By assuming that an uncharged drop situated in a uniform electric field E retains a spheroidal shape while oscillating about its equilibrium configuration, two approximate equations of motion are derived for the deformation ratio γ expressed as the ratio a/b of the major and minor axis of the drop. Solutions of these equations of motion indicate that the stability of a drop of undistorted radius R and surface tension T depends upon E(R/T)½ and the initial displacement of γ from its equilibrium value. The predictions of the two equations are compared to assess the accuracy of the spheroidal assumption as applied to such a dynamical situation. The analysis is used to determine the stability criterion of a drop subject to a step function field. Finally, the limit of validity of the spheroidal assumption is discussed in terms of Rayleigh's criterion for the stability of charged spherical drops. By applying Rayleigh's criterion to the poles of a spheroidal drop, the stage at which the drop departs from spheroidal form to form conical jets was approximately determined.

The problem of a viscous vortex core embedded in an unsteady outer potential swirling flow is considered. By introducing a suitable similarity variable, the full Navier–Stokes equations for the unsteady axisymmetric flow of an incompressible fluid are reduced to two ordinary differential equations. These are solved numerically. When the radial flux of a particular outer potential flow satisfies certain conditions a family of three-cell core structures is possible. This family is not represented by any known analytical solution.

This work is useful for studying meteorological flow systems such as tornadoes. In particular, it suggests how two- and three-cell structures can develop from a one-cell structure and also shows the sensitivity of the core flow to small changes in the outer potential flow.

A numerical solution of the Navier-Stokes equations of motion by means of finite-difference forms of the vorticity and continuity equations is presented. This is applied to the study of the flow of an incompressible fluid produced by the motion from rest of a piston in a cylindrical tube of circular cross-section.

Experiments at high Reynolds number indicated the presence in the starting flow of a ring vortex which was not reproduced by computation. Iteration to determine the stream function was not found to be necessary to achieve 1% accuracy. Omitting iteration is equivalent to only slightly disturbing the flow. An additional random disturbance applied to the flow at each time step was found to result in the production of the ring vortex.

Experiments are described in which velocities were measured ahead of a semi-infinite Rankine body moving parallel to a uniform magnetic field in a conducting fluid. The flow disturbance in front of the body is found to increase in length as N½, where N is the interaction parameter. In most of the experiments this parameter was varied from 4 to about 50. Measurements made along the axis of symmetry in the flow show that there is a relatively short region of stagnant fluid directly ahead of the body. The major part of the disturbance is found to consist of a much longer region in which the flow undergoes transition from conditions in the free stream to conditions near the body. Velocity profiles across the flow in this region show that for increased N, at a fixed distance ahead of the body, the velocity defect increases and the disturbance becomes more confined radially. Although the radial gradients in the flow increase with N, they are found to be much smaller than would be expected in a flow containing thin current layers. A physical model of the flow which has currents and pressures consistent with these results is discussed.

Flow visualization has been used quantitatively to determine the flow relative to a piston and a free surface started from rest. The discharge of water from a cylindrical reservoir was investigated. Flow with a free surface started from rest was found to have a critical Reynolds number (based on tube diameter and surface speed) of about 450 above which a ring vortex was produced just below the surface.

Measurements at Reynolds numbers of 525 and 1200 were compared with computations made by the methods described in Part 1. The computed drift of tracer particles agreed well with observed values. The largest discrepancies occurred in the radial component of the drift in the early stages of the motion and amounted to 2½% of the tube diameter.

An experimental study of boundary-layer turbulence in a free surface channel flow is described. Attention is concentrated on the effects of different surface roughness conditions on the turbulence structure in the boundary region. Hydrogen bubble flow tracers and medium high-speed motion photography were used to obtain an instantaneous visual and quantitative description of the flow field. In particular it proved possible to record instantaneous longitudinal and vertical velocity profiles from which distributions of the instantaneous Reynolds stress contribution were computed.

