Experiments are performed to measure the upward velocity U of the cavity resulting from the draining of a vertical liquid-filled tube of radius b, which rotates about its axis at various speeds. With no rotation the cavity moves so that the Froude number U/(gb)½ is constant, where g is the gravitational acceleration. For high angular velocities Ω, however, the cavity appears to adopt a constant value of the Rossby number U/Ωb, implying that gravitational forces are dominaked by centrifugal forces. The cavity velocity finally achieved is found to be the same as the maximum group velocity of infinitesimal waves, so that U/Ωb = 0.52. Approximate theoretical models which satisfactorily describe the development of the motion are constructed.

Here, we have studied the propagation of an arbitrary disturbance bounded in space on an arbitrary two- or three-dimensional transonic flow. First we have presented a general theory valid for an arbitrary system of n first-order quasi- linear partial differential equations and then used the theory for the special case of gasdynamic equations. If a disturbance is created in the neighbourhood of a sonic point, only a part of the disturbance stays in the transonic region and it is bounded by a wave front perpendicular to the streamlines. This part of the disturbance is governed by a very simple partial differential equation and the problem essentially reduces to the discussion of one-dimensional waves. The disturbance decays in the neighbourhood of the points where the flow acceler- ates from a subsonic state to a supersonic state and it attains a steady state where the flow is decelerating.

The paper describes a hydrodynamic stability analysis of the flow in an alluvial channel in which dunes have developed along the bed. The purpose is to develop a mathematical model describing the three-dimensional flow leading to instability of an originally straight channel. The model offers an explanation of the fact that some channels tend to meander, others to braid.

This paper investigates the Bénard problem in a binary mixture of dilute gases in which an imposed vertical temperature gradient induces a concentration gradient owing to the thermal diffusion effect. The transfer equations are derived by first-order perturbation theory which leads to instability criteria. Numerical results indicate that instability will set in only as stationary convection. This is distinctly different from the cases of liquids and concentrated gases, in which the thermal diffusion (or Soret) effect gives rise to oscillatory instability. It is disclosed in the study that the destabilization of the dilute gas-mixture layer is enhanced by an increase in the thermal diffusion ratio and/or the molecular weight ratio of the species.

This paper discusses the theory of the reflexion and scattering by an irregular coastline of a Poincaré-type wave on a rotating ocean. It is assumed that the coast is straight except for small deviations from the rectilinear form, and that these deviations may be regarded as a random function of position along the coast. The rigorous theory of energy-transfer processes in random media is applied to determine the power flux from the incident Poincard wave into the scattered Kelvin wave, which propagates in a unique direction along the coast, and into Poincard ocean wave noise. The relative efficiencies of generation of these waves is examined in some detail, and studied in particular for varying ranges of values of certain non-dimensional parameters characterizing the coastal configuration. Detailed estimates are given for a shoreline whose irregularities are specified by a Gaussian spectrum of Fourier components, and the results extra-polated in the concluding section of the paper to give a general qualitative discussion of the effects of an arbitrary coastline on an incident wave.

The nonlinear hydromagnetic dynamo problem is investigated for the case of convection in a layer of an electrically conducting fluid heated from below. It is shown that two-dimensional convection rolls in conjunction with a longitudinal mean flow are capable of amplifying a magnetic field in the form of a wave propagating in the longitudinal direction. The action of the Lorentz forces causes a reduction of the amplitude of convection with the consequence that the energy of the magnetic field cannot grow beyond an equilibrium value which is determined as a function of the parameters of the problem. The analysis is based on an expansion in powers of the longitudinal wavenumber β of the magnetic field and applies in the case of large values of the magnetic Prandtl number.

The pattern and propagation of hydromagnetic Rossby waves excited by travelling forcing effects on a rotating spherical shell of incompressible, inviscid, perfectly conducting fluid are studied using Lighthill's technique. The basic magnetic field H0, is assumed to be uniform and acting in the ‘beta-plane’ in an arbitrary direction. The two situations when the forcing effects travel along and perpendicular to H0, are considered.

The steady forcing effects travelling in the direction of an eastward or west-ward H0, excite two types of wave systems. The first consists of uiiattenuated signals directly behind or ahead of the forcing effect. The other system consists of semicircular waves travelling in all directions or waves travelling in a limited wedge.

When the forcing effect moves eastward, perpendicular to an H0, acting northward, the waves excited in the steady case are confined to certain wedges and trail behind with cusp-shaped wave crests. The magnetic field increases the semi-angle of the wedges, so that the region of disturbance is expanded. An oscillatory forcing effect generates various systems of waves. If the frequency of oscillation exceeds a certain critical frequency, excitement of waves in all directions is possible. The situation with a westward-moving forcing effect is also discussed. The effect of large rotation is to reduce the length of the waves.

