A detailed experimental study of flow in a rectangular sudden expansion using both active and passive forcing techniques has been made. The configuration consists of a 2:1-aspect-ratio rectangular channel which undergoes a sudden expansion such that the backward-facing step height (h) is uniform, and equal to the minor side of the inlet channel. Passive forcing was provided by the system geometry; the rectangular vortex rings formed in the shear layer of the expanding jet undergo self-induction, deforming the jet cross-section and introducing transverse velocities not found in plane or axisymmetric configurations. Active forcing was induced by periodic fluctuations in the system flow rate at the jet natural frequency. This served to enhance the unusual three-dimensional effects and phase-lock the flow for ensemble analysis. The results presented here include a description of the evolution of an isolated vortex in this configuration obtained from flow visualization of a suddenly started jet, as well as forced steady-state three-component velocity measurements which are used to characterize the flow field. The evolution of a rectangular vortex ring in the jet shear layer is traced, and both fluctuating and time-constant transverse velocities are related to the passage of the vortex structures.

We investigate the evolution of a vertically tilted geostrophic vortex of cylindrical shape and circular horizontal cross-section using the recently developed method of boundary surface dynamics. The vortex consists of a finite volume of constant potential vorticity immersed in a spatially unbounded fluid of uniform density stratification. The fully nonlinear three-dimensional problem is then reduced to the calculation of the Lagrangian evolution of the boundary surface of the vortex region, thus decreasing the dimensionality by one. In the numerical simulations presented here, the vortex shows a general tendency to attain vertical alignment and a horizontal axisymmetrical shape by wobbling about its centre and going through three basic stages of evolution: (a) the circular horizontal cross-sections of the upper and lower parts of the vortex distort and become elongated; (b) the upper and lower sections then become vertically aligned by reducing their horizontal intercentroid distances; and (c) the distorted horizontal cross-sections relax towards axisymmetry, often through the process of filamentation. For a given vortex height, if the horizontal scale of the flow is close to the internal radius of deformation, or equivalently, the density stratification is not too strong, the processes of filamentation and vertical alignment are enhanced. However, for stronger stratifications, both filamentation and vertical alignment are found to be greatly inhibited. For relatively small initial inclination angles, filamentation only occurs in the upper and lower sections of the vortex. Increasing the angle of tilt also increases the tendency of the surface to steepen and filament in the middle sections of the vortex. For a fixed value of the ratio of horizontal scale of the flow to the deformation radius, taller vortices have an increased tendency to align and axisymmetrize than shorter vortices of equal inclination angle.

Short water surface waves generated by wind in a water tunnel have been measured by an optical technique that provides a synoptic picture of the water surface gradient over an area of water surface (Zhang & Cox 1994). These images of the surface gradient can be integrated to recover the shape of the water surface and find the two-dimensional wavenumber spectrum. Waveforms and two-dimensional structures of short wind waves have many interesting features: short and steep waves featuring sharp troughs and flat crests are very commonly seen and most of the short waves are far less steep than the limiting wave forms; waveforms that resemble capillary–gravity solitons are observed with a close match to the form theoretically predicted for potential flows (Longuet-Higgins 1989); capillaries are mainly found as parasitics on the downwind faces of gravity waves, and the longest wavelengths of those parasitic capillaries found are less than 1 cm; the phenomenon of capillary blockage (Phillips 1981) on dispersive freely travelling short waves is also observed. The spectra of short waves generated by low winds show a characteristic dip at the transition wavenumber between the gravity and capillary regimes, and the dip becomes filled in as the wind increases. The spectral cut-off at high wavenumbers shows a power law behaviour with an exponent of about minus four. The wavenumber of the transition from the dip to the cut-off is not sensitive to the change of wind speed. The minus fourth power law of the extreme capillary wind wave spectrum can be explained through a model of energy balances. The concept of an equilibrium spectrum is still useful. It is shown that the dip in the spectrum of capillary–gravity waves is a result of blockage of both capillary–gravity wind waves and parasitic capillary waves.

We report the results of an experimental study of the transition to turbulence in a pipe under the condition of constant mass flux. The transition behaviour and structures observed in this experiment were qualitatively the same as those described in previous reported studies performed in pressure-driven systems. A variety of jet and suction devices were used to create repeatable disturbances which were then used to test the stability of developed Poiseuille flow. The Reynolds number (Re) and the parameters governing the disturbances were varied and the outcome, whether or not transition occurred some distance downstream of the injection point, was recorded. It was found that a critical amplitude of disturbance was required to cause transition at a given Re and that this amplitude varied in a systematic way with Re. This finite, critical level was found to be a robust feature, and was relatively insensitive to the form of disturbance. We interpret this as evidence for disconnected solutions which may provide a pointer for making progress in this fundamental, and as yet unresolved, problem in fluid mechanics.

