The stretching of line elements, surface elements and wave vectors by a random, isotropic, solenoidal velocity field in D dimensions is studied. The rates of growth of line elements and (D – 1)-dimensional surface elements are found to be equal if the statistics are invariant to velocity reversal. The analysis is applied to convection of a sparse distribution of sheets of passive scalar in a random straining field whose correlation scale is large compared with the sheet size. This is Batchelor's (1959) κ−1 spectral regime. Some exact analytical solutions are found when the velocity field varies rapidly in time. These include the dissipation spectrum and a joint probability distribution that describes the simultaneous effect of Stretching and molecular diffusivity κ on the amplitude profile of a sheet. The latter leads to probability distributions of the scalar field and its space derivatives. For a growing κ−1 range at zero κ, these derivatives have essentially lognormal statistics. In the steady-state κ−1 regime at κ > 0, intermittencies measured by moment ratios are much smaller than for lognormal statistics, and they increase less rapidly with the order of the derivative than in the κ = 0 case. The κ > 0 distributions have singularities a t zero amplitude, due to a background of highly diffused sheets. The results do not depend strongly on D. But as D → ∞, temporal fluctuations in the stretching rates become negligible and Batchelor's (1959) constant-strain dissipation spectrum is recovered.

Axisymmetric vortex rings which propagate steadily through an unbounded ideal fluid at rest at infinity are considered. The vorticity in the ring is proportional to the distance from the axis of symmetry. Recent theoretical work suggests the existence of a one-parameter family, [npar ]2 ≥ α ≥ 0 (the parameter α is taken as the non-dimensional mean core radius), of these vortex rings extending from Hill's spherical vortex, which has the parameter value α = [npar ]2, to vortex rings of small cross-section, where α → 0. This paper gives a numerical description of vortex rings in this family. As well as the core boundary, propagation velocity and flux, various other properties of the vortex ring are given, including the circulation, fluid impulse and kinetic energy. This numerical description is then compared with asymptotic descriptions which can be found near both ends of the family, that is, when α → [npar ]2 and α → 0.

We present an experimental study of a symmetric foil performing pitching oscillations in a vertically flowing soap film. By varying the frequency and amplitude of the oscillation we visualize a variety of wakes with up to 16 vortices per oscillation period, including von Kármán vortex street, inverted von Kármán vortex street, 2P wake, 2P+2S wake and novel wakes ranging from 4P to 8P. We map out the wake types in a phase diagram spanned by the width-based Strouhal number and the dimensionless amplitude. We follow the time evolution of the vortex formation near the round leading edge and the shedding process at the sharp trailing edge in detail. This allows us to identify the origins of the vortices in the 2P wake, to understand that two distinct 2P regions are present in the phase diagram due to the timing of the vortex shedding at the leading edge and the trailing edge and to propose a simple model for the vorticity generation. We use the model to describe the transition from 2P wake to 2S wake with increasing oscillation frequency and the transition from the von Kármán wake, typically associated with drag, to the inverted von Kármán wake, typically associated with thrust generation.

Direct numerical simulation of turbulent homogeneous shear flow is performed in order to clarify compressibility effects on the turbulence growth in the flow. The two Mach numbers relevant to homogeneous shear flow are the turbulent Mach number Mt and the gradient Mach number Mg. Two series of simulations are performed where the initial values of Mg and Mt are increased separately. The growth rate of turbulent kinetic energy is observed to decrease in both series of simulations. This ‘stabilizing’ effect of compressibility on the turbulent energy growth rate is observed to be substantially larger in the DNS series where the initial value of Mg is changed. A systematic comparison of the different DNS cases shows that the compressibility effect of reduced turbulent energy growth rate is primarily due to the reduced level of turbulence production and not due to explicit dilatational effects. The reduced turbulence production is not a mean density effect since the mean density remains constant in compressible homogeneous shear flow. The stabilizing effect of compressibility on the turbulence growth is observed to increase with the gradient Mach number Mg in the homogeneous shear flow DNS. Estimates of Mg for the mixing layer and the boundary layer are obtained. These estimates show that the parameter Mg becomes much larger in the high-speed mixing layer relative to the high-speed boundary layer even though the mean flow Mach numbers are the same in the two flows. Therefore, the inhibition of turbulent energy production and consequent ‘stabilizing’ effect of compressibility on the turbulence (over and above that due to any mean density variation) is expected to be larger in the mixing layer relative to the boundary layer, in agreement with experimental observations.

