The problem of whether a stream of microscopic particles may be concentrated into a focal point by entrainment within a carrier gas is considered for dilute particles linearly coupled to the velocity field of an incompressible gas. Typically, the dynamical behaviour of the particles is governed by a so-called Stokes number S, the product of their relaxation time and a characteristic value of the velocity gradient in the suspending fluid. An inequality due to Robinson (1956) is used to illustrate the natural tendency of potential flows to concentrate the particles. For geometries with planar or axial symmetry, with errors cubic in their initial distance to the axis, the trajectories of identical particles originating near an axis of symmetry are shown to cross it at a common focal point provided they have some initial convergence and their Stokes number is larger than a critical value S*. The position of the focal point of supercritical particles depends on their Stokes number, tending to infinity as S approaches S*. Particle trajectories originating far from the axis of symmetry are seen to cross the centreline at defocused positions, in analogy with the optical geometric aberration effect. The focusing phenomenon is illustrated numerically for two-dimensional potential flows through nozzles of several geometries and also analysed in the proximity of the axis of symmetry. For these examples, the threshold value S* of the Stokes number for focusing is of order one, over an order of magnitude larger than typical values of the familiar critical Stokes number marking the onset of particle impaction on solid surfaces. The focal width may be made over two orders of magnitude smaller than the nozzle diameter by restricting the region where particles are seeded to a moderate angle away from the axis. This angle may be higher than ¼π for the case of a jet exiting through a slit in an infinitely thin plate. There is also some discussion of the use of high-resolution focusing instruments.

The fully developed region of periodic flows through curved pipes of circular cross-section and arbitrary curvature has been simulated numerically. The volumetric flow rate, prescribed by a cosine function, remains positive throughout the entire cycle. Such flows are characterized by three parameters: the frequency parameter α, the amplitude ratio γ and the mean Dean number κm. We use the Projection Method to solve the finite-difference approximation of the Navier–Stokes equations in their primitive form. The effect of κm on the flow has been extensively studied for the range 0.7559 [les ] κm [les ] 756 for α = 15 and γ = 1, and the curvature ratio, δ, equal to $\frac{1}{7}$. Interactions between the Stokes layer and the interior are noted and a variety of pulsatile motions along with reversal of the axial-flow direction are revealed. The manner in which the secondary motions evolve with increasing Dean number, and how they change direction from outward to inward ‘centrifuging’ at the centre, is also explained. Reversal in the axial flow is observed for all values of Dean number studied and occupies a region ranging from the area immediately adjacent to the entire wall for low values of Dean number to the entire inner half of the cross-section for larger values. When reversal of the axial flow is present, the local maximum axial shear stress is found at the inner bend where the backflow region is located. The values of circumferential shear stress for κm = 0.7559 and 151.2 confirm the existence of a single-vortex structure in the half-cross-section, whereas the values for larger values of mean Dean number are indicative of more complicated vortical structures.

The temporal evolution of the flow patterns about a shallow bottom topography in a deep, rapidly rotating, stratified flow is studied on the basis of an inviscid Boussinesq model. The initial boundary-value problem, linearized for a thin three-dimensional obstacle, is solved for an impulsively started flow. The indicial response obtained reveals a window for the horizontal wavenumber spectra of the obstacle geometry. Only the portion of spectra within this window, which is shut at the start and widens linearly with increasing time, contributes to the solution. Thus, only the relatively large-scale, cyclonic feature associated with the wavenumber origin can dominate the flow in the early period, while the more familiar inertial wave system emerges much later.

Examples of solutions computed via an FFT algorithm confirm that, except in the two opposite limits for the zero and infinite stratification, the cyclonic disturbance and inertial waves coexist, but a solitary pressure hill associated with the cyclonic disturbance remains dominant throughout most evolution stages. For a sufficiently strong stratification, the solution to the linear pressure equation suggests the emergence of a secondary eddy in the lee; its significance and validity are discussed.

