Particle-size segregation within granular materials is of great technological significance yet it is still very poorly understood. There are several causes of segregation, but this paper focuses on kinetic sieving which is the dominant mechanism in dense gravity-driven shallow free-surface flows, or, granular avalanches. The segregation model is derived from a three-phase mixture theory composed of large particles, small particles and a passive interstitial fluid. Steady-state solutions are constructed for a normally graded inflow in a steady uniform flow field. This problem is of fundamental interest, because it shows how an unstably stratified layer readjusts into a stable configuration. Expansion fans and concentration shocks are generated and sufficiently far downstream inversely graded segregated layers form, with the larger particles overlying the finer ones. This a good approximation for segregation in flows with weak diffusive remixing. The distance for complete segregation to occur is shown to increase with rising fluid density and tends to infinity as its density approaches that of the grains. If the particles are buoyant then the initial configuration is stable. An exact time-dependent two-dimensional solution is constructed for plug flow, which exploits the uncoupling of material columns of grains in the absence of shear. This yields insight into the nature of more complex numerical solutions for strong shear, which are computed with a high-resolution shock-capturing numerical scheme.

The work of Benjamin, Mullin, Pfister and others on the non-uniqueness of solutions of the Navier–Stokes equation in Couette flow at small aspect ratio has revealed the existence of ‘anomalous’ 1-cell modes. A natural question to ask is whether these ‘anomalous’ modes are robust enough to survive the application of a body force, such as an externally applied magnetic field. We find that the answer is positive, although, with increasing magnetic field, steady 2-cell flows are generally more stable than 1-cell states. We also show that new time-dependent flows are easy to excite at relatively low Reynolds numbers compared to the case without a magnetic field, and we present two such flows.

In a recent paper (J. Fluid Mech. vol. 506, 2004, p. 207), B. R. Fabijonas and D. D. Holm claim that they have found a general method to construct new solutions of the Navier–Stokes equations from any known base flow solution. In this note, we argue that Fabijonas & Holm's solutions are very special in character. Although they can be defined in all space, they satisfy the Navier–Stokes equations on a single fixed trajectory of the chosen base flow. We show that this limits the usefulness of the solution and the applicability of the method. In particular, it is demonstrated that, in general, the iterative ‘WKB-bootstrapping’ algorithm designed by the authors cannot be applied after the first iteration. We also show that a second iteration is possible only if the base flow satisfies strong constraints. The consequence of these constraints is that no extension of the Craik–Criminale solutions to multiple frequencies is found to be possible. By applying Fabijonas & Holm's construction to a simple model equation, we demonstrate that their solution can also predict (unphysical) behaviours which cannot be reproduced by any global solution.

The $k^{-1}$ spectral law for near-wall turbulence has received only limited experimental support, the most convincing evidence being that of Nickels et al. (Phys. Rev. Lett. vol. 95, 2005, 074501.1). The real-space analogue of this law is a logarithmic dependence on $r$ of the streamwise longitudinal structure function. We show that, unlike the $k^{-1}$ law, the logarithmic law is readily seen in the experimental data. We argue that this difference arises from the finite value of Reynolds number in the experiments. Reducing the Reynolds number is equivalent to restricting the range of eddy sizes which contribute to the $k^{-1}$, or ln$r$, laws. While the logarithmic law is relatively insensitive to a truncation in the range of eddy sizes (it continues to hold over the relevant range of eddy sizes), it turns out that the $k^{-1}$ law is not. This is a direct consequence of the so-called aliasing problem associated with one-dimensional spectra, whereby energy is systematically and artificially displaced to small wavenumbers.

