Experiments were conducted to test the linear theory of internal gravity waves produced in a stably stratified liquid by the forced oscillations and the initial impulsive motion of a two-dimensional stationary disturbance. The measurements of the wave configuration in a medium whose density increased linearly with depth were made by means of a Toepler-schlieren system. The agreement between observation and prediction was found to be good.

This is a theoretical study of the experimental work in the succeeding paper by Partridge & Lyall (1967). This paper shows that the characteristic shape of typical bubbles in fluidized beds, with a large solids to fluid density ratio, will be reached after a time of order (r0/g)½ from the introduction, naturally or artificially, into the bed of a circular or spherical bubble, where r0 is the initial bubble radius and g the gravitation constant. The initial acceleration is g in two dimensions and 2g in three dimensions. A measure of the growth rate of the distortion from a circular or spherical shape is given as a tentative guide to wake formation and growth rate. It is shown that the wake growth rate in the three-dimensional case is faster than in the two-dimensional case, and, further, that bubbles formed naturally at the bottom of a fluidized bed will distort faster than bubbles starting from rest.

The method used is based on a convective term linearization process on equations of motion for a fluidized system given by Murray (1965a, b) and an extension of a method given by Walters & Davidson (1962, 1963) in the case of the initial motion of a gas bubble formed in an inviscid liquid and starting from rest.

Comparison with experiment is difficult since the bubble conditions discussed in the theory are very difficult to reproduce artificially in the laboratory. The results obtained by Partridge & Lyall (1967) are shown to be not inconsistent with those predicted here.

The flow in the wake of a two-dimensional blunt-trailing-edge body was investigated in the Reynolds number range, Reynolds number being referred to base height, 1·3 × 104 to 4·1 × 104. The effects of splitter plates and base bleed on the vortex street were examined. Measurements were made of the longitudinal spacing between vortices and the velocity of the vortices, and compared with values predicted by von Kármán's potential vortex street model. The lateral spacing was estimated by using both the von Kármán and Kronauer stability criteria. A new universal wake Strouhal number is devised, using the value of lateral spacing predicted by the Kronauer stability condition as the length dimension. A correlation of bluff-body data was found when pressure drag coefficient times Strouhal number was plotted against base pressure.

A study is made of the asymptotic equations governing gravity waves on water which cause a transition from one surface level to a slightly higher one and approach a steady wave-form as time increases, at least at the head of the wave. A two-parameter family of limit processes is surveyed in each of which the time scale and the horizontal length scale tend to infinity in a definite relation to the amplitude, as that tends to zero. Small-amplitude linearization is shown to be possible at most during a transitory stage of the wave development. Arbitrarily close approach to steadiness at the head of the wave is found to imply that a substantial part of the transition wave must be ultimately governed by a nearsteady variant of the non-linear equation of Korteweg & de Vries (1895) and must take the form of a train of cnoidal waves characterized by a parameter which changes slightly from crest to crest.

The investigation concerns the stability of an incompressible second-order fluid (visco-elastic) film flowing down an inclined plane under gravity with respect to two-dimensional disturbances. When the elastic parameter is negative as in the case of a solution of polyisobutylene in cetane, surface disturbances (‘soft’ waves) are found to be unstable. The analysis in this case also reveals the existence of growing shear waves (‘hard’ waves) which are highly damped in ordinary Newtonian fluids.

Experiments have been carried out to observe the motion of an initially station-ary circular bubble in a two-dimensional bed of spherical particles fluidized by air. Theoretical work is given in the preceding paper (part 1) by Murray (1967). The experimental problem is similar to that of Walters & Davidson (1962) who devised a technique for releasing a circular air bubble in a two-dimensional bed of water, but it is made more difficult by the requirement that the stationary void, and any boundary used to maintain it, should here be permeable to the fluidizing gas flow. Two different experimental techniques were used.

There are stationary solutions of finite amplitude convection in a layer of fluid heated from below which show increasing heat transport with decreasing Rayleigh number in the neighbourhood of its critical value. It is shown that those solutions are unstable and that convection with periodic time dependence can occur in these cases, when the heat flux is the given parameter instead of the temperature difference between the boundaries of the layer. The time dependence has been calculated explicitly for the case of convection with temperature variation of the material properties.

