A theoretical analysis is presented for the flow of a Herschel-Bulkley fluid around the inside surface of a rotating cylinder, at rotation speeds for which the fluid largely collects in a prominent pool in the lower part of the cylinder. The analysis, based on lubrication theory, predicts the steady states typically reached after a small number of rotations. The analysis is also modified to consider the drainage of the film around a stationary cylinder, which allows an exploration of the dynamics when a rotating drum is suddenly stopped. The predictions of the theory are compared with experiments in which a Carbopol suspension is rotated inside an acrylic drum.

In the present work, a numerical framework is developed for simulating gas-solid flows associated with interaction of a rocket plume on a planetary surface. For this investigation, a unique, two-way coupled, dusty-gas flow model has been developed in the coupled direct simulation Monte Carlo (DSMC)-discrete element method (DEM) framework, which has been applied for dust dispersion on lunar surface. In this model, the gas-gas collisions are modeled probabilistically, whereas grain-grain interactions are computed deterministically. Most importantly, a multiphase fluid-solid two-way coupling model is proposed where the fluid-particle interaction is modeled macroscopically by considering the momentum and energy exchange between the two phases. The use of the DEM approach allows us to model dense granular phase near a planetary regolith. Additionally, the framework allows us to calculate not only the particle trajectories, but also their temperatures. To make the framework efficient for dynamic calculation of drag and heat transfer from gas to grain phase, the same coefficients are precalculated (and used later as a look-up table) for gas flow over an isolated granular particle over a range of speed ratios for free molecular flow conditions. Using the developed framework, a comprehensive numerical study is performed to model dust dispersion associated with lunar landing. The influence of particle diameter on gas and grain phases and dust transportation has been analyzed in the coupled framework. Additionally, the impact of the two-way coupled gas-grain interaction model is analyzed and compared with one-way coupled model.

Turbulence in wall-bounded flows is characterized by stable statistics for the mean flow and the fluctuations both for the case of the ensemble and the time mean. Although in a substantial set of turbulent systems, this stable statistical state corresponds to a stable fixed point of an associated statistical state dynamics (SSD) closed at second order, referred to as S3T, this is not the case for wall turbulence. In wall turbulence the trajectory of the statistical state evolves on a transient chaotic attractor in the S3T statistical state phase space and the time-mean statistical state is neither a stable fixed point of this SSD nor, if the time-mean statistical state is maintained as an equilibrium state, is it stable. Nevertheless, sufficiently small perturbations from the ensemble or time-mean state of wall turbulence are expected to relax back to the mean statistical state following an effective linear dynamics. In this work the dynamics of spanwise uniform perturbations to the time-mean flow are studied using a linear inverse model to identify the linear operator governing the ensemble stability of the ensemble or time-mean state by obtaining the time mean stability properties over the transient attractor of the turbulence identified by the S3T SSD. The ensemble or time-mean stability of an unstable equilibrium can be understood by noting that, even when every member of an ensemble is unstable, the ensemble mean may be stable with perturbations following an identifiable stable dynamics. While simplifying insight into turbulent flows has commonly been obtained by identifying and studying ensemble mean statistical states, less attention has been accorded to identifying and studying the ensemble mean dynamics. We show that, in the case of wall turbulence, even though stable fixed-point SSD equilibria are not available to allow application of traditional perturbation analysis methods to identify the perturbation stability of the mean state, an effective linear stability analysis can be obtained to identify the perturbation dynamics of the ensemble or time-mean statistical state.

Using fluid displacement in a Hele-Shaw cell, we experimentally demonstrate the arresting effect of an imposed flow on a phase separation occurring in a growing liquid-liquid interfacial region, unlike in an initially homogeneous single-phase mixture quenched within the miscibility gap. Increasing the imposed flow rate reduces the exponent α in the form $\mathrm{\Delta }A\propto t^{\alpha }$, where $\mathrm{\Delta }A$ is the increase in the area occupied by the displacing fluid owing to phase separation. We show that α can be expressed as a function of the ratio of the phase separation rate to the flow rate on a single curve, indicating that the competition between the imposed flow and phase separation rates determines the degree of the interfacial phase separation. The arresting effect and the mechanism are verified by a numerical simulation.

