Document VersionPublisher's PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication:• A submitted manuscript is the author's version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.• Users may download and print one copy of any publication from the public portal for the purpose of private study or research.• You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal ? Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Spatial distributions of heavy particles suspended in an incompressible isotropic and homogeneous turbulent flow are investigated by means of high resolution direct numerical simulations. In the dissipative range, it is shown that particles form fractal clusters with properties independent of the Reynolds number. Clustering is there optimal when the particle response time is of the order of the Kolmogorov time scale . In the inertial range, the particle distribution is no longer scale invariant. It is, however, shown that deviations from uniformity depend on a rescaled contraction rate, which is different from the local Stokes number given by dimensional analysis. Particle distribution is characterized by voids spanning all scales of the turbulent flow; their signature in the coarse-grained mass probability distribution is an algebraic behavior at small densities.

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Key Words dynamical models, multifractalss Abstract We review the most important theoretical and numerical results obtained in the realm of shell models for the energy-turbulent cascade. We mainly focus here on those results that had or will have some impact on the fluid-dynamics community. In particular, we address the problem of small-scale intermittency by discussing energy-helicity interactions, energy-dissipation multifractality, and universality of intermittency, i.e., independence of anomalous scaling exponents from large-scale forcing and boundary conditions. A multifractal-based description of multiscale and multitime correlation functions in turbulence is also presented. Finally, we also briefly review the analytical difficulties, and hopes, of calculating anomalous exponents.where the dynamical (complex) variable, u n (t), represents the time evolution of a velocity fluctuation over a wavelength k n = k 0 λ n , with λ, the intershell ratio, usually set to 2. 0066-4189/03/0115-0441$14.00 441 Annu. Rev. Fluid Mech. 2003.35:441-468. Downloaded from www.annualreviews.org Access provided by Columbia University on 07/23/18. For personal use only.

We present the results of direct numerical simulations of heavy particle transport in homogeneous, isotropic, fully developed turbulence, up to resolution 512 3 (R λ ≈ 185). Following the trajectories of up to 120 million particles with Stokes numbers, St, in the range from 0.16 to 3.5 we are able to characterize in full detail the statistics of particle acceleration. We show that: (i) The root-mean-squared acceleration a rms sharply falls off from the fluid tracer value already at quite small Stokes numbers; (ii) At a given St the normalised acceleration a rms /(ǫ 3 /ν) 1/4 increases with R λ consistently with the trend observed for fluid tracers; (iii) The tails of the probability density function of the normalised acceleration a/a rms decrease with St. Two concurrent mechanisms lead to the above results: preferential concentration of particles, very effective at small St, and filtering induced by the particle response time, that takes over at larger St.

The statistical properties of velocity and acceleration fields along the trajectories of fluid particles transported by a fully developed turbulent flow are investigated by means of high resolution direct numerical simulations. We present results for Lagrangian velocity structure functions, the acceleration probability density function and the acceleration variance conditioned on the instantaneous velocity. These are compared with predictions of the multifractal formalism and its merits and limitations are discussed. Understanding the Lagrangian statistics of particles advected by a turbulent velocity field, u(x, t), is important both for its theoretical implications [1] and for applications, such as the development of phenomenological and stochastic models for turbulent mixing [2]. Recently, several authors have attempted to describe Lagrangian statistics such as acceleration by constructing models based on equilibrium statistics (see e.g. [3,4,5], critically reviewed in [6]). In this letter we show how the multifractal formalism offers an alternative approach which is rooted in the phenomenology of turbulence. Here, we derive the Lagrangian statistics from the Eulerian statistics without introducing ad hoc hypotheses.In order to obtain an accurate description of the particle statistics it is necessary to measure the positions, X(t), and velocities, v(t) ≡Ẋ(t) = u(X(t), t), of the particles with very high resolution, ranging from fractions of the Kolmogorov timescale, τ η , to multiples of the Lagrangian integral time scale, T L . The ratio of these timescales, T L /τ η , gives an estimate of the micro-scale Reynolds number, R λ , which may easily reach values of order 10 3 in laboratory experiments. Despite recent advances in experimental techniques for measuring Lagrangian turbulent statistics [7,8,9], direct numerical simulations (DNS) still offer higher accuracy albeit at a slightly lower Reynolds number [10,11,12,13]. In this letter we are concerned with single particle statistics, that is, the statistics of velocity and acceleration fluctuations along individual particle trajectories. Here, we analyse Lagrangian data obtained from a recent DNS of homogeneous isotropic turbulence [14] which was performed on 512 3 and 1024 3 cubic lattices with Reynolds numbers up to R λ ∼ 280. The Navier-Stokes equations were integrated using fully de-aliased pseudo-spectral methods for a total time T ≈ T L . Millions of Lagrangian particles (passive tracers) were released into the flow once a statistically stationary velocity field had been obtained. The positions and velocities of the particles were stored at a sampling rate of 0.07τ η . The Lagrangian velocity was calculated using linear interpolation. Acceleration was calculated both by following the particle and by direct computation from all three forces acting on the particle -the pressure gradients, viscous forces and the large scale forcing. The two measurements were found to be in very good agreement. The flow was forced by keeping the total energy constant in th...

