By what process can droplets be extracted out of the sea? This is an old problem, well-documented by precise field measurements of the size distribution of the spray aerosols. Lhuissier & Villermaux (J. Fluid Mech., this issue, vol. 696, 2012, pp. 5–44) study and characterize the bursting of an emerging bubble. They show that this single type of event can, by itself, generate the droplet size distributions in a sea spray. This is a remarkable result showing how, in a complex system, a statistical distribution can be entirely produced by the dynamics of one dominant phenomenon.

We depict and analyse the complete evolution of an air bubble formed in a water bulk, from the time it emerges at the liquid surface, up to its fragmentation into dispersed drops. To this end, experiments describing the drainage of the bubble cap film, its puncture and the resulting bursting dynamics determining the aerosol formation are conducted on tapwater bubbles. We discover that the mechanism of marginal pinching at the bubble foot and associated convection motions in the bubble cap, known as marginal regeneration, both drive the bubble cap drainage rate, and are responsible for its puncture. The resulting original film thickness evolution law in time, supplemented with considerations about the nucleation of holes piercing the film together culminate in a determination of the cap film thickness at bursting , where is the bubble cap radius of curvature, and a length which we determine. Subsequent to a hole nucleation event, the cap bursting dynamics conditions the resulting spray. The latter depends both on the bubble shape prescribed by , where is the capillary length based on gravity, and on . The mean drop size , the number of drops generated per bubble and the drop size distribution are derived, comparing well with measurements. Combined with known bubble production rates over the ocean, our findings offer an adjustable parameter-free prediction for the aerosol flux and spray structure caused by bubble bursting in this precise context.

In the present study, we investigate the scaling of relative velocity structure functions, of order two and higher, for inertial particles, both in the dissipation range and the inertial subrange using direct numerical simulations (DNS). Within the inertial subrange our findings show that contrary to the well-known attenuation in the tails of the one-point acceleration probability density function (p.d.f.) with increasing inertia (Bec et al., J. Fluid Mech., vol. 550, 2006, pp. 349–358), the opposite occurs with the velocity structure function at sufficiently large Stokes numbers. We observe reduced scaling exponents for the structure function when compared to those of the fluid, and correspondingly broader p.d.f.s, similar to what occurs with a passive scalar. DNS allows us to isolate the two effects of inertia, namely biased sampling of the velocity field, a result of preferential concentration, and filtering, i.e. the tendency for the inertial particle velocity to attenuate the velocity fluctuations in the fluid. By isolating these effects, we show that sampling is playing the dominant role for low-order moments of the structure function, whereas filtering accounts for most of the scaling behaviour observed with the higher-order structure functions in the inertial subrange. In the dissipation range, we see evidence of so-called ‘crossing trajectories’, the ‘sling effect’ or ‘caustics’, and find good agreement with the theory put forth by Wilkinson et al. (Phys. Rev. Lett., vol. 97, 2006, 048501) and Falkovich & Pumir (J. Atmos. Sci., vol. 64, 2007, 4497) for Stokes numbers greater than 0.5. We also look at the scaling exponents within the context of the model proposed by Bec et al. (J. Fluid Mech., vol. 646, 2010, pp. 527–536). Another interesting finding is that inertial particles at low Stokes numbers sample regions of higher kinetic energy than the fluid particle field, the converse occurring at high Stokes numbers. The trend at low Stokes numbers is predicted by the theory of Chun et al. (J. Fluid Mech., vol. 536, 2005, 219–251). This work is relevant to modelling the particle collision rate (Sundaram & Collins, J. Fluid Mech., vol. 335, 1997, pp. 75–109), and highlights the interesting array of phenomena induced by inertia.

