The effects of convection and refraction dominate the heart-shaped pattern of jet noise. These can be corrected out to yield the small ‘basic directivity’ of the eddy noise generators. The observed quasi-ellipsoidal pattern was predicted by Ribner (1963, 1964) in a variant of the Lighthill theory postulating isotropic turbulence superposed on a mean shear flow. This had the feature of dealing with the joint effects of the quadrupoles without displaying them individually. The present paper reformulates the theory so as to calculate the relative contributions of the different quadrupole self and cross-correlations to the sound emitted in a given direction. Some minor errors are corrected.

Of the thirty-six possible quadrupole correlations only nine yield distinct non-vanishing contributions to the axisymmetric noise pattern of a round jet. The correlations contribute either cos4θ, cos2θ sin2θ or sin4θ directional patterns, where θ is the angle with the jet axis. A separation into parts called ‘self noise’ (from turbulence alone) and ‘shear noise’ (jointly from turbulence and mean flow) may be made.

The nine self-noise patterns combine as $A\; cos^4\theta(1)+A\; cos^2sin^2\theta(\frac{7}{8}+\frac{7}{8}+\frac{1}{8}+\frac{1}{8})+A sin^2 \theta (\frac {12}{32} & + & \frac{12}{32}+\frac{7}{32}+\frac{1}{32})\\ & = & A(cos^2\theta+sin^\theta)^2 = A;$ this is uniform in all directions as it must be, arising from isotropic turbulence. The two non-vanishing shear-noise correlation patterns combine as $B\;cos^4\theta (1) + B\;cos^2\theta sin^2\theta(\frac{1}{2})=B(cos^2\theta+sin^2\theta)^2 = A;$

The overall ‘basic’ pattern (self noise plus shear noise) thus has the form A + B(cos2θ + cos4θ)/2; this is a slight change from the previous result. The dimensional constants A and B are of comparable magnitude; the pattern in any plane through the jet axis thus resembles an ellipse of modest eccentricity.

Frequency spectra are also discussed, following the earlier work. Since the self noise depends quadratically on turbulent velocity components and the shear noise only linearly, there is a relative shift of the self noise to higher frequencies. This in conjunction with refraction figures in the explanation of the deeper pitch of jet noise radiated at small angles to the axis.

Finally, the predictions are shown to be compatible with recent experimental results.

In this, the third, part we present a complete asymptotic analysis of the distribution of velocity and electric potential in an electrically conducting liquid between two circular electrodes of finite diameter, 2b, when a current is passed between them. The electrodes are set opposite to each other in insulating planes, a distance 2a apart, and a magnetic field is applied perpendicular to these planes.

The asymptotic solution is obtained under the restriction that the Hartmann number M satisfies both the conditions M [Gt ] 1 and M½l [Gt ] 1, where l = b/a. It enables us to calculate the distribution of velocity and electrical potential throughout the flow field, and provides an expansion for the resistance R between the electrodes in descending powers of M½l, which is correct provided terms of order M−1 and (lM½)−3 are neglected. Comparison of the theoretical results with experiment shows good agreement in the measurements of R and in the direct probe measurements of electrical potential within the fluid. This is one of the first experiments in which direct probe measurements of an MHD flow, as well as external measurements, have provided such a satisfactory confirmation of the theory. Direct measurements of the velocity, by means of a Pitot tube as in part 2, or by means of a hot-film anemometer undertaken by Malcolm (1968), agree less well with the theory.

Solutions of the diffusion equation are obtained for mass transfer in a fully developed turbulent flow in a plain circular tube in two axisymmetric situations. The cases studied are a point source positioned at the centre of the tube and a ring source in the tube wall in which there is a uniform mass flux along a short length of the tube. The purpose of the work is to establish the correctness of the descriptions of the velocity profile and radial eddy diffusivities of mass and momentum in order to provide a firm base from which consideration of the non-axisymmetric situation could proceed.

