Using a physical space (i.e. non-modal) approach, we investigate interactions between fast inertio-gravity (IG) waves and slow balanced flows in a shallow rotating fluid. Specifically, we consider a train of IG waves impinging on a steady, exactly balanced vortex. For simplicity, the one-dimensional problem is studied first; the limitations of one-dimensionality are offset by the ability to define balance in an exact way. An asymptotic analysis of the problem in the small-amplitude limit is performed to demonstrate the existence of interactions. It is shown that these interactions are not confined to the modification of the wave field by the vortex but, more importantly, that the waves are able to alter in a non-trivial way the potential vorticity associated with that vortex. Interestingly, in this one-dimensional problem, once the waves have traversed the vortex region and have propagated away, the vortex exactly recovers its initial shape and thus bears no signature of the interaction. Furthermore, we prove this last result in the case of arbitrary vortex and wave amplitudes. Numerical integrations of the full one-dimensional shallow-water equations in strongly nonlinear regimes are also performed: they confirm that time-dependent interactions exist and increase with wave amplitude, while at the final state the vortex bears no sign of the interaction. In addition, they reveal that cyclonic vortices interact more strongly with the wave field than anticyclonic ones.

We use the lubrication approximation to investigate the steady two-dimensional flow of a thin film of viscous fluid on the outside of a rigid circular cylinder that is rotating about its (horizontal) axis. Primarily we are concerned with the flow that ensues when fluid is supplied continuously as a ‘curtain’ from above the cylinder, so that it flows round the cylinder and eventually falls off near the bottom. This problem may be thought of as a ‘hybrid’ of the two classical problems studied by Nusselt (1916a, b) and Moffatt (1977), concerning, respectively, flow on a stationary cylinder with a prescribed supply flux, and flow on a rotating cylinder when the supply flux is zero. For all these problems there are indeterminacies in the steady lubrication solution; we present a variety of possible solutions, including both ‘full-film’ and ‘partial-film’ solutions, and solutions that involve smooth ‘jumps’ in the free-surface profile. We show, for example, that stagnation points can occur in the flow, that solutions exist that do not have top-to-bottom symmetry, that in curtain flows the curtain generally takes a characteristic ‘buckled’ shape, and that in full-film curtain flows there is always some fluid that is ‘trapped’ near the rotating cylinder, never escaping as part of the curtain that detaches at the bottom of the cylinder. Also we show that finite-thickness films involving jumps cannot occur in these coating flows (though they are known to occur in rimming flows).

A premixed ducted flame, burning in the wake of a bluff-body flame-holder, is considered. For such a flame, interaction between acoustic waves and unsteady combustion can lead to self-excited oscillations. The concept of a time-invariant turbulent flame speed is used to develop a kinematic model of the response of the flame to flow disturbances. Variations in the oncoming flow velocity at the flame-holder drive perturbations in the flame initiation surface and hence in the instantaneous rate of heat release. For linear fluctuations, the transfer function between heat release and velocity can be determined analytically from the model and is in good agreement with experiment across a wide frequency range. For nonlinear fluctuations, the model reproduces the flame surface distortions seen in schlieren films.

Coupling this kinematic flame model with an analysis of the acoustic waves generated in the duct by the unsteady combustion enables the time evolution of disturbances to be calculated. Self-excited oscillations occur above a critical fuel–air ratio. The frequency and amplitude of the resulting limit cycles are in satisfactory agreement with experiment. Flow reversal is predicted to occur during part of the limit-cycle oscillation and the flame then moves upstream of the flame-holder, just as in experimental visualizations. The main nonlinearity is identified in the rate of heat release, which essentially ‘saturates’ once the amplitude of the velocity fluctuation exceeds its mean. We show that, for this type of nonlinearity, describing function analysis can be used to give a good estimate of the limit-cycle frequency and amplitude from a quasi-nonlinear theory.

Multiphase flow modelling is still a major challenge in fluid dynamics and, although many different models have been derived, there is no clear evidence of their relevance to certain flow situations. That is particularly valid for bubbly flows, because most of the studies have considered the case of fluidized beds. In the present study we give a general formulation to five existing models and study their relevance to bubbly flows. The results of the linear analysis of those models clearly show that only two of them are applicable to that case. They both show a very similar qualitative linear stability behaviour. In the subsequent asymptotic analysis we derive an equation hierarchy which describes the weakly nonlinear stability of the models. Their qualitative behaviour up to first order with respect to the small parameter is again identical. A permanent-wave solution of the first two equations of the hierarchy is found. It is shown, however, that the permanent-wave (soliton) solution is very unlikely to occur for the most common case of gas bubbles in water. The reason is that the weakly nonlinear equations are unstable due to the low magnitude of the bulk modulus of elasticity. Physically relevant stabilization can eventually be achieved using some available experimental data. Finally, a necessary condition for existence of a fully nonlinear soliton is derived.

