Comptes Rendus
Structures and statistics of fluid turbulence / Structures et statistiques de la turbulence des fluides
The Lundgren–Monin–Novikov hierarchy: Kinetic equations for turbulence
[La hiérarchie de Lundgren–Monin–Novikov : Des équations cinétiques de la turbulence]
Comptes Rendus. Physique, Volume 13 (2012) no. 9-10, pp. 929-953.

Nous présentons un aperçu des travaux récents sur la description statistique des écoulements turbulents en terme de fonctions densité de probabilité (PDFs) dans le cadre de la hiérarchie de Lundgren–Monin–Novikov (LMN). Dans ce cadre, des équations dʼévolution pour les PDFs sont dérivées à partir des équations fondamentales décrivant la dynamique des fluides. Le problème de fermeture se pose soit sous la forme dʼune couplage aux PDFs multipoints, soit sous la forme de moyennes conditionnelles qui entrent dans les équations dʼévolution en tant que fonctions inconnues. Nous nous concentrons principalement sur le dernier cas et utilisons les données des simulations numériques directes (DNS) pour calculer les termes non fermés. Nous donnerons donc une introduction de base aux techniques analytiques. Ensuite, nous présenterons quelques applications, en particulier à la statistique de la vorticité en deux dimensions, aux statistiques de vitesse et de vorticité en un point en trois dimensions, à la statistique de la température dans la convection de Rayleigh–Bénard et à la turbulence de Burgers.

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.

Publié le :
DOI : 10.1016/j.crhy.2012.09.009
Keywords: Turbulence, Lundgren–Monin–Novikov hierarchy, Probability density functions
Mot clés : Turbulence, Hiérarchie de Lundgren–Monin–Novikov, Fonctions densité de probabilité

Rudolf Friedrich 1, 2 ; Anton Daitche 1 ; Oliver Kamps 2 ; Johannes Lülff 1 ; Michel Voßkuhle 3 ; Michael Wilczek 1

1 Institute for Theoretical Physics, Westfälische Wilhelms-Universität, Wilhelm-Klemm-Str. 9, 48149 Münster, Germany
2 Center for Nonlinear Science, Westfälische Wilhelms-Universität, Corrensstr. 2, 48149 Münster, Germany
3 Laboratoire de physique, ENS de Lyon, 46, allée dʼItalie, 69007 Lyon, France
@article{CRPHYS_2012__13_9-10_929_0,
     author = {Rudolf Friedrich and Anton Daitche and Oliver Kamps and Johannes L\"ulff and Michel Vo{\ss}kuhle and Michael Wilczek},
     title = {The {Lundgren{\textendash}Monin{\textendash}Novikov} hierarchy: {Kinetic} equations for turbulence},
     journal = {Comptes Rendus. Physique},
     pages = {929--953},
     publisher = {Elsevier},
     volume = {13},
     number = {9-10},
     year = {2012},
     doi = {10.1016/j.crhy.2012.09.009},
     language = {en},
}
TY  - JOUR
AU  - Rudolf Friedrich
AU  - Anton Daitche
AU  - Oliver Kamps
AU  - Johannes Lülff
AU  - Michel Voßkuhle
AU  - Michael Wilczek
TI  - The Lundgren–Monin–Novikov hierarchy: Kinetic equations for turbulence
JO  - Comptes Rendus. Physique
PY  - 2012
SP  - 929
EP  - 953
VL  - 13
IS  - 9-10
PB  - Elsevier
DO  - 10.1016/j.crhy.2012.09.009
LA  - en
ID  - CRPHYS_2012__13_9-10_929_0
ER  - 
%0 Journal Article
%A Rudolf Friedrich
%A Anton Daitche
%A Oliver Kamps
%A Johannes Lülff
%A Michel Voßkuhle
%A Michael Wilczek
%T The Lundgren–Monin–Novikov hierarchy: Kinetic equations for turbulence
%J Comptes Rendus. Physique
%D 2012
%P 929-953
%V 13
%N 9-10
%I Elsevier
%R 10.1016/j.crhy.2012.09.009
%G en
%F CRPHYS_2012__13_9-10_929_0
Rudolf Friedrich; Anton Daitche; Oliver Kamps; Johannes Lülff; Michel Voßkuhle; Michael Wilczek. The Lundgren–Monin–Novikov hierarchy: Kinetic equations for turbulence. Comptes Rendus. Physique, Volume 13 (2012) no. 9-10, pp. 929-953. doi : 10.1016/j.crhy.2012.09.009. https://comptes-rendus.academie-sciences.fr/physique/articles/10.1016/j.crhy.2012.09.009/

