The three-dimensional motion observed by Chen & Chen (1997) in the convection cells generated by sideways heating of a solute gradient is further examined by experiments and linear stability analysis. In the experiments, we obtained visualizations and PIV measurements of the velocity of the fluid motion in the longitudinal plane perpendicular to the imposed temperature gradient. The flow consists of a horizontal row of counter-rotating vortices within each convection cell. The magnitude of this secondary motion is approximately one-half that of the primary convection cell. Results of a linear stability analysis of a parallel double-diffusive flow model of the actual ow show that the instability is in the salt-finger mode under the experimental conditions. The perturbation streamlines in the longitudinal plane at onset consist of a horizontal row of counter-rotating vortices similar to those observed in the experiments.

An efficient algorithm for hydrodynamical interaction of many deformable drops subject to shear flow at small Reynolds numbers with triply periodic boundaries is developed. The algorithm, at each time step, is a hybrid of boundary-integral and economical multipole techniques, and scales practically linearly with the number of drops N in the range N < 1000, for NΔ ∼ 103 boundary elements per drop. A new near-singularity subtraction in the double layer overcomes the divergence of velocity iterations at high drop volume fractions c and substantial viscosity ratio γ. Extensive long-time simulations for N = 100–200 and NΔ = 1000–2000 are performed up to c = 0.55 and drop-to-medium viscosity ratios up to λ = 5, to calculate the non-dimensional emulsion viscosity μ* = Σ12/(μeγ˙), and the first N1 = (Σ11−Σ22)/(μe[mid ]γ˙[mid ]) and second N2 = (Σ22−Σ33)/(μe[mid ]γ˙[mid ]) normal stress differences, where γ˙ is the shear rate, μe is the matrix viscosity, and Σij is the average stress tensor. For c = 0.45 and 0.5, μ* is a strong function of the capillary number Ca = μe[mid ]γ˙[mid ]a/σ (where a is the non-deformed drop radius, and σ is the interfacial tension) for Ca [Lt ] 1, so that most of the shear thinning occurs for nearly non-deformed drops. For c = 0.55 and λ = 1, however, the results suggest phase transition to a partially ordered state at Ca [les ] 0.05, and μ* becomes a weaker function of c and Ca; using λ = 3 delays phase transition to smaller Ca. A positive first normal stress difference, N1, is a strong function of Ca; the second normal stress difference, N2, is always negative and is a relatively weak function of Ca. It is found at c = 0.5 that small systems (N ∼ 10) fail to predict the correct behaviour of the viscosity and can give particularly large errors for N1, while larger systems N [ges ] O(102)show very good convergence. For N ∼ 102 and NΔ ∼ 103, the present algorithm is two orders of magnitude faster than a standard boundary-integral code, which has made the calculations feasible.

In this paper we first describe the current method for obtaining the Camassa–Holm equation in the context of water waves; this requires a detour via the Green–Naghdi model equations, although the important connection with classical (Korteweg–de Vries) results is included. The assumptions underlying this derivation are described and their roles analysed. (The critical assumptions are, (i) the simplified structure through the depth of the water leading to the Green–Naghdi equations, and, (ii) the choice of submanifold in the Hamiltonian representation of the Green–Naghdi equations. The first of these turns out to be unimportant because the Green–Naghdi equations can be obtained directly from the full equations, if quantities averaged over the depth are considered. However, starting from the Green–Naghdi equations precludes, from the outset, any role for the variation of the flow properties with depth; we shall show that this variation is significant. The second assumption is inconsistent with the governing equations.)

