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.

We present an experimental study of capillary surfers, a new fluid-mediated active system that bridges the gap between dissipation- and inertia-dominated regimes. Surfers are wave-driven particles that self-propel and interact on a fluid interface via an extended field of surface waves. A surfer's speed and interaction with its environment can be tuned broadly through the particle, fluid, and vibration parameters. The wave nature of interactions among surfers allows for multistability of interaction modes and promises a number of novel collective behaviors.

Before a droplet can contact a surface during impact, it must first displace the air beneath it. Over a wide range of impact velocities, the droplet first squeezes the air into a thin film, enhancing its resistance to drainage; this slows the progress of the liquid toward the surface. Indeed, below a critical impact velocity, the air film remains intact, and the droplet rebounds off of the air film without making contact. For impact velocities exceeding this critical impact velocity, the droplet always makes contact. The initiation of contact formation requires a topological transition, whereby the initially connected gas domain is ruptured and a liquid capillary bridge forms, binding the droplet to the surface. Here we probe this transition in detail around the critical impact velocity using calibrated total internal reflection microscopy to monitor the air film thickness and profile at high speed during the impact process. Two air film rupture modalities are observed: nucleated contacts, which are isolated and do not correspond to the global minimum air film thickness, and spontaneous contacts, which occur always on a ring centered upon the impact axis where the air film reaches its global minimum. Our measurements show that for impact velocities exceeding the critical velocity for contact initiation, the air film ruptures at a nearly identical height $h_{min}\approx 20$ nm, for two fluids: silicone oil and a water-glycerol mixture. The height and time duration of the air film prior to contact are presented for over 180 droplet impact experiments. Impact events of water solution droplets show statistics for contact nucleation different from those for the silicone oil; this suggests that another mechanism may dominate contact nucleation during impact of the solution. Nevertheless, a critical impact velocity above which contact always occurs is identifiable for both liquids.

Internal waves are ubiquitous in the ocean and play an essential role in the transport of energy and mixing. Their peculiar reflection enables the concentration of energy on a limit cycle. With wave beams viewed as rays, this reflection on an inclined slope shrinks its width and generically brings closer two initially different trajectories eventually reaching a limit cycle called an attractor. Following previous studies, a ray-tracing algorithm is used to track the convergence of wave beams onto such a structure in a 3D axisymmetric domain. This information is used to design experiments using a truncated conical shaped tank in order to form an inertia-gravity waves attractor in a 3D axisymmetric geometry. By increasing the amplitude of the forcing, an evolution of the attractor characteristics can be observed. The occurrence of waves at frequencies lower than the forcing frequency ${\omega }_0$ suggests triadic resonant instability in a rotating or in a stratified case. Experiments performed in a stratification-only or a rotation-only case indicate two distinct behaviors. The existence of easily excited standing waves, resonant modes of the tank, at frequencies lower than the forcing one enables sharp triadic resonance instability for internal gravity waves, which is not possible for inertial waves. The effect of the symmetry axis is also investigated by adding a cylinder of sufficient diameter at the center of the domain for the wave to reflect on and thereby avoid the interaction on the singularity. Without it, the large amplitude of the waves on the axis triggers nonlinear effects and mixing, denying the access to the wave turbulence regime.

From microfluidics to fog-harvesting applications, tiny droplets are transported along various solid substrates including hairs, threads, grooves, and other light structures. Driven by gravity, a droplet sliding along a vertical fiber is a complex problem since it is losing volume and speed as it goes down. With the help of an original setup, we solve this problem by tracking in real-time droplet characteristics and dynamics. Single fibers as well as multiple fiber systems are studied to consider the presence of grooves. On both fibers and grooved threads, droplet speed and volume are seen to decay rapidly because the liquid entity is leaving a thin film behind. This film exerts a capillary force able to stop the droplet motion before it is completely drained. A model is proposed to capture the droplet dynamics. We evidence also that multiple vertical fibers are enhancing the droplet speed while simultaneously promoting increased liquid loss on grooves.

