A strong axial magnetic dipole field with magnetic energy 15 times larger than the kinetic energy of thermal convection is realized by a direct numerical simulation of the magnetohydrodynamic equation of an electrically conducting Boussinesq fluid in a rotating spherical shell which is driven by a temperature difference between the outer and inner boundaries against a gravity force pointed towards the system centre. Cyclonic and anticyclonic convection vortices are generated and play a primary role in the magnetic field intensification. The magnetic field is enhanced through the stretching of magnetic lines in four particular parts of the convection fields, namely inside anticyclones, between cyclones and their western neighbouring anticyclones at middle as well as low latitudes, and between anticyclones and the outer boundary. A ‘twist-turn’ loop of intense magnetic flux density is identified as a fundamental structure which yields dominant contributions both to the toroidal and poloidal components of the longitudinally averaged magnetic field. Various types of competitive interaction between the magnetic field and convection vortices are observed. Among these, a creation-and-annihilation cycle in a statistically equilibrium state is particularly important. It is composed of three sequentially recurrent dynamical processes: the generation of convection vortices by the Rayleigh–Bénard instability, the growth of anticyclones and the intensification of magnetic field by a concentrate-and-stretch mechanism, and the breakdown of vortices by the Lorentz force followed by diminution of the magnetic field. The energy transfer from the velocity to the magnetic fields takes place predominantly in this dynamical cycle.

The loss of axisymmetry in a swirling flow that is generated inside an enclosed cylindrical container by the steady rotation of one endwall is examined numerically. The two dimensionless parameters that govern these flows are the cylinder aspect ratio and a Reynolds number associated with the rotation of the endwall. This study deals with a fixed aspect ratio, height/radius = 2.5. At low Reynolds numbers the basic flow is steady and axisymmetric; as the Reynolds number increases the basic state develops a double recirculation zone on the axis, so-called vortex breakdown bubbles. On further increase in the Reynolds number the flow becomes unsteady through a supercritical Hopf bifurcation but remains axisymmetric. After the onset of unsteadiness, another two unsteady axisymmetric solution branches appear with further increase in Reynolds number, each with its own temporal characteristic: one is periodic and the other is quasi-periodic with a very low frequency modulation. Solutions on these additional branches are unstable to three-dimensional perturbations, leading to nonlinear modulated rotating wave states, but with the flow still dominated by the corresponding underlying axisymmetric mode. A study of the flow behaviour on and bifurcations between these solution branches is presented, both for axisymmetric and for fully three-dimensional flows. The presence of modulated rotating waves alters the structure of the bifurcation diagram and gives rise to its own dynamics, such as a truncated cascade of period doublings of very-low-frequency modulated states.

The sequence of transitions in going from steady to unsteady chaotic flow in a close-packed face-centred cubic array of spheres is examined using lattice-Boltzmann simulations. The transition to unsteady flow occurs via a supercritical Hopf bifurcation in which only the streamwise component of the spatially averaged velocity fluctuates and certain reflectional symmetries are broken. At larger Reynolds numbers, the cross-stream components of the spatially averaged velocity fluctuate with frequencies that are incommensurate with those of the streamwise component. This transition is accompanied by the breaking of rotational symmetries that persisted through the Hopf bifurcation. The resulting trajectories in the spatially averaged velocity phase space are quasi-periodic. At larger Reynolds numbers, the fluctuations are chaotic, having continuous frequency spectra with no easily identified fundamental frequencies. Visualizations of the unsteady flows in various dynamic states show that vortices are produced in which the velocity and vorticity are closely aligned. With increasing Reynolds number, the geometrical structure of the flow changes from one that is dominated by extension and shear to one in which the streamlines are helical. A mechanism for the dynamics is proposed in which energy is transferred to smaller scales by the dynamic interaction of vortices sustained by the underlying time-averaged flow.

