The axisymmetric flow of liquid in a rigidly bounded annular container of height H, rotating with angular velocity Ω and subjected to a temperature difference ΔT between its vertical cylindrical perfectly conducting side walls, whose distance apart is L, is analysed in the boundary-layer approximation for small Ekman number v/2ΩL2, with gαΔTHv/4Ω2L2K ∼ 1. The heat transfer across the annulus is then convection-dominated, as is characteristic of the experimentally observed ‘upper symmetric regime’. The Prandtl number v/k is assumed large, and H is restricted to be less than about 2L. The side wall boundary-layer equations are the same as in (non-rotating) convection in a rectangular cavity. The horizontal boundary layers are Ekman layers and the four boundary layers, together with certain spatial averages in the interior, are determined independently of the interior flow details. The determination of the latter comprises a ‘secondary’ problem in which viscosity and heat conduction are important throughout the interior; the meridional streamlines are not necessarily parallel to the isotherms. The secondary problem is discussed qualitatively but not solved. The theory agrees fairly well with an available numerical experiment in the upper symmetric regime, for v/k [bumpe ] 7, after finite-Ekmannumber effects such as finite boundary-layer thickness are allowed for heuris-tically.

The experiments reported here extend those of Townsend which form the basis of his model of free turbulence. Here straining is carried to a strain ratio of 6:1, while Townsend's straining went only to 4:1. Two kinds of distorting ducts are used to produce the uniform mean strain applied to initially nearly isotropic grid turbulence.

The results differ from Townsend's in that: (i) a considerably higher degree of anisotropy is achieved, Townsend's measure of anisotropy attaining values up to 0·6, rather than the maximum of 0·42 he found; (ii) there is no evidence that an equilibrium structure is attained; and (iii) the strained turbulence rapidly becomes less anisotropic when the straining ceases.

It is found to be possible to predict the variation of the total turbulence energy using rapid-distortion theory with a correction for decay. However, the individual components cannot be accurately predicted in this way.

A thin-film, heated-element probe is calibrated to measure fluctuating skin friction. The probe is mounted flush with the surface of a flat plate which is oscillated in its own plane, in the direction of the air stream. The relationship between fluctuating heat transfer from the film and fluctuating skin friction is discussed theoretically using Lighthill's (1954) technique.

The motion of puffs which move in the direction of the buoyancy force was considered in a previous paper (Richards 1965). We now consider cases in which the directions of motion and of buoyancy are inclined or opposed. The main assumption is that the distributions of velocity inside the puffs are similar throughout the motion. Although this assumption is thought to be accurate only when the surroundings are neutral, we have reason to think that the same assumption may sometimes be used as an approximation to the internal distribution of velocity in other types of environment.

The particular case of motion when the buoyancy force is constant is calculated in detail. The calculations are supported by observations from new laboratory experiments.

When the end of a long closed horizontal tube containing a stratified fluid is raised, a laminar accelerating flow begins. The flow is two-dimensional in the central portion of the tube, and, in this region, it is predictable, at least until the onset of instability. The occurrence of instability, its nature, and the subsequent transition to turbulence, are described qualitatively. The apparatus may be used for the study of a variety of other internal hydraulic phenomena with applications to meteorology and oceanography.

The pressure drop ΔP created by the motion of a ‘small’ spherical particle settling along the axis of a large-diameter circular cylinder filled with a quiescent liquid was measured in the particle Reynolds number range (based on diameter) from Re = 0·2 to 21,000. For Re < 125 it was found that ΔPA/D = 2·0 (A = cylinder cross-sectional area; D = particle drag), in agreement with existing theory in the Stokes and Oseen regimes. Beyond Re = 125 a fairly abrupt transition occurs, the ΔPA/D ratio decreasing asymptotically towards 1·0, the limiting value predicted by elementary momentum principles for an ‘unbounded’ fluid, with increasing Re. At Re ≈ 6000 the transition is essentially complete.

Stability to infinitesimal disturbances—when a parallel magnetic field is imposed—is investigated for the flow in the boundary layer set up by two-dimensional motion between parallel planes of a viscous, incompressible, electrically conducting fluid under the influence of a transverse magnetic field. The flow is assumed to take place at low magnetic Reynolds number. The usual asymptotic methods are employed for the solution, but, apart from the Tollmientype power series solution, an exact solution of the inviscid equation is obtained in terms of the hypergeometric function and its analytic continuation. Curves of neutral stability for two-dimensional disturbances are calculated and the results for critical Reynolds number modified to take into account three-dimensional disturbances. The parallel magnetic field is found to have a strong stabilizing influence.

An incompressible fluid fills a container of fixed shape and size and of uniform cross-section in the (x, y)-plane, the m rigid side walls and the two rigid end walls being in contact with the fluid. Here (x, y, z) are the Cartesian co-ordinates of a general point in the frame of reference in which the container is stationary. Fluid is withdrawn from the container at Q cm3/sec via certain permeable parts of the side walls and replaced at the same steady rate via other permeable parts of the side walls. As, by hypothesis, the vorticity of the entering and leaving fluid relative to the container is zero, the concomitant fluid motion within the container, Eulerian velocity u = −∇ϕ − ∇ × A, is irrotational when the container is stationary in an inertial frame. The present paper is concerned with the effects on u of uniform rotation of the whole system with angular velocity Ω about the z-axis when the normal component of u on the side walls is independent of z.

