The motion of a disk rising steadily along the axis in a rotating fluid between two infinite plates is considered. In the limit of zero Rossby number and with the disk in the middle position, the boundary value problem based on the linear, viscous equations of motion is reduced to a system of dual-integral equations which renders ‘exact’ solutions for arbitrary values of the Taylor number, Ta, and disk-to-wall distance, H (scaled by the radius of the disk). The investigation is focused on the drag and on the flow field when Ta is large (but finite) for various H. Comparisons with previous asymptotic results for ‘short’ and ‘long’ containers, and with the preceding unbounded-configuration ‘exact’ solution, provide both confirmation and novel insights.

In particular, it is shown that the ‘free’ Taylor column on the particle appears for H > 0.08 Ta and attains its fully developed features when H > 0.25 Ta (approximately). The present drag calculations improve the compatibility of the linear theory with Maxworthy's (1968) experiments in short containers, but for the long container the claimed discrepancy with experiments remains unexplained.

For thermosolutal convection in an enclosure of arbitrary vertical aspect ratio, mixed boundary conditions – with salt flux and temperature prescribed on a horizontal boundary – can lead to symmetry breaking via pitchfork bifurcation. In the present paper we consider an enclosure of very small height-to-length aspect ratio δ, as encountered in the world's oceans. In this case, if the ratios of the vertical to horizontal components of viscosity, [Pscr ], and of diffusivity, [Lscr ], are of order unity, advective transport cannot set in even at very high Rayleigh numbers. The ratios [Pscr ] and [Lscr ] must be substantially less than unity in order for convection to dominate the heat and solute transport.

We have investigated numerically the effects of monotonic and non-monotonic temperature and salinity boundary conditions in a two-dimensional domain at constant δ = 0.01 and constant [Pscr ] = [Lscr ] = 0.01. This ratio of eddy-mixing coefficients reflects the different scales of motion – vertical and horizontal – in the ocean, rather than a physically realizable laboratory fluid. It is found that when the salt-flux strength, γ, is sufficiently large, the system undergoes a second bifurcation for both types of boundary conditions. It is a Hopf bifurcation, leading from the asymmetric steady states produced by the first one to oscillatory solutions. These periodic solutions are stable and very robust. An approximate Hopf bifurcation diagram has been produced. We conclude that non-monotonic salt-flux conditions are neither necessary nor sufficient to induce the oscillations, while the strength of the salt flux is crucial.

We consider the nonlinear spatial evolution in the streamwise direction of slightly three-dimensional disturbances in the form of oblique travelling waves (with spanwise wavenumber kz much less than the streamwise one kx) in a mixing layer vx = u(y) at large Reynolds numbers. A study is made of the transition (with the growth of amplitude) to the regime of a nonlinear critical layer (CL) from regimes of a viscous CL and an unsteady CL, which we have investigated earlier (Churilov & Shukhman 1994). We have found a new type of transition to the nonlinear CL regime that has no analogy in the two-dimensional case, namely the transition from a stage of ‘explosive’ development. A nonlinear evolution equation is obtained which describes the development of disturbances in a regime of a quasi-steady nonlinear CL. We show that unlike the two-dimensional case there are two stages of disturbance growth after transition. In the first stage (immediately after transition) the amplitude A increases as x. Later, at the second stage, the ‘classical’ law A ∼ x2/3 is reached, which is usual for two-dimensional disturbances. It is demonstrated that with the growth of kz the region of three-dimensional behaviour is expanded, in particular the amplitude threshold of transition to the nonlinear CL regime from a stage of ‘explosive’ development rises and therefore in the ‘strongly three-dimensional’ limit kz = O(kx) such a transition cannot be realized in the framework of weakly nonlinear theory.

The amplitude equations governing the nonlinear interaction among normal modes are derived for a multilayer quasi-geostrophic channel. The set of normal modes can represent any wavy disturbance to a parallel shear flow, which may be stable or unstable. Orthogonality in the sense of pseudomomentum or pseudoenergy is used to obtain the amplitude equations in a direct fashion, and pseudoenergy and pseudomomentum conservation laws permit the properties of the interaction coefficients to be deduced. Particular attention is paid to triads exhibiting explosive resonant interaction, as they lead to nonlinear instability of the basic flow. The relationship between this mechanism and the most recently discovered nonlinear stability conditions is discussed.

