It is a well-known observation in fluidization technology, axial filters and the blood microcirculation that the discharge concentration of a particulate suspension through a small circular side pore which is fed by a large main tube can be significantly lower than the feed concentration. Two underlying mechanisms are believed to be responsible for this exit concentration defect: the fluid skimming from the particle-free layer at the main tube wall and the particle screening due to the hydrodynamic interaction with the pore entrance. In this paper we shall focus our attention only on the first mechanism and shall present a theory which relates the discharge concentration to the dimensionless volume discharge rate 2πQ through the side pore (scaled to the wall shear rate in the main tube and the pore radius) and the ratio of the particle to pore entrance diameters, under creeping flow conditions and for small particle concentrations. First, the shape of the capture tube cross-section upstream of the pore is computed on the basis of a simplified three-dimensional velocity field which neglects the disturbance produced by the orifice on the incoming shear flow. Surprisingly simple closed-form expressions for this shape are derived as Q → ∞ or as Q → 0. Also, using a recently developed exact solution for the simple shear flow past an orifice (Davis 1991), we are able to rigorously demonstrate that, even for small Q, the disturbance produced by the orifice on the shear flow has only a minor effect on the capture tube cross-section far upstream. This simplified flow field is then used to construct a. three-dimensional theory for the discharge concentration defect due to pure fluid skimming for a dilute suspension of spheres. The qualitative features of the theoretical predictions show the same trends as the experimental observations in the microcirculation, although the limits of this theory are well below the observed hematocrit concentrations and the particles are taken as rigid spheres.

We modify a recent theory of Longuet-Higgins (1989a, b) to study the resonant interaction between an isotropic mode and one or two distort ional modes of an oscillating bubble in water when the isotropic mode is forced by ambient sound. Gravity and buoyant rise are ignored. The energy exchange between modes is strong enough so that both (or all three) can attain comparable amplitudes after a long time. We show that for two-mode interactions the mode-coupling equations are similar to those studied in other physical contexts such as nonlinear optics, coupled oscillators and standing waves in a basin. Instability around fixed points is examined for various bubble radii, phase mismatch, and detuning of the external forcing. Numerical evidences of chaotic bubble oscillations and sound radiation are discussed. It is found that in a certain parameter domain, Hopf bifurcations are possible, and chaos is reached via a period-doubling sequence. However, when there are three interacting modes, each of the two distortion modes interacts with the breathing mode directly and the route to chaos is via a quasi-periodic 2-torus. Possible relevance of this theory to the observed erratic drifting of a bubble is discussed.

A two-dimensional acoustical waveguide described by two infinite parallel lines a distance 2d apart has a circle of radius a < d positioned symmetrically between them. The potential satisfies the two-dimensional Helmholtz equation in the fluid region between the circle and the lines, and the normal gradient of the potential vanishes on both. For motions which are antisymmetric about the centreline of the guide there exists a cutoff frequency below which no propagation down the guide is possible. It is proved that for a circle of sufficiently small radius there exists a trapped mode, having a frequency close to the cutoff frequency, which is antisymmetric about the centreline of the guide and symmetric about a line through the centre of the circle perpendicular to the centreline. The method used is due to Ursell (1951) who established the existence of a trapped surface wave mode in the vicinity of a long totally submerged horizontal circular cylinder of small radius in deep water. Numerical computations in the present work reveal that a single trapped mode appears to exist for all values of a ≤ d and not just when the circle is small. The present method, when used to attempt to construct a solution antisymmetric about both the centreline and a line perpendicular to it through the centre of the circle does not lead to a trapped mode. The trapped modes can equally well be regarded as surface-wave modes, as in an infinitely long tank of water with a free surface, into which has been placed symmetrically, a vertical rigid circular cylinder extending throughout the depth. Numerical evidence for the existence of such trapped modes when the cylinder is of rectangular cross-section was presented in Evans & Linton (1991).

