Using the continuum model of Pedley, Hill & Kessler (1988) for bioconvection in a suspension of swimming, gyrotactic micro-organisms, we investigate the existence and stability of a two-dimensional plume in tall, narrow chambers with stress-free sidewalls. The system is governed by the Navier–Stokes equations for an incompressible fluid coupled with a micro-organism conservation equation. These equations are solved numerically using a conservative finite-difference scheme. In sufficiently deep chambers, the plume is always unstable to both varicose and meandering modes. A linear stability analysis for an infinitely long plume predicts the growth rates of these instabilities, explains the mechanisms, and is in good agreement with the numerical results.

The influence of inertia and elasticity on the onset and stability of Taylor-vortex flow (TVF) is examined for an Oldroyd-B fluid. The Galerkin projection method is used to obtain the departure from Couette flow (CF). Only axisymmetric flow is examined. The solution is capable of capturing the dynamical behaviour observed experimentally for viscoelastic fluids in the inertio-elastic and purely elastic ranges. For flow with dominant inertia, the bifurcation picture is similar to that for a Newtonian fluid. However, transition from CF to TVF is oscillatory because of fluid elasticity. Steady TVF sets in, via supercritical bifurcation, as Re reaches a critical value, Rec. The critical Reynolds number decreases with fluid elasticity, and is strongly influenced by fluid retardation. As elasticity exceeds a critical level, a subcritical bifurcation emerges at Rec, similar to that predicted by the Landau–Ginzburg equation. It is found that slip along the axial direction tends to be generally destabilizing. The coherence of the formulation is established under steady and transient conditions through comparison with exact linear stability analysis, experimental measurements, and flow visualization. Good agreement is obtained between theory and the measurements of Muller et al. (1993) in the limit of purely elastic overstable TVF.

In this paper, an experimental study of laminar magnetohydrodynamic (MHD) buoyancy-driven flow in a cylindrical cell with axis horizontal is described. A steady uniform magnetic field is applied vertically to the mercury-filled cell, which is also subjected to a horizontal temperature gradient. The main features of this internal MHD thermogravitational flow are made experimentally evident from temperature and electric potential measurements. Whatever the level of convection, raising the Hartmann number Ha to a value of the order of 10 is sufficient to stabilize an initially turbulent flow. At much higher values of the Hartmann number (Ha∼100) the MHD effects cause a change of regime from boundary-layer driven to core driven. In this latter regime an inviscid inertialess MHD core flow is bounded by a Hartmann layer on the horizontal cylindrical wall and viscous layers on the endwalls. Since the Hartmann layer is found to stay electrically inactive along the cell, the relevant asymptotic (Ha[Gt ]1) laws for velocity and heat transfer are found from the balance between the curl of buoyancy and Lorentz forces in the core, together with the condition that the flow of electric current between core and Hartmann layer is negligible. A modified Rayleigh number RaG/Ha2, which is a measure of the ratio of thermal convection to diffusion when there is a balance between buoyancy and Lorentz forces, is the determining parameter for the flow.

We present simulation results of vortex-induced vibrations of an infinitely long flexible cylinder at Reynolds number Re = 1000, corresponding to a ‘young’ turbulent wake (i.e. exhibiting a small inertial subrange). The simulations are based on a new class of spectral methods suitable for unstructured and hybrid grids. To obtain different responses of the coupled flow–structure system we vary the structure's bending stiffness to model the behaviour of a vibrating inflexible (rigid) cylinder, a cable, and a beam. We have found that unlike the laminar flow previously studied, the amplitude of the cross-flow oscillation is about one diameter for the cable and the beam, close to experimental measurements, but is lower for the rigid cylinder. We have also found that for the latter case the flow response corresponds to parallel shedding, but for the beam and cable with free endpoints a mixed response consisting of oblique and parallel shedding is obtained, caused by the modulated travelling wave motion of the structure. This mixed shedding pattern which alternates periodically along the span can be directly related to periodic spatial variation of the lift force. In the case of structures with pinned endpoints a standing wave response is obtained for the cylinder; lace-like flow structures are observed similar to the ones seen in the laminar regime. Examination of the frequency spectra in the near wake shows that at Re = 1000 all cases follow a −5/3 law in the inertial range, which extends about half a decade in wavenumber. However, these spectra are different in all three cases both in low and high frequencies, with the exception of the beam and cable, for which the high-frequency portion is identical despite the differences in the displacement time history and the large-scale features of the corresponding flow.