The behaviour of an evolving, stably stratified turbulent shear flow was investigated in a ten-layer, closed-loop, salt-stratified water channel. Simultaneous single-point measurements of the mean and fluctuating density and longitudinal and vertical velocities were made over a wide range of downstream positions. For strong stability, i.e. a mean gradient Richardson number Ri greater than a critical value of Ricr ≈ 0.25, there is no observed growth of turbulence and the buoyancy effects are similar to those in the unsheared experiments of Stillinger, Helland & Van Atta (1983) and Itsweire, Helland & Van Atta (1986). For values of Richardson number less than Ricr the turbulence grows at a rate depending on Ri and for large evolution times the ratio between the Ozmidov and turbulent lengthscale approaches a constant value which is also a function of Richardson number.

Normalized velocity and density power spectra for the present experiments conform to normalized spectra from previous moderate- to high-Reynolds-number studies. With increasing $\tau = (x/\overline{U}) (\partial \overline{U}/\partial z)$ or decreasing stability, the stratified shear spectra exhibit greater portions of the universal non-stratified spectrum curve. The shapes of the shear-stress and buoyancy-flux cospectra confirm that they act as sources and sinks for the velocity and density fluctuations.

This paper examines a flow–aerofoil interaction problem that we believe is likely to be an important issue in advanced aircraft propulsion systems involving supersonic propellers. They are potentially noisy and it is important to identify the mechanism by which they generate noise so that they can be optimized for acceptable operation. One of the most potent sources of noise lies in the possibility that a second stage propeller blade cuts through the core of the tip vortex shed from a first stage blade. Then suddenly the streaming core flow must adjust to the boundary constraints of that second stage blade, and the adjustment will inevitably involve compressive waves that escape as sound. How strong these sound waves are, how long their life is and where they go to are important questions, the answers to some of which are obtained in this paper. We examine here what we think is a canonical problem and determine the level and directionality of the sound generated by the interruption of the axial vortex-core flow by a supersonic blade. The principal sound is launched in the Mach-wave direction, where the pressure pulse has an amplitude that decreases much more slowly than it would from spherical spreading. This pressure pulse can reach a distant observer with a very large amplitude, 160 dB or higher being typical. The peak sound pressures are found to be independent of blade speed at high supersonic tip velocity, while the energy radiated in the pulse, because of its reducing duration, attenuates as the supersonic speed increases. This aspect gives grounds for believing that the higher the speed, the quieter will be the stage interaction sound of a contrarotating supersonic propeller.

The flow field in a two-dimensional mixing layer, highly disturbed by a sinusoidally oscillating flap, was mapped in order to estimate the significance of the nonlinear processes associated with the large coherent structures existing in this flow. A mixing layer which does not diverge linearly in the direction of streaming is loosely defined as being highly disturbed. Two velocity components were measured throughout the flow field using a rake of X-wire probes. Streaklines were calculated from the phase-locked measured data and were compared to pictures of smoke injected into the flow, creating a link between flow visualization and quantitative experimental results. The phase-locked vorticity and the Reynolds stresses were calculated from these measurements.

It was determined that fluctuations, locked in phase with the disturbance frequency, were not only responsible for the fast initial growth of the mixing layer but also for its contraction farther downstream (the occurrence of regions I and II in the parlance of Oster & Wygnanski 1982). The resumption of the growth of the mixing layer in region III is not controlled by the phase-locked oscillations. The first subharmonic of the imposed frequency was insignificant everywhere, and vortex amalgamation was not observed by visual means.

Detailed comparisons between experimental results and theoretical calculations, based on a linear stability model, were carried out. The theory predicted very well the normalized, cross-flow distribution of any quantity that was measured, but failed to predict the amplification rates of these quantities in the direction of streaming.

The flow field in a plane, turbulent mixing layer disturbed by a small oscillating flap was investigated. Three experiments were carried out: one in which the flap oscillated sinusoidally at a single frequency: a second in which the flap oscillated at two frequencies, a fundamental and a subharmonic, but the ensuing motion was dominated by the fundamental perturbation; and a third in which the amplitude of the subharmonic perturbation was increased until a distortion in the mean flow was noticeable. Two velocity components were measured at all phase angles relative to the subharmonic oscillation of the flap at densely spaced intervals. The data were used to map the phase-locked spanwise vorticity component and the phase-locked streakline patterns for the purpose of assessing the relevance of the latter to the understanding of the dynamical process involved.

