An analytical expression is derived for the time-averaged radiation force induced by an acoustic wave field between N particles freely suspended in a fluid. Both the host fluid and the medium inside the particles are assumed to be ideal compressible fluids. The incident field is assumed to be moderate so that the scattered and refracted fields of the particles can be taken to linear approximation. Multiple re-scattering of sound between the particles and shape modes of all orders are allowed for. No restrictions are imposed on the size of the particles, the separation distances between them and their number. The present study substantially extends the existing theory, which is based on essential simplifications and valid only for pairwise interactions. In particular, the new theory allows one to follow continuously the evolution of the radiation interaction force from large to small separation distances. The general results are illustrated in the case of two air bubbles in water. It is shown that generally the interbubble force behaves in far more complicated way than is predicted by the classical Bjerknes theory.

We examine the gravitational dispersal of dense fluid through a horizontal permeable layer, which is separated from a second underlying layer by a narrow band of much lower permeability. We derive a series of analytical solutions which describe the propagation of the fluid through the upper layer and the draining of the fluid into the underlying region. The model predicts that the current initially spreads according to a self-similar solution. However, as the drainage becomes established, the spreading slows, and in fact the fluid only spreads a finite distance before it has fully drained into the underlying layer. We examine the sensitivity of the results to the initial conditions through numerical solution of the governing equations. We find that for sources of sufficiently large initial aspect ratio (defined as the ratio of height to length), the solution converges rapidly to the initially self-similar regime. For longer and shallower initial source conditions, this convergence does not occur, but we derive estimates for the run-out length of the current, which compare favourably with our numerical data. We also present some preliminary laboratory experiments, which support the model.

The action of a rising vortex pair on the thermal boundary layer at an air–water interface is studied both experimentally and numerically. The objective is to relate variations in the surface temperature field to the hydrodynamics of the vortex pair below. The existence of a thermal boundary layer on the water side of an air–water interface is well known; it is this boundary layer which is disrupted by the action of the vortex system. Experimentally, the vortices were generated via the motion of a pair of submerged flaps. The flow was quantified through simultaneous measurement of both the subsurface velocity field, via digital particle image velocimetry (DPIV), and the surface temperature field, via an infrared (IR) sensitive imager. The results of the physical experiments show a clearly defined disruption of the ambient thermal boundary layer which is well correlated with the vorticity field below. Numerical experiments were carried out in a parameter space similar to that of the physical experiments. Included in the numerical experiments was a simple surfactant model which enabled the exploration of the complex role surface elasticity played in the vortex–free surface interaction. The results of this combined experimental and numerical investigation suggest that surface straining rate is an important parameter in correlating the subsurface flow with the surface temperature field. A model based on surface straining rate is presented to explain the interaction.

A destabilizing vertical temperature gradient and a rotating magnetic field have been applied to a cylindrical column of liquid gallium. The convective flows which arise as a function of these parameters are identified. For small magnetic field strengths, a regime of stationary flow is observed. This regime is bounded by critical values of the Rayleigh and magnetic Taylor numbers. As the rotating magnetic field is increased, the critical Rayleigh number can increase by more than a factor of 10. The rotating magnetic field itself induces an instability at a critical value of the magnetic Taylor number independent of the Rayleigh number. The nature of the bifurcations (whether subcritical or supercritical) and the convective flows occurring at the critical Rayleigh numbers are dependent upon the magnetic Taylor number. For small magnetic Taylor numbers, the experimental observations are consistent with the occurrence of a single asymmetric meridional roll which is driven around the cylinder by the rotating magnetic field.

A class of unsteady vortex solutions of the Euler equations is investigated. The solutions satisfy the von Kármán–Bödewadt similarity scalings and correspond to free and forced oscillations of radially unbounded solid-body-type vortices with axially varying rotation rates. The vortices may be of unbounded vertical extent or confined by impermeable top and/or bottom plates. In the latter case the bounding plates may be stationary or oscillatory. A solution breakdown result tends to support the hypothesis that breakdown of viscous Bödewadt-type counter-rotating vortex flows is an essentially inviscid process.

