We consider the motion of a single quasi-geostrophic ellipsoid of uniform potential vorticity in equilibrium with a linear background shear flow. This motion depends on four parameters: the height-to-width aspect ratio of the vortex, $h/r$, and three parameters characterizing the background shear flow, namely the strain rate, $\gamma$, the ratio of the background rotation rate to the strain, $\beta$, and the angle from which the shear is applied, $\theta$. We generate the equilibria over a large range of these parameters and analyse their linear stability. For the second-order ($m\,{=}\,2$) modes which preserve the ellipsoidal form, we are able to derive equations for the eigenmodes and growth rates. For the higher-order modes we use a numerical method to determine the full linear stability to general disturbances ($m\,{>}\,2$).

Overall we find that the equilibria are stable over most of the parameter space considered, and where instability does occur the marginal instability is usually ellipsoidal. From these results, we determine the parameter values for which the vortex is most stable, and conjecture that these are the vortex characteristics which would be the most commonly observed in turbulent flows.

The changes in the evolution of the spatial and temporal distribution of the wall shear stresses (WSS) and gradients of wall shear stresses (GWSS) at different stages of the enlargement of an abdominal aortic aneurysm (AAA) are important in understanding the aetiology and progression of this vascular disease since they affect the wall structural integrity, primarily via the changes induced on the shape, functions and metabolism of the endothelial cells. Particle image velocimetry (PIV) measurements were performed in in vitro aneurysm models, while changing their geometric parameters systematically. It has been shown that, even at the very early stages of the disease, i.e. increase in the diameter $\leq\,$ 50%, the flow separates from the wall and a large vortex ring, usually followed by internal shear layers, is created. These lead to the generation of WSS that drastically differ in mean and fluctuating components from the healthy vessel. Inside the AAA, the mean WSS becomes negative along most of the aneurysmal wall and the magnitude of the WSS can be as low as 26% of the value in a healthy abdominal aorta.

Two regions with distinct patterns of WSS were identified inside the AAA: the proximal region of flow detachment, characterized by oscillatory WSS of very low mean, and the region of flow reattachment, located distally, where large, negative WSS and sustained GWSS are produced as a result of the impact of the vortex ring on the wall.

Comparison of the measured values of WSS and GWSS to an analytical solution, calculated for slowly expanding aneurysms shows a very good agreement, thus providing a validation of the PIV measurements.

Stereoscopic particle image velocimetry (PIV) measurements were taken simultaneously in streamwise–spanwise and streamwise–wall-normal planes in a zero-pressure-gradient turbulent boundary layer over a flat plate. Polarization techniques were employed to allow PIV to be taken in both planes simultaneously. Image preprocessing techniques were used to improve the quality of data near the line of intersection of the planes. Linear stochastic estimation was performed on these data, revealing the streamwise, spanwise, and wall-normal extent of swirl events primarily near the top of the log region of the boundary layer. Swirl events with rotation consistent with the mean vorticity are found to have a large footprint in the lower limit of the log region whereas swirls with opposite-signed vorticity are found to have little influence lower in the boundary layer. These long-time-averaged statistics contain features that are consistent with the hairpin packet model (or its kinematic equivalent). This model also seems to provide a reasonable description of many instantaneous events involving large-scale coherence, where long regions of streamwise momentum deficit are surrounded by vortex cores.

Building on our recent work on induced-charge electro-osmosis (ICEO) and electrophoresis (ICEP), as well as the Russian literature on spherical metal colloids, we examine the rich consequences of broken geometric and field symmetries upon the ICEO flow around conducting bodies. Through a variety of paradigmatic examples involving ideally polarizable (e.g. metal) bodies with thin double layers in weak fields, we demonstrate that spatial asymmetry generally leads to a net pumping of fluid past the body by ICEO, or, in the case of a freely suspended colloidal particle, translation and/or rotation by ICEP. We have chosen model systems that are simple enough to admit analysis, yet which contain the most important broken symmetries. Specifically, we consider (i) symmetrically shaped bodies with inhomogeneous surface properties, (ii) ‘nearly symmetric’ shapes (using a boundary perturbation scheme), (iii) highly asymmetric bodies composed of two symmetric bodies tethered together, (iv) symmetric conductors in electric-field gradients, and (v) arbitrarily shaped conductors in general non-uniform fields in two dimensions (using complex analysis). In non-uniform fields, ICEO flow and ICEP motion exist in addition to the more familiar dielectrophoretic forces and torques on the bodies (which also vary with the square of the electric field). We treat all of these problems in two and three dimensions, so our study has relevence for both colloids and microfluidics. In the colloidal context, we describe principles to ‘design’ polarizable particles which rotate to orient themselves and translate steadily in a desired direction in a DC or AC electric field. We also describe ‘ICEO spinners’ that rotate continuously in AC fields of arbitrary direction, although we show that ‘near spheres’ with small helical perturbations do not rotate, to leading order in the shape perturbation. In the microfluidic context, strong and steady flows can be driven by small AC potentials applied to systems containing asymmetric structures, which holds promise for portable or implantable self-powered devices. These results build upon and generalize recent studies in AC electro-osmosis (ACEO). Unlike ACEO, however, the inducing surfaces in ICEO can be physically distinct from the driving electrodes, increasing the frequency range and geometries available.

