A direct numerical simulation of spatially periodic wavy core flows is carried out under the assumption that the densities of the two fluids are identical and that the viscosity of the oil core is so large that it moves as a rigid solid which may nevertheless be deformed by pressure forces in the water. The waves which develop are asymmetric with steep slopes in the high-pressure region at the front face of the wave crest and shallower slopes at the low-pressure region at the lee side of the crest. The simulation gives excellent agreement with the experiments of Bai, Chen & Joseph (1992) on up flow in vertical core flow where axisymmetric bamboo waves are observed. We define a threshold Reynolds number and explore its utility; the pressure force of the water on the core relative to a fixed reference pressure is negative for Reynolds numbers below the threshold and is positive above. The wave length increases with the hold-up ratio when the Reynolds number is smaller than a second threshold and decreases for larger Reynolds numbers. We verify that very high pressures are generated at stagnation points on the wavefront. It is suggested that a positive pressure force is required to levitate the core off the wall when the densities are not matched and to centre the core when they are. A further conjecture is that the principal features which govern wavy core flows cannot be obtained from any theory in which inertia is neglected.

Results are presented from an assessment of the applicability of fractal scale-similarity in the spatio–temporal structure of Sc [Gt ] 1 conserved scalar fields ζ(x, t) and scalar energy dissipation rate fields ∇(x, t) in turbulent flows. Over 2 million spatial and temporal intersections were analysed from fully resolved three-dimensional (256) spatial measurements as well as fully resolved four-dimensional spatio–temporal measurements containing up to 3 million points. Statistical criteria were used to assess both deterministic and stochastic fractal scale-similarity and to differentiate between fractal and random sets. Results span the range of spatio–temporal scales from the scalar diffusion scales (ΛD, TD) to the viscous diffusion scales (Λv, Tv) and to the outer scales (δ, Tδ). Over this entire range of scales, slightly over 99.0% of all intersections with the scalar dissipation support geometry showed scale-similarity as fractal as stochastically self-similar fBm sets having the same record length. Dissipation values above the mean were found to have support dimension D = 0.66. The dissipation support dimension decreased sharply with increasing dissipation values. Virtually no intersections showed scaling as random as a random set with the same relative cover. In contrast, intersections with scalar isosurfaces showed scaling only approximately as fractal as a corresponding fBm set and only over the range of spatio–temporal scales between (ΛD, TD) and (Λv, Tv). On these inner scales the isosurface dimension was D = 0.48 and was largely independent of the isoscalar value. At larger scales, scalar isosurfaces showed no fractal scale-similarity. In contrast, isoscalar level crossing sets showed no fractal scale-similarity over any range of scales, even though the scalar dissipation support geometry for the same data is clearly fractal. These results were found to be unaffected by noise.

As a first step in a more general study of the influence of the body shape upon the initial time-development of wake vortices, we consider the case of a 5.20 aspect-ratio semicircular-section cylinder, fitted with two endplates, and with the rounded side and the flat side in turn facing the oncoming current. The flow structure is analysed by means of a detailed qualitative and quantitative analysis of numerous flow visualization pictures, for Reynolds numbers Re ranging between 60 and 600. Beyond the first phase of development, necessary for the vortex-shedding process to take place (t* [ges ] 6), a change in the flow evolution with Re is found for both body configurations, at a critical Reynolds number Rec whose final value is, within about 5%, 190 and 140 for the rounded and flat forebody respectively. However, this change varies with the body shape. Thus, above Re = 200, it is shown that the reversal of the body compared to the free stream (flat-forebody configuration) implies a clear difference in the wake development with a quasi-symmetrical shifting of the vortex cores to the rear of the recirculating zone and a complete annihilation of the process of vortex shedding, at least during the limited period of time corresponding to the present observation. The consequences of the modifications of the wake behaviour are quantitatively evaluated by considering the time- and Re-evolution of the wake geometrical parameters and of the axial velocity distribution; they are related to the body geometry.

A three-dimensional mode of spatial instability, related to the temporal algebraic growth that determines lift-up in parallel flow, is found to occur in the two-dimensional boundary layer growing over a flat surface. This unstable perturbation can be framed within the limits of Prandtl's standard boundary-layer approximation, and therefore develops at any Reynolds number for which the boundary layer exists, in sharp contrast to all previously known flow instabilities which only occur beyond a sharply defined Reynolds-number threshold. It is thus a good candidate for the initial linear amplification mechanism that leads to bypass transition.

