Turbulence of a unidirectional flow

Dedicated to Henry P. McKean, a mentor and a friend ABSTRACT. We discuss recent advances in the theory of turbulent solutions of the Navier–Stokes equations and the existence of their associated invariant measures. The statistical theory given by the invariant measures is described and associated with historically-known scaling laws. These are Hack’s law in one dimension, the Batchelor–Kraichnan law in two dimensions and the Kolmogorov’s scaling law in three dimensions. Applications to problems in turbulence are discussed and applications to Reynolds Averaged Navier Stokes (RANS) and Large Eddy Simulation (LES) models in computational turbu-lence. 1