The turbulent velocity profile is deduced from a two-part model based on a sublayer profile and the Von Kármán similarity hypothesis. The radial eddy diffusivity of momentum is described by an expression due to Reichardt and Van Driest and from this the radial eddy diffusivity of mass as a function of radius is obtained by use of a ratio which takes account of fluid properties and the value of the radial eddy diffusivity.

The analysis is substantiated by experiments carried out with nitrous oxide, Schmidt number = 0·77, for Reynolds numbers from 20,000 to 130,000. The concentration profiles measured at several axial positions downstream from the source are in good agreement with the analytical solutions in both cases. Direct measurements of the eddy diffusivity of mass and momentum were obtained as added confirmation and also gave good agreement with the theory.

Adiabatic disturbances propagating as transverse waves in an inviscid fluid rotating as a Rankine vortex about the axis of its cylindrical container are considered. The propagation of the first mode of the first two harmonic waves has been investigated. Relative to a fixed co-ordinate system, for each harmonic, there are three waves. Two waves are rotating in the same direction as the fluid, one faster and the other slower than the core of the fluid, and one wave rotates in the opposite direction. The latter is stable and relative to the core of the rotating fluid it is the fastest wave. Relative to the container, the other two waves are speeded up by rotation. However, relative to the rotating core, the angular velocity of the fast wave decreases when the fluid is speeded up, and when it is zero the wave breaks down. As the region of potential flow decreases the angular velocity of the slow wave increases and its amplitude decreases, and in the limit of vanishing potential flow, the wave rotates as fast as the fluid and its amplitude tends to zero.

Stability to infinitesimal disturbances, when a parallel magnetic field is imposed, is investigated for the free boundary-layer type flows, of low magnetic Reynolds number, between two unbounded parallel streams of a viscous, incompressible, electrically conducting fluid. Neutral stability curves are calculated for small wave-number making use of the limiting profile: previous results by another author are found to be incomplete. A qualitative neutral stability picture is conjectured for other values of the wave-number and, granted a certain part of this conjecture, the conclusion is that the critical Reynolds number remains zero until the parameter Q/R exceeds the value (Q/R)crit [eDot ] 0·0233. It is suggested that a sufficiently strong magnetic field can stabilize a flow of any finite Reynolds number.

An analysis of a hydrostatic thrust bearing with electrically conducting compressible lubricant under an axial-current-induced pinch is presented. It is shown that the load capacity of the bearing can be increased by the pinch effect and the magnitude of the pinch effect depends on the mass flow rate. It is also shown that a load proportional to the square of the axial current can be sustained even when there is no flow or external pressurization.

The paper examines the influence of a piston motion on the gaseous motion set up by an explosion. This is a problem of relevance not only to explosions in which the remnants of the original explosive are present, but also by the application of the hypersonic similarity law, to the steady flow at high Mach number past a blunt-nosed object with increasing thickness. In the vicinity of the piston perturbation theory breaks down and matching techniques have to be employed. A solution is given in terms of a small parameter which essentially depends on the ratio of the speed of the piston to that of the shock. Some particular cases are discussed in which the piston motion plays a similar role to the counter pressure of the undisturbed gas.

Theory and experiment are presented for mass transfer into a fully developed turbulent flow in a plain circular tube in two non-axisymmetric cases. The cases studied are a diametral line source and a discontinuous ring source, in which there is a uniform mass flux over rectangular areas of the tube wall. A comparison is made between the concentration profiles predicted by the solutions of the diffusion equation and experiments using nitrous oxide, Schmidt number S = 0·77, as a tracer gas in air. The range of experiments covers Reynolds numbers R from 20,000 to 120,000.

In the analysis, the assumption is made that the tangential and radial eddy diffusivities of mass are equal at a point. The radial diffusivity of mass, which is a function of radial position, is related to the radial eddy diffusivity of momentum by a ratio, which takes account of fluid properties and the value of the radial eddy diffusivity of momentum. The satisfactory agreement between analysis and experiment establishes the correctness of this assumption. Further confirmation was obtained by direct evaluation of the tangential eddy diffusivity of mass from the measured concentration profiles.

