Explicit stationary solutions to the equations of vorticity conservation on the $f$- and $\beta $-planes are considered, describing barotropic dipoles (modons) with elliptical frontiers. The far field – outside the elliptical separatrix demarcating the regions of closed and open streamlines – is given analytically, whereas the interior is solved numerically using a successive linearization algorithm. Both ellipses extended along the translation axis and those extended in the transverse direction are considered. In the latter case, it is shown that, among the possible solutions, there exist the so-called supersmooth modons marked by continuity of the vorticity derivatives at the separatrix. On the $\beta$-plane, the separatrix aspect ratio that allows a supersmooth solution varies depending on the modon translation speed and size, while on the $ f$-plane, there is only one such separatrix aspect ratio. In this context, the limiting transition from the $\beta $-plane to the $f$-plane is discussed. The dependence between the absolute (or relative) vorticity $q$ and the ‘co-moving’ streamfunction $\Psi $, which is nonlinear in the interior of non-circular modons, is analysed in detail for both $\beta $- and $f$-planes, the main concern being the relation between the separatrix form, on the one hand, and the shape of the $q$ against $\Psi $ scattergraph on the other. The stability of elliptical dipoles versus the separatrix aspect ratio is examined based on numerical simulations of the temporal evolution of the modons found. The supersmooth modons appear to be the most stable among all the elliptical dipoles.

We have developed a numerical package to simulate particle motions in fluid interfaces. The particles are moved in a direct simulation respecting the fundamental equations of motion of fluids and solid particles without the use of models. The fluid–particle motion is resolved by the method of distributed Lagrange multipliers and the interface is moved by the method of level sets. The present work fills a gap since there are no other theoretical methods available to describe the nonlinear fluid dynamics of capillary attraction.

Two different cases of constrained motions of floating particles are studied here. In the first case, we study motions of floating spheres under the constraint that the contact angle is fixed by the Young–Dupr´e law; the contact line must move when the contact angle is fixed. In the second case, we study motions of disks (short cylinders) with flat ends in which the contact line is pinned at the sharp edge of the disk; the contact angle must change when the disks move and this angle can change within the limits specified by the Gibbs extension to the Young–Dupré law. The fact that sharp edged particles cling to interfaces independent of particle wettability is not fully appreciated and needs study.

The numerical scheme presented here is at present the only one which can move floating particles in direct simulation. We simulate the evolution of single heavier-than-liquid spheres and disks to their equilibrium depth and the evolution to clusters of two and fours spheres and two disks under lateral forces, collectively called capillary attraction. New experiments by Wang, Bai & Joseph on the equilibrium depth of floating disks pinned at the edge are presented and compared with analysis and simulations.

This paper studies the trajectories and near field of round jets in crossflow. Incompressible direct numerical simulations are performed at velocity ratios of 1.5 and 5.7 and the effects of jet velocity profile and boundary layer thickness on the jet trajectory are examined. The ‘rd’ scaling used at present (Margason 1993) does not contain any information on these parameters, and trajectories scaled by rd do not collapse. The trajectory is strongly influenced by the near field which depends on both the jet velocity profile and the crossflow boundary layer. A length scale is proposed to describe the near field of the jet. An analytical expression is proposed for this length scale which is a measure of the relative inertia of the jet and the crossflow. Incorporating this length scale significantly improves the scaling of the trajectories.

When seawater becomes supercooled, collections of small ice crystals, known as frazil ice, form and grow. A model of frazil ice dynamics is presented that deals explicitly with the buoyant settling of frazil crystals onto an overlying surface. This yields further insight into transport associated with the ice pump mechanism, whereby ice is melted at depth and transferred to a shallower location as a result of the pressure variation of seawater's freezing temperature. The model is applied to a vertical cross-section through an Ice Shelf Water plume beneath Filchner–Ronne Ice Shelf, Antarctica, and helps to elucidate the depth-variation in its properties for the first time, as well as predicting the precipitation rate of frazil crystals. The model predicts that frazil ice should be preferentially located in a narrow layer near the ice shelf base as a result of the maximum supercooling there and an influx of crystals rising under their own buoyancy. The deposition of these crystals onto the ice shelf is governed by the balance between crystal rising and turbulent transfer of frazil away from the shelf, which is investigated in some detail.

