Numerical treatments for large eddy simulations of liquid-liquid dispersions via population balance equation

In this work, we employ the two-fluid model under the large eddy simulations (LES) framework to investigate liquid-liquid dispersions in stirred tanks. The population balance equation was solved by the one primary and one secondary particle method, which was proven as identical as one-node quadrature method of moments. First, Aiyer's break-age kernel was investigated for its capability in the context of chemical stirred tank applications [Aiyer et al., "A population balance model for large eddy simulation of polydisperse droplet evolution," J. Fluid Mech. 878, 700-739 (2019)]. Second, two new methods were proposed to handle the consistency problem and boundedness problem. These numerical problems were shown in our previous studies but had never been discussed in detail. Three test cases were launched, and results showed that our implementation ensures the moments' boundedness. The inconsistency problem was also treated properly. The predicted diameter also agrees well with experiments. Meanwhile, the phase segregation problem as observed in the unsteady Reynolds-Averaged Navier-Stokes simulations disappeared when a LES turbulence model was employed.