The problem of a cylindrically or spherically imploding and reflecting shock wave in a flow initially at rest is studied without the use of the strong-shock approximation. Dimensional arguments are first used to show that this flow admits a general solution where an infinitesimally weak shock from infinity strengthens as it converges towards the origin. For a perfect-gas equation of state, this solution depends only on the dimensionality of the flow and on the ratio of specific heats. The Guderley power-law result can then be interpreted as the leading-order, strong-shock approximation, valid near the origin at the implosion centre. We improve the Guderley solution by adding two further terms in the series expansion solution for both the incoming and the reflected shock waves. A series expansion, valid where the shock is still weak and very far from the origin, is also constructed. With an appropriate change of variables and using the exact shock-jump conditions, a numerical, characteristics-based solution is obtained describing the general shock motion from almost infinity to very close to the reflection point. Comparisons are made between the series expansions, the characteristics solution, and the results obtained using an Euler solver. These show that the addition of two terms to the Guderley solution significantly extends the range of validity of the strong-shock series expansion.

We study the wake of a cylinder performing rotary oscillations around its axis at moderate Reynolds number. We observe that the structure of the vortex shedding is strongly affected by the forcing parameters. The forced wake is characterized by a ‘lock-in’ region where the vortices are shed at the forcing frequency and a region where the vortices can be reorganized to give a second frequency close to those observed for the unforced wake. We show that these modifications of the wake structure change the dynamic of the fluctuations downstream from the cylinder. We vary the amplitude and the frequency of the oscillations and study the consequences of these modifications on the mean flow and the global drag applied on the cylinder. We then discuss the mechanism responsible for the modification of the fluctuations and the modification of the drag coefficient.

Multiple states of spatially localized steady convection are found in numerical simulations of water–ethanol mixtures in two dimensions. Realistic boundary conditions at the top and bottom are used, with periodic boundary conditions in the horizontal. The states form by a mechanism similar to the pinning region around a Maxwell point in variational systems, but are located in a parameter regime in which the conduction state is overstable. Despite this the localized states can be stable. The properties of the localized states are described in detail, and the mechanism of their destruction with increasing or decreasing Rayleigh number is elucidated. When the Rayleigh number becomes too large the fronts bounding the state at either end unpin and move apart, allowing steady convection to invade the domain. In contrast, when the Rayleigh number is too small the fronts move inwards, and eliminate the localized state which decays into dispersive chaos. Out of this state spatially localized states re-emerge at irregular times before decaying again. Thus an interval of Rayleigh numbers exists that is characterized by relaxation oscillations between localized convection and dispersive chaos.

An upper bound on the heat flux for infinite-Prandtl-number convection between two parallel plates is determined for the cases of no-slip and free-slip boundary conditions. For no-slip the large-Rayleigh-number ($\hbox{\it Ra}$) scaling for the Nusselt number is consistent with $\hbox{\it Nu}\,{<}\,c\, \hbox{\it Ra}^{1/3}$, as predicted by Chan (1971). However, his commonly accepted picture of an infinite hierarchy of multiple boundary layer solutions smoothly approaching this scaling is incorrect. Instead, we find a novel terminating sequence in which the optimal asymptotic scaling is achieved with a three-boundary-layer solution. In the case of free-slip, we find an asymptotic scaling of $\hbox{\it Nu}\,{<}\,c\, \hbox{\it Ra}^{5/12}$, corroborating the conservative estimate obtained in Plasting & Ierley (2005). Here the infinite hierarchy of multiple-boundary-layer solutions obtains, albeit with anomalous features not previously encountered. Thus for neither boundary condition does the optimal solution conform to the well-established models of finite-Prandtl-number convection (Busse 1969 b), plane Couette flow, and plane or circular Poiseuille flow (Busse 1970). We reconcile these findings with a suitable continuation from no-slip to free-slip, discovering that the key distinction – finite versus geometric saturation – is entirely determined by the singularity, or not, of the initial, single-boundary-layer, solution. It is proposed that this selection principle applies to all upper bound problems.

