Symmetry reduction of turbulent pipe flows

TL;DR

This paper introduces a Fourier-based symmetry reduction method for turbulent pipe flows, revealing how flow structures deform and advect at speeds different from the mean, and relates this to geometric phase velocity in state space.

Contribution

It presents a novel symmetry reduction scheme for turbulent flows and links physical advection speeds to geometric phase velocity in state space.

Findings

Flow structures deform and advect at speeds 1.43 times the comoving frame velocity.

Symmetry reduction reveals the shape-changing dynamics induce a geometric phase velocity.

The excess advection speed is explained by the geometric phase velocity U_g.

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 $\mathfrak{\mathsf{Re}}=3200$ 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 $u-U_{d}\approx0.43U_{d}$ 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.