A model for particle propulsion by an instantaneous heat discharge is presented. The flow is driven by a gravity-independent transient fluid dilatation, engendered by an unsteady temperature field which corresponds to heat emission from a localized source located within the particle. We focus on the highly eccentric case, where the heat is released in proximity to the particle surface. Solution of the Stokes equations and subsequent evaluation of the resulting hydrodynamic thrust yields a nonlinear non-autonomous ordinary differential equation governing the evolution of particle position with time. This equation depends upon a single parameter which represents the relative effects of heating magnitude and initial geometry.

Two-dimensional simulations based on the isothermal lattice-Boltzmann method have been undertaken on microchannels with a sudden expansion or contraction. The study provides insight into the analysis of flows in complicated microdevices. The flow is pressure driven, and computations are performed for several Knudsen numbers, and area and pressure ratios, allowing the effects of compressibility and rarefaction to be assessed. The pressure drop for both the converging and diverging channels shows a discontinuity in slope at the junction, and is accompanied by a jump in velocity. The pressure drop in each section can be predicted well by the theory for straight channels. The mass flow ratio between converging and diverging channels is close to unity, and the streamlines are attached in both cases. It is deduced that compressibility and rarefaction have opposite effects on the flow. These results suggest that complex channels of the type considered here can be understood in terms of their primary units, and they experience only small secondary losses.

We investigate the nonlinear dynamics of turbulent shear flows, with and without rotation, in the context of a simple but physically motivated closure of the equation governing the evolution of the Reynolds stress tensor. We show that the model naturally accounts for some familiar phenomena in parallel shear flows, such as the subcritical transition to turbulence at a finite Reynolds number and the occurrence of a universal velocity profile close to a wall at large Reynolds number. For rotating shear flows we find that, depending on the Rayleigh discriminant of the system, the model predicts either linear instability or nonlinear instability or complete stability as the Reynolds number is increased to large values. We investigate the properties of Couette–Taylor flows for varying inner and outer cylinder rotation rates and identify the region of linear instability (similar to Taylor's), as well as regions of finite-amplitude instability qualitatively compatible with recent experiments. We also discuss quantitative predictions of the model in comparison with a range of experimental torque measurements. Finally, we consider the relevance of this work to the question of the hydrodynamic stability of astrophysical accretion disks.

We consider an axisymmetric, freely suspended fluid drop with surface tension, whose viscosity is so large that both inertia and forcing by an external fluid can be ignored. We show that whatever the initial condition, pinchoff can never occur.

In this work, we present a detailed experimental study of the periodic formation of bubbles in an air–water coflowing stream, as well as a simple model to describe the process. The frequency of formation of bubbles was measured analysing a large number of images recorded with a high-speed camera for a wide range of experimental conditions and air-injection needle geometries. The analysis of the images indicated that the bubble-formation process consisted of two distinct stages, namely the ligament expansion stage, characterized by the radial growth of an air ligament left attached to the injection needle after the pinch-off of a bubble, and the ligament collapse stage, characterized by the formation of a neck at the tip of the injection needle which propagates downstream, at a velocity which is nearly the liquid velocity, until it collapses generating a new bubble. A simplified model, based on the Rayleigh–Plesset equation for a cylindrical geometry to determine the dynamics of the liquid stream and on Bernoulli's equation to determine the air pressure near the neck, has been proposed to estimate the duration of the ligament collapse stage, $t_{col}$. The experimental bubble-formation frequency, properly scaled with the breakup time given by the model, is shown to collapse onto the same curve for all the experimental conditions used here, indicating that our simple model seems to retain the main physical aspects of the process.

Thrust bearings have been the object of a considerable amount of research for many years. The attention that these bearings have received is primarily due to the important role they play in the design and operation of heavy equipment. The objectives of this work are to examine the effect of translation and bearing geometry, such as recess depth and width, on the performance of circular thrust bearings. A closed-form solution is obtained for laminar viscous flow that extends prior results which generally focus on stationary or hydrostatic bearings. Earlier studies have examined specific aspects of translation, such as recirculation in the recess, but the current study is a comprehensive analysis in the case of laminar flow. The analysis reveals a single dimensionless parameter that describes the influence of the bearing speed. Expressions that predict the load-carrying capacity of the bearing, the tilting moment exerted on the bearing, the volumetric leakage of the bearing, and power due to lubricant injection and translation are obtained. Streamline patterns under the bearing show conditions at higher speeds when the injected lubricant does not penetrate underneath the entire bearing surface.

