In this paper direct numerical evidence of flow-induced incipient cavitation is presented through lattice Boltzmann simulations of multiphase flows with a non-ideal thermodynamic equation of state. Cavitation emerges spontaneously as a result of the underlying non-ideal interactions, with no need for any modelling criteria based on the fluid variables, such as pressure or stress tensor. The onset of cavitation is well captured by Joseph’s minimum tension criteria, (Joseph, J. Fluid Mech., vol. 366, 1998, pp. 367–378; Dabiri, Sirignano & Joseph, Phys. Fluids, vol. 19, 2007, 072112), complemented with surface tension corrections, as proposed by Brennen (Cavitation and Bubble Dynamics, Oxford University Press, 1995). The simulations also show that the cavitation number (${C}_{N} $) proves to be a poor predictor of the onset of cavitation. Finally, strong dependence of the bubble morphology on the surface tension is also highlighted.

We consider rotating Rayleigh–Bénard convection of a fluid with a Prandtl number of $\mathit{Pr}=0.8$ in a cylindrical cell with an aspect ratio ${\it\Gamma}=1/2$. Direct numerical simulations (DNS) were performed for the Rayleigh number range $10^{5}\leqslant \mathit{Ra}\leqslant 10^{9}$ and the inverse Rossby number range $0\leqslant 1/\mathit{Ro}\leqslant 20$. We propose a method to capture regime transitions based on the decomposition of the velocity field into toroidal and poloidal parts. We identify four different regimes. First, a buoyancy-dominated regime occurring while the toroidal energy $e_{tor}$ is not affected by rotation and remains equal to that in the non-rotating case, $e_{tor}^{0}$. Second, a rotation-influenced regime, starting at rotation rates where $e_{tor}>e_{tor}^{0}$ and ending at a critical inverse Rossby number $1/\mathit{Ro}_{cr}$ that is determined by the balance of the toroidal and poloidal energy, $e_{tor}=e_{pol}$. Third, a rotation-dominated regime, where the toroidal energy $e_{tor}$ is larger than both $e_{pol}$ and $e_{tor}^{0}$. Fourth, a geostrophic regime for high rotation rates where the toroidal energy drops below the value for non-rotating convection.

This paper presents a comprehensive study of the near-bed hydrodynamics at non-moving streambeds based on laboratory experiments in open-channel flows. Pressure and velocity measurements were made with an array of up to 15 miniaturized piezo-resistive pressure sensors within the bed and slightly above it, and a two-dimensional particle-image-velocimetry (PIV) system measuring in streamwise vertical or horizontal planes. Three different types of bed materials were studied covering typical natural streambed conditions. The range of the global Reynolds number covered in the experiments was from 20000 to 200000. This study provides new insights into the flow structure over gravel beds based on the PIV measurements in both streamwise vertical and horizontal planes. In a streamwise vertical plane, large-scale wedge-like flow structures were observed where a zone of faster fluid over-rolled a zone with slower fluid. The resulting shear layer was inclined along the flow at an angle of 10°–25° to the bed, and was populated with clockwise rotating eddies. This mechanism occurred with sufficient frequency and shape to leave an ‘imprint’ in the velocity statistics. Typically, the described flow pattern is formed near the bed and is approximately scaled with the height of the logarithmic layer, although the biggest structures extended over the whole flow depth. In a horizontal near-bed plane, turbulent structures formed a patched ‘chessboard’ pattern with regions of lower and higher velocities that were elongated in the streamwise direction. Their lateral extension was typically two to four times the equivalent sand roughness with lengths up to several water depths. The dimensions of the regions were increasing linearly with the distance from the bed. These findings are consistent with conceptual models originally developed for smooth-wall flows. They also support observations made in rough-bed flume experiments, numerical simulations and natural rivers. Spatial fields of bed-pressure fluctuations were reconstructed by applying Taylor's frozen turbulence hypothesis on time data obtained with an array of pressure sensors. Based on the conditional sampling of velocity patterns associated with pressure-drop events a distinct bed-destabilizing flow-pressure pattern was identified. If a high-speed fluid in the wake of a large-scale wedge-like flow structure reaches the vicinity of the bed, a phenomenon akin to a Bernoulli effect leads to a distinctive low-pressure pattern. The resulting force may exceed the particles' submerged weight and is assumed to be able to give an initial lift to the particle. As a result, the exposed area of a particle is amplified and its angle of repose is reduced, increasing the probability for entrainment.