The current study investigates turbulence modification by particles in a backward-facing step flow with a fully developed channel flow inlet. This flow provides a range of flow regimes in which to compare turbulence modification under the same experimental conditions. Gas-phase velocities in the presence of 3–40% mass loadings of three different particle classes (90 and 150 μm diameter glass and 70 μm diameter copper spheres) were measured. Attenuation of the streamwise fluid turbulence of up to 35% was observed in the channel-flow extension region of the flow for a 40% mass loading of the largest particles. The level of attenuation decreased with decreasing particle Stokes number, particle Reynolds number and mass loading. No modification of the turbulence was found in the separated shear layer or in the redevelopment region behind the step, although there were significant particle loadings in these regions.

Experimental results from the Interface Configuration Experiment (ICE) performed aboard the Space Shuttle and the Mir Space Station are reported. The experiment concerns fluid interfaces in certain ‘exotic’ containers in a low-gravity environment. These containers are rotationally symmetric and have the property that for given contact angle and liquid volume, a continuum of distinct rotationally symmetric equilibrium configurations can appear, all of which have the same mechanical energy. These symmetric equilibrium configurations are unstable, in that deformations that are not rotationally symmetric can be shown mathematically to yield configurations with lower energy. It is found experimentally, in confirmation of mathematical results and of numerical computations, that distinct locally stable configurations can form that are not rotationally symmetric and have differing dynamic characteristics. It is found that this intriguing phenomenon of asymmetric local energy minimizers can occur even if conditions for an exotic container are not entirely met.

The dynamics of a passive tracer in the velocity field of four identical point vortices, moving under the influence of their self-induced advection, is investigated. Of interest is the change in mixing and transport properties of the tracer for the three different classes of vortex motion: periodic, quasi-periodic and chaotic. As a consequence of conservation laws, the vortex motion is confined to a finite region of phase space; therefore, the tracer phase space can be partitioned into an inner and an outer region. We find that in the case of quasi-periodic vortex motion the tracer phase space exhibits a well-defined barrier to transport between the central chaotic region and the outer region, where the trajectories are regular. In the case of chaotic vortex motion the barrier becomes permeable. The particle dynamics goes through an intermittent behaviour, where forays into the central region alternate with trapping in outer annular orbits. In the far field, an estimate of diffusion rates is made through a multipole expansion of the tracer velocity field. We make use of a specific stochastic model for the tracer velocity field which predicts no diffusion for the case of quasi-periodic vortex motion and, for the case of chaotic vortex motion, a diffusion rate that goes to zero at large distances from the vortex cluster.

Linear mode conversion is the partial transfer of wave energy from one wave type (a) to another (b) in a weakly non-uniform background state. For propagation in one dimension (x), the local wavenumber kjx of each wave (j = a, b) varies with x; if these are equal at some xR, the waves are locally in phase, and resonant energy transfer can occur. We model wave propagation in the Gulf of Guinea, where wave a is an equatorially trapped Rossby–gravity (Yanai) wave, and wave b is a coastal Kelvin wave along the (zonal) north coast of the Gulf, both propagating in zonal coordinate x. The coupling of the waves is due to the overlap of their eigenfunctions (normal modes in y, the meridional coordinate). We derive coupled mode equations from a variational principle, and obtain an analytic expression for the wave-energy conversion coefficient, in terms of the wave frequency and the scale length of the thermocline depth.

An experimental configuration has been found which allows detailed observation of three-dimensional vortex–vortex and associated vortex–surface interactions which appear similar to those observed in fully turbulent flow. Hydrogen bubble visualization illustrates a complicated intertwining, or braiding, of two initially co-rotating vortices. It is observed that the three-dimensional interactions of the braided vortices induces a pair of local surface-fluid eruptions reminiscent of the ‘bursting’ behaviour characteristic of the near-wall regeneration process of fully turbulent boundary layers.