For the infinite-Prandtl-number limit of the Boussinesq equations, the enhancement of vertical heat transport in Rayleigh–Bénard convection, the Nusselt number $\hbox{\it Nu}$, is bounded above in terms of the Rayleigh number $\hbox{\it Ra}$ according to $\hbox{\it Nu}\,{\le}\,0.644 \,{\times}\hbox{\it Ra}^{{1}/{3}} [\log{\hbox{\it Ra}}]^{{1}/{3}}$ as $\hbox{\it Ra}\,{\rightarrow}\,\infty$. This result follows from the utilization of a novel logarithmic profile in the background method for producing bounds on bulk transport, together with new estimates for the bi-Laplacian in a weighted $L^{2}$ space. It is a quantitative improvement of the best currently available analytic result, and it comes within the logarithmic factor of the pure 1/3 scaling anticipated by both the classical marginally stable boundary layer argument and the most recent high-resolution numerical computations of the optimal bound on $\hbox{\it Nu}$ using the background method.

Self-similar analytical nonlinear solutions to the hydrostatic Boussinesq equations are derived which describe unbalanced inertial pulsations of anticyclonic lens-like circular vortices in stably stratified rotating fluid. Any steady axisymmetric solution for a finite-volume anticyclonic vortex in the reduced-gravity approximation is shown to correspond to a set of time-periodic solutions with the amplitude of pulsations being within a range limited by the intensity of the stationary vortex. These solutions represent an extension of previous reduced-gravity analytical pulson solutions of particular forms with spatially uniform divergence of horizontal velocity oscillating in time within the vortex volume. In the self-similar form the pulson solution describes the expansion and contraction of a vortex which maintains the same spatial structure in the Lagrangian coordinates.

The interaction of acoustic waves with a planar counterflow flame is investigated numerically employing a detailed kinetic model and one-step global kinetic models. The mathematical formulation of quasi-one-dimensional fully unsteady laminar counterflow flames is presented and the governing equations are integrated numerically based on a MacCormack predictor–corrector scheme with second-order accuracy in space. Navier–Stokes characteristic boundary conditions are implemented to accurately represent perfect and partial reflection of acoustic waves at the boundaries. For well-resolved simulations, the occurrence of self-excited flame–acoustics instabilities is analysed in both non-premixed and premixed flames for a range of flow strain rates and flame locations, and employing two finite-rate kinetic models. Unlike the detailed kinetic model, one-step global models with large activation energy and overall reaction order greater than unity promote the amplification of acoustic pressure fluctuations in counterflow non-premixed flames. In contrast, premixed counterflow flames exhibit flame–acoustics instabilities with both kinetic models. While previous unsteady counterflow studies required external perturbations, the resonant unsteady phenomena predicted in this study are self-sustained under favourable boundary conditions. Detailed analyses of the characteristic time scales associated with convection, diffusion, chemistry and acoustics are presented to provide a better understanding of the exact coupling mechanisms.

In this paper we extend the familiar concept of spatial instability and growth of disturbances in the downstream direction to include spatial instability and growth in the wall-normal direction. The stability theory of boundary layers has generally been concerned with determining the evolution of disturbances inside a boundary layer (this is where disturbances have their largest amplitudes and can cause a laminar boundary layer to become turbulent). Outside a boundary layer, where the basic flow is uniform, normal-mode disturbances decay exponentially with distance from the wall to satisfy homogeneous boundary conditions. In this paper we present a surprising scenario where an impulsive disturbance, made up of a superposition of these normal modes, nonetheless grows exponentially with distance from the wall. While the usual convective instability with exponential growth in the downstream direction can be efficiently characterized by spatial modes with complex wavenumbers, the new convective instability can be efficiently characterized by modes with exponentially diverging ‘eigenfunctions’ obtained by moving certain branch-cuts in the complex wavenumber plane. The new instability is therefore associated with an interaction between the discrete spectrum and the continuous spectrum. We emphasize, however, that the homogeneous boundary conditions are always satisfied, and that at any finite time exponential growth only occurs over a finite distance from the wall, but this distance increases linearly with time. Interactions between poles and branch-cuts have been found before, but the results presented here provide a physical interpretation for this spectral behaviour. A further curiosity is that some of these divergent modes have been found to violate Howard's semi-circle theorem.

