Energy stability bounds on convective heat transport: Numerical study

The concept of nonlinear energy stability has recently been extended to deduce bounds on energy dissipation and transport in incompressible flows, even for turbulent flows. In this approach an effective stability condition on ''background'' flow or temperature profiles is derived, which when satisfied ensures that the profile produces a rigorous upper estimate to the bulk dissipation. Optimization of the test background profiles in search of the lowest upper bounds leads to nonlinear Euler-Lagrange equations for the extremal profile. In this paper, in the context of convective heat transport in the Boussinesq equations, we describe numerical solutions of the Euler-Lagrange equations for the optimal background temperature and present the numerical computation of the implied bounds.