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The Camassa-Holm equations and turbulence. (English) Zbl 1194.76069

Summary: We survey our results on the Camassa-Holm equations and their relation to turbulence as discussed in [The Camassa-Holm equations as a closure model for turbulent channel and pipe flow, Phys. Rev. Lett 81, 5338 (1998); Phys. Fluids 11, No. 8, 2343–2353 (1999; Zbl 1147.76357)]. In particular we will provide a more detailed mathematical treatment of those equations for pipe flows which yield accurate predictions of turbulent flow profiles for very large Reynolds numbers. There are many facts connecting the Camassa-Holm equations to turbulent fluid flows. The dimension of the attractor agrees with the heuristic argument based on the Kolmogorov statistical theory of turbulence. The statistical properties of the energy spectrum agree in numerical simulation with the Kolmogorov power law. Furthermore, comparison of mean flow profiles for turbulent flow in channels and pipes given by experimental and numerical data show acceptable agreement with the profile of the corresponding solution of the Camassa-Holm equations.

MSC:

76F20 Dynamical systems approach to turbulence
35Q35 PDEs in connection with fluid mechanics
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
76M25 Other numerical methods (fluid mechanics) (MSC2010)

Citations:

Zbl 1147.76357
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References:

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