When an electric field is applied to a drop suspended in another liquid the drop deforms. The relation between the applied field and the mode and magnitude of the deformation have been studied extensively. Nevertheless, Torza, Cox & Mason (1971) found that quantitative agreement between the leaky dielectric theory (Taylor 1966) and experiment is quite poor. Here we describe results from a new series of experiments. Drops suspended in weakly conducting liquids were deformed into spheroids with both steady and oscillatory fields. Drop deformation, interfacial tension, and the electrical properties of the fluids were measured for each system to provide a definitive test of the theory. The agreement between the leaky dielectric model and our results for drop deformations in steady fields is much improved over previous results, although discrepancies remain for some systems. Drop deformations in oscillatory fields consist of steady and oscillatory parts because of the quadratic dependence on the field strength. Measurements of the steady part at 60 Hz, where the oscillatory deformation is negligible, are in excellent agreement with the theory. The effects of frequency on the steady deformation were studied by measuring oblate deformations at a series of frequencies and field strengths; the agreement with theory is good. Finally, the time-dependent total deformation was measured under conditions where both parts of the deformation are commensurate. Good agreement was found between the measured and predicted maximum and minimum deformations. Nevertheless, only a small range of fluid properties could be studied owing to the need to avoid droplet sedimentation.

An oscillatory flow over a cohesionless bottom can produce regular three-dimensional bedforms known as brick-pattern ripples characterized by crests perpendicular to the direction of fluid oscillations joined by equally spaced bridges shifted by half a wavelength between adjacent sequences (a photo of brick-pattern ripples is shown in Sleath 1984, p. 141). In the present paper brick-pattern ripple formation is explained on the basis of a weakly nonlinear stability analysis of a flat cohesionless bottom subject to an oscillatory flow in which three-dimensional perturbations are considered. It is shown that brick-pattern ripples are generated by the simultaneous growth of two-dimensional and three-dimensional perturbations which interact with each other, according to a mechanism similar to that described by Craik (1971) in a different context, forming a resonant triad. A comparison between the present theoretical finding and experimental data by Sleath & Ellis (1978), concerning the region of existence of brick-pattern ripples in the parameter space and their geometrical characteristics, supports the validity of the present approach.

The stability of the flow resulting from the oscillations of a sphere in a viscous fluid is investigated. The calculation for the transverse oscillations of the sphere is performed in a linear regime and the result in the weakly nonlinear regime is described; the stability in the case of torsional oscillations is considered in the linear regime, where we take torsional oscillations to mean oscillations about a fixed axis through the centre of the sphere. In both cases we assume that the frequency of the oscillations is large, so that the unsteady boundary layer that results is thin. In the transverse case, the linear stability problem depends only on the radial variable and time. Employing Floquet theory we may reduce the system to a coupled infinite system of ordinary differential equations, with homogeneous boundary conditions, the eigenvalues of this system being found numerically. In the torsional case, the linear stability problem again depends only on the radial variable and time, although the angular variation is retained in a parametric form and is determined at higher order. A WKBJ perturbation solution is constructed and the evolution of the amplitude of the vortex is found. The solution is determined by finding a saddle point in the complex plane of the angular coordinate, and thus the critical Taylor number is derived.

In a non-vertical borehole light particles tend to rise towards the upper side of the borehole. The resulting non-uniform density distribution tends to induce an upwards contribution to the longitudinal flow along that upper side of the flow, with a compensating downflow elsewhere. On average the particles experience an extra upflow proportional to the cross-sectionally averaged concentration of particles. Mathematically this concentration-related change of speed corresponds to the nonlinearity of the Burgers equation. Such is the strength of the buoyancy effect that in realistic flow conditions the Burgers nonlinearity can be significant for particle volume fractions of only one part per thousand.

Strong forcing was used to produce vortex pairing in a submerged axisymmetric water jet. Phase-averaged hot-wire measurements were combined with phase-averaged flow visualization to identify the relevant nonlinear interactions. The leading resonant interaction was not a subharmonic resonance. Instead it was a triad resonance involving the subharmonic, the fundamental and the 3/2 harmonic. The profound influence of higher harmonics on the amplification of the fundamental and subharmonic was demonstrated in a systematic way by successive truncation of the Fourier series representation of the excitation waveform.