The incompressible boundary layer in the corner formed by two intersecting, semi-infinite planes is investigated, when the free-stream flow, aligned with the corner, is taken to be of the form U∞F(x), x representing the non-dimensional streamwise distance from the leading edge. In Dhanak & Duck (1997) similarity solutions for F(x) = xn were considered, and it was found that solutions exist for only a range of values of n, whilst for ∞ > n > −0.018, approximately, two solutions exist. In this paper, we extend the work of Dhanak & Duck to the case of non-90° corner angles and allow for streamwise development of solutions. In addition, the effect of transpiration at the walls of the corner is investigated. The governing equations are of boundary-layer type and as such are parabolic in nature. Crucially, although the leading-order pressure term is known a priori, the third-order pressure term is not, but this is nonetheless present in the leading-order governing equations, together with the transverse and crossflow viscous terms.

Particular attention is paid to flows which develop spatially from similarity solutions. It turns out that two scenarios are possible. In some cases the problem may be treated in the usual parabolic sense, with standard numerical marching procedures being entirely appropriate. In other cases standard marching procedures lead to numerically inconsistent solutions. The source of this difficulty is linked to the existence of eigensolutions emanating from the leading edge (which are not present in flows appropriate to the first scenario), analogous to those found in the computation of some two-dimensional hypersonic boundary layers (Neiland 1970; Mikhailov et al. 1971; Brown & Stewartson 1975). In order to circumvent this difficulty, a different numerical solution strategy is adopted, based on a global Newton iteration procedure.

A number of numerical solutions for the entire corner flow region are presented.

Using direct numerical simulations (DNS) and large-eddy simulations (LES) of velocity and passive scalar in isotropic turbulence (up to 5123 grid points), we examine directly and quantitatively the refined similarity hypotheses as applied to passive scalar fields (RSHP) with Prandtl number of order one. Unlike previous experimental investigations, exact energy and scalar dissipation rates were used and scaling exponents were quantified as a function of local Reynolds number. We first demonstrate that the forced DNS and LES scalar fields exhibit realistic inertial-range dynamics and that the statistical characteristics compare well with other numerical, theoretical and experimental studies. The Obukhov–Corrsin constant for the k−5/3 scalar variance spectrum obtained from the 5123 mesh is 0.87±0.10. Various statistics indicated that the scalar field is more intermittent than the velocity field. The joint probability distribution of locally-averaged energy dissipation εr and scalar dissipation χr is close to log-normal with a correlation coefficient of 0.25±0.01 between the logarithmic dissipations in the inertial subrange. The intermittency parameter for scalar dissipation is estimated to be in the range 0.43≈0.77, based on direct calculations of the variance of lnχr. The scaling exponents of the conditional scalar increment δrθ[mid ] χr,εr suggest a tendency to follow RSHP. Most significantly, the scaling exponent of δrθ[mid ] χr,εr over εr was shown to be approximately −⅙ in the inertial subrange, confirming a dynamical aspect of RSHP. In agreement with recent experimental results (Zhu et al. 1995; Stolovitzky et al. 1995), the probability distributions of the random variable βs = δrθ[mid ] χr,εr/ (χ1/2r ε−⅙rr1/3) were found to be nearly Gaussian. However, contrary to the experimental results, we find that the moments of βs are almost identical to those for the velocity field found in Part 1 of this study (Wang et al. 1996) and are insensitive to Reynolds number, large-scale forcing, and subgrid modelling.

The collisional flow of a slightly inelastic granular material down a rough inclined plane is usually described by kinetic theories. We present an experimental study aimed at analysing the assumptions and the quantitative predictions of such theories. A two-dimensional channel coupled to a model granular material and image analysis allow detailed and complete measurement of the kinematics and structure of the flows. We determine the range of inclination and particle flux for which the flow is stationary and uniform. The characteristic profiles of solid fraction, mean velocity and granular temperature are systematically measured. Both the true collisional and the dilute kinetic regimes are examined. We show that a quasi-hydrodynamic description of these regimes seems relevant, and that the pressure and the viscosity terms are in good qualitative agreement with the prediction of the kinetic theory. The profiles are well described by the kinetic theory near the top of the flow, at low solid fraction. Conversely there are large discrepancies near the rough plane, where the material is structured in layers.

A linear stability analysis is performed for the homogeneous state of a monodisperse gas-fluidized bed of spherical particles undergoing hydrodynamic interactions and solid-body collisions at small particle Reynolds number and finite Stokes number. A prerequisite for the stability analysis is the determination of the particle velocity variance which controls the particle-phase pressure. In the absence of an imposed shear, this velocity variance arises solely due to the hydrodynamic interactions among the particles. Since the uniform state of these suspensions is unstable over a wide range of values of particle volume fraction φ and Stokes number St, full dynamic simulations cannot be used in general to characterize the properties of the homogeneous state. Instead, we use an asymptotic analysis for large Stokes numbers together with numerical simulations of the hydrodynamic interactions among particles with specified velocities to determine the hydrodynamic sources and sinks of particle-phase energy. In this limit, the velocity distribution to leading order is Maxwellian and therefore standard kinetic theories for granular/hard-sphere molecular systems can be used to predict the particle-phase pressure and rheology of the bed once the velocity variance of the particles is determined. The analysis is then extended to moderately large Stokes numbers for which the anisotropy of the velocity distribution is considerable by using a kinetic theory which combines the theoretical analysis of Koch (1990) for dilute suspensions (φ [Lt ] 1) with numerical simulation results for non-dilute suspensions at large Stokes numbers. A linear stability analysis of the resulting equations of motion provides the first a priori predictions of the marginal stability limits for the homogeneous state of a gas-fluidized bed. Dynamical simulations following the detailed motions of the particles in small periodic unit cells confirm the theoretical predictions for the particle velocity variance. Simulations using larger unit cells exhibit an inhomogeneous structure consistent with the predicted instability of the homogeneous gas–solid suspension.

