The unsteady buoyancy-induced flow generated by an isothermal flat vertical surface enclosed in a long rectangular cavity is numerically analyzed. The enclosed fluid is assumed to be Newtonian and the walls are adiabatic. The transient process is initiated by suddenly and uniformly raising the temperature of the surface to a steady value higher than that of the fluid in the cavity. The full two-dimensional conservation equations, representing mass, momentum, and energy balance, are solved using a modified finite-difference scheme originally called the simple arbitrary La-grangian-Eulerian (SALE) technique. The resulting transport is found to be composed of several distinct regimes. At very short times a quasi-one-dimensional conduction regime occurs adjacent to the surface. The temperature distribution and heat transfer coefficient are found to closely follow the known one-dimensional conduction solution. At intermediate times the flow adjacent to the surface is observed to resemble the boundary-layer regime near a semi-infinite surface immersed in an infinite medium. Finally, at later times the adiabatic boundaries cause the decay of the temperature gradients. Eventually, the velocity field decays through viscous dissipation effects.