We perform a detailed experimental study of large-scale vortices propagating in the rotating shallow-water layer in a paraboloidal vessel. A specific data acquisition technique is used in order to ensure precise measurements of the free-surface elevation. We find two qualitatively different types of vortex behaviour controlled by the relative elevation value. For small elevations we observe a standard quasi-geostrophic pattern with an asymmetric secondary circulation around an initially symmetric vortex which leads to a meridional drift and Rossby wave radiation. This type of behaviour is exhibited by both cyclonic and anticyclonic vortices. For relative elevations larger than 1 (nonlinear regime) the necessarily anticyclonic vortices are drifting strictly zonally maintaining their circular symmetry during the viscous decay. By varying the initial latitude of the vortex we were able to check that in the nonlinear regime the vortex lifetime is not sensitive to the beta-effect, while it is the case in the quasi-geostrophic regime. In the same way we show that the observed difference in cyclone–anticyclone lifetimes is not influenced by the beta-effect.

Experiments were conducted to investigate the effect of compressibility on turbulent reacting mixing layers with moderate heat release. Side- and plan-view visualizations of the reacting mixing layers, which were formed between a high-speed high-temperature vitiated-air stream and a low-speed ambient-temperature hydrogen stream, were obtained using a combined OH/acetone planar laser-induced fluorescence imaging technique. The instantaneous images of OH provide two-dimensional maps of the regions of combustion, and similar images of acetone, which was seeded into the fuel stream, provide maps of the regions of unburned fuel. Two low-compressibility (Mc=0.32, 0.35) reacting mixing layers with differing density ratios and one high-compressibility (Mc=0.70) reacting mixing layer were studied. Higher average acetone signals were measured in the compressible mixing layer than in its low-compressibility counterpart (i.e. same density ratio), indicating a lower entrainment ratio. Additionally, the compressible mixing layer had slightly wider regions of OH and 50% higher OH signals, which was an unexpected result since lowering the entrainment ratio had the opposite effect at low compressibilities. The large-scale structural changes induced by compressibility are believed to be primarily responsible for the difference in the behaviour of the high- and low-compressibility reacting mixing layers. It is proposed that the coexistence of broad regions of OH and high acetone signals is a manifestation of a more biased distribution of mixture compositions in the compressible mixing layer. Other mechanisms through which compressibility can affect the combustion are discussed.

This paper deals with the theoretical and experimental investigation of acoustically levitated droplets. A method of calculation of the acoustic radiation pressure based on the boundary element method (BEM) is presented. It is applied to predict shapes of droplets levitated in an acoustic field (and as a result, deformed by it). The method was compared with several known exact and approximate analytical results for rigid spheres and shown to be accurate (and a widely used approximate formula for the acoustic levitation force acting on a rigid sphere was found to be inaccurate for sound wavelengths comparable with the sphere radius). The method was also compared with some other theoretical approaches known from the literature.

Displacement of the droplet centre relative to the pressure node is accounted for and shown to be significant. The results for droplet shapes and displacements are compared with experimental data, and the agreement is found to be rather good. Furthermore, the experimental investigations reveal a unique relationship between the aspect ratio of an oblate droplet and the sound pressure level in the levitator. This relationship agrees well with the predicted shapes. A practical link between droplet shape or droplet displacement and sound pressure level in a levitator is therefore now available.

The expansion and contraction of a bubble pinned at a submerged tube tip and driven by constant gas flow rate Q are studied both theoretically and experimentally for Reynolds number Re[Lt ]1. Bubble shape, gas pressure, surface velocities, and extrapolated detached bubble volume are determined by a boundary integral method for various Bond (Bo=ρga2/σ) and capillary (Ca=μQ/σa2) numbers, where a is the capillary radius, ρ and μ are the liquid density and viscosity, σ is the surface tension, and g is the gravitational acceleration.

Bubble expansion from a flat interface to near detachment is simulated for a full range of Ca (0.01–100) and Bo (0.01–0.5). The maximum gas pressure is found to vary almost linearly with Ca for 0.01[les ]Ca[les ]100. This correlation allows the maximum bubble pressure method for measuring dynamic surface tension to be extended to viscous liquids. Simulated detached bubble volumes approach static values for Ca[Lt ]1, and asymptote as Q3/4 for Ca[Gt ]1, in agreement with analytic predictions. In the limit Ca→0, two singular time domains are identified near the beginning and the end of bubble growth during which viscous and capillary forces become comparable.

Expansion and contraction experiments were conducted using a viscous silicone oil. Digitized video images of deforming bubbles compare well with numerical solutions. It is observed that a bubble contracting at high Ca snaps off.

An analysis is given, based on rapid distortion theory, of the effect on homogeneous turbulence when it passes through a linear cascade of thin flat plates or aerofoils and in particular the turbulence that emerges on the downstream side. The main results are evaluated for the limiting case in which the spacing ratio of the cascade is very small so that the cascade tends to completely suppress the normal component of velocity of the turbulence everywhere, both within the cascade and downstream of it, resulting in a ‘planar’ (quasi-two-dimensional) turbulence. Some comparisons with experimental measurements of grid generated turbulence passed through cascades of thin aerofoils and in the regions adjacent to the wakes of isolated streamwise plates in a wind tunnel are shown.

