Optimal Energy Dissipation Bounds for 2D and 3D Stress-Driven Shear Flows

The background method (Doering \& Constantin, 1995) allows the derivation of rigorous bounds on bulk turbulent quantities in a variety of wall-bounded flows as a function of the governing parameters. A classical example is to bound the energy dissipation $\epsilon$ in surface-driven shear flows as a function of the driving force, expressed by the Grashoff number $Gr$. Of particular interest is to compute the best bounds achievable within this framework. However, the variational problem determining the optimal bounds is difficult to solve when the flow is driven by a boundary flux. Tang et al. (2004) first resolved this difficulty by modelling a surface stress with a localised body force. Instead, we propose a novel numerical approach based on Semidefinite Programming that is able to handle fixed-flux boundary conditions directly, and thereby revisit the bounds on $\epsilon$ for surface-stress-driven shear flows. In the 2D case, we find that $\epsilon>8Gr^{3/2}$, improving the scaling law $\epsilon>4Gr^{3/2}$ proven by Hagstrom \& Doering (2014). In 3D, we confirm the results of Tang et al., suggesting that a surface stress can be modelled accurately by a body force. Finally, a careful analysis ensures that, in principle, our bounds hold analytically for a fixed $Gr$.