A macroscopic equation of mass conservation is obtained by ensemble-averaging the basic conservation laws in a porous medium. In the long-time limit this ‘macro-transport’ equation takes the form of a macroscopic Fick's law with a constant effective diffusivity tensor. An asymptotic analysis in low volume fraction of the effective diffusivity in a bed of fixed spheres is carried out for all values of the Péclet number ℙ = Ua/Df, where U is the average velocity through the bed. a is the particle radius and Df is the molecular diffusivity of the solute in the fluid. Several physical mechanisms causing dispersion are revealed by this analysis. The stochastic velocity fluctuations induced in the fluid by the randomly positioned bed particles give rise to a convectively driven contribution to dispersion. At high Péclet numbers, this convective dispersion mechanism is purely mechanical, and the resulting effective diffusivities are independent of molecular diffusion and grow linearly with ℙ. The region of zero velocity in and near the bed particles gives rise to non-mechanical dispersion mechanisms that dominate the longitudinal diffusivity at very high Péclet numbers. One such mechanism involves the retention of the diffusing species in permeable particles, from which it can escape only by molecular diffusion, leading to a diffusion coefficient that grows as ℙ2. Even if the bed particles are impermeable, non-mechanical contributions that grow as ℙ ln ℙ and ℙ2 at high ℙ arise from a diffusive boundary layer near the solid surfaces and from regions of closed streamlines respectively. The results for the longitudinal and transverse effective diffusivities as functions of the Péclet number are summarized in tabular form in §6. Because the same physical mechanisms promote dispersion in dilute and dense fixed beds, the predicted Péclet-number dependences of the effective diffusivities are applicable to all porous media. The theoretical predictions are compared with experiments in densely packed beds of impermeable particles, and the agreement is shown to be remarkably good.

Although models of the pressure–strain term explain many features of nearly uniform homogeneous shear flows, a discrepancy remains (Leslie 1980). It is suggested that the discrepancy is caused by use of an empirical expression for the fluctuation part of the pressure–strain term, the part usually denoted by ϕij,1. The discrepancy is eliminated by replacement of the empirical ϕij,1 with a recent theoretical expression. Relatedly, the Launder, Reece & Rodi (1975) model for the mean-field part ϕij,2 is shown to be a good approximation for both a strongly and weakly sheared flow. This model of ϕij,2 when combined with the theoretical ϕij,1 is found to provide an explanation for experiments of both Champagne, Harris & Corrsin (1970) and Harris, Graham & Corrsin (1977). Full correction requires that deviations from local isotropy be accounted for. Special emphasis is given to a theoretical demonstration that the pressure–strain term does not cause a retun to isotropy but, rather, it resists large anisotropy – a weaker effect.

A central role in the mechanism of the self-sustained oscillations of the flow about cavity-type bodies is played by the reattachment edge. Experimentally it has been found that periodic pressure pulses generated on this edge are fed back to the origin of the shear layer and cause the production of discrete vortices. The oscillations have been found to be suppressed or attenuated when the edge has the shape of a ramp of small angle, or when it is properly rounded. To clarify the role of the shape of the reattachment edge in the mechanism of the oscillations, a mathematical model is developed for the vortex–edge interaction. In this model the interaction of one discrete vortex, imbedded within a constant-speed parallel flow, with the reattachment edge is studied. Two typical shapes of the reattachment edge are examined; a ramp of variable angle and an ellipse. The main conclusion of the present analysis is the strong dependence of the pressure pulses, that are induced on the surface of the edge, on the specific shape of the edge. The pressure pulses on reattachment edges with shapes that give rise to steady flows have been found to be of insignificant amplitude. On the other hand, when the reattachment edge has a shape that is known to result in oscillating flow, the induced pressure pulses are of very large amplitude. Intermediate values of the pressure are found for configurations known to stabilize partially the flow. The present results indicate that, for the establishment of the oscillation, the feedback force generated by the vortex–edge interaction must have an appropriate value. The feedback force may be eliminated if the shape of the lip of the edge is properly designed.

This paper describes experiments concerning the structure of large-scale vortices and the unsteady reverse-flow properties in the reattaching zone of a nominally two-dimensional separation bubble formed at the leading edge of a blunt flat plate with right-angled corners. The experiment was performed in a wind tunnel with a constant Reynolds number 2.6 × 104 (based on the main-flow velocity and the thickness of the plate). Split-film probes, being sensitive to instantaneous reversals of flow direction, were extensively employed. An important feature of this study is a judicious use of surface-pressure fluctuations as a conditioning signal to educe the structure of the large-scale vortices.

Distributions of fluctuating-velocity vectors and contour lines of high-frequency turbulent energy in a few space–time domains are presented and discussed. The most economical interpretation of these space-time distributions is that the large-scale vortices in the reattaching zone are hairpin vortices whose configuration is sketched in the text. The unsteady flow in the reattaching zone is mainly governed by two agents; the motion of the large-scale vortices and the low-frequency unsteadiness. The unsteady flow is clarified in terms of the motion (in a space–time domain) of zeros of the longitudinal velocity close to the surface of the plate; the effects of the two agents on this motion are presented separately. On the basis of these results, a mathematical model of the unsteady flow in the reattaching zone is suggested and found to yield good comparison with measured reverse-flow intermittency and frequency of local-flow reversals. It appears that the separation bubble experiences shrinkage and enlargement in connection with the low-frequency unsteadiness and that the speed of shrinkage is much greater than that of enlargement. The strength of the large-scale vortices in the reattaching zone seems to be dependent on the phase of the low-frequency unsteadiness.

The conjecture of Vainshtein & Zel'dovich (1972) concerning the existence of a fast dynamo (i.e. one whose growth rate is independent of magnetic diffusivity η in the limit η → 0) is discussed with particular reference to (i) the stretch–twist–fold cycle which can double the strength of a magnetic flux tube, and (ii) the space-periodic Beltrami flow of maximal helicity, which has been shown to be capable of space-periodic dynamo action with the same period as the velocity field, by Arnold & Korkina (1983) and by Galloway & Frisch (1984). The topological constraint associated with conservation of magnetic helicity is shown to preclude fast dynamo action unless the scale of the magnetic field is almost everywhere of order η½ as η → 0; in this case, the field structure is severely singular in the limit. A steady incompressible velocity field, quadratic in the space variables, is shown to mimic the action of the stretch–twist–fold cycle, and is proposed as a plausible candidate for fast dynamo action.

By using the triple-deck scaling of Stewartson (1969) and Messiter (1970) we show that small but relatively sudden surface geometry variations that produce only very weak static pressure variations can nevertheless produce strong, i.e. 0(1), coupling between an externally imposed acoustic disturbance and a spatially growing Tollmien- Schlichting wave. The analysis provides a qualitative explanation of the Leehey & Shapiro (1979) boundary-layer receptivity measurements and is in good quantitative agreement with the Aizin & Polyakov (1979) experiment. It may also explain why small ‘trip wires’ can promote early transition.