The eddy structures and their mutual interactions in a three-dimensional channel with a backward-facing step for the transitional Reynolds number 900 were investigated numerically. The aim was to reveal the structural development of the entire vortical flow field, which could immensely enhance the knowledge about vortical structures occurring in the recirculation region near the step wall. Simulations were made to reproduce the experimental observations and provide clear indications about the strong interaction between the shear layer instabilities. Physical instabilities of this type were amplified by the shedding-type instabilities and induced by the interaction of coherent structures with the sidewalls of the duct. These interactions were responsible for the flapping motion of interior shear layer. Careful attempts were made to reveal the behaviour of these vortical structures by means of vortex stretching, roll-up of vortex lines and formation of vortex tubes. Also, the three-dimensional flow topology of the velocity field corresponding to stationary helical vortex (SHV) was analysed extensively. The SHV flow consisted of a pair of counter-rotating helical cells in a double helix structure wrapped around the vortex tube. The roll-up shear-layer hovering vortices were observed near the step to initiate the Kelvin–Helmholtz-like instability. The Kelvin–Helmholtz vortices were developed into lambda-shaped vortices which impinged on the step-wall and were elongated into the hairpin-like vortices.

The collision of axisymmetric turbulent plumes with buoyancy fluxes of opposite sign is examined experimentally. The total buoyancy flux loss of each plume as a result of the collision is measured. The measurements are made using a new experimental technique for measuring the buoyancy flux of a plume based on the ventilation theory of Linden, Lane-Serff & Smeed (J. Fluid Mech. vol. 212, 1990, p. 309). The experimental results are presented as functions of the buoyancy flux ratio $\psi$ and the ratio of radial to vertical separation $\sigma $. For axially aligned plumes we find that the lower-buoyancy-flux plume loses all its buoyancy flux when $\psi \,{<}\,$0.3, and that there is very little buoyancy flux loss for either plume when $\sigma \,{>}\,$0.25. This plume–plume collision is modelled using a modified set of entrainment equations. The model allows for the exchange of buoyancy and deflection of the plumes as they pass by each other. We present predictions of total buoyancy flux loss as a function of both plume strength and separation. The model predictions are compared to the experimental measurements of buoyancy flux loss, and show good agreement.

A new approach to studying ripple formation in steady flow is proposed, based on a model for the erosion and deposition of the particles. Higher shear stresses at the crests lead to a loss of particles from there, which can suppress the instability induced by fluid inertia. This model accounts for recent observations that ripples do not grow when the fluid viscosity is increased, a behaviour not present in the classical approach based on an algebraic dependence of the particle flux on the bottom shear stress.

A new model for the erosion and deposition of particles, developed in Part 1, is now applied to oscillating flows to give an analytic expression for the growth rate. We find that ripples can grow in oscillating flows when they do not in steady flows of the same magnitude, this being due to a suppression in rapidly oscillating flows of the erosion by higher shear rates on the crests.

The self-propulsion of a generally deformable body at low-Reynolds-number conditions is discussed. The translational and rotational velocities of the body relative to an inertial reference system are presented as surface quadratures using a Lagrangian ‘body-fixed’ shape description. The power dissipated into the fluid is obtained as a quadratic functional of the surface deformation rate. For symmetric strokes, the net displacement obtained by the execution of a single deformation cycle is provided by a functional of the intrinsic swimmer shape and its time derivative.

The single-domain Darcy–Brinkman model is applied to some analytically tractable flows through adjacent porous and pure-fluid domains and is compared systematically with the multiple-domain Stokes–Darcy model. In particular, we focus on the interaction between flow and solidification within the mushy layer during binary alloy solidification in a corner flow and on the effects of the chosen mathematical description on the resulting macrosegregation patterns. Large-scale results provided by the multiple-domain formulation depend strongly on the microscopic interfacial conditions. No satisfactory agreement between the single- and multiple-domain approaches is obtained when using previously suggested conditions written directly at the interface between the liquid and the porous medium. Rather, we define a viscous transition zone inside the porous domain, where the Stokes equation still applies, and we impose continuity of pressure and velocities across it. This new condition provides good agreement between the two formulations of solidification problems when there is a continuous variation of porosity across the interface between a partially solidified region (mushy zone) and the melt.

