The analysis of unsteady free convection has classically been made difficult because of the singularities which occur in the governing boundary-layer equations, and because anomalies often occur which are related to the occurrence of these singularities. In the present paper a semisimilar analysis of unsteady free convection in the vicinity of a vertical flat plate is presented, wherein a number of possible wall-temperature variations with time and position are derived. Unique scalings are formulated for the semisimilar equations that aid in the numerical solutions and in the physical interpretation of the results. These scalings collapse the infinite time and position coordinates into a finite region, and present the semisimilar problem in a format bounded by similarity equations. Solutions are carried out which indicate the occurrence of overshoots in the temperature profiles and heat transfer for a variety of conditions. Also, concepts such as the ‘limit-of-pure-conduction’ and ‘leading-edge penetration distance’ are shown to require special interpretation under variable wall-temperature conditions.

The breaking of periodic progressive two-layer interfacial waves at a Gaussian ridge is investigated through laboratory experiments. Length scales of the incident wave and topography are used to parameterize when and how breaking occurs. Qualitative observations suggest both shear and convection play a role in the instability of waves breaking at the ridge. Simultaneous particle image velocimetry (PIV) and planar laser-induced fluorescence (PLIF) measurements are used to calculate high resolution, two-dimensional velocity and density fields from which the local gradient Richardson number Rig is calculated. The transition to breaking occurred when 0.2 ≤ Rig ≤ 0.4. In these wave-ridge breaking events, the destabilizing effects of waves steepening in shallow layers may be responsible for breaking at higher Rig than for similar waves breaking through shear instability in deep water (Troy & Koseff, J. Fluid Mech., vol. 543, 2005b, p. 107). Due to the effects of unsteadiness, nonlinear shoaling and flow separation, the canonical Rig > 0.25 is not sufficient to predict the stability of interfacial waves. A simple model is developed to estimate Rig in waves between finite depth layers using scales of the incident wave scale and topography. The observed breaking transition corresponds with a constant estimated value of Rig from the model, suggesting that interfacial shear plays an important role in initial wave instability. For wave amplitudes above the initial breaking transition, convective breaking events are also observed.

The subgrid-scale (SGS) stress in the atmospheric surface layer is studied using measurement data. Field measurements employing a novel array technique were conducted to provide data for obtaining resolvable- and subgrid-scale variables. We analyse the conditional SGS stress and the conditional stress production rate conditional on the resolvable-scale velocity, which must be reproduced by the SGS model for large-eddy simulation (LES) to predict correctly the one-point resolvable-scale velocity statistics. The results show that both buoyancy and shear play important roles in the physics of the SGS stress. Strong buoyancy and vertical shear associated with updrafts and positive streamwise velocity fluctuations cause conditional forward energy transfer and strong anisotropy in the conditional SGS stress. Downward returning flows associated with large convective eddies result in conditional energy backscatter and much less anisotropic SGS stress. Predictions of the conditional SGS stress and the conditional stress production rate predicted using several SGS models are compared with measurements. None of those models are able to predict correctly the trends of both statistics. The Smagorinsky and one nonlinear model under-predict the anisotropy and the variations of the anisotropy, whereas the other nonlinear model and the mixed model over-predict both. The deficiencies of the SGS models that cause inaccurate LES statistics, such as the over-prediction of the mean shear and under-prediction of the vertical velocity skewness, are identified. The present study shows that analyses of conditional SGS stress and conditional SGS stress production provide a systematic approach for studying SGS physics and evaluating SGS models and can potentially be used to target specific aspects of LES that are important for a given application.

The linear stability of three-dimensional vortices in rotating, stratified flows has been studied by analysing the non-hydrostatic inviscid Boussinesq equations. We have focused on a widely used model of geophysical and astrophysical vortices, which assumes an axisymmetric Gaussian structure for pressure anomalies in the horizontal and vertical directions. For a range of Rossby numbers ($-0.5<Ro<0.5$) and Burger numbers ($0.02<Bu<2.3$) relevant to observed long-lived vortices, the growth rate and spatial structure of the most unstable eigenmodes have been numerically calculated and presented as a function of $Ro{-}Bu$. We have found neutrally stable vortices only over a small region of the $Ro{-}Bu$ parameter space: cyclones with $Ro\sim 0.02{-}0.05$ and $Bu\sim 0.85{-}0.95$. However, we have also found that anticyclones in general have slower growth rates compared to cyclones. In particular, the growth rate of the most unstable eigenmode for anticyclones in a large region of the parameter space (e.g. $Ro<0$ and $0.5\lesssim Bu\lesssim 1.3$) is slower than 50 turnaround times of the vortex (which often corresponds to several years for ocean eddies). For cyclones, the region with such slow growth rates is confined to $0<Ro<0.1$ and $0.5\lesssim Bu\lesssim 1.3$. While most calculations have been done for $f/\bar{N}=0.1$ (where $f$ and $\bar{N}$ are the Coriolis and background Brunt–Väisälä frequencies), we have numerically verified and explained analytically, using non-dimensionalized equations, the insensitivity of the results to reducing $f/\bar{N}$ to the more ocean-relevant value of 0.01. The results of our stability analysis of Gaussian vortices both support and contradict the findings of earlier studies with QG or multilayer models or with other families of vortices. The results of this paper provide a stepping stone to study the more complicated problems of the stability of geophysical (e.g. those in the atmospheres of giant planets) and astrophysical vortices (in accretion disks).

