EXACT RELATIONS IN THE THEORY OF DEVELOPED HYDRODYNAMIC TURBULENCE

Exact relations of two types in the statistical theory of fully develo ped homogeneous isotropic turbulence in an incompressible fluid were f ound. The relations of the first type connect two-point and three-poin t objects of the theory which are correlation functions and susceptibi lities. The second types of relations are the ''frequency sum rules'' which express some frequency integrals from ''fully dressed'' many-poi nt 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 a nd susceptibilities. The derivation of these relations uses no perturb ation expansions and no additional assumptions. This means that the re lations are exact in the framework of the statistical theory of turbul ence. We showed that ''a many-point scaling'' gives birth to the ''glo bal scaling.'' Here ''many-point scaling'' is the assumption that two- point, three-point, etc. objects of the theory of turbulence are unifo rm functions in the inertial interval and may be characterized by some scaling exponents. Under this assumption the only global scale-invari ant model of fully developed turbulence suggested by Kolmogorov [Dokl. Akad. Nauk SSSR 32, 19 (1941)] is consistent with the exact relations deduced.