A conceptual framework for analysing the energetics of density-stratified Boussinesq fluid flows is discussed. The concept of gravitational available potential energy is used to formulate an energy budget in which the evolution of the background potential energy, i.e. the minimum potential energy attainable through adiabatic motions, can be explicitly examined. For closed systems, the background potential energy can change only due to diabatic processes. The rate of change of background potential energy is proportional to the molecular diffusivity. Changes in the background potential energy provide a direct measure of the potential energy changes due to irreversible diapycnal mixing. For open systems, background potential energy can also change due to boundary fluxes, which can be explicitly measured. The analysis is particularly appropriate for evaluation of diabatic mixing rates in numerical simulations of turbulent flows. The energetics of a shear driven mixing layer is used to illustrate the analysis.

An asymptotic solution is found for the temperature and circulation structure for an unsteady two-dimensional model of the thermal bar phenomenon. The non-rotating case considered here is relevant to laboratory models of the thermal bar. The main implication of the asymptotic results is that the thermal bar might propagate out from the shore more slowly than predicted by a purely heat-balance-based estimate. The solution is discussed in the context of available experimental results.

Chatwin & Sullivan (1990) have demonstrated that, for a wide range of self-similar scalar fields, the moments of the probability density function of concentration have a very simple form. Here, an extension to this simple form which takes account of the source distribution is developed. This extension has two effects. Firstly it modifies the values of the two parameters appearing in the original theory and in particular explains the observed behaviour of these parameters very near to a line source of heat in grid tubulence. Secondly, it introduces an additional parameter in the description of each moment beyond the second. It is shown that these additional parameters are necessary in order to describe measurements of the first four central moments throughout the concentration field from a continuous line source of heat in grid-generated turbulence.

The evolution of a finite-amplitude three-dimensional localized disturbance embedded in external shear flows is addressed. Using the fluid impulse integral as a characteristic of such a disturbance, the Euler vorticity equation is integrated analytically, and a system of linear equations describing the temporal evolution of the three components of the fluid impulse is obtained. Analysis of this system of equations shows that inviscid plane parallel flows as well as high Reynolds number two-dimensional boundary layers are always unstable to small localized disturbances, a typical dimension of which is much smaller than a dimensional length scale corresponding to an O(1) change of the external velocity. Since the integral character of the fluid impulse is insensitive to the details of the flow, universal properties are obtained. The analysis predicts that the growing vortex disturbance will be inclined at 45° to the external flow direction, in a plane normal to the transverse axis. This prediction agrees with previous experimental observations concerning the growth of hairpin vortices in laminar and turbulent boundary layers. In order to demonstrate the potential of this approach, it is applied to Taylor-Couette flow, which has additional dynamical effects owing to rotation. Accordingly, a new instability criterion associated with three-dimensional localized disturbances is found. The validity of this criterion is supported by our experimental results.

The diffraction of water waves by a vertical circular cylinder is considered in the regime where the wave amplitude A and cylinder radius a are of the same order, and both are small compared to the wavelength. The wave slope is small, and a conventional linear analysis applies in the outer domain far from the cylinder. Significant nonlinear effects exist in the complementary inner domain close to the cylinder, associated with the free-surface boundary condition. Using inner coordinates scaled with respect to a, it is shown that the leading-order nonlinear contribution to the velocity potential includes terms proportional to both A2a and A3. The wave load which acts on the cylinder near the free surface includes second- and third-harmonic components which are proportional respectively to A2a2 and A3a. In a conventional perturbation analysis, where A [Lt ] a, these components would be ordered in magnitude corresponding to the different powers of A, but here they are of the same order. The second- and third-order components of the total force are of comparable magnitude for practical values of the wave slope.

Existing theories for calculating the energy transfer rates to gravity waves due to resonant nonlinear interactions among wave components whose lengths are long in comparison to wave elevations have been verified experimentally and are well accepted. There is uncertainty, however, about prediction of energy transfer rates within a set of waves having short to moderate lengths when these are present simultaneously with a long wave whose amplitude is not small in comparison to the short wavelengths. Here we implement both a direct numerical method that avoids small-amplitude approximations and a spectral method which includes perturbations of high order. These are applied to an interacting set of short- to intermediate-length waves with and without the presence of a large long wave. The same cases are also studied experimentally. Experimentally and numerical results are in reasonable agreement with the finding that the long wave does influence the energy transfer rates. The physical reason for this is identified and the implications for computations of energy transfer to short waves in a wave spectrum are discussed.

