The patterns of vortex formation from a cylinder oscillating in a horizontal plane, located at various depths of submergence beneath a free surface, are characterized using high-image-density particle image velocimetry (PIV). Instantaneous representations of the velocity field, streamline topology and vorticity patterns are referenced to the instantaneous velocity of the cylinder. In turn, these features are related to the magnitude and phase of the instantaneous transverse force, which can exhibit highly nonlinear, spike-like fluctuations. When a finite gap is maintained between the free surface and the cylinder, the patterns of vorticity concentrations are altered in such a fashion that both the peak magnitude and the degree of amplitude modulation of the negative transverse force spikes are attenuated, but shifted little in phase, relative to the case of the fully submerged cylinder. On the other hand, when the cylinder is sufficiently close to the free surface such that the gap region is eliminated, the patterns of vorticity concentrations are fundamentally different and the negative transverse force spikes are remarkably more pronounced and consistent than for deeper submergence.

An experimental and theoretical investigation of the flow and density distribution arising from the upward turbulent injection of a dense fluid into a stratified environment of finite extent is presented. Initially, the rising fluid reaches a maximum height before the flow reverses direction and intrudes either along the base of the tank or at an intermediate height in the environment. As more dense fluid is added through either a point or line source, both the fountain and the environment evolve with time. We determine expressions for the motion of the ascending and descending ‘fronts’ that mark the vertical extent of the spreading layer. We also consider the changes to the environmental density profile and determine an expression for the rate at which the top of the fountain rises due to these changes. Finally, we apply our results quantitatively to two physical problems: the replenishment of magma chambers and the heating or cooling of a room.

Richtmyer–Meshkov instability is investigated for negative Atwood number and two-dimensional sinusoidal perturbations by comparing experiments, numerical simulations and analytic theories. The experiments were conducted on the NOVA laser with strong radiatively driven shocks with Mach numbers greater than 10. Three different hydrodynamics codes (RAGE, PROMETHEUS and FronTier) reproduce the amplitude evolution and the gross features in the experiment while the fine-scale features differ in the different numerical techniques. Linearized theories correctly calculate the growth rates at small amplitude and early time, but fail at large amplitude and late time. A nonlinear theory using asymptotic matching between the linear theory and a potential flow model shows much better agreement with the late-time and large-amplitude growth rates found in the experiments and simulations. We vary the incident shock strength and initial perturbation amplitude to study the behaviour of the simulations and theory and to study the effects of compression and nonlinearity.

An experimental study of flow transition in a rotating Taylor–Couette system was made by investigating the spatio–temporal velocity field (axial velocity component) by the ultrasonic Doppler method. The flow fields for the range of Reynolds numbers 9<R*<40 (where R*=R/Rc; Rc is the critical Reynolds number for Taylor vortex flow) were decomposed by two-dimensional Fourier transform and the orthogonal decomposition technique, and intensities of coherent structural modes were quantitatively obtained. The variation of the intensities of various modes with respect to Reynolds number clearly shows a transition behaviour of the quasi-periodic state resulting from the wavy vortex mode and the modulating waves, which is found to disappear suddenly at about R*=21. A new mode was found after the disappearance of the quasi-periodic state, which in turn disappears at R*=36. Beyond this regime, there was no coherent structure found except for the stationary Taylor vortices and so-called broad-band component, which is attributed to chaos. The total energy occupation (the number of modes which occupy 90% of the total energy) and the global entropy support such transition behaviour quantitatively. After the disappearance of the newly found mode, the number of modes needed to compose the velocity field is still finite and small – about 40–50. We call this flow state ‘soft turbulence’.

A combined experimental and numerical investigation is presented of the multiple oscillatory states that exist in the flows produced in a completely filled, enclosed, circular cylinder driven by the constant rotation of one of its endwalls. The flow in a cylinder of height to radius ratio 2.5 is interrogated experimentally using flow visualization and digitized images to extract quantitative temporal information. Numerical solutions of the axisymmetric Navier–Stokes equations are used to study the same flow over a range of Reynolds numbers where the flow is observed to remain axisymmetric. Three oscillatory states have been identified, two of them are periodic and the third is quasi-periodic with a modulation frequency much smaller than the base frequency. The range of Reynolds numbers for which the quasi-periodic flow exists brackets the switch between the two periodic states. The results from the combined experimental and numerical study agree both qualitatively and quantitatively, providing unambiguous evidence of the existence and robustness of these multiple time-dependent states.

The phenomenon of a succession of upstream-advancing solitary waves generated by underwater disturbances moving steadily with a transcritical velocity in two- dimensional shallow water channels is investigated. The two-dimensional Navier–Stokes (NS) equations with the complete set of viscous boundary conditions are solved numerically by the finite-difference method to simulate the phenomenon. The overall features of the phenomenon illustrated by the present numerical results are unanimous with observations in nature as well as in laboratories. The relations between amplitude and celerity, and between amplitude and period of generation of solitary waves can be accurately simulated by the present numerical method, and are in good agreement with predictions of theoretical formulae. The dependence of solitary wave radiation on the blockage and on the body shape is investigated. It furnishes collateral evidence of the experimental findings that the blockage plays a key role in the generation of solitary waves. The amplitude increases while the period of generation decreases as the blockage coefficient increases. It is found that in a viscous flow the shape of an underwater object has a significant effect on the generation of solitary waves owing to the viscous effect in the boundary layer. If a change in body shape results in increasing the region of the viscous boundary layer, it enhances the viscous effect and so does the disturbance forcing; therefore the amplitudes of solitary waves increase. In addition, detailed information of the flow, such as the pressure distribution, velocity and vorticity fields, are given by the present NS solutions.

