Abstract: This is work done with Martine Le Berre on one of the most simple real turbulent flow: the plane Poiseuille flow between two parallel planes driven by a pressure gradient parallel to those planes in the limit of a very large pressure gradient. The theoretical understanding of wall bounded flows like this one often relies on the log-law of von-Karman-Prandtl valid close to the walls. This law introduces the Prandtl length which is itself extrapolated from Boussinesq theory of turbulence. I shall explain how a rational theory of turbulence (inhomogeneous and anisotropic) can be built different of Prandtl and Landau. Applied to Poiseuille flows, this covers both the neighborhood of the wall, and the rest of the channel. Because of the simplicity of the geometry this can be formulated in a rather simple way and the corresponding equations can be solved almost explicitly in the limit of a large Reynolds number, with different approximations in different domains that are matched near the boundaries. This yields in particular the explicit velocity profile with its very thin boundary layers near the walls.