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Small scale exact coherent structures at large Reynolds numbers in plane Couette flow. (English) Zbl 1388.37079
The authors study the transition to turbulence for a plane Couette flow, i.e., a flow between two parallel plates, and the formation of exact coherent structures. These structures are three-dimensional solutions of the Navier-Stokes equations, which exhibit a relatively simple temporal dynamics. Subsequently these structures undergo saddle-node bifurcations and for increasing Reynolds number also secondary bifurcations, which yield increasingly complex temporal dynamics.
In contrast the spatial complexity remains rather limited. Spatial scaling of stationary plane Couette flow solutions are presented and then two families of such structures, one in the mid plane, and one close to the wall, are computed numerically. Their scaling and localization properties are analyzed and bifurcation diagrams are discussed. The possible relation between these self-similar coherent structures and the formation of boundary layer profiles is also presented.

MSC:
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
76F06 Transition to turbulence
76F10 Shear flows and turbulence
76F20 Dynamical systems approach to turbulence
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