In the 1980s the incorporation of ideas from dynamical systems theory into theoretical fluid mechanics, reinforced by elegant experiments, fundamentally changed the way in which we view and analyse Lagrangian transport. The majority of work along these lines was restricted to two-dimensional flows and the generalization of the dynamical systems point of view to fully three-dimensional flows has seen less progress. This situation may now change with the work of Pouransari et al. (J. Fluid Mech., this issue, vol. 654, 2010, pp. 5–34) who study transport in a three-dimensional time-periodic flow and show that completely new types of dynamical systems structures and consequently, coherent structures, form a geometrical template governing transport.

Mixing under laminar flow conditions is key to a wide variety of industrial fluid systems of size extending from micrometres to metres. Profound insight into three-dimensional laminar mixing mechanisms is essential for better understanding of the behaviour of such systems and is in fact imperative for further advancement of (in particular, microscopic) mixing technology. This insight remains limited to date, however. The present study concentrates on a fundamental transport phenomenon relevant to laminar mixing: the formation and interaction of coherent structures in the web of three-dimensional paths of passive tracers due to fluid inertia. Such coherent structures geometrically determine the transport properties of the flow and thus their formation and topological structure are essential to three-dimensional mixing phenomena. The formation of coherent structures, its universal character and its impact upon three-dimensional transport properties is demonstrated by way of experimentally realizable time-periodic model flows. Key result is that fluid inertia induces partial disintegration of coherent structures of the non-inertial limit into chaotic regions and merger of surviving parts into intricate three-dimensional structures. This response to inertial perturbations, though exhibiting great diversity, follows a universal scenario and is therefore believed to reflect an essentially three-dimensional route to chaos. Furthermore, a first outlook towards experimental validation and investigation of the observed dynamics is made.

The oceanic general circulation is forced at large scales and is unstable to mesoscale eddies. Large-scale currents and eddy flows are approximately in geostrophic balance. Geostrophic dynamics is characterized by an inverse energy cascade except for dissipation near the boundaries. In this paper, we confront the dilemma of how the general circulation may achieve dynamical equilibrium in the presence of continuous large-scale forcing and the absence of boundary dissipation. We do this with a forced horizontal flow with spatially uniform rotation, vertical stratification and vertical shear in a horizontally periodic domain, i.e. a version of Eady's flow carried to turbulent equilibrium. A direct route to interior dissipation is presented that is essentially non-geostrophic in its dynamics, with significant submesoscale frontogenesis, frontal instability and breakdown, and forward kinetic energy cascade to dissipation. To support this conclusion, a series of simulations is made with both quasigeostrophic and Boussinesq models. The quasigeostrophic model is shown as increasingly inefficient in achieving equilibration through viscous dissipation at increasingly higher numerical resolution (hence Reynolds number), whereas the non-geostrophic Boussinesq model equilibrates with only weak dependence on resolution and Rossby number.

A novel flow configuration devised for investigation of active control of separated airfoil flows using synthetic jets is presented. The configuration consists of a flat plate, with an elliptic leading edge and a blunt trailing edge, at zero incidence in a free stream. Flow separation is induced on the upper surface of the airfoil at the aft-chord location by applying suction and blowing on the top boundary of the computational domain. Typical separated airfoil flows are generally characterized by at least three distinct frequency scales corresponding to the shear layer instability, the unsteadiness of the separated region and the vortex shedding in the wake, and all these features are present in the current flow. Two-dimensional Navier–Stokes simulations of this flow at a chord Reynolds number of 6 × 104 have been carried out to examine the nonlinear dynamics in this flow and its implications for synthetic-jet-based separation control. The results show that there is a strong nonlinear coupling between the various features of the flow, and that the uncontrolled as well as the forced flow is characterized by a variety of ‘lock-on’ states that result from this nonlinear coupling. The most effective separation control is found to occur at the highest forcing frequency for which both the shear layer and the separated region lock on to the forcing frequency. The effects of the Reynolds number on the scaling of the characteristic frequencies of the separated flow and its subsequent control are studied by repeating some of the simulations at a higher Reynolds number of 1 × 105.

