Mean velocity equation for fluctuating flow

Arguments are presented that the momentum equation derived by Zwanzig-Mori technique of N-particle statistical mechanics is identical to the Reynolds equation for the mean velocity field in fluctuating hydrodynamics; thus, calculating the non-linear stress of the momentum equation is perhaps a means to avoid the closure problem of turbulence theory. In this paper, the third-order term of the friction force is calculated. Multilinear mode coupling theory is applied in order to obtain formulas for the third- and fourth-order equilibrium time correlation functions appearing in the expression. The force term is obtained as a multilinear convolution integral containing higher order gradients of the velocity field.As an application, circular jet flow for low Reynolds numbers is investigated. Numerical solutions have been obtained from an iteration starting with the corresponding solution of the Navier-Stokes equation. Deviations become distinct for Re>30.