Free-surface deformations accompanying natural convection in shallow liquid pools heated from below by a straight wire are studied. Two-dimensional flow ensues as density gradients are set up in the bulk liquid and surface tension gradients are set up in the interface. Surface profiles are measured experimentally using a reflecting schlieren technique. Profiles concave above the rising warm current are produced in the shallower pools and convex profiles in the deeper pools. The profile for intermediate depths is a hybrid with a shallow crest rising from a shallow trough. A mathematical model of the system is developed and yields numerical results in qualitative agreement with experiment. The import- ance of various fluid properties and system parameters on the surface profile is examined using further calculations based on the model.

An inviscid model of two-dimensional vortex shedding behind a square-based section is developed. The model uses a discrete-vortex approximation for the free shear layers. The motion of the shear layers is computed from the velocities of the discrete vortices, which in turn are derived through a Schwartz-Christoffel transformation of the section. The flow round the body is impulsively started from rest and initially develops symmetrically. The introduction of a small asymmetric disturbance results in asymmetric interaction of the shear layers amplifying into steady vortex-shedding motion.

The model is shown to predict the form of vortex shedding, the Strouhal number and some other flow quantities to a good degree of agreement with experimental results.

Flow separation can be observed (1) at the leading edge of a spilling breaker or ‘white-cap’, (2) at the lower edge of a tidal bore or hydraulic jump and (3) upstream of an obstacle abutting a steady free-surface flow. At the point of flow separation there is a discontinuity in the slope of the free surface. The flow upstream of this point is relatively smooth; the flow downstream of the discontinuity is turbulent.

In this note, a local solution for the flow in the neighbourhood of the discontinuity is derived. The turbulence is represented by a constant eddy viscosity N, and the tangential stress across the interface between the laminar and turbulent zones is expressed in terms of a drag coefficient C. It is shown that the inclinations of the free surface of the two sides of the discontinuity depend on C only, and are independent of N and g. As C increases from zero to large values, so the inclination of the free surface in the turbulent zone increases from 10° 54′ to 30°. In the laminar zone the inclination of the free surface simultaneously decreases from 10° 54′ to 0°, the densities in the two zones being assumed equal.

Owing to the possible entrainment of air at the separation point, the effective density ρ′ in the turbulent zone may be less than the density ρ in the laminar zone. When these densities are allowed to be different it is found that the possible flows are of two distinct types. Flows of the first type, called ‘quasi-static’, are contiguous to a state of rest. Flows of the second type, called ‘dynamic’, are contiguous with the frictional flows described above, for which ρ′ = ρ At a given positive value of C there exists generally only one quasi-static solution. There is also just one dynamic solution provided ρ′/ρ > 0·50012. On the other hand, if ρ′/ρ < 0·5 there may be either two or no dynamic flows, depending on the value of C; and when 0·5 < ρ′/ρ > 0·50012 there may be three such flows.

The inclination of the free surface is studied as a function of C and ρ′/ρ.

A dilute suspension of rigid axisymmetric particles in a time-dependent shear flow is considered in circumstances where the shear flow alignment of the particles dominates small but not unimportant Brownian disorientations. Three cases are studied: stress relaxation on the cessation of a steady shear, the sudden application of a steady shear from a state of rest and the sudden change from one steady shear rate to another. The rheology exhibits effects on two basic time scales provided that the particle aspect ratio r is not extreme: oscillatory features with a frequency proportional to the shear rate γ which are due to the rotation of the particles about their Jeffery orbits, and an exponentially fading memory due to Brownian diffusion with a characteristic time praportional to the inverse diffusion rate D−1. When the particle aspect ratio r becomes large, the oscillation frequency is reduced to γ/r while the diffusion rate is enhanced to Dr2 for some features and to Dr4 for others.

The movement of a heavy spherical particle in vertically oscillating liquid is studied by means of an analog computer. Using the Oseen-Tchen-Houghton force-balance model, it is shown that in the case of asymmetrical oscillations the particle may be made quasi-stationary with a finite amplitude of oscillation. It is suggested that under such conditions a mean movement of the particle in the upward direction is also possible.

For flow of a viscous fluid in a divergent channel of small angle it is shown that small disturbances to the basic Jeffery-Hamel flow may grow, according to nonlinear theory, to produce a secondary (supercritical) flow, in which the main flow winds from side to side in the channel and vortices form, with the whole pattern moving slowly downstream.

