Matched asymptotic expansions, are used to describe turbulent Couette–Poiseuille flow (plane duct flow with a pressure gradient and a moving wall). A special modification of conventional eddy-diffusivity closure accounts for the experimentally observed non-coincidence of the locations of zero shear stress and maximum velocity. An asymptotic solution is presented which is valid as the Reynolds number tends to infinity for the whole family of Couette–Poiseuille flows (adverse, favourable, and zero pressure gradients in combination with a moving wall). It is shown that plane Poiseuille flow is a limiting case of Couette–Poiseuille flow. The solution agrees with experimental data for plane Couette flow, for the limiting plane Poiseuille flow, and for a special case having zero net flow and an adverse pressure gradient. The asymptotic analysis shows that conventional eddy diffusivity closures are inadequate for general Couette–Poiseuille flows.

An approximate set of equations is derived for a compressible liquid of infinite Prandtl number. These are referred to as the anelastic-liquid equations. The approximation requires the product of absolute temperature and volume coefficient of thermal expansion to be small compared to one. A single parameter defined as the ratio of the depth of the convecting layer, d, to the temperature scale height of the liquid, HT, governs the importance of the non-Boussinesq effects of compressibility, viscous dissipation, variable adiabatic temperature gradients and non-hydrostatic pressure gradients. When d/HT [Lt ] 1 the Boussinesq equations result, but when d/HT is O(1) the non-Boussinesq terms become important. Using a time-dependent numerical model, the anelastic-liquid equations are solved in two dimensions and a systematic investigation of compressible convection is presented in which d/HT is varied from 0·1 to 1·5. Both marginal stability and finite-amplitude convection are studied. For d/HT [les ] 1·0 the effect of density variations is primarily geometric; descending parcels of liquid contract and ascending parcels expand, resulting in an increase in vorticity with depth. When d/HT > 1·0 the density stratification significantly stabilizes the lower regions of the marginal state solutions. At all values of d/HT [ges ] 0·25, an adiabatic temperature gradient proportional to temperature has a noticeable stabilizing effect on the lower regions. For d/HT [ges ] 0·5, marginal solutions are completely stabilized at the bottom of the layer and penetrative convection occurs for a finite range of supercritical Rayleigh numbers. In the finite-amplitude solutions adiabatic heating and cooling produces an isentropic central region. Viscous dissipation acts to redistribute buoyancy sources and intense frictional heating influences flow solutions locally in a time-dependent manner. The ratio of the total viscous heating in the convecting system, ϕ, to the heat flux across the upper surface, Fu, has an upper limit equal to d/HT. This limit is achieved at high Rayleigh numbers, when heating is entirely from below, and, for sufficiently large values of d/HT, Φ/Fu is greater than 1·00.

An initial thermal stratification may be swept out of a valley if a cross-wind is strong enough: the valley is then said to be ventilated. Numerical and laboratory experiments indicate that the critical parameter is the Froude number, , where is the mean horizontal velocity above the crest of the valley sides, N is the Brunt–Väisälä or bouyancy frequency of the thermal stratification in the valley air and h is the height of the valley walls. Ventilation occurs whenever the Froude number exceeds a value of 1·3.

Two problems exhibiting breakdown in local similarity solutions are discussed, and the appropriate asymptotic form of the exact solution is determined in each case. The first problem is the very elementary problem of pressure driven flow along a duct whose cross-section has a sharp corner of angle 2α. When 2α < ½π, a local similarity solution is valid, whereas, when 2α > ½π, the solution near the corner depends on the global geometry of the cross-section. The transitional behaviour when 2α = ½π is determined.

The second problem concerns low-Reynolds-number flow in the wedge-shaped region |θ| < α when either a normal velocity proportional to distance from the vertex is imposed on both boundaries, or a finite flux 2Q is introduced or extracted at the vertex (the Jeffery–Hamel problem). It is shown that the similarity solution in either case is valid only when 2α < 2αc ≈ 257·5°; a modified problem is solved exactly revealing the behaviour when α > αc, and the transitional behaviour when α = αc (when the similarity solutions become infinite everywhere). An interesting conclusion for the Jeffery–Hamel problem is that when α > αc, inertia forces are of dominant importance throughout the flow field no matter how small the source Reynolds number 2Q/ν may be.

