In many industrial processes granular materials are mixed together in partially filled slowly rotating drums. In this paper a general theoretical framework is developed for the quasi-two-dimensional motion of granular material in a rotating drum. The key assumption is that the body can be divided into a fluid-like and a solid-like region, that are separated by a non-material singular surface at which discontinuities occur. Experiments show that close to the free surface there is a thin rapidly moving fluid-like avalanche that flows downslope, and beneath it there is a large region of slowly rotating solid-like material. The solid region provides a net transport of material upslope and there is strong mass transfer between the two regions. In the theory the avalanche is treated as a shallow incompressible Mohr–Coulomb or inviscid material sliding on a moving bed at which there is erosion and deposition. The solid is treated as a rigid rotating body, and the two regions are coupled together using a mass jump condition. The theory has the potential to model time-dependent intermittent flow with shock waves, as well as steady-state continuous flow. An exact solution for the case of steady continuous flow is presented. This demonstrates that when the base of the avalanche lies above the axis of revolution a solid core develops in the centre of the drum. Experiments are presented to show how a mono-disperse granular material mixes in the drum, and the results are compared with the predictions using the exact solution.

Steady, spatial, algebraically growing eigenfunctions are now known to occur in several important classes of boundary-layer flow, including two-dimensional hypersonic boundary layers and more recently in Blasius boundary layers subject to three-dimensional linearized disturbances, and in more general three-dimensional boundary layers. These spatial eigensolutions are particularly important and intriguing, given that they exist within the broad limits of the classical steady boundary-layer approximation, and as such are independent of Reynolds number.

In this paper we make the natural extension to these previous (stability) analyses by incorporating the effects of unsteadiness into the model for treating disturbances to a quite general class of similarity-type boundary-layer flows. The flow disturbances are inherently non-parallel, but this effect is properly incorporated into the analysis.

A further motivation for this paper is that Duck et al. (1999, 2000) have shown that by permitting a spanwise component of flow within a boundary layer of the appropriate form (in particular, growing linearly with the spanwise coordinate), it is found that new families of solutions exist – even the Blasius boundary layer has a three-dimensional ‘cousin’. Therefore a further aim of this paper is to assess the stability of the different solution branches, using the ideas introduced in this paper, to give some clues as to which of the solutions may be encountered experimentally.

Several numerical methods are presented for tackling various aspects of the problem. It is shown that when algebraically growing, steady eigensolutions exist, their effect remains important in the unsteady context. We show how even infinitesimal, unsteady flow perturbations can provoke extremely large-amplitude flow responses, including in some cases truly unstable flow disturbances which grow algebraically downstream without bound in the linear context. There are some interesting parallels suggested therefore regarding mechanisms perhaps linked to bypass transition in an important class of boundary-layer flows.

The temporal dynamics of large-scale structures in a plane turbulent mixing layer are studied through the development of a low-order dynamical system of ordinary differential equations (ODEs). This model is derived by projecting Navier–Stokes equations onto an empirical basis set from the proper orthogonal decomposition (POD) using a Galerkin method. To obtain this low-dimensional set of equations, a truncation is performed that only includes the first POD mode for selected streamwise/spanwise (k1/k3) modes. The initial truncations are for k3 = 0; however, once these truncations are evaluated, non-zero spanwise wavenumbers are added. These truncated systems of equations are then examined in the pseudo-Fourier space in which they are solved and by reconstructing the velocity field. Two different methods for closing the mean streamwise velocity are evaluated that show the importance of introducing, into the low-order dynamical system, a term allowing feedback between the turbulent and mean flows. The results of the numerical simulations show a strongly periodic flow indicative of the spanwise vorticity. The simulated flow had the correct energy distributions in the cross-stream direction. These models also indicated that the events associated with the centre of the mixing layer lead the temporal dynamics. For truncations involving both spanwise and streamwise wavenumbers, the reconstructed velocity field exhibits the main spanwise and streamwise vortical structures known to exist in this flow. The streamwise aligned vorticity is shown to connect spanwise vortex tubes.