Numerical solutions of the unsteady Navier–Stokes equations are considered for the flow induced by a thick-core vortex convecting along a surface in a two-dimensional incompressible flow. The presence of the vortex induces an adverse streamwise pressure gradient along the surface that leads to the formation of a secondary recirculation region followed by a narrow eruption of near-wall fluid in solutions of the unsteady boundary-layer equations. The locally thickening boundary layer in the vicinity of the eruption provokes an interaction between the viscous boundary layer and the outer inviscid flow. Numerical solutions of the Navier–Stokes equations show that the interaction occurs on two distinct streamwise length scales depending upon which of three Reynolds-number regimes is being considered. At high Reynolds numbers, the spike leads to a small-scale interaction; at moderate Reynolds numbers, the flow experiences a large-scale interaction followed by the small-scale interaction due to the spike; at low Reynolds numbers, large-scale interaction occurs, but there is no spike or subsequent small-scale interaction. The large-scale interaction is found to play an essential role in determining the overall evolution of unsteady separation in the moderate-Reynolds-number regime; it accelerates the spike formation process and leads to formation of secondary recirculation regions, splitting of the primary recirculation region into multiple corotating eddies and ejections of near-wall vorticity. These eddies later merge prior to being lifted away from the surface and causing detachment of the thick-core vortex.

In global ocean dynamics Rossby waves play a vital rôle in the long-term distribution of vorticity; knowledge of the interaction between these waves and topography is crucial to a full understanding of this process, and hence to the transportation of energy, mixing and ocean circulation. The interaction of baroclinic Rossby waves with abrupt topography is the focus of this study. In this paper we model the ocean as a continuously stratified fluid for which the linear theory predicts a qualitatively different structure for the wave modes than that predicted by barotropic or simple layered models, even if most of the density variation is confined to the thermocline. We consider the scattering of a westward-propagating baroclinic Rossby wave by a narrow ridge on the ocean floor, modelled by a line barrier of infinite extent, orientated at an arbitrary angle to the incident wave. Transmission and reflection coefficients for the propagating modes are found using both an algebraic method and, in the case where this breaks down, matched asymptotic expansions. The results are compared with recent analyses of satellite altimetry data.

Unsteady turbulence in stably and unstably stratified flow with system rotation around the vertical axis is analysed using the rapid distortion theory (RDT). Complete linear solutions for the spectra, variances and covariances are obtained analytically, and their characteristics, including the short- and long-time asymptotics and the effect of initial conditions, are examined in detail. It has been found that the rotation modifies the energy partition among the three kinetic energy components and the potential energy, and the ratio of the Coriolis parameter f to the Brunt–Väisälä frequency N, i.e. f/N, determines the final steady values of these components. The ratio also determines the phase of the energy/flux oscillation. Depending on whether f/N > 1 or f/N < 1, there is a phase shift of ±π/4. However, unsteady aspects are largely dominated by stratification. This occurs because the effects of the Coriolis parameter f appear only in the form of fk3, which vanishes for the horizontal wavenumber components (k3 = 0), which contribute most to the energies and the fluxes. For example, the oscillation frequency of the energies and the fluxes asymptotes to 2N over a long time, in agreement with the stratified non-rotating turbulence. The initial time development is also dominanted by the stratification, and the short-time asymptotics (Nt, ft [Lt ] 1) agree with those for non-rotating stratified fluids in the lowest-order approximation. In the special case of f = N, all the wavenumber components oscillate in phase, leading to no inviscid decay of oscillation. This is in contrast to the general case of f ≠ N, in which inviscid decay has been observed. For pure rotation (f ≠ 0, N = 0), analytical solutions showed that any turbulence that is initially axisymmetric around the rotation axis recovers exact three-dimensional isotropy in the kinetic energy components. Comparison with previous DNS and experiments shows that many of the unsteady aspects of the kinetic and potential energies and the vertical density flux can be explained by the linear processes described by RDT. Even the time development of the vertical vorticity, which would represent the small-scale characteristics of turbulence, agrees well with DNS. For unstably stratified turbulence, the initial processes observed in DNS and experiments, such as the initial decay of the kinetic energy due to viscosity and the subsequent rapid growth of the vertical kinetic energy compared to the horizontal kinetic energy, could be explained by RDT.

