Measurements were made of two components of the average and fluctuating velocities, and of the local self-diffusion coefficients in a flow of granular material. The experiments were performed in a 1 m-high vertical channel with roughened sidewalls and with polished glass plates at the front and the back to create a two-dimensional flow. The particles used were glass spheres with a nominal diameter of 3 mm. The flows were high density and were characterized by the presence of long-duration frictional contacts between particles. The velocity measurements indicated that the flows consisted of a central uniform regime and a shear regime close to the walls. The fluctuating velocities in the transverse direction increased in magnitude from the centre towards the walls. A similar variation was not observed for the streamwise fluctuations. The self-diffusion coefficients showed a significant dependence on the fluctuating velocities and the shear rate. The velocity fluctuations were highly anistropic with the streamwise components being 2 to 2.5 times the transverse components. The self-diffusion coefficients for the streamwise direction were an order-of-magnitude higher than those for the transverse direction. The surface roughness of the particles led to a decrease in the self-diffusion coefficients.

The formation of a thermocline in a water column, in which shear-free turbulence is generated both at the surface and bottom, and a stabilizing buoyancy flux is imposed at the surface, is studied using a laboratory experiment and a numerical model with the aim of understanding the formation of a tidal front in coastal seas. The results show that the formation of a thermocline, which always occurs in the absence of bottom mixing, is inhibited and the water column maintains a vertically mixed state, when bottom mixing becomes sufficiently strong. It is found from both experimental and numerical results that the criterion for the formation of a thermocline is determined by the balance between the rate of work that is necessary to maintain a mixed state against the formation of stratification by the buoyancy flux and the turbulent kinetic energy flux from the bottom supplied to the depth of thermocline formation. The depth of the thermocline, when it is formed, is found to decrease with bottom mixing.

Reconnection of two antiparallel vortex tubes is studied as a prototypical coherent structure interaction to quantify compressibility effects in vorticity dynamics. Direct numerical simulations of the Navier-Stokes equations for a perfect gas are carried out with initially polytropically related pressure and density fields. For an initial Reynolds number (Re = Γ /v, circulation divided by the kinematic viscosity) of 1000, the pointwise initial maximum Mach number (M) is varied from 0.5 to 1.45. At M=0.5, not surprisingly, the dynamics are essentially incompressible. As M increases, the transfer of Γ starts earlier. For the highest M, we find that shocklet formation between the two vortex tubes enhances early Γ transfer due to viscous cross-diffusion as well as baroclinic vorticity generation. The reconnection at later times occurs primarily due to viscous cross-diffusion for all M. However, with increasing M, the higher early Γ transfer reduces the vortices’ curvature growth and hence the Γ transfer rate; i.e. for the Re case studied, the reconnection timescale increases with M. With increasing M, reduced vortex stretching by weaker ‘bridges’ decreases the peak vorticity at late times. Compressibility effects are significant in countering the stretching of the bridges even at late times. Our observations suggest significantly altered coherent structure dynamics in turbulent flows, when compressible.

Chaotic natural convection in a differentially heated air-filled cavity of aspect ratio 4 with adiabatic horizontal walls is investigated by direct numerical integration of the unsteady two-dimensional equations. Time integration is performed with a spectral algorithm using Chebyshev spatial approximations and a second-order finite-difference time-stepping scheme. Asymptotic solutions have been obtained for three values of the Rayleigh number based on cavity height up to 1010. The time-averaged flow fields show that the flow structure increasingly departs from the well-known laminar one. Large recirculating zones located on the outer edge of the boundary layers form and move upstream with increasing Rayleigh number. The time-dependent solution is made up of travelling waves which run downstream in the boundary layers. The amplitude of these waves grows as they travel downstream and hook-like temperature patterns form at the outer edge of the thermal boundary layer. At the largest Rayleigh number investigated they grow to such a point that they result in the formation of large unsteady eddies that totally disrupt the boundary layers. These eddies throw hot and cold fluid into the upper and lower parts of the core region, resulting in thermally more homogeneous top and bottom regions that squeeze a region of increased stratification near the mid-cavity height. It is also shown that these large unsteady eddies keep the internal waves in the stratified core region excited. These simulations also give access to the second-order statistics such as turbulent kinetic energy, thermal and viscous dissipation, Reynolds stresses and turbulent heat fluxes.

