Understanding the force on a finite-sized particle immersed in a turbulent boundary layer close to a rough wall is of fundamental importance in predicting particle motion in a variety of industrial and environmental applications. Here we perform fully resolved direct numerical simulations with an immersed boundary method to investigate the forces on a stationary finite-sized particle in wall turbulence over a rough bed. Results show that for the particle sitting on the rough bed, lift is the main contributor to wall-normal force and can be well predicted with proper application of existing force models. The higher lift force is mainly due to the sweep events which temporarily increase the relative velocity as well as local shear magnitude. In contrast, for a particle located slightly away from the rough bed, the wall-normal force is mainly due to the wall-normal component of drag, so ejections and outward interactions are responsible for the increase in the wall-normal force. The standard drag law with near-wall correction can reasonably predict this wall-normal force on the particle, except for the large fluctuations due to self-induced vortex shedding.

Nonlinear internal waves propagating in a quasi-two-layer stratification with a shear background current are investigated by means of high-resolution direct numerical simulations. The simulations are performed in a two-dimensional rectangular domain, representing a laboratory-scale tank of finite length. The internal waves are generated using the lock-release mechanism, while the background shear currents are induced by basin-scale internal seiches created using a tilted tank suddenly returned to the untilted configuration. Both mechanisms are readily realizable in a laboratory environment. Depending on the configuration of the initial density profile, both internal solitary-like waves (ISWs) and finite-amplitude dispersive wave trains (DWTs) may be observed in the simulations. The ISWs are observed when the pycnocline is located relatively far away from the middepth and/or the background shear is oriented against the wave-induced velocity shear. Comparison of the waveforms observed in the simulations to those described by the fully nonlinear Dubreil-Jacotin–Long (DJL) theory suggests that these waves are indeed solitary-like, implying that laboratory experiments of ISWs propagating in a shear background current can be realized in a relatively straightforward manner. As the pycnocline approaches the middepth and/or the background shear reverses its polarity and increases its magnitude, the waves tend to have a smaller amplitude and a larger half-width. When a wave's amplitude is less than approximately 3% of its half-width, the dispersive effect becomes dominant and the wave loses its solitary-like form. In this case, a finite-amplitude DWT is formed instead, in which the leading wave has a reduced propagation speed from the ISW propagation speed. Analysis based on the DJL theory suggests that exact solitary waves of similar amplitude in such a background environment have an extremely large half-width that is on the same order of magnitude as the finite length of the simulation domain, implying that their formation cannot be supported in a laboratory environment.

Recent discoveries have allowed a mathematical and computational foundation for understanding the rheology of particles suspended in viscoelastic fluids. We employ these new tools to understand the extensional rheology of such suspensions. We accomplish this by first calculating the renormalized particle contribution to the extensional viscosity in such a suspension in the dilute particle limit over a wide range of extensional Weissenberg number and Hencky strain. The models we use for the suspending fluids are the simplest dumbbell models—the Oldroyd-B, FENE-P, and Giesekus models—such that our results are general for polymer solutions which exhibit strong strain hardening at values of the Weissenberg number above those which engender the coil-stretch transition, $\mathrm{Wi}\ge 0.5$. We demonstrate that the effect of particles on the “extra elongational viscosity” relative to the fluid is nonmonotonic in strain (increasing for small strain and then decreasing for large strain). Thus at a fixed strain, the particle “extra viscosity” relative to the fluid may increase or decrease with Wi. We demonstrate that this interesting behavior is due to the interplay between the two contributions of the “particle-induced fluid stress” (PIFS) and the “stresslet” to the extra viscosity. The contribution of the particle-induced fluid stress to the suspension viscosity increases at small strain but plateaus and then decreases at higher values of the strain. Thus, at small strain the local velocity gradients near a particle increase the polymer stretch, while for greater strain, polymers which have undergone the coil-stretch transition collapse in the neighborhood of a given particle. On the other hand, the stresslet contribution to the viscosity relative to the fluid decreases monotonically as the polymer stretch surrounding a given particle “shields” the particle and thus reduces the local surface tractions. Beyond Hencky strain of approximately 2 the decreasing value of the stresslet coupled with the plateauing of the PIFS, causes the overall reduction in the particle-induced extra viscosity relative to that of the fluid.