This paper investigates the influence of different three-dimensional multi-mode initial conditions on the rate of growth of a mixing layer initiated via a Richtmyer–Meshkov instability through a series of well-controlled numerical experiments. Results are presented for large-eddy simulation of narrowband and broadband perturbations at grid resolutions up to 3 × 109 points using two completely different numerical methods, and comparisons are made with theory and experiment. It is shown that the mixing-layer growth is strongly dependent on initial conditions, the narrowband case giving a power-law exponent θ ≈ 0.26 at low Atwood and θ ≈ 0.3 at high Atwood numbers. The broadband case uses a perturbation power spectrum of the form P(k) ∝ k−2 with a proposed theoretical growth rate of θ = 2/3. The numerical results confirm this; however, they highlight the necessity of a very fine grid to capture an appropriately broad range of initial scales. In addition, an analysis of the kinetic energy decay rates, fluctuating kinetic energy spectra, plane-averaged volume fraction profiles and mixing parameters is presented for each case.

Large-amplitude oscillations of incompressible viscous drops are studied at small capillary number. On the long viscous time scale, a formal perturbation scheme is developed to determine original modulation equations. These two ordinary differential equations comprise the averaged condition for conservation of energy and the averaged projection of the Navier–Stokes equations onto the vorticity vector. The modulation equations are applied to the free decay of axisymmetric oblate–prolate spheroid oscillations. On the long time scale, only the modulation equation for energy is required. In this example, the results compare well with linear viscous theory, weakly nonlinear inviscid theory and experimental observations. The new results show that previous experimental observations and numerical simulations are all manifestations of a single-valued relationship between dimensionless decay rate and amplitude. Moreover, if the amplitude of the oscillations does not exceed 30% of the drop radius, this decay rate may be approximated by a quadratic. The new results also show that, when the amplitude of the oscillations exceeds 20% of the drop radius, fluid in the inviscid bulk of the drop is undergoing abrupt changes in its acceleration in comparison to the acceleration during small-amplitude deformations.

The aeroacoustics of low-Mach-number boundary-layer flow over backward and forward facing steps is studied using large-eddy simulation and Lighthill's theory. The Reynolds number based on the step height and free-stream velocity ranges from 21000 to 328 as the step height is varied from 53% to 0.83% of the unperturbed boundary layer thickness at the step. For the largest step size, statistics of wall pressure fluctuations such as the root mean square values and frequency spectral density yield favourable comparisons with available experimental measurements. A low-frequency Green's function for the step geometry, valid for an acoustically compact step height, is employed to evaluate the volume integral in the solution to Lighthill's equation. Consistent with the result of previous theoretical studies, the steps act primarily as a dipole source aligned in the streamwise direction. The sound from the forward step is shown to be significantly louder than that from the backward step and the underlying reason is analysed in terms of source strength and distribution relative to the Green's function. The forward step generates stronger sources in regions closer to the step corner, which is heavily weighted by the Green's function. A detailed analysis of flow field and Green's function weighted sources reveals that the backward step generates sound mainly through a diffraction mechanism, while the forward step generates sound through a combination of diffraction and turbulence modification by the step. As the step height decreases, the difference in noise level between forward and backward steps is much reduced as turbulence modification becomes less significant.

The reflection of shock waves off cylindrical surfaces of different radii is examined with the help of time-resolved flow visualization. The primary diagnostic is a newly developed technique based on the tracking of deliberately introduced small perturbations in the flow. The main focus of the investigation is to determine at which position the shock wave receives information about the shape of the wall that it reflects off. In the pseudo-steady shock reflection off a plane wall, it is commonly accepted that the reflection changes from regular to irregular as soon as sonic signals generated behind the reflection point catch up with the reflection point, and the common interpretation is that this corresponds to the transition from regular to Mach reflection (the so-called sonic criterion). The results obtained here for convex circular surfaces show that this ‘catch-up’ condition occurs at wall angles considerably higher than in the plane wall case, while a visible Mach stem occurs only further along the surface at wall angles distinctively lower than those for plane walls. Tests are also conducted on a surface where a cylindrical portion is followed by a fixed angle plane section. The Mach numbers are chosen to be on either side of the plane wall transition condition so as to examine the adjustment from reflection on the cylindrical portion to that on the plane wall. These tests confirm that the wall angle at which sonic ‘catch-up’ to the reflection point occurs on the cylindrical surface is much higher than the corresponding wall angle predicted by the sonic criterion for a plane wall. While the transition from regular to irregular reflection is not the main concern of this contribution, the present results show that the transition criteria developed for steady and pseudo-steady flows are only of limited use in the fully unsteady flows such as the one investigated here.

