The response of a wall boundary layer to the motion of a convected vortex is investigated. The principal cases considered are for a rectilinear filament of strength –κ located a distance a above a plane wall and convected to the right in a uniform flow of speed U∞*. The inviscid solution predicts that such a vortex will remain at constant height a above the wall and be convected with constant speed αU∞*. Here α is termed the fractional convection rate of the vortex, and cases in the parameter range 0 [les ] α < 1 are considered. The motion is initiated at time t* = 0 and numerical calculations of the developing boundary-layer flow are carried out for α = 0, 0.2, 0.4, 0.55, 0.7, 0.75 and 0.8. For α < 0.75, a rapid lift-up of the boundary-layer streamlines and strong boundary-layer growth occurs in the region behind the vortex; in addition an unusual separation phenomenon is observed for α [les ] 0.55. For α [ges ] 0.75, the boundary-layer development is more gradual, but ultimately substantial localized boundary-layer growth also occurs. In all cases, it is argued that a strong inviscid–viscous interaction will take place in the form of an eruption of the boundary-layer flow. The generalization of these results to two-dimensional vortices with cores of finite dimension is discussed.

The prevalence in a turbulent mixing layer of dynamical events with a coherent history over substantial times suggests that it is profitable to study in detail entirely deterministic versions of this flow and to attempt to use a simplified synthesis of these solutions as the fundamental representation in a stochastic treatment of the layer. It is proposed that the deterministic representation of the flow be achieved by the embedding of a short hierarchy of motions which are studied in detail, though not exhaustively, in Parts 1, 2 and 3. Part 1 deals with the fundamental or first-order motion, which is the evolution of a layer constrained to be purely two-dimensional.

Experimental evidence suggests that in the turbulent mixing layer the fundamental mechanism of growth is two-dimensional and little affected by the presence of vigorous three-dimensional motion. To quantify this apparent property and study the growth of streamwise vorticity, we write for the velocity field \[ {\boldmath V}(x, t) = {\boldmath U}(x, z, t) + {\boldmath u}(x, y, z, t), \] where U is two-dimensional and u is three-dimensional. In a first version of the problem U is independent of u, while in the second U is the spanwise average of V. In both cases the equation for u is linearized around U. The equations for U and u are solved simultaneously by a finite-difference calculation starting with a slightly disturbed parallel shear layer.

The solutions provide a detailed description of the growth of the three-dimensional motion. They show that its characteristics are dictated by the distribution of spanwise vorticity which results from roll-up and pairing. Pairing inhibits its growth. The solutions also demonstrate that even when the three-dimensional flow attains large amplitudes it has a negligible effect on the interaction of spanwise vortices and thus on the growth of the layer.

Gravity–capillary standing waves in water of arbitrary uniform depth are considered. The classical perturbation calculation yields unbounded coefficients for some critical values of the depth. In the present paper solutions valid near the first critical value of the depth are derived. A problem of non-uniqueness is discovered and discussed. It is shown that two solutions exist, one with higher frequency and one with lower frequency than the zeroth-order solution. They are found analytically at the critical value of the depth and numerically in an interval around it. Graphs of the results are included.

In this paper, laminar condensation on a spherical drop in a forced flow is investigated. The drop experiences a strong, radial, condensation-induced velocity while undergoing slow translation. In view of the high condensation velocity, the flow field, although the drop experiences slow translation, is not in the Stokes-flow regime. The drop environment is assumed to consist of a mixture of saturated steam (condensable) and air (non-condensable). The study has been carried out in two different ways. In Part 1 the continuous phase is treated as quasi-steady and the governing equations for this phase are solved through a singular perturbation technique. The transient heat-up of the drop interior is solved by the series-truncation numerical method. The solution for the total problem is obtained by matching the results for the continuous and dispersed phases. In Part 2 both the phases are treated as fully transient and the entire set of coupled equations are solved by numerical means. Validity of the quasi-steady assumption of Part 1 is discussed. Effects due to the presence of the non-condensable component and of the drop surface temperature on transport processes are discussed in both parts. A significant contribution of the present study is the inclusion of the roles played by both the viscous and the inertial effects in the problem treatment.

