Fluid particle accelerations in fully developed turbulence

Superstatistical mechanics of tracer-particle motions in turbulence

The Lagrangian stochastic model of Reynolds [Phys. Fluids 15, L1-4 (2003)]] for the accelerations of fluid particles in turbulence is shown to predict precisely the observed Reynolds-number dependency of the distribution of Lagrangian accelerations and the exponents characterizing the observed extended self-similarity scaling of the Lagrangian velocity structure functions. Departures from superstatistics of the log-normal kind are accounted for and their impact upon model predictions is quantified.

Structure and kinematics in dense free-surface granular flow

We show that the structure of a dense, free-surface boundary layer granular flow is similar to the structure of a laminar liquid flow: There is a strong component of order (stratification parallel to the mean flow) superposed with a mild component of disorder (self-diffusion perpendicular to the mean flow). We also show that the self-diffusion coefficient scales with the mean velocity and propose a model that relates this scaling to the ordered structure of the flow. Last, we show that the structure of the flow imprints an oscillatory signature (similar to that found in confined granular flow) on the mean velocity profile.

Kinematic simulation of turbulent dispersion of triangles

As three particles are advected by a turbulent flow, they separate from each other and develop nontrivial geometries, which effectively reflect the structure of the turbulence. We investigate here the geometry, in a statistical sense, of three Lagrangian particles advected, in two dimensions, by kinematic simulation (KS). KS is a Lagrangian model of turbulent diffusion that makes no use of any delta correlation in time at any level. With this approach, situations with a very large range of inertial scales and varying persistence of spatial flow structure can be studied. We first demonstrate that the model flow reproduces recent experimental results at low Reynolds numbers. The statistical properties of the shape distribution at a much higher Reynolds number is then considered. The numerical results support the existence of nontrivial shape statistics, with a high probability of having elongated triangles. Even at the highest available inertial range of scales, corresponding to a ratio between large and small scale L/eta=17,000, a perfect self-similar regime is not found. The effects of the parameters of the synthetic flow, such as the exponent of the spectrum and the effect of the sweeping affect our results, are also discussed. Special attention is given to the effects of persistence of spatial flow structure.

Lévy flights in inhomogeneous media

We investigate the impact of external periodic potentials on superdiffusive random walks known as Lévy flights and show that even strongly superdiffusive transport is substantially affected by the external field. Unlike ordinary random walks, Lévy flights are surprisingly sensitive to the shape of the potential while their asymptotic behavior ceases to depend on the Lévy index mu. Our analysis is based on a novel generalization of the Fokker-Planck equation suitable for systems in thermal equilibrium. Thus, the results presented are applicable to the large class of situations in which superdiffusion is caused by topological complexity, such as diffusion on folded polymers and scale-free networks.

Critique of q-entropy for thermal statistics

During the past 12 years there have been numerous papers on a relation between entropy and probability which is nonadditive and has a parameter q that depends on the nature of the thermodynamic system under consideration. For q=1 this relation corresponds to the Boltzmann-Gibbs entropy, but for other values of q it is claimed that it leads to a formalism that is consistent with the laws of thermodynamics. However, it is shown here that the joint entropy for systems having different values of q is not defined in this formalism, and consequently fundamental thermodynamic concepts such as temperature and heat exchange cannot be considered for such systems. Moreover, for q not equal 1 the probability distribution for weakly interacting systems does not factor into the product of the probability distribution for the separate systems, leading to spurious correlations and other unphysical consequences, e.g., nonextensive energy, that have been ignored in various applications given in the literature.

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.

Digital holography of particle fields: reconstruction by use of complex amplitude

Digital holography appears to be a strong contender as the next-generation technology for holographic diagnostics of particle fields and holographic particle image velocimetry for flow field measurement. With the digital holographic approach, holograms are directly recorded by a digital camera and reconstructed numerically. This not only eliminates wet chemical processing and mechanical scanning, but also enables the use of complex amplitude information inaccessible by optical reconstruction, thereby allowing flexible reconstruction algorithms to achieve optimization of specific information. However, owing to the inherently low pixel resolution of solid-state imaging sensors, digital holography gives poor depth resolution for images, a problem that severely impairs the usefulness of digital holography especially in densely populated particle fields. This paper describes a technique that significantly improves particle axial-location accuracy by exploring the reconstructed complex amplitude information, compared with other numerical reconstruction schemes that merely mimic traditional optical reconstruction. This novel method allows accurate extraction of particle locations from forward-scattering particle holograms even at high particle loadings.

