Wall-bounded flows, in their transition from a laminar state to turbulence, pass through a set of particular stages characterized by different physical processes. Among wall-bounded flows, separated flows have a special place because their dynamics can either be noise amplifiers or oscillators. For several years Marxen and co-workers have been studying the evolution of two- and three-dimensional perturbations in the laminar part of a laminar separation bubble. In Marxen et al. (J. Fluid Mech., vol. 728, 2013, p. 58) they study vortex formation and its evolution in laminar–turbulent transition in a forced separation bubble. By the combined use of numerical and experimental methods, different mechanisms of secondary instabilities have been highlighted: elliptic instability of vortex cores and hyperbolic instability responsible for three-dimensionality in the braid region. This work shows, for the first time in laminar separation bubbles, the first nonlinear stages of transition to turbulence of such a flow. However, since this type of flow is very sensitive to various environmental stresses, several scenarios for transition to turbulence remain to be explored.

We study the classical Landau–Levich dip-coating problem in the case where the interface has significant elasticity. One aim of this work is to unravel the effect of surface-adsorbed hydrophobic particles on Landau–Levich flow. Motivated by recent findings (Vella, Aussillous & Mahadevan, Europhys. Lett., vol. 68, 2004, pp. 212–218) that a jammed monolayer of adsorbed particles on a fluid interface makes it respond akin to an elastic solid, we use the Helfrich elasticity model to study the effect of interfacial elasticity on Landau–Levich flow. We define an elasticity number, $\mathit{El}$, which represents the relative strength of viscous forces to elasticity. The main assumptions of the theory are that $\mathit{El}$ be small, and that surface tension effects are negligible. The shape of the free surface is formulated as a nonlinear boundary value problem: we develop the solution as an asymptotic expansion in the small parameter ${\mathit{El}}^{1/ 7} $ and use the method of matched asymptotic expansions to determine the film thickness as a function of $\mathit{El}$. The solution to the shape of the static meniscus is not as straightforward as in the classical Landau–Levich problem, as evaluation of higher-order effects is necessary in order to close the problem. A remarkable aspect of the problem is the occurrence of multiple solutions, and five of these are found numerically. In any event, the film thickness varies as ${\mathit{El}}^{4/ 7} $ in qualitative agreement with the experiments of Ouriemi & Homsy (Phys. Fluids, 2013, in press).

We present experimental results on the role of flexibility and aspect ratio in bio-inspired aquatic propulsion. Direct thrust and power measurements are used to determine the propulsive efficiency of flexible panels undergoing a leading-edge pitching motion. We find that flexible panels can give a significant amplification of thrust production of $\mathscr{O}(100{\unicode{x2013}} 200\hspace{0.167em} \% )$ and propulsive efficiency of $\mathscr{O}(100\hspace{0.167em} \% )$ when compared to rigid panels. The data highlight that the global maximum in propulsive efficiency across a range of panel flexibilities is achieved when two conditions are simultaneously satisfied: (i) the oscillation of the panel yields a Strouhal number in the optimal range ($0. 25\lt \mathit{St}\lt 0. 35$) predicted by Triantafyllou, Triantafyllou & Grosenbaugh (J. Fluid Struct., vol. 7, 1993, pp. 205–224); and (ii) this frequency of motion is tuned to the structural resonant frequency of the panel. In addition, new scaling laws for the thrust production and power input to the fluid are derived for the rigid and flexible panels. It is found that the dominant forces are the characteristic elastic force and the characteristic fluid force. In the flexible regime the data scale using the characteristic elastic force and in the rigid limit the data scale using the characteristic fluid force.

The behaviour of a nominally two-dimensional ventilated partial cavity was examined over a wide range of size scales and flow speeds to determine the influence of Froude, Reynolds, and Weber number on the cavity shape, dynamics, and gas entrainment rate. Two geometrically similar experiments were conducted with a 14:1 length scale ratio. The results were compared to a two-dimensional semi-analytical model of the cavity flow, and Froude scaling was found to be sufficient to match basic cavity shapes. However, the air flux required to maintain a stable cavity did not scale with Froude number alone, as the dynamics of the cavity closure changed with increasing Reynolds number. The required air flux differed over one order of magnitude between the lowest and highest Reynolds number flows. But, for sufficiently high Reynolds numbers, the rate of scaled entrainment appeared to approach Reynolds number independence. Modest changes in surface tension of the small-scale experiment suggested that the Weber number was important only at the lowest speeds and smaller length scale. Otherwise, the Weber numbers of the flows were sufficiently high to make the effects of interfacial tension negligible. We also observed that modest unsteadiness in the inflow to the large-scale cavity led to a significant increase in the required air flux needed to maintain a stable cavity, with the required excess gas flux nominally proportional to the flow’s perturbation amplitude. Finally, discussion is provided on how these results relate to model testing of partial cavity drag reduction (PCDR) systems for surface ships.

