The dynamics of the vortex structures appearing in an oscillatory boundary layer (Stokes boundary layer), when the flow departs from the laminar regime, is investigated by means of flow visualizations and a quantitative analysis of the velocity and vorticity fields. The data are obtained by means of direct numerical simulations of the Navier–Stokes and continuity equations. The wall is flat but characterized by small imperfections. The analysis is aimed at identifying points in common and differences between wall turbulence in unsteady flows and the well-investigated turbulence structure in the steady case. As in Jimenez & Moin (1991), the goal is to isolate the basic flow unit and to study its morphology and dynamics. Therefore, the computational domain is kept as small as possible.

The elementary process which maintains turbulence in oscillatory boundary layers is found to be similar to that of steady flows. Indeed, when turbulence is generated, a sequence of events similar to those observed in steady boundary layers is observed. However, these events do not occur randomly in time but with a repetition time scale which is about half the period of fluid oscillations. At the end of the accelerating phases of the cycle, low-speed streaks appear close to the wall. During the early part of the decelerating phases the strength of the low-speed streaks grows. Then the streaks twist, oscillate and eventually break, originating small-scale vortices. Far from the wall, the analysis of the vorticity field has revealed the existence of a sequence of streamwise vortices of alternating circulation pumping low-speed fluid far from the wall as suggested by Sendstad & Moin (1992) for steady flows. The vortex structures observed far from the wall disappear when too small a computational domain is used, even though turbulence is self-sustaining. The present results suggest that the streak instability mechanism is the dominant mechanism generating and maintaining turbulence; no evidence of the well-known parent vortex structures spawning offspring vortices is found. Although wall imperfections are necessary to trigger transition to turbulence, the characteristics of the coherent vortex structures, for example the spacing of the low-speed streaks, are found to be independent of wall imperfections.

A rapidly growing instability is observed to develop between unequal-strength counter- rotating vortex pairs. The vortex pairs are generated in a towing tank in the wakes of wings with outboard triangular flaps. The vortices from the wing tip and the inboard tip of the flap form the counter-rotating vortex pair on each side of the wing. The flow fields are studied using flow visualization and particle image velocimetry. Both chord- based and circulation-based Reynolds numbers are of O(105). The circulation strength ratios of the flap- to tip-vortex pairs range from −0.4 to −0.7. The initial sinuous stage of the instability of the weaker flap vortex has a wavelength of order one wing span and becomes observable in about 15 wing spans downstream of the wing. The nearly straight vortex filaments first form loops around the stronger wing-tip vortices. The loops soon detach and form rings and move in the wake under self-induction. These vortex rings can move to the other side of the wake. The subsequent development of the instability makes the nearly quasi-steady and two-dimensional wakes unsteady and three-dimensional over a distance of 50 to 100 wing spans. A rectangular wing is also used to generate the classical wake vortex pair with the circulation ratio of −1.0, which serves as a reference flow. This counter-rotating vortex pair, under similar experimental conditions, takes over 200 spans to develop visible deformations. Velocity, vorticity and enstrophy measurements in a fixed plane, in conjuction with the flow observations, are used to quantify the behaviour of the vortex pairs. The vortices in a pair initially orbit around their vorticity centroid, which takes the pair out of the path of the wing. Once the three-dimensional interactions develop, two-dimensional kinetic energy and enstrophy drop, and enstrophy dispersion radius increases sharply. This rapid transformation of the wake into a highly three-dimensional one offers a possible way of alleviating the hazard posed by the vortex wake of transport aircraft.

A theoretical model for internal solitary waves for stratification consisting of two layers of incompressible fluid with a constant Brunt–Väisälä frequency and a density jump at the boundary between layers (‘2.5-layer model’) is presented. The equation of motion for solitary waves in the case of a constant Brunt–Väisälä frequency N is linear, and nonlinearity appears due only to boundary conditions between layers. This allows one to obtain in the case of long waves a single ordinary differential equation for an internal solitary wave profile. In the case of nearly homogeneous layers the solitons obtained here coincide with the solitons calculated by Choi & Camassa (1999), and in the weakly nonlinear case they reduce to KdV solitons. In the general situation strong 2.5-layer solitons can correspond to higher modes. Sufficiently strong solitons could also possess a recirculating core (at least, as a formal solution).

