The mixing of a passive scalar in a three-dimensional, statistically stationary turbulent Navier-Stokes flow at a constant and moderate Taylor microscale Reynolds number $R_{\lambda }=42$ is studied by means of direct numerical simulations for Schmidt numbers between 1 and 64. The freely decaying passive scalar is represented in two different ways: (1) in the Lagrangian frame of reference as a cloud of up to 4.8 billion individually advected massless tracer particles subject to a stochastic Wiener process along the tracer tracks that describes scalar diffusion or (2) in the standard Eulerian frame of reference as an advection-diffusion equation of the continuum concentration field. In both cases, the scalar is initially seeded in a small cubic subvolume. The mean mixing time $\langle t_s\rangle$ is determined by the mean compressive strain rate $\langle {\lambda }_3\rangle <0$ which is obtained from the probability density functions of the local finite-time Lyapunov exponents in the Lagrangian frame, ${\lambda }_i\left(t\right)$ with $i=1,2$ and 3. The direct comparison of freely decaying Lagrangian and Eulerian passive scalars gives a good agreement of the scalar variance for times $t\lesssim 10\langle t_s\rangle$ and for the probability density functions $P(\mathrm{\Theta },t)$ taken with respect to the whole simulation domain. We also show how the multilayer aggregations of scalar filaments and sheets in the Lagrangian frame are increasingly influenced by the noise due to discreteness with progressing dilution of the initially high tracer particle concentration. This limits the Lagrangian approach in its present form and for the obtainable Schmidt numbers to studies of shorter time periods. A simple one-dimensional advection-diffusion model of a solitary strip is finally applied to the problem at hand to derive the probability density function of the scalar concentration, $P(\mathrm{\Theta },t)$, from the one of the compressive local finite-time Lyapunov exponent, $p({\lambda }_3,t)$. Model prediction with and without self-convolution and numerical data of the scalar distributions agree qualitatively, however with quantitative differences particularly for small scalar concentrations. The present Lagrangian approach to passive scalar mixing in turbulence opens the application of more flexible passive scalar injection and boundary conditions and allows to relax the resolution constraints for high-Schmidt number mixing studies.

The motion of a bubble sliding over an inclined wall from moderate to high bubble Reynolds number is studied experimentally for a wide range of liquid properties and bubbles sizes, considering wall inclination angles from nearly horizontal to nearly vertical. All experiments are restricted to sliding behavior, below the transition to steady bouncing motion. We study both the shape of the bubble and its drag coefficient. For small angles, the bubble shape is dominated by gravitational effects resulting in a flattened shape against the wall; for large angles, the bubble remains in constant contact with the wall but adopts a shape that is aligned perpendicularly to the wall, closer to that observed for an inertia-dominated free rising bubble. We model this transition of shape considering balances among surface tension, gravitational, and inertial forces; we observe good agreement with experiments. We found that the drag coefficient is strongly influenced by the shape that the bubble adopts as it slides over the wall. By considering the flow in the film and around the bubble, we propose a correlation to predict the drag coefficient for each regime of bubble shape. In the regime dominated by viscous effects, the drag of a single bubble is increased due to the mirror effect with the wall and by the friction in the film formed between the wall; conversely, for the case dominated by inertial effects, the drag coefficient is constant. The behavior for a single bubble is changed: no significant increase due to deformation. In both shape regimes the proposed expression agrees well with the experimental measurements.

We employ a rheological theory to show that circular Taylor-Couette flow of a suspension of non-Brownian spheres is less stable than that of a Newtonian fluid, at equal effective viscosity. The destabilization is related to the preferred orientation of the separation vector of the closely interacting spheres, in the compressive direction of the base flow. The results agree qualitatively with experimental observations from the literature.

