A multiple-scale analysis of the Kuramoto-Sivashinsky one-dimensional model of a flame front with 2π-periodic boundary conditions is presented. For arbitrary large values of the number M of linearly unstable modes there exist stable steady solutions of period 2π/N where N = O(M). These ‘cellular solutions’ exhibit elastic behaviour under perturbations of wavelength much larger than 2π/N. The results are illustrated by numerical experiments. Elasticity has its origin in the translation and Galilean invariances. Similar invariance properties are likely to be at the root of the viscoelastic behaviour of turbulent flows conjectured by many authors.

Results are given for experiments on water-lubricated pipelining of 6.01 P cylinder oil in a vertical apparatus in up- and downflow in regimes of modest flow rates, less than 3 ft/s. Measured values of the flow rates, holdup ratios, pressure gradients and flow types are presented and compared with theoretical predictions based on ideal laminar flow and on the predictions of the linear theory of stability. New flow types, not achieved in horizontal flows, are observed: bamboo waves in upflow and corkscrew waves in downflow. Nearly perfect core–annular flows are observed in downflows and these are nearly optimally efficient with values close to the ideal. The holdup ratio in upflow and fast downflow is a constant independent of the value and the ratio of values of the flow rates of oil and water. A vanishing holdup ratio can be achieved by fluidizing a long lubricated column of oil in the downflow of water. The results of experiments are compared with computations from ideal theory for perfect core–annular flow and from the linear theory of stability. Satisfactory agreements are achieved for the celerity and diagnosis of flow type. The wave is shown to be nearly stationary, convected with the oil core in this oil and all oils of relatively high viscosity. These results are robust with respect to moderate changes in the values of viscosity and surface tension. The computed wavelengths are somewhat smaller than the average length of the bamboo waves which are observed. This is explained by stretching effects of buoyancy and lubrication forces induced by the wave. Other points of agreement and disagreement are reviewed.

We study the generalized Bragg scattering of surface waves over a wavy bottom. We consider the problem in the general context of nonlinear wave–wave interactions, and write down and provide geometric constructions for the Bragg resonance conditions for second-order triad (class I) and third-order quartet (class II and class III) wave– bottom interactions. Class I resonance involving one bottom and two surface wave components is classical. Class II resonance manifests bottom nonlinearity (it involves two bottom and two surface wave components), and has been studied in the laboratory. Class III Bragg resonance is new and is a result of free-surface nonlinearity involving resonant interaction among one bottom and three surface wave components. The amplitude of the resonant wave is quadratic in the surface wave slope and linear in the bottom steepness, and, unlike the former two cases, the resonant wave may be either reflected or transmitted (relative to the incident waves) depending on the wave–bottom geometry. To predict the initial spatial/temporal growth of the Bragg resonant wave for these resonances, we also provide the regular perturbation solution up to third order. To confirm these predictions and to obtain an efficient computational tool for general wave–bottom problems with resonant interactions, we extend and develop a powerful high-order spectral method originally developed for nonlinear wave–wave and wave–body interactions. The efficacy of the method is illustrated in high-order Bragg resonance computations in two and three dimensions. These results compare well with existing experiments and perturbation theory for the known class I and class II Bragg resonance cases, and obtain and elucidate the new class III resonance. It is shown that under realistic conditions with moderate to small surface and bottom steepnesses, the amplitudes of third-order class II and class III Bragg resonant waves can be comparable in magnitude to those resulting from class I interactions and appreciable relative to the incident wave.

Sir Charles Darwin has advocated a study of the ‘drift’ of material surfaces in the classically investigated irrotational flows past bodies. This suggestion is followed up and given further support in the present paper. In particular, it is shown how secondary flows can be evaluated by use of the ‘drift function’ t for the primary flow. This is a function (§1) such that material surfaces initially at right angles to the stream drift into shapes expressible by equations t=constant.

The analysis leads to a simple expression (§4) for the secondary velocity field in the flow past an infinite cylinder of any cross-section, with the upstream velocity normal to its axis and increasing linearly with distance along the axis-a problem in which only the secondary vorticity field was previously known. The drift past a sphere is computed and illustrated (§6), and the secondary vorticity field in shear flow past a sphere is tabulated (§7). There is also a detailed study (§3) of the asymptotic form of the secondary velocity field in flow past any body, based on a result of Darwin concerning ‘hydrodynamic mass’.