This paper deals with the Soret separation of a binary mixture in a cylinder subjected to an axial temperature gradient. The study is connected to an experiment designed to measure the Soret coefficient of an AgI-KI mixture corresponding to a moderate Prandtl number (Pr = 0.6) and a high Schmidt number (Sc = 60). In such an experiment the species separation is often hidden by a mixing effect due to the buoyancy-driven convection generated by a horizontal temperature gradient induced by some defect of the heating system. Here, such a defect is simulated by a slight misorientation of the cell with respect to the vertical; a small inclination (γ = 1°) of the cell has been considered, but the results can be generalized for any other small γ. For situations corresponding to a top heating and a positive Soret parameter, S, two quite different regimes have been exhibited depending on the value of S. For moderate S, the induced solutal buoyancy balances the imposed thermal buoyancy, slowing down the flow and giving a good separation rate. For small S this balance does not exist (except in the centre), leading to a remixing of the species and thus to poor separation (the separation would be still worse for negative S). The smaller the (positive) Soret parameter is, the smaller the cell misorientation γ has to be to allow a good separation rate.

Groups of short waves within a narrow frequency band are known to be accompanied by second-order long waves travelling at the group velocity of the predominant short waves. When the short waves are refracted by bottom topography, new long waves can be further radiated and propagated away from the topography at the shallow-water speed. Since over a long submarine ridge there can be trapped modes of long-period waves, incident groups of short waves can excite the trapped waves through a second-order mechanism. In this paper we study such excitations over a rectangular shelf which scatters the first-order short waves. By employing asymptotic methods we examine the transient excitation of the trapped long wave by both sinusoidal wave groups and wave packets. The effects of a small angular spread of the incident waves are also included.

‘Bioconvection’ is the name given to pattern-forming convective motions set up in suspensions of swimming micro-organisms. ‘Gyrotaxis’ describes the way the swimming is guided through a balance between the physical torques generated by viscous drag and by gravity operating on an asymmetric distribution of mass within the organism. When the organisms are heavier towards the rear, gyrotaxis turns them so that they swim towards regions of most rapid downflow. The presence of gyrotaxis means that bioconvective instability can develop from an initially uniform suspension, without an unstable density stratification. In this paper a continuum model for suspensions of gyrotactic micro-organisms is proposed and discussed; in particular, account is taken of the fact that the organisms of interest are non-spherical, so that their orientation is influenced by the strain rate in the ambient flow as well as the vorticity. This model is used to analyse the linear instability of a uniform suspension. It is shown that the suspension is unstable if the disturbance wavenumber is less than a critical value which, together with the wavenumber of the most rapidly growing disturbance, is calculated explicitly. The subsequent convection pattern is predicted to be three-dimensional (i.e. with variation in the vertical as well as the horizontal direction) if the cells are sufficiently elongated. Numerical results are given for suspensions of a particular algal species (Chlamydomonas nivalis); the predicted wavelength of the most rapidly growing disturbance is 5–6 times larger than the wavelength of steady-state patterns observed in experiments. The main reasons for the difference are probably that the analysis describes the onset of convection, not the final, nonlinear steady state, and that the experimental fluid layer has finite depth.

We investigate numerically the time evolution of a two-dimensional flow submitted to a spatially periodic shear force. Initially, the flow is at equilibrium, the forcing balancing viscous stresses. At Reynolds numbers slightly above critical, a large-scale, linear instability drives the fluid towards a stable laminar state. At larger Reynolds number turbulence finally develops after several transient states. These transient states are described by measuring the divergence rate of linearized trajectories from the turbulent flow. This rate gives asymptotically a measure of the first Lyapunov exponent of the flow. We find that the first Lyapunov exponent scales as the characteristic frequency of the flow at large scale. We show here data on incompressible, isothermal and perfect gas (subsonic) two-dimensional flows with unit Prandtl number, and Reynolds number around 30.

