Particle image velocimetry (PIV) is used to investigate the three-dimensional wakes of rigid pitching panels with a trapezoidal geometry, chosen to model idealized fish caudal fins. Experiments are performed for Strouhal numbers from 0.17 to 0.56 for two different trailing edge pitching amplitudes. A Lagrangian coherent structure (LCS) analysis is employed to investigate the formation and evolution of the panel wake. A classic reverse von Kármán vortex street pattern is observed along the mid-span of the near wake, but the vortices realign and exhibit strong interactions near the spanwise edges of the wake. At higher Strouhal numbers, the complexity of the wake increases downstream of the trailing edge as the spanwise vortices spread transversely and lose coherence as the wake splits. This wake transition is shown to correspond to a qualitative change in the LCS pattern surrounding each vortex core, and can be identified as a quantitative event that is not dependent on arbitrary threshold levels. The location of this transition is observed to depend on both the pitching amplitude and free stream velocity, but is not constant for a fixed Strouhal number. On the panel surface, the trapezoidal planform geometry is observed to create additional vortices along the swept edges that retain coherence for low Strouhal numbers or high sweep angles. These additional swept-edge structures are conjectured to add to the complex three-dimensional flow near the tips of the panel.

Convective dynamos in a rotating spherical shell feature steady zonal flows. This process is studied numerically for Prandtl numbers of 0.1 and 1, Ekman numbers in the range $E=\te{-4}$–$\te{-5}$, magnetic Prandtl number from 0.5 to 10 and Rayleigh numbers up to 100 times supercritical. The zonal flow is mainly of thermal wind origin, and minimizes the shear of the axisymmetric poloidal magnetic field lines, according to Ferraro's law of corotation. The dissipation in the interior of the fluid is mainly ohmic, while the introduction of rigid velocity boundary conditions confines viscous dissipation in the Ekman boundary layers. The root-mean-square amplitude $U_\varphi$ of the zonal flow in the spherical shell scales as $U_\varphi=(F/\Omega)^{0.5}$, F being the buoyancy flux through the shell and $\Omega$ the rotation rate. As a consequence of the corotation law, this scaling relationship is remarkably independent of the magnetic field amplitude. It does not depend on thermal, kinematic and magnetic diffusivities, owing to the large-scale and steady nature of forcing and dissipative processes. The scaling law is in agreement with the zonal-flow amplitude at the external boundary of the Earth's liquid core.

In this study we consider the separated flow of an inviscid fluid around a moving flat plate. The motion of the plate, which is initially started from rest, is prescribed and unconstrained and we set ourselves the task of fully characterizing the resulting motion in the surrounding fluid. To do this we use a boundary integral representation for the complex-conjugate velocity field $\Phi(z,t)$ and require that the force and torque on the plate be determined as part of the solution. The flow solution is assumed to consist of a bound vortex sheet coincident with the plate and two free vortex sheets that emanate from each of the plate's two sharp edges. The time evolution of these vortex sheets is then considered in general. For physical reasons, the flow solution is required to satisfy the unsteady Kutta condition, which states that $\Phi(z,t)$ must be bounded everywhere, and the rigorous imposition of this condition then yields two types of additional constraint. The first governs the rate at which circulation is shed from the plate's edges and the second ensures that the free vortex sheets are shed tangentially. In fact, all the familiar flow characteristics associated with the imposition of the steady Kutta condition are rigorously shown to have exact parallels in the unsteady case. In addition, explicit expressions for the normal force and torque on the plate are derived. An asymptotic solution to the full system of evolution equations is developed for small times $t > 0$ and a simplified version of this solution is used as an initial condition for a desingularized numerical treatment of the full problem. A fast numerical algorithm is proposed and implemented and the results of several example calculations are presented. The featured examples are limited to high effective angles of attack due to the occurrence of a specific type of event that prevents further time-integration of the evolution equations using the current numerical method. The event corresponds physically to a situation in which a Lagrangian point placed at one of the plate's edges moves onto instead of away from the edge.

