Single Fluid Particle Research Articles

Time irreversibility and multifractality of power along single particle trajectories in turbulence

The irreversible turbulent energy cascade epitomizes strongly non-equilibrium systems. At the level of single fluid particles, time irreversibility is revealed by the asymmetry of the rate of kinetic energy change, the Lagrangian power, whose moments display a power-law dependence on the Reynolds number, as recently shown by Xu et al. [H. Xu et al, Proc. Natl. Acad. Sci. U.S.A. 111, 7558 (2014)]. Here Lagrangian power statistics are rationalized within the multifractal model of turbulence, whose predictions are shown to agree with numerical and empirical data. Multifractal predictions are also tested, for very large Reynolds numbers, in dynamical models of the turbulent cascade, obtaining remarkably good agreement for statistical quantities insensitive to the asymmetry and, remarkably, deviations for those probing the asymmetry. These findings raise fundamental questions concerning time irreversibility in the infinite-Reynolds-number limit of the Navier-Stokes equations.

A level-set model for thermocapillary motion of deformable fluid particles

A new level-set model is proposed for simulating immiscible thermocapillary flows with variable fluid-property ratios at dynamically deformable interfaces. The Navier–Stokes equations coupled with the energy conservation equation are solved by means of a finite-volume/level-set approach, adapted to a multiple marker methodology in order to avoid the numerical coalescence of the fluid particles. The temperature field is coupled to the surface tension through an equation of state. Some numerical examples including thermocapillary driven convection in two superimposed fluid layers, and thermocapillary motion of single and multiple fluid particles are computed using the present method. These results are compared against analytical solutions and numerical results from the literature as validations of the proposed model.

A level-set method for thermal motion of bubbles and droplets

A conservative level-set model for direct simulation of two-phase flows with thermocapillary effects at dynamically deformable interface is presented. The Navier-Stokes equations coupled with the energy conservation equation are solved by means of a finite-volume/level-set method. Some numerical examples including thermocapillary motion of single and multiple fluid particles are computed by means of the present method. The results are compared with analytical solutions and numerical results from the literature as validations of the proposed model.

The distribution of “time of flight” in three dimensional stationary chaotic advection

The distributions of “time of flight” (time spent by a single fluid particle between two crossings of the Poincaré section) are investigated for five different three dimensional stationary chaotic mixers. Above all, we study the large tails of those distributions and show that mainly two types of behaviors are encountered. In the case of slipping walls, as expected, we obtain an exponential decay, which, however, does not scale with the Lyapunov exponent. Using a simple model, we suggest that this decay is related to the negative eigenvalues of the fixed points of the flow. When no-slip walls are considered, as predicted by the model, the behavior is radically different, with a very large tail following a power law with an exponent close to −3.

A generalized equation for rheology of emulsions and suspensions of deformable particles subjected to simple shear at low Reynolds number

We present analyses to provide a generalized rheological equation for suspensions and emulsions of non-Brownian particles. These multiparticle systems are subjected to a steady straining flow at low Reynolds number. We first consider the effect of a single deformable fluid particle on the ambient velocity and stress fields to constrain the rheological behavior of dilute mixtures. In the homogenization process, we introduce a first volume correction by considering a finite domain for the incompressible matrix. We then extend the solution for the rheology of concentrated system using an incremental differential method operating in a fixed and finite volume, where we account for the effective volume of particles through a crowding factor. This approach provides a self-consistent method to approximate hydrodynamic interactions between bubbles, droplets, or solid particles in concentrated systems. The resultant non-linear model predicts the relative viscosity over particle volume fractions ranging from dilute to the the random close packing in the limit of small deformation (capillary or Weissenberg numbers) for any viscosity ratio between the dispersed and continuous phases. The predictions from our model are tested against published datasets and other constitutive equations over different ranges of viscosity ratio, volume fraction, and shear rate. These comparisons show that our model, is in excellent agreement with published datasets. Moreover, comparisons with experimental data show that the model performs very well when extrapolated to high capillary numbers (C a≫1). We also predict the existence of two dimensionless numbers; a critical viscosity ratio and critical capillary numbers that characterize transitions in the macroscopic rheological behavior of emulsions. Finally, we present a regime diagram in terms of the viscosity ratio and capillary number that constrains conditions where emulsions behave like Newtonian or Non-Newtonian fluids.

