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Prandtl–Batchelor theorem for flows with quasiperiodic time dependence

Published online by Cambridge University Press:  07 January 2019

Hassan Arbabi Open the ORCID record for Hassan Arbabi [Opens in a new window]  and
Igor Mezić
Hassan Arbabi*
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Igor Mezić*
Affiliation:
Department of Mechanical Engineering, University of California Santa Barbara, Santa Barbara, CA 93106, USA
*
†Email addresses for correspondence: arbabi@mit.edu, mezic@engineering.ucsb.edu
†Email addresses for correspondence: arbabi@mit.edu, mezic@engineering.ucsb.edu
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Abstract

The classical Prandtl–Batchelor theorem (Prandtl, Proc. Intl Mathematical Congress, Heidelberg, 1904, pp. 484–491; Batchelor, J. Fluid Mech., vol. 1 (02), 1956, pp. 177–190) states that in the regions of steady 2D flow where viscous forces are small and streamlines are closed, the vorticity is constant. In this paper, we extend this theorem to recirculating flows with quasiperiodic time dependence using ergodic and geometric analysis of Lagrangian dynamics. In particular, we show that 2D quasiperiodic viscous flows, in the limit of zero viscosity, cannot converge to recirculating inviscid flows with non-uniform vorticity distribution. A corollary of this result is that if the vorticity contours form a family of closed curves in a quasiperiodic viscous flow, then at the limit of zero viscosity, vorticity is constant in the area enclosed by those curves at all times.

JFM classification

Mixing: Chaotic advection Vortex Flows: Vortex dynamics
Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 862 , 10 March 2019 , R1
Copyright
© 2019 Cambridge University Press 

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References

Arbabi, H. & Mezić, I. 2017 Study of dynamics in post-transient flows using Koopman mode decomposition. Phys. Rev. Fluids 2, 124402.Google Scholar
Arnold, V. I. 1989 Mathematical Methods of Classical Mechanics, vol. 60. Springer.Google Scholar
Batchelor, G. K. 1956 On steady laminar flow with closed streamlines at large Reynolds number. J. Fluid Mech. 1 (02), 177–190.Google Scholar
Blennerhassett, P. J. 1979 A three-dimensional analogue of the Prandtl–Batchelor closed streamline theory. J. Fluid Mech. 93 (2), 319–324.Google Scholar
Burggraf, O. R. 1966 Analytical and numerical studies of the structure of steady separated flows. J. Fluid Mech. 24 (1), 113–151.Google Scholar
Childress, S., Landman, M. & Strauss, H. 1990 Steady motion with helical symmetry at large Reynolds number. In Proc. IUTAM Symp. on Topological Fluid dynamics (ed. Moffatt, H. K. & Tsinober, A.), pp. 216–224. Cambridge University Press.Google Scholar
Govindarajan, N., Mohr, R., Chandrasekaran, S. & Mezić, I.2018 On the approximation of Koopman spectra for measure preserving transformations. arXiv:1803.03920.Google Scholar
Grimshaw, R. 1969 On steady recirculating flows. J. Fluid Mech. 39 (4), 695–703.Google Scholar
Lee, J. M. 2003 Introduction to Smooth Manifolds. Springer.Google Scholar
Mane, R. 1987 Ergodic Theory and Differentiable Dynamics. Springer.Google Scholar
Masmoudi, N. 2007 Remarks about the inviscid limit of the Navier–Stokes system. Commun. Math. Phys. 270 (3), 777–788.Google Scholar
Mezić, I.1994 On geometrical and statistical properties of dynamical systems: theory and applications. PhD thesis, California Institute of Technology.Google Scholar
Mezić, I. 2002 An extension of Prandtl–Batchelor theory and consequences for chaotic advection. Phys. Fluids 14 (9), L61–L64.Google Scholar
Pan, F. & Acrivos, A. 1967 Steady flows in rectangular cavities. J. Fluid Mech. 28 (4), 643–655.Google Scholar
Petersen, K. E. 1989 Ergodic Theory, vol. 2. Cambridge University Press.Google Scholar
Prandtl, L. 1904 Uber Flüssigkeitsbewegung bei sehr kleiner Reibung. In Proceedings of the International Mathematical Congress, Heidelberg, pp. 484–491, see Gesammelte Abhandlungen II, 575 (1961).Google Scholar
Sandoval, M. & Chernyshenko, S. 2010 Extension of the Prandtl–Batchelor theorem to three-dimensional flows slowly varying in one direction. J. Fluid Mech. 654, 351–361.Google Scholar
Susuki, Y. & Mezić, I.2018 Uniformly bounded sets in qausiperiodically forced dynamical systems. arXiv:1808.08340.Google Scholar
Trefethen, L. N. 2000 Spectral Methods in MATLAB, vol. 10. SIAM.Google Scholar