Egolf Peter W, Hutter Kolumban
Thermal Sciences and Engineering Institute, University of Applied Sciences of Western Switzerland, CH-1401 Yverdon-les-Bains, Switzerland.
%Laboratory of Hydraulics, Hydrology and Glaciology, Swiss Federal Institute of Technology, ETH, Hönggerberg HIA 58D, CH 8093 Zurich, Switzerland.
Entropy (Basel). 2018 Feb 7;20(2):109. doi: 10.3390/e20020109.
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 . By applying this nonequilibrium thermodynamics, the Boltzmann-Gibbs thermodynamic approach of Kraichnan to turbulence is generalized. This physical modeling implies fractional calculus methods, obeying anomalous diffusion, described by Lévy statistics with < 5/3 (sub diffusion), = 5/3 (normal or Brownian diffusion) and > 5/3 (super diffusion). The generalized energy spectrum of Kraichnan, occurring at small wave numbers , now reveals the more general and precise result . This corresponds well for = 5/3 with the Kolmogorov-Oboukov energy spectrum and for > 5/3 to turbulence with intermittency. The enstrophy spectrum, occurring at large wave numbers , leads to a power law, suggesting that large wave-number eddies are in thermodynamic equilibrium, which is characterized by = 1, finally resulting in Kraichnan's correct 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.
详细回顾了Tsallis的扩展热力学，并将其应用于湍流。它基于对带有一个参数的指数函数和对数函数的推广。通过应用这种非平衡热力学，Kraichnan的玻尔兹曼 - 吉布斯热力学方法被推广到湍流。这种物理建模意味着分数微积分方法，服从反常扩散，由Lévy统计描述，其中 < 5/3（次扩散）， = 5/3（正常或布朗扩散）以及 > 5/3（超扩散）。Kraichnan在小波数处出现的广义能谱，现在揭示了更一般且精确的结果。对于 = 5/3，这与Kolmogorov - Oboukov能谱吻合良好，而对于 > 5/3，则与具有间歇性的湍流吻合。在大波数处出现的涡量谱导致幂律，表明大波数涡旋处于热力学平衡，其特征为 = 1，最终得到Kraichnan正确的涡量谱。该理论以自然的方式揭示了湍流的广义温度，它在非平衡能量传递域随波数降低，并且在平衡涡量传递域呈现具有恒定广义温度的能量均分定律。本文包含众多新结果；其中一些以八个新（已证明）命题的形式给出。
