Unstable modes of a linear shear flow in shallow water on an equatorial $\beta$-plane are obtained over a wide range of values of a non-dimensional parameter and are interpreted in terms of resonance between neutral waves. The non-dimensional parameter in the system is $E \,{\equiv}\, \gamma^{4} / (gH\beta^{2})$, where $\gamma$, $g$, $H$ and $\beta$ are the meridional shear of basic zonal flow, gravitational constant, equivalent depth and the north–south gradient of the Coriolis parameter, respectively. The value of $E$ is varied within the range $-2.50 \,{\le}\,\log E \,{\le}\,7.50$.

The problem is solved numerically in a channel of width $5\gamma/\beta$. The structures of the most unstable modes, and the combinations of resonating neutral waves that cause the instability, change according to the value of $E$ as follows. For $\log E \,{<}\, 2.00$, the most unstable modes have zonally non-symmetric structures; the most unstable modes for $\log E \,{<}\, 1.00$ are caused by resonance between equatorial Kelvin modes and continuous modes, and those for $1.00 \,{\le}\,\log E \,{<}\, 2.00$ are caused by resonance between equatorial Kelvin modes and westward mixed Rossby–gravity modes. The most unstable modes for $\log E \,{\ge}\, 2.00$ have symmetric structures and are identical with inertially unstable modes. Examinations of dispersion curves suggest that non-symmetric unstable modes for $1.00 \,{\le}\,\log E \,{<}\, 2.00$ and inertially unstable modes for $\log E \,{\ge}\, 2.00$ are the same kind of instability.

The initial multi-mode interfacial velocity and density perturbations present at the onset of a small-Atwood-number, incompressible, miscible Rayleigh–Taylor instability- driven mixing layer have been quantified using a combination of experimental techniques. The streamwise interfacial and spanwise interfacial perturbations were measured using high-resolution thermocouples and planar laser-induced fluorescence (PLIF), respectively. The initial multi-mode streamwise velocity perturbations at the two-fluid density interface were measured using particle-image velocimetry (PIV). It was found that the measured initial conditions describe an initially anisotropic state, in which the perturbations in the streamwise and spanwise directions are independent of one another. The evolution of various fluctuating velocity and density statistics, together with velocity and density variance spectra, were measured using PIV and high-resolution thermocouple data. The evolution of the velocity and density statistics is used to investigate the early-time evolution and the onset of strongly nonlinear, transitional dynamics within the mixing layer. The early-time evolution of the density and vertical velocity variance spectra indicate that velocity fluctuations are the dominant mechanism driving the instability development. The implications of the present experimental measurements on the initialization of Reynolds-averaged turbulent transport and mixing models and of direct and large-eddy simulations of Rayleigh–Taylor instability-induced turbulence are discussed.

Film drainage between two drops with viscosity equal to that of the matrix fluid is studied using a numerical method that can capture both the external problem of two touching drops and the inner problem of pressure-driven local film drainage, without assumptions about the dimensions of the film or the use of lubrication approximations. We use a non-singular boundary integral method that has sufficient stability and accuracy to simulate film thicknesses down to and smaller than $10^{-4}$ times the undeformed drop radius. After validation of the method we investigate the validity of various results obtained from simple film-drainage models and asymptotic theories. Our results for buoyancy-driven collisions are in agreement with a recently developed asymptotic theory. External-flow-driven collisions are different from buoyancy-driven collisions, which means that the internal circulation inside the drop plays a significant role in film drainage, even for small capillary numbers, as has been recently shown (Nemer et al., Phys. Rev. Lett., vol. 92, 2004, 114501). Despite that, we find excellent correspondence with simple drainage models when considering the drainage time only.

The three-dimensional stability of the family of two-dimensional inviscid vortex patches discovered by Abrashkin & Yakubovich (Sov. Phys. Dokl. vol. 29, 1984, p. 370) is explored. Generally unsteady and non-uniform, these bounded regions of vorticity evolve freely in a surrounding irrotational flow. This family of solutions includes the Rankine circular vortex, Kirchhoff's ellipse, and freely rotating polygonal vortices as special cases. Taking advantage of their Lagrangian description, the stability analysis is carried out with the theory of local instabilities. It is shown that, apart from the Rankine vortex, these flows are three-dimensionally unstable. Background rotation or density stratification may however be stabilizing.

