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.

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.

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.

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.

Benjamin (1962) introduced the idea that a given ‘primary’ swirling flow, with cylindrical stream surfaces, may have associated with it ‘conjugate flows’, also swirling and cylindrical, which in a certain sense are equivalent to the primary one. He deduced that, in cases where such conjugate flows exist and where the primary flow cannot support standing waves of small amplitude, the conjugate flow nearest the primary one (a) can support such waves, and (b) has a ‘flow force’ greater than that of the primary flow. In the present paper these two results are proved rigorously by a method which differs from Benjamin's.

The velocity of a large gas bubble rising along the axis of a cylindrical container filled with liquid, is derived to a first approximation from a simple flow model of the system. The resulting expressions involve ratios of certain infinite series, which are evaluated for a few cases, and the theory is then seen to agree with previously accepted results at its asymptotes. Experiments are performed which confirm the first approximation and which allow the theory to be recast as a semi-empirical theory relating bubble volume with velocity. In this form it agrees with the results of Uno & Kintner (1956), but indicates that the relation between volume and curvature which they employed was in error.

This paper describes a very general method for determining the steady two-dimensional potential flow about one or more bodies of arbitrary shape operating at arbitrary Froude number near a free surface. The boundary condition of zero velocity (solid wall) or prescribed velocity (suction or blowing) normal to the body surface is satisfied exactly, and the boundary condition of constant pressure on the free surface is satisfied using the classic small-wave approximation. Calculations made by the present method are compared with analytic results, other theoretical calculations and experimental data. Examples for which no comparison exists are also presented to illustrate the capability of the method.

Axially symmetric potential flow about an axially symmetric rigid body is considered. The potential due to the body is represented as a superposition of potentials of point sources distributed along a segment of the axis inside the body. The source strength distribution satisfies a linear integral equation. A complete uniform asymptotic expansion of its solution is obtained with respect to the slenderness ratio ε½, which is the maximum radius of the body divided by its length. The expansion contains integral powers of ε multiplied by powers of log ε. From it expansions of the potential, the virtual mass and the dipole moment of the body are obtained. The flow about the body in the presence of an axially symmetric stationary obstacle is also determined. The method of analysis involves a technique for the asymptotic solution of integral equations.

This paper shows the existence of a three-dimensional solution of the boundary-layer equations of viscous incompressible flow in the immediate neighbourhood of a two-dimensional stagnation point of attachment. The numerical solution has been obtained.

The measurement of the turbulent shear stresses and normal and bi-normal intensities with a hot-wire anemometer requires that the directional sensitivity of the hot-wire be known. Normal component or cosine law cooling is generally assumed, although for finite wire lengths the non-uniform wire temperature must cause a deviation from the cosine law.

Careful heat transfer measurements from wires inclined and normal to the flow were taken for several values of the Reynolds number, the length-to-diameter ratio of the wire, the overheat ratio and for several support configurations. All experiments were performed in air at low subsonic velocities, i.e. M < 0·1. The measurements indicate that the heat loss from an inclined wire is larger than that from a wire normal to the flow with the same normal component of velocity. The data were correlated by \[ U^2_E(\alpha) = U^2(0)(\cos^2\alpha + k^2\sin^2\alpha), \] where UE(α) is the effective cooling velocity at the angle α between the normal to the wire and the mean flow direction and U(0) is the velocity at α = 0. The value of k was found to depend primarily upon the length-to-diameter ratio ([lscr ]/d) of the wire. For platinum wires k is approximately 0·20 for [lscr ]/d = 200, decreases with increasing [lscr ]/d, and becomes effectively zero at [lscr ]/d = 600.

To aid in interpreting the heat transfer data, measurements of the temperature distribution along inclined and normal wires were made with a high sensitivity infra-red detector coupled to a high resolution microscrope with reflective optics. The measurements indicate that inclined wires and normal wires have nearly identical end conduction losses, although the temperature distribution on an inclined wire is slightly asymmetrical. Therefore, the deviation from the cosine law is caused by an increase in the convection heat loss, and this increase is attributed to the tangential component of velocity.

Hot-wire response equations to include the effects of the tangential velocity component as well as the non-linearities caused by high intensity turbulence are derived for linearized constant temperature operation. For low intensity turbulence similar equations are derived for constant current operation. The equations are applied to an X-wire array to determine the errors in selected turbulence quantities which arise from the assumption of cosine law cooling. The error depends upon the quantity measured, the method of operation, and [lscr ]/d. For [lscr ]/d = 200 the error ranges from 0 to 17%.

An analysis has been made of a two-dimensional turbulent wake which was subjected to a homogeneous strain. This was applied in such a way that the rate of growth of the wake tended to be accelerated but the main stream convective velocity remained constant. An attempt to describe the flow by classical self-preservation techniques was made. However, the results suggested that a detrainment of wake fluid would result. In lieu of this a more realistic approach was sought and it was found that in certain situations where the flow has had an opportunity to develop before the strain is applied, the growth of the wake in the strain field can be described successfully by a simple geometric distortion. It appears likely that in these cases the interaction between the external strain and the turbulent structure within the wake is negligible, the wake remaining essentially passive. Suggestions as to the mechanism of the turbulent mixing and transport under these conditions are made.

Experiments are described in which the velocity profiles of axisymmetric jets are measured in water and in dilute solutions of water-soluble polymers. The ambient fluid and that entering through the nozzle were homogeneous. The similarity characteristics of velocity profiles including a linear variation of centre-line velocity with the inverse of the distance from the nozzle were unaltered by these substances. On adding large polyethylene oxide molecules to the system the jet spread at an increased angle, possibly due to the entrainment rate being enhanced by an increase in turbulent elasticity for polyethylene oxide solutions. It is also postulated that additives which had no effect on the velocity were not large enough to influence the small microscale eddies in which the energy to activate an entrainment cycle is stored.