This work is aimed at understanding mechanisms which govern the growth of secondary three-dimensional modes of a particular type which feed from a resonant energy exchange with the primary Kármán instability in two-dimensional wakes. Our approach was to introduce controlled time-periodic three-dimensional (oblique) wave pairs of equal but opposite sign, simultaneously with a two-dimensional wave. The waves were introduced by an array of v-component-producing elements on the top and bottom surfaces of the body. These were formed by metallized electrodes which were vapour deposited onto a piezoelectrically active polymer wrapped around the surface. The amplitudes, streamwise and spanwise wavenumbers, and initial phase difference are all individually controllable. The initial work focused on a fundamental/subharmonic interaction, and the dependence on spanwise wave-number. The results include mode eigenfunction modulus and phase distributions in space, and stream functions for the phase-reconstructed flow field. Analysis of these shows that such a resonance mechanism exists and its features can account for characteristic changes associated with the growth of three-dimensional structures in the wake of two-dimensional bodies.

Measurements of the dissipation of short water waves in a wave tank are analysed and described. Monochromatic waves with lengths between 6 and 10 cm generated by an axisymmetric wavemaker propagated through a turbulent flow field generated by a submerged vertically oscillating grid below the wavemaker. The horizontal turbulence velocity was measured with a hot-film anemometer with the grid oscillating, but the wavemaker off. With the wavemaker operating, wave amplitude vs. distance from the wavemaker was measured with and without operation of the turbulence generator. Wave dissipation due to turbulence was measured and quantified. Much of the wave energy transfer to turbulence may not occur in the normal energy-containing depth of the waves. Rather, most of it may first be convected downward and out of the wave zone by the vertical turbulent velocities. The experimental data are consistent with this possibility.

We first recall the concepts of spectral eddy viscosity and diffusivity, derived from the two-point closures of turbulence, in the framework of large-eddy simulations in Fourier space. The case of a spectrum which does not decrease as $k^{-\frac{5}{3}}$ at the cutoff is studied. Then, a spectral large-eddy simulation of decaying isotropic turbulence convecting a passive temperature is performed, at a resolution of 1283 collocation points. It is shown that the temperature spectrum tends to follow in the energetic scales a k−1 range, followed by a $k^{-\frac{5}{3}}$ inertial–convective range at higher wavenumbers. This is in agreement with previous independent calculations (Lesieur & Rogallo 1989). When self-similar spectra have developed, the temperature variance and kinetic energy decay respectively like t−1.37 and t−1.85, with identical initial spectra peaking at ki = 20 and ∝ k8 for k → 0. In the k−1 range, the temperature spectrum is found to collapse according to the law ET(k, t) = 0.1η(〈u2〉/ε) k−1, where ε and η are the kinetic energy and temperature variance dissipation rates. The spectral eddy viscosity and diffusivity are recalculated explicitly from the large-eddy simulation: the anomalous ∝ ln k behaviour of the eddy diffusivity in the eddy-viscosity plateau is shown to be associated with the large-scale intermittency of the passive temperature: the p.d.f. of the velocity component u is Gaussian (∼ exp − X2), while the scalar T, the velocity derivatives ∂u/∂x and ∂u/∂z, and the temperature derivative ∂T/∂z are all close to exponential exp - |X| at high |X|. The pressure distribution is exponential at low pressure and Gaussian at high.

For stably stratified Boussinesq turbulence, the coupling between the temperature and the velocity fields leads to the disappearance of the ‘anomalous’ temperature behaviour (k−1 range, logarithmic eddy diffusivity and exponential probability density function for T). These are the highest-resolution calculations ever performed for this problem. We also split the eddy viscous coefficients into a vortex and a wave component. In both cases (unstratified and stratified), comparisons with direct numerical simulations are performed.

