Internal natural convective heat transfer from a thin walled, vertical cylinder with an exposed vertical surface is investigated numerically. The top and bottom end faces are assumed isothermal. This setup approximates the vertical wellbore of a Christmas tree for simulating cool-down of subsea oil and gas equipment during shutdown operations. The primary objective of this study is to determine the cooling rate of the interior fluid and the onset of fluid motion (rotation) caused by the two non-adiabatic surfaces as a function of Biot number (Bi) applied at the vertical cylindrical wall. The flow is assumed to be three-dimensional, non-steady, and transitional with constant fluid properties except for the density variation with temperature. This latter effect gives rise to the buoyancy force; treated using the Bousinessq approach. We solve the dimensionless governing equations numerically using COMSOL Multiphysics, a commercial finite-element method (FEM) based code. The specific application that motivated this investigation involved a range of Prandtl numbers (Pr) from 0.7 to 168. In addition, we consider small and moderate values of Rayleigh numbers (Ra). Steady-state solutions for temperature and velocity are stratified at the heat source region regardless of the Biot number.