Theory is developed to predict the growth and structure of “sibling” waves developing through triadic resonant instability of a vertically confined mode-1 internal “parent” wave in uniform stratification including the influence of background rotation. For a sufficiently hydrostatic parent wave, two branches for growth of sibling waves are dominant. The branch with largest growth rate corresponds to sibling waves having frequencies much larger than that of the parent; the other branch corresponds to sibling waves having frequencies close to half the frequency of the parent. Numerical simulations show that sibling waves corresponding to the subharmonic branch appear in practice. In the absence of rotation, the sibling waves corresponding to this branch are predicted to have near-constant growth rate as their horizontal wave number increases. With rotation, however, the growth rate peaks at moderate wave number. In all cases, as confirmed by numerical simulations, the e-folding time for the growth of the sibling waves can be thousands of buoyancy periods for parent waves having amplitudes typical of realistic oceanic internal modes. In nonuniform stratification, the parent wave self-interacts immediately to force superharmonics. Nonetheless, numerical simulations with symmetric top-hat stratification show that triadic resonant instability eventually emerges. Such emergence is not evident in simulations with stratification more representative of the ocean. The results suggest a reconsideration of the efficacy of parametric subharmonic instability in leading to the breakdown of low-mode internal tides in the ocean.

Performing global resolvent analysis for high-Reynolds-number turbulent flow calls for the handling of a large discrete operator. Even though such a large operator is required in the analysis, most applications of resolvent analysis extracts only a few dominant resolvent response and forcing modes. Here we consider the use of randomized numerical linear algebra to reduce the dimension of the resolvent operator for achieving computational speed up and memory saving compared to the standard resolvent analysis. To accomplish this goal, we utilize sketching of the linear operator with random test matrices with a Gaussian distribution and with insights from the base flow incorporated to perform singular value decomposition on a low-rank matrix holding dominant characteristics of the full resolvent operator. The strength of the randomized resolvent analysis is demonstrated on a turbulent separated flow over an airfoil. This randomized approach clears the path towards tackling resolvent analysis for higher-Reynolds-number bi- and triglobal base flows.

We revisit the tears of wine problem for thin films in water-ethanol mixtures and present a model for the climbing dynamics. The formulation includes a Marangoni stress balanced by both the normal and tangential components of gravity as well as surface tension which lead to distinctly different behavior. The prior literature did not address the wine tears but rather the behavior of the film at earlier stages and the behavior of the meniscus. In the lubrication limit we obtain an equation that is already well known for rising films in the presence of thermal gradients. Such models can exhibit nonclassical shocks that are undercompressive. We present basic theory that allows one to identify the signature of an undercompressive wave. We observe both compressive and undercompressive waves in new experiments, and we argue that, in the case of a preswirled glass, the famous “wine tears” emerge from a reverse undercompressive shock originating at the meniscus.

Vibrated substrates are useful tools to move and reshape drops. Experiments usually involve periodical excitation at small amplitude but new effects can be generated by using larger and/or quicker movements. Recently, impulsive motion of a plate has been shown to enhance drop takeoff. In this article we discuss the beautiful, elusive shapes obtained when a water droplet deposited on a nonwetting substrate is subjected to a strong vertical impulse. Drops are highly reshaped to form truncated cones that eventually collapse. We report and discuss the evolution of the geometrical features of these so-called “vase-shaped droplets”. Our understanding of the physical phenomena at stake in the experiment allows us to model the evolution of the shape of the droplet. In particular, we show that the top and the bottom of the vase follow dynamics with different timescales, leading to the truncated cone shape. We show that it is possible to play with one of these dynamics by changing the wetting of the liquid on the rising substrate, and are thus able to change the shape of the vase from cylindrical to completely flat.

When a dipolar spheroid is subjected to a shear flow in the presence of an external field, there is a torque due to the dipole-field interaction which tends to align the spheroid in the direction of the field, and a torque due to the shear flow which tends to rotate the particle in closed “Jeffrey orbits.” The presence of an external field is known to result in complex phase portraits containing a limit cycle along with multiple stationary points [Almog and Frankel, J. Fluid Mech. 289, 243 (1995); Sobecki et al., Phys. Rev. Fluids 3, 084201 (2018)]. When the external field is in the flow plane, depending on the strength and orientation of the field, the phase portrait in orientation space could have two to six stationary nodes and/or a limit cycle. When the external field strength is low, there are two stationary points off the flow plane which are both stable/unstable and an unstable/ stable limit cycle on the flow plane. When the external field strength is high, there is one stable node where the particle orientation is parallel to the field, and one unstable node where the particle orientation is anti-parallel to the field. As the external field strength is increased, the manner in which the phase portrait evolves depends on the external field orientation with respect to the flow direction and the particle shape factor. It is shown that complex phase portraits result from the relatively simple variations in the location of the stationary points in the orientation space as the parameter $\mathrm{\Sigma }$ is varied. Here $\mathrm{\Sigma }$ is the dimensionless ratio of the torques due to the external field and the shear flow. Depending on the aspect ratio of the particle, there are up to two saddle-node bifurcations, two subcritical bifurcations on the flow plane, and one reverse saddle-node merger of two stationary points off the flow plane. The limits of an ideal thin rod/disk are shown to be singular limits, where there are no stationary points off the flow plane even for a low external field, and the orientation of the particle changes discontinuously as the cross-stream component of the external field changes sign.

