A two-dimensional finite difference computer program in cylindrical coordinates has been developed to solve the case of laminar natural convection in a vertical tube open to a large reservoir. Such a device, the open thermosyphon, is used in a number of applications, including the cooling of gas turbines, geothermal energy extraction, and thermosyphon solar water heaters. The objectives of this work were to study the nature of fluid flow and the heat transfer rate along the tube wall. A semi-implicit, time marching, finite difference solution procedure was used, satisfying continuity, momentum, and energy equations for incompressible flow. Results show three well-defined flow regimes appearing as functions of the tube length-to-radius (aspect) ratio. Fluid motion in the tube and heat transfer rates became oscillatory at long time intervals. Plots of streamlines and isotherms at selected times for different aspect ratio tubes are also presented to show the transient behavior of fluid motion.