  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 and asymmetrical. Compared to the simple
convective roll, the branching flows have lower maximum speeds and thinner
boundary layers, but nearly the same maximum power density.
