2379
20181220113750.0
1099-4300
DOI
10.3390/e20020109
ARTICLE
eng
Tsallis extended thermodynamics applied to 2-d turbulence :
Lévy statistics and q-fractional generalized kraichnanian energy and enstrophy spectra
2018
2018-02
41 pages
This article belongs to the Special Issue Phenomenological Thermodynamics of Irreversible Processes
visible
The extended thermodynamics of Tsallis is reviewed in detail and applied to turbulence. It is based on a generalization of the exponential and logarithmic functions with a parameter q. By applying this nonequilibrium thermodynamics, the Boltzmann-Gibbs thermodynamic approach of Kraichnan to 2-d turbulence is generalized. This physical modeling implies fractional calculus methods, obeying anomalous diffusion, described by Lévy statistics with q < 5/3 (sub diffusion), q = 5/3 (normal or Brownian diffusion) and q > 5/3 (super diffusion). The generalized energy spectrum of Kraichnan, occurring at small wave numbers k, now reveals the more general and precise result k−q. This corresponds well for q = 5/3 with the Kolmogorov-Oboukov energy spectrum and for q > 5/3 to turbulence with intermittency. The enstrophy spectrum, occurring at large wave numbers k, leads to a k−3q power law, suggesting that large wave-number eddies are in thermodynamic equilibrium, which is characterized by q = 1, finally resulting in Kraichnan’s correct k −3 enstrophy spectrum. The theory reveals in a natural manner a generalized temperature of turbulence, which in the non-equilibrium energy transfer domain decreases with wave number and shows an energy equipartition law with a constant generalized temperature in the equilibrium enstrophy transfer domain. The article contains numerous new results; some are stated in form of eight new (proven) propositions.
eng
correct
English
HEIG-VD
IGT - Institut de Génie Thermique
Ingénierie et Architecture
Ingénierie
extended thermodynamics
eng
Tsallis entropy
eng
escort probability
eng
fractional calculus
eng
2-d turbulence
eng
spectra of Kraichnan
eng
Kolmogorov-Oboukov spectrum
eng
Lévy statistics
eng
intermittency
eng
scientifique
Egolf, Peter William
Haute école d’ingénierie et de gestion du canton de Vaud (HEIG-VD) HES-SO // Haute Ecole Spécialisée de Suisse Occidentale
Hutter, Kolumban
Laboratory of Hydraulics, Hydrology and Glaciology, Swiss Federal Institute of Technology, ETH
2018, 20(2), 109
Entropy
2520408
http://hesso.tind.io/record/2379/files/Egolf_2018_Tsallis.pdf
pdfa
2466569
http://hesso.tind.io/record/2379/files/Egolf_2018_Tsallis.pdf?subformat=pdfa
GREEN
oai:hesso.tind.io:2379
GLOBAL_SET
I2
scientifique