It is known that the mixing or spreading rate of free mixing layers decreases with an increase in the convective Mach number of the flow. At supersonic convective Mach number the natural rate of mixing of the shear layers is very small. It is believed that the decrease in mixing rate is directly related to the decrease in the rate of growth of the instabilities of these flows. In an earlier study (Tarn & Hu 1989) it was found that inside a rectangular channel supersonic free shear layers can support two families of instability waves and two families of acoustic wave modes. In this paper the possibility of driving these normal acoustic wave modes into resonant instability by using a periodic Mach wave system is investigated. The Mach waves can be generated by wavy walls. By properly choosing the wavelength of the periodic Mach wave system mutual secular excitation of two selected acoustic wave modes can be achieved. In undergoing resonant instability, the acoustic modes are locked into mutual simultaneous forcing. The periodic Mach waves serve as a catalyst without actually being involved in energy transfer. The resonant instability process is analysed by the method of multiple scales. Numerical results indicate that by using wavy walls with an amplitude-to-wavelength ratio of 1½% it is possible to obtain a total spatial growth of e9 folds over a distance of ten channel heights. This offers reasonable promise for mixing enhancement. The results of a parametric study of the effects of flow Mach numbers, temperature ratio, shear-layer thickness, modal numbers as well as three-dimensional effects on the spatial growth rate of the resonant instability are reported and examined so as to provide basic information needed for future feasibility analysis.

Results from laboratory experiments on oscillatory flows over topograph in a rapidly rotating cylinder of homogeneous liquid are presented and compared with weakly nonlinear and low-order theories. With periodic forcing, the motion can be either periodic or chaotic. In the periodic regime, linear Rossby waves excited by the sloshing flow over shallow bottom topography become resonant at forcing frequencies that are integer multiples of the natural free Rossby wave frequency. As the topographic effect or the forcing amplitude is increased, the maximum response is shifted away from the linearly resonant frequency; to higher periods for azimuthal topographic wavenumbers of 1 and to lower periods for topographic zonal wavenumbers exceeding 1, in agreement with theory. The simple theories which use slippery sidewalk do not describe the observed chaotic flows. These complex states are associated with the development of small-scale vortices in the sidewall boundary layer that are shed into the interior. For both periodic and chaotic flows, long-time particle paths can contain significant chaotic components which are revealed in direct Poincaré sections constructed from observations of surface floats.

The stability of a laminar mixed-convection boundary layer adjacent to a vertical isothermal surface is examined, using linear stability theory and the parallel-flow approximations. The analysis is valid when the imposed forced-convection effects are small compared to natural-convection effects. The stability equations are solved numerically for aiding and opposing forced-convection effects, for Pr = 0.733 (air) and 6.7 (water). For aiding mixed convection in air, a new feature was found. A small, separated region of instability arises upstream of the ‘conventional’, or ‘primary’, neutral curve. In this region, selective amplification of a narrow band of disturbance frequencies occurred, but disturbance growth was small. Further downstream, disturbance growth rates in flows with an aiding free stream are slower than in natural convection. The opposite is true for an opposing free stream in air. Selective disturbance amplification occurred downstream for all conditions, as in natural convection. In water, an aiding flow was destablizing compared to natural convection, and an opposing flow was stabilizing. Evidence of a separated upstream region of instability was also found for aiding mixed convection in water. However, converged solutions could not be obtained in this circumstance.

A formal derivation of evolution equations is given for viscous gravity waves and viscous capillary—gravity waves with surfactants in water of infinite depth. Multiple scales are used to describe the slow modulation of a wave packet, and matched asymptotic expansions are introduced to represent the viscous boundary layer at the free surface. The resulting dissipative nonlinear Schrödinger equations show that the largest terms in the damping coefficients are unaltered from previous linear results up to third order in the amplitude expansions. The modulational instability of infinite wavetrains of small but finite amplitude is studied numerically. The results show the effect of viscosity and surfactants on the Benjamin—Feir instability and subsequent nonlinear evolution. In an inviscid limit for capillary—gravity waves, a small-amplitude recurrence is observed that is not directly related to the Benjamin—Feir instability.