Fourier and wavelet transformation techniques are utilized in a complementary manner in order to characterize temporal aspects of the transition of a planar jet shear layer. The subharmonic is found to exhibit an interesting temporal amplitude and phase variation that has not been previously reported. This takes the form of intermittent π-shifts in subharmonic phase between two fixed phase values. These phase jumps are highly correlated with local minima of the subharmonic amplitude. In contrast, the fundamental amplitude and phase show no such behaviour. The temporal phase behaviour of the subharmonic has the effect of intermittently disrupting the phase lock with the fundamental. A dynamical systems model is developed which is based on a classic vortex representation of the shear layer. The Hamiltonian formulation of the problem is shown to provide remarkable agreement with the experimental results. All the essential aspects of the temporal amplitude and phase behaviour of the subharmonic are reproduced by the model including amplitude-dependent effects. The model is also shown to provide a dynamical systems based explanation for time-averaged amplitude and phase behaviour observed in these as well as earlier experiments. The results of experiments involving both bimodal forcing at fundamental and subharmonic frequencies with prescribed initial effective phase angle as well as experiments involving only fundamental excitation over an amplitude range extending two orders of magnitude are presented. The temporal subharmonic amplitude and phase behaviour is observed in bimodal forcing experiments in those regions of the flow characterized by subharmonic mode suppression and vortex tearing events (even if the forcing amplitudes are quite large). In addition, temporal subharmonic amplitude and phase behaviour is the rule in experiments involving low-amplitude forcing of the fundamental and the natural development of the subharmonic.

Using a control-volume method and the simpler algorithm, we computed steady-state and time-dependent solutions for two-dimensional convection in an open-top porous box, up to a Rayleigh number of 1100. The evolution of the convective system from onset to high Rayleigh numbers is characterized by two types of transitions in the flow patterns. The first type is a decrease in the horizontal aspect ratio of the cells. We observe two such bifurcations. The first occurs at Ra = 107.8 when the convective pattern switches from a steady one-cell roll to a steady two-cell roll. The second occurs at Ra ≈ 510 when an unsteady two-cell roll evolves to a steady four-cell roll. The second type of transition is from a steady to an unsteady pattern and we also observe two of these bifurcations. The first occurs at Ra ≈ 425 in the two-cell convective pattern; the second at Ra ≈ 970 in the four-cell pattern. Both types of bifurcations are associated with an increase in the average vertical convective heat transport. In the bi-cellular solutions, the appearance of non-periodic unsteady convection corresponds to the onset of the expected theoretical scaling Nu ∝ Ra and also to the onset of plume formation. Although our highest quadri-cellular solutions show signs of non-periodic convection, they do not reach the onset of plume formation. An important hysterisis loop exists for Rayleigh numbers in the range 425–505. Unsteady convection appears only in the direction of increasing Rayleigh numbers. In the decreasing direction, steady quadri-cellular flow patterns prevail.

This paper discusses numerical experiments in which an initially uniform columnar vortex is subject to several types of axisymmetric forcing that mimic the strain field of a turbulent flow. The mean value of the strain along the vortex axis is in all cases zero, and the vortex is alternately stretched and compressed. The emphasis is on identifying the parameter range in which the vortex survives indefinitely. This extends previous work in which the effect of steady single-scale non-uniform strains was studied. In a first series of experiments the effect of the unsteadiness of the forcing is analysed, and it is found that the vortex survives as a compact object if the ratio between the oscillation frequency and the strain itself is low enough. A theoretical explanation is given which agrees with the numerical results. The strain is then generalized to include several spatial scales and oscillation frequencies, with characteristics similar to those in turbulent flows. The largest velocities are carried by the large scales, while the highest gradients and faster time scales are associated with the shorter wavelengths. Also in these cases ‘infinitely long’ vortices are obtained which are more or less uniform and compact. Vorticity profiles averaged along their axes are approximately Gaussian. The radii obtained from these profiles are proportional to the Burgers' radius of the r.m.s. (small-scale) axial strain, while the azimuthal velocities are proportional to the maximum (large-scale) axial velocity differences. The study is motivated by previous observations of intense vortex filaments in turbulent flows, and the scalings found in the present experiments are consistent with those found in the turbulent simulations.

