Results of numerical simulations of the flow of a non-rotating, inviscid, Boussinesq fluid over smooth two-dimensional obstacles are described. The fluid has finite depth and a rigid lid. Far upstream of the obstacle the horizontal velocity and buoyancy frequency are uniform. Comparisons with linear theory for small-amplitude obstacles are made and the long-time behaviour is compared with steady-state Long's model solutions. Comparisons with the time-dependent results of Baines (1979) are done. For Froude numbers between ½ and 1 the obstacle amplitude is varied in order to determine the amplitudes needed to initiate wave breaking. These results are compared with the predictions of Long's model and with the experimental results of Baines (1977) showing good agreement with the latter. It is found that wave breaking occurs for amplitudes significantly lower than Long's model predicts for a large range of Froude numbers. This is shown to be the result of the generation of large-amplitude lee waves with wavelengths longer than that of stationary lee waves, but not long enough to propagate upstream. The behaviour of these waves is coupled to the generation of both longer mode-one waves which do propagate upstream from the obstacle and to mode-two waves which propagate against the flow as they are advected downstream. It is also coupled to oscillations in the wave drag. The periods of the wave drag oscillations are compared to experimental results showing good agreement with cases for which oscillations have been observed. The behaviour of these large transient lee waves is compared with the theoretical results contained in Grimshaw & Yi (1991), showing some similarities. As the Froude number approaches 0.5 the breaking behaviour is no longer due to these large waves, with the result that wave breaking occurs much later.

This paper builds on our claim that most vortical structures in transitional and turbulent flows are partially polarized. Polarization is inferred by the application of helical wave decomposition. We analyse initially polarized isolated viscous vortex rings through direct numerical simulation of the Navier-Stokes equations using divergence-free axisymmetric eigenfunctions of the curl operator. Integral measures of the degree of polarization, such as the fractions of energy, enstrophy, and helicity associated with right-handed (or left-handed) eigenfunctions, remain nearly constant during evolution, thereby suggesting that polarization is a persistent feature. However, for polarized rings an axial vortex (tail) develops near the axis, where the local ratio of right- to left-handed vorticities develops significant non-uniformities due to spatial separation of peaks of polarized components. Reconnection can occur in rings when polarized and is clearly discerned from the evolution of axisymmetric vortex surfaces; but interestingly, the location of reconnection cannot be inferred from the vorticity magnitude. The ring propagation velocity Up decreases monotonically as the degree of initial polarization increases. Unlike force-balance arguments, two explanations based on vortex dynamics provided here are not restricted to thin rings and predict reduction in Up correctly. These results reveal surprising differences among the evolutionary dynamics of polarized, partially polarized, and unpolarized rings.

The topology of vortex lines and surfaces is examined in incompressible viscous axisymmetric flows with swirl. We argue that the evolving topology of the vorticity field must be examined in terms of axisymmetric vortex surfaces rather than lines, because only the surfaces enjoy structural stability. The meridional cross-sections of these surfaces are the orbits of a dynamical system with the azimuthal circulation being a Hamiltonian H and with time as a bifurcation parameter μ. The dependence of H on μ is governed by the Navier–Stokes equations; their numerical solutions provide H. The level curves of H establish a time history for the motion of vortex surfaces, so that the circulation they contain remains constant. Equivalently, there exists a virtual velocity field in which the motion of the vortex surfaces is frozen almost everywhere; the exceptions occur at critical points in the phase portrait where the virtual velocity is singular. The separatrices emerging from saddle points partition the phase portrait into islands; each island corresponds to a structurally stable vortex structure. By using the flux of the meridional vorticity field, we obtain a precise definition of reconnection: the transfer of flux between islands. Local analysis near critical points shows that the virtual velocity (because of its singular behaviour) performs ‘cut-and-connect’ of vortex surfaces with the correct rate of circulation transfer - thereby validating the long-standing viscous ‘cut-and-connect’ scenario which implicitly assumes that vortex surfaces (and vortex lines) can be followed over a short period of time in a viscous fluid. Bifurcations in the phase portrait represent (contrary to reconnection) changes in the topology of the vorticity field, where islands spontaneously appear or disappear. Often such topology changes are catastrophic, because islands emerge or perish with finite circulation. These and other phenomena are illustrated by direct numerical simulations of vortex rings at a Reynolds number of 800.

