The transitions from the onset of convection to fully developed turbulence of a Rayleigh–Be´nard flow, in a low-aspect-ratio cell and in mercury, are studied through three-dimensional numerical simulation of the Navier–Stokes equations. The calculation of the growth rate of the azimuthal energy modes permitted the accurate determination of the critical Rayleigh number for the establishment of the convective regime(Rac=3750)which is in good agreement with analytical and other numerical results. Increasing the Rayleigh number, the flow remained steady up toRa≃2.11×104when an oscillatory instability was observed. Further increases in the Rayleigh produced a chaotic state through the period doubling mechanism and finally the turbulent state was achieved. It is shown that forRa⩾Racthe mean flow consists of a large-scale convective cell which persists in the whole range of studied Rayleigh numbers(Ra⩽106). The dependence of the Nusselt number over the Rayleigh number is also analyzed and, forRa⩾3.75×104, when the turbulent state is reached, a power law in quantitative agreement with previous results at higher Ra is observed. ©1997 American Institute of Physics.