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A numerical analysis is performed in this study to observe the phase change and natural convection problem occurring in the confined geometry. The unsteady, incompressible 3-D Navier-Stokes equation and energy equation have been considered as governing equation in this study. In addition, numerical tools based on finite volume method were developed and those were found to be tools to analyze phase change and natural convection problem with qualitative accuracy. For verification, phase interfacial lines and flow patterns were compared to previous studies and showed good agreements for the case of square cavity and extended cubic. For application, phase change and natural convection problem inside circular pipe were considered. With and without axial flow cases were compared with each other to observe the effect of axial flow on the flow pattern and phase change characteristics. It was found that the forced convective flow tends to disturb the formation of ice-plug and natural convective flow tends to promote the energy exchanges and expedite the formation of ice-plug.