High-resolution large-eddy simulation is used to investigate the mean and turbulence properties of a separated flow in a channel constricted by periodically distributed hill-shaped protrusions on one wall that obstruct the channel by 33% of its height and are arranged 9 hill heights apart. The geometry is a modification of an experimental configuration, the adaptation providing an extended region of post-reattachment recovery and allowing high-quality simulations to be performed at acceptable computing costs. The Reynolds number, based on the hill height and the bulk velocity above the crest is 10595. The simulated domain is streamwise as well as spanwise periodic, extending from one hill crest to the next in the streamwise direction and over 4.5 hill heights in the spanwise direction. This arrangement minimizes uncertainties associated with boundary conditions and makes the flow an especially attractive generic test case for validating turbulence closures for statistically two-dimensional separation. The emphasis of the study is on elucidating the turbulence mechanisms associated with separation, recirculation reattachment, acceleration and wall proximity. Hence, careful attention has been paid to resolution, and a body-fitted, low-aspect-ratio, nearly orthogonal numerical grid of close to 5 million nodes has been used. Unusually, the results of two entirely independent simulations with different codes for identical flow and numerical conditions are compared and shown to agree closely. Results are included for mean velocity, Reynolds stresses, anisotropy measures, spectra and budgets for the Reynolds stresses. Moreover, an analysis of structural characteristics is undertaken on the basis of instantaneous realizations, and links to features observed in the statistical results are identified and interpreted. Among a number of interesting features, a distinct ‘splatting’ of eddies on the windward hill side following reattachment is observed, which generates strong spanwise fluctuations that are reflected, statistically, by the spanwise normal stress near the wall exceeding that of the streamwise stress by a substantial margin, despite the absence of spanwise strain.

Aspect ratio effects on the vortex shedding flow from a circular cylinder have been studied by using moveable end plates. Experiments were carried out to measure fluctuating forces, shedding frequency and spanwise correlation whilst varying end plate separation and Reynolds number. The aspect ratio (0.25–12) was found to have a most striking effect on the fluctuating lift. Within a certain range of Reynolds number an increase of the sectional fluctuating lift was obtained for reduced aspect ratio, and showed a maximum for an aspect ratio of 1, where the fluctuating lift could be almost twice the value for very large aspect ratios. This increase of the lift amplitude was found to be accompanied by enhanced spanwise correlation of the flow. The measurements were carried out over the Reynolds number range 8 × 103 < Re < 1.4 × 105. The strong increase in fluctuating lift with small aspect ratio did not occur at the lower and upper boundaries of this range. In the lower Reynolds number range (Re < 2 × 104) the trend could be reversed, i.e. the fluctuating lift decreased with decreasing aspect ratio. Also, with small aspect ratio, a shedding breakdown was found in the upper Reynolds number range (Re = 1.3 × 105). The main three-dimensional feature observed was a spanwise variation in the phase of vortex shedding, accompanied by amplitude modulation in the lift signal. However, the level of three-dimensionality can be reduced by using a small aspect ratio. Three-dimensional vortex shedding features are discussed and comparison of the results with those from both two-dimensional numerical simulations and other experiments using large aspect ratios are presented.

Suspensions are composed of mixtures of particles and fluid and are omnipresent in natural phenomena and in industrial processes. The present paper addresses the rheology of concentrated suspensions of non-colloidal particles. While hydrodynamic interactions or lubrication forces between the particles are important in the dilute regime, they become of lesser significance when the concentration is increased, and direct particle contacts become dominant in the rheological response of concentrated suspensions, particularly those close to the maximum volume fraction where the suspension ceases to flow. The rheology of these dense suspensions can be approached via a diversity of approaches that the paper introduces successively. The mixture of particles and fluid can be seen as a fluid with effective rheological properties but also as a two-phase system wherein the fluid and particles can experience relative motion. Rheometry can be undertaken at an imposed volume fraction but also at imposed values of particle normal stress, which is particularly suited to yield examination of the rheology close to the jamming transition. The response of suspensions to unsteady or transient flows provides access to different features of the suspension rheology. Finally, beyond the problem of suspension of rigid, non-colloidal spheres in a Newtonian fluid, there are a great variety of complex mixtures of particles and fluid that remain relatively unexplored.