Adjacent oceanic water masses with the same density but different concentrations of heat and salt generate interleaving accompanied by double-diffusive processes. Laboratory experiments with salt and sugar concentrations are used to study the interleaving process. Most double-diffusive studies have treated vertical configurations in which one of the two components contains a destabilizing feature, salt above fresh water for salt fingers or warm underlying cold for the diffusive case. However, when the fluid lacks any gravitationally unstable feature, i.e. no gravitational potential energy is available in either component, the question arises as to what the source of energy is to drive the system. Such a case is discussed here and it is shown that the ultimate source of the energy is the chemical potential associated with the different property distributions. Diffusion creates a destabilizing property distribution and then enables the resulting potential energy to be released.

We analyse numerical solutions in the annulus model of rotating convection outside the tangent cylinder in a spherical shell. This model is capable of producing zonal flows with multiple jets. We investigate the conditions under which multi-jet solutions can be found. Although boundary friction reduces the strength of the zonal flow, it enhances the formation of multi-jets. More general models have a well-defined Ekman-layer term. In the annulus model, the Ekman-layer term has a similar form, but with variable strength. We have explored how the strength of the Ekman-layer term affects the form and strength of the zonal flows. We find that strong multi-jet zonal flows can be found for realistic values of the boundary friction, and hence have implications for convection in experiments and enclosed planetary cores. In addition, at higher Rayleigh numbers the importance of boundary friction is enhanced relative to bulk viscosity. Convection in the annulus model often occurs in the form of short-lived bursts as opposed to quasi-steady equilibriums. We have investigated when these events occur and their characteristics. In particular, we find precursors and afterglows of the convective bursts. We have obtained the $\beta$-scaling for a range of quantities when the thermal forcing is moderate. An examination of the components of the energy rate of change shows that the total Ekman-layer dissipation is of second order in the large $\beta$ limit. However, the $\beta$-scaling of the forces driving the zonal flow seems to suggest that the zonal Ekman-layer dissipation remains important. We have introduced the concept of flow Taylorization, an analogue to the Taylorization used in magnetohydrodynamics studies and find a $\beta$-scaling of this quantity compatible with the moderate strength of the zonal flow. We also determine the typical length scale on which convection operates and compare this to the numerically determined length scale.

Two rakes of cross-wire probes were used to capture the two-point velocity statistics in a flow through an axisymmetric sudden expansion. The expansion ratio of the facility is 3, and has a constant geometry. Measurements were acquired at a Reynolds number equal to 54 000, based on centreline velocity and inlet pipe diameter. The two-point velocity correlations were obtained along a plane normal to the flow ($r,\theta$), at eleven downstream step-height positions spanning from the recirculating region, through reattachment, and into the redeveloping region of the flow. Measurements were acquired by means of a flying-hot-wire technique to overcome rectification errors near the outer wall of the pipe where flow recirculations were greatest. A mixed application of proper orthogonal (in radius) and Fourier decomposition (in azimuth) was performed at each streamwise location to provide insight into the dynamics of the most energetic modes in all regions of the flow. This multi-point analysis reveals that the flow evolves from the Fourier-azimuthal mode $m\,{=}\,2$ (containing the largest amount of turbulent kinetic energy) in the recirculating region, to $m\,{=}\,1$ in the reattachment and redeveloping regions of the flow. An eigenvector reconstruction of the kernel, using the most energetic modes from the decomposition, displays the spatial dependence of the Fourier-azimuthal modes and the characteristics that govern the turbulent shear layer and recirculating regions of the flow.

A simplified model for the pulmonary alveolus that imitates the rhythmical expansion of the alveolus and the periodic shear flow in the adjacent airway is explored. The model consists of two eccentric cylinders and incompressible fluid that occupies the gap between them. The two cylinders undergo a simultaneous rhythmical expansion and contraction (mimicking the alveolus expansion) while the inner cylinder performs a periodic rotation about its axis (inducing shear flow mimicking airway ductal flow). An analytical solution is obtained for the creeping flow induced by the simultaneously expanding cylinders. It is shown that above a certain critical value of rotation to expansion velocity ratio, the flow exhibits characteristic features such as a saddle point and closed streamlines about a centre, similar to those existing inside a single alveolus during inhalation and exhalation. Poincaré maps of the trajectories of fluid particles demonstrate that, under various flow conditions, chaotic trajectories may exist, provided that expansion and rotation are slightly out of phase. This is similar to normal breathing conditions where the periodic expansion of the alveolus and the tidal flow (i.e. shear flow above the mouth of the alveolus) may be slightly out of phase. A novel definition of overall convective mixing efficiency is also suggested that inherently discounts reversible processes that do not contribute to mixing. It is demonstrated that two different convective mechanisms, related to the irreversibility of exhalation and inhalation and the onset of chaos, govern mixing efficiency in lung alveoli.

