Spatially developing vortices found in curved channels (Dean vortices) and in a concave boundary layer (Görtler vortices) asre studied numerically using Legendre spectral-element methods. The linear instability of these vortices with respect to spanwise perturbations (Eckhaus instability) is examined using parabolized spatial stability analysis. The nonlinear evolution of this instability is studied by solving the parabolized Navier–Stokes equations. When the energy level of Dean and Görtler vortices in the flow is low, the spatial growth of the vortices is governed by primary instability (Dean or Görtler instability). At this stage, vortices with different wavelengths can develop at the same time and do not interact with each other significantly. When certain vortices reach the nonlinear stage first and become the dominant wavelength, spatial Eckhaus instability sets in. For all cases studied, spatially developing Dean and Görtler vortices are found to be most unstable to spanwise disturbances with wavelength twice or $\frac{3}{2}$ times that of the dominant one. The nonlinear growth of these perturbations generates a small vortex pair in between two pairs of vortices with long wavelength, but forces two pairs of vortices with short wavelength to develop into one pair. For Görtler vortices, this is manifested mostly by irregular and deformed vortex structures. For Dean vortices, this is manifested by vortex splitting and merging, and spatial Eckhaus instability plays an important role in the wavenumber selection process.

The electrophoretic motion of an arbitrary prolate body of revolution perpendicular to an infinite conducting planar wall is investigated by a combined analytical–numerical method. The electric field is exerted normal to the conducting planar wall and parallel to the axis of revolution of the particle. The governing equations and boundary conditions are obtained under the assumption of electric double layer thin compared to the local particle curvature radius and the spacing between the particle and the boundary. The axisymmetrical electrostatic and hydrodynamic equations are solved by the method of distribution of singularities along a certain line segment on the axis of revolution inside the particle. The analytical expressions for fundamental singularities both of electrostatic and hydrodynamic equations in the presence of the infinite planar wall are derived. Employing a piecewise parabolic approximation for the density function and applying the boundary collocation method to satisfy the boundary conditions on the surface of the particle, a system of linear algebraic equations is obtained which can be solved by matrix reduction technique.

Solutions for the electrophoretic velocity of the colloidal prolate spheroid are presented for various values of a/b and a/d, where a and b are the major and minor axes of the particle respectively and d is the distance between the centre and the wall. Numerical tests show that convergence to at least four digits can be achieved. For the limiting cases of a = b or d→ ∞, our results agree quite well with the exact solutions of electrophoresis of a sphere moving perpendicularly to an infinite planar wall or of a prolate spheroid in an unbounded fluid. As expected, owing to the effect of the wall, the electrophoretic mobility of the particle decreases monotonically for a given spheroid as it gets closer to the wall. Another important feature is that the wall effect on electrophoresis will reduce with the increase of slenderness ratio of the prolate spheroid at the same value of a/d. The boundary effect on the particle mobility and flow pattern in electrophoresis differ significantly from those of the corresponding sedimentation problem and the wall effect on the electrophoresis is much weaker than that on the sedimentation. In order to demonstrate the generality of the proposed method, the convergent results for prolate Cassini ovals are also given in the present paper.

The first Hopf bifurcation of the infinite cylinder wake is analysed theoretically and by direct simulation. It is shown that a decomposition into a series of harmonics is a convenient theoretical and practical tool for this investigation. Two basic properties of the instability allowing the use and truncation of the series of harmonics are identified: the lock-in of frequencies in the flow and separation of the rapid timescale of the periodicity from the slow timescale of the non-periodic behaviour. The Landau model is investigated under weak assumptions allowing strong nonlinearities and transition to saturation of amplitudes. It is found to be rather well satisfied locally at a fixed position of the flow until saturation. It is shown, however, that no truncated expansion into a series of powers of amplitude can account correctly for this fact. The validity of the local Landau model is found to be related to the variation of the form of the unstable mode substantially slower than its amplification. Physically relevant characteristics of the Hopf bifurcation under the assumption of separation of three timescales – those of the periodicity, amplification and deformation of the mode – are suggested.