Returning to the full equations for the water-wave problem, we retain both parameters (amplitude, ε, and shallowness, δ) and then seek a solution as an asymptotic expansion valid for, ε → 0, δ → 0, independently. Retaining terms O(ε), O(δ2) and O(εδ2), the resulting equation for the horizontal velocity component, evaluated at a specific depth, is a Camassa–Holm equation. Some properties of this equation, and how these relate to the surface wave, are described; the role of this special depth is discussed. The validity of the equation is also addressed; it is shown that the Camassa–Holm equation may not be uniformly valid: on suitably short length scales (measured by δ) other terms become important (resulting in a higher-order Korteweg–de Vries equation, for example). Finally, we indicate how our derivation can be extended to other scenarios; in particular, as an example, we produce a two-dimensional Camassa–Holm equation for water waves.

This paper describes the interaction of symmetric vortices in a three-dimensional quasi-geostrophic fluid. The initial vortices are taken to be uniform-potential-vorticity ellipsoids, of height 2h and width 2R, and with centres at (±d/2; 0, 0), embedded within a background flow having constant background rotational and buoyancy frequencies, f/2 and N respectively. This problem was previously studied by von Hardenburg et al. (2000), who determined the dimensionless critical merger distance d/R as a function of the height-to-width aspect ratio h/R (scaled by f/N). Their study, however, was limited to small to moderate values of h/R, as it was anticipated that merger at large h/R would reduce to that for two columnar two-dimensional vortices, i.e. d/R ≈ 3.31. Here, it is shown that no such two-dimensional limit exists; merger is found to occur at any aspect ratio, with d ∼ h for h/R [Gt ] 1.

New results are also found for small to moderate values of h/R. In particular, our numerical simulations reveal that asymmetric merger is predominant, despite the initial conditions, if one includes a small amount of random noise. For small to moderate h/R, decreasing the initial separation distance d first results in a weak exchange of material, with one vortex growing at the expense of the other. As d decreases further, this exchange increases and leads to two dominant but strongly asymmetric vortices. Finally, for yet smaller d, rapid merger into a single dominant vortex occurs – in effect the initial vortices exchange nearly all of their material with one another in a nearly symmetrical fashion.

The particle pressure is the surface force in a particle/fluid mixture that is exerted solely by the particle phase. This paper presents measurements of the particle pressure on the faces of a two-dimensional gas-fluidized bed and gives insight into the mechanisms by which bubbles generate particle pressure. The particle pressure is measured by a specially designed ‘particle pressure transducer’. The results show that, around single bubbles, the most significant particle pressures are generated below and to the sides of the bubble and that these particle pressures steadily increase and reach a maximum value at bubble eruption. The dominant mechanism appears to be defluidization of material in the particle phase that results from the bubble attracting fluidizing gas away from the surrounding material; the surrounding material is no longer supported by the gas flow and can only be supported across interparticle contacts which results in the observed particle pressures. The contribution of particle motion to particle pressure generation is insignificant.

The magnitude of the particle pressure below a single bubble in a gas-fluidized bed depends on the bubble size and the density of the solid particles, as might be expected as the amount of gas attracted by the bubble should increase with bubble size and because the weight of defluidized material depends on the density of the solid material. A simple scaling of these quantities is suggested that is otherwise independent of the bed material.

In freely bubbling gas-fluidized beds the particle pressures generated behave differently. Overall they are smaller in magnitude and reach their maximum value soon after the bubble passes instead of at eruption. In this situation, it appears that the bubbles interact with one another in such a way that the de uidization effect below a leading bubble is largely counteracted by refluidizing gas exiting the roof of trailing bubbles.

A numerical exploration of the linear stability of a fluid confined between two coaxial cylinders rotating independently and with an imposed axial pressure gradient (spiral Poiseuille flow) is presented. The investigation covers a wide range of experimental parameters, being focused on co-rotation situations. The exploration is made for a wide gap case in order to compare the numerical results with previous experimental data available. The competition between shear and centrifugal instability mechanisms affects the topological features of the neutral stability curves and the critical surface is observed to exhibit zeroth-order discontinuities. These curves may exhibit disconnected branches which lower the critical values of instability considerably. The same phenomenon has been reported in similar fluid flows where shear and centrifugal instability mechanisms compete. The stability analysis of the rigid-body rotation case is studied in detail and the asymptotic critical values are found to be qualitatively similar to those obtained in rotating Hagen-Poiseuille and spiral Couette flows. The results are in good agreement with the previous experimental explorations.

