The dynamics and the three-dimensional structure of vortices in a linearly stratified, non-rotating fluid are investigated by means of laboratory experiments, an analytical model and through numerical simulations. The laboratory experiments show that such vortices have a thin pancake-like appearance. Due to vertical diffusion of momentum the strength of these vortices decreases rapidly and their thickness increases in time. Also it is found that inside a vortex the linear ambient density profile becomes perturbed, resulting in a local steepening of the density gradient. Based on the assumption of a quasi-two-dimensional axisymmetric flow (i.e. with zero vertical velocity) a model is derived from the Boussinesq equations that illustrates that the velocity field of the vortex decays due to diffusion and that the vortex is in so-called cyclostrophic balance. This means that the centrifugal force inside the vortex is balanced by a pressure gradient force that is provided by a perturbation of the density profile in a way that is observed in the experiments. Numerical simulations are performed, using a finite difference method in a cylindrical coordinate system. As an initial condition the three-dimensional vorticity and density structure of the vortex, found with the diffusion model, are used. The influence of the Froude number, Schmidt number and Reynolds number, as well as the initial thickness of the vortex, on the evolution of the flow are investigated. For a specific combination of flow parameters it is found that during the decay of the vortex the relaxation of the isopycnals back to their undisturbed positions can result in a stretching of the vortex. Potential energy of the perturbed isopycnals is then converted into kinetic energy of the vortex. However, when the stratification is strong enough (i.e. for small Froude numbers), the evolution of the vortex can be described almost perfectly by the diffusion model alone.

The physical mechanisms underlying the dynamics of the dissipation of passive scalar fluctuations with a uniform mean gradient in stationary isotropic turbulence are studied using data from direct numerical simulations (DNS), at grid resolutions up to 5123. The ensemble-averaged Taylor-scale Reynolds number is up to about 240 and the Schmidt number is from ⅛ to 1. Special attention is given to statistics conditioned upon the energy dissipation rate because of their important role in the Lagrangian spectral relaxation (LSR) model of turbulent mixing. In general, the dominant physical processes are those of nonlinear amplification by strain rate fluctuations, and destruction by molecular diffusivity. Scalar dissipation tends to form elongated structures in space, with only a limited overlap with zones of intense energy dissipation. Scalar gradient fluctuations are preferentially aligned with the direction of most compressive strain rate, especially in regions of high energy dissipation. Both the nature of this alignment and the timescale of the resulting scalar gradient amplification appear to be nearly universal in regard to Reynolds and Schmidt numbers. Most of the terms appearing in the budget equation for conditional scalar dissipation show neutral behaviour at low energy dissipation but increased magnitudes at high energy dissipation. Although homogeneity requires that transport terms have a zero unconditional average, conditional molecular transport is found to be significant, especially at lower Reynolds or Schmidt numbers within the simulation data range. The physical insights obtained from DNS are used for a priori testing and development of the LSR model. In particular, based on the DNS data, improved functional forms are introduced for several model coefficients which were previously taken as constants. Similar improvements including new closure schemes for specific terms are also achieved for the modelling of conditional scalar variance.

In this paper the methods of Parts 1 and 2 are extended to the case when the elliptic cylinder is executing angular oscillations about its centreline. At large distances from the cylinder the solution for a beam of waves tends to a similarity solution that decays more rapidly with distance than does the similarity solution for rectilinear oscillations described in Thomas & Stevenson (1972). Figure 4 shows in a remarkable way how the predicted wave profiles change with distance from the inviscid solution to the similarity one.

In the latter part of the paper the predictions of Parts 1 and 2 and other theories are compared with recent experimental observations. The results of the experiments are in good agreement with the predictions of Parts 1 and 2.

We study finite-inertia effects on the collision rate of bidisperse heavy particles in a turbulent gas, using direct numerical simulations and kinematic descriptions. As shown previously for a monodisperse system (Sundaram & Collins 1997; Wang, Wexler & Zhou 2000), a statistical mechanical description of the average collision kernel consists of two parts, namely a description of the relative velocity between two colliding particles (the turbulent transport effect) and of the non-uniform particle distribution due to dynamic interaction of particles with coherent vortex structures (the accumulation effect). We first show that this description remains valid and accurate for a bidisperse system involving two groups of particles of inertial response time τp1 and τp2, respectively. Numerical results for the turbulent transport effect and the accumulation effect have been obtained as a function of τp1 and τp2. Interestingly, the accumulation effect in a bidisperse system is bounded above by that of a monodisperse system. An explanation for this observation is given, in terms of the correlation between concentration fields of the two size groups. Simulations show that particles from two size groups were found in different regions of a vortex, thus reducing the net accumulation effect in a bidisperse system. The turbulent transport effect, on the other hand, is bounded below by the level in a monodisperse system, due to a differential inertia effect. The above observations imply that the size polydispersity enhances the turbulent transport effect but weakens the accumulation effect, relative to a monodisperse system.