Spectral analysis and flow visualization are presented for various velocity ratios and Reynolds numbers of a jet issuing perpendicularly from a developing pipe flow into a crossflow. The results are complete with conditional averages of various turbulent quantities for one jet-to-cross-flow velocity ratio R of 0.5. A unique conditional-sampling technique separated the contributions from the turbulent jet flow, the irrotational jet flow, the turbulent crossflow and the irrotational crossflow by using two conditioning functions simultaneously. The intermittency factor profiles indicate that irrotational cross-flow intrudes into the pipe but does not contribute to the average turbulent quantities, while the jet-pipe irrotational flow contributes significantly to them in the region above the exit where the interaction between the boundary-layer eddies and those of the pipe starts to take place. Further downstream, the contributions of the oncoming boundary-layer eddies to the statistical averages reduce significantly. The downstream development depends mainly on the average relative eddy sizes of the interacting turbulent fields.

The self-similar evolution to turbulence of a multi-mode Rayleigh–Taylor mix at small density differences ($A_{t} \sim 7.5 \times 10^{ - 4}$), is investigated through particle image velocimetry (PIV), and high-resolution thermocouple measurements. The density difference has been achieved through a temperature difference in the fluid. Cold fluid enters above the hot in a closed channel to form an unstable interface. This buoyancy-driven mixing experiment allows for long data collection times, short transients, and is statistically steady. First-, second-, and third-order statistics with spectra of velocity and temperature fields are presented. Analysis of the measurements has shed light on the structure of mixing as it develops to a self-similar regime in this flow. The onset of self-similarity is marked by the development of a self-preserving form of the temperature spectra, and the collapse of velocity profiles expressed in self-similar units. Vertical velocity fluctuations dominate horizontal velocity fluctuations in this experiment, with a ratio approaching 2:1 in the self-similar regime. This anisotropy extends to the Taylor microscales that undergo differential straining in the direction of gravity. Up to two decades of velocity spectra development, and four decades of temperature spectra, have been captured from the experiment. The velocity spectra consist of an inertial range comprised of anisotropic vertical and horizontal velocity fluctuations, and a more isotropic dissipative range. Buoyancy forcing occurs across the spectrum of velocity and temperature scales, but was not found to affect the structure of the spectra, resulting in a $-5/3$ slope, similar to other canonical turbulent flows. A scaling argument is presented to explain this observation. The net kinetic energy dissipation, as the flow evolves from an initial state to a final self-similar state was measured to be 49% of the accompanying loss in potential energy, and is in close agreement with values obtained from three-dimensional numerical simulations.

The dynamics of a thick layer of viscous liquid flowing down a thin vertical fibre is investigated. Three qualitatively different regimes of the interfacial patterns in the form of beads were observed experimentally. Two typical regimes at relatively small flow rate are described reasonably well by the creeping-flow model equation proposed here.

The coupling of Brownian displacements and shear-induced convection of spherical colloidal particles in dilute suspensions is examined using solutions of appropriate convective diffusion equations for the time-dependent probability density and also by calculation of relevant statistical quantities for an ensemble of diffusing particles from Langevin equations. Based on a fundamental solution for convective diffusion from a point in a general linear field, analytical expressions for the probability density fα(r; t) are given for the case of an arbitrary, two-dimensional linear flow field. The parameter α, which characterizes the flow, may range from − 1 (pure rotation), through zero (simple shear), to + 1 (pure elongation). The Langevin approach offers interesting insights into the physical mechanism of diffusive–convective coupling, and may also be used to obtain rigorous expressions for moments of the probability density appropriate to a particle diffusing in an unbounded quadratic (Poiseuille) flow. Preliminary experiments are described which qualitatively verify the theoretical predictions for Poiseuille flow, and which suggest a simple, direct method for measuring particle diffusivities. Finally the effect of bounding walls on convective diffusion is considered by means of Monte Carlo calculations. Results show that particle-wall interactions significantly affect the average behaviour of particles located initially within distances of a few particle radii of the wall, since the frictional force is no longer isotropic.

