The velocity distribution of dilute suspensions of heavy particles in gas–solid turbulent flows is investigated. A statistical approach – the mesoscopic Eulerian formalism (MEF) – is developed in which an average conditioned on a realization of the turbulent carrier flow is introduced and enables a decomposition of the instantaneous particle velocity into two contributions. The first is a contribution from an underlying continuous turbulent velocity field shared by all the particles – the mesoscopic Eulerian particle velocity field (MEPVF) – that accounts for all particle–particle and fluid–particle two-point correlations. The second contribution corresponds to a distribution – the quasi-Brownian velocity distribution (QBVD) – that represents a random velocity component satisfying the molecular chaos assumption that is not spatially correlated and identified with each particle of the system. The MEF is used to investigate properties of statistically stationary particle-laden isotropic turbulence. The carrier flow is computed using direct numerical simulation (DNS) or large-eddy simulation (LES) with discrete particle tracking employed for the dispersed phase. Particle material densities are much larger than that of the fluid and the force of the fluid on the particle is assumed to reduce to the drag contribution. Computations are performed in the dilute regime for which the influences of inter-particle collisions and fluid-turbulence modulation are neglected. The simulations show that increases in particle inertia increase the contribution of the quasi-Brownian component to the particle velocity. The particle velocity field is correlated at larger length scales than the fluid, with the integral length scales of the MEPVF also increasing with particle inertia. Consistent with the previous work of Abrahamson (1975), the MEF shows that in the limiting case of large inertia, particle motion becomes stochastically equivalent to a Brownian motion with a random spatial distribution of positions and velocities. For the current system of statistically stationary isotropic turbulence, both the DNS and LES show that the fraction of the kinetic energy residing in the mesoscopic field decreases with particle inertia as the square root of the ratio of the total particulate-phase kinetic energy to that of the fluid.

We investigate the time evolution of an impulsive start problem for arbitrary Knudsen numbers ($\hbox{\it Kn}$) using a linearized kinetic formulation. The early-time behaviour is described by a solution of the collisionless Boltzmann equation. The same solution can be used to describe the late-time behaviour for $\hbox{\it Kn}\,{\gg}\,1$. The late-time behaviour for $\hbox{\it Kn}\,{<}\,0.5$ is captured by a newly proposed second-order slip model with no adjustable parameters. All theoretical results are verified by direct Monte Carlo solutions of the nonlinear Boltzmann equation. A measure of the timescale to steady state, normalized by the momentum diffusion timescale, shows that the timescale to steady state is significantly extended by ballistic transport, even at low Knudsen numbers where the latter is only important close to the system walls. This effect is captured for $\hbox{\it Kn}\,{<}\,0.5$ by the slip model which predicts the equivalent effective domain size increase (slip length).

A representation of multicomponent-liquid (MC-liquid) composition as a linear combination of two single-Gamma probability distribution functions (PDFs) is used to describe a large number of MC-liquid drops evaporating in a gas flow. The PDF, called the double-Gamma PDF, depends on the molar mass. The gas-phase conservation equations are written in an Eulerian frame and the drops are described in a Lagrangian frame. Gas conservation equations for mass, momentum, species and energy are combined with differential conservation equations for the first four moments of the gas-composition PDF and coupled to the perfect gas equation of state. Source terms in all conservation equations account for the gas/drop interaction. The drop governing equations encompass differential conservation statements for position, mass, momentum, energy and four moments of the liquid-composition PDF. Simulations are performed for a three-dimensional mixing layer whose lower stream is initially laden with drops colder than the surrounding gas. Initial perturbations excite the layer to promote the double pairing of its four initial spanwise vortices to an ultimate vortex. During the layer evolution, the drops heat and evaporate. The results address the layer evolution, and the state of the gas and drops when layers reach a momentum-thickness maximum past the double vortex pairing. Of interest is the influence of the liquid composition on the development of the vortical features of the flow, on the vortical state reached after the second pairing, and on the gas temperature and composition. The MC-liquid simulations are initiated with a single-Gamma PDF composition so as to explore the development of the double-Gamma PDF. Examination of equivalent simulations with n-decane, diesel and three kerosenes as the liquid, permits assessment of the single-species versus the MC-liquid aspect, and of mixture composition specific effects. Global layer growth and global rotational characteristics are unaffected by liquid specificity; however, the global mixing is highly liquid-specific. Also liquid-specific is the evolution of the ensemble-averaged drop characteristics and of the volumetric averages representing the gas composition. Visualized rotational characteristics show that the small-scale vortical activity increases with increased fuel volatility, which is confirmed by analysis of the vorticity budgets. Homogeneous-plane-average budgets of the vorticity and vorticity-magnitude equations indicate that the stretching and tilting, and momentum-source terms are responsible for the difference among simulations. For all MC liquids, the gas displays a high level of composition heterogeneity, which can directly be traced to the original PDF representing the MC-liquid composition. Under most conditions, the single-Gamma PDF develops into a double-Gamma PDF; however, the extent of this transformation, indicative of vapour condensation onto drops, is not readily parametrized by the liquid volatility, initial carrier-gas temperature or trace vapour in the initial gas.