Situations in which the basic velocity is constant in each layer are treated in detail. A particular formulation of the stability condition is given that emphasizes the close connection between linear and nonlinear stability. It is established that this stability condition is also a necessary condition: when it is not satisfied, and when the flow is linearly stable, explosive resonant interaction of baroclinic Rossby waves acts as a destabilizing mechanism. Two- and three-layer models are specifically considered; their stability features are presented in the form of stability diagrams, and interaction coefficients are calculated in particular cases.

The behaviour of baroclinic waves growing from instability in a two-layer channel flow with rigid (no-slip) sidewalls is described and contrasted with that for the more traditional free-slip boundary conditions. The linear theory for the onset of small-amplitude disturbances shows that the change in lateral boundary conditions has only a modest effect for typical laboratory parameter values, although the no-slip case is slightly more unstable at very small values of bottom friction. On the other hand, the nonlinear evolution of no-slip modes is completely different. While the free-slip case becomes aperiodic only at large values of the supercriticality (F–Fc)/Fc, the rigid wall case can be subcritically chaotic. Aperiodic, highly nonlinear wave motions are possible for external control values set in the linearly stable region of parameter space. Weakly nonlinear analysis shows that the no-slip case has a negative Landau constant for moderately small values of bottom friction, and it is in this regime that high-resolution numerical simulations exhibit subcritical chaos. At larger values of bottom friction, the rigid-wall simulations undergo a supercritical quasi-periodic transition to chaos at modest, order one, supercriticality, which is substantially smaller than that required for chaos in the free-slip case.

During the meandering motion of a dipolar vortex on a β-plane mass is exchanged both between the dipole and the ambient fluid and between the two dipole halves. The mass exchange (as well as the meandering motion) is caused by variations of the relative vorticity of the vortices due to conservation of potential vorticity. Previous studies have shown that a modulated point-vortex model captures the essential features in the dipole evolution. For this model we write the equations of motion of passive tracers in the form of a periodically perturbed integrable Hamiltonian system and subsequently study transport using a ‘dynamical-systems theory’ approach. The amount of mass exchanged between different regions of the flow is evaluated as a function of two parameters: the gradient of ambient vorticity, β, and the initial direction of propagation of the dipole, α0. Mass exchange between the dipole and the surroundings increases with increasing both β and α0. The exchange rate (amount of mass exchanged per unit time) increases with β and has a maximum for a particular value of α0 (≈ 0.62π). Dipolar vortices in a rotating fluid (with a sloping bottom providing the ‘topographic’ β-effect) show, in addition to the relative vorticity variations, a second perturbation that leads to exchange of mass. The points where vorticity is extreme approach each other as the dipole moves to shallower parts of the fluid and separate as the couple moves to deeper parts. This mechanism is studied independently and it is shown to lead to a stronger exchange between the dipole halves and the ambient fluid but no exchange between the two dipole halves.

The temporal instability of a cylindrical capillary jet is analysed numerically for different liquid Reynolds numbers Re, disturbance wavenumbers k, and amplitudes ε0. The breakup mechanism of viscous liquid jets and the formation of satellite drops are described. The results show that the satellite size decreases with decreasing Re, and increasing k and ε0. Marginal Reynolds numbers below which no satellite drops are formed are obtained for a large range of wavenumbers. The growth rates of the disturbances are calculated and compared with those from the linear theory. These results match for low-Re jets, however as Re is increased the results from the linear theory slightly overpredict those from the nonlinear analysis. (At the wavenumber of k = 0.9, the linear theory underpredicts the nonlinear results.) The breakup time is shown to decrease exponentially with increasing the amplitude of the disturbance. The cut-off wavenumber is shown to be strongly dependent on the amplitude of the initial disturbance for amplitudes larger than approximately $\frac13$ of the initial jet radius. The stable oscillations of liquid jets are also investigated. The results indicate that liquid jets with Re ∼ O(1) do not oscillate, and the disturbances are overdamped. However, liquid jets with higher Re oscillate with a period which depends on Re and ε0. The period of the oscillation decreases with increasing Re at small ε0; however, it increases with increasing Re at large ε0. Marginal Reynolds numbers below which the disturbances are overdamped are obtained for a wide range of wavenumbers and ε0 = 0.05.