The temporal instability of a cylindrical capillary jet is analysed numerically for different liquid Reynolds numbers Re, disturbance wavenumbers k, and amplitudes ε0. The breakup mechanism of viscous liquid jets and the formation of satellite drops are described. The results show that the satellite size decreases with decreasing Re, and increasing k and ε0. Marginal Reynolds numbers below which no satellite drops are formed are obtained for a large range of wavenumbers. The growth rates of the disturbances are calculated and compared with those from the linear theory. These results match for low-Re jets, however as Re is increased the results from the linear theory slightly overpredict those from the nonlinear analysis. (At the wavenumber of k = 0.9, the linear theory underpredicts the nonlinear results.) The breakup time is shown to decrease exponentially with increasing the amplitude of the disturbance. The cut-off wavenumber is shown to be strongly dependent on the amplitude of the initial disturbance for amplitudes larger than approximately $\frac13$ of the initial jet radius. The stable oscillations of liquid jets are also investigated. The results indicate that liquid jets with Re ∼ O(1) do not oscillate, and the disturbances are overdamped. However, liquid jets with higher Re oscillate with a period which depends on Re and ε0. The period of the oscillation decreases with increasing Re at small ε0; however, it increases with increasing Re at large ε0. Marginal Reynolds numbers below which the disturbances are overdamped are obtained for a wide range of wavenumbers and ε0 = 0.05.

The circular hydraulic jump commonly forms on a horizontal plate struck by a vertical jet of liquid. New observations of this phenomenon are described.

A previously unreported instability of the jump is examined. This is shown to arise when the local Reynolds number Rj just ahead of the jump exceeds a critical value of 147. Prior to this instability, the flow behind the jump contains a closed eddy, the length of which decreases to zero as Rj increases towards its critical value. Physical explanations for this flow structure and instability are proposed.

Accurate measurements of liquid depths were made using a light-absorption technique, in which a laser was shone through water containing a strong dye. Liquid depths ahead of and behind the jump were so determined and depth profiles of the jump in the stable regime were obtained.

As the outer depth was increased, the jump closed in on the jet and eventually disappeared: this extinction of the jump is also investigated.

A new theory for the turbulent plane wall jet without external stream is proposed based on a similarity analysis of the governing equations. The asymptotic invariance principle (AIP) is used to require that properly scaled profiles reduce to similarity solutions of the inner and outer equations separately in the limit of infinite Reynolds number. Application to the inner equations shows that the appropriate velocity scale is the friction velocity, u∗, and the length scale is v/u∗. For finite Reynolds numbers, the profiles retain a dependence on the length-scale ratio, y+1/2 = u∗y1/2/v, where y1/2 is the distance from the wall at which the mean velocity has dropped to 1/2 its maximum value. In the limit as y+1/2 → ∞, the familiar law of the wall is obtained. Application of the AIP to the outer equations shows the appropriate velocity scale to be Um, the velocity maximum, and the length scale y1/2; but again the profiles retain a dependence on y+1/2 for finite values of it. The Reynolds shear stress in the outer layer scales with u2*, while the normal stresses scale with U2m. Also Um ∼ yn1/2 where n < −1/2 and must be determined from the data. The theory cannot rule out the possibility that the outer flow may retain a dependence on the source conditions, even asymptotically.

The fact that both these profiles describe the entire wall jet for finite values of y+1/2, but reduce to inner and outer profiles in the limit, is used to determine their functional forms in the ‘overlap’ region which both retain. The result from near asymptotics is that the velocity profiles in the overlap region must be power laws, but with parameters which depend on Reynolds number y+1/2 and are only asymptotically constant. The theoretical friction law is also a power law depending on the velocity parameters. As a consequence, the asymptotic plane wall jet cannot grow linearly, although the difference from linear growth is small.

It is hypothesized that the inner part of the wall jet and the inner part of the zero-pressure-gradient boundary layer are the same. It follows immediately that all of the wall jet and boundary layer parameters should be the same, except for two in the outer flow which can differ only by a constant scale factor. The theory is shown to be in excellent agreement with the experimental data which show that source conditions may determine uniquely the asymptotic state achieved. Surprisingly, only a single parameter, B1 = (Umv/Mo)/ (y+1/2Mo/v2)n = constant where n ≈ −0.528, appears to be required to determine the entire flow for a given source.

