A procedure for determining the flow which results from introducing a long two-dimensional obstacle of finite height into a unidirectional, two-dimensional, stable, but otherwise arbitrary stratified shear flow of finite depth is described. The method is based on a generalization of the known results for two-layer flows, described in Baines (1984). The flow is assumed to be hydrostatic with negligible mixing, and the stratified flow is represented by an arbitrary number of discrete layers, so that the model is hydraulic in character. The procedure involves the calculation of the changes to the steady-state flow resulting from successive increases in the height of the topography from zero. For a given initial flow, introduction of an obstacle only alters the flow in its vicinity for obstacle heights hm less than a height hc, where the flow is critical (implying zero wave speed) at the obstacle crest for some particular internal wave mode. Increasing the obstacle height further causes the flow to adjust to maintain a critical condition at the obstacle crest, and this causes disturbances with the structure of the critical mode to be propagated upstream. These may take the form of an upstream hydraulic jump or of a time-dependent rarefaction (implying a disturbance which becomes increasingly spread out with time), or both, depending on the nonlinear dispersive properties of the system. Their passage past a given upstream location results in a permanent change to the local velocity and density profiles. As the obstacle height is further increased these processes will continue until the flow becomes critical just upstream of the obstacle, or a fluid layer becomes blocked. For greater obstacle heights the above phenomena may be repeated with other modes. A numerical procedure which implements these processes has been developed, and examples of applicatons to two- and three-layer systems are given.

The general method described in Baines 1988 has been applied to stratified flows of finite depth over long obstacles where the flow initially has uniform horizontal velocity. The fluid consists of a finite number of homogeneous layers of equal thickness and with equal density increments. This represents the state of continuous stratification with constant density gradient as closely as possible, for a given number of layers. Two-, three-, four- and sixty-four-layered models are studied in detail. The results are expressed in terms of the initial Froude number F0 (F0 = U/ĉ1 where U is the fluid speed and ĉ1 is the speed of the fastest long internal wave mode in the fluid at rest) and the obstacle height. In general, introduction of an obstacle into the flow causes disturbances to propagate upstream (columnar disturbance modes) which alter the velocity and density profiles there. These may accumulate to cause upstream blocking of some of the fluid layers if F0 is sufficiently small. As the number of fluid layers increases, so does the range of F0 for which this upstream blocked flow occurs. There are no upstream disturbances for F0 > 1, and for F0 < 1 the upstream disturbances are of the rarefaction type if upstream blocking does not occur. The results for three and four layers show how several coexisting modes may interact to affect the upstream profiles. The results for sixty-four-layers provide theoretical support for the observational criterion (Baines 1979b) that blocking in initially uniformly stratified flow occurs when Nhm/U > 2 (N is the Brunt—Väisälä frequency and hm the obstacle height), provided that more than two modes are present. In some situations, layered models are found to be inadequate as a representation of continuous stratification when one or more layers thicken to the extent that their discreteness is significant.

Two-dimensional convection in a Boussinesq fluid with infinite Prandtl number, confined between rigid horizontal boundaries and stress-free lateral boundaries, has been investigated in a series of numerical experiments. In a layer heated from below steady convection becomes unstable to oscillatory modes caused by the formation of hot or cold blobs in thermal boundary layers. Convection driven by internal heating shows a transition from steady motion through periodic oscillations to a chaotic regime, owing to the formation of cold blobs which plunge downwards and eventually split the roll. The interesting feature of this idealized problem is the interaction between constraints imposed by nonlinear dynamics and the obvious spatial structures associated with the sinking sheets and changes in the preferred cell size. These spatial structures modify the bifurcation patterns that are familiar from transitions to chaos in low-order systems. On the other hand, even large-amplitude disturbances are constrained to show periodic or quasi-periodic behaviour, and the bifurcation sequences can be followed in considerable detail. There are examples of quasi-periodic behaviour followed by intermittency, of period-doubling cascades and of transitions from quasi-periodicity to chaos, associated with a preference for narrower rolls as the Rayleigh number is increased.