In this paper, we investigate the problem of transient laminar condensation on a moving drop by the semianalytical series-truncation method. The objectives are to assess the validity and the accuracy of the matched-asymptotic method employed in Part 1. The fluid flow and thermodynamic variables are expanded as complete series of Legendre polynomials. The resulting transient momentum, energy and species equations are integrated numerically. The numerical scheme basically involves a three-point central difference for the spatial derivatives and a backward difference expression for the temporal derivatives. The finite-difference equations have been solved by the strongly implicit procedure. Good agreement of the fully transient numerical results with the singular perturbation approximation results of Part 1 lends credibility to a quasi-steady treatment of the continuous phase. The computational time requirements for the fully numerical solutions increase with decreasing non-condensable gas mass fraction in the bulk environment.

Statistical studies of hydrodynamic interactions between many particles in a dilute dispersion raise a problem of divergent integrals. This problem arises in particular when calculating the average velocity of sedimentation of solid spheres in a viscous fluid. The solution to this problem was given by Batchelor (1972) for monodisperse suspensions of spheres, on the basis of an assumption of homogeneity. This assumption is removed here. The problem of divergent integrals is reconsidered. The solution treats as successive steps:

1. the average flow due to random statistically independent point forces;

2. the average flow due to random statistically independent solid spheres, without hydrodynamic interactions;

3. the average sedimentation velocity of random, pairwise-dependent solid spheres with hydrodynamic interactions, in a dilute suspension.

Considering the case of identical spheres, and assuming homogeneity in any horizontal plane, an expression is obtained for the average sedimentation velocity of a sphere in an otherwise inhomogeneous dispersion. The formula is written in terms of integrals involving probability distributions. It reduces, when the suspension is homogeneous, to a formula obtained by Batchelor.

The probability distributions are not calculated in this paper. In order to evaluate numerically the average velocity of sedimentation, a simple expression for the pair distribution function is assumed, and two different concentration profiles are considered, viz. a sinusoidal variation and a step function. In the case of sinusoidal concentration wave, it is found that the contribution of the inhomogeneity is, for small wavelengths, comparable in magnitude to that calculated for a homogeneous dispersion by Batchelor (1972), i.e. –6.55c.

The difference in velocity between the crest and trough of the wave is an increasing function of the wavelength. For a step function in concentration, particles at the top of the cloud start to fall faster, this effect being limited to a top layer about 10 radii thick.

For future study of the long-term behaviour of a sedimenting cloud, the evolution of the pair distribution function should be added to the present theory.

Miniature tubular pumps are used to emit droplets from ink-jet matrix heads. This paper deals with the simulation of the behaviour of a viscous and compressible liquid in such pumps. The response of the liquid in the frequency and time domain is analysed.

An approximate method is given to determine the droplet speed and size.

A second-order closure model describing the diffusion of a passive scalar from a small source is presented. The model improves upon the earlier work of Lewellen & Teske (1976) by ensuring the early stage of the release, the so-called meander phase, is accurately described. In addition to the mean concentration and scalar fluxes, a model equation for the evolution of the scalar variance is proposed. The latter introduces a new lengthscale which represents the scale of the concentration fluctuations. The model predictions are compared with the recent experimental data of Fackrell & Robins (1982a, b).

Unidirectional and periodically modulated short waves on a horizontal or very nearly horizontal bottom are known to be accompanied by long waves which propagate together with the envelope of the short waves at their group velocity. However, for variable depth with a horizontal lengthscale which is not too great compared with the group length, long waves of another kind are further induced. If the variation of depth is only one-dimensional and localized in a finite region, then the additional long waves can radiate away from this region, in directions which differ from those of the short waves and their envelopes. There are also critical depths which define caustics for these new long waves but not for the short waves. Thus, while obliquely incident short waves can pass over a topography, these second-order long waves may be trapped on a ridge or away from a canyon.

