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 $\mathfrak{\mathsf{Re}}=3200$ 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.Comment: arXiv admin note: substantial text overlap with arXiv:1407.748

Crest speeds of unsteady surface water waves

“…Fedele (2014a) calculates that the leaning coefficient at crest maxima of unsteady oceanic groups attains a slowdown of c/c 0 ≈ 0.8. Other fluid dynamical phenomena with the same underlying physics include self-propulsion at low Reynolds numbers (Wilczek & Shapere 1989) and the speed of fluctuating coherent structures in turbulent pipe flow that depends on both their dynamical inertia and also on their geometrical shape as it varies in time (Fedele, Abessi & Roberts 2015). Clearly, the unsteadiness of dispersive focusing is essential for crest slowdown to be observed.…”

The Shifted Proper Orthogonal Decomposition: A Mode Decomposition for Multiple Transport Phenomena

“…Note further that the approach also works with a non-constant transport velocity, since the shift can be set for each time step separately. In a similar manner, shifts have been used in other reduction frameworks, e. g., for the model reduction of a combustion [26] and in a symmetry reduction framework [6,17,37]. The latter one treats partial differential equations (PDEs) which are equivariant with respect to a group action which in turn induces symmetries in the solution space.…”

“…It is used in the framework of symmetry reduction (cf. [6,17,31,37]), where the translation is accounted for by applying a Lie group action to a symmetry-reduced or frozen solution. In [31], for instance, the framework has been analyzed for nonlinear parameterdependent evolution equations and applied to a numerical simulation of the Burgers equation.…”

Permuted proper orthogonal decomposition for analysis of advecting structures

“…2003). Furthermore, Fedele, Abessi & Roberts (2015) reduced translational symmetry in turbulent pipe flow measurements using a Fourier-based method, while Reiss et al. (2018) developed an iterative procedure which handles structures advecting at different velocities.…”