We present a similarity analysis of strong turbulent jets directed perpendicularly into a crossflow. The analysis neglects pressure terms in the governing equations, and assumes complete similarity in each of two intermediate-asymptotic regions of the flow: a jet region, where the jet is largely unaffected by the crossflow, and a wake-like region, where the jet has been deflected well into the crossflow. Scaling laws are derived for velocity, scalar concentration, and jet trajectory, and show good agreement with existing experimental data. The structure of the counter-rotating vortex pair implied by this analysis is significantly different from typical representations found in the literature.

Detailed measurements of the velocity field in the symmetry plane of two jets and two jet flames in a crossflow are obtained using particle image velocimetry. The jets issue into a wind tunnel at density-weighted jet-to-crossflow velocity ratios r = 10 and r = 21, with corresponding Reynolds numbers 6000 and 12 800. Ensemble statistics of the velocity field are presented, and some interesting features of the entrainment process in transverse jets are discussed. Deviations from the simple behaviour predicted by the similarity analysis presented in Part 1 are highlighted. Simultaneous planar laser-induced fluorescence imaging of the OH radical is performed in selected regions of the flames. Results suggests that flame/flow interaction is strong near the lifted flamebase, but increasingly weaker further downstream.

Direct numerical simulations (DNS) are presented of three-dimensional, Rayleigh–Taylor instability (RTI) between two incompressible, miscible fluids, with a 3:1 density ratio. Periodic boundary conditions are imposed in the horizontal directions of a rectangular domain, with no-slip top and bottom walls. Solutions are obtained for the Navier–Stokes equations, augmented by a species transport-diffusion equation, with various initial perturbations. The DNS achieved outer-scale Reynolds numbers, based on mixing-zone height and its rate of growth, in excess of 3000. Initial growth is diffusive and independent of the initial perturbations. The onset of nonlinear growth is not predicted by available linear-stability theory. Following the diffusive-growth stage, growth rates are found to depend on the initial perturbations, up to the end of the simulations. Mixing is found to be even more sensitive to initial conditions than growth rates. Taylor microscales and Reynolds numbers are anisotropic throughout the simulations. Improved collapse of many statistics is achieved if the height of the mixing zone, rather than time, is used as the scaling or progress variable. Mixing has dynamical consequences for this flow, since it is driven by the action of the imposed acceleration field on local density differences.

Using Stokesian dynamics simulations, we examine the flow of a monodisperse, neutrally buoyant, homogeneous suspension of non-Brownian solid spheres in simple shear, starting from a large number of independent hard-sphere distributions and ensemble averaging the results. We construct a novel method for computing the gradient diffusivity via simulations on a homogeneous suspension and, although our results are only approximate due to the small number of particles used in the simulations, we present here the first values of this important parameter, both along and normal to the plane of shear, to be obtained directly either experimentally or numerically. We show furthermore that, although the system of equations describing the particle motions is deterministic, the particle displacements in the two directions normal to the bulk flow have Gaussian distributions with zero mean and a variance which eventually grows linearly in time thereby establishing that the system of particles is diffusive. For particle concentrations up to 45%, we compute the corresponding tracer diffusivities both from the slope of the mean-square particle displacement and by integrating the corresponding velocity autocorrelations and find good agreement between the two sets of results.

The effect of mass loading and inter-particle collisions on the development of the polydispersed two-phase flow downstream of a confined bluff body is discussed. The bluff-body flow configuration, which is one of the simplest turbulent recirculating flows, is relevant for applications and forms the basis of numerous combustion devices. The present data are obtained for isothermal conditions by using a two-component phase-Doppler anemometer allowing size and velocity measurements. Polydispersed glass beads are introduced into the flow. The statistical properties of narrow particle size classes are displayed and analysed in order to allow for the wide range of particle relaxation times. The evolution of mass fluxes and mass concentration per size class is estimated from the PDA data. A correction is introduced to ensure that the mass flow rate of particles per size class from data integration is correct.

We show that the development of the continuous phase is very sensitive to initial mass loading of the inner jet. An increase in mass loading corresponds to an increase in momentum flux ratio between the central jet and annular flow. In the present situation, this implies a complete reorganization of the recirculation zone and the turbulent field. The importance of direct modulation of turbulence induced by particles is demonstrated in the inner jet. Moreover, our data confirm that the prediction of fluid/particle velocity correlation is essential to take these effects into account for partly responsive beads.

