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Many experimental studies have demonstrated that ducted premixed flames exhibit stable limit cycles in some regions of parameter space. Recent experiments have also shown that these (period-1) limit cycles subsequently bifurcate to period- $2^{n}$ , quasiperiodic, multiperiodic or chaotic behaviour. These secondary bifurcations cannot be found computationally using most existing frequency domain methods, because these methods assume that the velocity and pressure signals are harmonic. In an earlier study we have shown that matrix-free continuation methods can efficiently calculate the limit cycles of large thermoacoustic systems. This paper demonstrates that these continuation methods can also efficiently calculate the bifurcations from the limit cycles. Furthermore, once these bifurcations are found, it is then possible to isolate the coupled flame–acoustic motion that causes the qualitative change in behaviour. This information is vital for techniques that use selective damping to move bifurcations to more favourable locations in the parameter space. The matrix-free methods are demonstrated on a model of a ducted axisymmetric premixed flame, using a kinematic $G$ -equation solver. The methods find limit cycles and period-2 limit cycles, and fold, period-doubling and Neimark–Sacker bifurcations as a function of the location of the flame in the duct, and the aspect ratio of the steady flame.
This paper exploits the turbulent flow control method using streamwise travelling waves (Quadrio et al. J. Fluid Mech., vol. 627, 2009, pp. 161–178) to study the effect of Reynolds number on turbulent skin-friction drag reduction. Direct numerical simulations (DNS) of a turbulent channel flow subjected to the streamwise travelling waves of spanwise wall velocity have been performed at Reynolds numbers ranging from $\mathit{Re}_{{\it\tau}}=200$ to 1600. To the best of the authors’ knowledge, this is the highest Reynolds number attempted with DNS for this type of flow control. The present DNS results confirm that the effectiveness of drag reduction deteriorates, and the maximum drag reduction achieved by travelling waves decreases significantly as the Reynolds number increases. The intensity of both the drag reduction and drag increase is reduced with the Reynolds number. Another important finding is that the value of the optimal control parameters changes, even in wall units, when the Reynolds number is increased. This trend is observed for the wall oscillation, stationary wave, and streamwise travelling wave cases. This implies that, when the control parameters used are close to optimal values found at a lower Reynolds number, the drag reduction deteriorates rapidly with increased Reynolds number. In this study, the effect of Reynolds number for the travelling wave is quantified using a scaling in the form $\mathit{Re}_{{\it\tau}}^{-{\it\alpha}}$ . No universal constant is found for the scaling parameter ${\it\alpha}$ . Instead, the scaling parameter ${\it\alpha}$ has a wide range of values depending on the flow control conditions. Further Reynolds number scaling issues are discussed. Turbulent statistics are analysed to explain a weaker drag reduction observed at high Reynolds numbers. The changes in the Stokes layer and also the mean and root-mean-squared (r.m.s.) velocity with the Reynolds number are also reported. The Reynolds shear stress analysis suggests an interesting possibility of a finite drag reduction at very high Reynolds numbers.
The dynamics of viscous fluid flow over a circular flexible plate are studied numerically by an immersed boundary–lattice Boltzmann method for the fluid flow and a finite-element method for the plate motion. When the plate is clamped at its centre and placed in a uniform flow, it deforms by the flow-induced forces exerted on its surface. A series of distinct deformation modes of the plate are found in terms of the azimuthal fold number from axial symmetry to multifold deformation patterns. The developing process of deformation modes is analysed and both steady and unsteady states of the fluid–structure system are identified. The drag reduction due to the plate deformation and the elastic potential energy of the flexible plate are investigated. Theoretical analysis is performed to elucidate the deformation characteristics. The results obtained in this study provide physical insight into the understanding of the mechanisms on the dynamics of the fluid–structure system.