A general method for computing the hydrodynamic interactions among an infinite suspension of particles, under the condition of vanishingly small particle Reynolds number, is presented. The method follows the procedure developed by O'Brien (1979) for constructing absolutely convergent expressions for particle interactions. For use in dynamic simulation, the convergence of these expressions is accelerated by application of the Ewald summation technique. The resulting hydrodynamic mobility and/or resistance matrices correctly include all far-field non-convergent interactions. Near-field lubrication interactions are incorporated into the resistance matrix using the technique developed by Durlofsky, Brady & Bossis (1987). The method is rigorous, accurate and computationally efficient, and forms the basis of the Stokesian-dynamics simulation method. The method is completely general and allows such diverse suspension problems as self-diffusion, sedimentation, rheology and flow in porous media to be treated within the same formulation for any microstructural arrangement of particles. The accuracy of the Stokesian-dynamics method is illustrated by comparing with the known exact results for spatially periodic suspensions.

Two-dimensional steady surface waves on a shearing flow are computed for the special case where the flow has uniform vorticity, i.e. in the absence of waves the velocity varies linearly with height. A boundary-integral method is used in the computation which is similar to that of Simmen & Saffman (1985) who describe such waves on deep water. Particular attention is given to the effects of finite depth with descriptions of waves of limiting steepness, waves with eddies beneath their crests and extremely high waves on high-speed flows.

Many qualitative features of these waves are relevant to steep waves propagating in shallow water, or on a strong wind-induced drift current. An important practical point in the interpretation of wave measurements of wind driven waves is that mean kinetic energy and potential energy densities are unequal even for infinitesimal waves. This may mean that wave energy spectra deduced from surface-elevation measurements in the conventional way may sometimes be misleading.

Two like-signed vorticity regions can pair or merge into one vortex. This phenomenon occurs if the original two vortices are sufficiently close together, that is, if the distance between the vorticity centroids is smaller than a certain critical merger distance, which depends on the initial shape of the vortex distributions. Our conclusions are based on an analytical/numerical study, which presents the first quantitative description of the cause and mechanism behind the restricted process of symmetric vortex merger. We use two complementary models to investigate the merger of identical vorticity regions. The first, based on a recently introduced low-order physical-space moment model of the two-dimensional Euler equations, is a Hamiltonian system of ordinary differential equations for the evolution of the centroid position, aspect ratio and orientation of each region. By imposing symmetry this system is made integrable and we obtain a necessary and sufficient condition for merger. This condition involves only the initial conditions and the conserved quantities. The second model is a high-resolution pseudospectral algorithm governing weakly dissipative flow in a box with periodic boundary conditions. When the results obtained by both methods are juxtaposed, we obtain a detailed kinematic insight into the merger process. When the moment model is generalized to include a weak Newtonian viscosity, we find a ‘metastable’ state with a lifetime depending on the dissipation timescale. This state attracts all initial configurations that do not merge on a convective timescale. Eventually, convective merger occurs and the state disappears. Furthermore, spectral simulations show that initial conditions with a centroid separation slightly larger than the critical merger distance initially cause a rapid approach towards this metastable state.

The axial and radial velocity components w and u, and the concentration c of a Rhodamine 6G dye were measured simultaneously in a turbulent buoyant jet, using laser-Doppler anemometry combined with a recently developed laser-induced-fluorescence concentration measurement technique. These non-intrusive techniques enable measurements in a region of plume motion where conventional probe-based techniques have had difficulties. The results of the study show that the asymptotic decay laws for velocity and concentration of a tracer transported by the flow are verified experimentally in both jets and plumes. The momentum and volume fluxes and the mean dilution factor are determined in dimensionless form as a function of the normalized distance from the flow source. Contradictory results from earlier experimental plume investigations concerning the decay laws of w and c and the plume width ratio bc/bw are discussed. The turbulence properties and the transition from momentum-driven jets to buoyancy-driven plumes are presented. The turbulence is found to scale with the mean flow as predicted by dimensional analysis and self-similarity. Buoyancy-produced turbulence is found to transport twice as much tracer as jet turbulence. Although velocity statistics in jets and plumes are found to be highly self-similar there is a strong disparity in the distribution of tracer concentration in the two flows. This occurs in the time-average mean flows as well as the r.m.s. turbulent quantities. Instantaneous concentration fluctuations are found to exceed time averages by as much as a factor of 3. The experimental results should provide a reasonable basis for validation of computer models of axisymmetric plumes.