We simulated two particle-based fluid models, namely multiparticle collision dynamics and dissipative particle dynamics, under shear using reverse nonequilibrium simulations (RNES). In cubic periodic simulation boxes, the expected shear flow profile for a Newtonian fluid developed, consistent with the fluid viscosities. However, unexpected secondary flows along the shear gradient formed when the simulation box was elongated in the flow direction. The standard shear flow profile was obtained when the simulation box was longer in the shear-gradient dimension than the flow dimension, while the secondary flows were always present when the flow dimension was at least 25% larger than the shear-gradient dimension. The secondary flows satisfy the boundary conditions imposed by the RNES and give a total flow field with a lower rate of viscous dissipation than the corresponding unidirectional flows. This work highlights a previously unappreciated limitation of RNES for generating shear flow in simulation boxes that are elongated in the flow dimension, an important consideration when applying RNES to complex fluids like polymer solutions.

We study experimentally the isoviscous displacement flow of two immiscible Newtonian fluids in an inclined pipe. The less dense displacing fluid is placed above the denser displaced fluid in a density-stable configuration. The displacing and displaced solutions are oil- and water-based, respectively. The former exhibits nonwetting behavior in the vicinity of the pipe wall, whereas the latter is wetting. The pipe has a small diameter-to-length ratio. The mixing and interpenetration of two fluids have been studied over a wide range of controlling parameters, revealing remarkable results. Compared to the previously studied miscible limit, we observe behavior at the interface between the two fluids where the displaced fluid stays “pinned” to the lower wall of the pipe upon pumping the displacing one. This phenomenon, which is observed over the full range of investigated flow rates, tilt angles, and density contrasts, is associated with the wetting characteristic of the displacing liquid and is also present when light and heavy viscosity mineral oils are used as the displacing fluid. Ultrasonic Doppler velocimetry revealed a segmented velocity profile at the interface of the immiscible fluids. Due to pinning, the efficiency of the removal of displaced fluid in the immiscible limit can be lowered by 14% compared to the miscible case due to the combined effects of the density-stable configuration and the immiscibility of the flow. Within the family of immiscible fluids, the maximum efficiency is achieved at close-to-vertical tilt angles, large density contrasts, and counterintuitively low imposed flow rates, which is of great importance in industrial design.

Acoustically driven bubbles are nonlinear oscillators showing a wide range of behaviors such as period-doubling bifurcations, deterministic chaos, and synchronization to an external signal. Here we demonstrate that bubbles driven with a constant heat source can couple with each other and yield in-phase synchronization, even in the absence of an external sound field. Besides perfect synchronization, we resolve the parameter space where independent oscillations and bubble merging occur. The main finding that only weakly driven oscillating bubbles synchronize provides a path for experimental scale-up to study spatiotemporal synchronization. In the wider scope the findings are relevant for heat transfer applications from structured heaters where complex multibubble oscillations are expected.

Two-dimensional fluid dynamics is often approximated via laboratory experiments that drive a thin layer of fluid electromagnetically. That approximation would be most accurate if both the direction and magnitude of the flow were uniform over the depth of the layer. In practice, boundary conditions require the flow magnitude to drop to zero at the no-slip floor, but put no strong constraint on flow direction. We measure the velocity magnitude and direction simultaneously at the free surface and lower interface of a thin, two-layer vortex flow. We find that the flow direction is almost entirely independent of depth, though its slight misalignment grows as the Reynolds number (Re) increases. Similarly, we find that the ratio of speeds at the free surface and interface nearly matches an analytically derived profile based on idealized assumptions, even for complex flows, but deviates systematically as Re increases. We find that flows with thinner fluid layers are better aligned and more nearly match the predicted speed ratio than flows with thicker layers. Finally, we observe that in time-dependent flows, flow structures at the interface tend to follow flow structures at the free surface via complicated dynamics, moving along similar paths with a short time delay. Our results suggest that the depth-averaged equation of motion recently developed for thin-layer flows [Suri et al., Phys. Fluids 26, 053601 (2014)], which relies on flow alignment and idealized profiles and was previously tested for Kolmogorov flows with Re up to 30, is reasonably accurate for vortex flows with Re up to 470.

