Hopf fibrations for turbulent pipe flows

We propose a generalization of Hopf fibrations to quotient the streamwise translation symmetry of turbulent pipe flows viewed as dynamical systems. In particular, we exploit the geometric structure of the associate high dimensional state space, which is that of a principal fiber bundle. The relation between the comoving frame velocity $U_{d}$ associated with the dynamical phase of an orbit in the bundle and the Taylor's hypothesis is investigated. As an application, Laser-Induced-Fluorescence techniques are exploited to capture planar fluorescent dye concentration fields tracing a turbulent pipe flow at the bulk Reynolds number $\mathfrak{\mathsf{Re}}=3200$. The symmetry reduction analysis of the experimental data reveals that the speed $u$ of dye concentration bursts is associated with the dynamical and geometric phases of the corresponding orbits in the fiber bundle. In particular, in the symmetry-reduced frame we unveil a pattern-changing dynamics of the passive scalar structures, which explains the observed speed $u\approx U_{d}+U_{g}$ of intense bursting events in terms of the geometric phase velocity $U_{g}\approx0.43U_{d}$ associated with the orbits in the bundle.