[1] A. Tsinober An Informal Conceptual Introduction to Turbulence, Springer, 2009

[2] A.S. Monin; A.M. Yaglom; J.L. Lumley Statistical Fluid Mechanics: Mechanics of Turbulence, vol. 1, Dover Publications, 2007

[3] A.S. Monin; A.M. Yaglom; J.L. Lumley Statistical Fluid Mechanics: Mechanics of Turbulence, vol. 2, Dover Publications, 2007

[4] S. Pope Turbulent Flows, Cambridge University Press, Cambridge, England, 2000

[5] O. Reynolds On the dynamical theory of incompressible viscous fluids and the determination of the criterion, Philos. Trans. R. Soc. Lond. A, Volume 186 (1895), pp. 123-164

[6] L. Keller, A. Friedmann, Differentialgleichungen für die turbulente Bewegung einer kompressiblen Flüssigkeit, in: Proc. First Internat. Congress Appl. Mech., 1924, pp. 395–405.

[7] V. Yakhot Mean-field approximation and extended self-similarity in turbulence, Phys. Rev. Lett., Volume 87 (2001), p. 234501

[8] G. Falkovich; I. Fouxon; Y. Oz New relations for correlation functions in Navier–Stokes turbulence, J. Fluid Mech., Volume 644 (1895), pp. 465-472

[9] M.D. Millionshchikov On the theory of homogeneous isotropic turbulence, Dokl. Acad. Nauk SSSR, Volume 32 (1941), pp. 611-614

[10] T.S. Lundgren Distribution functions in the statistical theory of turbulence, Phys. Fluids, Volume 10 (1967) no. 5, pp. 969-975

[11] A.S. Monin Equations of turbulent motion, Prikl. Mat. Mekh., Volume 31 (1967) no. 6, p. 1057

[12] E.A. Novikov Kinetic equations for a vortex field, Sov. Phys. Dokl., Volume 12 (1968) no. 11, pp. 1006-1008

[13] N.N. Bogolyubova Foundations of Kinetic Theory. N.N. Bogolyubovʼs Method, Nauka, 1966

[14] E.M. Lifshitz; L.P. Pitaevskiĭ Physical Kinetics, Course of Theoretical Physics, Butterworth–Heinemann, 1981

[15] L.R. Ulinich; B.Y. Lyubimov The statistical theory of turbulence of an incompressible fluid at large Reynolds number, Sov. Phys. JETP, Volume 28 (1969) no. 3, pp. 494-500

[16] S.B. Pope PDF methods for turbulent reactive flows, Prog. Energ. Combust. Sci., Volume 11 (1985) no. 2, pp. 119-192

[17] Y.G. Sinai; V. Yakhot Limiting probability distributions of a passive scalar in a random velocity field, Phys. Rev. Lett., Volume 63 (1989), pp. 1962-1964

[18] V. Yakhot Probability distributions in high-Rayleigh number Bénard convection, Phys. Rev. Lett., Volume 63 (1989), pp. 1965-1967

[19] V. Yakhot Probability density and scaling exponents of the moments of longitudinal velocity difference in strong turbulence, Phys. Rev. E, Volume 57 (1998), pp. 1737-1751

[20] I. Hosokawa One-point velocity statistics in decaying homogeneous isotropic turbulence, Phys. Rev. E, Volume 78 (2008), p. 066312

[21] T. Tatsumi; T. Yoshimura Inertial similarity of velocity distributions in homogeneous isotropic turbulence, Fluid Dyn. Res., Volume 35 (2004) no. 2, pp. 123-158

