When a wave breaks, the tip forms a liquid sheet which impinges the base and creates an air cavity which breaks into bubbles. Gomez-Ledesma, Kiger & Duncan (J. Fluid Mech., this issue, vol. 680, 2011, pp. 5–30) have conducted a nice experiment on this problem, enabling them to discuss both the inclination of the jet and the effect of its translation. This work has interesting links with other transient cavities.

An experimental study on the impact of a translating two-dimensional transient jet on an initially quiescent liquid pool is studied experimentally using high-speed cinematic visualization and particle image velocimetry methods. Six jet conditions (covering a range of jet thicknesses, velocities and inclination angles relative to vertical) are considered, with measurements performed over a range of horizontal translation speeds for each jet condition. For all conditions studied herein, the jet penetrates into the pool and forms two craters – one upstream and one downstream of the jet. Gravity acts to close these craters, which after a short time pinch off at intermediate depths, thereby entrapping cavities of air. The translation speed of the jet is found to have a dramatic effect on the cavity shapes, pinch-off depths and pinch-off times. A simple theory based on a potential flow and a hydrostatically driven collapse is used to model this flow, and the resulting jet tip trajectories and cavity shapes compare favourably with the experimental data.

Mitchell originally proposed that an asymmetric ion flux across an organism's membrane could generate electric fields that drive locomotion. Although this locomotion mechanism was later rejected for some species of bacteria, engineered Janus particles have been realized that can swim due to ion fluxes generated by asymmetric electrochemical reactions. Here we present governing equations, scaling analyses and numerical simulations that describe the motion of bimetallic rod-shaped motors in hydrogen peroxide solutions due to reaction-induced charge auto-electrophoresis. The coupled Poisson–Nernst–Planck–Stokes equations are numerically solved using Frumkin-corrected Butler–Volmer equations to represent electrochemical reactions at the rod surface. Our simulations show strong agreement with the scaling analysis and experiments. The analysis shows that electrokinetic locomotion results from electro-osmotic fluid slip around the nanomotor surface. The electroviscous flow is driven by electrical body forces which are generated from a coupling of a reaction-induced dipolar charge density distribution and the electric field it creates. The magnitude of the electroviscous velocity increases quadratically with the surface reaction rate for an uncharged motor, and linearly when the motor supports a finite surface charge.

The four-dimensional (4D) incompressible Navier–Stokes equations are solved numerically for the plane channel geometry. The fourth spatial coordinate is introduced formally to be homogeneous and mathematically orthogonal to the others, similar to the spanwise coordinate. Exponential growth of small 4D perturbations superimposed onto 3D turbulent solutions was observed in the Reynolds number range from Re = 4000 to Re = 10000. The growth rate of small 4D perturbations expressed in wall units was found to be λ+4D = 0.016 independent of Reynolds number. Nonlinear evolution of 4D perturbations leads either to attenuation of turbulence and relaminarization or to establishment of a self-sustained 4D turbulent solution (4D turbulent flow). Both results on flow evolution were obtained at the lowest Reynolds number, depending on the grid resolution, pointing to the proximity of Re = 4000 as the critical Reynolds number for 4D turbulence. Self-sustained 4D turbulence appeared to be less intense compared with 3D turbulence in terms of mean wall friction, which is about 55% of that predicted by the empirical Dean law for turbulent channel flow at all Reynolds numbers considered. Thus, the law of resistance of 4D turbulent channel flow can be expressed as Cf = 0.04Re−0.25.

In this paper we present an experimental study of the stabilization of gas-fluidized beds of magnetic powders by application of a cross-flow magnetic field. The powders tested consist of magnetite and steel powders in a range of particle size dp between 35 and 110 μm, allowing us to investigate the effect of particle size and material properties on magnetic stabilization. In the operation mode employed by us the magnetic field is applied to the unstable bubbling bed and the gas velocity is slowly decreased. According to our observations, the bed is stabilized at a critical gas velocity by the jamming of particle chains formed during bubbling because of the attractive forces induced between the magnetized particles, which are thus responsible for stabilization. Although the magnetic field is applied in the horizontal direction, these chains are mechanically stable at orientations close to the gas flow direction, in agreement with the prediction of an unconfined chain model based on the balance between gas flow shear and interparticle magnetic force fm. Since fm is increased as dp is increased, the critical gas velocity at marginal stability vc for a fixed field strength B is seen to increase with dp. As the gas velocity v0 is decreased below vc, there is a rearrangement of the structure depending on particle size. Restructuring of the bed depends on particle size as derived from measurements of its permeability to the gas flow, which causes the yield stress to be a function of particle size. It is also inferred from our results that natural agglomeration of fine particles (in the absence of a magnetic field) due to van der Waals forces enhances the yield stress of the magnetically stabilized bed. From our experimental results it is concluded that structural effects, as affected by operating conditions and material properties, play a main role in the rheology of the stabilized magnetofluidized bed (MFB).