Finally we propose a generalization of the spectral eddy viscosity to highly intermittent situations in physical space: in this structure-function model, the spectral eddy viscosity is based upon a kinetic energy spectrum local in space. The latter is calculated with the aid of a local second-order velocity structure function. This structure function model is compared with other models, including Smagorinsky's, for isotropic decaying turbulence, and with high-resolution direct simulations. It is shown that low-pressure regions mark coherent structures of high vorticity. The pressure spectra are shown to follow Batchelor's quasi-normal law: $\alpha C^2_{\rm k}\epsilon^{\frac{4}{3}}k^{-\frac{7}{3}}$ (Ck is Kolmogorov's constant), with α ≈ 1.32.

The thermal boundary layer on the wall of a side-heated cavity at early time is known to exhibit a complex travelling wave during growth to steady state and a similar feature is observed on isolated heated semi-infinite plates. Direct numerical solutions of the Navier–Stokes equations together with a linearized stability analysis are used to study the character of the flow at early time in detail. It is demonstrated that the cavity flow is essentially identical to the plate flow, and that for early time the flow is one-dimensional. Using the stability results it has been possible to accurately describe the form of the observed instability, as well as to reconcile a previously unexplained discrepancy in the speed of development of the flow.

An experimental study has been conducted to investigate the three-dimensional structure of a plane, two-stream mixing layer through direct measurements. A secondary streamwise vortex structure has been shown to ride among the primary spanwise vortices in past flow visualization investigations. The main objective of the present study was to establish quantitatively the presence and role of the streamwise vortex structure in the development of a plane turbulent mixing layer at relatively high Reynolds numbers (Reδ ∼ 2.9 × 104). A two-stream mixing layer with a velocity ratio, U2/U1 = 0.6 was generated with the initial boundary layers laminar and nominally two-dimensional. Mean flow and turbulence measurements were made on fine cross-plane grids across the mixing layer at several streamwise locations with a single rotatable cross-wire probe. The results indicate that the instability, leading to the formation of streamwise vortices, is initially amplified just downstream of the first spanwise roll-up. The streamwise vortices first appear in clusters containing vorticity of both signs. Further downstream, the vortices re-align to form counter-rotating pairs, although there is a relatively large variation in the scale and strengths of the individual vortices. The streamwise vortex spacing increases in a step-wise fashion, at least partially through the amalgamation of like-sign vortices. For the flow conditions investigated, the wavelength associated with the streamwise vortices scales with the mixing-layer vorticity thickness, while their mean strength decays as approximately 1/X1.5. In the near field, the streamwise vortices grossly distort the mean velocity and turbulence distributions within the mixing layer. In particular, the streamwise vorticity is found to be strongly correlated in position, strength and scale with the secondary shear stress ($\overline{u^{\prime}w^{\prime}}$). The secondary shear stress data suggest that the streamwise structures persist through to what would normally be considered the self-similar region, although they are very weak by this point and the mixing layer otherwise appears to be two-dimensional.

By using asymptotic analysis, an eigensolution technique has been developed for predicting the flow of gas contained in a pie-shaped cylinder of finite length rotating rapidly about its vertex. This problem has application to a conventional cylindrical gas centrifuge with radial walls. Three different types of boundary layers exist in the flow: Ekman layers on the top and bottom, buoyancy layers on the radial walls, and a cylindrical ‘pancake’ layer on the outer wall of the cylinder. A single sixth-order partial differential equation is obtained for the axial velocity in the cylindrical layer, and the other layers provide matching conditions. The problem is formulated for no-slip and prescribed temperature conditions on the solid surfaces and for adiabatic no shear stress with zero pressure at the inner free surface. Eigenvalues are computed for this problem and compared with those for the open cylinder, and solutions are presented for flows induced by mass throughput and by differential temperature conditions.

Two-dimensional thermosolutal convection is perhaps the simplest example of an idealized fluid dynamical system that displays a rich variety of dynamical behaviour which is amenable to investigation by a combination of analytical and numerical techniques. The transition to chaos found in numerical experiments can be related to behaviour near a multiple bifurcation of codimension three. The resulting third-order normal form equations provide a rational approximation to the governing partial differential equations and thereby confirm that temporal chaos is present in thermosolutal convection. The complex dynamics is associated with a heteroclinic orbit in phase space linking a pair of saddle-foci with eigenvalues satisfying Shil'nikov's criterion. The same bifurcation structure occurs in a truncated fifth-order model and numerical experiments confirm that similar behaviour extends to a significant region of parameter space.

