An Eulerian label advection method for conservative volume-based tracking of bubbles/droplets

Abstract

We develop a robust volume-conservative framework for tracking blob evolution in complex two-phase flow that accurately and uniquely obtains the volume transfer among bubbles/droplets (blobs). This new framework is built on a volume-tracking matrix (VTM) that quantifies the volume transfer between any two blobs in two separated instances (snapshots) during the evolution, and an efficient Eulerian label advection (ELA) algorithm that explicitly provides the unique, consistent, volume-conservative VTM. Given a set of blobs defined at a snapshot by, say, a connected-component labeling (CCL) method and the grid-level volume-fraction flux from the conservative Volume of Fluid (cVOF) method [1], ELA gives the VTM by solving the Eulerian flow of each blob's fluid through time. Due to its grid-level Eulerian nature, ELA is independent of the complexity of the blob-level evolution, including high-arity (tertiary, quaternary, etc.) events and cycles which prevent previous methods from obtaining the VTM. We prove theoretically that ELA is volume-conservative to machine precision, with the same Courant restriction as cVOF. Furthermore, we show that, by allowing a diffusive error, multiplying the VTM obtains volume-conservative tracking over longer intervals without increasing the computational cost of ELA. We verify all these results using extensive simulations of evolving blob populations in flows with prescribed velocity and isotropic homogeneous turbulence (IHT).