We analyse the dynamics of small, rigid, dilute spherical particles in the far wake of a bluff body under the assumption that the background flow field is approximated by a periodic array of Stuart vortices that can be considered to be a regularization of the von Kármán vortex street. Using geometric singular perturbation theory and numerical methods, we show that when inertia (measured by a dimensionless Stokes number) is not too large, there is a periodic attractor in the phase space of the dynamical system governing the particle motion. We argue that this provides a simple mechanism to explain the unexpected ‘focusing’ effect that has been observed both numerically and experimentally in the far-wake flow past a bluff body by Tang et al. (1992). Their results show that over a range of Reynolds numbers and intermediate values of the Stokes number, particles injected into the wake of a bluff body concentrate near the edges of the vortex structures downstream, thus tending to ‘demix’ rather than disperse homogeneously.

The surface current generated by internal waves in the ocean affects surface gravity waves. The propagation of short surface waves is studied using both simple ray theory for linear waves and a fully nonlinear numerical potential solver. Attention is directed to the case of short waves with initially uniform wavenumber, as may be generated by a strong gust of wind. In general, some of the waves are focused by the surface current and in these regions the waves steepen and may break. Comparisons are made between ray theory and the more accurate solutions. For ray theory, the occurrence of focusing is examined in some detail and exact analytic solutions are found for rays on currents with linear and quadratic spatial variation – only the latter giving focusing for our initial conditions. With regard to interpretation of remote sensing of the sea surface, we find that enhanced wave steepness is not necessarily associated with a particular phase of the internal wave, and simplistic interpretations may sometimes be misleading.

A three-dimensional analysis is performed to investigate the effects of an electric field on the steady deformation and small-amplitude oscillation of a bubble in dielectric liquid. To deal with a general class of electric fields, an electric field near the bubble is approximately represented by the sum of a uniform field and a linear field. Analytical results have been obtained for steady deformation and modification of oscillation frequency by using the domain perturbation method with the angular momentum operator approach.

It has been found that, to the first order, the steady shape of a bubble in an arbitrary electric field can be represented by a linear combination of a finite number of spherical harmonics Yml, where 0[les ]l[les ]4 and [mid ]m[mid ][les ]l. For the oscillation about the deformed steady shape, the overall frequency modification from the value of free oscillation about a spherical shape is obtained by considering two contributions separately: (i) that due to the deformed steady shape (indirect effect), and (ii) that due to the direct effect of an electric field. Both the direct and indirect effects of an electric field split the (2l+1)-fold degenerate frequency of Yml modes, in the case of free oscillation about a spherical shape, into different frequencies that depend on m. However, when the average is taken over the (2l+1) values of m, the frequency splitting due to the indirect effect via the deformed steady shape preserves the average value, while the splitting due to the direct effect of an electric field does not.

The oscillation characteristics of a bubble in a uniform electric field under the negligible compressibility assumption are compared with those of a conducting drop in a uniform electric field. For axisymmetric oscillation modes, deforming the steady shape into a prolate spheroid has the same effect of decreasing the oscillation frequency in both the drop and the bubble. However, the electric field has different effects on the oscillation about a spherical shape. The oscillation frequency increases with the increase of electric field in the case of a bubble, while it decreases in the case of a drop. This fundamental difference comes from the fact that the electric field outside the bubble exerts a suppressive surface force while the electric field outside the conducting drop exerts a pulling force on the surface.

The existence of solitary waves near the minimum phase speed for waves in the gravity–capillary regime triggered our search for additional wave forms. We show that the governing Schrödinger-type equation also has a rich family of periodic solutions, and a preliminary study of these solutions is the objective of the present note.

The level-set approach is applied to a regime of premixed turbulent combustion where the Kolmogorov scale is smaller than the flame thickness. This regime is called the thin reaction zones regime. It is characterized by the condition that small eddies can penetrate into the preheat zone, but not into the reaction zone.