An experimental study to identify the structures present in a jet in crossflow has been carried out at a jet-to-crossflow velocity ratio U/Ucf = 3.8 and Reynolds number Re = UcfD/v = 6600. The hot-wire velocity data measured with a rake of eight X-wires at x/D = 5 and 15 and flow visualizations using planar laser-induced fluorescence (PLIF) confirm that the well-established pair of counter-rotating vortices is a feature of the mean field and that the upright, tornado-like or Fric's vortices that are shed to the leeward side of the jet are connected to the jet flow at the core. The counter-rotating vortex pair is strongly modulated by a coherent velocity field that, in fact, is as important as the mean velocity field. Three different structures – folded vortex rings, horseshoe vortices and handle-type structures – contribute to this coherent field. The new handle-like structures identified in the current study link the boundary layer vorticity with the counter-rotating vortex pair through the upright tornado-like vortices. They are responsible for the modulation and meandering of the counter-rotating vortex pair observed both in video recordings of visualizations and in the instantaneous velocity field. These results corroborate that the genesis of the dominant counter-rotating vortex pair strongly depends on the high pressure gradients that develop in the region near the jet exit, both inside and outside the nozzle.

Presented here are two results concerning the interaction between vorticity and strain. Both are obtained experimentally by investigating the hyperbolic flow created in Taylor's four-roll mill. It is first shown that this pure straining flow becomes intrinsically unstable through a supercritical bifurcation to form an array of counter-rotating vortices aligned in the stretching direction. The dimensionless parameter characterizing the flow is the internal Reynolds number Re = γΔ2/v based on the velocity gradient γ and on the gap between the rollers Δ, and the threshold value is Rec = 17. Near the threshold, the transverse velocity profiles of these vortices are in excellent agreement with those predicted by the theory of Kerr & Dold (1994) in the case of an infinite hyperbolic flow. A second result is obtained at high Reynolds number. Measurements of the velocity profile in the direction parallel to the vortices show that the velocity gradient (the stretching) is systematically weaker inside the vortices than elsewhere. This demonstrates experimentally the existence of a negative feedback of rotation on stretching. This effect is ascribed to the two-dimensionalization due to the vortex fast rotation. An implication of these results for turbulent flows is a nonlinear limitation of the vorticity stretching, an effect characterized recently by Ohkitani 1998.

Exact solutions are presented for an arbitrary number of steadily moving bubbles in a Hele-Shaw channel when surface tension is neglected. According to the symmetry displayed by the bubbles, the solutions are classified into two groups: (i) solutions where the bubbles are symmetrical about the channel centreline and (ii) solutions in which the bubbles have fore-and-aft symmetry. The general solutions are expressed in integral form but in some special cases analytic solutions in terms of elliptic integrals are found. The possible relevance of these exact solutions to experiments is also briefly discussed.

The temporal evolution of nonlinear wave fields of surface gravity waves is studied by large-scale direct numerical simulations of primitive equations in order to verify Hasselmann's theory for nonlinear energy transfer among component gravity waves. In the simulations, all the nonlinear interactions, including both resonant and non-resonant ones, are taken into account up to the four-wave processes. The initial wave field is constructed by combining more than two million component free waves in such a way that it has the JONSWAP or the Pierson–Moskowitz spectrum. The nonlinear energy transfer is evaluated from the rate of change of the spectrum, and is compared with Hasselmann's theory. It is shown that, in spite of apparently insufficient duration of the simulations such as just a few tens of characteristic periods, the energy transfer obtained by the present method shows satisfactory agreement with Hasselmann's theory, at least in their qualitative features.

