The present paper deals with the temperature field past an isothermal sphere. Acrivos & Taylor (1962) have obtained results for the case R [les ] 1 with no restriction on the Prandtl number. The results of the present paper are for the case R < 1 in which the Prandtl number is of order unity. An expansion for the Nusselt number is given up to and including the term of order R2.

From an analysis of small oscillations of a viscous fluid droplet immersed in another viscous fluid a general dispersion equation is derived by which frequency and rate of damping of oscillations can be calculated for arbitrary values of droplet size, physical properties of the fluids, and interfacial viscosity and elasticity coefficients. The equation is studied for two distinct extremes of interfacial characteristics: (i) a free interface between the two fluids in which only a constant, uniform interfacial tension acts; (ii) an ‘inextensible’ interface between the two fluids, that is, a highly condensed film or membrane which, to first order, cannot be locally expanded or contracted. Results obtained are compared with those previously published for various special cases.

When the viscosities of both fluids are low, the primary contribution to the rate of damping of oscillations is generally the viscous dissipation in a boundary layer near the interface, in both the free and inextensible interface situations. For this reason inviscid velocity profiles, which do not account for the boundarylayer flow, do not lead to good approximations to the damping rate. The two exceptions in which the approximation based on inviscid profiles is adequate occur when the interface is free and either the interior or exterior fluid is a gas of negligible density and viscosity.

In this paper a reservoir connected through a horizontal contraction to a channel is considered. Both the reservoir and the channel are considered to contain a stable multi-layered system of fluids. The conditions under which there is a flow in only one layer, and the depth in this flowing layer decreases continuously from its depth in the reservoir to its depth in the channel, give the maximum discharge that can be obtained with a flow only from this single layer. For this case the volume discharge calculations are carried out at a single section (the section of minimum width). Where there are velocities in only two layers and the depth in each of these layers decreases continuously from their depths in the reservoir to their depths in the channel, the theory involves computations at two sections in the flow. These are the section of minimum width and a section upstream of the position of minimum width (the virtual point of control). For this flow it is shown that the solution is the one in which the velocity and density distributions are self similar and that the depths of the layers at the point of maximum contraction are two-thirds of those far upstream. It is then shown that for any stable continuous or discrete density stratification in the reservoir a self similar solution will satisfy the conditions for the depths of the flowing layers to decrease smoothly from the reservoir to downstream of the contraction. Again the ratio of the depth at the contraction to that far upstream is two-thirds.

When there is a very large density difference between the fluid in the lower dead water and that in the lowest flowing streamline then this streamline becomes horizontal and may be considered as a frictionless bed. The flow when the bed is not horizontal but where there is a small rise in the channel at the position of maximum contraction is considered for the case where two discrete layers flow under a volume of dead water. In this case the velocity and density profiles are not self similar.

Experiments have been carried out with a contraction in a flume for the withdrawal of two discrete layers from a three layer system and the withdrawal from a fluid with a linear density gradient. In both cases the reservoir and channel bed and hence the lowest streamline was effectively horizontal. These experiments confirmed the theoretical predictions.

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.

Arguments are given which suggest that the mean flow of a turbulent fluid is irrotational far from the boundaries. Mathematical difficulties precluded a detailed theory so a simple experiment was performed. This experiment qualitatively confirmed the suggestion. Other well-known experiments on the mean flow of turbulent fluid may also be interpreted in terms of vorticity expulsion, an idea closely related to Scorer's hypothesis on turbulent mixing.

Theoretical papers on stellar rotation usually assume that convective regions rotate uniformly because of a high turbulent viscosity. However, if vorticity expulsion occurs, convective stellar cores will not rotate. The concept of a turbulent viscosity is criticized.

The turbulence problem is formulated using the Wiener stochastic expansion. The expansion is useful for processes which are in some sense nearly normal, and can be used for non-linear non-Gaussian processes such as many turbulent fluid flows. Here we present the general formulation for statistically inhomogeneous and anistropic processes.

The transfer term in the energy equation, or equivalently the third-order velocity correlation, forms a sensitive measure of the amount of non-Gaussianity present in real fluid flows. Experimental evidence shows that in many flows this component is small compared with the Gaussian part. It is shown that a homogeneous and isotropic flow which has but a small non-Gaussian part possesses a distribution at one point which is Gaussian to terms of second order. The experiments suggest that immediately behind a grid in a wind tunnel the flow is very nearly normal. The non-Gaussian part grows at a moderate rate, at least within the range of distance downstream (or decay time) available in the usual experiments. This growth is probably due to the relative increase in the amount of energy in the smallest eddies, which are non-normal.

A necessary criterion for the validity of the zero-fourth-cumulant approximation is suggested: the transfer term in dimensionless form should be small. It is shown that calculations using the zero-fourth-cumulant approximation have given negative energy spectra when this condition is violated, probably for the reason that the process is no longer nearly Gaussian. However, even when this condition is fulfilled, it is shown that that approximation must be corrected.

