An experimental study of a magnetohydrodynamic flow in a system of n (n[les ]5) U-bends is presented. The bends are electrically coupled via common electrically conducting walls parallel to the external magnetic field. In the test section the fluid flows perpendicular–parallel–perpendicular to the magnetic field. The Hartmann number M varies in the range 6×102[les ]M[les ]2.4×103, and the interaction parameter N in the range 102[les ]N[les ]4.3×104. The experimental data for the wall electric potentials and the pressure have been compared with the theoretical asymptotic values calculated for N[Gt ]M3/2[Gt ]1. This assumption in theory ensures the inertialess nature of the flow. For n=1 the agreement between the theory and the experiment is good. With increasing number of bends quantitative (for n=3) and then qualitative (for n=5) disagreement appears. For the first time in strong-field magnetohydrodynamics this disagreement has been observed on the Hartmann walls, i.e. walls perpendicular to the field. The experimental results for the wall potential indicate that for n=5 in some of the ducts parallel to the field qualitatively different flow patterns are established than those predicted by the asymptotic inertialess theory. The flow in the core depends on N, i.e. is of inertial nature. In the whole range of N investigated there is only a slight tendency of the wall potential to approach theoretical values. This demonstrates the stability of the new flow pattern and that even such high values of N as 4.3×104 are insufficient for the core flow to be inertialess. A strong dependence of the pressure drop on N has been observed in all the flow configurations investigated. The dependence of the inertial part of the pressure drop in each bend scales with N−1/3, as long as N−1/3[Lt ]1. This is characteristic of electromagnetic–inertia interaction in the boundary and internal layers parallel to the field. A linear increase of the pressure drop with the number of coupled bends has been observed, confirming qualitatively previous theoretical results. The effects of magnetic field inclination and different flow distribution between bends have also been studied.

Experimental verification of theoretical investigations into the behaviour of liquid metal convective flows is required to validate analytical models and numerical simulation codes. A real-time radioscopic density visualization system has been developed and is beginning to provide useful data. The X-ray facility for studying opaque low-Prandtl-number fluid flow is described. Density changes as low as 10−3 g cm−3 can be visualized by careful control of scatter radiation. The new capability is demonstrated with natural convection benchmark experiments in a narrow vertical layer of gallium melt of aspect ratio A=1.4. The density field in that cell is three-dimensional. Good agreement between calculations and experiments is obtained. Experiments with solidification are provided for further theoretical use.

Flow in a circular pipe rotating about its axis, at low Reynolds number, is investigated. The simulation is performed by a finite difference scheme, second-order accurate in space and in time. A non-uniform grid in the radial direction yields accurate solutions with a reasonable number of grid points. The numerical method has been tested for the non-rotating pipe in the limit ν→0 to prove the energy conservation properties. In the viscous case a grid refinement check has been performed and some conclusions about drag reduction have been reached. The mean and turbulent quantities have been compared with the numerical and experimental non-rotating pipe data of Eggels et al. (1994a, b). When the pipe rotates, a degree of drag reduction is achieved in the numerical simulations just as in the experiments. Through the visualization of the vorticity field the drag reduction has been related to the modification of the vortical structures near the wall. A comparison between the vorticity in the non-rotating and in the high rotation case has shown a spiral motion leading to the transport of streamwise vorticity far from the wall.

This paper reports the results of direct numerical simulation of the motion of a two-dimensional circular cylinder in Couette flow and in Poiseuille flow of an Oldroyd-B fluid. Both neutrally buoyant and non-neutrally buoyant cylinders are considered. The cylinder's motion and the mechanisms which cause the cylinders to migrate are studied. The stable equilibrium position of neutrally buoyant particles varies with inertia, elasticity, shear thinning and the blockage ratio of the channel in both shear flows. Shear thinning promotes the migration of the cylinder to the wall while inertia causes the cylinder to migrate away from the wall. The cylinder moves closer to the wall in a narrower channel. In a Poiseuille flow, the effect of elastic normal stresses is manifested by an attraction toward the nearby wall if the blockage is strong. If the blockage is weak, the normal stresses act through the curvature of the inflow velocity profile and generate a lateral force that points to the centreline. In both cases, the migration of particles is controlled by elastic normal stresses which in the limit of slow flow in two dimensions are compressive and proportional to the square of the shear rate on the body. A slightly buoyant cylinder in Couette flow migrates to an equilibrium position nearer the centreline of the channel in a viscoelastic fluid than in a Newtonian fluid. On the other hand, the same slightly buoyant cylinder in Poiseuille flow moves to a stable position farther away from the centreline of the channel in a viscoelastic fluid than in a Newtonian fluid. Marked effects of shear thinning are documented and discussed.

