Compound threads and jets consist of a core liquid surrounded by an annulus of a second immiscible liquid. Capillary forces derived from axisymmetric disturbances in the circumferential curvatures of the two interfaces destabilize cylindrical base states of compound threads and jets (with inner and outer radii R1 and aR1 respectively). The capillary instability causes breakup into drops; the presence of the annular phase allows both the annular- and core-phase properties to influence the drop size. Of technological interest is breakup where the core snaps first, and then the annulus. This results in compound drops. With jets, this pattern can form composite particles, or if the annular fluid is evaporatively removed, single drops whose size is modulated by both fluids.

This paper is a study of the linear temporal instability of compound threads and jets to understand how annular fluid properties control drop size in jet breakup, and to determine conditions which favour compound drop formation. The temporal dispersion equation is solved numerically for non-dimensional annular thicknesses a of order one, and analytically for thin annuli (a – 1 = ε [Lt ] 1) by asymptotic expansion in ε. There are two temporally growing modes: a stretching mode, unstable for wavelengths greater than the undisturbed inner circumference 2πR1, in which the two interfaces grow in phase; and a squeezing mode, unstable for wavelengths greater than 2πaR1, which grows exactly out of phase. Growth rates are always real, indicating that in jetting configurations disturbances convect downstream with the base velocity. For order-one thicknesses, the growth rate of the stretching mode is higher for the entire range of system parameters examined. The drop size scales with the wavenumber of the maximally growing wave (kmax). We find that for the dominant stretching mode and a = 2, variations from 0.1 to 10 in the ratios of the annulus to core viscosity, or the tension of the outer surface to that of the inner interface, can result in changes in kmax by a factor of approximately 2. However, for these changes in the system ratios, the growth rate (smax) and the ratio of the amplitude of the outer to the inner interface (Amax) for the fastest growing wave only change marginally, with Amax near one. The system appears most sensitive to the ratio of the density of the annulus to the core fluid. For a variation between 0.1 and 10, kmax again changes by a factor of 2, but Amax and smax vary more significantly with large amplitude ratios for low density ratios. The amplitude ratio of the stretching mode at the maximally growing wave (Amax) indicates whether the film or core will break first. When this ratio is near one, linear theory predicts that the core breaks with the annulus intact, forming compound drops. Except for low values of the density ratio, our results indicate that most system conditions promote compound drop formation.

For thin annuli, the growth rate disparity between modes becomes even greater. In the limit ε → 0, the squeezing growth rate is roughly proportional to ε2 while the stretching mode growth rate is roughly proportional to ε0 and asymptotes to a single jet with radius R1 and tension equal to the sum of the two tensions. Thus, in this limit the growth rate and kmax are independent of the film density and viscosity. The amplitude ratio of the stretching mode becomes equal to one for all wavenumbers; so thin films break as compound drops. Our results compare favourably with previously published measurements on unstable waves in compound jets.

An analytical solution to the nonlinear equations of motion and thermodynamic energy for gravity currents propagating in stable atmosphere is found. This solution differs from the previous analytical studies in several aspects. In our solution the head of the gravity current is a strong vortex and the dynamics are non-hydrostatic. The solution has two regimes: (i) a supercritical regime when the Froude number Fr = (c – U)/Na is larger than 1 – in this case the cold front is local; (ii) a subcritical regime when Fr is smaller than 1. Here, ahead of the front there is a disturbance of nonlinear gravity waves. The scale of the wave and its amplitude increase as the Froude number decreases.

We found that the square of the speed of the gravity current (relative to the synoptic wind) is proportional to the mean drop of potential temperature over the front area times the front height a. The constant of proportionality is function of the environmental conditions. The thermal, velocity and vorticity fields can be described by non-dimensional structure functions of two numbers: pa = 1/Fr and ka. The amplitude of the structure functions is proportional to (c – U) 2/a for the thermal field, to (c – U) for the velocity field, and to (c – U)/a for the vorticity field.

The propagation is studied in terms of the vorticity equation. The horizontal gradient of the buoyancy term always tends to propagate the cold front. The nonlinear advection term in most of the cases investigated here tends to slow the propagation of the gravity current. The propagation of the disturbance of nonlinear gravity waves ahead of the front in regime (ii) in most of the cases is due to the buoyancy term. The nonlinear advection term tends to slow the propagation when the synoptic wind blows in the direction opposite to that of the front propagation, and increase the propagation when the synoptic wind blows in the direction of propagation.

