Renormalization group in the theory of fully developed turbulence. The problem of infrared-essential corrections to the Navier-Stokes equation

Influence of compressibility on scaling regimes of strongly anisotropic fully developed turbulence

“…Therefore, the scaling dimensions of the operators F are greater than the dimensions of the nonlocal operators F 1 , F 2 , and the leading contribution to the IR asymptotic form of the spectrum is determined by the contributions of F 1 and F 2 . We note that due to renormalization, scaling dimension of an operator F does not coincide in general with a naive sum of scaling dimensions of the fields and derivatives entering into F. But, for the incompressible case, the hypothesis that scaling dimension of a nonlocal operator is the sum of scaling dimensions of its local parts and of the factors of type ∆ −1 ∂v has been confirmed in [25] by the explicit one-loop calculation of the scaling dimensions related to the local operators with the canonical dimension d + 4, and we also accept it in what follows.…”

“…The effective variables ā1 (s) and ā2 (s) satisfy equations like Eqs. (25). In the infrared asymptotic region k → 0 they take on the form āi ∼ k −γa i , and the infrared asymptotic form of the dimensionless arguments ( 30) is given by the expressions ūi ∼ k −∆a i with the scaling dimensions ∆ a i (for more details see, e.g., [8,25,26]).…”

“…(25). In the infrared asymptotic region k → 0 they take on the form āi ∼ k −γa i , and the infrared asymptotic form of the dimensionless arguments ( 30) is given by the expressions ūi ∼ k −∆a i with the scaling dimensions ∆ a i (for more details see, e.g., [8,25,26]). In the linear approximation with respect to the small parameters a 01 anda 02 in the functional (13) the leading correction to the scaling function R takes on the form (1 + const • k −∆max ) where ∆ max is the maximal dimension among ∆ a i .…”