Measuring intense rotation and dissipation in turbulent flows

Turbulent flows are highly intermittent--for example, they exhibit intense bursts of vorticity and strain. Kolmogorov theory describes such behaviour in the form of energy cascades from large to small spatial and temporal scales, where energy is dissipated as heat. But the causes of high intermittency in turbulence, which show non-gaussian statistics, are not well understood. Such intermittency can be important, for example, for enhancing the mixing of chemicals, by producing sharp drops in local pressure that can induce cavitation (damaging mechanical components and biological organisms), and by causing intense vortices in atmospheric flows. Here we present observations of the three components of velocity and all nine velocity gradients within a small volume, which allow us to determine simultaneously the dissipation (a measure of strain) and enstrophy (a measure of rotational energy) of a turbulent flow. Combining the statistics of all measurements and the evolution of individual bursts, we find that a typical sequence for intense events begins with rapid strain growth, followed by rising vorticity and a final sudden decline in stretching. We suggest two mechanisms which can produce these characteristics, depending whether they are due to the advection of coherent structures through our observed volume or caused locally.

Long time correlations in lagrangian dynamics: a key to intermittency in turbulence

Using a new experimental technique, based on the scattering of ultrasounds, we perform a direct measurement of particle velocities, in a fully turbulent flow. This allows us to approach intermittency in turbulence from a dynamical point of view and to analyze the Lagrangian velocity fluctuations in the framework of random walks. We find experimentally that the elementary steps in the walk have random uncorrelated directions but a magnitude that is extremely long range correlated in time. Theoretically, a Langevin equation is proposed and shown to account for the observed one- and two-point statistics. This approach connects intermittency to the dynamics of the flow.

Lagrangian statistics and temporal intermittency in a shell model of turbulence

We study the statistics of single-particle Lagrangian velocity in a shell model of turbulence. We show that the small-scale velocity fluctuations are intermittent, with scaling exponents connected to the Eulerian structure function scaling exponents. The observed reduced scaling range is interpreted as a manifestation of the intermediate dissipative range, as it disappears in a Gaussian model of turbulence.

Multifractal stationary random measures and multifractal random walks with log infinitely divisible scaling laws

We define a large class of continuous time multifractal random measures and processes with arbitrary log infinitely divisible exact or asymptotic scaling law. These processes generalize within a unified framework both the recently defined log-normal multifractal random walk [J.F. Muzy, J. Delour, and E. Bacry, Eur. J. Phys. B 17, 537 (2000), E. Bacry, J. Delour, and J.F. Muzy, Phys. Rev. E 64, 026103 (2001)] and the log-Poisson "product of cylindrical pulses" [J. Barral and B.B. Mandelbrot, Cowles Foundation Discussion Paper No. 1287, 2001 (unpublished)]. Our construction is based on some "continuous stochastic multiplication" [as introduced in F. Schmitt and D. Marsan, Eur. J. Phys. B. 20, 3 (2001)] from coarse to fine scales that can be seen as a continuous interpolation of discrete multiplicative cascades. We prove the stochastic convergence of the defined processes and study their main statistical properties. The question of genericity (universality) of limit multifractal processes is addressed within this new framework. We finally provide a method for numerical simulations and discuss some specific examples.

Length scales of acceleration for locally isotropic turbulence

Length scales are determined that govern the behavior at small separations of the correlations of fluid-particle acceleration, viscous force, and pressure gradient. The length scales and an associated universal constant are quantified on the basis of published data. The length scale governing pressure spectra at high wave numbers is discussed. Fluid-particle acceleration correlation is governed by two length scales: one arises from the pressure gradient, the other from the viscous force.

Clustering dynamics of Lagrangian tracers in free-surface flows

We study the formation of clusters of passive Lagrangian tracers in a nonsmooth turbulent flow in a flat free-slip surface as a model for particle dynamics on free surfaces. Single particle and pair dispersion show different behavior for short and large times: on short times particles cluster exponentially rapidly until patches of the size of the divergence correlation length are depleted; on larger times the pair dispersion is dominated by almost ballistic hopping between clusters. We also find that the distribution of particle density is close to algebraic and can trace this back to the exponential distribution of the divergence field of the surface flow.

