[Une théorie phénoménologique des fluctuations de vitesse Eulérienne et Lagrangienne dans un écoulement turbulent]
Nous présentons une théorie phénoménologique des fluctuations de vitesse dans un écoulement turbulent pleinement développé isotrope et homogène. Nous mettons lʼaccent sur les fluctuations des incréments spatiaux (Eulérien) et temporels (Lagrangien) de vitesse. La nature universelle du phénomène dʼintermittence observé sur les mesures expérimentales et les simulations numériques est complètement pris en compte par les arguments développés par le formalisme multifractal, et ses extensions aux échelles dissipatives et au cadre Lagrangien. Cet article présente les prédictions de cette description multifractale et les compare aux données empiriques. En particulier, des prédictions explicites sont obtenues pour des grandeurs statistiques, comme les fonctions de densité de probabilité et les moments dʼordres supérieurs, des gradients de vitesse et de lʼaccélération. Dans le cadre Eulérien, à un nombre de Reynolds donné, nous montrons que ces prédictions ne dépendent que dʼune fonction à paramètres, appelée spectre de singularités, et dʼune constante régissant la transition entre les régimes inertiels et dissipatifs. Le spectre des singularités Lagrangien est relié à son homologue Eulérien par une transformation basée sur la nature incompressible, homogène et isotrope de lʼécoulement, alors que la constante restante est difficile à estimer à partir des données. Nous montrons finalement que lʼincrément est inadapté à quantifier précisément la nature singulière de la vitesse Lagrangienne. Cela est confirmé par lʼutilisation dʼincréments dʼordres supérieurs non biaisés par la présence de comportements linéaires, comme nous lʼobservons sur la vitesse le long dʼune trajectoire.
A phenomenological theory of the fluctuations of velocity occurring in a fully developed homogeneous and isotropic turbulent flow is presented. The focus is made on the fluctuations of the spatial (Eulerian) and temporal (Lagrangian) velocity increments. The universal nature of the intermittency phenomenon as observed in experimental measurements and numerical simulations is shown to be fully taken into account by the multiscale picture proposed by the multifractal formalism, and its extensions to the dissipative scales and to the Lagrangian framework. The article is devoted to the presentation of these arguments and to their comparisons against empirical data. In particular, explicit predictions of the statistics, such as probability density functions and high order moments, of the velocity gradients and acceleration are derived. In the Eulerian framework, at a given Reynolds number, they are shown to depend on a single parameter function called the singularity spectrum and to a universal constant governing the transition between the inertial and dissipative ranges. The Lagrangian singularity spectrum compares well with its Eulerian counterpart by a transformation based on incompressibility, homogeneity and isotropy and the remaining constant is shown to be difficult to estimate on empirical data. It is finally underlined the limitations of the increment to quantify accurately the singular nature of Lagrangian velocity. This is confirmed using higher order increments unbiased by the presence of linear trends, as they are observed on velocity along a trajectory.
Mot clés : Turbulence, Intermittence, Eulérien et Lagrangien
Laurent Chevillard 1 ; Bernard Castaing 1 ; Alain Arneodo 1 ; Emmanuel Lévêque 1 ; Jean-François Pinton 1 ; Stéphane G. Roux 1
@article{CRPHYS_2012__13_9-10_899_0,
author = {Laurent Chevillard and Bernard Castaing and Alain Arneodo and Emmanuel L\'ev\^eque and Jean-Fran\c{c}ois Pinton and St\'ephane G. Roux},
title = {A phenomenological theory of {Eulerian} and {Lagrangian} velocity fluctuations in turbulent flows},
journal = {Comptes Rendus. Physique},
pages = {899--928},
publisher = {Elsevier},
volume = {13},
number = {9-10},
year = {2012},
doi = {10.1016/j.crhy.2012.09.002},
language = {en},
}
