A Stochastic Representation of the Local Structure of Turbulence

Based on the mechanics of the Euler equation at short time, we show that a Recent Fluid Deformation (RFD) closure for the vorticity field, neglecting the early stage of advection of fluid particles, allows to build a 3D incompressible velocity field that shares many properties with empirical turbulence, such as the teardrop shape of the R-Q plane. Unfortunately, non gaussianity is weak (i.e. no intermittency) and vorticity gets preferentially aligned with the wrong eigenvector of the deformation. We then show that slightly modifying the former vectorial field in order to impose the long range correlated nature of turbulence allows to reproduce the main properties of stationary flows. Doing so, we end up with a realistic incompressible, skewed and intermittent velocity field that reproduces the main characteristics of 3D turbulence in the inertial range, including correct vorticity alignment properties.Comment: 6 pages, 3 figures, final version, published

On the Rapid Increase of Intermittency in the Near-Dissipation Range of Fully Developed Turbulence

Intermittency, measured as log(F(r)/3), where F(r) is the flatness of velocity increments at scale r, is found to rapidly increase as viscous effects intensify, and eventually saturate at very small scales. This feature defines a finite intermediate range of scales between the inertial and dissipation ranges, that we shall call near-dissipation range. It is argued that intermittency is multiplied by a universal factor, independent of the Reynolds number Re, throughout the near-dissipation range. The (logarithmic) extension of the near-dissipation range varies as \sqrt(log Re). As a consequence, scaling properties of velocity increments in the near-dissipation range strongly depend on the Reynolds number.Comment: 7 pages, 7 figures, to appear in EPJ

Lagrangian time correlations of vorticity alignments in isotropic turbulence: observations and model predictions

Motivated by results from recent particle tracking experiments in turbulence (Xu et al., Nat. Phys. 7, 709 (2011)), we study the Lagrangian time correlations of vorticity alignments with the three eigenvectors of the deformation-rate tensor. We use data from direct numerical simulations (DNS), and explore the predictions of a Lagrangian model for the velocity gradient tensor. We find that the initial increase of correlation of vorticity direction with the most extensive eigen-direction observed by Xu et al. is reproduced accurately using the Lagrangian model, as well as the evolution of correlation with the other two eigendirections. Conversely, time correlations of vorticity direction with the eigen-frame of the pressure Hessian tensor show differences with the model.Comment: 4 pages, 2 figures, minor changes, final version published in Phys. Fluid

Multi-scale model of gradient evolution in turbulent flows

A multi-scale model for the evolution of the velocity gradient tensor in fully developed turbulence is proposed. The model is based on a coupling between a ``Restricted Euler'' dynamics [{\it P. Vieillefosse, Physica A, {\bf 14}, 150 (1984)}] which describes gradient self-stretching, and a deterministic cascade model which allows for energy exchange between different scales. We show that inclusion of the cascade process is sufficient to regularize the well-known finite time singularity of the Restricted Euler dynamics. At the same time, the model retains topological and geometrical features of real turbulent flows: these include the alignment between vorticity and the intermediate eigenvector of the strain-rate tensor and the typical teardrop shape of the joint probability density between the two invariants, $R-Q$, of the gradient tensor. The model also possesses skewed, non-Gaussian longitudinal gradient fluctuations and the correct scaling of energy dissipation as a function of Reynolds number. Derivative flatness coefficients are in good agreement with experimental data.Comment: 4 pages, 4 figure

Reynolds number effect on the velocity increment skewness in isotropic turbulence

Second and third order longitudinal structure functions and wavenumber spectra of isotropic turbulence are computed using the EDQNM model and compared to results of the multifractal formalism. At the highest Reynolds number available in windtunnel experiments, $R_\lambda=2500$, both the multifractal model and EDQNM give power-law corrections to the inertial range scaling of the velocity increment skewness. For EDQNM, this correction is a finite Reynolds number effect, whereas for the multifractal formalism it is an intermittency correction that persists at any high Reynolds number. Furthermore, the two approaches yield realistic behavior of second and third order statistics of the velocity fluctuations in the dissipative and near-dissipative ranges. Similarities and differences are highlighted, in particular the Reynolds number dependence

Une peinture aléatoire de la turbulence des fluides

This manuscript, for the French habilitation, is devoted to the physics of homogeneous and isotropic fluid turbulence. In the first part, we propose a probabilistic model for velocity fluctuations able to reproduce experimental and numerical observations. This model, built up from arguments developed in the context of the multifractal formalism and the probabilistic approach of the propagator, depends on two free parameters, related to the intermittency phenomenon and the physics of energy transfers. It does apply on both the Eulerian description of velocity fields and on velocity fluctuations along Lagrangian trajectories. The second part focuses on the velocity gradients dynamics along Lagrangian trajectories. We study the consequences of a proposed closure for the pressure term that governs mainly the dynamics. We explain several experimental facts such that the preferential vorticity alignments and the joint-density of the invariants. The last part concerns the construction and the mathematical analysis of a stochastic representation, or painting, of statistical stationary solutions of the Navier-Stokes equations, as it is observed in real and numerical flows. We show the realism of this representation that depends on a unique free parameter called the intermittency coefficient.Ce manuscrit d'habilitation à diriger des recherches est consacré à la physique de la turbulence des fluides homogènes et isotropes. La première partie se propose d'établir un modèle probabiliste des fluctuations de la vitesse d'un écoulement pleinement turbulent capable de reproduire les observations expérimentales et numériques. Le modèle, qui reprend à la fois les arguments du formalisme multifractal et de l'approche probabiliste du propagateur, dépend de deux paramètres libres liés au phénomène d'intermittence et à la physique des transferts d'énergie. Il s'applique à la fois à la description Eulérienne des champs de vitesse et à l'approche Lagrangienne qui s'attache à décrire les fluctuations de vitesse le long des trajectoires des particules fluides. La deuxième partie se propose d'étudier la dynamique des variations spatiales de la vitesse, ou tenseur des gradients, le long des trajectoires lagrangiennes. Nous étudions les conséquences de la fermeture du terme de pression, par nature non local, qui gouverne principalement la dynamique des gradients. Plusieurs faits expérimentaux sont ainsi expliqués (notamment l'alignement préférentiel de la vorticité et la densité conjointe des invariants). La dernière partie est consacrée à la construction et l'analyse mathématique d'une représentation, ou peinture, aléatoire de la solution statistiquement stationnaire des équations de Navier et Stokes, comme elle est observée dans les expériences et les simulations. Nous montrons le réalisme de cette représentation quant à décrire les écoulements turbulents et ramène la description de la turbulence des fluides à un unique paramètre libre, appelé coefficient d'intermittence