Possible way of closing the chain of equations for the statistical moments in turbulence theory

A statistical description of turbulent velocity pulsations by the characteristic functional method is considered. Equations for the velocity covariance and Green's function that describe the average velocity response to an external force are obtained. For the non-linear term in the equation for the velocity covariance, an exact expression in the form of two terms, which can be treated as a result of momentum transport due to turbulent viscosity and the action of effective random forces, is found (in the conventional phenomenological description, only the turbulent viscosity is taken into account). Using for higher-order statistical moments a lower approximation in perturbation theory, a scheme for closing the chain of equations for the statistical moments is proposed. As a result, a closed system of equations for the velocity covariance and Green's function is constructed. The solution of this system corresponds to summing a certain infinite subsequence of the total perturbation series.