The tenth anniversaries of the deaths of George Batchelor and David Crighton occurred, respectively, in March and April this year. In commemoration and celebration of their lives and works, an afternoon of talks was held in the Department of Applied Mathematics and Theoretical Physics (DAMTP), University of Cambridge on 19 April 2010. Three of those talks are summarised here: Keith Moffatt and Shon Ffowcs-Williams give impressions of the lives and spirits of these two prominent figures in the history of Jfm – George its founder and David its Editor from 1996, having been an associate editor since 1979; John Hinch gives insight into MicroHydrodynamics, a term coined by George to describe the research area that dominated the second half of his career.

Ten years have elapsed since the passing of George Keith Batchelor (8 March 1920–30 March 2000), formerly Professor of Fluid Dynamics at the University of Cambridge, and Founder Editor of the Journal of Fluid Mechanics. It is fitting to remind the readers of this Journal what a great scientist he was, both in respect of his own contributions to our subject, and even more in respect of his inspirational influence on generations of research students and younger colleagues, and also more widely on the international stage, on which he was a revered, if sometimes controversial, personality.

Batchelor made his name with research in turbulence in the 1940s and 1950s. He became disillusioned with turbulence at the Marseille meeting in 1961. At the end of the 1960s, he started his second wave of research on low-Reynolds-number suspensions of particles. Ten years after he died, I will describe his key results, what was before and what followed. Eight of his 10 most cited papers are in micro-hydrodynamics.

Apart from being extremely clever, David Crighton was a very nice person. He must have been well brought up and made his family very proud of him (see figures 1 and 2).

This paper extends an existing analytical model of the aeroacoustic response of a rectilinear cascade of flat-plate blades to three-dimensional incident vortical gusts, by providing closed-form expressions for the acoustic field inside the inter-blade channels, as well as for the pressure jump over the blades in subsonic flows. The extended formulation is dedicated to future implementation in a fan-broadband-noise-prediction tool. The intended applications include the modern turbofan engines, for which analytical modelling is believed to be a good alternative to more expensive numerical techniques. The initial model taken as a reference is based on the Wiener–Hopf technique. An analytical solution valid over the whole space is first derived by making an extensive use of the residue theorem. The accuracy of the model is shown by comparing with numerical predictions of benchmark configurations available in the literature. This full exact solution could be used as a reference for future assessment of numerical solvers, of linearized Euler equations for instance, in rectilinear or narrow-annulus configurations. In addition, the pressure jump is a key piece of information because it can be used as a source term in an acoustic analogy when the rectilinear-cascade model is applied to three-dimensional blade rows by resorting to a strip-theory approach. When used as such in a true rectilinear-cascade configuration, it reproduces the exact radiated field that can be derived directly. The solution is also compared to a classical single-airfoil formulation to highlight the cascade effect. This effect is found important when the blades of the cascade overlap significantly, but the cascade solution tends to the single-airfoil one as the overlap goes to zero. This suggests that both models can be used as the continuation of each other if needed.

This paper examines two related problems from liquid-film theory. Firstly, a steady-state flow of a liquid film down a pre-wetted plate is considered, in which there is a precursor film in front of the main film. Assuming the former to be thin, a full asymptotic description of the problem is developed and simple analytical estimates for the extent and depth of the precursor film's influence on the main film are provided. Secondly, the so-called drag-out problem is considered, where an inclined plate is withdrawn from a pool of liquid. Using a combination of numerical and asymptotic means, the parameter range where the classical Landau–Levich–Wilson solution is not unique is determined.

We examine the efficacy of streamwise travelling waves generated by a zero-net-mass-flux surface blowing and suction for controlling the onset of turbulence in a channel flow. For small-amplitude actuation, we utilize a weakly nonlinear analysis to determine base-flow modifications and assess the resulting net power balance. Receptivity analysis of the velocity fluctuations around this base flow is then employed to design the travelling waves. Our simulation-free approach reveals that, relative to the flow with no control, the downstream travelling waves with properly designed speed and frequency can significantly reduce receptivity, which makes them well suited for controlling the onset of turbulence. In contrast, the velocity fluctuations around the upstream travelling waves exhibit larger receptivity to disturbances. Our theoretical predictions, obtained by perturbation analysis (in the wave amplitude) of the linearized Navier–Stokes equations with spatially periodic coefficients, are verified using full-scale simulations of the nonlinear flow dynamics in the companion paper (Lieu et al., J. Fluid Mech., 2010, doi:10.1017/S002211201000340X).

