Certain medical treatments involve the introduction of exogenous liquids in the lungs. These liquids can form plugs within the airways. The plugs propagate throughout the branching network in the lungs being forced by airflow. They leave a deposited film on the airway walls and split at bifurcations. Understanding the resulting distribution of liquid throughout the lungs is important for the effective administration of the prescribed treatments. In this paper, we investigate numerically the splitting of a liquid plug by a two-dimensional pulmonary bifurcation under the influence of a transverse gravitational field. The splitting is characterized by the splitting ratio, which is the ratio of volume of the liquid plug in the daughter channels and depends on the capillary number and the orientation of the bifurcation plane with respect to a three-dimensional gravitational field. It is observed that gravity induces asymmetry in the splitting, causing the splitting ratio to be reduced. This effect is mitigated as the capillary number is increased. It is also observed that there exists a critical capillary number where the plug will not split and will instead propagate entirely into the gravitationally favoured daughter channel. We also compute the wall stresses on the bifurcation walls and observe the locations where stresses and their gradients are the highest in magnitude.

A similarity solution of the boundary layer equations for a wall jet on a heated horizontal surface at constant temperature taking into account the coupling of the temperature and velocity fields by buoyancy is described. This similarity solution exists for any value of ${\it\Lambda}=Gr/Re^{2}$, characterizing this coupling between natural and forced convection over the horizontal plate, where $Gr$ is a Grashof number and $Re$ is a Reynolds number, provided that the plate temperature is higher than the ambient temperature (${\it\Lambda}>0$, say). Two main qualitative differences are found in the flow structure in relation to the well-known Glauert’s similarity solution for a wall jet without natural convection effects (i.e. when ${\it\Lambda}=0$): the first is that the similarity variable and structure of the horizontal velocity and temperature have the same functional form for both a radially spreading jet and a two-dimensional jet; the second is that the maximum of the horizontal velocity increases as the jet spreads over the surface, instead of decreasing like in Glauert’s solution, as the radial or horizontal distance to the power $1/5$. To check this similarity solution we solve numerically the boundary layer equations for the particular case of a jet with constant velocity and temperature emerging from a slot of height ${\it\delta}$ and radius $r_{0}$ (in the radially spreading case). An approximate, analytical similarity solution near the jet exit is also found that helps to start the numerical integration. Far from the jet exit the numerical solution tends to the similarity solution for any set of values of the non-dimensional parameters governing the problem, provided that the plate is heated (${\it\Lambda}>0$). No similarity solution is found numerically for the case of a cooled plate (${\it\Lambda}<0$). For ${\it\Lambda}=0$ Glauert’s similarity solution is recovered.

Obtaining low-order models for unstable flows in a systematic and computationally tractable manner has been a long-standing challenge. In this study, we show that the Eigensystem Realisation Algorithm (ERA) can be applied directly to unstable flows, and that the resulting models can be used to design robust stabilising feedback controllers. We consider the unstable flow around a D-shaped body, equipped with body-mounted actuators, and sensors located either in the wake or on the base of the body. A linear model is first obtained using approximate balanced truncation. It is then shown that it is straightforward and justified to obtain models for unstable flows by directly applying the ERA to the open-loop impulse response. We show that such models can also be obtained from the response of the nonlinear flow to a small impulse. Using robust control tools, the models are used to design and implement both proportional and $\mathscr{H}_{\infty }$ loop-shaping controllers. The designed controllers were found to be robust enough to stabilise the wake, even from the nonlinear vortex shedding state and in some cases at off-design Reynolds numbers.

