We study steady flows that are optimal for heat transfer in a two-dimensional periodic domain. The flows maximize heat transfer under the constraints of incompressibility and a given energy budget (i.e., mean viscous power dissipation). Using an unconstrained optimization approach, we compute optima starting from 30–50 random initializations across several decades of Pe, the energy budget parameter. At Pe between ${10}^{4.5}$ and ${10}^{4.75}$, convective rolls with U-shaped branching near the walls emerge. They exceed the heat transfer of the simple convective roll optimum at Pe between ${10}^5$ and ${10}^{5.25}$. At larger Pe, multiple layers of branching occur in the optima and become increasingly elongated, asymmetrical, and heterogeneous. The rate of heat transfer scales as ${\mathrm{Pe}}^{0.575}$, which, in this range of Pe, is very close to the ${\mathrm{Pe}}^{2/3}{(ln\mathrm{Pe})}^{-4/3}$ behavior of self-similar branching solutions proposed earlier. Compared to the simple convective roll, the branching flows have lower maximum speeds and thinner boundary layers but nearly the same maximum power density.

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• Boundary layers
• Diffusion
• Optimization problems
• Thermal convection