TY - JOUR AU - Laurent Chevillard AU - Bernard Castaing AU - Alain Arneodo AU - Emmanuel Lévêque AU - Jean-François Pinton AU - Stéphane G. Roux TI - A phenomenological theory of Eulerian and Lagrangian velocity fluctuations in turbulent flows JO - Comptes Rendus. Physique PY - 2012 SP - 899 EP - 928 VL - 13 IS - 9-10 PB - Elsevier DO - 10.1016/j.crhy.2012.09.002 LA - en ID - CRPHYS_2012__13_9-10_899_0 ER -
%0 Journal Article %A Laurent Chevillard %A Bernard Castaing %A Alain Arneodo %A Emmanuel Lévêque %A Jean-François Pinton %A Stéphane G. Roux %T A phenomenological theory of Eulerian and Lagrangian velocity fluctuations in turbulent flows %J Comptes Rendus. Physique %D 2012 %P 899-928 %V 13 %N 9-10 %I Elsevier %R 10.1016/j.crhy.2012.09.002 %G en %F CRPHYS_2012__13_9-10_899_0
Laurent Chevillard; Bernard Castaing; Alain Arneodo; Emmanuel Lévêque; Jean-François Pinton; Stéphane G. Roux. A phenomenological theory of Eulerian and Lagrangian velocity fluctuations in turbulent flows. Comptes Rendus. Physique, Volume 13 (2012) no. 9-10, pp. 899-928. doi : 10.1016/j.crhy.2012.09.002. https://comptes-rendus.academie-sciences.fr/physique/articles/10.1016/j.crhy.2012.09.002/
[1] Weather Prediction by Numerical Process, Cambridge University Press, Cambridge, 1922
[2] The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers, Dokl. Akad. Nauk SSSR, Volume 30 (1941), p. 301 (in Russian). English translation: Proc. R. Soc. London, Ser. A, 434, 1991, pp. 9
[3] The Theory of Homogeneous Turbulence, Cambridge University Press, Cambridge, 1953
[4] A First Course in Turbulence, MIT Press, Cambridge, MA, 1972
[5] On Kolmogorovʼs inertial-range theories, J. Fluid Mech., Volume 62 (1974), p. 305
[6] Statistical Fluid Mechanics, MIT Press, Cambridge, MA, 1975
[7] Turbulence, Cambridge University Press, Cambridge, 1995
[8] Turbulent Flows, Cambridge University Press, Cambridge, 2000
[9] An Informal Introduction to Turbulence, Kluwer Academic, Dordrecht, 2001
[10] Universality of the Kolmogorov constant in numerical simulations of turbulence, Phys. Rev. E, Volume 56 (1997), p. 1746
[11] The bottleneck effect and the Kolmogorov constant in isotropic turbulence, J. Fluid Mech., Volume 657 (2010), p. 171
[12] Velocity probability density functions of high Reynolds number turbulence, Physica D, Volume 46 (1990), p. 177
[13] Multifractality in the statistics of the velocity gradients in turbulence, Phys. Rev. Lett., Volume 67 (1991), p. 2299
[14] Phys. Rev. Lett., 68 (1992), p. 2766
[15] Some specific features of atmospheric turbulence, J. Fluid Mech., Volume 13 (1962), p. 77
[16] A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number, J. Fluid Mech., Volume 13 (1962), p. 77
[17] Intermittency and Reynolds number, Phys. Fluids, Volume 10 (1998), p. 910
[18] Unified multifractal description of velocity increments statistics in turbulence: Intermittency and skewness, Physica D, Volume 218 (2006), p. 77
[19] Experimental Lagrangian acceleration probability density function measurement, Physica D, Volume 193 (2004), p. 245
[20] Lagrangian velocity statistics in turbulent flows: Effects of dissipation, Phys. Rev. Lett., Volume 91 (2003), p. 214502
[21] An experimental investigation of the relative diffusion of particle pairs in three-dimensional turbulent flow, J. Fluid Mech., Volume 422 (2000), p. 207
[22] Fluid particle accelerations in fully developed turbulence, Nature, Volume 409 (2001), p. 1017
[23] Measurement of particle accelerations in fully developed turbulence, J. Fluid Mech., Volume 469 (2002), p. 121
[24] Measurement of Lagrangian velocity in fully developed turbulence, Phys. Rev. Lett., Volume 87 (2001), p. 214501
[25] Long time correlations in Lagrangian dynamics: A key to intermittency in turbulence, Phys. Rev. Lett., Volume 89 (2002), p. 254502
[26] Lagrangian velocity fluctuations in fully developed turbulence: Scaling, intermittency, and dynamics, J. Stat. Phys., Volume 113 (2003), p. 701
[27] High order Lagrangian velocity statistics in turbulence, Phys. Rev. Lett., Volume 96 (2006), p. 024503
[28] Experimental investigation of Lagrangian structure functions in turbulence, Phys. Rev. E, Volume 80 (2009), p. 026316
[29] Lagrangian statistics from direct numerical simulations of isotropic turbulence, J. Fluid Mech., Volume 207 (1989), p. 531
[30] Lagrangian characteristics of turbulence and scalar transport in direct numerical simulations, J. Fluid Mech., Volume 427 (2001), p. 241
