Diffusiophoresis and diffusioosmosis are electrokinetic phenomena where relative motion is induced between a charged surface and a surrounding electrolyte due to a concentration gradient of ions. In the literature, a relative velocity between a surface and the electrolyte has been derived for a valence-symmetric $\left(z\text{:}z\right)$ electrolyte. In this article, we reformulate the governing equations in a convenient form based on a systematic generalization of the nonlinear Poisson-Boltzmann equations in the limit of a thin double layer, which allows us to derive results for diffusiophoretic and diffusioosmotic velocities for a mixture of electrolytes with a general combination of cation and anion valences. We find that the relative motion depends significantly on ion valences. We also provide analytical approximations for the diffusiophoretic and diffusioosmotic velocities and discuss their accuracy and applicability. Further, we tabulate diffusiphoretic velocities for some common cases, which highlights the importance of asymmetry in cation and anion valences. Finally, we discuss the validity of our assumptions and the importance of effects such as finite ion size, dielectric decrement, and surface conduction for typical experimental conditions.

Free-shear-flow turbulence with sufficiently fast advection speeds radiates Mach waves, with steepened and skewed pressure profiles. These form within about a mixing layer thickness and dominate the sound field. Their generation and propagation is investigated through comparison of numerical simulations of a temporally developing mixing layer with a series of model-flow simulations designed to isolate physical mechanisms. The first of these are numerical simulations of nonlinearly saturating instability waves, which despite being much simpler than corresponding turbulence reproduce key features of the sound. Motivated in part by this agreement, instability analysis is used to motivate the inclusion of artificial sources in turbulence simulations that are designed to induce specific alterations to the turbulence structures, leaving most of its broadband spectrum unchanged. Comparisons show how insensitive the radiation is to the particular structure. To assess how strongly the near-field sound is coupled to the turbulence, a high dilatational dissipation is imposed to suppress the waves. This significantly reduces radiated pressure intensity, but little changes the Reynolds stresses ($<8\%$), which supports a source-plus-sound perspective. Given this, a low-dimensional nonlinear gas-dynamic mechanism is proposed for the generation and near-field propagation of the waves. The analysis uses a second-order wavy-wall asymptotic solution, and it reproduces the key observations: the sound-field structure, pressure skewness, and even the radiated pressure levels to within a factor of 2.

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

We investigate, both by way of theory and by experiments, the mechanically forced Faraday instability in immiscible three-fluid systems that is two-interface fluid layers. The theoretical model suggests that two-interface fluid layers offer underlying physics quite distinct from the typical single-interface system due to the coupling of the fluid interfaces, resulting in alternating double-tongued stability curves. This allows for the possibility of unique codimension points unattainable in traditional two-fluid systems. In addition, the presence of a third fluid in the problem can lead to either enhanced or delayed destabilization of the system at target frequencies. Experimental results qualitatively support the theory, though precise agreement between the theory and the experiment is hindered due to the sidewall damping.

We study the effects of heterogeneity on interfacial pinning and hydrodynamic drag using molecular dynamics (MD) simulations of Janus nanospheres at a liquid-vapor interface. We construct the free-energy landscape for this system, both in the continuum approximation using surface tensions and the flat-interface approximation, and atomistically using MD and thermodynamic integration. The results of the two methods differ in detail due to interfacial distortion and finite width, as well as thermal fluctuations, and only the MD landscape is consistent with simulations of a nanosphere approaching the interface from the liquid or vapor side. When dragged along an interface, these Janus particles exhibit a velocity-dependent tilt accompanied by a weak variation in drag force, but never an enhancement of the drag force beyond the value when fully immersed. This velocity dependence arises when the interface is pinned at heterogeneities and prevents the particle from rotating, and similar behavior is observed for homogeneous but nonspherical particles. The occurrence of different particle orientations having different drag coefficients may lead to an apparent violation of the Stokes-Einstein relation.

The paper analyzes how the structure of the flow past solid (porosity $P=0\%$) and porous ($P=36\%$) plates containing parallel solid elements changes with increasing gap distance from the bottom surface. Fully three-dimensional large-eddy simulations are conducted in a straight channel with a horizontal smooth bottom surface and a vertical thin plate of height $H$ positioned at a distance $G$ from the channel bottom for $0\le G/H\le 1$. For small $G/H$ values, the main features of the wake flow are the formation of a bottom-attached recirculation region behind the plate and of corotating vortices originating from eddies shed inside the separated shear layer at the top edge of the plate. For sufficiently high $G/H$ values, the wake flow is dominated by the shedding of counter-rotating wake vortices behind the plate (antisymmetric vortex shedding). For $G/H\le 1$, the wake vortices originating in the bottom separated shear layer lose their coherence after they start interacting with the bottom surface. The effects of increasing $G/H$ on the turbulent kinetic energy, nondimensional streamwise drag force acting on the plate and friction velocity distributions on the bottom surface are analyzed. The paper also discusses how the bleeding flow present in the porous plate cases affects the dynamics of the large-scale coherent structures and its effect on the spectral content of the flow, streamwise drag force, and bed friction velocity distributions.