A series of hot-wire measurements has been carried out in a plane wake to investigate the structure of the turbulence boundary and the relation of its instantaneous position to the behaviour in the core of the flow. The principal measured quantities are as follows: mean velocity profile; intermittency factor; burst rate; mean of the longitudinal component of velocity conditioned upon various specified interface positions; autocorrelation of the intermittency signal; probability densities a t the half-intermittency point for the time between bursts and the duration of a burst; probability density for the longitudinal velocity component and its time derivative at various points across the wake; probability density a t the half-intermittency point for the same quantities in the turbulent and irrotational zones separately. In addition, the profile of the second moment of the probability density for the time between bursts has been obtained indirectly and part of the theory of Phillips (1955) has been shown to be applicable in the intermittent region.

The present measurements appear to indicate that the turbulence boundary in the wake resembles that in other plane flows more closely than has been supposed hitherto. The theory of normally distributed random noise was found to explain many of the observed statistical properties of the turbulence boundary.

This paper complements earlier papers by Van Dyke (1970) and by Wilson (1971) which have appeared under the same title. Second-order boundary-layer theory is used to examine the region near the entrance to a single channel placed in a uniform stream. It is found that there are additional effects to those present in the three models treated by Van Dyke and Wilson. In particular, the cascade model misses the leading term in the separation force while the irrotational-entry model misses that in the skin friction.

There are also two new effects far downstream: logarithmic terms appear (apparently for the first time in second-order theory); and a resonance with the first eigensolution occurs.

Finite amplitude, impulsively started spin-up and spin-down is analysed for axially symmetric flow of a viscous, incompressible, electrically conducting fluid confined between infinite, flat, parallel, insulating boundaries. A uniform axial magnetic field is present in the initial state, but is subsequently distorted by fluid motions. The method of matched asymptotic expansions reduces the problem to a first-order, ordinary, nonlinear, integro-differential equation for the transient development of the interior angular velocity on the time scale of spin- up, as driven by quasi-steady nonlinear Ekman-Hartmann boundary layers. This two-parameter equation is solved analytically in certain limits and numeric-ally in general. The solutions show that nonlinear non-magnetic spin-up and spin-down take longer than for linearized flow, spin-down occurring more rapidly in the early stages but requiring more time for completion than spin-up. A magnetic field promotes both spin-up and spin-down, but a weak field is relatively ineffective for spin-down yet very effective for spin-up. A strong magnetic field dominates nonlinear processes and gives identical spin-up and spin-down times, which coincide with that found from linear hydromagnetic theory.

A theoretical analysis is made of the effect described in the title. It is shown that both (i) the eigenfrequencies of the inertial waves and (ii) the moment produced by the inertial wave pressure distribution are independent of the location of the off-axis cylinders. Hence, one can use Stewartson's tables computed for a centrally located cylinder to calculate the frequencies and residues for an eccentrically located cylinder.

Howard's technique for examining the stability characteristics of a two-dimensional, inviscid, heterogeneous shear flow in the vicinity of a neutral curve is considered. Examples are presented for which erroneous conclusions would be obtained by a direct interpretation of the results of this technique. Examples for which instability would be deduced for a stable region; an example for which stability would be deduced for an unstable region; and an example for which the technique breaks down altogether are discussed.

The last example is plane Couette flow between two rigid walls, which is shown to be destabilized by the addition of stable stratification. This example thus proves that the existence of a relative extremum in the basic vorticityprofile is not a necessary condition for the instability of an inviscid, heterogeneous shear flow.

A numerical method for solving the time-dependent Navier–Stokes equations in two space dimensions at high Reynolds number is presented. The crux of the method lies in the numerical simulation of the process of vorticity generation and dispersal, using computer-generated pseudo-random numbers. An application to flow past a circular cylinder is presented.

An experimental study of fully developed uniform-density turbulent flow in a circular pipe containing one or two rods located off-centre is described. The friction factor in both cases was found to be approximately 5 % higher than the simple pipe friction factor. The shear stress distribution on the rod surface was determined using calibrated boundary-layer fences. The normalized shear stress distributions were independent of Reynolds number in the range 3·7 × 104 to 2·15 × 105. Mean-velocity measurements were obtained to check the validity of the universal law of the wall near the rod surface. Secondary-flow velocities were measured by a hot-wire anemometer and integrated to yield the secondary-flow stream function. Secondary-flow velocities of the order of 1 % of the mean velocity were observed. In the gap between the two pins, however, the secondary-flow velocities were only ½% of the mean velocity. It is demonstrated that the secondary flow cannot be neglected if a force balance is used to determine the shear stress distribution on the rod surface.