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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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