A mathematical model and numerical method for studying the collective dynamics of geotactic, gyrotactic and chemotactic micro-organisms immersed in a viscous fluid is presented. The Navier–Stokes equations of fluid dynamics are solved in the presence of a discrete collection of micro-organisms. These microbes act as point sources of gravitational force in the fluid equations, and thus affect the fluid flow. Physical factors, e.g. vorticity and gravity, as well as sensory factors affect swimming speed and direction. In the case of chemotactic microbes, the swimming orientation is a function of a molecular field. In the model considered here, the molecules are a nutrient whose consumption results in an upward gradient of concentration that drives its downward diffusion. The resultant upward chemotactically induced accumulation of cells results in (Rayleigh–Taylor) instability and eventually in steady or chaotic convection that transports molecules and affects the translocation of organisms. Computational results that examine the long-time behaviour of the full nonlinear system are presented.

The actual dynamical system consisting of fluid and suspended swimming organisms is obviously three-dimensional, as are the basic modelling equations. While the computations presented in this paper are two-dimensional, they provide results that match remarkably well the spatial patterns and long-time temporal dynamics of actual experiments; various physically applicable assumptions yield steady states, chaotic states, and bottom-standing plumes. The simplified representation of microbes as point particles allows the variation of input parameters and modelling details, while performing calculations with very large numbers of particles (≈104–105), enough so that realistic cell concentrations and macroscopic fluid effects can be modelled with one particle representing one microbe, rather than some collection of microbes. It is demonstrated that this modelling framework can be used to test hypotheses concerning the coupled effects of microbial behaviour, fluid dynamics and molecular mixing. Thus, not only are insights provided into the differing dynamics concerning purely geotactic and gyrotactic microbes, the dynamics of competing strategies for chemotaxis, but it is demonstrated that relatively economical explorations in two dimensions can deliver striking insights and distinguish among hypotheses.

The subject of the present work is the start-up vortex issuing from a sharp trailing edge accelerated from rest in still air. A numerical simulation of the flow has been performed in the case of a semi-infinite at plate by solving the Navier–Stokes equations in the ψ_ω formulation. The numerical algorithm is based on a fast multigrid implicit integration of the difference equations in an unstructured mesh that is dynamically built to minimize the computational costs. A local refinement of the mesh near the edge of the plate increases the accuracy of the simulation. The results show that the asymptotic stage of the vortex evolution is self-similar in the mean, but the appearance of instabilities produces a time-dependent flow which is not instantaneously self-similar.

A statistical ensemble of large-eddy simulations (LES) is run simultaneously for the same flow. The information provided by the different large-scale velocity fields is used in an ensemble-averaged version of the dynamic model. This produces local model parameters that only depend on the statistical properties of the flow. An important property of the ensemble-averaged dynamic procedure is that it does not require any spatial averaging and can thus be used in fully inhomogeneous flows. Also, the ensemble of LES provides statistics of the large-scale velocity that can be used for building new models for the subgrid-scale stress tensor. The ensemble-averaged dynamic procedure has been implemented with various models for three flows: decaying isotropic turbulence, forced isotropic turbulence, and the time-developing plane wake. It is found that the results are almost independent of the number of LES in the statistical ensemble provided that the ensemble contains at least 16 realizations.

Freely decaying turbulent flows in a stably stratified fluid are simulated with a pseudo-spectral numerical code solving the fully nonlinear Navier–Stokes equations under the Boussinesq approximation with periodic boundary conditions. The flow is decomposed into a turbulent field and a horizontal mean flow ū(z, t) defined as the average of the horizontal velocity component in a horizontal plane at height z and time t. Similarly, the density field is decomposed into a turbulent field and a (stable) mean density profile ρ(z, t) defined as the average of the density field in a horizontal plane at height z and time t. Attention is paid to the effect of the turbulent velocity field on an initial z-periodic horizontal mean flow (Simulation A) or an initial z-periodic perturbation of the mean density profile (Simulation B). Both A and B are performed under conditions of moderate and strong stratification and are compared to the non-stratified simulations.