A vertically vibrating liquid layer produces liquid ligaments that disintegrate to form a spray with drops of a controllable size. Previous experimental investigations of ultrasonic atomisation have shown that when such a spray forms, there exists a predominant surface-wave mode from which drops are generated with a mean diameter that follows Lang’s equation. In this paper, we determined this predominant surface-wave mode physically and, by utilising the coupled level-set and volume-of-fluid method, we numerically studied the threshold condition for spray formation based on a cell model of the predominant surface wavelength that excludes the effects of the container walls. We defined a condition whereby the broken drop holds a zero area-averaged vertical velocity in the laboratory reference frame as the criterion for the formation of a spray. The results of our calculations indicated that the onset of a spray occurs in the subharmonic unstable region for a threshold dimensionless forcing strength ${\it\beta}_{c}=({\it\rho}_{l}{\it\Delta}_{0}^{3}{\it\Omega}^{2})/{\it\sigma}\sim O(1)$ , where ${\it\rho}_{l}$ and ${\it\sigma}$ denote the liquid density and surface tension coefficient, respectively, ${\it\Delta}_{0}$ is the forcing displacement amplitude and ${\it\Omega}$ is the forcing angular frequency. Spray formation due to the Faraday instability can be considered as a process whereby the liquid layer absorbs energy from the inertial force, and releases it by producing drops that leave the surface of the liquid layer. We demonstrated that for a deep liquid layer, the threshold condition for the formation of a spray is determined only by the forcing strength, and is independent of the initial conditions of the liquid surface.
We have established a parallel, adaptive interface-tracking framework in order to conduct, based on the framework, direct simulation of binary head-on droplet collision in the high-Weber-number regime (from 200 to 1500) that exhibits complex topological changes and substantial length scale variations. The overall algorithms include a combined Eulerian and Lagrangian solver to track moving interfaces, conservative Lagrangian mesh modification and reconstruction, cell-based unstructured adaptive mesh refinement (AMR) in the Eulerian solver, and associated Eulerian and Lagrangian domain partitions to minimize communication overhead. Based on the combined computational and experimental efforts, we have resolved for the first time the free-surface instabilities of the colliding droplets at such high Weber number. We detail the characteristics of coalescence, stretch, end pinching, fingering, free-surface movement and drop breakup. The Taylor–Culick rim is present soon after the collision. Furthermore, we observe two types of longitudinal instabilities on the rim, namely, the Rayleigh–Taylor (RT)-type instability in the initial deceleration phase of the circular sheet right after droplet coalescence, and later the Rayleigh–Plateau (RP) instabilities. As the Taylor–Culick rim disintegrates in the retraction phase, fingering effect is profound and resulting in wider droplet size distribution.
In this paper, the viscous effects of shock reflection hysteresis in steady supersonic flow is studied analytically, experimentally and numerically, taking the boundary layer developed over the reflecting surface into consideration. Based on a hypothesis that the interaction origin keep invariable during the shock–boundary layer interaction, the separation region is replaced by a virtual wedge with a fixed angle. Combined the free-interaction theory with the shock reflection hysteresis theory, a detailed analysis describing the viscous flow structures of shock reflection configurations is proposed. It is illustrated by mean of further analysis that a shock reflection hysteresis which is similar to the one that exists in the reflection of symmetric shock waves is found theoretically. Experimental results verify the analytical interaction model as well as the existence of two shock reflection patterns, although the hysteresis is not conformed by experiments. This paper also presents results of simulations for the hysteresis process which results from keeping the wedge angle constant and changing the free-stream Mach number, for confirming the hysteresis phenomenon.
