UPPER-BOUNDS FOR TURBULENT COUETTE-FLOW INCORPORATING THE POLOIDAL POWER CONSTRAINT

The upper bound on momentum transport in the turbulent regime of plane Couette flow is considered. Busse (1970) obtained a bound from a vari ational formulation based on total energy conservation and the mean mo mentum equation. Two-dimensional asymptotic solutions of the resulting Euler-Lagrange equations for the system were obtained in the large-Re ynolds-number limit. Here we make a toroidal poloidal decomposition of the flow and impose an additional power integral constraint, which ca nnot be satisfied by two-dimensional flows. Nevertheless, we show that the additional constraint can be met by only small modifications to B usse's solution, which leaves his momentum transport bound unaltered a t lowest order. On the one hand, the result suggests that the addition of further integral constraints will not significantly improve bound estimates. On the other, our optimal solution, which possesses a weak spanwise roll in the outermost of Busse's nested boundary layers, appe ars to explain the three-dimensional structures observed in experiment s. Only in the outermost boundary layer and in the main stream is the solution three-dimensional. Motion in the thinner layers remains two-d imensional characterized by streamwise rolls.