Disintegration processes occurring with charged liquid jets of distilled water have been examined. High-speed photographic techniques were used to determine the effect of charging on disintegration of the jet, the size distribution of the drops formed, and the velocities of the drops. At relatively small currents, the jet remains undisplaced while the drops formed acquire components of velocity perpendicular to the jet axis. At sufficiently large currents, the jet develops kink or longitudinal instabilities which can cause the jet to be appreciably displaced, and the drops may have relatively large components of velocity perpendicular to the jet axis. The size distributions of the drops formed are significantly different from those resulting in the absence of electrification. Mean drop size decreases with increased charging in all cases. Drop speeds increase with increased charging as a result of both increased electrical repulsion and reduction in size.

The linear, normal mode instability of three-dimensional laminar and turbulent shear layers is studied. The flows consist of two streams semi-infinitely extended in the y-direction, flowing obliquely in the x-z plane. It is found that the stability of the flows depends on the main flow velocity components in the direction of the disturbance wave-number vector. Numerical calculations are performed to obtain the neutral stability curves. Under the usual parallel unperturbed flow assumption, the neutral stability curves pass through the origin of the α-R diagram, where α is the wave-number and R is the flow Reynolds number. It is also found that eddy viscosity has a destabilizing effect for small Reynolds numbers but a stabilizing effect at larger Reynolds numbers. Because any linear perturbation trajectory eventually leaves the unstable region of the α-R plane, it is probable that a Lin-Benney or Taylor-Goertler secondary instability ensues. Suitable components of the existing turbulence grow and develop into a large eddy system which causes rapid entrainment, giving rise to a turbulent burst. A first-order non-parallel correction is made to the neutral stability curves. The new curves have both upper and lower branches, and there exist minimum critical Reynolds numbers.

A simple laboratory model for the wind-driven ocean circulation is re-studied experimentally and theoretically. Introduced by Pedlosky & Greenspan (1967), the model consists of a rotating cylinder with sloping bottom, the fluid inside being driven by the steady relative rotation of the cylinder's lid. A linear theory is developed to illustrate the modification in the interior and Stewartson boundary layers caused by variation of the bottom slope from 0 to O(1); Stommel's (1948) model is obtained when the bottom slope tan α [Lt ] E¼, and the Munk & Carrier (1950) model is obtained for E¼ [Lt ] tan α [Lt ] 1 (E is the Ekman number). Measurements of the interior cross-contour ‘Sverdrup’ velocity agree well with theory when the Ekman-layer Reynolds number RE is ≈ 1 or less. The western boundarylayer azimuthal velocity agrees reasonably well with theory, although the observed variation with depth and bottom slope were not predicted. The western boundary layer shows downstream intensification when RE is increased from ≈ 1 until topographic Rossby waves appear in the transition region between western boundary layer and interior. The motion becomes unstable when a critical value of RE is reached, independent of the bottom slope, and a low-frequency two-dimensional flow oscillation is observed. A brief comparison is made with previous wind-driven ocean circulation studies.

In an experimental study the development of transition downstream of Görtler vortices was investigated. With the tellurium method it was possible to distinguish beyond the Görtler vortices to successive instability modes. The first deforms the vortex pattern in a steady way and produces between each vortex pair boundary-layer profiles with two points of inflexion. When this has been established another instability mode starts, consisting of regular three-dimensional oscillations. By detailed flow visualization a nearly complete picture of the different flow patterns can be obtained.