This paper describes experimentally, numerically and theoretically how the three-dimensional instabilities of a cylinder wake are modified by the presence of a linear density stratification. The first part is focused on the case of a cylinder with a small tilt angle between the cylinder’s axis and the vertical. The classical mode A well-known for a homogeneous fluid is still present. It is more unstable for moderate stratifications but it is stabilized by a strong stratification. The second part treats the case of a moderate tilt angle. For moderate stratifications, a new unstable mode appears, mode S, characterized by undulated layers of strong density gradients and axial flow. These structures correspond to Kelvin–Helmholtz billows created by the strong shear present in the critical layer of each tilted von Kármán vortex. The last two parts deal with the case of a strongly tilted cylinder. For a weak stratification, an instability (mode RT) appears far from the cylinder, due to the overturning of the isopycnals by the von Kármán vortices. For a strong stratification, a short wavelength unstable mode (mode L) appears, even in the absence of von Kármán vortices. It is probably due to the strong shear created by the lee waves upstream of a secondary recirculation bubble. A map of the four different unstable modes is established in terms of the three parameters of the study: the Reynolds number, the Froude number (characterizing the stratification) and the tilt angle.
Closed-loop control of an amplifier flow is experimentally investigated. A feed-forward algorithm is implemented to control the flow downstream of a backward-facing step (BFS) perturbed by upstream perturbations. Upstream and downstream data are extracted from real-time velocity fields to compute an ARMAX model used to effect actuation. This work, done at Reynolds number 430, investigates the practical feasibility of this approach which has shown great promise in a recent numerical study by Hervé et al. (J. Fluid Mech., vol. 702, 2012, pp. 26–58). The linear nature of the regime is checked, two-dimensional upstream perturbations are introduced, and the degree to which the flow can be controlled is quantified. The resulting actuation is able to effectively reduce downstream energy levels and fluctuations. The limitations and difficulties of applying such an approach to an experiment are also discussed.
We investigate in detail the problem of confined pressure-driven laminar flow of neutrally buoyant non-Brownian suspensions using a frictional rheology based on the recent proposal of Boyer et al. (Phys. Rev. Lett., vol. 107 (18), 2011, 188301). The friction coefficient (shear stress over particle normal stress) and solid volume fraction are taken as functions of the dimensionless viscous number $I$ defined as the ratio between the fluid shear stress and the particle normal stress. We clarify the contributions of the contact and hydrodynamic interactions on the evolution of the friction coefficient between the dilute and dense regimes reducing the phenomenological constitutive description to three physical parameters. We also propose an extension of this constitutive framework from the flowing regime (bounded by the maximum flowing solid volume fraction) to the fully jammed state (the random close packing limit). We obtain an analytical solution of the fully developed flow in channel and pipe for the frictional suspension rheology. The result can be transposed to dry granular flow upon appropriate redefinition of the dimensionless number $I$ . The predictions are in excellent agreement with available experimental results for neutrally buoyant suspensions, when using the values of the constitutive parameters obtained independently from stress-controlled rheological measurements. In particular, the frictional rheology correctly predicts the transition from Poiseuille to plug flow and the associated particles migration with the increase of the entrance solid volume fraction. We also numerically solve for the axial development of the flow from the inlet of the channel/pipe toward the fully developed state. The available experimental data are in good agreement with our numerical predictions, when using an accepted phenomenological description of the relative phase slip obtained independently from batch-settlement experiments. The solution of the axial development of the flow notably provides a quantitative estimation of the entrance length effect in a pipe for suspensions when the continuum assumption is valid. Practically, the latter requires that the predicted width of the central (jammed) plug is wider than one particle diameter. A simple analytical expression for development length, inversely proportional to the gap-averaged diffusivity of a frictional suspension, is shown to encapsulate the numerical solution in the entire range of flow conditions from dilute to dense.
While studying the problem of predicting freak waves it was realized that it would be advantageous to introduce a simple measure for such extreme events. Although it is customary to characterize extremes in terms of wave height or its maximum it is argued in this paper that wave height is an ill-defined quantity in contrast to, for example, the envelope of a wave train. Also, the last measure has physical relevance, because the square of the envelope is the potential energy of the wave train. The well-known representation of a narrow-band wave train is given in terms of a slowly varying envelope function ${\it\rho}$ and a slowly varying frequency ${\it\omega}=-\partial {\it\phi}/\partial t$ where ${\it\phi}$ is the phase of the wave train. The key point is now that the notion of a local frequency and envelope is generalized by also applying the same definitions for a wave train with a broad-banded spectrum. It turns out that this reduction of a complicated signal to only two parameters, namely envelope and frequency, still provides useful information on how to characterize extreme events in a time series. As an example, for a linear wave train the joint probability distribution of envelope height and period is obtained and is validated against results from a Monte Carlo simulation. The extension to the nonlinear regime is, as will be seen, fairly straightforward.