A two-field model of potential vorticity (PV) staircase structure and dynamics relevant to both beta-plane and drift-wave plasma turbulence is studied numerically and analytically. The model evolves averaged PV whose flux is both driven by and regulates a potential enstrophy field, $\varepsilon$. The model employs a closure using a mixing length model. Its link to bistability, vital to staircase generation, is analyzed and verified by integrating the equations numerically. Long-time staircase evolution consistently manifests a pattern of metastable quasiperiodic configurations, lasting for hundreds of time units, yet interspersed with abrupt ($\mathrm{\Delta }t\ll 1$) mergers of adjacent steps in the staircase. The mergers occur at the staircase lattice defects where the pattern has not completely relaxed to a strictly periodic solution that can be obtained analytically. Other types of stationary solutions are solitons and kinks in the PV gradient and $\varepsilon$-profiles. The waiting time between mergers increases strongly as the number of steps in the staircase decreases. This is because of an exponential decrease in interstep coupling strength with growing spacing. The long-time staircase dynamics is shown numerically be determined by local interaction with adjacent steps. Mergers reveal themselves through the explosive growth of the turbulent PV flux, which, however, abruptly drops to its global constant value once the merger is completed.

When a stable stratification between two miscible fluids is excited by a vertical and periodic forcing, a turbulent mixing zone can develop, triggered by the Faraday instability. The mixing zone grows and saturates to a recently predicted final value $L_{\mathrm{sat}}$ [Gréa and Ebo Adou, J. Fluid Mech. 837, 293 (2018)] when resonance conditions are no longer fulfilled. Notably, it is expected from the Mathieu stability diagram that the instability may evolve from a harmonic to a subharmonic regime for particular initial conditions. This transition is evidenced here in the full inhomogeneous system using direct numerical simulations with ${1024}^3$ points: the analysis of one-point statistics and spectra reveals that turbulence is greatly enhanced after the transition, while the global anisotropy of both the velocity and concentration fields is significantly reduced. Furthermore, using the concept of sorted density field, we compute the background potential energy $e_p^b$ of the flow, which increases only after the transition as a signature of irreversible mixing. While the gain in $e_p^b$ strongly depends on the control parameters of the instability, the cumulative mixing efficiency is more robust. At saturation of the instability, available potential energy is partially released in the flow as background potential energy. Finally, it is shown numerically that for fixed parameters, a multiple-frequency forcing can modify the duration of the harmonic regime without significantly altering the asymptotic state.

This work investigates the motion of neutrally buoyant, slightly deformable straight and curved prolate fluid-filled capsules in unbounded simple shear flow at zero Reynolds number using direct simulations. The curved capsules serve as a model for the typical crescent-shaped sickle red blood cells in sickle cell disease (SCD). The effects of deformability and curvature on the dynamics are revealed. We show that with low deformability, straight prolate spheroidal capsules exhibit tumbling in the shear plane as their unique asymptotically stable orbit. This result contrasts with that for rigid spheroids, where infinitely many neutrally stable Jeffery orbits exist. The dynamics of curved prolate capsules are more complicated due to a combined effect of deformability and curvature. At short times, depending on the initial orientation, slightly deformable curved prolate capsules exhibit either a Jeffery-like motion such as tumbling or kayaking, or a non-Jeffery-like behavior in which the director (end-to-end vector) of the capsule crosses the shear-gradient plane back and forth. At long times, however, a Jeffery-like quasiperiodic orbit is taken regardless of the initial orientation. We further show that the average of the long-time trajectory can be well approximated using the analytical solution for Jeffery orbits with an effective orbit constant $C_{eff}$ and aspect ratio ${\ell }_{eff}$. These parameters are useful for characterizing the dynamics of curved capsules as a function of given deformability and curvature. As the capsule becomes more deformable or curved, $C_{eff}$ decreases, indicating a shift of the orbit towards log-rolling motion, while ${\ell }_{eff}$ increases weakly as the degree of curvature increases but shows negligible dependency on deformability. These features are not changed substantially as the viscosity ratio between the inner and outer fluids is changed from 1 to 5. As cell deformability, cell shape, and cell-cell interactions are all pathologically altered in blood disorders such as SCD, these results will have clear implications on improving our understanding of the pathophysiology of hematologic disease.

