The Camassa-Holm equations and turbulence

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. Helm, E. Olson, E.S. Titi, S. Wynne, The Camassa-Holm equations as a closure mode l for turbulent channel and pipe flow, Phys. Rev. Lett 81 (1998) 5338. S. C hen, C. Foias, D.D. Helm, E. Olson, E.S. Titi, S. Wynne, A connection betwe en the Camassa-Holm equations and turbulent flows in channels and pipes, Ph ys. Fluids, in press. In particular we will provide a more detailed mathema tical treatment of those equations for pipe flows which yield accurate pred ictions of turbulent flow profiles for very large Reynolds numbers. There a re many facts connecting the Camassa-Holm equations to turbulent fluid flow s. The dimension of the attractor agrees with the heuristic argument based on the Kolmogorov statistical theory of turbulence. The statistical propert ies of the energy spectrum agree in numerical simulation with the Kolmogoro v power law. Furthermore, comparison of mean flow profiles for turbulent fl ow in channels and pipes given by experimental and numerical data show acce ptable agreement with the profile of the corresponding solution of the Cama ssa-Holm equations. (C) 1999 Elsevier Science B.V. All rights reserved.