A combined theoretical and experimental investigation was carried out with the objective of evaluating theoretical predictions relating to a two-dimensional airfoil subjected to high amplitude harmonic oscillation of the free stream at constant angle of attack. Current theoretical approaches were reviewed and extended for the purposes of quantifying the bound, unsteady vortex sheet strength along the airfoil chord. This resulted in a closed form solution that is valid for arbitrary reduced frequencies and amplitudes. In the experiments, the bound, unsteady vortex strength of a symmetric 18 % thick airfoil at low angles of attack was measured in a dedicated unsteady wind tunnel at maximum reduced frequencies of 0.1 and at velocity oscillations less than or equal to 50 %. With the boundary layer tripped near the leading edge and mid-chord, the phase and amplitude variations of the lift coefficient corresponded reasonably well with the theory. Near the maximum lift coefficient overshoot, the data exhibited an additional high-frequency oscillation. Comparisons of the measured and predicted vortex sheet indicated the existence of a recirculation bubble upstream of the trailing edge which sheds into the wake and modifies the Kutta condition. Without boundary layer tripping, a mid-chord bubble is present that strengthens during flow deceleration and its shedding produces a dramatically different effect. Instead of a lift coefficient overshoot, as per the theory, the data exhibit a significant undershoot. This undershoot is also accompanied by high-frequency oscillations that are characterized by the bubble shedding. In summary, the location of bubble and its subsequent shedding play decisive roles in the resulting temporal aerodynamic loads.

In the present paper, we combine numerical and experimental approaches to study the dynamics of stable and unstable internal wave attractors. The problem is considered in a classic trapezoidal set-up filled with a uniformly stratified fluid. Energy is injected into the system at global scale by the small-amplitude motion of a vertical wall. Wave motion in the test tank is measured with the help of conventional synthetic schlieren and particle image velocimetry techniques. The numerical set-up closely reproduces the experimental one in terms of geometry and the operational range of the Reynolds and Schmidt numbers. The spectral element method is used as a numerical tool to simulate the nonlinear dynamics of a viscous salt-stratified fluid. We show that the results of 3D calculations are in excellent qualitative and quantitative agreement with the experimental data, including the spatial and temporal parameters of the secondary waves produced by triadic resonance instability. Further, we explore experimentally and numerically the effect of lateral walls on secondary currents and spanwise distribution of velocity amplitudes in the wave beams. Finally, we test the assumption of a bidimensional flow and estimate the error made in synthetic schlieren measurements due to this assumption.

The early stage of three-dimensional laminar-to-turbulent transition behind a micro-ramp is studied in the incompressible regime using tomographic particle image velocimetry. Experiments are conducted at supercritical micro-ramp height $h$ based Reynolds number $Re_{h}=1170$. The measurement domain encompasses 6 ramp widths spanwise and 73 ramp heights streamwise. The mean flow topology reveals the underlying vortex structure of the wake flow with multiple pairs of streamwise counter-rotating vortices visualized by streamwise vorticity. The primary pair generates a vigorous upwash motion in the symmetry plane with a pronounced momentum deficit. A secondary vortex pair is induced closer to the wall. The tertiary and even further vortices maintain a streamwise orientation, but are produced progressively outwards of the secondary pair and follow a wedge-type pattern. The instantaneous flow pattern reveals that the earliest unstable mode of the wake features arc-like Kelvin–Helmholtz (K–H) vortices in the separated shear layer. Under the influence of the K–H vortices, the wake exhibits a high level of fluctuations with a pulsatile mode for the streamwise momentum deficit. The K–H vortices are lifted up due to the upwash induced by the quasi-streamwise vortex pair, while they appear to undergo pairing, distortion and finally breakdown. Immediately downstream, a streamwise interval of relatively low vortical activity separates the end of the K–H region from the formation of new hairpin vortices close to the wall. The latter vortex structures originate from the region of maximum wall shear, induced by the secondary vortex pair causing strong ejection events which transport low-speed flow upwards. The whole pattern features a cascade of hairpin vortices along a turbulent/non-turbulent interface. The wedge-shaped cascade signifies the formation of a turbulent wedge. The turbulent properties of the wake are inspected with the spatial distribution of the velocity fluctuations and turbulence production in the developing boundary layer. Inside the wedge region, the velocity fluctuations approach quasi-spanwise homogeneity, indicating the development towards a turbulent boundary layer. The wedge interface is characterized by a localized higher level of velocity fluctuations and turbulence production, associated to the deflection of the shear layer close to the wall and the onset of coherent hairpin vortices inducing localized large-scale ejections.