Controlled gliding is one of the most energetically efficient modes of transportation for natural and human powered fliers. Here we demonstrate that gliding and landing strategies with different optimality criteria can be identified through deep-reinforcement-learning without explicit knowledge of the underlying physics. We combine a two-dimensional model of a controlled elliptical body with deep-reinforcement-learning (D-RL) to achieve gliding with either minimum energy expenditure, or fastest time of arrival, at a predetermined location. In both cases the gliding trajectories are smooth, although energy/time optimal strategies are distinguished by small/high frequency actuations. We examine the effects of the ellipse's shape and weight on the optimal policies for controlled gliding. We find that the model-free reinforcement learning leads to more robust gliding than model-based optimal control strategies with a modest additional computational cost. We also demonstrate that the gliders with D-RL can generalize their strategies to reach the target location from previously unseen starting positions. The model-free character and robustness of D-RL suggests a promising framework for developing robotic devices capable of exploiting complex flow environments.

When wind blows at the surface of a liquid of sufficiently high viscosity, a wave packet of small amplitude is first generated, which sporadically forms large-amplitude fluid bumps that rapidly propagate downstream. These nonlinear structures, first observed by Francis [J. R. D. Francis, Philos. Mag. 42, 695 (1954)], have an almost vertical rear facing the wind and a weak slope at the front. We call them viscous solitons. We investigate their dynamics in a wind-tunnel experiment using silicon oil of kinematic viscosity $1000{\mathrm{mm}}^2{\mathrm{s}}^{-1}$ by means of laser sheet profilometry and particle image velocimetry. We give evidence of their subcritical nature: They are emitted in a region of large shear stress but, once formed, they are sustained by the wind and propagate in a region of lower stress. Their propagation velocity is given by the balance between aerodynamic drag in the air and viscous drag in the liquid. The stable soliton branch of the subcritical bifurcation diagram is reconstructed from the measured soliton amplitude at various wind velocities and distances along the channel. At large wind velocity, the emission frequency of solitons increases, resulting in a long-range sheltering of downstream mature solitons by newly formed upstream solitons, which limits their course.

Capillary fingering is a displacement process that can occur when a nonwetting fluid displaces a wetting fluid from a homogeneous disordered porous medium. Here, we investigate how this process is influenced by a pore size gradient. Using microfluidic experiments and computational pore-network models, we show that the nonwetting fluid displacement behavior depends sensitively on the direction and the magnitude of the gradient. The fluid displacement depends on the competition between a pore size gradient and pore-scale disorder; indeed, a sufficiently large gradient can completely suppress capillary fingering. By analyzing capillary forces at the pore scale, we identify a nondimensional parameter that describes the physics underlying these diverse flow behaviors. Our results thus expand the understanding of flow in complex porous media and suggest a new way to control flow behavior via the introduction of pore size gradients.

The motion of helicoidal swimmers moving in a pool filled with a granular medium is studied experimentally. The horizontal displacement through granular beads is measured considering geometrical modifications of the swimmer, the size, and frictional properties of the media. We found three main parameters which affect the swimming performance: the diameter, the wavelength, and the angle of the helix. The swimming speed scales with the rotation speed, $\omega R$. The size of particles does not affect the swimming speed significantly; the swimming speed is reduced when the particle's angle of repose increases. It was found that a maximum swimming speed is achieved when the helix angle is close to ${55}^{\circ }$. The experimental data are compared with predictions of the granular resistive force theory, which was extended to be applicable for a swimmer with a rigid helical tail, leading to good agreement.

We consider the steady flow of a viscoelastic liquid film over an inclined wall with sinusoidal corrugations of arbitrary wavelength and depth. We develop a computational model and carry out detailed numerical simulations based on the finite-element method to investigate this flow. To this end, we solve the two-dimensional momentum and mass conservation equations while employing the Phan-Thien-Tanner (PTT) constitutive model to account for the rheology of the viscoelastic material. An elliptic grid generation scheme is used to follow the large deformations of the liquid film. We perform a thorough parametric analysis to investigate the combined effects of elasticity, inertia, and capillary and viscous forces on the characteristics of the steady flow. The results of our simulations indicate that fluid elasticity suppresses interfacial deformation at low flow rates, whereas at moderate values of $\mathrm{Re}$ it enhances the deformation considerably. In the latter case, elastic forces may even give rise to the formation of a static hump and a cusp at the free surface, the size of which increases with the relaxation time of the liquid. The resonance of the liquid film with the substrate undulations is also enhanced by shear thinning. Interestingly, it is predicted that under certain conditions the transition to the inertia regime is not smooth and a hysteresis loop arises, which is the signature of an abrupt change of the film shape, since its high deformations cannot be sustained. Additionally, we have performed calculations for a wide range of different geometrical characteristics of the substrate. We find that viscoelastic effects become more pronounced in the case of long-wavelength wall undulations, while for substrates with short wavelengths the effect of shear thinning is less significant due to the presence of vortices inside the corrugations.

