The Camassa-Holm equations as a closure model for turbulent channel flow
  We propose the viscous Camassa-Holm equations as a closure approximation for the Reynolds-averaged equations of the incompressible Navier-Stokes fluid. This approximation is tested on turbulent channel flows with steady mean. Analytical solutions for the mean velocity and the Reynolds shear stress across the entire channel are obtained, showing good agreement with experimental measurements and direct numerical simulations. As Reynolds number varies, these analytical mean velocity profiles form a family of curves whose envelopes are shown to have either power law, or logarithmic behavior, depending on the choice of drag law. 
Shiyi Chen, Ciprian Foias, Darryl D. Holm, Eric Olson, Edriss S. Titi,   Shannon Wynne