Formulation of the Statistical Equations of Turbulent Flows with Variable Density

Abstract

The statistical equations of turbulent flows were derived by Boussinesq and Reynolds, in cases where density ρ is constant, using a separation of each random quantity w into its average value $overline w$ and a fluctuation w“, so $overline {w} = 0$. For the general case when the density is variable, the same “Reynolds average” may be used, but the formulae are not simple and the physical meanings of the terms are not clear. In order to express the principles of physics with the fundamental quantities per unit volume ρw, such as mass ρ, momentum ρu_i, internal energy ρe, entropy ρs, Favre retains the separation into the average value $\overline {rho w} = \overline rho \tilde w$ and a fluctuation ρw’, so $\overline {rho w} ' = 0$. The formulae are simpler and the physical meanings clearer. The main statistical equations are given.

This work was supported in part by the Air Force Office of Scientific Research through the European Office of Aerospace Research, United States Air Force, under contract F 61052-67-C-0025.