An SPH modeling of bubble rising and coalescing in three dimensions

The numerical simulation of bubbly flows is challenging due to the unstable multiphase interfaces with large density ratios and viscous ratios. The multiphase Smoothed Particle Hydrodynamics (SPH) method is applied in this paper to simulate the phenomena of bubbles rising and coalescing in three dimensions. Firstly, the multiphase SPH model is introduced in detail, including the derivation of the discretized governing equations based on the principle of virtual work, the viscous force, the multiphase interface treatment, the time-stepping scheme, the boundary implementation, etc. Considering the expensive computational cost in three-dimensional (3-D) SPH simulations, the effects of the scale of the computational domain and the density ratio on the multiphase interface are numerically investigated in order to decrease the amount of calculation. Afterwards, several cases of single bubbles rising through viscous fluids are tested and the SPH results are validated by both the experimental data and other numerical results in the literature. Furthermore, the phenomena of bubbles coalescing in both vertical and horizontal directions are simulated and the results agree well with the experimental data. It is found that the background pressure in the equation of state is essential to keep the multiphase interface smooth and stable when the Bond number is relatively small. The fair agreements between the results of SPH and other reference results demonstrate that the present multiphase SPH model is robust and stable enough to accurately simulate the dynamic phenomena of rising bubbles in different conditions, which can give a reference for engineering applications.