The linear hydrodynamic stability of a film of viscoelastic fluid flowing down an inclined wavy surface is studied. We investigate the stability of the flow with respect to infinitesimal two- and three-dimensional (2D and 3D) disturbances and employ the Floquet-Bloch theory to examine the effect of periodic disturbances of any wavelength. The study is based on the numerical solution of the momentum equations along with the Phan-Thien–Tanner (PTT) model to account for material viscoelasticity. The generalized eigenvalue problem is solved using Arnoldi's algorithm, in a Newton-like implementation in order to calculate faster the critical conditions for the onset of the instability. Our results are in excellent agreement with the previous experimental and theoretical results in the case of Newtonian liquids flowing over flat and undulating substrates and viscoelastic liquids over flat substrates. We present detailed stability maps for finite amplitude of the wall corrugations and a wide range of material parameters. Our calculations indicate that fluid elasticity is primarily stabilizing, while shear thinning of the fluid tends to destabilize the fluid flow. In order to investigate the mechanisms involved, we perform an energy analysis of the flow under long-wave disturbances indicating that the convection of stress-gradient disturbances provides an additional viscoelastic mechanism for the destabilization of the flow, in contrast to the base state stress gradients which contribute to stabilization of the flow. Besides the usual long-wave instability, conditions are identified which lead to unstable disturbances of wavelength equal or smaller than the wavelength of the substrate. Experimental observations for Newtonian liquids have indicated that these short-wave instabilities will dominate and a similar behavior is predicted for viscoelastic liquids. Sometimes, before the short-wave instabilities, a hysteresis loop in the steady flow can be identified, which leads to a sharp change in the critical frequency. Finally, we examine the stability of the flow when subjected to disturbances in the spanwise direction and show that for highly elastic liquids 3D instabilities may arise.

The passage of microfluidic droplets in miscible thread-forming flows is shown to improve the mixing efficiency of low- and high-viscosity fluids at the small scale. Hydrodynamic interactions between recirculating flow patterns and viscous core-annular flows are experimentally investigated in square microchannels. The deformation of thick segmented flows injected in thin fluids is examined through the evolution of droplet size, spacing, and velocity along a square microchannel. Droplets also help in probing the various states of lubrication of the viscous central stream in the bulging, tubing, and threading regimes at various capillary and Péclet numbers. A range of intriguing flow phenomena is revealed using a dual approach based on thread and droplet mutual behaviors, including droplet breakup, formation of low-viscosity currents, thread splitting, viscous fingering, and viscous buckling instabilities. The thread-forming ability of miscible fluids having large viscosity contrasts is discussed in conjunction with mixing applications where a thinner or a viscosifier is continuously added to a viscosity-differing fluid in confined microsystems.

Practically all flows are turbulent in nature and contain some kind of irregularly shaped particles, e.g., dirt, pollen, or life forms such as bacteria or insects. The effects of the particles on such flows and vice versa are highly nontrivial and are not completely understood, particularly when the particles are finite sized. Here, we report an experimental study of millimetric fibers in a strongly sheared turbulent flow. We find that the fibers show a preferred orientation of $-0.38\pi \pm 0.05\pi$ ($-68\pm 9^{\circ }$) with respect to the mean flow direction in high-Reynolds-number Taylor-Couette turbulence, for all studied Reynolds numbers, fiber concentrations, and locations. Despite the finite size of the anisotropic particles, we can explain the preferential alignment by using Jefferey's equation, which provides evidence of the benefit of a simplified point-particle approach. Furthermore, the fiber angular velocity is strongly intermittent, again indicative of point-particle-like behavior in turbulence. Thus large anisotropic particles still can retain signatures of the local flow despite classical spatial and temporal filtering effects.

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.