The influence of stratification on the merging of like-sign vortices of equal intensity and shape is investigated by numerical simulations in a quasi-geostrophic, two-layer stratified model. Two different types of vortices are considered: vortices defined as circular patches of uniform potential vorticity in the upper layer but no PV anomaly in the lower layer (referred to as PVI vortices), and vortices defined as circular patches of uniform relative vorticity in the upper layer but no motion in the lower layer (referred to as RVI vortices). In particular, it is found that, in the RVI case, the merging behaviour depends strongly on the magnitude of the stratification (i.e. the ratio of internal Rossby radius and vortex radius). The critical point here appears to be whether or not the initial eddies have a deep flow signature in terms of PV.

The specific phenomenon of scale-dependent merging observed is interpreted in terms of the competitive effects of hetonic interaction and vortex shape. In the case of weaker stratification, the baroclinic structure of the eddies can be seen as dominated by a mechanism of hetonic interaction in which bottom flow appears to counteract the tendency of surface eddies to merge. In the case of larger stratification, the eddy interaction mechanism is shown to be barotropically dominated, although interface deformation still determines the actual eddy vorticity profile during the initialization stage. Repulsion (hetonic) effect therefore oppose attraction (barotropic shape) effects in a competitive process dependent on the relationship between the original eddy lengthscale and the first internal Rossby radius.

A concluding discussion considers the implications of such analysis for real situations, in the ocean or in the laboratory.

The boundary layer generated by the harmonic oscillations of a wavy wall in a fluid otherwise at rest is studied. First the wall waviness is assumed to be of small amplitude and large values of the Reynolds number are considered. The results obtained by means of a linear analysis, where the time variable appears only as a parameter, show that resonance may occur. Indeed it is found that when the Reynolds number is larger than a critical value, an instant within the decelerating part of the cycle exists such that a waviness of infinitesimal amplitude induces unbounded perturbations of the flow in the Stokes layer. The passage through resonance is then studied by means of a multiple-timescale approach, taking into account the damping effect of local acceleration within a small time range around resonance. The asymptotic approach fails beyond a threshold value of the Reynolds number, because the damping effect of the local acceleration terms spreads over the whole cycle. The problem is then tackled by means of an approach that takes into account the above damping effect throughout the whole cycle. Finally, a numerical procedure is used that also allows the inclusion of nonlinear terms and the study of the interactions among forced and free modes. The numerical approach reveals that, even for relatively large values of the amplitude of the wall waviness, nonlinear effects are negligible and the damping of resonance is mainly due to local acceleration effects. The relevance of the results to the understanding of transition to turbulence in Stokes layers is discussed.

Asymptotically exact evolution equations for counterpropagating shallow-water edge waves are derived. The structure of the equations depends only on the symmetries of the problem and on the fact that the group velocity of the edge waves is of order one. As a result the equations take the form of parametrically forced Davey–Stewartson equations with mean-field coupling. The calculations extend existing work on parametric excitation of edge waves by normally incident waves to arbitrary beach profiles with asymptotically constant depth, and include coupling to wave-generated mean longshore currents. Dissipation arises generically from radiation damping, but we also consider heuristically the effects of linear boundary-layer damping. Spatially modulated waves do not couple to the parametric forcing due to the non-locality of the evolution equations and are damped. Thus only spatially uniform wavetrains are expected as stable solutions. If linear dissipation is included the parametric coupling selects standing waves, but in the undamped case travelling wave states are possible. Both classes of solutions are examined for modulational instabilities, and stability conditions for the generic evolution equations are presented. However, modulational instability is found to be excluded in the shallow-water formulation through the effects of the mean flow. Explicit numerical results for two experimentally relevant beach profiles, exponentially decaying and piecewise linear, are presented.

