We investigate Rayleigh-Bénard convection in a cubical cavity under the effect of wall and gas radiation. Coupled direct numerical simulations are carried out for a radiating air / \chemform{H\_2O} / \chemform{CO\_2} mixture at room mean temperature, using a Chebyshev spectral method for the flow and a ray-tracing method for the radiation field The global absorption distribution function (ADF) model is used to represent the spectral radiative properties of the mixture. Time-averaging is then applied to compare the results, regardless of the multiple flow configurations obtained. Three different Rayleigh numbers are studied, from Ra = 1E7 to Ra = 1E9. Under the Boussinesq approximation, solutions to the uncoupled simulations only depend on the Rayleigh and Prandtl numbers and are not affected by the size of the cavity. However, it goes otherway when radiation is taken into account ; two cavitiy sizes are then considerated with edges of 1 and 3 meters long. As the cavity size increases, at fixed Rayleigh number, so does the effects of radiation on the flow. Radiation has a stabilizing effect on the large-scale circulation. The large-scale circulation settles in vertical mid-planes or diagonal planes depending on the radiation conditions, and reorientations are occasionally observed in the cavity. In the same time, the convective flux in the core, as well as the conductive flux at the walls and the kinetic energy, are increased as radiation is taken into account, and this increase is more important for the larger size of the cavity. The contribution of radiation to the potential energy balance and to the "thermal energy" balance are presented. Proper orthogonal decomposition (POD) analysis is then used in order to better understand the evolution of the flow structures and the influence of radiation. The shape of the modes, as well as their symmetries and temporal dynamics, are discussed. The influence of the modes in the reorientation process is also studied.