We study drop impact on a deep pool of the same fluid, with an emphasis on the air layer trapped under the droplets from its formation to its rupture. The penetration velocity of the air layer at a very short time scale prior to its rupture is shown, using an energy argument and experimental verification, to be one-half of the impact velocity. We then deduce the dependence of the rupture position on the liquid viscosity and the impact velocity. We show that the volume of the resulting air bubbles can be related to both those resulting from droplets impacting on solid surfaces and those resulting from rigid spheres impacting on liquid surfaces.

Two-dimensional buoyant clouds moving along inclined boundaries under a gravitational force are investigated theoretically and experimentally. It is found that the ‘thermal theory’ gives a good description of the flow in the slope angle range 5° [lsim ] θ ≤ 90°. In this range the spatial growth rates of the cloud height and length are constant for a given slope angle and show a linear dependence on θ. For a cloud released with zero initial velocity the front velocity Uf first increases and then decreases, with the characteristic time of acceleration predicted by theory. In the decelerating state Uf/(g0′Q0/xf)½ is 2·6 ± 0·2 at θ ≃ 15°, and then reduces uniformly with increasing θ to a value of 1·5 ≃ 0·2 at 90° (where g0′Q0 is the released buoyancy and xf is the front position measured from a virtual origin). The shape of the cloud is well approximated by a half-ellipse. The variation of the ratio of the principal axes of the half-ellipse with slope angle is identical with that of the head of an inclined starting plume (Britter & Linden 1980). However, the cloud has a greater growth rate than the head of a starting plume.

Many wake flows exhibit self-excited flow oscillations which are sustained by the flow itself and are not caused by amplification of external noise. The archetypal example of a self-excited wake flow is the low Reynolds number flow past a circular cylinder. This flow exhibits self-sustained periodic vortex shedding above a critical Reynolds number. In general, control of such flows requires stabilization of many globally unstable modes; the present work describes a multiple-sensor control strategy for the cylinder wake which succeeds in controlling a simplified wake model at a Reynolds number above that at which single-sensor schemes fail.

Representation of the flow field by a finite set of coherent structures or modes, which are extracted by proper orthogonal decomposition and correspond to the large-scale wake components, allows the efficient design of a closed-loop control algorithm. A neural network is used to furnish an empirical prediction of the modal response of the wake to external control forcing. This model avoids the need for explicit representation of the control actuator–wake interaction. Additionally, the neural network structure of the model allows the design of a robust nonlinear control algorithm. Furthermore the controller does not necessarily require velocity field information, but can control the wake using other quantities (for example flow visualization pictures) which characterize the structure of the velocity field. Successful control of a simplified cylinder wake model is used to demonstrate the feasibility of the low-dimensional control strategy.

The detailed structure of the aqueous surface sublayer flow immediately adjacent to the wind-driven air–water interface is investigated in a laboratory wind-wave flume using particle image velocimetry (PIV) techniques. The goal is to investigate quantitatively the character of the flow in this crucial, very thin region which is often disrupted by microscale breaking events. In this study, we also examine critically the conclusions of Okuda, Kawai & Toba (1977), who argued that for very short, strongly forced wind-wave conditions, shear stress is the dominant mechanism for transmitting the atmospheric wind stress into the water motion – waves and surface drift currents. In strong contrast, other authors have more recently observed very substantial normal stress contributions on the air side. The availability of PIV and associated image technology now permits a timely re-examination of the results of Okuda et al., which have been influential in shaping present perceptions of the physics of this dynamically important region. The PIV technique used in the present study overcomes many of the inherent shortcomings of the hydrogen bubble measurements, and allows reliable determination of the fluid velocity and shear within 200 μm of the instantaneous wind-driven air–water interface.

The results obtained in this study are not in accord with the conclusions of Okuda et al. that the tangential stress component dominates the wind stress. It is found that prior to the formation of wind waves, the tangential stress contributes the entire wind stress, as expected. With increasing distance downwind, the mean tangential stress level decreases marginally, but as the wave field develops, the total wind stress increases significantly. Thus, the wave form drag, represented by the difference between the total wind stress and the mean tangential stress, also increases systematically with wave development and provides the major proportion of the wind stress once the waves have developed beyond their early growth stage. This scenario reconciles the question of relative importance of normal and tangential stresses at an air–water interface. Finally, consideration is given to the extrapolation of these detailed laboratory results to the field, where the present findings suggest that the sea surface is unlikely to become fully aerodynamically rough, at least for moderate to strong winds.

