Laminar-Turbulent Patterning in Transitional Flows

Abstract

Wall-bounded flows experience a transition to turbulence characterized by the coexistence of laminar and turbulent domains in some range of Reynolds number R, the natural control parameter. This transitional regime takes place between an upper threshold R t above which turbulence is uniform (featureless) and a lower threshold R g below which any form of turbulence decays, possibly at the end of overlong chaotic transients. The most emblematic cases of flow along flat plates transiting to/from turbulence according to this scenario are reviewed. The coexistence is generally in the form of bands, alternatively laminar and turbulent, and oriented obliquely with respect to the general flow direction. The final decay of the bands at R g points to the relevance of directed percolation and criticality in the sense of statistical-physics phase transitions. The nature of the transition at R t where bands form is still somewhat mysterious and does not easily fit the scheme holding for pattern-forming instabilities at increasing control parameter on a laminar background. In contrast, the bands arise at R t out of a uniform turbulent background at a decreasing control parameter. Ingredients of a possible theory of laminar-turbulent patterning are discussed.Keywords: transition to/from turbulence; wall-bounded shear flow; plane Couette flow; turbulent patterning; phase transitions; directed percolationThe present Special-Issue contribution deals with the transition to turbulence in wall-bounded flows, an important case of systems driven far from equilibrium where patterns develop against a turbulent background. This active field of research is rapidly evolving, and important results have been obtained recently. To set the frame, in Section 1, I will summarize a recent paper reviewing the subject from a more general standpoint [1], enabling me to focus on a specific feature of this transition: the existence of a statistically well-organized laminar-turbulent patterning of flows along planar walls in some intermediate range of Reynolds numbers [R g , R t ]. The Reynolds number is the main control parameter of the problem. Its generic expression reads R = V /ν, in which V and are typical velocity and length scales, and ν the fluid's kinematic viscosity. R compares the typical shear rate V/ to the viscous diffusion rate over the same length scale ν/ 2 . R g is a global stability threshold marking unconditional return to laminar flow and R t some upper threshold beyond which turbulence is essentially uniform. After having taken the cylindrical shear configuration as an illustrating case in Section 2, I will turn to strictly planar cases in Section 3. The best understood part of the transition scenario, pattern decay at R g is considered in Section 4. How patterns emerge as R is decreased from large values is next examined in Section 5 before a discussion of perspectives and questions that, in my view, remain open in Section 6. I have tried to limit the bibliography to contributions of specific significance, historical or...

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