During the Leidenfrost effect, a stable vapor film separates a hot solid from an evaporating liquid. Eventually, after formation and upon cooling, the vapor layer cannot be sustained and undergoes a violent collapse evidenced by explosive boiling. Computationally, modeling this instability involves an interplay between hydrodynamics, thermodynamics, rapid evaporation, and length scales from $\mu \mathrm{m}$ to cm. Selective assumptions, made to reduce computational costs, have limited most previous studies to steady-state investigations. Here, we combine two-phase laminar flow, heat transfer, and evaporation in a finite-element simulation to examine the failure of Leidenfrost vapor layers upon cooling. During periods of quiescence, the geometry of the vapor layer agrees well with steady-state lubrication theory. In the simulations, we report the local temperature of the solid at failure, $T_{-}$, which provides a lower bound for recent experimental work using the same geometric and material conditions. Surprisingly, we find that inertial forces, which are typically ignored in theoretical treatments of the vapor layer, are responsible for initiating the instability leading to failure.

The collapse dynamics and runout of columns of elongated grains in two dimensions are numerically investigated in dry and immersed conditions, by means of an unresolved finite elements/discrete elements model. The elongated grains are modeled as rigid aggregates of disks. The column aspect ratio is systematically varied from 0.125 to 16 in order to span short and tall columns. To analyze the effect of the initial grain orientation, columns with an initial grain orientation that is either random or aligned with a given direction are both considered. Collapse dynamics, both in dry and immersed cases, are found analogous to that previously observed for circular grain columns, particularly with respect to the power law dependency for the runout as a function of the column aspect ratio. The effect of the fluid mainly results in a decrease of the runout distance. Interestingly, the collapse dynamics and runout are not significantly affected by the initial orientation of the grains, except maybe in the extreme case where the grains are all horizontally oriented, which geometrically prevents the collapse. Finally, a scaling based on the front propagation energy is proposed allowing one to unify the runout of short to tall and dry to immersed columns in a single description, regardless of the initial grain orientation.

We study experimentally the dynamics of non-wetting blobs flowing simultaneously with a wetting fluid in a quasi-two-dimensional porous medium consisting of random obstacles. The blobs continuously merge, forming larger ones (coalescence), and break up into smaller ones (fragmentation), leading to an overall dynamic equilibrium between the two processes. We develop a clustering algorithm for the identification of fragmentation and coalescence events that records the size of the blobs prior to and immediately after each event from high-resolution videos of the immiscible flow experiments. The results provide significant insight into the main physical features of these two processes, such as blob size distributions, breakup and coalescence frequency as a function of total flow rate, and the size distributions of the blobs formed by either the fragmentation or coalescence of other ones. One of the salient features of the fragmentation process in our study is that blobs that are smaller than the typical pore size exhibit a higher probability of producing two almost identical children (in size), whereas larger blobs break up into two children of different sizes. In the latter case, one of the children is found to have a dimension that is practically equal to the typical pore size. Our experimental results are also interpreted in the framework of a mean-field approach, where the dynamics of the blob sizes is expressed through an integro-differential population balance equation that comprises terms for the description of the rates of size gains and losses by either fragmentation or coalescence. We recover appropriate expressions for the relevant coalescence and fragmentation kernels, as functions of the blob sizes that participate in each event. A rather surprising result is that for a given blob size population, we obtain an equilibrium between the gains by fragmentation and the losses by coalescence. Furthermore, the opposite is also true, as the population gains by coalescence are found to be equal to the losses by fragmentation.

The dynamics of a thin low-density polyethylene sheet subject to periodic forcing due to Bénard-Kàrmàn vortices in a long narrow water channel is investigated here. In particular, the time-averaged sheet deflection and its oscillation amplitude are considered. The former is first illustrated to be well-approximated by the static equilibrium between the buoyancy force, the elastic restoring force, and the profile drag based on the depth-averaged water speed. Our observations also indicate that the presence of upstream vortices hinder the overall reconfiguration effect, well-known in an otherwise steady flow. For the sheet-tip oscillations, a simple model based on a torsional-spring-mounted flat plate correctly captures the measured tip amplitude ${\delta }_b$ over a wide range of sheet physical properties and flow conditions. Furthermore, a rich phenomenology of structural dynamics including vortex-forced-vibration, lock-in with the sheet natural frequency, and flow-induced vibration due to the sheet wake, multiple-frequency, and modal response is reported.