This paper deals with the steady flow in a rectangular cavity where the motion is driven by the uniform translation of the top wall. Creeping flow solutions for cavities having aspect ratios from ¼ to 5 were obtained numerically by a relaxation technique and were shown to compare favourably with Dean & Montagnon's (1949) similarity solution, as extended by Moffatt (1964), in the region near the bottom corners of a square cavity as well as throughout the major portion of a cavity with aspect ratio equal to 5. In addition, for a Reynolds number range from 20 to 4000, flow patterns were determined experimentally by means of a photographic technique for finite cavities, as well as for cavities of effectively infinite depth. These experimental results suggest that, within finite cavities, the high Reynolds number steady flow should consist essentially of a single inviscid core of uniform vorticity with viscous effects being confined to thin shear layers near the boundaries, while, for cavities of infinite depth, the viscous and inertia forces should remain of comparable magnitude throughout the whole domain even in the limit of very large Reynolds number R.

An analysis is presented of the deformation of a solid-like, viscoelastic sphere suspended in the infinite Stokesian flow field of a Newtonian fluid undergoing an arbitrary time-dependent homogeneous deformation far from the particle. The results of the analysis are then used to deduce the macroscopic rheological behaviour of a dilute monodisperse suspension of slightly deformable spheres.

Even though inertial effects and second-order terms in the particle deformation are neglected, it is found that non-linear rheological effects can arise, because of the interaction between the deformed particle and the flow. As a consequence, the rheological relation obtained here differs from those presented earlier by Fröhlich & Sack (1946) and by Oldroyd (1955) through the appearance of certain terms which are non-linear in the deformation rate.

When the suspended particles are purely elastic in their behaviour the rheological equation presented here reduces for certain flows to a special case of Oldroyd's (1958) phenomenological model, with material constants which can be directly related to suspension properties.

This paper is an analysis of the steady incompressible, two-dimensional flow of conducting fluids through ducts of arbitrarily varying cross-section under the action of a strong, uniform, transverse, magnetic field. More precisely, the flow is such that the velocity is given by ${\bf u} = (u_x(\tilde{x}, \tilde{y}), u_y(\tilde{x}, \tilde{y}), 0)$, the position of the duct walls by $\tilde{y} = f_t(\tilde{x}), f_b(\tilde{x})$ and $\tilde{z} = \pm b $, where b [Gt ] ft−fb, and the magnetic field by B0 = (0, B0, O). It is assumed that the magnetic field is strong enough to satisfy the conditions that the interaction parameter, N(=M2/R) [Gt ] 1, where M is the Hartmann number and R is the Reynolds number, and also that M [Gt ] 1 and Rm [Lt ] 1, where Rm is the magnetic Reynolds number.

We examine the flow in three separate regions:

1. the ‘core’ region in which the pressure gradient is balanced by electromagnetic forces;

2. Hartmann boundary layers where electromagnetic forces are balanced by viscous forces;

3. thin layers parallel to the magnetic field in which electromagnetic forces, inertial forces, and the pressure gradient balance each other. These layers which have thickness $O(N^{-\frac{1}{3}})$ occur where the slope of the duct wall changes abruptly.

By expanding the solution as a series in descending powers of N we calculate the velocity distribution for regions (i) and (ii) for finite values of N attainable in the laboratory.

A solution is obtained for steady, cellular convection when the Rayleigh number and the Prandtl number are large. The core of each two-dimensional cell contains a highly viscous, isothermal flow. Adjacent to the horizontal boundaries are thin thermal boundary layers. On the vertical boundaries between cells thin thermal plumes drive the viscous flow. The non-dimensional velocities and heat transfer between the horizontal boundaries are found to be functions only of the Rayleigh number. The theory is used to test the hypothesis of large scale convective cells in the earth's mantle. Using accepted values of the Rayleigh number for the earth's mantle the theory predicts the generally accepted velocity associated with continental drift. The theory also predicts values for the heat flux to the earth's surface which are in good agreement with measurements carried out on the ocean floors.