Liquid nanofilms are ubiquitous in nature and technology, and their equilibrium and out-of-equilibrium dynamics are key to a multitude of phenomena and processes. We numerically study the evolution and rupture of viscous nanometric films, incorporating the effects of surface tension, van der Waals forces, thermal fluctuations, and viscous shear. We show that thermal fluctuations create perturbations that can trigger film rupture, but they do not significantly affect the growth rate of the perturbations. The film rupture time can be predicted from a linear stability analysis of the governing thin film equation, by considering the most unstable wavelength and the thermal roughness. Furthermore, applying a sufficiently large unidirectional shear can stabilize large perturbations, creating a finite-amplitude traveling wave instead of film rupture. In three dimensions, unidirectional shear does not inhibit rupture, as perturbations are not suppressed in the direction perpendicular to the applied shear. However, if the direction of shear varies in time, then the growth of large perturbations is prevented in all directions, and rupture can be impeded.

When a drop of a volatile liquid is deposited on a uniformly heated wettable, thermally conducting substrate, one expects to see it spread into a thin film and evaporate. Contrary to this intuition, due to thermal Marangoni contraction, the deposited drop contracts into a spherical-cap-shaped puddle, with a finite apparent contact angle. Strikingly, this contracted droplet, above a threshold temperature, well below the boiling point of the liquid, starts to spontaneously move on the substrate in an apparently erratic way. We describe and quantify this self-propulsion of the volatile drop. It arises due to spontaneous symmetry breaking of thermal Marangoni convection, which is induced by the nonuniform evaporation of the droplet. Using infrared imaging, we reveal the characteristic interfacial flow patterns associated with Marangoni convection in the evaporating drop. A scaling relation describes the correlation between the moving velocity of the drop and the apparent contact angle, both of which increase with the substrate temperature.

Active flow control can achieve substantial performance gains and meet the challenges of next-generation air vehicles and energy-harvesting devices. The use of active flow control techniques with the moving flap or morphing surfaces has been shown to be a viable path to regulating the flow-induced vibration of the foil. However, due to the complex nature of flow over morphing surfaces, all physical phenomena are intertwined, which prevents a clear understanding of the underlying flow physics and, therefore, a successful design of a controlling action to optimally modify them. In this research an active flow control framework with the model predictive control theory is proposed to modulate the flow-induced flutter of a foil using the morphing flap surface. The geometrically weighted dynamic-relevant modes are used to build surrogate models to achieve rapid model-based active control of complex systems. It is shown that the flap is capable of both facilitating and eliminating fluid-induced vibrations by regulating the lift forces exerted on the foil. Furthermore, the control framework provides full knowledge of how the structure modifies the flow and has the potential to identify the ambient environmental change simultaneously.

Fluid-structure interactions between active and passive components are important for many biological systems to function. A particular example is chromatin in the cell nucleus, where ATP-powered processes drive coherent motions of the chromatin fiber over micron lengths. Motivated by this system, we develop a multiscale model of a long flexible polymer immersed in a suspension of active force dipoles as an analog to a chromatin fiber in an active fluid—the nucleoplasm. Linear analysis identifies an orientational instability driven by hydrodynamic and alignment interactions between the fiber and the suspension, and numerical simulations show activity can drive coherent motions and structured conformations. These results demonstrate how active and passive components, connected through fluid-structure interactions, can generate coherent structures and self-organize on large scales.