Multiple mechanisms for the regeneration of hairpin-like coherent flow structures in transitional and turbulent boundary layers have been proposed in the published literature, but a complete understanding of the typical topologies of coherent structures observed in the literature has not yet been achieved. To contribute to this understanding, a numerical study is performed of a turbulent spot triggered in a zero-pressure-gradient laminar boundary layer by a pulsed, transverse jet. Two direct numerical simulations (DNS) capture the growth of the spot into a mature turbulent region containing a large number of coherent vortical flow structures. The boundary-layer Reynolds number based on the test-surface streamwise length is $\mathit{Re}_{L}=309\,200$ . The internal structure of the spot is characterized by densely spaced packets of hairpin vortices. Lateral growth of the spot occurs as new hairpin vortices form along the spanwise edges of the spot. The formation of these hairpin vortices is attributed to unstable shear layers that develop in the streamwise–spanwise plane due to the wall-normal motions induced by the streamwise oriented legs of hairpin vortices within the spot. Results are presented that highlight the mechanism by which the instability of such shear layers forms wavepackets of hairpin vortices; how the formation of these vortices produces a flow environment that promotes the creation of new hairpin vortices; and how the newly created hairpin vortices impact the production of turbulence kinetic energy in the flow region surrounding the spot. A quantitative description of the hairpin-vortex regeneration mechanism based on the transport of the instantaneous vorticity vector is presented to illustrate how the velocity and vorticity fields interact with the local strain rates to promote the growth of coherent vortical structures. The simulation results also shed light on a mechanism that seems to have a dominant influence on the formation of the calmed region in the wake of the turbulent spot.
Nonlinear interactions between the two wakes behind a pair of square cylinders, which are placed side by side in a uniform flow, are investigated by the linear and weakly nonlinear stability analyses and numerical simulations. It is known from the linear stability analysis that the flow past a pair of cylinders becomes unstable to a symmetric or an antisymmetric mode of disturbance, depending on the gap ratio, the ratio of the gap distance between the two cylinders to the cylinder diameter. The antisymmetric mode gives the critical condition for smaller gap ratios than a threshold value, and for larger gap ratios the symmetric mode becomes the most unstable. We focus on the flow pattern arising through the nonlinear interactions of the two modes of disturbance for gap ratios around the threshold value when both modes are growing. We derive a couple of amplitude equations for the two modes to properly describe the nonlinear interaction between them by applying the weakly nonlinear stability theory. The amplitude equations are shown to have three equilibrium solutions except the null solution such as a mixed-mode solution, symmetric and antisymmetric single-mode solutions. Examination of the stability of each equilibrium solution leads to a conclusion that the mixed-mode solution exchange its stability with both the symmetric and the antisymmetric single-mode solutions simultaneously. In the case where the mixed-mode solution is stable, both the symmetric and antisymmetric modes have finite amplitudes, and the resultant flow has an asymmetric flow pattern comprising of finite amplitudes of the two modes of disturbance superposed on the steady symmetric flow. While in the case where both the single-mode solutions are stable, either of the symmetric- and antisymmetric-mode solutions survives, overwhelming the other. Then, if the symmetric mode attains at an equilibrium finite amplitude and the antisymmetric mode vanishes, the resultant flow is symmetric, and if the antisymmetric mode survives and the symmetric mode decays out, the flow becomes asymmetric with the antisymmetric mode of disturbance superposed on the steady symmetric flow. Thus, the flow appearing due to instability differs depending on the initial condition, not uniquely determined, when both single-mode solutions are stable. We numerically delineated the region in the parameter space of the gap ratio and the Reynolds number where the mixed-mode solution is stable. The theoretical results obtained from the weakly nonlinear stability analyses are confirmed by numerical simulations. The conclusion derived from the stability analysis of the equilibrium solutions of the amplitude equations is widely applicable also to other double Hopf bifurcation problems.