Simulation A shows that the turbulence–mean flow interaction is strongly affected by the buoyancy forces. In the absence of a stratification, the mean flow perturbation decays rapidly due to the turbulent diffusion of momentum. When a moderate stratification is applied, the mean flow perturbation decays much more slowly whereas it oscillates and grows with time when the stratification is strong. These results are interpreted by defining a time-dependent eddy viscosity. Whereas the eddy viscosity coefficient has positive values in the non-stratified simulation, it is affected by the buoyancy forces and decreases after a period of order N−1. For large times, the eddy diffusivity oscillates and its time-averaged value over a few turnover timescales is positive but small when the stratification is moderate, and roughly zero when the stratification is strong. These results are interpreted by defining a time-dependent eddy viscosity. Whereas the eddy viscosity coefficient has positive values in the non-stratified simulation, it is affected by the buoyancy forces and decreases after a period of order N−1 in the stratified simulations (where N is the Brunt–Väisälä frequency associated with the background linear stratification). At large time, we find that the eddy viscosity remains roughly zero when the stratification is moderate, whereas it oscillates but remains persistently negative in the strongly stratified case, which causes the horizontal mean flow to accelerate.

We conclude that the presence of a stable stratification strongly affects the temporal behaviour of the mean quantities ū and ρ in turbulent flows and partly explains the formation of horizontal layers in stratified geofluids such as oceans and atmospheres.

The initial evolution of the momentum and buoyancy fluxes in a freely decaying, stably stratified homogeneous turbulent flow with r.m.s. velocity u′0 and integral lengthscale l0 is calculated using a weakly inhomogeneous and unsteady form of the rapid distortion theory (RDT) in order to study the growth of small temporal and spatial perturbations in the large-scale mean stratification N(z, t) and mean velocity profile ū(z, t) (here N is the local Brunt–Väisälä frequency and ū is the local velocity of the horizontal mean flow) when the ratio of buoyancy forces to inertial forces is large, i.e. Nl0/u′0[Gt ]1. The lengthscale L of the perturbations in the mean profiles of stratification and shear is assumed to be large compared to l0 and the presence of a uniform background mean shear can be taken into account in the model provided that the inertial shear forces are still weaker than the buoyancy forces, i.e. when the Richardson number Ri = (N/∂zū)2[Gt ]1 at each height.

When a mean shear perturbation is introduced initially with no uniform background mean shear and uniform stratification, the analysis shows that the perturbations in the mean flow profile grow on a timescale of order N-1. When the mean density profile is perturbed initially in the absence of a background mean shear, layers with significant density gradient fluctuations grow on a timescale of order N−10 (where N0 is the order of magnitude of the initial Brunt–Väisälä frequency) without any associated mean velocity gradients in the layers. These results are in good agreement with the direct numerical simulations performed by Galmiche et al. (2002) and are consistent with the earlier physically based conjectures made by Phillips (1972) and Posmentier (1977). The model also shows that when there is a background mean shear in combination with perturbations in the mean stratification, negative shear stresses develop which cause the mean velocity gradient to grow in the density layers. The linear analysis for short times indicates that the scale on which the mean perturbations grow fastest is of order u′0/N0, which is consistent with the experiments of Park et al. (1994).

We conclude that linear mechanisms are widely involved in the formation of shear and density layers in stratified flows as is observed in some laboratory experiments and geophysical flows, but note that the layers are also significantly influenced by nonlinear and dissipative processes at large times.