The stability of non-Newtonian jets was investigated. A linearized stability analysis shows that a liquid column of a viscoelastic fluid exhibits more rapid growth of axisymmetric wave disturbances than a Newtonian fluid of the same zero shear viscosity. This result is independent of the form of constitutive equation chosen. Experiments in weakly elastic fluids confirm this expectation, whereas data on fluids with more pronounced elastic properties indicate that non-linear phenomena are dominating. The disturbances appear as a series of droplets connected by random lengths of threads, which thin with distance and eventually lead to jet breakup. Even in dilute viscoelastic solutions, jet breakup does not occur by the growth of clearly defined waves.

Bounds for the complex wave velocity c, determined by the Orr-Sommerfeld equation and the boundary conditions for channel flow, have been given by Joseph (1968a,b). In these notes it is shown how two of Joseph's theorems can be uniformly improved.

A disk moves edgewise in a vertical plane with a sufficiently small constant velocity in a uniformly stratified inviscid fluid under gravity. The resultant hydrodynamic thrust on the disk is estimated. Configurations of some associated phase surfaces are shown. A transient Green's function is presented for the case when the Boussinesq approximation is not made, and also for an analogous case of inertial waves in a rotating fluid.

This paper is an investigation of MHD boundary layers in a transverse magnetic field when the magnetic Reynolds number (Rm) is small. The main purpose is to understand something about the suppression of separation by a strong magnetic field, with particular emphasis on the behaviour near a rear stagnation point. Given an O(1) inviscid flow it is shown that there is a critical value of N, the interaction parameter, to completely suppress separation. This value is one half that proposed by Leibovich (1967), a discrepancy that is due to the non-regularity of the boundary-layer equations at a rear stagnation point, a possibility that Leibovich did not consider in his solution. Model linear equations suggest the true role of Leibovich's solution. The possibility of a viscous wake leaving the rear stagnation point is considered and it is suggested that one does not arise from vorticity generated in the boundary layer.

The scattering of infinitesimal surface waves normally incident on a rectangular obstacle in a channel of finite depth is considered. A variational formulation is used as the basis of numerical computations. Scattering properties for bottom and surface obstacles of various proportions, including thin barriers and surface docks, are presented. Comparison with experimental and theoretical results by other investigators is also made.

The viscous, creeping flow through a cylindrical tube of a liquid, which contains rigid, spherical particles, is investigated analytically. The spheres are located on the axis of the cylinder and are equally spaced. Solutions are derived for particles in motion and fixed, with and without fluid discharge. Numerical results are presented for the drag on each sphere and the mean pressure drop for a wide range of sizes and spacings of the spheres. The study is motivated by possible application to blood flow in capillaries, where red blood cells represent particles of the same order of magnitude as the diameter of the capillary itself. The results may also be of interest in other applications, such as sedimentation and fluidized beds. It is shown that there is little interaction between particles if the spacing is more than one tube diameter, and that the additional pressure drop over that for Poiseuille flow is less than 50% if the sphere diameter is less than 0·8 of the tube diameter.

The main purpose of this work is to show how a continuous finite bandwidth of modes can be readily incorporated into the description of post-critical Rayleigh-Bénard convection by the use of slowly varying (in space and time) amplitudes. Previous attempts have used a multimodal discrete analysis. We show that in addition to obtaining results consistent with the discrete mode approach, there is a larger class of stable and realizable solutions. The main feature of these solutions is that the amplitude and wave-number of the motion is that of the most unstable mode almost everywhere, but, depending on external and initial conditions, the roll couplets in different parts of space may be 180° out of phase. The resulting discontinuities are smoothed by hyperbolic tangent functions. In addition, it is clear that the mechanism for propagating spatial nonuniformities is diffusive in character.

The stability and time-dependent deflexions of a thin flexible cylinder with zero bending rigidity set in a viscous stream are examined. The cylinder is fixed at one end and free at the other. Modal shapes are found based on solutions to linearized equations resulting from small deflexion assumptions; the dependence of cylinder motion upon the aerodynamic, elastic and physical size characteristics of the cylinder is exhibited. It is found that the cylinder motion is always unstable and that cylinder amplitude increases without bound as time is increased.