This paper is devoted to an experimental investigation of receptivity of a laminar swept-wing boundary layer due to scattering of free-stream vortices on localized (in the streamwise direction) surface vibrations. The experiments were conducted under completely controlled disturbance conditions by means of a hot-wire anemometer on a model of a swept wing with a sweep angle of 25°. Both the free-stream vortices and the surface vibrations were generated by disturbance sources; their frequency–wavenumber spectra were measured thoroughly. The free-stream vorticity vectors were directed perpendicular to the incident-flow velocity vector and parallel to the swept-wing-model surface. The linearity of the receptivity mechanism under investigation (in a sense that the corresponding receptivity coefficients are independent of the disturbances amplitudes) has been checked carefully. The main goal of this experiment was to estimate the vibration-vortex receptivity coefficients as functions of the disturbance frequency, spanwise wavenumber and vortex offset parameter. This goal has been attained. Being defined in Fourier space, the obtained receptivity coefficients are independent of the specific surface vibration shape and can be used for verification of various receptivity theories.

The Mach-number scaling of the individual azimuthal modes of jet mixing noise was studied for jets in flight conditions, i.e. with co-flow. The data were obtained via a series of direct numerical simulations (DNS), performed of fully turbulent jets with a target Reynolds number, based on nozzle diameter, of $Re_{jet}=8000$. The DNS included a pipe 25 diameters in length in order to ensure that the flow developed to a fully turbulent state before exiting into a laminar co-flow, and to account for all possible noise generation mechanisms. To allow for a detailed study of the jet mixing noise component of the combined pipe–jet configuration, acoustic liner boundary conditions on the inside of the pipe and a modification to the synthetic turbulent inlet boundary condition of the pipe were applied to minimize internal noise in the pipe. Despite these measures, the use of a phased-array source breakdown technique was essential in order to isolate the sources associated with jet noise mechanisms from additional noise sources that could be attributed to internal noise or unsteady flow past the nozzle lip, in particular for the axisymmetric mode. Decomposing the sound radiation from the pipe–jet configuration into its azimuthal Fourier modes, and accounting for the co-flow effects, it was found that at $90^{\circ }$ the individual azimuthal Fourier modes of far-field pressure for the jet mixing noise component exhibit the same $M^{8}$ scaling with the centreline jet Mach number as that experimentally documented for the overall noise field. Applying the phased-array source breakdown code to the DNS data at smaller angles to the jet axis, an increase of the velocity exponent of the jet noise source was found, approaching 10 at $30^{\circ }$. At this smaller angle the higher azimuthal modes again showed the same behaviour as the axisymmetric mode.

We report on the dynamics of the formation and growth of the leading-edge vortex and the corresponding unsteady aerodynamic torque induced by large-scale flow-induced oscillations of an elastically mounted flat plate. All experiments are performed using a high-bandwidth cyber-physical system, which enables the user to access a wide range of structural dynamics using a feedback control system. A series of two-dimensional particle image velocimetry measurements are carried out to characterize the behaviour of the separated flow structures and its relation to the plate kinematics and unsteady aerodynamic torque generation. By modulating the structural properties of the cyber-physical system, we systematically analyse the formation, strength and separation of the leading-edge vortex, and the dependence on kinematic parameters. We demonstrate that the leading-edge vortex growth and strength scale with the characteristic feeding shear-layer velocity and that a potential flow model using the measured vortex circulation and position can, when coupled with the steady moment of the flat plate, accurately predict the net aerodynamic torque on the plate. Connections to previous results on optimal vortex formation time are also discussed.

