Turbulence measurements were obtained in a three-dimensional supersonic turbulent boundary layer. A 20° curved fin was used to generate a three-dimensional compression of a boundary layer at Mach 3 in the absence of shock waves. Data include hot-wire measurements of five components of the Reynolds stress tensor. The results are interpreted in terms of the mean flow field history of the turbulence. It is demonstrated that in-plane curvature can have a strong stabilizing effect on the turbulence.

We examine the flow in a horizontal Hele-Shaw cell in which the undisturbed unidirectional flow at infinity is required to stream around a vertical cylinder spanning the gap between the two (horizontal) plates of the cell. A combination of matched asymptotic expansions and numerical methods is employed to elucidate the structure of the boundary layer near the surface of the cylinder. The two length scales of the problem are the gap, h, and the length of the body, l; it is assumed that h/l<<1. The characteristic Reynolds number based on l is O(1). The length scales associated with the boundary layer and the classical Hele-Shaw flow pattern are O(h) and O(l), respectively.

It is found that the boundary layer contains streamwise vorticity. This vorticity is generated at the three no-slip surfaces (the two plates and the cylinder wall) as a result of the cross-flow induced by the streamwise acceleration/deceleration of the flow around the curved cylinder. The strength of the secondary flow, hence the associated streamwise vorticity, is proportional to changes in body curvature. The validity of the classical Hele-Shaw flow is examined systematically, and higher-order corrections are worked out. This results in a displacement thickness that is roughly 30% of the gap. In other words, the lowest-order correction to the classical Hele-Shaw flow may be obtained by requiring the outer flow (on the scale O(l)) to satisfy the no-penetration boundary condition on a displaced cylinder surface. The boundary layer contains ‘corner’ vortices at the intersections of the horizontal plates and the vertical cylinder surface.

Planar steady gravity waves of finite amplitude at the surface of an ideal incompressible fluid above a flat bottom are studied theoretically. A new approach to the construction of some steady flows of heavy fluid with a partially free surface is proposed. The hypothesis is suggested and justified that these flows are close to gravity waves. For the case of the highest waves a one-parameter family of exact solutions describing free boundary flows above a flat bottom and under two uneven symmetrically located caps is derived. This family of solutions gives an approximation to the highest water waves in moderate to shallow water depths, enabling relatively simple calculation of their properties.

Alongshore propagating low-frequency O(0.01 Hz) waves related to the direction and intensity of the alongshore current were first observed in the surf zone by Oltman-Shay, Howd & Birkemeier (1989). Based on a linear stability analysis, Bowen & Holman (1989) demonstrated that a shear instability of the alongshore current gives rise to alongshore propagating shear (vorticity) waves. The fully nonlinear dynamics of finite-amplitude shear waves, investigated numerically by Allen, Newberger & Holman (1996), depend on α, the non-dimensional ratio of frictional to nonlinear terms, essentially an inverse Reynolds number. A wide range of shear wave environments are reported as a function of α, from equilibrated waves at larger α to fully turbulent flow at smaller α. When α is above the critical level αc, the system is stable. In this paper, a weakly nonlinear theory, applicable to α just below αc, is developed. The amplitude of the instability is governed by a complex Ginzburg–Landau equation. For the same beach slope and base-state alongshore current used in Allen et al. (1996), an equilibrated shear wave is found analytically. The finite-amplitude behaviour of the analytic shear wave, including a forced second-harmonic correction to the mean alongshore current, and amplitude dispersion, agree well with the numerical results of Allen et al. (1996). Limitations in their numerical model prevent the development of a side-band instability. The stability of the equilibrated shear wave is demonstrated analytically. The analytical results confirm that the Allen et al. (1996) model correctly reproduces many important features of weakly nonlinear shear waves.

We investigate the influence of rotational and vibrational energy relaxation on the stability of laminar boundary layers in supersonic flows by numerically solving the linearized equations of motion for a flow in thermal non-equilibrium. We model air as a mixture of nitrogen, oxygen and carbon dioxide, and derive accurate models for the relaxation rates from published experimental data in the field of physical chemistry. The influence of rotational relaxation is to dampen high-frequency instabilities, consistent with the well known damping effect of rotational relaxation on acoustical waves. The influence of rotational relaxation can be modelled with acceptable accuracy through the use of the bulk-viscosity approximation when the bulk viscosity is computed with a formula described herein. Vibrational relaxation affects the growth of disturbances by changing the characteristics of the laminar mean flow. The influence is strongest when the flow field contains a region at, or near, stagnation conditions, followed by a rapid expansion, such as inside wind tunnels and around bodies with a blunt leading edge, whereby the rapid expansion causes the internal energy to freeze in a distribution out of equilibrium. For flows at Mach 4.5 and stagnation temperature of 1000 K, the total amplification exhibited by boundary-layer disturbances over a sharp flat plate in wind-tunnel flows can reach a value that is fifty times as high as the value computed under the assumption of thermal equilibrium. The difference in amplification can be twice as high in the case of a blunt flat plate at atmospheric flight conditions.

