Taylor's hypothesis, relating temporal to spatial fluctuations in turbulent flows is investigated using powerful numerical computations by del Álamo & Jiménez (J. Fluid Mech., 2009, this issue, vol. 640, pp. 5–26). Their results cast doubt on recent interpretations of bimodal spectra in relation to very large-scale turbulent structures in experimental measurements in turbulent shear flows.

A new method is introduced for estimating the convection velocity of individual modes in turbulent shear flows that, in contrast to most previous ones, only requires spectral information in the temporal or spatial direction over which a modal decomposition is desired, while only using local derivatives in other directions. If no spectral information is desired, the method provides a natural definition for the average convection velocity, as well as a way to estimate the accuracy of the frozen-turbulence approximation. Existing data from numerical turbulent channels at friction Reynolds numbers Reτ ≤ 1900 are used to validate the new method against classical ones, and to characterize the dependence of the convection velocity on the eddy wavelength and wall distance. The results indicate that the small scales in turbulent channels travel at the local mean velocity, while large ‘global’ modes travel at a more uniform speed proportional to the bulk velocity. To estimate the systematic deviations introduced in experimental spectra by the use of Taylor's approximation with a wavelength-independent convection velocity, a semi-empirical fit to the computed convection velocities is provided. It represents well the data throughout the Reynolds number range of the simulations. It is shown that Taylor's approximation not only displaces the large scales near the wall to shorter apparent wavelengths but also modifies the shape of the spectrum, giving rise to spurious peaks similar to those observed in some experiments. To a lesser extent the opposite is true above the logarithmic layer. The effect increases with the Reynolds number, suggesting that some of the recent challenges to the kx−1 energy spectrum may have to be reconsidered.

The deformation of a drop as it flows along the axis of a circular capillary in low Reynolds number pressure-driven flow is investigated numerically by means of boundary integral computations. If gravity effects are negligible, the drop motion is determined by three independent parameters: the size a of the undeformed drop relative to the radius R of the capillary, the viscosity ratio λ between the drop phase and the wetting phase and the capillary number Ca, which measures the relative importance of viscous and capillary forces. We investigate the drop behaviour in the parameter space (a/R, λ, Ca), at capillary numbers higher than those considered previously. If the fluid flow rate is maintained, the presence of the drop causes a change in the pressure difference between the ends of the capillary, and this too is investigated. Estimates for the drop deformation at high capillary number are based on a simple model for annular flow and, in most cases, agree well with full numerical results if λ ≥ 1/2, in which case the drop elongation increases without limit as Ca increases. If λ < 1/2, the drop elongates towards a limiting non-zero cylindrical radius. Low-viscosity drops (λ < 1) break up owing to a re-entrant jet at the rear, whereas a travelling capillary wave instability eventually develops on more viscous drops (λ > 1). A companion paper (Lac & Sherwood, J. Fluid Mech., doi:10.1017/S002211200999156X) uses these results in order to predict the change in electrical streaming potential caused by the presence of the drop when the capillary wall is charged.

The electrical streaming potential generated by a two-phase pressure-driven Stokes flow in a cylindrical capillary is computed numerically. The potential difference ΔΦ between the two ends of the capillary, proportional to the pressure difference Δp for single-phase flow, is modified by the presence of a suspended drop on the centreline of the capillary. We determine the change in ΔΦ caused by the presence of an uncharged insulating neutrally buoyant drop at a small electric Hartmann number, i.e. when the perturbation to the flow field caused by electric stresses is negligible.

The drop velocity and deformation, and the consequent changes in the pressure difference Δp and streaming potential ΔΦ, depend upon three independent parameters: the size a of the undeformed drop relative to the radius R of the capillary; the viscosity ratio λ between the drop phase and the continuous phase; and the capillary number Ca which measures the ratio of viscous to capillary forces. We investigate how the streaming potential depends on these parameters: purely hydrodynamic aspects of the problem are discussed by Lac & Sherwood (J. Fluid Mech., doi:10.1017/S0022112009991212).

