Revealing the state space of turbulent pipe flow by symmetry reduction

Journal of Fluid Mechanics

Volumn 721, Issue , 2013, Pages 514-540

  Willis, A P ^a   Cvitanovic P ^b,c   Avila, M ^c,d  

^a UNIVERSITY OF SHEFFIELD   (United Kingdom)

^b GEORGIA INSTITUTE OF TECHNOLOGY   (United States)

^c MAX PLANCK INSTITUTE FOR DYNAMICS AND SELF ORGANIZATION   (Germany)

^d UNIVERSITY OF ERLANGEN NUREMBERG   (Germany)

Author keywords

Instability; Turbulence modelling; Turbulent flows

Indexed keywords

PIPE FLOW; PLASMA STABILITY; REYNOLDS NUMBER; TURBULENCE; TURBULENCE MODELS; TURBULENT FLOW; WAVE TRANSMISSION;

AZIMUTHAL ROTATION; DIMENSIONAL SUBSPACE; INFINITE DIMENSIONAL; RELATIVE PERIODIC ORBITS; SYMMETRY REDUCTION; TRAVELLING WAVE SOLUTION; TURBULENT DYNAMICS; TURBULENT PIPE FLOW;

SHEAR FLOW;

FLOW STABILITY; NUMERICAL MODEL; PIPE FLOW; REYNOLDS NUMBER; SHEAR FLOW; SYMMETRY; TURBULENCE; TURBULENT FLOW;

EID: 84878394941     PISSN: 00221120     EISSN: 14697645     Source Type: Journal    
DOI: 10.1017/jfm.2013.75     Document Type: Article

References (53)

