Exact relations in the theory of developed hydrodynamic turbulence

Abstract

Exact relations of two types in the statistical theory of fully developed homogeneous isotropic turbulence in an incompressible fluid were found. The relations of the first type connect two-point and three-point objects of the theory which are correlation functions and susceptibilities. The second type of relations are the ‘‘frequency sum rules’’ which express some frequency integrals from ‘‘fully dressed’’ many-point objects (like vertices) via corresponding bare values. Our approach is based on the Navier-Stokes equation in quasi-Lagrangian variables and on the generating functional technique for correlation functions and susceptibilities. The derivation of these relations uses no perturbation expansions and no additional assumptions. This means that the relations are exact in the framework of the statistical theory of turbulence. We showed that ‘‘a many-point scaling’’ gives birth to the ‘‘global scaling.’’ Here ‘‘many-point scaling’’ is the assumption that two-point, three-point, etc. objects of the theory of turbulence are uniform functions in the inertial interval and may be characterized by some scaling exponents. Under this assumption the only global scale-invariant model of fully developed turbulence suggested by Kolmogorov [Dokl. Akad. Nauk SSR 32, 19 (1941)] is consistent with the exact relations deduced.