Kolmogorov's similarity hypotheses and his 4/5 law are valid at very large Reynolds numbers. For flows encountered in the laboratory, the effect of a finite Reynolds number and of the non-stationarity or inhomogeneity associated with the large scales can affect the behaviour of the scales in the inertial range significantly. This paper focuses on the source of inhomogeneity in two types of flows, those dominated mainly by a decay of energy in the streamwise direction and those which are forced, through a continuous injection of energy at large scales. Results based on a parameterization of the second-order velocity structure function indicate that the normalized third-order structure function approaches 4/5 much more rapidly for forced than for decaying turbulence. This trend is supported by grid turbulence measurements and numerical data in a periodic box.

Active control of spatially evolving three-dimensional instability waves in the boundary layer of a flat plate with zero pressure gradient has been investigated both numerically and experimentally. The boundary layer was artificially excited by various computer-generated perturbations in a low-turbulence wind tunnel so as to create a three-dimensional field of instability waves. Sensors were used to detect the on-coming disturbances and appropriate control signals sent to the downstream actuators to generate counteracting disturbances. The key element of this type of active control is the determination of the transfer function of the control system connecting the sensors with the actuators. Modelling by linear theory showed that the amplitudes of the disturbances downstream were significantly suppressed when the causal transfer function was applied. To simplify the control problem different strategies using approximations to the transfer function were used. It was shown that with a very simple transfer function almost the same result could be achieved as with the full causal transfer function. The modelling was validated experimentally, first by an off-line control using different control modes, i.e. different transfer functions, and then by a real-time hardware implementation using one of the modes. Good agreement between the experimental measurements and numerical predictions indicated that a simple control strategy could be developed to inhibit the growth of the three-dimensional instability waves over extensive regions of the flow.

A cascade hypothesis for a strongly stratified fluid is developed on the basis of the Boussinesq equations. According to this hypothesis, kinetic and potential energy are transferred from large to small scales in a highly anisotropic turbulent cascade. A relation for the ratio, $ l_{v}/l_{h} $, between the vertical and horizontal length scale is derived, showing how this ratio decreases with increased stratification. Similarity expressions are formulated for the horizontal and vertical spectra of kinetic and potential energy. A series of box simulations of the Boussinesq equations are carried out and a good agreement between the proposed hypothesis and the simulations is seen. The simulations with strongest stratification give horizontal kinetic and potential energy spectra of the form $ E_{K_{h}} \,{=}\, C_{1} \epsilon_{K}^{2/3} k_{h}^{-5/3} $ and $ E_{P_{h}} \,{=}\, C_{2} \epsilon_{P} k_{h}^{-5/3}/\epsilon_{K}^{1/3} $, where $ k_{h} $ is the horizontal wavenumber, $ \epsilon_{K} $ and $ \epsilon_{P} $ are the dissipation of kinetic and potential energy, respectively, and $ C_{1} $ and $ C_{2} $ are two constants. Within the given numerical accuracy, it is found that these two constants have the same value: $ C_{1} \approx C_{2} \,{=}\, 0.51 \pm 0.02 $.

We consider generalizations of fluid models wherein the fluid is assumed to be incompressible, but with the viscosity depending on the pressure. We show that a natural setting for the development of such models is a class of implicit constitutive relations, which, in addition to the fluid model described here, provides a means for developing other complex models for viscoelastic fluids which cannot be set within the ambit of classical explicit constitutive relations for the stress in terms of the histories of appropriate kinematical variables.