This study builds on and confirms the theoretical findings of Part 1 of this paper (Moarref & Jovanović, J. Fluid Mech., 2010, doi:10.1017/S0022112010003393). We use direct numerical simulation of the Navier–Stokes equations to assess the efficacy of blowing and suction in the form of streamwise travelling waves for controlling the onset of turbulence in a channel flow. We highlight the effects of the modified base flow on the dynamics of velocity fluctuations and net power balance. Our simulations verify the theoretical predictions of Part 1 that the upstream travelling waves promote turbulence even when the uncontrolled flow stays laminar. On the other hand, the downstream travelling waves with parameters selected in Part 1 are capable of reducing the fluctuations' kinetic energy, thereby maintaining the laminar flow. In flows driven by a fixed pressure gradient, a positive net efficiency as large as 25 % relative to the uncontrolled turbulent flow can be achieved with downstream waves. Furthermore, we show that these waves can also relaminarize fully developed turbulent flows at low Reynolds numbers. We conclude that the theory developed in Part 1 for the linearized flow equations with uncertainty has considerable ability to predict full-scale phenomena.

Multiple-scale asymptotics is used to analyse the Euler equations for the dynamical situation of a gravity wave (GW) near breaking level. A simple saturation argument in combination with linear theory is used to obtain the relevant dynamical scales. As a small expansion parameter, the ratio of the inverse of the vertical wavenumber and potential temperature and pressure scale heights is used, which we allow to be of the same order of magnitude here. It is shown that the resulting equation hierarchy is consistent with that obtained from the pseudo-incompressible equations, both for non-hydrostatic and hydrostatic GWs, while this is not the case for the anelastic equations unless the additional assumption of sufficiently weak stratification is adopted. To describe vertical propagation of wavepackets over several atmospheric-scale heights, Wentzel–Kramers–Brillouin (WKB) theory is used to show that the pseudo-incompressible flow divergence generates the same amplitude equation that also obtains from the full Euler equations. This gives a mathematical justification for the use of the pseudo-incompressible equations in the study of GW breaking in the atmosphere for arbitrary background stratification. The WKB theory interestingly even holds at wave amplitudes close to static instability. In the mean-flow equations, we obtain in addition to the classic wave-induced momentum-flux divergences a wave-induced correction of hydrostatic balance in the vertical momentum equation, which cannot be obtained from Boussinesq or anelastic dynamics.

Recent simulations and experiments appear to show that the rotating-disk boundary layer is linearly globally stable despite the existence of local absolute instability. However, we argue that linear global instability can be created by local absolute instability at the edge of the disk. This argument is based on investigations of the linearized complex Ginzburg–Landau equation with weakly spatially varying coefficients to model the propagation of a wavepacket through a weakly inhomogeneous unstable medium. Therefore, this mechanism could operate in a variety of locally absolutely unstable flows. The corresponding nonlinear global mode has a front at the radius of onset of absolute instability when the disk edge is far from the front. This front moves radially outwards when the Reynolds number at the disk edge is reduced. It is shown that the laminar–turbulent transition front also behaves in this manner, possibly explaining the scatter in observed transitional Reynolds numbers.

A nanoscale thermal anemometry probe (NSTAP) has been developed to measure velocity fluctuations at ultra-small scales. The sensing element is a free-standing platinum nanoscale wire, 100 nm × 2 μm × 60 μm, suspended between two current-carrying contacts and the sensor is an order of magnitude smaller than presently available commercial hot wires. The probe is constructed using standard semiconductor and MEMS manufacturing methods, which enables many probes to be manufactured simultaneously. Measurements were performed in grid-generated turbulence and compared to conventional hot-wire probes with a range of sensor lengths. The results demonstrate that the NSTAP behaves similarly to conventional hot-wire probes but with better spatial resolution and faster temporal response. The results are used to investigate spatial filtering effects, including the impact of spatial filtering on the probability density of velocity and velocity increment statistics.

Transition to uniform turbulence in cylindrical pipe flow occurs experimentally via the spatial expansion of isolated coherent structures called ‘slugs’, triggered by localized finite-amplitude disturbances. We study this process numerically by examining the preferred route in phase space through which a critical disturbance initiates a ‘slug’. This entails first identifying the relative attractor – ‘edge state’ – on the laminar–turbulent boundary in a long pipe and then studying the dynamics along its low-dimensional unstable manifold, leading to the turbulent state. Even though the fully turbulent state delocalizes at Re ≈ 2300, the edge state is found to be localized over the range Re = 2000–6000, and progressively reduces in both energy and spatial extent as Re is increased. A key process in the genesis of a slug is found to be vortex shedding via a Kelvin–Helmholtz mechanism from wall-attached shear layers quickly formed at the edge state's upstream boundary. Whether these shedded vortices travel on average faster or slower downstream than the developing turbulence determines whether a puff or a slug (respectively) is formed. This observation suggests that slugs are out-of-equilibrium puffs which therefore do not co-exist with stable puffs.