By considering the iso-scalar surface of the deficient-species mass fraction Y immediately ahead of the reaction zone a field equation for the scalar quantity G(x, t) is derived, which describes the location of the thin reaction zone. It resembles the level-set equation used in the corrugated flamelet regime, but the resulting propagation velocity s*L normal to the front is a fluctuating quantity and the curvature term is multiplied by the diffusivity of the deficient species rather than the Markstein diffusivity. It is shown that in the thin reaction zones regime diffusive effects are dominant and the contribution of s*L to the solution of the level-set equation is small.

In order to model turbulent premixed combustion an equation is used that contains only the leading-order terms of both regimes, the previously analysed corrugated flamelets regime and the thin reaction zones regime. That equation accounts for non-constant density but not for gas expansion effects within the flame front which are important in the corrugated flamelets regime. By splitting G into a mean and a fluctuation, equations for the Favre mean [Gtilde]and the variance [Gtilde]″2 are derived. These quantities describe the mean flame position and the turbulent flame brush thickness, respectively. The equation for [Gtilde]″2 is closed by considering two-point statistics. Scaling arguments are then used to derive a model equation for the flame surface area ratio [rhotilde]. The balance between production, kinematic restoration and dissipation in this equation leads to a quadratic equation for the turbulent burning velocity. Its solution shows the ‘bending’ behaviour of the turbulent to laminar burning velocity ratio sT/sL, plotted as a function of v′/sL. It is shown that the bending results from the transition from the corrugated amelets to the thin reaction zones regimes. This is equivalent to a transition from Damköhler's large-scale to his small-scale turbulence regime.

Steady-state velocity and orientation distributions of sedimenting fibres were measured as a function of particle concentration and aspect ratio. Two different regimes of sedimentation were clearly identified. For dilute suspensions, the fibres tend to align in the direction of gravity with occasional flipping and clump together to form packets. In this regime, the vertical mean sedimentation speed is not hindered and can be larger than the Stokes' velocity of an isolated vertical fibre. Its scaling is a complex function of particle volume fraction and aspect ratio. As the concentration is increased, the fibres still tend to orient in the direction of gravity. The mean velocity becomes hindered and scales with particle volume fraction. The velocity fluctuations were found to be large and anisotropic. They were found to increase with increasing volume fraction. A similar substantial anisotropy of the orientation distribution was observed for all particle concentrations and aspect ratios studied.

Radial flow takes place in a heterogeneous porous formation of random and stationary log-conductivity Y(x), characterized by the mean 〈Y〉, the variance σ2Y and the two- point autocorrelation ρY which in turn has finite and different horizontal and vertical integral scales, I and Iv, respectively. The steady flow is driven by a head difference between a fully penetrating well and an outer boundary, the mean velocity U being radial. A tracer is injected for a short time through the well envelope and the thin plume spreads due to advection by the random velocity field and to pore-scale dispersion. Transport is characterized by the mean front r=R(t) and by the second spatial moment of the plume Srr. Under ergodic conditions, i.e. for a well length much larger than the vertical integral scale, Srr is equal to the radial fluid trajectory variance Xrr.

The aim of the study is to determine Xrr(t) for a given heterogeneous structure and for given pore-scale dispersivities. The problem is more complex than the similar one for mean uniform flow. To simplify it, the well is replaced by a line source, the domain is assumed to be infinite and a first-order approximation in σ2Y is adopted. The solution is still difficult, being expressed with the aid of a few quadratures. It is found, however, that it can be derived quite accurately for a sufficiently small anisotropy ratio e=Iv/I by retaining only one term of the velocity two-point covariance. This major simplification leads to simple calculations and even to analytical solutions in the absence of pore-scale dispersion.

To compare the results with those prevailing in homogeneous media, apparent and equivalent macrodispersivities are defined for convenience.

The major difference between transport in radial and uniform flow is that the asymptotic, large-time, apparent macrodispersivity in the former is smaller by a factor of 3 than in the latter. For a three-dimensional point source the reduction is by a factor of 5. This effect is explained by the rapid change of the mean velocity during the period in which the velocities of two particles injected at the source become uncorrelated.

In contrast, the equivalent macrodispersivity tends to its value in uniform flow far from the well, where the flow is slowly varying in space.

