General solutions are obtained for the long gravity wave equation from solutions of the wave equation in three dimensions. The method is applied to the theory of Tsunamis. It is also suggested that these waves are the cause of the shelf oscillations observed at Guadelupe.

The stability of the co-current stratified flow of oil and water was investigated experimentally in a horizontal rectangular conduit. Laminar-turbulent transitions were determined for both phases. With the two-phase system the transition to turbulence in the water phase occurred at a higher Reynolds number in the presence of a laminar oil layer provided the input water-to-oil ratio was relatively high, while the transition in the oil phase took place at a lower Reynolds number in the presence of a turbulent water layer. The appearance of first interfacial waves coincided with the transition to turbulence of the less viscous or water phase. This suggests that in the system investigated the resonance mechanism as proposed by Phillips (1957) was responsible for the generation of these first waves. However, at relatively high water flow rates and water-to-oil ratios more pronounced waves were observed which appeared to be generated by an instability in the mean flow.

A smoke plume that has become nearly horizontal at some distance from the source behaves much like a ‘line-thermal’, for which, using a perturbation method, a solution in laminar flow may be obtained, on the supposition that excess temperatures are small and buoyant movements slow, i.e. that the Rayleigh number of the problem is suitably low. In analogy with some other problems in turbulent flow and turbulent diffusion, the laminar solution is then assumed to approximate what is observed in the turbulent case, provided that the rate of growth of the diffusing cloud is assessed realistically.

The so-calculated pattern of streamlines in a cross-section of the plume agrees qualitatively with the observed behaviour of hot plumes and puffs, consisting of two vortex-like structures of opposite sense of rotation, lying on either side of the plume centre. The bodily upward movement of the plume is found to depend critically on the rate of growth of the plume. Thus when the plume diameter grows faster than linearly with distance (such behaviour characterizes the ‘quasi-asymptotic’ stage of relative diffusion predicted by Batchelor, 1952, for which Richardson's (4/3)-power law of eddy diffusivity holds) the plume tends to reach an asymptotic height. A crude theoretical estimate of the asymptotic height attained shows fair agreement with observations reported elsewhere. Although the plume nearly reaches this asymptotic height in the quasi-asymptotic phase, it retains a small gradient in the final phase which may be of importance at large distances from the source. The small-Rayleigh-number criterion restricts the validity of the solution to ‘weakly buoyant’ plumes.

In a barotropic fluid, a free turbulent flow induces a fluctuating potential flow which is determined by the instantaneous flow near the edge of the turbulent flow. If the surrounding fluid is stably stratified, internal wave-motions are possible and, in general, wave-energy accumulates until it is sufficient to modify the turbulent flow. Here the growth of wave-motion from rest is examined with particular reference to the atmospheric problem of wave excitation by the surface boundary layer. Wind shear is supposed negligible outside the turbulent flow and the disturbances from the boundary layer are assumed to travel with a convection velocity V relative to the upper air. For times large compared with {−g/ρ(dρ/dz)}−½ (ρ is the potential density), most of the wave-energy resides in components of phase-velocity near the convection velocity. For a model atmosphere with increased stability above a tropopause, this resonance mechanism leads to the formation of wave-groups with crests at right-angles to the convection velocity and wavelengths near 2πV[−g/ρ(dρ/dz)]−½. Using likely values for the surface disturbances, wave-amplitudes of order 100 m can develop within several hours of the initiation of the boundary layer but sufficient amplitude to produce overturning or breaking is unlikely within a reasonable time.

The problem considered is that of the diffraction of the field of a point source in a fluid of infinite depth by an infinite vertical cylinder. It is shown that the surface wave component of the velocity potential may be expressed in terms of the solution to a classical electromagnetic (or acoustic) diffraction problem.

This paper presents a linearized theory for the average decay of a tape-induced fully developed turbulent swirl in flow through a pipe. In the Reynolds number range between 104 and 105 the theoretical analysis was found to be in good agreement with experimental data obtained with water in a 1 in. pipe, provided the eddy diffusivity was chosen appropriately.

It was observed that a turbulent swirl decays to about 10–20% of its initial intensity in a distance of about 50 pipe diameters, the decay being more rapid at smaller than at larger Reynolds numbers. The theoretical swirl velocity distribution agreed qualitatively with experimental measurements at distances less than 20 diameters downstream from the outlet of the swirl inducer, but deviated from the experimental results further downstream.