The properties of turbulent mixing in a Richtmyer–Meshkov (RM) unstable fluid layer are studied under the impact of a single shock followed by a reshock wave using simultaneous velocity–density measurements to provide new insights into the physics of RM mixing. The experiments were conducted on a varicose fluid layer (heavy fluid) interposed in air (light fluid) inside a horizontal shock tube at an incident Mach number of 1.21 and a reflected reshock Mach number of 1.14. The light–heavy–light fluid layer is observed to develop a nonlinear growth pattern, with no transition to turbulence upon impact by a single shock (up to ). However, upon reshock, enhanced mixing between the heavy and light fluids along with a transition to a turbulent state characterized by the generation of significant turbulent velocity fluctuations () is observed. The streamwise and spanwise root-mean-squared velocity fluctuation statistics show similar trends across the fluid layer after reshock, with no observable preference for the direction of the shock wave motion. The measured streamwise mass flux () shows opposing signs on either side of the density peak within the fluid layer, consistent with the turbulent material transport being driven along the direction of the density gradient. Measurements of three of the six independent components of the general Reynolds stress tensor () show that the self-correlation terms and are similar in magnitude across much of the fluid layer, and much larger than the cross-correlation term . Most importantly, the Reynolds stresses () are dominated by the mean density, cross-velocity product term (), with the mass flux product and triple correlation terms being negligibly smaller in comparison. A lack of homogeneous mixing (and, possibly, a long-term imprint of the initial conditions) is observed in the spanwise turbulent mass flux measurements, with important implications for the simulation and modelling of RM mixing flows.

We study direct numerical simulations (DNS) of a jet in cross-flow at low values of the jet-to-cross-flow velocity ratio . We observe that, as the ratio increases, the flow evolves from simple periodic vortex shedding (a limit cycle) to more complicated quasi-periodic behaviour, before finally becoming turbulent, as seen in the simulation of Bagheri et al. (J. Fluid. Mech., vol. 624, 2009b, pp. 33–44). The value of at which the first bifurcation occurs for our numerical set-up is found, and shedding of hairpin vortices characteristic of a shear layer instability is observed. We focus on this first bifurcation, and find that a global linear stability analysis predicts well the frequency and initial growth rate of the nonlinear DNS at the critical value of and that good qualitative predictions about the dynamics can still be made at slightly higher values of where multiple unstable eigenmodes are present. In addition, we compute the adjoint global eigenmodes, and find that the overlap of the direct and the adjoint eigenmode, also known as a ‘wavemaker’, provides evidence that the source of the first instability lies in the shear layer just downstream of the jet.

Compressible large-eddy simulations are carried out to study the aero-optical distortions caused by Mach 0.5 flat-plate turbulent boundary layers at Reynolds numbers of , 1770 and 3550, based on momentum thickness. The fluctuations of refractive index are calculated from the density field, and wavefront distortions of an optical beam traversing the boundary layer are computed based on geometric optics. The effects of aperture size, small-scale turbulence, different flow regions and beam elevation angle are examined and the underlying flow physics is analysed. It is found that the level of optical distortion decreases with increasing Reynolds number within the Reynolds-number range considered. The contributions from the viscous sublayer and buffer layer are small, while the wake region plays a dominant role, followed by the logarithmic layer. By low-pass filtering the fluctuating density field, it is shown that small-scale turbulence is optically inactive. Consistent with previous experimental findings, the distortion magnitude is dependent on the propagation direction due to anisotropy of the boundary-layer vortical structures. Density correlations and length scales are analysed to understand the elevation-angle dependence and its relation to turbulence structures. The applicability of Sutton’s linking equation to boundary-layer flows is examined, and excellent agreement between linking equation predictions and directly integrated distortions is obtained when the density length scale is appropriately defined.

We present a theoretical and experimental study of a viscous fluid layer spreading over a deep layer of denser, inviscid fluid. Specifically, we study an axisymmetric flow produced by a vertical line source. Close to the source, the flow is controlled viscously, with a balance between radial compressive stresses and hoop stresses. Further out, the flow is driven by gradients in the buoyancy force and is resisted by viscous extensional and hoop stresses. An understanding of these different fluid-mechanical relationships is developed by asymptotic analyses for early times and for the near and far fields at late times. Confirmation of the late-time, far-field behaviour is obtained from a series of laboratory experiments in which golden syrup was injected into denser solutions of potassium carbonate. We use our mathematical solutions to discuss a physical mechanism by which horizontal viscous stresses in a spreading ice shelf, such as those in West Antarctica, can buttress the grounded ice sheet that supplies it.