The nonlinear interaction between the mean flow and a coherent disturbance in a two-dimensional turbulent mixing layer is addressed. Based on considerations from stability theory, previous experimental results, in particular the modification of the mean velocity profile, the peculiar growth of the forced shear-layer thickness and the spatial growth of the disturbance amplitude, are explained. A model that assumes a quasi-parallel mean flow having a self-similar mean velocity profile is developed. The model is capable of predicting the downstream evolution of turbulent mixing layers subjected to external excitations.

The existence of two-dimensional flows with an isotropic and negative eddy viscosity is demonstrated. Such flows, when subject to a very weak large-scale perturbation of wavenumber k will amplify it with a rate proportional to k2, independent of the direction.

Specifically, it is assumed that the basic (unperturbed) flow is space-time periodic, possesses a centre of symmetry (parity-invariance) and has three- or six-fold rotational invariance to ensure isotropy of the eddy-viscosity tensor.

The eddy viscosities emerging from the multiscale analysis are calculated by two different methods. First, there is an expansion in powers of the Reynolds number which can be carried out to large orders, and then extended analytically (thanks to a meromorphy property) beyond the disk of convergence. Secondly, there is a spectral method. The two methods typically agree within a fraction of 1%.

A simple example, the ‘decorated hexagonal flow’, of a time-independent flow with isotropic negative eddy viscosity is given. Flows with randomly chosen Fourier components and the required symmetry have typically a 30% chance of developing a negative eddy viscosity when the Reynolds number is increased.

For basic flow driven by a prescribed external force and sufficiently strong largescale flow, the analysis is extended to the nonlinear régime. It is found that the largescale dynamics is governed by a Navier-Stokes or a Navier-Stokes-Kuramoto-Sivashinsky equation, depending on the sign and strength of the eddy viscosity. When the driving force is not mirror-symmetric, a new ‘chiral’ nonlinearity appears. In special cases, the large-scale equation reduces to the Burgers equation. With chiral forcing, circular vortex patches are strongly enhanced or attenuated, depending on their cyclonicity.

Active circulation control of the two-dimensional unsteady separated flow past a semiinfinite plate with transverse motion is considered. The rolling-up of the separated shear layer is modelled by a point vortex whose time-dependent circulation is predicted by an unsteady Kutta condition. A suitable vortex shedding mechanism introduced. A control strategy able to maintain constant circulation when a vortex is present is derived. An exact solution for the nonlinear controller is then obtained. Dynamical systems analysis is used to explore the performance of the controlled system. The control strategy is applied to a class of flows and the results are discussed. A procedure to determine the position and the circulation of the vortex, knowing the velocity signature on the plate, is derived. Finally, a physical explanation of the control mechanism is presented.

When a liquid meniscus held at the exit of a metallic capillary tube is charged to a high voltage V, the free surface often takes the form of a cone whose apex emits a steady microjet, and thus injects a certain charge I and liquid volume Q per unit time into the surrounding gas. This work deals with liquids with relatively large conductivities K, for which the jet diameter dj is much smaller than the diameter dn of the capillary tube. In the limit dj/dn → 0, the structure of the jet (dj and I, in particular) becomes independent of electrostatic parameters such as V or the electrode configuration, being governed mostly by the liquid properties and flow rate Q. Furthermore, the measured current is given approximately by I = f(ε) (γQK/ε)½ for a wide variety of liquids and conditions (ε, and γ are, respectively, the dielectric constant of the liquid and the coefficient of interfacial tension; f(ε) is shown in figure 11). The following explanation is proposed for this behaviour. Convection associated with the liquid flow Q transports the net surface charge towards the cone tip. This upsets the electrostatic surface charge distribution slightly at distances r from the apex large compared to a certain charge relaxation length λ, but substantially when r ∼ λ. When the fluid motion is modelled as a sink flow, λ is of the order of r* = (Qεε0/K)$\frac13$ (ε0 is the electrical permittivity of vacuum). If, in addition, the surface charge density is described through Taylor's theory, the corresponding surface current convected towards the apex scales as Is ∼ (γQK/ε)½, as observed for the spray current. The sink flow hypothesis is shown to be realistic for sufficiently small jet Reynolds numbers. In a few photographs of ethylene glycol cone jets, we find the rough scaling dj ∼ 0.4r* for the jet diameter, which shows that the jet forms as soon as charge relaxation effects set in. In the limit ε [Gt ] 1, an upper bound is found for the convected current at the virtual cone apex, which accounts for only one-quarter of the total measured spray current. The rest of the charge must accordingly reach the head of the jet by conduction through the bulk.

