Modeling and simulations of high-density two-phase flows using projection-based Cahn-Hilliard Navier-Stokes equations

TL;DR

The paper addresses accurately simulating high-density-ratio two-phase flows by combining a thermodynamically consistent Cahn–Hilliard–Navier–Stokes model with a projection-based solver and variational multiscale stabilization. Adaptive octree AMR enables scalable 3D simulations at density ratios up to $10^5$:1, while a careful time-stepping and solver strategy preserves energy stability and mass conservation. The authors validate the framework through manufactured solutions, capillary waves, and bubble-rise benchmarks, and apply it to Oscillating Droplet Method–like ODM problems for molten metals in microgravity, achieving interfacial tensions within about $5\%$ of experimental values. The work demonstrates accurate, efficient resolution of interfacial dynamics in extreme density contrasts and provides a robust tool for material science and manufacturing applications, with clear directions for extending to more complex multi-phase and MHD scenarios.

Abstract

Accurately modeling the dynamics of high-density ratio ($\mathcal{O}(10^5)$) two-phase flows is important for many material science and manufacturing applications. This work considers numerical simulations of molten metal oscillations in microgravity to analyze the interplay between surface tension and density ratio, a critical factor for terrestrial manufacturing applications. We present a projection-based computational framework for solving a thermodynamically-consistent Cahn-Hilliard Navier-Stokes equations for two-phase flows with large density ratios. The framework employs a modified version of the pressure-decoupled solver based on the Helmholtz-Hodge decomposition presented in Khanwale et al. [{\it A projection-based, semi-implicit time-stepping approach for the Cahn-Hilliard Navier-Stokes equations on adaptive octree meshes.}, Journal of Computational Physics 475 (2023): 111874]. We validate our numerical method on several canonical problems, including the capillary wave and single bubble rise problems. We also present a comprehensive convergence study to investigate the effect of mesh resolution, time-step, and interfacial thickness on droplet-shape oscillations. We further demonstrate the robustness of our framework by successfully simulating three distinct physical systems with extremely large density ratios ($10^4$-$10^5:1$), achieving results that have not been previously reported in the literature.

Figures (13)

• Figure 1: Illustration of 1:2 load balancing constraint in a 2D spatial domain Sundar2008. (a) Before load balancing; (b) After load balancing (where the element $h$ is subdivided into $h_1,h_2,h_3,h_4$.)
• Figure 2: Illustration of top-down and bottom-up traversals in a 2D tree with quadratic element order Saurabh2021Scalable. The leftmost figure shows the unique shared nodes (color-coded by tree level). During top-down traversal, shared nodes across children of a parent node are recursively duplicated, propagating downward until the leaf nodes are reached. At the leaf level, any missing local elemental nodes are interpolated from their immediate parent (as shown in the rightmost figure). The bottom-up traversal then aggregates the duplicated nodes, merging them into a consistent global structure.
• Figure 3: Manufactured Solution: Panel (a) Temporal convergence using $h=1/2^8$; (b) spatial convergence using $\delta t = 5 \times 10^{-4}$.
• Figure 4: Schematic diagram of the capillary wave problem: computational domain and boundary conditions.
• Figure 5: Comparison of the numerical and analytical capillary wave amplitudes for $\rho_+ / \rho_- = 1$ at different Reynolds numbers. Panel a) shows results for $Re=50$; panel b) shows results for $Re=100$; panel c) shows results for $Re=200$; and panel d) shows results for $Re=500$.

Theorems & Definitions (5)

• Definition 1
• Definition 2
• Remark 1
• Remark 2
• Definition 3