[22] T. Tatsumi; T. Yoshimura Local similarity of velocity distributions in homogeneous isotropic turbulence, Fluid Dyn. Res., Volume 39 (2007) no. 1–3, p. 221

[23] T. Tatsumi Cross-independence closure for statistical mechanics of fluid turbulence, J. Fluid Mech., Volume 670 (2011), pp. 365-403

[24] G. Boffetta; M. Cencini; J. Davoudi Closure of two-dimensional turbulence: The role of pressure gradients, Phys. Rev. E, Volume 66 (2002), p. 017301

[25] E.S.C. Ching Probability densities of turbulent temperature fluctuations, Phys. Rev. Lett., Volume 70 (1993) no. 3, pp. 283-286

[26] S.B. Pope; E.S.C. Ching Stationary probability density functions: An exact result, Phys. Fluids A, Volume 5 (1993) no. 7, pp. 1529-1531

[27] E.S.C. Ching General formula for stationary or statistically homogeneous probability density functions, Phys. Rev. E, Volume 53 (1996) no. 6, pp. 5899-5903

[28] A.M. Polyakov Turbulence without pressure, Phys. Rev. E, Volume 52 (1995), pp. 6183-6188

[29] J. Béc; K. Khanin Burgers turbulence, Phys. Rep., Volume 447 (2007), pp. 1-66

[30] R. Friedrich; J. Peinke Description of a turbulent cascade by a Fokker–Planck equation, Phys. Rev. Lett., Volume 78 (1997) no. 5, pp. 863-866

[31] S. Lück; J. Peinke; R. Friedrich Uniform statistical description of the transition between near and far field turbulence in a wake flow, Phys. Rev. Lett., Volume 83 (1999), pp. 5495-5498

[32] C. Renner; J. Peinke; R. Friedrich Experimental indications for Markov properties of small-scale turbulence, J. Fluid Mech., Volume 433 (2001), pp. 383-409

[33] C. Renner; J. Peinke; R. Friedrich; O. Chanal; B. Chabaud Universality of small scale turbulence, Phys. Rev. Lett., Volume 89 (2002) no. 12, p. 124502

[34] R. Friedrich; J. Peinke; M. Sahimi; M.R.R. Tabar Approaching complexity by stochastic methods: From biological systems to turbulence, Phys. Rep., Volume 506 (2011) no. 5, pp. 87-162

[35] S. Lück; C. Renner; J. Peinke; R. Friedrich The Markov–Einstein coherence length – a new meaning for the Taylor length in turbulence, Phys. Lett. A, Volume 359 (2006) no. 5, pp. 335-338

[36] R. Friedrich, M. Voßkuhle, O. Kamps, M. Wilczek, Two-point vorticity statistics in the inverse turbulent cascade, Phys. Fluids, in press.

[37] U. Frisch Turbulence: The Legacy of A.N. Kolmogorov, Cambridge University Press, 1995

[38] H.P. Robertson The invariant theory of isotropic turbulence, Math. Proc. Camb. Philos. Soc., Volume 36 (1940), pp. 209-223

[39] S. Chandrasekhar The invariant theory of isotropic turbulence in magneto-hydrodynamics, Proc. R. Soc. Lond. A, Volume 204 (1951) no. 1079, pp. 435-449

[40] S. Chandrasekhar The invariant theory of isotropic turbulence in magneto-hydrodynamics II, Proc. R. Soc. Lond. A, Volume 207 (1951) no. 1090, pp. 301-306

[41] G.K. Batchelor The Theory of Homogeneous Turbulence, Cambridge Science Classics, Cambridge University Press, 1953

[42] T. de Kármán; L. Howarth On the statistical theory of isotropic turbulence, Proc. R. Soc. Lond. A, Volume 164 (1938) no. 917, pp. 192-215

[43] M. Wilczek; A. Daitche; R. Friedrich On the velocity distribution in homogeneous isotropic turbulence: correlations and deviations from Gaussianity, J. Fluid Mech., Volume 676 (2011), pp. 191-217

