A paradox concerning the flight of insects in two-dimensional space is identified: insects maintaining their bodies in a particular position (hovering) cannot, on average, generate hydrodynamic force if the induced flow is temporally periodic and converges to rest at infinity. This paradox is derived by using the far-field representation of periodic flow and the generalized Blasius formula, an exact formula for a force that acts on a moving body, based on the incompressible Navier–Stokes equations. Using this formula, the time-averaged force can be calculated solely in terms of the time-averaged far-field flow. A straightforward calculation represents the averaged force acting on an insect under a uniform flow, −〈V〉, determined by the balance between the hydrodynamic force and an external force such as gravity. The averaged force converges to zero in the limit 〈V〉 → 0, which implies that insects in two-dimensional space cannot hover under any finite external force if the direction of the uniform flow has a component parallel to the external force. This paradox provides insight into the effect of the singular behaviour of the flow around hovering insects: the far-field wake covers the whole space. On the basis of these assumptions, the relationship between this paradox and real insects that actually achieve hovering is discussed.

The stack-driven flow between two interconnected rooms produced by a single heat source is studied. In particular, the features of the transient flow for different positions and areas of two openings in the shared vertical wall are analysed. An analytical model provides the time evolution of the stratified flows in rooms of any size. The concept of an equivalent layer representing a non-uniform density profile, which is useful in other contexts, is included in the theoretical approach and provides physical insight and aids the mathematical solution of the problem. New salt-bath experiments are performed to simulate the thermal forcing between the rooms, to validate the model and to analyse the mixing generated and the effects of a source of volume in the configuration studied.

A two-dimensional horizontal flow is discussed, which is induced by other, buoyancy-driven flows elsewhere. It is an adaptation of the incompressible wall jet, which is driven by conditions a t the leading edge and has no streamwise pressure gradient. The relation of this flow to the classical buoyancy-driven boundary layers on inclined and horizontal surfaces is investigated, as well as its possible connexion with a two-dimensional buoyant plume driven by a line source of heat. Composite flows are constructed by patching various such solutions together. The composite flows exhibit $Gr^{\frac{1}{4}}$ scaling (Gr being the Grashof number).

This paper concerns feedback stabilization of point-vortex equilibria above an inclined thin plate and a three-plate configuration known as the Kasper wing in the presence of an oncoming uniform flow. The flow is assumed to be potential and is modelled by the two-dimensional incompressible Euler equations. Actuation has the form of blowing and suction localized on the main plate and is represented in terms of a sink–source singularity, whereas measurement of pressure across the plate serves as system output. We focus on point-vortex equilibria forming a one-parameter family with locus approaching the trailing edge of the main plate and show that these equilibria are either unstable or neutrally stable. Using methods of linear control theory we find that the system dynamics linearized around these equilibria is both controllable and observable for almost all actuator and sensor locations. The design of the feedback control is based on the linear–quadratic–Gaussian (LQG) compensator. Computational results demonstrate the effectiveness of this control and the key finding of this study is that Kasper wing configurations are in general not only more controllable than their single-plate counterparts, but also exhibit larger basins of attraction under LQG feedback control. The feedback control is then applied to systems with additional perturbations added to the flow in the form of random fluctuations of the angle of attack and a vorticity shedding mechanism. Another important observation is that, in the presence of these additional perturbations, the control remains robust, provided the system does not deviate too far from its original state. Furthermore, except in a few isolated cases, introducing a vorticity-shedding mechanism enhanced the effectiveness of the control. Physical interpretation is provided for the results of the controllability and observability analysis as well as the response of the feedback control to different perturbations.