A number of time-periodic flows have been found to have a property called RZIF: when a linear stability analysis is carried out about the temporal mean (rather than the usual steady state), an eigenvalue is obtained whose real part is zero and whose imaginary part is the nonlinear frequency. For two-dimensional thermosolutal convection, a Hopf bifurcation leads to traveling waves which satisfy the RZIF property and standing waves which do not. We have investigated this property numerically for counter-rotating Couette-Taylor flow, in which a Hopf bifurcation gives rise to branches of upwards and downwards traveling spirals and ribbons which are an equal superposition of the two. In the regime that we have studied, we find that both spirals and ribbons satisfy the RZIF property. As the outer Reynolds number is increased, the ribbon branch is succeeded by two types of heteroclinic orbits, both of which connect saddle states containing two axially stacked pairs of axisymmetric vortices. One heteroclinic orbit is nonaxisymmetric, with excursions that resemble the ribbons, while the other remains axisymmetric.

We study the effect of shear on the coherent structures near the hot surface, namely, line plumes, in turbulent Rayleigh-Bénard convection (RBC) and turbulent mixed convection (MC) for the range of near-surface Rayleigh numbers $5.75\times {10}^7\le {\text{Ra}}_w\le 2.17\times {10}^9$ and shear Reynolds numbers $8.02\times {10}^2\le \text{Re}\le 15\times {10}^3$ for a Prandtl number range of $10.1\ge \text{Pr}\ge 0.7$ in water and air. Plumes are visualized by particle scattering in MC in air while they are extracted from the particle image velocimetry fields in RBC in water. We also use the planforms of plume structure obtained by Gilpin et al. [J. Heat Transfer 100, 71 (1978)] in MC in water using electrochemical visualization, as well as those obtained by Pirozzoli et al. [J. Fluid Mech. 821, 482 (2017)] in simulations. The planforms of plume structure show that shear aligns the line plumes and increases their mean spacing $\lambda$. An increase in ${\text{Ra}}_w$ decreases $\lambda$, while the resulting increase in Re in RBC, due to the increase of larger large-scale flow strength, counteracts this effect. Further, plumes are seen more spaced and smeared in air, compared to that in water, due to the lower Pr. We show that these complex dependences of $\lambda$ on ${\text{Ra}}_w$, Re, and Pr in RBC and MC can be described by a common scaling law ${\lambda }^{*}=\lambda -{\lambda }_0=S{Z}_{\mathrm{sh}}/D$, where ${\lambda }_0$ is the mean plume spacing in the absence of shear, $S={\text{Re}}^3/{\text{Ra}}_w$ is a shear parameter, $Z_{\mathrm{sh}}=\nu /U_{\mathrm{sh}}$ is the viscous-shear length, and $D$ a function of Pr.

We derive precursors of extreme dissipation events in a turbulent channel flow. Using a recently developed method that combines dynamics and statistics for the underlying attractor, we extract a characteristic state that precedes laminarization events that subsequently lead to extreme dissipation episodes. Our approach utilizes coarse statistical information for the turbulent attractor, in the form of second-order statistics, to identify high-likelihood regions in the state space. We then search within this high-probability manifold for the state that leads to the most finite-time growth of the flow kinetic energy. This state has both high probability of occurrence and leads to extreme values of dissipation. We use the alignment between a given turbulent state and this critical state as a precursor for extreme events and demonstrate its favorable properties for prediction of extreme dissipation events. Finally, we analyze the physical relevance of the derived precursor and show its robust character for different Reynolds numbers. Overall, we find that our choice of precursor works well at the Reynolds number it is computed at and at higher Reynolds number flows with similar extreme events.

