Vortex-induced vibration of solid structures has been known since ancient times, and remains a very important problem of fluid mechanics. Experimental studies of the vortex-induced vibration of circular cylinders have provided many fundamental insights. Recent extremely carefully controlled forced vibration experiments by Morse & Williamson (J. Fluid Mech., 2009, this issue, vol. 634, pp. 5–39), generating finely resolved data sets, are providing a fresh understanding of the mechanisms involved, especially for ocean structures with very low mass and damping.

In order to predict response and wake modes for elastically mounted circular cylinders in a fluid flow, we employ controlled-vibration experiments, comprised of prescribed transverse vibration of a cylinder in the flow, over a wide regime of amplitude and frequency. A key to this study is the compilation of high-resolution contour plots of fluid force, in the plane of normalized amplitude and wavelength. With such resolution, we are able to discover discontinuities in the force and phase contours, which enable us to clearly identify boundaries separating different fluid-forcing regimes. These appear remarkably similar to boundaries separating different vortex-formation modes in the map of regimes by Williamson & Roshko (J. Fluids Struct., vol. 2, 1988, pp. 355–381). Vorticity measurements exhibit the 2S, 2P and P + S vortex modes, as well as a regime in which the vortex formation is not synchronized with the body vibration. By employing such fine-resolution data, we discover a high-amplitude regime in which two vortex-formation modes overlap. Associated with this overlap regime, we identify a new distinct mode of vortex formation comprised of two pairs of vortices formed per cycle, where the secondary vortex in each pair is much weaker than the primary vortex. This vortex mode, which we define as the 2POVERLAP mode (2PO), is significant because it is responsible for generating the peak resonant response of the body. We find that the wake can switch intermittently between the 2P and 2PO modes, even as the cylinder is vibrating with constant amplitude and frequency. By examining the energy transfer from fluid to body motion, we predict a free-vibration response which agrees closely with measurements for an elastically mounted cylinder. In this work, we introduce the concept of an ‘energy portrait’, which is a plot of the energy transfer into the body motion and the energy dissipated by damping, as a function of normalized amplitude. Such a plot allows us to identify stable and unstable amplitude-response solutions, dependent on the rate of change of net energy transfer with amplitude (the sign of dE*/dA*). Our energy portraits show how the vibration system may exhibit a hysteretic mode transition or intermittent mode switching, both of which correspond with such phenomena measured from free vibration. Finally, we define the complete regime in the amplitude–wavelength plane in which free vibration may exist, which requires not only a periodic component of positive excitation but also stability of the equilibrium solutions.

The dynamics of passive scalars and tracers during the formation and subsequent persistence of a laminar tripolar vortex, obtained through an unstable monopolar vortex seeded with a k = 2 azimuthal perturbation, is investigated. Two-dimensional direct numerical simulations of passive scalars with Schmidt numbers Sc = 0.1, 1, 10 and 100 are performed. The scalar variance for the four cases is analysed, as well as the different dispersion patterns up to 10 times greater than the time for formation of the tripolar vortex. During the formation of the tripole, an accelerated scalar dissipation is observed. That dissipation is connected to the advection-dominated processes associated with the growth of the perturbation mode. During that process, the patterns of mixing of the different passive scalars are very much the same as for vorticity. This stage of accelerated dissipation is preceded and followed by stages of diffusion-dominated scalar dissipation. Passive Lagrangian tracers are used to explore the transport of fluid elements during the evolution, and to provide a detailed view of the tripolar vortex formation and behaviour for longer times. Chaotic mixing was studied by examining patterns of spatial variation of finite-time Lyapunov exponents. As the perturbation grows and the tripolar vortex is formed, two large regions of regular flow, divided by a region of chaotic flow, form for each satellite. When the tripole is fully formed, it is composed of three distinct regular regions, corresponding to the core of negative vorticity and the two satellites of positive vorticity. The comparison between the evolution for vorticity, concentration, randomly distributed particles and Lyapunov exponents shows that transport occurs mainly in the regions of chaotic flow that surround the tripolar vortex after its formation. For longer times, both the chaotic/regular flow interfaces and the vorticity gradients are responsible for the integrity of the tripolar system.

Direct numerical simulation (DNS) is performed to investigate the vortex synchronization phenomena in the wake behind a circular cylinder at the Reynolds numbers, Re = 220 (mode-A regime) and 360 (mode-B regime). To generate vortex synchronization, a sinusoidal streamwise velocity perturbation, the frequency of which is about twice the natural shedding frequency, is superimposed on the free stream velocity. At Re = 360, vortex synchronization occurs when the perturbation frequency is exactly twice the natural shedding frequency. However, at Re = 220, it does not occur when the same perturbation frequency condition is imposed. Instead, it occurs when the perturbation frequency is near twice the hypothetical two-dimensional laminar vortex shedding frequency as if there were no wake transition at Re = 220.

