Numerical results are presented on oscillatory convection occuring in a binary-fluid mixture, in the vicinity of the codimension-two point. It is recalled that this corresponds to the case of a fluid heated from below and stabilized by a concentration gradient, where the transitional behaviors are known to be quite intricated, Our aim is to bring results for a relatively unexplored configuration, The fluid is confined in an axisymmetrical cylinder with rigid impermeable realistic boundaries and a small aspect ratio (Radius/height = 2). From direct numerical simulations (with a spectral Chebyshev method), preliminary results we given about (1) the extension, in the (Ra,psi) plane, of the domain where this oscillatory convection is present, (2) the temporal characteristics evolution in this domain (frequencies and traveling waves aspects).