The free-fall motion of a thin disk with small dimensionless moments of inertia (${I}^{\ast } \lt 1{0}^{- 3} $) was investigated experimentally. The transition from two-dimensional zigzag motion to three-dimensional spiral motion occurs due to the growth of three-dimensional disturbances. Oscillations in the direction normal to the zigzag plane increase with the development of this instability. At the same time, the oscillation of the nutation angle decreases to zero and the angle remains constant. The effects of initial conditions (release angle) were investigated. Two kinds of transition modes, zigzag–spiral transition and zigzag–spiral–zigzag intermittence transition, were observed to be separated by a critical Reynolds number. In addition, the solution of the generalized Kirchhoff equations shows that the small ${I}^{\ast } $ is responsible for the growth of disturbances in the third dimension (perpendicular to the planar motion).

This study considers the convective-type instability of the near-field flow of a planar, pure thermal plume with a finite area source. Previous studies revealed the existence of an off-axis thermal boundary-layer instability, driving a puffing instability in the central ascending column, and qualitatively showed correlations between instabilities in these two flow regions. This paper extends the analysis to examine the effect of Prandtl number on transitional near-field behaviours and reports on the stability characteristics of a near-field, pure thermal plume based on a direct stability analysis. The variations in flow behaviours in response to symmetric and asymmetric disturbances suggest the existence of coupled instability mechanisms in the off-axis thermal boundary layer and the central ascending column.

We propose a novel model of a pure liquid film evaporating into an inert gas, taking into account an effect of convection of the vapour by the evaporation flow of the gas. For the liquid phase, the long-wave approximation is applied to the governing equations. Assuming that fluctuations of the vapour concentration in the gas phase are localized in the vicinity of the liquid–gas interface, we consider only the limit of the mass transport equation at the interface. The diffusion term in the vertical direction of the mass transport equation is modelled by introducing the concentration boundary layer above the liquid film and solving the stationary convection–diffusion equation for the concentration inside the boundary layer. We investigate the linear stability of a flat film based on our model. The effect of vapour diffusion along the interface mitigates the Marangoni effect for short-wavelength disturbances. The local variation of vertical advection is found to be negligible. A critical thickness above which the film is stable exists under the presence of gravity. The effect of fluctuation of mass loss of the liquid induced by horizontal vapour diffusion becomes the primary instability mechanism in a very thin region. The effects of the resistance of phase change and the time derivative of the interface concentration are also examined.

The deepening of a shear-driven turbulent layer penetrating into a stably stratified quiescent layer is studied using direct numerical simulation (DNS). The simulation design mimics the classical laboratory experiments by Kato & Phillips (J. Fluid Mech., vol. 37, 1969, pp. 643–655) in that it starts with linear stratification and applies a constant shear stress at the lower boundary, but avoids sidewall and rotation effects inherent in the original experiment. It is found that the layers universally deepen as a function of the square root of time, independent of the initial stratification and the Reynolds number of the simulations, provided that the Reynolds number is large enough. Consistent with this finding, the dimensionless entrainment velocity varies with the bulk Richardson number as $R{i}^{- 1/ 2} $. In addition, it is observed that all cases evolve in a self-similar fashion. A self-similarity analysis of the conservation equations shows that only a square root growth law is consistent with self-similar behaviour.