The incompressible, inviscid and axisymmetric dynamics of perturbations on a solid-body rotation flow with a uniform axial velocity in a rotating, finite-length, straight, circular pipe are studied via global analysis techniques and numerical simulations. The investigation establishes the coexistence of both axisymmetric wall-separation and vortex-breakdown zones above a critical swirl level, ${\it\omega}_{1}$ . We first describe the bifurcation diagram of steady-state solutions of the flow problem as a function of the swirl ratio ${\it\omega}$ . We prove that the base columnar flow is a unique steady-state solution when ${\it\omega}$ is below ${\it\omega}_{1}$ . This state is asymptotically stable and a global attractor of the flow dynamics. However, when ${\it\omega}>{\it\omega}_{1}$ , we reveal, in addition to the base columnar flow, the coexistence of states that describe swirling flows around either centreline stagnant breakdown zones or wall quasi-stagnant zones, where both the axial and radial velocities vanish. We demonstrate that when ${\it\omega}>{\it\omega}_{1}$ , the base columnar flow is a min–max point of an energy functional that governs the problem, while the swirling flows around the quasi-stagnant and stagnant zones are global and local minimizer states and become attractors of the flow dynamics. We also find additional min–max states that are transient attractors of the flow dynamics. Numerical simulations describe the evolution of perturbations on above-critical columnar states to either the breakdown or the wall-separation states. The growth of perturbations in both cases is composed of a linear stage of the evolution, with growth rates accurately predicted by the analysis of Wang & Rusak (Phys. Fluids, vol. 8, 1996a, pp. 1007–1016), followed by a stage of saturation to either one of the separation zone states. The wall-separation states have the same chance of appearing as that of vortex-breakdown states and there is no hysteresis loop between them. This is strikingly different from the dynamics of vortices with medium or narrow vortical core size in a pipe.
We report on direct numerical simulations of two-dimensional, horizontally periodic Rayleigh–Bénard convection between free-slip boundaries. We focus on the ability of the convection to drive large-scale horizontal flow that is vertically sheared. For the Prandtl numbers ( $\mathit{Pr}$ ) between 1 and 10 simulated here, this large-scale shear can be induced by raising the Rayleigh number ( $\mathit{Ra}$ ) sufficiently, and we explore the resulting convection for $\mathit{Ra}$ up to $10^{10}$ . When present in our simulations, the sheared mean flow accounts for a large fraction of the total kinetic energy, and this fraction tends towards unity as $\mathit{Ra}\rightarrow \infty$ . The shear helps disperse convective structures, and it reduces vertical heat flux; in parameter regimes where one state with large-scale shear and one without are both stable, the Nusselt number of the state with shear is smaller and grows more slowly with $\mathit{Ra}$ . When the large-scale shear is present with $\mathit{Pr}\lesssim 2$ , the convection undergoes strong global oscillations on long timescales, and heat transport occurs in bursts. Nusselt numbers, time-averaged over these bursts, vary non-monotonically with $\mathit{Ra}$ for $\mathit{Pr}=1$ . When the shear is present with $\mathit{Pr}\gtrsim 3$ , the flow does not burst, and convective heat transport is sustained at all times. Nusselt numbers then grow roughly as powers of $\mathit{Ra}$ , but the growth rates are slower than any previously reported for Rayleigh–Bénard convection without large-scale shear. We find that the Nusselt numbers grow proportionally to $\mathit{Ra}^{0.077}$ when $\mathit{Pr}=3$ and to $\mathit{Ra}^{0.19}$ when $\mathit{Pr}=10$ . Analogies with tokamak plasmas are described.