Incompressible Newtonian flow in a two-dimensional channel with periodically placed sharp edged baffles has been studied both by numerical simulation and by experimental flow visualization. The flow was observed to be steady and symmetric at low Reynolds numbers, with recirculating eddies downstream of each baffle. At a critical Reynolds number (based on channel width and cross-sectional mean velocity) of approximately 100 the flow became asymmetric and unsteady. This transition to unsteadiness led to an eddy shedding regime, with eddies formed and shed successively from each baffle. A stability study suggested that the mechanism for transition to unsteady flow is a Kelvin–Helmholtz instability associated with the shear layer formed downstream of the sharp edged baffles. The frequency of the unsteadiness is, however, dependent on the full flow field, and not only the shear layer characteristics. Experimental observations show that the instability is followed by a secondary transition to three-dimensional disordered flow. Experimentally observed flows in the two-dimensional regime were found to be in close agreement with the numerical simulation for both the steady and unsteady flows.

In this paper the nonlinear evolution of two-dimensional shear-flow instabilities near the ocean surface is studied. The approach is numerical, through direct simulation of the incompressible Euler equations subject to the dynamic and kinematic boundary conditions at the free surface. The problem is formulated using boundary-fitted coordinates, and for the numerical simulation a spectral spatial discretization method is used involving Fourier modes in the streamwise direction and Chebyshev polynomials along the depth. An explicit integration is performed in time using a splitting scheme. The initial state of the flow is assumed to be a known parallel shear flow with a flat free surface. A perturbation having the form of the fastest growing linear instability mode of the shear flow is then introduced, and its subsequent evolution is followed numerically. According to linear theory, a shear flow with a free surface has two linear instability modes, corresponding to different branches of the dispersion relation: Branch I, at low wavenumbers; and Branch II, at high wavenumbers for low Froude numbers, and low wavenumbers for high Froude numbers. Our simulations show that the two branches have a distinctly different nonlinear evolution.

Branch I: At low Froude numbers, Branch I instability waves develop strong oval-shaped vortices immediately below the ocean surface. The induced velocity field presents a very sharp shear near the crest of the free-surface elevation in the horizontal direction. As a result, the free-surface wave acquires steep slopes, while its amplitude remains very small, and eventually the computer code crashes suggesting that the wave will break.

Branch II: At low Froude numbers, Branch II instability waves develop weak vortices with dimensions considerably smaller than their distance from the ocean surface. The induced velocity field at the ocean surface varies smoothly in space, and the free-surface elevation takes the form of a propagating wave. At high Froude numbers, however, the growing rates of the Branch II instability waves increase, resulting in the formation of strong vortices. The free surface reaches a large amplitude, and strong vertical velocity shear develops at the free surface. The computer code eventually crashes suggesting that the wave will break. This behaviour of the ocean surface persists even in the infinite-Froude-number limit.

It is concluded that the free-surface manifestation of shear-flow instabilities acquires the form of a propagating water wave only if the induced velocity field at the ocean surface varies smoothly along the direction of propagation.

This study examines the influence of coherent structures and attendant microfronts on scaling laws. Toward this goal, we analyse atmospheric observations of turbulence collected 45 m above a flat surface during the Lammefjord Experiment in Denmark. These observations represent more than 40 hours of nearly stationary strong wind conditions and include more than 1600 samples of the main coherent structures. These samples occupy about 40% of the total record and explain the majority of the Reynolds stress.