Time-resolved tracking of a sound scatterer in a complex flow: nonstationary signal analysis and applications

It is known that ultrasound techniques yield nonintrusive measurements of hydrodynamic flows. For example, the study of the echoes produced by a large number of particles insonified by pulsed wavetrains has led to a now-standard velocimetry device. In this paper, a new technique for the measurement of the velocity of individual solid particles moving in fluid flows is proposed. It relies on the ability to resolve in time the Doppler shift of the sound scattered by the continuously insonified particle. For this signal-processing problem two classes of approaches can be used: time-frequency analysis and parametric high-resolution methods. In the first class the spectrogram and reassigned spectrogram is considered, and applied to detect the motion of a small bead settling in a fluid at rest. In nonstationary flows, methods in the second class are more robust. An approximated maximum likelihood (AML) technique has been adapted, coupled with a generalized Kalman filter. This method allows for the estimation of rapidly varying frequencies; the parametric nature of the algorithm also provides an estimate of the variance of the identified frequency parameters.

Relative dispersion in fully developed turbulence: the Richardson's law and intermittency corrections

Relative dispersion in fully developed turbulence is investigated by means of direct numerical simulations. Lagrangian statistics is found to be compatible with Richardson description although small systematic deviations are found. The value of the Richardson constant is estimated as C2 approximately equal to 0.55, in a close agreement with recent experimental findings [S. Ott and J. Mann, J. Fluid Mech. 422, 207 (2000)]. By means of exit-time statistics it is shown that the deviations from Richardson's law are a consequence of Eulerian intermittency. The measured Lagrangian scaling exponents require a set of Eulerian structure function exponents zeta(p) which are remarkably close to standard ones known for fully developed turbulence.

Turbulence in nature and in the laboratory

Fluid turbulence has attracted the attention of physicists, mathematicians, and engineers for over 100 years, yet it remains an unsolved problem. The reasons for the difficulties are outlined and recent advances in describing its small-scale statistical structure are described. Contrary to traditional notions, new experimental evidence indicates that the small scales are anisotropic, reflecting the overall character of the flow. The consequences of this finding with regard to the long-held postulate of the universality of the small-scale turbulence structure are discussed.

Measurement of Lagrangian velocity in fully developed turbulence

We have developed a new experimental technique to measure the Lagrangian velocity of tracer particles in a turbulent flow, based on ultrasonic Doppler tracking. This method yields a direct access to the velocity of a single particle at a turbulent Reynolds number R(lambda) = 740, with two decades of time resolution, below the Lagrangian correlation time. We observe that the Lagrangian velocity spectrum has a Lorentzian form E(L)(omega) = u(2)(rms)T(L)/[1+(T(L)omega)(2)], in agreement with a Kolmogorov-like scaling in the inertial range. The probability density functions of the velocity time increments display an intermittency which is more pronounced than that of the corresponding Eulerian spatial increments.

"Strange" Fermi processes and power-law nonthermal tails from a self-consistent fractional kinetic equation

This study advocates the application of fractional dynamics to the description of anomalous acceleration processes in self-organized turbulent systems. Such processes (termed "strange" accelerations) involve both the non-Markovian fractal time acceleration events associated with a generalized stochastic Fermi mechanism, and the velocity-space Levy flights identified with nonlocal violent accelerations in turbulent media far from the (quasi)equilibrium. The "strange" acceleration processes are quantified by a fractional extension of the velocity-space transport equation with fractional time and phase space derivatives. A self-consistent nonlinear fractional kinetic equation is proposed for the stochastic fractal time accelerations near the turbulent nonequilibrium saturation state. The ensuing self-consistent energy distribution reveals a power-law superthermal tail psi(epsilon) proportional to epsilon(-eta) with slope 6 or =eta< or =7 depending on the type of acceleration process (persistent or antipersistent). The results obtained are in close agreement with observational data on the Earth's magnetotail.

Pressure spectrum in homogeneous turbulence

The pressure spectrum in homogeneous steady turbulence is studied using direct numerical simulation with resolution up to 1024(3) and the Reynolds number R(lambda) between 38 and 478. The energy spectrum is found to have a finite inertial range with the Kolmogorov constant K = 1.65+/-0.05 followed by a bump at large wave numbers. The pressure spectrum in the inertial range is found to be approximately P(k) = B(p)epsilon;(4/3)k(-7/3) with B(p) = 8.0+/-0.5, and followed by a bump of nearly k(-5/3) at higher wave numbers. Universality and a new scaling of the pressure spectrum are discussed.