The problem of thermoacoustic heating at the closed end of a tube, in which small gas oscillations are maintained, leads in the case of adiabatic walls, and within the framework of a linear theory of the oscillations and a second-order theory of the heating effect, to singular behaviour of the equilibrium temperature at the tube end. Two cases are discussed: one with vanishing viscosity and one with viscosity tending to infinity. The singularities turn out to be similar in character and integrable in both cases.

The interfacial shape of two immiscible simple fluids in a vertical cylinder which oscillates about its axis is investigated using the theory of domain perturbations. The perturbation stresses are expressed by integrals over the history of the deformation. At first order the azimuthal velocity field satisfies the requirements of continuity in velocity and shear stresses across the interface. At second order the solution consists of a mean part and a time-periodic part varying at twice the frequency of the cylinder. The mean problem is inverted for the mean secondary flow, pressure and interfacial shape. Experimental data for two polymeric oils (TLA227 and STP) show qualitative agreement with theoretical predictions for the mean interfacial shapes.

This paper analyses the coupling between an imposed disturbance and an instability wave that propagates downstream on a shear layer which emanates from a separation point on a smooth surface. Since the wavelengths of the most-amplified instability waves will generally be small compared with the streamwise body dimensions, the analysis is restricted to this ‘high-frequency’ limit and the solution is obtained by using matched asymptotic expansions. An ‘inner’ solution, valid near the separation point, is matched onto an outer solution, which represents an instability wave on a slowly diverging mean flow. The analysis relates the amplitude of this instability to that of the imposed disturbance.

A narrow flow passing over an obstacle in a rotating channel is analysed. When the upstream Froude number of the flow approaches unity and the obstacle height is sufficiently small, stationary Kelvin waves may appear in the channel. Under these conditions the usual nonlinear hydraulic theory (e.g. Gill 1977) must be replaced by a nonlinear dispersive theory. When the flow upstream of the obstacle is subcritical, the nonlinear dispersive theory produces three solutions, two of which resemble the solutions of hydraulic theory and a third which contains cnoidal lee waves. Upstream influence due to the obstacle becomes a function of obstacle shape as well as height. The ‘controlled’ solution is distinguished by the presence of a partial solitary wave in the lee of the obstacle.

We examine the steady axisymmetric source–sink flows of a stably stratified fluid in a rotating annulus, for which S ∼ O(1), E [Lt ] 1. Numerical methods are used to integrate the unsteady Navier–Stokes equations to obtain the approximate steady solutions. Results on the radial and vertical structures of the flow and temperature-field details are presented. Specific comparisons of the relative sizes of the terms in the equations are conducted to reveal the balance of the dynamic effects. The profiles of the vorticity components are displayed. In the linear flow regime of a homogeneous fluid, the transport of fluid in the meridional plane takes place entirely via boundary layers. As stratification increases, the meridional flows are less concentrated in the boundary layers, and an appreciable portion of the meridional fluid transport is carried through the main body of fluid. The distinction between the sidewall layers and the interior becomes less clear. The flows in the main body of fluid develop vertical velocity shear, resulting in a thermal-wind relation. In the nonlinear case, the source sidewall layer thickens and the sink layer thins. As stratification increases, the meridional fluid transport through the main body of fluid is more pronounced than in the linear case. The balance of terms indicates that the bulk of the flow field is still characterized by the thermal-wind relation.

In order to study the tidal generation and evolution of internal waves, both experimentally and theoretically, a two-dimensional two-layer model has been developed. The laboratory model consists of two immiscible fluid layers in a long wave tank, where time-dependent flow can be created by towing a suitably shaped obstacle with a sinusoidal motion. The effects of changing the Froude number and the ratio of the depths of the two fluid layers on the shape of the interface are studied. The first theoretical model considers the motion to be quasi-steady, allowing only a balance between buoyancy and inertial forces, and is in reasonable agreement with the experimental results for very slow motions and the initial phase of the tidal cycle only. A more complete numerical solution is based on the vortex-point method in which the interface is modelled by a set of discrete vortices, while the rigid boundaries, which include the obstacle, are modelled by a source distribution. The resulting governing equations are expressed in Lagrangian form, and the baroclinic generation of vorticity is related to the generation of source density, through two simultaneous integrodifferential equations. For small Froude numbers, the experimental and numerical results are in good quantitative agreement. At large Froude numbers, the numerical solution, which does not consider mixing between the two layers nor boundary-layer separation from the obstacle, gives a slight difference in the position of the maximum interface displacement from that found in the experiment.

