The deformation of a moving spherical viscous drop subject to axisymmetric perturbations is considered. The problem is formulated using two different variations of the boundary integral method for Stokes flow, one due to Rallison & Acrivos, and the other based on an interfacial distribution of Stokeslets. An iterative method for solving the resulting Fredholm integral equations of the second kind is developed, and is implemented for the case of axisymmetric motion. It is shown that in the absence of surface tension, a moving spherical drop is unstable. Prolate perturbations cause the ejection of a tail from the rear of the drop, and the entrainment of a thin filament of ambient fluid into the drop. Oblate perturbations cause the drop to develop into a nearly steady ring. The viscosity ratio plays an important role in determining the timescale and the detailed pattern of deformation. Filamentation of the drop emerges as a persistent but secondary mechanism of evolution for both prolate and oblate perturbations. Surface tension is not capable of suppressing the growth of perturbations of sufficiently large amplitude, but is capable of preventing filamentation.

The behaviour of a turbulent boundary layer on a flat surface as it encounters the nose of a cylindrical wing mounted normal to that surface is being investigated. A three-component laser anemometer has been developed to measure this highly turbulent three-dimensional flow. Measurements of all the non-zero mean-velocity and Reynolds-stress components have been made with this instrument in the plane of symmetry upstream of the wing. These data have been used to estimate some of the component terms of the turbulence kinetic energy equation. Histograms of velocity fluctuations and short-time cross-correlations between the laser anemometer and a hot-wire probe have also been measured in the plane of symmetry. In all, these results reveal much of the time-dependent and time-averaged turbulence structure of the flow here.

Separation occurs in the plane of symmetry because of the adverse pressure gradient imposed by the wing. In the time mean the resulting separated flow consists of two fairly distinct regions: a thin upstream region characterized by low mean backflow velocities and a relatively thick downstream region dominated by the intense recirculation of the mean junction vortex. In the upstream region the turbulence stresses develop in a manner qualitatively similar to those of a two-dimensional boundary layer separating in an adverse pressure gradient. In the vicinity of the junction vortex, though, the turbulence stresses are much greater and reach’ values many times larger than those normally observed in turbulent flows. These large stresses are associated with bimodal (double-peaked) histograms of velocity fluctuations produced by a velocity variation that is bistable. These observations are consistent with large-scale low-frequency unsteadiness of the instantaneous flow structure associated with the junction vortex. This unsteadiness seems to be produced by fluctuations in the momentum and vorticity of fluid from the outer part of the boundary layer which is recirculated as it impinges on the leading edge of the wing. Though we would expect these fluctuations to be produced by coherent structures in the boundary layer, frequencies of the large-scale unsteadiness are substantially lower than the passage frequency of such structures. It therefore seems that only a fraction of the turbulent structures are recirculated in this way.

The decay of grid-generated turbulence in the presence of strong thermal stratification is studied in a continuously stratified, open-loop wind tunnel at Brunt–Väisälä frequencies up to 2.5s−1. The data include one-point statistical measurements through moments of fourth order and associated power- and cross-spectra. Cross-channel phase measurements are used to analyse the scales of correlation of velocity and temperature. The present data are considerably more coherent than previous salt-stratified data, and the structural form of stratified turbulence is thus more clearly manifested. No internal wave effects are observed at any stage of the decay. Stratified turbulence is found to be a two-scale process dominated by buoyancy forces at large scales of motion and dissipative effects at small scales. The two-scale structure is used to develop universal buoyancy scalings for the decay of the vertical heat flux, the scalar variance, and the molecular dissipation rates, and, in particular, for the vertical velocity decay. Velocity and temperature spectra satisfy universal equilibrium scaling at high wavenumbers, but show buoyancy effects at small wavenumbers. The flow remains isotropic at high wavenumbers over the entire range of turbulent decay studied. Cospectral and phase data are used to validate the two-scale model of the turbulence. The flow may show large-scale restratification while active turbulence persists at smaller scales, so that the vanishing of the vertical transport does not represent extinction of turbulent motion. Additionally, an original universal equilibrium scaling is developed for the cross-spectrum. Lengthscale evolution is measured, and the overturning and buoyancy lengthscales (associated with potential and kinetic energy, respectively) are found to characterize flow development. The role of the Prandtl number is assessed by comparison to previous works, and the Prandtl number is found to have a significant influence upon stratified turbulence evolution.