The basilar artery is one of the three vessels providing the blood supply to the human brain. It arises from the confluence of the two vertebral arteries. In fact, it is the only artery of this size in the human body arising from a confluence instead of a bifurcation. Earlier work, concerning flow computations in simplified models of the basilar artery, has demonstrated that a junction causes distinctive flow phenomena. This paper presents three-dimensional finite-element computations of steady viscous flow in a rigid symmetrical junction geometry representing the anatomical situation in a more realistic way. The geometry consists of two round tubes merging into a single round outlet tube. The Reynolds number for the basilar artery ranges from 200 to 600, and both symmetrical and asymmetrical inflow from the two inlet tubes has been considered.

Just downstream of the confluence a ‘double hump’ axial velocity profile is found. In the transition zone the flow pattern appears to have a complicated structure. In the symmetrical case the axial velocity profile shows a sharp central ridge, whereas in the asymmetrical case the highest ‘hump’ crosses the centreline of the tube. The flow has a highly three-dimensional character with secondary velocities easily exceeding 25% of the mean axial flow velocity. The secondary flow pattern consists of four vortices. Under all simulated flow conditions, the inlet length turns out to be much larger than the average length of the human basilar artery.

To validate the computational model, a comparison is made between numerical and experimental results for a junction geometry consisting of tubes with a rectangular cross-section. The experiments have been performed in a Perspex model with laser Doppler velocimetry and dye injection techniques. Good agreement between experimental and computational results is found. Moreover, all essential flow phenomena turn out to be quite similar to those obtained for the circular tube geometry.

Consider the directional solidification of a binary alloy rejecting a heavy solute as it solidifies upward. If the solidification front is planar, the fluid melt ahead of the front is stably stratified and convection is not expected. In this paper we analyse the linear stability of planar solidification asymptotically in the limit of large solutal Rayleigh number, R. Three distinct linear modes are found which correspond to internal waves, buoyancy edge waves, or morphological modes. Of these three modes, only the morphological modes are subject to an instability. We find that for large Rayleigh number this instability first occurs at long wavelengths with wavenumbers that scale on R−1/14. The scalings derived from the linear analysis are used to construct a nonlinear theory for the morphological instability in the large Rayleigh number limit. Similarity solutions are found which describe steadily convecting, non-planar growth reminiscent of an observed phenomenon known as steepling.

A commonly observed bedform in wide erodible-bed channels consists of rows of streaks or stripes parallel to the flow. These stripes can be manifested in terms of transverse variation of bed elevation, characteristic grain size (and thus roughness or both. The former case is manifested most strongly in sediment with a nearly uniform size distribution and the latter most strongly in sediment with substantial heterogeneity in size. The amplitude of stripes is rarely larger than one or two grain diameters, and the transverse spacing is invariably of the order of the flow depth. They are closely linked to a pattern of paired cells of secondary flow in the flow cross-section.

An existing theory of streak formation for the case of uniform sediment relies on a second-order turbulence closure which explicitly links the streamwise flow to transverse variations in bed elevation. The theory successfully predicts the formation of streaks, but only at rather high values of the Shields stress, i.e. rather strong sediment transport. Streaks are commonly observed, however, at Shields stresses as low as only slightly above the threshold of motion.

In the present analysis the previous flow model is adapted to the case of transverse variation of roughness as well as elevation, and the constraint of uniform sediment is removed. The theory indicates that allowance for even slight heterogeneity of bed sediment results in the formation of streaks at any Shields stress above the threshold of motion. The resulting streaks are hybrid in the sense that they show transverse variation in both elevation and roughness. The model thus provides a general theory of streak formation.

