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.

We analyse the buckling stability of a thin, viscous sheet when subject to simple shear, providing conditions for the onset of the dominant out-of-plane modes using two models: (i) an asymptotic theory for the dynamics of a viscous plate, and (ii) the full Stokes equations. In either case, the plate is stabilized by a combination of viscous resistance, surface tension and buoyancy relative to an underlying denser fluid. In the limit of vanishing thickness, plates buckle at a shear rate independent of buoyancy, where is the plate thickness, is the average surface tension between the upper and lower surfaces, and is the fluid viscosity. For thicker plates stabilized by an equal surface tension at the upper and lower surfaces, at and above onset, the most unstable mode has moderate wavelength, is stationary in the frame of the centreline, spans the width of the plate with crests and troughs aligned at approximately to the walls, and closely resembles elastic shear modes. The thickest plates that can buckle have an aspect ratio (thickness/width) of approximately 0.6 and are stabilized only by internal viscous resistance. We show that the viscous plate model can only accurately describe the onset of buckling for vanishingly thin plates but provides an excellent description of the most unstable mode above onset. Finally, we show that, by modifying the plate model to incorporate advection and make the model material-frame-invariant, it is possible to extend its predictive power to describe relatively short, travelling waves.

We extend the stability analysis of incompressible Kolmogorov flow, induced by a spatially periodic external force in an unbounded domain, to a compressible hard-sphere gas confined between two parallel isothermal walls. The two-dimensional problem is studied by means of temporal stability analysis of a ‘slip flow’ continuum-limit model and the direct simulation Monte Carlo (DSMC) method. The neutral curve is obtained in terms of the Reynolds () and Knudsen () numbers, for a given non-dimensional wavenumber of the external force. In the incompressible limit (), the problem is governed only by the Reynolds number, and our neutral curve coincides with the critical Reynolds number () calculated in previous incompressible analyses. Fluid compressibility () affects the flow field through the generation of viscous dissipation, coupling flow shear rates with irreversible heat production, and resulting in elevated bulk-fluid temperatures. This mechanism has a stabilizing effect on the system, thus increasing (compared to its incompressible value) with increasing . When compressibility effects become strong enough, transition to instability changes type from ‘exchange of stabilities’ to ‘overstability’, and perturbations are dominated by fluctuations in the thermodynamic fields. Most remarkably, compressibility confines the instability to small () Knudsen numbers, above which the Kolmogorov flow is stable for all . Good agreement is found between ‘slip flow’ and DSMC analyses, suggesting the former as a useful alternative in studying the effects of various parameters on the onset of instability, particularly in the context of small Knudsen numbers considered.

We measured the correlation spectrum of pressure fluctuations in a driving mixing layer with a Taylor-scale Reynolds number up to by a newly developed pressure probe with spatial and temporal resolutions that are sufficient to analyse inertial-subrange statistics. The influence of the mean velocity gradient tensor in the mixing layer, which is almost constant near its centreline, is studied using an idea similar to that underlying the linear response theory developed in statistical mechanics for systems at or near thermal equilibrium. If we write the spectrum as , where is the isotropic Kolmogorov spectrum in the absence of mean shear, then for small the deviation due to the shear is approximately linear and is determined by a few non-dimensional universal constants in addition to , and the mean energy dissipation rate. We also measured the pressure–velocity and velocity–velocity correlation spectra. Deviations from isotropy due to shear are shown to be approximately proportional to at large .

A long-standing problem in non-Newtonian fluid mechanics, namely the relationship between drag experienced by a sphere settling in a tube filled with a dilute polymeric solution and the sphere sedimentation velocity, is investigated via self-consistent multiscale flow simulations. Comparison with experimental measurements by Arigo et al. (J. Non-Newtonian Fluid Mech., vol. 60, 1995, pp. 225–257) have revealed that the evolution of the drag coefficient as a function of fluid elasticity can be accurately predicted when the macromolecular dynamics is described by realistic micromechanical models that closely capture the transient extensional viscosity of the experimental fluid at high extension rates. Specifically, for the first time we have computed the drag coefficient on the sphere at high Weissenberg number utilizing multi-segment bead–spring chain models with appropriate molecular parameters and have demonstrated that a hi-fidelity multiscale simulation is not only capable of accurately describing the drag on the sphere as a function of at various sphere-to-tube diameter ratios but also it can closely reproduce the experimentally observed velocity and stresses in the wake of the sphere.