A double Hopf bifurcation has been found of the flow in a cylinder driven by the rotation of an endwall. A detailed analysis of the multiple solutions in a large region of parameter space, computed with an efficient and accurate three-dimensional Navier-Stokes solver, is presented. At the double Hopf point, an axisymmetric limit cycle and a rotating wave bifurcate simultaneously. The corresponding mode interaction generates an unstable two-torus modulated rotating wave solution and gives a wedge-shaped region in parameter space where the two periodic solutions are both stable. By exploring in detail the three-dimensional structure of the flow, we have identified the two mechanisms that compete in the neighbourhood of the double Hopf point. Both are associated with the jet that is formed when the Ekman layer on the rotating endwall is turned by the stationary sidewall.

The eigenvalue problem for slow oscillations of a liquid in a set of N cylindrical wells that are bounded above by free surfaces and below by a common, semi-infinite reservoir is formulated on the assumption that the depth of the wells is large compared with their width, so that the lowest mode in each well, for which the fluid moves as a rigid body, dominates the higher modes. Detailed results are presented for a single well, a pair of identical circular wells, and linear and equilateral triplets. Comparison with Molin's (2001) result for a rectangular well suggests that the present result for a circular well should provide a good approximation for the Helmholtz mode in any well of the same cross-sectional area and moderate aspect ratio.

The instability of the three-dimensional high-shear layer associated with a near-wall low-speed streak is investigated experimentally. A single low-speed streak, not unlike the near-wall low-speed streaks in transitional and turbulent flows, is produced in a laminar boundary layer by using a small piece of screen set normal to the wall. In order to excite symmetric and anti-symmetric modes separately, well-controlled external disturbances are introduced into the laminar low-speed streak through small holes drilled behind the screen. The growth of the excited symmetric varicose mode is essentially governed by the Kelvin–Helmholtz instability of the in ectional velocity profiles across the streak in the normal-to-wall direction and it can occur when the streak width is larger than the shear-layer thickness. The spatial growth rate of the symmetric mode is very sensitive to the streak width and is rapidly reduced as the velocity defect decreases flowing to the momentum transfer by viscous stresses. By contrast, the anti-symmetric sinuous mode that causes the streak meandering is dominated by the wake-type instability of spanwise velocity distributions across the streak. As far as the linear instability is concerned, the growth rate of the anti-symmetric mode is not so strongly affected by the decrease in the streak width, and its exponential growth may continue further downstream than that of the symmetric mode. As for the mode competition, it is important to note that when the streak width is narrow and comparable with the shear-layer thickness, the low-speed streak becomes more unstable to the anti-symmetric modes than to the symmetric modes. It is clearly demonstrated that the growth of the symmetric mode leads to the formation of hairpin vortices with a pair of counter-rotating streamwise vortices, while the anti-symmetric mode evolves into a train of quasi-streamwise vortices with vorticity of alternate sign.

Numerical simulations are used to investigate the resonant instabilities in two-dimensional flow past an open cavity. The compressible Navier–Stokes equations are solved directly (no turbulence model) for cavities with laminar boundary layers upstream. The computational domain is large enough to directly resolve a portion of the radiated acoustic field, which is shown to be in good visual agreement with schlieren photographs from experiments at several different Mach numbers. The results show a transition from a shear-layer mode, primarily for shorter cavities and lower Mach numbers, to a wake mode for longer cavities and higher Mach numbers. The shear-layer mode is characterized well by the acoustic feedback process described by Rossiter (1964), and disturbances in the shear layer compare well with predictions based on linear stability analysis of the Kelvin–Helmholtz mode. The wake mode is characterized instead by a large-scale vortex shedding with Strouhal number independent of Mach number. The wake mode oscillation is similar in many ways to that reported by Gharib & Roshko (1987) for incompressible flow with a laminar upstream boundary layer. Transition to wake mode occurs as the length and/or depth of the cavity becomes large compared to the upstream boundary-layer thickness, or as the Mach and/or Reynolds numbers are raised. Under these conditions, it is shown that the Kelvin–Helmholtz instability grows to sufficient strength that a strong recirculating flow is induced in the cavity. The resulting mean flow is similar to wake profiles that are absolutely unstable, and absolute instability may provide an explanation of the hydrodynamic feedback mechanism that leads to wake mode. Predictive criteria for the onset of shear-layer oscillations (from steady flow) and for the transition to wake mode are developed based on linear theory for amplification rates in the shear layer, and a simple model for the acoustic efficiency of edge scattering.