Molecular tagging is used to study the dispersion and deformation of patterns written in turbulent air. The writing is done by fusing ${\mathrm{O}}_2$ and ${\mathrm{N}}_2$ molecules into NO in the focus of a strong ultraviolet laser beam. By crossing several of these laser beams, patterns that have both small and large scales can be painted. The patterns are visualized a while later by inducing fluorescence of the NO molecules with a second UV laser and registering the image. The width of the lines that make the pattern is approximately $50\mu \mathrm{m}$, a few times the Kolmogorov length $\eta$, the smallest length scale in turbulence, while the largest size of the patterns ($\approx 4\mathrm{mm}$) is inside the inertial range of the used turbulent jet flow. At small scales molecular clouds disperse under the joint action of molecular diffusion and turbulence. The experiments reveal this highly nontrivial interaction. At inertial-range scales ($\approx 200\eta$) we verify the Batchelor dispersion of objects whose size is inside the inertial range. Patterns are compressible objects and spontaneously develop concentration fluctuations. We show for the first time the nontrivial statistical properties of these fluctuations. Finally, we use the information in written and deformed lines to quantify turbulent intermittency, obtaining results that agree with the established scaling anomaly of velocity structure functions.

We present the observation of an expanding vortex ring below a fast-spreading ethanol-water film over a deep substrate of water. The radius of the vortex ring ($r_v$) is shown to scale with time as $t^{1/2}$. The outer radius of the disturbed region ($r_o$), which comprises the expanding film of length $r_f$ and the plume of length $l_p$ at its periphery, scales in the same way as $r_v$. Based on this observation, contrary to the earlier proposal of the origin of such vortex rings to be due to evaporative cooling, we hypothesize that such vortex rings occur due to the separation of the shear layer below the plumes. Based on this hypothesis, we obtain scaling laws for the radius of the vortex ring ($r_v$) and its radial expansion velocity ($U_v$). These scaling laws are shown to match our measurements, thereby validating the phenomenology of such vortex rings.

We experimentally investigate the amplitude of the common sound heard when we pour water, tea, or coffee on a filled receptacle. Using water jets from circular nozzles, we find that pouring is audible when the nozzle distance from the free surface is above one-third of the jet breakup length. The sound amplitude increases with the steepness of the jet corrugation, informing that thin jets are louder than thick ones for a given pouring height. Since the jet corrugation is related to the air entrainment rate, the sound of pouring can serve as a practical noninvasive probe in aeration processes. After the jet breaks up into drops impacting on the free surface, the pouring sound amplitude increases with the nozzle diameter, unlike before breakup.

Turbulent scalar mixing occurs within a wide variety of natural and engineering flows. Predicting and controlling the scalar concentration(s) within these flows can yield immediate benefits to numerous applications across many fields. To that end, a better understanding of the effects of scalar-field initial conditions on the evolution(s) of scalar fields is required to either promote or delay the rate at which mixing occurs. Direct numerical simulations are employed herein to simulate the evolution in time of the hydrodynamic and (passive) scalar fields within a fully developed turbulent channel flow. The effects of the scalar field initial conditions are studied by analyzing the evolution of the scalar field subject to three different initial conditions with interfaces oriented normal to the streamwise, wall-normal, and transverse directions. Particular emphasis is placed on the scalar variance and dissipation rate budgets, including the evolutions of their constituent terms. The fastest mixing occurs for the initial condition in which the interface is aligned normal to the mean velocity vector. The rapid mixing in this case is associated with higher rates of production and destruction of the scalar dissipation, as well as strong advection and stretching of the interface by the mean flow. In addition to better mixing arising from the stronger turbulence near the wall, enhanced mixing is correlated with having the edge of an interface along a channel wall, such that a large distortion of the initial interface arises from the combined effects of the no-slip condition at the channel walls with the advection of the interface by the mean flow in the region between the walls. To maximize this effect, it is recommended that scalar interfaces be aligned normal to the mean velocity vector to promote mixing within internal flows.

