Summary: A direct numerical simulation is carried out to determine the transition to geostrophic turbulence in an air-filled differentially heated rotating annulus. The coupled Navier-Stokes and energy equations are solved numerically with a spectral method based on collocation Chebyshev and Fourier approximations associated with a second-order time accurate scheme. For a temperature difference fixed at \(\Delta T=5 K\), when increasing the rotation rate, we have obtained the successive occurrence of the different regimes as described in the literature. The route to turbulent flow, studied with respect to two principal dimensionless parameters, the Taylor number Ta* and the thermal Rossby number, presents steady axisymmetric solution, periodic flow, quasi-periodic flow, periodic flow, before transition to chaotic motion.