Owing to the high storage capacity of the latent heat energy storage unit (LHSU), it is preferred among other types of thermal energy storage to overcome the mismatch between energy supply and demand. The orientation of the LHSU affects the melting process of the phase change material (PCM), and this effect could vary according to the fin shape of the LHSU. Thus, the present study aims to numerically examine the impact of vertical and horizontal orientations on the longitudinal, spiral, and annular finned shell and tube LHSUs. The computational fluid dynamics simulation solved the time-dependent conservation equations of mass, momentum, and energy in the three-dimensional domain. The enthalpy-porosity and the Boussinesq approximation were used respectively to simulate the PCM phase change and the change in density. The results show that the LHSUs have completely different behavior in the vertical orientation than in the horizontal orientation due to the various obstructions to the natural convection flow induced by the fins, depending on the fin shape, and the direction of natural convection flow. In the vertical orientation, the annular finned LHSU has a faster melting rate than the longitudinal finned LHSU by 25.6%. In contrast, in the horizontal orientation, the longitudinal finned LHSU melted more rapidly than the annular finned LHSU by 52.5%. The spiral finned LHSU performance was moderate between the two other finned LHSUs in both orientations due to its spiral shape that lies between the shape of the annular and longitudinal fins.