In this paper the collective oscillations of a bubble cloud in an acoustic field are theoretically analysed with concepts and techniques of condensed matter physics. More specifically, we will calculate the eigenmodes and their excitabilities, eigenfrequencies, densities of states, responses, absorption and participation ratios to better understand the collective dynamics of coupled bubbles and address the question of possible localization of acoustic energy in the bubble cloud. The radial oscillations of the individual bubbles in the acoustic field are described by coupled linearized Rayleigh–Plesset equations. We explore the effects of viscous damping, distance between bubbles, polydispersity, geometric disorder, size of the bubbles and size of the cloud. For large enough clusters, the collective response is often very different from that of a typical mode, as the frequency response of each mode is sufficiently wide that many modes are excited when the cloud is driven by ultrasound. The reason is the strong effect of viscosity on the collective mode response, which is surprising, as viscous damping effects are small for single-bubble oscillations in water. Localization of acoustic energy is only found in the case of substantial bubble size polydispersity or geometric disorder. The lack of localization for a weak disorder is traced back to the long-range 1/r interaction potential between the individual bubbles. The results of the present paper are connected to recent experimental observations of collective bubble oscillations in a two-dimensional bubble cloud, where pronounced edge states and a pronounced low-frequency response had been observed, both consistent with the present theoretical findings. Finally, an outlook to future possible experiments is given.

The flow field of an incompressible viscous fluid in a precessing sphere is investigated by the asymptotic analysis for large Reynolds numbers and small Poincaré numbers. The long-standing unsolved equation (Roberts & Stewartson Astrophys. J., vol. 137, 1963, p. 777) for the velocity in the critical region of the boundary layer is solved for the first time in the literature, which enables us to describe explicitly the structure of the conical shear layers spawned from the critical regions into the interior inviscid region. Most of the flux between the boundary layer and the interior is taking place through these conical shear layers. The velocity field in the whole sphere, expanded in a power series of the Poincaré number, is quantitatively determined up to the first order, leaving the solid-body-rotation component to the next-order analysis.

An experimental study was conducted to examine the interfacial waves in two-layer exchange flows downslope of a bottom sill. The objective of the study was to understand the generation mechanisms of different interfacial waves and the generation conditions and development of Kelvin–Helmholtz (KH) instabilities in spatially varying flow field. During the flow regime of the steady maximal exchange, low-frequency oscillation of the density interface above the sill crest was observed, which resulted in periodic piling and releasing of the lower-layer fluid. This low-frequency oscillation led to the development of large-scale downslope waves that caused significant interfacial entrainment. The periodic time of those piling and releasing events was found to be inversely proportional to the square root of the reduced gravitational acceleration and to be related to the baroclinic forcing of the basin internal seiching. During late sub-maximal exchange, regular and frequent KH instabilities were generated at the interface down the slope at bulk Richardson number of approximately 0.07–0.21, which is considerably larger than previous theoretical predictions for parallel flows with a sharp density interface. Two different growth patterns of those KH instabilities were observed. Measurements of their initial growth rate and the wavelength were also obtained and compared to the theoretical predictions.

We study the propagation of premixed flames in two-dimensional homogeneous isotropic turbulence using a Navier–Stokes/front-capturing methodology within the context of hydrodynamic theory. The flame is treated as a thin layer separating burnt and unburnt gases, of vanishingly small thickness, smaller than the smallest fluid scales. The method is thus suitable to investigate the flame propagation in the wrinkled flamelet regime of turbulent combustion. A flow-control system regulates the mean position of the flame and the incident turbulence intensity. In this context we study the individual effects of turbulence intensity, turbulence scale, thermal expansion, hydrodynamic strain and hydrodynamic instability on the propagation characteristics of the flame. Results are obtained assuming positive Markstein length, corresponding to lean hydrocarbon–air or rich hydrogen–air mixtures. For stable planar flames we find a quadratic dependence of turbulent speed on turbulence intensity. Upon onset of hydrodynamic instability, corrugated structures replace the planar conformation and we observe a greater resilience to turbulence, the quadratic scaling being replaced by scaling exponents less than one. Such resilience is also confirmed by the observation of a threshold turbulence intensity below which the propagation speed of corrugated flames is indistinguishable from the laminar speed. Turbulent speed is found to increase and later plateau with increasing thermal expansion, this affecting the average flame displacement but not the mean flame curvature. In addition, turbulence integral scale is also observed to affect the propagation of the flame with the existence of an intermediate scale maximizing the turbulent speed. This maximizing scale is smaller for corrugated flames than it is for planar flames, implying that small eddies that will be unable to significantly perturb a planar front could be rather effective in perturbing a corrugated flame. Turbulent planar flames, and more so corrugated flames, were observed to experience a positive mean hydrodynamic strain, which was explained in terms of the overwhelming mean contribution of the normal component of strain. The positive straining causes a decrease in the mean laminar propagation speed which in turn can decrease the turbulent speed. The effect of the flame on the incident turbulent field was examined in terms of loss of isotropy and vorticity destruction by thermal expansion. The latter can be mitigated by a baroclinic vorticity generation which is enhanced for corrugated flames.

