Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-13T08:49:55.112Z Has data issue: false hasContentIssue false
  • English
  • Français

Invariant recurrent solutions embedded in a turbulent two-dimensional Kolmogorov flow

Published online by Cambridge University Press:  28 March 2013

Gary J. Chandler  and
Rich R. Kerswell
Gary J. Chandler
Affiliation:
School of Mathematics, University of Bristol, University walk, Bristol, UK
Rich R. Kerswell*
Affiliation:
School of Mathematics, University of Bristol, University walk, Bristol, UK
*
†Email address for correspondence: R.R.Kerswell@bristol.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider long-time simulations of two-dimensional turbulence body forced by $\sin 4y\hat {\boldsymbol{x}} $ on the torus $(x, y)\in \mathop{[0, 2\mathrm{\pi} ] }\nolimits ^{2} $ with the purpose of extracting simple invariant sets or ‘exact recurrent flows’ embedded in this turbulence. Each recurrent flow represents a sustained closed cycle of dynamical processes which underpins the turbulence. These are used to reconstruct the turbulence statistics using periodic orbit theory. The approach is found to be reasonably successful at a low value of the forcing where the flow is close to but not fully in its asymptotic (strongly) turbulent regime. Here, a total of 50 recurrent flows are found with the majority buried in the part of phase space most populated by the turbulence giving rise to a good reproduction of the energy and dissipation p.d.f. However, at higher forcing amplitudes now in the asymptotic turbulent regime, the generated turbulence data set proves insufficiently long to yield enough recurrent flows to make viable predictions. Despite this, the general approach seems promising providing enough simulation data is available since it is open to extensive automation and naturally generates dynamically important exact solutions for the flow.

JFM classification

Nonlinear Dynamical Systems: Chaos Nonlinear Dynamical Systems: Nonlinear Dynamical Systems Turbulent Flows: Turbulence theory
Type
Papers
Information
Journal of Fluid Mechanics , Volume 722 , 10 May 2013 , pp. 554 - 595
Copyright
©2013 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