The time-periodic phenomena occurring at low Reynolds numbers (Re [lsim ] 180) in the wake of a circular cylinder (finite-length section) are well modelled by a Ginzburg–Landau (GL) equation with zero boundary conditions (Albarède & Monkewitz 1992). According to the GL model, the wake is mainly governed by a rescaled length, based on the aspect ratio and the Reynolds number. However, the determination of coefficients is not complete: we correct a former evaluation of the nonlinear Landau coefficient, we show difficulties in obtaining a consistent set of coefficients for different Reynolds numbers or end configurations, and we propose the use of an ‘influential’ length. New two-point velocimetry results are presented: phase measurements show that a subtle property is shared by the three-dimensional wake and the GL model.

Two time-quasi-periodic phenomena – the second mode observed for smaller aspect ratios, and the dislocated chevron observed for larger aspect ratios – are presented and precisely related to the GL model. Only the linear characteristics of the second mode are readily explained; its existence depends on the end conditions. Moreover, through a quasi-static variation of the length, the second mode evolves continuously to end cells (and vice versa). Observations of the dislocated chevron are recalled. A very similar instability is found on the chevron solution of the GL equation, when the model parameters (c1, c2) move towards the phase diffusion unstable region. The early stages of this instability are qualitatively similar to the observed patterns.

Results of an experimental study of the interaction of a turbulent jet with a free surface when the jet issues parallel to the free surface are presented. Three different jets, with different exit velocities and jet-exit diameters, all located two jet-exit diameters below the free surface were studied. At this depth the jet flow, in each case, is fully turbulent before significant interaction with the free surface occurs. The effects of the Froude number (Fr) and the Reynolds number (Re) were investigated by varying the jet-exit velocity and jet-exit diameter. Froude-number effects were identified by increasing the Froude number from Fr = 1 to 8 at Re = 12700. Reynolds-number effects were identified by increasing the Reynolds number from Re = 12700 to 102000 at Fr = 1. Qualitative features of the subsurface flow and free-surface disturbances were examined using flow visualization. Measurements of all six Reynolds stresses and the three mean velocity components were obtained in two planes 16 and 32 jet diameters downstream using a three-component laser velocimeter. For all the jets, the interaction of vorticity tangential to the surface with its ‘image’ above the surface contributes to an outward flow near the free surface. This interaction is also shown to be directly related to the observed decrease in the surface-normal velocity fluctuations and the corresponding increase in the tangential velocity fluctuations near the free surface. At high Froude number, the larger surface disturbances diminish the interaction of the tangential vorticity with its image, resulting in a smaller outward flow and less energy transfer from the surface-normal to tangential velocity fluctuations near the surface. Energy is transferred instead to free-surface disturbances (waves) with the result that the turbulence kinetic energy is 20% lower and the Reynolds stresses are reduced. At high Reynolds number, the rate of evolution of the interaction of the jet with the free surface was reduced as shown by comparison of the rate of change with distance downstream of the local Reynolds and Froude numbers. In addition, the decay of tangential vorticity near the surface is slower than for low Reynolds number so that vortex filaments have time to undergo multiple reconnections to the free surface before they eventually decay.

The problem solved concerns a slender ship moving at a near-critical steady speed in a shallow channel, not necessarily in symmetric configuration, involving the special phenomenon of generation of solitary waves. By using the technique of matched asymptotic expansions along with nonlinear shallow-water wave theory, the problem is reduced to a Kadomtsev–Petviashvili equation in the far field, matched with a nearfield solution obtained by an improved slender-body theory, taking the local wave elevation and longitudinal disturbance velocity into account. The ship can be either fixed or free to squat. Besides wave pattern and wave resistance, the hydrodynamic lift force and trim moment are calculated by pressure integration in the fixed-hull case; running sinkage and trim, by condition of hydrodynamic equilibrium in the free-hull case. The numerical procedure for solving the KP equation consists of a finite-difference method, namely, fractional step algorithm with Crank–Nicolson-like schemes in each half step. Calculated results are compared with several published shipmodel experiments and other theoretical predictions; satisfactory agreement is demonstrated.