The stability properties and stationary statistics of inviscid barotropic flow over topography are examined. Minimum enstrophy states have potential vorticity proportional to the streamfunction and are nonlinearly stable; correspondingly, canonical equilibrium based on energy and enstrophy conservation predicts mean potential vorticity is proportional to the mean streamfunction. It is demonstrated that in the limit of infinite resolution the canonical mean state is statistically sharp, that is, without any eddy energy on any scale, and is identical to the nonlinearly stable minimum enstrophy state. Special attention is given to the interaction between small scales and a dynamically evolving large-scale flow. On the β-plane, these stable flows have a westward large-scale component. Possibilities for a general relation between inviscid statistical equilibrium and nonlinear stability theory are examined.

High-viscosity liquids hate to work. Low-viscosity liquids are the victims of the laziness of high-viscosity liquids because they are easy to push around.

The arrangement of components in steady flow of immiscible liquids is typically non-unique. The problem of selection of arrangements is defined here and is studied by variational methods under the hypothesis that the realized arrangements are the ones that maximize the speed on exterior boundaries for prescribed boundary tractions, or the ones that minimize the tractions for prescribed speeds. The arrangements which minimize tractions also minimize the dissipation by putting low-viscosity liquid in regions of high shear. The variational problem is used as a guide to intuition in the design and interpretation of experiments when results of analysis of stability are unavailable. In fact we always observe some kind of shielding of high-viscosity liquid. This can occur by sheet coating in which low-viscosity liquid encapsulates high-viscosity liquid, or through the formation of rigidly rotating masses of high-viscosity liquid which we call rollers. In other cases we get emulsions of low-viscosity liquid in a high-viscosity foam. The emulsions arise from a fingering instability. The low-viscosity liquid fingers into the high-viscosity liquid and then low-viscosity bubbles are pinched off the fingers. The emulsions seem to have a very low effective viscosity and they shield the high-viscosity liquid from shearing. In the problem of Taylor instability with two fluids, low-viscosity Taylor cells are separated by stable high-viscosity rollers.

Two problems representative of the gravity flow of granular materials are considered in the context of a theory presented by Goodman (1970). The problems consist of steady fully-developed flow of a granular material down an inclined plane and between vertical parallel plates. It is shown that the dynamical behaviour of these materials is quite different from that of a viscous fluid. For the inclined flow problem, the normal stresses are not only unequal but vary non-linearly with depth. Also the maximum value of the mass flux distribution does not necessarily occur at the upper surface. For the vertical channel-flow problem, the material behaves somewhat like a Bingham fluid in that a plug region exists in the central part of the channel. The interesting feature of this problem is that the concentration of material volume in the shearing region outside the plug may either increase or decrease from the plug to the channel wall, depending on the boundary conditions. Experimental evidence for these phenomena in real granular materials is cited.

The results of this investigation suggest that the gravity flow of granular materials is essentially governed by two factors-a material characteristic length, which is possibly related to the grain size, and the externally imposed constraints such as the gravity field or the pressure exerted upon the granular material from the confining plates.

This paper investigates the propagation of an air finger into a fluid-filled, axially uniform tube of elliptical or rectangular cross-section with transverse length scale a and aspect ratio α. Gravity is assumed to act parallel to the tube's axis. The problem is studied numerically by a finite-element-based direct solution of the free-surface Stokes equations.

In rectangular tubes, our results for the pressure drop across the bubble tip, Δp, are in good agreement with the asymptotic predictions of Wong et al. (1995b) at low values of the capillary number, Ca (ratio of viscous to surface-tension forces). At larger Ca, Wong et al.'s (1995b) predictions are found to underestimate Δp. In both elliptical and rectangular tubes, the ratio Δp(α)/Δp(α = 1) is approximately independent of Ca and thus equal to the ratio of the static meniscus curvatures.

In non-axisymmetric tubes, the air-liquid interface develops a noticeable asymmetry near the bubble tip at all values of the capillary number. The tip asymmetry decays with increasing distance from the bubble tip, but the decay rate becomes very small as Ca increases. For example, in a rectangular tube with α = 1.5, when Ca = 10, the maximum and minimum finger radii still differ by more than 10% at a distance 100a behind the finger tip. At large Ca the air finger ultimately becomes axisymmetric with radius r∞. In this regime, we find that r∞ in elliptical and rectangular tubes is related to r∞ in circular and square tubes, respectively, by a simple, empirical scaling law. The scaling has the physical interpretation that for rectangular and elliptical tubes of a given cross-sectional area, the propagation speed of an air finger, which is driven by the injection of air at a constant volumetric rate, is independent of the tube's aspect ratio.