We present a combined analytical and numerical study of the instabilities of a pair of parallel unequal-strength vortices. We extend the analyses of a vortex in an external strain field (Crow, AIAA J. vol. 8, 1970, p. 2172; Widnall et al., J. Fluid Mech. vol. 66, 1974, p. 35) to include the orbital motion of the vortex pair. For counter-rotating pairs, the classic Crow-type periodic displacement perturbations are unstable for all vortex strength ratios, with fastest-growing wavelengths several times the vortex spacing. For co-rotating pairs, the orbital motion acts to suppress instability due to displacement perturbations. Instabilities in this case arise for elliptic perturbations at wavelengths that scale with the vortex core size. We also examine the influence of a second vortex pair by extending Crouch's (J. Fluid Mech. vol. 350, 1997, p. 311) analysis. Numerical results from a spectral initial-value code with subgrid-scale modelling agree with the growth rates from the theoretical models. Computations reveal the nonlinear evolution at late times, including wrapping and ring-rejection behaviour observed in experiments. A pair of co-rotating Gaussian vortices perturbed by noise develops elliptic instabilities, leading to the formation of vorticity bridges between the two vortices. The bridging is a prelude to vortex merger. Analytic, computational and experimental results agree well at circulation Reynolds numbers of order $10^5$.

The paper examines the stability of the uniform flow which approaches a two-dimensional stagnation region formed when a cylinder or a two-dimensional blunt body of finite curvature is immersed in a crossflow. It is shown that such a flow is unstable with respect to three-dimensional disturbances. This conclusion is reached on the basis of a mathematical analysis of a simplified form of the disturbance equation for the stream-wise component of the vorticity vector. The ultimate, or stable, flow pattern is governed by a singular Sturm–Liouville problem whose solution possesses a single eigenvalue. The resulting flow is one in which a regularly distributed system of counter-rotating vortices is super-imposed on the basic, Hiemenz-like pattern of streamlines. The spacing of the vortices is a unique function of the characteristics of the flow, and a theoretical estimate for it agrees well with experimental results. The analysis is extended heuristically to include the effect of free-stream turbulence on the spacing.

The problem is similar to the classical Görtler–Hämmerlin study of the stability of stagnation flow against an infinite flat plate, which revealed the existence of a spectrum of eigenvalues for the disturbance equation. The present analysis yields the same result when an infinite radius of curvature is assumed for the blunt body.

Large-eddy simulations of a generic turbulent flow with discrete effusion are reported. The computational domain is periodic in both streamwise and spanwise directions and contains both the injection and the suction sides. The blowing ratio is close to 1.2 while the Reynolds number in the aperture is of order 2600. The numerical results for this fully developed bi-periodic turbulent flow with effusion are compared to available experimental data from a large-scale spatially evolving isothermal configuration. It is shown that many features are shared by the two flow configurations. The main difference is related to the mean streamwise velocity profile, which is more flat for the bi-periodic situation where the cumulative effect of an infinite number of upstream jets is accounted for. The necessity of considering both sides of the plate is also established by analysing the vortical structure of the flow and some differences with the classical jet-in-crossflow case are highlighted. Finally, the numerical results are analysed in terms of wall modelling for full-coverage film cooling. For the operating point considered, it is demonstrated that the streamwise momentum flux is dominated by non-viscous effects, although the area where only the viscous shear stress contributes is very large given the small porosity value (4%).

A theoretical examination is made of the mechanical quasi-equilibrium stability of a horizontal, binary-mixture layer with Soret effect in the presence of a high-frequency vibrational field. The boundaries of the layer are assumed to be rigid, isothermal and impermeable. The axis of vibration is longitudinal. The study is based on the system of equations describing the behaviour of mean fields. The conditions of quasi-equilibrium are formulated. A linear stability analysis for normal modes is carried out. In the limit of long-wave disturbances the regular perturbation method is used with the wavenumber as a small parameter. For the case of an arbitrary wavenumber, the calculations are made using straight forward numerical integration. The boundaries of stability and the critical disturbance characteristics are determined for representative parameter values. Different instability mechanism and forms are discussed.