We present an experimental study of ‘polygons’ forming on the free surface of a swirling water flow in a partially filled cylindrical container. In our set-up, we rotate the bottom plate and the cylinder wall with separate motors. We thereby vary rotation rate and shear strength independently and move from a rigidly rotating ‘Newton’s bucket’ flow to one where bottom and cylinder wall are rotating oppositely and the surface is strongly turbulent but flat on average. Between those two extremes, we find polygonal states for which the rotational symmetry is spontaneously broken. We investigate the phase diagram spanned by the two rotational frequencies at a given water filling height and find polygons in a regime, where the two frequencies are sufficiently different and, predominantly, when they have opposite signs. In addition to the extension of the family of polygons found with the stationary cylinder, we find a new family of smaller polygons for larger rotation rates of the cylinder, opposite to that of the bottom plate. Further, we find a ‘monogon’, a figure with one corner, roughly an eccentric circle rotating in the same sense as the cylinder. The case where only the bottom plate is rotating is compared with the results of Jansson et al. (Phys. Rev. Lett., vol. 96, 2006, art. 174502), where the same size of cylinder was used, and although the overall structure of the phase diagram spanned by water height and rotational frequency is the same, many details are different. To test the effect of small experimental defects, such as misalignment of the bottom plate, we investigate whether the rotating polygons are phase locked with the bottom plate, and although we find cases where the frequency ratio of figure and bottom plate is nearly rational, we do not find phase locking. Finally, we show that the system has a surprising multistability and excitability, and we note that this can cause quantitative differences between the phase diagrams obtained in comparable experiments.
We present a data-assimilation technique based on a variational formulation and a Lagrange multipliers approach to enforce the Navier–Stokes equations. A general operator (referred to as the measure operator) is defined in order to mathematically describe an experimental measure. The presented method is applied to the case of mean flow measurements. Such a flow can be described by the Reynolds-averaged Navier–Stokes (RANS) equations, which can be formulated as the classical Navier–Stokes equations driven by a forcing term involving the Reynolds stresses. The stress term is an unknown of the equations and is thus chosen as the control parameter in our study. The data-assimilation algorithm is derived to minimize the error between a mean flow measurement and the measure performed on a numerical solution of the steady, forced Navier–Stokes equations; the optimal forcing is found when this error is minimal. We demonstrate the developed data-assimilation framework on a test case: the two-dimensional flow around an infinite cylinder at a Reynolds number of $\mathit{Re}=150$ . The mean flow is computed by time-averaging instantaneous flow fields from a direct numerical simulation (DNS). We then perform several ‘measures’ on this mean flow and apply the data-assimilation method to reconstruct the full mean flow field. Spatial interpolation, extrapolation, state vector reconstruction and noise filtering are considered independently. The efficacy of the developed identification algorithm is quantified for each of these cases and compared with more traditional methods when possible. We also analyse the identified forcing in terms of unsteadiness characterization, present a way to recover the second-order statistical moments of the fluctuating velocities and finally explore the possibility of pressure reconstruction from velocity measurements.