Shear flow along a slotted plate can give rise to highly coherent, small-scale oscillations within each slot; the dimensionless frequency is an order of magnitude lower than the frequency of the classical Kelvin–Helmholtz instability. Within a given slot, the unsteady development of a swirl pattern, as well as the flow into and out of the slot, is compatible with the evolution of a large-scale structure, which shows increased agglomeration of vorticity along the surface of the plate. Moreover, the streamwise phase shift of the unsteady flow rates through the slots is in accord with the phase shift of the velocity fluctuation of the downstream propagating disturbance along either side of the plate. This phase shift is associated with development of the large-scale structure, and corresponds to a dimensionless frequency that scales on the plate length, as verified by experiments over ranges of length and inflow velocity. The value of this dimensionless frequency is close to that in the absence of the slotted plate i.e. the Kelvin–Helmholtz instability of the free shear layer.

The paper describes results of experiments of a converging spherical shock wave reflected from a spherical wall. In order to visualize the motion and the flow field behind the shock waves, an aspheric lens-shaped transparent test section made of acrylic PMMA (polymethyl methacrylate) with an inner spherical cavity was designed and constructed. This test section made optical flow visualization with collimated object beams possible. Spherical shock waves were produced at the centre of the spherical cavity by explosion of silver azide pellets ranging from 1.0 to 10.0 mg with corresponding energies of 1.9 to 19 J. The charges were ignited by irradiation of a pulsed Nd:YAG laser beam. Pressures were also measured at two points with pressure transducers mounted flush at the inner wall of the test section. The pellet was simultaneously ignited on two sides or was shaped to produce a uniform diverging spherical shock wave. This spherically diverging shock wave was reflected from the spherical inner wall of the test section to form a converging spherical shock wave. We visualized the shock-wave motion by using double exposure holographic interferometry and time-resolved high-speed video recording. The sequence of diverging and converging spherical shock-wave propagations and their interaction with gaseous explosion products were observed. The convergence, acceleration and stability of the imploding shock wave in the test section were studied.

The spatial development of planar incompressible countercurrent shear layers was investigated experimentally. A facility was constructed to establish countercurrent shear layers without the formation of global stagnation in the flow. Particle image velocimetry was employed to obtain detailed measurements within the region of self-preservation for velocity ratios $U_{2}/U_{1}$ between 0 and −0.3. The spatial growth rate of countercurrent shear layers was found to agree generally with simple analytical theory. At 30% counterflow, the growth rate was approximately twice as large as the case with no counterflow. Peak turbulence quantities, when normalized by the applied shear magnitude, $\Delta U$, were found to be nominally constant for low levels of counterflow, but at counterflow velocities above 13% of the primary stream velocity, peak turbulence levels increased. The observed transition is accompanied by the development of mean flow three-dimensionality. The deviation occurs at a counterflow level that is in agree- ment with theoretical predictions for transition from convective to absolute instability.

The laminar breakdown of a three-dimensional flat-plate boundary-layer flow with favourable and ensuing adverse pressure gradient (APG), generic for an infinite swept wing, is investigated in detail by means of spatial direct numerical simulations. Emphasis is on transition mechanisms in the region of adverse pressure gradient where the undistorted laminar base flow also becomes unstable with respect to Tollmien–Schlichting (TS) waves that are most amplified if two-dimensional with respect to the local streamwise direction. The influence of finite-amplitude crossflow vortices, coming from the region of favourable pressure gradient, on the TS instability properties is investigated. It turns out that crossflow-vortex-induced secondary instabilities are the most amplified disturbances even for low-amplitude vortex modes. The TS waves act as generators of fundamental low-frequency secondary modes, but are neither important for their growth nor for breakdown. As for active disturbance control, any method aiming at attenuating two-dimensional TS waves must fail. On the other hand, the passive upstream-flow-deformation technique also delays transition in the APG region.

We present experiments, using an ultra-high-speed video camera, which indicate flow-induced buckling of a surfactant layer on the free surface of a deforming water drop. This results in the ‘puncturing’ of the liquid surface, sending out a narrow jet. The phenomenon is produced when a water drop falls onto and passes through a hemispherical soap film. The soap film wraps around the drop forming a closed pocket, with a thin layer of air separating the two. This occurs without initial contact between the soap film and the drop liquid, which becomes clear during the subsequent dynamics, as the original film unwraps fully from the drop surface. Under particular conditions the soap film makes contact with the top of the drop, producing a toroidal bubble, which is driven down the drop to form a hemisphere at its bottom. We propose that the resulting rapid reduction in surface area of the bubble leads to overpacking of the surfactant molecules and buckling of the surface, which allows the jet to emerge, from the bottom tip of the drop. Marangoni-driven boundary layers, converging at the bottom, may also play a role in forcing out the jet.

