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 equationsto model turbulent flows. The Camassa-Holm equations, given their specialgeometric and physical properties, appear particularly well suited for studyingturbulent flows. We identify the steady solution of the Camassa-Holm equationwith the mean flow of the Reynolds equation and compare the results withempirical data for turbulent flows in channels and pipes. The data suggeststhat the constant $\alpha$ version of the Camassa-Holm equations, derived underthe assumptions that the fluctuation statistics are isotropic and homogeneous,holds to order $\alpha$ distance from the boundaries. Near a boundary, theseassumptions are no longer valid and the length scale $\alpha$ is seen to dependon the distance to the nearest wall. Thus, a turbulent flow is divided into tworegions: the constant $\alpha$ region away from boundaries, and the near wallregion. In the near wall region, Reynolds number scaling conditions imply that$\alpha$ decreases as Reynolds number increases. Away from boundaries, thesescaling conditions imply $\alpha$ is independent of Reynolds number. Given theagreement with empirical and numerical data, our current work indicates thatthe Camassa-Holm equations provide a promising theoretical framework from whichto understand some turbulent flows.