Numerical solutions have been obtained for the problem of steady, incompressible, viscous flow between two infinite concentric cylinders, the cross-sections of the inner and outer cylinders being circular and square respectively. The square cylinder is fixed and the flow is driven by the rotation of the circular cylinder. Solutions are given for Reynolds number in the range 1–1400 and for several values of the parameter B, defined as the ratio of the side of the square to the diameter of the circle.

A parabolic approximation to the reduced wave equation is investigated for the propagation of periodic surface waves in shoaling water. The approximation is derived from splitting the wave field into transmitted and reflected components.

In the case of an area with straight and parallel bottom contour lines, the asymptotic form of the solution for high frequencies is compared with the geometrical optics approximation.

Two numerical solution techniques are applied to the propagation of an incident plane wave over a circular shoal.

The fluctuating velocity components, u and w, and their gradients at the wall, \[ (\partial u/\partial y)|_{w}\quad {\rm and}\quad (\partial w/\partial y)|_{w}, \] were measured in a fully developed turbulent channel flow using hot-film probes. The data were taken in an oil channel at a low Reynolds number which allowed velocity measurements very close to the wall. From simultaneous measurements of these signals it could be deduced that coherent flow structures, inclined to the wall, travel downstream. Space-time correlations imply that these are rotating structures. The spanwise distance between these structures was found to be z+ ≈ 50 and they could be observed over a streamwise distance of Δx+ > 1000.

A submerged sphere is considered to be absorbing power from an incident wave through an integrated mooring and power take-off system. It is shown that the power absorbed (as characterized by the absorption length) depends on the hydrodynamic properties of the sphere; in particular on the added-mass and damping coefficients. These coefficients are determined and the results used to study the power absorption properties of the sphere. Curves are given showing the variation of the absorption length with wavenumber, for differing depths of submergence.

The flow of closely fitting neutrally buoyant elastic spheres through a circular cylindrical tube is considered under the assumptions that the Reynolds’ equation is valid in the fluid and equations of linear elasticity hold in the solid. Computations are carried out for several values of Poisson's ratio. The results are compared with the results of previous models on elastic compressible particles.

An experimental study of the flow past a thin finite length plate in a supersonic low density stream is reported. The paper discusses the corrections that are necessary for surface pressures measured under rarefied conditions. It is shown that the recent method of ‘orifice’ corrections due to Harbour & Bienkowski is versatile and reliable to use for both cold wall and insulated wall measurements. For the conditions of the experiment, the flow over the plate was found to be dominated by both leading-edge and trailing-edge interactions.

A numerical solution is obtained to describe the behaviour of a uniform bore over a sloping beach and the subsequent run-up and back-wash. The results exhibit features which have only previously been described in a qualitative manner. These include the formation’ of a landward-facing bore in the back-wash. A comprehensive set of results are presented for a typical initial subcritical bore height ratio.

The turbulent velocity fluctuations at eight positions on sections of a plane wake and a boundary layer have been sampled simultaneously and recorded in digital form on magnetic tape for subsequent numerical analysis. Two configurations have been used (lines of equally-spaced sensors in planes normal to the flow, and arrays with three rows) with sensors responsive both to streamwise and cross-stream components of the fluctuations. To the extent that the Taylor approximation of ‘frozen’ flow is valid, the recorded fluctuations may be interpreted as instantaneous values at grid points in the volume swept out by the array.

The records have been examined, (a) to find evidence for flow patterns with marked periodicity in one direction, and (b) to select dimensions and orientations for simple eddy flow patterns whose random superposition would lead to correlation functions with a close resemblance to those calculated from the recorded data. In the wake, clear evidence was found for periodic flow patterns that resemble the eddies of a von Kármán street, but, although the spacing of eddy centres in each group was uniform, it varied considerably from one group to another, suggesting that groups are being observed in different stages of development.

Two kinds of correlation were calculated from the records, (i) simple mean values of velocity products, and (ii) mean values of the products weighted by the total intensity or Reynolds stress in the effective volume swept by the array of sensors. For both kinds, the correlations are well described by simple inclined roller-type eddies, but the correspondence is greatly improved by weighting in favour of intensity or Reynolds stress. It appears that the eddies contributing most to intensity or Reynolds stress are less variable in form than all the eddies together, and that those contributing most to Reynolds stress are significantly different in shape and in orientation from those contributing most to turbulent energy.

