Three-wave resonant interactions constitute an essential nonlinear mechanism coupling capillary surface waves. In a previous work [Haudin et al. Phys. Rev. E 93, 043110 (2016)], we characterized experimentally the generation by this mechanism of a daughter wave, whose amplitude saturates due to the viscous dissipation. Here, we show experimentally the generation of a daughter wave verifying the resonant conditions, but not the dispersion relation. By modeling the response of the free surface at the lowest nonlinear order, we explain this observation as a forced interaction. The bandwidth of the linear transfer function of the free surface is indeed increased by the significant viscous dissipation. The observation of free surface excitations not following the linear dispersion relation then becomes possible. This forced three-wave interaction mechanism could have important consequences for wave turbulence in experimental or natural systems with nonnegligible dissipation.

This study investigates the influence of the surrounding gas on a droplet impacting a smooth dry glass surface at high Weber and Reynolds numbers. It was performed using a flywheel experiment and different gases at ambient pressure. We analyzed the splashing outcome by measuring the size, velocity, and angle of the secondary droplets and by calculating the total volume ejected. We show that gas entrapment is not the mechanism responsible for splashing at high Weber and Reynolds numbers. We demonstrate that splashing is influenced by the density, followed by the viscosity, and last by the mean free path of the surrounding gas. Furthermore, the surrounding gas primarily affects the number of secondary droplets ejected and their ejection angle, whereas the droplet size and horizontal velocity are independent of the surrounding gas properties. We provide the first theoretical expression for the total volume ejected using the theory of Riboux and Gordillo [Phys. Rev. Lett. 113, 024507 (2014)], which attributes the secondary droplet generation to a lift force experienced by spreading lamella. The relationship between the ejected volume and the splashing parameter is described by a power function.

To effectively apply the superhydrophobic surface for flow control, it is necessary to understand the effect of air-water interface shape on the diffusion characteristics of trapped air pockets. In the present work, using the reflection interference contrast microscopy, we directly measure the temporal variation of the three-dimensional shape of the air-water interface formed on submerged hydrophobic surfaces in a channel flow. The grate and the circular hole are considered a roughness feature, its size is varied between 80 and $200\mu \mathrm{m}$, and the Reynolds numbers of 1850 and 5000 are considered based on channel half-height and bulk velocity. The temporal variation of the height, the curvature of the air-water interface, and the corresponding diffusion length is measured, and the diffusion process is classified into three phases. For each phase, the diffusion characteristics are analyzed in relation to the roughness structure type, size, and Reynolds number. Based on the analysis, we propose a scaling relation for the Sherwood number (dimensionless diffusion length). For the first two phases when the interface is pinned at the roughness edge, the Sherwood number is determined by the interface curvature (i.e., the contact angle between the interface and the roughness sidewall), but it is scaled with the Peclet number (the ratio between the shear rate and the diffusion rate) for the last phase when the interface is depinned and moves downward. These relations are also validated with the data available in the literature, and we think this is useful as a simple guideline to design a roughness geometry for the superhydrophobic surfaces.

Under statically stable conditions in wall-bounded flows, turbulence is generated by shear and dissipated by buoyancy and viscosity. Most studies have focused on the steady-state balance of this budget, but the time evolution of the turbulence when the stabilizing buoyancy flux is first imposed is as important in many geophysical applications. In this paper, we utilize a new paradigm on the critical role of shear production damping by buoyancy to develop a quantitative model for the rate of decay of turbulent kinetic energy (TKE), at early and intermediate times, after buoyancy is abruptly imposed on a steady neutrally stratified Ekman boundary-layer flow. Scaling analyses and reduced models are developed to obtain an expression for this rate of decay, which is then validated using results from direct numerical simulation. We further show that the TKE production term persists as a parameter outside of the classic stability parameter term (the flux Richardson number) under unsteady conditions, and is therefore needed to describe the dynamics of evolving stable boundary layers. The long-time steady dissipation-production-buoyancy balance is also analyzed using the same modeling framework and confirms that the damping of shear TKE production by buoyancy is the main agent for the reduction in TKE, while the direct role of buoyancy destruction is secondary.

