This paper describes the flow of a homogeneous fluid contained in a rapidly rotating cylinder. The upper part of the cylinder rotates slightly faster, giving rise to a discontinuity in the sidewall velocity. The Stewartson-layer structure arising at the sidewall is essentially affected by this discontinuity. In contrast with previously studied problems, the E¼ layer (E is the Ekman number) is unable to perform the matching of the interior flow to the sidewall. It is shown that this matching is carried out partially by the E¼ layer and partially by the E1/3 layer, the latter accounting for the jump discontinuity. This paper also presents an analytical description of the flow in the singularity region near the sidewall discontinuity.

The effects of stable stratification on turbulence structure have been experimentally investigated in stratified open-channel flow and a theoretical spectral-equation model has been applied to the stably stratified flow. The measurements were made in the outer layer of open-channel flow with strongly stable density gradient, where the wall effect was small. Velocity and temperature fluctuations were simultaneously measured by a laser-Doppler velocimeter and a cold-film probe. Measurements include turbulent intensities, correlation coefficients of turbulent fluxes and coherence–phase relationships. These turbulent quantities were correlated with the local gradient Richardson number and compared with the values calculated using a spectral-equation model and with other laboratory measurements. In stable conditions, turbulent motions approach wavelike motions, and negative heat and momentum transfer against the mean temperature and velocity gradient occurs in strongly stable stratification.

When the flow over a submerged, round, upright cylinder, situated in a large ocean, is forced by a train of plane waves, linear theory (Yamamuro 1981) shows that the response can be abnormally large for certain forcing frequencies. The aim of this paper is to present a weakly nonlinear theory, where wave interactions, arising from the quadratic terms in the free-surface boundary conditions, can yield abnormally large responses.

A specific interaction will be considered between a flow at a subharmonic frequency and a flow at the driving frequency. The reason for considering such an interaction derived from a consideration of some experimental results of Barnard, Pritchard & Provis (1981).

The Gröwth of Gortler vortices in boundary layers on concave walls is investigated. It is shown that for vortices of wavelength comparable to the boundary-layer thickness the appropriate linear stability equations cannot be reduced to ordinary differential equations. The partial differential equations governing the linear stability of the flow are solved numerically, and neutral stability is defined by the condition that a dimensionless energy function associated with the flow should have a maximum or minimum when plotted as a function of the downstream variable X. The position of neutral stability is found to depend on how and where the boundary layer is perturbed, so that the concept of a unique neutral curve so familiar in hydrodynamic-stability theory is not tenable in the Gortler problem, except for asymptotically small wavelengths. The results obtained are compared with previous parallel-flow theories and the small-wavelength asymptotic results of Hall (1982a, b), which are found to be reasonably accurate even for moderate values of the wavelength. The parallel-flow theories of the growth of Gortler vortices are found to be irrelevant except for the small-wavelength limit. The main deficiency of the parallel-flow theories is shown to arise from the inability of any ordinary differential approximation to the full partial differential stability equations to describe adequately the decay of the vortex at the edge of the boundary layer. This deficiency becomes intensified as the wavelength of the vortices increases and is the cause of the wide spread of the neutral curves predicted by parallel-flow theories. It is found that for a wall of constant radius of curvature a given vortex imposed on the flow can grow for at most a finite range of values of X. This result is entirely consistent with, and is explicable by the asymptotic results of, Hall (1982a).

The consequence of imposing an axisymmetric travelling-wave disturbance on the Poiseuille flow between two concentric cylinders is examined. A nonlinear analysis is taken, using perturbed bifurcation and singular perturbation theory, to determine how resonant wall oscillations affect flow stability. Subcritical, stable, finite-amplitude perturbations to the basic Poiseuille flow are found and conjectures on their significance are given.

The results of an experimental study of the interaction between waves and a current propagating in the same direction, have been reported by Kemp & Simons (1982). This paper describes the second part of the study, and considers the case of waves propagating against the current. Tests were performed in a laboratory flume with smooth and rough beds, and velocity measurements were made with a directionally sensitive laser anemometer as described in the previous paper. Analysis, including ensemble averaging of velocities and surface elevation, was performed by an on-line computer.

Results indicate that the rate of wave attenuation is greatly increased by the addition of an opposing current, and reduced by a following current. Wave profiles remain closely described by Stokes second-order theory; orbital velocities are also found to be in agreement with a second-order wave theory modified to take account of the presence of the current.

