# Similarities between laminar-turbulence transition and others like BCS-BEC crossover, quark-hadron transition etc

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.

• I'm not familiar with BCS/BEC but my research area is in laminar to turbulent transition. The wording of your question makes it sound like the Reynolds number is dependent on the (laminar or turbulent) state of the system. The Reynolds number is usually viewed as a parameter of the system and provides a convenient nondimensionalization of the governing equations. – OSE May 8 '14 at 18:33
• It is true that turbulent flows are associated with higher Reynolds numbers but there is not a fixed line where this transition happens. Instead, transition occurs through the growth of instabilities whose initial amplitude is very system dependent. There are a variety of instabilities for different domain geometries which become unstable and have different growth rates at different Reynolds numbers. – OSE May 8 '14 at 18:36
• @OSE Coupling strength is a system parameter even in QCD and BCS/BEC, but it varies with some external driven parameter or the scale at which it is observed. May be this is true in fluid dynamics too? Page-4 of Gawedzki's lecture also talks of a scale-dependent Reynolds number. – GuSuku May 8 '14 at 18:44
• @OSE If I understand you right, onset of turbulence is similar to a crossover than a phase transition? Many readings seemed to suggest a critical point for this onset eg hyperphysics.phy-astr.gsu.edu/hbase/pturb.html – GuSuku May 8 '14 at 18:47
• For closed-systems (such as Taylor-Couette flow) there is a critical parameter beyond which the flow is turbulent. My previous comments were in reference to open-flow systems (such as air flowing over a wing). In this case there is a critical Reynolds number where instabilities first become unstable but these disturbances must grow several orders of magnitude before the disturbances lead to turbulence. This growth takes place over a not insignificant distance. In that regard it seems more similar to crossover than a phase transition. – OSE May 8 '14 at 19:32