A basis is developed for a similarity treatment of the coupled mean fields in turbulent open-channel flows over a flat sand bed in equilibrium with a suspension of sand. Hypotheses involving multiple scales, asymptotic similarity, and matching over an overlapping region are made instead of the traditional dependence on eddy diffusion and Reynolds analogies. Velocity measurements using the laser-Doppler technique in experiments with well-sorted natural sands showed that, contrary to the implications of previous models, the effect of suspended sediment on the mean velocity profile may, over a variety of laboratory conditions, be confined to a layer adjacent to the bed. This is interpreted within the similarity framework in terms of the existence of an inner lengthscale, differing from either the viscous or the grain scale, associated with the presence of sediment. The suggested lengthscales and concentration scales are examined in light of the experimental results.

The rotating flow of a separating mixture within an axisymmetric container is considered with emphasis on the pure fluid layer adjacent to the inclined ‘bottom’ boundary from which particles are removed by centrifugal buoyancy. Within the framework of ‘mixture’ (‘diffusion’) model and when the relative density difference, the Ekman number E and particle Taylor number β are small, it is shown that the behaviour of that layer is governed by Ξ = E½αI|cotγB|/β (where αI is the volume fraction of the dispersed particles and γB is the elevation angle of the bottom wall with respect to the centrifugal force). If Ξ is small the layer thickens quickly into an inviscid core, in accordance with previous studies. However, novel features show up for Ξ large or O(1), when the viscosity-induced Ekman suction is able to counteract the separation velocity. In the former case the pure fluid layer is thin and quasi-steady, and the remaining part of the interface is essentially perpendicular to the force field, in close apparent resemblance to the analog gravitational process. In the latter case, a thin quasi-steady layer and a continuously thickening core of pure fluid coexist in the same vessel.

A brief review of the relevant papers and an examination of the current status of research in the field of the acoustics of gas-particle suspensions are given. Non-stationary momentum, mass and energy transfer processes between a gas (vapour) and dispersed particles (droplets) under high-frequency acoustic perturbations are considered. A comparative evaluation of characteristic times and temperature differences for gas-particle and vapour-droplet mixtures subjected to acoustic perturbation is given. General dispersion equations to describe the propagation of weak monochromatic waves for a wide range of frequencies complying with the requirements of the acoustic homogeneity of the medium are derived. Frequency dependences of propagation velocity and attenuation coefficient of weak waves in water vapour-droplet mixtures are investigated. Frequency ranges are indicated over which different types of approximate theories are valid.

The form of the momentum equation for one-dimensional (vertical) unsteady mean motion of solid particles in a fluidized bed or a sedimenting dispersion is established from physical arguments. In the case of a fluidized bed that is slightly non-uniform this equation contains two dependent variables, the local mean particle velocity V and the local concentration ϕ, and several statistical parameters of the particle motion in a uniform bed. All these parameters are functions of ϕ with clear physical meanings, and the important ones are measurable. It is a novel feature of the equation that it contains two explicit contributions to the bulk modulus of elasticity of the particle configuration, one arising from the transfer of particle momentum by velocity fluctuations and one arising from the effective repulsive force exerted between particles in random motion. This latter contribution, which proves to be the more important of the two, is related to the gradient diffusivity of the particles, a key quantity in the new theory.

The equation of mean motion of the particles and the equation of particle conservation are sufficient to determine the behaviour of a small disturbance with sinusoidal variation of V and ϕ in the vertical direction. Particle inertia forces in such a propagating wavy disturbance may promote amplitude growth, whereas particle diffusion tends to suppress it, and instability occurs when the particle Froude number exceeds a critical value. Rough estimates of the relevant parameters allow the criterion for instability to be put in approximate numerical from for both gas-fluidized beds (for which the flow Reynolds number at marginal stability is small) and liquid-fluidized beds of solid spherical particles (for which the Reynolds number is well above unity), although more information about the particle diffusivity in particular is needed. The predictions of the theory appear to be in qualitative accord with the available observational data on instability of gas- and liquid-fluidized beds.

A theoretical model is developed for predicting the critical plate-orifice distances for viscous fluid buckling in plane and axisymmetric low-Reynolds-number bounded jets in stagnation flow. The theory is based on a hypothesis that treats the spatial growth of perturbation to the jet in a manner that distinguishes between the near-wall and far-field regions of the jet. This perturbation growth rate is shown to be the important parameter in the determination of the critical plate-orifice distances.

