Variational bound on energy dissipation in plane Couette flow

We present numerical solutions to the extended Doering-Constantin variational principle for upper bounds on the energy dissipation rate in turbulent plane Couette flow. Using the compound matrix technique in order to reformulate this principle's spectral constraint, we derive a system of equations that is amenable to numerical treatment in the entire range from low to asymptotically high Reynolds numbers. Our variational bound exhibits a minimum at intermediate Reynolds numbers, and reproduces the Busse bound in the asymptotic regime. As a consequence of a bifurcation of the minimizing wavenumbers, there exist two length scales that determine the optimal upper bound: the effective width of the variational profile's boundary segments, and the extension of their flat interior part.