This paper is a summary of the Fifth Beer-Sheva Seminar on Magnetohydrodynamic (MHD) Flows and Turbulence, held in Jerusalem during 2–6 March 1987, with 99 participants from 12 countries. Reviews and research papers were presented on general problems of turbulence, MHD turbulence, fundamental MHD, two-phase flows with and without magnetic fields, and on different applications of liquid-metal MHD, especially in power generation nuclear fission and fusion, and in metallurgy.

The input of a hot, turbulently convecting fluid to fill a chamber can result in the roof of the chamber melting. The rate of melting of the roof is here analysed experimentally and theoretically. Three separate cases are considered. The melt may be heavier than the fluid and initially sink through it. The intense motion in the fluid then mixes the falling melt in with it. Alternatively, the melt may be less dense than the fluid and form a separate layer between the roof and the fluid. This melt layer can itself be in quite vigorous convective motion. An intermediate case is shown to be possible, wherein the melt is initially denser than the fluid, and sinks. As its temperature increases and its density decreases, it becomes less dense than the surrounding fluid and rises. Experimental simulations of each of these three cases are described. The experiments employ a roof of either wax or ice which is melted by the aqueous salt solution beneath it. The second case, that of a light melt, has important geological applications. It describes the melting of the continental crust by the emplacement of a hot, relatively dense input of fluid basaltic rock. Both the basaltic layer and the resultant granitic melt layer crystallize and increase their viscosities as they cool. These effects are incorporated into the analysis and the rate of melting and the temperatures of the two layers are calculated as functions of time. The process is exemplified by the formation of the Cerro Galan volcanic system in Northwestern Argentina over the last 5 million years. An Appendix analyses the thermal history of the fluid in a chamber that does not melt and compares the results obtained with those derived previously.

A drop of fluid, initially held spherical by surface tension, will deform when an electric or magnetic field is applied. The deformation will depend on the electric/magnetic properties (permittivity/permeability and conductivity) of the drop and of the surrounding fluid. The full time-dependent low-Reynolds-number problem for the drop deformation is studied by means of a numerical boundary-integral technique. Fluids with arbitrary electrical properties are considered, but the viscosities of the drop and of the surrounding fluid are assumed to be equal.

Two modes of breakup have been observed experimentally: (i) tip-streaming from drops with pointed ends, and (ii) division of the drop into two blobs connected by a thin thread. Pointed ends are predicted by the numerical scheme when the permittivity of the drop is high compared with that of the surrounding fluid. Division into blobs is predicted when the conductivity of the drop is higher than that of the surrounding fluid. Some experiments have been reported in which the drop deformation exhibits hysteresis. This behaviour has not in general been reproduced in the numerical simulations, suggesting that the viscosity ratio of the two fluids can play an important role.

The effects of a magnetic field on core turbulence, mean-velocity boundary-layer profiles, turbulence-intensity boundary-layer profiles and turbulent spectral-energy distributions have been experimentally determined for combustion-driven magneto-hydrodynamic (MHD) flows. The turbulence suppression of the core was found to be similar to that of liquid-metal MHD flows, even though the turbulent structure was entirely different. The mean-velocity and turbulence-intensity boundary-layer profiles were affected much less than those of liquid-metal flows, primarily because the low-temperature thermal boundary layer reduced the electrical conductivity near the wall. No spectral dependence of turbulence suppression was observed in the core.

The flow of a jet of an inviscid, incompressible fluid emerging from a cylindrical pipe, of circular cross-section, into a stream of the same fluid is considered. The component of the external flow normal to the axis of the pipe is taken to be small compared with the speed of the jet. The results throw some light on the mechanisms that are responsible for the observed deflection of jets into the crossflow direction.

Mixed-layer deepening due to grid-generated turbulence is studied experimentally with the aim of explaining the contradictory results of previous studies. Entrainment rates are calculated at fixed distances from the grid in order to avoid the necessity of using an empirical expression for the decay of the turbulent velocity scale. It is shown that an incorrect form of this decay law can cause large errors in the predicted Richardson number dependence of the entrainment rate. For this study this dependence can be expressed as a power law of the form E = KRi−1,2. The spread of the results imply that an error of at least ± 10% is realistic in the determination of the exponent.

