Symmetry of vortices in transition of plane Couette flow at moderate Reynolds number

The shooting method, which was established originally as a tool to find out an unstable steady solution of subcritical shear flows (cf. J. Phys. Soc. Jpn. 70, 703 [1]), is applied to outline a structural aspect of laminar-turbulent transition of minimal plane Couette flow at the moderate Reynolds number. By adopt as initial conditions the points on the plane in phase space spanned by three distinct exact steady solutions, including the Hairpin Vortex Solution (HVS, cf. Phys.Rev.Lett.102,114501 (2009)), a plenty of trial calculations based on the method are carried out. The result implies that HVS is on the boundary separating the laminar and the turbulent attractors of the plane Couette flow, and that one of the unstable manifolds of HVS constitutes the boundary. Moreover, the asymptotic behaviour of HVS at higher Reynolds number is investigated, which enables us to quantify the role of HVS in the laminar-turbulent transition of Plane Couette flow at the moderate Reynolds number.