This paper contains theoretical and experimental results on the relative motion of two pulsating spherical bubbles along their line of centres, in a liquid subjected to an acoustic field. The motion is caused only by the secondary Bjerknes forces. The linear theory for the secondary Bjerknes forces is modified by introducing a model for the coupling between the pulsations of the interfaces. The secondary effects introduced by this model are determined by the frequency indices of the bubbles, defined as the ratio of the forcing frequency to the resonance frequency of each bubble. The equations of motion are set up with the conservative Lagrangian formalism. This approach allows an analytical study of all the possible patterns of motion and the identification of the set of governing parameters: total energy and interaction coefficient. A pair of bubbles driven far from their resonance frequencies may attract or repel, depending on whether their frequency indices are respectively on the same side or on either side of unity. For forcing frequencies close to resonance, the proposed model predicts a new pattern of relative motion, namely a periodic motion (oscillations) around an equilibrium bubble separation. The experimental study identifies this new periodic pattern of motion, for acoustically levitated bubbles of nearly equal sizes, forced near their resonance frequency. A quantitative study on the variation of the relative velocity with the separation between the bubbles shows that the conservative model for the motion holds for large and moderate separations. The following information is reported: (a) a classification of the pairs of bubbles, based upon their phase difference in oscillations; (b) a model for the coupling of the pulsations of two bubbles; (c) formulas for the interaction force field of two pulsating bubbles, for all of the categories; (d) a study of all possible patterns of relative motion (collisions, scattering and oscillations), with their conditions of occurrence; (e) experimental data for two attracting bubbles; (f) experimental data for two oscillating bubbles.

We consider the nonlinear spin-up/down of a rotating stratified fluid in a conical container. An analysis of axisymmetric similarity-type solutions to the relevant boundary-layer problem, Duck, Foster & Hewitt (1997), has revealed three types of behaviour for this geometry. In general, the boundary layer evolves to either a steady state, or a gradually thickening boundary layer, or a finite-time singularity depending on the Schmidt number, the ratio of initial to final rotation rates, and the relative importance of rotation and stratification.

In this paper we emphasize the experimental aspects of an investigation into the initial readjustment process. We make comparisons with the previously presented boundary-layer theory, showing good quantitative agreement for positive changes in the rotation rate of the container (relative to the initial rotation sense). The boundary-layer analysis is shown to be less successful in predicting the flow evolution for nonlinear decelerations of the container. We discuss the qualitative features of the spin-down experiments, which, in general, are dominated by non-axisymmetric effects. The experiments are conducted using salt-stratified solutions, which have a Schmidt number of approximately 700.

The latter sections of the paper present some stability results for the steady boundary-layer states. A high degree of non-uniqueness is possible for the system of steady governing equations; however the experimental results are repeatable and stability calculations suggest that ‘higher branch’ solutions are, in general, unstable. The eigenvalue spectrum arising from the linear stability analysis is shown to have both continuous and discrete components. Some analytical results concerning the continuous spectrum are presented in an appendix.

A brief appendix completes the previous analysis of Duck, Foster & Hewitt (1997), presenting numerical evidence of a different form of finite-time singularity available for a more general boundary-layer problem.

This paper presents the first experimental results on Marangoni–Bénard instability in a symmetrical three-layer system. A pure thermocapillary phenomenon has been observed by performing the experiment in a microgravity environment where buoyancy forces can be neglected. This configuration enables the hydrodynamic stability of two identical liquid–liquid interfaces subjected to a normal gradient of temperature to be studied. The flow is driven by one interface only and obeys the criterion based on the heat diffusivity ratio proposed by Scriven & Sternling (1959) and Smith (1966). The measured critical temperature difference for the onset of convection is compared to the value obtained from two-dimensional numerical simulations. The results of the simulations are in reasonable agreement with the velocimetry and the thermal experimental data for moderate supercriticality. Numerically and experimentally, the convective pattern exhibits a transition between different convective regimes for similar temperature gradients. Their common detailed features are discussed.

Orthonormal wavelet transformations are used to decompose velocity signals of grid turbulence into both space and scale. The transforms exhibit small-scale enhancements of (i) the spatial fluctuation, (ii) the correlation in space between the adjacent scales, and (iii) the correlation in space between the longitudinal and transverse components. The spatial fluctuation and the scale–scale correlation at small scales are more significant in the transverse component than in the longitudinal component. These features are the same for different families of wavelets.

Turbulence contains tube-like structures of vorticity. We demonstrate that wavelet transforms of velocities are enhanced at the positions of the tubes, by using a direct numerical simulation. Thus our wavelet analyses have captured the effects of those coherent structures on velocities measured in the experiment, which would be difficult for traditional analysis techniques such as those with velocity increments.