The turbulent boundary layer along a compression ramp with a deflection angle of 18° at a free-stream Mach number of M = 3 and a Reynolds number of Reθ = 1685 with respect to free-stream quantities and mean momentum thickness at inflow is studied by direct numerical simulation. The conservation equations for mass, momentum, and energy are solved in generalized coordinates using a 5th-order hybrid compact- finite-difference-ENO scheme for the spatial discretization of the convective fluxes and 6th-order central compact finite differences for the diffusive fluxes. For time advancement a 3rd-order Runge–Kutta scheme is used. The computational domain is discretized with about 15 × 106 grid points. Turbulent inflow data are provided by a separate zero-pressure-gradient boundary-layer simulation. For statistical analysis, the flow is sampled 600 times over about 385 characteristic timescales δ0/U∞, defined by the mean boundary-layer thickness at inflow and the free-stream velocity. Diagnostics show that the numerical representation of the flow field is sufficiently well resolved.

Near the corner, a small area of separated flow develops. The shock motion is limited to less than about 10% of the mean boundary-layer thickness. The shock oscillates slightly around its mean location with a frequency of similar magnitude to the bursting frequency of the incoming boundary layer. Turbulent fluctuations are significantly amplified owing to the shock–boundary-layer interaction. Reynolds-stress maxima are amplified by a factor of about 4. Turbulent normal and shear stresses are amplified differently, resulting in a change of the structure parameter. Compressibility affects the turbulence structure in the interaction area around the corner and during the relaxation after reattachment downstream of the corner. Correlations involving pressure fluctuations are significantly enhanced in these regions. The strong Reynolds analogy which suggests a perfect correlation between velocity and temperature fluctuations is found to be invalid in the interaction area.

In this paper, we study the transverse vortex-induced vibrations of an elastically mounted rigid cylinder in a fluid flow. We use simultaneous force, displacement and vorticity measurements (using DPIV) for the first time in free vibrations. There exist two distinct types of response in such systems, depending on whether one has a high or low combined mass–damping parameter (m*ζ). In the classical high-(m*ζ) case, an ‘initial’ and ‘lower’ amplitude branch are separated by a discontinuous mode transition, whereas in the case of low (m*ζ), a further higher-amplitude ‘upper’ branch of response appears, and there exist two mode transitions.

To understand the existence of more than one mode transition for low (m*ζ), we employ two distinct formulations of the equation of motion, one of which uses the ‘total force’, while the other uses the ‘vortex force’, which is related only to the dynamics of vorticity. The first mode transition involves a jump in ‘vortex phase’ (between vortex force and displacement), ϕvortex, at which point the frequency of oscillation (f) passes through the natural frequency of the system in the fluid, f ∼ fNwater. This transition is associated with a jump between 2S [harr ] 2P vortex wake modes, and a corresponding switch in vortex shedding timing. Across the second mode transition, there is a jump in ‘total phase’, phis;total , at which point f ∼ fNvacuum. In this case, there is no jump in ϕvortex, since both branches are associated with the 2P mode, and there is therefore no switch in timing of shedding, contrary to previous assumptions. Interestingly, for the high-(m*ζ) case, the vibration frequency jumps across both fNwater and fNvacuum, corresponding to the simultaneous jumps in ϕvortex and ϕtotal. This causes a switch in the timing of shedding, coincident with the ‘total phase’ jump, in agreement with previous assumptions.

For large mass ratios, m* = O(100), the vibration frequency for synchronization lies close to the natural frequency (f* = f/fN ≈ 1.0), but as mass is reduced to m* = O(1), f* can reach remarkably large values. We deduce an expression for the frequency of the lower-branch vibration, as follows:

formula here

which agrees very well with a wide set of experimental data. This frequency equation uncovers the existence of a critical mass ratio, where the frequency f* becomes large: m*crit = 0.54. When m* < m*crit, the lower branch can never be reached and it ceases to exist. The upper-branch large-amplitude vibrations persist for all velocities, no matter how high, and the frequency increases indefinitely with flow velocity. Experiments at m* < m*crit show that the upper-branch vibrations continue to the limits (in flow speed) of our facility.

Flow in a symmetric channel with a sudden expansion makes a transition from a symmetric flow to an asymmetric one due to a symmetry-breaking pitchfork bifurcation on a gradual increase of the Reynolds number if the system is perfectly symmetric. However, an unavoidable infinitesimal imperfection of the system may render the pitchfork bifurcation imperfect. A weakly nonlinear stability analysis is proposed to investigate the structural instability of the bifurcation for such a flow. As a result, an amplitude equation for a disturbance is derived by including the effect of the imperfection of the system, and its coefficients are evaluated numerically. The equilibrium amplitude of the disturbance is calculated from the amplitude equation and compared with the experimental results for the flow in a channel that is presumed symmetric and also with the numerical solution of the full nonlinear equations for the flow in a slightly asymmetric channel.

