The entrainment experiments of Kato & Phillips (1969) and Kantha, Phillips & Azad (1977) (hereafter KP and KPA) are analysed to demonstrate a more general and effective scaling of the entrainment observations. The preferred scaling is \[ V^{-1} dh/dt = E(R_v), \] where h is the mixed-layer depth, V is the mean velocity of the mixed layer, Rv = B/V2 and B is the total mixed-layer buoyancy. This scaling effectively collapses entrainment data taken at various h/L, where L is the tank width, and in cases in which the interior is density stratified (KP) or homogeneous (KPA). The entrainment law E(Rv) is computed from the KP and KPA observations using the conservation equations for mean momentum and buoyancy. A side-wall drag term is included in the momentum conservation equation. In the range 0·5 < Rv < 1·0, which includes nearly all of the KP, KPA data, E ≃ 5 × 10−4R−4v. This is very similar to the entrainment law followed by a surface half-jet (Ellison & Turner 1959) and by the wind-driven ocean surface mixed layer (Price, Mooers & Van Leer 1978).

The analysis shows that, when forcing is steady, Rv is quasi-steady and, provided that side-wall drag is not large, Rv ≃ 0·6 over a wide range of RT = B/U2*, where U* is the friction velocity of the imposed stress. In the absence of side-wall drag (vanishing h/L) the conservation of momentum then leads to U−1*dh/dt = n(0·6)½R−½T, where n = ½ or 1 if the interior is linearly stratified or homogeneous. The KP, KPA data show this dependence throughout the range 17 < RT < 160 where the effect of side-wall drag is negligible or can be removed by a linear extrapolation. This result, together with the form and magnitude of the observed side-wall effect, suggests that mean momentum conservation is a key constraint upon the entrainment rate in the KP, KPA experiments.

The formation of singularities in two-dimensional magnetohydrodynamic flow is investigated by direct numerical simulation. It is shown that two-dimensional magnetohydrodynamic turbulence is not as singular as three-dimensional hydrodynamic turbulence (in the sense that it has a less highly excited small-scale structure) but that it is more singular than two-dimensional hydrodynamic turbulence.

A rational theory is developed to describe the reattachment of a laminar shear layer in supersonic flow. In the neighbourhood of reattachment the flow develops a threetiered or ‘triple-deck’ structure analogous to that which occurs at a point of separation (Stewartson & Williams 1969) and, as in the separation problem, the local flow pattern may be found independently of the flow in the surrounding regions. The fundamental problem of the reattachment triple deck reduces to the solution of the incompressible boundary-layer equations in the lower deck, which is of streamwise and lateral dimensions O(R−⅜) and O(R−⅝), where R [Gt ] 1 is a representative Reynolds number for the flow. Pressure variations in this region are O(R−¼). Asymptotic solutions in terms of x, the scaled streamwise lower-deck variable, are derived to confirm the transition from a reverse flow profile at x = 0+, through reattachment, to a forward flow as x → ∞, the attainment of the required asymptotic form downstream (as x → ∞) being shown to depend crucially upon the correct choice of the finite part of the pressure in the lower deck at x = 0+. The lower-deck solution is singular at x = 0+ and assumes a complicated multi-structured form which is shown to match upstream with the solution in a largely inviscid region of dimension O(R−½) where the pressure is O(1) and the major part of the flow reversal takes place. Solutions are presented for reattachment at a wall and for symmetric reattachment behind a wedge or bluff body. In the former case the results also explain the apparent ignorance of upstream conditions in the expansive triple-deck solution formulated by Stewartson (1970) in the context of supersonic flow around a convex corner.

The unsteady large amplitude linearized motion of a flexible slender wing in inviscid incompressible fluid is analysed. This extension to large amplitudes is made by assuming small local angles of incidence during motion along a finite amplitude trajectory. As a specific example, the periodic propulsion produced by a flexible slender wing is analysed for large amplitude harmonic motion with small local angles of attack and results for the available thrust and efficiency are presented.

Axisymmetric high Reynolds number flows in tubes of slowly varying radius are shown to be governed to a first approximation and in suitable co-ordinates by a partial differential equation which, in a particular case, allows solutions independent of the streamwise co-ordinate. The solutions of the resulting ordinary differential equation give flows with inflexion points in the velocity profiles and reversed flow in some cases.

The axisymmetric flow of an incompressible fluid through a pipe (of radius a) suffering a severe constriction is studied for large Reynolds numbers R, the features of symmetric channel flows being virtually the same. Here ‘severe’ refers to a constriction whose typical dimensions are finite, and the oncoming velocity profile is taken to be of a realistic type, i.e. with no slip at the wall. The study adopts (Kirchhoff) free-streamline theory, which, for the mostly inviscid description, affords a rational basis consistent with viscous separation. The major (triple-deck) separation takes place on the constriction surface and is followed by a downstream eddy of length O(aR). Another, less familiar, separation is predicted to occur at a distance 0.087a In R + O(a) ahead of the finite obstacle. Free-streamline solutions are found in the two main extremes of moderately severe and very severe constriction. In both extremes, and in any slowly varying constriction, the major separation is sited near the maximum constriction point. The upstream separation point is also derived, to O(a) accuracy in each case. The upstream separation can be suppressed, however, if the constriction has no definite starting point and decaysslowly upstream, but then the upstream flow response extends over a much increased distance. Comparisons with Navier-Stokes solutions and with experiments tend to favour the predictions of the free-streamline theory.