The potential on the capillary wall is assumed sufficiently small so that the electrical double layer is described by the linearized Poisson–Boltzmann equation. The Debye length characterizing the thickness of the charge cloud is taken to be small compared with all other length scales, including the width of the gap between the drop and the capillary wall. The electric potential satisfies Laplace's equation, which we solve by means of a boundary integral method. The presence of the drop increases |ΔΦ| when the drop is more viscous than the surrounding fluid (λ > 1), though the change in |ΔΦ| can take either sign for λ < 1. However, the difference between ΔΦ and Δp (suitably non-dimensionalized) is always positive. Asymptotic predictions for the streaming potential in the case of a vanishingly small spherical droplet, and for large drops at high capillary numbers, agree well with computations.

This paper gives a new derivation and an analysis of long-wave model equations for the dynamics of the free surface of a body of water which has random bathymetry. This is a problem of hydrodynamical significance to coastal regions and to global-scale propagation of tsunamis, for which there may be imperfect knowledge of the detailed topography of the bottom. The surface motion is assumed to be in a long-wavelength dynamical regime, while the bottom of the fluid region is given by a stationary random process whose realizations vary over short length scales and are decorrelated on the longer principal length scale of the surface waves. Our basic conclusions are that coherent solutions propagating over a random bottom maintain basic properties of their structure over long distances, but however, the effect of the random bottom introduces uncertainty in the location of the solution profile and modifies the amplitude by random factors. It also gives rise to a random scattered component of the solution, but this does not result in the dispersion of the principal component of the solution, at least over length and time scales considered in this regime. We illustrate these results with numerical simulations.

The mathematical question is one of homogenization theory in the long-wave scaling regime, for which our work is a reappraisal of the paper of Rosales & Papanicolaou (Stud. Appl. Math., vol. 68, 1983, pp. 89–102). In particular, we derive appropriate Boussinesq and Korteweg–deVries type equations with random coefficients which describe the free-surface evolution in this regime. The derivation is performed from the point of view of perturbation theory for Hamiltonian partial differential equations with a small parameter, with a subsequent analysis of the random effects in the resulting solutions. In the analysis, we highlight the distinction between the effective equations for a fixed typical realization, for which there are coherent solitary-wave solutions, and their ensemble average, which may exhibit diffusive effects. Our results extend the prior analysis to the case of non-zero variance σ2β > 0, and furthermore the analysis identifies the canonical limit random process as a white noise with covariance σβ2δ(X − X′) and quantifies the variations in phase and amplitude of the principal and scattered components of solutions. We find that the random topography can give rise to an additional linear term in the KdV limit equations, which depends upon a skew property of the random process and whose sign affects the stability of solutions. Finally we generalize this analysis to the case in which the bottom has large-scale deterministic variations on which are superposed random fluctuations with slowly varying statistical properties.

The transport of momentum and a passive scalar (temperature) in a three-dimensional transitional wake of a heated square cylinder has been carried out through direct numerical simulations using the lattice Boltzmann method at a Reynolds number Rd = 200 (d is the cylinder diameter) and a Prandlt number of 0.7. The simulations shows that while momentum and heat are transported by vortical structures, heat is in general more effectively transported than momentum. It is argued that the nature of the structural flow is responsible for the longitudinal heat flux uθ being larger than the lateral one vθ in the wake region extending up to 45d. It was shown that a gradient transport model could, to a first-order approximation, be used to model uv but would be less accurate for modelling vθ. Also the Reynolds analogy between momentum and heat transports is not verified in this flow. The fluctuating temperature field presents thermal structures similar to the velocity structures with, however, a different spatial organization. In addition the analogy between fluctuating turbulent kinetic energy and the temperature variance is relatively well satisfied throughout the wake flow.

We study steep capillary-gravity waves that form at the interface between two stably stratified layers of immiscible liquids in a horizontally oscillating vessel. The oscillatory nature of the external forcing prevents the waves from overturning, and thus enables the development of steep waves at large forcing. They arise through a supercritical pitchfork bifurcation, characterized by the square root dependence of the height of the wave on the excess vibrational Froude number (W, square root of the ratio of vibrational to gravitational forces). At a critical value Wc, a transition to a linear variation in W is observed. It is accompanied by sharp qualitative changes in the harmonic content of the wave shape, so that trochoidal waves characterize the weakly nonlinear regime, but ‘finger’-like waves form for W ≥ Wc. In this strongly nonlinear regime, the wavelength is a function of the product of amplitude and frequency of forcing, whereas for W < Wc, the wavelength exhibits an explicit dependence on the frequency of forcing that is due to the effect of viscosity. Most significantly, the radius of curvature of the wave crests decreases monotonically with W to reach the capillary length for W = Wc, i.e. the lengthscale for which surface tension forces balance gravitational forces. For W < Wc, gravitational restoring forces dominate, but for W ≥ Wc, the wave development is increasingly defined by localized surface tension effects.

