The flow induced by a localized droplet of soluble surfactant on the surface of a thin film is analysed, motivated by an interest in the interaction between inhaled droplets and the lung's thin liquid lining after an aerosol lands on its surface. The spreading is driven by gradients of surface tension and results in flow of the droplet and underlying liquid film. This induced flow field plays an important role in the transport of dissolved species from the droplet, through the film, and to the tissue for absorption.

Evolution equations for the film thickness, surface and bulk liquid concentrations are derived using lubrication theory, since the depth of the film is much smaller than the characteristic radius of the droplet. Solutions are obtained numerically using the method of lines for a variety of surface Péclet numbers.

We find that the effect of solubility is to decrease both film disturbances and surface concentrations, and to induce an absorption-driven backflow. In addition, there is a gravity-driven backflow from hydrostatics. At large surface Péclet numbers, large film disturbances are obtained and more surfactant is able to diffuse across the rigid permeable wall, while surface diffusion causes more rapid spreading and decreases film disturbances. Gravity acts as a restoring force by creating a bidirectional flow, and hence disenhances the vertical flux of surfactant across the air-liquid interface. This model may have implications for the delivery of drugs by aerosol inhalation.

Matched asymptotic expansion methods are used to establish governing equations of Oseen type for a tethered sphere that describes a circular path and a stationary sphere subjected to a rotating fluid in an ‘antisedimentation’ tube. The two cases are shown to be significantly different, in contrast to an earlier presentation (Davis & Brenner 1986), because only the latter is subject to the Coriolis force. The evaluation of the force and torque coefficients is much improved, enabling better comparisons to be made with the classical rectilinear trajectory result of Proudman & Pearson (1957).

Trace impurities often collect on the upstream side of an obstacle in the surface of flowing liquid. The transition from practically free surface to surface sufficiently clogged to be treated as stationary can be quite sharp. The viscous flow underneath is nonlinearly coupled to the convective mass transfer of surface-active material. For two-dimensional flow at high Reynolds number the first observations were due to Thoreau, Langton and Reynolds over 100 years ago, and the theory was given by Harper & Dixon in 1974. If the whole problem is considered from a frame of reference moving with the stream instead of fixed to the downstream surface film, the solution refers to the leading edge of a slowly spreading oil slick.

The present work gives the theory corresponding to Harper & Dixon's for low Reynolds numbers (Stokes flow), for which there is a very simple leading approximation near the transition for a soluble surfactant, and a more complicated one, which can still be found exactly, for an insoluble surfactant which spreads onto clear liquid by surface diffusion. In both cases the surface remains flat: the ridge often observed is not a Stokes flow phenomenon.

The results are used to clarify the circumstances in which Savic's stagnant-cap approximation is useful for a bubble rising in a viscous liquid: the rear stagnation point now plays the role of the obstacle in the surface, and the flow near the surface transition can be treated locally as if it were two-dimensional instead of axisymmetric.

An experimental and analytical investigation is conducted into the dynamics of small, non-spherical, rigid particles suspended in a flow that is time-dependent from the point of view of the particle, which may be moving. The particles are unaffected by Brownian forces. Of special interest in this study are flows that are time-periodic in the Lagrangian frame; this allows for the use of mathematical tools that have been developed for periodically forced differential equations, and for precise and well-characterized experiments. For some classes of periodic flows, it is shown that there exists a global periodic attractor for the orientation dynamics of particles that follow a given particle path, i.e. there is 1:1 phase locking of the orientation dynamics with the forcing. This is an important situation because it leads to a strong ordering of an ensemble of particles that follow the same particle path as the flow; such order has significant ramifications for stress and birefringence. In other classes of periodic flows, no such attractor exists; therefore, an ensemble of random initial orientations of the particles on a particle path will not converge and disorder is maintained. Experiments are performed using a computer-controlled four-roll mill to create well-characterized flows in which these various types of dynamical behaviour are realized.

We study fixed-flux convection in a long, narrow slot which is inclined to the horizontal. (Gravity is in the vertical direction, and horizontal is perpendicular to this.) Because of the fixed-flux boundary conditions the convective modes have much larger lengthscales in the along-slot direction than in the transverse direction. In the case of a horizontal slot this disparity in scales has been previously exploited to obtain an amplitude equation for the single mode which first becomes unstable as the Rayleigh number is increased above critical. When the slot is tilted we show that there is a distinguished limit in which there are two active modes in the slightly supercritical regime. This new limit is when the horizontal wavenumber, the supercriticality, and the tilt of the slot away from vertical, are all small. A modification of the well-known expansion for fixed flux convection in a horizontal slot leads to a coupled system of partial differential equations for the amplitudes of the two modes.

