CFD simulations of heat transfer between two fluids in a parallel-flow vertical double-pipe heat exchanger, taking into consideration conduction in the walls, viscous dissipation, variable fluid properties, and axial diffusion in the fluids, illustrate the effects of buoyancy on the velocity and temperature profiles as well as on the Nusselt numbers and friction coefficients. In particular, the results show that flow reversal occurs at different positions depending on the flow direction. In the developing region, the fluid bulk temperatures, the wall temperature, and the local Nusselt numbers for both fluids depend on the flow direction. For the conditions under consideration, the Nusselt number for the fully developed flow in the inner pipe lies between the corresponding forced convection values for the constant heat flux and constant temperature conditions, while the one in the annulus is smaller than both these forced convection values. It is established that the flow and thermal fields in the inner cylinder and the annulus cannot be predicted correctly by studying each one separately since the thermal condition at the solid wall separating the two fluids depends on flow characteristics such as the Richardson number in the inner tube.