The present work is a numerical study of fluid flow and energy transport in single rock fractures. Each fracture is modelled as a parallel channel with a distribution of contact areas where the fracture aperture is zero. The pressure drop required to sustain a given rate of flow is calculated by solving the Stokes equations along with the incompressibility constraint. The permeability of the fracture is then determined as the ratio of mean flow and pressure drop. With flow distribution known, the advection–diffusion equation is solved for a thermal mixing problem. All differential equations have been solved by the finite-element technique.

Results show that the fracture permeability is affected by the presence of contact areas and the magnitude of h, the mean fracture aperture. The limiting case of zero aperture is realized at h/L [les ] 0.016, where L is the characteristic dimension of the fracture. The other limiting case of a two-dimensional fracture is seen for h/L [ges ] 1. The thermal dispersion profiles at steady state are unaffected by the size of aperture h and the presence of contact areas for Péclet numbers Pe less than 50 and a contact fraction of up to 20%. However, the transient dispersion problem is seen to be influenced by the contact areas for Pe > 10.

A weakly nonlinear theory is developed for convection in an infinite rigid horizontal rectangular channel uniformly heated from below. A combination of analytical and numerical techniques along and in the cross-section of the channel leads to the derivation of an amplitude equation governing the spatial and temporal evolution of the flow above the critical Rayleigh number. Results are obtained for general Prandtl numbers and a wide range of aspect ratios. Overall trends are confirmed by comparison with results for an idealized model with stress-free horizontal boundaries. For wide channels, where the aspect ratio is large, the limiting form of the amplitude equation is predicted by reference to the two-dimensional equation describing roll patterns in infinite layers. The connection with this well-developed theory is established for both rigid and stress-free horizontal boundary conditions.

Consider the flow of an incompressible fluid between two infinite concentric circular cylinders. The outer cylinder is at rest whilst the angular velocity of the inner cylinder has a steady part and also a harmonically oscillating component. We examine the situation where, for a suitable choice of parameters, two types of vortex instability can occur simultaneously; first a short-wavelength mode which is essentially trapped in a thin ‘Stokes’ layer near the inner cylinder and, secondly, a long-wavelength mode which fills the whole region between the cylinders. We investigate the problem in which two short-wavelength vortices and one long-wavelength vortex coexist and are such that each pair interacts to drive the third. Additionally, the short-wavelength disturbances are nonlinear in their own right. Coupled amplitude equations for the three modes are derived and their solution discussed.

This form of interaction may also take place in a boundary layer. Such a situation is more complex than that under consideration here as it would be necessary to take into account the growth of the boundary layer. However, this simplified problem gives an insight into the behaviour of the more difficult situation.

The scattering of water waves by an array of N bottom-mounted vertical circular cylinders is solved exactly (under the assumption of linear water wave theory) using the method proposed by Spring & Monkmeyer in 1974. A major simplification to this theory has been found which makes the evaluation of quantities such as the forces on the cylinders much simpler. New formulae are given for the first and mean second-order forces together with one for the free-surface elevation in the vicinity of a particular cylinder. Comparisons are made between the exact results shown here and those generated using the approximate method of McIver & Evans (1984). The behaviour of the forces on the bodies in the long-wave limit is also examined for the special case of two cylinders with equal radii.

The onset and the form of surface-tension-driven convection in three different small circular and one small square container has been studied experimentally. In the smallest circular container, with increasing aspect ratio, the pattern consisted of first a circular roll and then segments of a circle outlined by different numbers of azimuthal nodal lines, with up to six segments. Simple solutions in the square container were the one-cellular pattern and a pattern consisting of four square cells. Unexpected solutions formed when the number of the cells in the square container was not a square number. When the aspect ratio permitted two cells, two triangular cells were observed. With space for three cells, one square cell and two wedge-shaped cells formed. The onset of convection in all fluid layers was characterized by a steep increase of the critical Marangoni number with decreasing aspect ratio.