It is suggested that the present theory is valid for quite large times of decay if initial energy spectra are chosen which are not too far from the actual physical values for fluid in turbulent flow. Equations are given for the next-higher-order term in a nearly normal approximation. The expansion is also used in § 6 to describe turbulent mixing problems and is compared with the zero-fourth-cumulant approximation for these problems. Computational results are presented in § 7 and compared with experiments by Stewart and Townsend.

The heat transfer appropriate to low Reynolds number hot-wire anemometry is calculated from the full non-linear equations of motion and of heat transfer by an iterative method starting with the Oseen solution and its heat flux analogue. The second and third iterates yield close agreement with measured data.

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.

Theoretical results are obtained for forced heat convection from a circular cylinder at low Reynolds numbers. Consideration is given to the cases of a moderate and a large Prandtl number, the analysis in each case being based upon the method of matched asymptotic expansions. Comparison between the moderate Prandtl number theory and known experimental results indicates excellent agreement; no relevant experimental work has been found for comparison with the large Prandtl number theory.

This paper describes an experimental and theoretical study of the complicated disturbance (Taylor column) due to the slow relative motion between a spherical, or short cylindrical, rigid object and an incompressible fluid of low viscosity in which the object is immersed, when the motion of the object is that of steady revolution with angular speed Ω rad/see about an axis (the Z-axis) whose perpendicular distance from the centre of the object, $\overline{R}$, is much greater than a typical linear dimension of the object, L, and the undisturbed fluid motion is one of steady rotation about the same axis with angular speed (Ω+Uϑ/R) and zero relative vorticity (i.e. d(UϑR)/dR = 0). It extends earlier experimental work on Taylor columns to systems of sufficiently large axial dimensions for Z variations in the disturbance pattern to be perceptible. Over the ranges of Rossby and Ekman numbers (based on L) covered by the experiments, namely ε = 1·89 × 10−3 to 2·36 ×10−1 and γ = 1·30 × 10−3 to 2·03 × 10−2 respectively, the axis of the Taylor column is found to trail in the downstream direction at a small angle ϕ = tan−1 (Kε) to the line parallel to the Z-axis through the centre of the object, where K = (1·54 ± 0·04) for a sphere. The variation with Z of the amplitude of the disturbance is roughly linear and the scale-length of this variation, Zc, is close to L/γ¼ over the limited range of γ covered by the experiments.

The experimental value of K is remarkably close to the theoretical value derived by Prof. Lighthill in the appendix, where he applies his general linear theory of waves generated in a dispersive system by travelling forcing effects to the problem of describing a Taylor column at large distances from the moving object when the fluid is inviscid and unbounded.

It has been suggested on theoretical grounds that a vortex could be initiated in a cylindrical region of fluid, originally in solid rotation, by the horizontal mixing of angular momentum produced by external stirring. In this paper various arguments for and against the mechanism are examined and their difficulties exposed. No firm conclusion is reached. A series of mathematical models of the mixing motions has been used, to bring out the differences between mechanical stirring and the agitation of a gas by random molecular motions. They suggest the introduction of a diffusion coefficient for angular momentum, to be determined empirically. These theoretical ideas are then applied to the interpretation of the results of a laboratory experiment which has been designed to test the proposed mechanism directly.

A wide, flat tank of liquid was set up on a rotating table and stirred with a vertically oscillated grid, whose elements were much smaller than the width of the tank. A neutrally buoyant particle was used as a tracer of fluid motions, to measure relative circulation velocities and the properties of the turbulence. The motion observed was dominated by the loss of angular momentum to the walls and the grid, an effect which has not been taken into account in previous theoretical assessments of the effects of mixing of angular momentum. The relative circulation present was not significantly different from zero, and the limits of error of the measurements imply that the rate of diffusion of angular momentum is less than 5% of that for fluid particles, with 95% probability.

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.

A simple frictional wind-driven model of the Antarctic circumpolar current is examined. The geometry includes what are thought to be the main features effecting the current, namely a gap corresponding to Drake Passage and a partial barrier corresponding to the South American peninsula. Solutions obtained by both numerical and analytical methods are presented. The analytic solution, valid for small values of a friction parameter, enables the total rate of transport of water by the current to be calculated as a function of the friction parameter and the wind stress distribution. The width which controls the rate of transport by the current tends to be the narrowest encountered rather than the average width as assumed in earlier zonal models. However, the current spreads out to several times this width due to frictional effects. The values of the eddy viscosity required to give a rate of transport of the observed order are about 103 cm2/s for a bottom-friction model and of the order of 108 cm2/s for a lateral friction model.

