We perform a direct numerical simulation of three-dimensional Navier-Stokes equations for a Rayleigh-Bénard convection system in a stationary cylinder with the top cold lid rotating. This convection system with an imposed swirl flow is a canonical problem for investigating axial vortices under unstable thermal gradients. The base flow is established by rotating the top lid and the fluid moves azimuthally along the side vertical wall into a meridional flow in the r - z plane. This forms an axial vortex core at the axis of the cylinder. This axial core, under a certain rotational Reynolds number (Re), breaks down to a vortex breakdown bubble whose dynamics are modified under thermal convection. We study the effect of rotation on varying Re for a Rayleigh number Ra = 2 × 105. The equations are formulated in such a way that the rotating-lid cylinder and Rayleigh-Bénard convection are extreme cases of the same numerical set up. From the present study, we find that as the rotational rate is increased, the system dynamics shift from a convection-dominated flow regime to a rotation-dominated regime. This shift in dynamics is quantified using the volume-averaged and time-averaged temperature, the heat flux, the thickness of the Bödewadt boundary layer and the relative Nusselt number. These quantities are shown to demarcate the convection- and rotation-dominated regimes, as compared to the qualitative description of flow patterns from velocity and temperature contours.