Final equilibrium state of a two-dimensional shear layer

We test a new statistical theory of organized structures in two-dimensional turbulence by direct numerical stimulations of the Navier–Stokes equations, using a pseudo-spectral method. We apply the theory to the final equilibrium state of a shear layer evolving from a band of uniform vorticity: a relationship between vorticity and stream function is predicted by maximizing an entropy with the constraints due the constants of the motion. A partial differential equation for the stream function is then obtained. In the particular case of a very thin initial vorticity band, the Stuart's vortices appear to be a family of solutions for this equation. In more general cases we do not solve the equation, but we test the theory by inspecting the relationship between stream function and vorticity in the final equilibrium state of the numerical computation. An excellent agreement is obtained in regions with strong vorticity mixing. However, local equilibrium is obtained before a complete mixing can occur in the whole fluid domain.