The motion of two immiscible fluids with arbitrary viscosities flowing through a capillary with an almost flat fluid-fluid interface is investigated in the limit of small capillary and Reynolds numbers. A complete description of the dynamics of the fluids is presented. It is shown that the motion of the fluid away from the moving contact line can be completely determined in terms of one material parameter, and how the capillary can be viewed as a device for measuring it. The dynamic behaviour of various contact angles, measured by others, is calculated. It is shown that they all depend on the radius of the capillary; hence, they do not represent properties of only the materials of the system.

The response of a heated circular cylinder to impulsive and sinusoidal variations in the velocity of flow past it has been simulated by numerical integration of the governing equations. The fluid has been treated as viscous and incompressible and as having constant properties. The range of Reynolds number investigated was 1 [Lt ] R [Lt ] 40. Since vortex shedding normally does not occur in this range, the flows were treated as symmetrical. The thermal and flow transients are presented for the following cases.: (i)impulsive starts from rest to final steady state Reynolds numbers 1, 5, 10, 26·67; (ii)impulsive increases in velocities of 50% magnitude from steady state Reynolds numbers 1, 10 and 26·67; (iii)sinusoidal variation in velocity with amplitude of 10% impressed on a mean flow at Reynolds number 10.

Results are also given for the thermal transients associated with instantaneous changes in cylinder temperature at Reynolds numbers 1, 5 and 40. The results obtained for transient and steady state flow parameters are in agreement with those obtained numerically and experimentally by other workers and the results for steady state heat flux from the cylinder are in agreement with experimental values. The new results obtained for heat transfer in unsteady flows provides information which is relevant to the operation of hot-wire anemometers.

The velocity induced by a transversely waving infinite flexible sheet in a viscoelastic liquid is investigated by a method of successive approximation up to second order in the amplitude of oscillation of the sheet. The incompressible second-order fluid model has been used and it is found that the elastic property of the fluid augments self-propulsion (increases the induced velocity) in some range of Reynolds number (based on the phase velocity) and hampers it (reduces the induced velocity) in some other range with higher values of Reynolds number.

The horseshoe vortex formed around the base of a cylinder by a separating laminar boundary layer has been investigated experimentally. Smoke flow visualization shows that both steady and unsteady vortex systems exist. Pressure distributions beneath both types of vortex system have been measured and the variation of the horseshoe vortex position on the plane of symmetry upstream of the cylinder has been determined. Unsteady horseshoe vortex systems are shown to have a complex oscillatory behaviour and the nature of this oscillatory behaviour is described. Using smoke flow visualization techniques some measurements have been made of the velocity distributions within horseshoe vortex systems.

Monochromatic waves obliquely incident on a plane beach, and strongly reflected there, are unstable to perturbations by edge waves. Theory suggests the possible width of the resonant edge wave frequency band. Experiments on beaches with absorbers at both ends show that the excited waves have frequencies at the centre of the band, as predicted by Guza & Bowen (1975). Advection by mean longshore currents must be taken into account. If reflectors are placed at the beach ends, the additional boundary conditions apparently lead to resonances scattered across the resonant band.

The well-known class of self-similar solutions for an ideal polytropic gas sphere of radius R(t) expanding into a vacuum with velocity $u(r, t) = r\dot{R}/R$ is shown to be convectively unstable. The physical mechanism results from the buoyancy force experienced by anisentropic distributions in the inertial (effective gravitational) field. An equation for the perturbed displacement ξ(r, t), derived from the linearized fluid equations in Lagrangian co-ordinates, is solved by separation of variables. Because the basic state is non-steady, the perturbations do not grow exponentially, but can be expressed in terms of hypergeometric functions. For initial density profiles \[ \rho_0(r)\sim(1-r^2/r^2_0)^{\kappa}, \] modes with angular dependence Ylm(θ, ϕ) are unstable provided l > 0 and κ < 1/(γ − 1), where γ is the ratio of specific heats. For large l, the characteristic growth time of the perturbations varies as l−½ and the amplification increases exponentially as a function of l. The radial eigenfunctions are proportional to rl, and the compressibility and vorticity are both non-zero.

