A connection between the Camassa–Holm equations and turbulent flows in channels and pipes Academic Article uri icon

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 a 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 Near a boundary, these assumptions are no longer valid and the length scale α is seen to depend on the distance to the nearest wall.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. © 1999 American Institute of Physics.

author list (cited authors)

  • Chen, S., Foias, C., Holm, D. D., Olson, E., Titi, E. S., & Wynne, S.

citation count

  • 174

publication date

  • August 1999