The flow and density distribution produced by injecting dense fluid upwards at the bottom of a homogeneous fluid have been investigated experimentally and theoretically. Both axisymmetric and line sources have been studied using small-scale laboratory experiments in which salt water is injected into a tank of fresh water. The turbulent fountain formed in this way rises to a maximum height which can be related to the Froude number of the inflow, and then falls back and spreads out along the floor. Continuing the inflow builds up a stable stratification in a similar manner to that discussed earlier for the ‘plume filling box model’ of Baines & Turner (1969) which is complementary to the present work. The fountain flows considered here have the important new feature that the volume of the inflow is significant, so the total volume of fluid in the ‘open’ container increases with time. The evolution is determined by the rate of entrainment into the fountain from its surroundings, which is found directly by experiment. Re-entrainment of fluid into the fountain continually changes the density profile in the mixed fluid collecting at the bottom of the chamber below the level of the fountain top, and controls the rate of rise of a ‘front’ of marked fluid. The top of the fountain rises linearly in time, at a rate which, for axisymmetric fountains, has been shown both experimentally and theoretically to be close to half the rate of rise of the free surface due to the inflow. Thus at a certain time the front rises above the top of the fountain. Once the mixed fluid at the bottom of the chamber has risen above the fountain its density profile remains unchanged. The front velocity, the fountain height and the density profile have all been obtained as functions of time using a theory which is in good agreement with the experimental results for a large range of input Froude numbers. For line fountains the results are less precise owing to an instability which causes the flow to switch irregularly from a symmetrical state to one in which the downflow occurs on one side only, and with a smaller maximum height. In concluding we discuss the applications which motivated the work, particularly the development of a stratified hybrid layer in magma chambers replenished from below, and the dynamically identical, but inverted problem of heating large buildings through ducts located near the roof.

Sound transmission through straight circular ducts with a uniform inviscid mean flow and a constant acoustic lining (impedance wall) is classically described by a modal expansion. A natural extension for ducts with axially slowly varying properties (diameter and mean flow, wall impedance) is a multiple-scales solution. It is shown in the present paper that a consistent approximation of boundary condition and isentropic mean flow allows the multiple-scales problem to have an exact solution. Since the calculational complexities are no greater than for the classical straight duct model, the present solution provides an attractive alternative to a full numerical solution if diameter variation is relevant. A unique feature of the present solution is that it provides a systematic approximation to the hollow-to-annular cylinder transition problem in the turbofan engine inlet duct.

A liquid drop placed on a vibrating diaphragm will burst into a fine spray of smaller secondary droplets if it is driven at the proper frequency and amplitude. The process begins when capillary waves appear on the free surface of the drop and then grow in amplitude and complexity as the acceleration amplitude of the diaphragm is slowly increased from zero. When the acceleration of the diaphragm rises above a well-defined critical value, small secondary droplets begin to be ejected from the free-surface wave crests. Then, quite suddenly, the entire volume of the drop is ejected from the vibrating diaphragm in the form of a spray. This event is the result of an interaction between the fluid dynamical process of droplet ejection and the vibrational dynamics of the diaphragm. During droplet ejection, the effective mass of the drop–diaphragm system decreases and the resonance frequency of the system increases. If the initial forcing frequency is above the resonance frequency of the system, droplet ejection causes the system to move closer to resonance, which in turn causes more vigorous vibration and faster droplet ejection. This ultimately leads to drop bursting. In this paper, the basic phenomenon of vibration-induced drop atomization and drop bursting will be introduced, demonstrated, and characterized. Experimental results and a simple mathematical model of the process will be presented and used to explain the basic physics of the system.

The diffraction of gravity waves at a discontinuity in depth is described by a scattering matrix that relates the asymptotic, plane-wave fields (each of which may contain waves travelling towards and away from the discontinuity) on the two sides of the discontinuity. Plane-wave and variational approximations for the elements of this scattering matrix are developed. These approximate results are tested by comparison with the more accurate results obtained by Newman for an infinite step. The plane-wave approximation to the magnitude of the transmission coefficient is within 5% of Newman's result for all wavelengths, but the corresponding approximation to the reflexion coefficient is satisfactory only for rather long wavelengths. The variational approximations to the complex transmission and reflexion coefficients agree with Newman's results, within the accuracy with which his graphs can be read, for all wavelengths. The variational approximations also are used to determine the effects of trapped modes on the resonant width of a shelf that terminates at a vertical cliff.

Fluid motion driven by the combined effects of a moving wall and natura convection is examined for rectangular cavities with heightlwidth ratios of ½, 1 and 2. The Reynolds number and Prandtl number are held fixed at Re = 100 and Pr = 1; the Grashof number is varied over the range of values Gr = 0, ±104, ±106. Flow and temperature fields obtained from numerical solutions of the Navier-Stokes equations reveal a marked influence of buoyancy for the larger aspect ratios when Gr = ±106 and the dominance of buoyancy for all aspect ratios when Gr = ± 106. Results are compared with earlier work where possible and some observations are offered on the convergence of the numerical solutions.