A combined numerical/analytical study of the wide-vortex/wave interaction equations, describing boundary-layer instability, is presented. Depending on the obliqueness β of the wave input, different solution properties are obtained. For β = 1, oscillations in the wave amplitude lead to the evolution of a strongly three-dimensional mean flow, while for β = 2 the interaction is characterized by the development of a singularity in the wave pressure amplitude. This latter behaviour is modelled using an approximate form for the mean flow skin friction and the resulting amplitude equation is analysed using a combination of numerical and asymptotic techniques. A simple method is described for determining the singularity location for a given spanwise wavenumber, and the asymptotic behaviour of the pressure amplitude as the singularity is approached is deduced.

A generalized model of flow in meandering subaqueous and subaerial channels is developed. The conservation equations of mass and momentum are depth/layer integrated, normalized, and represented as deviations from a straight base state. This allows the determination of integrable forms which can be solved at both linear and nonlinear levels. The effects of various flow and geometric parameters on the flow dynamics are studied. Although the model is not limited to any specific planform, this study focuses on sine-generated curves. In analysing the flow patterns, the turbidity current of the subaqueous case is simplified to a conservative density flow with water entrainment from above neglected. The subaqueous model thus formally corresponds to a subcritical or only mildly supercritical mud-rich turbidity current. By extension, however the analysis can be applied to a depositional or erosional current carrying sand that is changing only slowly in the streamwise direction. By bringing the subaqueous and subaerial cases into a common form, flow behaviour in the two environments can be compared under similar geometric and boundary conditions. A major difference between the two cases is the degree of superelevation of channel flow around bends, which is modest in the subaerial case but substantial in the subaqueous case. Another difference concerns Coriolis effects: some of the largest subaqueous meandering systems are so large that Coriolis effects can become important. The model is applied to meander bends on the youngest channel in the mid-fan region of the Amazon Fan and a mildly sinuous bend of the North-West Atlantic Mid-Ocean Channel. In the absence of specific data on the turbid flows that created the channel, the model can be used to make inferences about the flow, and in particular the range of values of flow velocity and sediment concentration that would allow the growth and downfan migration of meander bends.

A three-dimensional mathematical model based on the Brinkman extended Darcy equation has been used to study double-diffusive natural convection in a fluid-saturated porous cubic enclosure subject to opposing and horizontal gradients of temperature and concentration. The flow is driven by conditions of constant temperature and concentration imposed along the two vertical sidewalls of the cubic enclosure, while the remaining walls are impermeable and adiabatic. The numerical simulations presented here span a wide range of porous thermal Rayleigh number, buoyancy ratio and Lewis number to identify the different steady-state flow patterns and bifurcations. The effect of the governing parameters on the domain of existence of the three-dimensional flow patterns is studied for opposing flows (N < 0). Comprehensive Nusselt and Sherwood number data are presented as functions of the governing parameters. The present results indicate that the double-diffusive flow in enclosures with opposing buoyancy forces is strictly three-dimensional for a certain range of parameters. At high Lewis numbers multiple dipole vortices form in the transverse planes near the horizontal top and bottom surfaces, which the two-dimensional models fail to detect. The dipolar vortex structures obtained are similar to those created in laboratory experiments by the injection of fluid into a stratified medium.

This paper presents the results of experimental studies of a collapsing cylindrical cavity (the convergence of a liquid shell) in a rotating fluid as well as the formation and propagation of a jump (bore) at the interface. The basic parameters of the liquid shell dynamics for a pulsed one-dimensional load are estimated using the equation of cylindrical cavity pulsation in an unbounded fluid. The theoretical model of a rotationally symmetric hydraulic jump moving along the free surface of a hollow vortex is constructed. The jump is simulated by a discontinuous solution of the equations in the long-wave approximation for tornado-like and hollow vortices. For comparison with the experimental data, basic theoretical results are obtained for flows in a hollow vortex with constant circulation and axial velocity varying along the radius of the rotating liquid shell.