is surfactant free with interfacial tension σ. The Weber number based on the drop density, Wed ≡ ρdU′2ca/σ [Lt ] 1 and the capillary number based on the gas viscosity, Cagg ≡ μgU′cσ [Lt ] 1. The Reynolds number characterizing flow inside the drops satisfies Red ≡ aU′cρd/μd [Gt ] We1/2d and the Stokes number characterizing the drop inertia, St ≡ 2Wed(9Cag)−1 ≡ 2(ρdU′caμ−1g)/9 is O(1) or larger.

We first analyse a simple model for the rebound process which is valid when St [Gt ] 1 and viscous dissipation in both the gas and in the drop can be neglected. We assume that the film separating the drops only serves to keep the interfaces from touching by supplying a constant excess pressure 2σ/a. A singular perturbation analysis reveals that when ln(We−1/4d) [Gt ] 1, rebound occurs on a time scale t′b = &23frac;1/2πaWe1/2dln1/2 (We−1/4d)U′−1c. Numerical results for Weber numbers in the range O(10−6) − O(10−1) compare very well to existing experimental and simulation results, indicating that the approximate treatment of the bounce process is applicable for Wed < 0:3.

In the second part of the paper we formulate a general theory that not only models the flow inside the drop but also takes into account the evolution of the gap width separating the drops. The drop deformation in the near-contact inner region is determined by solving the lubrication equations and matching to an outer solution. The resulting equations are solved numerically using a direct, semi-implicit, matrix inversion technique for capillary numbers in the range 10−8 to 10−4 and Stokes numbers from 2 to 200. Trajectories are mapped out in terms of Cag and the parameter χ = (Wed/Cag)1/2 so that St ≡ 2/9χ2. For small Stokes numbers, the drops behave as nearly rigid spheres and come to rest without any significant rebound. For O(1) Stokes numbers, the surfaces deform noticeably and a dimple forms when the gap thickness is approximately O(aCa1/2). The dimple extent increases, reaches a maximum and then decreases to zero. Meanwhile, the centroids of the two drops come to rest momentarily and then the drops rebound, executing oscillatory motions before finally coming to rest. As the Stokes number increases with Cag held fixed, more energy is stored as deformation energy and the maximum radial extent of the dimple increases accordingly. For St [Gt ] 1, no oscillations in the centroid positions are observed, but the temporal evolution of the minimum gap thickness exhibits two minima. One minimum occurs during the dimple evolution process and corresponds to the minimum attained by the dimple rim. The second minimum occurs along the axis of symmetry when the dimple relaxes, a tail forms and then retracts. A detailed analysis of the interface shapes, pressure profiles and the force acting on the drops allows us to obtain a complete picture of the collision and rebound process.

Digital particle image velocimetry (DPIV) measurements of the velocity field under breaking waves in the laboratory are presented. The region of turbulent fluid directly generated by breaking is too large to be imaged in one video frame and so an ensemble-averaged representation of the flow is built up from a mosaic of image frames. It is found that breaking generates at least one coherent vortex that slowly propagates downstream at a speed consistent with the velocity induced by its image in the free surface. Both the kinetic energy of the flow and the vorticity decay approximately as t−1. The Reynolds stress of the turbulence also decays as t−1 and is, within the accuracy of the measurements, everywhere negative, consistent with downward transport of streamwise momentum. Estimates of the mometum flux from waves to currents based on the measurements of the Reynolds stress are consistent with earlier estimates. The implications of the measurements for breaking in the field are discussed. Based on geometrical optics and wave action conservation, we suggest that the presence of the breaking-induced vortex provides an explanation for the suppression of short waves by breaking. Finally, in Appendices, estimates of the majority of the terms in the turbulent kinetic energy budget are presented at an early stage in the evolution of the turbulence, and comparisons with independent acoustical measurements of breaking are presented.

Consider the three-dimensional flow of a viscous Newtonian fluid upon an arbitrarily curved substrate when the fluid film is thin as occurs in many draining, coating and biological flows. We drive the lubrication model of the dynamics of the film expressed in terms of the film thickness. The comprehensive model accurately includes the effects of the curvature of the substrate, via a physical multiple-scale approach, and gravity and inertia, via more rigorous centre manifold techniques. This new approach theoretically supports the use of the model over a wide range of parameters and provides a sound basis for further development of lubrication models. Numerical simulations exhibit some generic features of the dynamics of such thin fluid films on substrates with complex curvature: we here simulate a film thinning at a corner, the flow around a torus, and draining of a film down a cylinder. The last is more accurate than other lubrication models. The model derived here describes well thin-film dynamics over a wide range of parameter regimes.

