 
Summary: Turbulence of a Unidirectional Flow
Bjšorn Birnir
Center for Complex and Nonlinear Science
and
Department of Mathematics
University of California, Santa Barbara
Abstract
Recent advances in the theory of turbulent solutions of the NavierStokes
equations are discussed and the existence of their associated invariant mea
sures. The statistical theory given by the invariant measures is described
and associated with historicallyknown scaling laws. These are Hack's law
in one dimension, the BachelorKraichnan law in two dimensions and the
Kolmogorov's scaling law in three dimensions. Applications to problems
in turbulence are discussed and applications to Reyolds Averaged Navier
Stokes (RANS) and Large Eddy Simulation (LES) models in computational
turbulence.
1 Introduction
Everyone is familiar with turbulence in one form or another. Airplane passengers
encounter it in wintertime as the plane begins to shake and is jerked in various
directions. Thermal currents and gravity waves in the atmosphere create turbu