Linear transient growth in a normal-mode-stable vortex column is studied by extracting ‘optimal modes’ of perturbations. Amplifications occur over a wide range of azimuthal wavenumbers $m$ and axial wavenumbers $k$, and can be more than three orders of magnitude even at moderate vortex Reynolds numbers $\hbox{\it Re}$ (checked up to $10^4$). Transient growth is unbounded in the inviscid limit. For given $\hbox{\it Re}$ and $k$, axisymmetric ($m\,{=}\,0$) modes undergo the largest volume-integrated energy growth, whereas maximum core energy growth occurs for bending waves ($|m|\,{=}\,1$). At fixed $m$ and $\hbox{\it Re}$, growth decreases with increasing $k$, due to the damping effect of viscosity. At fixed $m$ and $k$, growth increases rapidly with $\hbox{\it Re}$ – pointing to the significance of transient growth in high-$\hbox{\it Re}$ practical flows, such as the trailing vortex. Inviscid effects not only cause transient growth, but also its subsequent arrest. There are two distinct mechanisms for growth. First, two-dimensional perturbations amplify because the streamlines have ‘positive tilt’, contributing $uv\,{>}\,0$ stress necessary for growth; here $u$ and $v$ are the radial and azimuthal velocity perturbations, respectively. Second, three-dimensional perturbations grow through azimuthal stretching of spiral vortex filaments containing radial vorticity. Decay in both cases is due to the differential advection – of axial vorticity by the mean swirl – transforming the perturbation streamlines to predominantly ‘negative tilt’, producing $uv\,{<}\,0$ stress. The transient growth mechanism is explained in terms of the distinct effects of the strain and vorticity components of the mean flow, which play counteractive roles. While strain amplifies energy, vorticity limits transient growth by inducing wave motions, in which radial vorticity is depleted by vortex line coiling, i.e. by the tilting of radial vorticity into axial and azimuthal components. Since the strain-to-vorticity ratio varies with radius in the vortex, the competition between strain and vorticity selects a preferred radius of localization of an ‘optimal perturbation’. With increasing growth, axisymmetric optimal modes are localized at progressively larger radii and their growth rates progressively diminished – both limiting the physical significance of such modes in high-$\hbox{\it Re}$ practical flows. An optimal bending wave, on the other hand, is localized closer to the vortex column, where a vorticity perturbation external to the core can resonantly excite vortex core waves. This leads to substantial growth of core fluctuation energy and, probably, to core transition to turbulence. Such resonant growth may be the mechanism for the appearance of bending waves in a vortex in a turbulent field.

The stability of an incompressible parallel flow with a linear wall slip – where the amount of wall slip is proportional to the velocity gradient – is analysed. Numerical computations are performed for microchannel flows in the slip-flow region. The wall slip causes short-wave instability while the slip-flow model is stable for long waves. This instability disappears once the slip wall condition is replaced by the no-slip condition.

This paper is primarily concerned with acoustic radiation of instability waves as they undergo rapid distortion, which is one of the fundamental mechanisms by which instability modes generate sound in subsonic flows. To fix the idea, we consider the case where the abrupt distortion is associated with scattering of a Tollmien–Schlichting (T-S) wave by the mean flow induced by a localized surface roughness in a compressible subsonic boundary layer with an $O(1)$ free-stream Mach number. The sound field was calculated by extending the asymptotic approach based on the triple-deck formalism, developed previously. This approach allows us to identify and approximate the sound source systematically by seeking the solution for the near field hydrodynamics as an asymptotic series in ascending powers of $\epsilon=R^{-1/8}$, where $R$ is the Reynolds number at the roughness site. It is found that the first four terms in the expansion act as octupole, quadrupole, dipole and monopole, respectively, and they make equal order-of-magnitude contributions to the acoustic far field. Some rather delicate source cancellations are noted. As a by-product, the analysis also shows that a localized roughness also influences the energetics of the T-S wave, and that effect can be characterized by a transmission coefficient, defined as the ratio of the T-S wave amplitude after the scattering to that before the scattering.

We present the results of direct numerical simulations of heavy particle transport in homogeneous, isotropic, fully developed turbulence, up to resolution $512^3$ ($R_\lambda\approx 185$). Following the trajectories of up to 120 million particles with Stokes numbers, St, in the range from 0.16 to 3.5 we are able to characterize in full detail the statistics of particle acceleration. We show that: (i) the root-mean-squared acceleration $a_{\rm rms}$ sharply falls off from the fluid tracer value at quite small Stokes numbers; (ii) at a given St the normalized acceleration $a_{\rm rms}/(\epsilon^3/\nu)^{1/4}$ increases with $R_\lambda$ consistently with the trend observed for fluid tracers; (iii) the tails of the probability density function of the normalized acceleration $a/a_{\rm rms}$ decrease with St. Two concurrent mechanisms lead to the above results: preferential concentration of particles, very effective at small St, and filtering induced by the particle response time, that takes over at larger St.

