From my limited readings on fluid dynamics, my understanding is that as the system changes from near-laminar flows to full turbulence, the dimensionless Reynolds number changes from $ R << 1$ to $R >> 1$.
How is the perturbation expansion parameter in field theory formalisms of turbulence (See bottom of page-10 and top of page-11 in Gawedzki's chao-dyn/9610003 and Cardy's lecture in Non-equilibrium Statistical Mechanics and Turbulence) related to Reynolds number? If there is no direct relation, does the expansion parameter change values similar to $R$ as the system changes from laminar to fully developed turbulence?
Is there a conceptual similarity between this field theory formalisms of fluid dynamics (as the system changes from laminar to turbulence) and systems like BCS-BEC crossover and/or quark-hadron transitions, where too some expansion parameter (usually dimensionless coupling strengths) change from $<<1$ to $>>1$? Am I right in understanding onset of turbulence as closer to a phase transition than a cross-over?
Also, just like in these other systems (BCS/BEC and Quark/Hadron), does the region of fluid dynamics, most difficult to model, lie somewhere in-between (ie. $R \sim 1 $)?
note: I understand that these other systems are fermionic while turbulence is not, and that these other systems are quantum mechanical while turbulence is classical. I am not asking for an exact dictionary, but only a higher-level similarity between them.