The model was applied to the data collected during the COPE experiment. The results are in reasonable agreement with experimental data.

The effects of mass transfer (e.g. via evaporation) of surface-active solutes on the hydrodynamic stability of capillary liquid jets are studied. A linear temporal stability analysis is carried out yielding evolution equations for systems satisfying general non- linear kinetic adsorption relations and accompanying surface constitutive equations. The discussion of the instability mechanism associated with the Marangoni effect clarifies that solute transfer into the jet is destabilizing whereas transfer in the opposite direction reduces instability. The general analysis is illustrated by a system satisfying Langmuir-type kinetic relations. Contrary to a clean system (i.e. in the absence of surfactants), reduced jet viscosity may lead to a substantial reduction in perturbation growth. Furthermore, the Marangoni effect gives rise to an overstability mechanism whereby perturbations whose dimensionless wavenumbers exceed unity grow with time through oscillations of increasing amplitude. The common diffusion-control approximation constitutes an upper bound which substantially overestimates the actual growth of perturbations. Considering solutes belonging to the homologous series of normal alcohols in water–air systems, the intermediate cases (e.g. hexanol–water–air which is ‘mixed-control’) are the most susceptible to Marangoni instability.

The spin-up from rest of a two-layer fluid with a free surface in a cylindrical container rotating about a vertical axis is investigated for small Ekman numbers. Numerical results from the axisymmetric Navier–Stokes equations, supported by comparisons with improved boundary-layer approximations, show that the Ekman-type layer on the bottom pushes the dense fluid of the lower layer to the periphery, and consequently the interface between the layers curves upward near the sidewall and descends near the centre. When the lower layer of fluid is sufficiently thin a bare spot appears at the bottom, i.e. a region where the light fluid is in direct contact with the horizontal boundary. The lower-layer fluid is spun-up quickly by the bottom Ekman layer, but the angular motion in the upper layer is provided by the much weaker detached Ekman layer on the interface between the two fluids, and hence the global spin-up process is prolonged compared with the homogeneous fluid case. The influence of the various dimensionless parameters and the connection with the continuous stratified case are discussed.

The study of the motion of a fluid ellipsoid has a long and fascinating history stretching back originally to Laplace in the late 18th century. Recently, this subject has been revived in the context of geophysical fluid dynamics, where it has been shown that an ellipsoid of uniform potential vorticity remains an ellipsoid in a background flow consisting of horizontal strain, vertical shear, and uniform rotation. The object of the present work is to present a simple, appealing, and practical way of investigating the motion of an ellipsoid not just in geophysical fluid dynamics but in general. The main result is that the motion of an ellipsoid may be reduced to the evolution of a symmetric, 3×3 matrix, under the action of an arbitrary 3×3 ‘flow’ matrix. The latter involves both the background flow, which must be linear in the Cartesian coordinates at the surface of the ellipsoid, and the self-induced flow, which was given by Laplace.

The resulting simple dynamical system lends itself ideally to both numerical and analytical study. We illustrate a few examples and then present a theory for the evolution of a vortex within a slowly varying background flow. We show that a vortex may evolve quasi-adiabatically, that is, it stays close to an equilibrium form associated with the instantaneous background flow. The departure from equilibrium, on the other hand, is proportional to the rate of change of the background flow.

The present work discusses the most commonly occurring shape of the coherent vortical structures in rapidly rotating stably stratified turbulence, under the quasi-geostrophic approximation. In decaying turbulence, these vortices – coherent regions of the materially-invariant potential vorticity – dominate the flow evolution, and indeed the flow evolution is governed by their interactions. An analysis of several exceptionally high-resolution simulations of quasi-geostrophic turbulence is performed. The results indicate that the population of vortices exhibits a mean height-to-width aspect ratio less than unity, in fact close to 0.8.

