Using a method due to Davies & Taylor (1950), a simple model is employed to derive the velocity of a two-dimensional gas bubble rising in liquid along the axis of a channel of finite width. The asymptotes of the solution agree well with previous results and an experimental investigation confirms the effect of channel width on bubble velocity. Measured values of velocity are, however, approximately 9% higher than theoretical values due to the three-dimensional nature of the real flow.

Integral relations are derived for steady, incompressible recirculating motions with small viscous forces. The circuit time of a fluid particle on a closed streamline in steady, inviscid flow is shown to be the same for all the closed streamlines on a surface of constant total head.

The discontinuities of velocity and velocity gradient that occur in the motion of inviscid fluid filling a closed, rotating cylinder set in a rotating support with the two rotation axes slightly misaligned are then investigated.

The investigation concerns the stability of an interface between two inviscid fluids of different densities which flow parallel to each other in an oscillatory manner. When the difference in the mean speeds is below the steady, critical speed for instability but is large compared to the amplitude of the fluctuations, parametric amplification of waves at the interface occurs, and the interface exhibits a resonance of a subharmonic nature. The paper concludes with a discussion of the stability of unsteady shear layers on the basis of the results.

In the first part of this work the stability of the erodible bed of a stream with a free surface is studied within the framework of classical hydraulics, in which the velocity variation with depth is reduced to a single mean velocity and the bed friction is related in a general way to the local depth and mean velocity. Only two-dimensional motions can be studied in this way. In considering bed friction and a difference in phase between deposition and the mean velocity gradient along the channel, this work combines aspects of earlier studies of Exner and Kennedy.

In the absence of a phase difference between erosion and mean velocity, the analysis proceeds without linearization. When the development from an equili- brium flow down a uniform slope is considered, the bed waves formed under subcritical flows are found to move downstream, while those under super-critical flows upstream; in both régimes bed waves are damped. In each case the side of a wave facing in the direction of motion is the steeper.

When a phase shift is introduced as well, the analysis is carried forward only after linearization. The primary effect of bed slope and friction is a shift in the ranges of phase angle for which growth can take place and a corresponding alteration in the wavelengths for maximum growth. Friction also reduces bed-wave celerity. Consideration is given to the physical processes represented by the artifice of a phase difference between erosion and mean velocity gradient.

The second section of this investigation concerns two-dimensional potential flow over a wavy stream bed. This problem is considered from a point of view different from that adopted previously by Kennedy; a modified criterion is proposed for the maximum Froude number at which bed waves will form. It is in better agreement with measured data.

In the third part of this paper, the potential analysis is extended to include class of three-dimensional motions. The conditions are found for the formation of dunes (bed waves 180° out of phase with the surface waves above) and anti-dunes (the two in phase). The criterion separating dunes and antidunes for two dimensions is found to give a lower limit on the Froude number for antidunes of the more general three-dimensional class. In the antidune regime the streamwise perturbation to the velocity can change sign between surface and stream bed; the limit for this is determined.

The erosion equation relating changes in bed level to local stream speed is generalized to include sediment convection in two dimensions. With this equation, three distinct regimes of bed-wave motion are found: at low Froude numbers, dunes moving downstream; at higher Froude numbers, antidunes moving upstream; and, finally, antidunes moving downstream. The criterion separating the last two r6gimes is just that mentioned above as the limit of flows whose velocity perturbation has the same sign from surface to stream bed. It is argued that this fundamental change in the flow marks also the end of the region of growth of small bed waves. It is found that three-dimensional dunes and upstream-moving antidunes can exist beyond the two-dimensional limit, but that the latter applies for all bed waves such that the ratio of stream depth to wavelength (d/λ) is small. This explains why the modified criterion for two dimensions provides for small d/λ an envelope for data obtained from observations of a wide variety of bed forms, but fails to do so for larger d/λ. The celerity of bed waves beneath three-dimensional flows is discussed. It is suggested that a class of waves occurring in natural streams will move more slowly than would two-dimensional waves in similar conditions.

The work concludes with a comparison of several methods of modelling erosive flows.

The effect of a uniform and constant rate of strain imposed upon a turbulent wake which is generated behind a circular cylinder has been examined experimentally. The strain field, produced by passing the flow through a distorting duct of constant cross-sectional area, simultaneously compresses the wake in the plane of the cylinder and extends it in the other lateral direction but applies no strain in the direction of the main stream. The investigation shows the existence of a large-scale, periodic eddy structure within the strained wake, similar to, but more intense than, that which was observed for the undistorted wake. These large lateral convective motions appear to be the primary mechanism controlling the development of the flow, and are responsible for the failure of the wake to attain the theoretically predicted self-preservation state. It is shown that the strain rate, which acts favourably upon these large eddies, is sufficient to produce a net increase in vorticity in spite of the ordinary processes of turbulent viscous decay. At the same time, however, the rate of viscous dissipation of the smallest eddies in the flow is augmented by the strain, and the resulting wake structure is almost entirely dominated by the large-eddy motions. A possible means by which the large eddies are initiated is discussed.