The behaviour of a fluid system governed by a quadratic equation of state for the temperature is studied. The model consists of two well-mixed and interconnected vessels subjected to external thermal forcing. If inertial effects are neglected the temperature response of the fluid is governed by two autonomous ordinary differential equations in time. An investigation of these equations revealed that, depending upon the choice of parameters, the system has two possible final states: one stationary, the other a relaxation oscillation. The transitions between these states occur as sub- and supercritically unstable Hopf bifurcations. For certain parameter ranges, a stationary solution can thus coexist with a relaxation oscillation, the initial conditions determining which of these is realized.

Two unequal rigid spheres are immersed in unbounded fluid and are acted on by externally applied forces and couples. The Reynolds number of the flow around them is assumed to be small, with the consequence that the hydrodynamic interactions between the spheres can be described by a set of linear relations between, on the one hand, the forces and couples exerted by the spheres on the fluid and, on the other, the translational and rotational velocities of the spheres. These relations may be represented completely by either a set of 10 resistance functions or a set of 10 mobility functions. When non-dimensionalized, each function depends on two variables, the non-dimensionalized centre-to-centre separation s and the ratio of the spheres’ radii λ. Two expressions are given for each function, one a power series in s−1 and the other an asymptotic expression valid when the spheres are close to touching.

A theory describing two-phase displacement in the gap between closely spaced planes is developed. The main assumptions of the theory are that the displaced fluid wets the walls, and that the capillary number Ca and the ratio of gap width to transverse characteristic length ε are both small. Relatively mild restrictions apply to the ratio M of viscosities of displacing to displaced fluids; in particular the theory holds for M = o(Ca−1/3). We formulate the theory as a double asymptotic expansion in the small parameters ε and Ca1/3. The expansion in ε is uniform while that in Ca1/3 is not, necessitating the use of matched asymptotic expansions. The previous work of Bretherton (1961) is clarified and extended, and both the form and the constants in the effective boundary condition of Chouke, van Meurs & van der Poel (1959) and of Saffman & Taylor (1958) are determined.

Liquid-metal flows in rectangular ducts having electrically insulating tops and bottoms and perfectly conducting sides and in the presence of strong, polar, transverse magnetic fields are examined. Solutions are presented for the boundary layers adjacent to the sides that are parallel to the magnetic field. Overshoots in the radial velocity profiles show that the side layers have zero displacement thickness and do not perturb the inviscid core. Very weak secondary flows involve four significant vortices, as reflected in the polar velocity profiles.

The structure of turbulence, especially in the near-wall region, was studied with the variable-interval time-averaging (VITA) technique and the uv-quadrant method. Both methods were applied to the same set of data, measured in a fully developed turbulent channel flow, to detect events associated with turbulence production. A close correspondence was found between VITA events and ejection type of events detected with the uv-quadrant method. Conditional averages of the fluctuating component of the streamwise velocity (u) and the component normal to the wall (v), as well as of the product uv, were constructed with both methods, and the cause for some of the apparent differences was investigated. In contrast to previous findings it was concluded that the uv-pattern obtained with the VITA technique has only one peak, and hence is quite similar to that obtained with the uv-quadrant method. It was shown that large peaks in the uv-signal (with u < 0, v > 0) imply large instantaneous outflow angles. For typical VITA events the outflow angle was often found to exceed 10°. Some events with large uv-peaks did not correspond to any strong activity in the u-component, but could be detected by applying the VITA technique to the v-signal.

The single- and two-point measurements made in a high-Reynolds-number single-stream mixing layer growing to encounter a wind-tunnel floor on its high-velocity side that were described by Wood & Bradshaw (1982) have been extended to the wall-bounded flow. It is shown that the expected large amplification of the normal-stress components in the plane of the wall does not occur until after the mixing layer reaches the surface. There is some evidence that the double-roller component of the large-eddy structure of the original free shear layer is being re-established in the wall-bounded flow after having been stretched and weakened by the initial effect of the wall. The triple-product terms appearing in the turbulent-energy and shear-stress equations are altered in a way that cannot be reproduced by models used in current calculation methods. It appears that all the pressure-fluctuation terms in the individual normal-stress and shear-stress transport equations respond in a non-monotonic manner to the imposition of the wall. The implications for calculation methods suitable for predicting the change from an initially unaffected free shear layer to a wall-bounded flow are discussed.