The appearance of chaotic particle trajectories in steady, laminar, incompressible flow through a twisted pipe of circular cross-section is demonstrated using standard dynamical systems diagnostics and a model flow based on Dean's perturbation solutions. A study is performed to determine the parameters that control fluid stirring in this mixing device that has no moving parts. Insight into the chaotic dynamics are provided by a simple one-dimensional map of the pipe boundary onto itself. The results of numerical experiments illustrating the stretching of material lines, stirring of blobs of material, and the three-dimensional trajectories of fluid particles are presented. Finally, enhanced longitudinal particle dispersal due to the coupling between chaos in the transverse direction and the non-uniform longitudinal transport of particles is shown.

The evolution of unsheared grid-generated turbulence in a stably stratified fluid was investigated in a closed-loop salt-stratified water channel. Simultaneous single-point measurements of the horizontal and vertical velocity and density fluctuations were obtained, including turbulent mass fluxes central in understanding the energetics of the fluctuating motion. When the buoyancy lengthscale was initially substantially larger than the largest turbulent scales, the initial behaviour of the velocity and density fields was similar to that in the non-stratified case. With further downstream development, the buoyancy lengthscale decreased while the turbulence scale grew. Deviations from neutral behaviour occurred when these two lengthscales became of the same order, after the initially larger inertial forces associated with the initial kinetic energy had become weaker and buoyancy forces became important.

Buoyancy forces produced anisotropy in the largest scales first, preventing them from overturning, while smaller-scale isotropic turbulent motions remained embedded within the larger-scale wave motions. These small-scale motions exhibited classical turbulent behaviour and scaled universally with Kolmogorov length and velocity scales. Eventually even the smallest scales of the decaying turbulence were affected by buoyancy, all isotropic motions disappeared, and Kolmogorov scaling failed. The turbulent vertical mass flux decreased to zero for this condition, indicating that the original turbulent field had been completely converted to random internal wave motions.

The transition from a fully turbulent state to one of internal waves occurred rapidly in a time less than the characteristic time of the turbulence based on the largest-scale eddies found in the flow at transition. The dissipation rate for complete transition to a wave field was found to be of the order of εt = 24.5νN2, where ν is the kinematic viscosity and N the Brunt-Väisälä frequency. This is in fairly good agreement with the value 30νN2 predicted by Gibson (1980, 1981).

The present experiments have determined quantitative limits on the range of active turbulent scales in homogeneous stratified turbulence, in terms of an upper limit near the buoyancy lengthscale and a lower limit determined by viscosity in the usual way. This description has been used here to help explain and assimilate the results from the earlier stratified grid-turbulence experiments of Lin & Veenhuizen (1975) and Dickey & Mellor (1980). While some of the features of the present observations may be qualitatively seen in the numerical simulations of the problem of Riley, Metcalfe & Weissman (1981), there are fundamental differences, probably due in part to large differences in initial lengthscale ratios and in the limited range of scales attainable in numerical simulations. The present experiments may serve as a useful test case for future modelling and interpretation of the behaviour of turbulence in stratified flows observed in the oceans and atmosphere.

Results are presented from pressure measurements performed in the irrotational near field of unbounded co-axial jets. Measurements were made for a variety of velocity and temperature ratios, and configurations both with and without serrations on the secondary nozzle lip. The principal objective of the study is to better understand the near pressure field of the jet, what it can tell us regarding the underlying turbulence structure, and in particular how it can be related to the source mechanisms of the flow.

A preliminary analysis of the axial, temporal and azimuthal structure of the pressure field shows it to be highly organized, with axial spatial modes (obtained by proper orthogonal decomposition) which resemble Fourier modes. The effects of serrations on the pressure fluctuations comprise a global reduction in level, a change in the axial energy distribution, and a modification of the evolution of the characteristic time scales.

A further analysis in frequency–wavenumber space is then performed, and a filtering operation is used to separate the convective and propagative footprints of the pressure field. This operation reveals two distinct signatures in the propagating component of the field: a low-frequency component which radiates at small angles to the flow axis and is characterized by extensive axial coherence, and a less-coherent high-frequency component which primarily radiates in sideline directions. The serrations are found to reduce the energy of the axially coherent propagating component, but its structure remains fundamentally unchanged; the high-frequency component is found to be enhanced. A further effect of the serrations involves a relative increase of the mean-square pressure level of the acoustic component – integrated over the measurement domain – with respect to the hydrodynamic component. The effect of increasing the velocity and temperature of the primary jet involves a relative increase in the acoustic component of the near field, while the hydrodynamic component remains relatively unchanged: this shows that the additional acoustic energy is generated by the mixing region which is produced by the interaction of the inner and the outer shear layers, whereas the hydrodynamic component of the near field is primarily driven by the outer shear layer.