A small air bubble (radius $a$) is injected in water (kinematic viscosity $\nu$) in the vicinity (distance $r_0$) of a propeller (radius $r_p$, angular frequency $\omega$). We study experimentally and theoretically the conditions under which the bubble can be ‘captured’, i.e. deviated from its vertical trajectory (imposed by gravity $g$) and moved toward the centre of the propeller ($r\,{=}\,0$). We show that the capture frequency $\omega_{\hbox{\scriptsize\it capt}}$ follows the relationship \[\omega_{\hbox{\scriptsize\it capt}}=\left(\frac{2ga^2}{9\beta\nu r_p f(\hbox{\it Re}_b)}\right)\left(\frac{r_0}{r_p}\right)^2(1+\cos\varphi_0),\] where $\beta$ is a dimensionless parameter characterizing the propeller, $f(\hbox{\it Re}_b)$ is an empirical correction to Stokes' drag law which accounts for finite-Reynolds-number effects and $\pi/2-\varphi_0$ is the angle between the axis of the propeller and the line between the centre of the propeller and the point where the bubble is injected. This law is found to be valid as long as the distance $d$ between the propeller and the water surface is larger than $3r_0$. For smaller distances, the capture frequency increases; using an image technique, we show how the above expression is modified by the presence of the surface.

We consider in detail the Taylor–Couette problem for a Bingham fluid, presenting a range of analytical and computational results. First, for co-rotating cylinders it is known that the critical inner cylinder Reynolds number $\hbox{\it Re}_{1,c}$, does not increase monotonically with the Bingham number $B$, over a range of small to moderate $B$. It is the only situation that we know of where a yield stress fluid flow is less stable that the corresponding Newtonian fluid flow. This effect was discovered independently by Landry (2003, MSc thesis) and Peng & Zhu (J. Fluid Mech. vol. 512, 2004, p. 21), but the mechanism has not been explained. Here we show that the decrease in critical Reynolds number is due to an increase (at small $B$) in the rate of strain of the basic flow, which amplifies the transfer of energy from the basic flow to the perturbation, via the inertial terms in the energy equation. At larger $B$, the yielded region contracts and the inertial energy transfer is bounded by the yield stress dissipation.

We next consider the effects of large $B$. For fixed radius and Reynolds number ratios, we show that for sufficiently large $B$ all basic flows have an unyielded fluid layer attached to the outer wall. For these flows we show that there is a similarity mapping that maps both the basic solution and the linear stability problem onto the stability problem for an outer cylinder of radius equal to the yield surface radius. The Reynolds and Bingham numbers of the transformed problems are smaller than that of the original problem, as is the wavenumber $k$. As $B \,{\to}\, \infty$, the yield surface approaches the inner cylinder, defining a narrow gap limiting problem that differs from the classical narrow gap limit. Via the transformed problem we derive an energy estimate for stability: $\hbox{\it Re}_{1,c} k_c \,{\sim}\, B^{1.5}$ as $B \,{\to}\, \infty$, which compares well with our computed results for a stationary outer cylinder: $\hbox{\it Re}_{1,c} \,{\sim}\, B^{1.25}$ and $k_c \,{\sim}\, B^{0.375}$. We also show how $\hbox{\it Re}_{1,c} \,{\sim}\, B^{1.25}$ can be deduced from a simple order of magnitude analysis, for a stationary outer cylinder. Finally, we consider the second (classical) narrow gap limit in which the radius ratio $\eta$, approaches unity, for fixed $B$ and Reynolds number ratio. We show that $\hbox{\it Re}_{1,c} \,{\gtrsim}\, (k^2[1 + O(B)] + \pi^2)/(1-\eta)^{1/2}$ in this limit.

