Purpose - This study aims to carry out the analysis of Rayleigh-Bénard convection within enclosures with curved isothermal walls, with the special implication on the heat flow visualization via the heatline approach. Design/methodology/approach - The Galerkin finite element method has been used to obtain the numerical solutions in terms of the streamlines (ψ), heatlines (Π), isotherms (θ), local and average Nusselt number (Nut) for various Rayleigh numbers (103 ≤ Ra ≥ 105), Prandtl numbers (Pr = 0.015 and 7.2) and wall curvatures (concavity/convexity). Findings - The presence of the larger fluid velocity within the curved cavities resulted in the larger heat transfer rates and thermal mixing compared to the square cavity. Case 3 (high concavity) exhibits the largest Nut at the low Ra for all Pr. At the high Ra, Nut is the largest for Case 3 (high concavity) at Pr = 0.015, whereas at Pr = 7.2, Nut is the largest for Case 1 (high concavity and convexity). Practical implications - The results may be useful for the material processing applications. Originality/value - The study of Rayleigh-Bénard convection in cavities with the curved isothermal walls is not carried out till date. The heatline approach is used for the heat flow visualization during Rayleigh- Benard convection within the curved walled enclosures for the first time. Also, the existence of the enhanced fluid and heat circulation cells within the curved walled cavities during Rayleigh-Benard heating is illustrated for the first time. © Emerald Publishing Limited.