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Abstract The historical evolution of the averaging concept in turbulence is presented according to a two-fold analysis, taking into consideration the physical meaning and the mathematical formulation. After having placed in their historical context some symptomatic characteristics of turbulent flows, the question of their interpretation as a turbulence syndrome, leading to a unitary identification of the phenomenon, is discussed. We then deal with the emergence of the notion of mean as a tool of physical understanding of such a unitary approach, and its relation with the progression of the "experimental" evidence. We conclude by relating this notion to the theoretical aspects of a determinism on average of the turbulent regime, its finality and some attempts to develop a probabilistic theory of turbulence. [ABSTRACT FROM AUTHOR]

The equations for a self-similar solution to an inviscid incompressible fluid are mapped into an integral equation that hopefully can be solved by iteration. It is argued that the exponents of the similarity are ruled by Kelvin's theorem of conservation of circulation. The end result is an iteration with a nonlinear term entering a kernel given by a 3D integral for a swirling flow, likely within reach of present-day computational power. Because of the slow decay of the similarity solution at large distances, its kinetic energy diverges, and some mathematical results excluding non-trivial solutions of the Euler equations in the self-similar case do not apply. [ABSTRACT FROM AUTHOR]

We consider here the works of French, British, and German researchers in fluid mechanics from 1870 to the beginning of the twentieth century. Our aim is to understand how the term “turbulence” introduced by William Thomson in 1887, which was not used by the main researchers of the time, including Joseph Boussinesq, Osborne Reynolds, Lord Rayleigh, Horace Lamb in the first editions of his book, became classical in the 1920s. We trace the first introductions of the terms “turbulence”, “turbulent flow” in the works of relatively unknown researchers between 1889 and 1903, until it reaches the vocabulary of mainstream researchers in fluid mechanics and physics. Our result is that the shift was in 1906–1908, when the term was used in the 1906 edition of the book of Horace Lamb, and in Lanchester's book, followed by a series of papers of German researchers before the First World War. The use of the word “turbulence”, a word used for a long time for crowds or for children, in a scientific context, corresponds to the introduction of a new concept, a new understanding of a scientific phenomenon clearly identified as being different from laminar motion. The study of the use of this term is also the study of the diffusion of a new concept among researchers of the time. [ABSTRACT FROM AUTHOR]

Still images, photographs and drawings, as well as movies are widely used in fluid mechanics and this has been true since the very early developments of this discipline. The intrinsic geometrical complexity of fluid flows, in particular when they are turbulent, explains this necessity of using visual representations to gain a physical understanding of the phenomena involved. The aesthetic appeal of images in fluid mechanics research is another reason why their use is more prevalent than in other fields of the physical sciences. [ABSTRACT FROM AUTHOR]

Intermittencies are commonly observed in fluid mechanics, and particularly, in pipe flows. Initially observed by Reynolds (1883), it took one century for reaching a rather full understanding of this phenomenon whose irregular dynamics (apparently stochastic) puzzled hydrodynamicists for decades. In this brief (non-exhaustive) review, mostly focused on the experimental characterization of this transition between laminar and turbulent regimes, we present some key contributions for evidencing the two concomittant and antagonist processes that are involved in this complex transition and were suggested by Reynolds. It is also shown that a clear explicative model was provided, based on the nonlinear dynamical systems theory, the experimental observations in fluid mechanics only providing an applied example, due to its obvious generic nature. [ABSTRACT FROM AUTHOR]

Ludwig Prandtl (1875–1953) has been called the father of modern aerodynamics. His name is associated most famously with the boundary layer concept, but also with several other topics in 20th-century fluid mechanics, particularly turbulence (Prandtl's mixing length). Among his disciples are pioneers of modern fluid mechanics like Heinrich Blasius, Theodore von Kármán, and Walter Tollmien. Furthermore, Prandtl founded the Aerodynamische Versuchsanstalt (AVA) and the Kaiser-Wilhelm-Institut für Strömungsforschung in Göttingen, nuclei for the growth of fluid mechanics in Germany. In this article I trace this development on the basis of my recent biography of Prandtl. [ABSTRACT FROM AUTHOR]

We present in this article the work of Henri Bénard (1874–1939), a French physicist who began the systematic experimental study of two hydrodynamic systems: the thermal convection of fluids heated from below (the Rayleigh–Bénard convection and the Bénard–Marangoni convection) and the periodical vortex shedding behind a bluff body in a flow (the Bénard–Kármán vortex street). Across his scientific biography, we review the interplay between experiments and theory in these two major subjects of fluid mechanics. [ABSTRACT FROM AUTHOR]

