Abstract
The hydrodynamic time evolution is Hamiltonian in the inertial range (i.e., in the absence of viscosity). From this we obtain that the macroscopic study of hydrodynamic turbulence is equivalent, at an abstract level, to the microscopic study of a heat flow in a nonstandard geometry. In the absence of fluctuations this means that the Kolmogorov theory of turbulence is equivalent to a heat flow for a suitable mechanical system. Turbulent fluctuations (intermittency) correspond to thermal fluctuations for the heat flow. A relatively crude estimate of the thermal fluctuations, based on standard ideas of nonequilibrium statistical mechanics is presented: this agrees remarkably well with what is observed in several turbulence experiments. A logical relation with the lognormal theory of Kolmogorov and Obukhov is also indicated, which shows what fails in this theory, and what can be rescued.
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Notes
- 1.
Dimensional analysis says how various quantities (like velocity or energy) depend on certain variables (like spatial distance, and time): velocity is spatial distance divided by time, energy is mass times velocity squared, etc. Dimensional analysis appears somewhat trivial, but for the turbulent energy cascade it has led to spectacular predictions.
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Ruelle, D. (2016). Hydrodynamic Turbulence as a Nonstandard Transport Phenomenon. In: Skiadas, C. (eds) The Foundations of Chaos Revisited: From Poincaré to Recent Advancements. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-29701-9_3
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