This finding is justified here by a simple model, in which vortices are taken to be ellipsoids of uniform potential vorticity. The model focuses on steady ellipsoids within a uniform background strain flow. This background flow approximates the effects of surrounding vortices in a turbulent flow on a given vortex. It is argued that the vortices which are able to withstand the highest levels of strain are those most likely to be found in the actual turbulent flow. Our calculations confirm that the optimal height-to-width aspect ratio is close to 0.8 for a wide range of background straining flows.

This paper presents an analysis and numerical study of the relations between the small-scale velocity and scalar fields in fully developed isotropic turbulence with random forcing of the large scales and with an imposed constant mean scalar gradient. Simulations have been performed for a range of Reynolds numbers from Reλ = 22 to 130 and Schmidt numbers from Sc = 1/25 to 144.

The simulations show that for all values of Sc [ges ] 0.1 steep scalar gradients are concentrated in intermittently distributed sheet-like structures with a thickness approximately equal to the Batchelor length scale η/Sc½ with η the Kolmogorov length scale. We observe that these sheets or cliffs are preferentially aligned perpendicular to the direction of the mean scalar gradient. Due to this preferential orientation of the cliffs the small-scale scalar field is anisotropic and this is an example of direct coupling between the large- and small-scale fluctuations in a turbulent field. The numerical simulations also show that the steep cliffs are formed by straining motions that compress the scalar field along the imposed mean scalar gradient in a very short time period, proportional to the Kolmogorov time scale. This is valid for the whole range of Sc. The generation of these concentration gradients is amplified by rotation of the scalar gradient in the direction of compressive strain. The combination of high strain rate and the alignment results in a large increase of the scalar gradient and therefore in a large scalar dissipation rate.

These results of our numerical study are discussed in the context of experimental results (Warhaft 2000) and kinematic simulations (Holzer & Siggia 1994). The theoretical arguments developed here follow from earlier work of Batchelor & Townsend (1956), Betchov (1956) and Dresselhaus & Tabor (1991).

We develop a model to describe the spreading of a reacting liquid which is injected at a steady rate into a permeable rock. We focus on the case in which there is a density difference between the host reservoir fluid and the injected liquid. We examine reactions which lead to precipitation and a decrease in permeability or dissolution and an increase in permeability. In both cases, we assume the reaction is rapid compared to the speed of the flow. As the current spreads under gravity, we show that the interface between the injected fluid and the original fluid and also the reaction front, may be described by similarity solutions. The morphology of the two interfaces is controlled by two parameters: the permeability ratio across the reaction front, k, and the speed of the reaction front as a fraction of the interstitial speed, λ. For a precipitation reaction, the reaction front lags some distance behind the leading edge of the region occupied by the injected fluid, and tends to terminate in a sharp vertical front. In contrast, for a dissolution reaction, the reaction front migrates as a gravity-driven finger along the base of the formation. In the case of large changes in permeability, kλ > 1, this finger advances to the front of the flow, whereas for smaller increases in permeability, kλ < 1, the finger is overrun with injected fluid which has already reacted and passed through the reaction front. We illustrate how these results are affected if the density of the reacting fluid decreases across the reaction zone. In the case of precipitation, small changes in density smooth out the leading edge of the reaction front, whereas large changes in density lead to slumping of the reaction front along the base of the current, and ultimately it extends to the nose of the flow. For dissolution reactions, the decrease in density across the reaction front causes the lateral extent of the finger to increase. As a result the critical value of the permeability ratio, k, for which the reaction front reaches the nose of the current decreases.

The inertial instability of equatorial shear flows is studied, with a view to understanding observed phenomena in the Earth's stratosphere and mesosphere. The basic state is a zonal flow of stratified fluid on an equatorial β-plane, with latitudinal shear. The simplest self-consistent model of the instability is used, so that the basic state and the disturbances are zonally symmetric, and a vertical diffusivity provides the scale selection. We study the interaction between the inertial instability, which takes the form of periodically varying disturbances in the vertical, and the mean flow, where ‘mean’ is a vertical mean.