To study the mixing of a passive scalar in nearly isotropic turbulence, experiments have been made in isotropically mixed thermal fields with thermal mesh size Mθ (a) equal to the momentum mesh size M, (b) larger than M (obtained by heating only alternate rods of the turbulence generating grid), and (c) smaller than M. This last condition was achieved by inserting a fine heating screen with Mθ < M, at locations downstream of the turbulence grid. The heating screen was designed to produce negligible statistical change in the velocity field a short distance downstream. In all the heated grid experiments, for a given initial configuration of the thermal field, the intensity of temperature fluctuations θ normalized by the mean temperature rise ΔT, and the decay rate of $\overline{\theta^2} $ were both independent of the temperature of the grid. The principal effect of having Mθ > M was an increase in the relative intensity of temperature fluctuations compared with the Mθ = M case, and a marginal increase in their decay rate; contrary to expectation, the ratio R of temperature to velocity integral scales in the region of approximate homogeneity did not differ from that corresponding to Mθ = M. In heated screen experiments, the relative decay rate was independent of Mθ/M and ΔT. For the three locations of the heating screen used in these experiments, the decay rate was also independent of the relative distance xs of the heating screen from the turbulence generating grid; however, larger xs was associated with larger relative intensity of fluctuations. To a first approximation, the ratio R approached unity according to the empirical relation R = 1 − A exp [− αxθ/(UT0)], where xθ is downstream distance measured from the heating screen, and T0 is a characteristic turbulence decay time scale at x0 = 0. It was also verified that the skewness of the streamwise temperature derivative is approximately zero sufficiently downstream of the heating screen. Where the present study overlaps with previous measurements, an extensive comparison reveals several points of agreement as well as departure.

An investigation has been carried out on the behaviour of a thin sheet of liquid in connexion with the new method of lacquer application known as ’curtain coating’. A method is described for measuring the velocity of the sheet and an equation deduced for calculating this for a liquid of any viscosity. The minimum liquid flow rate required to maintain a stable sheet is discussed, and some effects of the impingement of the sheet on a rapidly moving surface are described.

Large-scale secondary flows can sometimes appear in turbulent boundary layers formed over rough surfaces, creating low- and high-momentum pathways along the surface (Barros & Christensen, J. Fluid Mech., vol. 748, 2014, R1). We investigate experimentally the dependence of these secondary flows on surface/flow conditions by measuring the flows over streamwise strips of roughness with systematically varied spanwise spacing. We find that the large-scale secondary flows are accentuated when the spacing of the roughness elements is roughly proportional to the boundary layer thickness ${\it\delta}$, and do not appear for cases with finer spacing. Cases with coarser spacing also generate ${\it\delta}$-scale secondary flows with tertiary flows in the spaces in between. These results show that the ratio of the spanwise length scale of roughness heterogeneity to the boundary layer thickness is a critical parameter for the occurrence of these secondary motions in turbulent boundary layers over rough walls.

We examine the buoyancy-driven rise of a bubble in a Newtonian or a viscoplastic fluid assuming axial symmetry and steady flow. Bubble pressure and rise velocity are determined, respectively, by requiring that its volume remains constant and its centre of mass remains fixed at the centre of the coordinate system. The continuous constitutive model suggested by Papanastasiou is used to describe the viscoplastic behaviour of the material. The flow equations are solved numerically using the mixed finite-element/Galerkin method. The nodal points of the computational mesh are determined by solving a set of elliptic differential equations to follow the often large deformations of the bubble surface. The accuracy of solutions is ascertained by mesh refinement and predictions are in very good agreement with previous experimental and theoretical results for Newtonian fluids. We determine the bubble shape and velocity and the shape of the yield surfaces for a wide range of material properties, expressed in terms of the Bingham Bn= Bond Bo = and Archimedes Ar= numbers, where ρ* is the density, μ*o the viscosity, γ* the surface tension and τ*y the yield stress of the material, g* the gravitational acceleration and R*b the radius of a spherical bubble of the same volume. If the fluid is viscoplastic, the material will not be deforming outside a finite region around the bubble and, under certain conditions, it will not be deforming either behind it or around its equatorial plane in contact with the bubble. As Bn increases, the yield surfaces at the bubble equatorial plane and away from the bubble merge and the bubble becomes entrapped. When Bo is small and the bubble cannot deform from the spherical shape the critical Bn is 0.143, i.e. it is a factor of 3/2 higher than the critical Bn for the entrapment of a solid sphere in a Bingham fluid, in direct correspondence with the 3/2 higher terminal velocity of a bubble over that of a sphere under the same buoyancy force in Stokes flow. As Bo increases allowing the bubble to squeeze through the material more easily, the critical Bingham number increases as well, but eventually it reaches an asymptotic value. Ar affects the critical Bn value much less.