The quadrant analysis of the intense tangential Reynolds stress in plane turbulent channels is generalized to three-dimensional structures (Qs), with special emphasis on the logarithmic and outer layers. Wall-detached Qs are background stress fluctuations. They are small and isotropically oriented, and their contributions to the mean stress cancel. Wall-attached Qs are larger, and carry most of the mean Reynolds stresses. They form a family of roughly self-similar objects that become increasingly complex away from the wall, resembling the vortex clusters in del Álamo et al. (J. Fluid Mech., vol. 561, 2006, pp. 329–358). Individual Qs have fractal dimensions of the order of , slightly fuller than the clusters. They can be described as ‘sponges of flakes’, while vortex clusters are ‘sponges of strings’. The number of attached Qs decays away from the wall, but the fraction of the stress that they carry is independent of their sizes. A substantial fraction of the stress resides in a few large objects extending beyond the centreline, reminiscent of the very large structures of several authors. The predominant logarithmic-layer structure is a side-by-side pair of a sweep (Q4) and an ejection (Q2), with an associated cluster, and shares dimensions and stresses with the conjectured attached eddies of Townsend (J. Fluid Mech., vol. 11, 1961, pp. 97–120). Those attached eddies tend to be aligned streamwise from each other, located near the side walls between the low- and high-velocity large-scale streaks, but that organization does not extend far enough to explain the very long structures in the centre of the channel.

We report an experimental study of the turbulent flow above a barchan dune in a channel, from particle image velocimetry measurements, for Reynolds numbers ranging from 9000, just below the threshold for particle motion, up to 24 000, where the dune moves. Two calculations of the speed-up over the dune are compared, the usual ‘same-elevation’ and the more relevant ‘Lagrangian’, showing that the latter is smaller by a factor of two. The two-layer structure of the flow disturbance – an essentially inviscid outer layer and a turbulent inner layer of thickness – is assessed. In the outer layer, streamline curvature is shown to be responsible for half of the Lagrangian speed-up, from the comparison of the velocity measurements with two Bernoulli calculations. In the inner layer, detailed measurements of the velocity and stresses are provided, down to , and the momentum budget is discussed. The Reynolds shear stress decreases monotonically towards the dune surface, according to the standard mixing-length closure, whereas the total shear stress increases strongly in the viscous sublayer. Along the dune surface, the shear stress increases up to the crest where it reaches twice its unperturbed value. A good estimate of the surface stress is provided by a parabolic fit of the inner velocity profile matching the outer flow at . Doubling the Reynolds number, the surface shear stress and the speed-up decrease by ∼30 %. The implications of these results on the dune motion, presented in Part 1 of this study (Franklin & Charru, J. Fluid Mech., vol. 675, 2011, pp. 199–222), are finally discussed.

Aquatic bacteria like Bacillus subtilis are heavier than water yet they are able to swim up an oxygen gradient and concentrate in a layer below the water surface, which will undergo Rayleigh–Taylor-type instabilities for sufficiently high concentrations. In the literature, a simplified chemotaxis–fluid system has been proposed as a model for bio-convection in modestly diluted cell suspensions. It couples a convective chemotaxis system for the oxygen-consuming and oxytactic bacteria with the incompressible Navier–Stokes equations subject to a gravitational force proportional to the relative surplus of the cell density compared to the water density. In this paper, we derive a high-resolution vorticity-based hybrid finite-volume finite-difference scheme, which allows us to investigate the nonlinear dynamics of a two-dimensional chemotaxis–fluid system with boundary conditions matching an experiment of Hillesdon et al. (Bull. Math. Biol., vol. 57, 1995, pp. 299–344). We present selected numerical examples, which illustrate (i) the formation of sinking plumes, (ii) the possible merging of neighbouring plumes and (iii) the convergence towards numerically stable stationary plumes. The examples with stable stationary plumes show how the surface-directed oxytaxis continuously feeds cells into a high-concentration layer near the surface, from where the fluid flow (recurring upwards in the space between the plumes) transports the cells into the plumes, where then gravity makes the cells sink and constitutes the driving force in maintaining the fluid convection and, thus, in shaping the plumes into (numerically) stable stationary states. Our numerical method is fully capable of solving the coupled chemotaxis–fluid system and enabling a full exploration of its dynamics, which cannot be done in a linearised framework.