This paper is concerned with the jet of liquid, open to the atmosphere, that emerges from a two-dimensional channel in which there is Poiseuille flow far upstream, the flow being driven by an applied pressure gradient. The problem is discussed with the aid of the method of matched asymptotic expansions; the small parameter involved is the inverse Reynolds number. A boundary layer forms adjacent to the free surface, and a classical boundary-layer analysis is applied to find the flow there (for moderate distances downstream); the influence of this boundary layer on the flow in the core of the jet is then investigated. Higher-order boundary-layer effects, such as indeterminacy and eigensolutions, are also discussed. The first few terms are found of an asymptotic expansion for the equation of the free surface, and considerations of momentum balance are applied to find the asymptotic contraction ratio of the jet.

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.

This paper presents some recent theoretical results on the energy exchange between a swimming flexible two-dimensional hydrofoil of finite profile thickness and the inviscid incompressible fluid in which the body swims. The rate at which kinetic energy is transferred to the fluid by the undulating hydrofoil, the power required to maintain the prescribed motion, and the resulting power available for propulsion are calculated in terms of the thickness to chord ratio and the displacement and rate of displacement of the hydrofoil. With a small unsteady perturbation theory, the analysis is decomposed to show separately the effects of the circulatory and non-circulatory flows, both depending on the first-order terms of the unsteady perturbation velocity components. In addition, an analysis is presented showing the effect of the non-linear unsteady pressure distribution on the surface of the hydrofoil. Contrary to what might be expected, this latter effect is of the same order of magnitude for a thick rounded-nose profile as for the flat plate where the effect is concentrated at the sharp leading edge and is related to the so-called suction force. However, except for small values of the reduced frequency, the non-linear contribution is negligible in comparison with the linear contribution.

New functions associated with the retarded flow in the wake are introduced and special techniques for their solution are presented, these being related to Theodorsen's function of unsteady airfoil theory for the special case of the undulating flat plate.

The numerical results reveal that the kinetic energy imparted to the fluid, the power required to maintain the motion, and the resulting propulsive power, follow closely those of an infinitesimal model for small values of the reduced frequency of oscillation, but diverge somewhat from the classical thin plate analysis for large reduced frequencies. Of particular interest is the fact that a very large percentage of the power required to maintain the motion is used in the generation of the wake, whereas a very small percentage of the power available for propulsion comes from the wake. This indicates that, if some mechanism could be devised to control the wake, very high swimming efficiencies could be attained. Fish, in all probability, have been succeeding in doing this for millions of years.

An experimental and analytical study of the separation of a turbulent boundary layer is reported. The turbulent boundary-layer separation model proposed by Sandborn & Kline (1961) is demonstrated to predict the experimental results. Two distinct turbulent separation regions, an intermittent and a steady separation, with correspondingly different velocity distributions are confirmed. The true zero wall shear stress turbulent separation point is determined by electronic means. The associated mean velocity profile is shown to belong to the same family of profiles as found for laminar separation. The velocity distribution at the point of reattachment of a turbulent boundary layer behind a step is also shown to belong to the laminar separation family.

Prediction of the location of steady turbulent boundary-layer separation is made using the technique employed by Stratford (1959) for intermittent separation.

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 equations governing high-frequency oscillatory viscous flows are investigated through the separation of the steady and the unsteady parts. All Reynolds number ranges are studied and the orders of magnitude of the steady streaming produced by the Reynolds stresses are established.

The oscillating circular cylinder at low Reynolds numbers is studied through the method of inner and outer expansions. Steady recirculating cells exist near the cylinder. The results compare very well with experiments. Analytic expressions for the streamfunction and the drag coefficient are obtained.

The oscillating flow towards an infinite plate is investigated in detail. The steady streaming is caused by the steady component of the Reynolds stress. The pressure gradient always causes reverse flow near the solid boundary.

Two-dimensional internal gravity waves in a rectangular container are examined theoretically and experimentally in (a) fluids which contain a single density discontinuity and (b) fluids in which the density gradient is everywhere continuous. The fractional density difference between the top and bottom of the fluid is small.

Good agreement is found between the observed and calculated wave profiles in case (a). Unlike surface standing waves, which tend to sharpen at their crests as the wave amplitude increases, and which eventually break at the crests when fluid accelerations become equal to that of gravity, internal wave crests are found to be flat and exhibit no instability. In the case (a) breaking is found to occur at the nodes of the interfacial wave, where the current shear, generated by the wave itself, is greatest. For sufficiently large wave amplitudes, a disturbance with the form of a vortex but with direction of rotation reversing twice every cycle, grows at the wave node and causes mixing. This instability is found to be followed by the generation of cross-waves, of which two different forms are observed.

Several modes of oscillation can be generated and are observed in a fluid with constant density gradient. The wave frequencies and shape are well predicted by theory. The experiments failed to establish any limitation of the possible wave amplitudes.