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.
Huang, Z., Lin, G., & Ardekani, A. (2021). A consistent and conservative volume distribution algorithm and its applications to multiphase flows using Phase-Field models. International Journal of Multiphase Flow, 142, 103727.