In the simplest conceivable case, D ≡ zu − zl is infinite (but D/Q remains finite). End effects are then negligible and u is everywhere independent of z. The solenoidal component of u, − ∇ × A, corresponds to j gyres, one for each of the j irreducible sets of circuits across which the net flow of fluid does not vanish that can be drawn within the m-ply connected region bounded by the side walls. While ∇ϕ, which satisfies ∇2ϕ = 0, depends on Q but not on Ω, j and v (the coefficient of kinematic viscosity), ∇ × A depends on all these quantities but vanishes identically when jΩ = 0. When jΩ ≠ 0 but v → 0, ∇2A + 2Ω, the absolute vorticity, tends to zero everywhere except in certain singular regions near the bounding surfaces, where boundary layers form.

End effects cannot be ignored when D is finite. When D is independent of x and y and equal to D0 (say) and Ω is sufficiently large for the boundary layers on the end walls to be of the Ekman type, 95% thickness δ = 3(v/Ω)½ (δ [Lt ] D0), the end effects that then arise are only confined to these boundary layers when j = 0. When j ≠ 0 boundary-layer suction influences the flow everywhere; thus ∇2A and ∇ϕ (but not ∇ × A) are reduced to zero in the main body of the fluid, the regions of non-zero ∇ϕ and ∇2A being the Ekman boundary layers on the end walls and boundary layers of another type, 95% thickness δs (typically greater than δ), on the side walls. A theoretical analysis of the structure of these boundary layers shows that non-linear effects, though unimportant in the end-wall boundary layers, can be significant and even dominant in the side-wall boundary layers. The analysis of an axisymmetric system, whose side walls are two coaxial cylinders, suggests an approximate expression for Δs. When D is not everywhere independent of x and y, non-viscous end effects arise which produce relative vorticity in the main body of the fluid even when j = 0.

Experiments using a variety of source-sink distribution generally confirm the results of the theory, show that instabilities of various kinds may occur under certain circumstances, and suggest several promising lines for future work.

The paper deals with the problem of estimating the sound field generated by a limited region of turbulence in an infinitely long, straight, hard-walled pipe. The field is analysed in a co-ordinate system moving with the assumed uniform mean flow, and the possibility of eddy convection relative to that reference system is considered. Large-scale turbulence is shown to induce plane acoustic waves of intensity proportional to the sixth power of flow velocity. The same is true of small-scale turbulence of low characteristic frequency. In both cases convective effects increase the acoustic output and distribute the bulk of the energy in a mode propagating upstream against the mean flow. Small-scale turbulence of higher frequency excites more modes, the sound increasing with very nearly the eighth power of velocity (U7.7) as soon as the second mode is excited. In the limit, when more than about 20 modes are excited, the energy output is unaffected by the constraint of the pipe walls, increasing with the eighth power of velocity, and being substantially amplified by convective motion.

An earlier paper (Howe 1967) considered a non-linear theory of open-channel steady flow of deep water past a slowly modulated wavy wall. The wave pattern on the free surface of the water was obtained as the solution of a stably posed elliptic Cauchy problem, the main feature of the solution being the appearance of a ‘shock’ across which there is an abrupt change of phase. Such phase jumps can occur in a wide range of similar problems, but the advantage of the present case is that it is rather well suited to experimental investigation. This paper is therefore a lead-in to the more general problem of phase jumps, and uses the principle of conservation of energy in conjunction with the earlier solution to predict the possible position of the discontinuity on the free surface of the water. The possible nature of the free surface in the vicinity of the phase jump is also discussed (figure 4). This is a region where the width of the wave troughs becomes dramatically shorter than that of the neighbouring troughs. An approximate method of determining the line along which the phase jump occurs, not depending on a knowledge of the solution of the Cauchy problem, is also presented.

The fine structure of the summer thermocline in the Mediterranean Sea around Malta, investigated by a new temperature-gradient meter and by photographs of dye tracers, is summarized. The principal internal feature of the thermocline is a series of thin, laminar-flow sheets of high static stability, separated by weakly turbulent layers of only moderate density gradient and a few metres thick. A mechanism for generating the patches of turbulence observed on these thermocline sheets is established by comparing dye photographs with a theory by Miles & Howard (1964).

In this paper we examine the stability of a two-dimensional wake profile of the form u(y) = U∞(1 – r e-sy2) with respect to a pulsed disturbance at a point in the fluid. The disturbed flow forms an expanding wave packet which is convected downstream. Far downstream, where asymptotic expansions are valid, the motion at any point in the wave packet is described by a particular three-dimensional wave having complex wave-numbers. In the special case of very unstable flows, where viscosity does not have a significant influence, it is possible to evaluate the three-dimensional eigenvalues in terms of two-dimensional ones using the inviscid form of Squire's transformation. In this way each point in the physical plane can be linked to a particular two-dimensional wave growing in both space and time by simple algebraic expressions which are independent of the mean flow velocity profile. Computed eigenvalues for the wake profile are used in these relations to find the behaviour of the wave packet in the physical plane.

The wind-driven circulation of a simple model of the oceanic circulation (linear and homogeneous) is investigated in detail to delineate the role of the Ekman layer mass flux in driving hitherto overlooked components of the oceanic circulation.

The role of upwelling boundary-layer regions in driving interior geostrophic circulations is discussed in detail. Several interesting circulations hidden in the earlier transport theories are described.