We present a numerical study of transient natural convection in a rectangular open thermosyphon having asymmetric thermal boundary conditions. One vertical wall of the thermosyphon is either heated by constant heat flux (“warmup”) or cooled by convection to the surroundings (“cooldown”). The top of the thermosyphon is open to a large reservoir of fluid at constant temperature. The vorticity, energy, and stream-function equations are solved by finite differences on graded mesh. The ADI method and iteration with overrelaxation are used. We find that the thermosyphon performs quite differently during cooldown compared with warmup. In cooldown, flows are mainly confined to the thermosyphon with little momentum and heat exchange with the reservoir. For warmup, the circulation resembles that for a symmetrically heated thermosyphon where there is a large exchange with the reservoir. The difference is explained by the temperature distributions. For cooldown, the fluid becomes stratified and the resulting stability reduces motion. In contrast, the transient temperature for warmup does not become stratified but generally exhibits the behavior of a uniformly heated vertical plate. For cooldown and Ra > 104, time-dependent heat transfer is predicted by a closed-form expression for one-dimensional conduction, which shows that Nu Bi1/2/A in the steady-state limit. For warmup, transient heat transfer behaves as one-dimensional conduction for early times and at steady state and for Ra* 105, can be approximated as that for a uniformly heated vertical plate.