The study and control of flow instabilities is a key problem in aerodynamics. Aircrafts are designed not only to generate the lift force needed to balance their weight but, more importantly, to be stable and reasonably steady when in cruise conditions. Similar flow stability properties are naturally achieved by biological flying objects such as the dandelion seeds that are transported by the wind owing to a disklike structure called a pappus. The pappus creates a parachute flow configuration and is a remarkable prototype of how the wake, which would be unsteady if the pappus was completely impermeable, can be stabilized by changing the body structure so as to allow the flow to pass through. We approach the problem using the approximation of an anisotropic and nonhomogeneous rigid porous disk, combined with the linear stability analysis technique. The results show the presence of a mean porosity threshold beyond which the flow is always characterized by a separated, steady, and axisymmetric recirculating vortex ring. We compare our results with those of real dandelion pappi. The threshold is very close to the experimentally observed values of porosity, explaining why the morphology of the pappus promotes a steady wake regime.

This paper investigates the flow of a solidifying liquid film on a solid surface subject to a complex kinematics, a process relevant to pancake making and surface coating. The flow is modeled using the lubrication approximation, with a gravity force whose magnitude and direction depend on the time-dependent orientation of the surface. Solidification is modeled with a temperature-dependent viscosity. Because the flow eventually ceases as the liquid film becomes very viscous, the key question this study aims to address is: what is the optimal surface kinematics for spreading the liquid layer uniformly? Two methods are proposed to tackle this problem. In the first one, the surface kinematics is assumed a priori to be harmonic and parameterized. The optimal parameters are inferred using the Monte Carlo method. This “brute-force” approach leads to a moderate improvement of the film uniformity compared to the reference case when no motion is imposed to the surface. The second method is formulated as an optimal control problem, constrained by the governing partial differential equation, and solved with an adjoint equation. Key benefits of this method are that no assumption is made on the form of the control, and that significant improvement in thickness uniformity are achieved with a comparatively smaller number of evaluations of the objective function.

Polyelectrolyte microgels find many uses as rheological modifiers and stimulus-responsive materials. Understanding their swelling and collapse dynamics therefore holds broad importance in science and technology. We report remarkably simple experiments, requiring little sophistication, that reveal the subtle physics of microgel collapse. Millimeter-scale bubbles, sugar grains, and other small particles remain suspended and supported by the yield stress of household hand sanitizer, which arises due to a jammed suspension of swollen microgels. By contrast, salt grains with almost identical physical properties sediment through the material, leaving milky “comet tails” behind. Remarkably, the settling speed of a salt crystal remains constant as it dissolves—completely independent of its size or shape until it completely dissolves. Because the settling speed does depend on the type of salt that sediments, we hypothesize that salt grains effectively bore holes through hand sanitizer, with a velocity that is limited by the salt-induced dynamic collapse of the individual microgel particles. A simple convection-diffusion-collapse model successfully relates sedimentation velocities to microgel collapse dynamics for various salts. This model and its predictions are consistent with other observations and with complementary microfluidic experiments.

The method of regularized Stokeslets, based on the divergence-free exact solution to the equations of highly viscous flow due to a spatially smoothed concentrated force, is widely employed in biological fluid mechanics. Many problems of interest are axisymmetric, motivating the study of the azimuthally integrated form of the Stokeslet which physically corresponds to a ring of smoothed forces. The regularized fundamental solution for the velocity (single-layer potential) and stress (double-layer potential) due to an axisymmetric ring of smoothed point forces, the regularized ringlet, is derived in terms of complete elliptic integrals of the first and second kind. The relative errors in the total drag and surrounding fluid velocity for the resistance problem on the translating, rotating unit sphere, as well as the condition number of the underlying resistance matrix, are calculated; the regularized method is also compared to three-dimensional regularized Stokeslets and the singular method of fundamental solutions. The velocity of Purcell's toroidal swimmer is calculated; regularized ringlets enable accurate evaluation of surface forces and propulsion speeds for nonslender tori. The benefits of regularization are illustrated by a model of the internal cytosolic fluid velocity profile in the rapidly growing pollen tube. Actomyosin transport of vesicles in the tube is modeled using forces immersed in the fluid, from which it is found that transport along the central actin bundle is essential for experimentally observed flow speeds to be attained. The effect of tube growth speed on the internal cytosolic velocity is also considered. For axisymmetric problems, the regularized ringlet method exhibits a comparable accuracy to the method of fundamental solutions while also allowing for the placement of forces inside of the fluid domain and having more satisfactory convergence properties.