The numerical results obtained by Mougin & Magnaudet (Phys. Rev. Lett. vol. 88, 2002a, 14502) for the flow past a freely moving spheroidal bubble with a prescribed spheroidal shape are processed to analyse the evolution of the forces and torques experienced by the bubble when it rises along a planar zigzag and a circular helix. It is found that, as soon as the wake becomes three-dimensional, a lateral force with a strength comparable with that of the buoyancy force occurs. This force, together with the corresponding torque, drives the horizontal movements of the bubble. The force and torque balances reveal how these wake-induced effects are balanced by added-mass effects to make possible the existence of zigzag and helical motions along which the angle between the velocity and the symmetry axis of the bubble remains small. The evolution of the wake during the zigzag indicates that the sign of the trailing vortices, and thus that of the wake-induced force and torque, is governed by the rotation of the bubble and reveals the sensitivity of the wake dynamics to the changes in the bubble velocity and rotation rate.

Based on a micropolar continuum of rough granular particles that takes into account the balance equations for the spin (/rotational) velocity and the spin granular temperature, the linear stability characteristics of an unbounded shear flow (${\bm u}\equiv (u_x, u_y, u_z) \,{=}\, (\dot\gamma y, 0, 0)$, where $x$, $y$ and $z$ are the streamwise, transverse and spanwise directions, respectively, and $\dot\gamma$ is the shear rate) are analysed. For pure spanwise perturbations ($k_z\neq 0$, with $k_x\,{=}\,0\,{=}\,k_y$, where $k_i$ is the wavenumber in the $i$th direction), we show that the streamwise translational velocity and the transverse spin velocity modes are subject to linear growths, owing to an inviscid ‘algebraic’ instability (that grows linearly with time). This algebraic instability is shown to be tied to a hidden mechanism of momentum transfer from the translational to the rotational modes, via pure spanwise perturbations to the transverse velocity – in short, we have uncovered an ‘instability-induced rotational-driving’ mechanism. Pure spanwise ($k_z\neq 0$, with $k_x\,{=}\,0\,{=}\,k_y$) and pure transverse ($k_y\neq 0$, with $k_x\,{=}\,0\,{=}\,k_z$) perturbations give rise to ‘exponential’ instabilities (that grow exponentially with time) which are related to similar stationary instabilities in the shear flow of smooth, inelastic particles. Both these instabilities also survive in the limiting case of perfectly elastic but rough particles. The scalings of hydrodynamic modes with wavenumbers have been obtained via the respective long-wave expansion. Perturbations with modulations in all three directions are shown to be stable in the asymptotic time limit, but there could be short-time ‘exponential’ growth of these general perturbations in the long-wave limit for both travelling and stationary waves. The growth rate of all instabilities is maximum at intermediate values of the tangential restitution coefficient ($\beta$), and decreases in both the perfectly smooth ($\beta\to -1$) and rough ($\beta\to 1$) limits; the associated instability length scale is minimum at intermediate $\beta$, and increases in both the perfectly smooth and rough limits. In the perfectly smooth limit, there is a window of particle volume fraction ($\phi$), $\phi_c^s <\phi < \phi_c^t$, over which the flow remains stable to all perturbations. With the inclusion of spin fields, the size of this window decreases and at moderate dissipations with $\beta\,{>}\,0.5$ the flow becomes unstable at all $\phi$.

A numerical treatment of the natural convection and passive dispersion in symmetrically interconnected tilted layers embedded in a rock which is subject to a constant vertical temperature gradient is presented. Such a system is a faithful model of configurations commonly found in the geophysical context. There, flow movements and temperature distributions are closely connected to phenomena of interest such as transport of contaminants and diagenesis. The important case of large thermal conductivity of the rock compared with that of the material filling the layer is discussed in order to show the decisive role of the temperature distribution and the geometrical parameters on the convective flow. The present analysis treats two cases, the fluid-filled layer and the saturated porous layer. Convective flows were calculated for small Rayleigh numbers and the resulting velocity fields were included in the analysis of the transport of a passive contaminant that was initially located where layers connect with each other. Transport of contaminants in the isotropic porous layer was studied by using a model which includes hydrodynamic dispersion terms. How far the tracer transports through the layers and the rate the tracer enters into the system were analysed. The influence of the angle of tilt has also been included. The molecular diffusive Péclet number which relates convective to diffusive species transport is closely associated to a considerable transporting rate, and for the porous layer the hydrodynamic dispersion appears to be an important effect to consider.