In this paper, we study the vibratory instability of a cellular flame, propagating downwards in a tube, which results from the coupling between the longitudinal acoustic modes of the tube and the modification of the cellular flame structure by the acceleration of the acoustic field. We assume that the wrinkling of the flame is of small amplitude a0, which is the case when the flame burning velocity is just above the critical velocity characterizing the Darrieus–Landau instability threshold. We demonstrate that, in this case, the growth rate of the corresponding thermoacoustic instability, non-dimensionalized with the acoustic frequency, is proportional to (kca0)2, where kc is the critical wavenumber of the cellular instability. If one extends the result up to amplitudes of the same order as the wavelength, then one obtains a relative growth rate of order unity which is much larger than the one obtained from the study of the vibratory instability of the planar flame. As is observed in experiments, the theory predicts that the primary sound is generated when the amplitude of the cells is sufficiently large that the fundamental tone becomes unstable first and that the vibratory instability for the fundamental tone occurs in the lower half of the tube. This suggests that the coupling between cellular flame and acoustic field studied here is the mechanism for primary sound generation.

A series of vertical density profiles was taken in a stratified tank in which a standing internal wave was forced to amplitudes at which it became unstable and, as a result of the instability, localized patches of mixing were generated within the fluid. By resorting the density profiles the available potential energy in the patches could be calculated and, by comparison with the average buoyancy flux in the tank (determined from density profiles taken before and after each mixing run), an average efficiency of utilization of available potential energy, ηAPE, was calculated. Along with previous measurements of the flux Richardson number, Rif, ηAPE was used to show that the mean value of the overturn Froude number, Frf, in the patches was 1, thus implying a balance between the rate of release of available potential energy and dissipation in the mixing patches. On the other hand, the patch-averaged overturn Reynolds number, Ret, was so low that, based on the results of previous laboratory experiments on stratified mixing in the wake of a biplanar grid, most of the patches cannot have been actively mixing at the time of sampling.

It is shown that the temperature and conductivity gradient spectra in different patches can be interpreted in a way consistent with the visualization of mixing events, that is, showing an evolution from the generation of an initially unstable density distribution, through the formation of coherent structures as the fluid restratifies and finally the degeneration of these structures into the finer scales of motion at which mixing occurs.

We analyse the nonlinear flow of a Newtonian fluid in an elastic tube when subjected to an oscillatory pressure gradient with motivation from the problem of blood flow in arteries. Two parameters: the unsteadiness, α = R0(ω/ν)½ and the diameter variation, ε = (Rmax − R0)/R0, are important in characterizing the flow problem. The diameter variation (ε) is taken to be small so that the perturbation method is valid, and asymptotic solutions for two limiting cases of the steady-streaming Reynolds number, Rs = (αε)2 (either small or large), are derived.

The results indicate that nonlinear convective acceleration induces finite mean pressure gradient and mean wall shear rate even when no mean flow occurs. The magnitude of this effect depends on the amplitude of the diameter variation and the flow rate waveforms and the phase angle difference between them, which can be related to the impedance (pressure/flow) phase angle. Changes in the impedance phase angle, which is indicative of the degree of wave reflection, can change the direction of the induced mean flow. It is also shown that the induced mean wall shear rate is proportional to α when α is large. In addition, it is observed that the steady flow structure in the core can be influenced by wave reflection. The streamlines in the core are always parallel to the tube wall when there is no reflection. However, with total reflection, the induced mean flow recirculates between the nodes and points of maximum amplitude in a closed streamline pattern. Implications of the steady-streaming phenomena for physiological flow applications are discussed in a concluding section.

Fast dynamo action in a chaotic time-periodic flow is investigated. Chaotic motion is created by perturbing a spatially periodic array of helical cells similar to Roberts’ cells, leading to an identifiable stretch–fold–shear fast dynamo mechanism. Using the stochastic Wiener bundle method to treat diffusion exactly, numerical results are presented suggesting fast dynamo action. A new numerical method for modelling the role of small magnetic diffusivity is introduced and results are compared with those calculated using the Wiener bundle method. Implications for the role of diffusion in the fast dynamo process are investigated. Finally the relation of the new method to a previously used ‘flux growth’ method are discussed.

