A two-dimensional numerical analysis of the laminar mixed convection flows in horizontal rectangular ducts heated from below (also called "Poiseuille-Rayleigh-Bénard flows") is presented. The algorithm is based on the augmented Lagrangian method and a finite volume type discretisation is used. An exhaustive bibliographical review of the studies on this subject is presented, and three PhD thesis are presented more in detail. They analyse the linear stability of the Poiseuille flow and the weakly non-linear stability of the thermoconvective rolls, transverse to the axis of the duct; they also present very fine experimental measures of the velocity fields in the linear and non-linear domains. The main objective of our work is to study in which degree the numerical simulations are able to reproduce the theoretical and experimental results, by means of systematic quantitative comparisons with these three thesis. The influence of the open and periodic boundary conditions upon the stability and space and time evolution of the transversal rolls is studied, taking into account the convective or absolute character of the instability. Five different open boundary conditions are tested. The amplitude of the perturbation upstream of the outlet boundary is shown to vary significantly from one open boundary condition to the other; on the other hand, the length of the perturbed zone and the structure of the flow outside this zone nearly not vary and can be considered as being constant. With the periodic boundary conditions, the transition between the transversal rolls and the Poiseuille flow occurs at the border between the convective instability and the linear stability regions. While with the open boundary conditions, this transition occurs between the absolute and convective instability regions. All the results of the weakly non-linear stability theory are reproduced by the 2D numerical simulations and some of the 3D experimental results are reproduced very precisely. The problems that have to be worken on are enlighted, some of them being able to be treated only with 3D numerical simulations.
