The effect of gravity on the dynamics of non-isothermic ultra-thin two-layer films is studied in this paper. The joint action of disjoining pressure and thermocapillary forces is taken into account. The problem is considered in a long-wave approximation. The linear stability of a quiescent state and thermocapillary flows is investigated. It has been found that the influence of the upper fluid density is significantly stronger than that of the difference of fluid densities. Nonlinear flow regimes are studied by means of numerical simulations. The gravity can lead to the formation of stripes or holes instead of droplets. The two-dimensional wavy patterns are replaced by one-dimensional waves with the fronts inclined or transverse to the direction of the horizontal temperature gradient.

In this paper, solutions of free, barotropic waves around axisymmetric seamounts are derived. Even though this type of oscillation has been studied before, we revisit this problem for two main reasons: (i) the linear, barotropic, shallow-water equations with a rigid lid are now solved with no further approximations, in contrast with previous studies; (ii) the solutions are applied to a wide family of seamounts with profiles proportional to exp(rs), with r being the radial distance from the centre of the mountain and s any positive real number. (Most previous works are restricted to the special case s = 2.) The resulting dispersion relation possesses a remarkable simplicity that reveals a number of wave characteristics, for instance, the discrete wave frequencies and the angular phase speed of the waves around the seamount are easily derived as a function of the seamount shape. By varying the shape parameter one can study trapped waves around flat-topped seamounts or guyots (s > 2) or sharp, cone-shaped topographies (s < 2).

The hybrid Euler–Lagrange (HEL) description of fluid mechanics, pioneered largely by Andrews & McIntyre (J. Fluid Mech., vol. 89, 1978, pp. 609–646), has had to face the fact, in common with all Lagrangian descriptions of fluid motion, that the variables used do not describe conditions at the coordinate x, upon which they depend, but conditions elsewhere at some displaced position xL(x, t) = x + ξ(x, t), generally dependent on time t. To address this issue, we employ ‘Lie dragging’ techniques of general tensor calculus to extend a method introduced by Moffatt (J. Fluid Mech., vol. 166, 1986, pp. 359–378) in the fluid dynamic context, whereby the point x is dragged to xL(x, t) by a ‘fictitious steady flow’ η*(x, t) in a unit of ‘fictitious time’. Whereas ξ(x, t) is a Lagrangian concept intimately linked to the location xL(x, t), the ‘dragging velocity’ η*(x, t) has an essentially Eulerian character, because it describes the fictitious velocity at x itself. For the case of constant-density fluids, we show, using solenoidal η*(x, t) instead of solenoidal ξ(x, t), how the HEL theory can be cast into Eulerian form. A useful aspect of this Eulerian development is that the mean flow itself remains solenoidal, a feature that traditional HEL theories lack. Our method realizes the objective sought by Holm (Physica D, vol. 170, 2002, pp. 253–286) in his derivation of the Navier–Stokes–α equation, which is the basis of one of the methods currently employed to represent the sub-grid scales in large-eddy simulations. His derivation, based on expansion to second order in ξ, contained an error which, when corrected, implied a violation of Kelvin's theorem on the constancy of circulation in inviscid incompressible fluid. We show that this is rectified when the expansion is in η* rather than ξ, Kelvin's theorem then being satisfied to all orders for which the expansion converges. We discuss the implications of our approach using η* for the Navier–Stokes–α theory.

We present a model for the relative velocity of inertial particles in turbulent flows that provides new physical insight into this problem. Our general formulation shows that the relative velocity has contributions from two terms, referred to as the ‘generalized acceleration’ and ‘generalized shear’, because they reduce to the well-known acceleration and shear terms in the Saffman–Turner limit. The generalized shear term represents particles' memory of the flow velocity difference along their trajectories and depends on the inertial particle pair dispersion backward in time. The importance of this backward dispersion in determining the particle relative velocity is emphasized. We find that our model with a two-phase separation behaviour, an early ballistic phase and a later tracer-like phase, as found by recent simulations for the forward (in time) dispersion of inertial particle pairs, gives good fits to the measured relative speeds from simulations at low Reynolds numbers. In the monodisperse case with identical particles, the generalized acceleration term vanishes and the relative velocity is determined by the generalized shear term. At large Reynolds numbers, our model gives a St1/2-dependence of the relative velocity on the Stokes number St in the inertial range for both the ballistic behaviour and the Richardson separation law. This leads to the same inertial-range scaling for the two-phase separation that well fits the simulation results. Our calculations for the bidisperse case show that, with the friction timescale of one particle fixed, the relative speed as a function of the other particle's friction time has a dip when the two timescales are similar. This indicates that similar-size particles tend to have stronger velocity correlation than different ones. We find that the primary contribution at the dip, i.e. for similar particles, is from the generalized shear term, while the generalized acceleration term is dominant for particles of very different sizes. Future numerical studies are motivated to check the accuracy of the assumptions made in our model and to investigate the backward-in-time dispersion of inertial particle pairs in turbulent flows.

