Turbulence in a wedge: the case of the mixing layerTurbulence in a wedge: the case of the mixing layer

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.