Upper bounds for turbulent Couette flow incorporating the poloidal power constraint

Abstract

The upper bound on momentum transport in the turbulent regime of plane Couette flow is considered. Busse (1970) obtained a bound from a variational formulation based on total energy conservation and the mean momentum equation. Two-dimensional asymptotic solutions of the resulting Euler-Lagrange equations for the system were obtained in the large-Reynolds-number limit. Here we make a toroidal poloidal decomposition of the flow and impose an additional power integral constraint, which cannot be satisfied by two-dimensional flows. Nevertheless, we show that the additional constraint can be met by only small modifications to Busse's solution, which leaves his momentum transport bound unaltered at 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, appears to explain the three-dimensional structures observed in experiments. Only in the outermost boundary layer and in the main stream is the solution three-dimensional. Motion in the thinner layers remains two-dimensional characterized by streamwise rolls.