A method is developed for statistical prediction of turbulent geophysical flows that is more efficient than ensemble integrations. We consider the evolution of low-order moments for inviscid quasi-geostrophic turbulence. Guided by statistical mechanics, equations are developed for predicting the mean and the variance about the mean as functions of position and time. These equations are consistent with the exact moment equations and contain irreversible (entropy producing) fluxes that must be specified in terms of known moments. Using simple choices for these dependences, the resulting scheme, involving just two spatial fields, typically outperforms 100-realization ensembles.

Oscillatory forcing of a porous medium may have a dramatic effect on the mean flow rate produced by a steady applied pressure gradient. The oscillatory forcing may excite nonlinear inertial effects leading to either enhancement or retardation of the mean flow. Here, in Part 1, we consider the effects of non-zero inertial forces on steady flows in porous media, and investigate the changes in the flow character arising from changes in both the strength of the inertial terms and the geometry of the medium. The steady-state Navier–Stokes equations are solved via a Galerkin finite element method to determine the velocity fields for simple two-dimensional models of porous media. Two geometric models are considered based on constricted channels and periodic arrays of circular cylinders. For both geometries, we observe solution multiplicity yielding both symmetric and asymmetric flow patterns. For the cylinder arrays, we demonstrate that inertial effects lead to anisotropy in the effective permeability, with the direction of minimum resistance dependent on the solid volume fraction. We identify nonlinear flow phenomena which might be exploited by oscillatory forcing to yield a net increase in the mean flow rate. In Part 2, we take up the subject of unsteady flows governed by the full time-dependent Navier–Stokes equations.

Numerical computations are employed to study the phenomenon of oscillatory forcing of flow through porous media. The Galerkin finite element method is used to solve the time-dependent Navier–Stokes equations to determine the unsteady velocity field and the mean flow rate subject to the combined action of a mean pressure gradient and an oscillatory body force. With strong forcing in the form of sinusoidal oscillations, the mean flow rate may be reduced to 40% of its unforced steady-state value. The effectiveness of the oscillatory forcing is a strong function of the dimensionless forcing level, which is inversely proportional to the square of the fluid viscosity. For a porous medium occupied by two fluids with disparate viscosities, oscillatory forcing may be used to reduce the flow rate of the less viscous fluid, with negligible effect on the more viscous fluid. The temporal waveform of the oscillatory forcing function has a significant impact on the effectiveness of this technique. A spike/plateau waveform is found to be much more efficient than a simple sinusoidal profile. With strong forcing, the spike waveform can induce a mean axial flow in the absence of a mean pressure gradient. In the presence of a mean pressure gradient, the spike waveform may be employed to reverse the direction of flow and drive a fluid against the direction of the mean pressure gradient. Owing to the viscosity dependence of the dimensionless forcing level, this mechanism may be employed as an oscillatory filter to separate two fluids of different viscosities, driving them in opposite directions in the porous medium. Possible applications of these mechanisms in enhanced oil recovery processes are discussed.

