  In this paper we discuss recent progress in using the Camassa-Holm equations
to model turbulent flows. The Camassa-Holm equations, given their special
geometric and physical properties, appear particularly well suited for studying
turbulent flows. We identify the steady solution of the Camassa-Holm equation
with the mean flow of the Reynolds equation and compare the results with
empirical data for turbulent flows in channels and pipes. The data suggests
that the constant $\alpha$ version of the Camassa-Holm equations, derived under
the assumptions that the fluctuation statistics are isotropic and homogeneous,
holds to order $\alpha$ distance from the boundaries. Near a boundary, these
assumptions are no longer valid and the length scale $\alpha$ is seen to depend
on the distance to the nearest wall. Thus, a turbulent flow is divided into two
regions: the constant $\alpha$ region away from boundaries, and the near wall
region. In the near wall region, Reynolds number scaling conditions imply that
$\alpha$ decreases as Reynolds number increases. Away from boundaries, these
scaling conditions imply $\alpha$ is independent of Reynolds number. Given the
agreement with empirical and numerical data, our current work indicates that
the Camassa-Holm equations provide a promising theoretical framework from which
to understand some turbulent flows.
