Hostname: page-component-7c8c6479df-ph5wq Total loading time: 0 Render date: 2024-03-28T20:53:36.432Z Has data issue: false hasContentIssue false

Spatiotemporal dynamics in two-dimensional Kolmogorov flow over large domains

Published online by Cambridge University Press:  10 June 2014

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: d.lucas@bris.ac.uk

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.

Type
Papers
Copyright
© 2014 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alonso, A., Sanchez, J. & Net, M. 2000 Transition to temporal chaos in an O(2)-symmetric convective system for low Prandtl numbers. Progr. Theoret. Phys. Suppl. 139, 315324.CrossRefGoogle Scholar
Armbruster, D., Nicolaenko, B., Smaoui, N. & Chossat, P. 1996 Symmetries and dynamics for 2D Navier–Stokes flow. Phys. D 95 (1), 8193.Google Scholar
Arnol’d, V. I. 1991 Kolmogorov’s hydrodynamic attractors. Proc. R. Soc. Lond. A 434, 1922.Google Scholar
Arnold, V. I. & Meshalkin, L. D. 1960 Seminar led by AN Kolmogorov on selected problems of analysis (1958–1959). Usp. Mat. Nauk 15 (247), 2024.Google Scholar
Avila, K., Moxey, D., de Lozar, A., Avila, M., Barkley, D. & Hof, B. 2011 The onset of turbulence in pipe flow. Science 333, 192196.Google Scholar
Balmforth, Neil. J. & Young, Y.-N. 2002 Stratified Kolmogorov flow. J. Fluid Mech. 450, 131168.CrossRefGoogle Scholar
Balmforth, N. J. & Young, Y.-N. 2005 Stratified Kolmogorov flow. Part 2. J. Fluid Mech. 528 (1), 2342.CrossRefGoogle Scholar
Batchaev, A. M. 2012 Laboratory simulation of the Kolmogorov flow on a spherical surface. Izv. Atmos. Ocean. Phys. 48 (6), 657662.CrossRefGoogle Scholar
Batchaev, A. M. & Ponomarev, V. M. 1989 Experimental and theoretical investigation of Kolmogorov flow on a cylindrical surface. Fluid Dyn. 24 (5), 675680.CrossRefGoogle Scholar
Beaume, C., Bergeon, A. & Knobloch, E. 2011 Homoclinic snaking of localized states in doubly diffusive convection. Phys. Fluids 23, 094102.Google Scholar
Berti, S. & Boffetta, G. 2010 Elastic waves and transition to elastic turbulence in a two-dimensional viscoelastic Kolmogorov flow. Phys. Rev. E 82 (3), 036314.Google Scholar
Boffetta, G., Celani, A., Mazzino, A., Puliafito, A. & Vergassola, M. 2005 The viscoelastic Kolmogorov flow: eddy viscosity and linear stability. J. Fluid Mech. 523 (1), 161170.Google Scholar
Bondarenko, N. F., Gak, M. Z. & Dolzhanskii, F. V. 1979 Laboratory and theoretical models of plane periodic flow. Akad. Nauk SSSR 15, 10171026.Google Scholar
Borue, V. & Orszag, S. A. 1996 Numerical study of three-dimensional Kolmogorov flow at high Reynolds numbers. J. Fluid Mech. 306, 293323.CrossRefGoogle Scholar
Burgess, J. M., Bizon, C., McCormick, W. D., Swift, J. B. & Swinney, H. L. 1999 Instability of the Kolmogorov flow in a soap film. Phys. Rev. E 60 (1), 715721.CrossRefGoogle Scholar
Chandler, G. J. & Kerswell, R. R. 2013 Invariant recurrent solutions embedded in a turbulent two-dimensional Kolmogorov flow. J. Fluid Mech. 722, 554595.CrossRefGoogle Scholar
Chapman, C. J. & Proctor, M. R. E. 1980 Nonlinear Rayleigh–Bénard convection between poorly conducting boundaries. J. Fluid Mech. 101 (4), 759782.Google Scholar
Chen, Z.-M. & Price, W. G. 2004 Chaotic behavior of a Galerkin model of a two-dimensional flow. Chaos 14 (4), 10561068.Google Scholar