This note presents measurements of radial and axial distributions of mean velocity, turbulent intensities and kinetic energy as well as radial distributions of the turbulent shear stress in the initial region of a turbulent air jet issuing from a long round pipe into still air. The pipe flow is transformed relatively smoothly into a jet flow. In the core subregion the mean centre-line velocity decreases slightly. The highest turbulence occurs at an axial distance of about 6d and radius of (0·7 to 0·8)d. On the axis the highest turbulent kinetic energy appears at a distance of (7·5 to 8·5)d. Normalized distributions of the turbulent quantities are in good agreement with known data on the developed region of jets issuing from short nozzles.

This analysis demonstrates theoretically that a lateral bending wave propagating along the walls of a two-dimensional channel filled with a viscous incompressible fluid will induce a mean flow. In addition to this ‘pure’ bending wave, another possible condition is investigated: that of the superposition of an area contraction wave propagating with the same speed as the lateral bending wave. This second condition, called complex wave motion, takes into account the slight occlusion which occurs naturally at the amplitude peaks when a finite amplitude bending wave propagates along the walls of a container. A perturbation solution is found which satisfies Navier—Stokes equations for the case in which wave amplitude/wavelength is ‘small’. However the wave amplitude is finite, in the sense that it is of the same order as the channel width. Under these conditions, the occlusion at the amplitude peaks is allowed to be of the same order as the channel width. For the case of a pure bending wave the motion induced by the peristaltis is found to be of second order in the perturbation parameter, whereas in the more realistic case a first-order pumping effect is obtained.

A Helmholtz velocity profile with velocity discontinuity 2U is embedded in an infinite continuously stratified Boussinesq fluid with constant Brunt—Väisälä frequency N. Linear theory shows that this system can support resonant over-reflexion, i.e. the existence of neutral modes consisting of outgoing internal gravity waves, whenever the horizontal wavenumber is less than N/2½U. This paper examines the weakly nonlinear theory of these modes. An equation governing the evolution of the amplitude of the interface displacement is derived. The time scale for this evolution is α−2, where α is a measure of the magnitude of the interface displacement, which is excited by an incident wave of magnitude O(α3). It is shown that the mode which is symmetrical with respect to the interface (and has a horizontal phase speed equal to the mean of the basic velocity discontinuity) remains neutral, with a finite amplitude wave on the interface. However, the other modes, which are not symmetrical with respect to the interface, become unstable owing to the self-interaction of the primary mode with its second harmonic. The interface displacement develops a singularity in a finite time.

Experiments were conducted in the free-piston shock tube and shock tunnel with dissociating nitrogen and carbon dioxide, ionizing argon and frozen argon to measure the transition condition in pseudosteady and steady flow. The transition condition in the steady flow, in which the wall was eliminated by symmetry, agrees with the calculated von Neumann condition. In the real gases this calculation assumed thermo-dynamic equilibrium after the reflected shock. In the pseudosteady flow of reflexion from a wedge the measured transition angle lies on the Mach-reflexion side of the calculated detachment condition by an amount which may be explained in terms of the displacement effect of the boundary layer on the wedge surface. A single criterion based on the availability of a length scale at the reflexion point explains the difference between the pseudosteady and steady flow transition condition and predicts a hysteresis effect in the transition angle when the shock angle is varied during steady flow. No significant effects on the transition condition due to finite relaxation length could be detected. However, new experiments in which interesting relaxation effects should be evident are suggested.

Seeley, Hummel & Smith (1975) reported the results of experiments to study the dynamics of flow around spheres at intermediate Reynolds numbers using a nondisturbing flow-visualization technique. The flow patterns were recorded on ciné photographs and the information stored was processed in order to obtain the velocity field. The position of fluid elements shown by the photochromic indicator traces were estimated by eye on a projection screen. In this paper, a new set of results based on the same films has been reduced and computed using the ‘POLLY’ film-reading system described by Esmail, Smith & Hummel (1976). Some numerical boundary-layer solutions are included to show the reliability of the data, and comparisons with the results previously reported by Seeley et al. (1975) are presented.J. W. Smith will be pleased to send a complete set of experimental data on request.

A comparison is made between the theoretically predicted and the observed stratification in a container which is traversed by a prescribed flux of fluid. Two different geometries were used illustrating respectively a useful procedure for the control of a stratified laboratory system and a mechanism which is believed to be geophysically significant, e.g. for the control of the stratification in certain estuaries. The behaviour of the fluid system was in all cases characterized by an almost stagnant interior with a boundary layer at the non-horizontal wall of buoyancy-layer type. Agreement between theory and experiment was satisfactory within experimental errors, say 10% of the overall temperature difference.