Numerical solution of this system suggests that all initial conditions eventually evolve into one of the two states, both of which consist of a single, steady roll in the cavity. The states are distinguished by the direction of circulation of the roll, and by the buoyancy fields, which are quite different in the two cases.

Direct numerical simulation is used to determine the flows that occur as the Reynolds number, Re, is increased in a plane channel undergoing system rotation about a spanwise axis. (Plane Poiseuille flow occurs for zero rotation rate and low Re.) A constant system rotation speed of 0.5, non-dimensionalized with respect to the bulk streamwise velocity and channel full width, is used throughout. The spectral numerical method solves the three-dimensional, time-dependent, incompressible Navier-Stokes equations using periodic boundary conditions in the streamwise and spanwise directions. On increasing the Reynolds number above the temporally periodic wavy vortex regime, near Re = 4Rec (Rec = 88.6 is the critical Re for development of vortices), a second temporal frequency, ω2, occurs in the flow that corresponds to slow, constant, spanwise motion of the vortices, superposed on the much faster, constant, streamwise motion of the wavy vortex waves. Curiously, ω2 is always frequency locked with the wavy vortex frequency ω1 for the parameter range explored, although the locking ratio varies. At the slightly higher Re of 4.1 Rec, ω2 is replaced by a new frequency ω′2 that corresponds to a modulation of the wavy vortices like that seen in modulated wavy Taylor vortex flow. However, unlike the Taylor-Couette geometry, the modulation frequency here can become frequency locked with the wavy vortex frequency. Increasing Re further to Re = 4.2 Rec results in the appearance of a second incommensurate modulation frequency ω3, yielding a quasi-periodic three-frequency flow, although there are only two frequencies (ωω′2 and ω3) present in the reference frame moving with the travelling wave associated with ω1. At still higher Re (Re = 4.5 Rec), weak temporal chaos occurs. This flow is not turbulent however. Calculations of the instantaneous largest Lyapunov exponent, λ(t), and the spatial structure of small perturbations to the flow show that the chaos is driven by spanwise shear instability of the streamwise velocity component. At the highest Re of 6.7 Rec considered, quasi-coherent turbulent boundary layer structures occur as transient, secondary streamwise-oriented vortices in the viscous sublayer near the inviscidly unstable (high-pressure) wall. Calculations of λ(t) and the spatial structure of small perturbations to the flow show that the coherent structures are not caused by the local growth of small disturbances to the flow.

An experimental investigation is described which has as its objectives the extension of the technical data base pertaining to roughness-induced transition and the advancement of the understanding of the physical processes by which three-dimensional roughness elements induce transition from laminar to turbulent flow in boundary layers. The investigation was carried out primarily with single hemispherical roughness elements surface mounted in a well-characterized zero-pressure-gradient laminar boundary layer on a flat plate. The critical roughness Reynolds number at which turbulence is regarded as originating at the roughness was determined for the roughness elements herein considered and evaluated in the context of data existing in the literature. The effect of a steady and oscillatory free-stream velocity on eddy shedding was also investigated. The Strouhal behaviour of the ‘hairpin’ eddies shed by the roughness and role they play in the evolution of a fully developed turbulent boundary layer, as well as whether their generation is governed by an inflexional instability, are examined. Distributions of mean velocity and intensity of the u-fluctuation demonstrating the evolution toward such distributions for a fully developed turbulent boundary layer were measured on the centreline at Reynolds numbers below and above the critical Reynolds number of transition. A two-region model is postulated for the evolutionary change toward a fully developed turbulent boundary layer: an inner region where the turbulence is generated by the complex interaction of the hairpin eddies with the pre-existing stationary vortices that lie near the surface and are inherent to a flow about a three-dimensional obstacle in a laminar boundary layer; and an outer region where the hairpin eddies deform and generate turbulent vortex rings. The structure of the resulting fully developed turbulent boundary layer is discussed in the light of the proposed model for the evolutionary process.

