The motion of a heavy sphere sedimenting through a dilute background suspension of neutrally buoyant spheres is analysed for small Reynolds number and large Péclet number. For this particular problem, it is possible not only to calculate the mean velocity of the heavy particle, but also the variance of the velocity and the coefficient of hydrodynamic diffusivity. Pairwise, hydrodynamic interactions between the heavy sphere and the background sphere are considered exactly using volume integrals and a trajectory analysis. Explicit formulae are given for the two limiting cases when the radius of the heavy sphere is much greater and much less than that of the background spheres, and numerical results are given for moderate size ratios. The mean velocity is relatively insensitive to the ratio of the radius of the background spheres to that of the heavy sphere, unless this ratio is very large, whereas the hydrodynamic diffusivity increases rapidly as the radius ratio is increased. The predictions are in reasonable agreement with the results of falling-ball rheometry experiments.

It is shown that the fundamental features of both thermal instabilities and the corresponding nonlinear convection in rapidly rotating spherical systems (in the range of the Taylor number 109 < T < 1012) are determined by the fluid properties characterized by the size of the Prandtl number. Coefficients of the asymptotic power law for the onset of convection at large Taylor number are estimated in the range of the Prandtl number 0.1 ≤ Pr ≤ 100. For fluids of moderately small Prandtl number, a new type of convective instability in the form of prograde spiralling drifting columnar rolls is discovered. The linear columnar rolls extend spirally from near latitude 60° to the equatorial region, and each spans azimuthally approximately five wavelengths with the inclination angle between a spirally elongated roll and the radial direction exceeding 45°. As a consequence, the radial lengthscale of the linear roll becomes comparable with the azimuthal lengthscale. A particularly significant finding is the connection between the new instability and the predominantly axisymmetric convection. Though non-axisymmetric motions are preferred at the onset of convection, the nonlinear convection (at the Rayleigh number of the order of (R—Rc)/Rc = O(0.1)) bifurcating supercritically from the spiralling mode is primarily dominated by the component of the axisymmetric zonal flow, which contains nearly 90% of the total kinetic energy. For fluids of moderately large Prandtl numbers, thermal instabilities at the onset of convection are concentrated in a cylindrical annulus coaxial with the axis of rotation; the position of the convection cylinder is strongly dependent on the size of the Prandtl number. The associated nonlinear convection consists of predominantly non-axisymmetric columnar rolls together with a superimposed weak mean flow that contains less than 10% of the total kinetic energy at (R—Rc)/Rc = O(0.1). A double-layer structure of the temperature field (with respect to the basic state) forms as a result of strong nonlinear interactions between the nonlinear flow and the temperature field. It is also demonstrated that the aspect ratio of the spherical shell does not substantially influence the fundamental properties of convection.

The evolution of the cellular structure of the two-dimensional creeping flow induced by a rotating circular cylinder set in the centre of a rectangular channel is studied numerically and experimentally when the aspect ratio A increases from 1 to 7. In the calculations, depending on the value of A, either only series in terms of polar coordinates, or both matched polar and Cartesian coordinates series are employed to represent the stream function and an efficient least-squares method, very easy to program, is selected to satisfy some of the boundary conditions. For the experiments, a special technique which visualizes intermittently the paths of solid tracers during long times of exposure permits us to observe the fluid motion in the whole domain, even in the regions where the velocities are very small. An excellent measure of agreement between the numerical and experimental results is found. Thus it is clearly shown how, in the region beyond the rotating flow directly driven by the cylinder, the two main corner cells visualized at A = 1, develop with increasing A and then coalesce, to finally merge and give rise to a single central cell. This central cell develops in its turn, tending finally to the unbounded channel reference cell, after passing through a maximum length however. Owing to the very high precision of the calculations, many details of the flow development have been clearly shown, in particular the periodicity, with increasing A, of all the different phases, progressively inducing a succession of cells. The prediction that the angle of separation of the fluid boundaries of the cells tends towards the theoretical limit of 58.61° when the aspect ratio becomes large is also confirmed.

