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

摘要

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 suggest 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 Near a boundary, these assumptions are no longer valid and the length scale #alpha# is seen to depend on the distance to the nearest wall. This, a turbulent flow is divided into two regions: the constant #alpha# region away from boundaries, and the near wall region. In the near well 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.