We study turbulent transport of passive tracers by random wave fields of a rather general nature. A formalism allowing for spatial inhomogeneity and anisotropy of an underlying velocity field (such as that caused by a latitudinally varying Coriolis parameter) is developed, with the aim of treating problems of large-scale ocean transport by long internal waves. For the special case of surface gravity waves on deep water, our results agree with the earlier theory of Herterich & Hasselmann (1982), though even in that case we discover additional, off-diagonal elements of the diffusion tensor emerging in the presence of a mean drift. An advective diffusion equation including all components of the diffusion tensor D plus a mean, Stokes-type drift u is derived and applied to the case of baroclinic inertia–gravity (BIG) waves. This application is of particular interest for ocean circulation and climate modelling, as the mean drift, according to our estimates, is comparable to ocean interior currents. Furthermore, while on the largest (100 km and greater) scales, wave-induced diffusion is found to be generally small compared to classical eddy-induced diffusion, the two become comparable on scales below 10 km. These scales are near the present limit on the spatial resolution of eddy-resolving ocean numerical models. Since we find that uz and Dzz vanish identically, net vertical transport is absent in wave systems of this type. However, for anisotropic wave spectra the diffusion tensor can have non-zero off-diagonal vertical elements, Dxz and Dyz, and it is shown that their presence leads to non-positive definiteness of D, and a negative diffusion constant is found along a particular principal axis. However, the simultaneous presence of a depth-dependent mean horizontal drift u(z) eliminates any potential unphysical behaviour.

Marginal convection in the form of Alfvén waves in an electrically conducting Bénard layer in the presence of a vertical magnetic field is investigated analytically using the Boussinesq model for the fluid. Small amplitude solutions are studied using the linearized magnetoconvection equations. These solutions are represented by double expansions in terms of two small parameters: a dimensionless viscosity and a dimensionless magnetic diffusivity. The leading-order problem corresponds to undamped Alfvén waves propagating between the boundaries of the fluid; buoyancy forces appear at higher order and can maintain the Alfvén waves against viscous and ohmic damping. The structure of the Alfvén waves is strongly dependent, even at leading order, on the physical nature of the walls. Four different types of boundary conditions are considered here: (A) illustrative, i.e. mathematically simple conditions, (B) solid, perfectly conducting walls, (C) vacuum external to the layer, and (D) solid, perfectly insulating walls. It is shown how in each case Alfvén waves are excited by a small, but sufficiently strong, thermal buoyancy but that, because of boundary layers, the solutions for the four sets of boundary conditions are very different.

Rectilinear homogeneous miscible displacements with gravity override are analysed by means of direct numerical simulations on the basis of the vorticity–streamfunction formulation of the governing equations. The vorticity-based point of view offers the advantage of clearly attributing the dominant flow characteristics to the effects of viscosity contrast, density difference, impermeable boundary conditions, or interactions among the above. Basic considerations regarding the vorticity field show that in an integral sense the coupling between viscosity and gravity vorticity is predominantly one way in nature, in that the gravity vorticity can amplify the viscous vorticity, but not vice versa. In particular, the vorticity point of view provides an explanation for the formation of the gravity tongue in terms of a focusing mechanism, which results from the combined action of the unfavourable viscosity gradient and the potential flow field generated by the interaction of the gravitational vorticity with the horizontal boundaries. This potential velocity field locally enhances the uniform global displacement velocity near the upper boundary, and thereby amplifies the viscous fingering instability along this section of the interface. In some parameter ranges, the gravity tongue exhibits interesting interactions with the viscous fingers next to it, such as pinching and partial merging. The influence of the Péclet number, the viscosity and density contrasts, and the aspect ratio on the dynamic evolution of the displacement is investigated quantitatively.

The effects of permeability heterogeneities on rectilinear displacements with viscosity contrast and density variations are investigated computationally by means of direct numerical simulations. Physical interpretations are given in terms of mutual interactions among the three vorticity components related to viscous, density and permeability effects. In homogeneous environments the combined effect of the unfavourable viscosity gradient and the potential velocity field generated by the horizontal boundaries was seen to produce a focusing mechanism that resulted in the formation of a strong vorticity layer and the related growth of a dominant gravity tongue (Ruith & Meiburg 2000). The more randomly distributed vorticity associated with the heterogeneities tends to ‘defocus’ this interaction, thereby preventing the formation of the vorticity layer and the gravity tongue. When compared to neutrally buoyant flows, the level of heterogeneity affects the breakthrough recovery quite differently. For moderate heterogeneities, a gravity tongue still forms and leads to early breakthrough, whereas the same result is accomplished for large heterogeneities by channelling. At intermediate levels of heterogeneity, these tendencies partially cancel each other, so that the breakthrough recovery reaches a maximum. Similarly, the dependence of the breakthrough recovery on the correlation length is quite different in displacements with density contrasts compared to neutrally buoyant flows. For neutrally buoyant flows the resonant interaction between viscosity and permeability vorticities typically leads to a minimal recovery at intermediate values of the correlation length. In contrast, displacements with density contrast give rise to a gravity tongue for both very small and very large values of this length, so that the recovery reaches a maximum at intermediate values.

