Hi,
Let us consider a large ensemble of vortices immersed in a flow domain. The Helmholtz free energy is given by F=E-Tk
F=E-Tk ln N,
where k is the Boltzman constant, and W is the number of the possible positions of a vortex.
However, the scale of turbulent vortices is much larger than the scale of molecular motion on which the measure of entropy and therefore the Helmholtz free energy is based. In particular, in incompressible Navier-Stokes turbulence, the temperature T is decoupled from the nonlinear dynamics underlying turbulence.
My question is whether it is possible to replace the temperature T by Reynolds number Re in defining the free energy for macroscopic vortices interactions in turbulence.
Thank you.
Wang Zhe
I have no final answers for this question, I can give my idea about the general approach to turbulence you are considering. Large vortex interaction lies in the inertial cascade where the viscosity is practically disregardable. Theoretically, you can substain the inertial cascade for Re-> +Inf. As you wrote, the evolution of the internal energy is decoupled for the incompressible flow model, indipendently from the Re number. Even if the viscosity depends on temperature, I cannot see your idea about an interchangin between Re and the temperature field
Dear Prof. Denaro,
Thank you for the answer!
The idea came from the Kosterlitz–Thouless transition, where a critical temperature is defined for the vortex unbindings. The low-temperature phase is visualised as a dilute vortex gas of closely bound vortex dipoles (favoring the minimum energy), with their size and density increasing with temperature. The high-temperature phase constitutes a plasma of unbound vortices (underlying the chaos in the velocity field).
The resemblance lets me think that maybe we can define a critical Reynolds number in the same manner in characterizing the laminar-turbulent transition.
Best regards,
Wang Zhe
I suppose you refer to this paper https://www.researchgate.net/profile/J_Kosterlitz/publication/210269742_Ordering_Metastability_and_Phase_Transitions_in_Two-Dimensional_Systems/links/561105ae08ae4833751a9d7e.pdf
but there the basic hypothesis is two-dimensionality of the vortices whilst vortical structure in turbulence have a 3d dynamics...
Article Ordering, Metastability and Phase Transitions in Two-Dimensi...
Dear Prof Denaro,
Awesome, you got the key point!
I study the vortices on a global cross section (resembles the Poincare section in the context of the dynamical system theory) intersecting transversely the vortex filaments. As vortex stretches it must be accompanied with a contaminant folding. It follows that the vortex stretching leads to the increases in the intensity of 2D vortices on the cross section, while the vortex folding leads to the proliferation of the vortices. The unbinding of vortices and their reorganization correpond to the twisting of vortex filaments, which seems to me a prerequisite of the vortex reconnection in vicious flows.
It is possible to define a topological entropy characterizing the complexity of the vortex filaments (Chorin's book 1994), which corresponds to the configurational entropy S of the 2D vortices as defined in the attached paper. Now the problem is problem is to find out a definition for the free energy.
Thank you.
Very kind wishes,
Wang Zhe
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