Adaptive momentum equation method for overcoming singularities of dispersed phases

The singularity issue arising from the phase fraction approaching zero in multiphase flow can significantly intensify the solution difficulty and lead to nonphysical results. By employing the conservative form of momentum equations in high-phase-fraction and discontinuity regions and the phase-intensive form of momentum equations in low-phase-fraction regions, computational reliability can be assured while avoiding the singularity issue. Regarding the proposed adaptive momentum equation method, the form of momentum equations for each cell is determined by a conversion bound and a phase fraction discontinuity detector. A comparative analysis is conducted on this method and other singularity-free methods. For discontinuities of dispersed phases, an error estimation method of the conversion bound is presented through theoretical analysis. Computational results demonstrate that the discontinuity detector accurately captures discontinuities in high-phase-fraction regions while disregarding pseudo-discontinuities in low-phase-fraction regions. Compared to the conservative form corrected by the terminal velocity method, the method yields higher-quality flow fields and potentially exhibits an efficiency improvement of over 10 times. Compared to the phase-intensive form, the method benefits from the physical quantity conservation, providing higher computational reliability. When encountering discontinuities, the expected error from the error estimation method aligns well with the actual error, indicating its effectiveness. When the conversion bound is below 1/10 000 of the inlet phase fraction, the errors of the adaptive method are essentially negligible.