The onset of natural convection is considered in a vertically oriented, thin, finite slab of saturated porous media when sidewall heat transfer exists. First, a linear stability analysis is carried out for a system with impermeable boundaries. The sidewall temperature increases linearly with depth while the smaller-area endwalls are insulated. Convection occurs when the Rayleigh number R is asymptotically large relative to the inverse square of the horizontal aspect ratio, H2 [Lt ] 1. The convection pattern is composed of an integer number of vertically oriented three-dimensional, finger-like cells. The wavelength of each cell, relative to the larger horizontal dimension of the slab, is proportional to H2½. This somewhat surprising type of modal configuration is also found when there is a specified vertical mass flux through the slab. In this second example one considers the characteristics of the 3-dimensional fully developed solution for the thin vertical-slab problem which is compatible with a linear temperature increase on the vertical walls. When R is like that found in the first problem, closely spaced finger-like cells are found superimposed on the generally upward fluid flow. It is concluded that sidewall heat loss has a very strong stabilizing effect on the initiation of buoyancy-induced convection relative to the more traditional situation where side- and endwalls are insulated. Furthermore the appearance of slender, finger-like convection cells is characteristic of motion in a narrow vertical-slab configuration. Finally it is noted that the precise modal configuration selected by a system is extremely sensitive to the value of the Rayleigh number.

The linearized boundary-value problem for surface waves of frequency ω in a closed basin of variable depth is reduced to a non-self-adjoint partial differential equation in the plane of the free surface. The corresponding variational form (which does not provide a definite upper or lower bound) for the eigenvalue κ = ω2/g is constructed. A self-adjoint partial differential equation, for which the variational form is the Rayleigh quotient (which provides an upper bound to κ), also is constructed; it offers significant advantages vis-à-vis the non-self-adjoint formulation, but at the expense of a more complicated operator. Three relatively simple variational approximations are constructed, two for a class of basins with sloping sides and the third for basins for which the variation of the depth relative to its mean is small. These general results are illustrated by comparison with Rayleigh's (1899) results for a semicircular channel, Sen's (1927) inverse results for a family of circular basins, and Lamb's (1932) results for a shallow circular paraboloid. The eigenvalue for the dominant mode in the paraboloid is determined through O(δ5), where δ = depth/radius.

Fully three-dimensional numerical solutions are presented for Rayleigh-Bénard convection subject to stress-free boundary conditions. A motion with square planform is studied for varying Rayleigh number R and Prandtl number σ. It may be understood partly in terms of a truncated modal representation (after Lorenz 1963). Thermal layers of unusual structure are found at high R. For small σ, steady solutions exist, but are not of ‘flywheel’ type, and the heat transport depends strongly on σ. The study also verifies that laminar convective flows may be ergodic.

The influence of periodic perturbations on the development of two-dimensional free shear layers generated by a splitter plate was investigated in cases where the ratios of the two velocities u1 and u2 either side of the splitter plate were such that 0 < u1/u2 < 1. Investigations were carried out in both a suction and a blower wind tunnel. Results show that even very weak periodic perturbations caused by the wind tunnel may cause significant nonlinear spreading in the downstream development of the shear layer, a behaviour which is also observed when the shear layer is deliberately excited. Other things being equal, the effect of the disturbance is greater when flow separation at the splitter plate is turbulent than when it is laminar.

No self-induced feedback frequencies were measured in the test section. All tonal components that were detected in the flow could be traced to external sources.

The influence of trailing-edge thickness on the shear-layer development is found to become significant when it exceeds 50% of the sum of boundary-layer displacement thickness at the point of separation. As the trailing edge becomes thicker, the range over which the shear layer is self-similar is shifted farther downstream. This behaviour may be crucial for predicting the evolution of shear layers in high-speed flows having thin boundary layers at separation.

The momentum thickness criterion for estimating the development length of the flow as suggested by Bradshaw is shown to be insufficient for two-stream layers, where additional parameters, e.g. the trailing-edge geometry, have to be taken into account. Discrepancies between previously published observations of shear layers, as well as the considerable scatter in reported measurements, may therefore, to a large extent, be attributable to contamination of the experimental facility.

