The linearized problem of Kelvin wave diffraction at the sharp edge of a thin semi-infinite barrier on a rotating earth is considered in the case when a discontinuity in depth stems from the edge in a direction parallel to the barrier. The solution in closed form is obtained using the Wiener-Hopf technique.

It is shown that the direct effect of the depth discontinuity is always to divert additional energy away from the barrier, in the form of outgoing cylindrical waves if the period is small, and provided conditions are suitable as double Kelvin waves if the period is sufficiently long. Results suggest that Kelvin waves which are the most persistent in following an irregular coastline bounding a sea of abruptly varying depth are of intermediate period.

The stability of a two-fluid vortex is studied as a step towards understanding the separation and containment problems in a gaseous-core nuclear rocket. In particular, the linear hydrodynamic stability of two incompressible, immiscible, viscous fluids occupying separate annular regions of a cylindrical Couette apparatus is considered. Neglecting surface tension and gravity, a conservative assumption, the governing equations for arbitrary jumps in fluid properties are derived and numerical solutions to the resultant eigenvalue problems obtained. Results are presented for the effect on neutral stability of density and viscosity jumps, varying gap widths, and differing fluid-fluid interfacial positions. The solutions are limited, however, to the case of stably stratified fluids and a stationary outer cylinder.

Two separate modes (multiple eigenvalues) have been discovered for all cases in which two fluids, differing in any property, are present. A rationale is presented for this phenomenon as well as for most of the other observed results.

While most results are believed to be manifestations of the Taylor cylindrical Couette instability phenomenon, evidence is presented for the existence of additional hidden eigenvalues attributable to the classical KelvinHelmholtz and/or the recently reported Yih viscosity-stratification instability phenomena.

Two problems representative of the gravity flow of granular materials are considered in the context of a theory presented by Goodman (1970). The problems consist of steady fully-developed flow of a granular material down an inclined plane and between vertical parallel plates. It is shown that the dynamical behaviour of these materials is quite different from that of a viscous fluid. For the inclined flow problem, the normal stresses are not only unequal but vary non-linearly with depth. Also the maximum value of the mass flux distribution does not necessarily occur at the upper surface. For the vertical channel-flow problem, the material behaves somewhat like a Bingham fluid in that a plug region exists in the central part of the channel. The interesting feature of this problem is that the concentration of material volume in the shearing region outside the plug may either increase or decrease from the plug to the channel wall, depending on the boundary conditions. Experimental evidence for these phenomena in real granular materials is cited.

The results of this investigation suggest that the gravity flow of granular materials is essentially governed by two factors-a material characteristic length, which is possibly related to the grain size, and the externally imposed constraints such as the gravity field or the pressure exerted upon the granular material from the confining plates.

Craik (1968) has previously examined the stability of horizontal liquid films when an air stream flows over the liquid surface and when the surface is contaminated by an insoluble surface-active agent. The present theoretical work extends this investigation to examine the role of soluble surface-active agents for cases where the liquid Reynolds number is small. This role is found to be a rather complex one which, under suitable conditions, may be either stabilizing or destabilizing.

Related experimental work is described which yields results in qualitative agreement with the theoretical analysis.

The effect of a temperature gradient in a gas inclined at an angle to a boundary wall has been investigated. For an infinite half-space of gas it is found that, in addition to the conventional temperature slip problem, the component of the temperature gradient parallel to the wall induces a net mass flow known as thermal creep. We show that the temperature slip and thermal creep effects can be decoupled and treated quite separately.

Expressions are obtained for the creep velocity and heat flux, both far from and at the boundary; it is noted that thermal creep tends to reduce the effective thermal conductivity of the medium.

An experiment with the pressure-driven flow down a long rotating channel is described. For zero rotation the flow is quasi-parabolic, laminar, and one-dimensional up the channel. With slight rotation Ω there is a weak double-vortex secondary circulation aligned with the channel. At intermediate Ω there exists an instability in the form of logitudinal rolls of non-dimensional wavenumber 5. The instability disappears at high rotation rates.