Solute transport in porous formations is governed by the large-scale heterogeneity of hydraulic conductivity. The two typical lengthscales are the local one (of the order of metres) and the regional one (of the order of kilometres). The formation is modelled as a random fixed structure, to reflect the uncertainty of the space distribution of conductivity, which has a lognormal probability distribution function. A first-order perturbation approximation, valid for small log-conductivity variance, is used in order to derive closed-form expressions of the Eulerian velocity covariances for uniform average flow. The concentration expectation value is determined by using a similar approximation, and it satisfies a diffusion equation with time-dependent apparent dispersion coefficients. The longitudinal coefficients tend to constant values in both two- and three-dimensional flows only after the solute body has travelled a few tens of conductivity integral scales. This may be an exceedingly large distance in many applications for which the transient stage prevails. Comparison of theoretical results with recent field experimental data is quite satisfactory.

The variance of the space-averaged concentration over a volume V may be quite large unless the lengthscale of the initial solute body or of V is large compared with the conductivity integral scale. This condition is bound to be obeyed for transport at the local scale, in which case the concentration may be assumed to satisfy the ergodic hypothesis. This is not generally the case at the regional scale, and the solute concentration is subjected to large uncertainty. The usefulness of the prediction of the concentration expectation value is then quite limited and the dispersion coefficients become meaningless.

In the second part of the study, the influence of knowledge of the conductivity and head at a set of points upon transport is examined. The statistical moments of the velocity and concentration fields are computed for a subensemble of formations and for conditional probability distribution functions of conductivity and head, with measured values kept fixed at the set of measurement points. For conditional statistics the velocity is not stationary, and its mean and variance vary throughout the space, even if its unconditional mean and variance are constant. The main aim of the analysis is to examine the reduction of concentration coefficient of variation, i.e. of its uncertainty, by conditioning. It is shown that measurements of transmissivity on a grid of points can be effective in reducing concentration variance, provided that the distance between the points is smaller than two conductivity integral scales. Head conditioning has a lesser effect upon variance reduction.

We consider a dilute dispersion containing small rigid particles in a Newtonian fluid. These spherical particles are of different size and density, and they settle relative to one another under the action of gravity. When the particles become close, they exert an attractive van der Waals force on each other, and doublets are formed when two particles come into contact as a result of this force. The rate at which doublets are formed is calculated using a trajectory analysis to follow the relative motion of pairs of particles.

We restrict our attention to dispersions where the Péclet number is large (negligible Brownian motion) and where the Reynolds number is small (negligible fluid inertia). However, the effects of the inertia of the particles on their trajectories are included, and these are measured by the Stokes number. A key dimensionless parameter is identified, denoted by Qij which provides a measure of the relative importance of gravity and the van der Waals force. An asymptotic solution to the trajectory equations is presented for large values of this parameter in the case of zero Stokes number. This asymptotic solution is then complemented by numerical computations of the particle trajectories. Application to typical hydrosol and aerosol dispersions is presented, and, in particular, a comparison is made between the effects of van der Waals forces and Maxwell slip in promoting collisions between aerosol particles.

The Stokes flow due to the motion of a small particle in arbitrary directions is investigated in the presence of a circular hole in an infinite thin plane wall separating a quiescent viscous fluid.

The solutions of the boundary-value problem are obtained in closed forms to the point-force approximation in toroidal coordinates, by the use of the Green and Neumann functions supplemented by the edge function to remove the singularity at the rim of the hole. The volume flux through the hole and the force and torque experienced by the small spherical particle are determined on the basis of this solution. The case of linear motion parallel to the plane of the wall is discussed in detail.