This is a study of turbulence which results from Kelvin—Helmholtz instability at the interface between two miscible fluids in a two-dimensional shear flow in the laboratory. The growth of two-dimensional ‘billows’, their disruption by turbulence, and the eventual decay of this turbulence and the re-establishment of a gravitationally and kinematically stable interface are described. Continuous measurements of density and horizontal velocity from both fixed and vertically moving probes have been made, and the records obtained are presented, together with photographs showing the simultaneous appearance of the flow, which serve to identify the physical nature of events seen in the records. The measurements show how the fine-structure of the density field described in earlier experiments is related to velocity fluctuations. The vertical length scales of the final mean velocity and density structure are found to be different, and to depend on the Richardson number at which instability first occurred. The eventual Richardson number at the centre of layer is, however, not dependent on the initial Richardson number and has a value of about one third. The implications of these results to the eddy diffusion coefficients, to the energy exchange, and to turbulence in the ocean and the atmosphere are discussed.

A large-eddy simulation technique is described for computing Rayleigh–Taylor instability. The method is based on high-wavenumber-preserving subgrid-scale models, combined with high-resolution numerical methods. The technique is verified to match linear stability theory and validated against direct numerical simulation data. The method is used to simulate Rayleigh–Taylor instability at a grid resolution of $1152^3$. The growth rate is found to depend on the mixing rate. A mixing transition is observed in the flow, during which an inertial range begins to form in the velocity spectrum and the rate of growth of the mixing zone is temporarily reduced. By measuring growth of the layer in units of dominant initial wavelength, criteria are established for reaching the hypothetical self-similar state of the mixing layer. A relation is obtained between the rate of growth of the mixing layer and the net mass flux through the plane associated with the initial location of the interface. A mix-dependent Atwood number is defined, which correlates well with the entrainment rate, suggesting that internal mixing reduces the layer's growth rate.

The effects of ‘crossing trajectories’ and inertia on the dispersion of particles suspended in a field of grid-generated turbulence were investigated experimentally. The effect of particle trajectories crossing the trajectories of fluid elements, under the influence of a potential field (usually gravity), is to force the particles from one region of highly correlated flow to another. In this manner, particles lose velocity correlation more rapidly than the corresponding fluid points and as a result disperse less.

A homogeneous decaying turbulent field was created behind a square biplanar grid in a wind tunnel. Particles were charged by a corona discharge then passed into the test section through a small plastic tube. A uniform electric field within the test section was used to simulate the effect of gravity, forcing the charged particles out of regions of correlated fluid at a higher than normal rate, therefore inducing the effects of crossing trajectories. Two sizes of glass beads (5 μm and 57 μm diameter) were employed in order to observe inertial effects. Laser-Doppler anemometry was used to measure particle mean-square displacement, autocorrelation coefficient, and mean-square velocity, from which dispersion coefficients were calculated.

For the two particle sizes used in the tests, it was found that the particle diffusion coefficient, after a suitably long time from their release, was influenced primarily by the effect of crossing trajectories. Only in the particle mean-square velocity was the particle inertia seen to have any effect. The ratio of the particle relaxation time to the Kolmogoroff timescale was found to be a good indicator for the effects of particle inertia.

This study investigates the two-way coupling effects of finite-size solid spherical particles on decaying isotropic turbulence using direct numerical simulation with an immersed boundary method. We fully resolve all the relevant scales of turbulence around freely moving particles of the Taylor length-scale size, 1.2≤d/λ≤2.6. The particle diameter and Stokes number in terms of Kolmogorov length- and time scales are 16≤d/η≤35 and 38≤τp/τk≤178, respectively, at the time the particles are released in the flow. The particles mass fraction range is 0.026≤φm≤1.0, corresponding to a volume fraction of 0.01≤φv≤0.1 and density ratio of 2.56≤ρp/ρf≤10. The maximum number of dispersed particles is 6400 for φv=0.1. The typical particle Reynolds number is of O(10). The effects of the particles on the temporal development of turbulence kinetic energy E(t), its dissipation rate (t), its two-way coupling rate of change Ψp(t) and frequency spectra E(ω) are discussed.

In contrast to particles with d < η, the effect of the particles in this study, with d > η, is that E(t) is always smaller than that of the single-phase flow. In addition, Ψp(t) is always positive for particles with d > η, whereas it can be positive or negative for particles with d < η.

