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From the laminar flow to turbulent flow, is it a kind of phase transition? If so, what is the critical point? And what about the correlation length behaviours and fluctuation? Any critical exponents? If not, why not? Why can't it be a phase transition?

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The transition from laminar to turbulent flow is not a thermodynamic phase transition. The reason is quite simple. Turbulent flow is intrinsically dissipative, and it is impossible to achieve an equilibrium state in the presence of dissipation. A stationary state is possible, but only in the presence of a continuous flux of energy.

With this caveat, it is possible to consider the transition as a dynamical transition between non-equilibrium states. In this respect, a strong analogy with the usual phase transitions has been found, including the equivalent of the critical behavior. A key ingredient is recognizing the role of spatial correlations between turbulent regions and the analogy with percolation models where the presence of critical phenomena without thermodynamics is a well-established result.

A recent paper where it is possible to find a few ideas about the analogy between phase transitions and phenomenon is here.

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