September  2014, 13(5): 1737-1757. doi: 10.3934/cpaa.2014.13.1737

The Kolmogorov-Obukhov-She-Leveque scaling in turbulence

1. 

Department of Mathematics, University of California, Santa Barbara

Received  September 2013 Revised  December 2013 Published  June 2014

We construct the 1962 Kolmogorov-Obukhov statistical theory of turbulence from the stochastic Navier-Stokes equations driven by generic noise. The intermittency corrections to the scaling exponents of the structure functions of turbulence are given by the She-Leveque intermittency corrections. We show how they are produced by She-Waymire log-Poisson processes, that are generated by the Feynmann-Kac formula from the stochastic Navier-Stokes equation. We find the Kolmogorov-Hopf equations and compute the invariant measures of turbulence for 1-point and 2-point statistics. Then projecting these measures we find the formulas for the probability distribution functions (PDFs) of the velocity differences in the structure functions. In the limit of zero intermittency, these PDFs reduce to the Generalized Hyperbolic Distributions of Barndorff-Nilsen.
Citation: Björn Birnir. The Kolmogorov-Obukhov-She-Leveque scaling in turbulence. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1737-1757. doi: 10.3934/cpaa.2014.13.1737
References:
[1]

F. Anselmet, Y. Gagne, E. J. Hopfinger and R. A. Antonia, High-order velocity structure function sin turbulent shear flows,, \emph{J. Fluid Mech.}, 14 (1984), 63. Google Scholar

[2]

A. Babin, A. Mahalov and B. Nicolaenko, Long-time averaged Euler and Navier-Stokes equations for rotation fluids,, \emph{In Structure and Dynamics of non-linear waves in Fluids, (1994). Google Scholar

[3]

A. Babin, A. Mahalov and B. Nicolaenko, Global splitting, integrability and regularity of 3d Euler and Navier-Stokes equation for uniformely rotation fluids,, \emph{Eur. J. Mech. B/Fluids}, 15 (1996). Google Scholar

[4]

O. E. Barndorff-Nilsen, Exponentially decreasing distributions for the logarithm of the particle size,, \emph{Proc. R. Soc. London A}, 353 (1977), 401. Google Scholar

[5]

O. E. Barndorff-Nilsen, Processes of normal inverse Gaussian type,, \emph{Finance and Stochastics}, 2 (1998), 41. doi: 10.1007/s007800050032. Google Scholar

[6]

O. E. Barndorff-Nilsen, P. Blaesild and Jurgen Schmiegel, A parsimonious and universal description of turbulent velocity increments,, \emph{Eur. Phys. J. B}, 41 (2004), 345. Google Scholar

[7]

R. Benzi, S. Ciliberto, C. Baudet, F. Massaioli, R. Tripiccione and S. Succi, Extended self-similarity in turbulent flow,, \emph{Phys. Rev. E}, 48 (1993), 401. Google Scholar

[8]

B. Birnir, Turbulence of a unidirectional flow,, \emph{Proceedings of the Conference on Probability, (2005). Google Scholar

[9]

B. Birnir, The existence and uniqueness and statistical theory of turbulent solution of the stochastic Navier-Stokes equation in three dimensions, an overview,, \emph{Banach J. Math. Anal.}, 4 (2010), 53. Google Scholar

[10]

B. Birnir, The Kolmogorov-Obukhov statistical theory of turbulence,, \emph{J. Nonlinear Sci.}, (2013), 00332. Google Scholar

[11]

B. Birnir, The Kolmogorov-Obukhov Theory of Turbulence,, Springer, (2013). doi: 10.1007/978-1-4614-6262-0. Google Scholar

[12]

B. Dubrulle, Intermittency in fully developed turbulence: in log-Poisson statistics and generalized scale covariance,, \emph{Phys. Rev. Letters}, 73 (1994), 959. Google Scholar

[13]

U. Frisch, Turbulence, Cambridge Univ. Press, (1995). Google Scholar

[14]

E. Hopf, Statistical hydrodynamics and functional calculus,, \emph{J. Rat. Mech. Anal.}, 1 (1953), 87. Google Scholar

[15]