The mortality of bacteria in sewage discharges into shallow seas is strongly influenced by light and varies with distance from the free surface. Here a criterion is derived for deciding between the opposing strategies of getting rapid bacterial decay by directing the sewage up to the surface at the sacrifice of initial dilution, or of achieving high initial dilution with a distributed source along the bed perpendicular to the flow direction.

A laser-Doppler anemometer (LDA) was used to measure turbulent velocities in drag-reducing fibre suspensions. Measurements of streamwise velocities (and, in one case, the circumferential velocity as well) were made in flow through a straight pipe at x/d = 190, and at Reynolds numbers in the range 1.4 × 104–5.3 × 104. The fibres used were chrysotile asbestos of high aspect ratio (∼ 106), at a concentration of 300 w.p.p.m. They were dispersed in an aqueous solution of a surfactant (0.5% by weight Aerosol OT). In some experiments, the fibre suspensions were supplemented by a drag-reducing polymer (Separan AP30) at a concentration of 150 w.p.p.m. A complete experiment involved passing a quantity of fibre suspension through the apparatus a number of times (at a given Reynolds number) and measuring the velocity distribution across the pipe during each pass. As the amount of drag reduction generally declined with the number of passes (i.e. due to fibre degradation), this provided a convenient way of varying the percentage drag reduction as an experimental parameter. Results were obtained for mean velocity and intensity profiles, autocorrelations, and one-dimensional energy spectra. The mean period of turbulent bursts was determined by measuring autocorrelations with short sampling times.

At the lowest Reynolds number (Re = 1.4 × 104), drag reductions of about 70% were obtained during the first two passes. This was accompanied by a reduction in the streamwise intensity below the level obtained in the surfactant solution alone. (Note: The opposite behaviour is found in drag-reducing polymer solutions, where intensity levels are larger than those in the solvent alone.) A measurement of the r.m.s. circumferential velocity showed an increased level (relative to surfactant alone) during this part of the experiment. During further passes, there was a transition to ‘polymer-like’ behaviour, with increased streamwise intensity, which subsequently declined with pass number (and hence drag reduction) towards the result for surfactant alone. This effect had previously been found in preliminary experiments at Re = 9 × 103 (McComb & Chan 1979). Repetition of the experiment a Re = 1.4 × 104, with the addition of Separan AP30, confirmed the existence of this transition from ‘fibre-like’ to ‘polymer-like’ drag reduction. In this case, the drag reduction was smaller (at about 60%), but the mixed suspension was much more resistant to degradation, with transition occurring at the ninth pass. However, such behaviour was not found at higher Reynolds numbers (Re = 3.2 × 104 and 5.3 × 104), in fibre suspensions where increased streamwise intensities occurred, even at high levels of drag reduction (about 70%).

Anomalous streamwise autocorrelations were found during ‘fibre-like’ drag reduction but in the ‘polymer-like’ regime they were very similar to those measured in polymer solution, and showed characteristically increased lengthscales. On the other hand, energy spectra were found to be anomalous in all cases and showed an energy deficit at lengthscales of the same order as the fibre length. Finally, mean bursting periods were found to be much increased, with the increases being about the same as those in polymer solutions at the same Reynolds number and percentage drag reduction.

The fundamental properties of nonlinear progressive edge waves are investigated experimentally in a physical model with a uniformly and mildly sloping beach with a straight shoreline. An evolution equation for the envelope of progressive edge waves is the nonlinear Schrödinger (NLS) equation. It is found that the timescale of viscous-dissipation effects in the experiments is comparable with the timescale of the theoretical evolution process for inviscid progressive edge waves. This lack of timescale separation indicates a major shortcoming of the NLS equation as a model of the laboratory experiments. Even with this limitation, a uniform train of edge waves is found to be unstable to a modulational perturbation as predicted by the NLS equation. However, behaviour of the evolution is both qualitatively and quantitatively different from the theoretical predictions. The evolution of the periodogram for the unstable wavetrain shows an asymmetric development of the sidebands about the fundamental frequency; instability growth is limited to the lower sideband. This behaviour leads to a sequential shift of wave energy to lower frequencies as the waves propagate. It is found that a locally soliton-shaped wave packet is unstable in the laboratory environment. It is estimated that a much-larger-scale experimental facility is required to achieve inviscid experiments for the NLS equation.