The wing structure of several insects, including dragonflies, is not smooth, but corrugated; its vertical cross section consists of a connected series of line segments. Some previous studies have reported that corrugated wings exhibit better aerodynamic performance than flat wings at low Reynolds numbers $[\text{Re}\simeq \mathit{O}\left({10}^3\right)]$. However, the mechanism remains unclear because of the complex wing structure and flow characteristics. A corrugated structure modifies the formation and behavior of aerodynamic flow features arising during unsteady wing motion, such as the leading-edge vortex (LEV). These modifications can be key to lift enhancement in many insects, though the details of these benefits remain imperfectly understood. In this study, we analyzed the flow around a two-dimensional corrugated wing model that started impulsively by direct numerical simulations. We focused on the period between the initial generation of LEVs and subsequent interactions before detachment. For the flat wing, it is known that a secondary vortex with a sign opposite to that of the LEV, the $\lambda$ vortex, develops and erupts to discourage lift enhancement. For corrugated wings, such an eruption of the $\lambda$ vortex can be suppressed by the corrugation structure, which enhances the lift. The detailed mechanism and its dependence on the angle of attack are also discussed.

Free-surface cusps appear as a generic feature in viscous flow with a free surface. However, a mathematical description has so far only been possible by constructing exact solutions to the Stokes equation in very specific and idealized geometries, using complex mapping techniques. Here we use the boundary integral formulation of the Stokes equation to show that cusps are local singular solutions to Stokes' equation. We recover Jeong and Moffatt's [J. Fluid Mech. 241, 1 (1992)] local cusp solution in the limit of diverging cusp curvature, demonstrating its universality across all viscous flows.

The displacement of a viscous liquid by a gas within a Hele-Shaw cell is a classical problem. The gas–liquid interface is hydrodynamically unstable, forming striking fingerlike patterns that have attracted research interest for decades. Generally, both the gas and liquid phases are taken to be incompressible, with the capillary number being the key parameter that determines the severity of the instability. Here, we consider a radially outward displacement driven by the steady compression of a gas reservoir. The associated gas-injection rate is then unsteady due to the compressibility of the gas. We identify a second nondimensional parameter, the compressibility number, that plays a strong role in the development of the fingering pattern. We use an axisymmetric model to study the impact of compressibility number on the unsteady evolution of injection rate and gas pressure. We use linear stability analysis to show that increasing the compressibility number delays the onset of finger development relative to the corresponding incompressible case. Finally, we present and compare a series of experiments and fully nonlinear simulations over a broad range of capillary and compressibility numbers. These results show that increasing the compressibility number systematically decreases the severity of the fingering pattern at high capillary number. Our results provide an unprecedented comparison of experiments with simulations for viscous fingering, a comprehensive understanding of the role of compressibility in unstable gas–liquid displacement flows, and insight into a new mechanism for controlling the development of fingering patterns.

We carry out direct numerical simulations of horizontally or vertically vibrated Rayleigh–Bénard (RB) convection over a wide range of Rayleigh number $\left(Ra\right)$ and dimensionless vibration frequency $\left(\omega \right)$ at fixed Prandtl number $Pr=4.38$ and dimensionless vibration amplitude $a=1.52\times {10}^{-3}$. It is shown that the global heat transport (measured by the Nusselt number $Nu$) is close to the value of standard RB convection in buoyancy-dominant regime at small $\omega$, whereas it is significantly enhanced by horizontal vibration or suppressed by vertical vibration in the vibration-dominant regime at large $\omega$. The division between the two regimes yields a critical vibration frequency ${\omega }^{*}$, which indicates the onset of vibration-induced $Nu$ enhancement or $Nu$ reduction. The values of ${\omega }^{*}$ are obtained based on the fitting between the numerical data and our proposed crossover functions. The dependence of ${\omega }^{*}$ on $Ra$ is then studied. It is found that the fitted critical frequency exhibits two close scaling relations: ${\omega }^{*}\sim R{a}^{-0.164}$ in horizontally vibrated RB convection and ${\omega }^{*}\sim R{a}^{-0.172}$ in vertically vibrated cases. Moreover, based on the competition of the kinetic energy production between buoyancy-dominant and vibration-dominant regimes, a physical model is proposed to predict the scaling behavior between ${\omega }^{*}$ and $Ra$, i.e., ${\omega }^{*}\sim R{a}^{-1/6}$, which agrees well with the measured scaling exponents of our numerical data.

