ar X iv : c ha o - dy n / 99 03 03 3 v 1 2 5 M ar 1 99 9 A connection between the Camassa - Holm equations and turbulent flows in channels and pipes
نویسندگان
چکیده
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 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 Near a boundary, these assumptions are no longer valid and the length scale α is seen to depend on the distance to the nearest wall.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. 1 Introduction. Laminar Poiseuille flow occurs when a fluid in a straight channel, or pipe, is driven by a constant upstream pressure gradient, yielding a symmetric parabolic stream
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