The general stability problem for a rotating zonal flow $\overline{U}(y)$ is considered theoretically. For perturbations independent of the co-ordinate in the direction of the flow, the problem is exactly analogous to the stability problem of a temperature-stratified fluid where the stratification $\overline{T}_z(z)$ corresponds to the quantity \[ (\partial\overline{U}/\partial y)(1/2\Omega)-1. \] This analogy extends to much more general mean fields (e.g. non-linear or time dependent) than does the oft-quoted analogy between thermal convection and cylindrical Couette flow. The instability theory is in qualitative agreement with the experiment.

The hydrodynamic theory of the resonant cylinder (Gans 1970) is extended to include the effects of a magnetic field parallel to the rotation axis. The linear response is modified by a change in boundary-layer suction and a change in the resonant length. These effects are of equal importance. The theory is valid for small container conductivity and for amplitudes such that the cube of the amplitude is less than the dimensionless precession rate. The importance of container conductivity is assessed. The free modes of the system are given in an appendix. These modes move both east and west.

Experimental apparatus capable of producing magnetic Reynolds numbers of the order of 20 for indefinite lengths of time is described. The apparatus was used to assess the linear theory, though not designed for this purpose. Experiments beyond the range of linear theory are described. The results show finite amplitude effects similar to those previously observed in precessing spheroids in the absence of magnetic effects (Malkus 1968). Additional structure attributable to magnetic effects is observed.

This paper describes some experiments in swirling flows in a diverging cylindrical tube in which various types of vortex breakdowns were observed.

In one set of experiments, the position of the breakdown, axial component of the velocity of the vortex core, swirl angle distribution ahead of the breakdown, and the pressure distribution along the tube were determined for various flow rates and for various values of circulation imparted to the fluid (water). Basically, three types of vortex breakdown were observed, viz. mild (double helix) breakdown, spiral breakdown (followed by turbulent mixing), and axisymmetric breakdown (followed by a thicker vortex core, then a spiral breakdown, and finally by turbulent mixing). The type and the location of the stationary breakdowns were found to be dependent, for the particular vortex tube used, upon the Reynolds and circulation numbers of the flow. In a spiral breakdown, the vortex core filament maintained the same sense of rotation as the upstream fluid elements. In an axisymmetric breakdown, the bubble included an inclined vortex-ring whose axis gyrated about the axis of the tube.

In a second set of experiments, the response of the abrupt structural change along the axis of flow to gradual and abrupt changes in the upstream and downstream flow conditions was examined. The axisymmetric breakdown responded in a manner analogous to the hydraulic jump in open-channel flow before if reached a new stationary position along the axis of the tube.

The observations reported and the evidence presented herein revealed that the axisymmetric breakdown is a finite transition between two sequent states of flow as proposed by Benjamin (1962, 1965, 1967) on theoretical grounds.

A linear theory is developed for the spin-up of a compressible fluid, stratified by a spherical gravity field. Numerical results are obtained for the case of strong stratification (BruntVisl frequency N much greater than the rotation frequency 0). The interior flow is solved in terms of a set of angular eigenfunctions which have been obtained numerically. The principal result is that the spin-up is limited to a layer adjacent to the spherical boundary, the thickness of the layer being of the order of L(0/N), where L is the radius of the boundary. The solution is qualitatively similar to that found by Holton (1965), Walin (1969), and Sakurai (1969a, b) for a stratified fluid in a cylinder. The thickness of the spin-up layer diminishes with latitude , the variation being described roughly by the formula L0| sin |/N. For the case of slow continuous spin-up, the Ekman suction velocity has been calculated, and the results show that || = 24 is the dividing angle between suction (|| > 24) and blowing (|| < 24).

Solutions are presented for creeping flows induced by two-and three-dimensional horizontal and vertical momentum jets in a linearly stratified unbounded diffusive viscous fluid. These linear problems are solved by replacing the momentum jet by a body force singularity represented by delta functions and solving the partial differential equations of motion by use of multi-dimensional Fourier transforms. The integral representations for the physical variables are evaluated by a combination of residue theory and numerical integration.