Experimental and numerical studies were made of the secondary flow induced in fully developed oscillatory laminar flow in a curved circular pipe. Photographs of traces of nylon particles suspended in water were taken systematically with Womersley numbers α = 5.5 ∼ 28 and oscillatory Dean numbers D = 40 ∼ 500. The secondary flow velocity component and the location of the vortex eye were obtained from the photographs. The experimental results were checked with the numerical ones and the variations of the secondary flow pattern with the Dean and Womersley numbers were analysed based on both results. These results suggest that secondary flows can be classified into five patterns.

In certain parameter regimes, it is possible to derive third-order sets of ordinary differential equations that are asymptotically exact descriptions of weakly nonlinear double convection and that exhibit chaotic behaviour. This paper presents a unified approach to deriving such models for two-dimensional convection in a horizontal layer of Boussinesq fluid with lateral constraints. Four situations are considered: thermosolutal convection, convection in an imposed vertical or horizontal magnetic field, and convection in a fluid layer rotating uniformly about a vertical axis. Thermosolutal convection and convection in an imposed horizontal magnetic field are shown here to be governed by the same sets of model equations, which exhibit the period-doubling cascades and chaotic solutions that are associated with the Shil'nikov bifurcation (Proctor & Weiss 1990). This establishes, for the first time, the existence of chaotic solutions of the equations governing two-dimensional magneto-convection. Moreover, in the limit of tall thin rolls, convection in an imposed vertical magnetic field and convection in a rotating fluid layer are both modelled by a new third-order set of ordinary differential equations, which is shown here to have chaotic solutions that are created in a homoclinic explosion, in the same manner as the chaotic solutions of the Lorenz equations. Unlike the Lorenz equations, however, this model provides an accurate description of convection in the parameter regime where the chaotic solutions appear.

We consider a nominally uniform flow over a semi-infinite flat plate. Our analysis shows how a small streamwise disturbance in the otherwise uniform flow ahead of the plate is amplified by leading-edge bluntness effects and eventually leads to a small-amplitude but nonlinear spanwise motion far downstream from the leading edge of the plate. This spanwise motion is then imposed on the viscous boundary-layer flow at the surface of the plate – causing an order-one change in its profile shape. This ultimately reduces the wall shear stress to zero – causing the boundary layer to undergo a localized separation, which may be characterized as a kind of bursting phenomenon that could be related to the turbulent bursts observed in some flat-plate boundary-layer experiments.

Thermal convection of a Boussinesq fluid in a layer confined between two infinite horizontal walls is investigated by direct numerical simulation (DNS) and by large-eddy simulation (LES) for zero horizontal mean motion. The lower-surface height varies sinusoidally in one horizontal direction while remaining constant in the other. Several cases are considered with amplitude δ up to 0.15H and wavelength λ of H to 8H (inclination up to 43°), where H is the mean fluid-layer height. Constant heat flux is prescribed at the lower surface of the initially at rest and isothermal fluid layer. In the LES, the surface is treated as rough surface (z0 = 10−4H) using the Monin-Oboukhov relationships. At the flat top an adiabatic frictionless boundary condition is applied which approximates a strong capping inversion of an atmospheric convective boundary layer. In both horizontal directions, the model domain extends over the same length (either 4H or 8H) with periodic lateral boundary conditions.

We compare DNS of moderate turbulence (Reynolds number based on H and on the convective velocity is 100, Prandtl number is 0.7) with LES of the fully developed turbulent state in terms of turbulence statistics and Characteristic large-scale-motion structures. The LES results for a flat surface generally agree well with the measurements of Adrian et al. (1986). The gross features of the flow statistics, such as profiles of turbulence variances and fluxes, are found to be not very sensitive to the variations of wavelength, amplitude, domain size and resolution and even the model type (DNS or LES), whereas details of the flow structure are changed considerably. The LES shows more turbulent structures and larger horizontal scales than the DNS. To a weak degree, the orography enforces rolls with axes both perpendicular and parallel to the wave crests and with horizontal wavelengths of about 2H to 4H. The orography has the largest effect for λ = 4H in the LES and for λ = 2H in the DNS. The results change little when the size of the computational domain is doubled in both horizontal directions. Most of the motion energy is contained in the large-scale structures and these structures are persistent in time over periods of several convective time units. The motion structure persists considerably longer over wavy terrain than over flat surfaces.