The coupling of Brownian displacements and shear-induced convection of spherical colloidal particles in dilute suspensions is examined using solutions of appropriate convective diffusion equations for the time-dependent probability density and also by calculation of relevant statistical quantities for an ensemble of diffusing particles from Langevin equations. Based on a fundamental solution for convective diffusion from a point in a general linear field, analytical expressions for the probability density fα(r; t) are given for the case of an arbitrary, two-dimensional linear flow field. The parameter α, which characterizes the flow, may range from − 1 (pure rotation), through zero (simple shear), to + 1 (pure elongation). The Langevin approach offers interesting insights into the physical mechanism of diffusive–convective coupling, and may also be used to obtain rigorous expressions for moments of the probability density appropriate to a particle diffusing in an unbounded quadratic (Poiseuille) flow. Preliminary experiments are described which qualitatively verify the theoretical predictions for Poiseuille flow, and which suggest a simple, direct method for measuring particle diffusivities. Finally the effect of bounding walls on convective diffusion is considered by means of Monte Carlo calculations. Results show that particle-wall interactions significantly affect the average behaviour of particles located initially within distances of a few particle radii of the wall, since the frictional force is no longer isotropic.

The streamline patterns of some simple two-dimensional Stokes flows are studied and the results used both to understand and to predict the streamlines of flows in more complicated geometries, in particular the streamlines of flows that contain eddies or regions of closed streamlines. Initially, streamline patterns are studied locally, either around some special point, such as a stagnation point or a point where a streamline meets a wall, or in a special region, such as a corner. The use of these local analyses is illustrated by finding the streamlines for shear flow around a rotating cylinder; the illustration also shows how fluid in Stokes flow can be turned back on itself (‘blocked’). The local analyses of flow in a corner are used to understand the eddy patterns that have been discovered in a variety of flows. The eddies occur in corner-like regions and in these regions the flow can be regarded as the superposition of two components. One component, the eddy flow, is the result of flow outside the corner stirring the fluid in the corner, while the other is the direct result of local conditions in the corner. The competition between these two components determines whether eddies actually appear in a given flow. Finally, the approach developed here is applied to a new flow situation, namely a shear flow which is bounded by a moving wall and which contains a stationary cylinder touching the wall. The streamlines deduced for different ratios of the shear strength to the wall velocity show both new eddy patterns and unexpected regions of blocked flow.

Measured values of mean velocity, Reynolds stresses and probability density distributions of fluctuating velocity are reported for the turbulent coaxial jets, with and without swirl, emerging into stagnant surroundings from a long pipe and an annulus concentric with the pipe. They were obtained using hot-wire anemometry and on-line data processing with the aid of a mini-computer. The results show that non-swirling coaxial flow configurations approach a self-similar state in a much smaller distance than that of the round jet, for velocity ratios ranging between 0·65 and 1·5; this is due to the mixing layer and vortex shedding that occur in the region downstream of the separation wall between the two streams. In the presence of swirl, the coaxial jet was found to develop at a faster rate. An assessment of turbulence models, based on Reynolds stress closures, suggests that previous assumptions for turbulent diffusion of turbulent kinetic energy are in error.

A global asymptotic solution for the linear stability of this flow with respect to axisymmetric disturbances is developed without invoking the usual quasi-parallel flow assumption. This solution predicts the occurrence of quasi-periodic spatially amplified disturbances relatively near the apex, which ultimately are stabilized further down the cone by the relative increase in viscous forces associated with the progressive thinning of the film. The wave speed and wavelength of these disturbances are found to decrease with increasing distance from the apex.

A theoretical and experimental study has been carried out on the flow of a liquid metal along a straight rectangular duct, whose pairs of opposite walls are highly conducting and insulating, situated in a planar non-uniform magnetic field parallel to the conducting walls. Magnitudes of the flux density and mean velocity are taken to be such that the Hartmann number M and interaction parameter N have very large values and the magnetic Reynolds number is extremely small.

