Towards a rational model for the triple velocity correlations of turbulence

The purpose of this paper is to present a rational approach to modelling th e triple velocity correlations that appear in the transport equations for t he Reynolds stresses. All existing models of these correlations have largel y been formulated on phenomenological grounds and are defective in one impo rtant aspect: they all neglect to allow for the dependence of these correla tions on the local gradients of mean velocity. The mathematical necessity f or this dependence will be demonstrated in the paper. The present contribut ion lies in the novel use of group representation theory to determine the m ost general tensorial form of these correlations in terms of all the second - and third-order tensor quantities which appear in the exact equations tha t govern their evolution. The requisite representation did not exist in the literature and had therefore to be developed specifically for this purpose . The outcome of this work is a mathematical framework for the construction of algebraic, explicit and rational models for the triple velocity correla tions that are theoretically consistent and include all the correct depende ncies. Previous models are reviewed, and all are shown to be incomplete sub sets of this new representation, even to lowest order.