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The time response of a linear time-invariant (LTI) system can be divided into transient and steady-state responses. The transient response represents the system's initial reaction to a change in input and diminishes to zero over time. In contrast, the steady-state response is the behavior that persists after the transient effects have faded.
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Capturing Turbulent Dynamics and Statistics in Experiments with Unstable Periodic Orbits.

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Turbulent fluid flows transiently follow unstable periodic orbits (UPOs). These UPOs are statistically significant, with energy input and dissipation rates closely matching those of the turbulent flow.

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Area of Science:

  • Fluid Dynamics
  • Turbulence Theory
  • Chaos Theory

Background:

  • Turbulent flows are complex and often appear chaotic.
  • Understanding the underlying dynamics of turbulence is a major challenge in physics.

Purpose of the Study:

  • To identify and validate the dynamical relevance of unstable periodic orbits (UPOs) in turbulent flows.
  • To investigate the statistical significance of UPOs in quasi-two-dimensional turbulent systems.

Main Methods:

  • Laboratory experiments and numerical simulations of quasi-two-dimensional turbulent flow.
  • Computation and analysis of unstable time-periodic solutions (UPOs).
  • Validation of UPOs by observing transient dynamics in turbulent flows.

Main Results:

  • Clear signatures of unstable time-periodic solutions were observed in moderately turbulent quasi-two-dimensional flow.
  • Turbulent flows were shown to transiently shadow these UPOs in both experiments and simulations.
  • Computed UPOs were statistically significant, with turbulent flows frequently visiting states near these orbits.
  • Average energy input and dissipation rates for turbulent flow and visited UPOs differed by only a few percent.

Conclusions:

  • Unstable periodic orbits play a dynamically relevant role in the behavior of turbulent flows.
  • Turbulent dynamics can be understood as a transient shadowing of these underlying UPOs.
  • The statistical significance and energetic similarity suggest UPOs are fundamental to understanding turbulent energy transfer.