It has been observed that the intensity of turbulence measured with a hot-wire anemometer close to the wall in a boundary layer can be substantially reduced by the presence of an obstacle nearby in a lateral direction. The obstacle in these experiments was another hot-wire probe. A brief survey was made of the positions of the two probes that could give a significant effect. However, the observation is reported mainly as an interesting phenomenon requiring further investigation. It is also of relevance to hot-wire anemometry; there may be a previously unsuspected and troublesome source of error due to a wire's own support.

Properties of turbulent thermal convection were measured in air between horizontal plates maintained at constant temperatures. Rayleigh numbers of 6·3 × 105, 2·5 × 106 and 1·0 × 107 were studied with a convection chamber designed to allow measurements to be taken along a horizontal path. Vertical profiles are presented of horizontally averaged temperature; r.m.s. fluctuations of temperature, horizontal velocity and vertical velocity; total heat flux; the correlation coefficient between vertical velocity and temperature; all terms in the thermal variance equation; and of terms in the turbulent kinetic energy balance.

One-dimensional spectra of temperature, horizontal velocity and vertical velocity at two different heights all disclose large intensity for waves of length 5 to 15 times the plate separation. Secondary peaks occur for scales of from 0·7 to 1·7 times the separation height.

Many of the observations are consistent with a thermal structure dominated by plumes extending most or all of the distance between plates.

The Hartmann oscillator problem is studied using the hydraulic analogy between compressible gasdynamics and incompressible flows with a free surface. Extensive photographs, taken of the oscillator in order to visualize the periodic flow, show that qualitative features of the flow agree well with the observations made with a cavity in an air jet.

The construction and operation of the water channel built and used at the University of California at Los Angeles for this study is also described. Continuous ‘shooting’ water flow was found to be analogous to supersonic isentropic gas flow; a static depth of approximately ¼ in. of water appeared to be satisfactory. It seems that the most valuable aspect of the hydraulic analogy is its ability to easily present excellent visualization of flow phenomena, both steady and unsteady.

This paper presents some new velocity correlation measurements in a two-dimensional zero-pressure-gradient turbulent boundary layer; these extend the work of Grant (1958). The most detailed of the new measurements concern: correlations between velocity fluctuations in the mean flow direction and fluctuations perpendicular to the wall at a different point, usually but not always in the same plane parallel to the wall; correlations between velocity fluctuations in the mean flow direction and fluctuations perpendicular to this but parallel to the wall, usually but not always close to the wall with the separation in the same direction as the latter velocity fluctuation; and various correlations with a fixed separation normal to the wall and a simultaneous variable separation in the mean flow direction.

The results are not consistent with any of the various models of the large eddy structure that have been proposed. The features of the present work most relevant to the formulation of a substitute are brought together in §9. Included are suggestions that the similarities between the wall and outer regions are more marked than the differences, and that the description of the large eddies in the wall region as a coherent eruption from the viscous sublayer is unsatisfactory.

The new experiments are considered in connexion with three questions posed by the previous work. These concern the large eddy contribution to the Reynolds stress; the asymmetry between upstream and downstream separations when there is also a separation normal to the wall; and the similarity between boundary layers and channel flow. A complicated situation surrounds the first point; there are arguments suggesting that the large eddy contribution might in some places be of opposite sign to the total Reynolds stress, but this no longer seems so likely. The upstream-downstream asymmetry is also more complicated than had been previously supposed, but the results can be systematically interpreted if the peaks of the correlation curves are considered in connexion with continuity and the tails of the curves are considered in connexion with the shearing action of the mean flow. Regarding the third point, the present experiments give no indication of any difference in the large eddy structure in the wall regions of boundary layer turbulence and channel flow turbulence.

Section 5 gives a brief account of some experiments with a vibrating ribbon in the turbulent boundary layer.

Results of calculations and experiments on the flow of a viscous liquid through an axisymmetric conduit expansion are reported. The streamlines and vorticity contours are presented as functions of the Reynolds number of the flow. The dynamic interaction between the main flow and the captive eddy between it and the walls is analysed, and it is concluded that, for laminar flow, the main role of the eddy is that of shaping the flow with a rather small energy exchange.