The dynamics of a near-surface vortex are examined in a two-layer setting on the beta-plane. Initially, the vortex is radially symmetric and localized in the upper layer. Two non-dimensional parameters govern its evolution and translation: the ratio δ of the thickness of the vortex to the total depth of the fluid, and the non-dimensional beta-effect number α = βL/f (f and β are the Coriolis parameter and its meridional gradient respectively, L is the radius of the vortex). We assume, as suggested by oceanic observations, that α < δ < 1: A simple set of asymptotic equations is derived, which describes the beta-induced translation of the vortex and a dipolar perturbation developing on and under the vortex (in both layers).

This set was solved numerically for oceanic lenses, and the following features were observed: (i) The meridional (southward) component of the translation speed of the lens rapidly ‘overtakes’ the zonal (westward) component. The former grows approximately linearly, whereas the latter oscillates about the Nof (1981) value (i.e. about the speed of translation of a vortex in a one-layer reduce-gravity fluid). (ii) Vortices of the same shape, but different radii and amplitudes, follow the same trajectory. The amplitude and radius affect only the absolute value, but not the direction, of the translation speed. (iii) In the lower layer below the vortex, a ‘region’ is generated where the velocity of the fluid is growing linearly with time. The velocity field in the region becomes more and more homogeneous (and equal to the translation speed of the vortex).

Steady and unsteady numerical simulations of two-dimensional flow in a collapsible channel were carried out to study the flow limitation which typically occurs when the upstream transmural pressure is held constant while flow rate and pressure gradient along the collapsible channel can vary independently. Multiple steady solutions are found for a range of upstream transmural pressures and Reynolds number using an arclength control method. The stability of these steady solutions is tested in order to check the correlation between flow limitation and self-excited oscillations (the latter being a consequence of unstable steady solutions). Both stable and unstable solutions are found when flow is limited. Self-excited oscillations and divergence instabilities are observed in certain solution branches. The instability of the steady solutions seems to depend on the unsteady boundary conditions used, i.e. on which parameters are allowed to vary. However, steady solutions associated with the solution branch before flow limitation where the membrane wall bulges are found to be stable for each of the three different boundary conditions employed. We conclude that there is no one to one correlation between the two phenomena in this two dimensional channel model.

In a reacting gas flow both gas-phase chemical activity and surface catalysis can increase the rate of heat transfer from the gas to a solid surface. In particular, when there is a discontinuous change in the catalytic properties of the surface, there can be a very large increase in the local heat transfer rate. In this study numerical simulations have been performed for the laminar high-speed flow of a high-temperature, non-equilibrium reacting gas mixture over a flat plate. The surface of the plate is partly catalytic, with the leading region non-catalytic, and a discontinuous change in the catalytic properties of the surface at the catalytic junction. The surface is assumed to be isothermal, and cold relative to the free stream. The gas is assumed to be a mixture of molecular and atomic forms of a diatomic gas in an inert gas forming a thermal bath, giving a three-species mixture with dissociation and recombination of the reactive species. The calculations are performed for a gas with atomic and molecular oxygen in an argon bath, but a full range of gas-phase chemical and surface catalytic effects is considered. Kinetic schemes with frozen gas-phase chemistry, and partial or full recombination of atomic oxygen in the boundary layer are investigated. The catalytic nature of the surface material is given by a catalytic recombination rate coeffcient, which varies from zero (non-catalytic) to one (fully catalytic), and the effects on the flow and the surface heat transfer of materials which are non-, partially, or fully catalytic are considered. A self-similar thin-layer analytical model of the change in the gas composition downstream of the catalytic junction is developed. For physically realistic (O(10−2)) values of the catalytic recombination rate coeffcient, the predictions from this model of the surface values of the atomic oxygen mass fraction and the catalytic surface heat transfer rate are excellent when the only change in the composition of the gas comes from the surface catalysis, and reasonable when there is partial recombination of the gas in the boundary layer due to the gas-phase chemistry. In contrast, when the surface is fully catalytic, the streamwise diffusion terms play a significant role, and the model is not valid. These results should apply to other situations with an attached boundary layer with recombination reactions. A comparison is made between the calculated and experimental measurements of the heat transfer rate at the catalytic junction. With a kinetic scheme which allows partial recombination in the boundary layer, good agreement is found between the experimental and predicted values for surface materials which are essentially non-catalytic. For a catalytic material (platinum), the experimental and numerical heat transfer rates are matched to estimate the value of the catalytic recombination rate coeffcient. The values obtained show a considerable amount of scatter, but are consistent with those found in the literature.