The problem of expansion into a vacuum of a semi-infinite layer composed of chaotically moving inelastic rough spherical particles is solved analytically. A variant of the matched asymptotic expansion scheme is used to obtain a matched composite solution, which is valid for both small and large times in the wave head, wave tail and intermediate domains of the disturbed part of the layer.

The effects of granular initial energy and particle collisonal properties on their hydrodynamic velocity, temperature, density and pressure are studied in the limit of low initial density of the granular gas. The total granular mass, M, within the disturbed region was found to change with time as log t. This is in contrast with the comparable classical result M ∼ t obtained for conservative (molecular) gases. This logarithmic dependence stems from the influence of kinetic energy losses, which reduce the granular temperature and speed of sound in the wave head region.

The ultimate escape energy and momentum (i.e. those achieved for long times by the expanding part of the layer) are shown to be finite quantities, dependent on the particle restitution coefficient, roughness and the initial granular temperature. The estimated mass of the escaping part of the layer, calculated here for dilute gases, serves as an upper bound on this quantity for all (also dense) comparable granular gases. This mass is determined by the collisional losses, as embodied within the particle restitution and roughness coefficients.

We present numerical solutions for two-dimensional laminar symmetric vortex systems descending in a stably stratified fluid, within the Boussinesq approximation. Three types of flows are considered: (I) tight vortices; (II) those deriving from an elliptical wing lift distribution; and (III) those deriving from a ‘high-lift’ distribution, with a part-span flap on the wing. The non-dimensional stratification ranges from zero to moderate, as it does for airliners. For Types I and II, with high Reynolds numbers and weak stratification, the solutions confirm the theory of Scorer & Davenport (1970) (their article lacks a crucial link which we provide, equivalent to one of Crow (1974)). Contrary to common conceptions and observations in small-scale experiments, the descent velocity increases exponentially with time, as the distance between vortices decreases and the circulation of the vortices proper is conserved. With moderate stratification, wakes with sufficient energy also attain the accelerating régime, until the vortex cores make contact. However, they first experience a rebound, which is both of practical importance and out of reach of simple formulas. Type III wakes produce two durable vortex pairs which tumble, and mitigate the buoyancy effect by exchanging fluid with the surroundings. These phenomena are obscured by low wing aspect ratios, Reynolds numbers below about 105, or appreciable surrounding turbulence; this may explain why neither a clear rebound nor an acceleration can be reconciled with experiments to date. We argue that airliner wakes have very little inherent diffusion, and that a rapid end to the wake's descent must reveal effects other than simple buoyancy. In particular, stratification promotes the Crow instability.

A new theory of the propagation of weak shocks into non-uniform, two-dimensional flows is introduced. The theory is based on a description of shock propagation in terms of a manifold equation together with compatibility conditions for shock strength and its normal derivatives behind the shock. This approach was developed by Ravindran & Prasad (1993) for shocks of arbitrary strength propagating into a medium at rest and is extended here to non-uniform media and restricted to moderately weak shocks. The theory is tested against known analytical solutions for cylindrical and plane shocks, and against a full direct numerical simulation (DNS) of a shock propagating into a sinusoidal shear flow. The test against DNS shows that the present theory accurately predicts the evolution of a moderately weak shock front, including the formation of shock-shocks due to shock focusing. The theory is then applied to the focusing of an initially parabolic shock, and to the propagation of an initially straight shock into a variety of simple flows (sinusoidal shear, vortex array, point-vortex array) exhibiting some fundamental properties of turbulent flows. A number of relations are deduced for the variation of shock quantities with initial shock strength MS0 and the Mach number of the flow ahead of the shock MU (e.g. separation of shock-shocks and maximum shock strength at a focus). It is found that shock-shocks are likely to form in turbulent flows with Mt/M1N > 0.14–0.25, where Mt is the average Mach number of the turbulence and M1N is the Mach number of the shock in a flow at rest. The shock moves up to 1.5% faster in a two-dimensional vortex array than in uniform flow.

Results are presented of an experimental study of the onset of time-dependent flows in a sample of liquid gallium subjected to a horizontal temperature gradient. The primary control parameter is the Grashof number which is set by the temperature difference. However, we have also taken the novel approach of varying the Prandtl number in a systematic way using the applied mean temperature. This has uncovered some surprising new dynamical states. Furthermore, the interaction between competing oscillatory flows has produced interesting dynamical behaviour including secondary Hopf bifurcations where both the frequency and amplitude grow from zero as the critical point is passed.

