Close-contact-melting heat-transfer characteristics, including natural convection in the liquid region in cylindrical capsules were studied theoretically. Elucidation of such heat-transfer mechanism is of great importance from the view-point of cost-effective ice storage and heat storage systems employing the latent heat-of-fusion thermal energy storage (TES) concept. The Growth Ring Method (GRM) was used as a numerical tool to handle the moving interface between the liquid and solid regions. The computed isotherms, streamlines and stored heat fractions were clarified in detail. In particular, a special attention was focused on the role of natural convection in the upper liquid region. Hitherto, no exact numerical analysis has been made to clarify the contribution of natural convection.