The paths followed by individual fluid particles can be extremely complicated even in smooth laminar flows. Such chaotic advection causes mixing of the fluid. This phenomenon is studied analytically for a class of spatially periodic flows comprising a basic flow of two-dimensional (or axisymmetric) counter-rotating vortices in a layer of fluid, and modulated by a perturbation which is periodic in time and/or space. Examples of this type of flow include Bénard convection just above the point of instability of two-dimensional roll cells, and Taylor vortex flow between concentric rotating cylinders. The transport of chaotically advected particles is modelled as a Markov process. This predicts diffusion-like mixing, and provides an expression for the diffusion coefficient. This expression explains some features of experimental results reported by Solomon & Gollub (1988): its accuracy is investigated through a detailed comparison with numerical results from a model of wavy Taylor vortex flow. The approximations used in the analysis are equivalent to those used to obtain the quasi-linear result for diffusion in the standard map.

A simple theoretical model of the thermal bar is derived on the basis of heat budget equations for the following three zones of a wedge-shaped water basin warmed from above: (i) stably stratified shallow warm-water zone; (ii) vicinity of the bar; (iii) convectively mixed deep cold zone. In contrast to the traditional approach, advective warming of the vicinity of the bar and associated facilitating of the thermal bar propagation are taken into account. Theoretical predictions are compared with the data of lacustrine and laboratory measurements taken from current literature. New laboratory experiments have been carried out to examine the laminar regime of the thermal bar.

A nozzle expansion into a vacuum chamber was used to investigate the evaporation of highly superheated liquid jets. The large molar specific heat of fluids with high molecular complexity — in this case C6F14 — is responsible for the new phenomena reported here. A model was developed to describe the basic physical effects. A cubic equation of state was used to describe the thermodynamic properties of the fluid. The evaporation was modelled as a sonic deflagration followed by an axisymmetric supersonic expansion. As in the case of hypersonic gas jets the final state is reached by a normal shock. For sufficiently high temperatures and expansion ratios a complete adiabatic evaporation of the liquid was found. At even higher temperatures the liquid evaporates completely within a rarefaction discontinuity. The predictions of the model are in good agreement with the experimental results.

In this paper we provide a detailed description of the interaction of solutal convection and morphological instability in the presence of a model boundary-layer flow. We present a detailed investigation of the structure of the marginal surfaces in Rayleigh-number, Sekerka-number, Reynolds-number space associated with a linear stability analysis. We give mathematical arguments and physical mechanisms to explain the results and present a coherent description of this complicated situation. We identify two new modes, one convective and one morphological. We show that the oscillatory so-called ‘mixed’ modes that result from the coupling of morphological and convective modes play a central role in the unfolding of the solution structure by the shear flow. This flow has the effect of decoupling the convective and morphological modes.

The concentration distribution resulting from a continuous point source in a fluid with a steady linear variation in velocity is distorted by the flow at distances greater than (K/Eb)½, where K is the molecular diffusion coefficient and Eb is a characteristic shear rate. The distribution has two distinct shapes depending on the number of principal axes of fluid strain that are expansive and the relative magnitude of irrotational and rotational shears. For irrotational flows a single expansive principal axis of strain results in a tube-like distribution, while two expansive axes results in a disk-like distribution. Approximate analytical solutions, derived by neglecting diffusion along the expansive axes, agree well with concentrations calculated by numerically convolving the exact instantaneous source solution. The effect of fluid vorticity is generally to reorient the distribution away from the principal axes of strain and to reduce the asymmetry of the concentration distribution. Aside from reorientation, the concentration distribution varies little until the vorticity approaches a critical value defined by a kinematic condition for equilibrium orientation in the presence of rotation. For vorticity greater than the critical value, the concentration distribution becomes axisymmetric around the axis of rotation. Application of these results to numerical simulations of isotropic turbulence suggests that tubes are more common than disks and that vorticity exceeds the critical value in at least 25% of the fluid.