We present an experimental study of the formation and stability of small-scale beads deposited onto a solid surface by sweeping a droplet stream over it. We are concerned particularly with beads formed from molten droplets deposited on a cold substrate, as in the work of Gao & Sonin (1994), but some results of isothermal deposition are also shown. We show that a molten bead forms with parallel contact lines which have been arrested by freezing while the bead itself is still largely in a liquid state, and that the still-molten material is stable when the contact angle is less than ½π and unstable when it exceeds ½π, consistent with Davis's (1980) theory. In addition, we present a relatively simple inviscid theory for small-scale (small Bond number) beads which allows us to compute the wavelength associated with the maximum growth rate of the instability, and show that it agrees with the dominant wavelength in the experiments.

Direct numerical simulations of passive scalars, with Prandtl numbers Pr=3, 5, and 7, advected by turbulence at three low Reynolds numbers were performed. The energy spectra are self-similar under the Kolmogorov scaling and exhibit behaviour consistent with many other investigations: a short inertial range for the highest Reynolds number and the universal exponential form of the spectrum for all Reynolds numbers in the dissipation range. In all cases the passive scalar spectra collapse to a single self-similar curve under the Batchelor scaling and exhibit the k−1 range followed by an exponential fall-off. We attribute the applicability of the Batchelor scaling to our low-Reynolds-number flows to the universality of the energy dissipation spectra. The Batchelor range is observed for wavenumbers in general agreement with experimental observations but smaller than predicted by the classical estimates. The discrepancy is caused by the fact that the velocity scales responsible for the generation of the Batchelor range are in the vicinity of the wavenumber of the maximum energy dissipation, which is one order of magnitude less than the Kolmogorov wavenumber used in the classical theory. Two different functional forms of passive scalar spectra proposed by Batchelor and Kraichnan were fitted to the simulation results and it was found that the Kraichnan model agrees very well with the data while the Batchelor formula displays systematic deviations from the data. Implications of these differences for the experimental procedures to measure the energy and passive scalar dissipation rates in oceanographic flows are discussed.

The flow of a constant-vorticity current past coastal topography is investigated in the long-wave weakly nonlinear limit. In contrast to other near-critical weakly nonlinear systems this problem does not exhibit hydraulically controlled solutions. It is shown that near criticality the evolution of the vorticity interface is governed by a forced BDA (Benjamin–Davis–Acrivos) equation. The solutions of this equation are discussed and two distinct near-critical flow regimes are identified. Owing to the non-local nature of the forcing, the first of these regimes is characterized by quasi-steady solutions controlled at the topography with some blocking of the upstream rotational fluid, while in the second regime steady nonlinear wavetrains form downstream of the obstacle with no upstream influence. In the hydraulic limit the velocity band for both of these critical regimes approaches zero.

This paper analyses the finite-amplitude flow of a constant-vorticity current past coastal topography in the long-wave limit. A forced finite-amplitude long-wave equation is derived to describe the evolution of the vorticity interface. An analysis of this equation shows that three distinct near-critical regimes occur. In the first the upstream flow is restricted, with overturning of the vorticity interface for sufficiently large topography. In the second quasi-steady nonlinear waves form downstream of the topography with weak upstream influence. In the third regime the upstream rotational fluid is partially blocked. Blocking and overturning are enhanced at headlands with steep rear faces and decreased at headlands with steep forward faces. For certain parameter values both overturning and partially blocked solutions are possible and the long-time evolution is critically dependent on the initial conditions. The reduction of the problem to a one-dimensional nonlinear wave equation allows solutions to be followed to much longer times and parameter space to be explored more finely than in the related pioneering contour-dynamical integrations of Stern (1991).

The interaction of a point vortex with a layer of constant vorticity, bounded below by a wall and above by an irrotational flow, is investigated as a model of vortex–boundary layer interaction. This model calculates both the evolution of the interface which separates the vortex layer from the irrotational flow and the trajectory of the vortex. In order to determine the conditions which lead to sustained unsteady interaction, three cases are investigated where the mutual interaction between the vortex and interface is initially assumed to be weak. (i) When a weak point vortex lies outside the layer, the vortex moves with a horizontal speed that is small relative to the long-wave phase speed of interfacial waves. A uniformly valid solution is found for the interface evolution. This solution shows that for long times the interface and the vortex approach an equilibrium state. (ii) When a weak vortex lies inside the layer, the vortex is convected by the mean flow and moves with a horizontal speed which matches the phase speed of an interfacial wave. This results in a strong interaction between the vortex and the interfacial wave. On the interface, a monochromatic wavetrain forms upstream of the vortex and acts to attract or repel the point vortex. The displacement of the vortex due to the wavetrain results in the modulation of the amplitude and wavelength of the wavetrain. If the point vortex is attracted toward the interface the horizontal speed of the vortex slows and disturbances directly above the vortex focus and grow leading to the ejection of vorticity. (iii) When the point vortex lies close to the wall and it is sufficiently strong it propagates downstream with a large horizontal velocity. In this case, the amplitude of the interfacial disturbance is independent of the vortex strength. Again, the vortex and the interface approach an equilibrium state. The results of this paper indicate that when the horizontal speed of the vortex matches the phase speed of the interfacial disturbance, it is necessary to account for the vertical displacement of the vortex in order to predict the behaviour of vortex–boundary layer interactions.

