Three-point method in the theory of homogeneous turbulence

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Abstract
A systematic method is presented, in some detail, for solving Navier–Stokes turbulence problems. This method is based on the use of a three‐point (or triple) Green’s function G(3). A new set of covariant turbulence equations is thereby derived from the Navier–Stokes equation, without having to make any ad hoc assumptions; in fact, these equations simply constitute the first order terms of an expansion in a new ordering parameter R. The derived set of equations preserve Galilean invariance, and reduce to the direct‐interaction approximation when certain definite terms are arbitrarily disregarded. It is the omission of these terms which is primarily responsible for the loss of Galilean invariance in several previous approximations. It is shown that random Galilean invariance is closely associated with a dependence of 〈G(3)〉 on the correlation length of the turbulence. A simple approximation is suggested which expresses 〈G(3)〉 in terms of a new (modified) single Green’s function. The modified Green’s function is random Galilean invariant, but does not involve an arbitrary constant as does the Green’s function used in the test‐field model.

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