The flow structures giving rise to the force on a circular cylinder in uniform, circular, orbital flows have been investigated for keulegan—Carpenter numbers (K) less than 2 and a Stokes parameter (β) of 483 using the random-vortex method. Comparisons with analysis using the method of inner and outer expansions are made and good agreement is found for K = 0.1. For higher K-values, the viscous force (the difference between the total force and the potential-flow force) acts mainly in opposition to the potential-flow force causing a substantial reduction in total force, in keeping with experimental measurements. Significant separation does not occur at K ≤ 1.5 and vorticity organizes itself asymmetrically about the line through the cylinder centre parallel to the incident velocity vector. Vorticity of one sense of rotation remains close to the half-surface lagging the velocity vector, while an area of vorticity of the opposite sense wraps itself around the cylinder. The net circulation in the flow (the circulation within a path encircling the cylinder at a large radius) is zero. Vortex shedding occurs at K > 1.5. Viscous forces due to non-uniform, orbital flows around a horizontal cylinder beneath waves were similar although vortex shedding tended to occur at lower K-values.

Extensive experimental studies are presented of the effects of prolonged streamline divergence on developing turbulent boundary layers. The experiment was arranged as source flow over a flat plate with a maximum divergence parameter of about 0.075. Mild, but alternating in sign, upstream-pressure-gradient effects on diverging boundary layers are also discussed.

It appears that two overlapping stages of development are involved. The initial stage covers a distance of about 20 initial boundary-layer thicknesses (δ0) from the start of divergence, where the coupled effects of pressure gradient and divergence are present. In this region there is a fairly large reduction in divergence parameter, Rθ (Reynolds number based on momentum thickness) remains constant (≈ 1400) and the boundary-layer properties change rapidly. In the second region, which lasts nearly to the end of the diverging section, the pressure-gradient effects are negligible, the rate of decrease in divergence parameter is very small and Rθ increases gradually. Up to the last measurement station (≈ 100δ0) the flow is still considered to be at a low Reynolds number (Rθ ≈ 2000). For almost the entire length of this region, the profiles of non-dimensional eddy viscosity appear to be self-similar, but have larger values than for the unperturbed flow. Also in this region, beyond 35δ0, the wake parameter, which has reduced significantly, becomes nearly constant and independent of Rθ. On the other hand the entrainment rate attains a constant value at around 50δ0. It appears that the boundary layer reaches a state of equilibrium. It is suggested that this is the result of an enhanced turbulent diffusion to the outer layer. Spectral measurements show that divergence affects mainly the low-wavenumber, large-scale motions. However, there is no change in large-eddy configurations, since the dimensionless structure parameters show only negligible deviations from the unperturbed values.

It is known that the breakup by surface tension of a cylindrical interface containing a viscous liquid can be dampeu by axial motion of the underlying liquid and that for an annular film the capillary instability can be completely suppressed (disturbances of all wavelengths decay) by certain axial velocity profiles. Here, using a linear stability analysis, it is shown that complete stabilization can also occur for thermocapillary-driven axial motions. However, the influence of thermocapillary instabilities typically shrinks the window in parameter space where stabilization is found, relative to the isothermal case. The influence of Reynolds, surface tension, Prandtl, and Biot parameters on limits of stabilization is calculated using continuation techniques. It is observed that windows of stabilization first open with topological changes of the neutral curves in parameter space. A long-wave analysis unfolds the nature of the singularities responsible for several of these topological changes. The analysis also leads to the physical mechanism responsible for (longwave) stabilization and in certain cases to necessary conditions for (long-wave) stabilization.

Instabilities and long-time evolution of gravity-capillary wavetrains (ripples) with moderate steepnesses (ε < 0.3) are studied experimentally and analytically. Wave-trains with frequencies of 8 ≤ f ≤ 25 Hz are generated mechanically in a channel containing clean, deep water; no artificial perturbations are introduced. Frequency spectra are obtained from in situ measurements; two-dimensional wavenumber spectra are obtained from remote sensing of the water surface using a high-speed imaging system. The analytical models are in viscid, uncoupled NLS (nonlinear Schrödinger) equations: one that describes the temporal evolution of longitudinal modulations and one that describes the spatial evolution of transverse modulations.