Direct numerical simulations (DNS) of particle-laden turbulent flow in straight, mildly curved and strongly bent pipes are performed in which the solid phase is modelled as small heavy spherical particles. A total of seven populations of dilute particles with different Stokes numbers, one-way coupled with their carrier phase, are simulated. The objective is to examine the effect of the curvature on micro-particle transport and accumulation. It is shown that even a slight non-zero curvature in the flow configuration strongly impact the particle concentration map such that the concentration of inertial particles with bulk Stokes number $0.45$ (based on bulk velocity and pipe radius) at the inner bend wall of mildly curved pipe becomes $12.8$ times larger than that in the viscous sublayer of the straight pipe. Near-wall helicoidal particle streaks are observed in the curved configurations with their inclination varying with the strength of the secondary motion of the carrier phase. A reflection layer, as previously observed in particle laden turbulent S-shaped channels, is also apparent in the strongly curved pipe with heavy particles. In addition, depending on the curvature, the central regions of the mean Dean vortices appear to be completely depleted of particles, as observed also in the partially relaminarised region at the inner bend. The turbophoretic drift of the particles is shown to be affected by weak and strong secondary motions of the carrier phase and geometry-induced centrifugal forces. The first- and second-order moments of the velocity and acceleration of the particulate phase in the same configurations are addressed in a companion paper by the same authors. The current data set will be useful for modelling particles advected in wall-bounded turbulent flows where the effects of the curvature are not negligible.

In this paper, the instability of layered two-phase flows caused by the presence of a soluble surfactant (or a surface-active solute) is studied. The fluids have different viscosities, but are density matched to focus on Marangoni effects. The fluids flow between two flat plates, which are maintained at different solute concentrations. This establishes a constant flux of solute from one fluid to the other in the base state. A linear stability analysis is performed, using a combination of asymptotic and numerical methods. In the creeping flow regime, Marangoni stresses destabilize the flow, provided that a concentration gradient is maintained across the fluids. One long-wave and two short-wave Marangoni instability modes arise, in different regions of parameter space. A well-defined condition for the long-wave instability is determined in terms of the viscosity and thickness ratios of the fluids, and the direction of mass transfer. Energy budget calculations show that the Marangoni stresses that drive long- and short-wave instabilities have distinct origins. The former is caused by interface deformation while the latter is associated with convection by the disturbance flow. Consequently, even when the interface is non-deforming (in the large-interfacial-tension limit), the flow can become unstable to short-wave disturbances. On increasing the Reynolds number, the viscosity-induced interfacial instability comes into play. This mode is shown to either suppress or enhance the Marangoni instability, depending on the viscosity and thickness ratios. This analysis is relevant to applications such as solvent extraction in microchannels, in which a surface-active solute is transferred between fluids in parallel stratified flow. It is also applicable to the thermocapillary problem of layered flow between heated plates.

We conduct experiments in a circular pipe with rotating Hagen–Poiseuille flow (RHPF) to which we apply spin-down or impulsive spin-down to rest, in order to analyse the threshold between convective and absolute instabilities through flow visualisations in the inlet region of the pipe. For a constant value of the Reynolds number, $Re$, the finite-amplitude wave packets generated by the arbitrary perturbation that results by reducing the swirl parameter, propagate upstream or downstream depending on the initial value of the swirl parameter, $L_{0}$. In fact, the main characteristic of the flow is that the velocity front of these wave packets changes from negative to positive when absolutely unstable modes are present in the initial state. The experimental results show that spin-down becomes a precise, reliable procedure to detect the onset of absolute instabilities. In addition, we give evidence of a gradual transition for Reynolds numbers ranging from 300 to 500 where a mode shift from $n=-1$ to $n=-2$ appears in the absolutely unstable region.