If a thick, turbulent boundary layer is disturbed near the rigid boundary, the flow changes are confined initially to a thin layer adjacent to the boundary. Elliott (1958) and Panofsky & Townsend (1964) have attempted to calculate the flow disturbance caused by an abrupt change in surface roughness by assuming special velocity distributions which are consistent with a logarithmic velocity variation near the boundary. Inspection of their distributions shows that the deviations from the upstream distribution are self-preserving in form, and it is shown that self-preserving development is dynamically possible if log l0/z0 (l0 being depth of modified flow, z0 roughness length) is fairly large and if l0 is small compared with the total thickness of the layer. Other kinds of surface disturbance may lead to self-preserving changes of the original flow and the theory is developed also for flow downstream of a line roughness, for the temperature distribution downstream of a boundary separating an upstream region of uniform roughness and heat-flux from a region of different or possibly varying roughness and heat-flux, and for the return of a complete boundary layer to self-preserving development after a disturbance. The requirement that the distributions of velocity and temperature should conform to the logarithmic, equilibrium forms near the surface makes the predictions of surface stress and surface flux nearly independent of the exact nature of the turbulent transfer process, and the profiles of velocity and temperature are determined within narrow limits by the surface fluxes. To provide explicit profiles, the mixing-length transfer relation is used. Its validity for the self-preserving flows is discussed in an appendix.

The distortion of a uniform magnetic field, aligned with the flow at infinity, by the potential flow of an inviscid conductor about a circular cylinder is determined. Potential flow of the fluid occurs when the interaction parameter is small; this is the case studied here. In the flow-potential and stream-function plane the problem may be formulated as a singular integral equation. Solutions of this equation show that for small fluid conductivities the magnetic field lines are distorted in the sense of being dragged along by the motion of the fluid. This process continues as the conductivity increases, with fewer and fewer of the magnetic field lines entering the body. For large conductivity this reduced flux of field lines enters over most of the body surface and exits in the neighbourhood of the rear stagnation point; behind the body there is a jet-like structure of magnetic field lines.

The paper considers the effect of turbulence-induced surface response on the sound radiated by a turbulent boundary layer. The analysis is confined to an infinite plane homogeneous surface and the conclusions may not be a good indication of the behaviour of more realistic structures. The main result of the analysis is that no fundamentally more efficient source of sound is introduced by the surface motion. The radiation remains quadrupole in character. The surface merely accounts for a reflexion of the turbulence-generated sound, with the reflexion coefficient being identical to that of plane acoustic waves. Dissipation in the surface reduces the magnitude of the image system. A brief discussion of the effect on the particular quadrupoles to be found in a turbulent boundary layer concludes the paper. There it is argued that the radiation will probably be increased by surface motion, but not by an order of magnitude.

An attempt has been made to predict the mean velocity field of turbulent jets immersed in secondary streams confined by constant-area ducts. The calculations have employed empirical data derived solely from the turbulent free jet. The conditions under which ducted jets may be considered approximately self-preserving have been examined by comparison of results of momentum-integral calculations with data from various sources on two-dimensional and axisymmetric jets immersed in secondary streams of constant velocity. For flows in which confining walls have significant effects, it is shown that the velocity field may be determined fairly well from free-jet data using the assumption of approximate self-preservation. Before the jet spreads to the wall, both the mean-velocity and turbulent shear stress are assumed self-preserving. Afterward only the shear stress is subject to this condition. Improvement of the methods used appears mainly to require a better model of the zone of recirculation.

This paper is concerned with the two-dimensional flow in a free waterfall, falling under the influence of gravity, the fluid being considered to be incompressible and inviscid. A parameter ε, such that 2/ε is the Froude number based on conditions far upstream, is defined and considered to be small. A flowline co-ordinate system is used to overcome the difficulty that the boundary geometry is not known in advance. An asymptotic expansion based on ε is constructed as an approximation valid upstream and near the edge, but singular far downstream. Another asymptotic expansion, based upon the thinness of the fall, is constructed as an approximation valid far downstream, but failing to satisfy the conditions upstream. The two expansions are then matched to give a solution covering the whole flow field. The shapes of the free streamlines are shown for a number of values of ε for which the solutions are seemingly valid.

The instability of a viscous fluid between two infinite vertical plates and heated from below in the presence of a magnetic field perpendicular to the plates is investigated, and the most critical stability boundary in the space of the Rayleigh number R, Hartmann number M, and the horizontal wave number a is determined. It is found that the most unstable mode is a symmetric mode with zero wave-number, and that for any M the fluid is unstable for any non-zero R, however small.