Two- and three-dimensional instabilities of the two-dimensional symmetric flow generated by a circular cylinder oscillating with simple harmonic motion in quiescent fluid, or, alternatively, by oscillatory flow past a stationary cylinder at low Keulegan-Carpenter and Stokes numbers are investigated via Floquet analysis and direct numerical simulation. Previous experimental visualization has found that the flows produced at low amplitudes and frequencies of motion can be grouped by their visual characteristics into a number of distinct regimes. At low values of Keulegan-Carpenter and Stokes numbers, the flow is two-dimensional and has a reflection symmetry about the axis of oscillation, in addition to a pair of spatio-temporal symmetries. This study isolates and classifies the symmetry-breaking instabilities from these two-dimensional basic states as functions of these control parameters. It is found that while the initial bifurcations produced by increasing the parameters can be to three-dimensional flows, much of the behaviour can be explained in terms of two-dimensional symmetry-breaking instabilities. These have two primary manifestations: at low Stokes numbers, the instability is synchronous with the imposed oscillation, and gives rise to a boomerang-shaped mode, while at higher Stokes numbers, the instability is quasi-periodic, with a well-defined second period, which becomes infinite as Stokes numbers are reduced along the marginal stability boundary, ‘freezing’ the quasi-periodic mode into a synchronous one. These two-dimensional modes are, with further small increase in control parameter, unstable to three-dimensional secondary instabilities, and these are the flows which have been reported in previous experimental studies. In contrast, the mode first reported by Honji (J. Fluid Mech. vol. 107, 1981, p. 509), which arises at high Stokes numbers, and lower Keulegan-Carpenter numbers than the two-dimensional quasi-periodic mode, has a three-dimensional primary instability arising directly from the symmetrical two-dimensional basic state.

We report an internal layer found in the turbulent flow through an asymmetric planar diffuser using large-eddy simulation; we discuss five issues relevant to the internal layer: definition and identification, conditions for occurrence, connection with its outer flow, similarity with other equilibrium flows, and growth. The present internal layer exists in a region with stabilized positive skin friction downstream of a sharp reduction. The streamwise pressure gradient changes suddenly from slightly favourable to strongly adverse at the diffuser throat, and relaxes in a prolonged mildly adverse region corresponding to the skin friction plateau. Development of the internal layer into the outer region is slow, in contrast to the internal layers previously identified from certain external boundary-layer flows where the sudden change in streamwise pressure gradient is from strongly adverse to mildly favourable. Signatures of the internal layer include an inflectional point in the wall-normal profiles of streamwise turbulence intensity, and a well-defined logarithmic slope in the mean streamwise velocity underneath a linear distribution extending to the core region of the diffuser. Some of these characteristics bear a certain resemblance to those existing in the C-type of Couette–Poiseuille turbulent flows. Frequency spectrum results indicate that application of strong adverse pressure gradient at the diffuser throat enhances the low-frequency content of streamwise turbulent fluctuations. Inside the internal layer, the frequency energy spectra at different streamwise locations, but with the same wall-normal coordinate, nearly collapse. Two-point correlations with streamwise, wall-normal and temporal separations were used to examine connections between fluctuations inside the internal layer and those in the core region of the diffuser where the mean streamwise velocity varies linearly with distance from the wall. Galilean decomposition of instantaneous velocity vectors reveals a string of well-defined spanwise vortices outside the internal layer. The internal layer discovered from this study provides qualified support for a conjecture advanced by Azad & Kassab some years ago (Phys. Fluids A, vol. 1, 1989, p. 564).