We investigate with direct numerical simulations the onset of unsteadiness, the route to chaos and the dynamics of fully chaotic natural convection in an upright square air-filled differentially heated cavity with adiabatic top and bottom walls. The numerical algorithm integrates the Boussinesq-type Navier–Stokes equations in velocity–pressure formulation with a Chebyshev spatial approximation and a finite-difference second-order time-marching scheme. Simulations are performed for Rayleigh numbers up to 1010, which is more than one order of magnitude higher than the onset of unsteadiness. The dynamics of the time-dependent solutions, their time-averaged structure and preliminary results concerning their statistics are presented. In particular, the internal gravity waves are shown to play an important role in the time-dependent dynamics of the solutions, both at the onset of unsteadiness and in the fully chaotic regime. The influence of unsteadiness on the local and global heat transfer coefficients is also examined.