A simple eddy–particle interaction (EPI) model was developed and shown to give a reasonable prediction of the collision kernel, except for a small parametric region where both τp1 and τp2 are on the order of the ow Kolmogorov time τk and thus the accumulation effect must be included. A more accurate model incorporating both the turbulent transport effect and the accumulation effect has also been developed. The model would provide an upper bound on the collision rates for a non-dilute bidisperse system, since turbulence modulation and particle-particle interactions are not considered in this model.

Finally, some consideration is given to the effect of nonlinear drag on the collision kernel. The results show that the drag nonlinearity can increase the collision kernel slightly (less than 10%) at large particle inertia.

We report on the structure of the scalar index-of-refraction field generated by turbulent, gas-phase, incompressible and compressible shear layers and incompressible jets, and on associated beam-propagation aero-optical phenomena. Using simultaneous imaging of the optical-beam distortion and the turbulent-flow index-of-refraction field, wavefront-phase functions were computed for optical beams emerging from the turbulent region in these free-shear flows, in an aero-optical regime producing weak wavefront distortions. Spatial wavefront-phase behaviour is found to be dominated by the large-scale structure of these flows. A simple level-set representation of the index-of-refraction field in high-Reynolds-number, incompressible shear layers is found to provide a good representation of observed wavefront-phase behaviour, indicating that the structure of the unsteady outer boundaries of the turbulent region provides the dominant contributions.

Thin viscous sheets occur frequently in situations ranging from polymer processing to global plate tectonics. Asympotic expansions in the sheet's dimensionless ‘slenderness’ ε [Lt ] 1 are used to derive two coupled equations that describe the deformation of a two-dimensional inertialess sheet with constant viscosity μ and variable thickness and curvature in response to arbitrary loading. Three model problems illustrate the partitioning of thin-sheet deformation between stretching and bending modes: (i) A sheet with fixed (hinged or clamped) ends, initially flat and of length L0 and thickness H0 ≡ εL0, inflated by a constant excess pressure ΔP applied to one side (‘film blowing’). The sheet deforms initially by bending on a time scale με4/ΔP ≡ τb, and thereafter by stretching except in bending boundary layers of width δ ∼ L0(t/τb−1/3 at the clamped ends. (ii) An initially horizontal ‘viscous beam’ with length L0 and thickness H0 ≡ εL0, clamped at one end, deforms by bending on a time scale τb = μH20/gδρL30 until it hangs nearly vertically. Thereafter it deforms by bending in a thin boundary layer at the clamped end, and elsewhere by stretching on a slow time scale ε−2τb. (iii) A sheet extruded horizontally at speed U0 from a slit of width H0 in a gravitational field deforms primarily by bending on a time scale (μH20/U30gδρ)1/4. The sheet's ‘hinge point’ moves in the direction opposite to the extrusion velocity, which may explain the observed retrograde motion of subducting oceanic lithosphere (‘trench rollback’).

A laboratory experiment involving rip currents generated on a barred beach with periodic rip channels indicates that rip currents contain energetic low-frequency oscillations in the presence of steady wave forcing. An analytic model for the time-averaged flow in a rip current is presented and its linear stability characteristics are investigated to evaluate whether the rip current oscillations can be explained by a jet instability mechanism. The instability model considers spatially growing disturbances in an offshore directed, shallow water jet. The effects of variable cross-shore bathymetry, non-parallel flow, turbulent mixing, and bottom friction are included in the model. Model results show that rip currents are highly unstable and the linear stability model can predict the scales of the observed unsteady motions.

We investigate the uptake of a passive solute through the walls of a pulsating, fluid-filled channel into an adjacent medium in which the solute diffuses and is consumed at a constant rate. One end of the channel is open to well-mixed fluid containing the solute. The channel walls oscillate periodically in time and this prescribed motion generates steady streaming within the channel. We determine how this flow enhances the overall solute consumption (i.e. the flux of solute into the channel), the solute dispersion along the channel and the quantity of solute in the adjacent medium. The solute disperses in the channel due to the interaction between advection and transverse diffusion. The time-mean solute distribution throughout the channel and the medium is determined for a wide range of parameters. The results are applied to a new surgical technique used to treat patients with severe coronary artery disease, in which narrow tubes are created within ischemic heart muscle in an attempt to reperfuse the area directly with oxygenated blood.

