Zinc-air batteries offer large specific energy densities, while relying on abundant and non-toxic materials. In this paper, we present the first multi-dimensional simulations of zinc-air batteries. We refine our existing theory based model of secondary zinc-air systems. The model comprises thermodynamically consistent multi-species transport in alkaline electrolytes, formation and dissolution of metallic zinc and passivating zinc oxide, as well as multi-phase coexistence in gas diffusion electrodes. For the first time, we simulate zinc shape-change during battery cycling by modeling convection of zinc solids. We validate our model with in-situ tomography of commercial button cells. Two-dimensional volume-averaged simulations of cell voltage and zinc electrode morphology during discharge agree with these measurements. Thus, we can study how electrolyte carbonation limits shelf-life and how zinc shape-change limits cycle-life. The charging current is found to be the major contributor to cycle-life limitations. Finally, we optimize initial anode structure and charge-discharge protocols for improved performance and cycle-ability. Furthermore, we extend our model to a flexible thin-film battery as example of alternative cell design.