We present a numerical study of the impact of a two-dimensional plunging wave on a rigid vertical wall in the context of potential flow. The plunging wave impinging the wall is generated using a mixed-Eulerian-Lagrangian (MEL) boundary-integral scheme. The initial stage of the impact is characterized by an oblique impact of a liquid wedge on the wall and is solved using a similarity solution. Following the initial impact, the MEL simulation is continued to capture the transient impact process. The effect of an air cushion trapped between the plunger and the wall is considered. In addition to details such as temporal evolutions and surface profiles, the main interests are the maximum impact pressure on the wall and its rise time. To arrive at appropriate scaling laws for these, simulations are performed and correlations are explored for a broad range of local plunging wave kinematic and geometric parameters. To assess the present results, direct comparisons are made with the experiment of Chan & Melville (1988). Reasonable quantitative agreement is obtained and likely sources for discrepancies are identified and discussed.

We study the shear-induced self-diffusion of both a liquid tracer and a tagged spherical particle along the directions perpendicular to the ambient flow in a dilute suspension of neutrally buoyant spheres undergoing a simple shearing motion in the absence of inertia and Brownian motion effects. The calculation of the liquid diffusivity requires the velocity of a fluid point under the influence of two spheres, which was determined via Lamb's series expansion; conversely, the calculation of the particle diffusivity involves the trajectories of three spheres, which were determined using far-field and near-field asymptotic expressions. The displacements of the liquid tracer and of the tagged sphere were then computed analytically when the spheres and the tracer are all far apart, and numerically for close encounters. After summing over all possible encounters, the leading terms of the lateral liquid diffusion coefficients, both within and normal to the plane of shear, were thereby found to be 0.12γac and 0.004γac, respectively, where γ is the applied shear rate, a the radius of the spheres and c their volume fraction. The analogous coefficients of the lateral particle diffusivity were found to be 0.11γac, and 0.005γac, respectively. Also, liquid and particle diffusivities in a monolayer, with the liquid tracer and all the particle centres lying on the same plane of shear, were found to be 0.067γyac, and 0.032γac, respectively, with c denoting the areal fraction occupied by the spheres on the plane.

The diffusional–thermal instability, which gives rise to striped quenching patterns that have been observed for diffusion flames, is analysed by studying the model of a one-dimensional convective diffusion flame in the diffusion-flame regime of activation-energy asymptotics. Attention is focused principally on near-extinction conditions with Lewis numbers less than unity, in which the reactants with high diffusivity diffuse into the strong segments of the reaction sheet, so that the regions between the strong segments become deficient in reactant and subject to the local quenching that leads to the striped patterns. This analysis differs from other flame stability analyses in that the complete description of the dispersion relation is obtained from a composite expansion of the results of an analysis with the conventional convective-diffusive scaling and one with reaction-zone scaling. The results predict that striped patterns will occur, for flames sufficiently close to quasi-steady extinction, with a finite wavenumber that in convective–diffusive scaling is proportional to the cube root of the Zel'dovich number. The convective–diffusive response contributes to the stabilization of long-wavelength disturbances by through positive excess enthalpies by which the flame becomes more resistant to instability, while the reaction-zone response provides stabilization of short-wavelength disturbances by transverse diffusion, within the reactive inner layer, which relaxes the perturbed scalar fields towards their unperturbed states. As quasi-steady extinction is approached, marginal stability arises first at an intermediate range between these two scalings. Parametric results for this bifurcation point are obtained through numerical solutions of the associated generalized eigenvalue problems. Comparisons with measured pattern dimensions for different sets of reactants and diluents reveal excellent qualitative agreement.

Active circulation control of the two-dimensional unsteady separated flow past a plate with a suction point on the downstream wall is considered. The rolling-up of the separated shear layer is modelled by a pair of point vortices whose time-dependent circulation is predicted by an unsteady Kutta condition. A nonlinear controller able to confine the wake to a single vortex pair of constant circulation is derived in closed form for any free-stream condition. Dynamical system analysis is used to explore the performance of the controlled system. Finally, the control strategy is applied to three different classes of unsteady flows and the results are discussed.