Mass transfer between a flowing region and an adjacent stationary medium can greatly alter the overall contaminant dispersion. Here, an extension is given of Taylor's (1953) method to encompass this class of complications. The only mathematical assumption made is that the mass flux transfer at the boundary depends linearly upon the concentration at earlier times. Expressions are derived for the longitudinal shear dispersion coefficient. Detailed results are presented for the effects both of reactions and of retention at the bed upon contaminant dispersion in turbulent open-channel flow.

The symmetry-plane laminar boundary layer over an impulsively-started prolate spheroid of axes ratio 1/4 at various incidence is calculated in detail. Results agree with the steady solutions at large times. The most important one is concerned with the similarity between the distribution of the leeside skin friction at a fixed incidence, but varying in time, and that of the leeside skin friction for steady flows varying in incidence. The latter patterns led previously to the concept of an open and closed separation sequence for steady flows, likewise the newly found similarity suggests an unsteady open and closed sequence; i.e. at low incidence, separation starts around the rear stagnation point and gradually expands upstream in time, but it is always of the closed type. At moderate to high incidence, closed separation prevails at small times, open separation develops at large times, but separation may either remain open at moderate incidence or return to closed at high incidence as the steady-state condition is approached. The rate of approach toward the steady-state condition increases with incidence. For a less slender spheroid there is no open separation involved; unsteady separation lines are all of the closed type. For bodies other than spheroids, similar ideas may be applied.

The consolidation or concentration of suspended particulate solids under the influence of gravitational forces is a problem of widespread practical and theoretical interest. The literature, which is scattered over several fields, contains most of the elements necessary for a complete understanding of gravity settling, but considerable controversy and confusion persists about their synthesis. Here we propose to construct a quantitative theory covering the full range of processes from transient settling of large, stable particles to the slow consolidation of flocculated suspensions of submicron particles. Conditions for the existence of shocks are identified and the basic equations describing the phenomena are solved numerically for several Péclet numbers.

We consider horizontal static liquid layers on planar solid boundaries and analyse their instabilities. The layers are either evaporating, when the plates are heated, or condensing, when the plates are cooled. Vapour recoil, thermocapillary, and rupture instabilities are discussed, along with the effects of mass loss (or gain) and non-equilibrium thermodynamic effects. Particular attention is paid to the development of dryout. We derive long-wave evolution equations for the interface shapes that govern the two-dimensional nonlinear stability of the layers subject to the above coupled mechanisms. These equations are analysed and their predictions discussed. Previous theoretical and experimental results are reviewed and compared with the present results. Finally, we discuss limitations of the modelling and extend our derivation to the case of three-dimensional disturbances.

We have investigated the modulated and unmodulated travelling azimuthal waves appearing on the toroidal Taylor–Görtler (TG) vortices in a fluid contained between two concentric spheres with the inner sphere rotating. For smaller-clearance cases, toroidal TG vortices appear at the equator, just as in the flow between two concentric cylinders with the inner cylinder rotating. When the Reynolds number of the flow increases quasi-statically, spiral TG vortices appear in addition to toroidal TG vortices, and no modulation occurs, even if the Reynolds number further increases quasi-statically. However, when the Reynolds number is increased from zero to a particular value with a specific acceleration of the inner sphere, modulated wavy toroidal TG vortices appear. We found that the necessary condition for occurrence of modulation is the prevention of spiral TG vortices. Using simultaneous flow-visualization and spectral techniques, and measuring the fluctuation of sinks and sources of vortex boundaries, we obtained the frequency f1 of travelling azimuthal waves passing a fixed point in the laboratory and the modulation frequencies f2 and f′2 of these waves, as determined by an observer in the laboratory and an observer fixed in a reference frame that rotates in phase with the travelling azimuthal waves, respectively. The relationship among the characteristic frequencies, f1, f2 and f′2, obtained by modal analysis and the experimental results, is (f′2 + kf1/m)/f2 = − 1, where k and m are a modulation parameter and the wavenumber of travelling azimuthal waves, respectively.