Coherent turbulent wave-packet structures in a jet at Reynolds number $460000$ and Mach number 0.4 are extracted from experimental measurements and are modeled as linear fluctuations around the mean flow. The linear model is based on harmonic optimal forcing structures and their associated flow response at individual Strouhal numbers, obtained from analysis of the global linear resolvent operator. These forcing-response wave packets (“resolvent modes”) are first discussed with regard to relevant physical mechanisms that provide energy gain of flow perturbations in the jet. Modal shear instability and the nonmodal Orr mechanism are identified as dominant elements, cleanly separated between the optimal and suboptimal forcing-response pairs. A theoretical development in the framework of spectral covariance dynamics then explicates the link between linear harmonic forcing-response structures and the cross-spectral density (CSD) of stochastic turbulent fluctuations. A low-rank model of the CSD at given Strouhal number is formulated from a truncated set of linear resolvent modes. Corresponding experimental CSD matrices are constructed from extensive two-point velocity measurements. Their eigenmodes (spectral proper orthogonal or SPOD modes) represent coherent wave-packet structures, and these are compared to their counterparts obtained from the linear model. Close agreement is demonstrated in the range of “preferred mode” Strouhal numbers, around a value of 0.4, between the leading coherent wave-packet structures as educed from the experiment and from the linear resolvent-based model.

The displacement of a viscous fluid by an air bubble in the narrow gap between two parallel plates can readily drive complex interfacial pattern formation known as viscous fingering. We focus on a modified system suggested recently by Zheng et al. [Phys. Rev. Lett. 115, 174501 (2015)], in which the onset of the fingering instability is delayed by introducing a time-dependent (power-law) plate separation. We perform a complete linear stability analysis of a depth-averaged theoretical model to show that the plate separation delays the onset of nonaxisymmetric instabilities, in qualitative agreement with the predictions obtained from a simplified analysis by Zheng et al. We then employ direct numerical simulations to show that in the parameter regime where the axisymmetrically expanding air bubble is unstable to nonaxisymmetric perturbations, the interface can evolve in a self-similar fashion such that the interface shape at a given time is simply a rescaled version of the shape at an earlier time. These novel, self-similar solutions are linearly stable but they only develop if the initially circular interface is subjected to unimodal perturbations. Conversely, the application of nonunimodal perturbations (e.g., via the superposition of multiple linearly unstable modes) leads to the development of complex, constantly evolving finger patterns similar to those that are typically observed in constant-width Hele-Shaw cells.

In dynamical systems theory, suppression of instabilities around a fixed point is generally achieved by controlling the linearized dynamics of infinitesimal perturbations, because considering small-amplitude disturbances allows for application of a range of celebrated techniques from linear control theory. In this paper, we consider the problem of design and implementation of a controller for fully nonlinear, high-dimensional, dynamical systems with the goal of steering trajectories to an unstable fixed point of the governing equations. Our control strategy is based on our previous work [A. Blanchard, S. Mowlavi, and T. P. Sapsis, Nonlinear Dynam. 95, 2745 (2019)] and takes advantage of the unique properties of the optimally time-dependent (OTD) modes, a set of global, time-evolving, orthonormal modes that track directions in phase space associated with transient growth and persistent instabilities. We show that the OTD control strategy introduced previously is robust with respect to perturbation amplitude even in cases in which the trajectory initially evolves on an attractor that lies far away from the target fixed point. In recognition of the fact that actuation capabilities are generally limited in practice, we also formulate a localized control strategy in which the OTD modes are computed in a spatially localized subdomain of the physical domain of interest. We suggest a strategy for selecting the optimal control domain based on a quantitative criterion derived from the OTD modes. We show that even when the range of the controller is reduced, OTD control is able to steer trajectories toward the target fixed point.

Ongoing development of fuel storage and delivery systems for space probes, interplanetary vehicles, satellites, and orbital platforms continues to drive interest in propellant management systems that utilize surface tension to retain, channel, and control flow in microgravity environments. Although it has been known for decades that capillary flows offer an ideal method of fuel management, there has been little research devoted to the general stability properties of such flows. In this work we demonstrate theoretically why capillary flows which channel wetting liquids in slender open triangular channels tend to be very stable against disturbances. By utilizing the gradient flow form of the governing fluid interface equation, we first prove that stationary interfaces in the presence of steady flow are asymptotically nonlinearly and exponentially stable in the Lyapunov sense. We then demonstrate that fluid interfaces exhibiting self-similar Washburn dynamics are transiently and asymptotically linearly stable to small perturbations. This second finding relies on a generalized nonmodal stability analysis due to the non-normality of the governing disturbance operator. Taken together, these findings reveal the robust nature of transient and steady capillary flows in open grooved channels and likely explain the prevalent use of capillary flow management systems in many emerging technologies ranging from CubeSats to point-of-care microfluidic diagnostic systems.