Nonlinear shallow water equations are employed to model the inviscid slumping of fluid along an inclined plane and analytical solutions for the motion are derived using the hodograph transformation to reveal the run-up and the inception of a bore on the backwash. Starting from rest, the fluid slumps along the inclined plane, attaining a maximum run-up, before receding and forming a relatively thin and fast moving backwash. This interacts with the less rapidly moving fluid within the interior to form a bore. The evolution of the bore and the velocity and height fields either side of it are also calculated to reveal that it initially grows in magnitude before diminishing and intersecting with the shoreline. This analytical solution reveals features of the solution, such as the onset of the bore and its growth and decline, previously known only through numerical computation and the method presented here may be applied quite widely to the run-up of other initial distributions of fluid.

We investigate the acoustic emission from wave packets to the far field. To this end, we develop a theory for one- and two-dimensional source fields in the shape of wave packets with Gaussian envelopes. This theory is based on an approximation to Lighthill's acoustic analogy for distant observers. It is formulated in the spectral domain in which a Gaussian wave packet is represented again by a Gaussian. This allows us to determine the directivity of the acoustic emission (e.g. superdirectivity and Mach waves) by simple geometric constructions in the spectral domain. It is shown that the character of the acoustic emission is mainly governed by the aspect ratio and the Mach number of the wave packet source. To illustrate the relevance of this theory we use it to study two prominent problems in subsonic jet aeroacoustics.

We consider shallow-water flow past a broad bottom ridge, localized in the flow direction, using the framework of the forced Su–Gardner (SG) system of equations, with a primary focus on the transcritical regime when the Froude number of the oncoming flow is close to unity. These equations are an asymptotic long-wave approximation of the full Euler system, obtained without a simultaneous expansion in the wave amplitude, and hence are expected to be superior to the usual weakly nonlinear Boussinesq-type models in reproducing the quantitative features of fully nonlinear shallow-water flows. A combination of the local transcritical hydraulic solution over the localized topography, which produces upstream and downstream hydraulic jumps, and unsteady undular bore solutions describing the resolution of these hydraulic jumps, is used to describe various flow regimes depending on the combination of the topography height and the Froude number. We take advantage of the recently developed modulation theory of SG undular bores to derive the main parameters of transcritical fully nonlinear shallow-water flow, such as the leading solitary wave amplitudes for the upstream and downstream undular bores, the speeds of the undular bores edges and the drag force. Our results confirm that most of the features of the previously developed description in the framework of the unidirectional forced Korteweg–de Vries (KdV) model hold up qualitatively for finite amplitude waves, while the quantitative description can be obtained in the framework of the bidirectional forced SG system. Our analytic solutions agree with numerical simulations of the forced SG equations within the range of applicability of these equations.

We propose a new unified model for the small, intermediate and large-scale evolution of freely decaying two-dimensional turbulence in the inviscid limit. The new model's centerpiece is a recent theory of vortex self-similarity (Dritschel et al., Phys. Rev. Lett., vol. 101, 2008, no. 094501), applicable to the intermediate range of scales spanned by an expanding population of vortices. This range is predicted to have a steep k−5 energy spectrum. At small scales, this gives way to Batchelor's (Batchelor, Phys. Fluids, vol. 12, 1969, p. 233) k−3 energy spectrum, corresponding to the (forward) enstrophy (mean square vorticity) cascade or, physically, to thinning filamentary debris produced by vortex collisions. This small-scale range carries with it nearly all of the enstrophy but negligible energy. At large scales, the slow growth of the maximum vortex size (~t1/6 in radius) implies a correspondingly slow inverse energy cascade. We argue that this exceedingly slow growth allows the large scales to approach equipartition (Kraichnan, Phys. Fluids, vol. 10, 1967, p. 1417; Fox & Orszag, Phys. Fluids, vol. 12, 1973, p. 169), ultimately leading to a k1 energy spectrum there. Put together, our proposed model has an energy spectrum ℰ(k, t) ∝ t1/3k1 at large scales, together with ℰ(k, t) ∝ t−2/3k−5 over the vortex population, and finally ℰ(k, t) ∝ t−1k−3 over an exponentially widening small-scale range dominated by incoherent filamentary debris.