Two well-defined intermittent features of the flow structure were visually identified close to the boundary. These consisted of fluid ejection phases, previously reported by Kline et al. (1967) for smooth boundary flow, and fluid inrush phases. Conditional averaging of the instantaneous velocity data yielded quantitative confirmation that ejection phases corresponded with ejection of low momentum fluid outwards from the boundary whilst inrush phases were associated with the transport of high momentum fluid inwards towards the boundary. Inrush and ejection events were present irrespective of the surface roughness condition.

Conditional averaging also indicated that both inrush and ejection sequences correlate with an extremely high contribution to Reynolds stress and hence turbulence production close to the boundary. Indeed the present results, taken with those from previous studies, suggest that turbulence production is dominated by the joint contribution from the inrush and ejection events. It is emphasized that these structural features are intermittent, forming important linked elements of a randomly repeating cycle of wall-region turbulence production which is apparently driven by some violent three-dimensional instability mechanism.

Whilst the most coherent effects of the observed inrush phases appear to be mainly confined to a region close to the boundary, the influence of the ejection phases is far more extensive. The ejected low momentum fluid elements, drawn from the viscous sublayer and from between the interstices of the roughness elements, travel outwards from the boundary into the body of the flow and give rise to very large positive contributions to Reynolds stress at points remote from the boundary. This effect is sufficiently strong to prompt the suggestion that the ejection process could represent a universal and dominant mode of momentum transport outside the immediate wall region and possibly extending across the entire thickness of the boundary layer.

A structural model based on the present observations is seen to exhibit consistency with many commonly visualized features and recorded average properties of turbulent boundary-layer flows in general.

An approximate expression is given for the thickness of weak fully dispersed shock waves. Using available data on the thermodynamic properties of air, it is shown that shocks of the strength expected in sonic bangs are fully dispersed. Estimated relaxation times for dry and humid air lead to wide variations in possible thickness, varying from millimetres to metres.

The factors bringing about the irreversible distortion or degeneration of a continuously forced standing internal gravity wave in a linearly stratified fluid are studied experimentally and theoretically. For a rectangular container there is strong evidence that the process is initiated by the unstable growth, from a subliminal level, of free wave modes forming triads in second-order resonant interaction with the original wave. These free modes grow by de-energizing the original wave and may collectively induce kinematical conditions sufficiently severe to create localized regions of density discontinuity within the fluid, leading to turbulence.

Although the possible free wave modes are doubly infinite in number, the geometrical constraints greatly reduce the number of possibilities for resonant triads. In many cases this permits critical wave amplitude to be predicted by consideration of one triad only, and the results are in excellent agreement with experiment.

It is speculated that a closely similar process explains observations by Malkus (1968), Aldridge & Toomre (1969), and McEwan (1970) in the analogous context of inertial oscillation of contained rotating fluids.

Second- and third-order structure functions were computed from velocity, temperature and humidity fluctuations in the wind over the ocean. Universal inertial-convective subrange constants (Kolmogoroff constants) were computed from these structure functions. The constant for velocity is consistent with other recent observations. The temperature and humidity constants are found to be equal within the measurement error and have values of about 0·8.

Time-dependent hydromagnetic phenomena in a rotating spherical cavity are investigated in the framework of an interior boundary-layer expansion. The interior problem is shown to contain waves whose frequencies are of order ω, Aω and A2ω, where ω is the rotation rate of the cavity and A2 = B2/4πρωA2 [Lt ] 1 is the Alfvén number. B is an imposed magnetic field, ρ the fluid density and R the radius of the cavity. The first type of wave is a modification of the hydro-dynamic inertial wave, the second is a pseudo-geostrophic wave and is involved in spin-up, and the third is related to the MAC waves of Braginskiy (1967).

It is shown that the MAC waves must satisfy more than the usual normal boundary conditions and that reference must be made to the boundary-layer solution to resolve the ambiguity regarding which conditions are to be taken. For normal liquid metals of small magnetic Prandtl number the MAC waves must satisfy full magnetic boundary conditions; only the no-slip conditions may be deferred to the boundary layers.

The boundary-layer structure is investigated in detail to display the interactions between applied field, viscosity, electrical conductivity, frequency and latitude. The decay of the pseudo-geostrophic modes, essentially the spin-up problem, is discussed for a non-axisymmetric constraining field and non-zero container conductivity. Three regimes exist, depending on container conductivity.