A novel approach to the study of the kinematics and dynamics of turbulent flows is presented. The method involves tracking in time coherent structures, and provides all of the information required to characterize eddies from birth to death. Spatially and temporally well-resolved DNSs of channel data at $\mathit{Re}_{{\it\tau}}=930{-}4200$ are used to analyse the evolution of three-dimensional sweeps, ejections (Lozano-Durán et al., J. Fluid Mech., vol. 694, 2012, pp. 100–130) and clusters of vortices (del Álamo et al., J. Fluid Mech., vol. 561, 2006, pp. 329–358). The results show that most of the eddies remain small and do not last for long times, but that some become large, attach to the wall and extend across the logarithmic layer. The latter are geometrically and temporally self-similar, with lifetimes proportional to their size (or distance from the wall), and their dynamics is controlled by the mean shear near their centre of gravity. They are responsible for most of the total momentum transfer. Their origin, eventual disappearance, and history are investigated and characterized, including their advection velocity at different wall distances and the temporal evolution of their size. Reinforcing previous results, the symmetry found between sweeps and ejections supports the idea that they are not independent structures, but different manifestations of larger quasi-streamwise rollers in which they are embedded. Spatially localized direct and inverse cascades are respectively associated with the splitting and merging of individual structures, as in the models of Richardson (Proc. R. Soc. Lond. A, vol. 97(686), 1920, pp. 354–373) or Obukhov (Izv. Akad. Nauk USSR, Ser. Geogr. Geofiz., vol. 5(4), 1941, pp. 453–466). It is found that the direct cascade predominates, but that both directions are roughly comparable. Most of the merged or split fragments have sizes of the order of a few Kolmogorov viscous units, but a substantial fraction of the growth and decay of the larger eddies is due to a self-similar inertial process in which eddies merge and split in fragments spanning a wide range of scales.
We present the first full-scale computational evidence of intermittent and synchronized dynamics of red blood cells in shear flow. These dynamics are characterized by the coexistence of a tumbling motion in which the cell behaves like a rigid body and a tank-treading motion in which the cell behaves like a liquid drop. In the intermittent dynamics, we observe sequences of tumbling interrupted by swinging, as well as sequences of swinging interrupted by tumbling. In the synchronized dynamics, the tumbling and membrane rotation are observed to occur simultaneously with integer ratios of the rotational frequencies. These dynamics are shown to be dependent on the stress-free state of the cytoskeleton, and are explained based on the cell membrane energy landscape.
The chaotic advection of passive tracers in a two-dimensional confined convection flow is addressed numerically near the onset of the oscillatory regime. We investigate here a differentially heated cavity with aspect ratio 2 and Prandtl number 0.71 for Rayleigh numbers around the first Hopf bifurcation. A scattering approach reveals different zones depending on whether the statistics of return times exhibit exponential or algebraic decay. Melnikov functions are computed and predict the appearance of the main mixing regions via the break-up of the homoclinic and heteroclinic orbits. The non-hyperbolic regions are characterised by a larger number of Kolmogorov–Arnold–Moser (KAM) tori. Based on the numerical extraction of many unstable periodic orbits (UPOs) and their stable/unstable manifolds, we suggest a coarse-graining procedure to estimate numerically the spatial fraction of chaos inside the cavity as a function of the Rayleigh number. Mixing is almost complete before the first transition to quasi-periodicity takes place. The algebraic mixing rate is estimated for tracers released from a localised source near the hot wall.
The motion of gravity-driven deformable drops (Bond number, $\mathit{Bo}\sim 0.8{-}11$ ) through a circular confining orifice (ratio of orifice diameter to drop diameter, $d/D<1$ ) was studied using high-speed imaging. Drops of water/glycerin, surrounded by silicone oil, fall toward and encounter the orifice plate after reaching terminal speed. The effects of surface wettability were investigated for both round-edged and sharp-edged orifices. For the round-edged case, a thin film of surrounding oil prevents the drop fluid from contacting the orifice surface, such that the flow outcomes of the drops are independent of surface wettability. For $d/D<0.8$ , the boundary between drop capture and release depends on a modified Bond number relating drop gravitational time scale to orifice surface tension time scale and is independent of viscosity ratio. Drops that release break into multiple fragments for larger $\mathit{Bo}$ and smaller $d/D$ . For the sharp-edged case, a contact is initiated at the orifice edge immediately upon impact, such that surface wettability influences the drop outcome. When the surface is hydrophobic, the contact line motion through the orifice enhances penetration of the drop fluid, but the trailing interface becomes pinned at the orifice edge, inhibiting drop release. When the surface is hydrophilic, a fraction of the drop fluid is always captured because the drop fluid spreads on both the upper and lower plate surfaces.