We study the modulational instability of geophysical Rossby and plasma drift waves within the Charney–Hasegawa–Mima (CHM) model both theoretically, using truncated (four-mode and three-mode) models, and numerically, using direct simulations of CHM equation in the Fourier space. We review the linear theory of Gill (Geophys. Fluid Dyn., vol. 6, 1974, p. 29) and extend it to show that for strong primary waves the most unstable modes are perpendicular to the primary wave, which correspond to generation of a zonal flow if the primary wave is purely meridional. For weak waves, the maximum growth occurs for off-zonal inclined modulations that are close to being in three-wave resonance with the primary wave. Our numerical simulations confirm the theoretical predictions of the linear theory as well as the nonlinear jet pinching predicted by Manin & Nazarenko (Phys. Fluids, vol. 6, 1994, p. 1158). We find that, for strong primary waves, these narrow zonal jets further roll up into Kármán-like vortex streets, and at this moment the truncated models fail. For weak primary waves, the growth of the unstable mode reverses and the system oscillates between a dominant jet and a dominate primary wave, so that the truncated description holds for longer. The two-dimensional vortex streets appear to be more stable than purely one-dimensional zonal jets, and their zonal-averaged speed can reach amplitudes much stronger than is allowed by the Rayleigh–Kuo instability criterion for the one-dimensional case. In the long term, the system transitions to turbulence helped by the vortex-pairing instability (for strong waves) and the resonant wave–wave interactions (for weak waves).

We report the multi-scale geometric analysis of Lagrangian structures in forced isotropic turbulence and also with a frozen turbulent field. A particle backward-tracking method, which is stable and topology preserving, was applied to obtain the Lagrangian scalar field φ governed by the pure advection equation in the Eulerian form ∂tφ + u · ∇φ = 0. The temporal evolution of Lagrangian structures was first obtained by extracting iso-surfaces of φ with resolution 10243 at different times, from t = 0 to t = Te, where Te is the eddy turnover time. The surface area growth rate of the Lagrangian structure was quantified and the formation of stretched and rolled-up structures was observed in straining regions and stretched vortex tubes, respectively. The multi-scale geometric analysis of Bermejo-Moreno & Pullin (J. Fluid Mech., vol. 603, 2008, p. 101) has been applied to the evolution of φ to extract structures at different length scales and to characterize their non-local geometry in a space of reduced geometrical parameters. In this multi-scale sense, we observe, for the evolving turbulent velocity field, an evolutionary breakdown of initially large-scale Lagrangian structures that first distort and then either themselves are broken down or stretched laterally into sheets. Moreover, after a finite time, this progression appears to be insensible to the form of the initially smooth Lagrangian field. In comparison with the statistical geometry of instantaneous passive scalar and enstrophy fields in turbulence obtained by Bermejo-Moreno & Pullin (2008) and Bermejo-Moreno et al. (J. Fluid Mech., vol. 620, 2009, p. 121), Lagrangian structures tend to exhibit more prevalent sheet-like shapes at intermediate and small scales. For the frozen flow, the Lagrangian field appears to be attracted onto a stream-surface field and it develops less complex multi-scale geometry than found for the turbulent velocity field. In the latter case, there appears to be a tendency for the Lagrangian field to move towards a vortex-surface field of the evolving turbulent flow but this is mitigated by cumulative viscous effects.

Consider uniform flow past four slender bodies with elliptical cross-section of constant ellipticity along the length of 0, 0.125, 0.25 and 0.375, respectively, for each body. Here, ellipticity is defined as the ratio of the semiminor axis of the ellipse to the semimajor axis. The bodies have a pointed nose which gradually increases in cross-section with a radius of curvature 419 mm to a mid-section which then remains constant up to a blunt end section with semimajor axis diameter 160 mm, the total length of all bodies being 800 mm. The bodies are side-mounted within a low-speed wind tunnel with an operational wind speed of the order 30 m s−1. The side force (or lift) is measured within an angle of attack range of −3° to 3° such that the body is rotated about the major axis of the ellipse cross-section. The lift slope is determined for each body, and how it varies with ellipticity. It is found that this variance follows a straight line which steadily increases with increasing ellipticity. It is shown that this result is predicted by a recently developed Oseen flow slender body theory, and cannot be predicted by either inviscid flow slender body theory or viscous crossflow theories based upon the Allen and Perkins method.

