The effect of surface tension on the onset of convection in horizontal double-diffusive layer was studied both experimentally and by linear stability analysis. The experiments were conducted in a rectangular tank with base dimension of 25×13 cm and 5 cm in height. A stable solute (NaCl) stratification was first established in the tank, and then a vertical temperature gradient was imposed. Vertical temperature and concentration profiles were measured using a thermocouple and a conductivity probe and the flow patterns were visualized by a schlieren system. Two types of experiments were carried out which illustrate the effect of surface tension on the onset of convection. In the rigid–rigid experiments, when the critical thermal Rayleigh number, RT, is reached, large double-diffusive plumes were seen simultaneously to rise from the heated bottom and descend from the cooled top. In the rigid–free experiments, owing to surface tension effects, the first instability onset was of the Marangoni type. Well-organized small plumes were seen to emerge and persist close to the top free surface at a relatively small RTM (where subscript M denotes ‘Marangoni’). At larger RTt > RTM (where subscript t denotes ‘top’) these plumes evolved into larger double-diffusive plumes. The onset of double-diffusive instability at the bottom region occurred at a still higher RTb > RTt (where subscript b denotes ‘bottom‘). A series of stability experiments was conducted for a layer with an initial top concentration of 2 wt% and different concentration gradients. The stability map shows that in the rigid–free case the early Marangoni instability in the top region reduces significantly the critical RT for the onset of double-diffusive convection. Compared with the rigid–rigid case, the critical RT in the top region is reduced by about 60% and in the bottom region by about 30%. The results of the linear stability analysis, which takes into account both surface tension and double-diffusive effects, are in general agreement with the experiments. The analysis is then applied to study the stability characteristics of such a layer as gravity is reduced to microgravity level. Results show that even at 10 −4g0, where g0 is the gravity at sea level, the double-diffusive effect is of equal importance to the Marangoni effect.

Interfacial stability of two-layer Couette flow was investigated experimentally in a channel bent into an annular ring. This paper is focused on the supercritical long-wave instability which arises for a broad range of flow parameters. Above the critical upper plate velocity, a slowly growing long wave appears with wavelength equal to the perimeter of the channel. Transients of this wave were studied within the theoretical frame of amplitude equations obtained from the long-wave interface equation. Near the onset of instability, the unstable fundamental harmonic is described by the Landau–Stuart equation, and the nonlinear dynamics of the harmonics closely follows the central and slaved modes analysis. For the higher upper plate velocity, harmonics gain some autonomy but they eventually are enslaved by the fundamental, through remarkable collapses of amplitudes and phase jumps leading to wave velocity and frequency locking. Dispersive effects play a crucial role in the nonlinear dynamics. Far from the threshold, the second harmonic becomes unstable and bistability appears: the saturated wave is dominated either by the fundamental harmonic, or by the even harmonics, after periodic energy exchange.

Using a two-fluid model of gas-fluidized beds, it is shown that periodic plane voidage waves travelling against gravity are unstable to perturbations with large transverse wavelength. This secondary instability sets in at arbitrarily small amplitudes of the plane wave and correspondingly small transverse wavenumbers of the two-dimensional perturbation. More precisely, if the bed is wide enough to accommodate sufficiently long horizontal waves, then the plane wave becomes unstable as soon as its amplitude has grown to the order of the square of the transverse wavenumber. The instability can be stationary or oscillatory in nature and has its origin in the interaction between the plane wave and four least-stable modes with small transverse wavenumber. Two of them represent a pair of bubble-like ‘mixed modes‘; the other two are initially, i. e. at the onset of the primary wave, pure transverse modes, one consisting only of an initially pure vertical velocity perturbation of the state of uniform fluidization. Depending on a relation between the eigenvalues of the least-stable modes at the primary bifurcation point, either one of these can be the dominant mode, which becomes (most) unstable along the growing vertically travelling plane wave. While the transverse modes gain longitudinal structure during this process, the mixed modes obtain a vertical component of the vertically averaged velocity as well, so that it appears that the secondary instability described here is a variant of Batchelor & Nitsche's (1991) ‘overturning’ instability found recently for unbounded stratified fluids, see also Batchelor (1993).