The solutions for vertical jets show the jet to be trapped within a layer of finite thickness and systems of rotors to be induced. The horizontal two-dimensional jet solution shows return flows above and below the jet and a pair of rotors. The three-dimensional horizontal jet has no return flow at finite distance and the diffusive contribution is found to be almost negligible in most situations, the primary character of the horizontal flows being given by the non-diffusive solution. Stokes's paradox is found to be non-existent in a density-stratified fluid.

The asymptotic stability of a rapidly rotating, horizontally bounded fluid, heated from below, is treated using boundary-layer methods. It is shown that the rotational constraint is so strong as to preclude instabilities, if the interior regions of the fluid are considered to be inviscid. The correct formulation allows this constraint to be broken by introducing horizontal diffusive effects into the interior, while vertical diffusion is confined to Ekman layers on the horizontal surfaces; no vertical layers exist. Moreover, the mechanism of instability is (to the lowest order) associated with energy conversions entirely within the interior region. The present formulation elucidates the role of the Ekman layers in producing high-order corrections to the limiting critical Rayleigh number, and the asymptotic results are extended to include higher-order terms. The effect of rigid side walls on the critical Rayleigh number, and on the azimuthal wavenumber, is considered. Except for very tall cylinders, the critical Rayleigh number is unaffected by the presence of side walls; the results for different azimuthal modes of convection are inconclusive, but indicate that no great error occurs if disturbances are assumed axisymmetric.

This paper discusses a general theory of wave propagation through a random medium whose random inhomogeneities are confined to small deviations from the mean. The theory is initially worked out in detail for the propagation of transverse waves along an infinite stretched string whose density is a random function of position. The manner in which the mean wave profile is modified by scattering from the density inhomogeneities is discussed in great detail, with particular emphasis on physical interpretation. The general theory of wave propagation in arbitrary dispersive or non-dispersive media is then discussed, and it is shown how the theory may be extended to wave propagation problems involving scattering from rough boundaries.

This paper considers the propagation and scattering of waves in dispersive and non-dispersive media containing random inhomogeneities. A detailed discussion of the propagation of the coherent component of the wave field is presented, and a rather general method for obtaining the absorption coefficient and scattering cross-section per unit volume of medium is described. A major objective is to derive macroscopic equations for the propagation of the coherent field in the sense that the damping due to scattering appears in the wave equation as pseudo-viscosity. The paper is concluded by a detailed application of the theory to the problem of the propagation of surface gravity waves over a stretch of shallow water with a ‘rough’ bed. By including finite amplitude effects a balance is achieved between the dispersion of the pseudo-viscous term and the non-linear convective terms, which enables the steady profile of a weak bore to be calculated.

When a layer of particles moves through a viscous liquid it experiences forces which tend to disrupt the layer into clusters of particles separated by open channels. A theoretical description of this process is presented and a viscous instability is predicted. The spatial growth of the instability is approximated by ez, where where a is the particle radius and d is the average distance between particles. This result implies that any initial irregularity in a uniform particle distribution will be amplified by viscous forces alone. Significant amplification will occur when the particle has drifted a small multiple of the separation distance, if this separation is not much greater than the particle diameter. Thus, any initially uniform particle layer will form clusters as it drifts through a viscous fluid. The distance in which this clustering occurs will be unaffected by changes in the particle velocity, as long as the Reynolds number remains small. The preferred form of irregularity will consist of small clusters separated by individual particles which trail some distance behind. Experimental verification of these conclusions is presented.

Wave-induced oscillations in harbours of constant depth but arbitrary shape in the horizontal plane connected to the open-sea are investigated both theoretically and experimentally. A theory termed the ‘arbitrary-shape harbour’ theory is developed. The solution of the Helmholtz equation is formulated as an integral equation which is then approximated by a matrix equation. The final solution is obtained by equating, at the harbour entrance, the wave amplitudes and their normal derivatives obtained from the solutions for the regions outside and inside the harbour. Special solutions using the method of separation of variables for the region inside circular and rectangular harbours are also obtained. Experiments were conducted to verify the theories. Four specific harbours were investigated: two circular harbours with 10° and 60° openings respectively, a rectangular harbour, and a model of the East and West Basins of Long Beach Harbour, California. In each case, the theoretical results agreed well with the experimental data.