Support for our model is provided in two parts. First, we address the evolution of large and ultra-large scales (much greater than any vortex) using a novel high-resolution vortex-in-cell simulation. This verifies equipartition, but more importantly allows us to better understand the approach to equipartition. Second, we address the intermediate and small scales by an ensemble of especially high-resolution direct numerical simulations.

The relaxation and breakup of an elongated droplet in a viscous and initially quiescent fluid is studied by solving the full Navier–Stokes equations using a three-dimensional finite volume method coupled with a moving mesh interface tracking (MMIT) scheme to locate the interface. The two fluids are assumed incompressible and immiscible. The interface is represented as a surface triangle mesh with zero thickness that moves with the fluid. Therefore, the jump and continuity conditions across the interface are implemented directly, without any smoothing of the fluid properties. Mesh adaptations on a tetrahedral mesh are employed to permit large deformation and to capture the changing curvature. Mesh separation is implemented to allow pinch-off. The detailed investigations of the relaxation and breakup process are presented in a more general flow regime compared to the previous works by Stone & Leal (J. Fluid Mech., vol. 198, 1989, p. 399) and Tong & Wang (Phys. Fluids, vol. 19, 2007, 092101), including the flow field of the both phases. The simulation results reveal that the vortex rings due to the interface motion and the conservation of mass play an important role in the relaxation and pinch-off process. The vortex rings are created and collapsed during the process. The effects of viscosity ratio, density ratio and length ratio on the relaxation and breakup are studied. The simulations indicate that the fluid velocity field and the neck shape are distinctly different for viscosity ratios larger and smaller than O(1), and thus a different end-pinching mechanism is observed for each regime. The length ratio also significantly affects the relaxation process and the velocity distributions, but not the neck shape. The influence of the density ratio on the relaxation and breakup process is minimal. However, the droplet evolution is retarded due to the large density of the suspending flow. The formation of a satellite droplet is observed, and the volume of the satellite droplet depends strongly on the length ratio and the viscosity ratio.

Explicit expressions for the added mass tensor of a bubble in strongly nonlinear deformation and motion near a plane wall are presented. Time evolutions and interconnections of added mass components are derived analytically and analysed. Interface dynamics have been predicted with two methods, assuming that the flow is irrotational, that the fluid is perfect and with the neglect of gravity. The assumptions that gravity and viscosity are negligible are verified by investigating their effects and by quantifying their impact in some cases of strong deformation, and criteria are presented to specify the conditions of their validity. The two methods are an analytical one and the boundary element method, and good agreement is found. It is explained why a strongly deforming bubble is decelerated. The classical Rayleigh–Plesset equation is extended with terms to account for arbitrary, axisymmetric deformation and to account for the proximity of a wall. An expression for the corresponding cycle frequency that is valid in the vicinity of the wall is derived. An equation similar to the Rayleigh–Plesset equation is presented for the most important anisotropic deformation mode. Well-known expressions for the angular frequencies of some periodic solutions without a wall follow easily from the equations presented. A periodically deforming bubble without initial velocity of the centroid and without a dominating isotropic deformation component is eventually always driven towards the wall. A simplified equation of motion of the centre of a deforming bubble is presented. If desired, full deformation computations can be speeded up by selecting an artificially low value of the polytropic constant Cp/Cv.