The basic property of equations describing dispersive waves is the existence of solutions representing uniform wave trains. In this paper a general theory is given for non-uniform wave trains whose amplitude, wave-number, etc., vary slowly in space and time, the length and time scales of the variation in amplitude, wave-number, etc., being large compared to the wavelength and period. Dispersive equations may be derived from a variational principle with appropriate Lagrangian, and the whole theory is developed in terms of the Lagrangian. Boussinesq's equations for long water waves are used as a typical example in presenting the theory.

The purpose of this paper is to provide some possible explantions for certain observed phenomena associated with the mean-velocity profile of a turbulent boundary layer which undergoes a rapid yawing. For the cases considered the yawing is caused by an obstruction attached to the wall upon which the boundary layer is developing. Only incompressible flow is considered.

§1 of the paper is concerned with the outer region of the boundary layer and deals with a phenomenon observed by Johnston (1960) who described it with his triangular model for the polar plot of the velocity distribution. This was also observed by Hornung & Joubert (1963). It is shown here by a first-approximation analysis that such a behaviour is mainly a consequence of the geometry of the apparatus used. The analysis also indicates that, for these geometries, the outer part of the boundary-layer profile can be described by a single vector-similarity defect law rather than the vector ‘wall-wake’ model proposed by Coles (1956). The former model agrees well with the experimental results of Hornung & Joubert.

In §2, the flow close to the wall is considered. Treating this region as an equilibrium layer and using similarity arguments, a three-dimensional version of the ‘law of the wall’ is derived. This relates the mean-velocity-vector distribution with the pressure-gradient vector and wall-shear-stress vector and explains how the profile skews near the wall. The theory is compared with Hornung & Joubert's experimental results. However at this stage the results are inconclusive because of the lack of a sufficient number of measured quantities.

The problem of the axially symmetric irrotational flow of an inviscid incompressible fluid with a free surface in a circular cylindrical container accelerating parallel to its axis is considered. A numerical procedure is developed and some interesting motions exhibiting the development of breakers, splashing and sustained oscillations of the surface are obtained by machine computation.

A theory is given to predict the shape and amplitude of a standing wave formed on a liquid film running down a vertical surface, and due to an upward flow of gas over the liquid surface. The wave is maintained in position by the pressure gradients induced within the gas stream by acceleration over the windward part of the wave; over the leeward part of the wave, the gas pressure is roughly constant due to breakaway of the gas flow.

The wave amplitude is found to be very sensitive to gas velocity so that the theory predicts a critical gas velocity beyond which the wave amplitude becomes very large; this critical velocity is confirmed by experiment, and the experiments confirm the predicted wave shape. The critical gas velocity also agrees reasonably well with published values of the flooding velocity in empty wetted-wall tubes; this velocity is defined as the point at which countercurrent flow of gas and liquid becomes unstable. The phenomenon of flooding, which has puzzled chemical engineers for many years, may thus be due to wave formation on the liquid film.

From the theory are derived three dimensionless groups, namely, Weber number $We \equiv \rho_g U_c^2t_0|T$, liquid-film Reynolds number $Re \equiv 4\rho_l Q|\mu, and Z \equiv T(\rho_l|\mu g)^{1/3}|\mu$. Here Uc is the critical gas velocity, Q is the liquid volume flow rate per unit wetted perimeter, ρg and ρl are the gas and liquid densities, μ is the liquid viscosity and T is its surface tension; $t_0 = (3\mu Q|\rho_lg)^{1/3}$ is the liquid film thickness in the absence of gas flow. We, Re and Z are uniquely related at the flooding point, and a diagram is presented to show this relation. This diagram will enable designers to predict flooding in wetted-wall tubes, though more experimental verification is required.

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 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.

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.

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.

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.

Plane potential flow past a circular cylinder beneath a free surface under gravity is investigated in order to determine the importance or otherwise of non-linear effects from the free-surface boundary condition. It is shown that non-linear second-order corrections to the first-order linearized expressions for the wave-induced forces on the cylinder are considerably larger than second-order effects which are present even with a linear free-surface condition. Further evidence for the importance of non-linearity is presented in the form of streamline plots of the first-order solution showing strange behaviour at wave crests.

A two-dimensional jet or wake is observed in a frame of reference moving with the fluid at infinity, so that the velocity w(y) in the x-direction tends to zero as y → ± ∞. The fluid is assumed to be an inviscid perfect gas, to undergo-adiabatic changes, and the local speed of sound to be a function of y such that a(y)→ a∞, as y → ±∞. If the disturbance pressure has the form