Abstract: We present an overview of recent works on the statistical description of turbulent flows in terms of probability density functions (PDFs) in the framework of the Lundgren–Monin–Novikov (LMN) hierarchy. Within this framework, evolution equations for the PDFs are derived from the basic equations of fluid motion. The closure problem arises either in terms of a coupling to multi-point PDFs or in terms of conditional averages entering the evolution equations as unknown functions. We mainly focus on the latter case and use data from direct numerical simulations (DNS) to specify the unclosed terms. Apart from giving an introduction into the basic analytical techniques, applications to two-dimensional vorticity statistics, to the single-point velocity and vorticity statistics of three-dimensional turbulence, to the temperature statistics of Rayleigh–Bénard convection and to Burgers turbulence are discussed. [Copyright &y& Elsevier]

Abstract: The notion of coherent structures in fluid mechanics – distinguishable regions of the flow field that share common properties and are correlated in space and time – has played a significant role in characterizing and modeling turbulent flows. They have not yet, however, truly become predictive tools, in part because there is no universally accepted way of extracting coherent structures from the flow field. A wide range of types of structures have been suggested, but there has been comparatively little work done to determine their role in the flow dynamics, and therefore to discern which structures are useful for more than flow visualization. Here, I review several common types of coherent structures, both Eulerian and Lagrangian, with a focus on two-dimensional turbulence far from boundaries. I also discuss a framework for pinpointing the dynamical role of coherent structures based on spatial localization of the spectral properties of the flow. Future work on coherent structures should focus on defining structures that play clear roles in the turbulence dynamics. [Copyright &y& Elsevier]

Abstract: The geometrical statistics of fluid deformation are analyzed theoretically within the framework of the restricted Euler approximation, and numerically using direct numerical simulations. The restricted Euler analysis predicts that asymptotically a material line element becomes an eigenvector of the velocity gradient regardless its initial orientation. The asymptotic stretching rate equals the intermediate eigenvalue of the strain rate tensor. Analyses of numerical data show that the pressure Hessian is the leading cause to destroy the alignment between the longest axis of the material element and the strongest stretching eigendirection of the strain rate. It also facilitates the alignment between the longest axis of the element and the intermediate eigendirection of the strain rate during initial evolution, but tends to oppose the alignment later. [Copyright &y& Elsevier]

Abstract: The insufficient understanding of the generation of small scales in turbulent flows results in serious impediment when trying to describe numerous physical problems, of natural or applied significance, and therefore, calls for new approaches. Here, we discuss the insight that can be gained by following the motion of a few points in a turbulent flow. This approach, which has shown its power in the context of the problem of dispersion of a passive scalar transported by turbulence, has led to new insight into some of the intriguing phenomena observed in turbulent flows, such as the alignment of vorticity with the eigenvectors of the rate of strain tensor. Recent work has focused on the motion of four points, forming initially a regular tetrad of size . In particular, the modeling perspective inspired by the tetrad approach will be discussed here. [Copyright &y& Elsevier]

Abstract: This Note presents two methods for mean streamwise velocity profiles of fully-developed turbulent pipe and channel flows near smooth walls. The first is the classical approach where the mean streamwise velocity is obtained by solving the momentum equation with an eddy viscosity formulation [R. Absi, A simple eddy viscosity formulation for turbulent boundary layers near smooth walls, C. R. Mecanique 337 (2009) 158–165]. The second approach presents a formulation of the velocity profile based on an analogy with an electric field distribution [C. Di Nucci, E. Fiorucci, Mean velocity profiles of fully-developed turbulent flows near smooth walls, C. R. Mecanique 339 (2011) 388–395] and a formulation for the turbulent shear stress. However, this formulation for the turbulent shear stress shows a weakness. A corrected formulation is presented. Comparisons with DNS data show that the classical approach with the eddy viscosity formulation provides more accurate profiles for both turbulent shear stress and velocity gradient. [Copyright &y& Elsevier]

Abstract: In the work presented in this Note, an Indirect Turbulence Model (ITM) is proposed to derive the mean velocity profiles in wall-bounded flows in hydraulically smooth channels having a very wide rectangular cross section. The analytical expression of the mean velocity distribution is given. The connection between the velocity distribution parameters and Reynoldsʼ number is indicated. The thickness of the viscous sublayer is evaluated. The skin friction coefficient is computed, and the analytical expression of the turbulent viscosity coefficient is provided. The validity of the proposed model is verified with reference to the velocity distributions, available in the literature, obtained with Direct Numerical Simulation (DNS) of Navier–Stokesʼ equations. [Copyright &y& Elsevier]