This paper extends the analysis of Sneyd (1985) on interfacial instabilities in aluminium reduction cells. The cell model consists of a plane fluid layer of relatively low electrical conductivity, sandwiched between an upper rigid wall and lower fluid layer, both of high conductivity. A steady current passes through the layers, and the magnetic field is assumed to be a linear function of position. The principal new effects introduced are (i) a horizontal current component in the aluminium; (ii) vertical magnetic field components, and vertical field gradients; (iii) an aluminium pool of finite depth; and (iv) uniform zeroth-order flow in the fluid layers, and mechanical dissipation. A dispersion relation for small-amplitude waves is derived and discussed. The destabilizing Kelvin—Helmholtz mechanism and electromagnetic forces compete with gravity, surface tension and mechanical dissipation. Electromagnetic destabilization is likely to occur in practice at wavelengths of 1 m or more, and becomes more intense with decreasing layer depths. The most dangerous mechanism appears to be driven by vertical gradients of the horizontal field.

Physical and numerical experiments were performed for a linearly stratified heat—salt system, uniformly heated at one endwall. The initial stratification was in the diffusive sense. Intrusions formed at the heated endwall and propagated out into the interior fluid. Three classes of flow were identified, based upon the gravitational stability ratio, Rp, and a lateral stability parameter, R1, For R1 > 1, a vertical lengthscale for the initial intrusion thickness was developed which agreed well with that observed in the physical experiments. In all cases, a region of salt fingering developed due to gradient reversal at the heated endwall. Two very distinct merging processes were observed depending on the specific flow class. The first process occurred under conditions of high gravitational and lateral stability, and appeared to be controlled by horizontal motions induced by the intrusions. The second process was observed under less stable conditions and was a result of vertical motions at the heated endwall within the intrusions themselves. In the least stable class of flow (low gravitational and lateral stability), the intrusions were found to be self-perpetuating in the sense that they continued to propagate following removal of the endwall heat flux.

We consider the flow of a viscous liquid film on the surface of a cylinder that is heated or cooled. Lubrication theory is used to study a thin film under the influence of gravity, capillary, thermocapillary, and intermolecular forces. We derive evolution equations for the interface shapes as a function of the azimuthal angle about the cylinder that govern the behaviour of the film subject to the above coupled forces. We use both analytical and numerical techniques to elucidate the dynamics and steady states of the thin layer over a wide range of thermal conditions and material properties. Finally, we extend our derivation to the case of three-dimensional dynamics and explore the stability of the film to small axial disturbances.

Turbulent air flow over a surface gravity wave of small amplitude is studied on the basis of a family of first-order closure models, of which the eddy viscosity model and Prandtl's mixing-length model are members. Results are obtained by the method of matched asymptotic expansions in three layers. The problem is modelled by taking into account the combined effects of turbulence and molecular viscosity, which accommodates a proper imposition of the boundary conditions at the wave surface. The detailed structure of the various wave-induced field variables throughout the flow is then investigated. In addition, it is found that the growth rate of the waves by wind depends on the turbulence model. In particular, the more sensitively the mixing length depends on the shear in the mean air flow, the higher the growth rate. The validity of the results we obtain is restricted to small drag coefficient and small phase speed. Comparisons are made with other theoretical studies and with recent laboratory and field observations.

The interaction of solid particles with the temporal features of a turbulent flow has direct relevance to problems in particle and spray combustion and the processing of particulate solids. The object of the present study was to examine the behaviour of particles in a jet dominated by vortex ring structures. An axisymmetric air jet laden with 55 μm glass particles was forced axially with an acoustic speaker to organize the vortex ring structures rolling up in the free shear layer downstream of the nozzle exit. Visualization studies of forced and unforced flow with Reynolds number of the order of 20000 were completed using a pulsed copper vapour laser. Instantaneous photographs and videotapes of strobed forced flow show that particles become clustered in the saddle regions downstream of the vortex rings and are propelled away from the jet axis by the outwardly moving flow in these regions. Phase-averaged spatial distribution of particle number density computed from digitized photographs and phase-averaged particle velocity measurements yield further evidence that local particle dispersion and concentration are governed by convection due to large-scale turbulence structures. The large-scale structures and convection mechanisms were shown to persist for particle-to-air mass loading ratios up to 0.65.

