Representation of interaction in the theory of turbulence

Abstract

The concept that in a turbulent flow energy exchanges only take place of pulsations of near scales is the basis of macroscopic theory of local turbulence structure. Universality and similarity of small-scale statistical pulsations are inferred from the assumption that the energy exchange is of random character. In the Eulerian equations of motion, together with the interactions which implement the energy exchange between pulsations, there are fictitious interactions related to the transfer of pulsations of a given scalel by the pulsations of scalesl′ >>l. It was emphasized in [2, 3] that in the Eulerian description of turbulence the effect of transfer results in a strong statistical dependence of pulsations of different scales. Therefore, the universality and similarity of small-scale pulsations can be observed only in these variables in which there are no effects of pure transfer of some pulsations by the others. Qualitative considerations were therefore given in [1–3] on the need for describing small-scale pulsations in a reference system which is in motion at each point with all large-scale pulsations. It is shown in the present article that such description of small-scale pulsations can be implemented with the aid of transfer representation similar to the representation of interaction in the quantum field theory [4]. Representation of interaction is of intermediate position between the Lagrangian and Eulerian descriptions of turbulence, since a transfer of a packet as an entity can be described in variables which are Lagrangian only as regards large-scale motions. Another way of eliminating transfer interactions is based on the introduction of nonsolenoid velocity as in [5]. From the physical point of view, the method employed in this article seems to be more appropriate.