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Turbulence noise

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Abstract

We show that the large-eddy motions in turbulent fluid flow obey a modified hydrodynamic equation with a stochastic turbulent stress whose distribution is a causal functional of the large-scale velocity field itself. We do so by means of an exact procedure of “statistical filtering” of the Navier-Stokes equations, which formally solves the closure problem, and we discuss the relation of our analysis with the “decimation theory” of Kraichnan. We show that the statistical filtering procedure can be formulated using field-theoretic path-integral methods within the Martin-Siggia-Rose (MSR) formalism for classical statistical dynamics. We also establish within the MSR formalism a “least-effective-action principle” for mean turbulent velocity profiles, which generalizes Onsager's principle of least dissipation. This minimum principle is a consequence of a simple realizability inequality and therefore holds also in any realizable closure. Symanzik's theorem in field theory—which characterizes the static effective action as the minimum expected value of the quantum Hamiltonian over all state vectors with prescribed expectations of fields—is extended to MSR theory with non-Hermitian Hamiltonian. This allows stationary mean velocity profiles and other turbulence statistics to be calculated variationally by a Rayleigh-Ritz procedure. Finally, we develop approximations of the exact Langevin equations for large eddies, e.g., a random-coupling DIA model, which yield new stochastic LES models. These are compared with stochastic subgrid modeling schemes proposed by Rose, Chasnov, Leith, and others, and various applications are discussed.

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  1. Department of Mathematics, University of Arizona, 85721, Tucson, Arizona

    Gregory L. Eyink

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Eyink, G.L. Turbulence noise. J Stat Phys 83, 955–1019 (1996). https://doi.org/10.1007/BF02179551

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