The characteristics of shallow-water waves in a linear shear flow are studied, and the relationship between waves and unstable modes is examined. Numerical integration of the linear shallow-water equations shows that over-reflection occurs when a wave packet is incident at the turning surface. This phenomenon can be explained by the conservation of momentum as discussed by Acheson (1976). The unstable modes of linear shear flow in a shallow water found by Satomura (1981) are described in terms of the properties of wave propagation as proposed by Lindzen and others. Ripas's (1983) theorem, which is the sufficient condition for stability of flows in shallow water, is also related to the wave geometry. The Orr mechanism, which is proposed by Lindzen (1988) as the primary mechanism of wave amplification, cannot explain the over-reflection of shallow-water waves. The amplification of these waves occurs in the opposite sense to that of Orr's solution.

The velocity field of homogeneous isotropic turbulence is simulated by a large number (38–1200) of random Fourier modes varying in space and time over a large number (> 100) of realizations. They are chosen so that the flow field has certain properties, namely (i) it satisfies continuity, (ii) the two-point Eulerian spatial spectra have a known form (e.g. the Kolmogorov inertial subrange), (iii) the time dependence is modelled by dividing the turbulence into large- and small-scales eddies, and by assuming that the large eddies advect the small eddies which also decorrelate as they are advected, (iv) the amplitudes of the large- and small-scale Fourier modes are each statistically independent and each Gaussian. The structure of the velocity field is found to be similar to that computed by direct numerical simulation with the same spectrum, although this simulation underestimates the lengths of tubes of intense vorticity.

Some new results and concepts have been obtained using this kinematic simulation: (a) for the inertial subrange (which cannot yet be simulated by other means) the simulation confirms the form of the Eulerian frequency spectrum $\phi^{\rm E}_{11} = C^{\rm E}\epsilon^{\frac{2}{3}}U^{\frac{2}{3}}_0\omega^{-\frac{5}{3}}$, where ε,U0,ω are the rate of energy dissipation per unit mass, large-scale r.m.s. velocity, and frequency. For isotropic Gaussian large-scale turbulence at very high Reynolds number, CE ≈ 0.78, which is close to the computed value of 0.82; (b) for an observer moving with the large eddies the ‘Eulerian—Lagrangian’ spectrum is ϕEL11 = CELεω−2, where CEL ≈ 0.73; (c) for an observer moving with a fluid particle the Lagrangian spectrum ϕL11 = CLεω−2, where CL ≈ 0.8, a value consistent with the atmospheric turbulence measurements by Hanna (1981) and approximately equal to CEL; (d) the mean-square relative displacement of a pair of particles 〈Δ2〉 tends to the Richardson (1926) and Obukhov (1941) form 〈Δ2〉 = GΔεt3, provided that the subrange extends over four decades in energy, and a suitable origin is chosen for the time t. The constant GΔ is computed and is equal to 0.1 (which is close to Tatarski's 1960 estimate of 0.06); (e) difference statistics (i.e. displacement from the initial trajectory) of single particles are also calculated. The exact result that Y2 = GYεt3 with GY = 2πCL is approximately confirmed (although it requires an even larger inertial subrange than that for 〈Δ2〉). It is found that the ratio [Rscr ]G = 2〈Y2〉/〈Δ2〉≈ 100, whereas in previous estimates [Rscr ]G≈ 1, because for much of the time pairs of particles move together around vortical regions and only separate for the proportion of the time (of O(fc)) they spend in straining regions where streamlines diverge. It is estimated that [Rscr ]G ≈ O(fc−3). Thus relative diffusion is both a ‘structural’ (or ‘topological’) process as well as an intermittent inverse cascade process determined by increasing eddy scales as the particles separate; (f) statistics of large-scale turbulence are also computed, including the Lagrangian timescale, the pressure spectra and correlations, and these agree with predictions of Batchelor (1951), Hinzc (1975) and George et al. (1984).