Pressure-driven flow through a tube with helical corrugations produced either by twisting a straight tube with arbitrary cross-section, or by embossing helical corrugations on a circular tube, or by inserting a helical fin inside a circular tube, is considered. The Stokes-flow problem is formulated in non-orthogonal curvilinear helical coordinates defined with respect to the helical pitch and azimuthal wavenumber, where the latter is determined by the tube cross-section rotational symmetry. In the first part of the paper, a perturbation analysis is carried out for a circular tube with small-amplitude sinusoidal corrugations, and the solutions of the first- and second-order perturbation problems are found by analytical methods. In the second part, an asymptotic analysis is performed for large-pitched helical corrugations and tubes with arbitrary cross-section, and the solutions of the zeroth-, first- and second-order problems are computed by finite-element methods for unidirectional and two-dimensional Stokes flow over the cross-sectional plane normal to the tube axis. The results illustrate the kinematic structure of the flow and demonstrate the dependence of the flow rate on the tube geometry.

The structures of separation on a smooth circular cylinder immersed in a sinusoidally oscillating flow are examined in detail for a constant value of the frequency parameter $\beta $ as the Carpenter–Keulegan number $K$ is systematically increased from the marginally stable to fully separated region in the ($K$, $\beta $)-plane. The positions of the separation points are measured using sublayer fences, flush-mounted hot-film sensors, and extensive high-speed video recordings. The variations of the length scales of the resulting structures, the irregularity of the dye concentration fields, and measurements with two sensors have shown that the separation is three-dimensional, time-dependent, often turbulent, and far from being an eruption of a double-sided single shear layer, or a self-contained bubble. The increase of the three-dimensionality of the flow, evolution of various sizes of structures, secondary separations within the primary separation zone, and the occasional eruption of multiple shear layers are quite similar to the first direct numerical simulation of a laminar separation bubble in the presence of an oscillating inlet flow.

The problem of irrotational inviscid incompressible free-surface flow is examined in the limit of small Froude number. Since this is a singular perturbation, singularities in the flow field (or its analytic continuation) such as stagnation points, or corners in submerged objects or on rough beds, lead to a divergent asymptotic expansion, with associated Stokes lines. Recent techniques in exponential asymptotics are employed to observe the switching on of exponentially small gravity waves across these Stokes lines.

As a concrete example, the flow over a step is considered. It is found that there are three possible parameter regimes, depending on whether the dimensionless step height is small, of the same order, or large compared to the square of the Froude number. Asymptotic results are derived in each case, and compared with numerical simulations of the full nonlinear problem. The agreement is remarkably good, even at relatively large Froude number. This is in contrast to the alternative analytical theory of small step height, which is accurate only for very small steps.

The coupling between topography, waves and currents in the surf zone may self-organize to produce the formation of shore-transverse or shore-oblique sand bars on an otherwise alongshore uniform beach. In the absence of shore-parallel bars, this has been shown by previous studies of linear stability analysis, but is now extended to the finite-amplitude regime. To this end, a nonlinear model coupling wave transformation and breaking, a shallow-water equations solver, sediment transport and bed updating is developed. The sediment flux consists of a stirring factor multiplied by the depth-averaged current plus a downslope correction. It is found that the cross-shore profile of the ratio of stirring factor to water depth together with the wave incidence angle primarily determine the shape and the type of bars, either transverse or oblique to the shore. In the latter case, they can open an acute angle against the current (up-current oriented) or with the current (down-current oriented). At the initial stages of development, both the intensity of the instability which is responsible for the formation of the bars and the damping due to downslope transport grow at a similar rate with bar amplitude, the former being somewhat stronger. As bars keep on growing, their finite-amplitude shape either enhances downslope transport or weakens the instability mechanism so that an equilibrium between both opposing tendencies occurs, leading to a final saturated amplitude. The overall shape of the saturated bars in plan view is similar to that of the small-amplitude ones. However, the final spacings may be up to a factor of 2 larger and final celerities can also be about a factor of 2 smaller or larger. In the case of alongshore migrating bars, the asymmetry of the longshore sections, the lee being steeper than the stoss, is well reproduced. Complex dynamics with merging and splitting of individual bars sometimes occur. Finally, in the case of shore-normal incidence the rip currents in the troughs between the bars are jet-like while the onshore return flow is wider and weaker as is observed in nature.