We derive a deterministic directional shoaling model and a stochastic directional shoaling model for a gravity surface wave field, valid for a beach with parallel depth contours accounting for refraction and nonlinear quadratic (three wave) interactions. A new phenomenon of non-resonant spectral evolution arises due to the medium inhomogeneity. The kernels of the kinetic equation depend on the bathymetry through an integral operator. Preliminary tests carried out on laboratory data for a unidirectional case indicate that the stochastic model also works rather well beyond the region where the waves may be regarded as nearly Gaussian. The limit of its applicability is decided by the dispersivity of the medium (relative to the nonlinearity). Good agreement with both laboratory data and the underlying deterministic model is found up to a value of about 1.5 for the spectral peak Ursell number. Beyond that only the deterministic model matches the measurements.

An experimental investigation of steep, high-frequency gravity waves (∼ 4 to 5 Hz) and the parasitic capillary waves they generate is reported. Spatial, as well as temporal, non-intrusive surface measurements are made using a new technique. This technique employs cylindrical lenses to magnify the vertical dimension in conjunction with an intensified, high-speed imaging system, facilitating the measurement of the disparate scales with a vertical surface-elevation resolution on the order of 10 μm. Thus, high-frequency parasitic capillary waves and the underlying gravity wave are measured simultaneously and accurately in space and time. Time series of spatial surface-elevation measurements are presented. It is shown that the location of the capillary waves is quasi-stationary in a coordinate system moving with the phase speed of the underlying gravity wave. Amplitudes and wavenumbers of the capillaries are modulated in space; however, they do not propagate with respect to the gravity wave. As capillary amplitudes are seen to decrease significantly and then increase again in a recurrence-like phenomenon, it is conjectured that resonance mechanisms are present. Measured surface profiles are compared to the theories of Longuet-Higgins (1963) and Crapper (1970) and the exact, two-dimensional numerical formulation of Schwartz & Vanden-Broeck (1979). Significant discrepancies are found between experimental and theoretical wavetrains in both amplitude and wavenumber. The theoretical predictions of the capillary wave amplitudes are much smaller than the measured amplitudes when the measured phase speed, amplitude, and wavelength of the gravity wave are used in the Longuet-Higgins model. In addition, this theory predicts larger wavenumbers of the capillaries as compared to experiments. The Crapper model predicts the correct order-of-magnitude capillary wave amplitude on the forward face of the gravity wave, but predicts larger amplitudes on the leeward face in comparison to the experiments. Also, it predicts larger capillary wavenumbers than are experimentally determined. Comparison of the measured profiles to multiple solutions of the stationary, symmetric, periodic solutions determined using the Schwartz & Vanden-Broeck numerical formulation show similar discrepancies. In particular, the assumed symmetry of the waveform about crest and trough in the numerical model precludes a positive comparison with the experiments, whose underlying waves exhibit significantly larger capillaries on their forward face than on their leeward face. Also, the a priori unknown multiplicity of numerical solutions for the same dimensionless surface tension and steepness parameters complicates comparison. Finally, using the temporal periodicity of the wave field, composite images of several successive wavelengths are constructed from which potential energy and surface energy are calculated as a function of distance downstream.

The results presented here are part of a long-term study in which we analyse the segregation of inertial particles in turbulent flows using the so called full Lagrangian method (FLM) to evaluate the ‘compressibility’ of the particle phase along a particle trajectory. In the present work, particles are advected by Stokes drag in a random flow field consisting of counter-rotating vortices and in a flow field composed of 200 random Fourier modes. Both flows are incompressible and, like turbulence, have structure and a distribution of scales with finite lifetime. The compressibility is obtained by first calculating the deformation tensor Jij associated with an infinitesimally small volume of particles following the trajectory of an individual particle. The fraction of the initial volume occupied by the particles centred around a position x at time t is denoted by |J|, where J ≡ det(Jij) and Jij ≡ ∂xi(x0, t)/∂x0,j, x0 denoting the initial position of the particle. The quantity d〈ln|J|〉/dt is shown to be equal to the particle averaged compressibility of the particle velocity field 〈∇ · v〉, which gives a measure of the rate-of-change of the total volume occupied by the particle phase as a continuum. In both flow fields the compressibility of the particle velocity field is shown to decrease continuously if the Stokes number St (the dimensionless particle relaxation time) is below a threshold value Stcr, indicating that the segregation of particles continues indefinitely. We show analytically and numerically that the long-time limit of 〈∇ · v〉 for sufficiently small values of St is proportional to St2 in the flow field composed of random Fourier modes, and to St in the flow field consisting of counter-rotating vortices. If St > Stcr, however, the particles are ‘mixed’. The level of mixing can be quantified by the degree of random uncorrelated motion (RUM) of particles which is a measure of the decorrelation of the velocities of two nearby particles. RUM is zero for fluid particles and increases rapidly with the Stokes number if St > Stcr, approaching unity for St ≫ 1. The spatial averages of the higher-order moments of the particle number density are shown to diverge with time indicating that the spatial distribution of particles may be very intermittent, being associated with non-zero values of RUM and the occurrence of singularities in the particle velocity field. Our results are consistent with previous observations of the radial distribution function in Chun et al. (J. Fluid Mech., vol. 536, 2005, p. 219).