The use of tank-treading as a means of propulsion for microswimmers in viscous shear flows is taken into account. We discuss the possibility of a vesicle to control the drift in an external shear flow, by locally varying the bending rigidity of its membrane. By analytical calculation in the quasi-spherical limit, the stationary shape and the orientation of the tank-treading vesicle in the external flow are determined, working to lowest order in the membrane inhomogeneity. The membrane inhomogeneity acts in the shape evolution equation as an additional force term, which can be used to balance the effect of the hydrodynamic stresses, thus allowing the vesicle to assume shapes and orientations that are impossible otherwise. The vesicle shapes and orientations required for migration transverse to the flow, together with the bending rigidity profiles leading to such shapes and orientations, are determined. Considering the variations in the concentration experienced during tank-treading, a simple model is presented, in which a vesicle is able to migrate up or down the gradient of a concentration field by stiffening or softening of its membrane.

The response of lean (ϕ ≤ 0.4) premixed hydrogen flames to maintained homogeneous isotropic turbulence is investigated using detailed numerical simulation in an idealised three-dimensional configuration over a range of Karlovitz numbers from 10 to 1562. In particular, a focus is placed on turbulence sufficiently intense that the flames can no longer be considered to be in the thin reaction burning regime. This transition to the so-called distributed burning regime is characterised through a number of diagnostics, and the relative roles of molecular and turbulent mixing processes are examined. The phenomenology and statistics of these flames are contrasted with a distributed thermonuclear flame from a related astrophysical study.

This paper presents an exact formula for calculating the fourth-moment tensor from the second-moment tensor for the three-dimensional Jeffery's equation. Although this approach falls within the category of a moment tensor closure, it does not rely upon an approximation, either analytic or curve fit, of the fourth-moment tensor as do previous closures. This closure is orthotropic in the sense of Cintra & Tucker (J. Rheol., vol. 39, 1995, p. 1095), or equivalently, a natural closure in the sense of Verleye & Dupret (Developments in Non-Newtonian Flow, 1993, p. 139). The existence of these explicit formulae has been asserted previously, but as far as the authors know, the explicit forms have yet to be published. The formulae involve elliptic integrals, and are valid whenever fibre orientation was isotropic at some point in time. Finally, this paper presents the fast exact closure, a fast and in principle exact method for solving Jeffery's equation, which does not require approximate closures nor the elliptic integral computation.

Although experimental and theoretical studies have been performed to identify the effects of elastic waves on multi-phase flow in porous structures, the literature lacks finely tuned experiments at the micro-scale. This paper reports observations and critical analysis of immiscible displacement in micro-scale porous media under ultrasonic energy. A number of experiments are performed on homogeneous and heterogeneous micromodels for varying wave frequency and power, initial water saturation, wettability and injection rates. We show that ultrasonic radiation influences the displacement pattern and yields lower residual non-wetting phase (oil) behind when low injection rates are applied. Higher wave frequency results in faster recovery of oil, but the ultimate recovery is controlled mainly by wave intensity. The presence of initial water saturation has a positive effect on the displacement, especially in an oil-wet medium. Of the possible mechanisms suggested for recovery enhancement under ultrasonic radiation, deformation of pore walls and change in fluid properties due to heating are not an issue in these experiments but other mechanisms including coalescence of oil droplets under oscillation, reduction of wetting films, adherence to grains and the peristaltic movement of fluids due to mechanical vibration were observed to be effective and are discussed in the analysis of the visual observations.