By considering the entrainment effect on the intermittency in the free boundary of shear layers, a set of turbulence model equations for the turbulent kinetic energy k, the dissipation rate ε, and the intermittency factor γ is proposed. This enables us to incorporate explicitly the intermittency effect in the conventional K–ε turbulence model equations. The eddy viscosity νt is estimated by a function of K, ε and γ. In contrast to the closure schemes of previous intermittency modelling which employ conditional zone averaged moments, the present model equations are based on the conventional Reynolds averaged moments. This method is more economical in the sense that it halves the number of partial differential equations to be solved. The proposed K–ε–γ model has been applied to compute a plane jet, a round jet, a plane far wake and a plane mixing layer. The computational results of the model show considerable improvement over previous models for all these shear flows. In particular, the spreading rate, the centreline mean velocity and the profiles of Reynolds stresses and turbulent kinetic energy are calculated with significantly improved accuracy.

Unsteady flow over a stationary sphere with a small fluctuation in the free-stream velocity is considered at small Reynolds number, Re. A matched asymptotic solution is obtained for the frequency-dependent (or the acceleration-dependent) part of the unsteady flow at very small frequency, ω, under the restriction St [Lt ] Re [Lt ] 1, where St is the Strouhal number. The acceleration-dependent part of the unsteady drag is found to be proportional to St ∼ ω instead of the ω½ dependence predicted by Stokes’ solution. Consequently, the expression for the Basset history force is incorrect for large time even for very small Reynolds numbers. Present results compare well with the previous numerical results of Mei, Lawrence & Adrian (1991) using a finite-difference method for the same unsteady flow at small Reynolds number. Using the principle of causality, the present analytical results at small Re, the numerical results at finite Re for low frequency, and Stokes’ results for high frequency, a modified expression for the history force is proposed in the time domain. It is confirmed by comparing with the finite-difference results at arbitrary frequency through Fourier transformation. The modified history force has an integration kernel that decays as t−2, instead of t½, at large time for both small and finite Reynolds numbers.

This paper describes laboratory experiments on the flow over a three-dimensional hill in a rotating fluid. The experiments were carried out in towing tanks, placed on rotating tables. Rotation is found to have a strong influence on the separation behind the hill. The topology of the separation is found to be the same for all the flows examined. The Rossby number R in the experiments is of order 1, the maximum value being 6. The separated flow is dominated by a single trailing vortex. In the majority of cases the surface stress field has a single separation line and there are no singular points. In a few experiments at the highest Rossby numbers the observations suggest more complex stress fields but the results are inconclusive.

A criterion for flow separation is sought. For values of D/L > 1, where D is the depth of the flow and L the lengthscale of the hill, separation is found to be primarily dependent on R. At sufficiently small values of R separation is suppressed and the flow remains fully attached.

Linear theory is found to give a good estimate for the critical value of R for flow separation. For hills with a moderate slope (slope ≤ 1) this critical value is around 1, decreasing with increasing slope. It is postulated that the existence of a single dominant trailing vortex is due to the uplifting and subsequent turning of transverse vorticity generated by surface pressure forces upstream of the separation line.

The problem of determining the profile of a plane diffuser (of given upstream width and length) that provides the maximum static pressure rise is solved. Two-dimensional, incompressible, laminar flow governed by the steady-state Navier-Stokes equations is assumed through the diffuser. Recent advances in computational resources and algorithms have made it possible to solve the ‘direct’ problem of determining such a flow through a body of known geometry. In this paper, a set of ‘adjoint’ equations is obtained, the solution to which permits the calculation of the direction and relative magnitude of change in the diffuser profile that leads to a higher pressure rise. The direct as well as the adjoint set of partial differential equations are obtained for Dirichlet-type inflow and outflow conditions. Repeatedly modifying the diffuser geometry with each solution to these two sets of equations with the above boundary conditions would in principle lead to a diffuser with the maximum static pressure rise, also called the optimum diffuser. The optimality condition, that the shear stress all along the wall must vanish for the optimum diffuser, is also recovered from the analysis. It is postulated that the adjoint set of equations continues to hold even if the computationally inconvenient Dirichlet-type outflow boundary condition is replaced by Neumann-type conditions. It is shown that numerical solutions obtained in this fashion do satisfy the optimality condition.