A closed flow circuit, comprising a smooth horizontal plastic pipe transporting a dilute suspension of plastic particles in water, was modified to produce a satisfactory environment for the detection of mean particle location across a fixed plane normal to the mean turbulent flow direction. It was established that mean concentration in the vicinity of the vertical diameter was a function of the vertical co-ordinate only, and this result enabled the particle diffusivity to be evaluated for a core region of the cross-section. For a particle relative density in the range 1.035 to 1.126 and a variety of flows, a suitable expression for the dimensionless particle diffusivity is A (1 - BW/u) SN, where u is the friction velocity, W the fall velocity for a particle in still water, S the relative density of the suspended material, and A, B and N are constants.

When a layer of liquid is heated from below at a rate which exceeds a certain critical value, a two- or three-dimensional motion is generated. This motion arises from the action of buoyancy and surface tension forces, the latter being due to variations in the temperature of the liquid surface.

The two-dimensional form of the flow has been studied by a numerical method. It consists of a series of rolls, rotating alternately clockwise and anticlockwise, which are shown to be symmetrical about the dividing streamlines. As well as a detailed description of the motion and temperature of the liquid, and of the effects on these characteristics of variations in the Rayleigh, Marangoni, Prandtl and Biot numbers, a study has been made of the conditions under which the motion first starts, the wavelength of the rolls and the rate of heat transfer across the liquid layer.

Two-dimensional isotropic turbulence is investigated in its development from an arbitrarily specified initial flow through its transformation into a statistically self-preserving decaying flow. Numerical simulation is the principal method of investigation. The early development is characterized by rapid growth of the mean squared vorticity gradient, and this growth is found to be predicted satisfactorily by the quasi-normal hypothesis. During the later states of decay the numerical results are found to be generally consistent with the predictions by Kraichnan, Leith and Batchelor of a k−3 inertial range spectrum. The dimensionless constant of the spectrum is found to be near 2, about half the value found earlier for turbulence maintained by a constant forcing amplitude. The results are also consistent with Batchelor's predictions of the time-dependent behaviour of certain quadratic moments: An inconsistency in those predictions is pointed out, however, which can be resolved by altering the inertial range spectrum by a logarithmic term, as suggested by Kraichnan. The most important two-point Eulerian correlation functions are exhibited. An investigation is made of the Gaussianity of the flow with results indicating a strong tendency toward intermittency in the enstrophy dissipation.

The investigation of laminar free convective plumes in an otherwise stationary environment has formed the basis of numerous investigations, initiated by Zeldovich (1937). For the non-rotating environment alone the authors have been able to locate twenty-nine papers: many of these repeat work previously undertaken. There are, however, two cases of some technological significance which have so far not been considered: (i) the plume in an otherwise quiescent environment for a fluid of very large Prandtl number, of importance in the heating of reservoirs of viscous fluid such as fuel oil; and (ii) the case of vanishingly small Prandtl number, of application to liquid metal-cooled nuclear reactors. Both of these cases have some theoretical interest, as will be shown. Their analysis leads to singular asymptotic perturbations and hence to matched-expansions techniques.

The axial and radial gradients of the tangential velocity distribution are calculated from prescribed secondary flow functions on the basis of a zero-order approximation to the momentum equations developed by Lewellen. It is shown that secondary flow functions may be devised which meet pertinent physical requirements and which at the same time lead to realistic tangential velocity gradients.

The total-temperature distribution in both the axial and radial directions is calculated from such secondary flow functions and corresponding tangential velocity results on the basis of an approximate turbulent energy equation. The method employed for the solution of this equation stresses the equivalence of the vortex tube to counter-current systems with transverse diffusion such as distillation columns and heat exchangers.

An availability function is derived that permits the evaluation of vortex tube performance on the basis of velocity data.

Turbulent diffusivities resulting from the quantitative use of the tangential velocity approximation are shown to agree with those derived from the total-temperature calculations.