A steady nonlinear dispersive wave theory is developed in terms of three important invariants of channel flow: discharge, energy, and momentum flux. As such, the work is an extension of Benjamin & Lighthill's approach for rectangular channels.

Considering the differential equation obtained, we examine the behaviour of flows and wave systems in arbitrary channels for changes of energy and momentum. In particular, the bore problem is studied, and previous approaches to this problem, using linear wave theory, are seen to be invalid. The present theory describes several phenomena of open-channel flow, explains a scatter in previously obtained experimental results, and enables simple design recommendations to be made for channels in which stationary or moving bores are expected.

While this work does describe the variation of physical quantities across the channel section, there are some important three-dimensional phenomena, noted experimentally, which remain unexplained.

The mechanisms which initiate secondary flow in developing turbulent flow along a corner are examined on the basis of both energy and vorticity considerations. This is done by experimentally evaluating the terms of an energy balance and vorticity balance applied to the mean motion along a corner bisector. The results show that a transverse flow is initiated and directed towards the corner as a direct result of turbulent shear stress gradients normal to the bisector. The results further indicate that anisotropy of the turbulent normal stresses does not play a major role in the generation of secondary flow. Possible extensions of the present results to other related flow situations are ahstrated and discussed.

It is suggested in this paper that axisymmetric holes in thin sheets of fluid in which surface tension forces predominate will open out if they are initially large in relation to the thickness of the sheet; but that small holes will close up. No exact criterion has been found for the critical hole size in a free falling sheet, but the behaviour of the sheet may be closely simulated by the suspension of a soap film between coaxial circular rings. Theoretical results and experimental observations on catenoid films so formed are described.

For a hole in a sheet standing under gravity on a horizontal plane an equilibrium configuration exists, which is shown to be unstable. It is suggested that in this case the equilibrium position serves to distinguish between holes which open and those which close. Experiments on the behaviour of holes in a mercury sheet reveal a well-defined critical size which is in good agreement with that predicted by the unstable equilibrium.

A further series of experiments on holes made in a sheet of water standing on paraffin wax gave no sharp distinction between opening and closing holes, and holes of a wide range of sizes could remain stationary. This behaviour is associated with changes in the angle of contact with the plane. Independent meniscus observations similar to those of Ablett for a steadily moving meniscus show that the angle of contact θa, for a meniscus about to advance is greater than the value θr for a meniscus on the point of receding. It is seen that this difference will produce a range of hole diameters within which a hole will be trapped and remain stationary. Observations on the minimum size of hole on a water sheet which will remain open are reported. But it was found that the largest holes which would remain stationary were too large in relation to the size of the sheet for reliable results to be obtained.

The research summarized in this paper is an experimental study of the velocity and turbulence fields in the gap between two parallel co-rotating or stationary disks with a source in the centre. Detailed hot-wire measurements over a wide range of source strengths and disk speeds are presented; the average velocity components are correlated in terms of dimensionless quantities; and the radial pressure distribution is analysed. Also, transition phenomena and conditions under which similarity in the velocity profiles can be expected are discussed.

For locally isotropic, homogeneous fluid turbulence, a digital Fourier analysis method of measuring directly the net scalar and velocity spectral transfer Tn(k) of scalar and kinetic energy to a pa;rticular wavenumber from all other wave-numbers is described and applied to heated-grid turbulence. The technique uses the imaginary part of a particular cross-spectrum to obtain the one-dimensional net spectral transfer function Ln(k1) of velocity and scalar turbulence, and is a refinement of that used previously by Van Atta & Chen for measuring the velocity kinetic energy transfer.

The detailed spectral transfer Tn(k, k′) from one wavenumber to any other is related to the imaginary part of a particular three-dimensional bispectrum. Tn(k, k′) can be, in principle, computed from a particular two-dimensional triple correlation. Unlike Tn(k), which can be obtained from Ln(k1), Tn(k, k′) cannot be determined from the measurable one-dimensional bippectrum B1, n, n (k1, k1′) nor the one-dimensional transfer spectrum Ln(k1, k1′).

The measured net transfer spectra Tn(k) have been used to determine the extent of validity for heated-grid turbulence of the dynamical equations for the three-dimensional power spectra of temperature and velocity in locally isotropic turbulence. The measured temperature transfer spectrum is also compared with those obtained from the power spectra of velocity and temperature by using various simple hypotheses.

The motion of bodies through fluid at low Reynolds number is appreciably affected by the container walls. Consequently the Stokes-flow theory due to Batchelor (1970) and others for a slender body falling in an unbounded fluid is difficult to test experimentally unless it is extended to take account of nearby boundaries. Theoretical expressions are given here for certain drag coefficients of a circular cylindrical slender rod of finite length falling close to a single plane wall or falling midway between two parallel plane walls. Experiments with a very viscous liquid are described in which cylinders of small thickness-to-length ratios (ranging from 1:10 to 1:100 approximately) are made to fall in suitable orientations. From their times of fall over a measured distance experimental drag coefficients are determined and compared with the corresponding theoretical value from the extension of Batchelor's theory. For rods falling in a horizontal orientation the theoretical and experimental results are consistent within the order of accuracy of the experiments. However, when results are compared for rods falling in a vertical orientation there is a significant difference for which possible explanations are presented.

