The multi-scale geometry of the near field in an axisymmetric jet

A characteristic feature of axisymmetric jets, and turbulent shear flows in general, is the entrainment of mass across the turbulent/non-turbulent interface (TNTI). The multi-scale nature of the TNTI surface area was recently observed to exhibit power-law scaling with a fractal dimension, $D_{f}$, between $D_{f}=2.3{-}2.4$, inferred from two-dimensional data, in both high Reynolds number boundary layers and the far field of axisymmetric jets. In this paper, we show that the fractal scaling previously observed in the far field of an axisymmetric jet is established at the end of the potential core. Simultaneous measurements of the velocity and scalar fields were obtained and coarse grain filtering was applied over two decades of scale separation, showing that $D_{f}$ evolves to ${\approx}2.35$ at $x/d=4.6$, which is similar to $D_{f}$ found in the far field between $x/d=40{-}60$. This is evidence that scale separation becomes sufficiently developed to achieve scale invariance of the TNTI surface area in the near field of the jet well before self-similarity is established. We also observe that the onset of this geometric scale invariance coincides with the onset of radial homogeneity shown by two-point velocity correlations. Finally, we present a simple theoretical basis for these results using an exact fractal construction based on the Koch curve and applying a coarse-grain filtering analysis.