In treating unsteady particle motions in creeping flows, a quasi-steady approximation is often used, which assumes that the particle's motion is so slow that it is composed of a series of steady states. In each of these states, the fluid is in a steady Stokes flow and the total force and torque on the particle are zero. This paper examines the validity of the quasi-steady method. For simple cases of sedimenting spheres, previous work has shown that neglecting the unsteady forces causes a cumulative error in the trajectory of the spheres. Here we will study the unsteady motion of solid bodies in several more complex flows: the rotation of an ellipsoid in a simple shear flow, the sedimentation of two elliptic cylinders and four circular cylinders in a quiescent fluid and the motion of an elliptic cylinder in a Poiseuille flow in a two-dimensional channel. The motion of the fluid is obtained by direct numerical simulation and the motion of the particles is determined by solving their equations of motion with solid inertia taken into account. Solutions with the unsteady inertia of the fluid included or neglected are compared with the quasi-steady solutions. For some flows, the effects of the solid inertia and the unsteady inertia of the fluid are importanty quantitatively but not qualitatively. In other cases, the character of the particles' motion is changed. In particular, the unsteady effects tend to suppress the periodic oscillations generated by the quasi-steady approximation. Thus, the results of quasi-steady calculatioins are never uniformly valid and can be completely misleading. The conditions under which the unsteady effects at small Reynolds numbers are important are explored and the implications for modelling of suspension flows are addressed.

We analyse the complete solidification from a side boundary of a finite volume of a binary alloy. Particular emphasis is placed upon the compositional stratification produced in the solid, the structure of which is determined by the competition between the rates of solidification and of laminar box filling by the fractionated fluid released at the solid/liquid interface. It is demonstrated by scaling arguments that numerical calculations performed at relatively low values of the Rayleigh and Lewis numbers may be used to describe equally well laboratory experiments previously performed at moderate Rayleigh and Lewis numbers and the high-Rayleigh-number, high-lewis-number convective regime expected during the solidification of a large magmatic body, provided that the balance between solidification and laminar box filling is maintained. This balance can be represented by a single dimensionless group of parameters. The boundary-layer analysis is extended to fluids whose viscosity is strongly dependent upon temperature and composition, and an effective viscosity is derived which may be used to describe both the magnitude and pattern of compositional stratification in the solid.

A full numerical simulation based on spectral methods is used to investigate linearly accelerating and decelerating flows past a rigid sphere. Although flow separation does not occur at Reynolds numbers below 20 for a steady flow, in the linearly decelerating flow separation is observed at much lower Reynolds numbers with complete detachment of vorticity possible in certain cases. The existence of a large recirculation region contributes to the result that a negative viscous force on the sphere is possible. The contribution of the pressure to the force includes a component that is well described by the inviscid added-mass term in both the accelerating and decelerating cases. The force on the sphere is found in general to initially decay in a power law manner after acceleration or deceleration ends followed by rapid convergence at later times to the steady state. For the cases examined this convergence is found to be exponential except for those in which the sphere is brought to rest in which case the convergence remains algebraic. This includes the special case of an infinite acceleration or deceleration where the free stream velocity is impulsively changed.

Uniformly sheared flows have been generated in a high-speed wind tunnel at shear rates higher than previously achieved, in an effort to approach those in the inner turbulent boundary layer. As at lower shear rates, the turbulence structure was found to attain a self-similar state with approximately constant anisotropies and exponential kinetic energy growth. The normal Reynolds stress anisotropies showed no systematic dependence upon the mean shear within the examined range; however, the shear stress anisotropy was significantly lower than the low-shear values, in conformity with boundary layer measurements and direct numerical simulations of homogeneous shear flow.

A popular method used to incorporate thermodynamic processes in a shallow water model (e.g. one used to study the upper layer of the ocean) is to allow for density variations in time and horizontal position, but keep all dynamical fields as depth independent. This is achieved by replacing the horizontal pressure gradient by its vertical average. These models have limitations, for instance they cannot represent the ‘thermal wind’ balance (between the horizontal density gradient and the vertical shear of the velocity) which dominates at low frequencies. A new model is now proposed which uses velocity and density fields varying linearly with depth, with coefficients that are functions of horizontal position and time. This model can explicitly represent the thermal wind balance, but its use is not restricted to low- frequency dynamics.

