In this paper we will survey our results on the Camassa–Holm equations and their relation to turbulence as discussed in S. Chen, C. Foias, D.D. Holm, E. Olson, E.S. Titi, S. Wynne, The Camassa–Holm equations as a closure model for turbulent channel and pipe flow, Phys. Rev. Lett 81 (1998) 5338. S. Chen, C. Foias, D.D. Holm, E. Olson, E.S. Titi, S. Wynne, A connection between the Camassa–Holm equations and turbulent flows in channels and pipes, Phys. Fluids, in press. 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.
