We study the effects of radiative transfer, due to small amounts of water vapor and carbon dioxide in air, on Rayleigh-Bénard flow structure and heat transfer in a cubic cavity up to a Rayleigh number Ra=10^10. Direct numerical simulations are carried out, with and without radiative transfer, up to Ra=10^9 and are used to assess the accuracy and to optimize the parameters of a spectral vanishing viscosity (SVV) LES model. Comparisons between SVV and DNS calculations are given in terms of first order and second order statistics, as well as the analysis of spatial auto-correlation functions, and POD eigenmodes. The model is then applied for the highest Rayleigh number Ra=10^10 considered in this study. Results with and without radiation show a weak influence of radiation on conductive flux, turbulent kinetic energy and variance of temperature fluctuations, while a significant increase of the kinetic energy of the mean flow and of the convective flux are observed when radiative transfer is taken into account. Finally, radiation effects on conductive and convective fluxes, and on the total kinetic energy are analyzed in the Rayleigh number range 10^6 ≤ Ra ≤ 10^10. It is shown that these effects decrease when the Rayleigh number increases.