The turbulent velocity decay law is also deduced from the data, and it is found that the decay cannot be represented by a simple power law. Indeed two distinct flow regions, with differing decay rates, are present.

The propagation of plane linear acoustic wave in a mixture of inert gases is considered by means of a variational formulation of the Boltzmann equations, through which the sound speed c is expressed with errors of order ε2 in terms of trial functions determined with errors of order ε. This feature allows the exact determination of the coefficient of sound dispersion d2 ≡ [dc/dω2] at zero frequency (ω = 0), in terms of trial functions known from the Chapman—Enskog theory. Explicit results for d2 are given for all combinations of noble gases from He to Xe, assumed to interact through the Lennard—Jones potential. Comparison with previous approximate descriptions and with experiments is made.

An instability-wave analysis is presented to describe the spatial evolution of a fundamental mode and its subharmonic on an inviscid parallel mixing layer. It incorporates explicitly the weakly nonlinear interaction between the two modes. The computational finding that the development of the subharmonic, leading eventually to pairing or shredding, crucially depends on its phase relation with the fundamental is fully confirmed. Furthermore it is shown that a critical fundamental amplitude has to be reached before the (spatial) subharmonic becomes phase locked with the fundamental and exhibits a modified growth rate. Then the analysis is exploited to explain the occurrence of amplitude modulations in ‘natural’ mixing layers and to estimate the width of the subharmonic spectral peaks. Also, the case of oblique subharmonic waves is briefly touched upon. In the last part, ways are explored to model non-parallel effects, i.e. to handle the saturation of the rapidly growing subharmonic. Using this wave description, the role of mode interaction in the ‘vortex pairing’ and ‘shredding’ process is assessed.

The evolution of unsteady boundary layers in the vicinity of the leading edge of a thin oscillating airfoil has been examined with a novel numerical method which is able to deal with the movement of the stagnation point and with regions of reverse and separated flow. Solutions to the unsteady boundary-layer equations, with a prescribed pressure distribution which causes flow reversal and separation, demonstrate the importance of numerical steps in distance and time and that a requirement similar to the stability criterion of Courant, Friedrichs and Lewy must be satisfied to avoid numerical errors. At the lower reduced frequencies of the investigation, solutions could not be obtained with this procedure and it was necessary to introduce interaction between the viscous and inviscid flows. The solutions obtained with the interactive method were increasingly different from those without interaction as the reduced frequency was decreased towards zero and, for some combinations of Reynolds number and frequency, exhibited behaviour consistent with the instability of separation bubbles.

The creeping flow of a liquid film along an inclined periodic wall of arbitrary geometry is considered. The problem is formulated using the boundary-integral method for Stokes flow. This method is extended to two-dimensional flows involving free surfaces, and is implemented in an iterative numerical procedure. Detailed calculations for flow along a sinusoidal wall are perfomed. The free-surface profile is studied as a function of flow rate, inclination angle, wave amplitude, and surface tension, and is compared with previous asymptotic solutions. The results include streamline patterns, velocity profiles and wall-shear-stress distributions, and establish criteria for flow reversal. For specified wall geometry, the asymptotic behaviour for very small flow rates is shown to be a strong function of surface tension. It is demonstrated that these results are valid in a qualitative sense for general wall geometries. The analogy between gravity-driven flow and the flow of a liquid layer on a rotating disk (spin coating) is also discussed.

The linear stability of two superimposed layers of fluid, heated from below and separated by a thin conducting plate, is investigated. It is shown that when the ratio of the depths of the layers is close to 1:2, two distinct modes of convection can occur with preferred horizontal wavenumber in the ratio 1:2. The nonlinear evolution of a disturbance consisting of both modes is considered, and it is shown that travelling waves are the preferred mode of nonlinear convection for a wide range of parameter values. Other possible types of behaviour, including modulated waves and an attracting homoclinic trajectory, are also described in detail.

