Unified description of turbulent entrainment

We present a mathematical description of turbulent entrainment that is applicable to free shear problems that evolve in space, time or both. Defining the global entrainment velocity $\overline V_g$ to be the fluid motion across an isosurface of an averaged scalar, we find that for a slender flow, $\overline V_g=\overline u_\zeta - \overline{D}h_t/\overline{D}t$, where $\overline D/\overline D t$ is the material derivative of the average flowfield and $\overline u_\zeta$ is the average velocity perpendicular to the flow direction across the interface located at $\zeta=h_t$. The description is shown to reproduce well-known results for the axisymmetric jet, the planar wake and the temporal jet, and provides a clear link between the local (small-scale) and global (integral) descriptions of turbulent entrainment. Application to unsteady jets/plumes demonstrates that, under unsteady conditions, the entrainment coefficient $\alpha$ no longer only captures entrainment of ambient fluid, but also time-dependency effects due to the loss of self-similarity.