Acoustic instabilities with frequencies roughly higher than 1 kHz remain among the most harmful instabilities, able to drastically affect the operation of engines and even leading to the destruction of the combustion chamber. By coupling with resonant transverse modes of the chamber, these pressure fluctuations can lead to a large increase of heat transfer fluctuations, as soon as fluctuations are in phase. To control engine stability, the mechanisms leading to the modulation of the local instantaneous rate of heat release must be understood. The commonly developed global approaches cannot identify the dominant mechanism(s) through which the acoustic oscillation modulates the local instantaneous rate of heat release. Local approaches are being developed based on processes that could be affected by acoustic perturbations. Liquid atomization is one of these processes. In the present paper, the effect of transverse acoustic perturbations on a coaxial air-assisted jet is studied experimentally. Here, five breakup regimes have been identified according to the flow conditions, in the absence of acoustics. The liquid jet is placed either at a pressure anti-node or at a velocity anti-node of an acoustic field. Acoustic levels up to 165 dB are produced. At a pressure anti-node, breakup of the liquid jet is affected by acoustics only if it is assisted by the coaxial gas flow. Effects on the liquid core are mainly due to the unsteady modulation of the annular gas flow induced by the acoustic waves when the mean dynamic pressure of the gas flow is lower than the acoustic pressure amplitude. At a velocity anti-node, local nonlinear radiation pressure effects lead to the flattening of the jet into a liquid sheet. A new criterion, based on an acoustic radiation Bond number, is proposed to predict jet flattening. Once the sheet is formed, it is rapidly atomized by three main phenomena: intrinsic sheet instabilities, Faraday instability and membrane breakup. Globally, this process promotes atomization. The spray is also spatially organized under these conditions: large liquid clusters and droplets with a low ejection velocity can be brought back to the velocity anti-node plane, under the action of the resulting radiation force. These results suggest that in rocket engines, because of the large number of injectors, a spatial redistribution of the spray could occur and lead to inhomogeneous combustion producing high-frequency combustion instabilities.

We calculate the electrophoretic mobility Me of a spherical colloidal particle, using modified Poisson–Nernst–Planck (PNP) equations that account for steric repulsion between finite sized ions, through Bikerman's mean-field model (Bikerman, Phil. Mag., vol. 33, 1942, p. 384). Ion steric effects are controlled by the bulk volume fraction of ions ν, and for ν = 0 the standard PNP equations are recovered. An asymptotic analysis in the thin-double-layer limit reveals at small zeta potentials (ζ < kBT/e ≈ 25 mV) Me to increase linearly with ζ for all ν, as expected from the Helmholtz–Smoluchowski (HS) formula. For larger ζ, however, it is well known that surface conduction of ions within the double layer reduces Me below the HS result. Crucially, however, in the PNP equations surface conduction becomes significant precisely because of the aphysically large and unbounded counter-ion densities predicted at large ζ. In contrast, ion steric effects impose a limit on the counter-ion density, thereby mitigating surface conduction. Hence, Me does not fall as far below HS for finite sized ions (ν ≠ 0). Indeed, at sufficiently large ν, ion steric effects are so dramatic that a maximum in Me is not observed for physically reasonable values of ζ(≤ 10 kBT/e), in stark contrast to the PNP-based calculations of O'Brien & White (J. Chem. Soc. Faraday Trans. II, vol. 74, 1978, p. 1607) and O'Brien (J. Colloid Interface Sci., vol. 92, 1983, p. 204). Finally, by calculating a Dukhin–Bikerman number characterizing the relative importance of surface conduction, we collapse Me versus ζ data for different ν onto a single master curve.

Embedding colloidal particles in polymeric hydrogels often endows the polymer skeleton with appealing characteristics for microfluidics and biosensing applications. This theoretical study provides a rigorous foundation for interpreting active electrical microrheology and electroacoustic experiments on such materials. In addition to viscoelastic properties of the composites, these techniques sense physicochemical characteristics of the particle–polymer interface. Wang & Hill (Soft Matter, vol. 4, 2008, p. 1048) studied the steady response of a rigid, impenetrable sphere in a compressible hydrogel skeleton. Here, we extend their analysis to arbitrary frequencies, showing, in general, how the frequency response depends on the particle size and charge, ionic strength of the electrolyte and elastic and hydrodynamic characteristics of the polymer skeleton. Our calculations capture the transition from quasi-steady compressible to quasi-steady incompressible dynamics as the frequency passes through the reciprocal draining time of the gel. Above the reciprocal draining time, the skeleton and fluid move in unison, so the dynamics are incompressible and, thus, given to an excellent approximation by the well-known dynamic electrophoretic mobility but with the Newtonian shear viscosity replaced by a complex, frequency-dependent value.