The instability of a partial cavity induced by the development of a re-entrant jet is investigated on the basis of experiments conducted on a diverging step. Detailed visualizations of the cavity behaviour allowed us to identify the domain of the re-entrant jet instability which leads to classical cloud cavitation. The surrounding regimes are also investigated, in particular the special case of thin cavities which do not oscillate in length but surprisingly exhibit a re-entrant jet of periodical behaviour. The velocity of the re-entrant jet is measured from visualizations, in the case of both cloud cavitation and thin cavities. The limits of the domain of the re-entrant jet instability are corroborated by velocity fluctuation measurements. By varying the divergence and the confinement of the channel, it is shown that the extent of the auto-oscillation domain primarily depends upon the average adverse pressure gradient in the channel. This conclusion is corroborated by the determination of the pressure gradient on the basis of LDV measurements which shows a good correlation between the domain of the cloud cavitation instability and the region of high adverse pressure gradient. A simple phenomenological model of the development of the re-entrant jet in an adverse pressure gradient confirms the strong influence of the pressure gradient on the development of the re-entrant jet and particularly on its thickness. An ultrasonic technique is developed to measure the re-entrant jet thickness, which allowed us to compare it with the cavity thickness. By considering an estimate of the characteristic height of the perturbations developing on the interface of the cavity and of the re-entrant jet, it is shown that cloud cavitation requires negligible interaction between both interfaces, i.e. a thick enough cavity. In the case of thin cavities, this interaction becomes predominant; the cavity interface breaks at many points, giving birth to small-scale vapour structures unlike the large-scale clouds which are periodically shed in the case of cloud cavitation. The low-frequency content of the cloud cavitation instability is investigated using spectral analysis of wall pressure signals. It is shown that the characteristic frequency of cloud cavitation corresponds to a Strouhal number of about 0.2 whatever the operating conditions and the cavity length may be, provided the Strouhal number is computed on the basis of the maximum cavity length. For long enough cavities, another peak is observed in the spectra, at lower frequency, which is interpreted as a surge-type instability. The present investigations give insight into the instabilities that a partial cavity may undergo, and particularly the re-entrant jet instability. Two parameters are shown to be of most importance in the analysis of the re-entrant jet instability: the adverse pressure gradient and the cavity thickness compared to the re-entrant jet thickness. The present results allowed us to conduct a qualitative phenomenological analysis of the stability of partial cavities on cavitating hydrofoils. It is conjectured that cloud cavitation should occur for short enough cavities, of the order of half the chordlength, whereas the instability often observed at the limit between partial cavitation and super-cavitation is here interpreted as a cavitation surge-type instability.

We study the motion of non-diffusive, passive particles within steady, three-dimensional vortex breakdown bubbles in a closed cylindrical container with a rotating bottom. The velocity fields are obtained by solving numerically the three-dimensional Navier–Stokes equations. We clarify the relationship between the manifold structure of axisymmetric (ideal) vortex breakdown bubbles and those of the three-dimensional real-life (laboratory) flow fields, which exhibit chaotic particle paths. We show that the upstream and downstream fixed hyperbolic points in the former are transformed into spiral-out and spiral-in saddles, respectively, in the latter. Material elements passing repeatedly through the two saddle foci undergo intense stretching and folding, leading to the growth of infinitely many Smale horseshoes and sensitive dependence on initial conditions via the mechanism discovered by šil'nikov (1965). Chaotic šil'nikov orbits spiral upward (from the spiral-in to the spiral-out saddle) around the axis and then downward near the surface, wrapping around the toroidal region in the interior of the bubble. Poincaré maps reveal that the dynamics of this region is rich and consistent with what we would generically anticipate for a mildly perturbed, volume-preserving, three-dimensional dynamical system (MacKay 1994; Mezić & Wiggins 1994a). Nested KAM-tori, cantori, and periodic islands are found embedded within stochastic regions. We calculate residence times of upstream-originating non-diffusive particles and show that when mapped to initial release locations the resulting maps exhibit fractal properties. We argue that there exists a Cantor set of initial conditions that leads to arbitrarily long residence times within the breakdown region. We also show that the emptying of the bubble does not take place in a continuous manner but rather in a sequence of discrete bursting events during which clusters of particles exit the bubble at once. A remarkable finding in this regard is that the rate at which an initial population of particles exits the breakdown region is described by the devil's staircase distribution, a fractal curve that has been already shown to describe a number of other chaotic physical systems.