We show that the sensitivity to mass loading greatly affects the dispersion of the glass beads. Particles recirculate at the lowest mass loading and the mass concentration of the dispersed phase in the recirculation zone and in the external shear layer is high. On the other hand, the memory of the initial jet is detected far downstream at the highest loading and the dispersion of particles is reduced. Axial or radial profiles of mean and r.m.s. velocity of the dispersed phase are displayed and analysed. The role of large-scale intermittency is discussed. Relevant Stokes numbers are introduced to account for different driving mechanisms in the turbulent field. Non-Stokesian effects are particularly important. We show that the anisotropy of the particle fluctuating motion is large and associated with production mechanisms via interaction with mean particle velocity gradients. A focus on the jet stagnation region proves that the particulate flow is very sensitive to inertia effects and that no local equilibrium with the fluid turbulence can be assumed when modelling such a configuration.

Finally, even at the small volume ratio considered here, we prove that it is highly probable that inter-particle collisions occur in the jet stagnation region at low mass loading and all along the inner jet flow at the highest mass loading. Redistribution of mean momentum and fluctuating kinetic energy between all colliding classes is therefore expected, which implies a fully coupled fluid and particle system.

The data and analysis presented provide a severe test case for the recent development in two-phase flow modelling and offer further challenges both to experimentation and model development. The validated data set has been selected for benchmarking and is available on the internet.

A new approach to ideal-fluid hydrodynamics based on the notion of continuous deformation of infinitesimal material elements is proposed. The matrix approach adheres to the Lagrangian (material) view of fluid motion, but instead of Lagrangian particle trajectories, it treats the Jacobi matrix of their derivatives with respect to Lagrangian variables as the fundamental quantity completely describing fluid motion.

A closed set of governing matrix equations equivalent to conventional Lagrangian equations is formulated in terms of this Jacobi matrix. The equation of motion is transformed into a nonlinear matrix differential equation in time only, where derivatives with respect to the Lagrangian variables do not appear. The continuity equation that requires constancy of the Jacobi determinant in time takes the form of an algebraic constraint on the Jacobi matrix. An accompanying linear consistency condition, which is responsible for the dependence on spatial variables and does not include time derivatives, ensures completeness of the system and reconstruction of the particle trajectories by the Jacobi matrix.

A class of exact solutions to the matrix equations that describes rotational non-stationary three-dimensional motions having no analogues in the conventional formulations is also found and investigated. A distinctive feature of these motions is precession of vortex lines (rectilinear or curvilinear) around a fixed axis in space. Boundary problems for the derived exact solutions including matching of rotational and potential motions across the boundary of a vortex tube are addressed. In particular, for the cylindrical vortex of elliptical cross-section involved in three-dimensional precession, the outer potential flow is constructed and shown to be a non-stationary periodic straining flow at a large distance from the vortex axis.

Free vibrations of two side-by-side cylinders with fixed support (no rotation and displacement) at both ends placed in a cross-flow were experimentally investigated. Two fibre-optic Bragg grating sensors were used to measure the dynamic strain, while a hot wire and flow visualization were employed to examine the flow field around the cylinders. Three T/d ratios, 3.00, 1.70 and 1.13, were investigated, where T is the centre-to-centre cylinder spacing and d is the diameter; they give rise to three different flow regimes. The investigation throws new light on the shed vortices and their evolution. A new interpretation is proposed for the two different dominant frequencies, which are associated with the narrow and the wide wake when the gap between the cylinders is between 1.5 and 2.0 as reported in the literature. The structural vibration behaviour is closely linked to the flow characteristics. At T/d = 3:00, the cross-flow root-mean-square strain distribution shows a very prominent peak at the reduced velocity Ur ≈ 26 when the vortex shedding frequency fs, coincides with the third-mode natural frequency of the combined fluid–cylinder system. When T/d < 3:00, this peak is not evident and the vibration is suppressed because of the weakening strength of the vortices. The characteristics of the system modal damping ratios, including both structural and fluid damping, and natural frequencies are also investigated. It is found that both parameters depend on T/d. Furthermore, they vary slowly with Ur, except near resonance where a sharp variation occurs. The sharp variation in the natural frequencies of the combined system is dictated by the vortex shedding frequency, in contrast with the lock-in phenomenon, where the forced vibration of a structure modifies the vortex shedding frequency. This behaviour of the system natural frequencies persists even in the case of the single cylinder and does not seem to depend on the interference between cylinders. A linear analysis of an isolated cylinder in a cross-flow has been carried out. The linear model prediction is qualitatively consistent with the experimental observation of the system damping ratios and natural frequencies, thus providing valuable insight into the physics of fluid–structure interactions.

There have been several attempts to introduce approximations into the exact form of Lilley's equation in order to express the source term as the sum of a quadrupole whose strength is quadratic in the fluctuating velocities and a dipole whose strength is proportional to the temperature fluctuations. The purpose of this note is to show that it is possible to choose the dependent (i.e. the pressure) variable so that this type of result can be derived directly from the Euler equations without introducing any additional approximations.