In the thermal convection problem with free boundaries, the interaction of two ‘roll’ disturbances is considered. The problem is reduced to a pair of non-linear ordinary differential equations, which should also provide a model for the interaction of two disturbances in more general situations than that for which these equations have been derived. The equations contain several parameters which necessitates a discussion of various possible types of solution. Some representative results are: (1) under certain circumstances, an equilibrium state may be composed of a mixture of a linearly stable disturbance and a linearly unstable disturbance; (2) for the thermal convection problem, when the Rayleigh number is slightly above the minimum critical value, the equilibrium state will contain only one of two linearly unstable disturbances. These and other results are compared with experimental observations.

The behaviour of dipolar vortices in a rotating fluid with a sloping bottom (simulating the variation of the Coriolis parameter on the Earth, with the direction of steepest bottom slope corresponding with the northern direction) has been investigated in the laboratory. Dipoles were generated by moving a vertical cylinder through the fluid. Dye photographs provided qualitative information, whereas quantitative information about the evolving flow field was obtained by streak photography. Dipoles initially directed under a certain angle relative to the west–east axis showed meandering or cycloid-like trajectories. Soeme symmetries between east-travelling dipoles (ETD's) and west-travelling dipoles (WTD's) were observed. ETD's are stable in the trajectory sense: a small deviation from zonal motion results in small oscillations around the equilibrium latitude. WTD's are unstable: small initial deviations produce large displacements in northern or southern directions. This asymmetry arises because the vorticity of a dipole moving westward is anticorrelated with the ambient vorticity, while the vorticities are correlated when the dipole moves eastward. ETD's increase in size and eventually split into two independent monopoles, the rate of growth depending on the gradient of planetary vorticity. WTD's are initially more compact but owing to the large displacements in the meridional direction strong asymmetries in the circulation of the two halves are produced, resulting in a large deformation of the weaker part. The experimental observations show good qualitative agreement with analytical and numerical results obtained using a modulated point-vortex model.

Flow and turbulence in a 50 Hz rotating-field MHD system are investigated using the hot-film constant-temperature anemometer. Factors affecting anemometer disturbances and response time are discussed. From measurements of the magnetic field at points within the liquid the distribution of MHD forces is estimated. The mean rotational velocity of the flow is of the expected order of magnitude but much less dependent on the axial co-ordinate than the corresponding MHD force. With the aid of a thermal transit-time anemometer, a weak secondary flow is detected. A note on scale-model studies of MHD systems envisaged in metallurgical applications of magnetohydrodynamics points out some basic difficulties in modelling large high-powered systems on a small scale.

Bubble oscillations play a crucial role in ultrasonic cleaning, a process by which micro- and nanoscale contaminant particles are removed from solid surfaces, such as semiconductor wafers, photomasks and membranes. Although it is well known that the ultrasonic cleaning may damage the functional patterns of ever-shrinking size in current manufacturing technology while removing dust and debris, the mechanisms leading to such damage have been elusive. Here we report observations of the dynamics of bubbles that yield microstructure damage under a continuous ultrasonic field via high-speed imaging. We find that the bubble behaviour can be classified into four types, namely volume oscillation, shape oscillation, splitting and chaotic oscillation, depending on the acoustic pressure and bubble size. This allows us to construct a regime map that can predict the bubble behaviour near a wall based on the experimental parameters. Our visualization experiments reveal that damage of microwalls and microcantilevers arises due to either splitting small bubbles or chaotically oscillating large bubbles in the ultrasonic field, with the forces generated by them quantitatively measured.

