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Transition in pipe flow: The saddle structure on the boundary of turbulence. (English) Zbl 1151.76495
Summary: The laminar-turbulent boundary \(\varSigma \) is the set separating initial conditions which relaminarize uneventfully from those which become turbulent. Phase space trajectories on this hypersurface in cylindrical pipe flow appear to be chaotic and show recurring evidence of coherent structures. A general numerical technique is developed for recognizing approaches to these structures and then for identifying the exact coherent solutions themselves. Numerical evidence is presented which suggests that trajectories on \(\varSigma \) are organized around only a few travelling waves and their heteroclinic connections. If the flow is suitably constrained to a subspace with a discrete rotational symmetry, it is possible to find locally attracting travelling waves embedded within \(\varSigma \). Four new types of travelling waves were found using this approach.

MSC:
76F06 Transition to turbulence
Software:
channelflow
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References:
[1] DOI: 10.1017/S0022112001004189 · Zbl 0987.76034 · doi:10.1017/S0022112001004189
[2] DOI: 10.1063/1.1566753 · Zbl 1186.76556 · doi:10.1063/1.1566753
[3] DOI: 10.1126/science.1100393 · doi:10.1126/science.1100393
[4] DOI: 10.1063/1.869185 · doi:10.1063/1.869185
[5] DOI: 10.1017/S0022112095000978 · Zbl 0867.76032 · doi:10.1017/S0022112095000978
[6] DOI: 10.1002/andp.18391220304 · doi:10.1002/andp.18391220304
[7] DOI: 10.1017/S0022112007005459 · Zbl 1175.76074 · doi:10.1017/S0022112007005459
[8] DOI: 10.1017/S0022112004008134 · Zbl 1116.76362 · doi:10.1017/S0022112004008134
[9] Toh, Proc. IUTAM Symp. on Geometry and Statistics of Turbulence (1999)
[10] DOI: 10.1103/PhysRevLett.91.224502 · doi:10.1103/PhysRevLett.91.224502
[11] DOI: 10.1103/PhysRevLett.96.174101 · doi:10.1103/PhysRevLett.96.174101
[12] DOI: 10.1137/0917003 · Zbl 0845.65021 · doi:10.1137/0917003
[13] Dennis, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (1995)
[14] DOI: 10.1103/PhysRevLett.99.034502 · doi:10.1103/PhysRevLett.99.034502
[15] DOI: 10.1137/0911026 · Zbl 0708.65049 · doi:10.1137/0911026
[16] Schneider, Phys. Rev. 75 pp 066313– (2007) · doi:10.1103/PhysRevB.75.174517
[17] DOI: 10.1137/0907058 · Zbl 0599.65018 · doi:10.1137/0907058
[18] DOI: 10.1098/rstl.1883.0029 · JFM 16.0845.02 · doi:10.1098/rstl.1883.0029
[19] DOI: 10.1063/1.1804549 · Zbl 1187.76429 · doi:10.1063/1.1804549
[20] DOI: 10.1103/PhysRevLett.99.074502 · doi:10.1103/PhysRevLett.99.074502
[21] Poiseuille, CR Acad. Sci. 11 pp 961– (1840)
[22] Pfenniger, Boundary Layer and Flow Control pp 970– (1961)
[23] DOI: 10.1017/S0022112007006398 · Zbl 1114.76304 · doi:10.1017/S0022112007006398
[24] DOI: 10.1103/PhysRevLett.96.094501 · doi:10.1103/PhysRevLett.96.094501
[25] DOI: 10.1063/1.2781596 · Zbl 1182.76562 · doi:10.1063/1.2781596
[26] DOI: 10.1017/S0022112090000829 · doi:10.1017/S0022112090000829
[27] Mellibovsky, Proc. 15th Intl Couette?Taylor Workshop (2007)
[28] Marqués, Phys. Fluids pp 729– (1990) · Zbl 0703.76028 · doi:10.1063/1.857726
[29] DOI: 10.1017/S0022112007006301 · Zbl 1123.76022 · doi:10.1017/S0022112007006301
[30] DOI: 10.1088/0951-7715/18/6/R01 · Zbl 1084.76033 · doi:10.1088/0951-7715/18/6/R01
[31] DOI: 10.1017/S0022112073001576 · doi:10.1017/S0022112073001576
[32] DOI: 10.1063/1.1890428 · Zbl 1187.76260 · doi:10.1063/1.1890428
[33] Joseph, Quart. Appl. Math. 26 pp 575– (1969)
[34] DOI: 10.1103/PhysRevLett.100.124501 · doi:10.1103/PhysRevLett.100.124501
[35] DOI: 10.1017/S0022112091002033 · Zbl 0721.76040 · doi:10.1017/S0022112091002033
[36] DOI: 10.1103/PhysRevLett.98.014501 · doi:10.1103/PhysRevLett.98.014501
[37] DOI: 10.1143/JPSJ.70.703 · doi:10.1143/JPSJ.70.703
[38] DOI: 10.1017/S0022112004009346 · Zbl 1065.76072 · doi:10.1017/S0022112004009346
[39] DOI: 10.1103/PhysRevLett.95.214502 · doi:10.1103/PhysRevLett.95.214502
[40] DOI: 10.1103/PhysRevLett.98.204501 · doi:10.1103/PhysRevLett.98.204501
[41] DOI: 10.1103/PhysRevLett.81.4140 · doi:10.1103/PhysRevLett.81.4140
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