A new stochastic model for the motion of particle pairs in isotropic high-Reynolds-number turbulence is proposed. The model is three-dimensional and its formulation takes account of recent improvements in the understanding of one-particle models. In particular the model is designed so that if the particle pairs are initially well mixed in the fluid, they will remain so. In contrast to previous models, the new model leads to a prediction for the particle separation probability density function which is in qualitative agreement with inertial subrange theory. The values of concentration variance from the model show encouraging agreement with experimental data. The model results suggest that, at large times, the intensity of concentration fluctuations (i.e. standard deviation of concentration divided by mean concentration) tends to zero in stationary conditions and to a constant in decaying turbulence.

A theoretical model for the spreading of viscous gravity currents over a rigid horizontal surface is derived, based on a lubrication theory approximation. The complete family of self-similar solutions of the governing equations is investigated by means of a phase-plane formalism developed in analogy to that of gas dynamics. The currents are represented by integral curves in the plane of two phase variables, Z and V, which are related to the depth and the average horizontal velocity of the fluid. Each integral curve corresponds to a certain self-similar viscous gravity current satisfying a particular set of initial and/or boundary conditions, and is obtained by solving a first-order ordinary differential equation of the form dV/dZ = f(Z, V), where f is a rational function. All conceivable self-similar currents can thus be obtained. A detailed analysis of the properties of the integral curves is presented, and asymptotic formulae describing the behaviour of the physical quantities near the singularities of the phase plane corresponding to sources, sinks, and current fronts are given. The derivation of self-similar solutions from the formalism is illustrated by several examples which include, in addition to the similarity flows studied by other authors, many other novel ones such as the extension to viscous flows of the classical problem of the breaking of a dam, the flows over plates with borders, as well as others. A self-similar solution of the second kind describing the axisymmetric collapse of a current towards the origin is obtained. The scaling laws for these flows are derived. Steady flows and progressive wave solutions are also studied and their connection to self-similar flows is discussed. The mathematical analogy between viscous gravity currents and other physical phenomena such as nonlinear heat conduction, nonlinear diffusion, and ground water motion is commented on.

The problem of nonlinear development of Gortler vortices and interaction with Tollmien–Schlichting waves is considered within the framework of incompressible Navier–Stokes equations which are solved by a Fourier–Chebyshev spectral method., It is shown that two-dimensional waves can be excited in the flow modulated by Gortler vortices. Owing to nonlinear effects, this interaction further leads to the development of oblique waves with spanwise wavelength equal to the Görtler vortex wavelength. Interaction is also considered of oblique waves with spanwise wavelength twice that of Gortler vortices.

One of G. I. Taylor's most famous papers concerns the large-time behaviour of a cloud of soluble matter which has been injected into a solvent in laminar flow in a pipe. In the past thirty years, a number of successful attempts have been made to derive differently or extend Taylor's result, which is that the cloud of solute eventually takes a Gaussian profile in the flow direction. The present paper is another examination of this well-worked problem, but this time from the viewpoint of a formal integral transform representation of the solution. This approach leads to a better understanding of the solution; it also enables efficient numerical computations, and leads to extended and new asymptotic expansions.

A Laplace transform in time and a Fourier transform in the flow direction leaves a complicated eigenvalue problem to be solved to give the cross-sectional behaviour. This eigenvalue problem is examined in detail, and the transforms are then inverted to give the concentration distribution. Both numerical and asymptotic methods are used. The numerical procedures lead to an accurate description of the concentration distribution, and the method could be generalized to compute dispersion in general parallel flows. The asymptotic procedures use two different classes of eigenvalues to give leading- and trailing-edge approximations for the solute cloud at small times. Meanwhile, at larger times, one eigenvalue branch dominates the solution and Taylor's result is recovered and extended using'the computer to generate extra terms in the approximation. Sixteen terms in the approximation are calculated, and a continued fraction expansion is deduced to enhance the accuracy.