The motion of a spherical particle undergoing electrophoresis in weakly inertial or viscoelastic shear flow is quantified via asymptotic analysis. We are motivated by several experimental studies reporting cross-streamline migration of electrophoretic colloids in Poiseuille microchannel flow. Specifically, particles migrate in a Newtonian liquid to the center (walls) of a channel when their electrophoretic velocity is in the opposite (same) direction to (as) the flow. Here, we calculate that weak fluid inertia causes a leading-order cross-streamline lift force of magnitude $5.50\varepsilon \left|\zeta \right|a^3\rho \dot{\gamma }{E}^{\infty }/\mu$ for electrophoresis along the velocity axis of an unbounded simple shear flow, where $\zeta$ and $a$ denote the particle zeta potential and radius, respectively; $\rho$, $\mu$, and $\varepsilon$ are the fluid density, viscosity, and permittivity, respectively; $\dot{\gamma }$ is the shear rate of the ambient flow; and $E^{\infty }$ is the strength of the imposed electric field. This force acts to propel the sphere to shear streamlines that, in a frame translating with the particle, are directed reverse to the electrophoretic motion, which is consistent with the above-mentioned experiments. Other recent experiments have observed migration of electrophoretic particles in Poiseuille flow of a viscoelastic polymer solution: the migration direction is opposite to that in a Newtonian liquid. Here, we calculate a leading-order cross-streamline lift force of magnitude $7.07(1-3.33{\mathrm{\Psi }}_2/{\mathrm{\Psi }}_1)\varepsilon \left|\zeta \right|a{\mathrm{\Psi }}_1\dot{\gamma }{E}^{\infty }/\mu$ for electrophoresis in simple shear flow of a second-order fluid, where ${\mathrm{\Psi }}_1$ and ${\mathrm{\Psi }}_2$ are the first and second normal stress coefficients, respectively. The lift is toward streamlines moving in the direction of electrophoresis for ${\mathrm{\Psi }}_2/{\mathrm{\Psi }}_1<0.3$ (a reasonable assumption for polymeric liquids, where ${\mathrm{\Psi }}_2$ and ${\mathrm{\Psi }}_1$ are negative and positive, respectively), which is consistent with the experiments. Finally, an estimation of the magnitude of the lift forces and associated drift velocities further suggests that the cumulative effect of weak instantaneous inertia or viscoelasticity is responsible, or at least contributes appreciably, to the observed migration.

Three-dimensional (3D) boundary layers are common flows in aircraft but present many problems to instability analysis and transition prediction. The difficulties of 3D boundary layers are reviewed and a method is proposed to predict the linear evolution of infinitesimal perturbations in 3D boundary layers, named as RTPSE, in which the line-marching parabolized stability equation (PSE) is improved by applying the ray tracing (RT) theory. Two major improvements are achieved. One is that the marching line is predefined along the direction of group velocity, which is related to both the characteristic line of local dispersion relation and the direction of energy propagation. Another is that the variation of the real part of the spanwise wave number is predicted by RT theory, while its imaginary part is determined based on the conservation relation of generalized growth rate. The implementation of RTPSE for 3D boundary layers is given in detail and involves linear stability theory, the PSE, and RT theory. Both the tracing ray and spanwise wave number are calculated in the real number space, only leading to a second-order error. Direct numerical simulation is performed to verify and validate the prediction by RTPSE in a 3D supersonic boundary layer on a blunt cone with a half angle of $7^{\circ }$ and an angle of attack of $9^{\circ }$. Results show that RTPSE can accurately predict the variation of spanwise wave number and linear evolution of disturbances for the whole wave packet, for stationary crossflow waves and for traveling crossflow waves, while the traditional PSE cannot. The application condition of RT theory is investigated numerically, and the caustic does not occur for unstable disturbances, implying that RTPSE is fully applicable to predict the linear evolution of disturbances in 3D boundary layers.