A statistically stationary and nearly homogeneous turbulent shear flow is established by an additional volume forcing in combination with stress-free boundary conditions in the shear direction. Both turbulent energy and enstrophy are stationary to a much better approximation than in previous simulations that use remeshing. The temporal fluctuations decrease with increasing Reynolds number. Energy spectra and shear-stress cospectra show that local isotropy is satisfactorily obeyed at the level of second-order moments. However, derivative moments of high order up to n = 7 yield increasing moments for n [ges ] 4 for the spanwise vorticity and the transverse derivative of the streamwise velocity in the range of Taylor Reynolds numbers 59 [les ] Rλ [les ] 99. These findings, which are in apparent violation of local isotropy, agree with recent measurements.

The large-eddy simulation (LES) equations are obtained from the application of two operators to the Navier-Stokes equations: a smooth filter and a discretization operator. The introduction ab initio of the discretization influences the structure of the unknown stress in the LES equations, which now contain a subgrid-scale stress tensor mainly due to discretization, and a filtered-scale stress tensor mainly due to filtering. Theoretical arguments are proposed supporting eddy viscosity models for the subgrid-scale stress tensor. However, no exact result can be derived for this term because the discretization is responsible for a loss of information and because its exact nature is usually unknown. The situation is different for the filtered-scale stress tensor for which an exact expansion in terms of the large-scale velocity and its derivatives is derived for a wide class of filters including the Gaussian, the tophat and all discrete filters. As a consequence of this generalized result, the filtered-scale stress tensor is shown to be invariant under the change of sign of the large-scale velocity. This implies that the filtered-scale stress tensor should lead to reversible dynamics in the limit of zero molecular viscosity when the discretization effects are neglected. Numerical results that illustrate this effect are presented together with a discussion on other approaches leading to reversible dynamics like the scale similarity based models and, surprisingly, the dynamic procedure.

Simultaneous excitation of a turbulent mixing layer by two frequencies, a fundamental and a subharmonic, was investigated experimentally. Plane perturbations were introduced to the flow at its origin by a small oscillating flap. The results describe two experiments that differ mainly in the amplitudes of the imposed perturbations and both are compared to the data acquired while the mixing layer was forced at a single frequency.

Conventional statistical quantities such as: mean velocity profiles, widths of the flow, turbulent intensities, spectra, phase-locked velocity and vorticity fields, as well as streaklines were computed. The rate of spread of the flow under concomitant excitation at the two frequencies was much greater than under a single frequency, although it remained dominated by two-dimensional eddies. The Reynolds stresses and turbulence production are associated with the deformation and orientation of the large coherent vortices. When the major axis of the coherent vortices starts leaning forward on the high-speed side of the flow, the production of turbulent energy changes sign (i.e. becomes negative) and this results in the flow thinning in the direction of streaming. It also indicates that energy is extracted from the turbulence to the mean motion. Resonance phenomena play an important role in the evolution of the flow. A vorticity budget showed that the change in mean vorticity was mainly caused by the nonlinear interaction between coherent vorticities. Nevertheless, the locally dominant frequency scales the mean growth rate, the inclination and distortion of the mean velocity profiles as well as the phase-locked vorticity contours.

A morphodynamic model is developed and analysed to gain fundamental understanding of the basic physical mechanisms responsible for the characteristics of shoreface-connected sand ridges observed in some coastal seas. These alongshore rhythmic bed forms have a horizontal lengthscale of order 5 km and are related to the mean current along the coast: the seaward ends of their crests are shifted upstream with respect to where they are attached to the shoreface. The model is based on the two-dimensional shallow water equations and assumes that the sediment transport only takes place during storms. The flux consists of a suspended-load part and a bed-load part and accounts for the effects of spatially non-uniform wave stirring as well as for the preferred downslope movement of sediment. The basic state of this model represents a steady longshore current, driven by wind and a pressure gradient. The dynamics of small perturbations to this state are controlled by a physical mechanism which is related to the transverse bottom slope. This causes a seaward deflection of the current over the ridges and the loss of sediment carrying capacity of the flow into deeper water. The orientation, spacing and shape of the modelled ridges agree well with field observations. Suspended-load transport and spatially non-uniform wave stirring are necessary in order to obtain correct e-folding timescales and migration speeds. The ridge growth is only due to suspended-load transport whereas the migration is controlled by bed-load transport.