It is elucidated that, as a result of vortex synchronization, the trajectory of the Kármán vortices and the vortex structure are changed. The Kármán vortices are formed along the mean separating streamline slightly inside the mean wake bubble at Re = 220, but slightly outside at Re = 360. Thus, the Reynolds shear stress force has different contribution to the streamwise force balance of the mean wake bubble depending on the Reynolds numbers: its magnitude is negligible at Re = 220, compared to other force components, while it reverses its sign at Re = 360. More importantly, at Re = 220, the mode-A instability is suppressed into two-dimensional laminar flow with strong Kármań vortices. At Re = 360, the dominant instability mode changes from mode B to mode A.

We reconsider exact solutions to the Navier–Stokes equations that describe a vortex in a viscous, incompressible fluid. This type of solution was first introduced by Long (J. Atmos. Sci., vol. 15 (1), 1958, p. 108) and is parameterized by an inverse Reynolds number ϵ. Long's attention (and that of many subsequent investigators) was centred upon the ‘quasi-cylindrical’ (QC) case corresponding to ϵ = 0. We show that the limit ϵ → 0 is not straightforward, and that it reveals other solutions to this fundamental exact reduction of the Navier–Stokes system (which are not of QC form). Through careful numerical investigation, supported by asymptotic descriptions, we identify new solutions and describe the full parameter space that is spanned by ϵ and the pressure at the vortex core. Some erroneous results that exist in the literature are corrected.

A theory on weakly rarefied low-Mach-number flows with surface reactions based on small sticking coefficients was formulated for a binary gas mixture with an irreversible surface reaction, and then extended to a multicomponent mixture with multi-step surface reactions for the situation when all chemically active species are small in concentration compared to a major inert species. Particular interest was placed on the interaction between the Knudsen layer and the surface reactions. Results show that the Knudsen layer modifies not only the incident flux of the molecules striking the surface but also the temperature-sensitive sticking coefficients, and consequently the surface reaction rates. The surface reactions in turn modify the flow structure in the Knudsen layer through the non-zero net flux at the surface. The rate expressions for the surface reactions based on sticking coefficients were derived, and the slip boundary conditions for the temperature and the species concentration suitable for application were established. The widely used Motz–Wise correction formula for the surface reaction rate was revised and the underlying assumptions leading to its derivation were shown to be inappropriate.

Theoretical investigation of acoustic wave interactions with turbulent premixed flames was conducted to evaluate the acoustic energy amplification and/or damping due to the interaction of low-frequency acoustic waves with turbulent flames in three-dimensional space. Such amplified or damped acoustic energy is either coherent or incoherent as wrinkled flames cause coherent energy of a monochromatic acoustic wave to be damped into incoherent energy of spatially diffused and spectrally broadened acoustic waves. Small perturbation method (SPM) up to the second order was utilized to analyse net coherent and incoherent acoustic energies of the reflected and transmitted waves scattered from a weakly wrinkled turbulent flame surface in random motion. General formulations for net coherent and incoherent energy budget of the scattered fields were derived that can be applied to any type of flame height statistics. Production and/or damping of acoustic energy scattered from a turbulent flame is attributed to two effects: one is the acoustic velocity jump due to flame's unsteady heat release and the other is the flame's wrinkling due to its unsteady motion. Dimensionless parameters that govern net acoustic energy budget were derived in case of Gaussian statistics of flame surface behaviour: the r.m.s. and correlation length of flame height, the frequency ratio of the incidence frequency to the flame's correlation frequency, the time ratio of the flame's diffusion to correlation time and the incidence angle. The results of the scattered acoustic energy budget showed that noticeable amplification of acoustic energy was obtained either for a small frequency ratio (≪1) at the critical incidence angle or for a large frequency ratio and time ratio (≫1), while damping was obtained for a small frequency ratio at off-critical incidence angles. The relative importance of unsteady heat release (the jump effect) and unsteady motion (the wrinkling effect) to net acoustic energy is controlled mainly by the frequency ratio: The unsteady heat release effect dominates the wrinkling effect at a large frequency ratio, and vice versa at a small frequency ratio. The energy transfer from coherent to incoherent energy is due to flame surface wrinkling and is enhanced with the square of the flame's r.m.s. height.

