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.

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.

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.

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.

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.

Asymptotic equations of long waves in a compressible medium of infinite height with arbitrary density and wind profiles, including cross-wind, are systematically derived to provide the mathematical description of certain pressure-front phenomena observed in the atmosphere. It is assumed that the final state of the wave is a plane wave moving with nearly constant velocity, even though its velocity field has a component transverse to the direction of propagation. The coefficients of the asymptotic long-wave equations are found to depend on the equilibrium profiles of the density and of the velocity component in the direction of propagation of the wave, but not on the profile of the transverse velocity. Any horizontal direction of propagation is found to be possible, even for waves of permanent type.

When a horizontal layer of viscous liquid with an adsorption film of varying concentration as its upper boundary is set in motion by a steady translation of its lower boundary, plane Couette flow with zero surface velocity is possible. In this paper the stability of this flow is considered, and it is found that the liquid layer can be unstable for long waves. The instability found for this flow, however, exists also for other laminar flows with an adsorbed film, and plane Couette flow has been chosen only as a simple means of demonstration.

The effect of the transverse motion of a cylinder on its natural vortex wake is examined at various driving frequencies giving special attention to the change in wake geometry along the span. Conditions are established for which the vortex wake frequency is controlled by the driving frequency of the cylinder.

An axisymmetric viscous-transonic equation is presented. A nozzle-type similarity solution of this equation has been found, which describes the initial stages in the development of shock waves downstream of a converging-diverging nozzle throat. This solution is an extension of a two-dimensional solution found previously (Sichel 1966). By an appropriate choice of an arbitrary scaling constant solutions were found such that there is essentially a weak normal shock near the axis with effects of wall and shock-wave curvature occurring only at a sufficiently large radius. The upstream and downstream asymptotic behaviour of these viscous-transonic nozzle solutions has been investigated.

When thin films of liquid are produced by rolling it is commonly observed that the liquid film is not uniform but has regularly spaced thick lines or ribs. This phenomenon has been studied experimentally and it was found that the onset of ribbing was governed by the relation $\eta S/\gamma = 10.3 (g/R)^{\frac{3}{4}}$. At higher values of the parameter η/Sγ the number of ribs per centimetre increased sharply. At values of ηS/γ > 2·4 the rib number became independent of this parameter and varied solely with the geometrical parameters of the system, i.e. roller radius and gap.

Laboratory measurements were made of the instabilities of the Ekman layer using hot wire anemometers. The apparatus consisted of two parallel circular rotating plates forming a spool; the air was admitted through screens at the outer edge and removed through a screen cage at the hub. In the Ekman layers formed on the inner surfaces of the plates, measurements were made of the mean velocities as functions of r and a; the velocity fluctuations were also measured.

It appears that the instability labelled type II by Faller always occurs first, and at zero Rossby number the critical Reynolds number is 56 ± 2. This instability originates in the boundary layer, but at slightly higher Reynolds number the fluctuations persist far into the geostrophic region, probably as inertial waves excited by the boundary-layer fluctuations.

At higher Reynolds number another instability appears of shorter wavelength and slower speed. This instability is confined to the boundary layer and is apparently the type I reported by Faller.

The phase speeds, frequencies, and wave-front orientations of both type instabilities have been measured.

A rigid cylinder is initially at relative rest in a uniformly rotating, inviscid, incompressible fluid, with its generators perpendicular to the axis of rotation. The fluid is accelerated suddenly to a small constant velocity parallel to the axis of rotation, which is maintained thereafter. The growth of the subsequent disturbance due to the cylinder is interpreted in terms of plane inertial waves, the disturbance energy propagating with the local group velocity, which is in the plane of the wave front and proportional to the wavelength. Taylor columns, in which the fluid moves with the cylinder rather than round it, grow indefinitely in both directions parallel to the rotation axis, the head of the column moving with finite speed.

In a slightly viscous fluid, an ultimate steady state is reached, in which the columns are of finite length.

If a cylinder is moved horizontally in a non-rotating uniformly stratified Boussinesq liquid, an identical analysis may be applied with a similar interpretation in terms of internal gravity waves rather than inertial waves.

The suppression of initially isotropic turbulence by the sudden application of a uniform magnetic field is considered. The problem is characterized by three dimensionless numbers, a Reynolds number R, a magnetic Reynolds number Rm and a magnetic interaction parameter N (for definitions, see equations (1.2) and (1.4)). It is supposed that R [Gt ] 1, Rm [Lt ] 1 and N [Gt ] 1. There are two important time scales, ta a time characteristic of magnetic suppression, and t0 = Ntd, the ‘turn-over’ time of the turbulent energy-containing eddies. For 0 < t [Lt ] t0 the response of the energy-containing components of the turbulence to the applied field is linear and the time dependence of the kinetic energy density K(t) and magnetic energy density M(t) are analysed. There are essentially two distinct contributions to each from two domains of wave-number space D1 and D2 (defined in figure 2). In D1 the response is severely anisotropic, while in D2 it is nearly isotropic. The relative importance of the contributions K1(t) (from D1) and K2(t) (from D2) to K(t) depends on the value of the Lundquist number S = (NRm)½. If S [Lt ] 1, then K1(t) dominates for all t [lsim ] t0 and K(t) ∝ t−½ for td [Lt ] t [Lt ] t0. If 1 [Lt ] S [Lt ] R−2m, then a changeover in the dominant contribution occurs when t = O(S½Rm) t0. Analogous results are obtained for the magnetic energy density.

The turbulent energy equation is converted into a differential equation for the turbulent shear stress by defining three empirical functions relating the turbulent intensity, diffusion and dissipation to the shear stress profile. This equation, the mean momentum equation and the mean continuity equation form a hyperbolic system. Numerical integrations by the method of characteristics with preliminary choices of the three empirical functions compare favourably with the results of conventional calculation methods over a wide range of pressure gradients. Nearly all the empirical information required has been derived solely from the boundary layer in zero pressure gradient.