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On freely decaying, anisotropic, axisymmetric Saffman turbulence

Published online by Cambridge University Press:  02 July 2012

P. A. Davidson ,
N. Okamoto  and
Y. Kaneda
P. A. Davidson*
Affiliation:
Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK
N. Okamoto
Affiliation:
Centre for Computational Science, Nagoya University, Nagoya 464-8603, Japan
Y. Kaneda
Affiliation:
Department of Computational Science & Engineering, Nagoya University, Nagoya 464-8603, Japan
*
†Email address for correspondence: [email protected]
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Abstract

We consider freely decaying, anisotropic, statistically axisymmetric, Saffman turbulence in which , where is the energy spectrum and the wavenumber. We note that such turbulence possesses two statistical invariants which are related to the form of the spectral tensor at small . These are and , where the subscripts and indicate quantities parallel and perpendicular to the axis of symmetry. Since and , and being integral scales, self-similarity of the large scales (when it applies) demands and . This, in turn, requires that is constant, contrary to the popular belief that freely decaying turbulence should exhibit a ‘return to isotropy’. Numerical simulations performed in large periodic domains, with different types and levels of initial anisotropy, confirm that and are indeed invariants and that, in the fully developed state, . Somewhat surprisingly, the same simulations also show that is more or less constant in the fully developed state. Simple theoretical arguments are given which suggest that, when and are both constant, the integral scales should evolve as and , irrespective of the level of anisotropy and of the presence of helicity. These decay laws, first proposed by Saffman (Phys. Fluids, vol. 10, 1967, p. 1349), are verified by the numerical simulations.

JFM classification

Turbulent Flows: Homogeneous turbulence Turbulent Flows: Turbulence simulation Turbulent Flows: Turbulence theory
Type
Papers
Information
Journal of Fluid Mechanics , Volume 706 , 10 September 2012 , pp. 150 - 172
Copyright
Copyright © Cambridge University Press 2012

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