The theory qualitatively predicts the integral features of the flow, namely the distributions along the duct of the potential difference between the conducting walls and the pressure. The experimental results indicate that the velocity profile is severely distorted by regions of non-uniform magnetic field with fluid moving towards the conducting walls; even though these walls are very good conductors the flow behaves more like that in a non-conducting duct than that predicted for a duct with perfectly conducting side walls.

A class of exact, self-similar, time-dependent solutions describing free surface flows under gravity is found which extends the self-propagating class of solutions discovered earlier by Freeman (1972) to those which decay with time.

An experimental study of the motion of small water droplets in a shock tube is reported. Droplet displacement data were obtained by means of reflected-light stroboscopic illumination for droplet diameters in the range 87–575 μm, and for shock strengths, ΔP/P1, in the range 0·0018–0·3. The displacement data are fitted by means of best-fit polynomials in time, which are used to compute droplet velocities, accelerations, and drag coefficients. All of our drag coefficient data have values which are larger than the steady drag at the same Reynolds numbers. The differences are attributed to time changes of the relative fluid velocity Ur. This may affect the size of the recirculating region and, therefore, the drag. In particular, it is argued that the drag is larger or smaller than the steady drag, depending on whether the dUr/dt is negative or positive, respectively. Our experiments, which were performed for dUr/dt < 0, confirm this expectation. Furthermore, it is shown that the difference between steady and transient drag coefficients, at the same Reynolds number, depends only on the value of a parameter A = (ρp/ρ0−1)(D/U2r)(dUr/dt). Here ρp and ρ0 are the densities of the droplets and of the surrounding gas, respectively, and D is the droplet diameter. In fact, in the Reynolds number range 3·2 < Re < 77, where multiple data are available having the same value of Re but having different values of A, the drag data can be expressed as CD = CDS(Re) – KA, where CDS(Re) is the steady drag at the instantaneous Reynolds number Re, and K is a constant of order 1.

Steady two-dimensional potential flow past a semi-infinite flat-bottomed body is considered. This stern flow is assumed to separate tangentially from the body. Gravity waves of finite amplitude occur on the free surface. An exact relation between the amplitude of these waves and the Fronde number F is derived. It shows that these waves can exist only for F greater than the value F* = 2·23. This is slightly less than the value Fc = 2·26 at which breaking occurs. For F slightly larger than F*, the steepness is a multi-valued function of F, indicating the existence of more than one solution for these values of F. In addition, a numerical scheme based on an integro-differential equation formulation is derived to solve the problem in the fully nonlinear case. The shape of the free surface profile is computed for different values of F. As a check on the numerical results, they are shown to satisfy the exact relation between steepness and the Froude number.

Direct numerical solutions of the three-dimensional time-dependent Navier-Stokes equations are presented for the evolution of three-dimensional finite-amplitude disturbances of plane Poiseuille and plane Couette flows. Spectral methods using Fourier series and Chebyshev polynomial series are used. It is found that plane Poiseuille flow can sustain neutrally stable two-dimensional finite-amplitude disturbances at Reynolds numbers larger than about 2800. No neutrally stable two-dimensional finite-amplitude disturbances of plane Couette flow were found.

Three-dimensional disturbances are shown to have a strongly destabilizing effect. It is shown that finite-amplitude disturbances can drive transition to turbulence in both plane Poiseuille flow and plane Couette flow at Reynolds numbers of order 1000. Details of the resulting flow fields are presented. It is also shown that plane Poiseuille flow cannot sustain turbulence at Reynolds numbers below about 500.

The intradiurnal heating and cooling cycle of the mixed layer of a tropical ocean is investigated through the use of a ‘pseudo-two-dimensional’ numerical model. Particular emphasis is given to two-component diffusion resulting from dynamic instabilities in the water column. The conservation equations for salt and heat include the effects of solar heating, horizontal advection and turbulent fluxes at the sea surface, while wind mixing enters through the use of depth-dependent eddy diffusion coefficients resulting from the wave-orbital shear model of Kitaigorodskiy (1961). All inputs are treated as functions of time of day, or calculated via the bulk aerodynamic method.

