We calculate the optimal upper and lower bounds, subject to the assumption of streamwise invariance, on the long-time-averaged mechanical energy dissipation rate $\varepsilon$ within the flow of an incompressible viscous fluid of constant kinematic viscosity $\nu$ and depth $h$ driven by a constant surface stress $\tau=\rho u^2_\star$, where $u_\star$ is the friction velocity. We show that $\varepsilon \leq \varepsilon_{\max}=\tau^2/(\rho^2\nu)$, i.e. the dissipation is bounded above by the dissipation associated with the laminar solution $\bu=\tau(z+h)/(\rho\nu) \xvec$, where $\xvec$ is the unit vector in the streamwise $x$-direction.

By using the variational ‘background method’ (due to Constantin, Doering and Hopf) and numerical continuation, we also generate a rigorous lower bound on the dissipation for arbitrary Grashof numbers $G$, where $G=\tau h^2/(\rho \nu^2)$. Under the assumption of streamwise invariance as $G \rightarrow \infty$, for flows where horizontal mean momentum balance and total power balance are imposed as constraints, we show numerically that the best possible lower bound for the dissipation is $\varepsilon \geq \varepsilon_{\min}=7.531 u_\star^3/h$, a bound that is independent of the flow viscosity. This scaling (though not the best possible numerical coefficient) can also be obtained directly by applying the same imposed constraints and restricting attention to a particular, analytically tractable, class of mean flows.