The effect of time-variant temperature on the dynamics of a single gas bubble in a liquid is investigated. With changes in temperature, several physical parameters controlling bubble behaviour change including surface tension, diffusivity, vapour pressure and gas solubility. A single-bubble model is formulated and a numerical simulation implemented to model the radius–time profile of a bubble, across a range of initial bubble sizes and rates of heating, taking into account the aforementioned parameter temperature dependences. The model is validated experimentally in a xanthan gum gel phantom, tracking the evolution of the bubbles using digital photography and an image analysis sizing algorithm. It is shown that the natural tendency for a bubble to dissolve can be reversed by an increase in temperature, but only above a certain radius-dependent threshold rate of heating.

Insects perform their multitude of flight skills at frequencies of tens to hundreds of Hertz, and the aerodynamics of these skills are fundamentally unsteady. Intuitively, unsteadiness may come from unsteady wing motion, unsteady surface vorticity or vorticity being shed into the rear and front wakes. In this study, we propose to investigate the aerodynamics of dragonfly using a simplified wing–wing model from the perspective of many-body force decomposition and the associated force elements. Insect flight usually operates at Reynolds numbers of the order of several hundreds, at which the surface vorticity is shown to play a substantial role. There are important cases where the added mass effect is non-negligible. Nevertheless, the major contribution to the forces comes from the vorticity within the flow. This study focused on the effects of mutual interactions due to phase differences between the fore- and hindwings in the translational as well as rotational motions. It is well known that the dynamic stall vortex is an important mechanism for an unsteady wing to gain lift. In analysing the life cycles of lift and thrust elements, we also associate some high lift and thrust with the mechanisms identified as ‘riding on’ lift elements, ‘driven by’ thrust elements and ‘sucked by’ thrust elements, by which a wing makes use of a shed or fused vortex below, in front of, and behind it, respectively. In addition, a shear layer attaching to each wing may also provide significant thrust elements.

As tidal turbine farms grow they interact with the larger scale flow along a channel by increasing the channel's drag coefficient. This interaction limits a channel's potential to produce power. A 1D model for a tidal channel is combined with a theory for turbines in a channel to show that the tuning of the flow through the turbines and the density of turbines in a channel's cross-section also interact with the larger scale flow, via the drag coefficient, to determine the power available for production. To maximise turbine efficiency, i.e. the power available per turbine, farms must occupy the largest fraction of a channel's cross-section permitted by navigational and environmental constraints. Maximising of power available with these necessarily densely packed farms requires turbines to be tuned for a particular channel and turbine density. The optimal through-flow tuning fraction varies from near 1/3 for small farms occupying a small fraction of the cross-section, to near 1 for large farms occupying most of the cross-section. Consequently, tunings are higher than the optimal through-flow tuning of 1/3 for an isolated turbine from the classic turbine theory. Large optimally tuned farms can realise most of a channel's potential. Optimal tunings are dependent on the number of turbines per row, the number of rows, as well as the channel geometry, the background bottom friction coefficient and the tidal forcing.

We consider freely decaying, statistically axisymmetric turbulence evolving in the presence of a background rotation, an imposed stratification or a uniform magnetic field. We focus on the case of Saffman turbulence in which E(k → 0) ~ k2 and show that, if the large scales evolve in a self-similar manner, then u⊥2 ℓ⊥2 ℓ∥ = constant in rotating, stratified and magnetohydrodynamic turbulence. This may be contrasted with E(k → 0) ~ k4 turbulence, in which u⊥2 ℓ⊥4 ℓ∥ ≈ constant. (Here the subscripts ⊥ and ∥ indicate directions perpendicular and parallel to the axis of symmetry, and ℓ⊥, ℓ∥ and u⊥ are suitably defined integral scales.) This constraint on the integral scales allows us to make simple, testable predictions for the temporal evolution of ℓ⊥, ℓ∥ and u⊥ in all three systems. There are only limited data sets against which to compare these predictions, but they are consistent with those data which are currently available.