Compound viscous jets composed of an inner core of one fluid surrounded by an annulus of another are studied here using long-wave theory. We investigate the limit of strongly differing viscosities in the neighbouring fluids: either a highly viscous core or annulus. Additionally, if inertia is present, the density ratio of the fluids is assumed to mirror that of the viscosities. The resulting asymptotic theory reduces to several earlier theories in the appropriate limits. Linear stability analysis and full numerical simulations of the one-dimensional set of equations allow an exploration of the dynamics of either the interior or exterior fluid. The results of our linear stability analysis demonstrate that, for both cases, a stretching and a squeezing mode exist, with the former being more dominant than the latter. Our numerical simulations show that in the highly viscous core case, the interfaces can move out-of-phase, leading to the breakup of the annular region; this is contrary to the linear theory predictions. In the highly viscous annulus case, our results demonstrate the possibility of breakup of either the core or the annulus, depending on the initial ratio of the radii.

In this paper, we report experimental results concerning a three-dimensional short-wave instability observed in a pair of equal co-rotating vortices. The pair is generated in water by impulsively started plates, and is analysed through dye visualizations and detailed quantitative measurements using particle image velocimetry. The instability mode, which is found to be stationary in the rotating frame of reference of the two-vortex system, consists of internal deformations of the vortex cores, which are characteristic of the elliptic instability occurring in strained vortical flows. Measurements of the spatial structure, wavelengths and growth rates are presented, as functions of Reynolds number and non-dimensional core size. The self-induced rotation of the vortex pair, which is not a background rotation of the entire flow, is found to lead to a shift of the unstable wavelength band to higher values, as well as to higher growth rates. In addition, a dramatic increase in the width of the unstable bands for large values of the rescaled core radius is found. Comparisons with recent theoretical results by Le Dizés & Laporte (2002) concerning elliptic instability of co-rotating vortices show very good agreement.

At later stages of the flow, when the perturbation amplitude becomes sufficiently large, the two vortices merge into a single structure. This happens for smaller cores sizes than in the case of two-dimensional merging. The three-dimensional merging leads to a final vortex characterized by turbulent small-scale motion, whose size appears to be larger than it would have been without instability. The vorticity profile of the final vortex is non-Gaussian after both two-dimensional and three-dimensional merging. The profile contains more vorticity outside the inner core than a Gaussian vortex, resulting from the ejection of vorticity filaments during the merging stage.

We consider the levitation of a drop of molten glass above a spherical porous mould, through which air is injected with constant velocity. The glass is assumed to be sufficiently viscous compared to air that motion in the drop is negligible. Thus static equilibrium shapes are determined by the coupling between the lubricating pressure in the supporting air cushion and the Young–Laplace equation. The upper surface of the drop is under constant atmospheric pressure; the static shape of the lower surface of the drop is computed using lubrication theory for the thin air film. Matching of the sessile curvature of the upper surface to the curvature of the mould gives rise to a series of capillary ‘brim’ waves near the edge of the drop which scale with powers of a modified capillary number. Several branches of static solutions are found, such that there are multiple solutions for some drop volumes, but no physically reasonable solutions for other drop volumes. Comparison with experiments and full Navier–Stokes calculations suggests that the stability of the process can be predicted from the solution branches for the static shapes, and related to the persistence of brim waves to the centre of the drop. This suggestion remains to be confirmed by a formal stability analysis.

Weakly nonlinear packets of surface gravity waves over topography are governed by a nonlinear Schrödinger equation with variable coefficients. Using this equation and assuming that the horizontal scale of topography is much larger than the width of the packet, we show that, counter-intuitively, the amplitude of a shoaling packet decays, while its width grows. Such behaviour is a result of the fact that the coefficient of the nonlinear term in the topography-modified Schrödinger equation decreases with depth. Furthermore, there exists a critical depth, $h_{cr}$, where this coefficient changes sign – if the packet reaches $h_{cr}$, it disperses.