The paper describes an interpretation of jet-noise theory and scale-model experiments to highlight physical properties of jet-noise sources at very high speed. The study is prompted by current efforts to suppress the noise of supersonic transport aircraft.

The principal noise sources are shown to be very large-scale wave-like undulations of the jet flow that travel downstream at supersonic speed for a distance of several jet diameters. These motions are relatively well ordered and are probably more akin to recognizable instabilities of a laminar flow than the confused small-scale turbulence. Because of this we postulate a model of the noise generating motions as the instability products of a jet flow of low equivalent Reynolds number. This Reynolds number is based on an eddy viscosity and can be further reduced by artificially increasing the small-scale turbulence level. This step would tend to stabilize the flow and inhibit the formation of large-scale noise producing eddies.

An exact periodic solution of the unsteady energy equation for an incompressible fluid with constant properties is derived to illustrate the effect of an oscillation through the viscous dissipation on a temperature field. The flow field used here is a generalization of the well-known Couette flow solution of steady flow, in which one wall is at rest and the other wall oscillates in its own plane about a constant mean velocity. The solution is subject to two boundary conditions that correspond to the heat-transfer and thermometer problems. In order to have some suggestions about the approximate solutions, the solution is compared with its own approximate form. The temperature field consists of a time-mean, first and second harmonic fluctuation. The time-mean temperature profiles show the large influence of oscillation. The time-mean heat flux into or the time-mean temperature of the oscillating wall increases with frequency, and is ultimately proportional to the square root of the frequency. In § 4 the present exact solution of the Couette flow is compared with the formerly obtained approximate solution of the flat plate boundary-layer flow in terms of the wall characteristic values at high frequencies.

The relaxation of an initially non-uniform gas to equilibrium is studied within the framework of the kinetic theory of gases. The macroscopic gas properties are taken to depend on one spatial dimension as well as the time. The amplitude of the non-uniformity is assumed to be small with a length scale large compared with the mean free path, and the Krook model of the Boltzmann collision integral is employed.

By applying multi-time scale perturbation methods to this reduced problem, uniformly valid analytical solutions for the macroscopic velocity, density and temperature are obtained. The macroscopic equations appropriate to each stage of the relaxation process are obtained in a straightforward and unambiguous manner. The distribution function obtained is shown to be a re-expansion of the Chapman–Enskog solution of the Krook equation, with additional terms accounting for the relaxation of the initial conditions to a near equilibrium form. The results indicate that the uniformly valid frst approximation to the macroscopic velocity, density and temperature can be obtained from the Navier–Stokes equations, but that no purely macroscopic set of equations will suffice for the determination of higher approximations.

The Brinkley–Kirkwood theory (1947) is modified to determine the law of propagation of a blast wave in an arbitrary inhomogeneous medium for spherically and cylindrically symmetric cases. The shock path is obtained in terms of a simple quadrature. The numerical results for the shock path and the entire flow region behind the shock, propagating in an exponential atmosphere, show excellent agreement with the exact numerical solution.

The method of conformal transformation is used to investigate the steady streaming generated by an oscillatory viscous flow over a wavy wall. By assuming that the amplitude of the wall is much smaller than the Stokes layer thickness, the equations are linearized and solved for large and small values of the parameter kR. This parameter is the ratio of the amplitude of oscillation of a fluid particle to the wavelength of the wall. When kR [Lt ] 1, the results due to Schlichting (1932) are recovered, and when kR [Gt ] 1 the equations resemble closely those derived in the theory of stability of plane parallel flows. With the aid of this theory the first-order steady streaming is found.

The problems commonly encountered in the measurement of the interfacial tension between two liquids by capillary tube or force measurement are described. In order to avoid such problems, a new method of measurement of the interfacial tension is developed here which is based on the details of axisymmetric capillary waves which can be generated on the interface. Analyses relating these details to the interfacial tension and showing how the details can be measured photographically are given. An apparatus for making these photographic measurements is described and photographs made with such an apparatus are presented. An analysis of these photographs is given which gives the interfacial tensions for the interfaces shown.