Certain results described occur regardless of the relative directions of current and wave. Mean velocities in the upper flow increase in the direction of the wave generator for increasing wave height. This suggests that the current is enhancing the wave-induced mass transport. Near the bed the velocity profiles so change that above the rough bed the current is retarded by the wave motion. In the logarithmic layer over the smooth bed velocities are increased with increasing wave height. However, all changes to velocity profiles have to be carefully interpreted, as the sidewall boundary layer decreases in thickness with even the smallest wave superimposed on the current.

Turbulence intensities and Reynolds stresses near the rough bed are increased by the presence of the waves, most strongly in a layer two roughness heights above bed level, where fluctuations are periodic and effected by vortices ejected from the roughness troughs. Above this level, and over the smooth bed, turbulence levels are similar to those for the currents alone.

Simultaneous measurements of the longitudinal and spanwise components sx and sz of the fluctuating velocity gradient at the wall of a pipe and the fluctuating velocity u and w at various distances from the wall have been analysed in order to provide information on the origin of flow-oriented wall eddies. Special conditional-averaging techniques, which use multiprobe measurements, are developed to capture the wall structures as they evolve in both space and time. It is found that wall eddies of wavelength λ+ ≈ 100, detected from measured patterns of sz(z), are closely related to spanwise variations of the velocity at y+ = 20 and 40. Analysis of measurements from probes arranged in the direction of the mean flow supports the notion that the wall structures result from the propagation of spanwise disturbances of the velocity at some distance from the wall, both downstream and towards the wall. A flow model, different from vortex models hypothesized by previous investigators, is proposed to account for the relation of flow phenomena in the viscous wall region to the low-speed wall streaks.

Free-surface and internal stationary waves in a meandering stream are treated, and analytical solutions given. It is shown that for each category there is an infinite number of Froude numbers, depending on the wavenumber of the meander, at which resonance occurs, and the amplitude of one of the wave components becomes infinite, according to the linear theory. These critical Froude numbers are interpreted physically. Furthermore, variable depth is treated for the case of free-surface waves, and in this treatment it is shown, incidentally, how the eigenvalues of a singular differential equation can be found under the requirement that the eigenfunction be non-singular.

Finally, an attempt is made to explain the self-induced, non-stationary waves in water flowing between corrugated vertical walls, found by Binnie (1960), by an instability mechanism proposed by Yih (1976). There is strong evidence that this mechanism is at work, at least when a sloshing mode is involved in the wave-triad interaction.

Liquids held by surface tension forces can bridge the gap between two solid bodies placed not too far apart from each other. The equilibrium conditions and stability criteria for static, cylindrical liquid bridges are well known. However, the behaviour of an unstable liquid bridge, regarding both its transition toward breaking and the resulting configuration, is a matter for discussion. The dynamical problem of axisymmetric rupture of a long liquid bridge anchored at two equal coaxial disks is treated in this paper through the adoption of one-dimensional theories which are widely used in capillary jet problems.

It is shown that the advection–diffusion equation for contaminant dispersion in water of variable depth admits of an inverse approach in which the stochastic mean concentration at a fixed sensitive location is calculated for arbitrary discharge sites. A ray method is used to obtain simple approximate solutions for the inverse problem when the sensitive location is situated at the shoreline. This makes explicit the considerable improvements in shoreline pollution levels which can be gained by siting effluent outfalls further away from the shore.

A nonlinear non-stationary problem of the development of two-dimensional convective motions in a plane horizontal fluid layer bounded by free surfaces and heated from below is studied. Horizontal dependences of the velocity and temperature are not assumed to be periodic or almost periodic; they have continuous wavenumber spectra and are represented by Fourier integrals. Vertical dependence of each variable is represented by several Fourier harmonics. Spectrum evolution is studied by means of the numerical integration of an initial-value problem. Initial disturbances of two qualitatively different classes are considered; viz those localized in horizontal extent within a narrow part of the layer as well as having the form of a roll set throughout a rather wide region. In both cases convective flows tend to evolve towards the arrays of well-established rolls with the same horizontal wavenumber ap, which apparently seems to be the physically optimal (preferred) one for two-dimensional convection at given Rayleigh number R and Prandtl number P. We see no indications that the deviation of the initial roll-disturbance wavenumber from ap must exceed some threshold value for giving rise to the flow readjustment in wavenumber. At sufficiently small P a decrease in ap with increasing R is observed which agrees with experiments. A comparison is made of various theoretical models and various experimental circumstances, whence it is seen that the less stable the flow (i.e. the easier it can readjust), the better the preferred wavenumber manifests itself. In particular, roll flows periodic in a horizontal direction all over the infinite layer are highly stable, and when only such flows are considered, as has most often been the case, the preferred wavenumber is not revealed.

