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 nonlocal 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 the mixing layer for which the average velocity field only depends on the angle of 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, one can satisfy the Cauchy-Schwarz inequalities for the turbulent stress, also called the realizability conditions. In the limit of small velocity differences between the two merging flows behind the splitter, we predict an angular spreading of the turbulent domain proportional to the square root of the velocity difference, in agreement with experiments.

• Received 23 December 2020
• Accepted 9 June 2021

DOI:https://doi.org/10.1103/PhysRevFluids.6.074603