A train of small-amplitude surface waves is incident normally on an arbitrary arrangement of thin barriers lying in a vertical plane in deep water. Each barrier is allowed to make small rolling or swaying oscillations of the same frequency as that of the incident wave. The boundary-value problem for the consequent fluid motion, assumed two-dimensional, is solved exactly using a technique which enables the amplitudes of the scattered waves far from the barriers to be readily determined. Reference is made to the associated wave radiation problem and to the calculation of forces and moments on the barriers.

Jets of water and of poly(ethylene oxide) solutions discharging in air were photographed using a novel image-motion compensating camera. Spray droplet formation is inhibited by low concentration polymer solutions. The effect of the polymer is to reduce, dampen, or eliminate small-scale surface disturbances in the jet, while not reducing but even amplifying larger scale motions. The initial laminar zone present in the jet efflux with water is eliminated with trace quantities of polymer. When substantial quantities of polymer are present (200 p.p.m.), the jet breakup is accompanied by filament formation linking all the drops together.

The equations which govern the flow at high Reynolds number in the vicinity of the trailing edge of a finite flat plate at incidence to a uniform supersonic stream are solved numerically using a finite-difference procedure. The critical order of magnitude of the angle of incidence α* for the occurrence of separation on one side of the plate is α* = O(R−¼) (Brown & Stewartson 1970), where R is a representative Reynolds number for the flow, and results are computed for three such values of α* which characterize the possible behaviour of the flow above the plate. The final set of computations leads to a numerical value for the trailing-edge stall angle α*s, the angle of incidence which just causes the flow to separate at the trailing edge of the plate. Analytic solutions are available in the form of asymptotic expansions near the trailing edge in terms of the scaled variable of order R−⅜. A multi-layer-type of expansion which occurs in the case α* = αs* is presented in detail for comparison with the computed solution.

The method of matched asymptotic expansions is used to solve the differential equation describing the shape of a meniscus on the outside of a circular cylinder. Since the perturbation quantity is proportional to the cylinder radius, the solution is valid basically for small oylinders. The predicted meniscus height is compared with numerical data to determine the accuracy of the two-term result; the third term is found but does not improve the estimate.

Magnetohydrodynamic flows generated in a semi-infinite viscous incompressible conducting fluid by the discharge of an electric current J0 from a point on the infinite plane bounding the fluid develop singularities when J0 exceeds a certain critical value. In practical applications sometimes currents much in excess of the critical value of J0 may be passed between electrodes before singularities appear in the velocity field. In this paper we consider the flow field associated with some current distributions and attempt to provide an explanation for the discrepancy between theory and experiment.

The perturbation of a turbulent flow by an organized wavelike disturbance is examined using a dynamical, rather than phenomenological, approach. On the basis of the assumption that an infinitesimal perturbation results in a linear change in the statistics of the turbulence, and that the turbulence is either weak or that the turbulent perturbations are quasi-Gaussian, a method for predicting the perturbation turbulent Reynolds stresses is developed. The novel aspect of the analysis is that all averaging is delayed until the dynamical equations have been solved rather than attempting to find, apriori, equations for averaged quantities. When applied to long-wave perturbations the analysis indicates that the perturbation shear stress is of primary dynamical importance, and that this stress is determined by the principal component of mean shear through a relation which depends on the spectrum of the turbulent velocity component parallel to the gradient of the undisturbed mean velocity (the component perpendicular to the wall in a turbulent boundary layer). Theoretical arguments and observations are used to estimate the form of this spectrum in a constant stress shear layer. This results in a prediction of the constitutive law relating turbulent stress and the mean flow. The law is visco-elastic in nature, and is in agreement with the known constitutive relation for stress perturbations to a constant stress boundary layer; it resembles the eddy viscosity relation used successfully by others in describing perturbations in turbulent flows. The details of the constitutive law depend on how well the turbulence obeys Taylor's hypothesis that phase velocity equals mean flow velocity, and some insight into this question is given.

The properties of finite-amplitude thermal convection for a Boussinesq fluid contained in a spherical shell are investigated. All nonlinear terms are retained in the equations, and both axisymmetric and non-axisymmetric solutions are studied. The velocity is expanded in terms of poloidal and toroidal vectors. Spherical surface harmonics resolve the horizontal structure of the flow, but finite differences are used in the vertical. With a few modifications, the transform method developed by Orszag (1970) is used to calculate the nonlinear terms, while Green's function techniques are applied to the poloidal equation and diffusion terms.