[31] Multifractal statistics of Lagrangian velocity and acceleration in turbulence, Phys. Rev. Lett., Volume 93 (2004), p. 064502
[32] Lagrangian investigations of turbulence, Ann. Rev. Fluid Mech., Volume 34 (2002), p. 115
[33] et al. Universal intermittent properties of particle trajectories in highly turbulent flows, Phys. Rev. Lett., Volume 100 (2008), p. 254504
[34] Lagrangian properties of particles in turbulence, Ann. Rev. Fluid Mech., Volume 41 (2009), p. 375
[35] Estimation of the Kolmogorov constant (C0) for the Lagrangian structure function, using a second-order Lagrangian model of grid turbulence, Phys. Fluids, Volume 7 (1995), p. 3083
[36] The Kolmogorov constant for the Lagrangian velocity spectrum and structure function, Phys. Fluids, Volume 14 (2002), p. 4456
[37] A prediction of the multifractal model: the intermediate dissipation range, Europhys. Lett., Volume 14 (1991), p. 439
[38] Superstatistics, Physics A, Volume 322 (2003), p. 267
[39] Statistics of Lagrangian velocities in turbulent flows, Phys. Rev. Lett., Volume 90 (2003), p. 084501
[40] Lagrangian and Eulerian velocity structure functions in hydrodynamic turbulence, Phys. Rev. Lett., Volume 104 (2010), p. 154501
[41] The Lundgren–Monin–Novikov hierarchy: Kinetic equations for turbulence, C. R. Phys., Volume 13 (2012), pp. 929-953
[42] Probability density functions, skewness, and flatness in large Reynolds number turbulence, Phys. Rev. E, Volume 53 (1996), p. 1613
[43] Transition at dissipative scales in large-Reynolds-number turbulence, Phys. Rev. E, Volume 65 (2002), p. 066301
[44] On the rapid increase of intermittency in the near-dissipation range of fully developed turbulence, Eur. Phys. J. B, Volume 45 (2005), p. 561
[45] Intermittency in a turbulent low temperature gaseous helium jet, Eur. Phys. J. B, Volume 17 (2000), p. 309
[46] Dynamical origins for non-Gaussian vorticity distributions in turbulent flows, Phys. Rev. E, Volume 80 (2009), p. 0160316
[47] On the velocity distribution in homogeneous isotropic turbulence: Correlations and deviations from Gaussianity, J. Fluid Mech., Volume 676 (2011), p. 191
[48] et al. Structure functions in turbulence, in various flow configurations, at Reynolds number between 30 and 5000, using extended self-similarity, Europhys. Lett., Volume 34 (1996), p. 411
[49] Universal scaling laws in fully developed turbulence, Phys. Rev. Lett., Volume 72 (1994), p. 336
[50] Intermittency in fully developed turbulence: Log-Poisson statistics and generalized scale covariance, Phys. Rev. Lett., Volume 73 (1994), p. 959
[51] Multifractal formalism for fractal signals: The structure-function approach versus the wavelet-transform modulus-maxima method, Phys. Rev. E, Volume 47 (1993), p. 875
[52] Bootstrap for empirical multifractal analysis, IEEE Signal Proc. Mag., Volume 24 (2007), p. 38
[53] Degrees of freedom of turbulence, Phys. Rev. A, Volume 35 (1987), p. 1971
[54] A statistical estimator of turbulence intermittency in physical and numerical experiments, Eur. Phys. J. B, Volume 16 (2000), p. 549
[55] Reynolds dependence of third-order velocity structure functions, Phys. Fluids, Volume 16 (2004), p. 482
[56] Multifractal scaling of velocity derivatives in turbulence, Phys. Rev. A, Volume 42 (1990), p. 7226
[57] Reynolds number dependence of skewness and flatness factors of turbulent velocity derivatives, Phys. Fluids, Volume 23 (1980), p. 252
[58] The phenomenology of small-scale turbulence, Ann. Rev. Fluid Mech., Volume 29 (1997), p. 435
[59] Intermittency, pressure and acceleration statistics from hot-wire measurements in wind–tunnel turbulence, J. Fluid Mech., Volume 501 (2004), p. 213
[60] Small-scale statistics in high-resolution direct numerical simulation of turbulence: Reynolds number dependence of one-point velocity gradient statistics, J. Fluid Mech., Volume 592 (2007), p. 335
[61] Reynolds number dependence of high-order moments of the streamwise turbulent velocity derivative, Boundary Layer Met., Volume 21 (1981), p. 159
[62] L. Chevillard, Unified multifractal description of the intermittency phenomenon in Eulerian and Lagrangian turbulence, PhD thesis, University of Bordeaux, 2004, unpublished, can be found online at http://tel.archives-ouvertes.fr/.