The primary aim of the analysis presented herein is to consolidate the ideas of the ‘conjugate-flow’ theory, which proposes that vortex breakdown is fundamentally a transition from a uniform state of swirling flow to one featuring stationary waves of finite amplitude. The original flow is assumed to be supercritical (i.e. incapable of bearing infinitesimal stationary waves), and the mechanism of the transition is explained on the basis of physical principles that are well established in relation to the analogous supercritical-flow phenomenon of the hydraulic jump or bore. In previous presentations of the theory the existence of appropriately descriptive solutions to the full equations of motion has only been inferred from these general principles, but here the solutions are demonstrated explicitly by means of a perturbation analysis. This has basically much in common with the classical theory of solitary and cnoidal waves, which is known to explain well the essential properties of weak bores.

In § 2 the basic equations of the problem are set out and the leading results of the original theoretical treatment are recalled. The new developments are mainly presented in § 3, where an analysis of finite-amplitude waves is completed by two different methods, each serving to illustrate points of interest. The effects of small energy losses and of small flow-force reductions (i.e. wave-resistance effects) are considered, and the analysis leads to a general classification of possible phenomena accompanying such changes of integral properties in either slightly supercritical or slightly subcritical vortex flows. The application to vortex breakdown remains the focus of attention, however, and § 3 includes a careful appraisal of some experimental observations on the phenomenon. In § 4 a summary is given of a variant on the previous methods which is required when the radial boundary of the flow is taken to infinity. The main analysis is developed without restriction to particular flow models, but in § 5 the results are applied to a specific example.

This paper describes an experimental technique which allows estimates of local turbulent properties of a shear layer to be obtained without the necessity of inserting a probe into the flow field. The probe is replaced by two beams of radiation, which pass through the entire flow field in two mutually perpendicular directions. It is shown that, although each beam independently reflects only an integral of the flow properties along the entire path between the source and detector, the covariance between the two detected signals does yield local turbulent information.

To verify the validity of the technique, experimental results are presented for the shear layer of a subsonic jet and are shown to be in good agreement with published hot-wire data.

The stress in a suspension of incompressible deformable particles exceeds that which would exist in the pure incompressible liquid undergoing the same flow by the product of the volume concentration of the particles and the tensor $\overline{p}_{ik} - 2\eta_0 e^{-(1)}_{ik}$ where η0 is the viscosity of the pure liquid, $\overline{p}_{ik}$ is the average stress and $e^{-(1)}_{ik}$ the average rate of strain in particles in the flowing suspension. For a dilute suspension of viscoelastic spheres these tensors can be determined by using an adaptation of Jeffery's solution of the problem of an isolated rigid ellipsoid. In steady laminar flow, the material of each sphere is continuously deformed and rotates within an ellipsoidal boundary of fixed dimensions and orientation. Approximate expressions are obtained for the steady-rate viscosity and normal stress differences in terms of the dynamic viscosity and dynamic rigidity functions of the suspension. These are valid either when the rate of shear is sufficiently small or when the ratio of the dynamic viscosity of the spheres to ηo is sufficiently large. The three normal stress components are all unequal. In slow steady elongation of the suspension, the spheres suffer static deformation into prolate spheroids so the elongational viscosity depends only on their static elastic properties. It appears from the investigation of special cases, however, that this static deformation is not possible for rates of elongation above a critical value.

The process by which a stratified, viscous fluid adjusts to small changes in the rotation rate of its container is studied. This paper treats the cases of homogeneous layers of different densities, as well as fluids which are continuously stratified.

It is shown that in several important cases the spin-up process, especially in the continuously stratified case, has a time scale which is very much longer than for homogeneous fluids, and that diffusion is the governing mechanism in the adjustment process.

In all cases the detailed problem, including a discussion of the side-wall boundary layers, is presented. Some novel features of the side-wall layers are discussed for the continuously stratified fluids, while in one case it is shown that no boundary layers appear during the transient approach to equilibrium.