The Poiseuille-Bénard flow ͑PBF͒ is studied by a two-dimensional numerical simulation for a Prandtl number equal to 6.4 ͑that of water at 23°C͒ and for a wide range of Rayleigh ͑Ra͒ and Reynolds ͑Re͒ numbers: Raр6000 and Reр3. The two observed flow configurations are ͑1͒ thermally stratified Poiseuille flow and ͑2͒ thermoconvective transversal rolls superimposed to the basic Poiseuille flow. The time evolution of the velocity components, the spatial development of the transversal rolls, their

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... equency, wavelength and velocity, the Nusselt number, together with the stability map in the Ra-Re plane, are studied in detail. Whenever possible, quantitative comparisons are made with published results: most of the experimental data, based on laser-Doppler anemometry ͑LDA͒, are recovered with amazing accuracy; a good agreement with results of convective stability deduced from a weakly nonlinear Ginzburg-Landau theory is also obtained.