twoWayGPBEFoam: An open-source Eulerian QBMM solver for monokinetic bubbly flows

twoWayGPBEFoam and oneWayGPBEFoam are an open-source meso-scale Eulerian-QBMM solvers for multiphase flows, implemented within the OpenFOAM software framework. Compared with the existing macroscopic Eulerian-Eulerian (E-E) solver twoPhaseEulerFoam, it can predict the size segregation phenomenon and the size-conditioned velocities of the disperse phase. On theoretically grounds, the evolution of the disperse phase in multiphase flows is controlled by the generalized population balance equation (GPBE). The GPBE can be transformed into the moments transport equations and solved by the finite-volume method with higher-order realizable spatial-discretization schemes and time-integration schemes. In order to address the closure problems of the size-conditioned spatial flux, the size-conditioned velocities need to be modelled. In previous works on CFD-PBE coupling, the size-conditioned velocities are assumed to be identical with the disperse phase velocity predicted by the E-E method. In this work, it was modelled by the velocity polynomial approximation (VPA), for which the velocity polynomial coefficients (VPCs) can be obtained from the moments themselves. Therefore, the disperse phase momentum equation in the E-E method is discarded in this approach. Meanwhile, the continuous phase is modelled by the Navier-Stokes equations, and is fully coupled with the moments transport equations of the disperse phase by the phase fraction and the interfacial momentum exchange terms. By several test cases with both one-way and two-way coupling, we show that the results predicted by the oneWayGPBEFoam and twoWayGPBEFoam agree well with the analytical solutions and the existing experimental data.