Abstract
A large conceptual gap separates the theory of low-dimensional chaotic dynamics from the infinite-dimensional nonlinear dynamics of turbulence. Recent advances in experimental imaging, computational methods, and dynamical systems theory suggest a way to bridge this gap in our understanding of turbulence. Recent discoveries show that recurrent coherent structures observed in wall-bounded shear flows (such as pipes and plane Couette flow) result from close passes to weakly unstable invariant solutions of the Navier-Stokes equations. These 3D, fully nonlinear solutions (equilibria, traveling waves, and periodic orbits) structure the state space of turbulent flows and provide a skeleton for analyzing their dynamics. We calculate a hierarchy of invariant solutions for plane Couette, a canonical wall-bounded shear flow. These solutions reveal organization in the flow's turbulent dynamics and can be used to predict directly from the fundamental equations physical quantities such as bulk flow rate and mean wall drag. All results and the code that generates them are disseminated through through our group's open-source CFD software and solution database Channelflow.org and the collaborative e-book ChaosBook.org.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Hopf, E.: A mathematical example displaying features of turbulence. Comm. Pure Appl. Math. 1 (1948) 303–322
Hof, B., van Doorne, C.W.H., Westerweel, J., Nieuwstadt, F.T.M., Faisst, H., Eckhardt, B., Wedin, H., Kerswell, R.R., Waleffe, F.: Experimental observation of nonlinear traveling waves in turbulent pipe flow. Science 305(5690) (2004) 1594–1598
Holmes, P., Lumley, J.L., Berkooz, G.: Turbulence, coherent structures, dynamical systems and symmetry. Cambridge University Press, Cambridge (1996)
Panton, R.L., ed.: Self-sustaining mechanisms of wall turbulence. Computational Mechanics Publications, Southhampton (1997)
Robinson, S.K.: Coherent motions in the turbulent boundary layer. Annual Review of Fluid Mechanics 23 (1991) 601–639
Gibson, J.F., Halcrow, J., Cvitanović, P.: Visualizing the geometry of state-space in plane Couette flow. J. Fluid Mech. 611 (2008) 107–130 arXiv:0705.3957.
Halcrow, J., Gibson, J.F., Cvitanović, P.: Equilibrium and traveling-wave solutions of plane Couette flow. arXiv:0808.3375, J. Fluid Mech., to appear (2009)
Halcrow, J., Gibson, J.F., Cvitanović, P., Viswanath, D.: Heteroclinic connections in plane Couette flow. J. Fluid Mech. 621 (2009) 365–376 arXiv:0808.1865.
Gibson, J.F., Cvitanović, P.: Periodic orbits of plane Couette flow. In preparation (2009)
Cvitanović, P., Artuso, R., Mainieri, R., Tanner, G., Vattay, G.: Chaos: Classical and quantum. Niels Bohr Institute, Copenhagen (2009) ChaosBook.org.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Cvitanović, P., Gibson, J.F. (2009). Geometry of state space in plane Couette flow. In: Eckhardt, B. (eds) Advances in Turbulence XII. Springer Proceedings in Physics, vol 132. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03085-7_17
Download citation
DOI: https://doi.org/10.1007/978-3-642-03085-7_17
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-03084-0
Online ISBN: 978-3-642-03085-7
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)