For two-dimensional flow there is a simple relation between the vorticity at a stress-free surface and the surface curvature. In this note the relation is generalized to flow in three dimensions. It is shown that in addition to a component of vorticity perpendicular to the flow there is also a component parallel to the direction of flow. The latter vanishes only at an umbilical point or when the flow is in one of the two principal directions of curvature.

The convective flow in a thin liquid layer with a free surface heated from below is studied using a combination of accurate experiments with silicone oil (v=0.1 cm2 s−1) and high-resolution direct numerical simulations of the time-dependent governing equations. It is demonstrated that above a certain value εs of the threshold of primary instability, ε=0, square convection cells rather than the seemingly all-embracing hexagons are the persistent dominant features of Bénard convection. The transition from hexagonal to square cells sets in via a subcritical bifurcation and is accompanied by a sudden rapid increase of the Nusselt number. This implies that square cells are the more efficient mode of heat transport. Their wavenumber exceeds that of hexagonal cells by about 8%. The transition depends on the Prandtl number and it is shifted towards higher εs if the Prandtl number is increased. The replacement of hexagonal by square cells is mediated by pentagonal cells. In the transitional regime from hexagonal to square cells, characterized by the presence of all three planforms, the system exhibits complex irregular dynamics on large spatial and temporal scales. The time dependence becomes more vivid with decreasing Prandtl number until finally non-stationary square cells appear. The simulations agree with the experimental observations in the phenomenology of the transition, and in the prediction of both the higher Nusselt number of square Bénard cells and the subcritical nature of the transition. Quantitative differences occur with respect to the values of εs and the Prandtl number beyond which the time dependence vanishes. These differences are the result of a considerably weaker mean flow in the simulation and of residual inhomogeneities in the lateral boundary conditions of the experiment which are below the threshold of control.

We present a numerical study of steady convection in a two-dimensional mushy layer during the directional solidification of a binary mixture. The calculations reveal the internal structure of strongly nonlinear states featuring upflow which has been focused into solid-free regions known as chimneys. The mushy layer is modelled as a porous medium whose permeability is a function of the local solid fraction. The mushy layer is coupled to a chimney that is modelled as a narrow vertical channel where lubrication scalings are used to simplify the Navier–Stokes equations. We use these methods to exhibit solutions which give the detailed structure of the temperature, solute, flow and solid-fraction fields within the mushy layer. A key finding of the numerics is that there are two distinct chimney solutions at low Rayleigh numbers, presumably corresponding to stable and unstable portions of a subcritical solution branch. We also explore the relationship between convective solutions with and without chimneys.

Arnol'd developed two distinct yet closely related approaches to the linear stability of Euler flows. One is widely used for two-dimensional flows and involves constructing a conserved functional whose first variation vanishes and whose second variation determines the linear (and nonlinear) stability of the motion. The second method is a refinement of Kelvin's energy principle which states that stable steady Euler flows represent extremums in energy under a virtual displacement of the vorticity field. The conserved-functional (or energy-Casimir) method has been extended by several authors to more complex flows, such as planar MHD flow. In this paper we generalize the Kelvin–Arnol'd energy method to two-dimensional inviscid flows subject to a body force of the form −ϕ∇f. Here ϕ is a materially conserved quantity and f an arbitrary function of position and of ϕ. This encompasses a broad class of conservative flows, such as natural-convection planar and poloidal MHD flow with the magnetic field trapped in the plane of the motion, flows driven by electrostatic forces, swirling recirculating flow, self-gravitating flows and poloidal MHD flow subject to an azimuthal magnetic field. We show that stable steady motions represent extremums in energy under a virtual displacement of ϕ and of the vorticity field. That is, d1E=0 at equilibrium and whenever d2E is positive or negative definite the flow is (linearly) stable. We also show that unstable normal modes must have a spatial structure which satisfies d2E=0. This provides a single stability test for a broad class of flows, and we describe a simple universal procedure for implementing this test. In passing, a new test for linear stability is developed. That is, we demonstrate that stability is ensured (for flows of the type considered here) whenever the Lagrangian of the flow is a maximum under a virtual displacement of the particle trajectories, the displacement being of the type normally associated with Hamilton's principle. A simple universal procedure for applying this test is also given. We apply our general stability criteria to a range of flows and recover some familiar results. We also extend these ideas to flows which are subject to more than one type of body force. For example, a new stability criterion is obtained (without the use of Casimirs) for natural convection in the presence of a magnetic field. Nonlinear stability is also considered. Specifically, we develop a nonlinear stability criterion for planar MHD flows which are subject to isomagnetic perturbations. This differs from previous criteria in that we are able to extend the linear criterion into the nonlinear regime. We also show how to extend the Kelvin–Arnol'd method to finite-amplitude perturbations.