We present the design and operation of a flow apparatus for investigations of mixing in time-periodic and spatially periodic chaotic flows. Uses are illustrated in terms of two devices operating in the Stokes regime: the partitioned-pipe mixer, a spatially periodic system consisting of sequences of flows in semicircular ducts, and the eccentric helical annular mixer, a time-periodic velocity field between eccentric cylinders with a superposed Poiseuille flow; other mixing flows can be implemented with relative ease. Fundamental differences between spatially periodic and time-periodic duct flows are readily apparent. Steady spatially periodic systems show segregated KAM-tubes coexisting with chaotic advection; such tubes are remarkably stable under a variety of experimental conditions. Time-periodic duct flows lead to complex streakline structures; since regular regions in the cross-sectional flow move through space, a streakline can find itself injected in a regular domain for some time then be trapped in a chaotic region, and so on, leading to ‘intermittent’ behaviour.

We study experimentally and theoretically the evolution of two-dimensional patterns in the Rayleigh—Taylor instability of a thin layer of viscous fluid spread on a solid surface. Various kinds of patterns of different symmetries are observed, with possible transition between patterns, the preferred symmetries being the axial and hexagonal ones. Starting from the lubrication hypothesis, we derive the nonlinear evolution equation of the interface, and the amplitude equation of its Fourier components. The evolution laws of the different patterns are calculated at order two or three, the preferred symmetries being related to the non-invariance of the system by amplitude reflection. We also discuss qualitatively the dripping at final stage of the instability.

An experimental investigation is conducted in order to quantify the nonlinear and parametric resonance mechanisms that are associated with the subharmonic growth in the transition to turbulence in plane mixing layers. Higher-order digital statistical analysis techniques are used to investigate the nonlinear and parametric mechanisms responsible for the energy transfer to the subharmonic. The results show that the dominant interaction is a parametric resonance mechanism between the fundamental and the subharmonic modes which leads to a pronounced growth of the subharmonic. Measurements also indicate that the fundamental, besides interacting with the subharmonic, is also engaged in redistributing its energy to the other Fourier components of the flow via nonlinear three-wave interactions. Local wavenumber measurements verify that frequency—wavenumber resonance matching conditions exist between the fundamental and subharmonic in the region where the subharmonic gains its energy by parametric resonance. The results are in general agreement with theoretical models by Kelly (1967), and Monkewitz (1988) on subharmonic growth.

A rigorous, kinematic description of the stretching and alignment of infinitesimal material elements in general flow fields is presented. An evolution equation is derived, in the Lagrangian frame, for the alignment angles between a material element and the principal axes of strain. The equation identifies the precise roles played by the local angular velocity and the rotation of the strain axes in the alignment process and provides the framework in which to investigate the extent to which the straining field is ‘persistent’. This general kinematical picture is specialized to study line and vortex stretching in fluid flows and analytically predicts the numerically observed alignment of the vorticity vector with the intermediate strain axis. The alignment equations are solved exactly for a number of special flow fields and investigated numerically for the ABC and STF flows. The kinematic formalism and numerical phenomenology suggests the use of new criteria to analyse the material element stretching properties of large-scale numerical simulations.