Asymptotic solutions are obtained for the rise of an axisymmetric hot plume from a localized source at the base of a half-space filled with very viscous fluid. We consider an effectively point source, generating a prescribed buoyancy flux $B$, and show that the length scale of the plume base is $z_0 \,{=}\, (32\upi \kappa^2\nu/B)$, where $\nu$ and $\kappa$ are the kinematic viscosity and thermal diffusivity. The internal structure of the plume for $z \,{\gg}\, z_0$ is found using stretched coordinates, and this is matched to a slender-body expansion for the external Stokes flow. Solutions are presented for both rigid (no-slip) and free-slip (no tangential stress) conditions on the lower boundary. In both cases we find that the typical vertical velocity in the plume increases slowly with height as $(B/\nu)^{1/2} [\ln (z/z_0)]^{1/2}$, and the plume radius increases as $(zz_0)^{1/2} [\ln (z/z_0)]^{-1/4}$.

Asymptotic solutions are obtained for an axisymmetric plume rising from a localized heat source at the base of a half-space filled with very viscous fluid. Specifically, we consider sources comprising a heated disk with either rigid (no-slip) or free-slip (no tangential stress) conditions on the lower boundary. The boundary layer which forms above the source is solved using stretched coordinates, and then matched to a slender plume which rises above it. At large Rayleigh numbers, the Nusselt number is given by $Nu \sim 4.06 Ra^{1/3}(\ln Ra)^{-1/3}$ (free-slip boundary) and $Nu \sim 2.90 Ra^{1/5}$ (rigid boundary), where the Rayleigh number is based on the radius of the source. Both these expressions have corrections arising from a slender-body expansion in powers of $(\ln Ra)^{-1}$.

The motion of an incompressible viscous rotating fluid in an oblate spheroidal container is studied by direct numerical simulation in an appropriate spheroidal coordinate system and in the linear approximation. The behaviour of a few eigenmodes is investigated as a function of the eccentricity $e$ of the container, for Ekman number $E \,{=}\, 10^{- 5}$. Viscous effects are shown through internal shear layers, the spatial structure of which strongly depends on the eccentricity. In particular, for the spin-over mode, a resonance occurs around a critical value $e_c\,{\approx}\,0.50$, where the decay rate deviates strongly from the predicted theoretical variation. This resonance is discussed in relation to the accidental coincidence between the spin-over frequency and two other frequencies corresponding to the (8,1,5) and (14,1,9) inertial eigenmodes.

The generation of internal gravity waves by an oscillatory tidal flow over a periodic array of thin vertical walls is calculated analytically. For small values of the non-dimensional height $B=2\pi H\!N/L\omega$, the radiated power per wall is the same as for a single thin wall, and proportional to $B^2$, in agreement with the linear scaling. (Here $H$ is the wall height, $N$ the buoyancy frequency, $L$ the wall spacing, and $\omega$ the tidal frequency.) The radiated power is periodic in $B$ with period $2\pi$. It diverges logarithmically for $B=(1+2n)\pi$, and vanishes for $B=2n\pi$.

The dynamics of spilling breakers in the presence of surfactants were studied experimentally. The spilling breakers were produced from Froude-scaled mechanically generated dispersively focused wave packets with average frequencies of 1.15, 1.26 and 1.42 Hz. Separate experiments were performed with the same wave-maker motions in clean water and in water with various bulk concentrations of the soluble surfactants sodium dodecyl sulfate (SDS) and Triton X-100 (TX). For nearly all surfactant conditions, the surface-pressure isotherm, equilibrium surface elasticity and surface viscosity were measured in situ in order to characterize the dynamic properties of the free surface. In clean water, all the waves considered herein break without overturning of the free surface. This breaking process begins with the formation of a bulge on the forward face of the wave crest and capillary waves upstream of the leading edge of the bulge (called the toe). After a short time, the flow separates under the toe and a turbulent flow is developed while the toe moves rapidly down the wave face. During the toe motion, a train of ripples appears between the toe and the crest and this train of ripples is swept downstream. In the presence of surfactants, the bulge shape is modified and its size generally decreases with increasing surfactant concentration. The capillary waves found upstream of the toe in the clean-water case are dramatically reduced at even the lowest concentrations of surfactants. With surfactants, the start of the breaking process is still initiated when the toe begins to move down the forward face of the wave. The pattern of ripples generated between the toe and the crest of the wave during this phase of the breaking process varies with the concentration of surfactant. It was found that the temporal history of the vertical distance between the toe and the wave crest scales with the nominal length $(\sigma_0/\rho g)^{1/2}$ while the bulge length from toe to crest scales with the nominal length $(\mu_s/\rho \sqrt{g})^{2/5}$, where $\sigma_0$ and $\mu_s$ are the ambient surface tension and the surface viscosity, respectively.