Boundary conditions are developed for rapid granular flows in which the rheology is dominated by grain–grain collisions. These conditions are $\overline{v}_0 = {\rm const}\,{\rm d}\overline{v}_0/{\rm d}y$ and u0 = const du0/dy, where $\overline{v}$ and u are the thermal (fluctuation) and flow velocities respectively, and the subscript indicates that these quantities and their derivatives are to be evaluated at the wall These boundary conditions are derived from the nature of individual grain–wall collisions, so that the proportionality constants involve the appropriate coefficient of restitution ew for the thermal velocity equation, and the fraction of diffuse (i.e. non-specular) collisions in the case of the flow-velocity equation. Direct application of these boundary conditions to the problem of Couette-flow shows that as long as the channel width h is very large compared with a grain diameter d it is permissible to set $\overline{v} = 0$ at the wall and to adopt the no-slip condition. Exceptions occur where d/h is not very small, when the wall is not rough, and when the grain–wall collisions are very elastic. Similar insight into other flows can be obtained qualitatively by a dimensional analysis treatment of the boundary conditions. Finally, the more difficult problem of self-bounding fluids is discussed qualitatively.

Recently, Leibovich & Stewartson (1983) developed a sufficient condition for the instability of columnar vortices with radial shears in both the azimuthal and axial velocities, while others (e.g. Staley & Gall 1984) have found instabilities in numerical simulations which conform exactly to expectations based on the Leibovich-Stewartson theory. The purpose of this brief note is to show that this three-dimensional stability problem is isomorphic to the classical two-dimensional inertialThe instability discussed here is sometimes referred to as ‘centrifugal instability’. stability problem when viewed in an appropriate local coordinate system. The instability is therefore clearly inertial in character, as suggested by Pedley (1969).

Heat-transfer measurements have been made in normal liquid He contained within a rotating, cylindrical, cryogenic Bénard cell with variable aspect ratio. Data are presented for a range of dimensionless angular velocities 0 ≤ Ω < 600 and Prandtl numbers 0.49 ≤ Pr < 0.76 and for three aspect ratios Γ of 7.81, 4.93 and 3.22. Where possible, comparisons are made with theoretical predictions and past experiments concerning heat transfer in rotating fluids.

Experiments on mixed-layer deepening in a straight wind flume initially containing a two-layer fluid were carried out. In order to simulate an effectively unlimited water body, or a situation where the presence of end walls is not yet noticeable, the water from the upper layer was withdrawn at the downstream end, and supplied again at the upstream end in part of the experiments. Upper-layer depths, and mean-density and mean-velocity profiles were measured for overall Richardson numbers (based on the surface friction velocity) ranging from 22 to 1090. Gradient Richardson numbers in the transition layer (between the mixed layer and the undisturbed layer) and in the mixed layer were estimated. The correction of the results for the effects of recirculation is discussed.

It is shown that the vortex sheet in a slot between two semi-infinite plates does not admit incompressible resonant perturbations. The semi-infinite vortex sheet entering a duct does admit incompressible resonance. These results indicate that the vortex-sheet approximation is less useful for impinging shear flows than for non-impinging flows. They also suggest an important role of downstream vortical disturbances in resonant flows.

The general solution for perturbations to flow with a vortex sheet and edges is written in terms of a Cauchy integral. Requirements on the behaviour of this solution at edges and at downstream infinity fix the criteria for resonance.

An array of closely spaced, aligned, turbulent, incompressible jets is effectively simulated by a line momentum source. In a shallow inviscid fluid, such a source induces a predominantly two-dimensional non-diffusive flow which is irrotational except along lines of velocity discontinuity downstream of the source. The flow can be generated by a distribution of line vortices of unknown strength along the unknown slipstreamlines. Based upon this vortex model, kinematic and dynamic conditions along the slipstreamlines are formulated, and the two resulting nonlinear singular integral equations are solved numerically using a Newton-Raphson-type iterative collocation method.

The flow field shows a marked resemblance to that induced by a nonlinear actuator disk. For the case of no ambient current, experimental results indicate that the slipstreamlines emanate from points which are close to, but not at the ends of, the source. As the ambient current strength increases, a dividing streamline appears in the induced sink flow upstream of the source, and the points from which the slipstreamlines emanate move closer to the ends of the source. Further increases in the current strength result in the smooth blending of this dividing streamline with the slipstreamline.

Laboratory experiments performed in a shallow water basin confirm all of the features predicted by the theory.