The dynamics of bubbles in upflow, in a vertical channel, is examined using direct numerical simulations (DNS), where both the flow and the bubbles are fully resolved. Two cases are simulated. In one case all the bubbles are of the same size and sufficiently small so they remain nearly spherical. In the second case, some of the small bubbles are coalesced into one large bubble. In both cases lift forces drive small bubbles to the wall, removing bubbles from the channel interior until the two-phase mixture is in hydrostatic equilibrium, and forming a bubble-rich wall layer. The same evolution has been seen in earlier DNS of bubbly upflows, but here the friction Reynolds number is higher (${\mathit{Re}}^{+ } = 250$). In addition to showing that the overall structure persists at higher Reynolds numbers, we show that the bubbles in the wall layer form clusters. The mechanism responsible for the clustering is explained and how bubbles move into and out of the wall layer is examined. The dynamics of the bubbles in the channel core is also compared with results obtained in fully periodic domains and found to be similar. The presence of the large bubble disrupts the wall layer slightly, but does not change the overall picture much, for the parameters examined here.

We study the flow of a thin liquid film along a flexible substrate. The flow is modelled using lubrication theory, assuming that gravity is the dominant driving force. The substrate is modelled as an elastic beam that deforms in two dimensions. Steady solutions are found using numerical and perturbation methods, and several different asymptotic regimes are identified. We obtain a complete characterization of how the length and stiffness of the beam and the imposed liquid flux determine the profile of the liquid film and the resulting beam deformation.

The rotating cylindrical annulus geometry was first developed by Busse (J. Fluid Mech., vol. 44, 1970, pp. 441–460) as a simplified analogue for studying convection in rapidly rotating spherical geometries. Although it has provided a more tractable two-dimensional model than the sphere, it is formally limited to asymptotically small slopes and thus weak velocities in the direction parallel to the rotation axis. We present an asymptotically reduced three-dimensional equation set to model quasi-geostrophic convection in the annulus geometry where order-one slopes are permissible; this model provides a closer analogue to quasi-geostrophic convection in spheres and spherical shells where steeply sloping boundaries are present. A linear stability analysis of the reduced equations shows that a new class of three-dimensional, convectively driven Rossby waves is present in this system. The gravest modes exhibit strong axial variations as the slope of the boundaries becomes large. In addition, higher-order eigenmodes showing increasingly complex axial dependence are found that possess critical Rayleigh numbers close to that of the gravest mode.

A novel analytical equation for the assessment of the accuracy of filters used for the interpolation and differentiation of scattered experimental data is presented. The equation takes into account the statistical nature of the filter output resulting from both the arbitrary positions of the data points and the randomness and noise present in the experimental data. Numerical estimation of the accuracy of the filter, using a Monte Carlo procedure, shows good agreement with the deduced analytical equation. This numerical procedure was also used to determine the accuracy of variance filters aimed at calculating the mean-square fluctuation of experimental data. The combination of the numerical results and analytical equations reveals the exact sources of inaccuracy arising in scattered point filters, namely: (i) the spectral inaccuracy of the weighting function; (ii) the noise or stochastic signal amplification; and (iii) the error arising from the random collocation of points within the filter window. The results also demonstrate that the use of the local mean in the calculation of the quadratic fluctuation leads to smaller estimation errors than the central mean. Finally, all these filters are used and critically evaluated in the framework of the stochastic position, diameter, and velocity of bubbles in a gas-fluidized bed. It is shown that the empirical coefficient of bubble coalescence in the two-dimensional bed tested, $\bar {\lambda } $, is in the range 2.0–2.4 when incorporating only the visible flow of bubbles. Here, the vertical distance over which a bubble survives without coalescing is $\bar {\lambda } {L}_{c} $, where ${L}_{c} $ is the characteristic separation between neighbouring bubbles in the horizontal direction prior to coalescence. It was also seen that the relative mean-square-root fluctuation of both bubble diameter and velocity is more than 50 % at the centre of the bed and remains nearly constant along the height of the bed.

This paper reports experimental measurements on the yield stress, the permeability to gas flow and the gas velocity at the jamming transition of gas-fluidized beds of magnetizable particles as affected by particle size and orientation and strength of an externally imposed magnetic field. Tested samples consisted of relatively monodisperse magnetite powders of $35$, $50$ and $65~\unicode[.5,0][STIXGeneral,Times]{x03BC} \mathrm{m} $ particle size. The permeability to gas flow and jamming transition velocity increase with particle size and in a specially marked way when the magnetic field is applied along the gas flow direction. The magnetic contribution to the yield stress is also particularly enhanced for co-flow magnetic fields. However, the effect of particle size on the yield stress shows a dependence on the microstructure packing as affected by particle size and orientation of the field. The magnetic yield stress increases with particle size for magnetic fields applied in the cross-flow configuration while the opposite trend is observed when the direction of the magnetic field is parallel to the gas flow. The observations reported in this paper are generally explained by the formation of chains of particles due to attractive magnetic forces between the magnetized particles and the orientation of these chains with respect to the magnetic field.