Our concern is with the evolution of large-amplitude Tollmien-Schlichting waves in boundary-layer flows. In fact, the disturbances we consider are of a comparable size to the unperturbed state. We shall describe two-dimensional disturbances which are locally periodic in time and space. This is achieved using a phase equation approach of the type discussed by Howard & Kopell (1977) in the context of reaction-diffusion equations. We shall consider both large and O(1) Reynolds number flows though, in order to keep our asymptotics respectable, our finite-Reynolds-number calculation will be carried out for the asymptotic suction flow. Our large-Reynolds-number analysis, though carried out for Blasius flow, is valid for any steady two-dimensional boundary layer. In both cases the phase-equation approach shows that the wavenumber and frequency will develop shocks or other discontinuities as the disturbance evolves. As a special case we consider the evolution of constant frequency/wavenumber disturbances and show that their modulational instability is controlled by Burgers equation at finite-Reynolds-number and by a new integro-differential evolution equation at large-Reynolds-numbers. For the large Reynolds number case the evolution equation points to the development of a spatially localized singularity at a finite time.

We consider two-dimensional problems based on linear water wave theory concerning the interaction of waves with horizontal cylinders in a fluid consisting of a layer of finite depth bounded above by a free surface and below by an infinite layer of fluid of greater density. For such a situation time-harmonic waves can propagate with two different wavenumbers K and k. In a single-layer fluid there are a number of reciprocity relations that exist connecting the various hydrodynamic quantities that arise. These relations are systematically extended to the two-fluid case. It is shown that for symmetric bodies the solutions to scattering problems where the incident wave has wavenumber K and those where it has wavenumber k are related so that the solution to both can be found by just solving one of them. The particular problems of wave scattering by a horizontal circular cylinder in either the upper or lower layer are then solved using multipole expansions.

When more than one wave is present in a system there exists the possibilty of a resonant interaction. Resonant modes become nonlinear at smaller amplitudes than nonresonant modes. If the nonlinearity causes increased growth rates then it may be that, for a time at least, the behaviour of the resonant modes will be the dominant feature. In shear layers resonant triads can be found where two oblique modes resonate with a plane wave and this case has received much attention in the literature. For a given plane wave, the resonance condition selects oblique modes of a certain wave angle and agreement has been found between predicted wave angles and those measured in experiments.

In this paper it is shown that resonance conditions can also be met between two planar waves in a Blasius boundary layer, where one of the waves is the usual unstable mode, and the other is a higher-order damped mode. The effects of wave modulation are modelled by performing a spatial analysis but allowing the frequency to become complex. It is found that for certain complex frequencies the strength of the nonlinear resonant interaction coefficients is greatly increased. Experiments have been performed in a low-turbulence wind tunnel in which disturbances with modulated and unmodulated sections were introduced into the boundary layer over a flat plate. It was found that disturbances with the frequency and modulation predicted by the theory do indeed show a much greater susceptibility to nonlinear breakdown than nonresonant disturbances.

The instability of an electrically conducting fluid of magnetic diffusivity λ and viscosity v in a rapidly rotating spherical container of magnetic diffusivity $\hat{\lambda}$ in the presence of a toroidal magnetic field is investigated. Attention is focused on the case of a toroidal magnetic field induced by a uniform current density parallel to the axis of rotation, which was first studied by Malkus (1967). We show that the internal ohmic dissipation does not affect the stability of the hydromagnetic solutions obtained by Malkus (1967) in the limit of small λ. It is solely the effect of the magnetic Hartmann boundary layer that causes instabilities of the otherwise stable solutions. When the container is a perfect conductor, $\hat{\lambda}$ = 0, the hydromagnetic instabilities grow at a rate proportional to the magnetic Ekman number of the fluid Eλ; when the container is a nearly perfect insulator, $\lambda/\hat{\lambda}\ll 1$, the hydromagnetic instabilities grow at a rate proportional to E1/2λ; when the container is a nearly perfect conductor, λ 1, the growth rates are proportional to λ, where λ is the magnetic Ekman number based on the diffusivity λ of the container. The main characteristics of the instabilities are not affected by varying magnetic properties of the container. In light of the destabilizing role played by the Hartmann boundary layer, we also examine the corresponding magnetoconvection in a rapidly rotating fluid sphere with the perfectly conducting container and stress-free velocity boundary conditions. Analytical magnetoconvection solutions in closed form are obtained and implications are discussed.