The experiments show that the evolution of wavetrains with sensible amplitudes and frequencies exceeding 9.8 Hz is dominated by modulational instabilities, i.e. resonant quartet interactions of the Benjamin–Feir type. These quartet interactions remain dominant even for wavetrains that are unstable to resonant triad interactions (f > 19.6 Hz) – if selective amplification does not occur (see Parts 1 and 2). The experiments further show that oblique perturbations with the same frequency as the underlying wavetrain, i.e. rhombus-quartet instabilities, amplify more rapidly and dominate all other modulational instabilities. The inviscid, uncoupled NLS equations predict the existence of modulational instabilities for wavetrains with frequencies exceeding 9.8 Hz, typically underpredict the bandwidth of unstable transverse modulations, typically overpredict the bandwidth of unstable longitudinal modulations, and do not predict the dominance of the rhombus-quartet instability. When the effects of weak viscosity are incorporated into the NLS models, the predicted bandwidths of unstable modulations are reduced, which is consistent with our measurements for longitudinal modulations, but not with our measurements for transverse modulations.

Both the experiments and NLS equations indicate that wavetrains in the frequency range 6.4–9.8 Hz are stable to modulational instabilities. However, in these experiments, wavetrains with sensible amplitudes excite one of the members of the Wilton ripples family. When second-harmonic resonance occurs, both the first-and second-harmonic wavetrains undergo rhombus-quartet instabilities. When third-harmonic resonance occurs, only the third-harmonic wavetrain undergoes rhombus-quartet instabilities.

Cavitation has been observed in the trailing vortex system of an elliptic planform hydrofoil. A complex dependence on Reynolds number and gas content is noted at inception. Some of the observations can be related to tension effects associated with the lack of sufficiently large-sized nuclei. Inception measurements are compared with estimates of pressure in the vortex obtained from LDV measurements of velocity within the vortex. It is concluded that a complete correlation is not possible without knowledge of the fluctuating levels of pressure in tip-vortex flows. When cavitation is fully developed, the observed tip-vortex trajectory shows a surprising lack of dependence on any of the physical parameters varied, such as angle of attack, Reynolds number, cavitation number, and dissolved gas content.

We explain the emergence of organized structures in two-dimensional turbulent flows by a theory of equilibrium statistical mechanics. This theory takes into account all the known constants of the motion for the Euler equations. The microscopic states are all the possible vorticity fields, while a macroscopic state is defined as a probability distribution of vorticity at each point of the domain, which describes in a statistical sense the fine-scale vorticity fluctuations. The organized structure appears as a state of maximal entropy, with the constraints of all the constants of the motion. The vorticity field obtained as the local average of this optimal macrostate is a steady solution of the Euler equation. The variational problem provides an explicit relationship between stream function and vorticity, which characterizes this steady state. Inertial structures in geophysical fluid dynamics can be predicted, using a generalization of the theory to potential vorticity.

A theory is derived for general motions of an inviscid vortex with circular crosssection and variable core area using a directed filament model of the vortex. The theory reduces in special cases to any of several previous vortex theories in the literature, but it is better suited than previous theories for handling nonlinear areavarying waves on the vortex core. Jump conditions across points of discontinuity and a variational form of the theory are also given. The theory is applied to obtain new solutions for several fundamental problems in vortex dynamics related to axisymmetric solitary waves on a vortex core, axisymmetric and helical vortex breakdowns and the buckling of a columnar vortex under compression.

The problem investigated is the stability of a flame anchored by recirculation within a channel with a cavity, acting as a two-dimensional approximation to a gas turbine combustion chamber. This is related to experiments of Vaneveld, Hom & Oppenheim (1982). The hypothesis studied is that hydrodynamic oscillations within the cavity can lead to flashback.

The method used is a semi-analytical-numerical technique where the conservation equations for enthalpy and fuel fraction are represented by the low-Mach-number combustion model of Ghoniem, Chorin & Oppenheim (1982). Burnt and unburnt gas are treated as incompressible fluids where the reaction zone acts as a source for volume expansion. The flame is modelled by a Lagrangian technique using a simple line interface calculation algorithm.