This paper describes computer simulation studies of granular materials under dense conditions where particles are in persistent contact with their neighbours and the elasticity of the material becomes an important rheological parameter. There are two regimes at this limit, one for which the stresses scale with both elastic and inertial properties (called the elastic–inertial regime), and a non-inertial quasi-static regime in which the stresses scale purely elastically (elastic–quasi-static). In these elastic regimes, the forces are generated by internal force chains. Reducing the concentration slightly causes a transition from an elastic to a purely inertial behaviour. This transition occurs so abruptly that a 2% concentration reduction can be accompanied by nearly three orders of magnitude of stress reduction. This indicates that granular flows near this limit are prone to instabilities such as those commonly observed in shear cells. Unexpectedly, there is no path between inertial (rapid) flow and quasi-static flow by varying the shear rate at a fixed concentration; only by reducing the concentration can one cause a transition from quasi-static to inertial flow. The solid concentrations at which this transition occurs as well as the magnitude of the stresses in the elastic regimes are strong functions of the particle surface friction, because the surface friction strongly affects the strength of the force chains. A parametric analysis of the elastic regime generated flowmaps showing the various regimes that might be realized in practice. Many common materials such as sand require such large shear rates to reach the elastic–inertial regime that it is unattainable for all practical purposes; such materials will demonstrate either an elastic–quasi-static behaviour or a pure inertial behaviour depending on the concentration – with many orders of magnitude of stress change between them. Finally, the effects of nonlinear contacts are investigated and an appropriate scaling is proposed that accounts for the nonlinear behaviour in the elastic–quasi-static regime.

Motion of a single fluid sphere is described by two theories, each characterized by different levels of Hill's vortex circulation within the sphere. An existing experimental data set giving measurements of vertical velocity along the major axis of the sphere is re-examined. Contrary to published discussions of that experiment, we find that the theory of Parlange agrees better with the laboratory data than that of Harper & Moore. This agreement supports the key difference between the two theories, i.e. that the fluid within the sphere is unlikely to have a singular (infinite) velocity as it moves upwards towards the stagnation region at the top of the sphere.

Asymptotic solutions for unsteady one-dimensional axisymmetric laminar flow in a pipe subject to rapid deceleration and/or acceleration are derived and their stability investigated using linear and weakly nonlinear analysis. In particular, base flow solutions for unsteady one-dimensional axisymmetric laminar flow in a pipe are derived by the method of matched asymptotic expansions. The solutions are valid for short times and can be successfully applied to the case of an arbitrary (but unidirectional) axisymmetric initial velocity distribution. Excellent agreement between asymptotic and analytical solutions for the case of an instantaneous pipe blockage is found for small time intervals. Linear stability of the base flow solutions obtained from the asymptotic expansions to a three-dimensional perturbation is investigated and the results are used to re-interpret the experimental results of Das & Arakeri (1998). Comparison of the neutral stability curves computed with and without the planar channel assumption shows that this assumption is accurate when the ratio of the unsteady boundary layer thickness to radius (i.e. δ1/R) is small but becomes unacceptable when this ratio exceeds 0.3. Both the current analysis and the experiments show that the flow instability is non-axisymmetric for δ1/R = 0.55 and 0.85. In addition, when δ1/R = 0.18 and 0.39, the neutral stability curves for n = 0 and n = 1 are found to be close to one another at all times but the most unstable mode in these two cases is the axisymmetric mode. The accuracy of the quasi-steady assumption, employed both in this research and in that of Das & Arakeri (1998), is supported by the fact that the results obtained under this assumption show satisfactory agreement with the experimental features such as type of instability and spacing between vortices. In addition, the computations show that the ratio of the rate of growth of perturbations to the rate of change of the base flow is much larger than 1 for all cases considered, providing further support for the quasi-steady assumption. The neutral stability curves obtained from linear stability analysis suggest that a weakly nonlinear approach can be used in order to study further development of instability. Weakly nonlinear analysis shows that the amplitude of the most unstable mode is governed by the complex Ginzburg–Landau equation which reduces to the Landau equation if the amplitude is a function of time only. The coefficients of the Landau equation are calculated for two cases of the experimental data given by Das & Arakeri (1998). It is shown that the real part of the Landau constant is positive in both cases. Therefore, finite-amplitude equilibrium is possible. These results are in qualitative agreement with experimental data of Das & Arakeri (1998).

