Transition of Planar Couette Flow at infinite Reynolds numbers

An outline of the state space of planar Couette flow at high Reynolds numbers ($Re < 10^5$) is investigated via a variety of efficient numerical techniques. It is verified from nonlinear analysis that the lower branch of {\it Hairpin Vortex State} (HVS) asymptotically approaches the primary (laminar) state with increasing $Re$. It is also predicted that the lower branch of HVS at high $Re$ belongs to the stability boundary that initiates transition to turbulence, and that one of the unstable manifolds of the lower branch of HVS lies on the boundary. These facts suggest HVS may provide a criterion to estimate a minimum perturbation arising transition to turbulent states at the infinite $Re$ limit.