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An autonomous dynamical system captures all LCSs in three-dimensional unsteady flows.

David Oettinger1, George Haller1

  • 1Institute of Mechanical Systems, ETH Zürich, Leonhardstrasse 21, 8092 Zürich, Switzerland.

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Summary

Researchers discovered that all Lagrangian coherent structures (LCSs) originate from a single dynamical system. This finding simplifies the detection of both hyperbolic and elliptic LCSs in complex fluid flows.

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Area of Science:

  • Fluid dynamics
  • Dynamical systems theory
  • Chaos theory

Background:

  • Lagrangian coherent structures (LCSs) are key to understanding tracer patterns in time-dependent flows.
  • Previous methods identified elliptic and hyperbolic LCSs using different variational principles and equations.
  • A unified approach for detecting LCSs in complex flows is needed.

Purpose of the Study:

  • To demonstrate that all variational LCSs in 3D unsteady flows share a common origin.
  • To introduce a novel method for detecting both elliptic and hyperbolic LCSs from a single computation.
  • To unify the study of LCSs within a single dynamical systems framework.

Main Methods:

  • Identified the intermediate eigenvector field (ξ₂(x₀)) of the Cauchy-Green strain tensor as the generator of a unified dynamical system.
  • Utilized classical dynamical systems methods, such as Poincaré maps, to analyze the ξ₂-system.
  • Applied the method to both steady and time-aperiodic flows to detect LCSs.

Main Results:

  • Demonstrated that the initial positions of all variational LCSs are invariant manifolds of the ξ₂-system.
  • Showcased the detection of both hyperbolic and elliptic LCSs from a single ξ₂-system computation.
  • Validated the approach on diverse flow types, including steady and time-aperiodic examples.

Conclusions:

  • The ξ₂-system provides a unified framework for understanding and detecting all types of LCSs in unsteady 3D flows.
  • This approach simplifies LCS detection by allowing the use of established dynamical systems tools.
  • The findings offer a new perspective on the geometry and dynamics of fluid flows.