Developing fluid flow in a curved duct of square cross-section is studied numerically by a factored ADI finite-difference method on a staggered grid. A central-difference scheme with primitive variables is used inside the computational domain to reduce numerical diffusion. Two Reynolds numbers, 574 and 790, based upon a bulk velocity and hydraulic diameter are chosen for curvature ratios of 1/6.45 and 1/2.3, respectively. It is found that the secondary flow is far more complicated than expected, with the appearance of at least two pairs of vortices. Main-flow separation is also observed for the higher curvature ratio. Furthermore, it is observed that the flow develops into two quite different states downstream, depending upon the inlet conditions.

Solution of the fully developed Navier-Stokes equations is shown to be not unique beyond a certain critical Reynolds number. Developing flow seems to evolve into the fully developed state along a particular branch, into which the fully developed solution bifurcates.

Using a sand-tank model and the salt-sugar system, double-diffusive fingers formed in a saturated porous medium. In contrast to the quasi-steady fingering typically observed in a viscous fluid, the fingering here was quite unsteady. The fingers’ structure was observed, and measurements of the sugar flux indicate that double-diffusive groundwater fingers can transport solutes at rates as much as two orders of magnitude larger than those associated with molecular diffusion in motionless groundwater. The buoyancy-flux ratio, r = αFT/βFS, increased from r = 0.65 ± 0.02 (at Rρ = 1.02) to r = 0.81 ± 0.06 (at Rρ = 1.50), where Rρ is the density-anomaly ratio. (Using the salt-sugar system in a viscous fluid, r was previously shown to decrease with increasing Rρ.) The buoyancy flux due to sugar varied approximately as R−5.6ρ, which is almost identical with the variation found for salt-sugar fingers in a viscous fluid. The model of Green (1984) was applied to the experiments and predicted buoyancy-flux ratios and finger widths that were in fairly good agreement with the measured values, although the predicted buoyancy fluxes due to sugar were significantly larger than the measured fluxes.

A general numerical scheme is developed to calculate the motion of shock waves in gases with non-uniform properties. The numerical scheme is based on the approximate theory of geometrical shock dynamics. The refracted shockfronts at both planar and curved gas interfaces are calculated. Both regular and irregular refraction patterns are obtained, and in particular, precursor-irregular refraction systems are found using the approximate theory. The numerical results are compared with recent theoretical and experimental investigations. It is shown that the shockfronts determined using geometrical shock dynamics are in good agreement with the actual shock waves.

The physical mechanism governing the centring of a hollow liquid shell in capillary oscillations, which has been observed in experiments, is investigated theoretically. First, the shell is assumed to be inviscid and to have a thickness that is much less than its spherical radius. A system of one-dimensional nonlinear equations of motion is derived using a thin-sheet model. From a numerical study the nonlinear effects of the wave are found to cause the core to oscillate slowly relative to the shell while the centre of mass of the whole system remains stationary. The effects of small viscosity are then considered in an approximation. Finally the strength of the centring mechanism is compared with that of the decentring effect due to buoyancy. The findings are consistent with the limited experimental information available.

It has been recently proved that lattice-gas models with Boolean particles can provide a very powerful method to study viscous flows at moderate Reynolds and small Mach numbers (d'Humières, Pomeau & Lallemand 1985; Frisch, Hasslacher & Pomeau 1986; d'Humières & Lallemand 1986). We present here algorithms for an extension of these models to provide a simple and efficient way to simulate a large variety of flow problems with free boundaries. This is done by introducing two different types of particles that can react following a specific kinetic scheme based on autocatalytic reactions. In order to check the powerful character and the reliability of the method we also present preliminary results of two-dimensional computer simulations concerning problems ranging from the competition between molecular diffusion and turbulent mixing in flows presenting a Kelvin-Helmholtz instability to the spontaneous generation of turbulence in premixed flame fronts subject to the Darrieus-Landau instability. The dynamics of an interface developing a Rayleigh-Taylor instability is also considered as well as some typical problems of phase transition such as spinodal decomposition and the nucleation process.

The method of matched asymptotic expansions is applied to several long-wave problems including the scattering of acoustic waves by a grating of cylinders and the scattering of water waves incident on horizontal cylinders. It is shown that a naïve application of the method can lead to incorrect results. A modified expansion procedure is developed and applied to a number of problems.