On the Wall Drag Term in the Averaged Momentum Equation for Dispersed Flows

When fluid particles such as bubbles and droplets are not in contact with the wall, one probably neglects the wall drag term in the one-dimensional momentum equation for the dispersed phase. This treatment however leads to an unphysical prediction of the motion of the dispersed phase. In the framework of the conventional two-fluid model, how to apply the wall drag to the dispersed phase is disputable. The interface force acting on a fluid particle results from the interaction between the fluid particle and the surrounding continuous fluid. To clarify the contributions to the forces acting on the dispersed phase, the volume-averaged momentum equations are formulated based on the equation of a single fluid particle motion. After that, one-dimensional momentum equations are newly obtained from the averaged equations. It is shown that the wall drag term in the dispersed phase is associated with the spatial gradient of the volume-averaged viscous stress of the continuous phase. The magnitude of the wall drag term for a phase is its volume fraction multiplied by the total two-phase pressure drop induced by the wall shear of the continuous phase.

Flight–crash events in turbulence

The statistical properties of turbulence differ in an essential way from those of systems in or near thermal equilibrium because of the flux of energy between vastly different scales at which energy is supplied and at which it is dissipated. We elucidate this difference by studying experimentally and numerically the fluctuations of the energy of a small fluid particle moving in a turbulent fluid. We demonstrate how the fundamental property of detailed balance is broken, so that the probabilities of forward and backward transitions are not equal for turbulence. In physical terms, we found that in a large set of flow configurations, fluid elements decelerate faster than accelerate, a feature known all too well from driving in dense traffic. The statistical signature of rare "flight-crash" events, associated with fast particle deceleration, provides a way to quantify irreversibility in a turbulent flow. Namely, we find that the third moment of the power fluctuations along a trajectory, nondimensionalized by the energy flux, displays a remarkable power law as a function of the Reynolds number, both in two and in three spatial dimensions. This establishes a relation between the irreversibility of the system and the range of active scales. We speculate that the breakdown of the detailed balance characterized here is a general feature of other systems very far from equilibrium, displaying a wide range of spatial scales.

Efficient computation of the flow around single fluid particles using an artificial boundary condition

SUMMARYTwo‐phase flows around fluid particles are often considered to be in infinite domains, to avoid influence of the domain walls. Numerical simulations, however, must be modeled with a bounded domain, thus introducing artificial boundaries. Modeling of fluid flow in a domain with such artificial boundaries requires a careful choice of suitable boundary conditions. Slip boundary conditions for example can have a large impact on the computational results if the domain is chosen to be too small, because they model impermeable walls. This paper introduces an artificial boundary condition for simulations of the flow around single rising or settling fluid particles based on the approximated decay behavior of the velocity and the pressure field in the surrounding liquid. This is applied to the simulation of rising gas bubbles in systems with a Reynolds number of up to 50, and the outcome is compared with experimental results and simulations with slip boundary condition. It is found that domain size can be reduced by a factor of about two compared with slip boundary conditions without loss of accuracy. Copyright © 2014 John Wiley & Sons, Ltd.

On Lagrangian single-particle statistics

In turbulence, ideas of energy cascade and energy flux, substantiated by the exact Kolmogorov relation, lead to the determination of scaling laws for the velocity spatial correlation function. Here we ask whether similar ideas can be applied to temporal correlations. We critically review the relevant theoretical and experimental results concerning the velocity statistics of a single fluid particle in the inertial range of statistically homogeneous, stationary and isotropic turbulence. We stress that the widely used relations for the second structure function, D2(t) ≡ ⟨[v(t) − v(0)]2⟩∝εt, relies on dimensional arguments only: no relation of D2(t) to the energy cascade is known, neither in two- nor in three-dimensional turbulence. State of the art experimental and numerical results demonstrate that at high Reynolds numbers, the derivative \documentclass[12pt]{minimal}\begin{document}$\frac{d D_2(t)}{dt}$\end{document}dD2(t)dt has a finite non-zero slope starting from t ≈ 2τη. The analysis of the acceleration spectrum ΦA(ω) indicates a possible small correction with respect to the dimensional expectation ΦA(ω) ∼ ω0 but present data are unable to discriminate between anomalous scaling and finite Reynolds effects in the second order moment of velocity Lagrangian statistics.

Motion of viscous drops in tubes filled with yield stress fluid

The dynamics of deformable viscous drops in yield stress medium is investigated experimentally. Tetrachloroethylene drops settle in a cylindrical tube filled with neutralized Carbopol solution under the action of gravity. For a single fluid particle, stopping conditions, established velocities and deformation patterns are examined for a variety of the physical parameters of the continuous medium determined by Carbopol concentration and the geometry of the system. For a pair of drops, we present time evolution of separation distance and of the shapes of the drops. It was found that if the initial separation between equal size drops does not exceed some critical value, the trailing particle moves with a considerably higher velocity than that of the leading one and eventually the two drops coalesce. Afterwards, a single large fluid particle is formed and it sometimes breaks up forming one large leading drop and one or several small trailing satellites. This behavior is qualitatively similar to that obtained previously in numerical simulations.