Armbruster, D., Nicolaenko, B., Smaoui, N. & Chossat, P. 1996 Symmetries and dynamics of 2-D Navier–Stokes flow. Physica D 95, 81–93.Google Scholar
Arnol’d, V. I. & Meshalkin, L. D. 1960 The seminar of A.N. Kolmogorov on selected topics in analysis (1958–1959). Usp. Mat. Nauk 15, 247–250.Google Scholar
Artuso, R., Aurell, E. & Cvitanović, P. 1990a Recycling of strange sets: I cycle expansions. Nonlinearity 3, 325–359.Google Scholar
Artuso, R., Aurell, E. & Cvitanović, P. 1990b Recycling of strange sets: II applications. Nonlinearity 3, 361–386.CrossRefGoogle Scholar
Auerbach, D., Cvitanović, P., Eckmann, J.-P., Gunaratne, G. & Procaccia, I. 1987 Exploring chaotic motion through periodic orbits. Phys. Rev. Lett. 58, 2387–2389.CrossRefGoogle ScholarPubMed
Bartello, P. & Warn, T. 1996 Self-similarity of decaying two-dimensional turbulence. J. Fluid Mech. 326, 357–372.CrossRefGoogle Scholar
Boffetta, G. & Musacchio, S. 2010 Evidence for a double cascade scenario in two-dimensional turbulence. Phys. Rev. E 82, 016307.Google Scholar
Boghosian, B. M., Fazendeiro, L. M., Lätt, J., Tang, H. & Coveney, P. V. 2011 New variational principles for locating periodic orbits of differential equations. Phil. Trans. R. Soc. A 369, 2211–2218.Google Scholar
Bondarenko, N. F., Gak, M. Z. & Dolzhanskii, F. V. 1979 Laboratory and theoretical models of plane periodic flow. Izv. Acad. Sci. USSR Atmos. Ocean. Phys. 15, 711–716.Google Scholar
Borue, V. & Orszag, S. A. 1996 Numerical study of three-dimensional Kolmogorov flow at high Reynolds numbers. J. Fluid Mech. 306, 293–323.Google 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, 715–721.Google Scholar
Christiansen, F., Cvitanović, P. & Putkaradze, V. 1997 Spatiotemporal chaos in terms of unstable recurrent patterns. Nonlinearity 10, 55–70.Google Scholar
Cvitanović, P. 1988 Invariant measurement of strange sets in terms of cycles. Phys. Rev. Lett. 61, 2729–2732.Google Scholar
Cvitanović, P. 1992 Periodic orbit theory in classical and quantum mechanics. Chaos 2, 1.Google Scholar
Cvitanović, P. 1995 Dynamical averaging in terms of periodic orbits. Physica D 83, 109–123.Google Scholar
Cvitanović, P. 2007 Continuous symmetry reduced trace formulas ChaosBook.org/~predrag/trace.pdf.Google Scholar
Cvitanović, P. 2012 Continuous symmetry reduced trace formulas. Preprint.Google Scholar
Cvitanović, P., Artuso, R., Dahlqvist, P., Mainieri, R., Tanner, G., Vattay, G., Whelan, N. & Wirzba, A. 2013 Classical and Quantum Chaos webbook available at http://chaosbook.org.Google Scholar
Cvitanović, P., Davidchack, R. & Siminos, E. 2010 On the state space geometry of the Kuramoto-Sivashinsky flow in a periodic domain. SIAM J. Appl. Dyn. Sys. 9, 1–33.CrossRefGoogle Scholar
Cvitanović, P. & Gibson, J. F. 2010 Geometry of turbulence in wall-bounded shear flows: periodic orbits. Phys. Scr. 142, 014007.Google Scholar
Dahlqvist, P. 1994 Determination of resonance spectra for bound cahotic systems. J. Phys. A: Math. Gen. 27, 763–785.Google Scholar
Dahlqvist, P. & Russberg, G. 1991 Periodic orbit quantization of bound chaotic systems. J. Phys. A 24, 4763–4778.Google Scholar
Dennis, J. E. & Schnabel, R. B. 1996 Numerical Methods for Unconstrained Optimisation and Nonlinear equations. In SIAM Classics. SIAM.Google Scholar
Dettmann, C. P. & Morriss, G. P. 1997 Stability ordering of cycle expansions. Phys. Rev. Lett. 78, 4201–4204.CrossRefGoogle Scholar
Duguet, Y., Pringle, C. C. T. & Kerswell, R. R. 2008 Relative periodic orbits in transitional pipe flow. Phys. Fluids 20, 114102.Google Scholar
Eckhardt, B., Faisst, H., Schmiegel, A. & Schumacher, J. 2002 Turbulence transition in shear flows. In Advances in Turbulence IX: Proceedings 9th European Turbulence Conference (Southampton) (ed. I. P. Castro et al.), p. 701, CISME.Google Scholar