A rarefied gas flow thermally induced around a heated (or cooled) flat plate, contained in a vessel, is considered in two different situations: (i) both sides of the plate are simultaneously and uniformly heated (or cooled); and (ii) only one side of the plate is uniformly heated. The former is known as the thermal edge flow and the latter, typically observed in the Crookes radiometer, may be called the radiometric flow. The steady behaviour of the gas induced in the container is investigated on the basis of the Bhatnagar–Gross–Krook (BGK) model of the Boltzmann equation and the diffuse reflection boundary condition by means of an accurate finite-difference method. The flow features are clarified for a wide range of the Knudsen number, with a particular emphasis placed on the structural similarity between the two flows. The limiting behaviour of the flow as the Knudsen number tends to zero (and thus the system approaches the continuum limit) is investigated for both flows. The detailed structure of the normal stress on the plate as well as the cause of the radiometric force (the force acting on the plate from the hotter to the colder side) is also clarified for the present infinitely thin plate.

A free-surface-induced morphological instability is studied in the laminar regime at large Reynolds numbers () and on sub-horizontal walls (). We analytically and numerically develop the stability analysis of an inclined melting–freezing interface bounding a free-surface laminar flow. The complete solution of both the linearized flow field and the heat conservation equations allows the exact derivation of the upper and lower temperature gradients at the interface, as required by the Stefan condition, from which the dispersion relationship is obtained. The eigenstructure is obtained and discussed. Free-surface dynamics appears to be crucial for the triggering of upstream propagating ice ripples, which grow at the liquid–solid interface. The kinematic and the dynamic conditions play a key role in controlling the formation of the free-surface fluctuations; these latter induce a streamline distortion with an increment of the wall-normal velocities and a destabilizing phase shift in the net heat transfer to the interface. Three-dimensional effects appear to be crucial at high Reynolds numbers. The role of inertia forces, vorticity, and thermal boundary conditions are also discussed.

The bed friction velocity distribution and sediment entrainment potential of Boussinesq compositional gravity currents propagating over a series of obstacles and over a smooth surface, respectively, are analysed based on high-resolution, three-dimensional large-eddy simulations. The investigation focuses on the parameter regime for which currents with a high volume of release go through an extended slumping phase with approximately constant front velocity (Tokyay, Constantinescu & Meiburg, J. Fluid Mech., vol. 672, 2011, 570–605). Under these conditions, a quasi-steady regime is reached between consecutive obstacles that is similar to the steady regime observed for constant-density channel flows over bottom obstacles. At a given location, this quasi-steady regime is reached in the tail of the current after the passage of the front and the associated hydraulic jumps reflected from the first few downstream obstacles. A double-averaging procedure is employed to characterize the global changes in the structure of the tail region between currents with a high volume of release propagating over smooth surfaces and over obstacles. Reynolds-number-induced scale effects on the flow and turbulence structure within the tail region are discussed in some detail. The presence of this quasi-steady regime is significant, since the simulations with obstacles show that most of the sediment is entrained by the tail of the current, rather than by its front. A detailed analysis of the effects of the obstacle shape on the quasi-steady mean flow and turbulence structure is presented, which provides insight into why gravity currents over dunes can entrain more sediment than gravity currents over ribs of comparable size. Finally, the bed friction velocity distributions and the potential to entrain sediment are compared for a compositional current with a high volume of release during the slumping phase, and a current with a low volume of release for which transition to the buoyancy–inertia phase occurs a short time after the release of the lock gate.