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

We present results from an asymptotic magnetohydrodynamic model that is suited for studying the rapidly rotating, low-viscosity regime typical of the electrically conducting fluid interiors of planets and stars. We show that the presence of sufficiently strong magnetic fields prevents the formation of large-scale vortices and saturates the inverse cascade at a finite length scale. This saturation corresponds to an equilibrated state in which the energetics of the depth-averaged flows are characterized by a balance of convective power input and ohmic dissipation. Quantitative criteria delineating the transition between finite-size flows and domain-filling (large-scale) vortices in electrically conducting fluids are found. By making use of the inferred and observed properties of planetary interiors, our results suggest that convection-driven large-scale vortices do not form in the electrically conducting regions of many bodies.

The global linear stability of the family of infinite rotating-cone boundary layers in an otherwise still fluid is investigated using a velocity-vorticity form of the linearized Navier-Stokes equations. The formulation is separable with respect to the azimuthal direction. Thus, disturbance development is simulated for a single azimuthal mode number. Numerical simulations are conducted for an extensive range of cone half-angles ($\psi \in [{20}^{\circ }:{80}^{\circ }]$) and stability parameters (Reynolds number, azimuthal mode number), where conditions are taken to be near those specifications necessary for the onset of absolute instability. A localized impulsive wall forcing is implemented that excites disturbances that form wave packets. This allows the disturbance evolution to be traced in the spatial-temporal plane. When a homogeneous flow approximation is utilised that neglects the spatial variation of the basic state, linear perturbations display characteristics consistent with local stability theory. For disturbances to the genuine spatially dependent inhomogeneous flow, global linear instability characterized by a faster than exponential temporal growth arises for azimuthal mode numbers greater than the conditions for critical absolute instability. Furthermore, a reasonable prediction for the azimuthal mode number needed to bring about a change in global behavior is achieved by coupling solutions of the Ginzburg-Landau equation with local stability properties. Thus, the local-global stability behavior is qualitatively similar to that found in the infinite rotating-disk boundary layer and many other globally unstable flows.

We consider the derivation and numerical solution of the flow of passive and active polar liquid crystals, whose molecular orientation is subjected to a tangential anchoring on an evolving curved surface. The underlying passive model is a simplified surface Ericksen-Leslie model, which is derived as a thin-film limit of the corresponding three-dimensional equations with appropriate boundary conditions. A finite element discretization is considered and the effect of hydrodynamics on the interplay of topology, geometric properties, and defect dynamics is studied for this model on various stationary and evolving surfaces. Additionally, we consider an active model. We propose a surface formulation for an active polar viscous gel and exemplarily demonstrate the effect of the underlying curvature on the location of topological defects on a torus.

We consider the unsteady regimes of an acoustically driven jet that forces a recirculating flow through successive reflections on the walls of a square cavity. The specific questions addressed are whether the system can sustain states of low-dimensional chaos when the acoustic intensity driving the jet is increased, and, if so, what are the pathway to it and the underlying physical mechanisms. We adopt two complementary approaches, both based on data extracted from numerical simulations: (i) We first characterize successive bifurcations through the analysis of leading frequencies. Two successive phases in the evolution of the system are singled out in this way, both leading to potentially chaotic states. The two phases are separated by a drastic simplification of the dynamics that immediately follows the emergence of intermittency. The second phase also features a second intermediate state where the dynamics is simplified due to frequency locking. (ii) Nonlinear time series analysis enables us to reconstruct the attractor of the underlying dynamical system and to calculate its correlation dimension and leading Lyapunov exponent. Both these quantities bring confirmation that the state preceding the dynamic simplification that initiates the second phase is chaotic. Poincaré maps further reveal that this chaotic state in fact results from a dynamic instability of the system between two nonchaotic states respectively observed at slightly lower and slightly higher acoustic forcing.

We explore the velocity fluctuations in a fluid due to a dilute suspension of randomly distributed vortex rings at moderate Reynolds number, for instance, those generated by a large colony of jellyfish. Unlike previous analysis of velocity fluctuations associated with gravitational sedimentation or suspensions of microswimmers, here the vortices have a finite lifetime and are constantly being produced. We find that the net velocity distribution is similar to that of a single vortex, except for the smallest velocities which involve contributions from many distant vortices; the result is a truncated $5/3$-stable distribution with variance (and mean energy) linear in the vortex volume fraction $\varphi$. The distribution has an inner core with a width scaling as ${\varphi }^{3/5}$, then long tails with power law ${\left|u\right|}^{-8/3}$, and finally a fixed cutoff (independent of $\varphi$) above which the probability density scales as ${\left|u\right|}^{-5}$, where $u$ is a component of the velocity. We argue that this distribution is robust in the sense that the distribution of any velocity fluctuations caused by random forces localized in space and time has the same properties, except possibly for a different scaling after the cutoff.