This work is concerned with the source of an important component of nonlinear loading on a horizontal cylinder beneath waves that is not present in conventional diffraction calculations. Earlier measurements (Chaplin 1984b) have suggested that circulation induced by steady streaming around the cylinder may be responsible for loading which in some cases reduces the perceived inertia force by 50%. The present work is aimed at studying the steady streaming around a cylinder in general non-uniform orbital flow, and determining whether in the particular case of wave-induced flow it could be related quantitatively to the loading.

The steady outer flow has been obtained numerically for cases where the steady streaming does not have a reversal, and for cases where a weak reversal is compatible with a uniform outer circulation. It is found that the outer circulation is closely related to the mean streaming velocity around the cylinder at the outer edge of the shear-wave layer. Results for conditions corresponding to previous measurements of force on a horizontal cylinder beneath waves suggest that separation, turbulence, transient effects and organized three-dimensional instabilities should also be considered.

Spin-up flows of a compressible gas in a finite, closed cylinder from an initial state of rest are studied, The flow is characterized by small reference Ekman numbers, and the peripheral Mach number is O(1). Comprehensive numerical solutions have been obtained for the full, time-dependent compressible Navier-Stokes equations. The details of the flow, temperature, and density evolution are described. In the early phase of spin-up, owing to the thermoacoustic disturbances caused by the compressible Rayleigh effect, the flows are oscillatory, and this oscillatory behaviour is pronounced at higher Mach numbers. The principal dynamical role of the Ekman layer is dominant over moderate times of orders of the homogeneous spin-up timescales. Owing to the density stratification in the radial direction, the Ekman layer is thicker in the central region of the interior. The interior azimuthal flows are mainly uniform in the axial direction. As the Mach number increases, the rate of spin-up in the interior becomes slower, and the propagating shear front is more diffusive. Explicit comparisons with the results for an infinite cylinder are made to ascertain the contributions of the endwall disks. In contrast to the usual incompressible spin-up from rest, the viscous effects are relatively more important for the case of a compressible fluid.

A variational formulation of water waves is developed, based on the Hamiltonian theory of surface waves. An exact and unified description of the two-dimensional problem in the vertical plane is obtained in the form of a Hamiltonian functional, expressed in terms of surface quantities as canonical variables. The stability of the corresponding canonical equations can be ensured by using positive definite approximate energy functionals. While preserving full linear dispersion, the method distinguishes between short-wave nonlinearity, allowing the description of Stokes waves in deep water, and long-wave nonlinearity, applying to long waves in shallow water. Both types of nonlinearity are found necessary to describe accurately large-amplitude solitary waves.

Direct numerical simulation is carried out to study the response of an oscillating cylinder in uniform flow and in the wake of an upstream cylinder. It is found that the response of the cylinder wake is either a periodic (lock-in) or a quasi-periodic (non-lock-in) state. In the lock-in state, the vortex shedding frequency equals the forcing frequency. In the non-lock-in state, the shedding frequency shows a smooth variation with the driving frequency. For a cylinder oscillating in uniform flow, a lock-in diagram of different forcing amplitude is computed. However, no clear chaotic behaviour is detected near the lock-in boundary. For a cylinder oscillating in the wake of an upstream cylinder, the response state is strongly influenced by the distance between the two cylinders. By changing cylinder spacing, two different flow regimes are identified. In the ‘vortex formation regime’, found at large spacings, the vortex street develops behind both the upstream and downstream cylinders. The strength of the naturally produced oscillation upstream of the second cylinder becomes important compared to the forced oscillation and dominates the flow, leading to a very small or even indistinguishable zone of synchronization. However, in the ‘vortex suppression regime’, observed at small spacings, the oncoming flow to the downstream cylinder becomes so weak that it hardly affects its vortex wake, and therefore a large zone of synchronization is obtained. The numerical results are in good agreement with available experimental data.

It is demonstrated experimentally and theoretically that through the use of an active (feedback) controller one can dramatically modify the nature of the flow in a toroidal thermal convection loop heated from below and cooled from above. In particular, we show how a simple control strategy can be used to suppress (laminarize) the naturally occurring chaotic motion or induce chaos in otherwise time-independent flow. The control strategy consists of sensing the deviation of fluid temperatures from desired values at a number of locations inside the loop and then altering the wall heating to either counteract or enhance such deviations.