Fully nonlinear model equations, including dispersive effects at one-order higher approximation than in the model of Choi & Camassa (J. Fluid Mech., vol. 396, 1999, pp. 1–36), are derived for long internal waves propagating in two spatial horizontal dimensions in a two-fluid system, where the lower layer is of infinite depth. The model equations consist of two coupled equations for the displacement of the interface and the horizontal velocity of the upper layer at an arbitrary elevation, and they are correct to O(μ2) terms, where μ is the ratio of thickness of the upper-layer fluid to a typical wavelength. For solitary waves propagating in one horizontal direction, the two coupled equations reduce to a single equation for the elevation of the interface. Solitary wave profiles obtained numerically from this equation for different wave speeds are in good agreement with computational results based on Euler's equations. A numerical approach for the propagation of solitary waves is provided in the weakly nonlinear case.

The conditions are examined in which stationary hydraulic jumps may occur in a continuously stratified layer of fluid of finite thickness moving over a horizontal boundary at z = 0 and beneath a deep static layer of uniform density. The velocity and density in the flowing layer are modified by turbulent mixing in the transition region. Entrainment of fluid from the overlying static layer is possible. Results are expressed in terms of a Froude number, Fr, characterizing the flow upstream of a transition. A Froude number, Fr*, is found that must be exceeded if conditions for the conservation of volume, mass and momentum fluxes across a hydraulic transition are satisfied. The condition Fr > Fr* is satisfied if the kinetic energy (KE) per unit area is greater than the potential energy per unit area, or if ∫0h [u2(z) − z2N2(z)]dz > 0, in a flow of speed u(z), in a layer of thickness h, with buoyancy frequency N(z). In the particular case (referred to as an ‘η profile’) of a flow with velocity and density that are constant if z ≤ ηh, decrease linearly if ηh ≤ z ≤ h, and in which u(z) = 0 and density is constant when z ≥ h, long linear internal waves can propagate upstream, ahead of a stationary hydraulic jump, for Fr in a range Fr* < Fr < Frc; here Frc is the largest Fr at which long waves, and wave energy, can propagate upstream in a flow with specified η. (Profiles other than the η profile exhibit similar properties.) It is concluded that whilst, in general, Fr > Fr* is a necessary condition for a hydraulic jump to occur, a more stringent condition may apply in cases where Fr* < Frc, i.e. that Fr > Frc.

Physical constraints are imposed on the form of the flow downstream of the hydraulic jump or transition that relate, for example, to its static and dynamic stability and its stability against a further hydraulic jump. A further condition is imposed that relates the rate of dissipation of turbulent KE within a transition to the loss in energy flux of the flow in passing through the transition from the upstream side to the downstream. The constraints restrict solutions for the downstream flow to those in which the flux of energy carried downstream by internal waves is negligible and in which dissipation of energy occurs within the transition region.

Although the problem is formulated in general terms, particular examples are given for η profiles, specifically when η = 0 and 0.4. The jump amplitude, the entrainment rate, the loss of energy flux and the shape of the velocity and density profiles in the flow downstream of a transition are determined when Fr > Fr* (and extending to those with Fr > Frc) in a number of extreme conditions: when the loss of energy flux in transitions is maximized, when the entrainment is maximized, when the jump amplitude is least and when loss of energy flux is maximized subject to the entrainment into the transitions being made zero. The ratio of the layer thickness downstream and upstream of a transition, the jump amplitude, is typically at least 1.4 when jumps are just possible (i.e. when Fr ~ Frc). The amplitude, entrainment and non-dimensionalized loss in energy flux increase with Fr in each of the extreme conditions, and the maximum loss in energy flux is close to that when the entrainment is greatest. The magnitude of the advective and diffusive fluxes across isopycnal surfaces, i.e. the diapycnal fluxes characterizing turbulent mixing in the transition region, also increase with Fr. Results are compared to those in which the velocity and density profiles downstream of the transition are similar to those upstream, and comparisons are also made with equivalent two-layer profiles and with a cosine-shaped profile with continuous gradients of velocity and density. If Fr is larger than a certain value (about 7 and > Frc, if η = 0.4), no solutions for flows downstream of a transition are found if there is no entrainment, implying that fluid must be entrained if a transition is to occur in flows with large Fr. Although the extreme conditions of loss of energy flux, jump amplitude or entrainment provide limits that it must satisfy, in general no unique downstream flow is found for a given flow upstream of a jump.