This paper considers the effect of an exponential variation in the background density field (as exists in compressible atmospheres) on the structure and dynamics of the quasi-geostrophic system, and compares the results with the corresponding Boussinesq limit in which background density variations are assumed small. The behaviour of the compressible system is understood via a closed-form analytic expression for the Green's function of the inversion operator relating potential vorticity and streamfunction. This expression makes explicit the anisotropy of the Green's function, inherited from the density profile, which has a slow, algebraic decay directly above the source and an exponential decay in all other directions. An immediate consequence for finite-volume vortices is a differential rotation of upper and lower levels that results in counterintuitive behaviour during the nonlinear evolution of ellipsoidal vortices, in which vortex destruction is confined to the lower vortex and wave activity is seen to propagate downwards. This is in contrast to the Boussinesq limit, which exhibits symmetric destruction of the upper and lower vortex, and in contrast to naive expectations based on a consideration of the mass distribution alone, which would lead to greater destruction of the upper vortex. Finally, the presence of a horizontal lower boundary introduces a strong barotropic component that is absent in the unbounded case (the presence of an upper boundary has almost no effect). The lower boundary also alters the differential rotation in the lower vortex with important consequences for the nonlinear evolution: for very small separation between the lower boundary and the vortex, the differential rotation is reversed leading to strong deformations of the middle vortex; for a critical separation, the vortex is stabilized by the reduction of the differential rotation, and remains coherent over remarkably long times.

A fully turbulent channel flow with smooth and rod-roughened walls has been investigated using hot-wire anemometry and direct numerical simulations (DNS). The mean flow follows the law of the wall for both surfaces and the velocity defect suggests that the outer layer is very little affected by the roughness. The Reynolds stresses appear to be very similar for the two surface geometries outside $y\,{\approx}\,5k$, where $k$ is the roughness height. A quadrant analysis shows that the structural differences close to the wall extend somewhat further out. The turbulence structure is further investigated using stress ratios and the anisotropy tensor, which corroborate the findings from the Reynolds stresses. Many of the recent investigations on boundary layers seem to find large differences between smooth and rough wall data in the outer layer also. A tentative explanation for the apparent dependence on flow type of the surface roughness effects is given.

In this paper experiments are reported on a condensing steam jet. Superheated steam is injected at the bottom centre of a cylindrical water vessel, resulting in a turbulent jet with Reynolds numbers varying between $7.9{\times}10^4$ and $18.1{\times}10.4$, depending on the bulk temperature of the water. Near the steam inlet, the flow is two-phase with rapidly condensing steam. Downstream of a development region the jet is essentially single-phase. Using particle image velocimetry in a vertical plane through the central axis, instantaneous velocity fields of the single-phase region have been measured at a rate of 15Hz. The velocity field in this region is found to be self-similar, i.e. the width of the jet, $r_{1/2}$, increases linearly with increasing distance to the virtual origin, and a Gaussian profile prevails if velocities and distances are properly scaled. The spreading rate is equal to the one usually found in single-phase jets, and temperature independent. The virtual origin of the jet is positioned at a temperature-dependent distance (3–7 nozzle diameters) upstream of the steam inlet, and this distance is shown to correlate with the length of the condensation region. The turbulence intensity is found to be similar to the intensities usually reported for single-phase jets, although full isotropy is only reached at a distance of 15 nozzle diameters from the nozzle. The jet exhibits a slow wobbling motion, which can be attributed to instability of the backflow resulting from the confinement. When the measurements are compensated for this wobble, a slightly smaller spreading rate is obtained, which indicates that unconditional averaging may conceal significant flow structuring.

The periodic generation of bubbles by injection of a gas at constant flow rate through an orifice at the bottom of a quiescent liquid of high viscosity is investigated numerically. The volume of the bubbles is determined as a function of a capillary number and a Bond number in the absence of inertial effects. Pairs of bubbles coalesce in the vicinity of the orifice when the capillary number is higher than a critical value that depends on the Bond number, but the final volume of the bubbles still follows a well-known scaling law at large capillary numbers. Qualitative experimental visualizations are presented that display the sequences of detachment and coalescence computed numerically.