Informed by large-eddy simulation (LES) data and resolvent analysis of the mean flow, we examine the structure of turbulence in jets in the subsonic, transonic and supersonic regimes. Spectral (frequency-space) proper orthogonal decomposition is used to extract energy spectra and decompose the flow into energy-ranked coherent structures. The educed structures are generally well predicted by the resolvent analysis. Over a range of low frequencies and the first few azimuthal mode numbers, these jets exhibit a low-rank response characterized by Kelvin–Helmholtz (KH) type wavepackets associated with the annular shear layer up to the end of the potential core and that are excited by forcing in the very-near-nozzle shear layer. These modes too have been experimentally observed before and predicted by quasi-parallel stability theory and other approximations – they comprise a considerable portion of the total turbulent energy. At still lower frequencies, particularly for the axisymmetric mode, and again at high frequencies for all azimuthal wavenumbers, the response is not low-rank, but consists of a family of similarly amplified modes. These modes, which are primarily active downstream of the potential core, are associated with the Orr mechanism. They occur also as subdominant modes in the range of frequencies dominated by the KH response. Our global analysis helps tie together previous observations based on local spatial stability theory, and explains why quasi-parallel predictions were successful at some frequencies and azimuthal wavenumbers, but failed at others.

The effects of Brownian motion alone and in combination with an interparticle force of hard-sphere type upon the particle configuration in a strongly sheared suspension are analysed. In the limit Pe→∞ under the influence of hydrodynamic interactions alone, the pair-distribution function of a dilute suspension of spheres has symmetry properties that yield a Newtonian constitutive behaviour and a zero self-diffusivity. Here, Pe=γ˛a2/2D is the Péclet number with γ˛ the shear rate, a the particle radius, and D the diffusivity of an isolated particle. Brownian diffusion at large Pe gives rise to an O(aPe−1) thin boundary layer at contact in which the effects of Brownian diffusion and advection balance, and the pair-distribution function is asymmetric within the boundary layer with a contact value of O(Pe0.78) in pure-straining motion; non-Newtonian effects, which scale as the product of the contact value and the O(a3Pe−1) layer volume, vanish as Pe−0.22 as Pe→∞.

Numerical simulations are used to investigate the effect of aspect ratio on the wake topology and hydrodynamic performance of thin ellipsoidal flapping foils. The study is motivated by the quest to understand the hydrodynamics of fish pectoral fins. The simulations employ an immersed boundary method that allows us to simulate flows with complex moving boundaries on fixed Cartesian grids. A detailed analysis of the vortex topology shows that the wake of low-aspect-ratio flapping foils is dominated by two sets of interconnected vortex loops that evolve into distinct vortex rings as they convect downstream. The flow downstream of these flapping foils is characterized by two oblique jets and the implications of this characteristic on the hydrodynamic performance are examined. Simulations are also used to examine the thrust and propulsive efficiency of these foils over a range of Strouhal and Reynolds numbers as well as pitch-bias angles.

Electric conduction from an electrolyte solution into a charge selective solid, such as ion exchange membrane or electrode, becomes unstable when the electrolyte concentration near the interface approaches zero owing to diffusion limitation. The sequence of events leading to instability is as follows: upon the decrease of the interface concentration, the electric double layer at the interface transforms from its common quasi-equilibrium structure to a different, non-equilibrium one. The key feature of this new structure is an extended space charge added to the usual one of the quasi-equilibrium electric double layer. The non-equilibrium electro-osmotic slip related to this extended space charge renders the quiescent conductance unstable. A unified asymptotic picture of the electric double-layer undercurrent, encompassing all regimes from quasi-equilibrium to the extreme non-equilibrium one, is developed and employed for derivation of a universal electro-osmotic slip formula. This formula is used for a linear stability study of quiescent electric conduction, yielding the precise parameter range of instability, compared with that in the full electroconvective formulation. The physical mechanism of instability is traced both kinematically, in terms of non-equilibrium electro-osmotic slip, and dynamically, in terms of forces acting in the electric double layer.