Using a time-periodic perturbation of a two-dimensional steady separation bubble on a plane no-slip boundary to generate chaotic particle trajectories in a localized region of an unbounded boundary layer flow, we study the impact of various geometrical structures that arise naturally in chaotic advection fields on the transport of a passive scalar from a local ‘hot spot’ on the no-slip boundary. We consider here the full advection-diffusion problem, though attention is restricted to the case of small scalar diffusion, or large Péclet number. In this regime, a certain one-dimensional unstable manifold is shown to be the dominant organizing structure in the distribution of the passive scalar. In general, it is found that the chaotic structures in the flow strongly influence the scalar distribution while, in contrast, the flux of passive scalar from the localized active no-slip surface is, to dominant order, independent of the overlying chaotic advection. Increasing the intensity of the chaotic advection by perturbing the velocity field further away from integrability results in more non-uniform scalar distributions, unlike the case in bounded flows where the chaotic advection leads to rapid homogenization of diffusive tracer. In the region of chaotic particle motion the scalar distribution attains an asymptotic state which is time-periodic, with the period the same as that of the time-dependent advection field. Some of these results are understood by using the shadowing property from dynamical systems theory. The shadowing property allows us to relate the advection-diffusion solution at large Péclet numbers to a fictitious zero-diffusivity or frozen-field solution, corresponding to infinitely large Péclet number. The zero-diffusivity solution is an unphysical quantity, but is found to be a powerful heuristic tool in understanding the role of small scalar diffusion. A novel feature in this problem is that the chaotic advection field is adjacent to a no-slip boundary. The interaction between the necessarily non-hyperbolic particle dynamics in a thin near-wall region and the strongly hyperbolic dynamics in the overlying chaotic advection field is found to have important consequences on the scalar distribution; that this is indeed the case is shown using shadowing. Comparisons are made throughout with the flux and the distributions of the passive scalar for the advection-diffusion problem corresponding to the steady, unperturbed, integrable advection field.

A previously unreported shock feature associated with the scouring of a horizontal granular bed by a dam-break wave is discussed. Near the wave centre, the present study shows, the free surface breaks backward and a hydraulic jump forms. This behaviour is described from the standpoint of shallow-water theory, suitably extended to deal with non-equilibrium sediment transport. The shock formation involves a particularly strong coupling between flow free-surface evolution and bed morphodynamics. Support for our conclusions is sought through experimental and numerical approaches. In order to magnify the observed phenomena, measurements were performed for the case of light bed particles moving in sheet and debris flow modes. A detailed picture of the transient two-phase flow is presented, based on whole field acquisition of the grain motions by particle tracking techniques. Corresponding shallow-water solutions are constructed numerically using a shock capturing scheme. Finally, an interpretation of the jump formation is proposed based on the theory of characteristics.

In this paper, air entrainment by a liquid jet is studied. The size of bubbles entrained by jets plunging into a liquid can be consistently decreased to the 50–100 μm range, and their number increased in a highly controllable fashion, by surrounding a mm-size jet by a hollow cap with a slightly larger inner diameter. When the right amount of air is supplied to the cap, small air bubbles detach from a steady annular cavity that forms around the jet and are entrained into the liquid. The fluid mechanical principles underlying this interesting and useful effect are investigated experimentally and theoretically in this paper. It is shown that a key aspect of the process is the jet surface roughness, which is studied quantitatively and explained in terms of the boundary layer instability inside the nozzle. The maximum bubble size is found to be nearly equal to one quarter of the wavelength of the jet surface disturbances, consistent with a breakup process of relatively large air pockets around the jet, as suggested by close-up pictures. The average bubble size downstream of the cap increases proportionally to the air to water flow ratio. Boundary integral simulations of the air pocket formation are carried out. The results are found to be useful in explaining important characteristics of the experiment such as the threshold for entrainment and cavity size.

An analysis of the quasi-steady streaming of the liquid in a vertically vibrated horizontal soap film is reported. The air around the soap film is seen to play a variety of roles: it transmits normal and tangential oscillatory stresses to the film, damps out Marangoni waves, and forces non-oscillatory deflection of the film and tangential motion of the liquid. Non-oscillatory volume forcing originating inside the liquid is also analysed. This forcing dominates the quasi-steady streaming when the excitation frequency is close to the eigenfrequency of a Marangoni mode of the soap film, while both volume forcing in the liquid and surface forcing of the gas on the liquid are important when no Marangoni mode resonates. Different manners by which the combined forcings can induce quasi-steady streaming motion are discussed and some numerical simulations of the quasi-steady liquid flow are presented.