We consider the linear stability and nonlinear evolution of two-dimensional shear flows that take the form of an unstratified plane Couette flow that is seeded with a localized ‘defect’ containing sharp density and vorticity variations. For such flows, matched asymptotic expansions furnish a reduced model that allows a straightforward and computationally efficient exploration of flows at sufficiently high Reynolds and Péclet numbers that sharp density and vorticity gradients persist throughout the onset, growth and saturation of instability. We are thereby able to study the linear and nonlinear dynamics of three canonical variants of stratified shear instability: Kelvin–Helmholtz instability, the Holmboe instability, and the lesser-considered Taylor instability, all of which are often interpreted in terms of the interactions of waves riding on sharp interfaces of density and vorticity. The dynamics near onset is catalogued; if the interfaces are sufficiently sharp, the onset of instability is subcritical, with a nonlinear state existing below the linear instability threshold. Beyond onset, both Holmboe and Taylor instabilities are susceptible to inherently two-dimensional secondary instabilities that lead to wave mergers and wavelength coarsening. Additional two-dimensional secondary instabilities are also found to appear for higher Prandtl numbers that take the form of parasitic Holmboe-like waves.

In order to analyse the geometric structure of turbulent flow patterns and their statistics for various scalar fields we adopt the dissipation element (DE) approach and apply it to turbulent channel flow by employing direct numerical simulations (DNS) of the Navier–Stokes equations. Gradient trajectories starting from any point in a scalar field in the directions of ascending and descending scalar gradients will always reach an extremum, i.e. a minimum or a maximum point, where . The set of all points and trajectories belonging to the same pair of extremal points defines a dissipation element. Extending previous DE approaches, which were only applied to homogeneous turbulence, we here focus on exploring the influence of solid walls on the dissipation element distribution. Employing group-theoretical methods and known symmetries of Navier–Stokes equations, we observe for the core region of the flow, i.e. the region beyond the buffer layer, that the probability distribution function (p.d.f.) of the DE length exhibits an invariant functional form, in other words, self-similar behaviour with respect to the wall distance. This is further augmented by the scaling behaviour of the mean DE length scale which shows a linear scaling with the wall distance. The known proportionality of the mean DE length and the Taylor length scale is also revisited. Utilizing a geometric analogy we give the number of DE elements as a function of the wall distance. Further, it is observed that the DE p.d.f. is rather insensitive, i.e. invariant with respect both to the Reynolds number and the actual scalar which has been employed for the analysis. In fact, a very remarkable degree of isotropy is observed for the DE p.d.f. in regions of high shear. This is in stark contrast to classical Kolmogorov scaling laws which usually exhibit a strong dependence on quantities such as shear, anisotropy and Reynolds number. In addition, Kolmogorov’s scaling behaviour is in many cases only visible for very large Reynolds numbers. This is rather different in the present DE approach which applies also for low Reynolds numbers. Moreover, we show that the DE p.d.f. agrees very well with the log-normal distribution and derive a log-normal p.d.f. model taking into account the wall-normal dependence. Finally, the conditional mean scalar differences of the turbulent kinetic energy at the extremal points of DE are examined. We present a power law with scaling exponent of known from Kolmogorov’s hypothesis for the centre of the channel and a logarithmic law near the wall.

The steady axisymmetric flow induced by a ring sink (or source) submerged in an unbounded inviscid fluid is computed and the resulting deformation of the free surface is obtained. Solutions are obtained analytically in the limit of small Froude number (and hence small surface deformation) and numerically for the full nonlinear problem. The small Froude number solutions are found to have the property that if the non-dimensional radius of the ring sink is less than , there is a central stagnation point on the surface surrounded by a dip which rises to the stagnation level in the far distance. However, as the radius of the ring sink increases beyond , a surface stagnation ring forms and moves outward as the ring sink radius increases. It is also shown that as the radius of the sink increases, the solutions in the vicinity of the ring sink/source change continuously from those due to a point sink/source () to those due to a line sink/source (). These properties are confirmed by the numerical solutions to the full nonlinear equations for finite Froude numbers. At small values of the Froude number and sink or source radius, the nonlinear solutions look like the approximate solutions, but as the flow rate increases a limiting maximum Froude number solution with a secondary stagnation ring is obtained. At large values of sink or source radius, however, this ring does not form and there is no obvious physical reason for the limit on solutions. The maximum Froude numbers at which steady solutions exist for each radius are computed.