This paper studies the large-scale low-frequency variability of the wind-driven midlatitude ocean gyres and their western boundary currents, such as the Gulf Stream or Kuroshio, simulated with the eddy-resolving quasi-geostrophic model. We applied empirical orthogonal functions analysis to turbulent flow solutions and statistically extracted robust and significant large-scale decadal variability modes concentrated around the eastward jet extension of the western boundary currents. In order to interpret these statistical modes dynamically, we linearized the governing quasi-geostrophic equations around the time-mean circulation and solved for the corresponding full set of linear eigenmodes with their eigenfrequencies. We then projected the extracted decadal variability on the eigenmodes and found that this variability is a multimodal coherent pattern phenomenon rather than a single mode or a combination of several modes as in the flow regimes preceding developed turbulence.

We determine optimal inflow boundary perturbations to steady flow through a straight inflexible tube with a smooth axisymmetric stenosis at a bulk-flow Reynolds number , for which the flow is asymptotically stable. The perturbations computed produce an optimal gain, i.e. kinetic energy in the domain at a given time horizon normalized by a measure of time-integrated energy on the inflow boundary segment. We demonstrate that similarly to the optimal initial condition problem, the gain can be interpreted as the leading singular value of the forward linearized operator that evolves the boundary conditions to the final state at a fixed time. In this investigation we restrict our attention to problems where the temporal profile of the perturbations examined is a product of a Gaussian bell and a sinusoid, whose frequency is selected to excite axial wavelengths similar to those of the optimal initial perturbations in the same geometry. Comparison of the final state induced by the optimal boundary perturbation with that induced by the optimal initial condition demonstrates a close agreement for the selected problem. Previous works dealing with optimal boundary perturbation considered a prescribed spatial structure and computed an optimal temporal variation of a wall-normal velocity component, whereas in this paper we consider the problem of a prescribed temporal structure and compute the optimal spatial variation of velocity boundary conditions over a one-dimensional inflow boundary segment. The methodology is capable of optimizing boundary perturbations in general non-parallel two- and three-dimensional flows.

We use LVDSMC (low-variance deviational Monte Carlo) simulations to calculate, under linearized conditions, the second-order temperature jump coefficient for a dilute gas whose temperature is governed by the Poisson equation with a constant forcing term, as in the case of homogeneous volumetric heating. Both the hard-sphere gas and the BGK model of the Boltzmann equation, for which slip/jump coefficients are not functions of temperature, are considered. The temperature jump relation and jump coefficient determined here are closely linked to the general jump relations for time-dependent problems that have yet to be systematically treated in the literature; as a result, they are different from those corresponding to the well-known linear and steady case where the temperature is governed by the homogeneous heat conduction (Laplace) equation.

Two Lagrangian-mean measures crucial to the accurate estimation of mean particle velocities in wavy or turbulent shear flows are considered. The measures are the pseudomomentum and generalized Stokes drift and of particular interest is their expression in terms of quantities directly measurable by fixed instruments. To proceed, the measures are first calculated for broad spectra of progressive symmetric rotational wave pairs of small amplitude. Both discrete and continuous spectra are considered and the waves may grow or decay. The expressions are then cast into a form composed of quantities that are measurable in a fixed reference frame, such as the surface slope spectrum of surface gravity waves or space–time velocity correlations in the interior of wavy shear flows. Finally, an example is given in which the measures are calculated for a plane channel flow subject to a broad spectrum of discrete progressive waves, specifically a numerical simulation of turbulent channel flow. It is seen that while the streamwise component of pseudomomentum is everywhere negative in the flow, the generalized Stokes drift changes sign, giving rise to an enhanced mass transport close to the boundary and a reduction in transport some distance from it. The sign change occurs 12.5 viscous units from the wall, near the centre of a 15 viscous units thick highly sheared layer of Stokes drift.

