×

zbMATH — the first resource for mathematics

Exact coherent states with hairpin-like vortex structure in channel flow. (English) Zbl 1415.76137
Summary: Hairpin vortices are widely studied as an important structural aspect of wall turbulence. The present work describes, for the first time, nonlinear travelling wave solutions to the Navier-Stokes equations in the channel flow geometry – exact coherent states (ECS) – that display hairpin-like vortex structure. This solution family comes into existence at a saddle-node bifurcation at Reynolds number \(\text{Re}=666\). At the bifurcation, the solution has a highly symmetric quasi-streamwise vortex structure similar to that reported for previously studied ECS. With increasing distance from the bifurcation, however, both the upper and lower branch solutions develop a vortical structure characteristic of hairpins: a spanwise-oriented ‘head’ near the channel centreplane where the mean shear vanishes connected to counter-rotating quasi-streamwise ‘legs’ that extend toward the channel wall. At \(\text{Re}=1800\), the upper branch solution has mean and Reynolds shear-stress profiles that closely resemble those of turbulent mean profiles in the same domain.

MSC:
76D05 Navier-Stokes equations for incompressible viscous fluids
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
76D17 Viscous vortex flows
76F06 Transition to turbulence
Software:
channelflow
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Adrian, R. J., Hairpin vortex organization in wall turbulence, Phys. Fluids, 19, 4, (2007) · Zbl 1146.76307
[2] Brand, E.; Gibson, J. F., A doubly-localized equilibrium solution of plane Couette flow, J. Fluid Mech., 750, R3, (2014)
[3] Chantry, M.; Willis, A. P.; Kerswell, R. R., Genesis of streamwise-localized solutions from globally periodic traveling waves in pipe flow, Phys. Rev. Lett., 112, 16, (2014)
[4] Cherubini, S.; Palma, P. D.; Robinet, J. C.; Bottaro, A., Edge states in a boundary layer, Phys. Fluids, 23, 5, (2011) · Zbl 1241.76246
[5] Duguet, Y.; Pringle, C. C. T.; Kerswell, R. R., Relative periodic orbits in transitional pipe flow, Phys. Fluids, 20, 11, (2008) · Zbl 1182.76222
[6] Duguet, Y.; Schlatter, P.; Henningson, D. S.; Eckhardt, B., Self-sustained localized structures in a boundary-layer flow, Phys. Rev. Lett., 108, (2012)
[7] Duguet, Y.; Willis, A. P.; Kerswell, R. R., Transition in pipe flow: the saddle structure on the boundary of turbulence, J. Fluid Mech., 613, 255-274, (2008) · Zbl 1151.76495
[8] Eckhardt, B.; Faisst, H.; Schmiegel, A.; Schneider, T. M., Dynamical systems and the transition to turbulence in linearly stable shear flows, Phil. Trans. R. Soc. Lond. A, 366, 1868, 1297-1315, (2008)
[9] Eckhardt, B.; Schneider, T. M.; Hof, B.; Westerweel, J., Turbulence transition in pipe flow, Annu. Rev. Fluid Mech., 39, 447-468, (2007) · Zbl 1296.76062
[10] Eitel-Amor, G.; Örlü, R.; Schlatter, P.; Flores, O., Hairpin vortices in turbulent boundary layers, Phys. Fluids, 27, 2, (2015)
[11] Gibson, J. F.2012 ChannelFlow: a spectral Navier-Stokes simulator in C\(++\). Tech. Rep., University of New Hampshire.
[12] Gibson, J. F.; Brand, E., Spanwise-localized solutions of planar shear flows, J. Fluid Mech., 745, 25-61, (2014)
[13] Gibson, J. F.; Halcrow, J.; Cvitanović, P., Visualizing the geometry of state space in plane Couette flow, J. Fluid Mech., 611, 107-130, (2008) · Zbl 1151.76453
[14] Gibson, J. F.; Halcrow, J.; Cvitanovic, P., Equilibrium and travelling-wave solutions of plane Couette flow, J. Fluid Mech., 638, 243-266, (2009) · Zbl 1183.76688
[15] Hof, B.; Van Doorne, C. W.; Westerweel, J.; Nieuwstadt, F. T.; Faisst, H.; Eckhardt, B.; Wedin, H.; Kerswell, R. R.; Waleffe, F., Experimental observation of nonlinear traveling waves in turbulent pipe flow, Science, 305, 5690, 1594-1598, (2004)