Abstract: The work proposes an indirect turbulence model to represent the mean streamwise velocity profile of fully-developed turbulent channel flows near smooth walls. The proposed turbulence model highlights that the parameters of the velocity distribution are functions of the friction Reynolds number. It is also shown that the proposed expression for the velocity distribution is in line with the principles of dimensional analysis; it allows one to satisfy the imposed boundary conditions; in the regions close to the wall it allows to reproduce (with good agreement) the velocity profiles available in literature obtained through a Direct Numerical Simulation (DNS). [Copyright &y& Elsevier]

Abstract: To enhance the temporal resolution of the PIV measurements of pseudo-periodic turbulent flows, Proper Orthogonal Decomposition (POD) has been previously used to time interpolate PIV database. In this note, it is demonstrated that such POD interpolation is equivalent to the classical mathematical interpolation when dealing with the whole POD eigenfunctions, since POD is a linear transform. In fact, the POD-based time interpolation is only valid for the large scale structures of the flow. The advantage of using POD procedure resides in its efficiency in extracting the dominant flow structures. In this sense, other interpolation methods such as turbulent filtering procedures could provide similar results. To cite this article: E. Bouhoubeiny, P. Druault, C. R. Mecanique 337 (2009). [Copyright &y& Elsevier]

Abstract: In this short review I argue that the progress in our understanding the mechanism of turbulent drag reduction is conditioned by obtaining experimental data on dynamics and statistics of polymer stretching and elastic stresses in inertial turbulence at high Reynolds numbers that is a technically challenging task. The suggested way out of the currently unresolved technical problem is to collect the same data in elastic turbulence, which is a smooth random flow similar to that found in inertial turbulence below the dissipation scale. Since the polymer stretching and elastic stresses in inertial turbulence are influenced only by small scales, it is appropriate to use information on the polymer stretching and elastic stresses obtained in elastic turbulence. The experimental data on the statistics of the polymer stretching, the coil–stretch transition, and elastic stresses together with spatial distribution and values of the rms of the velocity gradients were collected in elastic turbulence for the last several years. This information serves a basis for a new hypothesis of turbulent drag reduction. To cite this article: V. Steinberg, C. R. Physique 10 (2009). [Copyright &y& Elsevier]

Abstract: The entrainment hypothesis was introduced by G.I. Taylor to describe one-dimensionally the development of turbulent jets issuing into a stagnant or coflowing environment. It relates the mass flow rate of surrounding fluid entrained into the jet to the characteristic velocity difference between the jet and the coflow. A model based on this hypothesis along with axial velocity assumed to follow a realistic Gaussian distribution is presented. It possesses an implicit analytical solution, and its results are compared and shown to be fully equivalent to previously published models that are rather based on a spreading hypothesis. All of them are found to be in agreement with experimental results, on a wide range of downstream positions and for various coflow intensities. To cite this article: N. Enjalbert et al., C. R. Mecanique 337 (2009). [Copyright &y& Elsevier]

Abstract: The accurate simulation of the dynamics of polydisperse evaporating sprays in unsteady gaseous flows with large scale vortical structures is both a crucial issue for industrial applications and a challenge for modeling and scientific computing. The difficulties encountered by the usual Lagrangian approaches make the use of Eulerian models attractive, aiming at a lower cost and an easier coupling with the carrier gaseous phase. Among these models, the multi-fluid model allows the detailed description of polydispersity and size/velocity correlations for droplets of various sizes. The purpose of the present study is two-fold. First, we extend the multi-fluid model in order to cope with droplet trajectory crossings by using the quadrature method of moments in velocity phase space conditioned by size. We identify the numerical difficulties and provide dedicated numerical schemes in order to preserve the velocity moment space. Second, we conduct a comparison study and demonstrate the capability of such an approach to capture the dynamics of an evaporating polydisperse spray in a two-dimensional free jet configuration. We evaluate the accuracy and computational cost of Eulerian models and related discretization schemes versus Lagrangian solvers. It shows that, even for finite Stokes number, the standard Eulerian multi-fluid model is accurate at reasonable cost. To cite this article: S. de Chaisemartin et al., C. R. Mecanique 337 (2009). [Copyright &y& Elsevier]

Abstract: The aim of this study is to improve the prediction of near-wall mean streamwise velocity profile by using a simple method. The profile is obtained by solving the momentum equation which is written as an ordinary differential equation. An eddy viscosity formulation based on a near-wall turbulent kinetic energy function [R. Absi, Analytical solutions for the modeled k-equation, ASME J. Appl. Mech. 75 (2008) 044501] and the van Driest mixing length equation [E.R. van Driest, On turbulent flow near a wall, J. Aero. Sci. 23 (1956) 1007] is used. The parameters obtained from the profiles are used for the computation of (variables with the superscript of + are those nondimensionalized by the wall friction velocity and the kinematic viscosity ν). Comparisons with DNS data of fully-developed turbulent channel flows for show good agreement (where denotes the friction Reynolds number defined by , ν and the channel half-width δ). To cite this article: R. Absi, C. R. Mecanique 337 (2009). [Copyright &y& Elsevier]