The weakly nonlinear regime is investigated analytically, for flows with an arbitrary dependence on latitude. An amplitude equation of the form dA/dt = A−k2A∫[mid ]A[mid ]2dt is derived for the disturbances, and the evolving stability properties of the mean flow are discussed. In the final steady state, the disturbances vanish, but there is a persistent mean flow change that stabilizes the flow. However, the magnitude of the mean flow change depends strongly on the initial conditions, so that the system has a long memory. The analysis is extended to include the effects of Rayleigh friction and Newtonian cooling, destroying the long-memory property.

A more strongly nonlinear regime is investigated with the help of numerical simulations, extending the results up to the point where the instability leads to density contour overturning. The instability is shown to lead to a homogenization of fQ¯ around the initially unstable region, where f is the Coriolis parameter, and Q¯ is the vertical mean of the potential vorticity. As the instability evolves, the line of zero Q¯ moves polewards, rather than equatorwards as might be expected from a simple self-neutralization argument.

The overflowing cylinder (OFC) is an experimental apparatus designed to generate a controlled straining flow at a free surface, whose dynamic properties may then be investigated. Surfactant solution is pumped up slowly through a vertical cylinder. On reaching the top, the liquid forms a flat free surface which expands radially before over flowing down the side of the cylinder. The velocity, surface tension and surfactant concentration on the expanding free surface are measured using a variety of non-invasive techniques.

A mathematical model for the OFC has been previously derived by Breward et al. (2001) and shown to give satisfactory agreement with experimental results. However, a puzzling indeterminacy in the model renders it unable to predict one scalar parameter (e.g. the surfactant concentration at the centre of the cylinder), which must be therefore be taken from the experiments.

In this paper we analyse the OFC model asymptotically and numerically. We show that solutions typically develop one of two possible singularities. In the first, the surface concentration of surfactant reaches zero a finite distance from the cylinder axis, while the surface velocity tends to infinity there. In the second, the surfactant concentration is exponentially large and a stagnation point forms just inside the rim of the cylinder. We propose a criterion for selecting the free parameter, based on the elimination of both singularities, and show that it leads to good agreement with experimental results.

The weakly nonlinear dynamics of quasi-geostrophic flows over a one-dimensional, periodic or random, small-scale topography is investigated using an asymptotic approach. Averaged (or homogenized) evolution equations which account for the flow–topography interaction are derived for both homogeneous and continuously stratified quasi-geostrophic fluids. The scaling assumptions are detailed in each case; for stratified fluids, they imply that the direct influence of the topography is confined within a thin bottom boundary layer, so that it is through a new bottom boundary condition that the topography affects the large-scale flow. For both homogeneous and stratified fluids, a single scalar function entirely encapsulates the properties of the topography that are relevant to the large-scale flow: it is the correlation function of the topographic height in the homogeneous case, and a linear transform thereof in the continuously stratified case.

Some properties of the averaged equations are discussed. Explicit nonlinear solutions in the form of one-dimensional travelling waves can be found. In the homogeneous case, previously studied by Volosov, they obey a second-order differential equation; in the stratified case on which we focus they obey a nonlinear pseudodifferential equation, which reduces to the Peierls–Nabarro equation for sinusoidal topography. The known solutions to this equation provide examples of nonlinear periodic and solitary waves in continuously stratified fluid over topography.

The influence of bottom topography on large-scale baroclinic instability is also examined using the averaged equations: they allow a straightforward extension of Eady's model which demonstrates the stabilizing effect of topography on baroclinic instability.