It is argued that the heat transfer between a roughened surface and a stream of incompressible fluid flowing over it is dependent on both the viscosity and thermal conductivity of the fluid even when the roughness is large enough for viscosity to have ceased to affect the skin friction.

Concentrating on closely spaced roughness, sufficiently large for the skin friction to be independent of Reynolds number, a simple model is constructed of the flow near the surface. It consists of horseshoe eddies which wrap themselves round the individual excrescences and trail unsteadily downstream; the eddies are imagined to scour the surface and thereby to transport heat between the surface and the more vigorous flow in the neighbourhood of the roughness crests. Taken in conjunction with Reynolds analogy between temperature and velocity distributions in the fluid away from the surface, the model leads to an expression for the rate of heat transfer which contains a function of the roughness Reynolds number and the Prandtl number of the fluid whose detailed form is found by appeal to the limited experimental data available. An order-of-magnitude argument suggests that the functional form established empirically is consistent with the assumed model of the flow close to the surface.

The object of the work is to establish a basis for the analysis of experimental data and for their extrapolation with respect to Reynolds number and Prandtl number.

A premixed ducted flame, burning in the wake of a bluff-body flame-holder, is considered. For such a flame, interaction between acoustic waves and unsteady combustion can lead to self-excited oscillations. The concept of a time-invariant turbulent flame speed is used to develop a kinematic model of the response of the flame to flow disturbances. Variations in the oncoming flow velocity at the flame-holder drive perturbations in the flame initiation surface and hence in the instantaneous rate of heat release. For linear fluctuations, the transfer function between heat release and velocity can be determined analytically from the model and is in good agreement with experiment across a wide frequency range. For nonlinear fluctuations, the model reproduces the flame surface distortions seen in schlieren films.

Coupling this kinematic flame model with an analysis of the acoustic waves generated in the duct by the unsteady combustion enables the time evolution of disturbances to be calculated. Self-excited oscillations occur above a critical fuel–air ratio. The frequency and amplitude of the resulting limit cycles are in satisfactory agreement with experiment. Flow reversal is predicted to occur during part of the limit-cycle oscillation and the flame then moves upstream of the flame-holder, just as in experimental visualizations. The main nonlinearity is identified in the rate of heat release, which essentially ‘saturates’ once the amplitude of the velocity fluctuation exceeds its mean. We show that, for this type of nonlinearity, describing function analysis can be used to give a good estimate of the limit-cycle frequency and amplitude from a quasi-nonlinear theory.

Parametric excitation of surface waves via forced vertical oscillation of a container filled with fluid (the Faraday instability) is investigated experimentally in a small-depth large-aspect-ratio system, with a viscous fluid and with two simultaneous forcing frequencies. The asymptotic pattern observed just above the threshold for the first instability of the flat surface is found to depend strongly on the frequency ratio and the amplitudes and phases of the two sinusoidal components of the driving acceleration. Parallel lines, squares, and hexagons are observed. With viscosity 100 cS, these stable standing-wave patterns do not exhibit strong sidewall effects, and are found in containers of various shapes including an irregular shape. A ‘quasi-pattern’ of twelvefold symmetry, analogous to a two-dimensional quasi-crystal, is observed for some even/odd frequency ratios. Many of the experimental phenomena can be modelled via cubic-order amplitude equations derived from symmetry arguments.

This work reports results of numerical simulations of viscous incompressible flow past a sphere. The primary objective is to identify transitions that occur with increasing Reynolds number, as well as their underlying physical mechanisms. The numerical method used is a mixed spectral element/Fourier spectral method developed for applications involving both Cartesian and cylindrical coordinates. In cylindrical coordinates, a formulation, based on special Jacobi-type polynomials, is used close to the axis of symmetry for the efficient treatment of the ‘pole’ problem. Spectral convergence and accuracy of the numerical formulation are verified. Many of the computations reported here were performed on parallel computers. It was found that the first transition of the flow past a sphere is a linear one and leads to a three-dimensional steady flow field with planar symmetry, i.e. it is of the ‘exchange of stability’ type, consistent with experimental observations on falling spheres and linear stability analysis results. The second transition leads to a single-frequency periodic flow with vortex shedding, which maintains the planar symmetry observed at lower Reynolds number. As the Reynolds number increases further, the planar symmetry is lost and the flow reaches a chaotic state. Small scales are first introduced in the flow by Kelvin–Helmholtz instability of the separating cylindrical shear layer; this shear layer instability is present even after the wake is rendered turbulent.