A fundamental characteristic of granular flows is that they are typically restricted to thin layers of rapid surface flow. Thus, a complete understanding of surface flows is key for an accurate representation of the dynamics of the entire flow. Experiments were conducted in three-dimensional tumblers: cylindrical tumblers of various diameters, a double-cone tumbler, and a spherical tumbler, the Froude number for the last two being a function of the local geometry and ranging from $2.6\times10^{-5}$ to $7.5\times10^{-4}$. Surface velocity measurements for 1 mm and 2 mm glass particles were obtained using particle tracking velocimetry. Results indicate that the streamwise surface velocity at the midpoint of the flowing layer is a linear function of local flowing layer length, regardless of tumbler shape, particle size, rotation rate, and fill fraction. In addition, the axial velocity of particles at the free surface is negligible. These results are key for the development of three-dimensional models of granular flows.

A weakly nonlinear analysis of one-dimensional viscous Faraday waves in two-dimensional large-aspect-ratio containers is presented. The surface wave is coupled to a viscous long-wave mean flow that is slaved to the free-surface deformation. The relevant Ginzburg–Landau-like amplitude equations are derived from first principles, and can be of three different types, depending on the ratio between wavelength, depth and the viscous length. These three equations are new in the context of Faraday waves. The coefficients of these equations are calculated for arbitrary viscosity and compared with their counterparts in the literature for small viscosity; a discrepancy in the cubic coefficient is due to a dramatic sensitivity of this coefficient on a small wavenumber shift due to interplay between viscous effects and parametric forcing.

A linear stability analysis of the Rayleigh–Taylor problem for an incompressible fluid undergoing a liquid–liquid phase transformation is presented. Both inviscid and linearly viscous fluids are considered and interfacial tension is taken into account. Instability is possible only when the phase with the higher density is above that with the lower density. Study of the inviscid case shows that the exchange of mass between the phases decreases significantly both the range of unstable wavenumbers and the maximum growth rate for unstable perturbations as compared to those arising classically. For a linearly viscous fluid, the shear and dilational viscosities of the interface are taken into account as are the migrational viscosities associated with the motion of the interface relative to the underlying fluid. When no mass exchange occurs between the phases in the base state and the interfacial viscosities are neglected, the growth rates exceed by at least an order of magnitude those for the classical Rayleigh–Taylor problem. The various interfacial viscosities slow the growth rates of disturbances, but do not influence the range of unstable wavenumbers. For both the inviscid and viscous cases, interfacial tension plays the same stabilizing role as it does classically.

An insoluble particle, a solid sphere or a spherical bubble, submerged in a liquid and approached by an advancing solidification front, may be captured by the front or rejected. The particle behaviour is determined by an interplay among van der Waals interactions, thermal conductivity differences between the particle and the melt, solid–liquid interfacial energy, the density change caused by the liquid–solid phase transition, and in the case of a bubble, the Marangoni effect at the liquid–gas interface. We calculate the particle velocity and the deformation of the front when the particle is close to the front, using the lubrication approximation, and investigate how the particle speed, relative to the front, depends on the parameters that characterize the described effects.

We have studied the evolution of grid turbulence in a planar contraction by focusing on the flow at the centre symmetry plane. Measurements are carried out in water with inlet Taylor-microscale Reynolds number varying from 51 to 99. Detailed laser-Doppler anemometry measurements show that the streamwise fluctuating velocity component for contraction ratio $C\,{<}\,2.5$ closely follows the decay of grid turbulence in a straight channel. Furthermore, the turbulent kinetic energy reaches a minimum value in the range of contraction ratio $1.5\,{<}\,C\,{<}\,2.5$. Turbulent intensity, independent of contraction angle and Reynolds number, decays exponentially. The results show that the flow reaches its peak of anisotropy at $2.5\,{<}\,C\,{<}\,3.5$ and then returns to a nearly fully isotropic state inside the contraction. The return to isotropy within the contraction is attributed to the rapid part of the pressure–strain correlation term in the transport equation of the Reynolds-stress anisotropy tensor.

This paper presents the results of investigations into particular features of laminar-to-turbulent transition of pipe flows. The first part considers transitional flows that occur ‘naturally’, i.e. without any forcing, when a critical Reynolds number is reached. Measurements are reported that were carried out to study the intermittent nature of pipe flows before they become fully turbulent. The second part of the paper concentrates on forced laminar-to-turbulent transition where the forcing was achieved by ring-type obstacles introduced into the flow close to the pipe inlet. The influence of the ring height was investigated and the results showed a dependence of the critical Reynolds number on the normalized height of the disturbances. The laminar-to-turbulent transition was also investigated when caused by partially closing an iris diaphragm that permitted the flow to be forced to turbulence over short time intervals. Investigations of controlled intermittency became possible in this way and corresponding results are presented.