A. N. Kolmogorov, Dissipation of energy under locally istotropic turbulence,, \emph{Dokl. Akad. Nauk SSSR}, 32 (1941), 16. Google Scholar

[16]

A. N. Kolmogorov, The local structure of turbulence in incompressible viscous fluid for very large Reynolds number,, \emph{Dokl. Akad. Nauk SSSR}, 30 (1941), 9. doi: 10.1098/rspa.1991.0075. Google Scholar

[17]

A. N. Kolmogorov, A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number,, \emph{J. Fluid Mech.}, 13 (1962), 82. Google Scholar

[18]

R. H. Kraichnan, Turbulent cascade and intermittency growth,, \emph{In Turbulence and Stochastic Processes, (1991), 65. doi: 10.1098/rspa.1991.0080. Google Scholar

[19]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace,, \emph{Acta Math.}, 63 (1934), 193. doi: 10.1007/BF02547354. Google Scholar

[20]

H. P. McKean, Turbulence without pressure: Existence of the invariant measure,, \emph{Methods and Applications of Analysis}, 9 (2002), 463. doi: 10.4310/MAA.2002.v9.n3.a10. Google Scholar

[21]

A. M. Obukhov, On the distribution of energy in the spectrum of turbulent flow,, \emph{Dokl. Akad. Nauk SSSR}, 32 (1941). Google Scholar

[22]

A. M. Obukhov, Some specific features of atmospheric turbulence,, \emph{J. Fluid Mech.}, 13 (1962), 77. Google Scholar

[23]

B. Oksendal and A. Sulem, Applied Stochastic Control of Jump Diffusions,, Springer, (2005). Google Scholar

[24]

L. Onsager, Statistical hydrodynamics,, \emph{Nuovo Cimento.}, 6 (1945), 279. Google Scholar

[25]

G. Da Prato, An Introduction of Infinite-Dimensional Analysis,, Springer Verlag, (2006). Google Scholar

[26]

Z-S She and E. Leveque, Universal scaling laws in fully developed turbulence,, \emph{Phys. Rev. Letters}, 72 (1994), 336. Google Scholar

[27]

Z-S She and E. Waymire, Quantized energy cascade and log-poisson statistics in fully developed turbulence,, \emph{Phys. Rev. Letters}, 74 (1995), 262. Google Scholar

[28]

Z-S She and Zhi-Xiong Zhang, Universal hierarchial symmetry for turbulence and general multi-scale fluctuation systems,, \emph{Acta Mech Sin}, 25 (2009), 279. Google Scholar

[29]

J. B. Walsh, An Introduction to Stochastic Differential Equations,, Springer Lecture Notes, (1984). Google Scholar

show all references

References:
[1]

F. Anselmet, Y. Gagne, E. J. Hopfinger and R. A. Antonia, High-order velocity structure function sin turbulent shear flows,, \emph{J. Fluid Mech.}, 14 (1984), 63. Google Scholar

[2]

A. Babin, A. Mahalov and B. Nicolaenko, Long-time averaged Euler and Navier-Stokes equations for rotation fluids,, \emph{In Structure and Dynamics of non-linear waves in Fluids, (1994). Google Scholar

[3]

A. Babin, A. Mahalov and B. Nicolaenko, Global splitting, integrability and regularity of 3d Euler and Navier-Stokes equation for uniformely rotation fluids,, \emph{Eur. J. Mech. B/Fluids}, 15 (1996). Google Scholar

[4]

O. E. Barndorff-Nilsen, Exponentially decreasing distributions for the logarithm of the particle size,, \emph{Proc. R. Soc. London A}, 353 (1977), 401. Google Scholar

[5]

O. E. Barndorff-Nilsen, Processes of normal inverse Gaussian type,, \emph{Finance and Stochastics}, 2 (1998), 41. doi: 10.1007/s007800050032. Google Scholar

[6]

O. E. Barndorff-Nilsen, P. Blaesild and Jurgen Schmiegel, A parsimonious and universal description of turbulent velocity increments,, \emph{Eur. Phys. J. B}, 41 (2004), 345. Google Scholar

[7]

