Assessing the ability of the Eulerian-Eulerian and the Eulerian-Lagrangian frameworks to capture meso-scale dynamics in bubbly flows

Abstract

Bubbly flows are characterised by instabilities that appear in the form of elongated meso-scale structures aligned along the direction of gravity. The instabilities result in a non-homogeneous distribution of the gas fraction in the system where high-cluster and high-voidage regions coexist. The correct prediction of the meso-scale dynamics is fundamental to formulate more accurate closure models for coarse-grained simulations applied to design systems of industrial scale. Two different frameworks are compared to test their capability of capturing the characteristic meso-scale structures (understood as hetereogeneities in the gas fraction distribution with dimensions larger than the bubble size and smaller than the typical lengths of the system): the Eulerian-Eulerian two-fluid model and a Eulerian-Lagrangian approach. This kind of benchmark has been already applied to gas-particle flows, but based to the best of our knowledge, this is the first time when the comparison between different frameworks is used to investigate the meso-scale flow structures of bubbly suspensions. We show that the Eulerian-Eulerian simulations are affected at low bubble loadings by unphysical numerical instabilities appearing due to the lack of hyperbolicity of the governing equation system. Unfortunately, the occurrence of the numerical instabilities cannot be predicted a priori, but when they are not present in the solution, the two frameworks are able to predict the same meso-scale dynamics. Our analysis suggests that, concerning meso-scale simulations, the Eulerian-Lagrangian approach produces physically faithful results and represents an ideal framework to formulate new closure models. The well-known limit of this methodology to extract parameter-independent Eulerian statistics has been addressed linking the post-processing technique to physical phenomenologies. In particular, we show that a characteristic post-processing length scale smaller than two bubble diameters is needed to correctly capture the phenomenologies associated with bubble small-scale clusters.