The turbulent flow field is determined using conformal mapping theory and the hybrid random vortex method. The vorticity generation takes place at the walls to achieve no slip, and is influenced by boundary-layer separation. To avoid locating the separation points a priori the numerical viscous sublayer is extended continuously past the corners, and their singularities are in effect cut off by using locally a corner rounding technique within the conformal mapping.

The computed unsteady boundary-layer separation and reattachment of the non-reacting flow field agrees with unsteady boundary-layer theory. On the basis of the numerical simulations of the flame stability problem it is concluded that hydrodynamic oscillations within the cavity, combined with unsteady boundary-layer separation and reattachment can cause a flashback.

A viscous-liquid drop spreads on a smooth horizontal surface, which is uniformly heated or cooled. Lubrication theory is used to study thin drops subject to capillary, thermocapillary and gravity forces, and a variety of contact-angle-versus-speed conditions. It is found for isothermal drops that gravity is very important at large times and determines the power law for unlimited spreading. Predictions compare well with the experimental data on isothermal spreading for both two-dimensional and axisymmetric configurations. It is found that heating (cooling) retards (augments) the spreading process by creating flows that counteract (reinforce) those associated with isothermal spreading. For zero advancing contact angle, heating will prevent the drop from spreading to infinity. Thus, the heat transfer serves as a sensitive control on the spreading.

We examine the stability to superharmonic disturbances of finite-amplitude two-dimensional travelling waves of permanent form in plane Poiseuille flow. The stability characteristics of these flows depend on whether the flux or pressure gradient are held constant. For both conditions we find several Hopf bifurcations on the upper branch of the solution surface of these two-dimensional waves. We calculate the periodic orbits which emanate from these bifurcations and find that there exist no solutions of this type at Reynolds numbers lower than the critical value for existence of two-dimensional waves (≈2900). We confirm the results of Jiménez (1987) who first detected a stable branch of these solutions by integrating the two-dimensional equations of motion numerically.

The linear stability of the zonal shear flow $\overline{u} = - \sec h^2 y$ is investigated in the framework of the beta-plane approximation. This retrograde jet is known to be more unstable than its eastward-propagating counterpart and has some surprising characteristics. First, this is a rare example of a flow in which barotropically unstable modes occur that do not have a critical point. Secondly, singular neutral modes exist in which the critical point occurs at the centre of the jet, where $\overline{u}^{\prime}_{\rm c}=0 $. It is shown in this paper that such singular modes form part of the stability boundary both for the varicose mode and also for the radiating sinuous mode.

A lifting-line theory is developed for wings of large aspect ratio oscillating in an inviscid fluid. The theory is unified in the sense that the wing may be curved or inclined to the flow, and the asymptotic expansion is uniformly valid with respect to the frequency. The method is based on the integral equation formulation of the problem. The technique, pioneered by Kida & Miyai (1978). consists of asymptotically solving the Fredholm equation of the first kind which links the unknown pressure jump and the normal velocity imposed on the wing. Use of the finite-part integral theory introduced by Hadamard (1932) and of a technique developed in Guermond (1987, 1988, 1990) yields an asymptotic expansion of the surface integral in terms of the inverse of the aspect ratio. At each approximation order, the problem reduces to a classical two-dimensional integral equation, whose unknown is the pressure jump, and whose right-hand side depends only on the previous approximation orders of the solution. The first finite-span correction is explicitly calculated. An extensive numerical study is carried out, and comparisons with published results are made.

Although the jet momentum flux has been traditionally accepted as constant, this is not in general true because a weak pressure field is induced in the ambient fluid with positive gradient and because the induced flow field carries momentum flux to the jet. The angle ϕ, at which the induced flow streamlines enter the jet, is the basic parameter which determines whether the jet momentum flux increases, remains constant or decreases. A theoretical solution is presented for the variation of the jet momentum flux in turbulent submerged jets in stationary ambient fluid. The solution presented in this paper generalizes previous theoretical solutions and is in good agreement with existing experimental results. The contribution of the induced pressure field relative to the induced velocity field in varying the jet momentum flux is investigated. The induced flow streamlines are calculated using non-constant jet momentum flux and are compared with Taylor's solution (where constant jet momentum flux was assumed).