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Functional Integration Theory for Incompressible Fluid Turbulence
Journal article   Open access   Peer reviewed

Functional Integration Theory for Incompressible Fluid Turbulence

Gerald Rosen
The Physics of fluids (1958), v 10(12), pp 2614-2619
01 Jan 1967
DOI: https://doi.org/10.1063/1.1762083
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https://doi.org/10.1063/1.1762083View
Published, Version of Record (VoR) Open

Abstract

A purely deductive approximation theory for incompressible fluid turbulence is presented, based exclusively on the Hopf characteristic functional space‐time formulation and without any additive statistical postulate. Certain real symmetric solenoidal tensors called ``stationary functionalities'' are defined in terms of the characteristic functional by functional integration, quantities which vanish in the neighborhood of an exact physical characteristic functional, hence, for an approximate physical characteristic functional. By explicit functional integration, the first and second tensorial rank stationary functionalities are evaluated with a zero‐mean velocity field Gaussian approximation for the characteristic functional. Equating the resulting expressions to zero produces a subsidiary equation and a dynamical equation for the two‐point velocity correlation tensor, equations which are identical to the Navier‐Stokes expectation value equation for a zero‐mean velocity field and to a specific two‐point Navier‐Stokes expectation value equation with a zero‐mean velocity field probability distribution such that the fourth‐order velocity field product expectation values are related to lower‐order product expectation values in the same way as for a zero‐mean Gaussian probability distribution. Specialized forms of the nonlinear integrodifferential dynamical equation are worked out for the cases of homogeneous and isotropic homogeneous turbulence.

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10 citations in Web of Science
7 citations in Scopus

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