A recent magnetic resonance imaging study [Boyce et al., Phys. Rev. Fluids 4, 034303 (2019)] found that when two bubbles were injected side-by-side into an incipiently fluidized bed, below a critical injection volume, one bubble collapsed, while the other bubble survived, reaching the bed surface. Limitations in the experimental measurements left open questions about the apparent collapse: (1) did the bubble actually collapse or rather move out of the imaging plane and (2) if the bubble collapsed, was it due to gas transfer between bubbles or a preference for gas to channel toward the surviving bubble. Here we demonstrate that computational fluid dynamics–discrete element method (CFD-DEM) simulations can recreate this phenomenon. Simulation results reveal that the bubble does in fact collapse and that this collapse occurs because the slight size difference between the two bubbles causes gas flow to channel preferentially to the larger bubble, leaving the smaller bubble without enough gas flow to support its roof.

This work employs invariant solutions of the Navier-Stokes equations to study the interaction between finite-size particles and near-wall coherent structures. We consider horizontal plane Couette flow and focus on Nagata's upper-branch equilibrium solution at low Reynolds numbers where this solution is linearly stable. When adding a single heavy particle with a diameter equivalent to 2.5 wall units ($\frac{1}{12}$ of the gap width), we observe that the solution remains stable and is essentially unchanged away from the particle. This result demonstrates that it is technically feasible to utilize exact coherent structures in conjunction with particle-resolved direct numerical simulation. While translating in the streamwise direction, the particle migrates laterally under the action of the quasistreamwise vortices until it reaches the region occupied by the low-speed streak, where it attains a periodic state of motion, independent of its initial position. As a result of the ensuing preferential particle location, the time-average streamwise particle velocity differs from the plane-average fluid-phase velocity at the same wall distance as the particle center, as previously observed in experiments and in numerical data for fully turbulent wall-bounded flows. Additional constrained simulations where the particle is maintained at a fixed spanwise position while freely translating in the other two directions reveal the existence of two equilibria located in the low-speed and the high-speed streak, respectively, the former being an unstable point. A parametric study with different particle to fluid density ratios is conducted which shows how inertia affects the spanwise fluctuations of the periodic particle motion. Finally, we discuss a number of potential future investigations of solid particle dynamics which can be conducted with the aid of invariant solutions (exact coherent structures) of the Navier-Stokes equations.

When two drops are slowly brought together and first touch, a microscopic liquid neck or a bridge forms between them. The expansion of the neck is controlled by the capillary (Laplace) pressure which diverges when the curvature of the interface is infinite at the point where the drops first touch. The change in topology and the flows that ensue as time advances and the bridge grows from microscopic to macroscopic scales, and the two drops merge into one, are intimately coupled to this singularity in the dynamics. Despite the large volume of work dedicated to this problem, currently experiment, theory, and computation are not in complete agreement with respect to the earliest times following the initial contact of the two drops. Experiments, supported by simulations, report an initial regime where the radius of the connecting bridge grows linearly in time before a transition to either a Stokes regime or an inertial regime where either viscous or inertial force balances capillary (surface tension) force. In the initial linear regime, referred to as the inertially limited viscous (ILV) regime, all three forces are thought to be important. This is in contrast to theory which predicts that all coalescence events begin in the Stokes regime where inertia is negligible. Here we use high-accuracy numerical simulations to show that the ILV regime is only realized when the two coalescing drops are initially separated by a finite distance. Moreover, for two drops that initially just touch at a point, coalescence always begins in the Stokes regime. It is demonstrated that the linear ILV regime is more akin to a Taylor-Culick-type regime whose existence and duration are purely consequences of the use of an initial bridge of finite size that poorly approximates the point contact condition that is a cardinal feature of the coalescence singularity.

Granular jumps—the change in height and depth-averaged velocity during granular flows—occur during transitions from thin and fast flows (supercritical) to thick and slow flows (subcritical). The present paper describes discrete element method simulations inspired by recent laboratory experiments, which produce standing jumps in two-dimensional free-surface dry granular flows down a slope. Special attention is paid to characterizing and measuring the finite length of those standing granular jumps, as well as to deciphering their internal structure. By varying macroscopic quantities, such as slope angle and mass discharge, and microscopic properties, such as interparticle friction and grain diameter, a rich variety of granular jump patterns is observed. Hydraulic-like granular jumps with an internal waterlike roller are identified, in addition to diffuse laminar granular jumps and steep colliding granular jumps. Moreover, a recently established depth-averaged relation for the prediction of jump heights is fed with the measured jump length and fitted against the numerical simulations to examine the effective friction in the granular medium for each type of jump. The dominant component of the general friction law is found to be different when transitioning from one jump pattern to the others. This study demonstrates that the granular jump pattern, and particularly its geometry and internal structure, can offer a stringent test for addressing the dissipation mechanisms that govern the flowability of granular media.