This study also uses a one-dimensional model of the fluid column when it is displaced from equilibrium to determine the frequency at which buckling is first initiated in the case of the plane jet.

The macroscopic dispersion of tracer in microscopically disordered fluid flow can ultimately, at large times, be described by an advection-diffusion equation. But before this asymptotic regime is reached there is an intermediate regime in which first and second spatial moments of the distribution are proportional to tν. Conventional advection-diffusion (which applies at large times) has ν = 1 but in the intermediate regime ν < 1. This phenomenon is referred to as ‘anomalous diffusion’ and this article discusses the special case ν = ½ in detail. This particular value of ν results from tracer dispersion in a central pipe with many stagnant side branches leading away from it. The tracer is “held up” or ‘arrested’ when it wanders into the side branches and so the dispersion in the central duct is more gradual than in conventional advection-diffusion (i.e. ν = ½ < 1).

This particular example serves as an entry point into a more general class of models which describe tracer arrest in closed pockets of recirculation, permeable particles, etc. with an integro-differential equation. In this view tracer is arrested and detained at a particular site for a random period. A quantity of fundamental importance in formulating a continuum model of this interrupted random walk is the distribution of stopping times at a site. Distributions with slowly decaying tails (long sojourns) produce anomalous diffusion while the conventional model results from distributions with short tails.

Two-dimensional flow of a surface-active monolayer on a thin viscous film is considered. Simplifications of negligible gravity and pressure forces are made. Interfacial properties are described by simple model equations of state. Solutions are obtained for when the monolayer is scraped along the interface by a barrier and a steady state exists where surface advection is balanced by surface diffusion. Surface velocity, film thickness and spreading rate dependence on surface diffusivity are examined.

A series of visualizations of non-cavitating and cavitating unsteady flows around an oscillating hydrofoil has been carried out in order to investigate the effect of unsteadiness on attached cavitation. The major conclusion of the present experimental analysis is that the strong interaction that was previously pointed out in the case of steady cavitation between an attached cavity and the boundary layer which develops upstream cavity detachment, still plays a prominent role in unsteady cavitation. We propose to generalize for the case of unsteady attached cavitation the two following points which were initially established under steady conditions and which constitute a cavitation detachment criterion:

1. a cavity detaches behind laminar separtion of the boundary layer;

2. transition to turbulence sweeps away an attached cavity.

Unsteady effects in a separated recirculating eddy beneath a subsonic or supersonic mainstream are considered, with emphasis on nonlinear properties. The eddy is slender and predominantly inviscid, its length being comparable with, or less than, the chord of an airfoil, for instance. It is found, from the study of a family of integro-differential equations, that the planar eddy can break up nonlinearly within a finite time, causing an eruption into the main stream and setting up a subsequent Euler stage in the unsteady motion. Comparisons with recent experiments and further applications of the theory are discussed.

A fluid-mechanical approach for the cleavage of biological cells is presented. The equations of motion were combined with concentration and orientation distribution balances, for active contractile filaments on the cell surface, to provide a dynamic evolution of interfacial forces and deformation. The resulting flow and the simultaneously developing surface-tension anisotropy provided a mechanism that facilitates the generation of a contractile ring at the cell equator: a major organelle in the establishment of cell furrow and the ultimate cleavage. The moving-boundary problem was solved numerically using boundary-integral representation for the Stokes equations which was modified to incorporate the anisotropic interfacial tensions.

The development of Görtler vortices in boundary layers over curved walls in the nonlinear regime is investigated. The growth of the boundary layer makes a parallel-flow analysis impossible except in the high-wavenumber regime so in general the instability equations must be integrated numerically. Here the spanwise dependence of the basic flow is described using a Fourier series expansion whilst the normal and streamwise variations are taken into account using finite differences. The calculations suggest that a given disturbance imposed at some position along the wall will eventually reach a local equilibrium state essentially independent of the initial conditions. In fact the equilibrium state essentially independent of the initial conditions. In fact the equilibrium state reached is qualitatively similar to the large-amplitude high-wavenumber solution described asymptotically by Hall (1982b). In general it is found that the nonlinear interactions are dominated by a ‘mean field’ type of interaction between the mean flow and the fundamental. Thus, even though higher harmonics of the fundamental are necessarily generated, most of the disturbance energy is confined to the mean flow correction and the fundamental. A major result of our calculations is the finding that the downstream velocity field develops a strongly inflexional character as the flow moves downstream; the latter result suggests that the major effect of Görtler vortices on boundary layers of practical importance might be to make them highly receptive to rapidly growing Rayleigh modes of instability.