Steady linear three-dimensional disturbances are investigated in a two-dimensional laminar boundary layer. The boundary layer is subject to a streamwise favourable-to-adverse pressure gradient and eventually undergoes separation. The separating flow corresponds to the first part of a pressure-induced laminar-separation bubble on a flat plate. Streamwise disturbance development in such a flow is studied by means of direct numerical simulation, a water-tunnel experiment and an adjoint-based parabolic theory suited to study spatial optimal growth. A complete overview of the disturbance evolution in various areas of the favourable-to-adverse pressure gradient laminar boundary layer is given. Results from all investigation methods show overall good agreement with respect to disturbance growth and shape within the entire domain. In the favourable pressure-gradient region and, again, slightly downstream of separation, transient growth caused by the lift-up effect dominates disturbance behaviour. In the adverse pressure-gradient region, a modal instability is observed. Evidence is presented that this instability is of Görtler type.

The stability of electrohydrodynamic flow between two horizontal plates with a vertical electrical conductivity gradient has been investigated in the presence of an imposed weak shear flow. The weak shear flow is driven by the horizontal pressure gradient, and the electrical conductivity gradient is generated by the concentration variation of the charge-carrying solute. An external electric field is applied across the fluid layer, and then the interaction between the unstable stratification of electrohydrodynamic flow and the shear arising from the plane Poiseuille flow is studied. A linear stability analysis has been implemented by considering both the longitudinal and transverse modes. Unlike the thermally stratified plane Poiseuille flow in which the longitudinal mode always dominates the onset of instability and is virtually unaffected by the superimposed shear flow, the instability of this mixed electrohydrodynamic–Poiseuille flow system is found to depend heavily on the shear flow, and the transverse mode may prevail over the longitudinal mode when the momentum of shear flow is sufficiently small. Particularly, an oscillatory longitudinal mode is found to exist, and it may become the critical mode when the conductivity gradient is small enough. The present results verify that an imposed weak shear flow may enhance the electrohydrodynamic instability in a fluid layer with electrical conductivity gradient.

Divergent ship bow waves were simulated experimentally with a two-dimensional wavemaker that employs a flexible wave board. The wavemaker was programmed so that the wave board created a time sequence of shapes that simulated the line of intersection between one side of the hull of a slender ship model moving at constant speed and an imaginary vertical plane oriented normal to the ship model track. The time history of the water surface shape was measured with a cinematic laser-induced fluorescence technique for eight Froude numbers (FD = U/, where U is the forward speed of the equivalent three-dimensional ship model, g the acceleration of gravity and D the ship model draft). The waves produced ranged from small-amplitude non-breaking waves at the lowest Froude numbers to plunging breakers at the highest Froude numbers. These waves are strongly forced and at the higher Froude numbers begin breaking before leaving the wave board. The time histories of various geometric characteristics of the water surface shape including the hull contact line, the wave crest, the plunging jet and the splash zone, which is here defined as both the turbulent zone on the front face of the wave in the spilling breakers and the turbulent zone generated ahead of the jet impact point in the plunging breakers, were measured. The phase speed of the primary wave generated during each run ranged from 2.56Uwl (where Uwl is the maximum speed of the wave board at the undisturbed water level in the tank) at the lowest Froude number to about 1.7Uwl at the three highest Froude numbers. The maximum heights of the primary wave, the contact point on the wavemaker and the splash zone increased in a nearly linear fashion with increasing FD. In the cases with plunging jets, the jet tip trajectory was parabolic with a vertical acceleration ranging from 0.6g at FD = 1.467 to 0.8g at FD = 1.817 (the highest Froude number).

We present a family of steadily rotating equilibrium states consisting of helically symmetric vortices in an incompressible inviscid irrotational unbounded fluid. These vortices are described by contours bounding regions of uniform axial vorticity. Helical symmetry implies material conservation of axial vorticity (in the absolute frame of reference) when the flow field parallel to vortex lines is proportional to (1+ϵ2r2)−1/2, where ϵ is the pitch and r is the distance from the axis. This material conservation property enables equilibria to be calculated simply by a restriction on the helical stream function. The states are parameterized by their mean radius and centroid position. In the case of a single vortex, parameter space cannot be fully filled by our numerical approach. We conjecture multiply connected contours will characterize equilibria where the algorithm fails. We also consider multiple vortices, evenly azimuthally spaced about the origin. Stability properties are investigated numerically using a helical CASL algorithm.

