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Objective barriers to the transport of dynamically active vector fields

Published online by Cambridge University Press:  27 October 2020

George Haller Open the ORCID record for George Haller [Opens in a new window] ,
Stergios Katsanoulis Open the ORCID record for Stergios Katsanoulis [Opens in a new window] ,
Markus Holzner ,
Bettina Frohnapfel Open the ORCID record for Bettina Frohnapfel [Opens in a new window]  and
Davide Gatti Open the ORCID record for Davide Gatti [Opens in a new window]
George Haller*
Affiliation:
Institute for Mechanical Systems, ETH Zürich, Zürich, Switzerland
Stergios Katsanoulis
Affiliation:
Institute for Mechanical Systems, ETH Zürich, Zürich, Switzerland
Markus Holzner
Affiliation:
WSL Swiss Federal Research Institute, Birmensdorf, Switzerland
Bettina Frohnapfel
Affiliation:
Institute of Fluid Mechanics, Karlsruhe Institute of Technology, Karlsruhe, Germany
Davide Gatti
Affiliation:
Institute of Fluid Mechanics, Karlsruhe Institute of Technology, Karlsruhe, Germany
*
†Email address for correspondence: georgehaller@ethz.ch
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Abstract

We derive a theory for material surfaces that maximally inhibit the diffusive transport of a dynamically active vector field, such as the linear momentum, the angular momentum or the vorticity, in general fluid flows. These special material surfaces (Lagrangian active barriers) provide physics-based, observer-independent boundaries of dynamically active coherent structures. We find that Lagrangian active barriers evolve from invariant surfaces of an associated steady and incompressible barrier equation, whose right-hand side is the time-averaged pullback of the viscous stress terms in the evolution equation for the dynamically active vector field. Instantaneous limits of these barriers mark objective Eulerian active barriers to the short-term diffusive transport of the dynamically active vector field. We obtain that in unsteady Beltrami flows, Lagrangian and Eulerian active barriers coincide exactly with purely advective transport barriers bounding observed coherent structures. In more general flows, active barriers can be identified by applying Lagrangian coherent structure (LCS) diagnostics, such as the finite-time Lyapunov exponent and the polar rotation angle, to the appropriate active barrier equation. In comparison to their passive counterparts, these active LCS diagnostics require no significant fluid particle separation and hence provide substantially higher-resolved LCS and Eulerian coherent structure boundaries from temporally shorter velocity data sets. We illustrate these results and their physical interpretation on two-dimensional, homogeneous, isotropic turbulence and on a three-dimensional turbulent channel flow.

JFM classification

Mathematical Foundations: General fluid mechanics Mathematical Foundations: Computational methods Mixing: Turbulent mixing
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 905 , 25 December 2020 , A17
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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Haller et al. Supplementary Material

Computation of Eulerian active barriers to momentum transport from aFTLE for under increasing barrier time s in the turbulent channel flow example

Download Haller et al. Supplementary Material(Video)
Video 2.9 MB

Haller et al. Supplementary Material

Computation of Eulerian active barriers to vorticity transport from aFTLE for under increasing barrier time s in the turbulent channel flow example

Download Haller et al. Supplementary Material(Video)
Video 3 MB

Haller et al. Supplementary Material

A moving cross section of Eulerian, momentum-based aFTLE along the channel, with level surfaces of the lambda2 parameter superimposed

Download Haller et al. Supplementary Material(Video)
Video 36 MB

Haller et al. Supplementary Material

A moving cross section of Eulerian, vorticity-based aFTLE along the channel, with level surfaces of the lambda2 parameter superimposed
Download Haller et al. Supplementary Material(Video)
Video 34.5 MB