According to the Prandtl–Batchelor theorem for a steady two-dimensional flow with closed streamlines in the inviscid limit the vorticity becomes constant in the region of closed streamlines. This is not true for three-dimensional flows. However, if the variation of the flow field along one direction is slow then it is possible to expand the solution in terms of a small parameter characterizing the rate of variation of the flow field in that direction. Then in the leading-order approximation the projections of the streamlines onto planes perpendicular to that direction can be closed. Under these circumstances the extension of the Prandtl–Batchelor theorem is obtained. The resulting equations turned out to be a three-dimensional analogue of the equations of the quasi-cylindrical approximation.

Results from experiments on wave interaction with a rigid vertical plate are reported. The 5m long plate is set against the wall of a 30m wide basin, at 100m from the wavemaker. This set-up is equivalent to a 10m plate in the middle of a 60m wide basin. Regular waves are produced, with wavelengths of 1.6m, 1.8m and 2m, and steepnesses H/L (H being the double amplitude and L being the wavelength) ranging from 2% to 5%. Free-surface elevations along the plate are measured with a row of 20 gauges. The focus is on the time evolution of the free-surface profile along the plate. At all steepnesses, strong deviations from the predictions of linear theory gradually take place as the reflected wave field develops in the basin. This phenomenon is attributed to third-order interactions between the incoming and reflected wave systems, on the weather side of the plate. The measured profiles along the plate are compared with the predictions of two numerical models: an approximate model based on the tertiary interaction theory of Longuet-Higgins & Phillips (J. Fluid Mech., vol. 12, 1962, p. 333) for plane waves, which provides a steady-state solution, and a fully nonlinear numerical wavetank based on extended Boussinesq equations. In most of the experimental tests, despite the large distance from the wavemaker to the plate and the small amplitude of the incident wave, no steady state is attained by the end of the exploitable part of the records.

The interaction of a laser-induced plasma with isotropic turbulence is studied using numerical simulations. The simulations use air as the working fluid and assume local thermodynamic equilibrium. The numerical method is fully spectral and uses a shock-capturing scheme in a corrector step. A model problem involving the effect of energy deposition on an isolated vortex is studied as a first step towards plasma/turbulence interaction. Turbulent Reynolds number Reλ = 30 and fluctuation Mach numbers Mt = 0.001 and 0.3 are considered. A tear-drop-shaped shock wave is observed to propagate into the background, and progressively become spherical in time. The turbulence experiences strong compression due to the shock wave and strong expansion in the core. This behaviour is spatially inhomogeneous and non-stationary in time. Statistics are computed as functions of radial distance from the plasma axis and angular distance across the surface of the shock wave. For Mt = 0.001, the shock wave propagates on a much faster time scale compared to the turbulence evolution. At Mt of 0.3, the time scale of the shock wave is comparable to that of the background. For both cases the mean flow is classified into shock formation, shock propagation and subsequent collapse of the plasma core, and the effect of turbulence on each of these phases is studied in detail. The effect of mean vorticity production on the turbulent vorticity field is also discussed. Turbulent kinetic energy budgets are presented to explain the mechanism underlying the transfer of energy between the mean flow and background turbulence.

Tracking the motion of lipid domains on a vesicle is a rheological technique allowing the measurement of surface shear viscosities of vesicular lipid phases. The ratio of surface to bulk viscosity defines a viscous length scale. Hydrodynamic interactions split the motion of the domains into different modes of diffusion. The measurability of surface shear viscosities from any mode of diffusion is limited to viscous length scales between the radius of the domains and the radius of the vesicle. The measurability of the surface shear viscosity results from the sensitivity of the diffusion to surface shear viscosities and from sufficient spatial resolution to resolve the diffusive motion. Switching between the various modes of diffusion is a trade between sensitivity gained and resolution lost by the hydrodynamic interactions leaving the measurability unchanged. Measurability drops with the number of domains making single-domain rheology the best technique to measure surface shear viscosities. Ultimately confinement of the domains to small vesicles renders measurements of surface rheological properties with domain-tracking rheology impossible. Experiments on domains in vesicles of a mixture of dioleoylphosphatidylcholine (DOPC), dipalmytoylphosphatidylcholin (DPPC) and cholesterol (Chol) exhibit diffusion that is entirely controlled by dissipation into the water. The diffusion is suppressed compared to the diffusion of isolated domains in a flat membrane due to confinement to the curved vesicle and by hydrodynamic interactions between the domains. Effects of surface shear viscosity can be neglected.