Direct numerical simulation is performed to investigate the flow physics of a supersonic turbulent boundary layer subjected to a longitudinal concave surface. Two physically consistent approaches are exploited to determine the boundary layer edge, and it is found that this turbulent boundary layer becomes noticeably thinned on the concave surface, which reflects the pronounced role of rise in density during this flow compression. In general the boundary layer is highly distorted, as manifested by wall pressure properties. Examinations of velocity statistics reveal that the scaling well established in canonical turbulence is violated in this distorted case. Mean streamwise velocity neither conforms to the universal log law nor obeys the velocity-defect law. Considerable increase is observed for Reynolds stresses throughout the concave surface, pointing to the effect of turbulence amplification. Nevertheless, the stress-bearing turbulent motions have barely changed in character, as evidenced by the quadrant analysis and structure parameter. The turbulence amplification is then understood by inspecting the production term of turbulent kinetic energy. We demonstrate that the outer boundary layer holds an increasingly important contribution to the total turbulence production throughout the concave surface. Accordingly, the amplification of turbulent kinetic energy is prominent in the outer layer. More insights are provided by inspecting the energy spectra. We find the outer-layer large structures are highly energized, even with an energy peak appearing in the lower-wake region, and they superimpose substantial large-scale energies on the near-wall region. Structural analyses demonstrate the organized turbulent motions, which are well scaled in canonical turbulence, have generally changed in their characteristic lengthscales under the influence of concave surface. Importantly, the flow visualization reveals stronger footprints overlaid onto the near-wall region, which suggests enhanced inner-outer interactions. This perspective, aided by the spanwise two-point correlations, is moreover supported by the quantification of amplitude modulation through a mathematical diagnostic tool. Results demonstrate that turbulence modulation is still governed by log-region superstructures, whereas the modulation strength has noticeably increased throughout the concave surface.

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

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

Recent work has identified persistent cluster states which were shown to be assembled and held together by hydrodynamic interactions alone [Driscoll et al. Nat. Phys. 13, 375 (2017)]. These states were seen in systems of colloidal microrollers; microrollers are colloidal particles which rotate about an axis parallel to the floor and generate strong, slowly decaying, advective flows. To understand these bound states, we study a simple, yet rich, model system of two microrollers. Here we show that pairs of microrollers can exhibit hydrodynamic bound states whose nature depends on a dimensionless number, denoted $B$, that compares the relative strength of gravitational forces and external torques. Using a dynamical system framework, we characterize these various states in phase space and analyze the bifurcations of the system as $B$ varies. In particular, we show that there is a critical value, $B^{*}$, above which active flows can beat gravity and lead to stable motile orbiting, or “leapfrog,” trajectories, reminiscent of the self-assembled motile structures, called “critters,” observed by Driscoll et al. We identify the conditions for the emergence of these trajectories and study their basin of attraction. This work shows that a wide variety of stable bound states can be obtained with only two particles. Our results aid in understanding the mechanisms that lead to spontaneous self-assembly in hydrodynamic systems, such as microroller suspensions, as well as how to optimize these systems for particle transport.

In numerical simulations with highly turbulent flows, the smallest scales are filtered; thereby, the effects of intermittency on those scales are neglected. When the flow is loaded by heavy small particles, the decimation of rapid changes in the velocity may lead to wrong results. This paper provides an approach to take account of subfiltered events of strong velocity jumps on the motion of heavy small particles. The idea is to force the filtered Navier-Stokes equations by a stochastic acceleration term with statistical properties identified by experiments and DNS. To this end, the stochastic model for supplemented acceleration contains the lognormal stochastic process for its norm (with long-range correlations) and the new stochastic model for the acceleration direction (with short-range correlations). The latter represents the Ornstein-Uhlenbeck process in Cartesian coordinates with relaxation to the local direction of the resolved vorticity; thereby, the geometry of highly stretched vortical structures is introduced in the designed model. Both stochastic processes depend on the local Reynolds number. The proposed flow model is applied for simulation of the background turbulence in which heavy particles are released and tracked. The assessment of single and two-time statistics of the particle acceleration and velocity clearly illustrates the advantage of the proposed flow model.