It is now believed that the scaling exponents of moments of velocity increments are anomalous, or that the departures from Kolmogorov's (1941) self-similar scaling increase nonlinearly with the increasing order of the moment. This appears to be true whether one considers velocity increments themselves or their absolute values. However, moments of order lower than 2 of the absolute values of velocity increments have not been investigated thoroughly for anomaly. Here, we discuss the importance of the scaling of non-integer moments of order between +2 and $-1$, and obtain them from direct numerical simulations at moderate Taylor microscale Reynolds numbers $R_\lambda\le$ 450, and experimental data at high Reynolds numbers $(R_\lambda \approx 10\,000)$. The relative difference between the measured exponents and Kolmogorov's prediction increases as the moment order decreases towards $-1$, thus showing that the anomaly is manifested in low-order moments as well.

We consider the two-dimensional inviscid flow that occurs when a fluid, initially at rest around a slender wedge-shaped void, is allowed to recoil under the action of surface tension. As noted by Keller & Miksis (1983), a similarity scaling is available, with lengths scaling like $t^{2/3}$. We find that an asymptotic balance is possible when the wedge semi-angle, $\alpha$, is small, in an inner region of $O(\alpha^{4/9})$, a distance of $O(\alpha^{-2/9})$ from the origin, which leads to a simpler boundary value problem at leading order. Although we are able to reformulate the inner problem in terms of a complex potential and reduce it to a single nonlinear integral equation, we are unable to find a solution numerically. This is because, as noted by Vanden-Broeck, Keller & Milewski (2000), and reproduced here numerically using a boundary integral method, the free surface is self-intersecting for $\alpha\,{<}\, \alpha_0\,{\approx}\,2.87^\circ$. Since disconnected solutions are only possible when there is a void inside the initial wedge, we consider the effect of an inviscid low-density fluid inside the wedge. In this case, a solution is available for a slightly smaller range of wedge semi-angles, since the flow of the interior fluid sucks the free surfaces together, with the exterior flow seeing pinch-off at a finite angle.

We conclude that for $\alpha\,{<}\,\alpha_0$, we must introduce the effect of viscosity at small times in order to regularize the initial value problem. Since the solution for $t\,{\ll}\,1$ in the presence of viscosity is simply connected (Billingham 2005), the free surface must first pinch off at some finite time and then continue to do so at a sequence of later times. We investigate this using boundary integral solutions of the full inviscid initial value problem, with smooth initial conditions close to those of the original problem. In addition, we show that the inner asymptotic scalings that we developed for the steady problem can also be used in this time-dependent problem. The unsteady inner equations reduce to those for steady unidirectional flow outside a region of constant pressure, and can be solved numerically.

We also show how the slender wedge solution can be related to the small-time behaviour of two coalescing drops, and describe the relationship between our solutions and those found by Duchemin, Eggers & Josserand (2003), for which a similar unsteady inner region exists. In each case, the free surface pinches off repeatedly, and no similarity solution exists.

We study size segregation in two classes of granular systems in time-periodically forced quasi-two-dimensional tumblers. In dry granular systems (DGS), particles are immersed in air, whereas in liquid granular systems (LGS) or slurries, particles are wholly immersed in a less dense liquid. Experimental results show lobe formation and classical radial segregation, depending on the frequency of the forcing and fill fraction of the system. Both DGS and LGS exhibit similar multi-lobe patterns when subjected to the same forcing. A simple model captures the experimental results and suggests that the symmetry of the forcing plays a crucial role in the emergence of the patterns. Lobes form via a reinforcement mechanism that depends on the period of the regular islands in the Poincaré map. The mechanism can be controlled by changing the forcing period and/or fill fraction.

The stability of vertically falling viscous liquid jets issuing from a circular orifice with a low fluid velocity where the flow is dominated by viscous effects torn down by gravity leading to a flow dominated by inertia far downstream, is studied by means of linear stability analysis as well as direct numerical simulation of the one-dimensional model. Jet stability is also studied experimentally. Results for marginal stability and critical frequencies are in excellent agreement with our theoretical predictions. A new global instability of a viscous jet is found. This instability is observable whenever a falling liquid jet exhibits a long enough viscosity dominated region. The source of instability is surface tension. In contrast to the Rayleigh capillary instability that is always present, but leads to very long disintegration lengths in highly viscous fluid jets, the instability described here only occurs when the volume flux falls below a certain limit depending on the fluid properties and the nozzle diameter.