It has been recognized that particle inertia throws dense particles out of regions of high vorticity and leads to an accumulation of particles in the straining-flow regions of a turbulent flow field. However, recent direct numerical simulations (DNS) indicate that the tendency to cluster is evident even at particle separations smaller than the size of the smallest eddy. Indeed, the particle radial distribution function (RDF), an important measure of clustering, increases as an inverse power of the interparticle separation for separations much smaller than the Kolmogorov length scale. Motivated by this observation, we have developed an analytical theory to predict the RDF in a turbulent flow for particles with a small, but non-zero Stokes number. Here, the Stokes number ($\hbox{\it St}$) is the ratio of the particle's viscous relaxation time to the Kolmogorov time. The theory approximates the turbulent flow in a reference frame following an aerosol particle as a local linear flow field with a velocity gradient tensor and acceleration that vary stochastically in time. In monodisperse suspensions, the power-law dependence of the pair probability is seen to arise from a balance of an inward drift caused by the particles' inertia that scales linearly with the particle separation distance and a pairwise diffusion owing to the random nature of the flow with a diffusivity that scales quadratically with the particle separation distance. The combined effect leads to a power law behaviour for the RDF with an exponent, $c_1$, that is proportional to $\hbox{\it St}^2$. Predictions of the analytical theory are compared with two types of numerical simulation: (i) particle pairs interacting in a local linear flow whose velocity varies according to a stochastic velocity gradient model; (ii) particles interacting in a flow field obtained from DNS of isotropic turbulence. The agreement with both types of simulation is very good. The theory also predicts the RDF for unlike particle pairs (particle pairs with different Stokes numbers). In this case, a second diffusion process occurs owing to the difference in the response of the pair to local fluid accelerations. The acceleration diffusivity is independent of the pair separation distance; thus, the RDF of particles with even slightly different viscous relaxation times undergoes a transition from the power law behaviour at large separations to a constant value at sufficiently small separations. The radial separation corresponding to the transition between these two behaviours is predicted to be proportional to the difference between the Stokes numbers of the two particles. Once again, the agreement between the theory and simulations is found to be very good. Clustering of particles enhances their rate of coagulation or coalescence. The theory and linear flow simulations are used to obtain predictions for the rate of coagulation of particles in the absence of hydrodynamic and colloidal particle interactions.

Parallel flows with step function velocity and density profiles can support waves which have negative energy, in the sense that exciting them lowers the total energy of the system. A number of instabilities can occur because of the coexistence of positive and negative energy waves, or because of the damping of negative energy waves; some particular examples are discussed to show how appreciation of this role of negative energy waves allows one to predict the existence of instability before doing any detailed analysis, and to gain insight into the instability mechanism.

An investigation of topological features of the velocity gradient field of turbulent channel flow has been carried out using results from a direct numerical simulation for which the Reynolds number based on the channel half-width and the centreline velocity was 7860. Plots of the joint probability density functions of the invariants of the rate of strain and velocity gradient tensors indicated that away from the wall region, the fine-scale motions in the flow have many characteristics in common with a variety of other turbulent and transitional flows: the intermediate principal strain rate tended to be positive at sites of high viscous dissipation of kinetic energy, while the invariants of the velocity gradient tensor showed that a preference existed for stable focus/stretching and unstable node/saddle/saddle topologies. Visualization of regions in the flow with stable focus/stretching topologies revealed arrays of discrete downstream-leaning flow structures which originated near the wall and penetrated into the outer region of the flow. In all regions of the flow, there was a strong preference for the vorticity to be aligned with the intermediate principal strain rate direction, with the effect increasing near the walls in response to boundary conditions.

We propose a novel cluster-based reduced-order modelling (CROM) strategy for unsteady flows. CROM combines the cluster analysis pioneered in Gunzburger’s group (Burkardt, Gunzburger & Lee, Comput. Meth. Appl. Mech. Engng, vol. 196, 2006a, pp. 337–355) and transition matrix models introduced in fluid dynamics in Eckhardt’s group (Schneider, Eckhardt & Vollmer, Phys. Rev. E, vol. 75, 2007, art. 066313). CROM constitutes a potential alternative to POD models and generalises the Ulam–Galerkin method classically used in dynamical systems to determine a finite-rank approximation of the Perron–Frobenius operator. The proposed strategy processes a time-resolved sequence of flow snapshots in two steps. First, the snapshot data are clustered into a small number of representative states, called centroids, in the state space. These centroids partition the state space in complementary non-overlapping regions (centroidal Voronoi cells). Departing from the standard algorithm, the probabilities of the clusters are determined, and the states are sorted by analysis of the transition matrix. Second, the transitions between the states are dynamically modelled using a Markov process. Physical mechanisms are then distilled by a refined analysis of the Markov process, e.g. using finite-time Lyapunov exponent (FTLE) and entropic methods. This CROM framework is applied to the Lorenz attractor (as illustrative example), to velocity fields of the spatially evolving incompressible mixing layer and the three-dimensional turbulent wake of a bluff body. For these examples, CROM is shown to identify non-trivial quasi-attractors and transition processes in an unsupervised manner. CROM has numerous potential applications for the systematic identification of physical mechanisms of complex dynamics, for comparison of flow evolution models, for the identification of precursors to desirable and undesirable events, and for flow control applications exploiting nonlinear actuation dynamics.