At the beginning of the last century Worthington and Cole discovered that the high-speed jets ejected after the impact of an axisymmetric solid on a liquid surface are intimately related to the formation and collapse of an air cavity created in the wake of the impactor. In this paper, we combine detailed boundary-integral simulations with analytical modelling to describe the formation of such Worthington jets after the impact of a circular disk on water. We extend our earlier model in Gekle et al. (Phys. Rev. Lett., vol. 102, 2009a, 034502), valid for describing only the jet base dynamics, to describe the whole jet. We find that the flow structure inside the jet may be divided into three different regions: the axial acceleration region, where the radial momentum of the incoming liquid is converted to axial momentum; the ballistic region, where fluid particles experience no further acceleration and move constantly with the velocity obtained at the end of the acceleration region; and the jet tip region, where the jet eventually breaks into droplets. From our modelling of the ballistic region we conclude that, contrary to the case of other physical situations where high-speed jets are also ejected, the types of Worthington jets studied here cannot be described using the theory of hyperbolic jets of Longuet-Higgins (J. Fluid Mech., vol. 127, 1983, p. 103). Most importantly, we find that the velocity and the shape of the ejected jets can be well predicted at any instant in time with the only knowledge of quantities obtained before pinch-off occurs. This fact allows us to provide closed expressions for the jet velocity and the sizes of the ejected droplets as a function of the velocity and the size of the impactor. We show that our results are also applicable to Worthington jets emerging after the collapse of a bubble growing from an underwater nozzle, although this system creates thicker jets than the disk impact.

The capillary breakup of the high-speed Worthington jets ejected after a cavity collapse in water occurs due to the high-Reynolds-number version of the capillary end-pinching mechanism first described, in the creeping flow limit, by Stone & Leal (J. Fluid Mech., vol. 198, 1989, p. 399). Using potential flow numerical simulations and theory, we find that the resulting drop ejection process does not depend on external noise and can be described as a function of a single dimensionless parameter, WeS = ρ R30S20/σ, which expresses the ratio of the capillary time to the inverse of the local strain rate, S0. Here, ρ and σ indicate the liquid density and the interfacial tension coefficient, respectively, and R0 is the initial radius of the jet. Our physical arguments predict the dimensionless size of the drops to scale as Ddrop/R0 ~ We−1/7S and the dimensionless time to break up as TS0 ~ We2/7S. These theoretical predictions are in good agreement with the numerical results.

In this paper, we present new experimental measurements of the turbulent transport of salt across an interface between two layers of fluid of equal depth but different salinities. The fluid is confined to a cylindrical annulus with a vertical axis. The outer cylinder is stationary and the inner cylinder rotates to produce a turbulent flow field consisting of an approximately irrotational mean azimuthal flow, with narrow boundary layers on the inner and outer cylinders. We focus on the limit of high-Richardson-number flow, defined as Ri = gΔρH/(ρ0u2rms), where ρ0 is a reference density, Δρ is the time-dependent difference of the layers' mean densities, urms is the root mean square of the turbulent velocity fluctuations and H is the layer depth. The mean flow has Reynolds number of the order of 104−105, and the turbulent fluctuations in the azimuthal and radial directions have root-mean-square speed of order 10% of the mean azimuthal flow. Measurements based on our experimental system show that when the Richardson number is in the range 7 < Ri < 200, the interface between the two layers remains sharp, each layer remains well mixed, and the vertical flux of salt between the layers, Fs ~(1.15 ± 0.15)Ri−1𝒜(H/ΔR)urmsΔS, where ΔS is the spatially-averaged time-dependent salinity difference between the layers and in general 𝒜(H/ΔR) is a dimensionless function of the tank aspect ratio, here taken to be unity, with ΔR being the gap width of the annulus. The salt transport appears to be caused by turbulent eddies scouring and sharpening the interface and implies a constant rate of conversion of the turbulent kinetic energy to potential energy, independent of the density contrast between the layers. For smaller values of Ri, the flow regime changes qualitatively, with eddies penetrating the interface, causing fluid in the two layers to co-mingle and rapidly homogenize.

The effect of applied magnetic fields on the deformation of a biocompatible hydrophobic ferrofluid drop suspended in a viscous medium is investigated numerically and compared with experimental data. A numerical formulation for the time-dependent simulation of magnetohydrodynamics of two immiscible non-conducting fluids is used with a volume-of-fluid scheme for fully deformable interfaces. Analytical formulae for ellipsoidal drops and near-spheroidal drops are reviewed and developed for code validation. At low magnetic fields, both the experimental and numerical results follow the asymptotic small deformation theory. The value of interfacial tension is deduced from an optimal fit of a numerically simulated shape with the experimentally obtained drop shape, and appears to be a constant for low applied magnetic fields. At high magnetic fields, on the other hand, experimental measurements deviate from numerical results if a constant interfacial tension is implemented. The difference can be represented as a dependence of apparent interfacial tension on the magnetic field. This idea is investigated computationally by varying the interfacial tension as a function of the applied magnetic field and by comparing the drop shapes with experimental data until a perfect match is found. This estimation method provides a consistent correlation for the variation in interfacial tension at high magnetic fields. A conclusion section provides a discussion of physical effects which may influence the microstructure and contribute to the reported observations.