An asymptotic analysis of laminar free convection in a boundary layer over an isothermal semi-infinite flat plate inclined at some angle to the vertical has been performed. Existing analytical solutions show no difference in the heat-transfer rate between the upper and lower surfaces of the plate, contrary to observations. To investigate this, higher-order perturbations of the non-dimensional temperature, velocity and pressure across the boundary layer were computed and found to show only small variations from first-order perturbations previously reported. Unexpectedly, third-order perturbations of all functions were found to be identical to those of the vertical plate, indicating that differences in temperature between both sides of the plate are limited to exceedingly small terms of order ${x}^{- 9/ 4} $ or smaller, $x$ being the distance from the leading edge, non-dimensionalized by the buoyancy length scale. Dominant differences between heat-transfer rates on the upper and lower surfaces were therefore concluded to be due to near-leading-edge effects. In applying an integral form of the conservation equations to the near-leading-edge region, it was found that, up to terms of order unity in $x$, the total heat-exchange rate for the inclined plate is identical to that of the vertical plate, so that the heat-transfer gain on one side balances exactly the loss occurring on the other. This simplification allowed determination of an upper bound for differences in heat-transfer rates between the upper and lower sides, even though complete determination of the differences would require a numerical integration of the full Navier–Stokes equations near the leading edge of the plate.

The issue of intermittency in numerical solutions of the 3D Navier–Stokes equations on a periodic box ${[0, L] }^{3} $ is addressed through four sets of numerical simulations that calculate a new set of variables defined by ${D}_{m} (t)= {({ \varpi }_{0}^{- 1} {\Omega }_{m} )}^{{\alpha }_{m} } $ for $1\leq m\leq \infty $ where ${\alpha }_{m} = 2m/ (4m- 3)$ and ${[{\Omega }_{m} (t)] }^{2m} = {L}^{- 3} \int \nolimits _{\mathscr{V}} {\vert \boldsymbol{\omega} \vert }^{2m} \hspace{0.167em} \mathrm{d} V$ with ${\varpi }_{0} = \nu {L}^{- 2} $. All four simulations unexpectedly show that the ${D}_{m} $ are ordered for $m= 1, \ldots , 9$ such that ${D}_{m+ 1} \lt {D}_{m} $. Moreover, the ${D}_{m} $ squeeze together such that ${D}_{m+ 1} / {D}_{m} \nearrow 1$ as $m$ increases. The values of ${D}_{1} $ lie far above the values of the rest of the ${D}_{m} $, giving rise to a suggestion that a depletion of nonlinearity is occurring which could be the cause of Navier–Stokes regularity. The first simulation is of very anisotropic decaying turbulence; the second and third are of decaying isotropic turbulence from random initial conditions and forced isotropic turbulence at fixed Grashof number respectively; the fourth is of very-high-Reynolds-number forced, stationary, isotropic turbulence at up to resolutions of $409{6}^{3} $.

The three-dimensional flow field and instantaneous forces are measured on pitching rectangular, lunate and truncate planforms of aspect-ratio four. The leading-edge vortex on the rectangular planform is compressed as it grows, and subsequently forms an arch-shaped vortex. For the lunate and truncate planforms, which both have identical spanwise leading-edge curvature but differ in planform area, outboard-directed convection of vorticity, rather than vortex stretching, mitigates arch-vortex formation. The vortical near wake that is formed by the planforms with spanwise leading-edge curvature is found to be strongly correlated with a favourable lift-to-drag ratio during the force-relaxation phase.