External forces, such as gravity, play significant role in the swimming properties of autonomous biological microswimmers as well as artificial swimming robots. Here we have studied the influence of the external forces on the transport characteristics of the triangular bead-spring microswimmers. The microswimmer, formed by connecting three beads using three springs in an equilateral triangular arrangement, is capable of performing autonomous translational (“mover”) and rotational (“rotor”) motions. We show that for a mover triangle the application of a small external force results in the alignment of swimming direction with that of the external force, a phenomenon known as “gravitaxis.” We demonstrate that this gravitactic nature of the active triangle is purely due to the hydrodynamic interaction among the beads. Under large external force, however, the gravitactic nature is lost. This transition from gravitactic to nongravitactic motion of the microswimmer is characterized by a saddle node or pitchfork bifurcations (depending on nature of active forces), where the strength of the critical external force scales linearly with the active force amplitude, $f_e^c\sim f_a$. However, for the rotor triangle only saddle node bifurcation is observed, which results in a vanishing angular velocity as the strength of the external force is increased. The critical value of the external force for the rotor, however, scales as $f_e^c\sim f_a^{2/3}$. These findings will provide insights into the nature of biological swimming under gravity, especially the gravitactic microorganisms such as Chlamydomonas, as well as help in the design of underwater vehicles.

Particles are today the main tool to study superfluid turbulence and visualize quantum vortices. In this Rapid Communication, we study the dynamics and the spatial distribution of particles in coflow and counterflow superfluid helium turbulence in the framework of the two-fluid Hall-Vinen-Bekarevich-Khalatnikov (HVBK) model. We perform three-dimensional numerical simulations of the HVBK equations along with the particle dynamics that depends on the motion of both fluid components. We find that, at low temperatures, where the superfluid mass fraction dominates, particles strongly cluster in vortex filaments regardless of their physical properties. At higher temperatures, as viscous drag becomes important and the two components become tightly coupled, the clustering dynamics in the coflowing case approach those found in classical turbulence, while under strong counterflow, the particle distribution is dominated by the quasi-two-dimensionalization of the flow.

We present experimental evidence that, together with the integral fluctuation theorem, which is fulfilled with high accuracy, a detailed-like fluctuation theorem holds for large entropy values in cascade processes in turbulent flows. Based on experimental data, we estimate the stochastic equations describing the scale-dependent cascade process in a turbulent flow by means of Fokker-Planck equations, and from the corresponding individual cascade trajectories an entropy term can be determined. Since the statistical fluctuation theorems set the occurrence of positive and negative entropy events in strict relation, we are able to verify how cascade trajectories, defined by entropy consumption or entropy production, are linked to turbulent structures: Trajectories with entropy production start from large velocity increments at large scale and converge to zero velocity increments at small scales; trajectories with entropy consumption end at small scale velocity increments with finite size and show a lower bound for small scale increments. A linear increase with the magnitude of the negative entropy value is found. This indicates a tendency to local discontinuities in the velocity field. Our findings show no lower bound of negative entropy values and thus for the corresponding piling up velocity differences of the small scale structures.

The energy cascade is the most significant feature that separates turbulence from other unsteady flows, and results from the behavior of the nonlinear term in the Navier-Stokes equations. The mathematical form of this term, however, places constraints on exactly how it can act. Here we consider the action of the nonlinear term in physical space rather than in Fourier space, where the energy transfer between scales can be interpreted as a mechanical process where some scales do work on others. This formulation reveals the fundamental role played by geometry, as work can only be done when the eigenframes of the turbulent stress and strain rate are appropriately aligned. By comparing a direct numerical simulation of the Navier-Stokes equations, an ensemble of random solenoidal vector fields, and a random sampling of uniform eigenframe alignments, we show that this geometric alignment plays a much stronger role in determining the flux between scales than do the magnitudes of the stress and strain rate. We also show that when the alignment is effectively two dimensional, even when embedded in a three-dimensional flow, the energy flux is typically inverse, suggesting that the inverse cascade in two-dimensional turbulence may have a kinematic origin. Our results point to some potentially fruitful directions for turbulence modeling.