We consider the motion of the mass of fluid ejected through a sharp-edged orifice by the motion of a piston. The vorticity formed by viscous forces within the separated flow at the sharp edge rolls up to form a concentrated vortex which, after a development period, consists of a core of very fine scale turbulence surrounded by a co-moving bubble of much larger scale turbulence. This bubble entrains outer fluid, mixes with it, and deposits the majority into a wake together with some small fraction of the total vorticity of the ring. Enough fluid is retained to account for the slow growth of the whole fluid mass. A theory which takes account of both the growth process and the loss of vorticity is proposed. By comparison with experimental measurements we have determined that the entrainment coefficient for turbulent vortex rings has a value equal to 0.011 ± 0.001, while their effective drag coefficient is 0.09 ± 0.01. These results seem to be independent of Reynolds number to within experimental accuracy.

A model for the instantaneous wall-shear-stress distribution is presented for zero-pressure-gradient turbulent boundary layers. The model, based on empirical and theoretical considerations, is able to reconstruct a statistically representative fluctuating wall-shear-stress time-series, ${ \tau }_{w}^{\ensuremath{\prime} } (t)$, using only the low-frequency content of the streamwise velocity measured in the logarithmic region, away from the wall. Results, including spectra and second-order moments, show that the model is capable of successfully capturing Reynolds number trends as observed in other studies.

The motion of the free surface of a viscous droplet is investigated. By using lubrication theory a model is developed for the motion of the free surface which includes both the effect of slip and the dependence of the contact angle on the slip velocity. We solve the resulting nonlinear partial differential equation in several ways. First we investigate the initial motion of the drop at a non-equilibrium contact angle using the method of matched asymptotics. Then we develop a pseudo-spectral method to numerically solve the full nonlinear system. The dependence of the spreading rate of the drop on the various physical parameters and for different slip models is determined.

The flow-induced vibrations of a circular cylinder, free to oscillate in the cross-flow direction and subjected to a forced rotation about its axis, are analysed by means of two- and three-dimensional numerical simulations. The impact of the symmetry breaking caused by the forced rotation on the vortex-induced vibration (VIV) mechanisms is investigated for a Reynolds number equal to $100$, based on the cylinder diameter and inflow velocity. The cylinder is found to oscillate freely up to a rotation rate (ratio between the cylinder surface and inflow velocities) close to $4$. Under forced rotation, the vibration amplitude exhibits a bell-shaped evolution as a function of the reduced velocity (inverse of the oscillator natural frequency) and reaches $1.9$ diameters, i.e. three times the maximum amplitude in the non-rotating case. The free vibrations of the rotating cylinder occur under a condition of wake–body synchronization similar to the lock-in condition driving non-rotating cylinder VIV. The largest vibration amplitudes are associated with a novel asymmetric wake pattern composed of a triplet of vortices and a single vortex shed per cycle, the ${\rm T} + {\rm S}$ pattern. In the low-frequency vibration regime, the flow exhibits another new topology, the U pattern, characterized by a transverse undulation of the spanwise vorticity layers without vortex detachment; consequently, free oscillations of the rotating cylinder may also develop in the absence of vortex shedding. The symmetry breaking due to the rotation is shown to directly impact the selection of the higher harmonics appearing in the fluid force spectra. The rotation also influences the mechanism of phasing between the force and the structural response.