Forced two-dimensional turbulence lives on the balance between the energy input and two dissipative mechanisms, viscosity and linear friction, resulting in a double cascade of energy and enstrophy. While it is known that the energy cascade is governed by the Reynolds number ${\mathrm{Re}}_{\alpha }=u_{\mathrm{rms}}/\left(\alpha L_{\mathrm{f}}\right)$, it has been more convenient to report $\mathrm{Re}=u_{\mathrm{rms}}{L}_{\mathrm{f}}/\nu$ (where $u_{\mathrm{rms}}$ is the fluctuating velocity, $L_{\mathrm{f}}$ the forcing scale, $\alpha$ the friction coefficient, and $\nu$ the kinematic viscosity). Therefore, it is unclear for which range of parameters the various hallmarks of fully developed turbulence will emerge. Here we use multiple laboratory setups in which a quasi-two-dimensional flow is generated in electromagnetic layers of fluids, over a wide range of $\mathrm{Re}$ and ${\mathrm{Re}}_{\alpha }$. The friction coefficient measured during turbulence decay is correctly estimated by a linear shear assumption, allowing us to readily estimate ${\mathrm{Re}}_{\alpha }$. We consider several observables characterizing the turbulence development: the fraction of energy input converted to fluctuating energy, the correlation scale of the flow, the velocity structure functions, the probability distribution of the velocity fluctuations and velocity differences, the single-particle diffusivity, and the separation time between particle pairs. All descriptors collapse on master curves against ${\mathrm{Re}}_{\alpha }$, providing a criterion for fully developed turbulence for this class of flows. Moreover, dedicated experiments in which the local $\mathrm{Re}$ and ${\mathrm{Re}}_{\alpha }$ are spatially decoupled show that only the latter is correlated with the growth of turbulent energy. Finally, a scaling relation is proposed that relates the amount of energy going to the large scales to the forcing scale-to-layer thickness ratio.

Convection and strong magnetic fields interact in the Sun's envelope and produce many magnetic features. In sunspots, the most intense of these structures, magnetic fields are strong enough to inhibit convection. In this work, we study simulations of magnetohydrodynamic Rayleigh-Bénard convection (RBC) using the Dedalus pseudospectral framework. We perform a suite of two- and three-dimensional simulations in which we vary the strength of the background magnetic field and the strength of convective driving (quantified, respectively, by the Chandrasekhar number, Q, and the Rayleigh number, Ra). We directly measure and report force balances between the Lorentz, buoyancy, and inertial forces to quantify magnetic constraint. In addition, we measure the solenoidal portion of the force by removing contributions to the pressure gradient to directly study the acceleration responsible for modifying the fluid velocity. We find three simulation regimes: a “constrained” regime where the background magnetic field dominates, a “magnetically influenced” regime where nonlinear Lorentz and inertial forces balance, and a transitional regime between the two. We find the constrained regime only becomes turbulent for large $Q\gtrsim {10}^{10}$. Magnetic field fluctuations scale as $B\propto \text{Ra}{Q}^{-7/6}$ in constrained regimes and scale as $B\propto {\text{Ra}}^{7/16}{Q}^{-1/2}$ in the magnetically influenced regime. The efficiency of convection quantified by the Nusselt number scales as the expected Nu $\propto$ Ra in the constrained regime, but a classical hydrodynamic RBC scaling of Nu $\propto {\text{Ra}}^{2/7}$ is found in the magnetically influenced regime even when the nonlinear Lorentz force heavily influences dynamics.