To study the dependence of the scaling laws on the choice of basis set, the time series of velocity fluctuations are decomposed into Fourier modes, the local Haar basis set and eigenvectors of the lagged covariance matrix. The three decompositions are compared by formulating joint projections. The decompositions are first applied to the samples of phased-locked coherent structures centred about eddy microfronts. The eigenvector decomposition is able to partially separate the small-scale variances due to the coherent eddy microfronts from that due to the small-scale structure with random phase. In the Fourier spectrum, both of these contributions to the variance appear together at the higher wavenumbers and their individual contributions cannot be separated. This effect is relatively minor for the scale distribution of energy but exerts an important influence on higher-moment statistics. Deviations from the −$\frac53$ scaling are observed to be slight and depend on choice of basis set.

The microfronts strongly influence the higher-order statistics such as the sixth-order structure function traditionally used to estimate the energy transfer variance. The intermittency of fine-scale structure, energy transfer variance and dissipation are not completely characterized by random phase, as often assumed, but are partly associated with microfronts characterized by systematic phase with respect to the main transporting eddies. These conclusions are supported by both the higher-order structure function and the higher-order Haar transform.

The Fourier and Haar spectra are also computed for the entire record. The peak of the Haar energy spectrum occurs at smaller scales than those of the Fourier spectrum. The Haar transform is local and emphasizes the width of the events. The Fourier spectrum peaks at the scale of the main periodicity, if it exists, which includes the spacing between the events.

The scattering of plane sound waves by a vortex is investigated by solving the compressible Navier–-Stokes equations numerically, and analytically with asymptotic expansions. Numerical errors associated with discretization and boundary conditions are made small by using high-order-accurate spatial differentiation and time marching schemes along with accurate non-reflecting boundary conditions. The accuracy of computations of flow fields with acoustic waves of amplitude five orders of magnitude smaller than the hydrodynamic fluctuations is directly verified. The properties of the scattered field are examined in detail. The results reveal inadequacies in previous vortex scattering theories when the circulation of the vortex is non-zero and refraction by the slowly decaying vortex flow field is important. Approximate analytical solutions that account for the refraction effect are developed and found to be in good agreement with the computations and experiments.

The flow structures generated and drag experienced by a rigid cylinder moving in an arbitrary direction through a rotating electrically conducting fluid in the presence of an applied magnetic field are investigated, with he aim of understanding better the nature of the small-scale flow in the core of the Earth which may be responsible for maintaining the geomagnetic field through dynamo action. Three cases are considered in the limit of small Rossby and magnetic Reynolds numbers. In the case of very weak rotation, the possible flow structures consist of a thin Hartmann layer and a long wake extending in the direction of the magnetic field, in which Lorentz and viscous forces balance, but only the long wake plays a dynamical role. The dominant drag force is experienced for motion that cuts magnetic lines of force. Motion of the cylinder parallel to its axis induces a much weaker drag, while that in the direction of the magnetic field induces none to dominant order. The cylinder also experiences weak lateral forces due to the Coriolis effect. In the case of weak rotation, the balance in the long wake is now magnetostrophic: between Lorentz and Coriolis forces. The drag is qualitatively identical to that in the first case, but the drag induced by motion parallel to the axis of the cylinder is increased, though still smaller than that for motions cutting magnetic lines of force. In the case of strong rotation, the flow structures consist of a thin Ekman layer and a foreshortened Taylor column extending in the direction of the rotation axis. In this column, the force balance is again magnetostrophic. Again only the large-scale structure plays a dynamical role. Motion of the cylinder perpendicular to its axis induces a larger drag than does motion parallel to its axis. The cylinder also experiences large lateral Coriolis forces.