The entrainment fluxes of salt and heat due to the mechanical stirring of the wind and the fluxes due to molecular diffusion are treated as separate, their respective contributions being added to form the diffusion coefficients used in an alternating-direction explicit scheme to integrate the heat and salt equations. Near the surface, in the absence of strong solar heating (i.e. during the night), these two fluxes alone are insufficient to remove the near-surface static instabilities; thus the presence of some additional process is suggested. A dynamic stability analysis is carried out, based on the temperature and salinity gradients. The resulting Rayleigh numbers indicate the possibility of double-diffusive convection, whereby the vertical transfers of salt and heat may proceed at rates far greater than can be accounted for by molecular diffusion alone. Therefore, the molecular diffusions in the model are increased by a factor roughly proportional to the one-third power of the ratio of the local effective Rayleigh number to a critical Rayleigh number. The modified molecular diffusivities are then added to the eddy diffusion coefficients due to the wind, to form the total diffusion coefficients used in the numerical integrations.

Comparisons are made between the model-generated profiles of temperature and the profiles observed in the ocean. The comparisons show reasonable agreement in the diurnal cycle of the heat wave at 1 m vertical resolution (except for the model-generated surface layer being too deep during the late afternoon hours). (Previous models typically predict only the temperature and thickness of a homogeneous layer.) The results obtained with the model are instructive in estimating the relative importance of the various mixing processes in the upper ocean.

Results from experiments with four different ducts are reported when magnitudes of the field strength and mean velocity are such that the Hartmann number and interaction parameter are large. The first is a straight, circular, highly conducting wall duct situated in a non-uniform transverse magnetic field. Results suggest that as a first approximation the flow may be regarded as being fully developed throughout. In fact there is a slight distortion of the flow in the non-uniform field region revealed by hot-film probe measurements of the streamwise velocity which varies in a novel but readily explicable manner. The second duct is similar except that its wall is weakly conducting. A pressure drop across the non-uniform field region suggests that the behaviour of the flow is weakly reminiscent of that in a non-conducting duct. The two other ducts also have weakly conducting walls but contain either one or two 90° bends and are situated in a uniform field. Symmetry of each duct about its mid-point leads to symmetric potential distributions which indicate the existence of two symmetrically arranged recirculating current flows and these lead to pressure drops across the bends. In the duct with two bends, part of it, the offset, lies parallel to the field lines and a surprising prediction relating the pressure drop across the offset to N finds some support.

A small region (called the surface tension region) where pressure differences across the oil–water and oil–air interfaces are important is shown to exist between the gravity viscous and monolayer regions in a spreading oil slick (Di Pietro, Huh & Cox 1978). The importance of this new region is that (i) it is necessary in order to connect the gravity–viscous and monolayer regions and (ii) it is a region where slopes of interfaces are large. This idea is used to find the thickness profile of an oil layer contained upstream of a barrier (an oil boom) placed across a channel in which water is flowing at a constant velocity. The assumption is also made that the velocity difference across the oil layer is small compared with the water velocity. The general conditions for the validity of the results are then discussed together with the modifications to the theory which are necessary if the boundary layer in the water below the oil should be turbulent rather than laminar. Good agreement is found to exist between experimental results for unsteady spreading on quiescent water in a channel and the results of the theory applied to this situation assuming quasi-steady spreading.

A random superposition of waves in a rotating, stratified, electrically conducting fluid leads to dynamo action in the sense that it yields a mean electric field having a component parallel to the mean magnetic field (‘α-effect’). Using Fourier analysis methods, we derive an explicit expression for the mean electric field. The α-effect has tensor form. We obtain a finite α-tensor even in a case of vanishing mean helicity. The result is discussed in the context of the solar turbulent dynamo.

The formalism required to determine the criterion for the onset of convection in a multi-layered porous medium heated from below is developed using a straightforward linear stability analysis. Detailed results for two- and three-layer configurations are presented. These results show that large permeability differences between the layers are required to force the system into an onset mode different from a homogeneous system.