Two similar problems are considered: what is the effect of applying a uniform and constant rate of strain (i) to the two-dimensional thermal mixing region in a homogeneous grid-generated turbulent field, and (ii) to the two-dimensional velocity mixing region formed between two uniform streams moving with different mean velocities? The imposed strain field is orientated so as to compress or separate the isothermal and isokinetic surfaces in the plane of interest.

Two theoretical models are presented; in the first, the profiles of temperature and velocity are assumed to be self-preserving and an assumption is made about the velocity scales; in the second, the statistical, rapid-distortion approach to dispersion due to Hunt & Mulhearn (1973) is applied. The circumstances in which these models differ and those where the simpler self-preserving model can be applied are determined. The measurements presented here indicate that the widths of both mixing layers decrease within the strain field, the width of the thermal mixing layer decreasing at a greater rate than that of the velocity mixing layer. However, the measured length scales were found to be 5% larger than the scales predicted by either of the analyses, which differed from each other by 5%. It is suggested that selective amplification of the energy-containing eddies by the strain field is responsible.

The aeroacoustic sound generated in a flow past two cylinders, one of which is oscillating and the other is fixed, is studied by direct numerical simulation. This problem involves key ingredients of the aeroacoustic noise generated from wind turbines, helicopters, axial flow fans and other turbomachinery: flow, a moving body and a fixed body. The corrected volume penalization method is successfully applied to resolve the sound pressure of aeroacoustic waves as a solution of the compressible Navier–Stokes equations. The sound pressure was shown to be in good agreement with the prediction by the Ffowcs Williams–Hawkings aeroacoustic analogy, which takes account of the cylinder motion, confirming the accuracy of the corrected volume penalization method. Prior to the case of two cylinders, sound generation in flow past a single oscillating cylinder is considered. The fluid motion can be either periodic or non-periodic depending on the frequency and the amplitude of cylinder oscillation. The acoustic power is significantly reduced when the fluid motion locks in to a frequency lower than the natural frequency of vortex shedding from a fixed cylinder. When a fixed cylinder is added, the acoustic power depends strongly on the distance between the cylinders, since that determines whether synchronization occurs and the phase difference between the three forces: the lift forces exerted on the two cylinders and the inertial force due to volume displacement effect of the oscillating cylinder. In particular, significant sound reduction is observed when the fixed cylinder is placed upstream and the frequency of the cylinder oscillation is set to the frequency for which the acoustic power is minimized in the single-cylinder case.

A three-dimensional linear stability analysis has been carried out to understand the origin of vortices and related density patterns in bounded uniform-shear flow of granular materials, using a kinetic-theory constitutive model. This flow is found to be unstable to pure spanwise stationary perturbations ($k_z\,{\neq}\, 0$, $k_x\,{=}\,0$ and $\partial/\partial y(.)\,{=}\,0$, where $k_i$ is the wavenumber for the $i$th direction) if the solid fraction is below some critical value $\nu\,{<}\, \nu_{3D}$. The growth rates of these spanwise instabilities are an order of magnitude larger than those of the two-dimensional ($k_z\,{=}\,0$) streamwise-independent ($k_x\,{=}\,0$) instabilities that occur if the solid fraction is above some critical value $\nu\,{>}\,\nu_{2D}$ (${>}\nu_{3D}$). The spanwise instabilities give birth to new three-dimensional travelling wave instabilities at non-zero values of the streamwise wavenumber ($k_x\,{\neq}\, 0$) in dilute flows ($\nu \,{<}\, \nu_{3D}$). For moderate-to-large densities with $k_x\,{\neq}\, 0$, there are additional three-dimensional instability modes in the form of both stationary and travelling waves, whose origin is tied to the corresponding two-dimensional instabilities.

While the two-dimensional streamwise-independent modes lead to the formation of stationary streamwise vortices for moderately dense flows ($\nu\,{>}\,\nu_{2D}$), the pure spanwise modes are responsible for the origin of such vortices in the dilute limit ($\nu\,{<}\,\nu_{3D}$). For more general kinds of perturbations ($k_x\,{\neq}\, 0$ and $k_z\,{\neq}\, 0$), ‘modulated’ streamwise vortices are born which could be either stationary or travelling depending on control parameters. The rolling motion of vortices will lead to a major redistribution of the streamwise velocity and hence such vortices can act as potential progenitors for the mixing of particles. The effect of non-zero wall slip has been investigated, and it is shown that some dilute-flow instabilities can disappear with the inclusion of the wall slip. Even though the streamwise granular vortices have similarities to the well-known stationary Taylor–Couette vortices (which are ‘hydrodynamic’ in origin), their origin is, however, tied to ‘constitutive’ instabilities, and hence they belong to a different class.