R. Benzi, S. Ciliberto, C. Baudet, F. Massaioli, R. Tripiccione and S. Succi, Extended self-similarity in turbulent flow,, \emph{Phys. Rev. E}, 48 (1993), 401. Google Scholar

[8]

B. Birnir, Turbulence of a unidirectional flow,, \emph{Proceedings of the Conference on Probability, (2005). Google Scholar

[9]

B. Birnir, The existence and uniqueness and statistical theory of turbulent solution of the stochastic Navier-Stokes equation in three dimensions, an overview,, \emph{Banach J. Math. Anal.}, 4 (2010), 53. Google Scholar

[10]

B. Birnir, The Kolmogorov-Obukhov statistical theory of turbulence,, \emph{J. Nonlinear Sci.}, (2013), 00332. Google Scholar

[11]

B. Birnir, The Kolmogorov-Obukhov Theory of Turbulence,, Springer, (2013). doi: 10.1007/978-1-4614-6262-0. Google Scholar

[12]

B. Dubrulle, Intermittency in fully developed turbulence: in log-Poisson statistics and generalized scale covariance,, \emph{Phys. Rev. Letters}, 73 (1994), 959. Google Scholar

[13]

U. Frisch, Turbulence, Cambridge Univ. Press, (1995). Google Scholar

[14]

E. Hopf, Statistical hydrodynamics and functional calculus,, \emph{J. Rat. Mech. Anal.}, 1 (1953), 87. Google Scholar

[15]

A. N. Kolmogorov, Dissipation of energy under locally istotropic turbulence,, \emph{Dokl. Akad. Nauk SSSR}, 32 (1941), 16. Google Scholar

[16]

A. N. Kolmogorov, The local structure of turbulence in incompressible viscous fluid for very large Reynolds number,, \emph{Dokl. Akad. Nauk SSSR}, 30 (1941), 9. doi: 10.1098/rspa.1991.0075. Google Scholar

[17]

A. N. Kolmogorov, A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number,, \emph{J. Fluid Mech.}, 13 (1962), 82. Google Scholar

[18]

R. H. Kraichnan, Turbulent cascade and intermittency growth,, \emph{In Turbulence and Stochastic Processes, (1991), 65. doi: 10.1098/rspa.1991.0080. Google Scholar

[19]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace,, \emph{Acta Math.}, 63 (1934), 193. doi: 10.1007/BF02547354. Google Scholar

[20]

H. P. McKean, Turbulence without pressure: Existence of the invariant measure,, \emph{Methods and Applications of Analysis}, 9 (2002), 463. doi: 10.4310/MAA.2002.v9.n3.a10. Google Scholar

[21]

A. M. Obukhov, On the distribution of energy in the spectrum of turbulent flow,, \emph{Dokl. Akad. Nauk SSSR}, 32 (1941). Google Scholar

[22]

A. M. Obukhov, Some specific features of atmospheric turbulence,, \emph{J. Fluid Mech.}, 13 (1962), 77. Google Scholar

[23]

B. Oksendal and A. Sulem, Applied Stochastic Control of Jump Diffusions,, Springer, (2005). Google Scholar

[24]

L. Onsager, Statistical hydrodynamics,, \emph{Nuovo Cimento.}, 6 (1945), 279. Google Scholar

[25]

G. Da Prato, An Introduction of Infinite-Dimensional Analysis,, Springer Verlag, (2006). Google Scholar

[26]

Z-S She and E. Leveque, Universal scaling laws in fully developed turbulence,, \emph{Phys. Rev. Letters}, 72 (1994), 336. Google Scholar

[27]

Z-S She and E. Waymire, Quantized energy cascade and log-poisson statistics in fully developed turbulence,, \emph{Phys. Rev. Letters}, 74 (1995), 262. Google Scholar

[28]

Z-S She and Zhi-Xiong Zhang, Universal hierarchial symmetry for turbulence and general multi-scale fluctuation systems,, \emph{Acta Mech Sin}, 25 (2009), 279. Google Scholar

[29]

J. B. Walsh, An Introduction to Stochastic Differential Equations,, Springer Lecture Notes, (1984). Google Scholar

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