Volume, mass, buoyancy variance, energy and momentum are conserved in the new model. Furthermore, these integrals of motion have the same dependence on the dynamical fields as the exact (continuously stratified) case. The evolution of the three components of the absolute vorticity field are correctly represented. Conservation of density–potential vorticity is not fulfilled, though, owing to artificial removal of the vertical curvature of the velocity field.

The integrals of motion are used to construct a ‘free energy’ [Escr ]f, which is quadratic to the lowest order in the deviation from a steady state with (at most) a uniform velocity field. [Escr ]f is positive definite, and therefore the free evolution of the system cannot lead to an ‘explosion’ of the dynamical fields. (This is not the case if the velocity shear and/or the density vertical gradient is excluded in the model, which results in a non-negative definite free energy.)

In a model with one active layer, linear waves on top of a steady state with no currents are, to a very good approximation, those of the first two vertical modes of the continuously stratified model. These are the familiar geophysical gravity and vortical waves (e.g. Poincaré, Rossby, and coastal Kelvin waves at mid-latitudes, equatorial waves, etc.).

Finally, baroclinic instability is well represented in the new model. For long perturbations (wavelengths of the order of the deformation radius of the first mode) the agreement with more precise calculations is excellent. On the other hand, the comparison with the eigenvalues of Eady's problem (which corresponds to wavelengths of the order of the deformation radius of the second mode) shows differences of the order of 40%. Nevertheless, the new model does have a high-wavenumber cutoff, even though it is constrained to linear profiles in depth and therefore cannot reproduce the exponential trapping of Eady's problem eigensolutions.

In sum, the integrals of motion, vorticity dynamics, free waves and baroclinic instability results all give confidence in the new model. Its main novelty, however, lies in the ability to incorporate thermodynamic processes.

The stability or instability of various linear shear flows in shallow water is considered. The linearized equations for waves on the surface of each flow are solved exactly in terms of known special functions. For unbounded shear flows, the exact reflection and transmission coefficients R and T for waves incident on the flow, are found. They are shown to satisfy the relation |R|2= 1+ |T|2, which proves that over reflection occurs at all wavenumbers. For flow bounded by a rigid wall, R is found. The poles of R yield the eigenvalue equation from which the unstable mides can be found. For flow in a channel, with two rigid walls, the eigenvalue equation for the modes is obtained. The results are compared with previous numerical results.

The transient flow field caused by an infinitely long circular cylinder placed in an unbounded viscous fluid oscillating in a direction normal to the cylinder axis, which is at rest, is considered. The flow is assumed to be started suddenly from rest and to remain symmetrical about the direction of motion. The method of solution is based on an accurate procedure for integrating the unsteady Navier–Stokes equations numerically. The numerical method has been carried out for large values of time for both moderate and high Reynolds numbers. The effects of the Reynolds number and of the Strouhal number on the laminar symmetric wake evolution are studied and compared with previous numerical and experimental results. The time variation of the drag coefficients is also presented and compared with an inviscid flow solution for the same problem. The comparison between viscous and inviscid flow results shows a better agreement for higher values of Reynolds and a Strouhal numbers. The mean flow for large times is calculated and is found to be in good agreement with previous predictions based on boundary-layer theory.

Results are Presented from an experimental study of fluid in a rotating cylinder which was subjected to precessional forcing. The primary objective was to determine the validity of the linear and inviscid approximations which are commonly adopted in numerical models of the problem. A miniature laser Doppler velocimeter was used to make quantitative measurements of the flow dynamics under a variety of forcing conditions. These ranged from impulsive forcing to continuous forcing at the fundamental resonance of the system. Inertial waves were excited in the fluid in each case, with the extent of nonlinear behaviour increasing from one forcing regime to the next. Good agreement was found with the predictions of linear theory in the weaker forcing regimes. For stronger forcing, it was possible to determine the approximate duration of linear behaviour before the onset of nonlinear dynamics. Viscous effects were found to be relatively weak when the frequency of precessional forcing was away from resonance. However, there was evidence of strong boundary-layer phenomena when conditions of resonance were approached.