Numerical methods are discussed for computing the pressure inside a two- or three-dimensional inviscid bubble with negligible density suspended in Stokes flow, subject to a specified rate of expansion. In the case of flow past a solitary two- or three-dimensional bubble, the bubble pressure is found by solving an integral equation of the first kind for the normal derivative of the pressure on the side of the liquid over the free surface, while requiring that the pressure field decays at a rate that is faster than the potential due to a point source. In another approach, an explicit expression for the bubble pressure is derived by applying the reciprocal theorem for the flow around the bubble and the flow due to a point source situated inside the bubble. In the case of flow past, or due to the expansion or shrinkage of, a periodic lattice of bubbles, the bubble pressure is found by solving an integral equation of the second kind for the density of an interfacial distribution of point-source dipoles, while ensuring existence and uniqueness of solution by spectrum deflation. The new methods considerably simplify the computation of the bubble pressure by circumventing the evaluation of the finite part of hypersingular integrals. Results of numerical simulations illustrate the pressure developing inside a solitary two- and three-dimensional incompressible bubble suspended in simple shear flow, and the pressure developing inside a doubly periodic array of gaseous inclusions representing shrinking pores trapped in a sintered medium.

A numerical model is developed to study the effects of friction on the steady exchange flow that evolves when a barrier is removed from a constriction separating two reservoirs of slightly different densities. The model has excellent agreement with an analytical solution and laboratory measurements of exchange flows through channels of constant width and depth. The model reveals three viscous flow regimes for a convergent–divergent contraction of constant depth, and three additional viscous flow regimes when an offset sill is introduced. Each regime is characterized by a different set of internal hydraulic control locations. Examination of the predicted interface profiles reveals that it is not possible to distinguish between different flow regimes on the basis of these profiles alone.

Statistical characteristics of turbulence in the near-surface region of a steady open- channel flow are examined using new data obtained in a high-Reynolds-number large-eddy simulation using a dynamic subgrid-scale model. These data, which correspond to a Reynolds number Re* = 1280 based on the total depth and shear velocity at the bottom wall, are systematically compared with those found in available direct numerical simulations in which Re* is typically one order of magnitude smaller. Emphasis is put on terms involved in the turbulent kinetic energy budget (dominated by dissipation and turbulent transport), and on the intercomponent transfer process by which energy is exchanged between the normal velocity component and the tangential ones. It is shown that the relative magnitude of the pressure–strain correlations depends directly on the anisotropy of the turbulence near the bottom of the surface-influenced layer, and that this anisotropy is a strongly decreasing function of Re*. This comparison also reveals the Re*-scaling laws of some of the statistical moments in the near-surface region, especially those involving vorticity fluctuations. Velocity variances, length scales and one-dimensional spectra are then compared with predictions of the rapid distortion theory elaborated by Hunt & Graham (1978) to predict the effect of the sudden insertion of a flat surface on a shearless turbulence. A very good agreement is found, both qualitatively and quantitatively, outside the thin viscous sublayer attached to the surface. As the present high-Reynolds-number statistics have been obtained after a significant number of turnover periods, this agreement strongly suggests that the validity of the Hunt & Graham theory is not restricted to short times after surface insertion.

We consider the unsteady three-dimensional Kármán flow induced by the impulsive rotation of an infinite rotating plane immersed in an incompressible viscous fluid with a dilute suspension of small solid monodisperse spherical particles. The flow is described in terms of a ‘dusty gas’ model, which treats the discrete phase (particles) and the continuous phase (fluid) as two continua occupying the same space and interacting through a Stokes drag mechanism. The model is extended to allow for a local gravitational acceleration in a direction parallel to the axis of rotation, and is valid for cases in which gravity acts either in the same direction as or in the opposite direction to the Ekman axial flow induced by the rotation of the plane.

Analysis based on the theory of characteristics shows that the role of gravity is crucial to the treatment of the discrete-phase equations, particularly in regard to the appropriate boundary conditions to be applied at the solid surface. Other notable features include the presence of an essential singularity in the solution when gravity is absent; indeed this phenomenon may help to explain some of the difficulties encountered in previous studies of this type. If the gravitational force is directed away from the rotating surface, a number of other interesting features arise, including the development of discontinuities in the particle distribution profiles, with corresponding particle-free regions contained between the interface and the rotating boundary. These ‘shock’ features can be associated with a critical axial location in the boundary layer at which a balance is achieved between Ekman suction induced by the rotating boundary and the influence of gravitational effects acting to move particles away from the boundary.