Results are presented from an experimental study of shallow flow in a channel partially obstructed by an array of circular cylinders. The cylinder array is a model for emergent vegetation in an open channel, but also represents a simple sparse porous medium. A shear layer with regular vortex structures forms at the edge of the array, evolving downstream to an equilibrium width and vortex size. The vortices induce nearly periodic oscillations with a frequency that matches the most unstable linear mode for a parallel shear flow. The shear layer is asymmetric about the array interface and has a two-layer structure. An inner region of maximum shear near the interface contains a velocity inflection point and establishes the penetration of momentum into the array. An outer region, resembling a boundary layer, forms in the main channel, and establishes the scale of the vortices. The vortex structure, educed by conditional sampling, shows strong crossflows with sweeps from the main channel and ejections from the array, which create significant momentum and mass fluxes across the interface. The sweeps maintain the coherent structures by enhancing shear and energy production at the interface. A linear stability analysis is consistent with the experimental results and demonstrates that the instability is excited by the differential drag between the channel and the array.

The streamwise and spanwise velocity components and the gradients of these components normal to the wall were examined by using hot-film sensors and flush-mounted wall elements to study the vortex structures associated with the bursting phenomenon. Quadrant probability analysis and conditional sampling techniques indicated that pairs of counter-rotating streamwise vortices occur frequently in the wall region of a bounded turbulent shear flow. A streamwise momentum defect occurred between the vortices as low-speed fluid was ‘pumped’ away from the wall by the vortex pair. The defect region was long and narrow and possibly forms the low-speed streak as observed in visualization studies. The velocity defect was terminated by a strong acceleration followed by a high speed region.

A layer of volatile viscous liquid drains down a uniformly heated inclined plate. Long-wave instabilities of the uniform film are studied by deriving an evolution equation for two-dimensional disturbances. This equation incorporates viscosity, gravity, surface tension, thermocapillarity, and evaporation eifects. The linear theory derived from this describes the competition among the instabilities. Numerical solution of the evolution equation describes the finite-amplitude behaviour that determines the propensity for dryout of the film. Among the phenomena that appear are the tendency to wave breaking, the creation of secondary structures, and the preemption of dryout by mean flow.

This paper attempts to explain theoretically the observed results that (1) the cells in steady convection approach a hexagonal form, and (2) the occurrence of ascent or descent in the middle of the cell depends on how the kinematical viscosity varies with temperature. The theory is based on non-linear equations and, of course, a variable coefficient of viscosity.

It is found that, due to the variation of viscosity with temperature, the non-linear terms contain a second-order term which is destabilizing. This second-order term regulates the development and leads to a final motion composed of regular hexagons with ascent or descent in the middle of the cell according as the viscosity decreases or increases with temperature.

The influence of a variable viscosity on Rayleigh's result concerning the initiation of convection is obtained as a by-product.

Conservable quantities measuring ‘wave activity’ are discussed. The equation for the most fundamental such quantity, wave-action, is derived in a simple but very general form which does not depend on the approximations of slow amplitude modulation, linearization, or conservative motion. The derivation is elementary, in the sense that a variational formulation of the equations of fluid motion is not used. The result depends, however, on a description of the disturbance in terms of particle displacements rather than velocities. A corollary is an elementary but general derivation of the approximate form of the wave-action equation found by Bretherton & Garrett (1968) for slowlyvarying, linear waves.

The sense in which the general wave-action equation follows from the classical ‘energy-momentum-tensor’ formalism is discussed, bringing in the concepts of pseudomomentum and pseudoenergy, which in turn are related to special cases such as Blokhintsev's conservation law in acoustics. Wave-action, pseudomomentum and pseudoenergy are the appropriate conservable measures of wave activity when ‘waves’ are defined respectively as departures from ensemble-, space- and time-averaged flows.

The relationship between the wave drag on a moving boundary and the fluxes of momentum and pseudomomentum is discussed.