We focus attention on an idealized granular material comprised of identical, smooth, imperfectly elastic, spherical particles which is flowing at such a density and is being deformed at such a rate that particles interact only through binary collisions with their neighbours. Using general forms of the probability distribution functions for the velocity of a single particle and for the likelihood of binary collisions, we derive local expressions for the balance of mass, linear momentum and fluctuation kinetic energy, and integral expressions for the stress, energy flux and energy dissipation that appear in them. We next introduce simple, physically plausible, forms for the probability densities which contain as parameters the mean density, the mean velocity and the mean specific kinetic energy of the velocity fluctuations. This allows us to carry out the integrations for the stress, energy flux and energy dissipation and to express these in terms of the mean fields. Finally, we determine the behaviour of these fields as solutions to the balance laws. As an illustration of this we consider the shear flow maintained between two parallel horizontal plates in relative motion.

The flow in a partially filled, strongly rotating cylinder with a differentially rotating endcap was studied both experimentally and numerically. The cylindrical container was mounted with a vertical axis of rotation, partially filled with an incompressible fluid, and rotated at a sufficiently high angular velocity that the fluid formed a film of essentially uniform thickness on the sidewall of the container. An axial circulation in this fluid film was induced by the differential rotation of one of the container endcaps. A laser-Doppler velocimeter was used to measure the axial and azimuthal velocity components. The experimental results were compared with a finite-difference model of the flow, and the agreement between the two was good. Boundary layers of thickness proportional to E1/3, where E = ν/ΩL2 is the Ekman number, are found both at the lateral wall and at the vertical free surface. The existence of an E1/3 boundary layer along the free surface is due to the invariant structure of the E½ Ekman layers on the horizontal surfaces with respect to a free surface. The radial transport in the Ekman layers of a partially filled rotating cylinder is essentially the same as that in a completely filled container. The axial transport, which in a completely filled container would have occurred in the volume now occupied by an empty core, is instead confined to a thin boundary layer along the free surface.

Exact solutions of the equations of motion for an inviscid fluid are rare. Using the formalism of John (1953), this paper presents a class of exact zero-gravity flows in which the free surface assumes the form of an ellipse having arbitrary but time-constant aspect ratio. The dynamically important region beneath the overturning crest of a breaking gravity wave is examined and the profile is found to be remarkably well approximated by a √3 aspect-ratio ellipse. The range of examples presented includes high-resolution computations in both deep and shallow water, and also the plunger-generated laboratory waves of Miller (1976).

The ellipse solution is shown to model qualitatively certain essential features of the numerical waves. A recent self-similar solution due to Longuet-Higgins (1981, 1982), in which the free surface is a parametric cubic curve, is also discussed.

A function ϕ is derived which is constant along the orthogonal trajectories of streamlines in two-dimensional flow. In irrotational flows, ϕ reduces to the velocity potential. The pair of functions ϕ and ψ, where ψ is the stream function, are used to define a coordinate system in rotational fluid flows. Tensor methods are used to transform the equations of motion of a turbulent fluid and the equations for second moments of turbulent fluctuations to this coordinate system. Explicit extra terms appear in the transformed equations embodying the effects of streamline curvature and mean flow acceleration. These extra terms are characterized by two lengthscales which arise naturally from the transformation: the local radius of curvature of the streamline and the ‘e-folding’ distance of the mean streamwise velocity.

An analytical solution of a system of linearized equations for a gas–particle mixture is obtained for steady periodic motions. A finite volume fraction of particles and a continuous distribution of particle sizes are taken into consideration. It is shown that the effect of a continuous distribution of particle radii on the acoustic motions of a dusty gas is incorporated through only four integral quantities containing the relaxation times of the particle velocity and temperature, integrated over all particle sizes. This solution is applied to the problem of acoustic reflection, absorption, and transmission by a screen of dusty gas.