Subject to uniform strain, an elliptical patch of vorticity in an in viscid, incompressible, two-dimensional fluid generally rotates or nutates and extends or compresses while retaining a precisely elliptical shape (the Kida solutions). This result is of interest because the uniform strain idealizes the leading-order distortional influence of distant vortices in a flow with many vortices. Because of the unsteady motion of the distant vortices, both the strain rate and the rotation rate of the strain axes typically vary with time. In the special case that the strain rate and rotation rate are steady, and when the strain rate is not too large, periodic motion of an elliptical vortex is possible. Larger strain rates lead to indefinite extension of the vortex.

Uniform strain, however, only approximately mimics the effect of distant vortices. The local variations- in the strain field around a vortex disturb the vortex, preventing it from retaining a simple, elliptical shape. These disturbances may amplify because of instabilities. In this paper, we examine the stability of periodic elliptical motion to small boundary disturbances, for the case of steady, uniform strain and rotation rate, first by linear Floquet theory and then by direct, high-resolution, nonlinear numerical integrations. It is discovered that a significant portion of the periodic solutions are linearly unstable. Instability can occur even when the strain rate is arbitrarily small and the basic motion arbitrarily close to circular. Extended nonlinear calculations exhibit recurrence, in some cases, and attrition of the vortex by repeated wave amplification, steepening, and breaking in others.

When buoyant fluid is released into the base of a crack in an elastic medjura the crack will propagate upwards, driven by the buoyancy of the fluid. Viscous fluid flow in such a fissure is described by the equations of lubrication theory with the pressure given by the sum of the hydrostatic pressure of the fluid and the elastic pressures exerted by the walls of the crack. The elastic pressure and the width of the crack are further coupled by an integro-differential equation derived from the theory of infinitesimal dislocations in an elastic medium. The steady buoyancy-driven propagation of a two-dimensional fluid-filled crack through an elastic medium is analysed and the governing equations for the pressure distribution and the shape of the crack are solved numerically using a collocation technique. The fluid pressure in the tip of an opening crack is shown to be very low. Accordingly, a region of relatively inviscid vapour or exsolved volatiles in the crack tip is predicted and allowed for in the formulation of the problem. The solutions show that the asymptotic width of the crack, its rate of ascent and the general features of the flow are determined primarily by the fluid mechanics; the strength of the medium and the vapour pressure in the crack tip affect only the local structure near the advancing tip of the crack. When applied to the transport of molten rock through the Earth's lithosphere by magma-fracture, this conclusion is of fundamental importance and challenges the geophysicist's usual emphasis on the controlling influence of fracture mechanics rather than that of fluid mechanics.

A numerical and experimental study is described for the two-dimensional steady flow through a uniform cascade of normal flat plates. The Navier–Stokes equations are written in terms of the stream function and vorticity and are solved using a second-order-accurate finite-difference scheme which is based on a modified procedure to preserve accuracy and iterative convergence at higher Reynolds numbers. The upstream and downstream boundary conditions are discussed and an asymptotic solution is employed both upstream and downstream. A frequently used method for dealing with corner singularities is shown to be inaccurate and a method for overcoming this problem is described. Numerical solutions have been obtained for blockage ratio of 50 % and Reynolds numbers in the range 0 [les ] R [les ] 500 and results for both the lengths of attached eddies and the drag coefficients are presented. The calculations indicate that the eddy length increases linearly with R, at least up to R = 500, and that the multiplicative constant is in very good agreement with the theoretical prediction of Smith (1985a), who considered a related problem. In the case of R = 0 the Navier–Stokes equations are solved using the finite-difference scheme and a modification of the boundary-element method which treats the corner singularities. The solutions obtained by the two methods are compared and the results are shown to be in good agreement. An experimental investigation has been performed at small and moderate values of the Reynolds numbers and there is excellent agreement with the numerical results both for flow streamlines and eddy lengths.

When a viscous liquid is displaced by a long air bubble in a capillary, it leaves behind a wetting liquid film. A lubrication analysis by Bretherton (1961), which assumes a mobile surface, underpredicts the film thickness at low bubble speeds. In this investigation, the Marangoni effect of small amounts of impurities is shown to be capable of explaining this discrepancy. We carry out an asymptotic analysis for different convective, diffusive and kinetic timescales and show that, if transport in the film is mass-transfer limited such that a bulk concentration gradient exists in the film, the film thickness increases by a maximum factor of 4 2/3; over Bretherton's mobile result at low bubble speeds. Moreover, at large bubble speeds, Bretherton's mobile prediction is approached for all ranges of timescales. For intermediate bubble speeds, the film thickness varies with respect to the bubble speed with an exponent smaller than 2/3 of the mobile theory. These results are favourably compared to literature data on film thickness.

