The influence of thermocapillary forces on buoyancy-driven convection is numerically simulated for shallow open cavities with differentially heated endwalls and filled with low-Prandtl-number fluid. Calculations are carried out by solving two-dimensional Navier-Stokes equations coupled to the energy equation, for three aspects ratios A = (length/height) = 4, 12.5 and 25, and several values of the Grashof number (up to 6 × 104) and Reynolds number (|Re| ≤ 1.67 × 104). Thermocapillarity can have a quite significant effect on the stability of a primarily buoyancy-driven flow. The result of the combination of the two basic mechanisms (thermo-capillarity) and buoyancy) depends on whether their effects are additive (positive Re) or opposing (negative Re); counter-acting mechanisms yield more complex flow patterns. The critical Grashof number Grc for the onset of the unsteady regime is found to decrease substantially within a small range of negative Re, and to increase for positive Re (and also for large negative Re). For Gr = 4 × 104, A = 4 and small negative Reynolds numbers, −2.4 × 103 ≤ Re < 0, mono-periodic and bi- or quasi-periodic regimes are shown to exist successively, followed by a reverse transition. The development of the instabilities from an initial steady-state regime has been investigated by varying Re for Gr = 1.5 × 104 (below Grc at Re = 0); the onset of buoyant instabilities is enhanced in a narrow range of Re only (-1200 < Re < -200). It is also noteworthy that for small enough Grashof numbers (e.g. Gr = 3 × 103), a steady-state solution prevails over the whole range of Reynolds numbers investigated. This means that a critical Grashof number exists below which the effect of the thermocapillary forces is no longer destabilizing.

Controlled resonant interactions between two spinning waves in a turbulent, axisymmetric air jet are documented. Interactions betwen two helical waves with spinning mode numbers of +m and −m induced a cos(2mϕ) distortion of the mean cross-section. The shape and orientation of the distortion were predictable based on the standing wave pattern. Square and elliptical jets were produced in this way and the spatial distribution of the coherent large-scale motion is documented. The elliptical distortion was comparable in magnitude to a jet issuing from a 2:1 elliptical nozzle. A near-resonance case produced from spinning mode numbers of m = 0 and + 2 was also examined.

Incoherent turbulent motion modulated by coherent large-scale motion contributes to second-order coherent stresses. The spatial distribution of wave-induced stress was measured in a jet whose cross-section had been distorted through controlled resonant interactions between two forced, helical waves spinning in opposite directions. The transfer of energy from the coherent motion to broadband turbulence is documented. Shape assumptions are examined by comparing radial distributions to predictions from linear, inviscid stability theory. Control over small-scale mixing is examined by demodulating the coherent envelope of small-scale turbulence and by correlating it with features of the coherent, large-scale motion. Coherent production is shown to be associated with the roll-up process and there is evidence of secondary, inflexional instabilities.

We study the rise of a spherical cap bubble in both two- and three-dimensional unbounded regions. In particular we focus on the problem of finding steady state-solutions. We assume that the fluid is incompressible, inviscid and irrotational, and use two different models to approximate the turbulent wake behind the bubble. We demonstrate numerically that in the case of zero surface tension we have a continuous spectrum of rise velocities. When we add small surface tension to the problem, the degeneracy is broken via a solvability mechanism, and we obtain velocity selection. Our results are in good agreement with the existing experimental studies.

The first flow visualization experimental results of transition in plane Couette flow are reported. The Couette flow water channel was of an infinite-belt type with counter-moving walls. The belt and channel walls were transparent making it possible to visualize the flow pattern in the streamwise-spanwise plane by utilizing fluid-suspended reflective flakes. Transition was triggered by a high-amplitude pointwise disturbance. The transitional Reynolds number, i.e. the lowest Reynolds number for which turbulence can be sustained, was determined to be 360 ± 10, based on half-channel height and half the velocity difference between the walls. For Reynolds numbers above this value a large enough amplitude of the initial disturbance gave rise to a growing turbulent spot. Its shape and spreading rate was determined for Reynolds numbers up to 1000.