We investigate the development of instability in a fluid with density locally of the form ρ0[1 −(N2 / g)z + A sin Kz], composed of an overall stable uniform gradient of buoyancy frequency, N, but with a superimposed sinusoidal variation of vertical wavenumber, K, and amplitude, A [Lt ] 1; g is the acceleration due to gravity and z is the upward vertical coordinate. Layers exist in which the fluid is statically unstable when the parameter r = N2 / gKA, is less than unity.

When r is zero, the density is sinusoidal in z and the problem reduces to one studied by Batchelor & Nitsche (1991). Their solution, which finds a gravest mode of linear instability with terms having vertical motions independent of z and with horizontal scales large in comparison with K−1, is extended to non-zero r. An effect of a small, but finite, r is to stabilize the fluid, increasing the critical Rayleigh number and the corresponding non-dimensional horizontal wavenumber. The vertical scale of the mode which first becomes unstable is reduced as r increases. A small sinusoidal shear destabilizes the fluid.

When r approaches unity, the density field contains regions of static instability which are of thickness small compared to K−1. The problem then approximates to one studied by Matthews (1988). Consistent solutions for the growth of disturbances are obtained by truncated series and, following Matthews, by the solution of a Fourier-transformed equation. A small uniform shear, characterized by a flow Reynolds number, Re > O, is found to stabilize the fluid, in that it increases the critical Rayleigh number of the onset of instability. This suggests that convective Rayleigh–Taylor instability, with constant phase lines parallel to the flow, is then the favoured mode of onset of instability. At very large Rayleigh numbers and at a Prandtl number of 700, however, the growth rate of the most rapidly growing linear disturbances may increase as Re increases from zero, and the form of the evolving flow is then less certain.

The theory is used to estimate the scale and growth rates of instability in overturning internal gravity waves in the laboratory experiment described in a companion paper (Thorpe 1994).

Internal waves in a uniformly stratified fluid of sufficiently large amplitude develop tilted layers in which the fluid is statically unstable. To investigate the evolution and subsequent development of this structure, experiments are made in which a horizontal rectangular tube containing a fluid of uniform density gradient is gently rocked at a selected frequency about a horizontal axis normal to the tube length. Large-amplitude standing internal gravity waves of the first mode are generated, and these steepen and overturn, the isopycnal surfaces folding to produce a vertically thin and horizontally extensive layer in which the fluid is statically unstable. In experiments with relatively small forcing, the layer persists for some 6 buoyancy periods, with no detected evidence of secondary instability, and static stability is re-established as the periodic flow reverses. The layer however breaks down, with consequent diapycnal mixing, when greater forcing is applied.

The scale and growth rates of instability in the overturning internal gravity waves are estimated using the theory developed in a companion paper by Thorpe (1994a). For the parameters of the laboratory experiments with relatively small forcing, the growth rates are small, consistent with the absence of signs of secondary instability. Larger growth rates and disturbance amplification factors of about 70 are predicted for the conditions in the experiment in which mixing was observed to occur. The experimental observations are consistent with an instability having a longitudinal structure.

We conclude that the form and development of breaking in internal gravity waves will vary according to the circumstances in which waves break, but depend on the Prandtl number of the fluid and, in particular, on the Rayleigh and Reynolds numbers of regions of static instability which develop as the waves overturn.

Using a data base generated by a numerical simulation, the three-dimensional coherent structures of a transitional, spatially evolving boundary layer are determined and their spatio-temporal behaviour is investigated in detail. The coherent structures are calculated by the proper orthogonal decomposition method (POD), which leads to an expansion of the flow field variables into Karhunen-Loéve eigenfunctions. It is shown that the dynamical coherent structures of the flat-plate boundary layer can be described by pairs of eigenfunctions that contain complete information on the spatial evolution of the structures. It is further demonstrated that first-order coherent structures determined by POD correspond to structures that are observed in experiments. In the region of the boundary layer where the spike signals of transition occur, higher-order coherent structures also play an essential role. By considering these higher-order structures as well as their dynamical behaviour in time, a compact description of the flow phenomena in the boundary layer can be obtained. The description of the events occurring at the spike stages of the transitional boundary layer shows, from a coherent structures point of view, striking similarities to the bursting event of fully turbulent boundary layers.