On Functional Formulation of the Statistical Theory of Homogeneous Turbulence and the Method of Sceleton Feynman Diagrams

Abstract

The paper reviews the application of the formalism of a characteristic functional for statistical description of a random velocity field obeying the Navier-Stokes equation for incompressible fluids in the presence of regular and random external forces. The equation in functional derivatives for the characteristic functional is obtained using a representation of the characteristic functional in the form of a functional integral over two fields. From this equation one can obtain equations for various statistical characteristics of the velocity field such as the variance of velocity pulsations (the pair correlation function) or the mean response of velocity field to external forces (Green’s function). The method of skeleton Feynman diagrams is used in the analysis of the equations and of the solution structures. This fact follows directly from the functional formulation of the theory without referring to the commonly used methods of perturbation theory. The vertices of three types arising in the theory formulation appear to be linked. This enables considering the vertex of only one type and simplify the diagrammatic representations of various quantities.