We examine the statistical mechanisms by which energy and scalar variance are cascaded to small scales for isotropic, three-dimensional turbulence. Two avenues are explored: (i) the traditional transfer function (defined by the nonlinear cascade that gives the time rate of change of the energy spectrum), and (ii) the bispectrum (the elementary triple-point correlation, averaged over directions perpendicular to three co-linear observation points). Our tools are direct numerical simulations (DNS), and the statistical theory of turbulence, here in the form of the test field model (TFM) (Kraichnan 1971). Comparison of the results indicates a fairly good quantitative agreement between DNS and the TFM at large Prandtl numbers (Pr ≥ 0.25), but substantial disagreement at lower Pr, where the transfer to small scales becomes too small. This disparity we trace to the Markovian aspect of the TFM; the more fundamental direct interaction approximation (DIA) (Kraichnan 1959) compares more favourably to DNS as Pr → 0. For Pr ∼ 1, we compare DNS and TFM bispectra for velocity and scalar fields in both Fourier and physical space. The physical space representation of bispectra serves as a useful means of discriminating between velocity and scalar transfer.

The buoyancy-driven motion of bubbles or drops towards a liquid-liquid interface at small and moderate Reynolds numbers is studied. Solutions of the unsteady nonlinear mathematical problem are performed by means of a general finite-element technique of Lagrangian type. Data for the development of the interface shapes and the film thickness are presented, and comparisons with previous theories and experiments are performed, supporting the reliability of our results. Two interesting phenomena are observed: a transient concavity at the bottom of the particle and particle elongation in the direction of motion. The drainage of the film formed between the particle and the interface, and the tailing mode are studied. Occurrence of a transient surface wave at the liquid-liquid interface and a toroidal dimpling in the film zone are observed in the tailing mode.

An experimental investigation of the dispersion of small particles in a developing, high-Reynolds-number, turbulent, plane mixing layer is presented. Flow visualizations, laser attenuation and diffraction techniques as well as hot-wire anemometry are used to describe the evolution of the mean and instantaneous particle and gas flow fields. It is shown that the large scale turbulent motion existing in the mixing layer plays a central role in the dispersion of the particles. The mean particle concentration field is shown to be the result of a statistical distribution of streaks produced by the large-scale coherent component of the turbulent motion in the mixing layer. Furthermore, for every particle size, the spreading of the particle concentration thickness is found to occur at a smaller rate than the one characterizing the momentum of the turbulent carrier gas. Large particles are shown to initially disperse into the mixing layer less effectively than the small ones. However, when both downstream and cross-stream coordinates are non-dimensionalized with a characteristic length proportional to the square of the droplet diameter, a universal, particle-size independent dispersion field is found to exist.

In this study we analyse the dispersion mechanisms of small water particles in an acoustically forced plane, turbulent mixing layer. When compared to the naturally developing flow, the excited mixing layer is shown to exhibit drastic changes in the cross-stream particle concentration evolution, with the particles now dispersing laterally at larger rates than that of the longitudinal momentum of the turbulent gas glow. The particle dispersion is shown to occur as a size-selective process characterized by the existence of an intermediate particle size range for which the lateral dispersion is maximized. Unlike in the natural flow evolution, the forced shear layer does not possess a non-dimensionalization rendering particle size independent dispersion properties. It is demonstrated that this behaviour results from the non-similarity of the developing gas motion. The mixing layer is shown to have inhomogeneities both in the droplet concentration and in the droplet-size probability density distribution function. Instantaneous flow visualizations as well as spectral analysis of laser extinction measurements show the presence of a coherent organization in the particle concentration field resulting from the large-scale eddies characterizing the underlying turbulent gas flow. Conditional, phase-average sample techniques are used to analyse the structure of this coherent particle dispersion field. The dispersion is shown to be controlled by an array of large streaks that emanate from the undisturbed spray, engulfing areas which are almost depleted of droplets. The data from the conditional sampling measurements are in good agreement with preliminary results from a simplified Eulerian model of the particle motion, showing the potential that this formulation can have for analysing this type of flow.