Low-Reynolds-number effects are observed in the inner region of a fully developed turbulent channel flow, using data obtained either from experiments or by direct numerical simulations. The Reynolds-number influence is observed on the turbulence intensities and to a lesser degree on the average production and dissipation of the turbulent energy. In the near-wall region, the data confirm Wei & Willmarth's (1989) conclusion that the Reynolds stresses do not scale on wall variables. One of the reasons proposed by these authors to account for this behaviour, namely the ‘geometry’ effect or direct interaction between inner regions on opposite walls, was investigated in some detail by introducing temperature at one of the walls, both in experiment and simulation. Although the extent of penetration of thermal excursions into the opposite side of the channel can be significant at low Reynolds numbers, the contribution these excursions make to the Reynolds shear stress and the spanwise vorticity in the opposite wall region is negligible. In the inner region, spectra and co-spectra of the velocity fluctuations u and v change rapidly with the Reynolds number, the variations being mainly confined to low wavenumbers in the u spectrum. These spectra, and the corresponding variances, are discussed in the context of the active/inactive motion concept and the possibility of increased vortex stretching at the wall. A comparison is made between the channel and the boundary layer at low Reynolds numbers.

The supersonic flow past a fin mounted on a flat plate is simulated numerically by solving the Reynolds averaged Navier—Stokes equations. The results agree well with the experimental data. Post-processing of the numerical solution provides the missing flow-field evidence for confirming the currently accepted flow model, whose conception was based mainly on surface data. It is found that the flow is dominated by a large vortical structure, which lies on the plate and whose core has a remarkably conical shape with flattened elliptical cross-section. Along the fin and close to the corner, a slowly growing smaller vortex develops. On top of the conical vortex and along it a λ-shock is formed. Quantitative data are presented, which show that the flow is not actually purely conical but a small deviation exists, especially at the part between the separation shock and the plate. This deviation is detected when the stream wise extent of the flow is more than 20–30 initial boundary-layer thicknesses. Owing to the rather quasi-conical nature of the flow, the various flow variables do not remain constant along rays that start at the origin of the conical flow field, but they vary slowly. Data are presented which support the view that this deviation from conical behaviour is mainly due to the effect of the smaller rate of development of the boundary later of the plate, compared to the conical vortex.

A two-dimensional disturbance evolving from a strictly linear, finite-growth-rate instability wave with nonlinear effects first becoming important in the critical layer is considered. The analysis is carried out for a general weakly non-parallel mean flow using matched asymptotic expansions. The flow in the critical layer is governed by a nonlinear vorticity equation which includes a spatial-evolution term. As in Goldstein & Hultgren (1988), the critical layer ages into a quasi-equilibrium one and the initial exponential growth of the instability wave is converted into a weak algebraic growth during the roll-up process. This leads to a next stage of evolution where the instability-wave growth is simultaneously affected by mean-flow divergence and nonlinear critical-layer effects and is eventually converted to decay. Expansions for the various streamwise regions of the flow are combined into a single composite formula accounting for both shear-layer spreading and nonlinear critical-layer effects and good agreement with the experimental results of Thomas & Chu (1989), Freymuth (1966), and C.-M. Ho & Y. Zohar (1989, private communication) is demonstrated.

Planar motion produced when a viscous fluid is forced from an initial state of rest is studied. We consider a vortex dipole produced by the action of a point force (Cantwell 1986), and a vortex quadrupole produced by the action of two equal forces of opposite direction. We also present results from an experimental investigation into the dynamics of the interactions between vortex dipoles as well as between vortex dipoles and a vertical wall in a stratified fluid. Theoretical consideration reveals that the dynamics of two-dimensional vortex-dipole interactions are determined by two main governing parameters: the dipolar intensity of the vorticity distribution (momentum) and the quadrupolar intensity of the vorticity distribution of the flow. We document details of different basic types of interactions and present a physical interpretation of the results obtained in terms of vortex multipoles: dipoles, quadrupoles and their combinations.