An existence theorem for localized stationary vortex solutions in an external shear flow is proved. The flow is three-dimensional and quasi-geostrophic in an unbounded domain. The external flow is unidirectional, with linear horizontal and vertical shear. The flow conserves an infinite family of Casimir integrals. Flows that have the same value of all Casimir integrals are called isovortical flows, and the potential vorticity (PV) fields of isovortical flows are stratified rearrangements of one another. The theorem guarantees the existence of a maximum-energy flow in any family of isovortical flows that satisfies the following conditions: the PV-anomaly must have compact support, it must have the same sign everywhere, and this sign must be the same as the sign of the external horizontal shear over the vertical interval to which the support of the PV-anomaly is confined. This flow represents a stationary and localized vortex, and the maximum-energy property implies that the vortex is stable. The PV-anomaly decreases monotonically outward from the vortex centre in each horizontal plane, but apart from this the profile is arbitrary.

Extended mild-slope equations for the propagation of small-amplitude water waves over variable bathymetry regions, recently proposed by Massel (1993) and Porter & Staziker (1995), are shown to exhibit an inconsistency concerning the sloping-bottom boundary condition, which renders them non-conservative with respect to wave energy. In the present work, a consistent coupled-mode theory is derived from a variational formulation of the complete linear problem, by representing the vertical distribution of the wave potential as a uniformly convergent series of local vertical modes at each horizontal position. This series consists of the vertical eigenfunctions associated with the propagating and all evanescent modes and, when the slope of the bottom is different from zero, an additional mode, carrying information about the bottom slope. The coupled-mode system obtained in this way contains an additional equation, as well as additional interaction terms in all other equations, and reduces to the previous extended mild-slope equations when the additional mode is neglected. Extensive numerical results demonstrate that the present model leads to the exact satisfaction of the bottom boundary condition and, thus, it is energy conservative. Moreover, it is numerically shown that the rate of decay of the modal-amplitude functions is improved from O(n−2), where n is the mode number, to O(n−4), when the additional sloping-bottom mode is included in the representation. This fact substantially accelerates the convergence of the modal series and ensures the uniform convergence of the velocity field up to and including the boundaries.

This paper presents a theoretical and experimental investigation into saline and particle-driven intrusions along the interface between two layers of different densities. The conditions at the nose of an intrusion are described in an analysis similar to that applied by Benjamin (1968) to boundary gravity currents. Equations for propagation velocity and front position as functions of relative density are derived. These are used in an integral model for intrusions, which also includes the effects of sedimentation of particles and detrainment of interstitial fluid. The model describes the time-evolution of the length of the intrusion and the sediment distribution it produces. Laboratory experiments were carried out with lock-releases of a fixed volume of saline or particle-laden fluid into a two-layer stratification. Measurements were taken of the intrusion propagation, intrusion position and sediment distribution, and are found to be in good agreement with the solutions of the integral model.

Numerical experiments on modified turbulent channels at moderate Reynolds numbers are used to differentiate between several possible regeneration cycles for the turbulent fluctuations in wall-bounded flows. It is shown that a cycle exists which is local to the near-wall region and does not depend on the outer flow. It involves the formation of velocity streaks from the advection of the mean profile by streamwise vortices, and the generation of the vortices from the instability of the streaks. Interrupting any of those processes leads to laminarization. The presence of the wall seems to be only necessary to maintain the mean shear. The generation of secondary vorticity at the wall is shown to be of little importance in turbulence generation under natural circumstances. Inhibiting its production increases turbulence intensity and drag.

We consider the free boundary problem for the evolution of a nearly straight slender fibre of viscous fluid. The motion is driven by prescribing the velocity of the ends of the fibre, and the free surface evolves under the action of surface tension, inertia and gravity. The three-dimensional Navier–Stokes equations and free-surface boundary conditions are analysed asymptotically, using the fact that the inverse aspect ratio, defined to be the ratio between a typical fibre radius and the initial fibre length, is small. This first part of the paper follows earlier work on the stretching of a slender viscous fibre with negligible surface tension effects. The inclusion of surface tension seriously complicates the problem for the evolution of the shape of the cross-section. We adapt ideas applied previously to two-dimensional Stokes flow to show that the shape of the cross-section can be described by means of a conformal map which depends on time and distance along the fibre axis. We give some examples of suitable relevant conformal maps and present numerical solutions of the resulting equations. We also use analytic methods to examine the coupling between stretching and the evolution of the cross-section shape.

Steady simple shear flows of smooth inelastic spheres are studied by means of a model kinetic equation and also of a direct Monte Carlo simulation method. Both approaches are based on the Enskog equation and provide for each other a test of consistency. The dependence of the granular temperature and of the shear and normal stresses on both the solid fraction and the coefficient of restitution is analysed. Quite a good agreement is found between theory and simulations in all cases. Also, simplified expressions based on the analytical solution of the model for small dissipation are shown to describe fairly well the simulation results even for not small inelasticity. A critical comparison with previous theories is carried out.