Symmetry reduction of turbulent pipe flows

Abstract

We propose and apply a Fourier-based symmetry-reduction scheme to remove, or quotient, the streamwise translation symmetry of laser-induced-fluorescence measurements of turbulent pipe flows that are viewed as dynamical systems in a high-dimensional state space. We also explain the relation between Taylor’s hypothesis and the comoving frame velocity $U_{d}$ of the turbulent orbit in state space. In particular, in physical space we observe flow structures that deform as they advect downstream at a speed that differs significantly from $U_{d}$. Indeed, the symmetry-reduction analysis of planar dye concentration fields at Reynolds number $Re=3200$ reveals that the speed $u$ at which high-concentration peaks advect is roughly 1.43 times $U_{d}$. In a physically meaningful symmetry-reduced frame, the excess speed $u-U_{d}\approx 0.43U_{d}$ can be explained in terms of the so-called geometric phase velocity $U_{g}$ associated with the orbit in state space. The ‘self-propulsion velocity’ $U_{g}$ is induced by the shape-changing dynamics of passive scalar structures observed in the symmetry-reduced frame, in analogy with that of a swimmer at low Reynolds numbers.