The paper presents the comparative study of the vortex-induced cooling of a heated channel for the four different cross-sections of the rigid cylinder, i.e., circular, square, semi-circular, and triangular, with or without the rigid/flexible splitter plate at the Reynolds number (based on the hydraulic diameter) of 200. The study presents a comprehensive analysis of the flow and thermal performance for all the cases. For flexible plate cases, a partitioned approach is invoked to solve the coupled fluid-structure-convection problem. The simulations show the reduction in the thermal boundary layer thickness at the locations of the vortices resulting in the improved Nusselt number. Furthermore, the thin plate's flow-induced motion significantly increases the vorticity field inside the channel, resulting in improved mixing and cooling. It is observed that the plate-motion amplitude is maximum when the plate is attached to the cylinder with the triangular cross-section. The power requirement analysis shows that the flexible plate reduces the power required to pump the channel's cold fluid. Thus, based on the observations of the present study, the authors recommend using the flexible plate attached to the cylinder for improved convective cooling.