The drag and lift forces acting on a rotating rigid sphere in a homogeneous linear shear flow are numerically studied by means of a three-dimensional numerical simulation. The effects of both the fluid shear and rotational speed of the sphere on the drag and lift forces are estimated for particle Reynolds numbers of 1[les ]Rep[les ]500.

The results show that the drag forces both on a stationary sphere in a linear shear flow and on a rotating sphere in a uniform unsheared flow increase with increasing the fluid shear and rotational speed. The lift force on a stationary sphere in a linear shear flow acts from the low-fluid-velocity side to the high-fluid-velocity side for low particle Reynolds numbers of Rep<60, whereas it acts from the high-velocity side to the low-velocity side for high particle Reynolds numbers of Rep>60. The change of the direction of the lift force can be explained well by considering the contributions of pressure and viscous forces to the total lift in terms of flow separation. The predicted direction of the lift force for high particle Reynolds numbers is also examined through a visualization experiment of an iron particle falling in a linear shear flow of a glycerin solution. On the other hand, the lift force on a rotating sphere in a uniform unsheared flow acts in the same direction independent of particle Reynolds numbers. Approximate expressions for the drag and lift coefficients for a rotating sphere in a linear shear flow are proposed over the wide range of 1[les ]Rep[les ]500.

Vortex connections at the surface are fundamental and prominent features in free-surface vortical flows. To understand the detailed mechanism of such connection, we consider, as a canonical problem, the laminar vortex connections at a free surface when an oblique vortex ring impinges upon that surface. We perform numerical simulations of the Navier–Stokes equations with viscous free-surface boundary conditions. It is found that the key to understanding the mechanism of vortex connection at a free surface is the surface layers: a viscous layer resulting from the dynamic zero-stress boundary conditions at the free surface, and a thicker blockage layer which is due to the kinematic boundary condition at the surface. In the blockage layer, the vertical vorticity component increases due to vortex stretching and vortex turning (from the transverse vorticity component). The vertical vorticity is then transported to the free surface through viscous diffusion and vortex stretching in the viscous layer leading to increased surface-normal vorticity. These mechanisms take place at the aft-shoulder regions of the vortex ring. Connection at the free surface is different from that at a free-slip wall owing to the generation of surface secondary vorticity. We study the components of this surface vorticity in detail and find that the presence of a free surface accelerates the connection process. We investigate the connection time scale and its dependence on initial incidence angle, Froude and Reynolds numbers. It is found that a criterion based on the streamline topology provides a precise definition for connection time, and may be preferred over existing definitions, e.g. those based on free-surface elevation or net circulation.

Transition from convective to absolute instability in Rayleigh–Bénard convection in the presence of a uni-directional Poiseuille flow is studied. The evaluation of the long-time behaviour of the Green function in the horizontal plane allows the determination of regions of convective and absolute instability in the Rayleigh–Reynolds number plane as a function of Prandtl number. It is found that the mode reaching zero group velocity at the convective–absolute transition always corresponds to transverse rolls, while the system remains convectively unstable with respect to pure streamwise (longitudinal) rolls for all non-zero Reynolds numbers. Finally, the roll pattern within the entire wave packet and in particular near its centre is elucidated and possible connections between experiments and our findings are discussed.

The velocity field in the immediate vicinity of a curved vortex comprises a circulation around the vortex, a component due to the vortex curvature, and a ‘remainder’ due to the more distant parts of the vortex. The first two components are relatively well understood but the remainder is known only for a few specific vortex geometries, most notably, the vortex ring. In this paper we derive a closed form for the remainder that is valid for all values of the pitch of an infinite helical vortex. The remainder is obtained firstly from Hardin's (1982) solution for the flow induced by a helical line vortex (of zero thickness). We then use Ricca's (1994) implementation of the Moore & Saffman (1972) formulation to obtain the remainder for a helical vortex with a finite circular core over which the circulation is distributed uniformly. It is shown analytically that the two remainders differ by 1/4 for all values of the pitch. This generalizes the results of Kuibin & Okulov (1998) who obtained the remainders and their difference asymptotically for small and large pitch. An asymptotic analysis of the new closed-form remainders using Mellin transforms provides a complete representation by a residue series and reveals a minor correction to the asymptotic expression of Kuibin & Okulov (1998) for the remainder at small pitch.