The nonlinear evolution of waves propagating on a spatially varying opposing current has been observed in a wave–current flume. Regular waves with different initial periods and different initial steepness, s (0.05 < s < 0.19), were generated and observed. Frequency downshift, even with very small initial steepness, was identified. As expected, it was found that opposing currents can have significant interactions with wavetrains. The ultimate frequency downshift increases with the increase in initial steepness. The evolution of frequency modulation was observed via the instantaneous frequency extracted by the Morlet-wavelet transform. The instantaneous frequency showed that often the process of frequency downshift can be local in time and gradual, but abrupt changes of local frequency were also detected. The presence of an opposing current can gradually block the primary wave energy and destroy the conservation of the wave action at downwave locations, thus increasing the asymmetric modulation and accelerating the effective frequency downshift.

In this paper, we present new theoretical and experimental results on the elliptical instability in a liquid metal contained in a rotating deformable cylinder in the presence of an imposed magnetic field. The imposed field, which is aligned with the rotating axis, has a double interest. On the one hand, it permits an analysis of the inertial waves excited by the elliptical instability by measuring their induced magnetic fields. On the other hand, it permits the control of the instability by acting on the Joule damping. In this paper, firstly an analytical calculation of the magnetic field induced by the flow and its associated Joule damping is presented. Also, the linear and weakly nonlinear theories of the elliptical instability are extended to include magnetic field effects. Then, the description of the experiments starts by the presentation of the effect of the imposed magnetic field strength. Close to the instability threshold, both super- and subcritical bifurcations are identified. When the imposed field is decreased, we observe a transition towards complex nonlinear evolutions that we describe with the help of two-dimensional phase diagrams. In a second set of experiments, we vary the eccentricity of the elliptic deformation over a large range in order to demonstrate that far from the instability threshold, the mean inertial wave amplitude is uncorrelated to the eccentricity. We show that, for a given eccentricity, this mean amplitude decreases when the rotation rate increases. In a last series of experiments, we focus on the description of the nonlinear evolution of an oscillatory eigenmode which is different from the principal stationary mode.

In this paper we describe the restrictions that the probability density function (p.d.f.) of the size of particles resulting from the rupture of a drop or bubble must satisfy. Using conservation of volume, we show that when a particle of diameter, D0, breaks into exactly two fragments of sizes D and D2 = (D30−D3)1/3 respectively, the resulting p.d.f., f(D; D0), must satisfy a symmetry relation given by D22f(D; D0) = D2f(D2; D0), which does not depend on the nature of the underlying fragmentation process. In general, for an arbitrary number of resulting particles, m(D0), we determine that the daughter p.d.f. should satisfy the conservation of volume condition given by m(D0) ∫0D0 (D/D0)3f(D; D0) dD = 1. A detailed analysis of some contemporary fragmentation models shows that they may not exhibit the required conservation of volume condition if they are not adequately formulated. Furthermore, we also analyse several models proposed in the literature for the breakup frequency of drops or bubbles based on different principles, g(ϵ, D0). Although, most of the models are formulated in terms of the particle size D0 and the dissipation rate of turbulent kinetic energy, ϵ, and apparently provide different results, we show here that they are nearly identical when expressed in dimensionless form in terms of the Weber number, g*(Wet) = g(ϵ, D0) D2/30 ϵ−1/3, with Wet ~ ρ ϵ2/3D05/3/σ, where ρ is the density of the continuous phase and σ the surface tension.

The relationship between asymptotic descriptions of vortex–wave interactions and more recent work on ‘exact coherent structures’ is investigated. In recent years immense interest has been focused on so-called self-sustained processes in turbulent shear flows where the importance of waves interacting with streamwise vortex flows has been elucidated in a number of papers. In this paper, it is shown that the so-called ‘lower branch’ state which has been shown to play a crucial role in these self-sustained processes is a finite Reynolds number analogue of a Rayleigh vortex–wave interaction with scales appropriately modified from those for external flows to Couette flow, the flow of interest here. Remarkable agreement between the asymptotic theory and numerical solutions of the Navier–Stokes equations is found even down to relatively small Reynolds numbers, thereby suggesting the possible importance of vortex–wave interaction theory in turbulent shear flows. The relevance of the work to more general shear flows is also discussed.