Slender sharp-edged flexible beams such as flapping wings of micro air vehicles (MAVs), piezoelectric fans and insect wings typically oscillate at moderate-to-high values of non-dimensional frequency parameter β with amplitudes as large as their widths resulting in Keulegan–Carpenter (KC) numbers of order one. Their oscillations give rise to aerodynamic damping forces which vary nonlinearly with the oscillation amplitude and frequency; in contrast, at infinitesimal KC numbers the fluid damping coefficient is independent of the oscillation amplitude. In this article, we present experimental results to demonstrate the phenomenon of nonlinear aerodynamic damping in slender sharp-edged beams oscillating in surrounding fluid with amplitudes comparable to their widths. Furthermore, we develop a general theory to predict the amplitude and frequency dependence of aerodynamic damping of these beams by coupling the structural motions to an inviscid incompressible fluid. The fluid–structure interaction model developed here accounts for separation of flow and vortex shedding at sharp edges of the beam, and studies vortex-shedding-induced aerodynamic damping in slender sharp-edged beams for different values of the KC number and the frequency parameter β. The predictions of the theoretical model agree well with the experimental results obtained after performing experiments with piezoelectric fans under vacuum and ambient conditions.

The onset of compressible convection in rapidly rotating spherical shells is studied in the anelastic approximation. An asymptotic theory valid at low Ekman number is developed and compared with numerical solutions of the full equations. Compressibility is measured by the number of density scale heights in the shell. In the Boussinesq problem, the location of the onset of convection is close to the tangent cylinder when there is no internal heating only a heat flux emerging from below. Compressibility strongly affects this result. With only a few scale heights or more of density present, there is onset of convection near the outer shell. Compressibility also strongly affects the frequencies and preferred azimuthal wavenumbers at onset. Compressible convection, like Boussinesq convection, shows strong spiralling in the equatorial plane at low Prandtl number. We also explore how higher-order linear modes penetrate inside the tangent cylinder at higher Rayleigh numbers and compare modes both symmetric and antisymmetric about the equator.

By adapting the general flame-acoustic interaction theory developed in Wu et al. (J. Fluid Mech., vol. 497, 2003, pp. 23–53), a systematic analysis is carried out for the interaction of a stable premixed flame in a duct with vortical disturbances superimposed on the oncoming mixture. A small-amplitude vortical perturbation, assumed to be a convecting gust with a frequency ω, induces a hydrodynamic field in the vicinity of the flame, causing an initially planar flame to wrinkle. The unsteady heat release resulting from the increased surface area of the wrinkling flame then generates a sound wave with frequency 2ω. When 2ω coincides with the natural frequency of an acoustic mode of the duct, a flame-acoustic resonance takes place, through which the flame-induced sound may attain an amplitude sufficiently large to modulate the flame through the unsteady Rayleigh–Taylor effect. A novel evolution system is derived to describe this two-way coupling for two cases: (a) a flame with a fixed mean position and (b) a moving flame. Numerical solutions show that for (a), the mutual flame-acoustic interaction initiates a violent subharmonic parametric instability, and the flame-acoustic system quickly evolves into a fully nonlinear regime, which probably corresponds to a state of self-sustained oscillation. This finding presents a peculiar instability scenario: a small-amplitude vortical perturbation may, by initiating acoustic-flame resonance, completely destabilize an otherwise stable planar flame. For a moving flame, the flame-acoustic resonance is of transient nature. The acoustic pressure gains substantially, but the parametric flame instability is induced only when the vortical disturbance exceeds a finite threshold.

A canonical flat-plate turbulent boundary layer with Reθ = 4590 is exposed to a favourable mean streamwise pressure gradient sufficient to cause relaminarization. The favourable pressure gradient is generated by a linear contraction, yielding a peak value of the acceleration parameter of K = 4.4 × 10−6 which is sustained for approximately 13 local boundary layer thicknesses. The relaminarization process is characterized by an extensive series of mean flow and turbulence measurements obtained at several representative streamwise locations. In anticipation of the loss of standard log-law behaviour, the local wall shear stress is directly measured using the oil-film interferometry technique. Mean flow measurements show a systematic variation in the Kármán and additive constants with applied streamwise strain rate. The series of measurements also indicate an apparent decoupling of the outer and near-wall regions of the accelerating boundary layer. In accord with this, conditional measurements show that fourth-quadrant sweep events are virtually eliminated, while much less frequent but larger-amplitude near-wall second-quadrant ejection events remain. The reduction in fourth-quadrant sweep events is matched by an observed increase in near-wall third-quadrant events. The consequent reduction in near-wall Reynolds stress correlation and associated cross-stream momentum transport results in a large reduction in cf for the relaminarized flow.

In this work a two-dimensional laminar flow past a square cylinder is considered. Actuators placed on the cylinder enable active control by blowing and suction. Proportional feedback control is then applied using velocity measurements taken in the cylinder wake. Projection onto an empirical subspace is combined with a calibration technique to build a low-order model of the incompressible Navier–Stokes equations. This model is used within an optimization method to determine a set of feedback gains which reduces the unsteadiness of the wake at Re = 150. The resulting controlled flows are further characterized by computing the critical Reynolds numbers for the onset of the vortex shedding instability.

