The present study is concerned with the buoyancy-driven airflow in an cumulus between two asymmetrically heated horizontal eccentric elliptical cylinders. Flow patterns and heat transfer characteristics for various geometric configurations and heating conditions are predicted. The governing equations are discretized with the finite volume method on a curvilinear grid system generated numerically by the body-fitted coordinate transformation. Dependence of the equivalent conductivity on the physical and geometric parameters, such as the Rayleigh number, the dimensionless eccentricity, and the ratio of the areas of the cylinders, has been evaluated. Results show that, for Ra > 104, the strength of the buoyancy-driven fluid motion as well as the enhancement in the heat transfer becomes appreciable. Among the nine possible geometric configurations considered in this study, case Vv exhibits the highest heat transfer performance. For some special cases, the numerical solutions are compared with existing information, and close agreement has been found.