Axial and azimuthal turbulence intensities in the rod-gap region are shown, for developed turbulent flow through parallel rod arrays, to increase strongly with decreasing rod spacing. Two array geometries are reported: one was constructed from a rectangular cross-section duct containing four rods and spaced at five pitch-to-diameter or wall-to-diameter ratios; the second was a test section containing six rods set in a regular square-pitch array to represent the interior flow region of a large array.

Measurements were made of the mean axial velocity, wall-shear-stress variation, axial-pressure distribution and Reynolds stresses. Techniques for resolving secondary-flow velocities to within ±1% of the local axial velocity failed to detect significant non-zero mean secondary-flow components. Analysis of the turbulent flow structure showed an energetic azimuthal turbulent-velocity component in the open rod gap for both geometries. The axial turbulent velocity has a coupled large-scale semiperiodic structure, with an antiphase relationship across the rod-gap centre or subchannel boundary. This structure is considered to be generated by an incompressible-flow parallel-channel instability, and, for closely spaced rod arrays, is the dominant process for intersubchannel mass, heat or momentum exchange.

Extensive single-point turbulence measurements made in a heated wall jet on a convex wall, and in an equivalent plane flow, show that the turbulence structure and the transfer of heat and momentum are affected by wall curvature. In the curved wall jet the points of zero shear stress and zero streamwise heat flux are displaced further from the point of maximum velocity, the Stanton number for heat transfer from the surface is reduced more than the skin-friction coefficient, and temperature profiles in the inner layer depart from the flat-flow wall law. The shear stress, turbulent intensities and turbulent heat fluxes are reduced by stabilizing curvature in the inner layer and increased by destabilizing curvature in the outer flow. The largest changes occur in the triple products, which are nearly doubled by destabilizing curvature.

The concept of adding a harbour, consisting of two parallel projections to a wave-energy device was first brought to the attention of the wave-energy community at a Symposium in Trondheim, Norway, in June 1982. The proponents of the idea claim that the performance of the device is considerably improved by the addition of the harbour, thereby reducing costs. In this paper two theoretical techniques are described for predicting the performance of the harbour system. First, a relatively simple approximate method using the theory of long thin harbours is described. Secondly, numerical techniques used for rigid-body interaction with waves are adapted to cope with harbour systems with no restrictions on dimensions. It is shown that the simpler approach gives results that agree closely with numerical calculations over a wide range of configurations. Hydrodynamic theory is used to evaluate the performance of the device, assuming that it can absorb energy through a resistive damper. The results are encouraging, demonstrating that the addition of a harbour can be very beneficial and confirming that the concept is worthy of closer scrutiny.

Results are presented of an experimental study on the transition to geostrophic turbulence, and the detailed behaviour within the turbulence regime, in a rotating, laterally heated annulus of fluid. Both spatial and temporal characteristics are examined, and the results are presented in the form of wavenumber and frequency spectra as a function of a single external parameter, the rotation rate.

The transition to turbulence proceeds in a sequence of steps from azimuthally symmetric (no waves present) to chaotic flow. The sequence includes doubly periodic flow (amplitude vacillation), semiperiodic flow (structural vacillation), and a transition zone where the characteristics undergo a gradual change to chaotic behaviour. The spectra in the transition zone are characterized by a gradual merging of the background signal with the spectral peaks defining regular wave flow as the rotation rate is increased.

Within the geostrophic turbulence regime, the wavenumber spectra are characterized by a broad peak at the baroclinic scale and a power dependence of energy density on wavenumber at the high-wavenumber end of the spectrum. Our data reveal a significant dependence of the slope on the thermal Rossby number, ranging from −4.8 at RoT = 0.17 to −2.4 at RoT = 0.02. The frequency spectra also show a power dependence of the energy density on frequency at the high-frequency end of the Spectrum. We find a nominal −4 power which does not appear to be sensitive to changes in Rossby or Taylor number.

The problem of capillary–gravity waves generated by certain moving oscillatory surface-pressure distributions is investigated. The main difficulty of the problem lies in finding the real roots of the modified frequency equations. This is dealt with by the use of certain geometric considerations. The critical condition that results from the formation of double roots of the modified frequency equations is represented as a surface. This surface divides the whole space into several distinct regions. For points in different regions the propagation of waves is different. The waves are determined in all cases.