Short-crested wave systems, as produced by two progressive waves propagating at an oblique angle to each other, have an extremely important effect on a sedimentary bed. The complex water-particle motions are conducive to lifting material into suspension and sustaining it in motion. In order to study this phenomenon rigorously, the variables of this wave system are derived to a third-order approximation by a perturbation method. The case of waves reflecting obliquely from a vertical wall is examined under the assumptions of full reflexion, uniform finite depth and an inviscid incompressible fluid. The new formulation reduces to standing or Stokes waves at the limiting angles of approach. Expressions for kinematic quantities are also presented.

This paper develops an alternative approach to evaluating the arbitrary constants found in Gill's solution for the boundary-layer free-convection regime in a vertical rectangular enclosure. The new method consists of calculating the net upward flow of energy through the enclosure and setting it equal to zero near the top and bottom boundaries of the cavity. The present method takes into account the impermeable and adiabatic properties of the horizontal end walls. The overall Nusselt number derived on this new basis is shown to agree well with available experimental and numerical heat-transfer correlations.

A series of experiments is described in which a fully submerged parallel-plate enclosure is subjected to regular incident waves. The resulting wave-interaction effects are defined in terms of a reflexion coefficient derived from wave-height measurements and a pressure coefficient derived from measurements of pressure in the depths of the enclosure. The latter parameter describes the pressure induced in the enclosure by the passage of regular waves above it and is of particular interest, at present, in having a bearing on the operation and performance of a sub-sea wave-energy converter currently under development. The outcome of these experiments is the partial verification of some recent theoretical work, including the demonstration of a pressure intensification at certain wave frequencies.

These experiments form part of a wider study concerned generally with the performance of immersed, pressure-driven, wave-energy converters.

A model equation is derived for the combined longitudinal and lateral dispersion of a buoyant contaminant in open-channel flow. The central hypotheses are that the water is shallow and that to a first approximation the effluent is vertically well mixed. The model includes allowance for the reduction in turbulent intensity due to weak vertical stratification, a buoyancy-driven secondary flow, and the redistribution of longitudinal momentum by the secondary flow. For a plume the theoretical results for the excess spreading due to buoyancy agree well with Prych's (1970) experimental results.

In a recent paper (Evans 1976) a theory was presented for the behaviour of an oscillating two-dimensional cylinder of any shape which was capable of absorbing energy from a given regular sinusoidal wave. In particular an expression was derived for the efficiency of power absorption of the cylinder when oscillating in a single mode in terms of properties of the solution of the so-called radiation problem in which the cylinder is forced to oscillate in the appropriate mode in the absence of the incident wave train.

In the present paper this theory is extended to two independent cylinders of arbitrary shape each oscillating in a single mode and capable of absorbing energy in that mode. A general expression for the efficiency is derived which depends on properties of the solution to a new radiation problem, in which one cylinder is forced to oscillate in the presence of the other cylinder, which is held fixed in its equilibrium position. In this case, the efficiency also depends on cross-coupling coefficients related to the force on the fixed cylinder due to the motion of the oscillating cylinder.

It is shown that the cylinders can be tuned to absorb all the incident wave energy at a given frequency even for symmetric cylinders, in contrast to the single symmetric cylinder, for which the maximum efficiency has been shown to be 50%.

The general solution to the new radiation problem is derived in terms of the solution to the radiation problem for a single cylinder, by assuming that the cylinders are far enough apart for local wave effects to be negligible.

The special case of two widely spaced rolling vertical plates is considered in detail and curves showing the variation of efficiency with wavelength are given for a variety of plate spacings and points of rotation.

It is well known that vortex sheets are diffused by viscosity. For a plane sheet this diffusion is described by a linear diffusion equation. Here we consider a rolled vortex sheet, and show that the viscous diffusion of the vorticity concentrated at the turns of the sheet, when they are very closely spaced, can also be described by a linear equation, in appropriately transformed variables.

The mixing of rain with otherwise calm near-surface water is investigated experimentally, using both uniform drops and crude drop-size distributions. The results are interpreted in terms of mixed-layer depths and the upward entrainment of quiescent water. Large drops, even in quite small numbers, are shown to play a very important role in the mixing process. Entrainment coefficients vary as the inverse of a suitably defined bulk Richardson number, and behave similarly in both fresh and salt receiving water.

A container of viscous incompressible liquid is bounded by rigid parallel planes and rotates steadily about an axis normal to these planes. A rigid sphere moves steadily parallel to the rotation axis and the Rossby and Ekman numbers characterizing the motion are both small. The drag on the sphere is calculated in the case when the length of the Taylor column is comparable to the axial dimension of the container. Viscous effects are allowed for in the boundary of the Taylor column, but the Ekman layers on the sphere and on the bounding planes are shown not to affect the drag to leading order. The determination of the drag involves solving dual integral equations. This is done numerically and, for the limiting cases of long and short containers, analytically. The interaction of the Taylor column and the ends of the container leads to an increase in the drag over its value in an unbounded fluid, but the increase is smaller than that measured by Maxworthy (1970).