There is a growing body of literature in which turbulent boundary layer flow along a mixed-boundary corner formed by a vertical solid wall and a horizontal free surface has been examined. While there is consensus regarding the existence of weak secondary flows in the near corner region, there is some disagreement as to the exact nature and origin of these flows. In two earlier works by the authors, evidence was presented supporting the existence of a weak streamwise vortex which rotates in toward the wall at the free surface and down away from the surface along the wall. This ‘inner secondary vortex’ is accompanied by an ‘outer secondary flow’ which transports low-momentum boundary layer fluid up along the wall and outward at the free surface. The magnitudes of the cross-stream velocities associated with these secondary flows were measured to be on the order of 1% of the free-stream speed. In this paper, high-resolution DPIV measurements made in the cross-stream plane are presented. These clearly show the inner and outer secondary flows. The cross-stream vector fields allow computation of terms in the turbulent streamwise vorticity transport equation. These terms indicate mean vorticity transport at the free surface associated with the outer secondary flow. In addition there appears to be a balance between the wall-normal and free-surface-normal fluctuating vorticity reorientation terms.

Recent studies on vortex ring generation, e.g. Rosenfeld et al. (1998), have highlighted the subtle effect of generation geometry on the final properties of rings. Experimental generation of vortex rings often involves moving a piston through a tube, resulting in a vortex ring being generated at the tube exit. A generation geometry that has been cited as a standard consists of the tube exit mounted flush with a wall, with the piston stroke ending at the tube exit, Glezer (1988). We employ this geometry to investigate the effect of the vortex that forms in front of the advancing piston (piston vortex) on the primary vortex ring that is formed at the tube exit. It is shown that when the piston finishes its stroke flush with the wall, and hence forms an uninterrupted plane, the piston vortex is convected through the primary ring and then ingested into the primary vortex. The ingestion of the piston vortex results in an increased ring impulse and an altered trajectory, when compared to the case when the piston motion finishes inside the tube. As the Reynolds number of the experiments, based on the piston speed and piston diameter, is the order of 3000, transition to turbulence is observed during the self-induced translation phase of the ring motion. Compared to the case when the piston is stopped inside the tube, the vortex ring which has ingested the piston vortex transitions to turbulence at a significantly reduced distance from the orifice exit and suggests the transition map suggested by Glezer (1988) is under question. A secondary instability characterized by vorticity filaments with components in the axial and radial directions, is observed forming on the piston vortex. The structure of the instability appears to be similar to the streamwise vortex filaments that form in the braid regions of shear layers. This instability is subsequently ingested into the primary ring during the translation phase and may act to accelerate the growth of the Tsai–Widnall instability. It is suggested that the origin of the instability is Görtler in nature and the result of the unsteady wall jet nature of the boundary layer separating on the piston face.

The formation and development of transverse and crescentic sand bars in the coastal marine environment has been investigated by means of a nonlinear numerical model based on the shallow-water equations and on a simplified sediment transport parameterization. By assuming normally approaching waves and a saturated surf zone, rhythmic patterns develop from a planar slope where random perturbations of small amplitude have been superimposed. Two types of bedforms appear: one is a crescentic bar pattern centred around the breakpoint and the other, herein modelled for the first time, is a transverse bar pattern. The feedback mechanism related to the formation and development of the patterns can be explained by coupling the water and sediment conservation equations. Basically, the waves stir up the sediment and keep it in suspension with a certain cross-shore distribution of depth-averaged concentration. Then, a current flowing with (against) the gradient of sediment concentration produces erosion (deposition). It is shown that inside the surf zone, these currents may occur due to the wave refraction and to the redistribution of wave breaking produced by the growing bedforms. Numerical simulations have been performed in order to understand the sensitivity of the pattern formation to the parameterization and to relate the hydro-morphodynamic input conditions to which of the patterns develops. It is suggested that crescentic bar growth would be favoured by high-energy conditions and fine sediment while transverse bars would grow for milder waves and coarser sediment. In intermediate conditions mixed patterns may occur.