The mini-tsunami phenomenon denotes a newly observed upstream wave generated by a ship traveling across a depth change at subcritical speed. In this study, a high-order spectral model with variable bathymetry (HOS-VB) is developed to study the effects of bottom and free-surface nonlinearities on the generation and propagation of mini-tsunamis over a range of depth-based Froude number ${\text{Fr}}_0$, average water depth to ship draft ratio $h_0/d$, bathymetric slope variation $\gamma$, and channel width to ship width ratio $W/B$. The analyses show that the wave amplitude of the mini-tsunamis is increased by both bottom and free-surface nonlinearities and that the influence of free-surface nonlinearities on the wave amplitude is more significant in a narrower channel. Second-order free-surface nonlinearities provide the most important contribution. Compared with linear results, second-order free-surface nonlinearities can more than double the maximum wave amplitude and induce a substantial increase in the mean free-surface level for a narrow channel at large Froude number. The wavelength at half-crest height is also increased by second-order free-surface nonlinearities, and this effect is enhanced as $h_0/d$ decreases. Moreover, the increasing trend of wave amplitude with ${\text{Fr}}_0$ and the decreasing trend of wave amplitude with $h_0/d$ are more significant as the free-surface nonlinearity increases.

We introduce similarity relations, along with their corresponding symmetry groups, for examining the velocity profile and friction factor of laminar Bingham fluid flows using friction coordinates. Specifically, we provide a valuable expression for calculating the friction factor of Bingham plastic fluids when pressure gradient data are accessible. Our findings reveal that there are no clear similarity relations for the mean velocity profile and friction factor formulas in bulk coordinates. Consequently, friction coordinates serve as the most suitable framework for describing this problem.

Turbulent flows laden with large, deformable drops are ubiquitous in nature and in a wide range of industrial processes. Prediction of the interactions between drops, which deform under the action of turbulence, exchange momentum via surface tension, and that can also exchange heat or mass, are complicated due to the wide range of scales involved: from the largest scales of the flow, down to the Kolmogorov scales of turbulence, and further down to the molecular scale of the interface. Due to this wide range of scales, the numerical description of these flows is challenging and requires robust and accurate numerical schemes that are able to capture both the turbulence characteristics and the dynamics of ever-moving and deforming interfaces including their topological changes (i.e., coalescence and breakage). In the past decades, various numerical methods have been proposed for simulating two-phase flows, from interface-tracking methods, where the interface is explicitly tracked with the use of marker points to interface-capturing methods, where the interface is identified as the isovalue of a color/marker function. Phase-field methods belong to the category of interface-capturing methods, and have emerged as promising approaches to simulate complex two-phase flows. In phase-field methods, the transport equation to describe the drop motion is obtained from first thermodynamics principles, and phenomena acting at the interface scale can be conveniently modeled. Although in realistic case scenarios, the physical thickness of the interface cannot be directly simulated, this family of methods offers desirable properties that have attracted the interest of researchers in recent years. In this work, we describe the fundamentals of the phase-field modeling associated with the direct numerical simulation of turbulence in the context of drop-laden flows. We discuss the potentials of the phase-field method with reference to breakage and coalescence phenomena, and to the corresponding drop size distribution; we examine how to model surface tension changes due to surfactant distribution, and we outline the framework to model heat and mass transfer fluxes. Finally, we present our perspectives for future developments of phase-field modeling of drop-laden turbulent flows in the context of the current available literature.