In experiments and numerical simulations we measure angles between the symmetry axes of small spheroids advected in turbulence (passive directors). Since turbulent strains tend to align nearby spheroids, one might think that their relative angles are quite small. We show that this intuition fails in general because angles between the symmetry axes of nearby particles are anomalously large. We identify two mechanisms that cause this phenomenon. First, the dynamics evolves to a fractal attractor despite the fact that the fluid velocity is spatially smooth at small scales. Second, this fractal forms steps akin to scar lines observed in the director patterns for random or chaotic two-dimensional maps.

We study the deformation and breakup of an axisymmetric electrolyte drop which is freely suspended in an infinite dielectric medium and subjected to an imposed electric field. The electric potential in the drop phase is assumed to be small, so that its governing equation is approximated by a linearized Poisson-Boltzmann or modified Helmholtz equation (the Debye-Hückel regime). An accurate and efficient boundary integral method is developed to solve the low-Reynolds-number flow problem for the time-dependent drop deformation, in the case of arbitrary Debye layer thickness. Extensive numerical results are presented for the case when the viscosity of the drop and surrounding medium are comparable. Qualitative similarities are found between the evolution of a drop with a thick Debye layer (characterized by the parameter $\chi \ll 1$, which is an inverse dimensionless Debye layer thickness) and a perfect dielectric drop in an insulating medium. In this limit, a highly elongated steady state is obtained for sufficiently large imposed electric field, and the field inside the drop is found to be well approximated using slender-body theory. In the opposite limit $\chi \gg 1$, when the Debye layer is thin, the drop behaves as a highly conducting drop, even for moderate permittivity ratio $Q={\epsilon }_1/{\epsilon }_2$, where ${\epsilon }_1,{\epsilon }_2$ is the dielectric permittivity of drop interior and exterior, respectively. For parameter values at which steady solutions no longer exist, we find three distinct types of unsteady solution or breakup modes. These are termed conical end formation, end splashing, and open end stretching. The second breakup mode, end splashing, resembles the breakup solution presented in a recent paper [R. B. Karyappa et al., J. Fluid Mech. 754, 550 (2014)]. We compute a phase diagram which illustrates the regions in parameter space in which the different breakup modes occur.

Using direct numerical simulations, we examine the effects of the Taylor Reynolds number, $R_{\lambda }\equiv u^{\prime }\lambda /\nu$ (where $u^{\prime }$, $\lambda$, and $\nu$ denote the fluid root-mean-squared velocity, the Taylor microscale and the fluid kinematic viscosity, respectively), and Froude number, $\text{Fr}\equiv a_{\eta }/g$ (where $a_{\eta }$ is the Kolmogorov acceleration, and $g$ is the acceleration due to gravity), on the motion of small, spherical, settling, bidisperse inertial particles (characterized by their Stokes number $\text{St}\equiv {\tau }_p/{\tau }_{\eta }$, which is the ratio of the particle response time to the Kolmogorov timescale) in isotropic turbulence. Particle accelerations play a key role in the relative motion of bidisperse particles, and we find that reducing Fr leads to an enhancement of the accelerations but a suppression of their intermittency. For Stokes numbers $\text{St}>1$, the effect of $R_{\lambda }$ on the accelerations is enhanced by gravity, since settling causes the particle accelerations to be affected by a larger range of flow scales. The results for the probability density function (PDF) of the particle relative velocities show that even when the particles are settling very fast, turbulence continues to play a key role in their vertical relative velocities, and increasingly so as $R_{\lambda }$ is increased. This occurs because although the settling velocity may be much larger than typical velocities of the turbulence, due to intermittency, there are significant regions of the flow where the turbulence contribution to the particle motion is of the same order as that from gravitational settling. Increasing $R_{\lambda }$ enhances the non-Gaussianity of the relative velocity PDFs, while reducing Fr has the opposite effect, and for fast settling particles, the PDFs become approximately Gaussian. Finally, we observe that low-order statistics such as the radial distribution function and the particle collision kernel are strongly affected by Fr and St, and especially by the degree of bidispersity of the particles. However, we also find that these low-order statistics are very weakly affected by $R_{\lambda }$ when $\text{St}\le O\left(1\right)$, irrespective of the degree of bidispersity. Therefore, although the mechanisms controlling the collision rates of monodisperse and bidisperse particles are different, they share the property of a weak sensitivity to $R_{\lambda }$ when $\text{St}\le O\left(1\right)$.