When a flat-plate airfoil at zero angle of attack encounters a vertical gust in an otherwise uniform flow, it experiences a force along the chord. This leading-edge suction force is examined for compressible flow with a time-dependent gust. A simple derivation of the thrust force is based on the fact that the leading edge is a singular point so that the flow here is dominated by the leading-edge dipole strength. From the viewpoint of a fluid-fixed observer the fluid does work on the airfoil, and this energy must come from the incident gust. Demonstrating energy conservation is not surprising, but it gives a better understanding of the relationship between the individual energy terms. The derivation shows that the acoustic energy can be calculated using compact assumptions at low frequency, but that it must be calculated non-compactly at high frequency. For a general gust the work done on the airfoil is shown to equal the energy taken from the fluid, the energy transfer occurring at the leading edge. For a sinusoidal gust the energy contained in the incident gust is shown to equal the sum of the energy remaining in the wake, the work done on the airfoil and the acoustic energy radiated away. The relative proportions of the incident energy going to these three energy types depends on the gust frequency, the acoustic radiation becoming more efficient as the frequency increases. For a fixed gust frequency, the thrust force goes to zero at a Mach number of one, and for an incident gust consisting of vorticity on the airfoil axis, the entire energy of the gust is radiated as acoustic energy at this Mach number.

The respective effects of an external field and departure of suspended particles from spherical shape on the rotary motion of axisymmetric dipolar particles placed in a homogeneous shear flow are studied. The analysis shows that, owing to cumulative effects, even a weak external field or a small deviation from spherical shape can significantly modify the resulting motion relative to that found in the corresponding classical problems of torque-free particles or dipolar spheres in homogeneous shear. Thus, unlike the latter problems, there are in the present problem cases when all particles approach a single limit cycle; in other cases multiple stable equilibria simultaneously coexist and the orientation space is appropriately divided into corresponding domains of attraction; in some situations possessing appropriate symmetry properties, particles may, depending upon their respective initial orientations, either move along a family of periodic orbits or else converge to a stable equilibrium orientation.

Many flows, including those containing suspended particles, are kept turbulent by the action of the bottom stress, and this turbulence is also responsible for maintaining sedimenting particles in suspension and in some cases entraining more particles from the bed. A convenient one-dimensional analogue of these processes is provided by laboratory experiments conducted in a mixing box, where a characterizable turbulence is generated by the vertical oscillation of a horizontal grid. In the present paper we report the results of a series of experiments with a grid located close to the bottom boundary to simulate the action of stresses acting at a rough boundary, and compare the results with those obtained using the more extensively studied geometry in which a similar grid is located in the interior of a stirred fluid layer. Experiments have been conducted both with dense, particle-free fluid layers and with layers containing sufficiently high concentrations of dense particles to have a significant effect on the bulk density. In the fluid case, the interface at the top of the stirred dense layer continues to rise as lighter fluid is entrained across the interface. Sediment layers are distinctly different, because the particles responsible for the density difference between the layers can fall out of the suspension as it changes in thickness. The work done in keeping particles in suspension and the effect of this on the turbulence above the grid must be taken into account. The mechanism of resuspension of particles depends on the level of turbulence near the bottom boundary, below the grid. As the stirring rate, and thus the intensity of turbulence, are increased three possible equilibrium states can be attained sequentially: the particles eventually all precipitate; or some particles precipitate while the remainder are held indefinitely in suspension; or all the particles are suspended. In the last two cases a stable, self-limited suspension layer is produced, separated from the overlying fluid by a sharp density interface at a fixed height. Theoretical arguments are presented which provide a satisfactory scaling of the experimental data. These are compared with previous theories and numerical experiments aimed at modelling both the one-dimensional problem and the corresponding processes in turbulent gravity currents. Comparisons are also made with sediment-laden channel flows and convecting layers containing sedimenting particles. Similar results will hold for light, positively buoyant particles or non-coalescing bubbles.

A new theoretical formulation is developed for the effects of surfactants on mass transport across the dynamic interface of a bubble which undergoes spherically symmetric volume oscillations. Owing to the presence of surfactants, the Henry's law boundary condition is no longer applicable; it is replaced by a flux boundary condition that features an interfacial resistance that depends on the concentration of surfactant molecules on the interface. The driving force is the disequilibrium partitioning of the gas between free and dissolved states across the interface. As in the clean surface problem analysed recently (Fyrillas & Szeri 1994), the transport problem is split into two parts: the smooth problem and the oscillatory problem. The smooth problem is treated using the method of multiple scales. An asymptotic solution to the oscillatory problem, valid in the limit of large Péclet number, is developed using the method of matched asymptotic expansions. By requiring that the outer limit of the inner approximation match zero, the splitting into smooth and oscillatory problems is determined unambiguously in successive powers of [weierp ]−1/2, where [weierp ] is the Péclet number. To leading order, the clean surface solution is recovered. Continuing to higher order it is shown that the concentration field depends on RI[weierp ]−1/2, where RI is the (dimensionless) interfacial resistance associated with the presence of surfactants. Although the influence of surfactants appears at higher order in the small parameter, surfactants are shown to have a very significant effect on bubble growth rates owing to the fact that the magnitude of RI is approximately the same as the magnitude of [weierp ]1/2 at conditions of practical interest. Hence the higher-order ‘corrections’ happen numerically to be of the same magnitude as the leading-order, clean surface problem. This is the fundamental reason for major increases in the bubble growth rates associated with the addition of surfactants. This is in contrast to the case of a still, surfactant-covered bubble, in which the first-order correction to the growth rate is of order RI[weierp ]−1 and presents a [weierp ]−1/2 correction. Finally, although existing experimental results have shown only enhancement of bubble growth in the presence of a surfactant the present theory suggests that it is possible for a surfactant, characterized by weak dependence of interfacial resistance on surface concentration, to inhibit rather than enhance the growth of bubbles by rectified diffusion.