A numerical investigation of an asymptotically reduced model for quasigeostrophic Rayleigh-Bénard convection is conducted in which the depth-averaged flows are numerically suppressed by modifying the governing equations. At the largest accessible values of the Rayleigh number $Ra$, the Reynolds number and Nusselt number show evidence of approaching the diffusion-free scalings of $Re\sim RaE/Pr$ and $Nu\sim P{r}^{-1/2}R{a}^{3/2}{E}^2$, respectively, where $E$ is the Ekman number and $Pr$ is the Prandtl number. For large $Ra$, the presence of depth-invariant flows, such as large-scale vortices, yield heat and momentum transport scalings that exceed those of the diffusion-free scaling laws. The Taylor microscale does not vary significantly with increasing $Ra$, whereas the integral length scale grows weakly. The computed length scales remain $O\left(1\right)$ with respect to the linearly unstable critical wave number; we therefore conclude that these scales remain viscously controlled. We do not find a point-wise Coriolis-inertia-Archimedean (CIA) force balance in the turbulent regime; interior dynamics are instead dominated by horizontal advection (inertia), vortex stretching (Coriolis) and the vertical pressure gradient. A secondary, subdominant balance between the Archimedean buoyancy force and the viscous force occurs in the interior and the ratio of the root mean square (rms) of these two forces is found to approach unity with increasing $Ra$. This secondary balance is attributed to the turbulent fluid interior acting as the dominant control on the heat transport. These findings indicate that a pointwise CIA balance does not occur in the high Rayleigh number regime of quasigeostrophic convection in the plane layer geometry. Instead, simulations are characterized by what may be termed a nonlocal CIA balance in which the buoyancy force is dominant within the thermal boundary layers and is spatially separated from the interior Coriolis and inertial forces.

Sufficiently thin liquid films on solid surfaces are often unstable to intermolecular forces and it is commonly assumed that their rupture occurs via a linear instability mechanism in the so-called spinodal regime. Here, a theoretical framework is created for the experimentally observed thermal regime, in which fluctuation-induced nanowaves rupture linearly stable films. Molecular simulations in a quasi-2D geometry identify these regimes and are accurately reproduced by stochastic simulations based on fluctuating hydrodynamics. Rare-event theory is then applied to, and developed for, this field to provide exceptional computational efficiency and accuracy that allows us to extend calculations deep into the thermal regime. Analysis of the rare-event theory reveals a picture of how and when “rogue nanowaves” are able to provide a route to film rupture. Finally, future applications of the new theoretical framework and experimental verification is discussed.

We analyze the effects of the fine structure of the electric double layer and of the electrode reaction rate upon wavelength selection of the growing perturbation mode in morphological instability in cathodic electrodeposition from a dilute electrolyte solution. The metal cathode is modeled as a cation exchanger with a fixed charge density equal to the average concentration of free electrons that is several orders of magnitude higher than the typical ionic concentration in a solution. This invokes the need to consider the steric modifications in the modeling of the electric double layer at the solution/electrode interface. Accordingly, the results for Bikerman's steric model are compared to those for the classical Nernst-Planck-Poisson-Stokes model valid for pointlike ions. For both models, the diffuse electric double layer splits into a thin inner portion commensurate with the width of the Stern layer and the steric exclusion width, and a much thicker outer portion scaling with the Debye length. For quiescent cathodic electrodeposition, this fine structure combined with a finite electrode reaction rate regularizes the short-wave singularity in morphological instability of the electrodeposition front. This regularization selects a finite range of unstable perturbation modes, with critical wavelength corresponding to the fastest-growing mode. This wavelength scales with the geometric average of the width of the highly charged inner portion of the electric double layer and the reaction–diffusion length, defined as the ratio of cation diffusivity to the electrode reaction rate. The fluid flow induced by the motion of the solid/liquid interface has a negligible effect on morphological instability. At the same time, the emerging electroconvective (electroosmotic) flow in the nonequilibrium regime with its related electroosmotic instability, has a major effect, selecting the width of the cathodic diffusion layer as the dominant length scale for morphological instability and the emerging dendrites. It is observed that the steric correction considerably lowers the voltage threshold for the onset of the nonequilibrium electroosmotic instability.

