Numerical calculations are carried out for the natural convection induced by temperature difference between a cold outer square cylinder and a hot inner circular cylinder. A two-dimensional solution for unsteady natural convection is obtained, using the immersed boundary method (IBM) to model an inner circular cylinder based on finite volume method, for different Rayleigh numbers varying over the range of $10^4;to;10^6$. The study goes further to investigate the effect of an inner cylinder location on the heat transfer and fluid flow. The location of inner circular cylinder is changed vertically along the center-line of square enclosure. The number, size and formation of cell strongly depend on Rayleigh number and the position of inner circular cylinder. The changes in heat transfer quantities have been presented.