This paper is devoted to the theoretical description of the dynamics of a rim formed by capillary forces at the edge of a free, thin liquid sheet. The rim dynamics are described using a quasi-one-dimensional approach accounting for the inertia of the liquid in the rim and for the liquid flow entering the rim from the sheet, surface tension and viscous stresses. The governing equations are derived from the mass, momentum and moment-of-momentum-balance equations of the rim. The theory provides a basis from which to analyse the linear stability of a straight line rim bounding a planar liquid sheet. The combined effect of the axisymmetric disturbances of the radius of the rim cross-section as well as of the transverse disturbances of the rim centreline is considered. The effect of the viscosity, relative film thickness and rim deceleration are investigated. The predicted wavelength of the most unstable mode is always very similar to the Rayleigh wavelength of the instability of an infinite cylindrical jet. This prediction is confirmed by various experimental data found in the literature. The maximum rate of growth of rim disturbances depends on all the parameters of the problem; however, the most pronounced effect can be attributed to the rim deceleration. This conclusion is confirmed by nonlinear simulations of rim deformation.

The approach to a wall of a non-Brownian rigid spherical particle, settling in a viscous fluid with a Reynolds number of the order of unity, is studied experimentally. Far from the wall, the fluid motion around the particle is driven by inertia and viscosity forces. The particle Stokes number is also of the order of unity, so that the particle motion far from the wall is driven by inertia. In the close vicinity of the wall, however, the particle–wall hydrodynamic interaction decelerates the particle significantly. An interferometric device is used to measure the vertical displacement of a millimetric size spherical bead at distances from the wall smaller than 0.1 sphere radius, with a spatial resolution of 100 nm. For the range of impact Stokes number (St*, based on the limit velocity of the sphere in an unbounded fluid) explored here (up to St* ≅ 5), the measurements reveal that a small region of negligible particle inertia still exists just prior to contact of the sphere with the wall. In this lubrication-like region, the particle velocity decreases linearly with decreasing particle–wall distance and vanishes at contact, ruling out the possibility of a rebound. The vertical extent of this region decreases with increasing Stokes number and is e.g. only 10 μm large at impact Stokes number St* ≅ 5.

A high-order direct numerical simulation code (The Pencil Code) has been used together with the immersed boundary method on a Cartesian grid to simulate particle impaction on a cylinder in a crossflow. The direct numerical scheme concerns only the fluid flow, into which the particles are subsequently coupled through a one-way drag-coefficient law. The immersed boundary method is extended to work with high-order discretization, and the particle impaction efficiency has been measured for Stokes numbers ranging from 0.001 to 40 for a range of different Reynolds numbers. Three modes of impaction on the front side of the cylinder are identified, where, for the large-Stokes-number mode (St > 0.3), an alternative to the traditional Stokes number is presented that provides better scaling. The intermediate impaction mode has a very steep decrease in impaction efficiency as the Stokes number is decreased, and this is identified as the range of Stokes numbers where the viscous boundary layer starts to take effect. The third mode of front-side impaction is for the very small particles with St < 0.1 exactly following the flow but impacting on the cylinder due to their finite radii. There will not be any capture on the front side of the cylinder for impact angles larger than ~56° for this mode. Finally, it is found that the particle impaction on the back side of the cylinder is strongly dependent on the flow Reynolds number, where large Reynolds numbers lead to larger impaction efficiencies. The upper limiting Stokes number of back-side impaction is around 0.13, apparently irrespective of the Reynolds number.

We introduce a coupled multiscale, multiphysics method (CM3) for solving for the behaviour of rarefied gas flows. The approach is to solve the kinetic equation for rarefied gases (the Boltzmann equation) over a very short interval of time in order to obtain accurate estimates of the components of the stress tensor and heat-flux vector. These estimates are used to close the conservation laws for mass, momentum and energy, which are subsequently used to advance continuum-level flow variables forward in time. After a finite time interval, the Boltzmann equation is solved again for the new continuum field, and the cycle is repeated. The target applications for this type of method are transition-regime gas flows for which standard continuum models (e.g. Navier–Stokes equations) cannot be used, but solution of Boltzmann's equation is prohibitively expensive. The use of molecular-level data to close the conservation laws significantly extends the range of applicability of the continuum conservation laws. In this study, the CM3 is used to perform two proof-of-principle calculations: a low-speed Rayleigh flow and a thermal Fourier flow. Velocity, temperature, shear-stress and heat-flux profiles compare well with direct-simulation Monte Carlo solutions for various Knudsen numbers ranging from the near-continuum regime to the transition regime. We discuss algorithmic problems and the solutions necessary to implement the CM3, building upon the conceptual framework of the heterogeneous multiscale methods.