The universal velocity log law proposed by von Kármán in wall-bounded turbulent flows is one of the cornerstones of turbulence theory. When buoyancy effects are important, the universal velocity log law is typically believed to break down according to Monin-Obukhov similarity theory (MOST), which has been used in almost all global weather and climate models to describe the dependence of the mean velocity profiles on buoyancy near the earth's surface and to characterize the surface-atmosphere exchange of momentum, heat, water vapor, and carbon dioxide. In contrast to MOST, we propose logarithmic profiles of near-wall mean velocity in the stably stratified atmospheric boundary layers based on direct numerical simulations and field observations across a wide range of buoyancy effects. We find that buoyancy does not seem to change the logarithmic nature of velocity profiles but instead modifies the slope of the log law in stably stratified conditions. This paper provides a perspective on wall turbulence and can be applied to numerical simulations of turbulence, weather, and climate.

The propulsion of mammalian spermatozoa relies on the spontaneous periodic oscillation of their flagella. These oscillations are driven internally by the coordinated action of ATP-powered dynein motors that exert sliding forces between microtubule doublets, resulting in bending waves that propagate along the flagellum and enable locomotion. We present an integrated chemomechanical model of a freely swimming spermatozoon that uses a sliding-control model of the axoneme capturing the two-way feedback between motor kinetics and elastic deformations while accounting for detailed fluid mechanics around the moving cell. We develop a robust computational framework that solves a boundary integral equation for the passive sperm head alongside the slender-body equation for the deforming flagellum described as a geometrically nonlinear internally actuated Euler-Bernoulli beam, and captures full hydrodynamic interactions. Nonlinear simulations are shown to produce spontaneous oscillations with realistic beating patterns and trajectories, which we analyze as a function of sperm number and motor activity. Our results indicate that the swimming velocity does not vary monotonically with dynein activity, but instead displays two maxima corresponding to distinct modes of swimming, each characterized by qualitatively different wave forms and trajectories. Our model also provides an estimate for the efficiency of swimming, which peaks at low sperm number.

A transformation that relates a compressible wall-bounded turbulent flow with nonuniform fluid properties to an equivalent incompressible flow with uniform fluid properties is derived and validated. The transformation accounts for both variable-property and intrinsic compressibility effects, the latter being the key improvement over the current state of the art. The importance of intrinsic compressibility effects contradicts the renowned Morkovin's hypothesis.

We present and analyze a theoretical model for the dynamics and interactions of “capillary surfers,” which are millimetric objects that self-propel while floating at the interface of a vibrating fluid bath. In our companion paper [I. Ho et al., Phys. Rev. Fluids 8, L112001 (2023)], we reported the results of an experimental investigation of the surfer system, which showed that surfer pairs may lock into one of seven bound states, and that larger collectives of surfers self-organize into coherent flocking states. Our theoretical model for the surfers' positional and orientational dynamics approximates a surfer as a pair of vertically oscillating point sources of weakly viscous gravity-capillary waves. We derive an analytical solution for the associated interfacial deformation and thus the hydrodynamic force exerted by one surfer on another. Our model recovers the bound states found in experiments and exhibits good agreement with experimental data. Moreover, we conduct a linear stability analysis of bound state solutions and compute numerically the associated eigenvalues. We find that the spacings of the bound states are quantized on the capillary wavelength, with stable branches of equilibria separated by unstable ones. Generally, our work shows that self-propelling objects coupled by capillary waves constitute a promising platform for studying active matter systems in which both inertial and viscous effects are relevant.