1. AUERBACH, D., CVITANOVIĆ, P., ECKMANN, J.-P., GUNARATNE, G. & PROCACCIA, I. 1987 Exploring chaotic motion through periodic orbits. Phys. Rev. Lett. 58, 23.
2. AVILA, K., MOXEY, D., DE LOZAR, A., AVILA, M., BARKLEY, D. & HOF, B. 2011 The onset of turbulence in pipe flow. Science 333, 192-196.
3. AVILA, M., WILLIS, A. P. & HOF, B. 2010 On the transient nature of localized pipe flow turbulence. J. Fluid Mech. 646, 127-136.
4. BEYN, W.-J. & THÜMMLER, V. 2004 Freezing solutions of equivariant evolution equations. SIAM J. Appl. Dyn. Syst. 3, 85-116.
5. CARTAN, É. 1935 La méthode du repère mobile, la théorie des groupes continus, et les espaces géné ralisés. In Exposés de Géométrie, vol. 5. Hermann.
6. CHRISTIANSEN, F., CVITANOVIĆ, P. & PUTKARADZE, V. 1997 Spatio-temporal chaos in terms of unstable recurrent patterns. Nonlinearity 10, 55-70.
7. CVITANOVIĆ, P. 2007 Continuous symmetry reduced trace formulas. www.cns.gatech.edu/predrag/Cvio7.pdf.
8. CVITANOVIĆ, P., ARTUSO, R., MAINIERI, R., TANNER, G. & VATTAY, G. 2012 Chaos: Classical and Quantum. Niels Bohr Inst., www.chaosbook.org.
9. CVITANOVIĆ, P., BORRERO-ECHEVERRY, D., CARROLL, K., ROBBINS, B. & SIMINOS, E. 2012 Cartography of high-dimensional flows: a visual guide to sections and slices. Chaos 22, 047506.
10. CVITANOVIĆ, P., DAVIDCHACK, R. L. & SIMINOS, E. 2009 On the state space geometry of the Kuramoto-Sivashinsky flow in a periodic domain. SIAM J. Appl. Dyn. Syst. 9, 1-33.
11. CVITANOVIĆ, P. & GIBSON, J. F. 2010 Geometry of turbulence in wall-bounded shear flows: periodic orbits. Phys. Scr. T 142, 014007.
12. DUGUET, Y., PRINGLE, C. C. T. & KERSWELL, R. R. 2008a Relative periodic orbits in transitional pipe flow. Phys. Fluids 20, 114102.
13. DUGUET, Y., WILLIS, A. P. & KERSWELL, R. R. 2008b Transition in pipe flow: the saddle structure on the boundary of turbulence. J. Fluid Mech. 613, 255-274.
14. FAISST, H. & ECKHARDT, B. 2003 Traveling waves in pipe flow. Phys. Rev. Lett. 91, 224502.
15. FELS, M. & OLVER, P. J. 1998 Moving coframes: I. A practical algorithm. Acta Appl. Math. 51, 161-213.
16. FELS, M. & OLVER, P. J. 1999 Moving coframes: II. Regularization and theoretical foundations. Acta Appl. Math. 55, 127-208.
17. FRISCH, U. 1996 Turbulence. Cambridge University Press.
18. FROEHLICH, S. & CVITANOVIĆ, P. 2011 Reduction of continuous symmetries of chaotic flows by the method of slices. Commun. Nonlinear Sci. Numer. Simul. 17, 2074-2084.
19. GIBSON, J. F., HALCROW, J. & CVITANOVIĆ, P. 2008 Visualizing the geometry of state space in plane Couette flow. J. Fluid Mech. 611, 107-130.
20. GIBSON, J. F., HALCROW, J. & CVITANOVIĆ, P. 2009 Equilibrium and travelling-wave solutions of plane Couette flow. J. Fluid Mech. 638, 1-24.
21. GREENE, J. M. & KIM, J.-S. 1988 The steady states of the Kuramoto-Sivashinsky equation. Physica D 33, 99-120.
22. HALCROW, J., GIBSON, J. F., CVITANOVIĆ, P. & VISWANATH, D. 2009 Heteroclinic connections in plane Couette flow. J. Fluid Mech. 621, 365-376.
23. HAMILTON, J. M., KIM, J. & WALEFFE, F. 1995 Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech. 287, 317-348.
24. HOF, B., DE LOZAR, A., KUIK, D. J. & WESTERWEEL, J. 2008 Repeller or attractor? Selecting the dynamical model for the onset of turbulence in pipe flow. Phys. Rev. Lett. 101, 214501.
25. 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.
26. HOPF, E. 1948 A mathematical example displaying features of turbulence. Comm. Pure Appl. Maths 1, 303-322.
27. HUYGENS, C. 1967 L'Horloge ápendule de 1673. Swets & Zeitlinger.
28. JIMÉNEZ, J. & MOIN, P. 1991 The minimal flow unit in near-wall turbulence. J. Fluid Mech. 225, 213-240.
29. KAWAHARA, G. & KIDA, S. 2001 Periodic motion embedded in plane Couette turbulence: regeneration cycle and burst. J. Fluid Mech. 449, 291-300.
30. KAWAHARA, G., UHLMANN, M. & VAN VEEN, L. 2012 The significance of simple invariant solutions in turbulent flows. Annu. Rev. Fluid Mech 44, 203-225.
31. KERSWELL, R. R. & TUTTY, O. R. 2007 Recurrence of travelling waves in transitional pipe flow. J. Fluid Mech. 584, 69-102.
32. KIM, K. C. & ADRIAN, R. J. 1999 Very large-scale motion in the outer layer. Phys. Fluids 11, 417-422.
33. KLINE, S. J., REYNOLDS, W. C., SCHRAUB, F. A. & RUNSTADLER, P. W. 1967 The structure of turbulent boundary layers. J. Fluid Mech. 30, 741-773.
34. KREILOS, T. & ECKHARDT, B. 2012 Periodic orbits near onset of chaos in plane Couette flow. Chaos 22, 047505.
35. LAN, Y. & CVITANOVIĆ, P. 2008 Unstable recurrent patterns in Kuramoto-Sivashinsky dynamics. Phys. Rev. E 78, 026208.
36. LOMBARDI, M., CAULFIELD, C. P., COSSU, C., PESCI, A. I. & GOLDSTEIN, R. E. 2011 Growth and instability of a laminar plume in a strongly stratified environment. J. Fluid Mech. 671, 184-206.
37. MELLIBOVSKY, F. & ECKHARDT, B. 2012 From travelling waves to mild chaos: A supercritical bifurcation cascade in pipe flow. J. Fluid Mech. 709, 149-190.
38. MELLIBOVSKY, F. & ECKHARDT, B. 2011 Takens-Bogdanov bifurcation of travelling-wave solutions in pipe flow. J. Fluid Mech. 670, 96-129.
39. MULLIN, T. & KERSWELL, R. R. 2005 Non-uniqueness of Solutions to the Navier-Stokes Equations and their Connection with Laminar-Turbulent Transition. Kluwer.
40. OLVER, P. J. 1999 Classical Invariant Theory. Cambridge University Press.
41. POINCARÉ, H. 1896 Sur les solutions périodiques et le principe de moindre action. C. R. Acad. Sci. Paris 123, 915-918.
42. PRINGLE, C. C. T., DUGUET, Y. & KERSWELL, R. R. 2009 Highly symmetric travelling waves in pipe flow. Phil. Trans. R. Soc. A 367, 457-472.
43. PRINGLE, C. C. T. & KERSWELL, R. R. 2007 Asymmetric, helical, and mirror-symmetric travelling waves in pipe flow. Phys. Rev. Lett. 99, 074502.
44. RAND, D. 1982 Dynamics and symmetry-predictions for modulated waves in rotating fluids. Arch. Rational Mech. Anal. 79, 1-3.
45. RECKE, L., SAMOILENKO, A., TKACHENKO, V. & YANCHUK, S. 2011 Frequency locking by external forcing in systems with rotational symmetry. Arxiv preprint. arXiv:1108.5990.
46. ROWLEY, C. W. & MARSDEN, J. E. 2000 Reconstruction equations and the Karhunen-Loéve expansion for systems with symmetry. Physica D 142, 1-19.
47. SCHNEIDER, T. M., ECKHARDT, B. & VOLLMER, J. 2007 Statistical analysis of coherent structures in transitional pipe flow. Phys. Rev. E 75, 066313.
48. SCHNEIDER, T. M., GIBSON, J. F. & BURKE, J. 2010 Snakes and ladders: localized solutions of plane Couette flow. Phys. Rev. Lett. 104, 104501.
49. SIMINOS, E. & CVITANOVIĆ, P. 2011 Continuous symmetry reduction and return maps for high-dimensional flows. Physica D 240, 187-198.
50. TEMPELMANN, D., HANIFI, A. & HENNINGSON, D. S. 2010 Spatial optimal growth in three-dimensional boundary layers. J. Fluid Mech. 646, 5-37.
51. VISWANATH, D. 2007 Recurrent motions within plane Couette turbulence. J. Fluid Mech. 580, 339-358.
52. WEDIN, H. & KERSWELL, R. R. 2004 Exact coherent structures in pipe flow: traveling wave solutions. J. Fluid Mech. 508, 333-371.
53. WILLIS, A. P. & KERSWELL, R. R. 2008 Coherent structures in localised and global pipe turbulence. Phys. Rev. Lett. 100, 124501.