In stratified turbulence, buoyancy forces inhibit vertical motion and lead to anisotropy over a wide range of length scales, which is characterized by layerwise pancake vortices, thin regions of strong shear, and patches of small-scale turbulence. It has long been known that stratified turbulence becomes increasingly isotropic as one moves to smaller length scales, as the eddy timescale decreases towards and below the buoyancy period. This paper investigates the anisotropy of stratified turbulence across scales and the transition towards isotropy at small scales, using a variety of techniques. Direct numerical simulations of strongly stratified turbulence, with buoyancy Reynolds numbers ${\text{Re}}_b$ up to 50, are analyzed. We examine the relative contributions of different components of the strain rate tensor to the kinetic energy dissipation, the invariants of the isotropy tensor, directional kinetic energy spectra, and the subfilter energy flux across different length scales. At small scales, the degree of isotropy is determined by ${\text{Re}}_b$, while at the Ozmidov and larger scales, the anisotropy also depends on the Froude number. The change in the anisotropy with scale and with the parameters is examined in detail. Interestingly, Ozmidov-scale eddies are found to become increasingly isotropic as ${\text{Re}}_b$ increases, as characterized by the isotropy tensor invariants and the subfilter energy flux. At larger scales, the energy spectra for near-vertical wave vectors have a spectral slope around $-3$, which shallow towards $-1$ for near-horizontal wave vectors. These spectra converge beyond the Ozmidov scale, increasingly so for large ${\text{Re}}_b$. These results suggest that ${\text{Re}}_b\gtrsim 500$ would be necessary to obtain the same degree of small-scale isotropy that is found in similarly sized simulations of unstratified turbulence.

We study the flow past a finite-length yawed three-dimensional cylinder by a finite-volume–fictitious-domain method. We validate our non-boundary-fitted method against boundary-fitted numerical results for a finite-length cylinder whose axis is parallel to the streamwise direction. Drag and lift forces exerted on the cylinder and vortex shedding onset and frequency are carefully analyzed. Satisfactory agreement with published results gives strong confidence in the numerical methodology provided the boundary layer is accurately resolved. Then we carry out a detailed study of the flow past a yawed cylinder of aspect ratio $L/D=3$ (where $L$ is the cylinder length and $D$ is the cylinder diameter) at moderate Reynolds numbers $\left(25\le \text{Re}\le 250\right)$. We show that the wake patterns and the associated transitions depend strongly on Re and the yaw angle $\theta$ with respect to the streamwise direction. Various regimes are encountered including a standing-eddy pattern, steady shedding of one or two pairs of counterrotating vortices, periodic shedding of counterrotating vortices and unsteady shedding of hairpin-shaped vortices. The steady shedding of one or two pairs of counterrotating vortices prevails in the range of parameters studied. Hydrodynamic forces exerted on the cylinder are well approximated by laws derived in the Stokes flow regime, even for moderate Reynolds numbers. For the highest Reynolds numbers $\left(\text{Re}=150,200,250\right)$ we show that the forces slowly depart from the Stokes-based laws. Simple modification of these laws is proposed, yielding a satisfactory match with the numerical results. An accurate way to compute the torque is also proposed based on the normal force to the cylinder.

Cilia and flagella are ubiquitous in nature and are known to help in transport and swimming at the cellular scale by performing oscillations. Fundamental to these periodic waveforms is the core internal structure of the filaments known as the axoneme, consisting of an array of microtubule doublets, protein linkers, and dynein motors. In the presence of ATP, the collective action of the molecular motors drives internal sliding motions that are converted to spontaneous oscillations by a mechanism that still remains elusive. A sliding controlled axonemal feedback mechanism has recently been proposed and explored in the limit of small deformations, where it was shown to result in nonlinear amplitude selection through a mechanical regulation of dynein kinetics. Here, we build on that model to derive a more complete set of planar nonlinear governing equations that retains all the geometric nonlinearities, incorporates intrinsic biochemical noise and accounts for long-range, nonlocal hydrodynamic interactions. For a clamped filament, motor activity drives a Hopf bifurcation leading to traveling wave solutions that propagate from tip to base, in agreement with previous weakly nonlinear studies. Quite remarkably, our results demonstrate the existence of a second transition far from equilibrium, where nonlinearities cause a reversal in the direction of wave propagation and produce a variety of waveforms that resemble the beating patterns of swimming spermatozoa. We further extend the model to account for asymmetric ciliary beats and also allow for generalized dynein regulation mechanisms that can qualitatively reproduce Chlamydomonas reinhardtii flagellar dynamics. In the spirit of dimensional reduction, limit cycle representations are obtained for various waveforms and highlight the role of biochemical noise. We also analyze the velocity fields generated by the filaments and apply principal component analysis to derive low-order flow representations in terms of fundamental Stokes singularities that could be of use for constructing minimal models of swimming microorganisms.