Observations of the flow of dense fluid into uniformly density-stratified environments down plane slopes with small inclination to the horizontal ([les ] 20°) are described, and a quantitative model for such flows is presented. In these experiments the dense fluid is released at the top of the slope for a finite period of time. The resulting downslope gravity current, or downflow, has uniform thickness with a distinct upper boundary, until it approaches its level of neutral density where the fluid leaves the proximity of the slope. Turbulent transfers of mass and momentum occur across the upper boundary, causing a continuous loss of fluid from the downflow in most cases, and associated loss of momentum. The flow may be characterized by the local values of the Richardson number Ri, the Reynolds number Re (generally large), and of M = QN3/g′2, where Q is the (two-dimensional) volume flux, N the buoyancy frequency and g′ the (negative) buoyancy of the dense fluid. The model for the downflow describes the turbulent transfers in terms of entrainment, detrainment and drag coefficients, Ee, Ed and k respectively, and the observations enable the determination of these coefficients in terms of the local values of M and Ri. The model may be regarded as an extension of that Ellison & Turner (1959) to stratified environments, describing the consequent substantial changes in mixing and distribution of the inflow. It permits the modelling of the bulk properties of these flows in geophysical situations, including shallow and deep flows in the ocean.

The coupling between a bulk vortical flow and a surfactant-influenced air/water interface has been examined in a canonical flow geometry through experiments and computations. The flow in an annular region bounded by stationary inner and outer cylinders is driven by the constant rotation of the floor and the free surface is initially covered by a uniformly distributed insoluble monolayer. When driven slowly, this geometry is referred to as the deep-channel surface viscometer and the flow is essentially azimuthal. The only interfacial property that affects the flow in this regime is the surface shear viscosity, μs, which is uniform on the surface due to the vanishingly small concentration gradient. However, when operated at higher Reynolds number, secondary flow drives the surfactant film towards the inner cylinder until the Marangoni stress balances the shear stress on the bulk fluid. In general, the flow can be influenced by the surface tension, σ, and the surface dilatational viscosity, κs, as well as μs. However, because of the small capillary number of the present flow, the effects of surface tension gradients dominate the surface viscosities in the radial stress balance, and the effect of μs can only come through the azimuthal stress. Vitamin K1 was chosen for this study since it forms a well-behaved insoluble monolayer on water and μs is essentially zero in the range of concentration on the surface, c, encountered. Thus the effect of Marangoni elasticity on the interfacial stress could be isolated. The flow near the interface was measured in an optical channel using digital particle image velocimetry. Steady axisymmetric flow was observed at the nominal Reynolds number of 8500. A numerical model has been developed using the axisymmetric Navier–Stokes equations to examine the details of the coupling between the bulk and the interface. The nonlinear equation of state, σ(c), for the vitamin K1 monolayer was measured and utilized in the computations. Agreement was demonstrated between the measurements and computations, but the flow is critically dependent on the nonlinear equation of state.

The eigenvalue problem for gravity waves on a shear flow of depth h and non-inflected velocity profile U(y) (typically parabolic) is revisited, following Burns (1953) and Yih (1972). Complementary variational formulations that provide upper and lower bounds to the Froude number F as a function of the wave speed c and wavenumber k are constructed. These formulations are used to improve Burns's long-wave approximation and to determine Yih's critical wavenumber k∗, for which the wave is stationary (c = 0) and to which k must be inferior for the existence of an upstream running wave.

Starting from stationary bifurcations in Couette–Dean flow, we compute stationary nontrivial solutions in the circular Couette geometry for an inertialess finitely extensible nonlinear elastic (FENE-P) dumbbell fluid. These solutions are isolated from the Couette flow branch arising at finite amplitude in saddle–node bifurcations as the Weissenberg number increases. Spatially, they are strongly localized axisymmetric vortex pairs embedded in an arbitrarily long ‘far field’ of pure Couette flow, and are thus qualitatively, and to some extent quantitatively, similar to the ‘diwhirl’ (Groisman & Steinberg 1997) and ‘flame’ patterns (Baumert & Muller 1999) observed experimentally. For computationally accessible parameter values, these solutions appear only above the linear instability limit of the Couette base flow, in contrast to the experimental observations. Correspondingly, they are themselves linearly unstable. Nevertheless, extrapolation of the trend in the bifurcation points with increasing polymer extensibility suggests that for sufficiently high extensibility the diwhirls will come into existence before the linear instability, as seen experimentally.