This paper describes the fluid mechanics of the natural ventilation of a space connected to a large body of stationary ambient fluid. The flows are driven by buoyancy differences between the interior and exterior fluids. Connections with the ambient fluid are high level and low level openings. Two main forms of ventilation are identified: mixing ventilation and displacement ventilation. Mixing ventilation occurs when the incoming ambient fluid mixes with the fluid within the space, as is, the case if dense fluid enters through a high level inlet. In this case vertical stratification is weak. Displacement ventilation occurs when dense fluid enters at low levels and displaces the lighter fluid within the space out through high level openings. A strong stable stratification develops in this case, and there is little mixing between the incoming fluid and that in the interior. Both of these modes of ventilation are studied theoretically and the results are compared with laboratory experiments. Transient draining flows which occur when a space initially contains fluid of a density different from the ambient are examined.

The presence of internal sources of buoyancy allows steady states to be established, and the effects of point, line and vertically distributed sources are studied. These steady states are extensions of filling box models, with the addition of continuous exchange of fluid with the environment outside the space. A major result of this work is that the form of the stratification within the space depends on the entrainment caused by the convective elements (plumes) produced by the buoyancy sources, but is independent of the strength of the sources. The strength of the stratification and the magnitudes of the velocities do, however, depend on the source strength. The effects of opening size(s) and configurations are determined, and criteria for producing a particular stratification within the space are established. Applications of this work to the ventilation of buildings are presented.

Three-dimensional travelling wave solutions are found for pressure-driven fluid flow through a circular pipe. They consist of three well-defined flow features – streamwise rolls and streaks which dominate and streamwise-dependent wavy structures. The travelling waves can be classified by the $m$-fold rotational symmetry they possess about the pipe axis with $m\,{=}\,1,2,3,4,5$ and $6$ solutions identified. All are born out of saddle-node bifurcations with the lowest corresponding to $m\,{=}\,3$ and traceable down to a Reynolds number (based on the mean velocity) of 1251. The new solutions are found using a constructive continuation procedure based upon key physical mechanisms thought generic to wall-bounded shear flows. It is believed that the appearance of these new alternative solutions to the governing equations as the Reynolds number is increased is a necessary precursor to the turbulent transition observed in experiments.

A high resolution, two component laser-Doppler anemometer has been used for turbulence measurements at a high data rate in a channel flow of water. Measurements of the velocity components in the stream direction and in a direction normal to the wall are reported over the Reynolds number range of 3000–40000. The combination of high spatial resolution and high data rates enabled accurate reconstruction of time dependent velocity traces. Long-time statistical averages of these signals clearly show that profiles of the dimensionless turbulence quantities such as turbulence intensities and Reynolds stress are strongly Reynolds-number dependent over a large part of the channel flow. For instance, in the Reynolds-number range of this investigation, it is shown that the fluctuating turbulence quantities do not scale with wall variables even as close as 15 viscous lengths from the wall. The velocity traces and associated power spectra exposed two phenomena which may explain the Reynolds number dependencies.

The growth, breakdown, and transition to turbulence of counter-rotating streamwise vortices, generated via a Görtler instability mechanism, was used to experimentally model the eddy structures found in transitional and turbulent flat-plate boundary layers. The naturally occurring vortices have been studied using smoke-wire visualization and multiple-probe hot-wire rakes. Results show that low-speed regions are formed between the vortices as low-momentum fluid is removed away from the wall. The low-speed regions grow in the normal direction faster than a nominally Blasius boundary layer and create strongly inflexional normal and spanwise profiles of the streamwise velocity component. Instability oscillations develop on these unstable profiles that scale with the local shear-layer thickness and velocity difference. Contrary to expectations however, the spatial scales of the temporal velocity fluctuations correlate better with the velocity gradient in the spanwise direction than with the normal velocity gradient. The nonlinear growth of the oscillations is quite rapid and breakdown into turbulence occurs within a short timescale.

