This study presents an efficient algebraic scheme known as MULES for sharp interface advection, verified against various schemes including first-order upwind, second-order central, van Leer flux limiter, and Geometric Volume-of-Fluid (VOF). Two problems involving a droplet in a two-dimensional (2D) vortex and a stationary droplet were examined. The model assessed the effects of the Interface Compression (IC) coefficient, ranging from 0 to 2, analyzing parameters such as Interface Advection Error (IAE) and Mass Conservation Error (MCE). Results indicated that increasing IC values enhanced interface tracking accuracy but introduced non-physical instabilities at higher values, compromising mass conservation. Specifically, the IAE decreased from 4.8% to 3.95% as IC increased from 0 to 2, showing a favorable effect until IC surpassed 1.4, where IAE fluctuated around 4%. Conversely, the MCE rose steeply from 0% to 23.19%, driven by parasitic currents and numerical instabilities. Additionally, MULES and van Leer flux limiter schemes evaluated volume fraction smoothing effects. Initial filtering reduced Dimensionless Pressure Difference (DPD) and Capillary Number (Ca), stabilizing the solution, but excessive filtering reintroduced numerical errors and instabilities. With one filtering step, DPD reduced by 0.23 and Ca dropped significantly by 73.31%, improving solution stability. However, further filtering increased DPD and Ca, reflecting the reintroduction of numerical errors. The maximum velocity of parasitic flow around the droplet initially decreased by almost 75% but increased by 30.92% with excessive filtering. IAE increased from 0.7 to 0.9 with initial filtering, then decreased to 0.63 with additional steps, indicating improved solver performance on smoother interfaces.

翻译：本研究提出了一种用于锐利界面平流的高效代数方案，称为MULES，并通过一阶迎风、二阶中心、van Leer通量限制器以及几何流体体积法等多种方案进行了验证。研究考察了二维涡流中的液滴和静止液滴两个问题。该模型评估了界面压缩系数在0到2范围内的影响，分析了界面平流误差和质量守恒误差等参数。结果表明，增加IC值提高了界面追踪精度，但在较高值时引入了非物理不稳定性，损害了质量守恒。具体而言，当IC从0增加到2时，IAE从4.8%降至3.95%，显示出积极效果，直到IC超过1.4后，IAE在4%左右波动。相反，由于寄生流和数值不稳定性，MCE从0%急剧上升至23.19%。此外，MULES和van Leer通量限制器方案评估了体积分数平滑效应。初始滤波降低了无量纲压差和毛细数，稳定了求解，但过度滤波重新引入了数值误差和不稳定性。经过一步滤波，DPD降低了0.23，Ca显著下降了73.31%，提高了求解稳定性。然而，进一步滤波增加了DPD和Ca，反映了数值误差的重新引入。液滴周围寄生流的最大速度最初降低了近75%，但过度滤波后增加了30.92%。IAE在初始滤波后从0.7增加到0.9，随后在额外滤波步骤后降至0.63，表明求解器在更平滑界面上的性能得到改善。