This paper describes an experimental investigation of mixing due to Rayleigh–Taylor instability between two miscible fluids. Attention is focused on the gravitationally driven instability between a layer of salt water and a layer of fresh water with particular emphasis on the internal structure within the mixing zone. Three-dimensional numerical simulations of the same flow are used to give extra insight into the behaviour found in the experiments.

The two layers are initially separated by a rigid barrier which is removed at the start of the experiment. The removal process injects vorticity into the flow and creates a small but significant initial disturbance. A novel aspect of the numerical investigation is that the measured velocity field for the start of the experiments has been used to initialize the simulations, achieving substantially improved agreement with experiment when compared with simulations using idealized initial conditions. It is shown that the spatial structure of these initial conditions is more important than their amplitude for the subsequent growth of the mixing region between the two layers. Simple measures of the growth of the instability are shown to be inappropriate due to the spatial structure of the initial conditions which continues to influence the flow throughout its evolution. As a result the mixing zone does not follow the classical quadratic time dependence predicted from similarity considerations. Direct comparison of external measures of the growth show the necessity to capture the gross features of the initial conditions while detailed measures of the internal structure show a rapid loss of memory of the finer details of the initial conditions.

Image processing techniques are employed to provide a detailed study of the internal structure and statistics of the concentration field. These measurements demonstrate that, at scales small compared with the confining geometry, the flow rapidly adopts self-similar turbulent behaviour with the influence of the barrier-induced perturbation confined to the larger length scales. Concentration power spectra and the fractal dimension of iso-concentration contours are found to be representative of fully developed turbulence and there is close agreement between the experiments and simulations. Other statistics of the mixing zone show a reasonable level of agreement, the discrepancies mainly being due to experimental noise and the finite resolution of the simulations.

We have investigated the role of elasticity in the stability of air–fluid interfaces during fluid displacement flows. Our investigations of the stability of coating flows with an eccentric cylinder geometry for both a viscous Newtonian fluid and ideal elastic Boger fluids are discussed in terms of three classes of phenomena. To begin, we have documented several new features in traditional fingering instabilities in elastic displacement flows. These include a very strong elastic destabilization of forward roll coating: a destabilization which can be correlated directly with the elasticity of the coating fluid and which appears to be present even in the absence of diverging channel walls. Moreover, elastic effects are shown to create a novel saw-toothed cusped pattern in the eccentric cylinder roll-and-plate geometry. Secondly, we have found that purely elastic bulk flow instabilities in the neighbourhood of air–fluid interfaces can cause surface deformations if the secondary flow is of sufficient strength. Finally, flows created by the displacement of less viscous air by a more viscous elastic fluid are found to display a new class of purely elastic instabilities which appear to be independent of traditional viscous fingering instabilities and elastic bulk flow instabilities. Thus interfaces which are stable for Newtonian fluids are unstable via purely elastic mechanisms. We have found that indeed elasticity has a dramatic effect on the stability of interfaces, not only changing the critical conditions, but also changing the manifestation of traditional fingering instabilities, and causing new purely elastic interfacial instabilities.

Crossflow instability of a three-dimensional boundary layer is a common cause of transition in swept-wing flows. The boundary-layer flow modified by the presence of finite-amplitude crossflow modes is susceptible to high-frequency secondary instabilities, which are believed to harbinger the onset of transition. The role of secondary instability in transition prediction is theoretically examined for the recent swept-wing experimental data by Reibert et al. (1996). Exploiting the experimental observation that the underlying three-dimensional boundary layer is convectively unstable, non-linear parabolized stability equations are used to compute a new basic state for the secondary instability analysis based on a two-dimensional eigenvalue approach. The predicted evolution of stationary crossflow vortices is in close agreement with the experimental data. The suppression of naturally dominant crossflow modes by artificial roughness distribution at a subcritical spacing is also confirmed. The analysis reveals a number of secondary instability modes belonging to two basic families which, in some sense, are akin to the ‘horseshoe’ and ‘sinuous’ modes of the Görtler vortex problem. The frequency range of the secondary instability is consistent with that measured in earlier experiments by Kohama et al. (1991), as is the overall growth of the secondary instability mode prior to the onset of transition (e.g. Kohama et al. 1996). Results indicate that the N-factor correlation based on secondary instability growth rates may yield a more robust criterion for transition onset prediction in comparison with an absolute amplitude criterion that is based on primary instability alone.