Abstract: In recent investigations, finite amplitude streamwise streaks, generated with roughness elements, have been used to delay transition to turbulence. The maximum height of these roughness elements is limited by the appearance of instabilities in their near wake, therefore putting a limit on the maximum streak amplitude they can produce. Here we prove that large amplitude streaks can be generated by ‘adding’ lower amplitude streaks with multiple arrays of roughness elements. To cite this article: M. Hollands, C. Cossu, C. R. Mecanique 337 (2009). [Copyright &y& Elsevier]

Abstract: The aim of this note is an evaluation of a cubic eddy-viscosity turbulent model. The study is carried out in square duct. This configuration presents a secondary flow and a significant anisotropy between the Reynolds stress components. In order to handle wall-proximity effects in the near-wall region, damping functions are introduced. The comparison of the modified model results with DNS and LES simulations shows good agreements. In particular, the mean secondary velocity and the mean streamwise vorticity are well predicted. To cite this article: H. Gnanga et al., C. R. Mecanique 337 (2009). [Copyright &y& Elsevier]

Abstract: 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 the 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. To cite this article: F.G. Schmitt, C. R. Mecanique 335 (2007). [Copyright &y& Elsevier]

Abstract: The spatially averaged balance equation of the turbulent kinetic energy is derived for flows in media laden with solid structures. Two additional contributions are highlighted: sub-filter production and dispersion. The sub-filter production is tightly linked to the drag force and to the wake dissipation. Within the framework of the plane channel flow, we then analyse magnitudes of the various contributions to the spatially averaged equation of the turbulent kinetic energy. We postulate a balance equation for the spatially averaged turbulent dissipation rate. Finally, we propose a model for the characteristic time scale of the sub-filter production. To cite this article: F. Pinson et al., C. R. Mecanique 335 (2007). [Copyright &y& Elsevier]

Abstract: The transport of a passive scalar within a turbulent plane channel flow has been theoretically analyzed by assuming that the Schmidt number Sc, associated to the molecular diffusivity of the passive scalar, is a large parameter. Throughout most of the channel cross-section the mean passive scalar density is constant, but adjacent to the walls a thin boundary layer develops embedded in the viscous sublayer, with a relative thickness of order . In this narrow region a passive scalar profile arises due to the non-vanishing flux normal to the wall. This profile is parameter independent (universal) and leads to a constant flux of passive scalar that results from the addition of both the molecular diffusion flux and the turbulent transport one. The Sc-asymptotic matching of this profile with the constant core value provides an analytical expression for the wall-normal flux that depends on the fluid dynamics of the carrier flow. By using a DNS code to solve the external turbulent flow, the analytical expression has been quantified and compared with empirical expressions based on experimental data, showing excellent agreement. To cite this article: P.L. Garcia-Ybarra, A. Pinelli, C. R. Mecanique 334 (2006). [Copyright &y& Elsevier]

Abstract: This introduction presents the main instabilities responsible for turbulence in tokamak plasmas, and the prominent features of the resulting transport. The usual techniques to construct reduced transport models are described. These models can be tested by analysing steady state and transient regimes. Another way to test the theory is to use a similarity principle, similar to the one used in fluid mechanics. Finally, the physics involved in the formation and sustainment of transport barriers is presented. To cite this article: X. Garbet, C. R. Physique 7 (2006). [Copyright &y& Elsevier]

Abstract: This work is focused on the study of the impingement of a turbulent plane jet on a moving film. A computational fluid dynamics code has been used to simulate the interaction between the turbulent plane jet and the moving film. Since the problem of coupling between turbulence and free surface flow is poorly understood and experiments in this problem are difficult to carry out, this new numerical tool has been designed to give insight into global and local parameters of the free surface flow. To cite this article: D. Lacanette et al., C. R. Mecanique 333 (2005). [Copyright &y& Elsevier]

Abstract: To allow for a reliable examination of the interaction between velocity fluctuations, acoustics and combustion, a novel numerical procedure is discussed in which a spectral solution of the Navier–Stokes equations is directly associated to a high-order finite difference fully compressible DNS solver (sixth order PADE). Using this combination of high-order solvers with accurate boundary conditions, simulations have been performed where a turbulent premixed V-shape flame develops in grid turbulence. In the light of the DNS results, a sub-model for premixed turbulent combustion is analyzed. To cite this article: R. Hauguel et al., C. R. Mecanique 333 (2005). [Copyright &y& Elsevier]