We study buoyant displacement flows with two miscible fluids of equal viscosity in the regime of low Atwood number and in ducts that are inclined close to horizontal. Using a combination of experimental, computational and analytical methods, we characterize the transitions in the flow regimes between inertial- and viscous-dominated regimes, and as the displacement flow rate is gradually increased. Three dimensionless groups largely describe these flows: densimetric Froude number , Reynolds number and duct inclination . Our results show that the flow regimes collapse into regions in a two-dimensional plane. These regions are qualitatively similar between pipes and plane channels, although viscous effects are more extensive in pipes. In each regime, we are able to give a leading-order estimate for the velocity of the leading displacement front, which is effectively a measure of displacement efficiency.

The Leidenfrost effect occurs when a drop of liquid (or a sublimating solid) is levitated above a sufficiently hot surface through the action of an insulating vapour layer flowing from its bottom surface. When such a drop is levitated above a surface with parallel, asymmetric sawtooth-shaped ridges it is known to be propelled in a unique direction, or ratcheted, by the interaction of the vapour layer with the surface. Here we exploit this effect to construct a ‘ratchet trap’ for Leidenfrost drops: a surface with concentric circular ridges, each asymmetric in cross-section. A combination of experiment and theory is used to study the dynamics of drops in these traps, whose centre is a stable fixed point. Numerical analysis of the evaporating flows over a ratchet surface suggests new insights into the mechanism of motion rectification that are incorporated into the simplest equations of motion for ratchet-driven motion of a Leidenfrost body; these resemble a central force problem in celestial mechanics with mass loss and drag. A phase-plane analysis of experimental trajectories is used to extract more detailed information about the ratcheting phenomenon. Orbiting drops are found to exhibit substantial deformations; those with large internal angular momentum can even undergo binary fission. Such ratchet traps may thus prove useful in the controlled study of many properties of Leidenfrost drops.

The fundamental mechanisms responsible for the creation of electrohydrodynamically driven roll structures in free electroconvection between two plates are analysed with reference to traditional Rayleigh–Bénard convection (RBC). Previously available knowledge limited to two dimensions is extended to three-dimensions, and a wide range of electric Reynolds numbers is analysed, extending into a fully inherently three-dimensional turbulent regime. Results reveal that structures appearing in three-dimensional electrohydrodynamics (EHD) are similar to those observed for RBC, and while two-dimensional EHD results bear some similarities with the three-dimensional results there are distinct differences. Analysis of two-point correlations and integral length scales show that full three-dimensional electroconvection is more chaotic than in two dimensions and this is also noted by qualitatively observing the roll structures that arise for both low () and high electric Reynolds numbers (up to ). Furthermore, calculations of mean profiles and second-order moments along with energy budgets and spectra have examined the validity of neglecting the fluctuating electric field in the Reynolds-averaged EHD equations and provide insight into the generation and transport mechanisms of turbulent EHD. Spectral and spatial data clearly indicate how fluctuating energy is transferred from electrical to hydrodynamic forms, on moving through the domain away from the charging electrode. It is shown that is not negligible close to the walls and terms acting as sources and sinks in the turbulent kinetic energy, turbulent scalar flux and turbulent scalar variance equations are examined. Profiles of hydrodynamic terms in the budgets resemble those in the literature for RBC; however there are terms specific to EHD that are significant, indicating that the transfer of energy in EHD is also attributed to further electrodynamic terms and a strong coupling exists between the charge flux and variance, due to the ionic drift term.

In this paper we describe flow through a porous medium in a two-dimensional rectangular cavity driven by differential heating of the impermeable lower surface. The upper surface is held at constant pressure and at a constant temperature equal to the minimum temperature of the lower surface, while the sidewalls are impermeable and thermally insulated. Numerical results for general values of the Darcy–Rayleigh number and the cavity aspect ratio are compared with theoretical predictions for the small Darcy–Rayleigh number limit where the temperature field is conduction-dominated, and with a boundary-layer theory for the large Darcy–Rayleigh number limit where convection is significant. In the latter case a horizontal boundary layer near the lower surface conveys fluid to the hot end of the cavity where it rises to the upper surface in a narrow plume. Predictions are made of the vertical heat transfer through the cavity.