Dispersed two-phase flows often involve interfacial activities such as chemical reaction and phase change, which couple the fluid dynamics with heat and mass transfer. As a step toward understanding such problems, we numerically simulate the sedimentation of solid bodies in a Newtonian fluid with convection heat transfer. The two-dimensional Navier–Stokes and energy equations are solved at moderate Reynolds numbers by a finite-element method, and the motion of solid particles is tracked using an arbitrary Lagrangian–Eulerian scheme. Results show that thermal convection may fundamentally change the way that particles move and interact. For a single particle settling in a channel, various Grashof-number regimes are identified, where the particle may settle straight down or migrate toward a wall or oscillate laterally. A pair of particles tend to separate if they are colder than the fluid and aggregate if they are hotter. These effects are analysed in terms of the competition between the thermal convection and the external flow relative to the particle. The mechanisms thus revealed have interesting implications for the formation of microstructures in interfacially active two-phase flows.

The random motion of nearly spherical bubbles in the turbulent flow behind a grid is studied experimentally. In quiescent water these bubbles rise at high Reynolds number. The turbulence is generated by an active grid of the design of Makita (1991), and can have turbulence Reynolds number Rλ of up to 200. Minor changes in the geometry of the grid and in its mode of operation improves the isotropy of the turbulence, compared with that reported by Makita (1991) and Mydlarski & Warhaft (1996). The trajectory of each bubble is measured with high spatial and temporal resolution with a specially developed technique that makes use of a position-sensitive detector. Bubble statistics such as the mean rise velocity and the root-mean-square velocity fluctuations are obtained by ensemble averaging over many identical bubbles. The resulting bubble mean rise velocity is significantly reduced (up to 35%) compared with the quiescent conditions. The vertical bubble velocity fluctuations are found to be non-Gaussian, whereas the horizontal displacements are Gaussian for all times. The diffusivity of bubbles is considerably less than that of fluid particles. These findings are qualitatively consistent with results obtained through theoretical analysis and numerical simulations by Spelt & Biesheuvel (1997).

A numerical investigation of vortex breakdown has been undertaken in an attempt to understand its properties, and the mechanisms responsible for it. Solutions of the full steady axisymmetric Navier—Stokes equations for breakdown in an unconfined viscous vortex have been obtained for core Reynolds numbers up to 200, for a two-parameter family of assumed upstream velocity distributions. Diffusion and convection of vorticity away from the vortex core, and the strong coupling between the circumferential and axial velocity fields in highly-swirling flows, are shown to lead to stagnation and reversal of the axial flow near the axis. The various theories of vortex breakdown are considered in light of the present numerical solutions.

By introducing small controlled perturbations prior to the onset of motion, two-dimensional convection cells of arbitrary width-depth ratios are produced in a horizontal fluid layer heated from below. The conditions employed correspond to the finite-amplitude Rayleigh stability problem for a constant viscosity, large Prandtl number, Boussinesq fluid with rigid, conducting boundaries. It was found that two-dimensional cells with width-depth ratios close to unity are stable at all Rayleigh numbers investigated (Rc [Lt ] R [Lt ] 2·5Rc). Cells whose widthdepth ratios are moderately too large or too small tend to undergo size adjustments toward a preferred value of about 1·1. If the width-depth ratios are much too large or too small, they tend to develop three-dimensional instabilities in the form of cell boundary distortions and transverse secondary cells, respectively. Eventually the flow settles into a new family of rolls with a more preferred widthdepth ratio. It is suggested that these observations may have implications on nonlinear interchange instability problems and geophysical flows.

The dynamics of the evolution of turbulence statistics depend on the structure of the turbulence. For example, wavenumber anisotropy in homogeneous turbulence is known to affect both the interaction between large and small scales (Kida & Hunt 1989), and the non-local effects of the pressure–strain-rate correlation in the one-point Reynolds stress equations (Reynolds 1989; Cambon et al. 1992). Good quantitative measures of turbulence structure are easy to construct using two-point or spectral data, but one-point measures are needed for the Reynolds-averaged modelling of engineering flows. Here we introduce a systematic framework for exploring the role of turbulence structure in the evolution of one-point turbulence statistics. Five one-point statistical measures of the energy-containing turbulence structure are introduced and used with direct numerical simulations to analyse the role of turbulence structure in several cases of homogeneous and inhomogeneous turbulence undergoing diverse modes of mean deformation. The one-point structure tensors are found to be useful descriptors of turbulence structure, and lead to a deeper understanding of some rather surprising observations from DNS and experiments.