The transition mechanism in a spatially developing two-dimensional wake is studied by means of direct numerical simulations. Five different types of forcing of the inlet are investigated: fundamental mode, fundamental and one or two subharmonics, fundamental mode and random noise, and random noise only. The effects of the amplitude levels of the perturbations on the development of the layer are also investigated. Statistical analyses are performed and some numerical results are compared with experimental measurements. When only a fundamental mode is forced, the energy spectra show amplification of the fundamental frequency and its higher harmonics, and the development of a stable vortex street. When the inlet flow is forced by a fundamental mode and two subharmonics, a vortex street also appears downstream, but the shape of the vortices is distorted. The amplitude of the subharmonic grows only after the saturation of the fundamental. Amplification of modes close to the fundamental mode is observed when random noise of large amplitude is added to the fundamental mode. The phase jitter around the fundamental frequency plays a critical role in generating vortices of random shape and spacing. Large- and small-scale distortions of the flow structure are observed. Pairing of vortices of the same sign is observed, as well as coupling of vortices of opposite sign. When the inlet profile is forced by random noise of amplitude 10−5 times the free-stream velocity, one frequency close to the most unstable one is amplified more than the others. The energy spectrum is otherwise full. When the same low amplitude (10−5) is used to force the fundamental mode and its two subharmonics, bands of energy develop around the forced modes and their harmonics. Finally, we find that large-deficit wakes are globally unstable when the size of the absolutely unstable region is greater than about three times the half-width of the wake

The generation of internal waves in a submarine rectangular trench by normally incident surface waves has been investigated through laboratory experiments and theory. A linear model was developed for small-amplitude, simple harmonic wave motions. In this model, the fluid outside the trench is homogeneous, and the fluid in the trench is composed of two homogeneous layers of different densities separated by a transition region of linear density variation; viscous dissipation is treated based on the assumption of a laminar boundary layer. In the experiments, the stratification in the trench was created using fresh water and salt water, and a scanning laser beam and detector system was used to measure the amplitude of internal waves. The study shows that, when the frequency of the surface waves corresponds to the natural frequency of internal waves, the amplitude of internal waves becomes large compared to the amplitude of surface waves. The natural frequency of oscillation of internal waves decreases as the thickness of the density interface increases and the depth of the lower fluid decreases. Two distinct classes of internal waves were observed, namely, standing internal waves when the lower fluid was deep, and travelling internal waves when the lower fluid was shallow. The linear model predicted the response curve for internal waves quite well in all the cases investigated. It was also found that the internal waves were strongly damped when the depth of the lower fluid was small compared to the wavelength of internal waves.

The effects of finite gap and various dilute solution properties on the previously studied purely elastic Taylor-Couette instability reported by Muller et al. (1989) and Larson et al. (1990) are investigated. The dilute solution properties which we consider are the ratio of the second to the first normal stress coefficient, ψ2/ψ1, and the ratio of the solvent to the polymer contribution to the shear viscosity, S. Linear stability predictions for the flow of an Oldroyd-B fluid are presented over a wide range of Deborah number, De, gap ratio, ε, and S. In addition, the Oldroyd-B model is modified to include second normal stress differences, and new stability predictions are presented for small negative and small positive ψ2/ψ1. Both the critical conditions and changes in the flow structure are presented. It is demonstrated that finite-gap effects are stabilizing even for relatively small gap ratios (0 < ε < 0.35). Furthermore, it is shown that there are two possible flow structures which can be chosen near the onset of instability: a standing wave structure (i.e. radially propagating vortices) or a travelling wave (i.e. vortices propagating up or down the coaxial cylinders). However, the strength and both the axial and radial dimensions of these vortices depend markedly on the gap, with both dimensions decreasing as the gap ratio increases. Thus, the number of vortices filling the gap increases with the gap ratio.

In a second study, we show that the instability is sensitive to the presence of second normal stress differences. Positive second normal stress differences are shown to be destabilizing, while negative differences are strongly stabilizing. Furthermore, when both finite-gap effects and small negative second normal stress differences are included, the predicted gap dependence of the critical De is in good agreement with previous measurements on the flow of a dilute polyisobutylene solution. Finally, we present new measurements of the critical values of the De for a series of dilute, viscous polystyrene solutions, for which ψ2 was found to be near zero. We find that as the polymer concentration increases (and therefore S decreases) the critical Deborah number decreases, in qualitative agreement with the theoretical predictions.

A tractable theory for weakly damped, nonlinear Stokes waves on deep water was recently formulated by Ruvinsky & Friedman (1985a, b; 1987). In this paper we show how the theory can be simplified, and that it is equivalent to a boundary-layer model for surface waves proposed by Longuet-Higgins (1969), when the latter is generalized to include surface tension and nonlinearity. The potential part of the flow is determined by boundary conditions applied at the base of the vortical boundary layer. The theory may be of use in discussing the generation of waves by wind.