This paper is concerned with the rate of transfer of heat or mass from a force-free couple-free particle immersed in fluid whose velocity far from the particle is steady and varies linearly with position. Asymptotic results for both small and large Péclet numbers are considered. There is at least a four-parameter family of different linear ambient velocity distributions, but nevertheless a comprehensive set of results for the transfer rate may be compiled by combining previously published work with some new developments. Some of these are exact results for particular linear ambient flow fields and some are approximate results for classes of linear flow fields.

For small Péclet number (P), the non-dimensional additional transfer rate due to convection is equal to αN20P½, where N0 is the Nusselt number for P = 0 and the proportionality constant α is a parameter of the concentration distribution due to a steady point source in the given linear ambient flow field. A general method of determining α is developed, and numerical values are found for some particular linear ambient flow fields. It is estimated that the value of α for any linear ambient flow field in which the vorticity does not dominate the straining motion lies within 10% of 0·34 when P is defined in terms of a particular invariant of the ambient rate-of-strain tensor E.

At large Péclet number the transfer rate N depends on the velocity distribution near the particle, and attention is restricted to the case of a sphere in low-Reynolds-number flow. For a rigid sphere $N = \beta P^{\frac{1}{3}}$ for any ambient pure straining motion, and the Levich concentration-boundary-layer method may be used to show that β = 0·90 for both axisymmetric and two-dimensional ambient pure straining, and probably for any other pure straining motion, when P is suitably defined. When the ambient vorticity ω is non-zero, the sphere rotates, and the Levich method cannot be used. However, it is shown that the part of the velocity distribution that varies sinusoidally with the azimuthal angle around the rotation axis does not affect the transfer rate and that N is asymptotically the same as for an ambient axisymmetric pure straining motion with rate of extension in the direction of the axis of symmetry equal to Eω(= ω. E. ω/ω2). In the exceptional case Eω = 0, N approaches a constant as P → ∞.

It is possible to interpolate between the asymptotic relations for small and large Péclet number with comparatively little uncertainty for any ambient pure straining motion and for any linear ambient flow field in which ω and Eω are non-zero.

A plane horizontal layer of a fluid with depth d is considered into which heat is introduced. Within the Boussinesq approximation an exact expression is obtained for the efficiency of convection γ in transforming the rate of heat supplied into the generation of kinetic energy. It agrees with results of numerical and laboratory experiments whose data can be used to calculate this value. In laboratory experiments γ is usually of order 10−8 to 10−6. Using this quantity estimates are obtained for the r.m.s. velocity of convective motions $\overline{u}$ and their time scale $\tau = d/\overline{u}$ for a regime where viscosity is important. These estimates agree with the results of a number of previous numerical experiments over a wide range of Rayleigh, R, and Prandtl numbers. Dimensionless convection equations normalized by these scales show the existence of thermal boundary layers and of almost isothermal regions within the bulk of the fluid. From this, the main regimes of heat transfer follow immediately: the Nusselt number N ∼ (R − Rcr)¼ for moderate R and $N \sim R^{\frac{1}{3}}$ for sufficiently large R.

A number of simple experiments have been carried out to measure $\overline{u}$ and τ for convection in water. Their results confirm the theoretical dependences of $\overline{u}$ and τ on external parameters and show that a smooth transition region exists from the viscous regime of convection to the more fully-developed turbulent one. The latter regime is considered by a scaling analysis. The results are compared with the author's measurements and other experimental data.

Similarly density convection is considered which arises by the separation of a medium into light and heavy fractions. Differences and analogies with the thermal convection are established. Elementary experiments confirm qualitatively the predicted dependences for $\overline{u}$ and τ. Applications of the results obtained are briefly discussed for studies of heat and mass transfer in the ocean and of convection in the Earth's mantle.

In the last section some general properties are considered for various forced flows, convection, turbulence and some types of atmospheric circulation, that allow one to formulate a ‘rule’ of the fastest response, which asserts that the kinetic energy of a fluid system is of order of the supplied power times the fastest relaxation time which the system possesses.