A numerical method employing an upwind finite-difference technique is adopted for an investigation of peristaltic pumping in circular cylindrical tubes. such as some organs in the living body. Various peristaltic flows are calculated under conditions of finite wave amplitudes, finite wavelengths and finite Reynolds numbers, and the influence of the magnitude of these quantities on the flow is investigated. The fluid mechanics of peristaltic mixing and transport are studied in detail by analysing the reflux and the trapping phenomena. The mechanical efficiency of peristaltic pumping is also discussed, with reference to engineering and physiological applications. It is shown that quantitative differences are observed between the results obtained for flows in a circular cylindrical tube and a two-dimensional plane channel. However, for both cases the appearance of peristaltic reflux depends upon the Reynolds number and the wavenumber (mean tube radius/wavelength). Much greater peristaltic mixing and transport are realized in a circular tube than in a plane channel.

An experimental technique for visualizing small-scale film flows is presented. The method utilizes hydrogen bubbles, dye injection, and optical sectioning. Specific aspects of applying this technique to coating processes, distinguished by two free surfaces, liquid-liquid interfaces, static and dynamic wetting lines, and small characteristic dimensions, are discussed in detail. This experimental approach is essential in providing the necessary boundary conditions for modern computer-aided theoretical computations of wetting-related flow fields. The power of the visualization method is demonstrated by examples showing the flow pattern of two liquid films merging on an inclined plane and the flow field of a liquid film being transferred from an inclined plane onto a moving surface.

The structure of the pressure and velocity field in the air above progressive, mechanically generated water waves was investigated in order to evaluate the influence of a mobile and deformable boundary on turbulence production and the related bursting phenomena. The Reynolds stress fluctuations were measured in a transformed Eulerian wave-following frame of reference, in a wind-wave research facility at Stanford University.

The structure of the wave-coherent velocity field was found to be very sensitive to the height of the critical layer below which the waves travel faster than the wind. Because the critical-layer height changes rapidly with the ratio (c / u*) of the wave speed to the wind friction velocity, the structure of the wave-coherent velocities depends strongly on the parameter c/Uδ0, where Uδ0 is the mean free-stream wind velocity. When the critical height is large enough that most of the flow in the turbulent boundary layer is below the critical height, the structure of the wave-coherent velocities is strongly affected by the Stokes layer (in the air), which under the influence of turbulence can have thickness comparable with the wave amplitude. In contrast, when the critical height is small enough that most of the flow in the boundary layer is above the critical height, the structure of the wave-coherent velocities is strongly affected by the critical layer. The latter was found to be nonlinear and turbulently diffusive.

The dependence of the structural behaviour of the wave-coherent velocity field upon the critical and Stokes layers results in considerable modifications of the turbulence-generating mechanism during the bursting-cycle, as the dimensionless wave speed c/Uδ0 changes. Such modifications are manifested by an enhancement of the contributions to the mean Reynolds stress of the bursting events (relative to their solid-wall counterparts), and their dependence on the dimensionless wave speed. For c/Uδ0 [ges ] 0.68 (or c/u* > 20), the nonlinear critical-layer thickness is large compared to the wave amplitude (except when c/Uδ0 = 0.68), and the diffused Stokes layer stimulates the wave-associated stress production. In the water proximity, the bursting contributions remain nearly constant with dimensionless wave speed; ejections account for 90% of the mean Reynolds stress, whereas sweeps provide 77%, the excess over 100% being balanced by the outward and inward interactions. For c/Uδ0 < 0.68, the critical-layer thickness is smaller than the wave amplitude and all contributions increase gradually with c/Uao. However, the ratio of ejection to sweep contributions remains unaltered and ≈ 1.15, indicating that sweeps are nearly as energetic as ejections a t all dimensionless wave speeds. The value of c/Uδ0 ≈ 0.68 appears to separate the flow regimes of high and low critical level, respectively, where significant and weak production of the wave-associated stresses have been found. Near the water surface the height distribution of the fractional contributions of the bursting events is also sensitive to the ratio c/UJO. In the equilibrium region of the boundary layer it remains uniform and in the free stream rises sharply, independent of dimensionless wave speed.

The mean time period between ejections or sweeps depends on both the wave and wind field characteristics and does not scale with either the inner or the outer flow variables. The former can be determined from the time between the first two largest consecutive peaks of the phase-averaged Reynolds stress distribution.