New nonlinear stability theorems are derived for disturbances to steady basic flows in the context of the multilayer quasi-geostrophic equations. These theorems are analogues of Arnol’d's second stability theorem, the latter applying to the two-dimensional Euler equations. Explicit upper bounds are obtained on both the disturbance energy and disturbance potential enstrophy in terms of the initial disturbance fields. An important feature of the present analysis is that the disturbances are allowed to have non-zero circulation. While Arnol’d's stability method relies on the energy–Casimir invariant being sign-definite, the new criteria can be applied to cases where it is sign-indefinite because of the disturbance circulations. A version of Andrews’ theorem is established for this problem, and uniform potential vorticity flow is shown to be nonlinearly stable. The special case of two-layer flow is treated in detail, with particular attention paid to the Phillips model of baroclinic instability. It is found that the short-wave portion of the marginal stability curve found in linear theory is precisely captured by the new nonlinear stability criteria.

A series of experiments was conducted to determine the conditions under which streamwise vortices can cause transition to turbulence in shear flows. A specially designed obstacle was used to produce a single vortex in a water-table flow, and the design of this obstacle is discussed. Laser-Doppler velocimetry measurements of the streamwise and crossflow velocity fields were made in transitional and non-transitional flows, and flow visualization was also used. It was found that strong vortices (vortices with large circulation) lead to turbulence while weaker vortices do not. Determination of a critical value of vortex strength for transition, however, was complicated by ambiguities in calculating the vortex circulation. The profiles of streamwise velocity were found to be inflexional for both transitional and non-transitional flows. Transition in single-vortex and multi-vortex flows was compared, and no qualitative differences were observed, suggesting no significant vortex interactions affecting transition.

A numerical model is presented for investigating control of the three-dimensional boundary-layer transition process. Control of a periodically forced, spatially evolving boundary layer in water is studied using surface heating techniques. The Navier–Stokes and energy equations are integrated using a fully implicit finite difference/spectral method. The Navier–Stokes equations are used in vorticity–velocity form and are coupled with the energy equation through the viscosity dependence on temperature. Passive control of small amplitude two-dimensional waves and three-dimensional oblique waves is numerically simulated with either uniform or non-uniform wall heating applied. Both amplitude levels and amplification rates are strongly reduced with heating applied. Comparison is made with parallel and non-parallel linear stability theory and experiments. Control of the early stages of the nonlinear breakdown process is also investigated using uniform wall heating. Both control of the fundamental and subharmonic routes to turbulence are investigated. For both breakdown processes, a strong reduction in amplitude levels and growth rates results. In particular, the high three-dimensional growth rates that are characteristic of the secondary instability process are significantly reduced below the uncontrolled levels.

The linear stability of granular material in an unbounded uniform shear flow is considered. Linearized equations of motion derived from kinetic theories are used to arrive at a linear initial-value problem for the perturbation quantities. Two cases are investigated: (a) wavelike disturbances with time constant wavenumber vector, and (b) disturbances that will change their wave structure in time owing to a shear-induced tilting of the wavenumber vector. In both cases, the stability analysis is based on the solution operator whose norm represents the maximum possible amplification of initial perturbations. Significant transient growth is observed which has its origin in the non-normality of the involved linear operator. For case (a), regions of asymptotic instability are found in the two-dimensional wavenumber plane, whereas case (b) is found to be asymptotically stable for all physically meaningful parameter combinations. Transient linear stability phenomena may provide a viable and fast mechanism to trigger finite-amplitude effects, and therefore constitute an important part of pattern formation in rapid particulate flows.

The dynamics of a Gaussian isolated barotropic eddy on a β-plane is considered. The analytical solution of the evolution of an isolated vortex is constructed by analogy to the theory of a point vortex. The results of a numerical experiment are compared with the conclusions of the theory for the case of the Gaussian vortex. Characteristics of the vortex such as its radius, trajectory of movement, kinetic energy, residual vorticity, and the structure of the vortex are discussed. The analysis of the numerical results shows that the experimentally determined radius of the vortex, its energy, and residual vorticity are in good agreement with the theory. On the other hand there is a difference between analytical and experimental values of velocity components, and hence in the trajectory of the centre of the vortex. The location of the separatrix of the streak function and its saddle point are considered as important characteristics of the structure of the vortex. We consider the phenomenon of the generation of the vortex sheet connected with the separatrix location as a cause of the difference between the experimental and analytical estimates of the velocity of the vortex.