Ensemble-averaged statistics at constant phase of the turbulent near-wake flow (Reynolds number ≈ 21400 around a square cylinder have been obtained from two-component laser-Doppler measurements. Phase was defined with reference to a signal taken from a pressure sensor located at the midpoint of a cylinder sidewall. The distinction is drawn between the near wake where the shed vortices are ‘mature’ and distinct and a base region where the vortices grow to maturity and are then shed. Differences in length and velocity scales and vortex celerities between the flow around a square cylinder and the more frequently studied flow around a circular cylinder are discussed. Scaling arguments based on the circulation discharged into the near wake are proposed to explain the differences. The relationship between flow topology and turbulence is also considered with vorticity saddles and streamline saddles being distinguished. While general agreement with previous studies of flow around a circular cylinder is found with regard to essential flow features in the near wake, some previously overlooked details are highlighted, e.g. the possibility of high Reynolds shear stresses in regions of peak vorticity, or asymmetries near the streamline saddle. The base region is examined in more detail than in previous studies, and vorticity saddles, zero-vorticity points, and streamline saddles are observed to differ in importance at different stages of the shedding process.

When an air bubble rises in a viscoelastic fluid there is a critical capillary number for cusping and jump in velocity: when the capillary number is below critical, which is about 1 in our data, there is no cusp at the tail of a (smooth) air bubble. For larger volumes, a two-dimensional cusp, sharp in one view and broad in the orthogonal view, is in evidence. Measurements suggest that the cusp tip is in the generic form y = ax2/3 satisfied by analytic cusps. The intervals of volumes for which dramatic changes in air bubble shape take place is very small and the two to ten fold increase in the rise velocity which accompanies the small change of volume could be modelled as a discontinuity. A second drag transition and an orientational transition occurred when U/c > 1 where U is the rise velocity of an air bubble and c is the shear wave speed. For U/c < 1, U is proportional to d2, where d is the equivalent diameter for a sphere of diameter d having the same volume, and when U/c > 1 then U is proportional to d and the Deborah number does not change with U. Moreover the bubble shapes when U/c < 1 are overall prolate (with or without a cusped tail) with the long side parallel to gravity, in contrast to the oblate shapes which are always observed in Newtonian fluids and in viscoelastic fluids with U/c > 1 when inertia is dominant. The formation of cusps occurs in all kinds of columns of different sizes and shapes. Cusping is generic but the orientation of the broad edge with respect to the sidewalls is an issue. There is no preferred orientation in columns with round cross-sections, or in the case of walls far away from the rising bubble. In columns with rectangular cross-sections, three relatively stable configurations can be observed: the cusp can be observed in the wide window and the broad edge in the narrow window; the cusp can be observed in the narrow window and the broad edge in the wide window or, less frequently, the broad edge lies along a diagonal. These orientational and drag alternatives are directly analogous to those which are observed in the settling of long or broad solid bodies (Liu & Joseph 1993).

The subharmonic resonance phenomenon is studied using hot-wire measurements and flow visualization in an initially laminar shear layer forced with two-frequencies for various choices of the fundamental frequency f and its subharmonic f/2 with controlled initial phase difference ϕin between them. We explore the effects of the controlling parameters, namely: (i) forcing frequencies and their initial amplitudes, (ii) initial phase difference ϕin, and (iii) detuning (i.e. when the second forcing frequency is slightly different from f/2). While several of our experimental observations support predictions based on weakly nonlinear theory, others do not. We explain our data in terms of vortex dynamics concepts.