An experimental study was conducted to measure the growth rates of mechanically generated surface water waves when subjected to a fully developed turbulent channel airflow. The study was designed to test the accuracy of the growth rates predicted by Miles's (1962b) theory. For a series of wave frequencies (from 2·04 to 6·04 Hz at 0·50 Hz increments) and centre-line wind velocities (0·20, 1·12 and 1·84 m/s) wave amplitudes were measured at three stations (2–21, 3–43 and 4·65 m) downwind from a wave generator. In addition, for centre-line velocities of 1–12 and 1·84 m/s, U* (the velocity at the outer edge of the viscous sublayer) and U1, (the shear velocity) were obtained from measured mean velocity and Reynolds stress profiles. The wave amplitude measurements at the wind velocity of 0·20 m/s provided attenuation rate estimates which agreed reasonably well with theoretical attenuation rates based on viscous effects both on the walls and in the bulk of the water. The amplitude measurements at the wind velocities of 1·12 and 1·84m/s provided growth rate estimates which were compared with theoretical growth rates (computed using the wave frequency, U1 and U* predicted by Miles's (1962b) theory. At 1·12m/s Miles's growth rateswere two to five times larger than those measured; at 1·84 m/s Miles's growth rates were about two times larger.

A recent investigation of hydromagnetic waves in a rotating fluid has revealed certain ‘valve’-like critical levels associated with each wave which can be effectively penetrated from one side only. This effect is illustrated in the present paper by means of two further examples, namely (a) the propagation of hydromagnetio-gravity waves in a non-uniform magnetic field, and (b) the propagation of internal gravity waves in a wind which, though unidirectional, is both horizontally and vertically sheared.

The nonlinear stability of a liquid film adjacent to a supersonic gas stream is investigated. The gas is assumed to exert a mean shear stress at the liquid/gas interface which in turn establishes a linear mean velocity profile in the liquid. The analysis takes into account the pressure perturbation exerted by the gas on the liquid assuming the disturbed gas motion to be inviscid and the mean gas velocity profile to be uniform. The problem is formulated within the long wave approximation, and solutions are obtained for finite amplitude waves by using the method of multiple scales. The results predict the existence of finite amplitude periodic waves, in qualitative agreement with recent experimental observations.

This paper examines the possible thickening of an initially sharp sonic boom by the turbulence it encounters in passing to the ground. Three apparently different viewpoints, all indicating substantial thickening, are shown to be actually identical and to give an irrelevant upper bound on wave thickness. All three approaches describe only the apparent mean diffusion induced by random convection of a sharp wave about its nominal position. Although a wave-front folding mechanism ultimately accounts for an apparent thickening as individual rays are weakened and tangled by turbulence, this process is too slow to be effective in the practical boom situation. The paper then considers what linear thickening of a wave packet results from propagation trough atmospheric turbulence and concludes that, in the relevant limit, a wave may be thickened by a factor of about 2 at the most. The conclusion is therefore reached that atmospheric turbulence cannot be the cause of the thousandfold discrepancy between the measured wave fronts and their Taylor thickness.

In recent years many problems concerned with the dispersion of a passive contaminant along pipes and channels have been investigated, and this paper is concerned with one such problem which arises in diverse applications. This is the study of the longitudinal dispersion of a contaminant whose concentration is prescribed as a harmonic function of time at one cross-section. On the basis of physical arguments and of detailed calculations for two laminar flows it is shown that for high frequencies the concentration pattern is transported downstream at the maximum fluid velocity but that for low frequencies it is transported at the discharge velocity, and that the fluctuations in concentration decay to zero in a much shorter downstream distance for high frequencies than for low frequencies. It is shown further that at high frequencies the concentration is exponentially small except near the places where the fluid velocity attains its maximum, whereas for low frequencies the variation in concentration over the cross-section is small. Some of these conclusions are compared with those made by others, and the agreement is in general satisfactory.

The steady flow of an ideal gas with finite conductivity past a thin wedge section in an aligned magnetic field is considered within the linearized theory. The wedge is assumed to be an insulator or a perfect conductor. The flow field around the obstacle is found by numerical integration of the Fourier transform for various values of the Mach number, the Alfven Mach number and the magnetic Reynolds number in the main stream. The outer flow is estimated by the method of asymptotic expansions. These results disclose how the inner flow including gas-dynamic features changes to the magnetohydrodynamic outer flow.

A similarity solution is found for the distribution of layer thickness in viscous source flow down an inclined plane. With this soldion a t lowest order, an asymptotic expansion in inverse powers of the downstream co-ordinate x allows a small correction for upstream influence. A simple experiment confirms the major features of the similarity solution: (i) a parabolic cross-stream variation in the layer thickness; (ii) spreading of the flow according to an $x^{\frac{3}{7}}$ power law; (iii) thin-ning of the layer along streamlines like $x^{-\frac{1}{7}}$; and (iv) surface velocities which vary as the square of the layer thickness. Deviations of the layer-thickness measurements from the parabolic profile follow the trend predicted by the first-order corrections, whereas systematically high measured values are explained qualitatively in terms of waves at the free surface.