Contour dynamics methods are used to determine the shapes and speeds of planar, steadily propagating, solitary waves on a two-dimensional layer of uniform vorticity adjacent to a free-slip plane wall in an, otherwise irrotational, unbounded incompressible fluid, as well as of axisymmetric solitary waves propagating on a tube of azimuthal vorticity proportional to the distance to the symmetry axis. A continuous family of solutions of the Euler equations is found in each case. In the planar case they range from small-amplitude solitons of the Benjamin–Ono equation to large-amplitude waves that tend to one member of the touching pair of counter-rotating vortices of Pierrehumbert (1980), but this convergence is slow in two small regions near the tips of the waves, for which an asymptotic analysis is presented. In the axisymmetric case, the small-amplitude waves obey a Korteweg–de Vries equation with small logarithmic corrections, and the large-amplitude waves tend to Hill's spherical vortex.

We consider the flow resulting from the interaction between the trailing edge of a supersonic splitter plate and sound waves incident on the trailing edge from upstream, as a model problem of relevance to understanding the unsteady flow in the vicinity of a supersonic jet nozzle. Morgan has previously shown that there is only one plausible solution to the outer potential-flow problem in this supersonic system, in which the vortex-sheet deflection close to the trailing edge varies linearly with the distance from the trailing edge, and that, in contrast to the subsonic version of the problem, it is not possible to construct an outer solution in which the vortex sheet leaves the plate smoothly (i.e. with zero gradient). In this paper our aim is to establish that this supersonic potential-theory solution is consistent with the equations governing the viscous flow close to the plate, and to provide a description of the nature of this inner flow, and we proceed by applying asymptotic analysis in the limit of large Reynolds number. For appropriate choices of the incident-wave amplitude and frequency, the canonical triple-deck structure at the trailing edge is realized, and the governing equations are then simplified by linearizing about the steady base flow in the lower deck; upstream of the trailing edge the unsteady flow is calculated analytically, whilst downstream a two-region parabolic scheme is employed. Our inner viscous flow is seen to match onto the outer potential-theory solution, and in particular we verify that the downstream evolution of the lower-deck flow as it emerges into the outer region corresponds exactly to the behaviour of the vortex sheet at the trailing edge in the outer flow. Once the consistency of the outer solution has been established, the dependence on the various flow parameters can be investigated, and we demonstrate in particular that significant unsteady shear-layer disturbances can be generated at the trailing edge over a wide range of values of the incidence angle, and that the amplitude of these disturbances decreases with increasing supersonic flow speed.

This paper is concerned with the downstream evolution of a resonant triad of initially non-interacting linear instability waves in a boundary layer with a weak adverse pressure gradient. The triad consists of a two-dimensional fundamental mode and a pair of equal-amplitude oblique modes that form a subharmonic standing wave in the spanwise direction. The growth rates are small and there is a well-defined common critical layer for these waves. As in Goldstein & Lee (1992), the wave interaction takes place entirely within this critical layer and is initially of the parametric-resonance type. This enhances the spatial growth rate of the subharmonic but does not affect that of the fundamental. However, in contrast to Goldstein & Lee (1992), the initial subharmonic amplitude is assumed to be small enough so that the fundamental can become nonlinear within its own critical layer before it is affected by the subharmonic. The subharmonic evolution is then dominated by the parametric-resonance effects and occurs on a much shorter streamwise scale than that of the fundamental. The subharmonic amplitude continues to increase during this parametric-resonance stage – even as the growth rate of the fundamental approaches zero – and the subharmonic eventually becomes large enough to influence the fundamental which causes both waves to evolve on the same shorter streamwise scale.