Chemically active droplets exhibit complex avoiding trajectories. While heterogeneity is inevitable in active matter experiments, it is mostly overlooked in their modeling. Exploiting its geometric simplicity, we fully resolve the head-on collision of two swimming droplets of different radii and demonstrate that even a small contrast in size critically conditions their collision and subsequent dynamics. We identify three fundamentally different regimes. The resulting high sensitivity of pairwise collisions is expected to profoundly affect their collective dynamics.

Although droplet retraction is involved in many applications, it receives less attention than droplet spreading. We present a detailed study of the retraction dynamics for water droplets with varying velocities impacting onto surfaces ranging from hydrophilic to superhydrophobic. The morphologies of retracting droplets are compared in detail. Two distinct retraction modes are classified: the inertial mode and the capillary mode. The retracting droplet with an inertial mode features a rim-lamella structure with a slowly decreasing dynamic receding contact angle while the capillary mode shows a collapsed rim and lamella and capillary wave propagation. During the retraction process, the inertial mode first takes place, and then the capillary mode occurs at a later stage. The inertial mode becomes more dominant with increasing impacting velocities. The dependencies of the retraction rate on impacting velocities and surface wettabilities are justified from the experimental measurements. The inertial-mode retraction rate increases with increasing velocity, while the capillary-mode retraction rate decreases. The retraction rate is generally higher for more hydrophobic surfaces, and its dependence on impacting velocities becomes stronger. An improved shape assumption for the retracting droplet is proposed by introducing an averaged dynamic receding contact angle and a shape factor. This semiempirical model can well explain the dependency of the inertial-mode retraction rate on impacting velocities for water droplets. Last but not the least, the normalized retraction curves generally collapse into a universal curve, revealing certain similarity behaviors for the droplet retraction on solid surfaces.

When a bubble bursts at the surface of a liquid, it creates a jet that may break up and produce jet droplets. This phenomenon has motivated numerous studies due to its multiple applications, from bubbles in a glass of champagne to ocean/atmosphere interactions. We simulate the bursting of a single bubble by direct numerical simulations of the axisymmetric two-phase liquid-gas Navier-Stokes equations. We describe the number, size, and velocity of all the ejected droplets, for a wide range of control parameters, defined as nondimensional numbers, the Laplace number which compares capillary and viscous forces and the Bond number which compares gravity and capillarity. The total vertical momentum of the ejected droplets is shown to follow a simple scaling relationship with a primary dependency on the Laplace number. Through a simple evaporation model, coupled with the dynamics obtained numerically, it is shown that all the jet droplets (up to 14) produced by the bursting event must be taken into account as they all contribute to the total amount of evaporated water. A simple scaling relationship is obtained for the total amount of evaporated water as a function of the bubble size and fluid properties. This relationship is an important step toward building a physics-based model of the ocean-atmosphere water vapor fluxes controlled by bubbles bursting at the surface.

We study droplet dynamics and breakup in generic time-dependent flows via a multicomponent lattice Boltzmann algorithm, with emphasis on flow startup conditions. We first study droplet breakup in a confined oscillatory shear flow via two different protocols. In one setup, we start from an initially spherical droplet and turn on the flow abruptly (“shock method”); in the other protocol, we start from an initially spherical droplet as well, but we progressively increase the amplitude of the flow, by allowing the droplet to relax to the steady state for each increase in amplitude, before increasing the flow amplitude again (“relaxation method”). The two protocols are shown to produce substantially different breakup scenarios. The mismatch between these two protocols is also studied for variations in the flow topology, the degree of confinement, and the inertia of the fluid. All results point to the fact that under extreme conditions of confinement the relaxation protocols can drive the droplets into metastable states, which break only for very intense flow amplitudes, but their stability is prone to external perturbations, such as an oscillatory driving force.

This study compares the predicted synthetic flow fields generated based on the representative structures of the attached eddy model to experimental data captured using Particle Image Velocimetry of a turbulent boundary layer. To this end, wall-parallel and cross-stream planar fields are analyzed qualitatively and quantitatively by examining instantaneous flow features and by statistical two-point correlation functions, respectively. Our results reveal that although single-point flow statistics are in good agreement with the experimental data, a comparison of instantaneous flow fields and multipoint statistics between the attached eddy model and experiments shows differences in the spatial coherence. Based on these observations, a modification to the placement of the representative eddies in the attached eddy model is proposed that incorporates the meandering of these flow structures, which has been extensively reported in turbulent boundary layers. Our results reveal that this subtle modification provides a superior spatial representation of a turbulent boundary layer from the attached eddy model by reducing periodic effects and the overestimated spatial coherence. Similar improvements are also reported for the spatial representation of the spanwise velocity component.