This paper describes how the flows around three equal circular cylinders arranged in an equilateral-triangular manner interact at different angles of incidence α and spacing ratios l/d. Some vortex-shedding-frequency data evaluated from flow visualization experiments conducted at Reynolds numbers of 2.1 × 103 and 3.5 × 103, based on the diameter of a single cylinder, using a dye -injection technique, are presented. In order to provide additional insight to the understanding of the flow structure around this particular cylinder array, some photographs indicating the typical flow patterns for various arrangements are also presented. The investigation indicates that the flows interact in a complex fashion for spacing ratios smaller than 2.29 and it also reveals that, at this range of spacing ratios and at α = 0°, bistable flow characteristic exists. Moreover, for l/d approximately smaller than 4.65 there always exists an angle at which the vortex shedding behind an upstream cylinder is suppressed by a nearest downstream cylinder. This angle is found not to remain constant but increases as the spacing ratio increases. For illustration and comparisons, some numerical results obtained from the application of the surface-vorticity method have also been presented.

A photographic study of the wakes of slugs rising in tubes of 19 mm and 52 mm internal diameter is presented. The dependence of the flow pattern in the wake upon the Reynolds number of the rising slug, R, is established for different slug lengths. Values of R covered in this study are in the range 25 to 1.3 × 104. For low values of R the flow pattern in the wake is laminar and axisymmetric and values of wake length and wake volume could be determined from the photographs: these values were correlated with the other variables in the system by means of dimensional analysis.

Disturbances in the form of pressure fields, source distributions and time-dependent bottom topographies are discussed and found to produce similar wave patterns. Results obtained for wide channels are discussed in the light of the features of soliton reflection at a wall. Comparison with experiments shows excellent agreement. The introduction of radiation conditions enables long-time simulation of the development of wave patterns in infinite and semi-infinite fluids. A stationary wave pattern is also found to emerge for slightly supercritical Froude numbers, but contrary to linear results the leading divergent waves may originate ahead of the disturbance. This behaviour is due to nonlinear interactions similar to those governing collisions between solitons. This study on wave generation by a moving disturbance is based on numerical solutions of Boussinesq-type equations. The equations in their most general form are integrated by an implicit difference method. Strongly supercritical cases are described by a simplified set of equations which is solved by a semi-implicit difference scheme.

The paper studies the radiation and diffraction by floating bodies of deep-water bichromatic and bidirectional surface waves subject to the second-order free-surface condition. A theory is developed for the evaluation of the second-order velocity potential and wave forces valid for bodies of arbitrary geometry, which does not involve the evaluation of integrals over the free surface or require an increased accuracy in the solution of the linear problem. Explicit sum- and difference-frequency ‘Green functions’ are derived for the radiation and diffraction problems, obtained from the solution of initial-value problems that ensure they satisfy the proper radiation condition at infinity. The second-order velocity potential is expressed as the sum of a particular and a homogeneous component. The former satisfies the non-homogeneous free-surface condition and is expressed explicitly in terms of the second-order Green functions. The latter is subject to the homogeneous free-surface condition and enforces the body boundary condition by the solution of a linear problem. An analysis is carried out of the singular behaviour of the second-order potential near the intersection of the body boundary with the free surface.

The interaction between a wave packet of small-amplitude short internal waves, and a finite-amplitude inertial wave field is described to second order in the short-wave amplitude. The discussion is based on the principle of wave action conservation and the equations for the wave-induced Lagrangian mean flow. It is demonstrated that as the short internal waves propagate through the inertial wave field they generate a wave-induced train of trailing inertial waves. The contribution of this wave-induced mean flow to the total energy balance is described. The results obtained here complement the finding of Broutman & Young (1986) that the short internal waves undergo a net change in energy after their encounter with the inertial wave field.

The interface at the top of a dilute sedimenting suspension of small particles which are not identical does not remain sharp but instead becomes increasingly diffuse as the sedimentation proceeds. For more concentrated suspensions, the self-sharpening effect of hindered settling leads to a considerable reduction in the observed spreading of the sedimenting interface. In order to quantify this spreading, a light extinction technique was used to measure the concentration profile in the interface of a suspension of particles with a small spread of sizes as it fell past a thin sheet of light. A particle volume-fraction range of 0.002 ≤ Φ0 ≤ 0.15 was examined, and each fluid-particle system had a particle Reynolds number less than 10−3 and a Péclet number greater than 107 so that inertia and colloidal effects were negligible. Calculations of the spreading arising from the small degree of polydispersity in particle sizes and the self-sharpening effect are presented. Surprisingly, the measured vertical thickness of the interface was found to be several times that predicted from this theory.