The thermal bar, a descending plane plume of fluid at the temperature of maximum density (3.98° C in water), is analysed as a laminar free-convection boundary layer, following the example of Kuiken & Rotem (1971) for the plume above a line source of heat. Numerical integration of the similarity form of the boundary-layer equations yields values of the vertical velocity and temperature gradient on the centre line and the horizontal velocity induced outside the thermal bar as functions of Prandtl number σ. The asymptotic behaviour of these parameters for both large and small σ is also obtained; in these cases, the thermal bar has a two-layer structure, and the method of matched asymptotic expansions is used. For the intermediate case σ= 1, an analytical calculation using approximate velocity and temperature profiles in the integrated boundary-layer equations yields good agreement with the numerical results. The applicability of the results to naturally occurring thermal bars (e. g. in lakes) is limited, but the laminar-flow analysis is likely to relate more closely to the phenomenon on a laboratory scale.

The thermal bar phenomenon is modelled numerically by the natural convection of a fluid contained in a two-dimensional triangular domain. The non-rotating case considered here is appropriate to laboratory models of the thermal bar. Three sets of results are presented reflecting varying degrees of nonlinearity. The results are discussed in relation to available theoretical and experimental results.

The work is concerned with the problem of the linear instability of symmetric short-crested water waves, the simplest three-dimensional wave pattern. Two complementary basic approaches were used. The first, previously developed by Ioualalen & Kharif (1993, 1994), is based on the application of the Galerkin method to the set of Euler equations linearized around essentially nonlinear basic states calculated using the Stokes-like series for the short-crested waves with great precision. An alternative analytical approach starts with the so-called Zakharov equation, i.e. an integro-differential equation for potential water waves derived by means of an asymptotic procedure in powers of wave steepness. Both approaches lead to the analysis of an eigenvalue problem of the type {\rm det}|{\boldmath A}-\gamma{\boldmath B}|=0 where A and B are infinite square matrices. The first approach should deal with matrices of quite general form although the problem is tractable numerically. The use of the proper canonical variables in our second approach turns the matrix B into the unit one, while the matrix A gets a very specific ‘nearly diagonal’ structure with some additional (Hamiltonian) properties of symmetry. This enables us to formulate simple necessary and sufficient a priori criteria of instability and to find instability characteristics analytically through an asymptotic procedure avoiding a number of additional assumptions that other authors were forced to accept.

A comparison of the two approaches is carried out. Surprisingly, the analytical results were found to hold their validity for rather steep waves (up to steepness 0.4) for a wide range of wave patterns. We have generalized the classical Phillips concept of weakly nonlinear wave instabilities by describing the interaction between the elementary classes of instabilities and have provided an understanding of when this interaction is essential. The mechanisms of the relatively high stability of short-crested waves are revealed and explained in terms of the interaction between different classes of instabilities. A helpful interpretation of the problem in terms of an infinite chain of interacting linear oscillators was developed.

As is well known, most gas-fluidized beds of solid particles bubble; that is, they are traversed by rising regions containing few particles. Most liquid-fluidized beds, on the other hand, do not. The aim of the present paper is to investigate whether this distinction can be accounted for by certain equations of motion which have commonly been used to describe both types of bed. For the particular case of a bed of 200 μm diameter glass beads fluidized by air at ambient conditions it is demonstrated, by direct numerical integration, that small perturbations of the uniform bed grow into structures resembling the bubbles observed in practice. When analogous computations are performed for a water-fluidized bed of 1 mm diameter glass beads, using the same equations, with parameters modified only to account for the greater density and viscosity of water and to secure the same bed expansion at minimum fluidization, it is found that bubble-like structures cannot be grown. The reasons for this difference in behaviour are discussed.

Asymptotic theory of high-aspect-ratio wings in steady incompressible flow is extended to a case where the wing forms either an open or closed circular arc. The generalization is based on an integral formulation of the problem, which resembles the one used by Guermond (1990) for a plane curved wing. A second-order approximation is obtained for the load distribution on two model wings, one resembling that of a gliding parachute, and the other resembling a short duct.