Using a nonlinear critical layer analysis, Goldstein & Leib (1988) derived a set of nonlinear evolution equations governing the spatial growth of a two-dimensional instability wave on a homogeneous incompressible tanh y mixing layer. In this study, we extend this analysis to the temporal growth of the García model of an incompressible stratified shear layer. We consider the stage of the evolution in which the growth first becomes nonlinear, with the nonlinearity appearing inside the critical layer. The Reynolds number is assumed to be just large enough so that the unsteady, nonlinear and viscous terms all enter at the same order of magnitude inside the critical layer. The equations are solved numerically for the inviscid case.

A plane turbulent wake generated by a flat plate is subjected to a uniform distortion. It is observed that nearly two-dimensional, quasi-periodic coherent structures dominate the distorted wake. Rapid distortion theory, applied to a kinematic vortex model of the coherent structures in the undistorted far wake, predicts many of the effects revealed by a hot-wire anemometry/pattern-recognition analysis of these structures. Specifically, rapid distortion theory predicts reasonably well the observed changes in the ensemble-averaged velocity patterns and the disproportionate amplification of the large-scale coherent structures relative to the smaller-scale ‘isotropic’ eddies. These results are consistent with the view that self-preservation of the distorted wake is not possible because of the selective amplification of the coherent structures, which control the development of the wake. As well, the entrainment rate in the distorted wake increases at a rate greater than that predicted by the self-preservation theory.

We present an experimental study of high-aspect-ratio Faraday waves. We have measured the dispersion relation and the damping rate, together with the critical amplitude for the primary instability for a wide range of frequencies. We find that our results are well explained by the linear theory, if damping from the moving contact line is considered in addition to the bulk damping. Just above the primary instability a seemingly disordered stationary state is observed. We argue that this state is a superposition of normal modes. Approximately 5% above the primary instability this state breaks down in favour of a quasi-crystalline state. This result is discussed, partly in the light of the recent third-order nonlinear theory.

A numerical investigation of the stability of an axisymmetric magnetic field is discussed. The magnetic field permeates a finitely conducting fluid contained within a rapidly rotating cylindrical annulus. The fluid is incompressible and viscid. The evolution of a non-axisymmetric perturbation to the axisymmetric magnetic field is governed by the momentum and induction equations which are integrated using a spectral timestep method. We follow the growth of the perturbation to finite amplitude and find that the character of the solution is dominated by the most unstable axially dependent mode found from the linear theory.

The spatial evolution of three-dimensional disturbances in an attachment-line boundary layer is computed by direct numerical simulation of the unsteady, incompressible Navier–Stokes equations. Disturbances are introduced into the boundary layer by harmonic sources that involve unsteady suction and blowing through the wall. Various harmonic-source generators are implemented on or near the attachment line, and the disturbance evolutions are compared. Previous two-dimensional simulation results and nonparallel theory are compared with the present results. The three-dimensional simulation results for disturbances with quasi-two-dimensional features indicate growth rates of only a few percent larger than pure two-dimensional results; however, the results are close enough to enable the use of the more computationally efficient, two-dimensional approach. However, true three-dimensional disturbances are more likely in practice and are more stable than two-dimensional disturbances. Disturbances generated off (but near) the attachment line spread both away from and toward the attachment line as they evolve. The evolution pattern is comparable to wave packets in flat-plate boundary-layer flows. Suction stabilizes the quasi-two-dimensional attachment-line instabilities, and blowing destabilizes these instabilities; these results qualitatively agree with the theory. Furthermore, suction stabilizes the disturbances that develop off the attachment line. Clearly, disturbances that are generated near the attachment line can supply energy to attachment-line instabilities, but suction can be used to stabilize these instabilities.

A modified version of the mild-slope equation is derived and its predictions of wave scattering by two-dimensional topography compared with those of other equations and with experimental data. In particular, the modified mild-slope equation is shown to be capable of describing known scattering properties of singly and doubly periodic ripple beds, for which the mild-slope equation fails. The new equation compares favourably with other models of scattering which improve on the mild-slope equation, in that it is widely applicable and computationally cheap.