Coherent features of a turbulent Mach 0.9, Reynolds number $10^{6}$ jet are educed from a high-fidelity large eddy simulation. Besides the well-known Kelvin–Helmholtz instabilities of the shear layer, a new class of trapped acoustic waves is identified in the potential core. A global linear stability analysis based on the turbulent mean flow is conducted. The trapped acoustic waves form branches of discrete eigenvalues in the global spectrum, and the corresponding global modes accurately match the educed structures. Discrete trapped acoustic modes occur in a hierarchy determined by their radial and axial order. A local dispersion relation is constructed from the global modes and found to agree favourably with an empirical dispersion relation educed from the simulation data. The product between direct and adjoint modes is then used to isolate the trapped waves. Under certain conditions, resonance in the form of a beating occurs between trapped acoustic waves of positive and negative group velocities. This resonance explains why the trapped modes are prominently observed in the simulation and as tones in previous experimental studies. In the past, these tones were attributed to external factors. Here, we show that they are an intrinsic feature of high-subsonic jets that can be unambiguously identified by a global linear stability analysis.

The interaction of two laser-induced bubbles in bulk water is investigated. The strength and direction of the emerging liquid jets can be controlled by adjusting the relative bubble positions, the time difference between bubble generation, and the laser pulse energies determining the bubble sizes. Experimental and numerical studies are performed for millimetre-sized bubble pairs. Taking bubbles of equal energy, a maximum jet velocity is found for close anti-phase bubbles, i.e. when the second bubble is produced at the maximum volume of the first one and the bubble walls are almost touching and not merging. Under these conditions, one bubble produces a fast jet with a peak velocity of about $150~\text{m}~\text{s}^{-1}$ that reaches a distance into the surrounding liquid of at least three times the maximum bubble radius. Collapse of the other bubble results in a slow jet of large mass that rapidly converts into a ring vortex. Correspondingly, the interaction with adjacent structures is dominated either by localized jet impact or by shear stresses extending over a larger area. Furthermore, interactions between micrometre-sized bubble pairs are investigated numerically to understand and predict how the effects of the physical parameters on bubble dynamics would change when the bubbles become smaller. The results are discussed with respect to micropumping and opto-injection.

Comparisons of numerical and experimental results for transient two-dimensional natural convection initiated by instantaneously heating and cooling the opposing vertical walls of a square cavity containing a stationary and isothermal fluid are presented. The good comparisons indicate that the simulation is capturing the important features of the flow. Several features are identified and discussed in detail; in particular, the presence of travelling wave instabilities on the vertical-wall boundary layers and horizontal intrusions, the existence of a rapid flow divergence in the region of the outflow of the intrusions, and the presence of cavity-scale oscillations, caused by the interaction of the intrusions with the opposing vertical boundary layer. The utilization of both numerical and experimental investigations has allowed a more complete exploitation of the available resources than would have been possible had each been conducted separately.

Self-sustained convective dynamos in planetary systems operate in an asymptotic regime of rapid rotation, where a balance is thought to hold between the Coriolis, pressure, buoyancy and Lorentz forces (the MAC balance). Classical numerical solutions have previously been obtained in a regime of moderate rotation where viscous and inertial forces are still significant. We define a uni-dimensional path in parameter space between classical models and asymptotic conditions from the requirements to enforce a MAC balance and to preserve the ratio between the magnetic diffusion and convective overturn times (the magnetic Reynolds number). Direct numerical simulations performed along this path show that the spatial structure of the solution at scales larger than the magnetic dissipation length is largely invariant. This enables the definition of large-eddy simulations resting on the assumption that small-scale details of the hydrodynamic turbulence are irrelevant to the determination of the large-scale asymptotic state. These simulations are shown to be in good agreement with direct simulations in the range where both are feasible, and can be computed for control parameter values far beyond the current state of the art, such as an Ekman number $E=10^{-8}$ . We obtain strong-field convective dynamos approaching the MAC balance and a Taylor state to an unprecedented degree of accuracy. The physical connection between classical models and asymptotic conditions is shown to be devoid of abrupt transitions, demonstrating the asymptotic relevance of classical numerical dynamo mechanisms. The fields of the system are confirmed to follow diffusivity-free, power-based scaling laws along the path.

