Batchelor entrainment hypothesis

The Batchelor entrainment hypothesis takes the inward mean edge velocity of a turbulent plume to be $v_e=\alpha w$ in the Boussinesq approximation. Its density-corrected non-Boussinesq form is $v_e=\alpha w\sqrt{\rho /{\rho }_0}$, where $\rho$ and ${\rho }_0$ are plume and ambient mass densities and $\alpha$ is the entrainment coefficient.

The entrainment coefficient is the dimensionless constant in the Batchelor entrainment hypothesis, so the mean speed at which ambient fluid crosses a plume edge is $\alpha$ times the plume's characteristic axial speed.