A numerical study of flow across a row of transversely oscillating square cylinders (of diameter d) has been undertaken using the lattice Boltzmann method, for a better understanding of fluid–structure interaction problems. The effects of cylinder oscillation frequency ratio (fe/fo, where fe is the cylinder oscillation frequency and fo is the corresponding vortex shedding frequency for stationary row of cylinders), amplitude ratio (A/d), non-dimensional spacing between the cylinders (s/d) and Reynolds number (Re) on ensuing flow regimes and flow parameters have been studied to understand the flow physics. Six different flow regimes observed in this study are the quasi-periodic non-lock-on-I, synchronous lock-on, quasi-periodic lock-on, quasi-periodic non-lock-on-II, synchronous non-lock-on and chaotic non-lock-on. It is observed that the range of the lock-on regime depends upon the relative dominance of incoming flow and cylinder motion. Although the lock-on regime in the case of Re = 80, s/d = 4 and A/d = 0.2 is substantially larger as compared to that for a single oscillating cylinder, the range of the lock-on regime shrinks with a reduction in the cylinder spacing, increase in the Reynolds number or decrease in the oscillation amplitude. It is also observed that the wake interaction behind the cylinders weakens with an increase in fe/fo, Re, A/d or s/d, leading to the formation of independent wakes and synchronous nature of the flow. For fe/fo ≥ 1.2, independent and intact oscillating wakes are noted and an additional frequency (wake oscillation frequency) is obtained in the time series of the lift coefficient. Although it was expected that the complexity in the wake interaction would increase with cylinder oscillation or amplitude ratio, an opposite effect (that is, formation of independent wakes) is noted from the results.

The effects of viscosity on Kelvin–Helmholtz instability in a channel are studied using three different theories; a purely irrotational theory based on the dissipation method, an exact rotational theory and a hybrid irrotational–rotational theory. These new results are compared with previous results from a viscous irrotational theory. An analysis of the neutral state is conducted and its predictions are compared with experimental results related to the transition from a stratified-smooth to a stratified-wavy or slug flow. For values of the gas fraction greater than about 0.20, there is an interval of velocity differences for which the flow is unstable for an interval of wavenumbers between two cutoff wavenumbers, k− and k+. For unstable flows with a velocity difference above that interval or with gas fractions less than 0.20, k− = 0. The maximum critical relative velocity that determines the onset of instability can be found when the kinematic viscosity of the gas and liquid are the same. This critical value is surprisingly achieved when both fluids are inviscid. The neutral curves from the analyses of potential flow of viscous fluids and the hybrid method, the only theories that account for the viscosity of both fluids in this work, indicate that the critical velocity does not change with the viscosity ratio when the kinematic viscosity of the liquid is greater than a critical value. For smaller liquid viscosities, the critical relative velocity decreases.

We investigate wind-tunnel turbulence generated by both conventional and multi-scale grids. Measurements were made in a tunnel which has a large test section, so that possible side wall effects are very small and the length ensures that the turbulence has time to settle down to a homogeneous shear-free state. The conventional and multi-scale grids were all designed to produce turbulence with the same integral scale, so that a direct comparison could be made between the different flows. Our primary finding is that the behaviour of the turbulence behind our multi-scale grids is virtually identical to that behind the equivalent conventional grid. In particular, all flows exhibit a power-law decay of energy, u2 ~ t−n, where n is very close to the classical Saffman exponent of n = 6/5. Moreover, all spectra exhibit classical Kolmogorov scaling, with the spectra collapsing on the integral scales at small k, and on the Kolmogorov microscales at large k. Our results are at odds with some other experiments performed on similar multi-scale grids, where significantly higher energy decay exponents and turbulence levels have been reported.

A linear stability analysis is carried out for the interaction of an erodible sediment bed with a sediment-laden, stratified flow above the bed, such as a turbidity or bottom current. The fluid motion is described by the full, two-dimensional Navier–Stokes equations in the Boussinesq approximation, while erosion is modelled as a diffusive flux of particles from the bed into the fluid. The stability analysis shows the existence of both Tollmien–Schlichting and internal wave modes in the stratified boundary layer. For the internal wave mode, the stratified boundary layer acts as a wave duct, whose height can be determined analytically from the Brunt–Väisälä frequency criterion. Consistent with this criterion, distinct unstable perturbation wavenumber regimes exist for the internal wave mode, which are associated with different numbers of pressure extrema in the wall-normal direction. For representative turbidity current parameters, the analysis predicts unstable wavelengths that are consistent with field observations. As a key condition for instability to occur, the base flow velocity boundary layer needs to be thinner than the corresponding concentration boundary layer. For most of the unstable wavenumber ranges, the phase relations between the sediment bed deformation and the associated wall shear stress and concentration perturbations are such that the sediment waves migrate in the upstream direction, which again is consistent with field observations.