We report herein the quantitative data and physical insights acquired on the self-acceleration and global pulsation of spherically expanding ${\mathrm{H}}_2\text{−}{\mathrm{O}}_2\text{−}{\mathrm{N}}_2$ flames, propagating in a constant-pressure environment and subjected to hydrodynamic and diffusional-thermal instabilities over a wide range of pressures and equivalence ratios. Results show that the critical radii for the onset of the transition stage and global pulsation stage have a similar variation with pressure and equivalence ratio and can be collapsed by plotting the nondimensional values normalized by the planar flame thickness. Furthermore, through experiments with fixed flame temperature achieved by adjusting the amount of ${\mathrm{N}}_2$ in air, it is demonstrated that the global pulsation frequencies dominated by the diffusional-thermal instability increase with its intensity, which is consistent with the hypothesis that the global pulsation behavior of cellular flames arises from the continuous cell growth and splitting during the flame propagation. The global pulsation frequencies of ${\mathrm{H}}_2$-air flames, subjected to the coupled hydrodynamic and diffusional-thermal instabilities, show a nonmonotonic trend with the equivalence ratio; while their nondimensional values, normalized by the flame time, collapse and decrease with increasing equivalence ratio, in that the pressure and flame temperature effects are properly scaled out through the normalization. The acceleration exponents of the transition stage and global pulsation stage are also determined, with the latter slightly smaller than the critical value of 1.5 suggested for self-turbulization.

Thermal convection in a cylindrical annulus with solid-body rotation and subject to an inward radial heating is investigated by the linear stability analysis and direct numerical simulation using the periodic boundary conditions. The Archimedean buoyancy is neglected in order to investigate the properties of thermal convection induced solely by the centrifugal buoyancy. The Coriolis buoyancy, which is usually neglected in similar studies, is included in the flow equations. The critical state occurs in the form of columnar vortices drifting in the retrograde direction. The critical parameters of these convective columnar vortices are determined for different values of the radius ratio and the Prandtl number (Pr). Instantaneous flow and temperature fields are computed for $\mathrm{Pr}=1$ in order to investigate higher instability modes and chaotic states. The kinetic energy and the coefficient of heat transfer by columnar vortices are evaluated for a large range of values of the Rayleigh number.

We investigate the electrophoresis of an ion-selective microgranule in an electrolyte solution. A semianalytical analysis of a small length of the electric double layer as well as overlimiting and extreme overlimiting currents is complemented by the direct numerical study of the full nonstationary Nernst-Planck-Poisson-Stokes system, with the corresponding boundary and initial conditions. Our results are in reasonably good agreement with the available experimental data. Moreover, they can be used to modify Dukhin's formula for the electrophoretic velocity. A steady-state solution is observed for moderate electric fields. Three boundary layers, nested inside each other, are formed in this solution as follows: an electric double layer, a space-charge region, and a thin diffusion layer. Only the electric double layer is present in the area of the outgoing ion flux. This flux generates a jet of high electric conductivity. Increasing the external field makes this jet narrower, but its conductivity increases. At the point on the granule surface where the ion flux vanishes, a separation of the diffusion boundary layer occurs. For sufficiently strong fields, the steady-state solution loses stability. Instability arises in the diffusion layer region but manifests itself in other regions. In particular, it generates electrokinetic microvortices. Two kinds of microvortices are found: large steady Dukhin-Mishchuk vortices and electrokinetic vortices that propagate from the pole of the particle towards the Dukhin-Mishchuk vortices. At small supercriticality, the oscillations of the unknowns are periodic, but increasing the external field makes the flow chaotic in the Feigenbaum scenario fashion.

We consider Newtonian liquid films on a horizontal substrate with a free and deformable surface. The substrate is subjected to oscillatory accelerations in the normal or in the horizontal direction. An algorithm based on a nonlinear coordinate transformation is presented that allows for direct numerical solutions of the fully nonlinear Navier-Stokes equations and appropriate boundary conditions. No surface tracking is necessary. Normal oscillations generate the traditional subharmonic and harmonic Faraday patterns. Lateral oscillations cause a pattern formation scenario qualitatively similar to spinodal dewetting, namely the disintegration of the film into isolated drops followed by coarsening or fusion, the stabilization of a “precursor” film, and no rupture. Ratchet-like lateral excitations break the horizontal mirror symmetry $x\to -x$ and give the patterns a preferred direction. We show that drops formed due to instability of the flat film start to travel in a distinguished direction. For thin films, the results are in good agreement to those of a recently studied lubrication-based dimension-reduced model [Bestehorn et al., Phys. Rev. E 88, 023025 (2013); Bestehorn, Phys. Fluids 25, 114106 (2013)].