The boundary-value problem for irrotational surface waves is derived from a variational integral I with the Lagrangian density [Lscr ] = Ξ ηt - [Hscr ] where Ξ (X, t) is the value of the velocity potential at the free surface, y = η(x, t), and [Hscr ] is the energy density in x space. [Hscr ] then is expressed as a functional of Ξ and η, qua canonical variables, by solving a reduced boundary-value problem for the potential, after which the requirement that I be stationary with respect to independent variations of Ξ and η yields a pair of evolution equations for Ξ and η. The Fourier expansions Ξ = pn(t)ϕn*(x) and η = qn(t)ϕn(x), where {ϕn} is an orthogonal set of basis functions, reduce I to Hamilton's action integral, in which the complex amplitudes pn and qn appear as canonically conjugate co-ordinates, and yield canonical equations for pn and qn that are the spectral transforms of the evolution equations for Ξ and η. The evolution equations are reduced (asymptotically) to partial differential equations for pn and qn by expanding [Hscr ] in powers of α = a/d and β = (d/l)2, where a and l are scales of amplitude and wavelength. Explicit third approximations are developed for β = O(α).

A comparison of laboratory experiments in a shallow-water tank driven by an oscillating piston and numerical solutions of the Korteweg-de Vries (KdV) equation show that the latter can accurately describe slightly dissipative wavepropagation for Ursell numbers (h1L2/h03) up to 800. This is an input-output experiment, where the initial condition for the KdV equation is obtained from upstream (station 1) data. At a downstream location, the number of crests and troughs and their phases (or relative locations within a period) agree quantitatively with numerical solutions. The crest-to-trough amplitudes disagree somewhat, as they are more sensitive to dissipative forces. This work firmly establishes the soliton concept as necessary for treating the propagation of shallow-water waves of moderate amplitude in a low-dissipation environment.

A laminar Mach 6 flat plate boundary layer is perturbed using three different types of disturbances introduced through blowing and suction. The linear and nonlinear development and eventual breakdown to turbulence are investigated using direct numerical simulation. The three different transition mechanisms compared are first mode oblique breakdown, second mode oblique breakdown and second mode fundamental resonance. The focus of the present work is to compare the nonlinear development and breakdown to turbulence for the different transition mechanisms and explain the heat transfer overshoot observed in experiments. First mode oblique breakdown leads to the shortest transition length and a clear peak in wall heat transfer in the transitional region. For all three transition mechanisms, the development of streamwise streaks precedes the breakdown to fully turbulent flow. The modal linear and nonlinear development are analysed including the breakdown of the streaks. The effect of wall cooling is investigated for second mode fundamental resonance and no qualitative differences in the nonlinear processes are observed. Finally, the development towards fully turbulent flow including mean flow, turbulent spectra, and turbulent fluctuations is shown and the first mode oblique breakdown simulation shows the furthest development towards a fully turbulent flow.

A set of complete two- and three-dimensional direct numerical simulations (DNS) in a differentially heated air-filled cavity of aspect ratio 4 with adiabatic horizontal walls is presented in this paper. Although the physical phenomenon is three-dimensional, owing to its prohibitive computational costs the majority of the previous DNS of turbulent and transition natural convection flows in enclosed cavities assumed a two-dimensional behaviour. The configurations selected here (Rayleigh number based on the cavity height 6.4 × 108, 2 × 109 and 1010, Pr = 0.71) are an extension to three dimensions of previous two-dimensional problems.

An overview of the numerical algorithm and the methodology used to verify the code and the simulations is presented. The main features of the flow, including the time-averaged flow structure, the power spectra and probability density distributions of a set of selected monitoring points, the turbulent statistics, the global kinetic energy balances and the internal waves motion phenomenon are described and discussed.

As expected, significant differences are observed between two- and three-dimensional results. For two-dimensional simulations the oscillations at the downstream part of the vertical boundary layer are clearly stronger, ejecting large eddies to the cavity core. In the three-dimensional simulations these large eddies do not persist and their energy is rapidly passed down to smaller scales of motion. It yields on a reduction of the large-scale mixing effect at the hot upper and cold lower regions and consequently the cavity core still remains almost motionless even for the highest Rayleigh number. The boundary layers remain laminar in their upstream parts up to the point where these eddies are ejected. The point where this phenomenon occurs clearly moves upstream for the three-dimensional simulations. It is also shown that, even for the three-dimensional simulations, these eddies are large enough to permanently excite an internal wave motion in the stratified core region. All these differences become more marked for the highest Rayleigh number.