The stability of the Rayleigh–Bénard–Poiseuille flow in a channel with large transverse aspect ratio (ratio of width to vertical channel height) is studied experimentally. The onset of thermal convection in the form of ‘transverse rolls’ (rolls with axes perpendicular to the Poiseuille flow direction) is determined in the Reynolds–Rayleigh number plane for two different working fluids: water and mineral oil with Prandtl numbers of approximately 6.5 and 450, respectively. By analysing experimental realizations of the system impulse response it is demonstrated that the observed onset of transverse rolls corresponds to their transition from convective to absolute instability. Finally, the system response to localized patches of supercriticality (in practice local ‘hot spots’) is observed and compared with analytical and numerical results of Martinand, Carrière & Monkewitz (J. Fluid Mech., vol. 502, 2004, p. 175 and vol. 551, 2006, p. 275). The experimentally observed two-dimensional saturated global modes associated with these patches appear to be of the ‘steep’ variety, analogous to the one-dimensional steep nonlinear modes of Pier, Huerre & Chomaz (Physica D, vol. 148, 2001, p. 49).

The classical problem of Taylor (Proc. Lond. Math. Soc., vol. 20, 1921, pp. 148–181) of Kelvin wave reflection in a semi-enclosed rectangular basin of uniform depth is extended to account for horizontally viscous effects. To this end, we add horizontally viscous terms to the hydrodynamic model (linearized depth-averaged shallow-water equations on a rotating plane, including bottom friction) and introduce a no-slip condition at the closed boundaries.

In a straight channel of infinite length, we obtain three types of wave solutions (normal modes). The first two wave types are viscous Kelvin and Poincaré modes. Compared to their inviscid counterparts, they display longitudinal boundary layers and a slight decrease in the characteristic length scales (wavelength or along-channel decay distance). For each viscous Poincaré mode, we additionally find a new mode with a nearly similar lateral structure. This third type, entirely due to viscous effects, represents evanescent waves with an along-channel decay distance bounded by the boundary-layer thickness.

The solution to the viscous Taylor problem is then written as a superposition of these normal modes: an incoming Kelvin wave and a truncated sum of reflected modes. To satisfy no slip at the lateral boundary, we apply a Galerkin method. The solution displays boundary layers, the lateral one at the basin's closed end being created by the (new) modes of the third type. Amphidromic points, in the inviscid and frictionless case located on the centreline of the basin, are now found on a line making a small angle to the longitudinal direction. Using parameter values representative for the Southern Bight of the North Sea, we finally compare the modelled and observed tide propagation in this basin.

Dissolution of carbon dioxide (CO2) injected into saline aquifers causes an unstable high-density diffusive front. Understanding how instability fingers develop has received much attention because they accelerate dissolution trapping, which favours long-term sequestration. The time for the onset of convection as the dominant transport mechanism has been traditionally studied by neglecting dispersion and treating the CO2–brine interface as a prescribed concentration boundary by analogy to a thermal convection problem. This work explores the effect of these simplifications. Results show that accounting for the CO2 mass flux across the prescribed concentration boundary has little effect on the onset of convection. However, accounting for dispersion causes a reduction of up to two orders of magnitude on the onset time. This implies that CO2 dissolution can be accelerated by activating dispersion as a transport mechanism, which can be achieved adopting a fluctuating injection regime.

Direct numerical simulation (DNS) of the near field of a three-dimensional spatially developing turbulent lifted hydrogen jet flame in heated coflow is performed with a detailed mechanism to determine the stabilization mechanism and the flame structure. The DNS was performed at a jet Reynolds number of 11,000 with over 940 million grid points. The results show that auto-ignition in a fuel-lean mixture at the flame base is the main source of stabilization of the lifted jet flame. A chemical flux analysis shows the occurrence of near-isothermal chemical chain branching preceding thermal runaway upstream of the stabilization point, indicative of hydrogen auto-ignition in the second limit. The Damköhler number and key intermediate-species behaviour near the leading edge of the lifted flame also verify that auto-ignition occurs at the flame base. At the lifted-flame base, it is found that heat release occurs predominantly through ignition in which the gradients of reactants are opposed. Downstream of the flame base, both rich-premixed and non-premixed flames develop and coexist with auto-ignition. In addition to auto-ignition, Lagrangian tracking of the flame base reveals the passage of large-scale flow structures and their correlation with the fluctuations of the flame base. In particular, the relative position of the flame base and the coherent flow structure induces a cyclic motion of the flame base in the transverse and axial directions about a mean lift-off height. This is confirmed by Lagrangian tracking of key scalars, heat release rate and velocity at the stabilization point.