This paper describes an experimental study of the influence of a low-frequency alternating magnetic field on a liquid-metal pool with a free surface. A 200 mm cylinder containing mercury is located in a solenoidal coil supplied with a single phase a.c. current of frequency 2–20 Hz. It is shown that, in that frequency range, the motion may be split into two parts: (i) a bulk motion driven by the mean Lorentz forces; (ii) a surface wave motion driven by the alternating part of the Lorentz forces.

The turbulent bulk flow is quite similar to those observed in previous electromagnetic stirring experiments at higher frequency. The peculiar feature, observed here, is the rapid decay of the mean characteristic velocity. That phenomenon seems to be related to the presence of fluctuating velocities forced by the alternating electromagnetic force.

The alternating part of the Lorentz forces is globally responsible for a surface motion whose pattern and amplitude depend on the applied electric current I and its frequency f. The (I, f)-parameter space may be split into four regions corresponding to four regimes, namely (i) concentric harmonic standing waves driven by the Lorentz forces, (ii) harmonic azimuthal waves, (iii) strong-amplitude subharmonic azimuthal waves, (iv) chaotic free-surface motion. The wave motion becomes negligible when the frequency is greater than 10 Hz.

This paper analyses the effect of an alternating magnetic field of low frequency ω on a cylindrical tank of liquid metal. Previous work with higher-frequency fields has focused attention on the mean recirculating motion, but in the low-frequency limit periodic motion and surface waves become important. We show that a system of forced standing axisymmetric waves of frequency 2ω is established, and that the growth of non-axisymmetric modes is governed by a coupled system of Mathieu-type equations. The stability regions associated with this system are discussed and it is shown that the most easily excited transition to a non-axisymmetric mode is subharmonic, with frequency ω. Comparison with experiment shows that the theory gives qualitatively correct predictions.

The diffusion-governed melting that occurs when a binary melt is placed in contact with a pure solid is described. It is shown that if the melt superheat is much greater than the solid supercooling, the melt composition at the interface equals that of the solid and so the solid will melt at a rate determined by the thermal diffusivity. However, as the liquid superheat decreases, chemical disequilibrium may lower the interface temperature and so the melt composition at the interface increases above that of the solid, according to the liquidus relation. In this case the solid will dissolve into the liquid at a rate determined by the solutal diffusivity. These diffusion-governed solutions are used to infer the different modes of convection which may rise when the interface between the solid and the melt is horizontal.

The theory is generalized to investigate the diffusion-governed melting of a binary solid solution placed in contact with a binary melt. If the melt superheat is sufficient then the rate of phase change is again determined by the thermal diffusivity. In this case, owing to the very small solutal diffusivity in both the solid and the liquid, the melt composition at the interface is nearly equal to that of the solid. This corresponds to the melting regime. As the liquid superheat decreases, the rate of phase change decreases to values determined by the solutal diffusivity in the liquid, and the melt composition at the interface evolves towards that of the far-field liquid. This corresponds to the dissolving regime. As the melt superheat decreases further, with the solid still changing phase into liquid, then the melt composition at the interface remains approximately equal to that of the far-field melt. In each case, a compositional boundary layer develops in the solid, just ahead of the interface, in order to restore the solid at the interface to thermodynamic equilibrium. These different phase change regimes may arise if the composition of the solid is either higher or lower than that of the liquid.

Effects of incomplete surface accommodation in rarefied gas flows have been studied using the direct simulation Monte Carlo (DSMC) method in conjunction with the Cercignani–Lampis gas–surface interaction model. Two different flows have been studied, both of which have previously been simulated in DSMC calculations for the case of complete surface accommodation. These are (a) the flow over a sharp, slender circular cone at Mach 5.1 and (b) the flow over a flat-faced circular cylinder at Mach 25. In each case, the gas simulated was nitrogen. It was found that in case (a) the accommodation coefficient of the kinetic energy due to the tangential component of velocity has the greatest influence, whereas in case (b) that of the normal component is also important, having a drastic effect on the rarefied flow field.