We examine the spatial evolution of an instability wave excited by an external source in a free, nearly non-dissipative, stably stratified shear flow with a small Richardson number Ri[Lt ]1. It turns out that at the nonlinear stage of evolution even so small a stratification modifies greatly the evolution behaviour compared with the case of a homogeneous flow which was studied in detail by Goldstein & Hultgren (1988).

We have investigated (analytically and numerically) different stages of evolution corresponding to different critical layer regimes, and determined the formation conditions and structure of a quasi-steady nonlinear critical layer.

It is shown that the stratification influence upon the nonlinear evolution is governed by the parameter (Pr−1)Ri/γ2L, where Pr is the Prandtl number and γL is the wave's linear growth rate (which is a measure of supercriticality), and this effect is important only when γL<Ri1/2, Pr≠1. The character of this influence radically depends on the sign of (Pr−1). Thus, when Pr<1 the amplitude in the course of the evolution varies in a limited range and either reaches saturation, when the supercriticality is small enough or, at higher supercriticality, performs quasi-periodic oscillations, whose structure becomes increasingly complicated with increasing γL. When Pr>1 stratification leads to the appearance of new evolutionary stages, namely the stage of explosive growth in the unsteady critical layer regime, and the stage of essentially unsteady evolution in the nonlinear critical layer regime, and to a modification of the power-law growth in the regime of a quasi-steady nonlinear critical layer.

There has recently been a surge in activity concerning the development of three-dimensionality in the wakes of nominally two-dimensional bluff bodies, yielding the realization that end effects can influence the wake vortex shedding pattern over long spanlengths. Much of this work has been focused on low Reynolds numbers (Re), but virtually no studies have investigated to what extent it is possible to control shedding patterns at higher Reynolds numbers, through the use of end manipulation. In the present paper, we demonstrate that it is possible to induce parallel shedding, oblique shedding and vortex dislocations, by manipulation of the end conditions, over a large range of Reynolds number. Such patterns affect the frequency of primary wake instability and its amplitude of fluctuation, as they do at low Reynolds number, although distinct differences are found at the higher Reynolds numbers.

We find that imposition of oblique shedding conditions at high Reynolds number leads to a spatial variation of both the oblique shedding angle and shedding frequency across the span, and to sparse dislocations which are not restricted to the spanwise end regions, as they are at low Reynolds numbers (under similar geometrical conditions). In the wake transition regime (Re=190–250), it is confirmed that the spontaneous appearance of vortex dislocations in mode-A shedding precludes the control of shedding patterns using end manipulation. However, it has proven possible to extend the regime of Reynolds number where dislocations ‘naturally’ exist to Re>250, by introducing them artificially through end control, where they would otherwise not occur. The possibility of introducing dislocations and of inducing oblique vortex shedding at higher Reynolds numbers has practical significance, if one can deliberately decorrelate the vortex shedding, and hence reduce the spanwise-integrated unsteady fluid forces on the body.

We confirm the existence of a transition in the mode of shedding at Re≈5000 (originally found by Norberg 1987) under conditions where parallel shedding is attempted. This mode transition displays similarities to an inverse of the mode A→mode B transition that is found in the wake transition regime. It is clear that vortex dislocations occur beyond Re=5000, although it is not clear why the flow is unstable to such a mode. Furthermore, there appears to be some support for the suggestion that vortex dislocations may be a feature of the flow for Re at least up to 30×103, as evidenced by the work of Norberg (1994).

Some recent studies have considered the stability of unbounded rapid granular shear flow, with the sole mechanism for stress generation being instantaneous inelastic collisions between grains. This paper extends these studies by presenting a linear stability analysis in which stress generation due to grain friction is also accounted for. This is accomplished by using the ‘frictional–kinetic’ model, which integrates in a simple manner the stress arising from the two mechanisms. Solution of the linearized equations of motion is obtained by allowing the wavenumber vector of the disturbances to rotate as a function of time. As in the case of a purely kinetic stress, it is found that the flow is stable to non-layering disturbances. Disturbances in the form of layering modes may lead to instability, depending on the solids fraction and material parameters. Instability is absent altogether if the balance of fluctuational energy is not considered or if the material is assumed to be incompressible. Friction may stabilize or destabilize the flow, depending on the inelasticity of grain collisions and the effective roughness of the medium. When a purely frictional stress is considered, it is found that the system is always neutrally stable. Even if the flow is asymptotically stable, there may be significant transient growth of disturbances due to the non-normality of the associated linear operator. The initial transient growth rate, as well as the temporal maximum of transient growth is enhanced by friction.