Atmospheric dissociation will be appreciable in the neighbourhood of projectiles travelling at speeds greater than 2 km/sec. This introductory paper on possible effects of dissociation on the airflow, and hence on the drag, stability and aerodynamic heating of such projectiles, is intended mainly as a source of ideas for later, more comprehensive investigations.

Dispersion of solid particles in decaying isotropic turbulence is studied numerically. The three-dimensional, time-dependent velocity field of a homogeneous, non-stationary turbulence was computed using the method of direct numerical simulation (DNS). A numerical grid containing 963 points was sufficient to resolve the turbulent motion at the Kolmogorov lengthscale for a range of microscale Reynolds numbers starting from Rλ = 25 and decaying to Rλ = 16. The dispersion characteristics of three different solid particles (corn, copper and glass) injected in the flow, were obtained by integrating the complete equation of particle motion along the instantaneous trajectories of 223 particles for each particle type, and then performing ensemble averaging. The three different particles are those used by Snyder & Lumley (1971), referred to throughout the paper as SL, in their pioneering wind-tunnel experiment. Good agreement was achieved between our DNS results and the measured time development of the mean-square displacement of the particles.

The simulation results also include the time development of the mean-square relative velocity of the particles, the Lagrangian velocity autocorrelation and the turbulent diffusivity of the particles and fluid points. The Lagrangian velocity frequency spectra of the particles and their surrounding fluid, as well as the time development of all the forces acting on one particle are also presented. In order to distinguish between the effects of inertia and gravity on the dispersion statistics we compare the results of simulations made with and without the buoyancy force included in the particle motion equation. A summary of the significant results is provided in §7 of the paper.

The main objective of the paper is to enhance the understanding of the physics of particle dispersion in a simple turbulent flow by examining the simulation results described above and answering the questions of how and why the dispersion statistics of a solid particle differ from those of its corresponding fluid point and surrounding fluid and what influences inertia and gravity have on these statistics.

Existing experiments on the low-speed flow downstream of steps and fences, and some new measurements downstream of a backward-facing step, are used to demonstrate the complicated nature of the flow in the reattachment region and its effect on the slow non-monotonic return of the shear layer to the ordinary boundary-layer state. A key feature of the flow is found to be the splitting of the shear layer at reattachment, where part of the flow is deflected upstream into the recirculating flow region to supply the entrainment; the part of the flow that continues downstream suffers a pronounced decrease in eddy length scale, evidently because the larger eddies are torn in two. This phenomenon will occur in all cases where a shear layer reattaches after a prolonged region of separation, either at low speed or in supersonic flow. For simplicity, the discussion in the present paper is confined to low-speed flows.

The horseshoe vortex formed around the base of a cylinder by a separating laminar boundary layer has been investigated experimentally. Smoke flow visualization shows that both steady and unsteady vortex systems exist. Pressure distributions beneath both types of vortex system have been measured and the variation of the horseshoe vortex position on the plane of symmetry upstream of the cylinder has been determined. Unsteady horseshoe vortex systems are shown to have a complex oscillatory behaviour and the nature of this oscillatory behaviour is described. Using smoke flow visualization techniques some measurements have been made of the velocity distributions within horseshoe vortex systems.

Motivated by our interest in unsteady aerodynamics of insect flight, we devise a computational tool to solve the Navier–Stokes equation around a two-dimensional moving wing, which mimics biological locomotion. The focus of the present work is frequency selection in forward flapping flight. We investigate the time scales associated with the shedding of the trailing- and leading-edge vortices, as well as the corresponding time-dependent forces. We present a generic mechanism of the frequency selection as a result of unsteady aerodynamics.