We present measurements of mean and conditional number densities, radial distribution functions (r.d.f.s), velocities and accelerations of sub-Kolmogorov-scale water droplets entraining at a shearless turbulence–turbulence interface (TTI) and a turbulence–non-turbulence interface (TNI). We thus look at statistics of an inhomogeneous inertial particle field in both homogeneous and inhomogeneous turbulence. As in a previous communication (Gerashchenko, Good & Warhaft J. Fluid Mech., vol. 818, 2011, pp. 293–303), an active grid produces high-Reynolds number turbulence on either one or both sides of a splitter plate in a wind tunnel. Sprays seed droplets on one side of the splitter plate, while screens dampen turbulence in the adjacent flow for the TNI. Gravitational and inertial effects are isolated by turning of the apparatus with respect to gravity. We parameterize the droplets under homogeneous conditions, where it is demonstrated that both the sweeping and loitering effects on the droplet settling velocities are present. In the inhomogeneous conditions, we show that the droplets are entrained in bulk, resulting in large-scale clusters and preserving the droplet-ambient conditions of the seeded side of the flows.

A new approach to the modelling of wetting fronts in porous media on the Darcy scale is developed, based on considering the types (modes) of motion the menisci go through on the pore scale. This approach is illustrated using a simple model case of imbibition of a viscous incompressible liquid into an isotropic porous matrix with two modes of motion for the menisci, the wetting mode and the threshold mode. The latter makes it necessary to introduce an essentially new technique of conjugate problems that allows one to link threshold phenomena on the pore scale with the motion on the Darcy scale. The developed approach (a) makes room for incorporating the actual physics of wetting on the pore scale, (b) brings in the physics associated with pore-scale thresholds, which determine when sections of the wetting front will be brought to a halt (pinned), and, importantly, (c) provides a regular framework for constructing models of increasing complexity.

Head-on collisions of two equal-sized viscous drops in a biaxial extensional flow were simulated using the boundary integral method in the Stokes flow limit, for capillary numbers of the order of –, typical of flow-induced coalescence experiments. At a certain point in time, during the drainage process, the flow was abruptly stopped and the time-dependent dynamics of drop deformation (relaxation) was followed to discern whether the pair of drops would eventually coalesce. The concept of coalescence probability was used to study the evolution of probable collisions. The polymeric system of polybutadiene (PBd) drops undergoing head-on collisions in a polydimethylsiloxane (PDMS) matrix, previously well-characterized both experimentally and numerically by Yoon et al. (Phys. Fluid, vol. 19, 2007, 102102), in which both fluids were Newtonian under the experimental conditions, was used as our reference. Film shapes, velocity profiles and pressure distributions were studied for initially parabolic or dimpled thin film shapes. It was shown that micrometre-sized drops undergoing relaxation can coalesce in the capillary number range studied, which also included cases of hindered coalescence and cases in which the flow interaction time for the collision was smaller than the drainage time; thus, this phenomenon could influence the final drop size distribution of blends. Further, these findings could be of interest in interpreting stop–strain experiments, in the case of a sudden change in flow conditions and in population balance studies of drops in blends.