In the water proximity, the height distribution of the normalized energy production is sensitive to c/Uδ0; only when c/Uδ0 [ges ] 0.68 does it show a peak of increasing magnitude with increasing dimensionless wave speed.

In oscillatory flows through systems of branched or curved tubes, Taylor dispersion is modified both by the oscillation and by the induced secondary motions. As a model for this process, We examine axial transport in an annular region containing an oscillatory axial and steady secondary (circumferential) flow. Two complementary approaches are used: an asymptotic analysis for an annulus with a narrow gap (δ) and for large values of the secondary flow Peclet number (P); and a numerical solution for arbitrary values of δ and P. The results exhibit a form of resonance when the secondary-flow time equals the oscillation period, giving rise to a prominent maximum in the transport rate. This observation is consistent with preliminary numerical results for oscillatory flow in a curved tube, and can be explained physically.

The ordinary differential equation $\epsilon^2 \ddot{y} + \epsilon^2 \delta \dot{y} = y^2 - \cos t - 1 - c$ which represents forced water waves on shallow water near resonance is considered when the dispersion ε and the constant c are small. Asymptotic and numerical methods are used to show that solutions which are bounded for all finite t can exist only if $c > - 2^{-\frac{2}{3}}\epsilon^{\frac{4}{3}}\times 1.4664 \ldots $.

This study is devoted to the onset of convection in differentially heated cylinders under gravity modulation. It specifically concerns the case of a vertical cylinder of infinite length, when a negative temperature gradient is maintained in the upward direction. The effect of modulation on the stability limits given by linear theory in the standard steady case is analysed. A method based on Floquet theory is proposed in the case of small values of the modulation amplitude ε, for a fixed value of the frequency ω. A general technique, called matrix method, which can easily be adapted to various kinds of geometries and boundary conditions, has been developed. Analytical approaches have been derived in some cases. Finally, an asymptotic analysis is presented for large ω, under very general boundary conditions and periodic constraints, for finite ε. An asymptotic relation is established for the onset of convection under periodic gravity modulation for large ω values, when ε [Lt ] ω; the mathematical and physical foundations of this inequality are discussed.

Reheat buzz is a combustion instability which occurs in the afterburners of jet aeroengines. Similar oscillations have been observed on a laboratory rig in which a confined flame is stabilized in the wake of a conical gutter. The source of energy for the instability is the unsteady heat release as the flame responds to velocity fluctuations in the approach flow. The amount of energy fed into the instability is determined primarily by the phase relationship between the unsteady heat release rate and pressure fluctuations. Two types of relationship between pressure and heat release have been observed. In the first kind, perturbations in heat release rate convect downstream from the lip of the flame stabilizer at the axial velocity of the cold reactants. In the second type, which occurs downstream of the first, the phase of the heat release rate at the buzz frequency is constant and close to the phase of the unsteady pressure. The characteristics of the resulting instability depend on which of these occupies the larger portion of the duct. Consequently two types of instability exist and the transition between them occurs sharply, with little change in mean flow conditions. The transition is associated with an abrupt change in buzz amplitude and frequency as well as with a change in the shape of the modal distributions.

Reheat buzz is a low-frequency instability of afterburners. It is caused by the interaction of longitudinal acoustic waves and unsteady combustion. Similar combustion instabilities occur in laboratory rigs. A theory is developed to determine the frequency and mode shape of the instability and is tested by comparison with the experimental results described in Part1. The predicted and measured frequencies are found to be within 6 Hz (7%) of each other. The theory is able to predict the observed variation of frequency with equivalence ratio, inlet Mach number and geometry.

Saffman argues that in decaying two-dimensional turbulence approximate dis-continuities of vorticity will form, and the energy spectrum will fall off as k−4 Saffman assumes that these discontinuities are well separated; in this paper, we examine how accumulation points of such discontinuities may give an energy spectrum of between k−4 and k−3. In particular we examine the energy spectra of spiral structures which form round the coherent vortices that are observed in numerical simulations of decaying two-dimensional turbulence. If the filaments of the spiral are assumed to be passively advected, the instantaneous energy spectrum has a $k^{-11/3}$ range. Thus we come some way to reconciling the argument of Saffman and the k−3 energy spectrum predicted by models of quasi-equilibrium two-dimensional turbulence based on a cascade of enstrophy in Fourier space.