Radiometric force on a 0.12 m circular vane is studied experimentally and numerically over a wide range of pressures that cover the flow regimes from near free molecular to near continuum. In the experiment, the vane is resistively heated to about 419 K on one side and 394 K on the other side, and immersed in a rarefied argon gas. The radiometric force is then measured on a nano-Newton thrust stand in a 3 m vacuum chamber and compared with the present numerical predictions and analytical predictions proposed by various authors. The computational modelling is conducted with a kinetic approach based on the solution of ellipsoidal statistical Bhatnagar–Gross–Krook (ES-BGK) equation. Numerical modelling showed the importance of regions with elevated pressure observed near the edges of the vane for the radiometric force production. A simple empirical expression is proposed for the radiometric force as a function of pressure that is found to be in good agreement with the experimental data. The shear force on the lateral side of the vane was found to decrease the total radiometric force.

Since insight into entropy generation is a key to increasing efficiency and thereby reducing fuel consumption and/or waste and – for wall-bounded flows – most entropy is generated in the viscous layer, we examine the transient behaviour of its dominant contributor there for a non-canonical flow. New measurements in oil flow are presented for the effects of favourable streamwise mean pressure gradients on temporal entropy generation rates and, in the process, on key Reynolds-stress-producing events such as sweep front passage and on the deceleration/outflow phase of the overall bursting process. Two extremes have been considered: (1) a high pressure gradient, nearing ‘laminarization’, and (2), for comparison, a low pressure gradient corresponding to many earlier experiments. In both cases, the peak temporal entropy generation rate occurs shortly after passage of the ejection/sweep interface. Whether sweep and ejection rates appear to decrease or increase with the pressure gradient depends on the feature examined and the manner of sampling. When compared using wall coordinates for velocities, distances and time, the trends and magnitudes of the transient behaviours are mostly the same. The main effects of the higher pressure gradient are (a) changes in the time lag between detections – representing modification of the shape of the sweep front and the sweep angle with the wall, (b) modification of the magnitude of an instantaneous Reynolds shear stress with wall distance and (c) enlarging the sweeps and ejections. Results, new for both low and high pressure gradients, are the temporal behaviours of the dominant contribution to entropy generation; it is found to be much more sensitive to distance from the wall than to streamwise pressure gradient.

The quasi-steady migration and deformation of bubbles rising in a wall-bounded linear shear flow are investigated experimentally in the low-but-finite-Reynolds-number regime. A travelling optical device that follows the bubble is used for this purpose. This apparatus allows us to determine accurately the bubble radius, contour and rising speed, together with the distance between the bubble and the wall. Thereby the transverse component of the hydrodynamic force is obtained for Reynolds numbers Re (based on the bubble diameter and slip velocity of the bubble in the undisturbed shear flow) less than 5. The results indicate that in the range 0.5 < Re < 1.5, the transverse force acting on a spherical bubble agrees well with an extension of the theoretical solution obtained by McLaughlin (J. Fluid Mech., vol. 246, 1993, pp. 249–265) for rigid spheres, whereas it becomes larger than the theoretical prediction for Re > 1.5. In the regime in which bubble deformation is significant, the shape of the bubble and the deformation-induced transverse force are determined both experimentally and computationally, using a spectral boundary element method. Both estimates are found to be in good agreement with each other, while the theory of Magnaudet, Takagi & Legendre (J. Fluid Mech., vol. 476, 2003, pp. 115–157) is found to predict accurately the deformation but fails to predict quantitatively the deformation-induced transverse force.

By identifying the stratification which leads to maximal buoyancy flux in a stably-stratified plane Couette flow, we make a prediction of what bulk stratification (as a function of the shear) is optimal for turbulent mixing. A previous attempt to do this (Caulfield, Tang & Plasting, J. Fluid Mech., vol. 498, 2004, p. 315) failed due to an unexpected degeneracy in the variational problem. Here, we overcome this issue by parameterizing the variational problem implicitly with the overall mixing efficiency which is then optimized across to return a rigorous upper bound on the buoyancy flux. We find that the bulk Richardson number quickly approaches 1/6 in the asymptotic limit of high shear with the associated mixing efficiency tending to 1/3. The predicted mean profiles associated with the bound appear to have a layered structure, with the gradient Richardson number being low both in the interior, and in boundary layers near the walls, with a global maximum, also equal to 1/6, occurring at the edge of the boundary layers.