Following a novel realization of low-Reynolds-number swimming (Dreyfus et al., Nature, vol. 436, 2005, p. 862), in which self-assembled filaments of paramagnetic micron-sized beads are tethered to red blood cells and then induced to swim under crossed uniform and oscillating magnetic fields, the dynamics of magnetoelastic filaments is studied. The filament is modelled as a slender elastica driven by a magnetic body torque. The model is applied to experiments of Goubault et al. (Phys. Rev. Lett., vol. 91, 2003, art. 260802) to predict the lifetimes of metastable static filament conformations that are known to form under uniform fields. A second experimental swimming scenario, complementary to that of Dreyfus et al. (2005), is described: filaments are capable of swimming even if not tethered to red blood cells. Yet, if both ends of the filament are left free and the material and magnetic parameters are uniform along its length then application of an oscillating transverse field can only generate homogeneous torques, and net translation is prohibited by symmetry. It is shown that fore–aft symmetry is broken when variation of the bending stiffness along the filament is accounted for by including elastic defects, which produces results consistent with the swimming phenomenology.

The stability of almost fully developed viscous flow in a rotating pipe is considered. In cylindrical polar co-ordinates (r, ø, z) this flow has the velocity components \[ \{W_0o(1),\quad\Omega r[1+o(\epsilon)],\quad W_0[1-r^2/r^2_0+o(1)]\},_{+}^{+} \] where ε = Wo/2Ωr0 and is bounded externally by the rigid cylinder r = r0, which rotates about its axis with angular velocity Ω. In the limit of small ε, the disturbance equations can be solved in terms of Bessel functions and it is shown that, in that limit, the flow is unstable for Reynolds numbers R = Wor0/v greater than Rc [asymp ] 82[sdot ]9. The unstable disturbances take the form of growing spiral waves, which are stationary relative to the rotating cylinder and the critical disturbance at R = Rc has azimuthal wave-number 1 and axial wavelength 2πr0/ε. Furthermore, it is shown that the most rapidly growing disturbance for R > Rc has an azimuthal wave-number which increases with R. Some of the problems involved in testing the results by experiment are discussed and a possible application to the theory of vortex breakdown is mentioned. In an appendix this instability is shown to be an example of inertial instability.

Stratification patterns are formed when a bidisperse mixture of large rough grains and smaller more mobile particles is poured between parallel plates to form a heap. At low flow rates discrete avalanches flow down the free surface and are brought to rest by the propagation of shock waves. Experiments performed in this paper show that the larger particles are segregated to the top of the avalanche, where the velocity is greatest, and are transported to the flow front. Here the particles are overrun but may rise to the free surface again by size segregation to create a recirculating coarse-grained front. Once the front is established composite images show that there is a steady regime in which any additional large grains that reach the front are deposited. This flow is therefore analogous to finger formation in geophysical mass flows, where the larger less mobile particles are shouldered aside to spontaneously form static lateral levees rather than being removed by basal deposition in two dimensions. At the heart of all these phenomena is a dynamic feedback between the bulk flow and the evolving particle-size distribution within the avalanche. A fully coupled theory for such segregation–mobility feedback effects is beyond the scope of this paper. However, it is shown how to derive a simplified uncoupled travelling-wave solution for the avalanche motion and reconstruct the bulk two-dimensional flow field using assumed velocity profiles through the avalanche depth. This allows a simple hyperbolic segregation theory to be used to construct exact solutions for the particle concentration and for the recirculation within the bulk flow. Depending on the material composition and the strength of the segregation and deposition, there are three types of solution. The coarse-particle front grows in length if more large particles arrive than can be deposited. If there are fewer large grains and if the segregation is strong enough, a breaking size-segregation wave forms at a unique position behind the front. It consists of two expansion fans, two shocks and a central ‘eye’ of constant concentration that are arranged in a ‘lens-like’ structure. Coarse grains just behind the front are recirculated, while those reaching the head are overrun and deposited. Upstream of the wave, the size distribution resembles a small-particle ‘sandwich’ with a raft of rapidly flowing large particles on top and a coarse deposited layer at the bottom, consistent with the experimental observations made here. If the segregation is weak, the central eye degenerates, and all the large particles are deposited without recirculation.