We determine solutions of the Euler equation representing isolated vortices (monopoles, dipoles) in an infinite domain, for arbitrary values of energy, circulation, angular momentum and impulse. A linear relationship between vorticity and stream function is assumed inside the vortex (while the flow is irrotational outside). The emergence of these solutions in a turbulent flow is justified by the statistical mechanics of continuous vorticity fields. The additional restriction of mixing to a ‘maximum-entropy bubble’, due to kinetic constraints, is assumed. The linear relationship between vorticity and stream function is obtained from the statistical theory in the limit of strong mixing (when constraints are weak). In this limit, maximizing entropy becomes equivalent to a kind of enstrophy minimization. New stability criteria are investigated and imply in particular that, in most cases, the vorticity must be continuous (or slightly discontinuous) at the vortex boundary. Then, the vortex radius is automatically determined by the integral constraints and we can obtain a classification of isolated vortices such as monopoles and dipoles (rotating or translating) in terms of a single control parameter. This article generalizes the classification obtained in a bounded domain by Chavanis & Sommeria (1996).

We study the generalized Bragg scattering of surface waves over a wavy bottom. We consider the problem in the general context of nonlinear wave–wave interactions, and write down and provide geometric constructions for the Bragg resonance conditions for second-order triad (class I) and third-order quartet (class II and class III) wave– bottom interactions. Class I resonance involving one bottom and two surface wave components is classical. Class II resonance manifests bottom nonlinearity (it involves two bottom and two surface wave components), and has been studied in the laboratory. Class III Bragg resonance is new and is a result of free-surface nonlinearity involving resonant interaction among one bottom and three surface wave components. The amplitude of the resonant wave is quadratic in the surface wave slope and linear in the bottom steepness, and, unlike the former two cases, the resonant wave may be either reflected or transmitted (relative to the incident waves) depending on the wave–bottom geometry. To predict the initial spatial/temporal growth of the Bragg resonant wave for these resonances, we also provide the regular perturbation solution up to third order. To confirm these predictions and to obtain an efficient computational tool for general wave–bottom problems with resonant interactions, we extend and develop a powerful high-order spectral method originally developed for nonlinear wave–wave and wave–body interactions. The efficacy of the method is illustrated in high-order Bragg resonance computations in two and three dimensions. These results compare well with existing experiments and perturbation theory for the known class I and class II Bragg resonance cases, and obtain and elucidate the new class III resonance. It is shown that under realistic conditions with moderate to small surface and bottom steepnesses, the amplitudes of third-order class II and class III Bragg resonant waves can be comparable in magnitude to those resulting from class I interactions and appreciable relative to the incident wave.

A numerical study of turbulent channel flow at various Reynolds numbers (Reτ=115, 210, 300) is conducted in order to examine the requirements for a reliable subgrid modelling in large-eddy simulations of wall-bounded flows. Using direct numerical simulation data, the interactions between large and small scales in the near-wall flow are analysed in detail which sheds light on the origin of the inverse cascade of turbulent kinetic energy observed in the buffer layer. It is shown that the correlation of the wall-normal subgrid stress and the wall-normal derivative of the streamwise grid-scale velocity plays the key role in the occurrence of the inverse cascade. A brief a priori test of several subgrid models shows that currently applied models are not capable of accounting properly for the complex interactions in the near-wall flow. A series of large-eddy simulations gives evidence that this deficiency may cause significant errors in important global quantities of the flow such as the mean wall shear stress. A study of the eddy-viscosity ansatz is conducted which reveals that the characteristic scales usually employed for the definition of the eddy viscosity are inappropriate in the vicinity of a wall. Therefore, a novel definition of the eddy viscosity is derived from the analysis of the near-wall energy budget. This new definition, which employs the wall-normal subgrid stress as a characteristic scale, is more consistent with the near-wall physics. No significant Reynolds-number effects are encountered in the present analysis which suggests that the findings may be generalized to flows at higher Reynolds numbers.

A detailed investigation of the effects of the Coriolis force on the three-dimensional linear instabilities of Stuart vortices is proposed. This exact inviscid solution describes an array of co-rotating vortices embedded in a shear flow. When the axis of rotation is perpendicular to the plane of the basic flow, the stability analysis consists of an eigenvalue problem for non-parallel versions of the coupled Orr–Sommerfeld and Squire equations, which is solved numerically by a spectral method. The Coriolis force acts on instabilities as a ‘tuner’, when compared to the non-rotating case. A weak anticyclonic rotation is destabilizing: three-dimensional Floquet modes are promoted, and at large spanwise wavenumber their behaviour is predicted by a ‘pressureless’ analysis. This latter analysis, which has been extensively discussed for simple flows in a recent paper (Leblanc & Cambon 1997) is shown to be relevant to the present study. The basic mechanism of short-wave breakdown is a competition between instabilities generated by the elliptical cores of the vortices and by the hyperbolic stagnation points in the braids, in accordance with predictions from the ‘geometrical optics’ stability theory. On the other hand, cyclonic or stronger anticyclonic rotation kills three-dimensional instabilities by a cut-off in the spanwise wavenumber. Under rapid rotation, the Stuart vortices are stabilized, whereas inertial waves propagate.