A connection between the Camassa–Holm equations and turbulent flows in channels and pipes
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In this paper, the steady solution of the Camassa-holm equation with the mean flow of the Reynolds equation is compared with empirical data for turbulent flows in channels and pipes.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 Near a boundary, these assumptions are no longer valid and the length scale α is seen to depend on the distance to the nearest wall.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 increas...read more
Citations
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On the scattering problem for the camassa-Holm equation
TL;DR: The Camassa-Holm equation as mentioned in this paper has a number of constants of motion arising as eigenvalues of an associated spectral problem, and the spectral picture is described and discussed in detail.
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The Three Dimensional Viscous Camassa–Holm Equations, and Their Relation to the Navier–Stokes Equations and Turbulence Theory
TL;DR: In this article, it was shown that the global, in time, regularity of the three dimensional viscous Camassa-Holm (Navier-Stokes-alpha) (NS-α) equations is bounded by (L/l�� ∈ )3, where L is a typical large spatial scale (e.g., the size of the domain).
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On a Leray-α model of turbulence
Alexey Cheskidov,Alexey Cheskidov,Darryl D. Holm,Darryl D. Holm,Eric Olson,Edriss S. Titi,Edriss S. Titi +6 more
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References
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