A unified approach for assessing and characterizing both the non-equilibrium and equilibrium states of planar homogeneous flows is analysed within the framework of single-point turbulence closure equations. The underlying methodology is based on the replacement of the modelled evolution equation for the Reynolds stress anisotropy tensor by an equivalent set of three equations for characteristic scalar invariants or state variables. For stress anisotropy evolution equations which use modelled pressure–strain rate correlations that are quasi-linear, this equivalence then leads to an analytic solution for the time evolution of the Reynolds stress anisotropy. With this analysis, the transient system characteristics can be studied, including the dependence on initial states, the occurrence of limit-cycle behaviour, and the system global stability. In the fixed-point asymptotic limit, these results are consistent with and unify previous equilibrium studies, and provide additional information allowing the resolution of some questions that could not be answered in the framework of previous developments. A new result on constraints applicable to the development of realizable pressure–strain rate models is obtained from a re-examination of the stress anisotropy invariant map. With the analytic solution for the transient behaviour, some recent non-equilibrium models, which incorporate relaxation effects, are evaluated in a variety of homogeneous flows in inertial and non-inertial frames.

The rate of heat and mass transfer at the surface of acoustically levitated pure liquid droplets is predicted theoretically for the case where an acoustic boundary layer appears near the droplet surface resulting in an acoustic streaming. The theory is based on the computation of the acoustic field and squeezed droplet shape by means of the boundary element method developed in Yarin, Pfaffenlehner & Tropea (1998). Given the acoustic field around the levitated droplet, the acoustic streaming near the droplet surface was calculated. This allowed calculation of the Sherwood and Nusselt number distributions over the droplet surface, as well as their average values. Then, the mass balance was used to calculate the evolution of the equivalent droplet radius in time.

The theory is applicable to droplets of arbitrary size relative to the sound wavelength λ, including those of the order of λ, when the compressible character of the gas flow is important. Also, the deformation of the droplets by the acoustic field is accounted for, as well as a displacement of the droplet centre from the pressure node. The effect of the internal circulation of liquid in the droplet sustained by the acoustic streaming in the gas is estimated. The distribution of the time-average heat and mass transfer rate over the droplet surface is found to have a maximum at the droplet equator and minima at its poles. The time and surface average of the Sherwood number was shown to be described by the expression Sh = KB/√ω[Dscr ]0, where B = A0e/(ρ0c0) is a scale of the velocity in the sound wave (A0e is the amplitude of the incident sound wave, ρ0 is the unperturbed air density, c0 is the sound velocity in air, ω is the angular frequency in the ultrasonic range, [Dscr ]0 is the mass diffusion coefficient of liquid vapour in air, which should be replaced by the thermal diffusivity of air in the computation of the Nusselt number). The coefficient K depends on the governing parameters (the acoustic field, the liquid properties), as well as on the current equivalent droplet radius a.

For small spherical droplets with a[Lt ]λ, K = (45/4π)1/2 = 1.89, if A0e is found from the sound pressure level (SPL) defined using A0e. On the other hand, if A0e is found from the same value of the SPL, but defined using the root-mean-square pressure amplitude (prms = A0e/√2), then Sh = KrmsBrms/ √ω[Dscr ]0, with Brms = √2B and Krms = K/√2 = 1.336. For large droplets squeezed significantly by the acoustic field, K appears always to be greater than 1.89. The evolution of an evaporating droplet in time is predicted and compared with the present experiments and existing data from the literature. The agreement is found to be rather good.

We also study and discuss the effect of an additional blowing (a gas jet impinging on a droplet) on the evaporation rate, as well as the enrichment of gas at the outer boundary of the acoustic bondary layer by liquid vapour. We show that, even at relatively high rates of blowing, the droplet evaporation is still governed by the acoustic streaming in the relatively strong acoustic fields we use. This makes it impossible to study forced convective heat and mass transfer under the present conditions using droplets levitated in strong acoustic fields.