Superhydrophobic surfaces have been shown to produce significant drag reduction for both laminar and turbulent flows of water through large- and small-scale channels. In this paper, a series of experiments were performed which investigated the effect of superhydrophobic-induced slip on the flow past a circular cylinder. In these experiments, circular cylinders were coated with a series of superhydrophobic surfaces fabricated from polydimethylsiloxane with well-defined micron-sized patterns of surface roughness. The presence of the superhydrophobic surface was found to have a significant effect on the vortex shedding dynamics in the wake of the circular cylinder. When compared to a smooth, no-slip cylinder, cylinders coated with superhydrophobic surfaces were found to delay the onset of vortex shedding and increase the length of the recirculation region in the wake of the cylinder. For superhydrophobic surfaces with ridges aligned in the flow direction, the separation point was found to move further upstream towards the front stagnation point of the cylinder and the vortex shedding frequency was found to increase. For superhydrophobic surfaces with ridges running normal to the flow direction, the separation point and shedding frequency trends were reversed. Thus, in this paper we demonstrate that vortex shedding dynamics is very sensitive to changes of feature spacing, size and orientation along superhydrophobic surfaces.

When a drop is deposited on a superhydrophilic micropillar array, the upper part of the drop (referred to as the bulk) collapses while the bottom part penetrates into the gaps of the array, forming a fringe film. Here we quantify the early stage dynamics of this process using a combination of experiment and theory. We show that the circular front of the fringe film spreads like t1/2, t being time, when coupled to the bulk flow. However, the film is found to advance like t1/3 through faceted zippering in the absence of the bulk. We then show that the spreading of the bulk and the entire drop footprint follows a power law (t1/4) that is different from Washburn's law. This work can be a starting point to completely understand the spreading of liquids on superhydrophilic surfaces and opens questions specific to superwetting behaviour including the criteria to determine whether the fringe film will expand through lateral zipping or advance radially outwards.

The radial distribution function (RDF, a statistical measure of preferential concentration), and the relative velocity measured along the line-of-centres of two particles are the key statistical inputs to the collision kernel for finite-inertia particles suspended in a turbulent flow Sundaram & Collins (J. Fluid Mech., vol. 335, 1997, p. 75). In this paper, we investigate the behaviour of these two-particle statistics using direct numerical simulation (DNS) of homogeneous isotropic turbulence. While it is known that the RDF for particles of any Stokes number (St) decreases with separation distance Sundaram & Collins (J. Fluid Mech., vol. 335, 1997, p. 75), Reade & Collins (Phys. Fluids, vol. 12, 2000, p. 2530), Salazar et al. (J. Fluid Mech., vol. 600, 2008, p. 245), we observe that the peak in the RDF versus St curve shifts to higher St as we increase the separation distance. Here, St is defined as the ratio of the particle's viscous relaxation time to the Kolmogorov time-scale of the flow. Furthermore, as found in a previous study Wang, Wexler, & Zhou (J. Fluid Mech., vol. 415, 2000, p. 117), the variance of the radial relative velocity (wr) is found to increase monotonically with increasing separation distance and increasing Stokes number; however, we show for the first time that the parameteric variation of the skewness of wr with St and r/η is qualitatively similar to that of the RDF, and points to a connection between the two. We then apply low-pass filters (using three different filter scales) on the DNS velocity field in wavenumber space in order to produce ‘perfect’ large-eddy simulation (LES) velocity fields without any errors associated with subgrid-scale modelling. We present visual evidence of the effect of sharp-spectral filtering on the flow structure and the particle field. We calculate the particle statistics in the filtered velocity field and find that the RDF decreases with filtering at low St and increases with filtering at high St, similar to Fede & Simonin (Phys. Fluids, vol. 18, 2006, p. 045103). We also find that the variation of the RDF with St shifts towards higher St with filtering at all separation distances. The variance of wr is found to decrease with filtering for all St and separation distances, but the skewness of wr shows a non-monotonic response to filtering that is qualitatively similar to the RDF. We consider the variation of the RDF and moments of wr with filter scale and find that they are approximately linear in the inertial range. We demonstrate that a simple model consisting of a redefinition of the St based on the time-scale of the filtered velocity field cannot recover the unfiltered statistics. Our findings provide insight on the effect of subgrid-scale eddies on the RDF and wr, and establish the requirements of a LES model for inertial particles that can correctly predict clustering and collisional behaviour.