The paper addresses the mathematical modelling of the formation of a pointed drop in a four-roller mill, observed by Taylor (1934) in the Cavendish Laboratory. Since the experiments were carried out with drops of small diameter compared to the mill size, the method of matched asymptotic expansions is applicable. A two-dimensional Stokes flow generated by the rotating rollers in the mill but with no drop effect (outer problem) is computed numerically by a boundary-element method. The local expansion of that flow at the centre of the mill, where the drop is to be positioned, is used as a far field for the flow around the drop in unbounded fluid (inner problem). Employing a plane-flow model and using complex-variable techniques, the explicit solutions previously obtained by the author are adapted to the inner problem. It is proved that, with an increasing rotation rate of the rollers, the drop does develop two apparent cusps on the interface, and its shapes have striking similarities with Taylor's experiments. Response diagrams showing the drop distortion versus the elongational strain demonstrate that these are one-to-one function of each other if the drop diameter is greater than a critical value determined by the size of the mill but cease to be one-to-one otherwise. This behaviour is identified with a sudden transition from a rounded drop to a cusped one at a critical strain.

In this paper we develop a new generic model for phototaxis in a suspension of microscopic swimming algae. Phototaxis is a directed swimming response dependent upon the light conditions sensed by the microorganisms. Positive phototaxis consists of motions directed toward the source of illumination and negative phototaxis is motion directed away from it. The model also incorporates the effect of shading whereby microorganisms nearer the light source absorb and scatter the light before it reaches those further away. This model of phototaxis and shading is then used to analyse the linear stability of a suspension of phototactic algae, uniformly illuminated from above, that swim in a fluid which is slightly less dense then they are. In the basic state there are no fluid motions and the up and down swimming caused by positive and negative phototaxis is balanced by diffusion, with the result that a horizontal, concentrated layer of algae forms, the vertical position of which depends on the light intensity. This leads to a bulk density stratification with a gravitationally stable layer of fluid overlying an unstable layer. When the resulting density gradient becomes large enough, a Rayleigh–Taylor type instability initiates fluid motions in the lower, unstable region that subsequently penetrate the upper, locally stable region. The behaviour of the suspension is characterized by four parameters: a layer depth parameter d, a Rayleigh number R, a Schmidt number Sc, and a sublayer position parameter C that specifies the vertical position of the sublayer in the fluid. Linear stability analysis of the basic state indicates that initial pattern wavelengths and the critical value of R at which the suspension becomes unstable are affected by the position of the sublayer, which is determined by the intensity of the light source, and by the type of boundary at the top of the fluid. Oscillatory modes of disturbance are also predicted for certain parameter ranges, driven by cells swimming vigorously upwards towards darker, concentrated downwelling regions and thus creating lower, positively buoyant regions which oppose the fluid motion in the convection patterns.

A mathematical analysis is carried out to model the series of processes following the occurrence of an electron avalanche in a liquid right through to the emission of a pressure transient and the formation of a bubble. The initial energy distribution is chosen to be Gaussian and it is assumed that the electrical energy injected into the system is transformed into thermal and mechanical components. From the mechanical point of view, an outgoing spherical pressure transient is formed at the edge of the plasma region, and at a later time a bubble is also formed. Theoretically, the pressure transient accounts for about 15% of the total injected energy, while it is necessary to revert to experimental results to fix the energy associated with the bubble (about 2%). A minimum such value can, however, be estimated. The maximum pressure amplitude is calculated. Concerning the thermal component of the energy, some is absorbed as internal energy by the liquid, while the remainder is stocked as latent heat of vaporization. The maximum temperature difference is derived as are the different energies as functions of the total injected energy. The advantage of this type of model is that the gas/vapour temperature in the bubble can continue to rise after the phase change takes place. The maximum bubble size following a given energy injection is calculated assuming an adiabatic expansion process. A mathematical expression for the liquid flow induced by the outgoing pressure transient is also found. Comparison between experimental and theoretical results is particularly good.

Solid bodies immersed in hot fluids may melt. The molten material produced can then mix with, and be assimilated into, the fluid influencing its compositional and thermal states. Compositional convection of melt and thermal convection of cooled fluid around the solid determine the heat flux from the fluid to the solid's surface. This, together with the thermal properties of the solid, controls the rate of melting. Experiments on melting wax spheres into water are described; these have shown how variations in the nature of melt flow round the sphere cause differing melting rates and hence the development of a distinctive melting morphology. Melting rates are calculated by a simple theoretical analysis which estimates melt layer thickness and the heat flux from the fluid. Melting rate predictions agree well with the experimental data. A geological application occurs when magma incorporates blocks of its surrounding wall rock. Relatively rapid melting rates are estimated, typically in the order of a half metre per day. Such fast rates indicate that this method of contamination may be an important influence on magmatic evolution in continental environments.