We present a combined experimental and numerical study of an idealized model of the propulsive stroke of the turning manoeuvre in fish. Specifically, we use the framework of Lagrangian coherent structures (LCSs) to describe the kinematics of the flow that results from a thin plate performing a large angle rotation about its tip in still fluid. Temporally and spatially well-resolved velocity fields are obtained using a two-dimensional, incompressible finite-volume solver, and are validated by comparisons with experimentally measured velocity fields and alternate numerical simulations. We then implement the recently proposed variational theory of LCSs to extract the hyperbolic and elliptic LCSs in the numerically generated velocity fields. Detailed LCS analysis is performed for a plate motion profile described by $\dot{\unicode[STIX]{x1D703}}(t)=\unicode[STIX]{x1D6FA}_{max}\sin ^{2}(\unicode[STIX]{x1D714}t)$ during $0\leqslant t\leqslant t_{o}$ and zero otherwise. The stopping time $t_{o}$ is given by $t_{o}=\unicode[STIX]{x03C0}/\unicode[STIX]{x1D714}=10~\text{s}$, the value of $\unicode[STIX]{x1D6FA}_{max}$ chosen to give a stopping angle of $\unicode[STIX]{x1D703}_{max}=90^{\circ }$, resulting in a Reynolds number $Re=c^{2}\unicode[STIX]{x1D6FA}_{max}/\unicode[STIX]{x1D708}=785.4$, where $c$ is the plate chord length and $\unicode[STIX]{x1D708}=10^{-6}~\text{m}^{2}~\text{s}^{-1}$ the kinematic viscosity of water. The flow comprises a starting and a stopping vortex, resulting in a pair of oppositely signed vortices of unequal strengths that move away from the plate in a direction closely aligned with the final plate orientation at $t/t_{o}\approx 2$. The hyperbolic LCSs are shown to encompass the fluid material that is advected away from the plate for $t>t_{o}$, henceforth referred to as the advected bulk. The starting and stopping vortices, identified using elliptic LCSs and hence more objective than Eulerian vortex detection methods, constitute only around two thirds of the advected bulk area. The advected bulk is traced back to $t=0$ to identify five distinct lobes of fluid that eventually form the advected bulk, and hence map the long-term fate of various regions in the fluid at $t=0$. The five different lobes of fluid are then shown to be delineated by repelling LCS boundaries at $t=0$. The linear momentum of the advected bulk region is shown to account for approximately half of the total impulse experienced by the plate in the direction of its final orientation, thus establishing its dynamical significance. We provide direct experimental evidence for the kinematic relevance of hyperbolic and elliptic LCSs using novel dye visualization experiments, and also show that attracting hyperbolic LCSs provide objective characterization of the spiral structures often observed in vortical flows. We conclude by showing that qualitatively similar LCSs persist for several other plate motion profiles and stopping angles as well.

Slow droplet motion on chemically heterogeneous substrates is considered analytically and numerically. We adopt the long-wave approximation which yields a single partial differential equation for the droplet height in time and space. A matched asymptotic analysis in the limit of nearly circular contact lines and vanishingly small slip lengths yields a reduced model consisting of a set of ordinary differential equations for the evolution of the Fourier harmonics of the contact line. The analytical predictions are found, within the domain of their validity, to be in good agreement with the solutions to the governing partial differential equation. The limitations of the reduced model when the contact line undergoes stronger deformations are partially lifted by proposing a hybrid scheme which couples the results of the asymptotic analysis with the boundary integral method. This approach improves the agreement with the governing partial differential equation, but at a computational cost which is significantly lower compared to that required for the full problem.

An experiment was performed to measure near-wall velocity and Reynolds stress profiles ina pressure-driven three-dimensional turbulent boundary layer. An initially two-dimensionalboundary layer (Reθ≈4000) was exposed to a strong spanwisepressure gradient. At the furthest downstream measurement locations there was also a fairlystrong favourable streamwise pressure gradient.

Measurements were made using a specially designed near-wall laser-Doppler anemometer(LDA), in addition to conventional methods. The LDA used short focal length optics, a mirrorprobe suspended in the flow, and side-scatter collection to achieve a measuring volume 35 μmin diameter and approximately 65 μm long.

The data presented include mean velocity measurements and Reynolds stresses, all extendingwell below y+=10, at several profile locations. Terms of theturbulent kinetic energy transport equation are presented at two profile locations. The meanflow is nearly collateral (i.e. W is proportional to U) atthe wall. Turbulent kinetic energy is mildly suppressed in the near-wall region and theshear stress components are strongly affected by three-dimensionality. As a result, theratio of shear stress to turbulent kinetic energy is suppressed throughout most of theboundary layer. The angles of stress and strain are misaligned, except very near the wall(around y+=10) where the angles nearly coincide with the meanflow angle. Three-dimensionality appears to mildly reduce the production of turbulentkinetic energy.