It is proposed that the observed spreading may be attributed to hydrodynamic interactions between particles that lead to fluctuations in particle settling velocities about the mean. An analysis of the data shows that the measured interface thickness, after subtracting off that predicted from polydispersity and self-sharpening, increases approximately with the square root of the settling distance and may therefore be described as a diffusion process, termed ‘self-induced hydrodynamic diffusion’. By sealing the hydrodynamic diffusivity as $D = au_{\frac{1}{2}}\hat{D}(\Phi_0)$, where u½ is the median hindered settling velocity, a is the median particle radius, and Φ0 is the volume fraction of particles well below the interface, an approximate analysis of the data was used to infer that the dimensionless scaled diffusion coefficient, $\hat{D}$, is between 1 and 2 for the smaller particle volume fractions examined, increases very rapidly with increasing concentration to a value between 10 and 15 for particle concentrations of a few percent by volume, and then levels off or declines slightly as the particle concentration is increased further.

The dynamics and structure of double-diffusive layers were studied experimentally by heating a linear stable solute gradient from a sidewall in a wide tank. The formation and subsequent development of the layers were investigated by various flow-visualization techniques, namely fluorescent dye, fluorescent particles and shadowgraph. Experiments were performed in order to determine the stability diagram of the flow, following in each experiment the phase trajectory of the system in the phase plane of thermal and solute Rayleigh numbers. The experimentally obtained stability diagram appears to be similar to that obtained numerically by Thangam et al. (1981) and by Hart (1971) for a vertical narrow slot and a steady basic flow. It is shown that if the temperature of the sidewall rises slowly to its prescribed value, the thickness of the initial layers, formed at the onset of instability, is a function of the ambient density gradient and fluid properties only. On the other hand, if the wall temperature rises quickly (almost impulsive heating), the thickness of the initial layers is proportional to the imposed temperature difference, provided that the Rayleigh number, based on this difference, is larger than some critical value which is associated with the first merging of the layers. A criterion for the first merging of the initial layers is obtained, and it is suggested that this merging is due to subsequent instability of the system. The subsequent merging process, following the first merging, is not a simple successive doubling of the layer thickness and in each of five nearly identical experiments a different dependence of the average layer thickness on the instantaneous Rayleigh number is obtained. This indicates that the system of layers behaves chaotically after the first merging. The final thickness of the layers depends on the prescribed lateral temperature difference, and the ratio between the final and the initial thickness of the layers is a linear function of the final Rayleigh number. Flow-visualization experiments indicate that the layers consist of vortices with vertical scale of the layer thickness and various horizontal scales.

Various types of moments of velocity and scalar fluctuations of the first to the fourth order have been measured and analysed. First, an orthogonal series expansion for the three-dimensional joint probability density function (p.d.f.) is developed using the cumulants and Hermite polynomials. This p.d.f. is found to provide satisfactory predictions for the statistical characteristics, including triple products, of turbulent momentum and scalar transfer. Next, the conditional sampling and averaging technique is employed to investigate the statistical characteristics of coherent turbulent transfer processes of momentum and scalar. Conditional p.d.f.s are developed for various moments of velocity and scalar up to the third order. It is shown that the present p.d.f.s can predict the detailed role of coherent motions in the dynamics of wall turbulent shear flows and in the relevant process of scalar transport by turbulence. In particular, the importance of coherent motions in the turbulent diffusion process of Reynolds-stress components and scalar fluxes is demonstrated for the first time by the present theory.

Over the past decades a large amount of work has been done on inviscid, steady progressive, gravity waves on a free surface. However, waves which occur in nature are never, in fact, free-surface waves, since they are always beneath a fluid of finite density, if only air. In spite of this, very little work has been done on finite-amplitude interfacial waves. Tsuji & Nagata (1973) carried out a perturbation expansion in wave amplitude to fifth order, for interfacial waves between two stationary fluids. Holyer (1979) extended the calculation using the computer and resorted to Pade approximants to sum the resulting series. Meiron & Saffman (1983) investigated numerically the limiting highest wave and demonstrated the existence of over-hanging gravity waves of permanent form.

