Scale invariant theory of fully developed hydrodynamic turbulence-Hamiltonian approach

Abstract

The statistical theory of fully developed homogeneous turbulence of an incompressible fluid presented here is based on the Hamiltonian equations for an ideal fluid in the Clebsch variables using the Wyld diagram technique. This theory is formulated in terms of the local Green function G( r, k, ω) and the local pair correlation function N( r, k, ω) describing the statistical properties of k-eddies in the vicinity of point r. One of the major difficulties arising from the masking effect of the sweeping interaction is effectively solved by transforming to a moving reference system associated with the fluid velocity in some reference point r 0. This change of coordinates eliminates the sweeping of k-eddies in a region of scale 1 k surrounding the reference point r 0. The convergence of all the integrals in the diagrams of arbitrary order of perturbation theory both in the IR and UV regions, is proved. This gives a diagrammatic proof of the Kolmogorov-Obukhov hypothesis that the dynamic interaction of eddies is local. In the inertial interval, the scale invariant solution of the Dyson-Wyld diagram equations has been obtained, which is consistent with the known Richardson-Kolmogorov-Obukhov concept of fully developed uniform turbulence. This new theory provides techniques for calculating the statistical characteristics of turbulence. For the purpose of illustration the asymptotic form of the simultaneous many-point velocity correlation functions when one of the wave vectors or the sum of a group of wave vectors tends to zero, is calculated.