An experimental study of the spatial features of Rayleigh-Bénard convection in a sall box is presented. Experiments are carried out in a rectangular cell (aspect ratios Γx = 2.03, Γy = 1.19) filled with silicone oil (Prandtl number, Pr = 130) for different Rayleigh numbers, Ra (up to Ra = 75Rac, Rac1707). The basic structure of the flow field for this range of Ra consists of two rolls with their axes parallel to the shorter horizontal side. Both senses of rotation for the rolls are observed, corresponding to the two branches of the bifurcation. Particle image velocimetry, with a 5 mW He-Ne laser as the illuminating source, is used to measure the velocity field in the midplane of the cell. From it the vorticity field (out of plane component) and two-dimensional streamlines are calculated. The flow has been measured to be three-dimensional, even for very low Ra, owing to the sidewall influence. The spatial features of the flow are shown to be dependent on both Ra and the sense of rotation of the rolls. Finally, a Fourier analysis of the velocity field is presented. The spatial and thermal dependences of the different Fourier terms are reported. The velocity field, in a first-order approximation, is quantitatively described.

We consider the statistically homogeneous motion that is generated by buoyancy forces after the creation of homogeneous random fluctuations in the density of infinite fluid at an initial instant. The mean density is uniform, and density fluctuations are smoothed by molecular diffusion. This turbulent flow system has interesting properties, and shows how self-generated motion contributes to the rate of mixing of an ‘active’ scalar contaminant.

If nonlinear terms in the governing equations are negligible, there is an exact solution which shows that the history of the motion depends crucially on the form of the buoyancy spectrum near zero wavenumber magnitude (κ). According to this solution the Reynolds number of the motion increases indefinitely, so the linear equations do not remain valid. There are indications of similar behaviour when the nonlinear terms are retained. The value of the three-dimensional buoyancy spectrum function at κ = 0 is shown to be independent of time, and this points to the existence of a similarity state of turbulence with decreasing mean-square velocity but increasing Reynolds number at large times.

We have made a numerical simulation of the flow field and have obtained the mean-square velocity and density fluctuations and the associated spectra as functions of time for various initial conditions. An estimate of the time required for the mean-square density fluctuation to fall to a specified small value is found. The expected similarity state at large times is confirmed by the numerical simulation, and there are indications of a second similarity state which develops asymptotically when the buoyancy spectrum is zero at κ = 0. The analytical and numerical results together give a comprehensive description of the birth, life and lingering death of buoyancy-generated turbulence.

Evolution equations and conservation laws are derived for a quite general layered quasi-geostrophic model: with arbitrary thickness and stratification structure and with either a free or a rigid (including the possibility of topography) boundary condition, at the top and bottom. The system is shown to be Hamiltonian, and Arnol'd stability conditions are derived, in the sense of both the first and second theorem, i.e. for pseudowestward and pseudoeastward basic flows, respectively, and for arbitrary perturbations of potential vorticity and Kelvin circulations.

Two examples of parallel basic flow in a channel are analysed: the sine profile in the so-called equivalent barotropic model (one layer with a free boundary) and Phillips’ problem (uniform flow in each of two layers with rigid boundaries). Using the second theorem with the optimum combination of pseudoenergy and pseudomomentum it is shown that, in both cases, the basic state is nonlinearly stable if the channel width L is small enough, namely, ΛL < π and $2(f_0L/\pi)^2 < g^{\prime}(H_1H_2)^{\frac{1}{2}} $, respectively. (In the first problem, Λ is the wavenumber of the sine profile; in the second one, g′ is the reduced gravity, H1 and H2 are the layer thicknesses, and f0 is the Coriolis parameter). The stability condition of either problem is found to be also a necessary one: as soon as it is violated a grave mode becomes unstable. It is shown explicitly that the second variation of the pseudoenergy and pseudomomentum of a growing (decaying) normal mode is identically zero, defining the direction of the unstable (stable) manifold.