We numerically investigate the influence of pulsed tangential jets on the flow past a circular cylinder. To this end a spectral-Lagrangian dual approach is used on the basis of time-series data. The analysis reveals that the flow response to unsteady forcing is driven by strong interactions between shear layers and pulsed jets. The latter preferentially lead to either the lock-on regime or the quasi-steady vortex feeding regime whether the excitation frequency is of the order of, or significantly greater than, the frequency of the natural instability. The intensity of the wake vortices is mainly influenced by the momentum coefficient through the introduction of opposite-sign vorticity in the shear layers. This feature is emphasized using a modal-based time reconstruction, i.e. by reconstructing the flow field upon a specific harmonic spectrum associated with a characteristic time scale. The quasi-steady regime exhibits small-scale counter-rotating vortices that circumscribe the separated region. In the lock-on regime, atypical wake patterns such as 2P or can be observed, depending on the forcing frequency and the momentum coefficient, highlighting remarkable analogies with oscillating cylinders.

Elliptic and hyperbolic geopotential solutions, for a homogeneous distribution of potential vorticity (PV), are obtained via PV inversion in geophysical vortices. The flow in the axisymmetrical three-dimensional vortices is steady and horizontal, where the centripetal acceleration plus the Coriolis acceleration equals the pressure anomaly gradient term (gradient wind or cyclo-geostrophic balance). It is found that the family of geopotential solutions in the vortex interior is completely parameterized by the PV density in the vortex and the squared aspect ratio between the horizontal and vertical semi-axes of the ellipsoidal or hyperbolic geopotential surfaces. Thus, the PV inversion task consists of obtaining, via solution of algebraic cubic equations, the absolute vertical vorticity and vertical stratification as a function of PV and aspect ratio. It is found that there is always a critical aspect ratio, which depends on PV, beyond which the PV inversion solutions are multi-valued. The complete vorticity and stratification solutions for the different regions in the PV and aspect ratio space are obtained and analysed with emphasis on the inertial and static instability of the vortex flow.

A novel scheme for strong coupling between inertial Lagrangian point particles and a continuous Eulerian fluid phase has been developed. A full mechanical coupling can only be achieved if torque-coupling is applied along with the more conventional force-coupling. The torque vector acting from the particles on the fluid is expressed in terms of a new antisymmetric particle stress tensor which adds to the Stokes stress tensor. A strongly-coupled simulation of a turbulent channel flow laden with prolate spheroidal particles with aspect ratio 5:1 demonstrated that the inclusion of torque-coupling reduced the modulation of the turbulent flow field observed in a two-way force-coupled simulation. The spin and orientation of the spheroids were significantly affected.

Motivated by recent experimental observations, we consider a steady-state boundary layer flow with polymers in forced convection above a heated plate and study how the heat transport might be affected by the polymers. We discuss how a set of equations can be derived for the problem and how these equations can be solved numerically by an iterative scheme. By carrying out such a scheme, we find that the effect of the polymers is equivalent to producing a space-dependent effective viscosity that first increases from the zero-shear value at the plate then decreases rapidly back to the zero-shear value far from the plate. We further show that such an effective viscosity leads to a decrease in the streamwise velocity near the plate, which in turn leads to a reduction in heat transport.

The equilibrium shape of an axisymmetric liquid drop or gas bubble in an immiscible supporting liquid held under gravity against a horizontal plane rigid surface is derived. A thin film of supporting liquid remains between the drop/bubble surface and the rigid substrate at equilibrium, of thickness determined by the balance of the disjoining pressure between the drop and the substrate and the internal Laplace pressure of the drop/bubble. The interface is macroscopically flat around the drop axis out to a radius but matches smoothly into the outer shape of radius through a boundary layer region of width where is a small parameter. The outer drop shape is determined by a balance of buoyancy forces and local Laplace pressure and is roughly spherical if , where is the capillary length in the interface with a logarithmic correction due to the action of the disjoining pressure across the flattened region. With the shape determined, we calculate the drag force on this flattened bubble to lowest order in the velocity as it moves across the rigid substrate using a lubrication approximation valid to terms of as an integral over the flattened bubble surface of the hydrodynamic pressure. The lubrication theory of itself is not sufficient to determine the drag due to the divergence of that integral if the outer flow field properties are neglected. By using the known exact result for the drag force on an undistorted bubble, the drag on the flattened bubble can be computed as an integral over the lubrication region alone. We derive the drag as a series expansion in the small parameter by means of a fairly intricate boundary layer analysis. The logarithmic divergence of the translational drag with film thickness for the undistorted bubble is replaced by the stronger divergence to leading order for the flattened bubble case. We present explicit numerical results for the first few terms in the expansion for the case of an exponentially repulsive disjoining pressure, and analytic expressions for these terms in the limit of very short-range disjoining pressure forces. The results of this calculation are compared with recent work (Hodges, Jensen & Rallison, J. Fluid Mech., vol. 512, 2004, p. 95) where disjoining pressure is neglected and hydrodynamic pressure is balanced against buoyancy and Laplace forces. The limits of validity of this linear drag theory are also presented.