[44] G. Boffetta; S. Musacchio Evidence for the double cascade scenario in two-dimensional turbulence, Phys. Rev. E, Volume 82 (2010), p. 016307

[45] H. Risken The Fokker–Planck Equation: Methods of Solutions and Applications, Springer Series in Synergetics, Springer, 1996

[46] C.W. Gardiner Handbook of Stochastic Methods: For Physics, Chemistry and the Natural Sciences, Springer Series in Synergetics, Springer, 2004

[47] E.A. Novikov Functionals and the random-force method in turbulence theory, Sov. Phys. JETP, Volume 20 (1965), p. 1290

[48] R. Courant; D. Hilbert Methods of Mathematical Physics, vol. II, Wiley–Interscience, 1962

[49] M. Chertkov; A. Pumir; B.I. Shraiman Lagrangian tetrad dynamics and the phenomenology of turbulence, Phys. Fluids, Volume 11 (1999) no. 8, pp. 2394-2410

[50] A. Naso; A. Pumir Scale dependence of the coarse-grained velocity derivative tensor structure in turbulence, Phys. Rev. E, Volume 72 (2005), p. 056318

[51] L. Chevillard; C. Meneveau Lagrangian dynamics and statistical geometric structure of turbulence, Phys. Rev. Lett., Volume 97 (2006), p. 174501

[52] L. Chevillard; C. Meneveau; L. Biferale; F. Toschi Modeling the pressure Hessian and viscous Laplacian in turbulence: Comparisons with direct numerical simulation and implications on velocity gradient dynamics, Phys. Fluids, Volume 20 (2008) no. 10, p. 101504

[53] R. Friedrich Statistics of Lagrangian velocities in turbulent flows, Phys. Rev. Lett., Volume 90 (2003), p. 084501

[54] R. Friedrich Lagrangian probability distributions of turbulent flows | arXiv

[55] E.A. Novikov Lagrangian infinitesimal increments of vorticity in 2d turbulence, Phys. Lett. A, Volume 236 (1997), pp. 65-68

[56] S. Ott; J. Mann An experimental test of Corrsinʼs conjecture and some related ideas, New J. Phys., Volume 7 (2005), p. 142

[57] M.S. Borgas The multifractal Lagrangian nature of turbulence, Philos. Trans. Roy. Soc., Volume 342 (1993) no. 1665, pp. 379-411

[58] G. Boffetta; F. De Lillo; S. Musacchio Lagrangian statistics and temporal intermittency in a shell model of turbulence, Phys. Rev. E, Volume 66 (2002) no. 6, p. 066307

[59] L. Chevillard; S.G. Roux; E. Levêque; N. Mordant; J.-F. Pinton; A. Arneodo Lagrangian velocity statistics in turbulent flows: Effects of dissipation, Phys. Rev. Lett., Volume 91 (2003) no. 21, p. 214502

[60] O. Kamps; R. Friedrich; R. Grauer Exact relation between Eulerian and Lagrangian velocity increment statistics, Phys. Rev. E, Volume 79 (2009) no. 6, p. 066301

[61] H. Homann; O. Kamps; R. Friedrich; R. Grauer Bridging from Eulerian to Lagrangian statistics in 3d hydro- and magnetohydrodynamic turbulent flows, New J. Phys., Volume 11 (2009) no. 7, p. 073020

[62] J. Lülff; M. Wilczek; A. Daitche ‘Turbulence Team Münster’ YouTube channel, 2012 http://www.youtube.com/user/turbulenceteamms

[63] P.K. Newton The N-Vortex Problem: Analytical Techniques, Springer, 2001

[64] H. Aref Point vortex dynamics: A classical mathematics playground, J. Math. Phys., Volume 48 (2007) no. 6, p. 065401

[65] J. Friedrich; R. Friedrich Vortex-model for the inverse cascade of 2d-turbulence | arXiv

[66] A. Vincent; M. Meneguzzi The spatial structure and statistical properties of homogeneous turbulence, J. Fluid Mech., Volume 225 (1991), pp. 1-20

