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Spatiotemporal dynamics in two-dimensional Kolmogorov flow over large domains

Published online by Cambridge University Press:  10 June 2014

Dan Lucas  and
Rich Kerswell
Dan Lucas*
Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK
Rich Kerswell
Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK
*
†Email address for correspondence: [email protected]
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Abstract

Kolmogorov flow in two dimensions – the two-dimensional (2D) Navier–Stokes equations with a sinusoidal body force – is considered over extended periodic domains to reveal localised spatiotemporal complexity. The flow response mimics the forcing at small forcing amplitudes but beyond a critical value develops a long wavelength instability. The ensuing state is described by a Cahn–Hilliard-type equation and as a result coarsening dynamics is observed for random initial data. After further bifurcations, this regime gives way to multiple attractors, some of which possess spatially localised time dependence. Co-existence of such attractors in a large domain gives rise to interesting collisional dynamics which is captured by a system of 5 (1-space and 1-time) partial differential equations (PDEs) based on a long wavelength limit. The coarsening regime reinstates itself at yet higher forcing amplitudes in the sense that only longest-wavelength solutions remain attractors. Eventually, there is one global longest-wavelength attractor which possesses two localised chaotic regions – a kink and antikink – which connect two steady one-dimensional (1D) flow regions of essentially half the domain width each. The wealth of spatiotemporal complexity uncovered presents a bountiful arena in which to study the existence of simple invariant localised solutions which presumably underpin all of the observed behaviour.

JFM classification

Instability: Instability Nonlinear Dynamical Systems: Nonlinear Dynamical Systems Nonlinear Dynamical Systems: Pattern formation
Type
Papers
Information
Journal of Fluid Mechanics , Volume 750 , 10 July 2014 , pp. 518 - 554
Copyright
© 2014 Cambridge University Press 

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Supplementary material: Image

Lucas and Kerswell supplementary movie

This movie shows the vorticity field for a DNS of the chaotic kink-antikink pair at Re=70 as discussed in section 4. Total integration is T=500 with timestep dt=0.005 and individual frames are separated by 5 time units. The dynamics are mostly limited to the central `eyes' of the kink and antikink which oscillate aperiodically. This is in stark contrast to the non-localised chaos observed for unit aspect ratio.

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Supplementary material: PDF

Lucas and Kerswell supplementary movie

Captions

Download Lucas and Kerswell supplementary movie(PDF)
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Supplementary material: Image

Lucas and Kerswell supplementary movie

This movie shows the vorticity field for the solution P1 at Re = 20 and corresponds directly with figure 16. Total integration is T = 20.7 with timestep dt = 0.05 and individual frames are separated by 0.2 time units. This movie clearly shows the standing wave-like motion of the central vorticity distribution, while the outer kink (right) and antikink (left) remain steady.

Download Lucas and Kerswell supplementary movie(Image)
Image 2.4 MB
Supplementary material: Image

Lucas and Kerswell supplementary movie

This movie shows the vorticity field for the chaotic repeller version of P1 at Re = 24.1 as discussed in section 4.5 and corresponding directly with figures 21 and 22. Total integration is T = 105 with timestep dt = 0.05 and individual frames are separated by 200 time units. This movie indicates the chaotic motions of the central region, eventually the motions result in a catastrophic collision with the antikink

Download Lucas and Kerswell supplementary movie(Image)
Image 2.1 MB
Supplementary material: Image

Lucas and Kerswell supplementary movie

This movie shows the vorticity field for the solution P2 at Re = 20 and corresponds directly with figure 22. Total integration is T = 21.4 with timestep dt = 0.05 and individual frames are separated by 0.2 time units. Here we see the largely the same behaviour as for P1, only now with two periodic regions.

Download Lucas and Kerswell supplementary movie(Image)
Image 2.9 MB
Supplementary material: Image

Lucas and Kerswell supplementary movie

This movie shows the vorticity field for the chaotic saddle at Re = 20.75 corresponding with figure 23. Total integration is T = 105 with timestep dt = 0.05 and individual frames are separated by 1000 time units. Striking in this movie is the uniform translation of the flow while two vortical patches oscillate chaotically between two kinks. Eventually these collide and in doing so one is annihilated leaving the solution P1.

Download Lucas and Kerswell supplementary movie(Image)
Image 2.6 MB
Supplementary material: Image

Lucas and Kerswell supplementary movie

This movie shows the vorticity field for aspect ratio one eighth at Re = 22 as discussed in section 5 and corresponding directly with figure 24. Total integration is T = 105 with timestep dt = 0.05 and individual frames are separated by 500 time units. Given a randomised initial condition this movie demonstrates the emergence of stable propagating kink-antikink bound states. The flow rapidly localises from the initial conditions to an assortment of kinks and antikinks and an isolated, translating P1-like structure in the left had portion of the domain.

Download Lucas and Kerswell supplementary movie(Image)
Image 2.6 MB
Supplementary material: Image

Lucas and Kerswell supplementary movie

This movie shows the vorticity field aspect ratio one eighth at Re = 19 as discussed in section 5 and corresponding directly with figure 26. Total integration is T = 105 with timestep dt = 0.05 and individual frames are separated by 500 time units. Given an initial condition comprised of kink-antikink travelling waves, this movie demonstrates several of the collisional behaviours we encounter. Beginning with an elastic swapping collision, then two rebounding collisions and finally a merging collision which results in 2 kinks and 1 antikink forming a localised chaotic oscillatory structure.

Download Lucas and Kerswell supplementary movie(Image)
Image 2.4 MB