The theory of an oscillating, high-aspect-ratio, lifting surface with a curved centreline (Cheng & Murillo 1984) is applied to a performance analysis of lunate-tail swimming propulsion. Thrust, power and propulsive efficiency are calculated for model lunate tails with various combinations of mode shapes and morphological features to ascertain the viability of the proportional-feathering concept, and to determine the influence of sweep and centreline curvature. One of the principal conclusions concerns the interchangeability of the heaving amplitude of the peduncle (identified with the major pitching axis) with the centreline sweep, and its effect on the propulsive efficiency, while maintaining the same thrust. Hydrodynamic reasons are also offered for the apparent preference for the crescent-moon fin shape over the V-shape at moderate sweep angles, and for the large sweep angles often found in V-shaped fins.

In a previous paper we reported on the effect of Dean number, κm, on the fully developed region of periodic flows through curved tubes. In this paper we again consider a sinusoldally varying volumetric flow rate in a curved pipe of arbitrary curvature ratio, δ, and investigate the effect of frequency parameter α, and Reynolds number Rem on the flow. Specifically, we report on the flow-field development for the range 7.5 [les ] α [les ] 25, and 50 [les ] Rem [les ] 450. The results, obtained by numerical integration of the full Navier–Stokes equations, reveal a number of characteristics of the flow previously unreported. For low values of Rem the secondary flow consists of a single vortex (Dean-type motion) in the half-cross-section at all times and for all values of α studied. For higher Rem we observe inward ‘centrifuging’ (Lyne-type motion) at the centre. This motion always occurs during the accelerating period of the volumetric flow rate. It appears at lower α for higher Rem and, for the given Rem at which it appears, it occurs at earlier times in the cycle for lower a. A striking feature is observed for α = 15 for the range 315 [les ] Rem [les ] 400: period tripling. The flow field varies periodically with time for the duration of three volumetric-flow-rate cycles then repeats for the subsequent three cycles, and so on. The computed axial pressure gradient also varies periodically with time but with the same period as the volumetric flow rate.

A pattern recognition technique has been applied to simultaneously sampled multipoint hot-wire anemometry data obtained in the far wake of a circular cylinder. Data from both the streamwise fluctuating velocity field and the temperature field have been analysed employing a computer code that uses a correlation approach to automatically detect and ensemble average flow patterns and patterns for mean-square fluctuations. Statistical tests then allow the significance and contribution to the turbulence intensity of the detected structures to be evaluated. This procedure has been used to infer the three-dimensional topology of the double-roller eddies previously identified in the far-wake region and to relate these to the motions responsible for entrainment. It appears that the two types of motion are not independent, but are linked together, forming parts of horseshoe vortex structures which account for at least 40% of the total turbulence energy. These structures originate near the centre of the flow, may extend across the centreline and typically occur in groups of about three. The resulting picture of the flow dynamics is related to the conclusions drawn from similar data by other workers and a possible regeneration mechanism is presented. The addition to the code of a fine-scale activity indicator, the choice of which is discussed in some detail, has allowed the relationship between these energetic large-scale motions and smaller eddies to be investigated. It seems that the most intense fine-scale activity is associated with the vortical cores of the double-roller eddies. It is shown that this observation is consistent with the concepts of ‘isotropy’ and ‘spotiness’ of the dissipative scales. It also suggests that the horseshoe vortices loose energy both to their own secondary instabilities and to smaller scales resulting from the breakup of other highly strained large eddies.