The flow in a split cylinder with each half in exact counter rotation is studied numerically. The exact counter rotation, quantified by a Reynolds number $\mathit{Re}$ based on the rotation rate and radius, imparts the system with an $O(2)$ symmetry (invariance to azimuthal rotations as well as to an involution consisting of a reflection about the mid-plane composed with a reflection about any meridional plane). The $O(2)$ symmetric basic state is dominated by a shear layer at the mid-plane separating the two counter-rotating bodies of fluid, created by the opposite-signed vortex lines emanating from the two endwalls being bent to meet at the split in the sidewall. With the exact counter rotation, the additional involution symmetry allows for steady non-axisymmetric states, that exist as a group orbit. Different members of the group simply correspond to different azimuthal orientations of the same flow structure. Steady states with azimuthal wavenumber $m$ (the value of $m$ depending on the cylinder aspect ratio $\unicode[STIX]{x1D6E4}$) are the primary modes of instability as $\mathit{Re}$ and $\unicode[STIX]{x1D6E4}$ are varied. Mode competition between different steady states ensues, and further bifurcations lead to a variety of different time-dependent states, including rotating waves, direction-reversing waves, as well as a number of slow–fast pulse waves with a variety of spatio-temporal symmetries. Further from the primary instabilities, the competition between the vortex lines from each half-cylinder settles on either a $m=2$ steady state or a limit cycle state with a half-period-flip spatio-temporal symmetry. By computing in symmetric subspaces as well as in the full space, we are able to unravel many details of the dynamics involved.

A solidification boundary layer, where vorticity is concentrated, is shown to scale as δ ∼ (v/V) sin θ where θ is the angle that the solidification front makes with the oncoming flow, v is the kinematic viscosity of the melt, and V is the solidification rate. Key similarities with the suction boundary layer are outlined. Transport of vorticity, by diffusion and advection, in flows that solidify is discussed in the context of the planar-flow spin casting process, with special attention paid to how vorticity leaves the flow. Direct experimental evidence for a solidification layer in that application is reported.

This article continues an exploration of instabilities of jets in two-dimensional, inviscid fluid on the beta-plane. At onset, for particular choices of the physical parameters, the normal modes responsible for instability have critical levels that coalesce along the axis of the jet. Matched asymptotic expansion (critical layer theory) is used to derive a reduced model describing the dynamics of these modes. Because the velocity profile is locally parabolic in the vicinity of the critical levels the dynamics is richer than in standard critical layer problems. The model captures the inviscid saturation of unstable modes, the excitation of neutral Rossby waves, and the decay of disturbances when there are no discrete normal modes. Inviscid saturation occurs when the vorticity distribution twists up into vortical structures that take the form of either a pair of ‘cat's eye’ patterns straddling the jet axis, or a single row of vortices. The addition of weak viscosity destroys these cat's eye structures and causes the critical layer to spread diffusively. The results are compared with numerical simulations of the governing equations.

The boundary layers associated with gravity waves in a fluid with a linear variation of density are discussed in order to examine the steady Eulerian velocities induced by the Reynolds stress. For the case of a standing wave, the induced steady motion is shown to decay in an outer boundary layer which represents a balance between buoyancy and diffusion when the wave slope is sufficiently small but when viscous decay effects are even smaller. When the wave slope is larger, it would appear that two outer regions must be considered. Results for progressive waves are discussed only briefly, as they are qualitatively similar to the surface wave case.

We consider the flow which develops when two separate spaces maintained at different temperatures, both in excess of the exterior temperature, are connected through high and low level openings to a central atrium in which there is negligible heat load but which can naturally ventilate through high and low level openings to the exterior. We show that with a small temperature contrast between the spaces or large openings from the atrium to the exterior, upflow displacement ventilation develops in each of the spaces, with air entering from the atrium at low level and exiting at high level. However, with a larger temperature contrast between the spaces or small openings between the atrium and the exterior, a convective circulation develops between the spaces, with upflow in the warmer space and downflow in the colder space. Exterior air, which may enter the atrium at low level, flows into the warmer space along with the air from the colder space. At high level, air flows back into the atrium from the warmer space, and then either vents from the building or flows into the colder space. In this convection dominated flow regime, the colder space is a net heat sink, whereas with the upward displacement ventilation, this space acts as a net heat source. This can have significant implications for energy usage and on the build up of contaminants in each of the spaces. We also show that in both steady flow regimes, the air at mid-level in the atrium is unventilated and stagnant. We discuss the relevance of our model for controlled natural ventilation in large public buildings such as shopping malls where individual shops often maintain temperatures independently of the central atrium-space.