Observations of entrainment into natural gravity-driven flows on sloping surfaces are described. It is shown that the laboratory-based entrainment law of Ellison & Turner (1959), which is often used for modelling of atmospheric and oceanic flows, underestimates the entrainment rates substantially, arguably due to the fact that the laboratory flows have been conducted at Reynolds numbers $(Re \lesssim 10^3)$ below what is required for mixing transition ($Re\sim10^3$–10^4) whereas natural flows occur at much higher Reynolds numbers $(Re\sim10^7)$. A new entrainment law of the form $E\sim Ri^{-3/4}$ is proposed for the atmospheric Richardson number range $0.15 \lt Ri \lt 1.5$. In contrast to the laboratory observation that entrainment ceases at $Ri = 0.8$, field observations show continuous entrainment over the entire Richardson number range.

Two-dimensional flow past a cylinder close to a free surface at a Reynolds number of 180 is numerically investigated. The wake behaviour for Froude numbers between 0.0 and 0.7 and for gap ratios between 0.1 and 5.0 is examined. For low Froude numbers, where the surface deformation is minimal, the simulations reveal that this problem shares many features in common with flow past a cylinder close to a no-slip wall. This suggests that the flow is largely governed by geometrical constraints in the low-Froude-number limit.

At Froude numbers in excess of 0.3–0.4, surface deformation becomes substantial. This can be traced to increases in the local Froude number to unity or higher in the gap between the cylinder and the surface. In turn, this is associated with supercritical to subcritical transitions in the near wake resulting in localized free-surface sharpening and wave breaking. Since surface vorticity is directly related to surface curvature, such high surface deformation results in significant surface vorticity, which can diffuse and then convect into the main flow, altering the development of Strouhal vortices from the top shear layer, affecting wake skewness and suppressing the absolute instability. The variations of parameters such as Strouhal number and formation length are provided for Froude numbers spanning the critical range.

At larger Froude numbers, good agreement is obtained with recently published experimental investigations. The previously seen metastable wake states are observed to occur for similar system parameters to the experiments despite the difference in Reynolds numbers by a factor of about 40. The wake state switching appears to be controlled by a feedback loop. Important elements of the feedback loop include the cyclic generation and suppression of the absolute instability of the wake, and the role of surface vorticity and vortices formed from the bottom shear layer in controlling vortex formation from the top shear layer. The proposed mechanism is presented. Shedding ceases at very small gap ratios (${\sim}\, 0.1$–0.2). This behaviour can be explained in terms of the fluid flux through the gap, vorticity diffusion into the surface and opposite-signed surface vorticity from the strong surface deformation.

A straight tube with a smooth axisymmetric constriction is an idealized representation of a stenosed artery. We examine the three-dimensional instabilities and transition to turbulence of steady flow, steady flow plus an oscillatory component, and an idealized vascular pulsatile flow in a tube with a smooth 75 % stenosis using both linear stability analysis and direct numerical simulation. Steady flow undergoes a weak Coanda-type wall attachment and turbulent transition through a subcritical bifurcation, leading to hysteretic behaviour with respect to changes in Reynolds number. The pulsatile flows become unstable through a subcritical period-doubling bifurcation involving alternating tilting of the vortex rings that are ejected from the throat with each pulse. These tilted vortex rings rapidly break down through a self-induction mechanism within the confines of the tube. While the linear instability modes for pulsatile flow have maximum energy well downstream of the stenosis, we have established using direct numerical simulation that breakdown can gradually propagate upstream until it occurs within a few tube diameters of the constriction, in agreement with previous experimental observations. At the Reynolds numbers employed in the present study, transition is localized, with relaminarization occurring further downstream. A non-exhaustive investigation has also been undertaken into the receptivity of the axisymmetric shear layer in the idealized physiological pulsatile flow, with the results suggesting it has localized convective instability over part of the pulse cycle.

We examine the dynamics of turbulent lazy plumes rising from horizontal area sources and from vertically distributed line sources into a quiescent environment of uniform density. First, we consider plumes with internal buoyancy flux gain and, secondly, plumes from horizontal area sources that have significant momentum flux deficits. We re-cast the conservation equations of Morton et al. (1956) for a constant entrainment coefficient $(\alpha)$ in terms of three dimensionless parameters: the plume radius $\beta$; a parameter $\Gamma$ characterizing the local balance of momentum, buoyancy and volume fluxes; and a parameter $\Lambda$ that characterizes the rate of internal buoyancy flux gain with height. For a plume with a linear internal buoyancy flux gain with height the flow is shown to be a constant-velocity lazy plume. For highly lazy area sources we derive exact solutions for the key plume parameters in terms of $\Gamma$ and an approximate solution for the variation of $\Gamma$ with height. We show that near the source there is a region of zero entrainment.