Motion and Deformation of Drops in Bingham Fluid

The axisymmetric dynamics of single and interacting deformable fluid inclusions in yield stress fluid are investigated. The drops move in a cylindrical tube under the action of gravity. The velocities of the drops, deformation patterns, and the evolution of yield domains were examined for a variety of physical parameters including the interfacial tension, the yield stress, and the geometry of the system. Particular attention is given to the evolution of the yielded regions in the ambient matrix and to the deformation of the interfaces of the inclusions. The results of simulations show that, for a single drop, a quasistationary state is established after a relatively long transient period. The interaction of two drops results in a substantial extension of the yielded domain and in the acceleration of the motion. Two equal-size drops tend to approach each other and coalesce. Afterward, a single large fluid particle is formed that sometimes breaks up forming one large leading drop and one or several small trailing satellites.

A Fully Mass and Volume Conserving Implementation of a Characteristic Method for Transport Problems

The characteristics-mixed method considers the transport not of a single point or fluid particle, but rather the mass in an entire region of fluid. This mass is transported along the characteristic curves of the hyperbolic part of the transport equation, and the scheme thereby produces very little numerical dispersion, conserves mass locally, and can use long time steps. However, since the shape of a characteristic trace-back region must be approximated in numerical implementation, its volume may be incorrect, resulting in inaccurate concentration densities and, further, inaccurate reaction dynamics. We present a simple modification to the characteristics-mixed method that conserves both mass and volume of the transported fluid regions. Our algorithm also handles boundary conditions through a space-time change of variables in the trace-back routines, which allows the boundary to be treated as if it were interior to the domain. Nearly point sources, such as wells, present special difficulties, since characteristic trace-back curves converge in their vicinity. We also present techniques that allow one to conservatively implement wells. The techniques are illustrated in five numerical examples.

Investigation of a generalized Obukhov model for turbulence

We introduce a generalization of Obukhov's model [A.M. Obukhov, Adv. Geophys. 6 (1959) 113] for the description of the joint position-velocity statistics of a single fluid particle in fully developed turbulence. In the presented model the velocity is assumed to undergo a continuous time random walk. This takes into account long time correlations. As a consequence the evolution equation for the joint position-velocity probability distribution is a Fokker–Planck equation with a fractional time derivative. We determine the solution of this equation in the form of an integral transform and derive a relation for arbitrary single time moments. Analytical solutions for the joint probability distribution and its moments are given.

Colloidal Hard-Sphere Crystal Growth Frustrated by Large Spherical Impurities

Impurities affect the nucleation, growth, and structure of crystals. Here we report the effect of large, spherical, polymethylmethacrylate impurities on the crystal growth of monodisperse, hard, polymethylmethacrylate colloids in a density- and optically matching apolar solvent mixture. Crystal growth, initiated at the bottom of the sample, was studied by imaging sequences of two-dimensional xy slices in the plane of the impurity's center with a laser scanning confocal microscope. Impurities form the center of grain boundaries, and a single fluid particle layer around the impurity persists in all cases. The growth rate sensitively depends on the impurity's size. Crystal growth is inhibited to a greater extent near smaller impurities, pointing to local crystal frustration induced by the curvature of the impurity.

STOCHASTIC MODELS OF LAGRANGIAN ACCELERATION OF FLUID PARTICLE IN DEVELOPED TURBULENCE

Modeling statistical properties of motion of a Lagrangian particle advected by a high-Reynolds-number flow is of much practical interest and complement traditional studies of turbulence made in Eulerian framework. The strong and nonlocal character of Lagrangian particle coupling due to pressure effects makes the main obstacle to derive turbulence statistics from the three-dimensional Navier–Stokes equation; motion of a single fluid-particle is strongly correlated to that of the other particles. Recent breakthrough Lagrangian experiments with high resolution of Kolmogorov scale have motivated growing interest to acceleration of a fluid particle. Experimental stationary statistics of Lagrangian acceleration conditioned on Lagrangian velocity reveals essential dependence of the acceleration variance upon the velocity. This is confirmed by direct numerical simulations. Lagrangian intermittency is considerably stronger than the Eulerian one. Statistics of Lagrangian acceleration depends on Reynolds number. In this review we present description of new simple models of Lagrangian acceleration that enable data analysis and some advance in phenomenological study of the Lagrangian single-particle dynamics. Simple Lagrangian stochastic modeling by Langevin-type dynamical equations is one the widely used tools. The models are aimed particularly to describe the observed highly non-Gaussian conditional and unconditional acceleration distributions. Stochastic one-dimensional toy models capture main features of the observed stationary statistics of acceleration. We review various models and focus in a more detail on the model which has some deductive support from the Navier–Stokes equation. Comparative analysis on the basis of the experimental data and direct numerical simulations is made.