Buckling instabilities of thin sheets or plates of viscous fluid occur in situations ranging from food and polymer processing to geology. Slim, Teichman & Mahadevan (J. Fluid Mech., this issue, vol. 694, 2012, pp. 5–28) study numerically the buckling of a sheared viscous plate floating on a denser fluid using three approaches: a classical ‘thin viscous plate’ model; full numerical solution of the three-dimensional Stokes equations; and a novel ‘advection-augmented’ thin-plate model that accounts (in an asymptotically inconsistent way) for the advection of perturbations by the background shear flow. The advection-augmented thin-plate model is markedly superior to the classical one in its ability to reproduce the predictions of the Stokes solution, illustrating the utility of judicious violations of asymptotic consistency in fluid-mechanical models.

Seeley, Hummel & Smith (1975) reported the results of experiments to study the dynamics of flow around spheres at intermediate Reynolds numbers using a nondisturbing flow-visualization technique. The flow patterns were recorded on ciné photographs and the information stored was processed in order to obtain the velocity field. The position of fluid elements shown by the photochromic indicator traces were estimated by eye on a projection screen. In this paper, a new set of results based on the same films has been reduced and computed using the ‘POLLY’ film-reading system described by Esmail, Smith & Hummel (1976). Some numerical boundary-layer solutions are included to show the reliability of the data, and comparisons with the results previously reported by Seeley et al. (1975) are presented.J. W. Smith will be pleased to send a complete set of experimental data on request.

The near wakes of flows past single- and multi-scale arrays of bars are studied by means of planar laser induced fluorescence (PLIF) and particle image velocimetry (PIV). The aim of this research is to better understand dispersion of passive scalar downstream of the multi-scale turbulence generator. In particular, the focus is on plausible manifestations of the space-scale unfolding (SSU) mechanism, which is often considered in the literature as the reason for the enhancement of the turbulent scalar flux in flows past fractal grids (i.e. specific multi-scale turbulence generators). The analysis of qualitative and quantitative PLIF results, as well as the simultaneously acquired PIV results, confirms the appearance of a physical scenario resembling the SSU mechanism. Unlike the anticipation of the literature, however, this scenario applies to some extent also to the flow past the single-scale obstacle. Application of a triple decomposition technique (which splits the acquired fields into their means, a number of coherent fluctuations and their stochastic parts) and a conditional-averaging technique reveals that the SSU mechanism is active in the vicinity of an intersection point between two adjacent wakes and is driven almost exclusively by coherent fluctuations associated with the larger of the intersecting wakes. This suggests that the SSU mechanism is related to the coherent fluctuations embedded in the flow rather than to the fine-scale turbulence and its underlying integral length scale, as proposed in previous works.

We study the mechanisms affecting the viscous-fingering instability in an elastic-walled Hele-Shaw cell by considering the stability of steady states of unidirectional peeling-by-pulling and peeling-by-bending. We demonstrate that the elasticity of the wall influences the steady base state but has a negligible direct effect on the behaviour of linear perturbations, which thus behave like in the ‘printer’s instability’ with rigid walls. Moreover, the geometry of the cell can be very well approximated as a triangular wedge in the stability analysis. We identify four distinct mechanisms – surface tension acting on the horizontal and the vertical interfacial curvatures, kinematic compression in the longitudinal base flow, and the films deposited on the cell walls – that each contribute to stabilizing the system. The vertical curvature is the dominant stabilizing mechanism for small capillary numbers, but all four mechanisms have a significant effect in a large region of parameter space.