Reynolds, Kolmogorov: irony strikes twice

In his landmark 1883 paper, Osborne Reynolds studied qualitatively different kinds of motion of fluid in a straight pipe. His aim was to demonstrate the transition to “sinuous”, in current terminology turbulent, motion as the flow rate increases. As it turns out, he had picked a set up that is practically simple but mathematically very complicated. The flow state that is void of any turbulence remains asymptotically stable for any experimentally reachable flow rate, and swirling flows occur suddenly and intermittently. Their onset is hysterectic and strongly depends on the way the flow rate is varied. Intrigued by this partially explained result, A. N. Kolmogorov introduced a more abstract model of fluid motion, restricting it to two spatial dimensions and discarding material boundaries. However, it soon turned out to be overly restrictive and exclude transitions like the one Reynolds had observed. In this presentation, I will show that extending Kolmogorov’s model to three spatial dimensions places it in the same category as elementary shear flows such as that of fluids in pipes and channels. I will study the hysteretic onset of turbulence and the structure of phase space by numerical bifurcation analysis as well as energy methods. This is joint work with Susumu Goto of Osaka University.