The stability of axisymmetric steady thermocapillary convection of electrically conducting fluids in half-zones under the influence of a static axial magnetic field is investigated numerically by linear stability theory. In addition, the energy transfer between the basic state and a disturbance is considered in order to elucidate the mechanics of the most unstable mode. Axial magnetic fields cause a concentration of the thermocapillary flow near the free surface of the liquid bridge. For the low Prandtl number fluids considered, the most dangerous disturbance is a non-axisymmetric steady mode. It is found that axial magnetic fields act to stabilize the basic state. The stabilizing effect increases with the Prandtl number and decreases with the zone height, the heat transfer rate at the free surface and buoyancy when the heating is from below. The magnetic field also influences the azimuthal symmetry of the most unstable mode.

Simultaneous velocity and temperature measurements were made with rakes of sensors that sliced a slightly heated turbulent wake in the spanwise direction, at different lateral positions 150 diameters downstream of the cylinder. A pattern recognition analysis of hotter-to-colder transitions was performed on temperature data measured at the mean velocity half-width. The velocity data from the different ‘slices’ was then conditionally averaged based on the identified temperature events. This procedure yielded the topology of the average three-dimensional large-scale structure which was visualized with iso-surfaces of negative values of the second eigenvector of [S2+Ω2]. The results indicate that the average structure of the velocity fluctuations (using a triple decomposition of the velocity field) is found to be a shear-aligned ring-shaped vortex. This vortex ring has strong outward lateral velocities in its symmetry plane which are like Grant's mixing jets. The mixing jet region extends outside the ring-like vortex and is bounded by two foci separated in the spanwise direction and an upstream saddle point. The two foci correspond to what has been previously identified in the literature as the double rollers.

The ring vortex extracts energy from the mean flow by stretching in the mixing jet region just upstream of the ring boundary. The production of the small-scale (incoherent) turbulence by the coherent field and one-component energy dissipation rate occur just downstream of the saddle point within the mixing jet region. Incoherent turbulence energy is extracted from the mean flow just outside the mixing jet region, but within the core of the structure. These processes are highly three-dimensional with a spanwise extent equal to the mean velocity half-width.

When a double decomposition is used, the coherent structure is found to be a tube-shaped vortex with a spanwise extent of about 2.5l0. The double roller motions are integral to this vortex in spite of its shape. Spatial averages of the coherent velocity field indicate that the mixing jet region causes a deficit of mean streamwise momentum, while the region outside the foci of the double rollers has a relatively small excess of streamwise momentum.

The instability of a two-dimensional moving contact line is studied for a thin liquid film flowing down an inclined plane, leading to the formation of rivulets. A fluorescent imaging method was developed to facilitate accurate measurement of the spacing between rivulets, tip velocity, three-dimensional shape and dynamic contact line. A fluid circulation system produced steady films at constant volumetric flux, in contrast to time-varying films at constant total volume, as in previous measurements. Comparisons are made with the existing data for constant-volume films, and with theoretical predictions for the wavelength of the rivulets formed at constant inlet flow rate. Data were also obtained for rivulet shapes, tip speeds and contact angles as functions of the angle of inclination of the plate and liquid Reynolds number.

Ogilvie (1963) obtained the solution for the interaction between gravity waves and a horizontal submerged circular cylinder on deep water. It is shown herein that in this solution the expressions for the velocity potential (both for the moving cylinder and for the fixed cylinder) lack one term: a time-periodic space-independent term, which does not modify the load, but has some important consequence on the pressure distribution.

Experiments have been performed to study the co-rotating wing-tip vortex pair produced by a pair of rectangular wings in a split-wing configuration. Detailed measurements made in cross-sections upstream and downstream of merger reveal, for the first time, the complex turbulence structure of this flow. The vortices spiral around each other and merge some 20 chordlengths downstream of the wings. As merger is approached the vortices lose their axisymmetry – their cores develop lopsided tangential velocity fields and the mean vorticity field is convected into filaments. The cores also become part of a single turbulence structure dominated by a braid of high turbulence levels that links them together. The braid, which quite closely resembles the structure formed between adjacent spanwise eddies of transitional mixing layers, grows in intensity with downstream distance and extends into the vortex cores. Unlike a single tip vortex, the unmerged cores appear turbulent.

The merging of the vortices wraps the cores and the flow structure that surrounds them into a large turbulent region with an intricate double spiral structure. This structure then relaxes to a closely axisymmetric state. The merged core appears stable and develops a structure similar to the laminar core of a vortex shed from a single wing. However, the turbulent region formed around the vortex core during the merger process is much larger and more axisymmetric than that found around a single wing-tip vortex.