Scale-dependent entrainment velocity and scale-independent net entrainment in a turbulent axisymmetric jet

The net entrainment in a jet is the product of the mean surface area ($\overline {S}$) and the mean entrainment velocity, $\overline {V} ~\overline{S}$, where, $\overline {V}=\alpha U_c$ with $\alpha$ the entrainment coefficient and $U_c$ the mean centreline velocity. Instantaneously, however, entrainment velocity ($v$) at a point on the interface is the difference between the interface and the fluid velocities, and the total entrainment $\int v ~{\rm d}s=V~ S$, where $S$ is the corrugated interface surface area and $V$ the area averaged entrainment velocity. Using time-resolved multi-scale PIV/PLIF measurements of velocity and scalar in an axisymmetric jet at $Re=25000$, we evaluate $V$ and $S$ directly at the smallest resolved scales, and by filtering the data at different scales ($\Delta$) we find their multi-scales counterparts, $V_\Delta$ and $S_\Delta$. We show that $\overline {V} ~\overline{S} = V_\Delta ~S_\Delta = V ~S$, independent of the scale. Furthermore, $S$ is found to have a fractal dimension $D_3 \approx 2.32\pm0.1$. Independently, we find that $V_\Delta\sim \Delta^{0.31}$, indicating increasing entrainment velocity with increasing length scale. This is consistent with a constant net entrainment across scales, and suggests $\alpha$ as a scale-dependent quantity.