In a recent paper (Megerian et al., J. Fluid Mech., vol. 593, 2007, pp. 93–129), experimental exploration of the behaviour of transverse-jet near-field shear-layer instabilities suggests a significant change in the character of the instability as jet-to-crossflow velocity ratios R are reduced below a critical range. The present study provides a detailed exploration of and additional insights into this transition, with quantification of the growth of disturbances at various locations along and about the jet shear layer, frequency tracking and response of the transverse jet to very strong single-mode forcing, creating a ‘lock-in’ response in the shear layer. In all instances, there is clear evidence that the flush transverse jet's near-field shear layer becomes globally unstable when R lies at or below a critical range near 3. These findings have important implications for and provide the underlying strategy by which active control of the transverse jet may be developed.

We investigate swash on an erodible beach using the one-dimensional shallow-water equations fully coupled to a bed-evolution (Exner) equation. In particular, the dam-break/bore-collapse initial condition of Shen & Meyer (J. Fluid Mech., vol. 16, 1963, pp. 113–125) and Peregrine & Williams (J. Fluid Mech., vol. 440, 2001, pp. 391–399) is investigated using a numerical model based on the method of characteristics. A sediment-transport formula (cubic in velocity u: Au3) is used here; this belongs to a family of sediment-transport formulae for which Pritchard & Hogg (Coastal Engng, vol. 52, 2005, pp. 1–23) showed that net sediment transport under the Shen & Meyer (1963) bore collapse is offshore throughout the swash zone when a non-erodible bed is considered. It is found that full coupling with the beach, although still resulting in the net offshore transport of sediment throughout the swash zone, leads to a large reduction in the net offshore transport of sediment from the beach face. This is particularly true for the upper third of the swash zone. Moreover, in contradistinction to swash flows over non-erodible beds, flows over erodible beaches are unique to the bed mobility and porosity under consideration; this has very important implications for run-up predictions. The conclusion is that it is essential to consider full coupling of water and bed motions (i.e. full morphodynamics) in order to understand and predict sediment transport in the swash, regardless of other physical effects (e.g. turbulence, infiltration, pre-suspended sediment, etc.).

We investigate statistics of large-scale structures from large-eddy simulation (LES) of turbulent channel flow at friction Reynolds numbers Reτ = 2K and 200K (where K denotes 1000). In order to capture the behaviour of large-scale structures properly, the channel length is chosen to be 96 times the channel half-height. In agreement with experiments, these large-scale structures are found to give rise to an apparent amplitude modulation of the underlying small-scale fluctuations. This effect is explained in terms of the phase relationship between the large- and small-scale activity. The shape of the dominant large-scale structure is investigated by conditional averages based on the large-scale velocity, determined using a filter width equal to the channel half-height. The conditioned field demonstrates coherence on a scale of several times the filter width, and the small-scale–large-scale relative phase difference increases away from the wall, passing through π/2 in the overlap region of the mean velocity before approaching π further from the wall. We also found that, near the wall, the convection velocity of the large scales departs slightly, but unequivocally, from the mean velocity.

The mechanism of wake-induced vibrations (WIV) of a pair of cylinders in a tandem arrangement is investigated by experiments. A typical WIV response is characterized by a build-up of amplitude persisting to high reduced velocities; this is different from a typical vortex-induced vibration (VIV) response, which occurs in a limited resonance range. We suggest that WIV of the downstream cylinder is excited by the unsteady vortex–structure interactions between the body and the upstream wake. Coherent vortices interfering with the downstream cylinder induce fluctuations in the fluid force that are not synchronized with the motion. A favourable phase lag between the displacement and the fluid force guarantees that a positive energy transfer from the flow to the structure sustains the oscillations. If the unsteady vortices are removed from the wake of the upstream body then WIV will not be excited. An experiment performed in a steady shear flow turned out to be central to the understanding of the origin of the fluid forces acting on the downstream cylinder.

A fluid droplet located on a superhydrophobic surface makes contact with the surface only at small isolated regions, and is mostly in contact with the surrounding air. As a result, a fluid in motion near such a surface experiences very low friction, and superhydrophobic surfaces display strong drag reduction in the laminar regime. Here we consider theoretically a superhydrophobic surface composed of circular posts (so-called fakir geometry) located on a planar rectangular lattice. Using a superposition of point forces with suitably spatially dependent strength, we derive the effective surface-slip length for a planar shear flow on such a fakir-like surface as the solution to an infinite series of linear equations. In the asymptotic limit of small surface coverage by the posts, the series can be interpreted as Riemann sums, and the slip length can be obtained analytically. For posts on a square lattice, our analytical prediction of the dimensionless slip length, in the low surface coverage limit, is in excellent quantitative agreement with previous numerical computations.