A stereoscopic particle image velocimetry (PIV) system for use in shallow (${\sim }$0.5 m deep) rivers was developed and deployed in the Urie River, Scotland, to study the interactions between turbulent flow and a Ranunculus penicillatus plant patch in its native environment. Statistical moments of the velocity field were calculated utilizing a new method of reducing the contribution of measurement noise, based on the measurement redundancy inherent in the stereoscopic PIV method. Reynolds normal and shear stresses, their budget terms, and higher-order moments of the velocity probability distribution in the wake of the plant patch were found to be dominated by the presence of a free shear layer induced by the plant drag. Plant motion, estimated from the PIV images, was characterized by travelling waves that propagate along the plant with a velocity similar to the eddy convection velocity, suggesting a direct coupling between turbulence and the plant motion. The characteristic frequency of the plant velocity fluctuations (${\sim }$1 Hz) may suggest that the plant motion is dominated by large eddies with scale similar to the flow depth or plant length. Plant and fluid velocity fluctuations were, in contrast, found to be strongly correlated only over a narrow (${\sim }$30 mm) elevation range above the top of the plant, supporting a contribution of the shear layer turbulence to the plant motion. Many aspects of flow–aquatic plant interactions remain to be clarified, and the newly developed stereoscopic field PIV system should prove valuable in future studies.

Direct numerical simulation is used to investigate the nonlinear evolution of a horizontally oriented mixing layer with uniform stable stratification and coordinate system rotation about the vertical axis. The important dimensional parameters governing inviscid dynamics are maximum shear $S(t)$, buoyancy frequency $N$, angular velocity of rotation $\Omega $ and characteristic shear thickness $L(t)$. The effect of rotation rate, $\Omega $, on the development of fluctuations in the shear layer is systematically studied in a regime of strong stratification. An instability mechanism, qualitatively distinct from the inertial instability, is found to deform columnar vortex cores in vertical planes for a strongly stratified rotating mixing layer. This mechanism emerges when centreline absolute vertical vorticity, $\langle {\omega }_{3} \rangle (t)+ 2\Omega $, is nearly zero as predicted by the linear stability analysis in Part 1 (J. Fluid. Mech., vol. 703, 2012, pp. 29–48). When the initial rotation rate is moderately anticyclonic, strong destabilization and a cascade to small scales is observed, consistent with prior studies involving horizontally sheared flow in the presence of rotation. Examination of enstrophy budgets in cases which are initially inertially unstable reveal the importance of baroclinic torque in maintaining lateral enstrophy fluctuations substantially beyond the time when the flow becomes inertially stable. The cyclonic stratified cases show weak nonlinearity in vortex dynamics. At high Reynolds number, despite the strong stratification, the flow exhibits three-dimensional, nonlinear dynamics and significant vertical mixing except for cases where the rotation is stabilizing.

A numerical investigation of the flow dynamics around a two-dimensional high-lift configuration was carried out by means of a zonal detached eddy simulation (ZDES) technique for flow conditions corresponding to aircraft approach. Both slat and flap regions have been scrutinized and compared with experimental data available in the literature. It is shown that slat and flap coves behave like shallow cavities. The distance between the upstream cusp and the downstream edge is the relevant length scale for each cove taken separately. Consistently with previous findings, this study indicates that the maximum of the broadband spectrum of slat (respectively flap) pressure fluctuations occurs for Strouhal numbers $0. 5\leq \mathit{St}\leq 4$ when based on slat chord (respectively on flap chord) and free-stream velocity. It is shown that mode $(n)$ of the slat cove and mode $(n+ 1)$ of the flap cove are very close making a coherent phase relationship possible. A large-scale coupled self-sustained oscillations mechanism between slat and flap cavities, evidenced by spectral analysis, occurs at a Strouhal number $\mathit{St}= 3{\unicode{x2013}} 6$ based on the main wing chord and free-stream velocity. This yields to an acoustic feedback mechanism characterized by a normalized frequency depending on the free stream Mach number like $\mathit{St}= (1- { M}_{0}^{2} )/ 2{M}_{0} $. The present result appears to line up with the findings by Hein et al. (J. Fluid Mech., vol. 582, 2007, pp. 179–202) who showed that two types of resonance could exist: surface waves ones, scaling with the total aerofoil length and longitudinal cavity-type resonances, scaling with the slat cove length.

A numerical database (Bernardini & Pirozzoli, Phys. Fluids, vol. 23, 2011, 085102) is analysed to investigate cross-statistics of wall-pressure fluctuations induced by supersonic turbulent boundary layers developing over a rigid smooth wall without a pressure gradient. The Mach number spans from 2 to 4 and a relatively large range of Reynolds numbers is considered. It is shown, for the first time, that the cross-statistics of wall-pressure fluctuations at such high Mach numbers is very similar to that usually observed in incompressible flow conditions. Furthermore, it is demonstrated that theoretical predictions provided by the well-known Corcos and Efimtsov models continue to apply.