Taylor's entrainment equation contains the entrainment function E that has traditionally been treated heuristically, as attested by 30 different expressions for E(Ri) available in the literature. Using a model independent procedure, we first derive the new relation: E = 2Psh $overline{rm u}$-3 which expresses E in terms of Ps, the shear production (of turbulent kinetic energy) averaged across the interface of the gravity current whose thickness and mean velocity are denoted by h and $overline{rm u}$. Second, using a turbulence model for the turbulence kinetic energy K and its rate of dissipation $varepsilon$ (K-$varepsilon$ model, integrated across the flow), we compute Ps to express E in terms of the Richardson number Ri and the density ratio Rρ characterizing double-diffusion. Third, we show that in the local (along the flow) case, the model reproduces the Ellison and Turner (1959) data while the non-local case reproduces the data by Princevac et al. (2005) which are up to ten times larger than the ET data.