We present a series of similarity solutions to describe the temperature field as liquid spreads from a line source into a porous rock saturated with liquid of higher temperature. We identify slow and fast flow regimes. In the slow flow regime, the liquid is heated to the far-field temperature by conduction of heat from the far field. In the fast flow regime, there is negligible conduction of heat from the far field. Instead, the liquid is heated to the far-field temperature by cooling a region of the host rock near the source, and an internal boundary layer develops within the newly injected liquid. We successfully test our quantitative theoretical predictions with a series of laboratory experiments in which water was injected into a consolidated bed of sand filled with liquid of different temperature. We extend our model to describe the vaporization of liquid as it spreads slowly from a central source into a superheated porous rock. A further family of similarity solutions shows that the rate of vaporization depends upon the injection rate as well as upon the initial superheat of the reservoir. For high injection rates, the liquid is typically heated to the interface temperature long before reaching the interface. The rate of vaporization then becomes independent of the initial liquid temperature, and depends mainly on the reservoir superheat. For lower injection rates, heat is conducted from ahead of the boiling front into the liquid. As a result, for progressively smaller injection rates, an increasing fraction of the liquid vaporizes, until virtually all the liquid boils, and only a very small liquid zone develops in the rock. Again, we successfully test our theoretical predictions with a laboratory experiment in which liquid water was injected into a superheated layer of permeable sandstone.

The stability of axially symmetric cone-and-plate flow of an Oldroyd-B fluid at non-zero Reynolds number is analysed. We show that stability is controlled by two parameters: [Escr ]1≡DeWe and [Escr ]2≡Re/We, where De, We, and Re are the Deborah, Weissenberg and Reynolds numbers respectively. The linear stability problem is solved by a perturbation method for [Escr ]2 small and by a Galerkin method when [Escr ]2=O(1). Our results show that for all values of the retardation parameter β and for all values of [Escr ]2 considered the base viscometric flow is stable if [Escr ]1 is sufficiently small. As [Escr ]1 increases past a critical value the flow becomes unstable as a pair of complex-conjugate eigenvalues crosses the imaginary axis into the right half-plane. The critical value of [Escr ]1 decreases as [Escr ]2 increases indicating that increasing inertia destabilizes the flow. For the range of values considered the critical wavenumber kc first decreases and then increases as [Escr ]2 increases. The wave speed on the other hand decreases monotonically with [Escr ]2. The critical mode at the onset of instability corresponds to a travelling wave propagating inward towards the apex of the cone with infinitely many logarithmically spaced toroidal roll cells.

Under fairly general assumptions, this paper shows that for periodic porous media, whose period is of the same order as that of the inclusion, the nonlinear correction to Darcy's law is quadratic in terms of the Reynolds number, i.e. cubic with respect to the seepage velocity. This claim is substantiated by reinspection of well-known experimental results, a mathematical proof (restricted to periodic porous media), and numerical calculations.

The influence of a small amount of gas within the saturating liquid of a porous medium on acoustic wave propagation is investigated. It is assumed that the gas volumes are spherical, homogeneously distributed, and that they are within a very narrow range of bubble sizes. It is shown that the compressibility of the saturating fluid is determined by viscous, thermal, and a newly introduced Biot-type damping of the oscillating gas bubbles, with mean gas bubble size and concentration as important parameters. Using a super-saturation technique, a homogeneous gas–liquid mixture within a porous test column is obtained. Gas bubble size and concentration are measured by means of compressibility experiments. Wave reflection and propagation experiments carried out in a vertical shock tube show pore pressure oscillations, which can be explained by incorporating a dynamic gas bubble behaviour in the linear Biot theory for plane wave propagation.

We investigate the nonlinear equilibration of magnetic fields in a smooth helical flow at large Reynolds number Re and magnetic Reynolds number Rm with Re[Gt ]Rm[Gt ]1. We start with a smooth spiral Couette flow driven by boundary conditions. Such flows act as dynamos, that is are unstable to growing magnetic fields; here we disregard purely hydrodynamic instabilities such as Taylor–Couette modes. The dominant feedback from a magnetic field mode is only on the mean flow and this yields a simplified ‘mean-flow system’ consisting of one magnetic mode and the mean flow, which we solve numerically. We also obtain the asymptotic structure of the equilibrated fields for weakly and strongly nonlinear regimes. In particular the field tends to concentrate in a cylindrical shell where all stretching and differential rotation is suppressed by the Lorentz force, and the fluid is in solid-body motion. This shell is bounded by thin diffusive layers where the stretching that maintains the field against diffusive decay is dominant.