[63] Pressure fluctuations in isotropic turbulence, Proc. Cambridge Philos. Soc., Volume 47 (1951), p. 359
[64] Transition between viscous and inertial-range scaling of turbulence structure functions, Phys. Rev. E, Volume 54 (1996), p. 3657
[65] Reynolds number effect on the velocity increment skewness in isotropic turbulence, Phys. Fluids, Volume 24 (2012), p. 015108
[66] Lagrangian structure functions in turbulence: A quantitative comparison between experiment and direct numerical simulation, Phys. Fluids, Volume 20 (2008), p. 065103
[67] Inertial range Eulerian and Lagrangian statistics from numerical simulations of isotropic turbulence, J. Fluid Mech., Volume 653 (2010), p. 221
[68] Intermittency of 1D velocity spatial profiles in turbulence: A magnitude cumulant analysis, Eur. Phys. J. B, Volume 23 (2001), p. 243
[69] The multifractal Lagrangian nature of turbulence, Phil. Trans. R. Soc. Lond. A, Volume 342 (1993), p. 379
[70] Lagrangian statistics and temporal intermittency in a shell model of turbulence, Phys. Rev. E, Volume 66 (2002), p. 066307
[71] Bridging from Eulerian to Lagrangian statistics in 3D hydro- and magnetohydrodynamic turbulent flows, New J. Phys., Volume 11 (2009), p. 73020
[72] Exact relation between Eulerian and Lagrangian velocity increment statistics, Phys. Rev. E, Volume 79 (2009), p. 066301
[73] Joint multifractal measures: Theory and applications to turbulence, Phys. Rev. A, Volume 41 (1990), p. 894
[74] The multifractal nature of turbulent energy dissipation, J. Fluid Mech., Volume 224 (1991), p. 429
[75] Singularities of the equations of fluid motion, Phys. Rev. A, Volume 38 (1988), p. 6287
[76] Velocity-gradient statistics along particle trajectories in turbulent flows: The refined similarity hypothesis in the Lagrangian frame, Phys. Rev. E, Volume 80 (2009), p. 066318
[77] Lagrangian refined Kolmogorov similarity hypothesis for gradient time evolution and correlation in turbulent flows, Phys. Rev. Lett., Volume 104 (2010), p. 084502
[78] Exit time of turbulent signals: A way to detect the intermediate dissipative range, Phys. Rev. E, Volume 60 (1999), p. R6295
[79] Scaling of acceleration in locally isotropic turbulence, J. Fluid Mech., Volume 452 (2002), p. 361
[80] Observation of intermittency in wave turbulence, Phys. Rev. Lett., Volume 98 (2007), p. 154501
[81] On the origin of intermittency in wave turbulence, Europhys. Lett., Volume 90 (2010), p. 34005
[82] Sub-Kolmogorov-scale fluctuations in fluid turbulence, Europhys. Lett., Volume 80 (2007), p. 54001
[83] Internal motion of a small element of fluid in an inviscid flow, Physica A, Volume 125 (1984), p. 150
[84] Alignment of vorticity and scalar gradient with strain rate in simulated Navier–Stokes turbulence, Phys. Fluids, Volume 30 (1987), p. 2343
[85] A diffusion model for velocity gradients in turbulence, Phys. Fluids A, Volume 2 (1990), p. 242
[86] Exact solution of a restricted Euler equation for the velocity gradient tensor, Phys. Fluids A, Volume 4 (1992), p. 782
[87] Lagrangian tetrad dynamics and the phenomenology of turbulence, Phys. Fluids, Volume 11 (1999), p. 2394
[88] Velocity-gradient dynamics in turbulence: Effect of viscosity and forcing, Theor. Comput. Fluid Dyn., Volume 16 (2003), p. 421
[89] Lagrangian dynamics and statistical geometric structure of turbulence, Phys. Rev. Lett., Volume 97 (2006), p. 174501
[90] Intermittency and universality in a Lagrangian model of velocity gradients in three-dimensional turbulence, C. R. Mecanique, Volume 335 (2007), p. 187
[91] Lagrangian dynamics and models of the velocity gradient tensor in turbulent flows, Ann. Rev. Fluid Mech., Volume 43 (2011), p. 219
[92] Autocorrelation and spectrum of dissipation fluctuations in a turbulent jet, Phys. Fluids, Volume 24 (1981), p. 554
[93] A stochastic representation of the local structure of turbulence, Europhys. Lett., Volume 89 (2010), p. 54002
[94] Probability, Random Variables and Stochastic Processes, McGraw-Hill Inc., New York, 1991
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