Scale- and Structure-Dependent Fractal Dimensions in a Two-Dimensional Atomizing Liquid Jet

TL;DR

This study uses two-dimensional numerical simulations to analyze how fractal dimensions of a liquid jet depend on scale and structure, revealing multiple scaling ranges and a hierarchy of interface features.

Contribution

It demonstrates that fractal dimensions in atomizing jets are scale- and structure-dependent, challenging the notion of a single universal fractal exponent.

Findings

Two distinct scaling ranges are identified separated by a crossover near Lbox 7.

Different interface structures (droplets, ligaments, main body) have different effective dimensions.

The hierarchy of dimensions persists across a wide range of Reynolds numbers.

Abstract

Atomization stretches and folds the liquid-gas interface before fragmenting it into ligaments and droplets, making fractal measures a natural descriptor of the breakup state. We examine this idea in two-dimensional volume-of-fluid direct numerical simulations, VOF-DNS, of a liquid jet with adaptive mesh refinement in Basilisk. Box counting of the full resolved interface does not yield a single scale-independent exponent. Instead, two scaling ranges appear, separated by a crossover near box-counting level Lbox about 7: coarser boxes measure the folded connected jet envelope, whereas finer boxes increasingly sample ligaments, droplets, and nearly smooth local interface segments. Decomposing the interface into detached droplets, ligaments, and the connected main body shows that the relevant effective dimension is structure dependent. Droplets remain near Euclidean at fine scales, ligaments occupy an intermediate level, and the main body carries the largest coarse-scale dimension. This hierarchy persists for liquid Reynolds numbers from 100 to 10000 at fixed gas Weber number 200. Thus, in this two-dimensional VOF-DNS setting, fractal dimension is best interpreted not as a single global exponent, but as a scale- and structure-resolved state variable for interfacial folding and breakup.