Mean Velocity Equation for Turbulent Fluid Flow: An Approach via Classical Statistical Mechanics

The possibility to derive an equation for the mean velocity field in turbulent flow by using classical statistical mechanics is investigated. An application of projection operator technique available in the literature is used for this purpose. It is argued that the hydrodynamic velocity defined there, in situations where the fluid is turbulent, is to be interpreted as the mean velocity field; in that case, the momentum component of the generalized transport equation derived there is the mean velocity equation. In this paper, stationary incompressible flow for constant mass density and temperature is considered. The stress tensor is obtained as a nonlinear functional of the mean velocity field, the linear part of which is the Stokes tensor. The formula contains a time correlation function in local equilibrium. Presently, there exists a microscopic theory for time correlations in total equilibrium only. For this reason and as a preliminary measure, the formula has been expanded into a power series in the mean velocity; though this limits the applicability to low Reynolds number flow. The second order term has been evaluated in a former paper of the author. For the third order term, the form of the kernel function is derived. Its calculation with the aid of the mode-coupling theory is completed; it will be reported in an separate paper. An numerical application with the data of the circular jet is under way.