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.

We investigated the generation and propagation of ultrasonic pressure waves produced by focused x-ray free-electron laser pulses in 14 to $30\mu \mathrm{m}$ diameter liquid water microjets. The pressure waves formed through reflections, at the surface of the microjets, of the initial shock launched in the liquid by the x-ray pulse. These waves developed a characteristic geometric pattern which is related to, but different from, the shock structures of supersonic gas jets. Fully developed waves had initial peak pressures ranging from less than –24 MPa to approximately 100 MPa, which exceed the compressive and tensile strengths of many materials, and correspond to extreme sound intensities on the order of $1\mathrm{GW}/\mathrm{m}{}^2$ and sound pressure levels above 270 dB (re: $1\mu \mathrm{Pa}$). The amplitudes and intensities were limited by the wave destroying its own propagation medium though cavitation, and therefore these ultrasonic waves in jets are one of the most intense propagating sounds that can be generated in liquid water. The pressure of the initial shock decayed exponentially, more rapidly in thinner jets, and the decay length was proportional to the jet diameter within the accuracy of measurements. Extrapolating our results to thinner jets, we find that the pressure waves may damage protein crystals carried by liquid jets in x-ray laser crystallography experiments conducted at megahertz repetition rates.

Reynolds-averaged Navier-Stokes (RANS) models are the primary numerical recourse to investigate complex engineering turbulent flows in industrial applications. However, to establish RANS models as reliable design tools, it is essential to provide estimates for the uncertainty in their predictions. In the recent past, an uncertainty estimation framework relying on eigenvalue and eigenvector perturbations to the modeled Reynolds stress tensor has been widely applied with satisfactory results. However, the methodology for the eigenvector perturbations is not well established. Evaluations using only eigenvalue perturbations do not provide comprehensive estimates of model form uncertainty, especially in flows with streamline curvature, recirculation, or flow separation. In this article, we outline a methodology for the eigenvector perturbations using a predictor-corrector approach, which uses the incipient eigenvalue perturbations along with the Reynolds stress transport equations to determine the eigenvector perturbations. This approach was applied to benchmark cases of complex turbulent flows. The uncertainty intervals estimated using the proposed framework exhibited substantial improvement over eigenvalue-only perturbations and are able to account for a significant proportion of the discrepancy between RANS predictions and high-fidelity data.

The objective of this work is to develop the framework and governing equations needed to analyze the interactions and spectral distribution of internal energy in a manner similar to that of a kinetic energy analysis. To emulate the role of velocity in a kinetic energy analysis, a variable ($\varphi \sim \sqrt{p}$, where $p$ is pressure) is introduced to enable the examination of internal energy dynamics. Evolution equations for the mean and fluctuating components of $\varphi$ are derived. These equations enable the precise examination of mean-turbulent flow internal energy interactions, internal-kinetic energy exchange, and the spectral distribution of internal energy.

At mesoscopic scales electrolyte solutions are modeled by the fluctuating generalized Poisson-Nernst-Planck (PNP) equations [J.-P. Péraud et al., Phys. Rev. Fluids 1, 074103 (2016)]. However, at length and time scales larger than the Debye scales, electrolytes are effectively electroneutral and the charged-fluid PNP equations become too stiff to solve numerically. Here we formulate the isothermal incompressible equations of fluctuating hydrodynamics for reactive multispecies mixtures involving charged species in the electroneutral limit and design a numerical algorithm to solve these equations. Our model does not assume a dilute electrolyte solution but rather treats all species on an equal footing, accounting for cross diffusion and nonideality using Maxwell-Stefan theory. By enforcing local electroneutrality as a constraint, we obtain an elliptic equation for the electric potential that replaces the Poisson equation in the fluctuating PNP equations. We develop a second-order midpoint predictor-corrector algorithm to solve either the charged-fluid or electroneutral equations with only a change of the elliptic solver. We use the electroneutral algorithm to study a gravitational fingering instability, triggered by thermal fluctuations, at an interface where an acid and base react to neutralize each other. Our results demonstrate that, because the four ions diffuse with very different coefficients, one must treat each ion as an individual species and cannot treat the acid, base, and salt as neutral species. This emphasizes the differences between electrodiffusion and classical Fickian diffusion, even at electroneutral scales.