Based on the computed stress and velocity fields, we propose a fully nonlinear self-sustaining mechanism for these flows. The mechanism is related to that for viscoelastic Dean flow vortices and arises from a finite-amplitude perturbation giving rise to a locally unstable profile of the azimuthal normal stress near the outer cylinder at the symmetry plane of the vortex pair. The unstable stress profile, in combination with a ‘tubeless siphon’ effect, nonlinearly sustains the patterns. We propose that these solitary, strongly nonlinear structures comprise fundamental building blocks for complex spatiotemporal dynamics in the flow of elastic liquids.

This experimental study is devoted to the description of the different patterns resulting from instabilities which appear in the flow between a rotating and a stationary disk enclosed by a stationary sidewall. With the help of visualizations we describe the different flow regimes as functions of two control parameters: the Reynolds number and the aspect ratio of the gap separating the disks, which are varied over large continuous ranges. Moreover, visualizations and ultrasonic anemometry lead to the description of the different instabilities and to the construction of a transition diagram that summarizes the domains of existence of the various patterns. Two different scenarios of transition are mainly followed by the flow. When the gap between the two disks is more than the thickness of the two disk boundary layers, circular and spiral waves destabilize the stationary disk boundary layer. Transition occurs in this case by the mixing of these waves. On the other hand, when the two boundary layers are merged, finite-size turbulent structures can appear. They consist of turbulent spots or turbulent spirals which invade the laminar domains as the Reynolds number of the flow is increased.

We consider the non-stationary dynamics of an intense localized vortex on a β-plane using a shallow-water model. An asymptotic theory for a vortex with piecewise-continuous potential vorticity is developed assuming the Rossby number to be small and the free surface elevation to be small but finite. Analogously to the well-known quasi-geostrophic model, the vortex translation is produced by a secondary dipole circulation (β-gyres) developed in the vortex vicinity and consisting of two parts. The first part (geostrophic β-gyres) coincides with the β-gyres in the geostrophic model, and the second (ageostrophic β-gyres) is due to ageostrophic terms in the governing equations. The time evolution of the ageostrophic β-gyres consists of fast and slow stages. During the fast stage the radiation of inertia–gravity waves results in the rapid development of the β-gyres from zero to a dipole field independent of the fast time variable. Correspondingly, the vortex accelerates practically instantaneously (compared to the typical swirling time) to some finite value of the translation speed. At the next slow stage the inertia–gravity wave radiation is insignificant and the β-gyres evolve with the typical swirling time. The total zonal translation speed induced by the geostrophic and ageostrophic β-gyres tends with increasing time to the speed of a steadily translating monopole exceeding (not exceeding) the drift velocity of Rossby waves for anticyclones (cyclones). This cyclone/anticyclone asymmetry generalizes the well-known finding about the greater longevity of anticyclones compared to cyclones to the case of non-stationary evolving monopoles. The influence of inertia–gravity waves upon the vortex evolution is analysed. The main role of these waves is to provide a ‘fast’ adjustment to the ‘slow’ vortex evolution. The energy of inertia–gravity waves is negligible compare to the energy of the geostrophic β-gyres. Yet another feature of the ageostrophic vortex evolution is that the area of the potential vorticity patch changes in the course of time, the cyclonic patch contracting and the anticyclonic one expanding.

The effect of a weak convective heat transfer on the thermocapillary interaction of two bubbles migrating in an externally imposed temperature gradient is examined. It is shown that, for short and moderate separation distances, the corrections to the individual migration velocities of the bubbles are of O(Pe), where Pe is Péclet number. For separation distances larger than O(Pe−1/2) the correction is of O(Pe2) as previously found for an isolated drop. The perturbations to the bubble velocities have opposite signs: the motion of the leading bubble is enhanced while the motion of the trailing one is retarded. A newly found feature is that equal-sized bubbles, which otherwise would move with equal velocities, acquire a relative motion apart from each other under the influence of convection. For slightly unequal bubbles there are three different regimes of large-time asymptotic behaviour: attraction up to the collision, infinite growth of the separation distance, and a steady migration with equal velocities, the steady motion separation distance being a function of the parameters of the problem. Sufficient conditions for the realization of each regime are given in terms of the Péclet number, initial separation and radii ratio.

It is shown that low Reynolds number fluid flows can cause suspended particles to respond as though they were in an equilibrium system with an effective potential. This general result follows naturally from the fact that different methods of moving particles in viscous fluids give rise to very different long-range flows. Two examples are discussed: electrophoretic ‘levitation’ of a heavy charged sphere, for which a hydrodynamic ‘pseudo-potential’ can be written in closed form, and quasi-two-dimensional crystals of like-charged colloidal spheres which form near charged walls, whose apparent attraction arises not from a force but from persistent fluid flows.