In view of applying our process to bilayer, and multilayer flows, we present
here a systematic and general methodology to derive two moments approximate models of shallow water type for a single thin lm down an incline in a turbulent regime. Some recent related works are [4, 5]. If the laminar case is now well understood and rigorously justified, few turbulent models exist. We consider a model of Mixing Length law  to describe turbulence. We perform a long wave asymptotic of the Navier-Stokes equations and construct a Hilbert expansion up to the second order with respect to the small aspect ratio (or film parameter) ? of the solution. This expansion appears to be singular because
of the turbulence model: two distinct behaviors of the fluid are identified, the so-called external and internal expansions of the solution (we refer to  for the vocabulary about singular expansions). The derivation of Shallow Water models from the previous expansions is then classical, but furnishes here new corrective terms of consistency. In particular they differ from more generally used SW systems with friction . After presenting the one free surface layer case, we will then briefly present the generalization for multilayer flows.
Joint work with Jean-Paul Vila
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