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Minimax principle on energy dissipation of incompressible shear flow

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Abstract

The energy dissipation rate is an important concept in the theory of turbulence. Doering-Constantin’s variational principle characterizes the upper bounds (maximum) of the time-averaged rate of viscous energy dissipation. In the present study, an optimization theoretical 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 Kakutani minimax theorem in the game theory is applied to obtain a set of conditions, under which the maximization and the minimization in the minimax principle are commutative. The results explain the spectral constraint of Doering-Constantin, and confirm the equivalence between Doering-Constantin’s variational principle and Howard-Busse’s statistical turbulence theory.

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Authors and Affiliations

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

    Bo Chen  (陈 波), Xiao-wei Li  (李孝伟) & Gao-lian Liu  (刘高联)

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  1. Bo Chen  (陈 波)
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  2. Xiao-wei Li  (李孝伟)
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  3. Gao-lian Liu  (刘高联)
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Corresponding author

Correspondence to Bo Chen  (陈 波).

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Contributed by Gao-lian LIU

Project supported by the National Natural Science Foundation of China (No. 10772103) and the Shanghai Leading Academic Discipline Project (No. Y0103)

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Chen, B., Li, Xw. & Liu, Gl. Minimax principle on energy dissipation of incompressible shear flow. Appl. Math. Mech.-Engl. Ed. 31, 805–814 (2010). https://doi.org/10.1007/s10483-010-1315-6

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  • DOI: https://doi.org/10.1007/s10483-010-1315-6

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