The lift force on a circular particle in plane Poiseuille flow perpendicular to gravity is studied by direct numerical simulation. The angular slip velocity Ωs=Ωp+½γ˙, where −½γ˙ is the angular velocity of the fluid at a point where the shear rate is γ˙ and Ωp is the angular velocity of the particle, is always positive at an equilibrium position at which the hydrodynamic lift balances the buoyant weight. The particle migrates to its equilibrium position and adjusts Ωp so that Ωs > 0 is nearly zero because Ωp ≈ −1/2γ˙ No matter where the particle is placed, it drifts to an equilibrium position with a unique, slightly positive equilibrium angular slip velocity. The angular slip velocity discrepancy defined as the difference between the angular slip velocity of a migrating particle and the angular slip velocity at its equilibrium position is positive below the position of equilibrium and negative above it. This discrepancy is the quantity that changes sign above and below the equilibrium position for neutrally buoyant particles, and also above and below the lower equilibrium position for heavy particles. The existence and properties of unstable positions of equilibrium due to newly identified turning-point transitions and those near the centreline are discussed.

The long particle model of Choi & Joseph (2001) that gives rise to an explicit formula for the particle velocity and the velocity profile across the channel through the centreline of the particle is modified to include the effect of the rotation of the particle. In view of the simplicity of the model, the explicit formula for Up and the velocity profile are in surprisingly good agreement with simulation values. The value of the Poiseuille flow velocity at the point at the particle's centre when the particle is absent is always larger than the particle velocity; the slip velocity is positive at steady flow.

The derivation of effective equations for pressure wave propagation in a bubbly fluid at very low void fractions is examined. A Vlasov-type equation is derived for the probability distribution of the bubbles in phase space instead of computing effective equations in terms of averaged quantities. This provides a more general description of the bubble mixture and contains previously derived effective equations as a special case. This Vlasov equation allows for the possibility that locally bubbles may oscillate with different phases or amplitudes or may have different sizes. The linearization of this equation recovers the dispersion relation derived by Carstensen & Foldy. The initial value problem is examined for both ideal bubbly flows and situations where the bubble dynamics have damping mechanisms. In the ideal case, it is found that the pressure waves will damp to zero whereas the bubbles continue to oscillate but with the oscillations becoming incoherent. This damping mechanism is similar to Landau damping in plasmas. Nonlinear effects are considered by using the Hamiltonian structure. It is proven that there is a damping mechanism due to the nonlinearity of single-bubble motion. The Vlasov equation is modified to include effects of liquid viscosity and heat transfer. It is shown that the pressure waves have two damping mechanisms, one from the effects of size distribution and the other from single-bubble damping effects. Consequently, the pressure waves can damp faster than bubble oscillations.

The effect of spatial variability of the hydraulic conductivity upon free-surface flow is investigated in a stochastic framework. We examine the three-dimensional free-surface gravitational flow problem for a sloped mean uniform flow in a randomly heterogeneous porous medium. The model also describes the interface between two fluids of differing densities, e.g. freshwater/saltwater and water/oil with the denser fluid at rest. We develop analytic solutions for the variance and integral scale of free-surface fluctuations and of specific discharge on the free surface. Additionally, we obtain semi-analytic solutions for the statistical moments of the head and the specific discharge beneath the free surface. Statistical moments are derived using a first-order approximation and then compared with their counterpart in an unbounded medium. The effect of anisotropy and angle of mean uniform flow on the statistical moments is analysed. The solutions can be used for solving more complex flows, slowly varying in the mean.

Numerical finite-difference results from the full axisymmetric incompressible Navier–Stokes equations are presented for the problem of the slow axial motion of a disk particle in an incompressible, rotating fluid in a long cylindrical container. The governing parameters are the Ekman number, E = ν*/(Ω*a*2), Rossby number, Ro = W*/(Ω*a*), and the dimensionless height of the container, 2H (the scaling length is the radius of the particle, a*; Ω* is the container angular velocity, W* is the particle axial velocity and ν* the kinematic viscosity). The study concerns the flow field for small values of E and Ro while HE is of order unity, and hence the appearance of a free Taylor column (slug) of fluid ‘trapped’ at the particle is expected. The numerical results are compared with predictions of previous analytical approximate studies. First, developed (quasi-steady-state) cases are considered. Excellent agreement with the exact linear (Ro = 0) solution of Ungarish & Vedensky (1995) is obtained when the computational Ro = 10−4. Next, the time-development for both an impulsive start and a start under a constant axial force is considered. A novel unexpected behaviour has been detected: the flow field first attains and maintains for a while the steady-state values of the unbounded configuration, and only afterwards adjusts to the bounded container steady state. Finally, the effects of the nonlinear momentum advection terms are investigated. It is shown that when Ro increases then the dimensionless drag (scaled by μ*a*W*) decreases, and the Taylor column becomes shorter, this effect being more pronounced in the rear region (μ* is the dynamic viscosity). The present results strengthen and extend the validity of the classical drag force predictions and therefore the issue of the large discrepancy between theory and experiments (Maxworthy 1970) concerning this force becomes more acute.