Convective flow phenomena in slender vertical slots with larger vertical than horizontal dimension, i.e. Hele Shaw slots, heated from below and subject to specified lateral boundary conditions, are investigated experimentally. Temperature fields in the liquid were visualized by holographic interferometry. Power-density spectra of local time-dependent thermocouple signals are calculated.

In these slender slots different steady and time-dependent convection patterns develop with increasing Rayleigh number. The range of oscillatory convective flow exhibits periodic, quasiperiodic or non-periodic structures including possible frequency locking, subharmonics and intermittency. Quasiperiodic and periodic oscillations reappear at higher Rayleigh numbers, with non-periodic flows occurring in-between. These time-dependent flows appear to be caused by an instability of thermal boundary layers at the horizontal walls. Finally, at still higher Rayleigh numbers a reverse transition to a steady-state flow pattern is observed. The transitions between steady and non-steady flows are characterized by hysteresis.

Measurements of the velocity characteristics and wall pressure are reported for the axisymmetric turbulent flow downstream of three bluff bodies (disks of 25% and 50% area blockage and a cone of 25% blockage) confined by a long pipe. The dimensions of the recirculation regions were found from the mean-velocity components, which were determined by a laser-Doppler velocimeter: the corresponding components of Reynolds stress were also recorded. The lengths and maximum widths of the recirculation bubbles (in bluff-body diameters), recirculating mass-flow rates (normalized by the average velocity in the plane of the baffle, U0, and the baffle diameter) and maximum turbulent kinetic energy (normalized by U02) were as follows: cone 1.55, 0.55, 0.19, 0.11; disk (25% blockage) 1.75, 0.62, 0.31, 0.19; disk (50% blockage) 2.20, 0.55, 0.26, 0.16. The increase in recirculation length with blockage is opposite to the trend in unconfined, annular jets. The distribution of Reynolds stresses is strongly dependent on blockage: for the smaller blockage both the disk and the cone have the maximum value of kinetic energy near the rear stagnation point. It is proposed that this is because the generation of turbulence by normal stresses is more important in the flow consequent on the smaller blockage.

The measurements include profiles of the velocity characteristics at, as well as upstream of, the trailing edges of the baffles for use as boundary conditions in numerical solutions of the equations of motion.

Measurements of air and solid-particle velocities were made in a vertical pipe two-phase flow by the use of a laser-Doppler velocimeter (LDV). Five kinds of plastic particles, diameters of which ranged from about 3 mm to 200 μm, were transported in a vertical pipe of 30 mm inner diameter. It was found that, the smaller the particle size, the flatter was the mean air velocity distribution for the same mass flow ratio of solids to air. Large particles increased air turbulence throughout the pipe section, while small particles reduced it. Both effects of promotion and suppression of turbulence were observed at the same time in the presence of particles of medium size, that is, the turbulence was increased around the pipe centre and reduced near the wall. The frequency spectrum of air turbulence normalized by the turbulence intensity was not changed by the large particles. In the presence of the small particles, the higher-frequency parts of the spectrum increased.

A description is given of the principle and a preliminary embodiment of a flow-visualization technique in which the test body is pulled through water.

Initially, dyed and undyed parts of the water are separated by a plane horizontal interface; as the body emerges from below this interface, it carries dyed fluid upwards in its wake. A succession of photographs reveals the nature of the flow.

In principle, the flow is steady in a frame fixed relative to the body, and unsteady in one relative to the tank and the initial interface. Interpretation of the experimental results must therefore proceed by way of an analysis in which the steady-state equations are solved for velocity and turbulence quantities but an unsteady-state equation is solved for the dye concentration.

Preliminary results are reported for the flows behind a wedge and behind a bar. They reveal that these particular flows are unsteady even in the body-fixed coordinate frame, being strongly oscillatory.