A Batchelor vortex represents the asymptotic solution of a trailing vortex in an aircraft wake. In this study, an unequal-strength, counter-rotating Batchelor vortex pair is employed as a model of the wake emanating from one side of an aircraft wing; this model is a direct extension of several prior investigations that have considered unequal-strength Lamb–Oseen vortices as representations of the aircraft wake problem. Both solution of the linearized Navier–Stokes equations and direct numerical simulations are employed to study the linear and nonlinear development of a vortex pair with a circulation ratio of . In contrast to prior investigations considering a Lamb–Oseen vortex pair, we note strong growth of the Kelvin mode coupled with an almost equal growth rate of the Crow instability. Three stages of nonlinear instability development are defined. In the initial stage, the Kelvin mode amplitude becomes sufficiently large that oscillations within the core of the weaker vortex are easily observable and significantly affect the profile of the weaker vortex. In the secondary stage, filaments of secondary vorticity emanate from the weaker vortex and are convected around the stronger vortex. In the tertiary stage, a transition in the dominant instability wavelength is observed from the short-wavelength Kelvin mode to the longer-wavelength Crow instability. Much of the instability growth is observed on the weaker vortex of the pair, although small perturbations in the stronger vortex are observed in the tertiary nonlinear growth phase.

We study the linear stability of the exact deep-water capillary wave solution of Crapper (J. Fluid Mech., vol. 2, 1957, pp. 532–540) subject to two-dimensional perturbations (both subharmonic and superharmonic). By linearizing a set of exact one-dimensional non-local evolution equations, a stability analysis is performed with the aid of Floquet theory. To validate our results, the exact evolution equations are integrated numerically in time and the numerical solutions are compared with the time evolution of linear normal modes. For superharmonic perturbations, contrary to Hogan (J. Fluid Mech., vol. 190, 1988, pp. 165–177), who detected two bubbles of instability for intermediate amplitudes, our results indicate that Crapper’s capillary waves are linearly stable to superharmonic disturbances for all wave amplitudes. For subharmonic perturbations, it is found that Crapper’s capillary waves are unstable, and our results generalize to the highly nonlinear regime the analysis for small amplitudes presented by Chen & Saffman (Stud. Appl. Maths, vol. 72, 1985, pp. 125–147).

The laminar free surface boundary layer beneath a solitary wave is investigated using numerical simulations. Across the boundary layer and are comparable in magnitude, where is the velocity, position and subscripts and refer to components tangential and normal to the free surface. In this region is approximately constant across the boundary layer while varies with and outside the boundary layer tends to . The numerical results are compared to analytical models and good agreement is found.

Using direct numerical simulation, we investigate stationary and homogeneous shear-driven turbulence in various stratifications, ranging from neutral to very stable. To attain and maintain a stationary flow, we throttle the mean shear so that the net production stays constant for all times. This results in a flow that is characterized solely by its mean shear and its mean buoyancy gradient, independent of initial conditions. The method of throttling is validated by comparison with experimental spectra in the case of neutral stratification. With increasing stratification comes the emergence of vertically sheared large-scale horizontal motions that preclude a straightforward interpretation of flow statistics. However, once these motions are excluded, simply by subtracting the horizontal average, the underlying flow appears amenable to the standard methods of turbulence analysis. It is shown that a direct acknowledgement of the confining influence of the periodic simulation box can lead to a meaningful physical interpretation of the large scales. Once an appropriate confinement scale is identified, many features, including horizontal spectra, flux–gradient relationships and length scales, of stratified sheared turbulence can be readily understood, both qualitatively and quantitatively, in terms of Monin–Obukhov similarity theory. Finally, the similarity-theory framework is used to interpret the scaling of the vertical diapycnal diffusivity in stratified turbulence.