Cvitanović, P. & Gibson, J. F. 2010 Geometry of the turbulence in wall-bounded shear flows: periodic orbits. Phys. Scr. 142, 014007.Google Scholar
Feudel, F. & Seehafer, N. 1995 Bifurcations and pattern formation in a two-dimensional Navier–Stokes fluid. Phys. Rev. E 52 (4), 35063511.Google Scholar
Fortova, S. V. 2013 Numerical simulation of the three-dimensional Kolmogorov flow in a shear layer. Comput. Math. Math. Phys. 53 (3), 311319.Google Scholar
Fukuta, H. & Murakami, Y. 1998 Side-wall effect on the long-wave instability in Kolmogorov flow. J. Phys. Soc. Japan 67 (5), 15971602.Google Scholar
Gallet, B. & Young, W. R. 2013 A two-dimensional vortex condensate at high Reynolds number. J. Fluid Mech. 715, 359388.CrossRefGoogle Scholar
Gelens, L. & Knobloch, E. 2011 Traveling waves and defects in the complex Swift–Hohenberg equation. Phys. Rev. E 056203.Google Scholar
Gotoh, K. & Yamada, M. 1987 The instability of rhombic cell flows. Fluid Dyn. Res. 1 (3), 165176.Google Scholar
van Hecke, M., Storm, S. & van Saarloos, W. 1999 Sources, sinks and wavenumber selection in coupled CGL equations and experimental implications for counter-propagating wave systems. Physica D 134, 147.Google Scholar
Kawahara, G. & Kida, S. 2001 Periodic motion embedded in plane Couette turbulence: regeneration cycle and burst. J. Fluid Mech. 449, 291300.CrossRefGoogle Scholar
Kazantsev, E. 1998 Unstable periodic orbits and attractor of the barotropic ocean model. Nonlinear Process. Geophys. 5 (4), 193208.Google Scholar
Kim, S. C. & Okamoto, H. 2003 Bifurcations and inviscid limit of rhombic Navier–Stokes flows in tori. IMA J. Appl. Maths 68 (2), 119134.CrossRefGoogle Scholar
Kreilos, T. & Eckhardt, B. 2012 Periodic orbits near onset of chaos in plane Couette flow. Chaos 22 (4), 047505.Google Scholar
Landsberg, A. S. & Knobloch, E. 1991 Direction-reversing travelling waves. Phys. Lett. A 159, 1720.Google Scholar
Lorenz, E. N. 1972 Barotropic instability of Rossby wave motion. J. Atmos. Sci. 29 (2), 258265.2.0.CO;2>CrossRefGoogle Scholar
Manela, A. & Zhang, J. 2012 The effect of compressibility on the stability of wall-bounded Kolmogorov flow. J. Fluid Mech. 694, 2949.Google Scholar
Manfroi, A. J. & Young, W. R. 1999 Slow evolution of zonal jets on the beta plane. J. Atmos. Sci. 56 (5), 784800.Google Scholar
Marchioro, C. 1986 An example of absence of turbulence for any Reynolds number. Commun. Math. Phys. 105 (1), 99106.Google Scholar
Meshalkin, L. D. & Sinai, Y. G. 1961 Investigation of the stability of a stationary solution of a system of equations for the plane movement of an incompressible viscous liquid. J. Appl. Math. Mech. 25, 17001705.CrossRefGoogle Scholar
Musacchio, S. & Boffetta, G. 2014 Turbulent channel without boundaries: the periodic Kolmogorov flow. Phys. Rev. E 89, 023004.Google Scholar
Nepomniashchii, A. A. 1976 On stability of secondary flows of a viscous fluid in unbounded space. Prikl. Mat. Mekh. 40, 886891.Google Scholar
Obukhov, A. M. 1983 Kolmogorov flow and laboratory simulation of it. Russian Math. Surveys 38 (4), 113126.Google Scholar
Okamoto, H. 1996 Nearly singular two-dimensional Kolmogorov flows for large Reynolds numbers. J. Dynam. Differential Equations 8 (2), 203220.CrossRefGoogle Scholar