We study the two-dimensional, laminar interactions between a contaminated free surface and a vortical flow below. Two canonical vortical flows are considered: a pair of vortex tubes impinging onto the free surface; and an unstable shear wake behind a body operating on the surface. A quantitative model for free-surface viscous flows in the presence of soluble or insoluble surfactants is developed. For the low to moderate Froude numbers considered here, for which weakly nonlinear free-surface boundary conditions are valid, the surface boundary layer and vorticity production are weak for clean water and the vortical flow evolution does not differ qualitatively from that under a free-slip boundary. When even a small amount of contamination is present, the flow can be dramatically affected. The vortical flow creates gradients in the surfactant surface concentration which leads to Marangoni stresses, strong surface vorticity generation, boundary layers, and even separation. These significantly influence the underlying flow which itself affects surfactant transport in a closed-loop interaction. The resulting flow features are intermediate between but qualitatively distinct from those under either a free- or no-slip boundary. Surfactant effects are most prominent for insoluble surface contamination with likely development of surfactant shocks and associated surface features such as Reynolds ridges. For soluble surfactant with initially uniform bulk concentration, surface concentration variations are moderated by sorption kinetics between the surface and bulk phases, and the overall effects are generally diminished. For initially stratified bulk concentrations, however, the evolution dynamics becomes more varied and surfactant effects may be amplified relative to the insoluble case. The dependence of these results on the properties of the contamination is studied.

The stability of an immiscible layer of fluid bounded by two other fluids of different viscosities and migrating through a porous medium is analysed, both theoretically and experimentally. Linear stability analyses for both one-dimensional and radial flows are presented, with particular emphasis upon the behaviour when one of the interfaces is highly stable and the other is unstable. For one-dimensional motion, it is found that owing to the unstable interface, the intermediate layer of fluid eventually breaks up into drops.

However, in the case of radial flow, both surface tension and the continuous thinning of the intermediate layer as it moves outward may stabilize the system. We investigate both of these stabilization mechanisms and quantify their effects in the relevant parameter space. When the outer interface is strongly unstable, there is a window of instability for an intermediate range of radial positions of the annulus. In this region, as the basic state evolves to larger radii, the linear stability theory predicts a cascade to higher wavenumbers. If the growth of the instability is sufficient that nonlinear effects become important, the annulus will break up into a number of drops corresponding to the dominant linear mode at the time of rupture.

In the laboratory, a Hele-Shaw cell was used to study these processes. New experiments show a cascade to higher-order modes and confirm quantitatively the prediction of drop formation. We also show experimentally that the radially spreading system is stabilized by surface tension at small radii and by the continual thinning of the annulus at large radii.

In this work, we treat the problem of small-scale, small-amplitude, internal waves interacting nonlinearly with a vigorous, large-scale, undulating shear. The amplitude of the background shear can be arbitrarily large, with a general profile, but our analysis requires that the amplitude vary on a length scale longer than the wavelength of the undulations. In order to illustrate the method, we consider the ray-theoretic model due to Broutman & Young (1986) of high-frequency oceanic internal waves that trap and detrap in a near-inertial wavepacket as a prototype problem. The near-inertial wavepacket tends to transport the high-frequency test waves from larger to smaller wavenumber, and hence to higher frequency. We identify the essential physical mechanisms of this wavenumber transport, and we quantify it. We also show that, for an initial ensemble of test waves with frequencies between the inertial and buoyancy frequencies and in which the number of test waves per frequency interval is proportional to the inverse square of the frequency, a single nonlinear wave–wave interaction fundamentally alters this initial distribution. After the interaction, the slope on a log-log plot is nearly flat, whereas initially it was -2. Our analysis captures this change in slope. The main techniques employed are classical adiabatic invariance theory and adiabatic separatrix crossing theory.