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

Abstract

Abstract The goal of this paper is to derive the equation for the turbulence dissipation rate ɛ for a shear-driven flow. In 1961, Davydov used a one-point closure model to derive the ɛ 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 ɛ 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 ɛ equation that contains three empirical ingredients: two constants, c1,ɛ and c2,ɛ, and a diffusion term Dɛ. In this work, a two-point closure is employed, yielding the following results: 1) the empirical constants get replaced by c1, c2, which are now functions of K and ɛ; 2) c1 and c2 are not independent because a general relation between the two that are valid for any K and ɛ are derived; 3) c1, c2 become constant with values close to the empirical values c1,ɛ, c2,ɛ (i.e., homogenous flows); and 4) the empirical form of the diffusion term Dɛ is no longer needed because it gets substituted by the K–ɛ dependence of c1, c2, which plays the role of the diffusion, together with the diffusion of the turbulent kinetic energy DK, which now enters the new ɛ equation (i.e., inhomogeneous flows). Thus, the three empirical ingredients c1,ɛ, c2,ɛ, Dɛ are replaced by a single function c1(K, ɛ) or c2(K, ɛ), plus a DK term. Three tests of the new equation for ɛ are presented: one concerning channel flow and two concerning the shear-driven planetary boundary layer (PBL).