Okamoto, H. 1998 A study of bifurcation of Kolmogorov flows with an emphasis on the singular limit. Doc. Math. 3, 523532; Proc. Int. Congress Math.Google Scholar
Okamoto, H. & Shōji, M. 1993 Bifurcation diagrams in Kolmogorov’s problem of viscous incompressible fluid on 2D flat tori. Jpn. J. Ind. Appl. Math. 10 (2), 191218.Google Scholar
Platt, N., Sirovich, L. & Fitzmaurice, N. 1991 An investigation of chaotic Kolmogorov flows. Phys. Fluids A 3 (4), 681696.Google Scholar
Roeller, K., Vollmer, J. & Herminghaus, S. 2009 Unstable Kolmogorov flow in granular matter. Chaos 19 (4), 041106.Google Scholar
Rollin, B., Dubief, Y. & Doering, C. R. 2011 Variations on Kolmogorov flow: turbulent energy dissipation and mean flow profiles. J. Fluid Mech. 670, 204213.Google Scholar
Sarris, I. E., Jeanmart, H., Carati, D. & Winckelmans, G. 2007 Box-size dependence and breaking of translational invariance in the velocity statistics computed from three-dimensional turbulent Kolmogorov flows. Phys. Fluids 19 (9), 095101.Google Scholar
She, Z. S. 1987 Metastability and vortex pairing in the Kolmogorov flow. Phys. Lett. A 124 (3), 161164.CrossRefGoogle Scholar
She, Z. S. 1988 Large-scale dynamics and transition to turbulence in the two-dimensional Kolmogorov flow. In Current Trends in Turbulence Research; Proceedings of the Fifth Beersheba International Seminar on Magnetohydrodynamics Flow and Turbulence, pp. 374396. Nice.Google Scholar
Shebalin, J. V. & Woodruff, S. L. 1997 Kolmogorov flow in three-dimensions. Phys. Fluids 9 (1), 164170.Google Scholar
Sivashinsky, G. I. 1985 Weak turbulence in periodic flows. Physica D 17 (2), 243255.Google Scholar
Sommeria, J. 1986 Experimental study of the two-dimensional inverse energy cascade in a square box. J. Fluid Mech. 170, 139168.Google Scholar
Suri, B., Tithof, J., Mitchell, R. Jr, Grigoriev, R. O. & Schatz, M. F. 2013 Velocity profile in a two-layer Kolmogorov-like flow. arXiv:1307.6247v1.Google Scholar
Thess, A. 1992 Instabilities in two-dimensional spatially periodic flows. Part I: Kolmogorov flow. Phys. Fluids A 4 (7), 13851395.Google Scholar
Tsang, Y.-K. & Young, W. R. 2008 Energy-enstrophy stability of -plane Kolmogorov flow with drag. Phys. Fluids 20 (8), 084102.CrossRefGoogle Scholar
Tsang, Y.-K. & Young, W. 2009 Forced-dissipative two-dimensional turbulence: A scaling regime controlled by drag. Phys. Rev. E 79 (4), 045308.Google Scholar
van Veen, L., Kida, S. & Kawahara, G. 2006 Periodic motion representing isotropic turbulence. Jpn. Soc. Fluid Mech. Fluid Dyn. Res. Int. J. 38 (1), 1946.Google Scholar
Viswanath, D. 2007 Recurrent motions within plane Couette turbulence. J. Fluid Mech. 580, 339.Google Scholar
Viswanath, D. 2009 The critical layer in pipe flow at high Reynolds number. Philos. Trans. R. Soc. A 367 (1888), 561576.Google Scholar
Willis, A. P., Cvitanović, P. & Avila, M. 2013 Revealing the state space of turbulent pipe flow by symmetry reduction. J. Fluid Mech. 721, 514540.CrossRefGoogle Scholar
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.

Download Lucas and Kerswell supplementary movie(Image)
Image 2 MB
Supplementary material: PDF

Lucas and Kerswell supplementary movie

Captions

Download Lucas and Kerswell supplementary movie(PDF)
PDF 228.5 KB
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