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 $\bar {V}_g$ to be the fluid motion across an isosurface of an averaged scalar, we find that for a slender flow, $\bar {V}_g=\bar {u}_\zeta - \bar {\textrm {D}}h_t/\bar {\textrm {D}}t$, where $\bar {\textrm {D}}/\bar {\textrm {D}} t$ is the material derivative of the average flow field and $\bar {u}_\zeta$ 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.