This paper describes a numerical study of the flow around flat plates with square leading and trailing edges on the basis of a finite-difference analysis of the two-dimensional Navier—Stokes equations. The chord-to-thickness ratio of a plate, d/h, ranges from 3 to 9 and the value of the Reynolds number based on the plate's thickness is constant and equal to 103. The numerical computation confirms the finding obtained in our previous experiments that vortex shedding from flat plates with square leading and trailing edges is caused by the impinging-shear-layer instability. In particular, the Strouhal number based on the plate's chord increases stepwise with increasing d/h in agreement with the experiment. Numerical analyses also provide some crucial information on the complicated vortical flow occurring near the trailing edge in conjunction with the vortex shedding mechanism. Finally, the mechanism of the impinging-shear-layer instability is discussed in the light of the experimental and numerical findings.

The problem of a vortex pair, rising obliquely at an angle of 45° toward a deformable free surface in a viscous, incompressible fluid, is solved with the aid of the Navier—Stokes equations. The full nonlinear boundary conditions at the free surface are applied. The oblique interaction of the vortex pair with the free surface results in a number of novel features that have not been observed for the special case of a vertical rise, reported earlier. These features include the directional change of trajectories near the free surface and the occurrence of waves driven by the vortex pair. Moreover, surface tension can completely change the flow characteristics such as the direction of the trajectories and the generation of secondary vortices. Numerical solutions are presented for selected Reynolds, Froude, and Weber numbers.

Fluidized beds are widely used in industry for carrying out a variety of chemical reactions and physical processes. Applications are frequently impeded by a lack of fundamental understanding of the mechanical behaviour of fluidized beds. Despite intensive experimental and theoretical study over the last four decades, there are still many aspects of fluidized beds and related fluid—particle systems that remain obscure. Further work is needed to understand interactions between the particles, influence of particle physical properties, development of non-obtrusive experimental techniques, and study of high-velocity beds and of novel-geometry beds in which particulate solids interact with interstitial fluid.

An international symposium was held at Stanford University on 1–4 July 1991 to discuss recent developments and the current state of knowledge and understanding of the mechanical behaviour of fluidized beds and related fluid—particle systems. The symposium was sponsored by the International Union of Theoretical and Applied Mechanics, and co-funded by the US Department of Energy, National Science Foundation and Electric Power Research Institute. The symposium was attended by 58 specialists representing academic institutions, industry and government research organizations in 11 countries. The diversity of background, coupled with differences in approach, ranging from purely theoretical to fully experimental, led to interesting exchanges where participants were often groping to understand the viewpoint of those involved. The result was frequently rewarding, occasionally perplexing, but certainly stimulating of thought and encouraging for further meetings of this nature.

The scientific committee for the Symposium were G. K. Batchelor (Cambridge University), J. J. H. Brouwers (Trent University), J. Gibilaro (University College London), J. R. Grace (University of British Columbia), G. M. Homsy (Stanford University) (Chairman), R. Jackson (Princeton University), R. I. Nigmatulin (Moscow University) and W. Schneider (Techische Universität Wien).

Each session began with an invited talk for one hour. This was followed by a series of 20-minute presentations. Participants were able at the end of each session to give brief (5 minute) unscheduled mini-presentations. Except for the latter, abstracts were submitted for the presentations, compiled by the organizers and distributed to participants. The meeting also included an informal workshop and a series of video and cinephotographic presentations not reported here. No formal proceedings of the meeting are being published; instead, this report is intended to summarize key findings and areas of discussion.

The titles of all formal presentations are identified with asterisks in the list of References. The report is organized around common themes, and the sequence of presentations may not be that in which they were given. The authors have endeavoured to interpret and extract the essence of the presentations, and apologize if any of the presentations have been misinterpreted or overlooked.

Depending on the flow conditions, the liquid film in annular two-phase flow in coiled tubes may be pushed towards the outer or the inner side by the centrifugal force. It is important to understand the mechanism of this ‘film inversion’ in order to develop a predictive model for the film thickness distribution. In this paper, this phenomenon is studied analytically, and a new criterion, based on the secondary flow in the thin liquid film, is proposed to predict its occurrence. The criterion shows good agreement with available experimental data. It is suggested that the analytical model can readily be extended to predict the distribution of the film thickness and film flow rate in coiled tubes.