Experiments with layers of salt and sugar solution separated by a diffusive interface are described. Interfacial waves were spontaneously generated by the convection once the system evolved to a critical value of the density-anomaly ratio Rρ≡βΔS/αΔT. The waves locally modulated the interfacial fluxes by modifying the interface thickness and thereby organized otherwise random convective motions into large-scale circulations. In turn, the waves themselves persisted for unusually long times owing to energy input from the organized convection. The dependence of the wave speed on the layer properties and channel dimensions was successfully predicted by assuming that coupling requires a matching of the wave and convection speeds, and that the system selects waves of an amplitude for which this resonance can occur. This ‘wave–convection coupling’ also appeared to increase the interfacial fluxes at low Rρ. The interaction of waves and convection may be important for oceanic thermohaline staircases and other systems where convection is driven by interfacial fluxes.

The leading edge of a localized, insoluble surfactant monolayer, advancing under the action of surface-tension gradients over the free surface of a thin, viscous, fluid layer, behaves locally like a rigid plate. Since lubrication theory fails to capture the integrable stress singularity at the monolayer tip, so overestimating the monolayer length, we investigate the quasi-steady two-dimensional Stokes flow near the tip, assuming that surface tension or gravity keeps the free surface locally at. Wiener–Hopf and matched-eigenfunction methods are used to compute the ‘stick-slip’ flow when the singularity is present; a boundary-element method is used to explore the nonlinear regularizing effects of weak ‘contaminant’ surfactant or surface diffusion. In the limit in which gravity strongly suppresses film deformations, a spreading monolayer drives an unsteady return flow (governed by a nonlinear diffusion equation) beneath most of the monolayer, and a series of weak vortices in the fluid ahead of the tip. As contaminant or surface diffusion increase in strength, they smooth the tip singularity over short lengthscales, eliminate the local stress maximum and ultimately destroy the vortices. The theory is readily extended to cases in which the film deforms freely over long lengthscales. Limitations of conventional thin-film approximations are discussed.

A localized, insoluble, surfactant monolayer, spreading under the action of surface-tension gradients over a thin liquid film, has at its leading edge an integrable stress singularity which renders conventional thin-film approximations locally non-uniform. Here high-Reynolds-number asymptotics are used to explore the quasi-steady two-dimensional developing flow near the monolayer tip, assuming that gravity keeps the free surface almost flat, that weak ‘contaminant’ surfactant regularizes the singularity and that the monolayer spreads fast enough for inertial effects to be important in a region which is long compared to the film depth but which is short compared to the length of the monolayer. It is shown how downward displacement of the inviscid core flow by the subsurface viscous boundary layer yields a non-uniform pressure distribution which, when the monolayer is spreading fast enough for cross-stream pressure gradients to be significant at its tip, creates a short free-surface hump which is the thin-film version of a Reynolds ridge. The ridge and other singular flow structures are smoothed as the monolayer slows and levels of contaminant are increased. The conditions under which lubrication theory provides a uniformly accurate approximation for this class of surfactant-spreading flows are established.

This work is an experimental study of the deformation and breakup of a bubble in a turbulent flow. A special facility was designed to obtain intense turbulence without significant mean flow. The experiments were performed under microgravity conditions to ensure that turbulence was the only cause of bubble deformation. A scalar parameter, characteristic of this deformation, was obtained by video processing of high-speed movies. The time evolution and spectral representation of this scalar parameter showed the dynamical characteristics of bubble deformation. The signatures of the eigenmodes of oscillation predicted by the linear theory were clearly observed and the predominance of the second mode was proved. In addition, numerical simulations were performed by computing the response of a damped oscillator to the measured turbulence forcing. Simulations and experiments were found to be in good agreement both qualitatively, from visual inspections of the signals, and quantitatively, from a statistical analysis. The role of bubble dynamics in the deformation process has been clarified. On the one hand, the time response of the bubble controls the maximum amount of energy which can be extracted from each turbulent eddy. On the other hand, the viscous damping limits the energy that the bubble can accumulate during its fluctuating deformation. Moreover, two breakup mechanisms have been identified. One mechanism results from the balance between two opposing dominant forces, and the other from a resonance oscillation. A new parameter, the mean efficiency coefficient, has been introduced to take into account the various aspects of bubble dynamics. Used together with the Weber number, this parameter allows the prediction of the occurrence of these two mechanisms. Finally, the influence of the residence time of the bubble on the statistics of the deformation has been analysed and quantified.

This paper studies the impinging of two ideal fluid jets. The usual two-dimensional model of jet flow uses an ideal, incompressible, weightless fluid to describe these impinging jets, so that the problem becomes one of complex analysis which seems to have an infinite number of analytical solutions, except for direct jet impacts. The new approach presented here is based on the construction of a dividing line between the two jets. It gives an efficient procedure for solving this problem numerically when the jets flow in arbitrary directions and the solution obtained seems to be unique.