To investigate the effectiveness of the dispersed-phase continuum (DPC) approximation to model the airflow in the print gap of inkjet printers, three-dimensional simulations using the DPC model were compared against those using the classic particle-in-cell (P-in-C) approach. The DPC approximation, due to the separation of time scales, models the dispersed phase with a momentum source that depends on a predefined temporally averaged particle number density field. The results demonstrated that the steady DPC model correlated well to the time-averaged P-in-C solution when the former's formulation accounted for the droplet deceleration. The steady DPC model requires less than 0.1% of the computational resources used by the transient P-in-C approach to compute the mean flow field. Further analyses indicated that the DPC model captured a supercritical pitchfork bifurcation, where the airflow shifted from a steady spanwise uniform regime to a standing wave regime at a critical number density. The P-in-C model computed a smooth continuous transition that characterizes an excited supercritical pitchfork bifurcation. An excellent correlation between the models was observed at print densities above the transition point, but at low number densities a certain level of discrepancy was observed. This was a result of the pseudoturbulence or spottiness induced by the local and instantaneous motion of droplets that excited the standing wave solution even at low number densities. The results, thus, demonstrated that the DPC model is effective at estimating the mean flow field and approximating the bifurcation diagram, while being simultaneously more computationally efficient than the P-in-C model.

Direct numerical simulations are used to elucidate the interplay of wettability and fluid viscosities on immiscible fluid displacements in a heterogeneous porous medium. We classify the flow regimes using qualitative and quantitative analysis into viscous fingering (low $M$, $M$ being the viscosity ratio—described as ratio of fluid viscosity of injected by defending phases), compact displacement (high $M$), and an intermediate transition regime $(M\approx 1)$. We use stability analysis to obtain theoretical phase boundaries between these regimes, which agree well with our analyses. At the macroscopic (sample) scale, we find that wettability strongly controls the threshold $M$ (at which the regimes change). At the pore scale, wettability alters the dominant pore-filling mechanism. At very small $M$ (viscous fingering regime), smaller pore spaces are preferentially invaded during imbibition, with flow of films of invading fluid along the pore walls. In contrast, during drainage, bursts result in filling of pores irrespective of their size. As $M$ increases, the effect of wettability decreases as cooperative filling becomes the dominant mechanism regardless of wettability. This suggest that for imbibition at a given contact angle, decreasing $M$ is associated with change in effective wetting from neutral-wet (cooperative filling) to strong-wet (film flow).

We study experimentally the propagation of an air finger through the Y-bifurcation of an elastic, liquid-filled Hele-Shaw channel as a benchtop model of airway reopening. With channel compliance provided by an elastic upper boundary, we can impose collapsed channel configurations into which we inject air with constant volumetric flow rate. We typically observe steady finger propagation in the main channel, which is lost ahead of the Y-bifurcation but subsequently recovered in the daughter channels. At low levels of initial collapse, steady finger shapes and bubble pressure in the daughter channels map onto those in the main channel. However, at higher levels of initial collapse where the elastic sheet almost touches the bottom boundary of the channel, experimentally indistinguishable fingers in the main channel can lead to multiple states of reopening of the daughter channels. The downstream distance at which steady propagation is recovered in the daughter channels also varies considerably with injection flow rate and initial collapse because of a transition in the mechanics regulating finger propagation. We find that the characteristic time and length scales of this recovery are largest in the regime where viscous and surface tension forces dominate finger propagation and that they decrease towards a constant plateau in the limit where elastic forces supersede viscous forces. Our findings suggest that practical networks are unlikely to comprise long-enough channels for steady-state propagation to be recovered between bifurcations.

The effect of liquid droplets generated from air-assisted atomization on gas flow characteristics was studied experimentally. A phase/Doppler particle analyzer was used to measure velocity and size distributions of continuous and dispersed phases in the droplet-laden two-phase flow. A comparison of mean gas velocity with and without droplets indicates the expected influence of dispersed phase on the carrier phase, i.e., two-way coupling. The flow characterization result shows the presence of liquid droplets contributes to the increase of gas-phase flow velocity in the spray field. The effect of liquid droplets on gas-phase turbulence is manifested in three ways. First, the presence of droplets leads to the increase in fluctuation velocity of gas-phase flow. Subsequently, it is observed that the range of fluctuation velocities in the gas phase is expanded in two-phase flow compared with single-phase flow. In the region characterized by a steep velocity gradient, the initial gas fluctuation velocities in two-phase flow demonstrate a notable enhancement of 20% compared with single-phase flow. Furthermore, the presence of droplets induces axial stretching within the shear region of the gas phase, and this stretching effect is particularly pronounced in cases of higher fuel-injection durations, primarily due to the influence of droplet gravity. The data obtained from the analysis of velocity gradient and fluctuation velocity within the two-phase flow field reveal a distinct segmental linear relationship, deviating from previous findings reported in the literature and highlighting a deeper understanding of the underlying mechanisms in current two-phase flow systems.

