Numerical methods for multiphase flows

Multiphase flows are ubiquitous in both nature and engineering. Over the past two to three decades, substantial progress has been made in developing numerical methods for simulating these complex flows. Yet, significant challenges persist in accurately capturing intricate interfacial dynamics and the multi-scale interactions inherent to multiphase systems. This review focuses on several key numerical approaches that have proven particularly relevant from both practical and theoretical perspectives. In particular, we discuss Volume-Of-Fluid techniques, level set methods, diffuse interface models, and front tracking methods, along with immersed boundary strategies designed for particle-laden flows. We also examine multi-fluid Eulerian frameworks, population balance models for reactive processes, and sub-grid scale techniques for handling unresolved dynamics. Furthermore, emerging hybrid strategies that integrate conventional numerical methods with data-driven machine learning techniques are highlighted as promising directions. In conclusion, while current methodologies offer valuable insights into multiphase flow behavior, continued interdisciplinary efforts are essential to enhance predictive accuracy, computational efficiency, and the overall applicability of these simulations to real-world challenges.