We report new experiments with the ‘sliced-cylinder’ β-plane model of Pedlosky & Greenspan (1967) and Beardsley (1969), but with a much wider basin such that the western boundary current and its eddies occupy a small fraction of the basin width. These experiments provide new insights into nonlinear aspects of the flow: the critical conditions for boundary current separation and the transition from stable to unstable flow are redefined, and a further transition from periodic to chaotic eddy shedding under strong anticyclonic forcing is also found. In the nonlinear regimes the western boundary current separates from the western wall and shoots into the interior as a narrow jet that undergoes a rapid adjustment to join with the broad slow interior flow. In the unstable regimes this adjustment involves eddy shedding. Each transition occurs at a fixed critical value of a Reynolds number Reγ based on the velocity and width scales for a purely viscous boundary current: the flow is unstable for Reγ > 123±4 and aperiodic for Reγ > 231±5. The results provide evidence that the mechanism causing instability is shear in the separated jet rather than the breaking of a large-amplitude Rossby wave. A quasi-geostrophic numerical model applied to the laboratory conditions yields a stability boundary and detailed characteristics of the flow largely consistent with those determined from the experiments. It also reveals a strong dependence of the circulation pattern on basin aspect ratio, and shows that an adverse higher-order pressure gradient is responsible for western boundary current separation in this model. Eddy–eddy interactions and feedback of fluctuations from the eddy formation region to upstream parts of the boundary current contribute to aperiodic behaviour. As a result of eddy shedding, passive tracer from each streamline in the boundary current can be stirred across much of the width of the basin.

The effect of non-uniformity on the development of a modulated, weakly nonlinear wavepacket is studied. The non-uniformity, characterized by slowly varying wavenumber and frequency of the primary wave, may lead to significant modification of the stability properties compared with the uniform case. As a specific example we consider a modulated Stokes wave on deep water. In the uniform case such a wave proves to be definitely unstable (Benjamin & Feir 1967). In the non-uniform case, on the other hand, the wave may become stable under certain conditions. One of these is an increase of the local group velocity in the direction of wave propagation. Then the Benjamin-Feir instability mechanism is quenched on a time scale determined by the degree of non-uniformity. In addition, a sufficient degree of non-uniformity leads to stability of the wave to linear perturbations. However, when the local group velocity decreases in the direction of wave propagation, non-uniformity has a destabilizing effect. A comparison is made with experiments. It is also shown that the analysis, based on this specific example, is readily applied to a greater variety of non-uniform, dispersive waves.

Surface-tension-driven Bénard convection in low-Prandtl-number fluids is studied by means of direct numerical simulation. The flow is computed in a three-dimensional rectangular domain with periodic boundary conditions in both horizontal directions and either a free-slip or no-slip bottom wall using a pseudospectral Fourier–Chebyshev discretization. Deformations of the free surface are neglected. The smallest possible domain compatible with the hexagonal flow structure at the linear stability threshold is selected. As the Marangoni number is increased from the critical value for instability of the quiescent state to approximately twice this value, the initially stationary hexagonal convection pattern becomes quickly time-dependent and eventually reaches a state of spatio-temporal chaos. No qualitative difference is observed between the zero-Prandtl-number limit and a finite Prandtl number corresponding to liquid sodium. This indicates that the zero-Prandtl-number limit provides a reasonable approximation for the prediction of low-Prandtl-number convection. For a free-slip bottom wall, the flow always remains three-dimensional. For the no-slip wall, two-dimensional solutions are observed in some interval of Marangoni numbers. Beyond the Marangoni number for onset of inertial convection in two-dimensional simulations, the convective flow becomes strongly intermittent because of the interplay of the flywheel effect and three-dimensional instabilities of the two-dimensional rolls. The velocity field in this intermittent regime is characterized by the occurrence of very small vortices at the free surface which form as a result of vortex stretching processes. Similar structures were found with the free-slip bottom at slightly smaller Marangoni number. These observations demonstrate that a high numerical resolution is necessary even at moderate Marangoni numbers in order to properly capture the small-scale dynamics of Marangoni convection at low Prandtl numbers.

