Assessment of an energy-based surface tension model for simulation of two-phase flows using second-order phase field methods

Second-order phase field models have emerged as an attractive option for capturing the advection of interfaces in two-phase flows. Prior to these, state-of-the-art models based on the Cahn-Hilliard equation, which is a fourth-order equation, allowed for the derivation of surface tension models through thermodynamic arguments. In contrast, the second-order phase field models do not follow a known energy law, and deriving a surface tension term for these models using thermodynamic arguments is not straightforward. In this work, we justify that the energy-based surface tension model from the Cahn-Hilliard context can be adopted for second-order phase field models as well and assess its performance. We test the surface tension model on three different second-order phase field equations; the conservative diffuse interface model of Chiu and Lin [1], and two models based on the modified Allen-Cahn equation introduced by Sun and Beckermann [2]. Additionally, we draw the connection between the energy-based model with a localized variation of the continuum surface force (CSF) model. Using canonical tests, we illustrate the lower magnitude of spurious currents, better accuracy, and superior convergence properties of the energy-based surface tension model compared to the CSF model, which is a popular choice used in conjunction with second-order phase field methods, and the localized CSF model. Importantly, in terms of computational expense and parallel efficiency, the energy-based model incurs no penalty compared to the CSF models.Comment: 13 pages, 5 figures, Revision submitted to Journal of Computational Physic

Inverse asymptotic treatment: capturing discontinuities in fluid flows via equation modification

A major challenge in developing accurate and robust numerical solutions to multi-physics problems is to correctly model evolving discontinuities in field quantities, which manifest themselves as interfaces between different phases in multi-phase flows, or as shock and contact discontinuities in compressible flows. When a quick response is required to rapidly emerging challenges, the complexity of bespoke discretization schemes impedes a swift transition from problem formulation to computation, which is exacerbated by the need to compose multiple interacting physics. We introduce "inverse asymptotic treatment" (IAT) as a unified framework for capturing discontinuities in fluid flows that enables building directly computable models based on off-the-shelf numerics. By capturing discontinuities through modifications at the level of the governing equations, IAT can seamlessly handle additional physics and thus enable novice end users to quickly obtain numerical results for various multi-physics scenarios. We outline IAT in the context of phase-field modeling of two-phase incompressible flows, and then demonstrate its generality by showing how localized artificial diffusivity (LAD) methods for single-phase compressible flows can be viewed as instances of IAT. Through the real-world example of a laminar hypersonic compression corner, we illustrate IAT's ability to, within just a few months, generate a directly computable model whose wall metrics predictions for sufficiently small corner angles come close to that of NASA's VULKAN-CFD solver. Finally, we propose a novel LAD approach via "reverse-engineered" PDE modifications, inspired by total variation diminishing (TVD) flux limiters, to eliminate the problem-dependent parameter tuning that plagues traditional LAD. We demonstrate that, when combined with second-order central differencing, it can robustly and accurately model compressible flows