Eckhardt, B., Schneider, T. M., Hof, B. & Westerweel, J. 2007 Turbulence transition in pipe flow. Annu. Rev. Fluid Mech. 39, 447–468.CrossRefGoogle Scholar
Eckmann, P. & Ruelle, D. 1985 Ergodic theory of chaotic systems. Rev. Mod. Phys. 57, 617–656.Google Scholar
Fazendeiro, L., Boghosian, B. M., Coveney, P. V. & Lätt, J. 2010 Unstable periodic orbits in weak turbulence. J. Comput. Sci. 1, 13–23.Google Scholar
Gaspard, P. 1997 Chaos Scattering and Statistical Mechanics. CUP.Google Scholar
Gibson, J. F., Halcrow, J. & Cvitanović, P. 2008 Visualizing the geometry of state space in plane Couette flow. J. Fluid Mech. 611, 107–130.Google Scholar
Gibson, J. F., Halcrow, J. & Cvitanović, P. 2009 Equilibrium and travelling-wave solutions of plane Couette flow. J. Fluid Mech. 638, 243–266.Google Scholar
Gotoh, K. & Yamada, M. 1986 The instability of rhombic cell flows. Fluid Dyn. Res. 1, 165–176.Google Scholar
Gutzwiller, 1990 Chaos in Classical and Quantum Mechanics. Springer.Google Scholar
Halcrow, J., Gibson, J. F., Cvitanović, P. & Viswanath, D. 2009 Heteroclinic connections in plane Couette flow. J. Fluid Mech. 621, 365–376.CrossRefGoogle Scholar
Hamilton, J. M., Kim, J. & Waleffe, F. 1995 Regeneration mechanism of near-wall turbulence structures. J. Fluid Mech. 287, 317–348.Google Scholar
Hof, B., van Doorne, C. W. H., Westerweel, J., Nieuwstadt, F. T. M., Faisst, H., Eckhardt, B., Wedin, H., Kerswell, R. R. & Waleffe, F. 2004 Experimental observation of nonlinear travelling waves in turbulent pipe flow. Science 305, 1594–1598.Google Scholar
Holmes, P., Lumley, J. L. & Berkooz, G. 1996 Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press.Google Scholar
Hopf, E. 1948 A mathematical example displaying features of turbulence. Commun. Appl. Maths 1, 303–322.Google Scholar
Kawahara, G. & Kida, S. 2001 Periodic motion embedded in plane Couette turbulence: regenerative cycle and burst. J. Fluid Mech. 449, 291–300.Google Scholar
Kawahara, G., Uhlmann, M. & van Veen, L. 2012 The significance of simple invariant solutions in turbulent flows. Annu. Rev. Fluid Mech. 44, 203–225.CrossRefGoogle Scholar
Kazantsev, E. 1998 Unstable periodic orbits and attractor of the barotropic ocean model. Nonlinear Process. Geophys. 5, 193–208.Google Scholar
Kazantsev, E. 2001 Sensitivity of the attractor of the barotropic ocean model to external influences: approach by unstable periodic orbits. Nonlinear Process. Geophys. 8, 281–300.Google Scholar
Kerswell, R. R. 2005 Recent Progress in understanding the transition to turbulence in a pipe. Nonlinearity 18, R17–R44.Google Scholar
Kim, S.-C. & Okamoto, H. 2003 Bifurcations and inviscid limit of rhombic Navier–Stokes flows in tori. IMA J. Appl. Maths 68, 119–134.Google Scholar
Kreilos, T. & Eckhardt, B. 2012 Periodic orbits near onset of chaos in plane Couette flow. Chaos 22, 047505.Google Scholar
Lan, Y. 2010 Cycle expansions: from maps to turbulence. Commun. Nonlinear Sci. Numer. Simul. 15, 502–526.Google Scholar
Lan, Y. & Cvitanović, P. 2004 Variational method for finding periodic orbits in a general flow. Phys. Rev. E 69, 016217.Google Scholar
Lan, Y. & Cvitanović, P. 2008 Unstable recurrent patterns in Kuramoto-Sivashinsky dynamics. Phys. Rev. E 78, 026208.CrossRefGoogle ScholarPubMed
Lopez, V., Boyland, P., Heath, M. T. & Moser, R. D. 2005 Relative periodic solutions of the complex Ginzburg–Landau equation. SIAM J. Appl. Dyn. Syst. 4, 1042–1075.Google Scholar
MacKay, R. S. & Miess, J. D. 1987 Hamiltonian Dynamical Systems. Adam Hilger.Google Scholar
Marchioro, C. 1986 An example of absence of turbulence for any Reynolds number. Commun. Math. Phys. 105, 99–106.Google Scholar