We use the lubrication approximation to investigate the steady flow of a thin rivulet of viscous fluid with prescribed volume flux draining down a planar or slowly varying substrate that is either uniformly hotter or uniformly colder than the surrounding atmosphere, when the surface tension of the fluid varies linearly with temperature. Utilizing the (implicit) solution of the governing ordinary differential equation that emerges, we undertake a comprehensive asymptotic and numerical analysis of the flow. In particular it is shown that the variation in surface tension drives a transverse flow that causes the fluid particles to spiral down the rivulet in helical vortices (which are absent in the corresponding isothermal problem). We find that a single continuous rivulet can run from the top to the bottom of a large horizontal circular cylinder provided that the cylinder is either warmer or significantly cooler than the surrounding atmosphere, but if it is only slightly cooler then a continuous rivulet is possible only for a sufficiently small flux (though a rivulet with a discontinuity in the free surface is possible for larger values of the flux). Moreover, near the top of the cylinder the rivulet has finite depth but infinite width, whereas near the bottom of the cylinder it has finite width and infinite depth if the cylinder is heated or slightly cooled, but has infinite width and finite depth if the cylinder is significantly cooled.

To the long established idea of bounding turbulent convective heat transport by a variational method based on energetic constraints, we now add a richer class of ‘z-constraints’ with the hope of tightening bounds considerably. We establish that only certain moments of the governing equations are effective for this purpose. We explore the initial consequences of groups of such constraints by use of perturbation theory, which clarifies the need that a given set of elements be mutually congruent.

Granular surface flows are important in industrial practice and natural systems, but the understanding of such flows is at present incomplete. We present a combined theoretical and experimental study of quasi-two-dimensional heap formation by pouring particles continuously at a point. Two cases are considered: open systems and closed systems. Experimental results show that the shear rate in the flowing layer is nearly independent of the mass flow rate, and the angle of static friction at the bed–layer interface increases with flow rate. Predictions of the model for the flowing layer thickness and interface angles are in good agreement with experiments.

The particle tracking flow visualization method (PTFV) and particle image velocimetry (PIV) are used to obtain a clear picture of vortex evolution on the suction surface of an impulsively started NACA 0012 wing. The experiments are conducted in a towing water tank. The formation, evolution, and shedding of the vortex system on the suction surface are observed and analysed by streak pictures of particle images. Five characteristic vortex evolution regimes are identified in the parameter domain of angle of attack and chord Reynolds number. The pathline patterns, instantaneous streamlines, and vorticity of various vortex evolution processes are presented. Stable vortex shedding in the wake is eventually established after the initial period of complex vortex evolution on the suction surface of the wing. Various types of instabilities in the wake, e.g. instability wave, surface vortex shedding, and bluff-body vortex shedding, are found to correspond to different evolution processes of the surface flow. The shedding frequency of the vortices is correlated and compared with several conventional results. Topological critical points, separatrices, and alleyways are identified and discussed to elucidate the unsteady structure of the instantaneous streamline patterns. The topological rule for the number of singular points is verified.

The natural ventilation of a room, both with a heated floor and connected to a cold exterior through two openings, is investigated by combining quantitative models with analogue laboratory experiments. The heated floor generates an areal source of buoyancy while the openings allow displacement ventilation to operate. When combined, these produce a steady state in which the air in the room is well-mixed, and the heat provided by the floor equals the heat lost by displacement. We develop a quantitative model describing this process, in which the advective heat transfer through the openings is balanced with the heat flux supplied at the floor. This model is successfully tested with observations from small-scale analogue laboratory experiments. We compare our results with the steady-state flow associated with a point source of buoyancy: for a given applied heat flux, an areal source produces heated air of lower temperature but a greater volume flux of air circulates through the room. We generalize the model to account for the effects of (i) a cooled roof as well as a heated floor, and (ii) an external wind or temperature gradient. In the former case, the direction of the flow through the openings depends on the temperature of the exterior air relative to an averaged roof and floor temperature. In the latter case, the flow is either buoyancy dominated or wind dominated depending on the strength of the pressure associated with the wind. Furthermore, there is an intermediate multiple-solution regime in which either flow regime may develop.