Dynamical systems with extreme events are difficult to capture with data-driven modeling due to the relative scarcity of data within extreme events compared to the typical dynamics of the system and the strong dependence of the long-time occurrence of extreme events on short-time conditions. A recently developed technique [D. Floryan and M. D. Graham, Nat. Mach. Intell. 4, 1113 (2022)], here denoted as Charts and Atlases for Nonlinear Data-Driven Dynamics on Manifolds, or CANDyMan, overcomes these difficulties by decomposing the time series into separate charts based on data similarity, learning dynamical models on each chart via individual time-mapping neural networks, then stitching the charts together to create a single atlas to yield a global dynamical model. We apply CANDyMan to a nine-dimensional model of turbulent shear flow between infinite parallel free-slip walls under a sinusoidal body force [J. Moehlis, H. Faisst, and B. Eckhardt, New J. Phys. 6, 56 (2004)], which undergoes extreme events in the form of intermittent quasi-laminarization and long-time full laminarization. The multichart model created by the CANDyMan technique is compared with both a standard data-driven model (i.e., the “single-chart” limit of the CANDyMan method) and a Koopman-based model created through extended dynamic mode decomposition-dictionary learning. We demonstrate that the CANDyMan method allows the trained dynamical models to more accurately forecast the evolution of the model coefficients than both a single-chart model and a Koopman model, reducing the error in the predictions as the model evolves forward in time. The technique exhibits more accurate predictions of extreme events than either a single-chart model or Koopman model, capturing the frequency of quasi-laminarization events and predicting the time until full laminarization more accurately than a single neural network.

The dissolution and subsequent mass transfer of carbon dioxide gas into liquid barriers plays a vital role in many environmental and industrial applications. In this work, we study the downward dissolution and propagation dynamics of ${\mathrm{CO}}_2$ into a vertical water barrier confined to a narrow vertical glass cylinder, using both experiments and direct numerical simulations. Initially, the dissolution of ${\mathrm{CO}}_2$ results in the formation of a ${\mathrm{CO}}_2$-rich water layer, which is denser in comparison to pure water, at the top gas-liquid interface. Continued dissolution of ${\mathrm{CO}}_2$ into the water barrier results in the layer becoming gravitationally unstable, leading to the onset of buoyancy-driven convection and, consequently, the shedding of a buoyant plume. By adding sodium fluorescein, a pH-sensitive fluorophore, we directly visualize the dissolution and propagation of the ${\mathrm{CO}}_2$ across the liquid barrier. Tracking the ${\mathrm{CO}}_2$ front propagation in time results in the discovery of two distinct transport regimes, a purely diffusive regime and an enhanced diffusive regime. Using direct numerical simulations, we are able to successfully explain the propagation dynamics of these two transport regimes in this laterally strongly confined geometry, namely by disentangling the contributions of diffusion and convection to the propagation of the ${\mathrm{CO}}_2$ front.

This paper is devoted to a specific motion observed for glycerin droplets sliding on a horizontal hydrophobic substrate under the influence of a shear flow. In this regime, the droplet elongates in the flow direction, adopting a rivulet shape. Waves develop on the droplet sheared surface, resulting in a wavy contracting and stretching motion mechanism, similar to the movement of a caterpillar. If long enough, the droplet can break up into several droplets that can be submitted to a pearling instability. Furthermore, these droplets can also coalesce.