The dislodging by dynamic pressure forces of a drop adhered by surface tension to a plane is analysed. An integro-differential equation describing the drop shape is solved numerically and the critical Weber number as a function of contact angle hysteresis is found.

The breaking of relativistic simple waves, for one-dimensional flow in the space-time of special relativity, is investigated. The cases of relativistic acoustic and magnetoacoustic waves are treated in detail and the critical time for breaking is evaluated.

In this paper we derive one-space-dimensional, reduced systems of equations (one-dimensional closure models) for viscoelastic free jets. We begin with the three-dimensional system of conservation laws and a Maxwell-Jeffreys constitutive law for an incompressible viscoelastic fluid. First, we exhibit exact truncations to a finite, closed system of one-dimensional equations based on classical velocity assumptions of von Kármán (1921). Next, we demonstrate that the three-dimensional free-surface boundary conditions overconstrain these truncated systems, so that only a very limited class of solutions exist. We then proceed to derive approximate one-dimensional closure theories through a slender-jet asymptotic scaling, combined with appropriate definitions of velocity, pressure and stress unknowns. Our non-axisymmetric one-dimensional slender-jet models incorporate the physical effects of inertia, viscoelasticity (viscosity, relaxation and retardation), gravity, surface tension, and properties of the ambient fluid, and include shear stresses and time dependence. Previous special one-dimensional slender-jet models correspond to the lowest-order equations in the present asymptotic theory by an a posteriori suppression to leading order of some of these effects, and a reduction to axisymmetry. We thereby: (i) derive existing one-dimensional models from the three-dimensional free surface boundary-value problem; (ii) clarify the sense of the one-dimensional approximation; (iii) deduce new one-dimensional closure models for non-axisymmetric viscoelastic free jets.

We resolve the disturbance structures that destabilize steady convection rolls in favour of a time-periodic pattern in two-dimensional containers of fluid-saturated porous material. Analysis of these structures shows that instability occurs as a travelling wave propagating in a closed loop outside the nearly motionless core. The travelling wave consists of five pairs of thermal cells and four pairs of vorticity disturbances in the case of a square container. The wave speed of the thermal disturbances is determined by an average base-state velocity and their structure by a balance between convection and thermal diffusion. Interpretation of the ‘exact’ solution is aided by a one-dimensional convection-loop model which correlates (i) point of transition, (ii) disturbance wavenumber, and (iii) oscillation frequency given the base-state temperature and velocity profiles. The resulting modified Mathieu-Hill equation clarifies the role of the vertical pressure gradient, induced by the impenetrable walls, and the role of the base-state thermal layer.

A novel method is presented for obtaining rigorous upper bounds on the finite-amplitude growth of instabilities to parallel shear flows on the beta-plane. The method relies on the existence of finite-amplitude Liapunov (normed) stability theorems, due to Arnol'd, which are nonlinear generalizations of the classical stability theorems of Rayleigh and Fjørtoft. Briefly, the idea is to use the finite-amplitude stability theorems to constrain the evolution of unstable flows in terms of their proximity to a stable flow. Two classes of general bounds are derived, and various examples are considered. It is also shown that, for a certain kind of forced-dissipative problem with dissipation proportional to vorticity, the finite-amplitude stability theorems (which were originally derived for inviscid, unforced flow) remain valid (though they are no longer strictly Liapunov); the saturation bounds therefore continue to hold under these conditions.

Motivated by considerations of the solar toroidal magnetic field we have studied the behaviour of a layer of uniform magnetic field embedded in a convectively stable atmosphere. Since the field can support extra mass, such a configuration is top-heavy and thus instabilities of the Rayleigh-Taylor type can occur. For both static and rotating basic states we have followed the evolution of the interchange modes (no bending of the field lines) by integrating numerically the nonlinear compressible MHD equations. The initial Rayleigh-Taylor instability of the magnetic field gives instabilities to strong shearing motions, thereby exciting secondary Kelvin-Helmholtz instabilities which wrap the gas into regions of intense vorticity. The subsequent motions are determined primarily by the strong interactions between vortices which are responsible for the rapid disruption of the magnetic layer.