The eigenvalue equation for three-dimensional waves in parallel and cross flows parallel to a fluid discontinuity has been considered for spatially growing waves. The discontinuity plane (x, y plane) is perpendicular to the gravitational acceleration and consists in general of a jump in speed, in flow direction and in density. With the assumption of waves which are periodic in time and periodic in the y direction, the eigenvalue equation is solved for the complex wavenumber α in the x direction. These waves are used to Fourier synthesize the wave trails generated by a time-periodic disturbance with a Gaussian amplitude distribution e−δy2 along the y axis. Lines of constant phase and lines of constant amplitude within the wave trail have been illustrated for some examples.

The interaction of weakly nonlinear waves with slowly varying boundaries is considered. Special emphasis is given to rotating fluids, but the analysis applies with minor modifications to waves in stratified fluids and shallow-water aves. An asymptotic solution of a variant of the Korteweg–de Vries equation with variable coefficients is developed that produces a ‘Green's law’ for the amplification of waves of finite amplitude. For shallow-water waves in water of variable depth, the result predicts wave growth proportional to the $-\frac{1}{3}$ power of the depth.

Steady, supersonic, dissipative, three-dimensional, axisymmetric flow is considered. A system of Burgers-type equations is shown to govern the flow field. In inviscid regions the Whitham theory gives the limiting form. Dissipative effects ultimately engulf the inviscid zone and at sufficiently large distances from the body the flow is governed by linear dissipative theory. The flow field is divided into zones based on the presence or absence of nonlinearity and dissipation. Estimates and criteria which describe the extent of these zones are given.

A model of vortex breakdown is presented and its predictions compared with the experiments of Sarpkaya (1971). The model is cntred about a theory of long, weakly nonlinear waves propagating on critical flows in tubes of variable cross-section. Although the weakly nonlinear theory must be extended beyond its domain of formal validity, many of the experimentally observed features of vortex breakdown are reproduced by the model. The description of the time evolution of the flow field that is presented requires numerical calculations that are not simple, but some important conclusions may be determined by easy computations. In particular, the axial position of a breakdown may be found from a very simple equation (10).

The structure of shear layers due to bottom topography in a rotating stratified fluid is obtained under the restriction σS [Lt ] E½, where σS = ναgΔT/KΩ2L is a measure of the stratification and E = ν/Ω2L is the Ekman number. The layers are found to be similar to the side-wall layers discussed by Barcilon & Pedlosky (19673) if σS [Gt ] E½ and are Stewartson layers if $\sigma S \ll E^{\frac{2}{3}}$. Some comments are made on the possibility of Taylor column formation in a stratified fluid.

Two-dimensional convection in a Boussinesq fluid confined between free boundaries is studied in a series of numerical experiments. Earlier calculations by Fromm and Veronis were limited to a maximum Rayleigh number R 50 times the critical value R, for linear instability. This range is extended to 1000Rc. Convection in water, with a Prandtl number p = 6·8, is systematically investigated, together with other models for Prandtl numbers between 0·01 and infinity. Two different modes of nonlinear behaviour are distinguished. For Prandtl numbers greater than unity there is a viscous regime in which the Nusselt number $N \approx 2(R/R_c)^{\frac{1}{3}}$, independently of p. The heat flux is a maximum for cells whose width is between 1·2 and 1·4 times the layer depth. This regime is found when $5 \leqslant R/R_c \lesssim p^{\frac{3}{2}}$. At higher Rayleigh numbers advection of vorticity becomes important and N ∞ R0·365. When p = 6·8 the heat flux is a maximum for square cells; steady convection is impossible for wider cells and finite amplitude oscillations appear instead, with periodic fluctuations of temperature and velocity in the layer. For p < 1 it is also found that N ∞ R0·365, with a constant of proportionality equal to 1·90 when p [Lt ] 1 and decreasing slowly as p is increased. The physical behaviour in these regimes is analysed and related to astrophysical convection.

The low Mach number sound field induced by the motion of line vortex filaments coupled to a two-dimensional semi-infinite duct is determined by means of a singular perturbation technique. Using the method of matched asymptotic expansions, solutions for the sound field are obtained by matching with an ‘inner’ region of incompressible flow. The radiation field induced by the emergence of a single vortex from the channel exhibits the edge scattering effects typical of half-plane problems. The sound field intensity is found to have angular dependence on sin2½ θ, where θ = 0 defines the exterior axis of symmetry. When a vortex pair propagates out of the duct however, the special symmetry of the fluid motion causes cancellation of the scattered field from the duct edges. In that case the sound field is driven from sources located at the duct exit. We show that this result is consistent with the general theories of both Curle and Powell. The sound field is essentially induced by a dipole at the exit plane of the duct, part of which drives a coupled weak monopole, while the remainder corresponds to an axial ‘edge force’ originating in the r−½ velocity singularities at the duct edges.