A physics-preserving tool for multiphase flow modeling

A consistent and conservative volume distribution algorithm


Multiphase flows study various interactions of fluids and exhibit a wide range of applications such as oil recovery. For example, it is useful in predicting oil spread during spillage to devise remedial strategies for reducing its environmental impact. Developing such strategies requires effective models capable of capturing air, water, and oil interactions. Under the framework of simulating and modeling multiphase flows, several models like the front-tracking method have been developed to determine the interface locations and to model surface tension.

Currently, Phase-Field Models comprising of a set of order parameters governing the volume fraction of the individual phases are commonly used to determine the locations of various phases due to their simplicity and effectiveness. Many Phase-Field Models, such as the conservative Allen-Cahn model, require the volume distribution functions of every phase at every individual location of a fixed domain to account for mass conservation. When there are more than three fluid phases, the volume distribution functions require satisfying additionally the consistency of reduction and the summation constraint to avoid producing local voids, overfilling, or fictitious phases.

While solving the volume distribution of two-phase problems is relatively straightforward, solving general multiphase problems and satisfying all the aforementioned requirements is often non-trivial and complex. As such, there are limited studies on solving the volume distribution problem in multiphase setups. Herein, Dr. Ziyang Huang, Professor Guang Lin, and Professor Arezoo M. Ardekani from Purdue University addressed the multiphase volume distribution problem that can accommodate an arbitrary number of phases, and the algorithm is physically consistent and conservative and has been applied to different Phase-Field Models. Their work is currently published in the International Journal of Multiphase Flow.

After theoretically analyzing their approach, the authors presented two applications of the proposed algorithm in modeling multiphase flows. In the first application, physical and general Lagrange multipliers were designed to enforce mass conservation for different Phase-Field models, while a multiphase conservative Allen-Cahn model was first developed to satisfy the consistency of reduction. In the second application, a boundedness mapping technique was developed to map the out-of-bound order parameters in the Phase-Field Models back to their physical intervals while preserving their physical properties. Furthermore, the Phase-Field Models were consistently coupled with the momentum equation during multiphase flow simulation to prevent any possible defects due to unphysical velocity fluctuations, interface deformations, etc.

The research team showed that the proposed algorithm satisfied both the summation and conservation constraints as well as the consistency of reduction. Thus, no overfilling voids or fictitious phases were observed after the volume distribution. The multiphase conservative Allen-Cahn model exhibited better performance in preserving the under-resolved structures than the multiphase Cahn-Hilliard model. Additionally, with the established numerical scheme to solve the governing equations, the resulting numerical solutions were bounded, conservative, and reduction consistent. These observations were analyzed theoretically and validated numerically. Furthermore, the consistency of mass and momentum transport and consistency of mass conservation were also satisfied.

In summary, Purdue University scientists reported the consistent and conservative study of the general multiphase volume distribution problem using the proposed algorithm and demonstrated their two applications. The authors successfully demonstrated the ability of the proposed models and their schemes to capture the complicated multiphase dynamics even in the presence of a large density and viscosity ratio. It is worth noting that the multiphase volume distribution problem plays a crucial role in developing multiphase Phase-Field models. In a statement to Advances in Engineering, the authors explained that the proposed volume distribution algorithm would advance multiphase flow modeling and simulation and provide physics-preserving data for natural phenomena and industrial applications.

About the author

Dr. Ziyang Huang is a postdoctoral research fellow at Mechanical Engineering, University of Michigan, Ann Arbor. He is interested in developing numerical models and physics-preserved schemes for multi-phase, multi-material, and multi-physics problems. He has contributed to the development of the consistent and conservative Phase-Field method for multi-phase flows including heat and mass transfer and phase change, and published a series of articles in leading journals including Journal of Computational Physics, Journal of Computational and Applied Mathematics, and International Journal of Multiphase Flow. He also served as a reviewer of Journal of Computational Physics and Computer Methods in Applied Mechanics and Engineering. Before joining the Scientific Computing and Flow Physics Laboratory with Prof. Eric Johnsen at University of Michigan, Ziyang received his Ph.D. from Purdue University in 2021, co-advised by Profs. Arezoo M. Ardekani and Guang Lin.

About the author

Guang Lin is a Full Professor in the School of Mechanical Engineering and Department of Mathematics at Purdue University. Prof. Guang Lin is the Director of Data Science Consulting Service that performs cutting-edge research on data science and provides hands-on consulting support for data analysis and business analytics. He is also the Chair of the Initiative for Data Science and Engineering Applications at the College of Engineering. Lin received his Ph.D. from Brown University in 2007 and worked as a Research Scientist at DOE Pacific Northwest National Laboratory before joining Purdue in 2014. Prof. Lin has received various awards, such as the NSF CAREER Award, Mid-Career Sigma Xi Award, University Faculty Scholar, Mathematical Biosciences Institute Early Career Award, and Ronald L. Brodzinski Award for Early Career Exception Achievement.

About the author

Dr. Ardekani is a Professor of Mechanical Engineering at Purdue University. Her research focuses on suspensions of particles and swimmers, biological flows, and complex fluids. Honored with the Presidential Early Career Award for Scientists and Engineers (PECASE) from President Obama, Arezoo has also received an NSF CAREER Award, the Arthur B. Metzner Early Career Award from the Society of Rheology, the Society of Engineering Science Young Investigator Medal, the Sigma Xi Mid-career Research Award, and is named a Purdue University Faculty Scholar. A Fellow of American Society of Mechanical Engineers, Arezoo has also received the College of Engineering Faculty Excellence Awards for Graduate Student Mentorship and Early Career Research, the Amelia Earhart Award, and the Society of Women Engineers Award. She received her Ph.D. from University of California Irvine in 2009 and was a Shapiro Postdoctoral Fellow at MIT. Arezoo has published 110 articles in leading journals including Proceedings of the National Academy of Sciences, Physical Review Letters, Journal of Computational Physics, Journal of Fluid Mechanics, Physical Review Fluids, and presented more than 70 invited/keynote talks. Arezoo is an Associate Editor of ASME Applied Mechanics Review, an Editorial Advisory Board Member of International Journal of Multiphase Flow, Journal of Non-Newtonian Fluid Mechanics, Physics of Fluids, and Physical Review Fluids, and a member of the American Physical Society-Division of Fluid Dynamics Executive Committee.


Huang, Z., Lin, G., & Ardekani, A. (2021). A consistent and conservative volume distribution algorithm and its applications to multiphase flows using Phase-Field modelsInternational Journal of Multiphase Flow142, 103727.

Go To International Journal of Multiphase Flow

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