We study the statistical properties of homogeneous and isotropic three-dimensional (3D) turbulent flows. By introducing a novel way to make numerical investigations of Navier-Stokes equations, we show that all 3D flows in nature possess a subset of nonlinear evolution leading to a reverse energy transfer: from small to large scales. Up to now, such an inverse cascade was only observed in flows under strong rotation and in quasi-two-dimensional geometries under strong confinement. We show here that energy flux is always reversed when mirror symmetry is broken, leading to a distribution of helicity in the system with a well-defined sign at all wave numbers. Our findings broaden the range of flows where the inverse energy cascade may be detected and rationalize the role played by helicity in the energy transfer process, showing that both 2D and 3D properties naturally coexist in all flows in nature. The unconventional numerical methodology here proposed, based on a Galerkin decimation of helical Fourier modes, paves the road for future studies on the influence of helicity on small-scale intermittency and the nature of the nonlinear interaction in magnetohydrodynamics.

The problem of anisotropy and its effects on the statistical theory of high Reynoldsnumber (Re) turbulence (and turbulent transport) is intimately related and intermingled with the problem of the universality of the (anomalous) scaling exponents of structure functions. Both problems had seen tremendous progress in the last five years. In this review we present a detailed description of the new tools that allow effective data analysis and systematic theoretical studies such as to separate isotropic from anisotropic aspects of turbulent statistical fluctuations. Employing the invariance of the equations of fluid mechanics to all rotations, we show how to decompose the (tensorial) statistical objects in terms of the irreducible representation of the SO(d) symmetry group (with d being the dimension, d = 2 or 3). This device allows a discussion of the scaling properties of the statistical objects in well defined sectors of the symmetry group, each of which is determined by the "angular momenta" sector numbers (j, m). For the case of turbulent advection of passive scalar or vector fields, this decomposition allows rigorous statements to be made: (i) the scaling exponents are universal, (ii) the isotropic scaling exponents are always leading, (iii) the anisotropic scaling exponents form a discrete spectrum which is strictly increasing as a function of j. This emerging picture offers a complete understanding of the decay of anisotropy upon going to smaller and smaller scales. Next we explain how to apply the SO(3) decomposition to the statistical Navier-Stokes theory. We show how to extract information about the scaling behavior in the isotropic sector. Doing so furnishes a systematic way to assess the universality of the scaling exponents in this sector, clarifying the anisotropic origin of the many measurements that claimed the opposite. A systematic analysis of Direct Numerical Simulations (DNS) of the Navier-Stokes equations and of experiments provides a strong support to the proposition that also for the non-linear problem there exists foliation of the statistical theory into sectors of the symmetry group. The exponents appear universal in each sector, and again strictly increasing as a function of j. An approximate calculation of the anisotropic exponents based on a closure theory is reviewed. The conflicting experimental measurements on the decay of anisotropy are explained and systematized, showing agreement with the theory presented here.