[67] A. Noullez; G. Wallace; W. Lempert; R.B. Miles; U. Frisch Transverse velocity increments in turbulent flow using the relief technique, J. Fluid Mech., Volume 339 (1997) no. 1, pp. 287-307

[68] T. Gotoh; D. Fukayama; T. Nakano Velocity field statistics in homogeneous steady turbulence obtained using a high-resolution direct numerical simulation, Phys. Fluids, Volume 14 (2002) no. 3, pp. 1065-1081

[69] G. Falkovich; V. Lebedev Single-point velocity distribution in turbulence, Phys. Rev. Lett., Volume 79 (1997) no. 21, pp. 4159-4161

[70] M. Wilczek; A. Daitche; R. Friedrich Theory for the single-point velocity statistics of fully developed turbulence, Europhys. Lett., Volume 93 (2011) no. 3, p. 34003

[71] M. Wilczek, Statistical and numerical investigations of fluid turbulence, Ph.D. thesis, Institute for Theoretical Physics, University of Münster, Germany, 2011.

[72] R.C.Y. Mui; D.G. Dommermuth; E.A. Novikov Conditionally averaged vorticity field and turbulence modeling, Phys. Rev. E, Volume 53 (1996) no. 3, pp. 2355-2359

[73] E.A. Novikov A new approach to the problem of turbulence, based on the conditionally averaged Navier–Stokes equations, Fluid Dyn. Res., Volume 12 (1993) no. 2, pp. 107-126

[74] C. Meneveau Lagrangian dynamics and models of the velocity gradient tensor in turbulent flows, Annu. Rev. Fluid Mech., Volume 43 (2011) no. 1, pp. 219-245

[75] M. Wilczek; R. Friedrich Dynamical origins for non-Gaussian vorticity distributions in turbulent flows, Phys. Rev. E, Volume 80 (2009), p. 016316

[76] M. Wilczek; B. Kadoch; K. Schneider; R. Friedrich; M. Farge Conditional vorticity budget of coherent and incoherent flow contributions in fully developed homogeneous isotropic turbulence, Phys. Fluids, Volume 24 (2012) no. 3, p. 035108

[77] E.A. Novikov Statistical balance of vorticity and a new scale for vortical structures in turbulence, Phys. Rev. Lett., Volume 71 (1993), pp. 2718-2720

[78] J. Lülff; M. Wilczek; R. Friedrich Temperature statistics in turbulent Rayleigh–Bénard convection, New J. Phys., Volume 13 (2011) no. 1, p. 015002

[79] S. Grossmann; D. Lohse Scaling in thermal convection: a unifying theory, J. Fluid Mech., Volume 407 (2000), pp. 27-56

[80] P. Angot; C.-H. Bruneau; P. Fabrie A penalization method to take into account obstacles in incompressible viscous flows, Numer. Math., Volume 81 (1999) no. 4, pp. 497-520

[81] K. Schneider Numerical simulation of the transient flow behaviour in chemical reactors using a penalisation method, Comput. Fluids, Volume 34 (2005) no. 10, pp. 1223-1238

[82] G.H. Keetels; U. DʼOrtona; W. Kramer; H.J.H. Clercx; K. Schneider; G.J.F. van Heijst Fourier spectral and wavelet solvers for the incompressible Navier–Stokes equations with volume-penalization: Convergence of a dipole-wall collision, J. Comput. Phys., Volume 227 (2007) no. 2, pp. 919-945

[83] M.S. Emran; J. Schumacher Fine-scale statistics of temperature and its derivatives in convective turbulence, J. Fluid Mech., Volume 611 (2008), pp. 13-34

[84] Y. Gasteuil; W.L. Shew; M. Gibert; F. Chillá; B. Castaing; J.-F. Pinton Lagrangian temperature, velocity, and local heat flux measurement in Rayleigh–Bénard convection, Phys. Rev. Lett., Volume 99 (2007) no. 23, p. 234302

[85] S. Eule; R. Friedrich A note on the forced Burgers equation, Phys. Lett. A, Volume 351 (2006), pp. 238-241

Cité par Sources :

Commentaires - Politique