In a previous paper, it was shown that abrupt changes in the surface conditions under a very deep boundary layer cause changes of mean velocity and temperature that satisfy the dynamical conditions for self-preserving development. Here the theory is extended to predict the development of the modified flow in the moderately deep layers that occur in nature and the laboratory. The problems considered are the changes in the velocity profile produced by an abrupt change of surface roughness and also by a line of concentrated roughness such as a fence, the changes in temperature produced by change of roughness combined with changes of heat flux at the surface, and diffusion of heat or a scalar pollutant from a line source at or near ground level. The predictions are compared with observations by Rider (1952) of the flow downwind of a hedge, by Rider, Philip & Bradley (1963) of temperature and humidity downwind of a change in surface, and of vertical diffusion from a line-source at ground level.

An attempt is made to explain the formation of vortices in free boundary layers by means of stability theory using a hyperbolic-tangent velocity profile. The vorticity distribution of the disturbed flow, as obtained by the inviscid linearized stability theory, is discussed. The path lines of particles which are initially placed along straight lines parallel to the x-axis are calculated. Lines connecting the positions of these particles give an impression of the instant shape of the disturbed flow. With increasing time the boundary layer becomes thinner in certain regions and thicker in others. A special line—originally positioned at the critical layer—shows in the thicker region a tendency to roll up. Also extrema of the vorticity are located there. Finally, these results are compared with those which can be expected from the non-linear Helmholtz equation. Disagreement is found in the neighbourhood of the critical layer. Using the non-linear stability theory of Stuart up to the third-order terms, the vorticity distribution shows the tendency expected from the non-linear equation.

A series expansion is derived for the three-dimensional boundary-layer flow over a flat plate, arising from a general main-stream flow over the plate. The series involved are calculated as far as terms of order ξ2, where ξ is a non-dimensional parameter defining distance measured from the leading edge of the plate. The results are applied to an example in which the main stream arises from the disturbance of a uniform stream by a circular cylinder mounted downstream from the leading edge of the plate, the axis of the cylinder being normal to the plate. Calculations are made for shear stress components on the plate, and for the deviation of direction of the limiting streamlines from those in the main stream.

The modification of a turbulent wind profile owing to momentum transfer from wind to surface waves is calculated in terms of the power spectrum of the surface-wave slope. The reduction in profile curvature, and hence of the wind-to-wave energy-transfer coefficient, is estimated on the basis of a Neumann spectrum. It appears that this reduction is likely to be small for typical wind speeds.

The change by an order of magnitude of the oscillating properties of the flow past a circular cylinder in the Reynolds number range 2*103 to 5*104 is demonstrated experimentally. It is shown that in this range these properties are highly susceptible to small disturbances of the frequency of the transition waves which precede turbulence in the shear layers just downstream of the cylinder. It is suggested that this susceptibility is responsible for the different lift coefficient values measured by various workers.

The effect of disturbances on the mean flow properties is also described.

The frequency-determining mechanism and the characteristic length of the oscillating flow are discussed.

Howarth's transformation is applied to the compressible laminar boundary-layer equations for the flow at a point of attachment on a general curved surface. It is shown that the boundary-layer equations yield similarity solutions for the case when the viscosity varies linearly with the temperature, the Prandtl number is unity and the surface is maintained at constant temperature. The resulting eighth-order boundary-value problem is solved numerically for various surface temperature conditions and for various values of b/a, where a and b are constants representing the local velocity gradients in the principal directions of flow at the edge of the boundary layer.

Flow and heat transfer properties of the similar solutions at both nodal and saddle-points of attachment are given in graphical and tabular form.

The transient pressure field produced behind a plane shock wave of arbitrary strength, which encounters a slender supersonic cone head on, is theoretically predicted. The analysis is restricted to small cone angles (in order that Mach-reflexion will occur at the surface) and to inviscid flow.

A conical-flow technique is utilized to transform the small-perturbation equations of time-dependent rotational motion of the fluid behind the deflected plane shock into time-independent equations. From these equations, a boundary-value problem for pressure alone, in the Mach-reflexion region, is derived which is solved numerically. Results for the small perturbation pressure field of the cone are contrasted with Smyrl's (1963) results, which hold for a moving wedge, for a range of values of cone and plane shock speeds.

It is shown that a sphere moving through a very viscous liquid with velocity V relative to a uniform simple shear, the translation velocity being parallel to the streamlines and measured relative to the streamline through the centre, experiences a lift force 81·2μVa2k½/v½ + smaller terms perpendicular to the flow direction, which acts to deflect the particle towards the streamlines moving in the direction opposite to V. Here, a denotes the radius of the sphere, κ the magnitude of the velocity gradient, and μ and v the viscosity and kinematic viscosity, respectively. The relevance of the result to the observations by Segrée & Silberberg (1962) of small spheres in Poiseuille flow is discussed briefly. Comments are also made about the problem of a sphere in a parabolic velocity profile and the functional dependence of the lift upon the parameters is obtained.