[16] Itano, T.; Generalis, S., Hairpin vortex solution in planar couette flow: a tapestry of knotted vortices, Phys. Rev. Lett., 102, (2009)
[17] Kawahara, G.; Uhlmann, M.; Van Veen, L., The significance of simple invariant solutions in turbulent flows, Annu. Rev. Fluid Mech., 44, 203-225, (2012) · Zbl 1352.76031
[18] Khapko, T.; Kreilos, T.; Schlatter, P.; Duguet, Y.; Eckhardt, B.; Henningson, D. S., Localized edge states in the asymptotic suction boundary layer, J. Fluid Mech., 717, R6, (2013) · Zbl 1284.76106
[19] Nagata, M.; Deguchi, K., Mirror-symmetric exact coherent states in plane Poiseuille flow, J. Fluid Mech., 735, R4, (2013) · Zbl 1294.76115
[20] Neelavara, S. A.; Duguet, Y.; Lusseyran, F., State space analysis of minimal channel flow, Fluid Dyn. Res., 49, 3, (2017)
[21] Park, J. S.; Graham, M. D., Exact coherent states and connections to turbulent dynamics in minimal channel flow, J. Fluid Mech., 782, 430-454, (2015) · Zbl 1381.76097
[22] Perry, A. E.; Chong, M. S., On the mechanism of wall turbulence, J. Fluid Mech., 119, 106-121, (1982) · Zbl 0517.76057
[23] Rawat, S.; Cossu, C.; Rincon, F., Travelling-wave solutions bifurcating from relative periodic orbits in plane Poiseuille flow, Comptes Rendus Mécanique, 344, 6, 448-455, (2016)
[24] Robinson, S. K., Coherent motions in the turbulent boundary-layer, Annu. Rev. Fluid Mech., 23, 1, 601-639, (1991)
[25] Schlatter, P.; Li, Q.; Örlü, R.; Hussain, F.; Henningson, D. S., On the near-wall vortical structures at moderate Reynolds numbers, Eur. J. Mech. (B/Fluids), 48, 75-93, (2014) · Zbl 06931934
[26] Skufca, J. D.; Yorke, J. A.; Eckhardt, B., Edge of chaos in a parallel shear flow, Phys. Rev. Lett., 96, 17, (2006)
[27] Smits, A. J.; Mckeon, B. J.; Marusic, I., High-Reynolds number wall turbulence, Annu. Rev. Fluid Mech., 43, 1, 353-375, (2011) · Zbl 1299.76002
[28] Theodorsen, T.1952Mechanism of Turbulence. In Second International Midwest Conference on Fluid Mechanics, pp. 1-19. Ohio State University.
[29] Toh, S.; Itano, T., A periodic-like solution in channel flow, J. Fluid Mech., 481, 67-76, (2003) · Zbl 1034.76014
[30] Townsend, A. A., Equilibrium layers and wall turbulence, J. Fluid Mech., 11, 1, 97-120, (1961) · Zbl 0127.42602
[31] Waleffe, F., On a self-sustaining process in shear flows, Phys. Fluids, 9, 4, 883-900, (1997)
[32] Waleffe, F., Three-dimensional coherent states in plane shear flows, Phys. Rev. Lett., 81, 19, 4140-4143, (1998)
[33] Waleffe, F., Exact coherent structures in channel flow, J. Fluid Mech., 435, 93-102, (2001) · Zbl 0987.76034
[34] Waleffe, F., Homotopy of exact coherent structures in plane shear flows, Phys. Fluids, 15, 6, 1517-1534, (2003) · Zbl 1186.76556
[35] Wall, D. P.; Nagata, M., Exact coherent states in channel flow, J. Fluid Mech., 788, 444-468, (2016) · Zbl 1381.76093
[36] Wang, J.; Gibson, J.; Waleffe, F., Lower branch coherent states in shear flows: Transition and control, Phys. Rev. Lett., 98, 20, (2007)
[37] Wedin, H.; Kerswell, R. R., Exact coherent structures in pipe flow: travelling wave solutions, J. Fluid Mech., 508, 333-371, (2004) · Zbl 1065.76072
[38] Woodcock, J. D.; Marusic, I., The statistical behaviour of attached eddies, Phys. Fluids, 27, 1, (2015)
[39] Wu, X.; Moin, P., Direct numerical simulation of turbulence in a nominally zero-pressure-gradient flat-plate boundary layer, J. Fluid Mech., 630, 5-41, (2009) · Zbl 1181.76084
[40] Zammert, S.; Eckhardt, B., Harbingers and latecomers – the order of appearance of exact coherent structures in plane Poiseuille flow, J. Turbul., 18, 2, 103-114, (2016)
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.