Theory and numerical simulations show that the nonlinear self-interaction of internal modes in non-uniform stratification results in energy being transferred to superharmonic disturbances forced at twice the horizontal wavenumber and frequency of the parent mode. These disturbances are not in themselves a single mode, but a superposition of modes such that the disturbance amplitude is largest where the change in the background buoyancy frequency with depth is largest. Through weakly nonlinear interactions with the parent mode, the disturbances evolve to develop vertical-scale structures that distort and modulate the parent mode. Because pure resonant wave triads do not exist in non-uniformly stratified fluid, parametric subharmonic instability (PSI) is not evident even though noise is superimposed upon the initial state. The results suggest a new mechanism for energy transfer to dissipative scales (from large to small vertical scale and with frequencies larger and smaller than that of the parent mode) through forcing superharmonic rather than subharmonic disturbances.

External turbulence-induced axial flow in an incompressible, normal-mode stable Lamb–Oseen (two-dimensional) vortex column is studied via direct numerical simulations of the Navier–Stokes equations. Azimuthally oriented vorticity filaments, formed from external turbulence, advect radially towards or away from the vortex axis (depending on the filament’s swirl direction), resulting in a net induced axial flow in the vortex core; axial flow increases with increasing vortex Reynolds number ($Re=$ vortex circulation/viscosity). This contrasts the viscous mechanism for axial flow generation downstream of a lifting body, wherein an axial pressure gradient is produced by viscous diffusion of the swirl (Batchelor, J. Fluid Mech., vol. 20, 1964, pp. 645–658). Analysis of the self-induced motion of an arbitrarily curved external filament shows that any non-axisymmetric filament undergoes radial advection. We then studied the evolution of a vortex column starting with an imposed optimal transient growth perturbation. For a range of Re values, axial flow develops and initially grows as (time)$^{5/2}$ before decreasing after two turnover times; for $Re=10\,000$ – the highest computationally achievable – axial flow at late times becomes sufficiently strong to induce vortex instability. Contrary to a prior claim of a parent–offspring mechanism at the outer edge of the core, vorticity tilting within the core by axial flow is the underlying mechanism producing energy growth. Thus, external perturbations in practical flows (at $Re\sim 10^{7}$) produce destabilizing axial flow, possibly leading to the sought-after vortex breakup.

Laboratory experiments in which uniformly stratified fluids are stirred by horizontally moving vertical grids, or arrays of vertical rods, are reviewed to examine their consistency and to compare their findings, particularly those relating to the generation of layers. Selected experiments are of three types, those in which (a) turbulence spreads from a horizontally confined region where it is continuously generated by an oscillating grid; (b) grid stirring is maintained throughout a rectangular tank; or (c) a ‘cloud’ of turbulence decays after a short period of horizontally localized grid mixing. In all the experiments turbulence is generated over the full vertical extent of the tank. In the experiments of types (a) and (c) layers of comparable scale are observed to intrude into the ambient fluid from the turbulent region. In the type (b) experiments, layers form only when the time interval between the passage of the grid through the stratified fluid is sufficiently long, allowing turbulence to decay substantially between grid strokes. Two mechanisms are found to be dominant in the production of layers. In experiments of type (a) and (c) overturning eddies in the turbulent region of scale significantly larger than the Ozmidov length scale collapse and spread, intruding and forming layers in the adjoining laminar region. Internal shear waves propagating ahead of the intrusions have a vertical wavelength that is approximately twice the layer height. In type (b) experiments, layers are formed through a process described by Holford & Linden (Dyn. Atmos. Oceans, vol. 30, 1999a, pp. 173–198): the bending of vortices shed by the grid bars. The height of the layers is approximately half the vertical wavelength of internal shear waves that travel at the speed of the grid. It is proposed that the flow field of the shear waves bends the vortices, resulting in diapycnal mixing that forms the layers. The relationship of layers and internal shear waves in the experiments is therefore as follows: in type (a) and (c) experiments internal waves are generated by layers intruding from the turbulent region into the quiescent stratified region, but in experiments of type (b) internal waves drive and dictate the height of the layers; layers are generated as a consequence of vortex bending by internal waves. There is insufficient evidence to establish whether zigzag instability, either of vortex pairs or of vortex streets shed by a moving grid, accounts for the layers in any of the three types of grid experiments. The Phillips and Posmentier instability may reinforce layers formed by other processes. The generation of pancake vortices or vortical mode motion is left for later review.