Fluidized beds are widely used in industry for carrying out a variety of chemical reactions and physical processes. Applications are frequently impeded by a lack of fundamental understanding of the mechanical behaviour of fluidized beds. Despite intensive experimental and theoretical study over the last four decades, there are still many aspects of fluidized beds and related fluid—particle systems that remain obscure. Further work is needed to understand interactions between the particles, influence of particle physical properties, development of non-obtrusive experimental techniques, and study of high-velocity beds and of novel-geometry beds in which particulate solids interact with interstitial fluid.

An international symposium was held at Stanford University on 1–4 July 1991 to discuss recent developments and the current state of knowledge and understanding of the mechanical behaviour of fluidized beds and related fluid—particle systems. The symposium was sponsored by the International Union of Theoretical and Applied Mechanics, and co-funded by the US Department of Energy, National Science Foundation and Electric Power Research Institute. The symposium was attended by 58 specialists representing academic institutions, industry and government research organizations in 11 countries. The diversity of background, coupled with differences in approach, ranging from purely theoretical to fully experimental, led to interesting exchanges where participants were often groping to understand the viewpoint of those involved. The result was frequently rewarding, occasionally perplexing, but certainly stimulating of thought and encouraging for further meetings of this nature.

The scientific committee for the Symposium were G. K. Batchelor (Cambridge University), J. J. H. Brouwers (Trent University), J. Gibilaro (University College London), J. R. Grace (University of British Columbia), G. M. Homsy (Stanford University) (Chairman), R. Jackson (Princeton University), R. I. Nigmatulin (Moscow University) and W. Schneider (Techische Universität Wien).

Each session began with an invited talk for one hour. This was followed by a series of 20-minute presentations. Participants were able at the end of each session to give brief (5 minute) unscheduled mini-presentations. Except for the latter, abstracts were submitted for the presentations, compiled by the organizers and distributed to participants. The meeting also included an informal workshop and a series of video and cinephotographic presentations not reported here. No formal proceedings of the meeting are being published; instead, this report is intended to summarize key findings and areas of discussion.

The titles of all formal presentations are identified with asterisks in the list of References. The report is organized around common themes, and the sequence of presentations may not be that in which they were given. The authors have endeavoured to interpret and extract the essence of the presentations, and apologize if any of the presentations have been misinterpreted or overlooked.

We study high-Reynolds-number flow in a two-dimensional collapsible channel in the asymptotic limit of wall deformations confined to the viscous boundary layer. The system is modelled using interactive boundary-layer equations for a Newtonian incompressible fluid coupled to the freely moving elastic wall under constant tension and external pressure. The deformation of the membrane is assumed to have small amplitude and long wavelength, whereas the flow comprises the inviscid core and the viscous boundary layers on both walls coupled to each other and to the membrane deformation. Firstly, by linking the interactive boundary-layer model to the small-amplitude, long-wavelength inviscid analysis, we conclude that the model is valid only when the pressure perturbations are fixed downstream from the wall indentation, contrary to the common assumption of classical boundary-layer theory. Next we explore possible steady states of the system, showing that a unique steady solution exists when the pressure is fixed precisely at the downstream end of the membrane, but there are multiple states possible if the pressure is specified further downstream. We examine the stability of these states by solving the generalized eigenvalue problem for perturbations to the nonlinear steady solutions and also by performing time integration of the full boundary-layer equations. Surprisingly, we find that no self-excited oscillations develop in the collapsible channel systems with finite-amplitude deformations. Instead, for each point in the parameter space, with the exception of points subject to numerical instabilities associated with the boundary-layer equations, exactly one of the steady states is predicted to be stable. We discuss these findings in relation to the results reported previously in the literature.

We study how heat transport is affected by finitely extensible polymers in a laminar boundary layer flow within the framework of the Prandtl–Blasius–Pohlhausen theory. The polymers are described by the finitely extensible nonlinear elastic-Peterlin model with a parameter $b^{2}$, which is the ratio of the maximum to the equilibrium value of the trace of the polymer conformation tensor. For very large $b^{2}$, heat transport is reduced. When $b^{2}$ is small, heat transport is enhanced. We investigate the transition from heat reduction to heat enhancement as a function of the polymer relaxation time and concentration, and show that the transition can be explained in terms of the functional shape of the space-dependent effective viscosity due to the polymers.