Tollmien-Schlichting route to elastoinertial turbulence in channel flow

Direct simulations of two-dimensional channel flow of a viscoelastic fluid have revealed the existence of a family of Tollmien-Schlichting (TS) attractors that is nonlinearly self-sustained by viscoelasticity.  Here, we describe the evolution of this branch in parameter space and its connections to the Newtonian TS wave and to elastoinertial turbulence (EIT). At Reynolds number Re=3000, there is a solution branch with TS-wave structure but which is not connected to the Newtonian solution branch.  At fixed Weissenberg number, Wi and increasing Reynolds number from 3000-10000, this attractor goes from displaying a striation of weak polymer stretch localized at the critical layer to an extended sheet of very large polymer stretch. This transition can be attributed to a coil-stretch transition when the local Weissenberg number at the hyperbolic stagnation points of the Kelvin cat's eye structure of the TS wave exceed 1/2. At Re=10000, unlike 3000, the Newtonian TS attractor evolves continuously into EIT as Wi is increased. We describe how the structure of the flow and stress fields changes, highlighting a ``sheet-shedding" process by which the individual sheets associated with the critical layer structure break up to form the layered multisheet structure characteristic of EIT.