Axisymmetric solutions become unstable to non-axisymmetric perturbations at values of the Rayleigh number that depend on Prandtl number and shell thickness. However, even when stable, axisymmetric solutions are not a preferred solution to the full equations; steady non-axisymmetric solutions are obtained for the same parameter values. Initial conditions determine the characteristics of the finite-amplitude solutions, including, in the cases of non-axisymmetry, whether or not a steady state is achieved. Transitions in horizontal flow structure can occur, accompanied by a transition in functional dependence of heat flux on Rayleigh number. The dominant modes in the solutions are usually the modes most unstable to the onset of convection, but not always.

In this paper we assume the existence of a nonlinear boundary layer centred on the critical point, and explore its effect on the development of unstable parallel shear flows. A velocity matching condition derived in a qualitative discussion suggests a growth of harmonics which differs from that predicted by previous theories; however, the prediction is in excellent agreement with experimental data. A hyperbolic-tangent velocity profile, subjected to perturbations with wavenumbers and frequencies close to marginal values, is then chosen as a mathematical model of the nonlinear development, both temporal and spatial instability growth being considered.

A singularity in the analysis which has been treated in previous theories by the introduction of viscosity is dealt with in the present work by the introduction of a growth boundary layer. The asymptotics are non-uniform and the time-dependent solution does not resemble the steady viscous solutions, even as the growth rate tends to zero. The theory suggests that the instability will develop as a series of temporally growing spiral vortices, a description differing from that of a cat's-eye pattern predicted by existing theories, but in accord with experimental and field observations.

The inviscid stability of swirling flows with mean velocity profiles similar to that obtained by Batchelor (1964) for a trailing vortex from an aircraft is studied with respect to infinitesimal non-axisymmetric disturbances. The flow is characterized by a swirl parameter q involving the ratio of the magnitude of the maximum swirl velocity to that of the maximum axial velocity. It is found that, as the swirl is continuously increased from zero, the disturbances die out quickly for a small value of q if n = 1 (n is the azimuthal wavenumber of the Fourier disturbance of type exp{i(αx + nϕ − αct)}); but for negative values of n, the amplification rate increases and then decreases, falling to negative values at q slightly greater than 1·5 for n = −1. The maximum amplification rate increases for increasingly negative n up to n = −6 (the highest mode investigated), and corresponds to q ≃ 0·85. The applicability of these results to attempts at destabilizing vortices is briefly discussed.

Finite-amplitude disturbances in plane Poiseuille flow are studied by a method involving Fourier expansion with numerical solution of the resulting partial differential equations in the coefficient functions. A number of solutions are developed which extend to relatively long times so that asymptotic stability or instability can be established with a degree of confidence. The amplitude for neutral stability is established for a fixed wavenumber for two values of the Reynolds number. Details of the neutral velocity fluctuation are presented. These and earlier results are expressed in terms of the asymptotic amplitude and compared with results obtained by prior workers. The results indicate that the expansion methods used by prior workers may be valid only for amplitudes considerably smaller than 0·01.

The perturbation of pre-existing surface gravity waves caused by the presence of an internal wave was studied both experimentally and analytically. An extensive series of experiments was performed, and quantitative results were obtained for the one-dimensional monochromatic interaction of internal waves and surface gravity waves. Internal wave-induced surface slope, amplitude and wavenumber modulations were measured for a wide range of interaction conditions. A complementary theoretical analysis, based on the conservation approach of Whitham (1962) and Longuet-Higgins & Stewart (1960,1961), was performed and a closed form solution obtained for the one-dimensional wave interaction. Both the theory and the experiment demonstrate that the effect increases with interaction distance. The maximum interaction effect is found to occur when the phase speed of the internal wave and the group velocity of the surface wave are matched. The phase of the internal wave at which maximum surface-wave modulation occurs is found to be a sensitive and continuous function of the relative wave speeds. The experimental data are in good agreement with the present theoretical analysis.

A laboratory study has been undertaken to measure the energy transfer from two surface waves to one internal gravity wave in a nonlinear, resonant interaction. The interacting waves form triads for which \[ \sigma_{1s} - \sigma_{2s} \pm\sigma_1 = 0\quad {\rm and}\quad \kappa_{1s} - \kappa_2s} \pm \kappa_I = 0; \] σj and κj being the frequency and wavenumber of the jth wave. Unlike previously published results involving single triplets of interacting waves, all waves here considered are standing waves. For both a diffuse, two-layer density field and a linearly increasing density with depth, the growth to steady state of a resonant internal wave is observed while two deep water surface eigen-modes are simultaneously forced by a paddle. Internal-wave amplitudes, phases and initial growth rates are compared with theoretical results derived assuming an arbitrary Boussinesq stratification, viscous dissipation and slight detuning of the internal wave. Inclusion of viscous dissipation and slight detuning permit predictions of steady-state amplitudes and phases as well as initial growth rates. Satisfactory agreement is found between predicted and measured amplitudes and phases. Results also suggest that the internal wave in a resonant triad can act as a catalyst, permitting appreciable energy transfer among surface waves.