Just after formation, optical fibres are wetted stably with acrylate at capillary numbers routinely exceeding 1000. It is hypothesized that this is possible because of dissolution of air into the liquid coating. A lubrication/boundary integral analysis that includes gas diffusion and solubility is developed. It is applied using conservatively estimated solubility and diffusivity coefficients and solutions are found that are consistent with industry practice and with the hypothesis. The results also agree with the claim of Deneka, Kar & Mensah (1988) that the use of high-solubility gases to bathe a wetting line allows significantly greater wetting speeds. The solutions indicate a maximum speed of wetting which increases with gas solubility and with reduction in wetting-channel diameter.

The relative motion of drops in shear flows is responsible for collisions leading to the creation of larger drops. The collision of liquid drops in a gas is considered here. The drops are small enough for the Reynolds number to be low (negligible fluid motion inertia), yet large enough for the Stokes number to be possibly of order unity (non-negligible inertia in the motion of drops). Possible concurrent effects of Van der Waals attractive forces and drop inertia are taken into account.

General expressions are first presented for the drag forces on two interacting drops of different sizes embedded in a general linear flow field. These expressions are obtained by superposition of solutions for the translation of drops and for steady drops in elementary linear flow fields (simple shear flows, pure straining motions). Earlier solutions adapted to the case of inertialess drops (by Zinchenko, Davis and coworkers) are completed here by the solution for a simple shear flow along the line of centres of the drops. A solution of this problem in bipolar coordinates is provided; it is consistent with another solution obtained as a superposition of other elementary flow fields.

The collision efficiency of drops is calculated neglecting gravity effects, that is for strongly sheared linear flow fields. Results are presented for the cases of a simple linear shear flow and an axisymmetric pure straining motion. As expected, the collision efficiency increases with the Stokes numbers, that is with drop inertia. On the other hand, the collision efficiency in a simple shear flow becomes negligible below some value of the ratio of radii, regardless of drop inertia. The value of this threshold increases with decreasing Van der Waals forces. The concurrence between drop inertia and attractive van der Waals forces results in various anisotropic shapes of the collision cross-section. By comparison, results for the collision efficiency in an axisymmetric pure straining motion are more regular. This flow field induces axisymmetric sections of collision and strong inertial effects resulting in collision efficiencies larger than unity. Effects of van der Waals forces only appear when one of the drops has a very low Stokes number.

Motivated by industrial and geophysical solidification problems such as segregation in metallic castings and brine expulsion from growing sea ice, we present and solve a model for steady convection in a two-dimensional mushy layer of a binary mixture. At sufficiently large amplitudes of convection, steady states are found in which plumes emanate from vertical chimneys (channels of zero solid fraction) in the mushy layer. The mush–liquid interface, including the chimney wall, is a free boundary whose shape and location we determine using local equilibrium conditions. We map out the changing structure of the system as the Rayleigh number varies, and compute various measures of the amplitude of convection including the flux of solute out of the mushy layer, through chimneys. We find that there are no steady states if the Rayleigh number is less than a global critical value, which is less than the linear critical value for convection to occur. At larger values of the Rayleigh number we find, in agreement with experiments, that the width of chimneys and the height of the mushy layer both decrease relative to the thermal-diffusion length, which is the scale height of the mushy layer in the absence of convection. We find evidence to suggest that the spacing between neighbouring chimneys at high Rayleigh numbers is smaller than the critical wavelengths of both the linear and global stability modes.