Meshalkin, L. D. & Sinai, Ya. G. 1961 Investigation of stability of a steady-state solution of a system of equations for the plane motion of an incompressible viscous liquid. Prikl. Mat. Mekh. 25, 1140–1143.Google Scholar
Obukhov, A. M. 1983 Kolmogorov flow and laboratoty simulation of it. Usp. Mat. Nauk 38, 101–111.Google Scholar
Okamoto, H. & Shoji, M. 1993 Bifurcation diagrams in Kolmogorov’s problem of viscous incompressible fluid on 2-D flat tori. Japan. J. Indust. Appl. Math. 10, 191–218.Google Scholar
Panton, R.L. (Ed.) 1997 Self-Sustaining Mechanisms of Wall Turbulence. Computational Mechanics.Google Scholar
Platt, N., Sirovich, L. & Fitzmaurice, N. 1991 An investigation of chaotic Kolmogorov flows. Phys. Fluids 3, 681–696.Google Scholar
Poincaré, H. 1892 Les méthodes nouvelles de la méchanique céleste. Guthier-Villars.Google Scholar
Robinson, S. K. 1991 Coherent motions in the turbulent boundary layer. Annu. Rev. Fluid Mech. 23, 601–639.Google Scholar
Ruelle, D. 1978 Statistical Mechanics, Thermodynamic Formalism. Addison-Wesley.Google Scholar
Saad, Y. & Schultz, M. H. 1986 GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7, 856–869.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, 095101.CrossRefGoogle Scholar
She, Z. S. 1988 Large-scale dynamics and transition to turbulence in the two-dimensional Kolmogorov flow. In Proceedings on Current Trends in Turbulence Research (ed. Branover, H., Mond, M. & Unger, Y.), vol. 117, pp. 374–396. American Institute of Aeronautics and Astronautics.Google Scholar
Shebalin, J. V. & Woodruff, S. L. 1997 Kolmogorov flow in three dimensions. Phys. Fluids 9, 164–170.CrossRefGoogle Scholar
Sommeria, J. 1986 Experimental study of the two-dimensional inverse energy cascade in a square box. J. Fluid Mech. 170, 139–168.Google Scholar
Trefethen, L. N. & Bau, D. 1997 Numerical Linear Algebra. SIAM.Google Scholar
Tsang, Y.-K. & Young, W. R. 2008 Energy-enstrophy stability of beta-plane Kolmogorov flow with drag. Phys. Fluids 20, 084102.Google Scholar
van Veen, L., Kawahara, G. & Kida, S. 2006 Periodic motion representing isotropic turbulence. Fluid Dyn. Res. 38, 19–46.Google Scholar
Viswanath, D. 2007 Recurrent motions within plane Couette turbulence. J. Fluid Mech. 580, 339–358.Google Scholar
Viswanath, D. 2009 The critical layer in pipe flow at high Reynolds number. Phil. Trans R. Soc. A 367, 561–576.Google Scholar
Waleffe, F. 1997 On a self-sustaining process in shear flows. Phys. Fluids 9, 884–900.CrossRefGoogle Scholar
Waleffe, F. 1998 Three-dimensional coherent states in plane shear flows. Phys. Rev. Lett. 81, 4140–4143.Google Scholar
Waleffe, F. 2001 Exact coherent structures in channel flow. J. Fluid Mech. 435, 93–102.Google Scholar
Waleffe, F. 2003 Homotopy of exact coherent structures in plane shear flows. Phys. Fluids 15, 1517–1534.Google Scholar
Willis, A. P., Cvitanović, P. & Avila, M. 2013 Revealing the state space of turbulent pipe flow by symmetry reduction. Preprint, arXiv: 1203.3701.Google Scholar
Zoldi, S. & Greenside, H. S. 1998 Spatially localised unstable periodic orbits of a high-dimensional chaotic system. Phys. Rev. E 57, R2511.Google Scholar
Supplementary material: Image

Chandler and Kerswell supplementary movie

This is a video of the unstable periodic orbit P4 (as listed in Table 3 of Chandler & Kerswell, 2013) found from DNS carried out at Re=100.

Download Chandler and Kerswell supplementary movie(Image)
Image 176.3 KB
Supplementary material: Image

Chandler and Kerswell supplementary movie

This is a video of the DNS segment at Re=100 which suggested the presence of the periodic orbit P4 and from which the initial velocity field guess was taken to converge P4 as an exact recurrent flow

Download Chandler and Kerswell supplementary movie(Image)
Image 2.7 MB