  The ultimate goal of a sound theory of turbulence in fluids is to close in a
rational way the Reynolds equations, namely to express the time averaged
turbulent stress tensor as a function of the time averaged velocity field. This
closure problem is a deep and unsolved problem of statistical physics whose
solution requires to go beyond the assumption of a homogeneous and isotropic
state, as fluctuations in turbulent flows are strongly related to the geometry
of this flow. This links the dissipation to the space dependence of the average
velocity field. Based on the idea that dissipation in fully developed
turbulence is by singular events resulting from an evolution described by the
Euler equations, it has been recently observed that the closure problem is
strongly restricted, and that it implies that the turbulent stress is a non
local function (in space) of the average velocity field, an extension of
classical Boussinesq theory of turbulent viscosity. The resulting equations for
the turbulent stress are derived here in one of the simplest possible physical
situation, the turbulent Poiseuille flow between two parallel plates. In this
case the integral kernel giving the turbulent stress, as function of the
averaged velocity field, takes a simple form leading to a full analysis of the
averaged turbulent flow in the limit of a very large Reynolds number. In this
limit one has to match a viscous boundary layer, near the walls bounding the
flow, and an outer solution in the bulk of the flow. This asymptotic analysis
is non trivial because one has to match solution with logarithms. A non trivial
and somewhat unexpected feature of this solution is that, besides the boundary
layers close to the walls, there is another "inner" boundary layer near the
center plane of the flow.