For smaller Ca (Ca < Ca), the air finger is always non-axisymmetric and the persisting draining flows in the thin film regions far behind the bubble tip ultimately lead to dry regions on the tube wall. Ca increases with increasing α and for α > αˆ dry spots will develop on the tube walls at all values of Ca.

It is shown that islands can trap long-wave energy in a way similar to the capture of a particle by an atomic nucleus. The frequencies of the captured waves form a discrete set, being determined by the shape of the island and the contours of the surrounding sea bed. If the depth at great distances tends to a constant value, the trapped modes must leak some energy to infinity, though the consequent rate of decay may be exceedingly small. The initial energy of the trapped modes may be absorbed from incident radiation of the same frequency or from a sharp pulse. The particular example of a rectilinear pulse incident on a circular island is discussed in some detail.

The effect of the rotation of the Earth is to split the frequencies of a pair of waves progressing in opposite directions round the island. The splitting of the frequencies produces slow beats in the waves as seen at any fixed point. Slight asymmetry in the island induces a slow exchange of energy between each pair of progressive modes.

The present investigation was suggested by the occurrence of regular oscillations having a period of 6 min and a beat period of about 3 h in long-wave records taken at Macquarie Island, in the Southern Ocean.

A combination of cross-wire probes with an array of flush-mounted skin-friction sensors are used to study the three-dimensional conditional organisation of large-scale structures in a high-Reynolds-number turbulent boundary layer. Previous studies have documented the amplitude modulation of small-scale motions in response to conditionally averaged large-scale events, but the data are largely restricted to the streamwise component of velocity alone. Here, we report results based on all three components of velocity and find that the small-scale spanwise and wall-normal fluctuations ($v$ and $w$) and the instantaneous Reynolds shear stress ($-{uw}$) are modulated in a very similar manner to that previously noted for the streamwise fluctuations ($u$). The envelope of the small scale fluctuations for all velocity components is well described by the large-scale component of the $u$ fluctuation. These results also confirm the conditional existence of roll modes associated with the very large-scale or ‘superstructure’ motions.

A solution has been obtained for steady propagation of a two-dimensional fluid fracture driven by buoyancy in an elastic medium. The problem is formulated in terms of an integro-differential equation governing the elastic deformation, coupled with the differential equation of lubrication theory for viscous flow in the crack. The numerical treatment of this system is carried out in terms of an eigenfunction expansion of the cavity shape, in which the coefficients are found by use of a nonlinear constrained optimization technique. When suitably non-dimensionalized, the solution appears to be unique. It exhibits a semi-infinite crack of constant width following the propagating fracture. For each value of the stress intensity factor of the medium, the width and propagation speed are determined. The results are applied to the problem of the vertical ascent of magma through the earth's mantle and crust. Values obtained for the crack width and ascent velocity are in accord with observations. This mechanism can explain the high ascent velocities required to quench diamonds during a Kimberlite eruption. The mechanism can also explain how basaltic eruptions can carry large mantle rocks (xenoliths) to the surface.

We consider the motion of a liquid film falling down a heated planar substrate. Using the integral-boundary-layer approximation of the Navier–Stokes/energy equations and free-surface boundary conditions, it is shown that the problem is governed by two coupled nonlinear partial differential equations for the evolution of the local film height and temperature distribution in time and space. Two-dimensional steady-state solutions of these equations are reported for different values of the governing dimensionless groups. Our computations demonstrate that the free surface develops a bump in the region where the wall temperature gradient is positive. We analyse the linear stability of this bump with respect to disturbances in the spanwise direction. We show that the operator of the linearized system has both a discrete and an essential spectrum. The discrete spectrum bifurcates from resonance poles at certain values of the wavenumber for the disturbances in the transverse direction. The essential spectrum is always stable while part of the discrete spectrum becomes unstable for values of the Marangoni number larger than a critical value. Above this critical Marangoni number the growth rate curve as a function of wavenumber has a finite band of unstable modes which increases as the Marangoni number increases.

An axisymmetric vortex-sheet model is applied to simulate an experiment of Didden (1979) in which a moving piston forces fluid from a circular tube, leading to the formation of a vortex ring. Comparison between simulation and experiment indicates that the model captures the basic features of the ring formation process. The computed results support the experimental finding that the ring trajectory and the circulation shedding rate do not behave as predicted by similarity theory for starting flow past a sharp edge. The factors responsible for the discrepancy between theory and observation are discussed.