Turbulent intermittency plays a fundamental role in fields ranging from combustion physics and chemical engineering to meteorology. There is a rather general agreement that multifractals are being very successful at quantifying this intermittency. However, we argue that cascade processes are the appropriate and necessary physical models to achieve dynamical modeling of turbulent intermittency. We first review some recent developments and point out new directions which overcome either completely or partially the limitations of current cascade models which are static, discrete in scale, acausal, purely phenomenological and lacking in universal features. We review the debate about universality classes for multifractal processes. Using both turbulent velocity and temperature data, we show that the latter are very well fitted by the (strong) universality, and that the recent (weak, log-Poisson) alternative is untenable for both strong and weak events. Using a continuous, space-time anisotropic framework, we then show how to produce a causal stochastic model of intermittent fields and use it to study the predictability of these fields. Finally, by returning to the origins of the turbulent "shell models" and restoring a large number of degrees of freedom (the Scaling Gyroscope Cascade, SGC models) we partially close the gap between the cascades and the dynamical Navier–Stokes equations. Furthermore, we point out that beyond a close agreement between universal parameters of the different modeling approaches and the empirical estimates in turbulence, there is a rather common structure involving both a "renormalized viscosity" and a "renormalized forcing". We conclude that this gives credence to the possibility of deriving analytical/renormalized models of intermittency built on this structure.

The morphology of fractured rock surfaces is studied in terms of their scaling invariance. Fresh brittle fractures of granite and gneiss were sampled with a mechanical laboratory profilometer, and (1 + 1)‐dimensional parallel profiles were added to build actual maps of the surfaces. A first step in the scaling invariance description is a self‐affine analysis using three independent methods. The root‐mean‐square and the maximum‐minimum difference of the height are shown to follow a power law with the sample length. The return probability and the Fourier spectrum are also computed. All these approaches converge to a unique self‐affine exponent: ζ = 0.80. Analysis over a broad statistical set provides a reproducibility error of ±0.05. No significant differences between the isotropic granite and the markedly anisotropic gneiss appear for the scaling exponents. An analysis of the profilometer shows that the two main drawbacks of the setup are not significant in these analyses. The systematic errors of the scaling analysis are estimated for the different methods. Isotropy of the scaling invariance within the mean fracture plane is shown either with the result obtained from different fracture orientations or with the two‐dimensional Fourier spectrum of the surface topography itself. The analysis is brought further into the multifractal framework. The structure functions are shown to have power law behavior, and their scaling exponent varies nonlinearly with the moment order. Finally, the corresponding conserved process belongs to a universal multifractal class with α = 1.5 for the Levy index and C1 = 0.3 for the fractal codimension of the mean singularities. The three indices (ζ, α and C1) completely characterize the scale invariance. The multifractal behavior is significant for physical properties which depend on high‐order moments like contact. According to this study and that of other groups, the self‐affine exponent ζ is constant over a large range of scales and for different fracture modes and various materials. This opens the possibility that there exists a form of universality in the cracking process. It appears that only the prefactor of the roughness is dependent on material and mode.

Hilbert-Huang transform is a method that has been introduced recently to decompose nonlinear, nonstationary time series into a sum of different modes, each one having a characteristic frequency. Here we show the first successful application of this approach to homogeneous turbulence time series. We associate each mode to dissipation, inertial range and integral scales. We then generalize this approach in order to characterize the scaling intermittency of turbulence in the inertial range, in an amplitude-frequency space. The new method is first validated using fractional Brownian motion simulations. We then obtain a 2D amplitude-frequency representation of the pdf of turbulent fluctuations with a scaling trend, and we show how multifractal exponents can be retrieved using this approach. We also find that the log-Poisson distribution fits the velocity amplitude pdf better than the lognormal distribution. p-6

Boussinesq's hypothesis is at the heart of eddy viscosity models, which are used in many different fields to model turbulent flows. In its present time formulation, this hypothesis corresponds to an alignment between Reynolds stress and mean strain tensors. We begin with historical remarks on Boussinesq's results and recall that he introduced a local averaging twenty years before Reynolds, but using an approach that prevented him from discovering Reynolds' stress tensor. We then introduce an indicator that characterizes the validity of this hypothesis. For experimental and numerical databases, when the tensors are known, this can be used to directly estimate the validity of this hypothesis. We show, using several different databases, that this hypothesis is almost never verified. We address in conclusion the analogy with kinetic theory, and the reason why this analogy cannot be applied in general for turbulent flows. RésuméA propos de l'hypothèse de viscosité turbulente de Boussinesq : rappels historiques etévaluation directe. L'hypothèse de Boussinesq est au coeur des modèles de viscosité, utilisés dans un grand nombre de contextes pour modéliser desécoulements turbulents. Dans sa formulation moderne, cette hypothèse correspond a un alignement entre tenseur de contrainte de Reynolds et tenseur de déformation moyen. Nous rappelons le contexte historique de l'énoncé de cette hypothèse, en soulignant que Boussinesq avait introduit une moyenne locale vingt ans avant Reynolds, mais en effectuant une erreur qui l'a privé de la mise enévidence du tenseur de Reynolds. Nous introduisons ensuite un indicateur, compris entre 0 et 1, indiquant le degré de validité de cette hypothèse. Pour des bases de données expérimentales et numériques, lorsque les différents tenseurs sont connus, ceci permet de tester directement, "a priori", cette hypothèse. Nous montrons ainsi, utilisant différentes bases de données d'écoulements turbulents, que l'hypothèse n'est presque jamais vérifiée. Nous discutons en conclusion de la théorie cinétique des gaz et de la raison pour laquelle cette analogie est discutable pour lesécoulements turbulents. Version française abrégéeLe "problème de la turbulence" touche un grand nombre de domaines, incluant l'ingénierie automobile, chimique, la combustion, l'aéronautique, la météorologie, l'océanologie, l'hydrologie, l'hydraulique fluviale, etc. Ces domaines sontà fort potentiel industriel et environnemental, et demandent souvent des réponses pratiques et quantitatives, faisant appelà des modèles. Parmi les différentes familles de modèles existant, beaucoup utilisent une moyenne de Reynolds, qui font intervenir les fluctuations instantannéesà petiteś echelles via le tenseur de Reynolds. La modélisation intervient ici, et permet de fournir, via une hypothèse, une "fermeture" exprimant le tenseur de Reynolds en fonction de quantités moyennes. La fermeture la plus courante, fournissant le tenseur de Reynolds en fonction du champ de vitesse moyen, est celle qui est utilisée dans les modèles de viscos...