A theroetical analysis is given for potential flow over, around and under a vehicle of general shape moving close to a plane ground surface. Solutions are given both in the form of a small-gap asymptotic expansion and a direct numerical computation, with close agreement between the two for two-dimensional flows with and without circulation. Some results for three-dimensional bodies are discussed.

Spatial ‘intermittency’ in the velocity field fine-structure of fully turbulent flow regions, first observed by Batchelor & Townsend (1949), is studied further here in grid-generated nearly isotropic turbulence and on the axis of a round jet. At large enough Reynolds numbers, appropriately filtered hot-wire anemometer signals appear intermittent as the turbulent patterns are convected past the hot wire by the mean flow. Measurements show that there is a decrease in the relative fluid volume (equal to the ‘intermittency factor’) occupied by fine-structure of given size as the turbulence Reynolds number is increased. They show also that, for a fixed Reynolds number, the relative volume is smaller for smaller fine-structure. The average linear dimension of the fine-structure regions turns out to be much larger than the sizes of fine-structure therein. At Rλ, = 110, for example, the ratio ranges from 15 to 30, decreasing with decreasing ‘eddy’ size. It appears to be approaching an asymptote with increasing Rλ.

The flatness factors and probability distributions of the first derivative, the second derivative, band-passed and high-passed velocity fluctuation signals were also measured. The turbulence Reynolds numbers Rλ ranged from 12 to 830. The flatness factors of the first and the second derivatives increase monotonically with Rλ. Those of the second derivative vary with Rλ0.25 for Rλ < 100, and with Rλ0.75 for Rλ > 300. No indication of asymptotic constant values was observed for Rλ up to the order of one thousand.

The probability distributions of velocity fluctuations and large-scale signals are nearly normal, while the small-scale signals are not. The flatness factor of the filtered band-pass velocity signal increases with increasing frequency.

At the larger Reynolds numbers, the square of the signal associated with large wave-numbers may be approximated by a log-normal probability distribution for amplitudes when probabilities fall between 0·3 and 0·95, in limited agreement with the theory of Kolmogorov (1962), Oboukhov (1962), Gurvich & Yaglom (1967).

This investigation describes a detailed study of the near pressure field within the potential cone of a subsonic circular turbulent jet.

The components of the near pressure field in the potential cone in which the potential flow condition exists within the fist four and a half diameter downstream appear to be moving with a phase velocity equal to the local speed of sound. The direction of propagation is roughly normal to the shear layer surrounding the cone. Some components of the hot-wire signal can be associated with the jet structure as a simple complex source, while others are related to the local characteristics of turbulence. Differences in the characteristics of the pressure field within the potential cone exist between the vortex generated noise at very low jet velocity and the eddy generated noise at higher velocity.

The power spectra obtained in the potential cone show the peak which is due to the pressure fluctuations and the flat portion due to the turbulence. The frequencies of the dominant components, in terms of the Strouhal number, are functions of both the axial and radial positions.

Microphone measurements were made in the near field outside for detailed comparison with the potential cone results.

The flow of a rotating homogeneous incompressible fluid over a step is investigated. In the physical system considered the rotation axis is vertical and the step, which is assumed to be infinitely long, is located on a horizontal plane surface. Upstream of the step the fluid is in a uniform free stream motion at an angle α to a line perpendicular to the step axis. The analysis is restricted by the following: E [Lt ] 1, Ro ∼ E½, h/D ∼ E½, H/D ∼ E0, and cos α ∼ E0 where Ro and E are the Rossby and Ekman numbers and h/D and H/D are the step height to step width and water depth to step width ratios respectively. The flow field is shown to consist of interior geostrophic regions, Ekman layers on the horizontal surfaces and vertical shear layers located in the vicinity of vertical planes defined by the edges of the step. In the vertical layers there is a balance between the inertial, Coriolis, and pressure terms in the momentum equations while the effects of viscosity are found to be negligible. Downstream of the step the streamlines are shifted to the right (positive or Northern Hemisphere rotation) of their upstream locations by a distance of S = 2½(h/D) E−½ cos α. Experiments are presented which are in good agreement with the theory advanced.