Roll-to-roll printing processes require formation and stretching of a liquid bridge to transfer liquid from one surface to another. Since inadequate liquid transfer can produce defects that are detrimental to printed products, electric fields are sometimes applied to enhance transfer, a method known as electrostatic assist (ESA). Because the physical mechanisms underlying ESA are not well-understood, we examine here the influence of electric fields on liquid transfer in two model geometries, both of which involve liquid bridges with moving contact lines. The bridges are axisymmetric and confined between two electrodes, one of which is flat and moves vertically upward, and the other which either is flat or has a cavity and is stationary. An electric field is applied in the axial direction, both perfect and leaky dielectric liquids are considered, and the governing equations are solved with the Galerkin finite-element method. For liquid transfer between two flat plates, application of an electric field stabilizes the liquid bridge. This allows more time for the contact line to retract on the less wettable surface and leads to an increase in liquid transfer to the more wettable surface. Tangential stresses due to surface charge can significantly enhance liquid transfer, even to the less wettable surface if the tangential stresses point toward that surface. For liquid transfer between a flat plate and a cavity, the electric field increases the pressure gradient near the contact line on the cavity wall, causing the contact line to slip and more liquid to be transferred from the cavity. Notably, the effect is more pronounced for a deep cavity, resulting in a larger percentage of liquid transferred compared to a shallow cavity. In contrast to the case of liquid transfer between two flat plates, surface charge does not have as significant an influence on liquid transfer due to the way the cavity and electric field modify the interface shape near the contact line. The results of this work illustrate the physical mechanisms through which electric fields can improve liquid transfer, and they provide guidance for optimizing ESA in industrial printing processes.

The ring-sheared drop is a flow configuration for microgravity, where surface tension provides containment and shear in the bulk is driven primarily by the action of surface shear viscosity. A drop is constrained by two thin contact rings, i.e., one stationary at a southern latitude and the other at the same latitude but in the north and rotating. Since we consider a microgravity setting, the drop is not restricted to being small. Furthermore, we allow for arbitrarily small surface shear viscosity, so that in general the interfacial and bulk flows are viscously coupled. Our numerical simulations show that even small surface shear viscosity (quantified nondimensionally by a Boussinesq number) can produce a significant meridional bulk flow at moderate ring rotation rates (quantified by a Reynolds number Re). At very low Re, the bulk flow is viscously dominated and surface viscosity makes very little difference. At high Re, the secondary flow is very weak if the surface viscosity is negligible and the flow tends toward solid-body rotation.

The early stages of the shock-driven explosion of liquid-water microdrops are studied numerically with a high-resolution discretization of the axisymmetric Euler equations. A level-set based conservative interface-interaction method is extended to allow for phase transition. The method is applied to a configuration that has been investigated in recent experiments [Stan et al., Nat. Phys. 12, 966 (2016); Stan et al., J. Phys. Chem. Lett. 7, 2055 (2016)]. The presented results show that the numerical model predicts the initial stages of the violent liquid-drop explosion dynamics accurately. Our results indicate that the deformation of the cylindrical vapor cavity within the droplet is not induced by a torus-shaped negative-pressure wave as was implied from the experimental data. We rather find that this torus-shaped wave is a shock wave, and that the observed vapor-cavity deformation is caused by interaction with a negative-pressure region preceding the torus shock. The simulation results show deviations from experimental results at later stages when the drop deformation is dominated by off-center cavitation, i.e., by effects that require extension of the underlying model to take into account generalized nucleation and recondensation.