The large-scale/long-time transport of inertial particles of arbitrary mass density under gravity is investigated by means of a formal multiple-scale perturbative expansion in the scale-separation parameter between the carrier flow and the particle concentration field. The resulting large-scale equation for the particle concentration is determined, and is found to be diffusive with a positive definite eddy diffusivity. The calculation of the latter tensor is reduced to the resolution of an auxiliary differential problem, consisting of a coupled set of two differential equations in a -dimensional coordinate system (three space coordinates plus three velocity coordinates plus time). Although expensive, numerical methods can be exploited to obtain the eddy diffusivity, for any desirable non-perturbative limit (e.g. arbitrary Stokes and Froude numbers). The aforementioned large-scale equation is then specialized to deal with two different relevant perturbative limits: (i) vanishing of both Stokes time and sedimenting particle velocity; (ii) vanishing Stokes time and finite sedimenting particle velocity. Both asymptotics lead to a greatly simplified auxiliary differential problem, now involving only space coordinates and thus easily tackled by standard numerical techniques. Explicit, exact expressions for the eddy diffusivities have been calculated, for both asymptotics, for the class of parallel flows, both static and time-dependent. This allows us to investigate analytically the role of gravity and inertia on the diffusion process by varying relevant features of the carrier flow, such as the form of its temporal correlation function. Our results exclude a universal role played by gravity and inertia on the diffusive behaviour: regimes of both enhanced and reduced diffusion may exist, depending on the detailed structure of the carrier flow.

In this study, we consider various contributions to the forces on an impulsively started finite plate from the perspective of a diagnostic vorticity force theory. The wing plate has an aspect ratio (AR) between 1 and 3, and is placed at low and high angles of attack ( and ), while the Reynolds number is either 100 or 300. The theory enables us to quantify the contributions to the forces exerted on the plate in terms of all of the fluid elements with non-zero vorticity, such as in the tip vortices (TiVs), leading- and trailing-edge vortices (LEV and TEV) as well on the plate surface. This line of force analysis has been pursued for two-dimensional flow in our previous studies. In contrast to the pressure force analysis (PFA), the vorticity force analysis (VFA) reveals new salient features in its applications to three-dimensional flow by examining sectional force contributions along the spanwise direction. In particular, at a large aspect ratio (), the force distributions of PFA and VFA show close agreements with each other in the middle sections, while at a lower aspect ratio (), the force distribution of PFA is substantially larger than that of VFA in most of the sections. The difference is compensated for by the contributions partly by the edge sections and mainly by the vortices in the outer regions. Further investigation is made fruitful by decomposing the vorticity into the spanwise (longitudinal) component (the only one in two-dimensional flow) and the other two orthogonal (transverse) components. The relative importance of the force contributions credited to the transverse components in the entire flow regions as well as in the two outer regions signifies the three-dimensional nature of the flow over a finite plate. The interplay between the LEV and the TiVs at various time stages is shown to play a key role in distinguishing the force contributions for the plate with a smaller aspect ratio and that with a larger aspect ratio. The present VFA provides a better perspective for flow control by relating the forces directly to the various sources of vorticity (or vortex structures) on or near the wing plate.

In the Nastrom–Gage spectrum of atmospheric turbulence, we observe a energy spectrum that transitions into a spectrum, with increasing wavenumber . The transition occurs near a transition wavenumber , located near the Rossby deformation wavenumber . The Tung–Orlando theory interprets this spectrum as a double downscale cascade of potential enstrophy and energy, from large scales to small scales, in which the downscale potential enstrophy cascade coexists with the downscale energy cascade over the same length scale range. We show that, in a temperature-forced two-layer quasi-geostrophic model, the rates with which potential enstrophy and energy are injected place the transition wavenumber near . We also show that, if the potential energy dominates the kinetic energy in the forcing range, then the Ekman term suppresses the upscale cascading potential enstrophy more than it suppresses the upscale cascading energy, a behaviour contrary to what occurs in two-dimensional turbulence. As a result, the ratio of injected potential enstrophy over injected energy, in the downscale direction, decreases, thereby tending to decrease the transition wavenumber further. Using a random Gaussian forcing model, we reach the same conclusion, under the modelling assumption that the asymmetric Ekman term predominantly suppresses the bottom layer forcing, thereby disregarding a possible entanglement between the Ekman term and the nonlinear interlayer interaction. Based on these results, we argue that the Tung–Orlando theory can account for the approximate coincidence between and . We also identify certain open questions that require further investigation via numerical simulations.