A Minimax Principle on Energy Dissipation of Incompressible Shear Flow

CHEN Bo; LI Xiao-wei; LIU Gao-lian

doi:10.3879/j.issn.1000-0887.2010.07.002

Volume 31 Issue 7

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CHEN Bo, LI Xiao-wei, LIU Gao-lian. A Minimax Principle on Energy Dissipation of Incompressible Shear Flow[J]. Applied Mathematics and Mechanics, 2010, 31(7): 772-780. doi: 10.3879/j.issn.1000-0887.2010.07.002

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A Minimax Principle on Energy Dissipation of Incompressible Shear Flow

doi: 10.3879/j.issn.1000-0887.2010.07.002

Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, P. R. China

Received Date: 1900-01-01
Rev Recd Date: 2010-05-28
Publish Date: 2010-07-15

Abstract

Abstract

Energy dissipation rate is one of the most important concepts in turbulence theory. Doering-Constantin's variational principle characterizes the upper bounds(maximum) of the tmie-averaged rate of viscous energy dissipation. In present study, an optmiization theoretic point of view was adopted to recast Doering-Constantin's formulation into a minimax principle for the energy dissipation of an incompressible shear flow. Then the Kakutaniminimax theorem in game theory was applied to obtain a set of conditions under which the maximization and the minimization in the minmiax principle are commutative. The results not only elucidate the spectral constraint of Doering-Constantin, but also confirm the equivalence of Doering-Constantin's variational principle and Howard-Busse's statistical turbulence theory.
- minimax theorem,
- variational method,
- gametheory,
- turbulence,
- dissipation rate,
- upper bound

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References(21)

References

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