Experimental studies of low-Reynolds number, pressure-driven core–annular flow in a straight capillary tube are reported. The annular film is thin compared with the radius of the tube, and the viscosity of the film fluid is much larger than the viscosity of the core fluid. Photographs show that the film is unstable under all conditions investigated in the experiment. The film fluid collects in axisymmetric lobes that are spaced periodically along the capillary wall. The spacing of the lobes and their translational velocity correspond closely with the wavelength of the most unstable disturbance and phase velocity calculated from linear stability theory. Eventually, the continued growth of the lobes results in the formation of a fluid lens that breaks the inner core.

We investigate the evolution of grid turbulence under the action of a stable (negatively buoyant), linear, temperature profile. The experiment was carried out in a large, open circuit, low-speed wind tunnel 0.91 × 0.91 m2 and 9.14 m in length specially designed for the study of stratified turbulence. The temperature gradient, formed at the entrance to the plenum chamber of the tunnel by means of an array of 72 horizontal, differentially heated elements was varied from 0 to 55 °C/m, giving a maximum Brunt–Väisälä frequency N of 1.3 s−1. The grid mesh M was 2.54 cm and the mean velocity U was varied from 2.8 to 4.2 m/s. Thus the mesh Froude number FrM = U/(NM) was varied from ∞ to 84.8 for the passive and most stable case respectively. We show that there are distinct stages in the evolution of the flow and these are determined by the turbulence Richardson number Riu, which is the square of the ratio of the turbulence to stratification timescales (Nτ)2. For Riu < 0.1 the flow is dynamically passive and the vertical heat flux correlation coefficient ρwθ is constant with a value of −0.7. From Riu ≈ 0.1 (where the turbulent potential energy is approximately 5% of the vertical kinetic energy) to Riu ≈ 1 the heat flux begins to decay under the action of the buoyancy forces and the vertical velocity variance $overline{\omega ^2}$ decays more rapidly than for the neutral case. By Riu ≈ 1 the turbulent potential energy is nearly as large as the vertical kinetic energy and the magnitude of the ratio of the buoyancy to mechanical dissipation terms in the kinetic energy budget reaches a maximum and then rapidly diminishes as the heat flux completely collapses (by Riu ≈ 2). For Riu > 2 the heat flux remains small except for the most stable case studied where a significant net counter gradient heat flux occurs. Here the vertical velocity variance increases as it draws its energy from the potential energy field. For all experiments the mechanical to thermal timescale ratio remains relatively constant up until Riu ≈ 1 and then increases from 1.1 to a maximum value of 2. The ratio of the r.m.s. longitudinal to vertical velocity fluctuations $(\overline {u^2}^{\frac{1}{2}}/(\overline{W^2}^{\frac{1}{2}}$, increases from the passive value (of 1.1) to a maximum of approximately 1.6 at Riu ≈ 1 and then decreases as the heat flux collapses. No direct evidence of internal waves is found from the single-point measurements. The results are presented in terms of relevant lengthscale ratios determined from auto-correlation functions, spectra and from integrated quantities such as variance and dissipation rates. Particular attention is paid to a comparison with previous work, especially that of Lienhard & Van Atta (1990), the only other wind-tunnel experiment on strongly stratified grid-generated turbulence without shear. It is shown that the evolution of the flow is very sensitive to initial conditions and in order to collapse the data of different experiments the turbulence time (τ) rather than the elapsed clock time (t) must be used, i.e. the data collapse well with Riu but not with Nt (which is more traditionally used).

The statistical characteristics of third-order moments (triple products) of velocity and scalar have been investigated from both experimental and theoretical points of view. The third-order moments have a highly intermittent nature and are dominated by coherent motions in shear-generated turbulence. Using a previously developed statistical method (Nagano & Tagawa 1988), the similarity between the Reynolds stress and scalar flux transport was analysed. On the basis of the experimental results, we have developed an entirely new approach to the modelling of triple products, and constructed a ‘structural’ turbulence model for triple velocity and scalar products. This model has a simple form and universal applicability, and its effectiveness has been tested by application to various types of flow.