The evolution of material lines, $l$, and vorticity, $\omega$, is investigated experimentally through three-dimensional particle-tracking velocimetry (3D-PTV) in quasi-homogeneous isotropic turbulence at $Re_{\lambda }\,{=}\,50$. Through 3D-PTV data the full set of velocity derivatives, $\partial u_{i}/\partial x_{j}$, is accessible. This allows us to monitor the evolution of various turbulent quantities along fluid particle trajectories. The main emphasis of the present work is on the physical mechanisms that govern the Lagrangian evolution of $l$ and $\omega$ and the essential differences inherent in these two processes. For example, we show that vortex stretching is smaller than material lines stretching, i.e. $\langle\omega_{i}\omega_{j}s_{ij}/\omega^{2}\rangle \,{<}\,\langle l_{i}l_{j}s_{ij}/l^{2}\rangle$, and expand on how this issue is closely related to the predominant alignment of $\omega$ and the intermediate principal strain eigenvector $\lambda_{2}$ of the rate of strain tensor, $s_{ij}$. By focusing on Lagrangian quantities we discern whether these alignments are driven and maintained mainly by vorticity or by strain. In this context, the tilting of $\omega$ and the rotation of the eigenframe $\lambda_{i}$ of the rate of strain tensor $s_{ij}$ are investigated systematically conditioned on different magnitudes of strain, $s^{2}$, and enstrophy, $\omega^{2}$. Further, we infer that viscosity contributes through the term $\nu\omega_{i}\nabla^2\omega_{i}$ to ${\rm D}\omega^{2}/{\rm D}t$, whereas ${\rm D}l^{2}/{\rm D}t$ has no diffusive term. This difference plays a key role in defining the mutual orientation between $\omega$ and $\lambda_{i}$. Viscosity thus contributes significantly to the difference in growth rates of $\langle\omega_{i}\omega_{j}s_{ij}\rangle$ and $\langle l_{i}l_{j}s_{ij}\rangle$.

The primary instability of salt fingers in an unbounded fluid with uniform vertical salinity/temperature $(\skew4\bar{S}_z^\ast/\skew2\bar{T}_z^\ast)$ gradients consists of a vertically $(z)$ uniform ‘elevator’ mode, amplifying exponentially in time. We compute the $z$-wavelength $(h_{0})$ of the fastest growing secondary instability by integrating a system of linear partial differential equations (PDEs) with time-dependent and horizontally periodic coefficients. It is suggested that this instability limits the amplitude of the primary mode and $h_{0}\skew4\bar{S}_z^\ast$ determines the value of the statistically steady r.m.s. salinity fluctuation $(S^\prime)$. Comparisons of $h_{0}$ with ${\langle{\overline{({S}')^2}}\rangle^{1/2}}/{\skew4\bar{S}_z^\ast}$ are made using direct numerical simulations (DNS), and our theory for the vertical salt/heat flux is also given and compared with the DNS.

As two fluid particles separate in time, the entire spectrum of eddy motions is being sampled from the smallest to the largest scales. In large-scale geophysical systems for which the Earth rotation is important, it has been conjectured that the relative diffusivity should vary respectively as $D^{2}$ and $D^{4/3}$ for distances respectively smaller and larger than a well-defined forcing scale of the order of the internal Rossby radius (with $D$ the r.m.s. separation distance). Particle paths data from a mid-latitude float experiment in the central part of the North Atlantic appear to support these statements partly: two particles initially separated by a few km within two distinct clusters west and east of the mid-Atlantic ridge, statistically dispersed following a Richardson regime ($D^{2}\,{\sim}\,t^{3}$ asymptotically) for r.m.s. separation distances between 40 and 300 km, in agreement with a $D^{4/3}$ law. At early times, and for smaller separation distances, an exponential growth, in agreement with a $D^{2}$ law, was briefly observed but only for the eastern cluster (with an e-folding time around 6 days). After a few months or separation distances greater than 300 km, the relative dispersion slowed down naturally to the Taylor absolute dispersion regime.