We investigate the convergence behaviour of a cylindrical, fast magnetohydrodynamic (MHD) shock wave in a neutrally ionized gas collapsing onto an axial line current that generates a power law in time, azimuthal magnetic field. The analysis is done within the framework of a modified version of ideal MHD for an inviscid, non-dissipative, neutrally ionized compressible gas. The time variation of the magnetic field is tuned such that it approaches zero at the instant that the shock reaches the axis. This configuration is motivated by the desire to produce a finite magnetic field at finite shock radius but a singular gas pressure and temperature at the instant of shock impact. Our main focus is on the variation with shock radius $r$, as $r\rightarrow 0$, of the shock Mach number $M(r)$ and pressure behind the shock $p(r)$ as a function of the magnetic field power-law exponent ${\it\mu}\geqslant 0$, where ${\it\mu}=0$ gives a constant-in-time line current. The flow problem is first formulated using an extension of geometrical shock dynamics (GSD) into the time domain to take account of the time-varying conditions ahead of the converging shock, coupled with appropriate shock-jump conditions for a fast, symmetric MHD shock. This provides a pair of ordinary differential equations describing both $M(r)$ and the time evolution on the shock, as a function of $r$, constrained by a collapse condition required to achieve tuned shock convergence. Asymptotic, analytical results for $M(r)$ and $p(r)$ are obtained over a range of ${\it\mu}$ for general ${\it\gamma}$, and for both small and large $r$. In addition, numerical solutions of the GSD equations are performed over a large range of $r$, for selected parameters using ${\it\gamma}=5/3$. The accuracy of the GSD model is verified for some cases using direct numerical solution of the full, radially symmetric MHD equations using a shock-capturing method. For the GSD solutions, it is found that the physical character of the shock convergence to the axis is a strong function of ${\it\mu}$. For $0\leqslant {\it\mu}<4/13$, $M$ and $p$ both approach unity at shock impact $r=0$ owing to the dominance of the strong magnetic field over the amplifying effects of geometrical convergence. When ${\it\mu}\geqslant 0.816$ (for ${\it\gamma}=5/3$), geometrical convergence is dominant and the shock behaves similarly to a converging gas dynamic shock with singular $M(r)$ and $p(r)$, $r\rightarrow 0$. For $4/13<{\it\mu}\leqslant 0.816$ three distinct regions of $M(r)$ variation are identified. For each of these $p(r)$ is singular at the axis.

Puffing of a decane/ethanol emulsion droplet and a droplet group under convective heating and its effects on fuel/air mixing are investigated by direct numerical simulation that resolves all of the liquid/gas and liquid/liquid interfaces. With distinct differences in the boiling point between decane and ethanol, the embedded ethanol sub-droplets can be superheated and boil explosively. Puffing, i.e. ejection of ethanol vapour, occurs from inside the parent decane droplet, causing secondary breakup of the droplet. The ejected ethanol vapour mixes with the outer gas mixture composed of air and vapour of the primary fuel decane, and its effects on fuel/air mixing can be characterised by the scalar dissipation rates (SDRs). For the primary-fuel SDR, the cross-scalar diffusion due to ethanol vapour puffing plays a dominant role in enhancing the micromixing. When the vapour ejection direction is inclined towards the wake direction, the wake is elongated, but the shape of the stoichiometric mixture fraction isosurface is not changed much, indicating a limited effect on droplet grouping in a spray. On the other hand, when the ejection direction is inclined towards the transverse direction, the stoichiometric surface is pushed further away in the transverse direction, and its topology is changed by the puffing. The trajectories of ejected ethanol vapour pockets can be predicted by the correlation obtained for a jet in cross-flow, and the vapour pockets may reach a few diameters away from the droplet. Therefore, in a multiple-droplet configuration, the transverse ethanol vapour ejection due to puffing may transiently change the droplet grouping characteristics. In simulation cases with multiple droplets, the interaction changing the droplet grouping due to puffing has been confirmed, especially for droplets in the most upstream position in a spray. This implies that puffing should be accurately included in the mixing and combustion modelling of such a biofuel-blended diesel spray process.

