Turbulent plane Poiseuille flow

The understanding of fully developed turbulence remains a major unsolved problem of statistical physics. A challenge there is how to use what we understand of this problem to build-up a closure method, that is to express the time-averaged turbulent stress tensor as a function of the time-averaged velocity field u ( x ). We have shown that the closure problem is strongly restricted due to constraints on the time-averaged quantities, and to scaling laws derived from the idea that dissipation in fully developed turbulence is by singular events resulting from an evolution described by the Euler equations. It implies that the turbulent stress is a non-local function in space of the time-averaged velocity u ( x ), involving an integral kernel, an extension of classical Boussinesq theory of turbulent viscosity. We treat one of the simplest possible physical situation, the turbulent Poiseuille flow between two parallel plates. In this case, the integral kernel takes a simple form leading to full analysis of the time-averaged turbulent flow. In the limit of a very large Reynolds number, one has to match a viscous boundary layer near the walls bounding the flow and an outer solution in the bulk of the flow, a non-trivial asymptotic analysis because of logarithms. Besides the boundary layers close to the walls, there is another “inner” boundary layer near the center plane of the flow. Our expression for the turbulent stress tensor yields ultimately the complete structure of the boundary layer, including in locations where viscosity becomes important.