Abstract
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 α version of the Camassa–Holm equations, derived under the assumptions that the fluctuation statistics are isotropic and homogeneous, holds to order α distance from the boundaries. Near a boundary, these assumptions are no longer valid and the length scale α is seen to depend on the distance to the nearest wall. Thus, a turbulent flow is divided into two regions: the constant α region away from boundaries, and the near wall region. In the near wall region, Reynolds number scaling conditions imply that α decreases as Reynolds number increases. Away from boundaries, these scaling conditions imply α 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.
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