The behaviour of converging spherical shock waves is considered using Whitham's theory of geometrical shock dynamics. An analysis of converging shocks whose initial shape takes the form of regular polyhedra is presented. The analysis of this problem is motivated by the earlier work on converging cylindrical shocks discussed in Schwendeman & Whitham (1987). In that paper, exact solutions were reported for converging polygonal shocks in which the initial shape re-forms repeatedly as the shock contracts. For the polyhedral case, the analysis is performed both analytically and numerically for an equivalent problem involving shock propagation in a converging channel with triangular cross-section. It is found that a repeating sequence of shock surfaces composed of nearly planar pieces develops, although the initial planar surface does not re-form, and that the increase in strength of the shock at each iterate in the sequence follows the same behaviour as for a converging spherical shock independent of the convergence angle of the channel. In this sense, the shocks are stable and the result is analogous to that found in the two-dimensional case. A numerical study of converging spherical shocks subject to smooth initial perturbations in strength shows a strong tendency to form surfaces composed of nearly planar pieces suggesting that the stability result is fairly general.

The spectral theory of a thin plate floating on shallow water is derived and used to solve the time-dependent motion. This theory is based on an energy inner product in which the evolution operator becomes unitary. Two solution methods are presented. In the first, the solution is expanded in the eigenfunctions of a self-adjoint operator, which are the incoming wave solutions for a single frequency. In the second, the scattering theory of Lax–Phillips is used. The Lax–Phillips scattering solution is suitable for calculating only the free motion of the plate. However, it determines the modes of vibration of the plate–water system. These modes, which both oscillate and decay, are found by a complex search algorithm based contour integration. As well as an application to modelling floating runways, the spectral theory for a floating thin plate on shallow water is a solvable model for more complicated hydroelastic systems.

This paper extends the numerical results of Hunter & Vanden-Broeck (1983) and Vanden-Broeck (1991) which were concerned with studies of solitary waves on the surface of fluids of finite depth under the action of gravity and surface tension. The aim of this paper is to answer the question of whether small-amplitude elevation solitary waves exist. Several analytical results have proved that bifurcating from Froude number F = 1, for Bond number τ between 0 and 1/3, there are families of ‘generalized’ solitary waves with periodic tails whose minimum amplitude is an exponentially small function of F−1. An open problem (which, for τ sufficiently close to 1/3, was recently proved by Sun 1999 to be false) is whether this amplitude can ever be zero, which would give a truly localized solitary wave.

The problem is first addressed in terms of model equations taking the form of generalized fifth-order KdV equations, where it is demonstrated that if such a zero-tail-amplitude solution occurs, it does so along codimension-one lines in the parameter plane. Moreover, along solution paths of generalized solitary waves a topological distinction is found between cases where the tail does vanish and those where it does not. This motivates a new set of numerical results for the full problem, formulated using a boundary integral method, namely to probe the size of the tail amplitude as τ varies for fixed F > 1. The strong conclusion from the numerical results is that true solitary waves of elevation do not exist for the steady gravity–capillary water wave problem, at least for 9/50 < τ < 1=3. This finding confirms and explains previous asymptotic results by Yang & Akylas.

The effect of stretching on the three-dimensional stability of a viscous unsteady vortex is addressed. The basic flow, which satisfies the Navier–Stokes equations, is a vortex with axial flow subjected to a time-dependent strain field oriented along its axis. The linear equations for the three-dimensional perturbations of the stretched vortex are first reduced by using successive changes of variables to equations which are almost identical to those of the unstretched vortex but with time-dependent parameters. These equations are then numerically solved in the particular case of the Batchelor vortex with a strain field which first compresses then stretches the vortex. Through this simulation, it is qualitatively demonstrated how the simultaneous action of stretching and azimuthal vorticity may destabilize a vortex. It is also argued that it provides a possible mechanism for the vortex bursts observed in turbulence experiments.