The problem of whether a spontaneous singularity can occur in finite time in an incompressible inviscid fluid flow is addressed. As suggested by previous numerical simulations, candidate flows are restricted to be invariant under the octahedral group of symmetries and to have a compact vortex tube in the fundamental domain. It is shown that in such a flow the image vorticity contributes strongly to the axial strain rate on the fundamental in a way which is only weakly proportional to the curvature of the vortex lines. Analysis of a model flow shows that axial strain rate scales as the inverse square of the distance to the origin, and that the velocity field forms a topological trap in which the vortex tube is accelerated towards the origin – a degenerate critical point. Evidence from simulations supports these findings. These features suggest that linear strain rate/vorticity coupling can occur in a finite-time pointwise collapse of such symmetric flows.

The work is devoted to stationary streaming flows resulting from standing capillary waves at interfaces between two immiscible liquids and their effect on the mass transfer rate of a passive scalar. In particular, oscillating liquid droplets immersed in another immiscible liquid are considered. Secondary streaming flows in the Stokes layers near the interface are calculated, as well as the corresponding vortical flows arising in the bulk. It is shown that the vortices can drastically enhance the mass transfer rate of a passive scalar which is to be extracted by one liquid from the other. The corresponding Sherwood number is of the order of [[mid ]uint[mid ]a/[Dscr ]1]1/2, where [mid ]uint[mid ] is the magnitude of the interfacial streaming velocity, a is the droplet radius, and [Dscr ]1 is the diffusion coefficient in liquid 1 (inside the droplet). This means that the effective diffusion coefficient is of the order of [Dscr ]1[[mid ]uint[mid ]a/ [Dscr ]1]1/2, which is two orders of magnitude higher than [Dscr ]1. The results obtained show that such flows can be of potential interest for novel bioseparator devices.

The main focus is the Reynolds number dependence of Kolmogorov normalized low-order moments of longitudinal and transverse velocity increments. The velocity increments are obtained in a large number of flows and over a wide range (40–4250) of the Taylor microscale Reynolds number Rλ. The Rλ dependence is examined for values of the separation, r, in the dissipative range, inertial range and in excess of the integral length scale. In each range, the Kolmogorov-normalized moments of longitudinal and transverse velocity increments increase with Rλ. The scaling exponents of both longitudinal and transverse velocity increments increase with Rλ, the increase being more significant for the latter than the former. As Rλ increases, the inequality between scaling exponents of longitudinal and transverse velocity increments diminishes, reflecting a reduced influence from the large-scale anisotropy or the mean shear on inertial range scales. At sufficiently large Rλ, inertial range exponents for the second-order moment of the pressure increment follow more closely those for the fourth-order moments of transverse velocity increments than the fourth-order moments of longitudinal velocity increments. Comparison with DNS data indicates that the magnitude and Rλ dependence of the mean square pressure gradient, based on the joint-Gaussian approximation, is incorrect. The validity of this approximation improves as r increases; when r exceeds the integral length scale, the Rλ dependence of the second-order pressure structure functions is in reasonable agreement with the result originally given by Batchelor (1951).

The leading-edge receptivity to acoustic waves of two-dimensional parabolic bodies was investigated using a spatial solution of the Navier–Stokes equations in vorticity/streamfunction form in parabolic coordinates. The free stream is composed of a uniform flow with a superposed periodic velocity fluctuation of small amplitude. The method follows that of Haddad & Corke (1998) in which the solution for the basic flow and linearized perturbation flow are solved separately. We primarily investigated the effect of frequency and angle of incidence (−180° [les ] α2 [les ] 180°) of the acoustic waves on the leading-edge receptivity. The results at α2 = 0° were found to be in quantitative agreement with those of Haddad & Corke (1998), and substantiated the Strouhal number scaling based on the nose radius. The results with sound waves at angles of incidence agreed qualitatively with the analysis of Hammerton & Kerschen (1996). These included a maximum receptivity at α2 = 90°, and an asymmetric variation in the receptivity with sound incidence angle, with minima at angles which were slightly less than α2 = 0° and α2 = 180°.