The occurrence of new unstable modes of quasigeostrophic baroclinic oscillation of rotating stratified shear flow over a wavy bottom is examined. To obtain a tractable mathematical problem, the bottom topography is considered as a perturbation modifying the oscillations. It is found that combinations of a top-intensified and a bottom-intensified Eady mode, each stable without topography, can be destabilized by topography if certain resonant conditions are met. These are that (i) the two modes possess the same wavefrequency, and (ii) topography possesses a wavenumber c bridging the gap between the wavenumbers of the modes a, b, i.e. c = a – b. Growth rate of this instability (called type A) is proportional to the amplitude of the topographic component. There are two special cases: (i) when one of the basic modes is a marginally neutral mode – according to the classical analysis without topography – the instability is stronger (type M) with growth rate proportional to the 2/3-power of topographic amplitude; (ii) when both modes are marginally neutral the instability is even stronger (type M2) with growth rate proportional to the square root of topographic amplitude.

These topographic instabilities, like classical baroclinic instability, draw their energy from the available potential by transporting buoyancy down the mean gradient associated with the geostrophic shear flow.

A general expression is obtained for the longitudinal dispersion coefficient for a passive contaminant in a varying channel. This expression reveals the upstream memory character of the dispersion coefficient. Simple examples are used to illustrate the effects of: a sudden change in breadth, centrifugally driven secondary flows, and changes in the depth profile.

Experiments are performed on steady and impulsively started flow in an approximately two-dimensional closed channel, with one wall locally indented. In plan the indentation is a long trapezium which halves the channel width: the inclination of the sloping walls is approximately 5.7°, and these tapered sections merge smoothly into the narrowest section via rounded corners. The Reynolds number $ Re = a_0\overline{u}_0/\nu $ (a0 = unindented channel width, $\overline{u}_0$ = steady mean velocity in the unindented channel) lies in the range 300 [les ] Re [les ] 1800. In steady flow, flow visualization reveals that separation occurs on the lee slope of the indentation, at a distance downstream of the convex corner which decreases (tending to a non-zero value) as Re increases. There is no upstream separation, and there is some evidence of three-dimensionality of the flow in the downstream separated eddy. Pressure measurements agree qualitatively but not quantitatively with theoretical predictions. Unsteady flow visualization reveals that, as in external flow, wall-shear reversal occurs over much of the lee slope (at dimensionless time $\tau = \overline{u}_0t/a_0 \approx 4$) before there is any evidence of severe boundary-layer thickening and breakaway. Then, at τ ≈ 5.5, a separated eddy develops, and its nose moves gradually upstream from the downstream end of the indentation to its eventual (τ ≈ 75) steady-state position on the lee slope. At about the same time as the wall-shear reversal, wavy vortices appear at the edge of the boundary layer on both walls of the channel, and (for Re < 750) subsequently disappear again; these are interpreted as manifestations of inflection-point instability and not as intrinsic aspects of boundary-layer separation. Pressure measurements are made to investigate the discrepancy between the actual pressure drop across the lee slope and that predicted on the assumption that energy dissipation is quasi-steady. This discrepancy has a maximum value of approximately $1.5\rho \overline{u}^2_0$ (ρ = fluid density), and decays to zero by the time τ ≈ 7.

The steady motion of a Newtonian fluid in a rectangular enclosure open on its upper side is considered under the action of thermocapillary forces due to surface-tension gradients along the free surface. An asymptotic solution, in the limiting case of the aspect ratio A → 0, is found and discussed for the cases where the surface deformation may be neglected, that is for contact angles at the lateral walls equal to ½π and very small values of the crispation number. The flow field has also been analysed in a wide range of the governing parameters A, Mg, Cr, by a computational model particularly appropriate to simulate free-surface flows. For A [Lt ] 1 the numerical results confirm the behaviour predicted by the asymptotic theory, while for A [ges ] 1 several characteristic features of the flow-field structure are emphasized. For increasing Mg, the surface layer under the free surface maintains in the mid-section a constant value, dependent only on A, and decreases together with the thermal boundary-layer thickness near the lateral walls. For increasing A, the motion remains confined in a region near the free surface; hence the overall Nu, starting from the pure conduction value (Nu → 1 as A → 0) inccreases with A, reaching a maximum, to tend again to unity as A → ∞. The surface deformation, at least for very small values of the crispation number, seems to have a negligible influence on the qualitative aspects of the flow-field structure.