The relative dispersion of a scalar plume is examined experimentally. A passive fluorescent tracer is continuously released from a flush-bed mounted source into the turbulent boundary layer of a laboratory-generated open channel flow. A two-dimensional particle image velocimetry–laser-induced florescence (PIV–LIF) technique is applied to measure the instantaneous horizontal velocity and concentration fields. Measured results are used to investigate the relationship between the boundary-layer turbulence and the evolution of the distance-neighbour function, namely the probability density distribution of the separation distance between two marked fluid particles within a cloud of particles. Special attention is paid to the hypothesis that a diffusion equation can describe the evolution of the distance-neighbour function. The diffusion coefficient in such an equation, termed the ‘relative diffusivity’, is directly calculated based on the concentration distribution. The results indicate that the relative diffusivity statistically depends on particle separation lengths instead of the overall size of the plume. Measurements at all stages of the dispersing plume collapse onto a single curve and follow a 4/3 power law in the inertial subrange. The Richardson–Obukhov constant is estimated from the presented dataset. The relationship between the one-dimensional (1D) representation of the distance-neighbour function and its three-dimensional (3D) representation is discussed. An extended model for relative diffusivity beyond the inertial subrange is proposed based on the structure of the turbulent velocity field, and it agrees well with measurements. The experimental evidence implies that, while the diffusion of the distance-neighbour function is completely determined by the underlying turbulence, the overall growth rate of the plume is affected by both the turbulent flow and its actual concentration distribution.

For a strictly inviscid barotropic flow with conservative body forces, the Helmholtz vorticity theorem shows that material or Lagrangian surfaces which are vortex surfaces at time t = 0 remain so for t > 0. In this study, a systematic methodology is developed for constructing smooth scalar fields φ(x, y, z, t = 0) for Taylor–Green and Kida–Pelz velocity fields, which, at t = 0, satisfy ω·∇φ = 0. We refer to such fields as vortex-surface fields. Then, for some constant C, iso-surfaces φ = C define vortex surfaces. It is shown that, given the vorticity, our definition of a vortex-surface field admits non-uniqueness, and this is presently resolved numerically using an optimization approach. Additionally, relations between vortex-surface fields and the classical Clebsch representation are discussed for flows with zero helicity. Equations describing the evolution of vortex-surface fields are then obtained for both inviscid and viscous incompressible flows. Both uniqueness and the distinction separating the evolution of vortex-surface fields and Lagrangian fields are discussed. By tracking φ as a Lagrangian field in slightly viscous flows, we show that the well-defined evolution of Lagrangian surfaces that are initially vortex surfaces can be a good approximation to vortex surfaces at later times prior to vortex reconnection. In the evolution of such Lagrangian fields, we observe that initially blob-like vortex surfaces are progressively stretched to sheet-like shapes so that neighbouring portions approach each other, with subsequent rolling up of structures near the interface, which reveals more information on dynamics than the iso-surfaces of vorticity magnitude. The non-local geometry in the evolution is quantified by two differential geometry properties. Rolled-up local shapes are found in the Lagrangian structures that were initially vortex surfaces close to the time of vortex reconnection. It is hypothesized that this is related to the formation of the very high vorticity regions.

The approach of combining computational fluid dynamics (CFD) for continuum fluid and the discrete element method (DEM) for discrete particles has been increasingly used to study the fundamentals of coupled particle–fluid flows. Different CFD–DEM models have been used. However, the origin and the applicability of these models are not clearly understood. In this paper, the origin of different model formulations is discussed first. It shows that, in connection with the continuum approach, three sets of formulations exist in the CFD–DEM approach: an original format set I, and subsequent derivations of set II and set III, respectively, corresponding to the so-called model A and model B in the literature. A comparison and the applicability of the three models are assessed theoretically and then verified from the study of three representative particle–fluid flow systems: fluidization, pneumatic conveying and hydrocyclones. It is demonstrated that sets I and II are essentially the same, with small differences resulting from different mathematical or numerical treatments of a few terms in the original equation. Set III is however a simplified version of set I. The testing cases show that all the three models are applicable to gas fluidization and, to a large extent, pneumatic conveying. However, the application of set III is conditional, as demonstrated in the case of hydrocyclones. Strictly speaking, set III is only valid when fluid flow is steady and uniform. Set II and, in particular, set I, which is somehow forgotten in the literature, are recommended for the future CFD–DEM modelling of complex particle–fluid flow.