Conditionally sampled hot-wire and ‘cold-wire’ (resistance-thermometer) measure- ments confirm the general flow picture advanced by Falco (1974, 1977, 1980; see also Smith & Abbott 1978) and by Head & Bandyopadhyay (1981; see also Smith & Abbott) on the basis of smoke observations and more limited hot-wire measurements. The probability density function of turbulent-zone lengths in the intermittent region varies rapidly with Reynolds number, supporting the above authors’ finding that the hairpin-vortex ‘typical eddies’ in the viscous superlayer scale on the viscous length ν/uτ, rather than on boundary-layer thickness. However the average turbulent-zone length, deduced as an integral moment of the probability distribution, tends to a constant fraction of the boundary-layer thickness above a momentum-thickness Reynolds number of 5000, which strongly suggests that at high Reynolds numbers the overall shape of the turbulent irrotational interface is controlled by the classical ‘large eddies’ and not by the viscosity-dependent small eddies. The intermittency profile is practically independent of Reynolds number. The second-order structural parameter $\overline{u^2}/\overline{v^2}$ increases strongly with increasing Reynolds number but the triple-product parameters, with the exception of the u-component skewness, vary only slowly with Reynolds number. This behaviour of the intermittency and velocity statistics is most simply explained by supposing that the lengthscale of the large eddies is nearly independent of Reynolds number while their intensity is somewhat lower at low Reynolds number. ‘Typical eddies’ evidently contribute to the Reynolds stresses at low Reynolds number, but it is probable that the large eddies carry most of the triple products at any Reynolds number. Our results confirm the usual finding that the mixing length and dissipation length parameter increase, while the wake component of the velocity profile decreases, as Reynolds number decreases.

A series of experiments has been conducted in a low-speed wind tunnel in which measurements were performed in a two-dimensional turbulent shear layer experiencing the mixing of both a passive and a chemically reacting species. The low-temperature air in the jet's primary flow was seeded with dilute concentrations of N2O4 so that the dissociation reaction N2 + N2O4 [harr ] 2NO2 + N2 occurred in a near-equilibrium manner within the mixing layer owing to the turbulent mixing properties and the imposed temperature gradient. Mean and fluctuating values of velocity, temperature and NO2 concentration were measured up to axial distances of 25 in. for jet velocities of 23 and 50ft/s (Rex [les ] 7 × 105) and for three primary temperatures (252, 273 and 305°K). Velocity and temperature measurements were performed with hot-wire probes, whereas a fibre optics light sensor probe was used to measure NO2 concentrations. Local correlations between species and other fluid properties were obtained by positioning a hot-wire sensor within the light gap of the fibre optics probe and simultaneously recording output signals from both probes. A relatively complete set of turbulent statistics was measured for the non-reacting shear layer, including such results as temperature/species correlations, probability densities, filtered and unfiltered moving-frame velocities, skewness and flatness factors, spectra, velocity and temperature integral scales, intermittency factors for velocity, temperature and passive species, and conventional intensities. Some typical results from the investigation are as follows: the turbulent Schmidt and Lewis numbers were 0·5 and 1·0 respectively; the correlation between passive NO2 concentration and temperature was approximately 0·95; dramatic changes consistent with equilibrium chemistry occurred in NO2 concentration profiles with chemical reaction; velocity, temperature and concentration spectra were comparable over a 2½-decade range in wavenumber (k−2); spectra, probability densities, time-trace data and smoke-seeded shear-layer photographs indicate that, for axial locations x = x0 [ges ] 18·5 in. and for speeds u1 [ges ] 23ft/s, undisturbed edge fluid rarely penetrates completely across the mixing region. Although not specifically addressed during the current study, measured results herein suggest that the turbulent motion for the present shear layer is characterized more by random and/or three-dimensionality effects than by large-scale two-dimensional coherent structures, as has been observed recently in other shear-layer investigations.

The absolute/convective instability properties of the Batchelor vortex are determined by direct numerical simulation of the linear impulse response. A novel decomposition procedure is applied to the computed wavepacket in order to retrieve the complex wavenumber and frequency prevailing along each spatio-temporal ray. In particular, the absolute wavenumber and frequency observed in the laboratory frame are determined as a function of swirl parameter and external flow. The introduction of a moderate amount of swirl is found to strongly promote absolute instability. In the case of wakes, the transitional helical mode that first undergoes a switch-over to absolute instability is found to be m=−1 without requiring any external counterflow. In the case of jets, the transitional helical mode is very sensitive to swirl and varies in the range −5[les ]m[les ]−1. Only a slight amount of external counterflow (1.5% of centreline velocity) is then necessary to trigger absolute instability. The results of this numerical procedure are in good qualitative and quantitative agreement with those obtained by direct application of the Briggs–Bers criterion to the inviscid dispersion relation (Olendraru et al. 1996). Implications for the dynamics of swirling jets and wakes are discussed.

A finite-difference procedure is employed to predict the development of turbulent flow in curved pipes. The turbulence model used involves the solution of two differential equations, one for the kinetic energy of the turbulence and the other for its dissipation rate. The predicted total-velocity contours for the developing flow in a 180° bend are compared with the experimental data. Predictions of fully developed velocity profiles for long helically wound pipes are also presented and compared with experimental measurements.