The uniform flow past a sphere undergoing steady rotation about an axis transverse to the free stream flow was investigated numerically. The objective was to reveal the effect of sphere rotation on the characteristics of the vortical wake structure and on the forces exerted on the sphere. This was achieved by solving the time-dependent, incompressible Navier–Stokes equations, using an accurate Fourier–Chebyshev spectral collocation method. Reynolds numbers Re of 100, 250 and 300 were considered, which for a stationary sphere cover the axisymmetric steady, non-axisymmetric steady and vortex shedding regimes. The study identified wake transitions that occur over the range of non-dimensional rotational speeds Ω* = 0 to 1.00, where Ω* is the maximum velocity on the sphere surface normalized by the free stream velocity. At Re = 100, sphere rotation triggers a transition to a steady double-threaded structure. At Re = 250, the wake undergoes a transition to vortex shedding for Ω* ≥ 0.08. With an increasing rotation rate, the recirculating region is progressively reduced until a further transition to a steady double-threaded wake structure for Ω* ≥ 0.30. At Re = 300, wake shedding is suppressed for Ω* ≥ 0.50 via the same mechanism found at Re = 250. For Ω* ≥ 0.80, the wake undergoes a further transition to vortex shedding, through what appears to be a shear layer instability of the Kelvin–Helmholtz type.

We consider a classic boundary-value problem for deep-water gravity-capillary waves in a shear flow, composed of the Rayleigh equation and the standard linearized kinematic and dynamic inviscid boundary conditions at the free surface. We derived the exact solution for this problem in terms of an infinite series in powers of a certain parameter e, which characterizes the smallness of the deviation of the wave motion from the potential motion. For the existence and absolute convergence of the solution it is sufficient that e be less than unity.

The truncated sums of the series provide approximate solutions with a priori prescribed accuracy. In particular, for the short-wave instability, which can be interpreted as the Miles critical-layer-type instability, the explicit approximate expressions for the growth rates are derived. The growth rates in a certain (very narrow) range of scales can exceed the Miles increments caused by the wind.

The effect of thin boundary layers on the dispersion relation was also investigated using an asymptotic procedure based on the smallness of the product of the layer thickness and wavenumber. The criterion specifying when and with what accuracy the boundary-layer influence can be neglected has been derived.

A canonical study is developed to investigate the unsteady interactions of a streamwise-oriented vortex impinging upon a finite surface using high-fidelity simulation. As a model problem, an analytically defined vortex superimposed on a free stream is convected towards an aspect-ratio-six ($\mathit{AR}=6$) plate oriented at an angle of ${\it\alpha}=4^{\circ }$ and Reynolds number of $\mathit{Re}=20\,000$ in order to characterize the unsteady modes of interaction resulting from different spanwise positions of the incoming vortex. Outboard, tip-aligned and inboard positioning are shown to produce three distinct flow regimes: when the vortex is positioned outboard of, but in close proximity to, the wingtip, it pairs with the tip vortex to form a dipole that propels itself away from the plate through mutual induction, and also leads to an enhancement of the tip vortex. When the incoming vortex is aligned with the wingtip, the tip vortex is initially strengthened by the proximity of the incident vortex, but both structures attenuate into the wake as instabilities arise in the pair’s feeding sheets from the entrainment of opposite-signed vorticity into either structure. Finally, when the incident vortex is positioned inboard of the wingtip, the vortex bifurcates in the time-mean sense with portions convecting above and below the wing, and the tip vortex is mostly suppressed. The time-mean bifurcation is actually a result of an unsteady spiralling instability in the vortex core that reorients the vortex as it impacts the leading edge, pinches off, and alternately attaches to either side of the wing. The increased effective angle of attack inboard of impingement enhances the three-dimensional recirculation region created by the separated boundary layer off the leading edge which draws fluid from the incident vortex inboard and diminishes its impact on the outboard section of the wing. The slight but remaining downwash present outboard of impingement reduces the effective angle of attack in that region, resulting in a small separation bubble on either side of the wing in the time-mean solution, effectively unloading the tip outboard of impingement and suppressing the tip vortex. All incident vortex positions provide substantial increases in the wing’s lift-to-drag ratio; however, significant sustained rolling moments also result. As the vortex is brought inboard, the rolling moment diminishes and eventually switches sign as the reduced outboard loading balances the augmented sectional lift inboard of impingement.