An image analysis method is developed and applied to shadowgraph images of supersonic jet flow to measure shock front propagation angles at numerous interrogation points distributed throughout the quiescent region outside of the jet shear layer. These shock fronts manifest in acoustic measurements of jet noise as steepened temporal waveforms that have been linked to the perception of crackle. The analysis method uses the Radon transform to quantitatively determine a local shock front propagation angle at each point. The dataset of angles is subsequently used to determine the locations and convection velocities of the sources inside the jet shear layer. The results indicate that the shock-like waves emerge immediately from the jet shear layer and are created by the supersonic convection of coherent structures. The statistical distribution of convection velocities follows an extreme value distribution, indicating that the shock front emitting sources are maxima of the underlying turbulence. A noise reduction method known to reduce the convection velocities in the jet shear layer is applied to the jet to investigate the effect on the shock front emission. The shock front angles change in concert with the reduction in convection velocity giving further evidence that the source of crackle is a flow field event.

The relation between the flux of temperature (or buoyancy), the vertical temperature gradient and the height above the bottom is investigated in an oceanographic context, using high-resolution temperature measurements. The model for the evolution of a stratified layer by Balmforth et al. (J. Fluid Mech., vol. 355, 1998, pp. 329–358) is reviewed and adapted to the case of a turbulent flow above a wall. Model predictions are compared with the average observational estimates of the flux, exploiting a flux estimation method proposed by Winters & D’Asaro (J. Fluid Mech., vol. 317, 1996, pp. 179–193). This estimation method enables the disentanglement of the dependence of the average flux on the height above the bottom and on the background temperature gradient. The classical N-shaped flux–gradient relation is found in the observations. The model and the observations show similar qualitative behaviour, despite the strong simplifications used in the model. The results shed light on the modulation of the temperature flux by the presence of the boundary, and support the idea of a turbulent flux following a mixing-length argument in a stratified flow. Furthermore, the results support the use of Thorpe scales close to a boundary, if sufficient averaging is performed, suggesting that the Thorpe scales are affected by the boundary in a similar way to the mixing length.

The dynamics of a cantilevered elastic sheet, with a uniform steady flow impinging on its clamped end, have been studied widely and provide insight into the stability of flags and biological phenomena. Recent measurements by Kim et al. (J. Fluid Mech., vol. 736, 2013, R1) show that reversing the sheet’s orientation, with the flow impinging on its free edge, dramatically alters its dynamics. In contrast to the conventional flag, which exhibits (small-amplitude) flutter above a critical flow speed, the inverted flag displays large-amplitude flapping over a finite band of flow speeds. The physical mechanisms giving rise to this flapping phenomenon are currently unknown. In this article, we use a combination of mathematical theory, scaling analysis and measurement to establish that this large-amplitude flapping motion is a vortex-induced vibration. Onset of flapping is shown mathematically to be due to divergence instability, verifying previous speculation based on a two-point measurement. Reducing the sheet’s aspect ratio (height/length) increases the critical flow speed for divergence and ultimately eliminates flapping. The flapping motion is associated with a separated flow – detailed measurements and scaling analysis show that it exhibits the required features of a vortex-induced vibration. Flapping is found to be periodic predominantly, with a transition to chaos as flow speed increases. Cessation of flapping occurs at higher speeds – increased damping reduces the flow speed range where flapping is observed, as required. These findings have implications for leaf motion and other biological processes, such as the dynamics of hair follicles, because they also can present an inverted-flag configuration.