The paper presents an exact analytical solution to the problem of finding the optimum profile of a two-dimensional plate which planes on a water surface without spray formation and maximizes the lift force. The lift is maximized under the only isoperimetric constraint of fixed total arclength of the plate. The exact solution is compared with approximate analytical and numerical results by Wu & Whitney (1972). The shape of the optimum plate turns out to be technically unrealizable because of small, tightly wound spirals near the end points. It was shown numerically that cutting off small segments near the end points leads on the one hand to insignificant change in the lift force and on the other hand to a non-separating boundary layer along the remaining part of the optimum plate.

Direct numerical simulations are performed of a confined three-dimensional, temporally developing, initially isothermal gas mixing layer with one stream laden with as many as 7.3×105 evaporating hydrocarbon droplets, at moderate gas temperature and subsonic Mach number. Complete two-way phase couplings of mass, momentum and energy are incorporated which are based on a thermodynamically self-consistent specification of the vapour enthalpy, internal energy and latent heat of vaporization. Effects of the initial liquid mass loading ratio (ML), initial Stokes number (St0), initial droplet temperature and flow three-dimensionality on the mixing layer growth and development are discussed. The dominant parameter governing flow modulation is found to be the liquid mass loading ratio. Variations in the initial Stokes number over the range 0.5[les ]St0[les ]2.0 do not cause significant modulations of either first- or second-order gas phase statistics. The mixing layer growth rate and kinetic energy are increasingly attenuated for increasing liquid loadings in the range 0[les ]ML[les ]0.35. The laden stream becomes saturated before evaporation is completed for all but the smallest liquid loadings owing to: (i) latent heat effects which reduce the gas temperature, and (ii) build up of the evaporated vapour mass fraction. However, droplets continue to be entrained into the layer where they evaporate owing to contact with the relatively higher-temperature vapour-free gas stream. The droplets within the layer are observed to be centrifuged out of high-vorticity regions and to migrate towards high-strain regions of the flow. This results in the formation of concentration streaks in spanwise braid regions which are wrapped around the periphery of secondary streamwise vortices. Persistent regions of positive and negative slip velocity and slip temperature are identified. The velocity component variances in both the streamwise and spanwise directions are found to be larger for the droplets than for the gas phase on the unladen stream side of the layer; however, the cross-stream velocity and temperature variances are larger for the gas. Finally, both the mean streamwise gas velocity and droplet number density profiles are observed to coincide for all ML when the cross-stream coordinate is normalized by the instantaneous vorticity thickness; however, first-order thermodynamic profiles do not coincide.

We consider a set of nonlinear boundary-layer equations that have been derived by Duck, Foster & Hewitt (1997a, DFH), for the swirling flow of a linearly stratified fluid in a conical container. In contrast to the unsteady analysis of DFH, we restrict attention to steady solutions and extend the previous discussion further by allowing the container to both co-rotate and counter-rotate relative to the contained swirling fluid. The system is governed by three parameters, which are essentially non-dimensional measures of the rotation, stratification and a Schmidt number. Some of the properties of this system are related (in some cases rather subtly) to those found in the swirling flow of a homogeneous fluid above an infinite rotating disk; however, the introduction of buoyancy effects with a sloping boundary leads to other (new) behaviours. A general description of the steady solutions to this system proves to be rather complicated and shows many interesting features, including non-uniqueness, singular solutions and bifurcation phenomena.

We present a broad description of the steady states with particular emphasis on boundaries in parameter space beyond which steady states cannot be continued.

A natural extension of this work (motivated by recent experimental results) is to investigate the possibility of solution branches corresponding to non-axisymmetric boundary-layer states appearing as bifurcations of the axisymmetric solutions. In an Appendix we give details of an exact, non-axisymmetric solution to the Navier–Stokes equations (with axisymmetric boundary conditions) corresponding to the flow of homogeneous fluid above a rotating disk.