The influence of tilt on flow reversals in two-dimensional thermal convection in rectangular cells with two typical aspect ratios, $\unicode[STIX]{x1D6E4}=\text{width/height}=1$ and 2, are investigated by means of direct numerical simulations. For $\unicode[STIX]{x1D6E4}=1$, tilt tends to suppress flow reversals. However, it is found that flow reversals characterized by two main rolls are promoted by tilt for $\unicode[STIX]{x1D6E4}=2$, which are even observed for some cases of small Prandtl numbers ($Pr$) and large tilt angles ($\unicode[STIX]{x1D6FD}$). Different from level cases where the four corner rolls all have opportunities to grow and trigger a flow reversal, the reversals in an anticlockwise tilted cell with $\unicode[STIX]{x1D6E4}=2$ are always led by the growth of the bottom-right or the top-left corner roll. Tilt is favourable for the growth of the bottom-right or the top-left corner roll and thus for breaking the balance between the two main rolls and triggering a flow reversal. The mode decomposition analysis shows that the appearance of the intermediate single-roll mode is crucial for reversals, and flow reversals in a tilted cell with $\unicode[STIX]{x1D6E4}=2$ can be viewed as a mode competition process between single-roll mode and horizontally adjacent double-roll mode. They can only occur in a limited range of $\unicode[STIX]{x1D6FD}$ where the two modes have comparable strength. Furthermore, the Nusselt numbers at the hot plate $Nu_{h}$ and at the cold plate $Nu_{c}$ behave differently during a flow reversal for $\unicode[STIX]{x1D6E4}=2$ due to the preference of single corner roll growth.

The linear and nonlinear spatio-temporal stability of an interface separating two Newtonian fluids in pressure-driven channel flow at moderate Reynolds numbers is analysed both theoretically and numerically. A linear, Orr–Sommerfeld-type analysis shows that most of such systems are unstable. The transition to an absolutely unstable regime is investigated, and is shown to occur in an intermediate range of Reynolds numbers and ratios of the thicknesses of the two layers, for near-density matched fluids with a viscosity contrast. A critical Reynolds number is found for transition from convective to absolute instability of relatively thin films. Results obtained from direct numerical simulations (DNSs) of the Navier–Stokes equations for long channels using a diffuse-interface method elucidate that waves generated by random noise at the inlet show that, near the inlet, waves are formed and amplified strongly, leading to ligament formation. Successive waves coalesce with each other further downstream, resulting in longer larger-amplitude waves further downstream. In the linearly absolute regime, the characteristics of the spatially growing wave near the inlet agree with that of the saddle point as predicted by the linear theory. The transition point from a convective to an absolute regime predicted by linear theory is also in agreement with a sharp change in the value of a healing length obtained from the DNSs.

The problem of a periodic planar arrangement of a large number of co-flowing, interacting jets is investigated. It is shown that this interaction gives rise to strong nearfield oscillations of large-scale spatial coherence and to far-field inhomogeneities. In the experiments performed, the jets were produced behind a flat plate perforated by holes arranged in a square or triangular periodic pattern and placed perpendicular to a uniform flow. At moderate Reynolds numbers, the interaction results in a remarkable low-frequency oscillation of the merging distance of the jets downstream of the plate. A detailed description of the recirculating flow in the cavities between the jets emphasizes the role of the backflow in the cavities on the oscillatory behaviour. This description is supported by measurements of the local fluctuating velocity and pressure, two-point correlation measurements and quantitative flow visualizations. These experimental observations suggest a new formulation for the instability dynamics of such unstable recirculating flows. This formulation, based on the nonlinear delayed saturation of the jet's shear layer instability (NLDS model) predicts successfully the dependence of the oscillation of the merging distance on the jet Reynolds number and on the local geometrical features of the confinement of the jets. Furthermore, it is shown that the diffusion of mass coming from one jet, seeded with an inert dye, gives rise to an exponential diffusion front over a distance corresponding to a few mesh sizes indicating a strong local coupling of the jets. At the scale of the whole jet assembly, the oscillations are organized as large-scale travelling waves, propagating from the boundaries of the domain to its centre. This symmetry-breaking property is discussed and supplemented by a spatio-temporal simulation of an array of coupled oscillators.