The onset of thermosolutal convection and finite-amplitude flows, due to vertical gradients of heat and solute, in a horizontal rectangular enclosure are investigated analytically and numerically. Dirichlet or Neumann boundary conditions for temperature and solute concentration are applied to the two horizontal walls of the enclosure, while the two vertical ones are assumed impermeable and insulated. The cases of stress-free and non-slip horizontal boundaries are considered. The governing equations are solved numerically using a finite element method. To study the linear stability of the quiescent state and of the fully developed flows, a reliable numerical technique is implemented on the basis of Galerkin and finite element methods. The thresholds for finite-amplitude, oscillatory and monotonic convection instabilities are determined explicitly in terms of the governing parameters. In the diffusive mode (solute is stabilizing) it is demonstrated that overstability and subcritical convection may set in at a Rayleigh number well below the threshold of monotonic instability, when the thermal to solutal diffusivity ratio is greater than unity. In an infinite layer with rigid boundaries, the wavelength at the onset of overstability was found to be a function of the governing parameters. Analytical solutions, for finite-amplitude convection, are derived on the basis of a weak nonlinear perturbation theory for general cases and on the basis of the parallel flow approximation for a shallow enclosure subject to Neumann boundary conditions. The stability of the parallel flow solution is studied and the threshold for Hopf bifurcation is determined. For a relatively large aspect ratio enclosure, the numerical solution indicates horizontally travelling waves developing near the threshold of the oscillatory convection. Multiple confined steady and unsteady states are found to coexist. Finally, note that all the numerical solutions presented in this paper were found to be stable.

The interaction of a laser-induced cavitation bubble with an elastic boundary and its dependence on the distance between bubble and boundary are investigated experimentally. The elastic boundary consists of a transparent polyacrylamide (PAA) gel with 80% water concentration with elastic modulus E = 0.25 MPa. At this E-value, the deformation and rebound of the boundary is very pronounced providing particularly interesting features of bubble dynamics. It is shown by means of high-speed photography with up to 5 million frames s−1 that bubble splitting, formation of liquid jets away from and towards the boundary, and jet-like ejection of the boundary material into the liquid are the main features of this interaction. The maximum liquid jet velocity measured was 960 m s−1. Such high-velocity jets penetrate the elastic boundary even through a water layer of 0.35 mm thickness. The jetting behaviour arises from the interaction between the counteracting forces induced by the rebound of the elastic boundary and the Bjerknes attraction force towards the boundary. General principles of the formation of annular and axial jets are discussed which allow the interpretation of the complex dynamics. The concept of the Kelvin impulse is examined with regard to bubble migration and jet formation. The results are discussed with respect to cavitation erosion, collateral damage in laser surgery, and cavitation-mediated enhancement of pulsed laser ablation of tissue.

The interaction of a laser-induced cavitation bubble with an elastic boundary is investigated experimentally by high-speed photography and acoustic measurements. The elastic material consists of a polyacrylamide (PAA) gel whose elastic properties can be controlled by modifying the water content of the sample. The elastic modulus, E, is varied between 0.017 MPa and 2.03 MPa, and the dimensionless bubble–boundary distance, γ, is for each value of E varied between γ = 0 and γ = 2.2. In this parameter space, jetting behaviour, jet velocity, bubble migration and bubble oscillation time are determined. The jetting behaviour varies between liquid jet formation towards or away from the elastic boundary, and formation of an annular jet which results in bubble splitting and the subsequent formation of two very fast axial liquid jets flowing in opposite directions. The liquid jet directed away from the boundary reaches a maximum velocity between 300 ms−1 and 600 ms−1 (depending on the elastic modulus of the sample) while the peak velocity of the jet directed towards the boundary ranges between 400 ms−1 and 800 ms−1 (velocity values averaged over 1 μs). Penetration of the elastic boundary by the liquid jet is observed for PAA samples with an intermediate elastic modulus between 0.12 and 0.4 MPa. In this same range of elastic moduli and for small γ-values, PAA material is ejected into the surrounding liquid due to the elastic rebound of the sample surface that was deformed during bubble expansion and forms a PAA jet upon rebound. For stiffer boundaries, the bubble behaviour is mainly characterized by the formation of an axial liquid jet and bubble migration directed towards the boundary, as if the bubble were adjacent to a rigid wall. For softer samples, the bubble behaviour becomes similar to that in a liquid with infinite extent. During bubble collapse, however, material is torn off the PAA sample when bubbles are produced close to the boundary. We conclude that liquid jet penetration into the boundary, jet-like ejection of boundary material, and tensile-stress-induced deformations of the boundary during bubble collapse are the major mechanisms responsible for cavitation erosion and for cavitation-enhanced ablation of elastic materials as, for example, biological tissues.

