An Attempt to Derive the ε Equation from a Two-Point Closure

The goal of this paper is to derive the equation for the turbulence dissipation rate epsilon for a shear-driven flow. In 1961, Davydov used a one-point closure model to derive the epsilon equation from first principles but the final result contained undetermined terms and thus lacked predictive power. Both in 1987 (Schiestel) and in 2001 (Rubinstein and Zhou), attempts were made to derive the epsilon equation from first principles using a two-point closure, but their methods relied on a phenomenological assumption. The standard practice has thus been to employ a heuristic form of the epsilon equation that contains three empirical ingredients: two constants, c(1,epsilon) and c(2,epsilon), and a diffusion term D-epsilon. In this work, a two-point closure is employed, yielding the following results: 1) the empirical constants get replaced by c(1), c(2), which are now functions of Kand epsilon; 2) c(1) and c(2) are not independent because a general relation between the two that are valid for any K and epsilon are derived; 3) c(1), c(2) become constant with values close to the empirical values c(1,epsilon), c(2,epsilon) (i.e., homogenous flows); and 4) the empirical form of the diffusion term D-epsilon is no longer needed because it gets substituted by the K-epsilon dependence of c(1), c(2), which plays the role of the diffusion, together with the diffusion of the turbulent kinetic energy D-K, which now enters the new epsilon equation (i.e., inhomogeneous flows). Thus, the three empirical ingredients c(1,epsilon), c(2,epsilon), D-epsilon are replaced by a single function c(1)(K, epsilon) or c(2)(K, epsilon), plus a D-K term. Three tests of the new equation for epsilon are presented: one concerning channel flow and two concerning the shear-driven planetary boundary layer (PBL).