Statistics of Lagrangian velocities in turbulent flows.

We present a generalized Fokker-Planck equation for the joint position-velocity probability distribution of a single fluid particle in a turbulent flow. Based on a simple estimate, the diffusion term is related to the two-point two-time Eulerian acceleration-acceleration correlation. Dimensional analysis yields a velocity increment probability distribution with normal scaling v approximately t(1/2). However, the statistics need not be Gaussian.

Mesoscopic dynamics of Voronoi fluid particles

We compare and contrast two recently reported mesoscopic fluid particle models based on a two-dimensional Voronoi tessellation. Both models describe a Newtonian fluid at mesoscopic scales where fluctuations are important. From the requirement of thermodynamic consistency, the equilibrium distribution function is given through the Einstein distribution function. We compute from the Einstein distribution the equilibrium distribution function for a single fluid particle. We observe excellent agreement between the simulation results for the proposed models and the theoretical distribution function.

Wake‐Phänomene hinter festen und fluiden Partikeln

AbstractWake phenomena behind solid and fluid particles. While detailed information is available about the nature and the phenomology of the wake, there is a distinct lack of practically useful and quantitative information about the size of the wake behind single fluid particles and in a swarm thereof. Many studies have been performed on the wake behind individual particles, yet the results are not only fragmentary, i. e. of little relevance for the operating regime of bubble columns, but also beset with surprisingly large discrepancies. These considerable differences are of a systematic nature and due partly to insufficient elimination of surfactant effects, partly to different boundary effects of the experimental equipment, and partly to differing measuring techniques. Experimental results for the wake in swarms of droplets and bubbles in the relevant operating regime are still completely lacking.

Evolution of flux-conserving tokamak equilibria with preprogrammed cross sections

The evolution of MHD equilibria toward high β is modeled by magnetic flux conservation with a given q( ψ) and by single fluid particle and energy balances which determine p( ψ, t). These one-dimensional flux surface averaged equations, written with magnetic flux ψ as the independent variable, are coupled to the two-dimensional MHD equilibrium equation through ψ, p( ψ, t), and q( ψ). The location and evolution of the plasma cross section boundary are precisely specified through the use of a fixed boundary equilibrium technique. In moving boundary studies (e.g., plasma compression) the resulting system of equations is advanced in time from an initial state by a procedure which utilizes two nested predictor-corrector loops together with an implicit time-stepping technique. The inner predictor-corrector loop advances the transport equations subject to a given equilibrium configuration while the outer loop evolves the equilibrium. For fixed plasma boundaries this procedure is modified for greater computational speed. These techniques provide satisfactory numerical convergence together with complete consistency between the coupled one-dimensional system of equations and the two-dimensional equilibrium. This method can be applied to the study of equilibrium evolution involving dramatic changes of plasma position, shape, and profiles while prescribing the evolution of the plasma boundary. As such, it can serve as a useful tool in the design of poloidal field systems or as a source of equilibria in high-β MHD stability studies. As an example, the compressional scaling laws of Furth and Yoshikawa are found to be modified for small aspect ratio.

An exact theory of nonlinear waves on a Lagrangian-mean flow

An exact and very general Lagrangian-mean description of the back effect of oscillatory disturbances upon the mean state is given. The basic formalism applies to any problem whose governing equations are given in the usual Eulerian form, and irrespective of whether spatial, temporal, ensemble, or ‘two-timing’ averages are appropriate. The generalized Lagrangian-mean velocity cannot be defined exactly as the ‘mean following a single fluid particle’, but in cases where spatial averages are taken can easily be visualized, for instance, as the motion of the centre of mass of a tube of fluid particles which lay along the direction of averaging in a hypothetical initial state of no disturbance.The equations for the Lagrangian-mean flow are more useful than their Eulerian-mean counterparts in significant respects, for instance in explicitly representing the effect upon mean-flow evolution of wave dissipation or forcing. Applications to irrotational acoustic or water waves, and to astrogeophysical problems of waves on axisymmetric mean flows are discussed. In the latter context the equations embody generalizations of the Eliassen-Palm and Charney-Drazin theorems showing the effects on the mean flow of departures from steady, conservative waves, for arbitrary, finite-amplitude disturbances to a stratified, rotating fluid, with allowance for self-gravitation as well as for an external gravitational field.The equations show generally how the pseudomomentum (or wave ‘momentum’) enters problems of mean-flow evolution. They also indicate the extent to which the net effect of the waves on the mean flow can be described by a ‘radiation stress’, and provide a general framework for explaining the asymmetry of radiation-stress tensors along the lines proposed by Jones (1973).