We investigated experimentally the flows induced by a localized surfactant (oleic acid) on thin glycerol films. The oleic acid creates surface-tension gradients, which drive convention on the surface and within the film. Qualitative descriptions of the Lagrangian flow field were provided by flow-visualization experiments. Quantitative measurements of surface flows were conducted using dyed glycerol markers, where the initial motion of these markers is used to define the position of the time-dependent ‘convection front’. The flow characteristics were found to depend largely upon the magnitude of a gravitational parameter, G, representing the ratio of gravitational to surface-tension gradient (Marangoni) forces. Small G (G < 0.5) caused net outflow of the film leading to this thinning and, in some cases, to film rupture. When G < 1, bi-directional flows were caused by hydrostatic pressure gradients which served to stabilize the film. Additionally, the position of a surface convection front was found to differ significantly from that of the surfactant's leading edge for all G > 0. For this reason, surface markers may not be used to measure accurately the position of the droplet's leading edge. Finally, simulations of the Lagrangian flows conducted using the theory of Gaver & Grotberg (1990) compare favourably with these experimental results in the limit of dilute surfactant concentrations, and thus experimental verification of that theory is provided by this work. The results of this study may be useful for understanding the behaviour of the lung's thin-film lining after an aerosol droplet of insoluble exogenous surfactant lands upon its surface.

To facilitate the theoretical prediction of the evolution of a short gravity wave on a long wave of finite amplitude, we consider a model where the long wave is represented by Gerstner's exact but rotational solution in Lagrangian coordinates. Analytical formulae for the modulation of an infinitesimal irrotational short wave are shown to be qualitatively accurate in comparison with the numerical results by Longuet-Higgins (1987) and with the analytical results by Henyey et al. (1988) for irrotational long waves. Discrepancies are generally of second order in the long-wave steepness, consistent with the vorticity in Gerstner's solution. Weakly nonlinear short waves are shown to be parametrically excited by the long wave over a long time. In particular, multiple bands of modulational instability appear in the parameter space. Numerical calculations of the nonlinear evolution equation show the onset of chaos for sufficiently large parameter $\alpha = \epsilon (k\overline{A})^2/2\Omega/\sigma $, where $\epsilon k\overline{A} $ is the short-wave steepness and (εΩ, σ) the frequency of the (long, short) wave. Furthermore, if the short-wave amplitude A is approximated by a two-mode truncated Fourier series, the evolution equation reduces to a non-autonomous Hamiltonian system. The numerical solutions confirm that the onset of chaos is an inherent feature of the parametrically excited nonlinear system.

Experiments on the propagation of linear and weakly nonlinear gravity waves over a rectangular submerged bar were undertaken through very careful measurements in a wave tank. Effects arising from the finite amplitude of the surface wave and those coming from the generation of vortices around bar edges were examined. Experimental data are compared with results of two theoretical models. The first model was derived from Takano (1960) and Kirby & Dalrymple's (1983) work and the second model was developed by Devillard, Dunlop & Souillard (1988) using the renormalized transfer matrix introduced by Miles (1967).

Axisymmetric equilibrium shapes and stability of linearly polarizable dielectric (ferrofluid) drops of fixed volume which are pendant/sessile on one plate of a parallel-plate capacitor and are subjected to an applied electric (magnetic) field are determined by solving simultaneously the free boundary problem comprised of the Young-Laplace equation for drop shape and the Laplace equation for electric (magnetic) field distribution. When the contact angle that the drop makes with the plate is fixed to be 90° and the distance between the plates is infinite, the results are identical to those of a free drop immersed in a uniform field and resolve discrepancies between previously reported theoretical predictions and experimental measurements. Remarkably, regardless of the value of the ratio of the permittivity (permeability) of the drop to that of the surrounding fluid, κ, drop shapes develop conical tips as drop deformation increases. However, three types of behaviour are found, depending on the value of κ. When κ < κ1, the drop deformation grows without bound as field strength rises. On the other hand, when κ > κ2 > κ1, families of equilibrium drop shapes become unstable at turning points with respect to field strength. Beyond the turning points, the unstable families terminate: the mean curvature at the virtually conical drop tip grows without bound. However, in the range κ1 < κ < κ2, the new results predict that drop deformation exhibits hysteresis, in accord with experiments of Bacri, Salin & Massart (1982) and Bacri & Salin (1982, 1983). Such hysteresis phenomena have been surmized previously on the basis of approximate theories, though they have not been calculated systematically until now. Moreover, detailed computations reveal the importance of varying the drop size and plate spacing, and whether, along the three-phase contact line, the contact line is fixed or the contact angle is prescribed.