A connection between the Camassa-Holm equations and turbulent flows in channels and pipes

In this paper we discuss recent progress in using the Camassa-Holm equation s to model turbulent flows. The Camassa-Holm equations, given their special geometric and physical properties, appear particularly well suited for stu dying turbulent flows. We identify the steady solution of the Camassa-Holm equation with the mean flow of the Reynolds equation and compare the result s with empirical data for turbulent flows in channels and pipes. The data s uggest that the constant alpha version of the Camassa-Holm equations, deriv ed under the assumptions that the fluctuation statistics are isotropic and homogeneous, holds to order alpha distance from the boundaries. Near a boun dary, these assumptions are no longer valid and the length scale alpha is s een to depend on the distance to the nearest wall. Thus, a turbulent flow i s 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 f rom boundaries, these scaling conditions imply alpha is independent of Reyn olds number. Given the agreement with empirical and numerical data, our cur rent work indicates that the Camassa-Holm equations provide a promising the oretical framework from which to understand some turbulent flows. (C) 1999 American Institute of Physics. [S1070-6631(99)00508-5].