The characteristics of sublayer bursting in turbulent pipe flow through a 180° curved bend at a pipe Reynolds number of 50000 were investigated. In particular, the effects of bend curvature on the bursting frequency were studied in detail. A flush-mounted shear stress gauge was used to measure the wall shear stress and the VITA technique was applied to isolate the bursting events in the wall shear signals. The results show that the bursting frequency starts to drop rather dramatically at the inlet region of the inner bend (closest to the centre of curvature of the bend). Contrary to expectation, the bursting frequency remains fairly constant along the outer bend, but increases sharply as the flow comes out of the bend. Possible explanations for this behaviour are proposed based on the important external effects present in a pipe bend; that is, those due to centrifugal force, streamline curvature and the superimposed secondary flow. Even though wall pressure measurements show that the flow recovers to a fully developed straight pipe condition at a short distance downstream of the bend exit, the circumferential wall shear stress distribution, the spectral content of the wall shear signal and the associated bursting frequency suggest that the near-wall flow takes a much longer distance to return to an unperturbed straight-pipe condition.

The behaviour of the stream wise velocity u in the near-wall region of a turbulent boundary layer is obtained by analysing the data from an array of hot wires aligned in the span wise direction. Conventional and conditional statistics are presented, relative to the occurrence of bursts and sweeps detected using a modified u-level method. Sweeps have an average stream wise length which is twice as large as that of bursts while the average span wise extent of sweeps is about 25% larger than that of bursts. Both instantaneous and conditionally averaged information is presented and discussed in the context of bursts and sweeps in the (x, z)-plane. Dependence on y+ is significant, and important differences are observed between instantaneous and conditionally averaged results. Conventional and conditional statistics of the velocity derivatives ∂u/∂x and ∂u/∂z provide some insight into the anisotropy of the mean-square velocity derivatives in the near-wall region. Conditionally averaged patterns of u compare favourably with the numerical simulations of Kim (1985) in the near-wall region of a turbulent channel flow, at a comparable Reynolds number.

The steady streaming outside the Stokes shear-wave layer, which forms on the surface of a sphere when placed close to an oscillatory point source, is considered. Particular attention is devoted to the case of high streaming Reynolds-number flow. Thin circular jets, analogous to the plane jets known to occur in two-dimensional flow, are predicted and visualized by means of a simple experiment.

An experimental study was conducted to investigate the’ generation process of random small-scale turbulence in an originally laminar mixing layer. The evolutions of the two types of deterministic structures, the spanwise and streamwise vortices, were first clarified in order to determine their roles in the transition process. A scaling rule for the streamwise distance from the trailing edge of the splitter plate to the vortex merging position was found for various velocity ratios. After this streamwise lengthscale was determined, it became clear that the spanwise wavelength of the streamwise vortices doubled after the merging of the spanwise structures which nominally doubled streamwise wavelengths. The most interesting finding was that the random small-scale eddies were produced by the interactions between the merging spanwise structures and the streamwise vortices.

Measurements of the relation between mass hold-up and flow rate have been made for glass beads in fully developed flow down an inclined chute, over the whole range of inclinations for which such flows are possible. Velocity profiles in the flowing material have also been measured. For a given inclination it is found that two different flow regimes may exist for each value of the flow rate in a certain interval. One is an ‘energetic’ flow, and is produced when the particles are dropped into the chute from a height, while the other is relatively quiescent and occurs when entry to the chute is regulated by a gate. At some values of the inclination jumps in the flow pattern occur between these branches, and it is even possible for both branches to coexist in the same chute, separated by a shock. A theoretical treatment of chute flow has been based on a rheological model of the material which takes into account both collisional and fractional mechanisms for generating stress. Its predictions include most aspects of the observed behaviour, but quantitative comparison of theory and experiment is difficult because of the uncertain values of some parameters appearing in the theory.

In a curved channel streamwise vortices, often called Dean vortices, may develop above a critical Reynolds number owing to centrifugal effects. Similar vortices can occur in a rotating plane channel due to Coriolis effects if the axis of rotation is normal to the mean flow velocity and parallel to the walls. In this paper the flow in a curved rotating channel is considered. It is shown from linear stability theory that there is a region for which centrifugal effects and Coriolis effects almost cancel each other, which increases the critical Reynolds number substantially. The flow visualization experiments carried out show that a complete cancellation of Dean vortices can be obtained for low Reynolds number. The rotation rate for which this occurs is in close agreement with predictions from linear stability theory. For curved channel flow a secondary instability of travelling wave type is found at a Reynolds number about three times higher than the critical one for the primary instability. It is shown that rotation can completely cancel the secondary instability.