Free surface flows of a rotational fluid past a two-dimensional semi-infinite body are considered. The fluid is assumed to be inviscid, incompressible, and of finite depth. A boundary integral method is used to solve the problem for the case where the free surface meets the body at a stagnation point. Supercritical solutions which satisfy the radiation condition are found for various values of the Froude number and the dimensionless vorticity. Subcritical solutions are also found; however these solutions violate the radiation condition and are characterized by a train of waves upstream. It is shown numerically that the amplitude of these waves increases as each of the Froude number, vorticity and height of the body above the bottom increases.

The nonlinear evolution of wavepackets in a laminar boundary layer has been studied experimentally. The packets were generated by acoustic excitations injected into the boundary layer through a small hole in the plate. Various packets with different phases relative to the envelope were studied. It was found that for all the packets the nonlinearity involved the appearance of oblique modes of frequency close to the subharmonic of the dominant two-dimensional wave. Moreover, the results confirmed that the phase had a strong influence on the strength of the nonlinear interaction. The experimental observations also indicated that although a subharmonic resonance appeared to be present in the process, it alone could not explain the nonlinear behaviour. The experiment demonstrated that the process must also involve a mechanism that generates oblique waves of frequency lower than the Tollmien–Schlichting band.

An infinite-order, Boussinesq-type differential equation for wave shoaling over variable bathymetry is derived. Defining three scaling parameters – nonlinearity, the dispersion parameter, and the bottom slope – the system is truncated to a finite order. Using Padé approximants the order in the dispersion parameter is effectively doubled. A derivation is made systematic by separately solving the Laplace equation in the undisturbed fluid domain and then addressing the nonlinear free-surface conditions. We show that the nonlinear interactions are faithfully captured. The shoaling and dispersion components are time independent.

By comparison with both experimental and numerical data, Dysthe's (1979) O(ε4) modified nonlinear Schrödinger; equation has been shown to model the evolution of a slowly varying wavetrain well (here ε is the wave steepness). In this work, we extend the equation to include a prescribed, large-scale, O(ε2) surface current which varies about a mean value. As an introduction, a heuristic derivation of the O(ε3) current-modified equation, used by Bakhanov et al. (1996), is given, before a more formal approach is used to derive the O(ε4) equation. Numerical solutions of the new equations are compared in one horizontal dimension with those from a fully nonlinear solver for velocity potential in the specific case of a sinusoidal surface current, such as may be due to an underlying internal wave. The comparisons are encouraging, especially for the O(ε4) equation.

A global stability investigation of two-dimensional vertical liquid sheet flows is experimentally carried out. The motivation is that previous investigations addressed the study of the local absolute/convective character of such flows, thus they are not able to predict the actual critical flow Weber number corresponding to sheet rupture. The objective of the paper is twofold: first, the link between local absolute and global instabilities is investigated and the measured length of the absolute instability region is correlated with the non-parallelism parameter (sheet slenderness ratio which is the reciprocal of the Froude number); then, a criterion to predict the flow Weber number value at sheet rupture is given for which the critical Weber number is correlated with Froude and Reynolds numbers. Tests are carried out on liquid (low-concentration water solutions of surfactants and low-viscosity motor oil) sheets issuing from a nozzle with a long horizontal exit section in still air under the gravitational field. A major goal of the experiments is the determination of the vertical location where the local Weber number equals unity, because this yields the length of the absolute instability region. This location is determined by observing the standing sinuous waves generated by an obstacle placed normally to the sheet, and by measuring the angle between the tangent to the wave at the obstacle and the vertical direction for the minimum liquid flow rate necessary to maintain the sheet stable (global instability onset).

The flickering candle: transition to a global oscillation in a thermal plume

Journal of Fluid Mechanics, vol. 390 (1999), pp. 297–323

On page 302, the fourth and third lines from the bottom of the page, for the word ‘important’ read ‘unimportant’.