Steady, inviscid, axisymmetric vortex flows past a sphere are obtained numerically as solutions of a partial differential equation for the stream function. The solutions found include vortex rings, bounded vortices attached to the sphere and infinite vortex tubes. Four families of attached vortices are described: vortex wakes behind the sphere, spherically annular vortices surrounding the spherical obstacle (which can be given analytically), bands of vorticity around the sphere and symmetric pairs of vortices fore and aft of the sphere. Each attached vortex leads to a one-parameter family of vortex rings, analogous to the connection between Hill's spherical vortex and the vortex rings of Norbury.

This paper presents experimental measurements of the approach and rebound of a particle colliding with a wall in a viscous fluid. The particle's trajectory was controlled by setting the initial inclination angle of a pendulum immersed in a fluid. The resulting collisions were monitored using a high-speed video camera. The diameters of the particles ranged from 3 to 12 mm, and the ratio of the particle density to fluid density varied from 1.2 to 7.8. The experiments were performed using a thick glass or Lucite wall with different mixtures of glycerol and water. With these parameters, the Reynolds number defined using the velocity just prior to impact ranged from 10 to approximately 3000. A coefficient of restitution was defined from the ratio of the velocity just prior to and after impact.

The experiments clearly demonstrate that the rebound velocity depends on the impact Stokes number (defined from the Reynolds number and the density ratio) and weakly on the elastic properties of the material. Below a Stokes number of approximately 10, no rebound of the particle occurred. For impact Stokes number above 500 the coefficient of restitution appears to asymptote to the values for dry collisions. The coefficients of restitution were also compared with previous experimental studies. In addition, the approach of the particle to the wall indicated that the particle slowed prior to impacting the surface. The distance at which the particle's trajectory varied due to the presence of the wall was dependent on the impact Stokes number. The particle surface roughness was found to affect the repeatability of some measurements, especially for low impact velocities.

The formation of segregation patterns in initially homogeneous, fluidized, binary mixtures of particles has been studied. The adjustment of the bed depends on the proportions of fine and coarse particles in the mixture and the gas flow rate relative to the minimum fluidization velocities of the two components. The particles are immobile until the gas flow rate is sufficiently large to fluidize the mixture of particles. When the gas flow rate exceeds this critical value, alternating vertical bands of coarse and fine particles form. At a second critical gas velocity this pattern breaks down and the more familiar pattern of a mixed horizontal band on top of a layer of coarse particles forms. A phase diagram, constructed from experimental observations, shows the conditions for which each of these regimes exists. Its structure is explained in terms of the fluidization and consequent mobility of the mixture components. When horizontal bands are present, the thickness of the lower layer of coarse particles decreases with increasing gas flow rate depending on the proportion of fine particles in the bed. This, and its development, can be understood by analogy with the sedimentation of particles through a turbulent fluid. The experiments imply that the efficiency of mixing by the bubbles in the fluidized bed is very much less than that expected from gas bubbles in a liquid.

This paper describes steady and unsteady flows in pipes with small, slowly varying curvature and torsion. Four new pipe shapes are studied, using Germano's extension of the Dean equations. Analytic and numerical solutions are obtained for flows driven by a steady pressure gradient. Oscillatory flows in pipes with non-uniform curvature are obtained by numerical methods. The effects of the non-uniformities in curvature and torsion are discussed, with particular reference to wall shear stress.

We present a stochastic analysis of a data set consisting of 1.25 × 107 samples of the local velocity measured in the turbulent region of a round free jet. We find evidence that the statistics of the longitudinal velocity increment v(r) can be described as a Markov process. This new approach to characterize small-scale turbulence leads to a Fokker–Planck equation for the r-evolution of the probability density function (p.d.f.) of v(r). This equation for p(v, r) is completely determined by two coefficients D1(v, r) and D2(v, r) (drift and diffusion coefficient, respectively). It is shown how these coefficients can be estimated directly from the experimental data without using any assumptions or models for the underlying stochastic process. The solutions of the resulting Fokker–Planck equation are compared with experimentally determined probability density functions. It is shown that the Fokker–Planck equation describes the measured p.d.f.(s) correctly, including intermittency effects. Furthermore, knowledge of the Fokker–Planck equation also allows the joint probability density of N increments on N different scales p(v1, r1, …, vN, rN) to be determined.