×

zbMATH — the first resource for mathematics

Equilibrium and travelling-wave solutions of plane Couette flow. (English) Zbl 1183.76688
Summary: We present 10 new equilibrium solutions to plane Couette flow in small periodic cells at low Reynolds number \(Re\) and two new travelling-wave solutions. The solutions are continued under changes of \(Re\) and spanwise period. We provide a partial classification of the isotropy groups of plane Couette flow and show which kinds of solutions are allowed by each isotropy group. We find two complementary visualizations particularly revealing. Suitably chosen sections of their three-dimensional physical space velocity fields are helpful in developing physical intuition about coherent structures observed in low-\(Re\) turbulence. Projections of these solutions and their unstable manifolds from their \(\infty \)-dimensional state space on to suitably chosen two- or three-dimensional subspaces reveal their interrelations and the role they play in organizing turbulence in wall-bounded shear flows.

MSC:
76D33 Waves for incompressible viscous fluids
Software:
channelflow
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] DOI: 10.1017/S0022112097005818 · Zbl 0898.76028 · doi:10.1017/S0022112097005818
[2] DOI: 10.1063/1.2753982 · Zbl 1182.76634 · doi:10.1063/1.2753982
[3] DOI: 10.1017/S0022112092000892 · Zbl 0744.76052 · doi:10.1017/S0022112092000892
[4] Peyret, Spectral Methods for Incompressible Flows (2002) · doi:10.1007/978-1-4757-6557-1
[5] DOI: 10.1088/0951-7715/10/1/004 · Zbl 0907.58042 · doi:10.1088/0951-7715/10/1/004
[6] DOI: 10.1103/PhysRevE.55.2023 · doi:10.1103/PhysRevE.55.2023
[7] DOI: 10.1017/S0022112097005661 · Zbl 0903.76027 · doi:10.1017/S0022112097005661
[8] DOI: 10.1017/S0022112090000829 · doi:10.1017/S0022112090000829
[9] Canuto, Spectral Methods in Fluid Dynamics (1988) · doi:10.1007/978-3-642-84108-8
[10] Marsden, Introduction to Mechanics and Symmetry (1999) · doi:10.1007/978-0-387-21792-5
[11] DOI: 10.1103/PhysRevE.78.026208 · doi:10.1103/PhysRevE.78.026208
[12] DOI: 10.1017/S0022112071002490 · doi:10.1017/S0022112071002490
[13] DOI: 10.1063/1.1825451 · Zbl 1187.76248 · doi:10.1063/1.1825451
[14] DOI: 10.1143/JPSJ.70.703 · doi:10.1143/JPSJ.70.703
[15] DOI: 10.1103/PhysRevLett.102.114501 · doi:10.1103/PhysRevLett.102.114501
[16] Hoyle, Pattern Formation: An Introduction to Methods (2006) · Zbl 1087.00001 · doi:10.1017/CBO9780511616051
[17] DOI: 10.1126/science.1100393 · doi:10.1126/science.1100393
[18] Harter, Principles of Symmetry, Dynamics, and Spectroscopy (1993)
[19] DOI: 10.1017/S0022112095000978 · Zbl 0867.76032 · doi:10.1017/S0022112095000978
[20] DOI: 10.1017/S0022112004009346 · Zbl 1065.76072 · doi:10.1017/S0022112004009346
[21] DOI: 10.1017/S0022112008005065 · Zbl 1171.76383 · doi:10.1017/S0022112008005065
[22] DOI: 10.1103/PhysRevLett.98.204501 · doi:10.1103/PhysRevLett.98.204501
[23] Golubitsky, The Symmetry Perspective (2002) · Zbl 1031.37001 · doi:10.1007/978-3-0348-8167-8
[24] DOI: 10.1063/1.1566753 · Zbl 1186.76556 · doi:10.1063/1.1566753
[25] Gilmore, The Symmetry of Chaos (2007) · Zbl 1137.37017
[26] DOI: 10.1017/S002211200800267X · Zbl 1151.76453 · doi:10.1017/S002211200800267X
[27] DOI: 10.1017/S0022112001004189 · Zbl 0987.76034 · doi:10.1017/S0022112001004189
[28] DOI: 10.1103/PhysRevLett.81.4140 · doi:10.1103/PhysRevLett.81.4140
[29] DOI: 10.1063/1.869185 · doi:10.1063/1.869185
[30] Frisch, Turbulence (1996)
[31] Waleffe, Stud. Appl. Math. 95 pp 319– (1995) · Zbl 0838.76026 · doi:10.1002/sapm1995953319
[32] DOI: 10.1103/PhysRevLett.91.224502 · doi:10.1103/PhysRevLett.91.224502
[33] Viswanath, Mathematics and Computation, a Contemporary View. The Abel Symposium 2006 (2008)
[34] DOI: 10.1017/S0022112007005459 · Zbl 1175.76074 · doi:10.1017/S0022112007005459
[35] DOI: 10.1007/s00162-002-0064-y · Zbl 1081.76023 · doi:10.1007/s00162-002-0064-y
[36] DOI: 10.1103/PhysRevLett.96.174101 · doi:10.1103/PhysRevLett.96.174101
[37] DOI: 10.1063/1.2943675 · Zbl 1182.76226 · doi:10.1063/1.2943675
[38] DOI: 10.1103/PhysRevE.78.037301 · doi:10.1103/PhysRevE.78.037301
[39] DOI: 10.1063/1.3009874 · Zbl 1182.76222 · doi:10.1063/1.3009874
[40] DOI: 10.1103/PhysRevLett.99.074502 · doi:10.1103/PhysRevLett.99.074502
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.