The receptivity of hypersonic boundary layers to free-stream disturbances, which is the process of environmental disturbances initially entering the boundary layers and generating disturbance waves, is altered considerably by the presence of bow shocks in hypersonic flow fields. This paper presents a numerical simulation study of the generation of boundary layer disturbance waves due to free-stream waves, for a two-dimensional Mach 15 viscous flow over a parabola. Both steady and unsteady flow solutions of the receptivity problem are obtained by computing the full Navier–Stokes equations using a high-order-accurate shock-fitting finite difference scheme. The effects of bow-shock/free-stream-sound interactions on the receptivity process are accurately taken into account by treating the shock as a discontinuity surface, governed by the Rankine-Hugoniot relations. The results show that the disturbance waves generated and developed in the hypersonic boundary layer contain both first-, second-, and third-mode waves. A parametric study is carried out on the receptivity characteristics for different free-stream waves, frequencies, nose bluntness characterized by Strouhal numbers, Reynolds numbers, Mach numbers, and wall cooling. In this paper, the hypersonic boundary-layer receptivity is characterized by a receptivity parameter defined as the ratio of the maximum induced wave amplitude in the first-mode-dominated region to the amplitude of the free-stream forcing wave. It is found that the receptivity parameter decreases when the forcing frequency or nose bluntness increase. The results also show that the generation of boundary layer waves is mainly due to the interaction of the boundary layer with the acoustic wave field behind the bow shock, rather than interactions with the entropy and vorticity wave fields.

Herein we present a simplified theory for the behaviour of a vortex embedded in a growing external straining flow. Such a flow arises naturally as a vortex moves relative to other vortices. While the strain may generally exhibit a complex time dependence, the salient features of the vortex evolution can be understood in the simpler context, studied here, of a linearly growing strain. Then, all of the typical stages of evolution can be seen, from linear deformation, to the stripping or erosion of low-lying peripheral vorticity, and finally to the breaking or rapid elongation of the vortex into a thin filament.

When, as is often the case in practice, the strain growth is slow, the vortex adjusts itself to be in approximate equilibrium with the background flow. Then, the vortex passes through, or near, a sequence of equilibrium states until, at a critical value of the strain, it suddenly breaks. In the intermediate period before breaking, the vortex continuously sheds peripheral vorticity, thereby steepening its edge gradients. This stripping is required to keep the vortex in a near equilibrium configuration.

We show that this behaviour can be captured, quantitatively, by a reduced model, the elliptical model, which represents the vortex by a nested set of elliptical vorticity contours, each having a (slightly) different aspect ratio and orientation. Here, we have extended the original elliptical model by allowing for edge vorticity levels to be shed when appropriate (to represent stripping) and by incorporating the flow induced by the vorticity being stripped away. The success of this model proves that the essential characteristics of vortex erosion are captured simply by the leading-order, elliptical shape deformations of vorticity contours.

Finally, we discuss the role of viscosity. Then, there is a competition between gradient steepening by stripping and smoothing by viscosity. If the strain grows too slowly, the vortex is dominated by viscous decay, and the edge gradients become very smooth. On the other hand, for sufficiently rapid strain growth (which can still be slow, depending on the viscosity), the vortex edge remains steep until the final breaking.

We revisit the classical problem of dispersion of a point discharge of tracer in laminar pipe Poiseuille flow. For a discharge at the centre of the pipe we show that in the limit of small non-dimensional diffusion, D, tracer dispersion can be divided into three regimes. For small times (t [Lt ] D−1/3), diffusion dominates advection yielding a spherically symmetric Gaussian dispersion cloud. At large times (t [Gt ] D−1), the flow is in the classical Taylor regime, for which the tracer is homogenized transversely across the pipe and diffuses with a Gaussian distribution longitudinally. However, in an intermediate regime (D−1/3 [Gt ] t [Gt ] D−1), the longitudinal diffusion is anomalous with a width proportional to t [Lt ] Dt2 and a distinctly asymmetric longitudinal distribution. We present a new solution valid in this regime and verify our results numerically. Analogous results are presented for an off-centre release; here the distribution width scales as D1/2t3/2 in the anomalous regime. These results suggest that anomalous diffusion is a hallmark of the shear dispersion of point discharges at times earlier than the Taylor regime.