The stroke-averaged lift forces due to various vortex rings and their mutual interactions are studied using a flapping flight vortex model (Rayner, J. Fluid Mech., vol. 91, 1979, p. 697; Ellington, Phil. Trans. R. Soc. Lond. B, vol. 305, 1984b, p. 115). The vortex system is decomposed into the wing plane (wing-linked) vortex ring, a loop closed by the bound vortex and (arc-shaped) trailing vortex and the wake (the vortex rings shed previously). Using the vorticity moment theory (Wu, AIAA J., vol. 19, 1981, p. 432) we are able to identify the roles of vortex rings in lift production or reduction and express the lift as function of areal contraction or expansion of vortex rings. The wake vortex rings induce areal contraction of the trailing vortex, which should decrease the lift, but this decrease is exactly compensated by the inducing effect of the trailing arc on the wake. The wake reduces the lift through inducing a downwash velocity on the wing plane. The lift force is shown to drop to a minimum at the second half stroke, and then increases to an asymptotic value slightly below the lift at the first half stroke, in such a way following the experimental observation of Birch & Dickinson (Nature, vol. 412, 2001, p. 729). The existence of the negative peak of lift is due to the first shed vortex ring which, just at the second half stroke, lies in the close vicinity to the wing plane, leading to a peak of the wing plane downwash velocity.

Homogeneity of turbulence generated by static grids is investigated with the help of hot-wire measurements in a wind-tunnel and direct numerical simulations based on the Lattice Bolztmann method. It is shown experimentally that Reynolds stresses and their anisotropy do not become homogeneous downstream of the grid, independent of the mesh Reynolds number for a grid porosity of 64%, which is higher than the lowest porosities suggested in the literature to realize homogeneous turbulence downstream of the grid. In order to validate the experimental observations and elucidate possible reasons for the inhomogeneity, direct numerical simulations have been performed over a wide range of grid porosity at a constant mesh Reynolds number. It is found from the simulations that the turbulence wake behind the symmetric grids is only homogeneous in its mean velocity but is inhomogeneous when turbulence quantities are considered, whereas the mean velocity field becomes inhomogeneous in the wake of a slightly non-uniform grid. The simulations are further analysed by evaluating the terms in the transport equation of the kinetic energy of turbulence to provide an explanation for the persistence of the inhomogeneity of Reynolds stresses far downstream of the grid. It is shown that the early homogenization of the mean velocity field hinders the homogenization of the turbulence field.

Mixing is studied in open-flow channels with conformally mapped wavy-wall profiles, using a point-vortex model in two-dimensional irrotational, incompressible mean flow. Unsteady dynamics of the separation bubble induced by oscillatory motion of point vortices located in the trough region produces chaotic mixing in the Lagrangian sense. Significant mass exchange between passive tracer particles inside and outside of the separation bubble forms an efficient mixing region which evolves in size as the vortex moves in the unsteady potential flow. The dynamics closely resembles that obtained by previous authors from numerical solutions of the unsteady Navier–Stokes equations for oscillatory unidirectional flow in a wavy channel. Of the wavy channels considered, the skew-symmetric form is most efficient at promoting passive mixing. Diffusion via gridless random walks increases lateral particle dispersion significantly at the expense of longitudinal particle dispersion due to the opposing effect of mass exchange at the front and rear of the particle ensemble. Active mixing in the wavy channel reveals that the fractal nature of the unstable manifold plays a crucial role in singular enhancement of productivity. Hyperbolic dynamics dominate over non-hyperbolicity which is restricted to the vortex core region. The model is simple yet qualitatively accurate, making it a potential candidate for the study of a wide range of vortex-induced transport and mixing problems.