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A geometric interpretation of coherent structures in Navier–Stokes flows

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The pressure in the incompressible three-dimensional Navier–Stokes and Euler equations is governed by Poisson's equation: this equation is studied using the geometry of three-forms in six dimensions. By studying the linear algebra of the vector space of three-forms Λ3W* where W is a six-dimensional real vector space, we relate the characterization of non-degenerate elements of Λ3W* to the sign of the Laplacian of the pressure—and hence to the balance between the vorticity and the rate of strain. When the Laplacian of the pressure, Δp, satisfies Δp>0, the three-form associated with Poisson's equation is the real part of a decomposable complex form and an almost-complex structure can be identified. When Δp<0, a real decomposable structure is identified. These results are discussed in the context of coherent structures in turbulence.

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https://hal.univ-angers.fr/hal-03054038
Contributeur : Okina Université d'Angers <>
Soumis le : vendredi 11 décembre 2020 - 11:47:05
Dernière modification le : samedi 12 décembre 2020 - 03:33:59
Archivage à long terme le : : vendredi 12 mars 2021 - 19:22:18

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Ian Roulstone, Bertrand Banos, J. Gibbon, Vladimir Roubtsov. A geometric interpretation of coherent structures in Navier–Stokes flows. Proceedings of Royal Society A Mathematical, physical and engineering sciences, Royal Society, 2009, 465 (2107), pp.2015 - 2021. ⟨10.1098/rspa.2008.0483⟩. ⟨hal-03054038⟩

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