This paper presents a general method for deriving approximate dynamical equations that satisfy a prescribed constraint typically chosen to filter out unwanted high frequency motions. The approximate equations take a simple general form in arbitrary phase-space coordinates. The family of semigeostrophic equations for rotating flow derived by Salmon (1983, 1985) fits this general form when the chosen constraint is geostrophic balance. More precisely, the semigeostrophic equations are equivalent to a Dirac-bracket projection of the exact Hamiltonian ønto the phase-space manifold corresponding to geostrophically balanced states. The more widely used quasi-geostrophic equations do not fit the general form and are instead equivalent to a metric projection of the exact dynamics on to the same geostrophic manifold. The metric, which corresponds to the Hamiltonian of the linearized dynamics, is an artificial component of the theory, and its presence explains why the quasi-geostrophic equations are valid only near a state isopycnals.

In this paper, we are concerned with the linear stability of zero pressure-gradient laminar boundary-layer flow over compliant walls which are composed of one or more layers of isotropic viscoelastic materials and backed by a rigid base. Wall compliance supports a whole host of new instabilities in addition to the Tollmien-Schlichting mode of instability, which originally exists even when the wall is rigid. The perturbations in the flow and the compliant wall are coupled at their common interface through the kinematic condition of velocity continuity and the dynamical condition of stress continuity. The disturbance modes in the flow are governed by the Orr-Sommerfeld equation using the locally-parallel flow assumption, and the response of the compliant layers is described using a displacement-stress formalism. The theoretical treatment provides a unified formulation of the stability eigenvalue problem that is applicable to compliant walls having any finite number of uniform layers; inclusive of viscous sublayer. The formulation is well suited to systematic numerical implementation. Results for single- and multi-layer walls are presented. Analyses of the eigenfunctions give an insight into some of the physics involved. Multi-layering gives a measure of control over the stability characteristics of compliant walls not available to single-layer walls. The present study provides evidence which suggests that substantial suppression of disturbance growth may be possible for suitably tailored compliant walls.

Viscoelastic theory is used to describe the response of a floating ice sheet to a moving vehicle. We adopt a two-parameter memory function to describe the behaviour of the ice, subjected to a steadily moving line or point load. The viscoelastic dissipation produces an asymmetric quasi-static response at subcritical speed, renders a finite response at the critical speed, and damps the shorter leading waves rather more severely than the longer trailing waves at supercritical speed. We extend earlier asymptotic theory to consider the anisotropic damping of the flexural waves. There is enhanced agreement between theory and experiment.

The wakes of a pair of circular cylinders are grossly unsteady when the cylinders are separated in a direction normal to the approaching flow by less than one cylinder diameter. The wakes flop randomly between two asymmetric states. The time-scale for the flopping is several orders of magnitude longer than the timescale of vortex shedding, and also several orders of magnitude longer than the timescale for instability of the separating shear layers. When a splitter plate is positioned suitably on the centreline of the cylinders, the flopping can be stopped and the flow made to assume either of the asymmetric states, or a symmetric steady state. For a range of plate positions a new, periodic oscillation occurs. Acoustic excitation can also destroy the flopping mean flow, replacing it by a symmetric flow.

A similarity transform is used to analyse the flow of an upper-convected Maxwell fluid in an infinitely long cylinder whose surface has a velocity that increases in magnitude linearly with axial coordinate. Two types of problem are considered, the accelerated surface flow - when the surface velocity is outward towards the tube ends, and the decelerated surface flow - when it is inward. For the accelerated surface flow, the introduction of elasticity prevents the loss of similarity solution that occurs without elasticity at a Reynolds number (Re) of 10.25: with elasticity, solutions up to a Reynolds number of 95 were computed. As elasticity is introduced, normal stress gradients in an elastic boundary layer near the accelerated surface help offset inertially generated negative axial pressure gradients; with sufficient elasticity the turning point in the non-elastic solution family at Re = 10.25 disappears. For the decelerated surface flow, solutions could not be computed beyond a critical Re that depends on the level of elasticity considered, because at this critical Re, the axial velocity profile at the centreline becomes infinitely blunt.