We present an experimental study of penetrative convection in gases, which allows reaching a lower Prandtl number $\mathrm{Pr}=0.7$ than classical water experiments. A heavy gas (${\mathrm{SF}}_6$) fills the bottom of a meter-sized rectangular tank. Diffusion of ${\mathrm{SF}}_6$ in air establishes a stratified vertical profile in part of the domain. Electric resistors heat the bottom plate of the tank up to a chosen temperature while water circulation fixes the top plate temperature at room temperature. After a short transient, the ${\mathrm{SF}}_6$-rich lower layer convects and progressively invades the upper stably-stratified layer while exciting propagating internal gravity waves. Density and particle image velocimetry (PIV) measurements are performed, allowing us to follow the growth of the convective region as well as the characteristics of the internal waves and convective fields. While mixing at the interface is often modeled in classical penetrative convection with an eddy diffusivity, we find that the convective layer growths here quadratically in time, which contradicts a stationary and homogeneous turbulent diffusion. The analysis of the wave spectra in the stratified domain reveals a peak frequency at the buoyancy frequency, with a clear cutoff above. Running PIV in both layers simultaneously allows us to compute the kinetic energy transfer from the convective zone into the stratified one, which amounts to about $4\%$. We also define a scaling law for the erosion rate as a function of the Froude number.

The triple decomposition of a velocity gradient tensor, which extracts local fluid motions of shear, rigid-body rotation, and irrotational strain, is extended to the decomposition of velocity vectors into shear and nonshear components. The present approach adapts the Biot-Savart law to reconstruct shear and nonshear velocities from the vorticity vectors of shear and rigid-body rotation, respectively. These velocities are related to the flows induced by small-scale shear layers or vortex tubes. The decomposed velocities are investigated with direct numerical simulations of isotropic turbulence and temporally evolving planar jets. The r.m.s. values of shear and nonshear velocities are about $70\%$ and $30\%$ of the r.m.s. value of total velocity fluctuations, and shear layers have a greater contribution to velocity fluctuations than vortex tubes. The shear and nonshear velocities are positively correlated at large scales, and the momentum transfer due to their interaction actively occurs at scales greater than 20 times the Kolmogorov scale. The contributions of shear and nonshear velocities to the Reynolds stress hardly depend on flows. The energy spectra of these velocities collapse well at small scales under Kolmogorov normalization. The present analysis of the turbulent jet confirms that shearing motion has dominant contributions to the production and diffusion of turbulent kinetic energy and the turbulent transport of a passive scalar. In addition, the energy transfer across scales is shown to be dominated by the large-scale velocity gradients arising from shearing motion and the small-scale stresses due to the shear velocity and its interaction with the nonshear component.

While various aerodynamic noise generation mechanisms and characteristics have been extensively studied for drone research, the noise features during the descending flight state have not been well addressed. In this work, we investigate the aerodynamics and aeroacoustics of a two-bladed drone rotor operating in descent using both experimental and numerical approaches. First, we measure the rotor thrust and torque in an anechoic wind tunnel at various rotational speeds and descent rates. The results reveal a significant loss of mean thrust and strong thrust fluctuations with the increase of descent rates due to the formation of highly unsteady vortex rings. The measured acoustic spectra show multiple humps occurring at the blade passing frequency and its harmonics. Next, we perform numerical simulations based on the delayed detached eddy simulations to gain more understanding of the noise generation mechanisms. The computed integrated aerodynamic forces and acoustic spectra agree well with the experimental results. We present nearfield flow structures that demonstrate the gradual formation process of the vortex ring and the complex turbulent wake structures. Notably, the computed aeroacoustic characteristics exhibit a similar tendency with respect to the descent rate to the experimental results. Moreover, the simulation results suggest that the additional noise has a radiation directivity in the axial direction perpendicular to the rotation plane. Finally, we conduct a noise source analysis on the blade surface based on the simulation results, revealing that the strong leading-edge sources are likely related to the blade-wake interaction process during descending flight.

The interaction between two passive human swimmer scale models is investigated both experimentally and numerically. The Froude number, comparing the swimming velocity to the characteristic wave velocity, is conserved in the study. The interaction is quantified for a large range of relative positions and for three speeds corresponding to cruising, average, and sprint swimming. The associated computational fluid dynamics study using OpenFoam allows us to determine the relative positions that optimize the drafting during an open-water race: just behind a lead swimmer or at the level of the hip of a neighbor, with reductions of drag of 40% and 30%, respectively.

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.

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.

We study the drag on a centimetric sphere in a uniform flow in the presence of a free surface as a function of submergence depth. Through direct force measurements in a custom benchtop recirculating flume, we demonstrate that the drag can significantly exceed the corresponding drag in a single-phase flow and achieves a peak at submergence depths just prior to complete immersion. The additional drag in the partially immersed state is rationalized by considering hydrostatic effects associated with the asymmetric surface height profile induced by the obstacle in the flow direction which persists for flow speeds below the minimum capillary-gravity wave speed. At these scales, the sphere's wettability plays a pronounced role in determining the maximum possible drag and results in hysteretic behaviors near touchdown and complete immersion. The influence of flow speed, sphere size, and surface tension on the drag characteristics are additionally explored through a combination of experiments and numerical simulations.