  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 tensor of turbulent
stress as a function of the time average of the 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, a kind of extension of classical Boussinesq theory of
turbulent viscosity. This leads to rather complex nonlinear integral
equation(s) for the time averaged velocity field. This one satisfies some
symmetries of the Euler equations. Such symmetries were used by Prandtl and
Landau to make various predictions about the shape of the turbulent domain in
simple geometries. We explore specifically the case of mixing layer for which
the average velocity field only depends on the angle in the wedge behind the
splitter plate. This solution yields a pressure difference between the two
sides of the splitter which contributes to the lift felt by the plate.
Moreover, because of the structure of the equations for the turbulent stress,
one can satisfy the Cauchy-Schwarz inequalities, also called the realizability
conditions, for this turbulent stress. Such realizability conditions cannot be
satisfied with a simple turbulent viscosity.
