Transitions of turbulent superstructures in generalized Kolmogorov flow

Letter
Open Access

Transitions of turbulent superstructures in generalized Kolmogorov flow

Cristian C. Lalescu and Michael Wilczek

Phys. Rev. Research 3, L022010 – Published 5 May 2021

Abstract

Self-organized large-scale flow structures occur in a wide range of turbulent flows. Yet, their emergence, dynamics, and interplay with small-scale turbulence are not well understood. Here, we investigate such self-organized turbulent superstructures in three-dimensional turbulent Kolmogorov flow with large-scale drag. Through extensive simulations, we uncover their low-dimensional dynamics featuring transitions between several stable and metastable large-scale structures as a function of the damping parameter. The main dissipation mechanism for the turbulent superstructures is the generation of small-scale turbulence, whose local structure depends strongly on the large-scale flow. Our results elucidate the generic emergence and low-dimensional dynamics of large-scale flow structures in fully developed turbulence and reveal a strong coupling of large- and small-scale flow features.

Received 8 September 2020
Accepted 23 March 2021

DOI:https://doi.org/10.1103/PhysRevResearch.3.L022010

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by the Max Planck Society.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Astrophysical fluid dynamics Flow instability Geophysical fluid dynamics Pattern formation Turbulence

Coherent structures Dynamical systems

Chaos & nonlinear dynamics Navier-Stokes equation

Nonlinear DynamicsFluid DynamicsStatistical Physics & Thermodynamics

Authors & Affiliations

Cristian C. Lalescu ^1,2 and Michael Wilczek ^1,3,*

¹Max Planck Institute for Dynamics and Self-Organization, Am Faßberg 17, 37077 Göttingen, Germany
²Max Planck Computing and Data Facility, Gießenbachstraße 2, 85748, Garching b. München, Germany
³Institute for the Dynamics of Complex Systems, University of Göttingen, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany

^*michael.wilczek@ds.mpg.de

Article Text

Click to Expand

Supplemental Material

Click to Expand

References

Click to Expand

Issue

Vol. 3, Iss. 2 — May - July 2021

Subject Areas

Fluid Dynamics

Reuse & Permissions

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Turbulent Kolmogorov flow on an aspect-ratio-three domain at $R e \approx 7180$ . (a)–(c): The system exhibits different large-scale flow patterns as a function of the damping parameter $μ$ . The visualization shows the $z$ -averaged vorticity $Ω$ along with streamlines of the 2D flow, averaged over a time window in which the large-scale pattern is persistent (see text for details, $T = L / U$ ). (d)–(f): Volume renderings of the instantaneous vorticity field show that the spatial distribution of small-scale vortex structures and the corresponding vorticity component flatness depend on the large-scale flow state (Supplemental Videos 1 and 2).
Reuse & Permissions
Figure 2
The large-scale flow structures exhibit a low-dimensional dynamics. Time series of the large-scale flow states characterized by their relative intensity (a) and their mean wave number (b) show that the large-scale flow states depend on the damping parameter. For $μ = 0$ , the flow organizes into a single vortex pair. For intermediate damping, the flow dynamically alternates between different large-scale flow states, whereas for $μ T = 3.5$ , the three-vortex-pair state is the only stable structure.
Reuse & Permissions
Figure 3
Energy budget of turbulent Kolmogorov flow. (a) Different contributions to the mean total energy. The energy contained in the 2D flow is comparable to the energy in the 3D turbulent fluctuations. (b) Energy budget of the 2D flow. The energy input into the 2D flow $ɛ^{f}$ increases with increased damping. The energy injection rate is almost completely balanced by the transfer of energy $Π$ from the 2D component to the 3D fluctuations. Dissipation through large-scale damping and viscous diffusion are comparably negligible. Error bars span 1 SD computed with respect to temporal variations.
Reuse & Permissions
Figure 4
The energy transfer and turbulent kinetic energy are strongly coupled to the large-scale flow (cf. Fig. 1). (a)–(c): time-averaged energy transfer term. Energy is predominantly transferred to 3D fluctuations at saddle points of the averaged 2D flow (indicated by streamlines). (d)–(f): time- and $z$ -averaged energy of 3D fluctuations. The energy of the fluctuations shows strong similarities to the energy transfer term. The relative standard deviation $ρ$ of the fluctuation energy as a measure of inhomogeneity depends on the large-scale state.
Reuse & Permissions
Figure 5
Cross-correlations between large-scale forcing, energy transfer, and small-scale dissipation. Temporal evolution of energy budget terms (top) and their cross-correlations (bottom) for $R e \approx 7180$ and $μ T = 1.1$ . Cross-correlations are computed to quantify the temporal offsets between the different signals (measured as the location of the maximum correlation). The inset shows the temporal offsets for different values of $μ$ .
Reuse & Permissions

Physical Review Research