[Paper Review] Comparison of free-surface and conservative Allen-Cahn phase-field lattice Boltzmann method

Christoph Schwarzmeier, Markus Holzer+4 more|arXiv (Cornell University)|Jun 23, 2022

Lattice Boltzmann Simulation Studies66 references25 citations

TL;DR

This study compares the free-surface lattice Boltzmann method (FSLBM) and the conservative Allen–Cahn phase-field lattice Boltzmann method (PFLBM) for simulating two-phase flows dominated by the heavier phase. The FSLBM uses a sharp interface and neglects the lighter phase’s dynamics, while the PFLBM employs a diffuse interface via the conservative Allen–Cahn equation. Key findings show FSLBM achieves higher computational efficiency with lower resolution, while PFLBM offers accurate surface tension modeling but is sensitive to mobility and interface width parameters.

ABSTRACT

This study compares the free-surface lattice Boltzmann method (FSLBM) with the conservative Allen-Cahn phase-field lattice Boltzmann method (PFLBM) in their ability to model two-phase flows in which the behavior of the system is dominated by the heavy phase. Both models are introduced and their individual properties, strengths and weaknesses are thoroughly discussed. Six numerical benchmark cases were simulated with both models, including (i) a standing gravity and (ii) capillary wave, (iii) an unconfined rising gas bubble in liquid, (iv) a Taylor bubble in a cylindrical tube, and (v) the vertical and (vi) oblique impact of a drop into a pool of liquid. Comparing the simulation results with either analytical models or experimental data from the literature, four major observations were made. Firstly, the PFLBM selected was able to simulate flows purely governed by surface tension with reasonable accuracy. Secondly, the FSLBM, a sharp interface model, generally requires a lower resolution than the PFLBM, a diffuse interface model. However, in the limit case of a standing wave, this was not observed. Thirdly, in simulations of a bubble moving in a liquid, the FSLBM accurately predicted the bubble's shape and rise velocity with low computational resolution. Finally, the PFLBM's accuracy is found to be sensitive to the choice of the model's mobility parameter and interface width.

Motivation & Objective

• To evaluate and compare the performance of the free-surface lattice Boltzmann method (FSLBM) and the conservative Allen–Cahn phase-field lattice Boltzmann method (PFLBM) in simulating two-phase flows dominated by the heavier phase.
• To assess the accuracy and computational efficiency of both models across a range of benchmark cases involving gravity, surface tension, and complex interfacial dynamics.
• To investigate the sensitivity of the PFLBM to model parameters such as mobility and interface width in capturing interfacial behavior.
• To validate simulation results against analytical solutions and experimental data for key test cases including standing waves, rising bubbles, and drop impacts.
• To determine the optimal model choice based on resolution requirements, accuracy, and computational cost in high-contrast liquid–gas systems.

Proposed method

• The FSLBM employs a volume-of-fluid approach with a sharp interface captured via an indicator field, modeling only the heavier phase and neglecting dynamics in the lighter phase.
• The PFLBM uses the conservative Allen–Cahn equation to model interfacial dynamics through a diffuse interface, with force terms added to the standard lattice Boltzmann equation.
• Both models are implemented in a parallel computing framework optimized for GPU architectures, enabling large-scale simulations.
• Six benchmark cases were simulated: standing gravity and capillary waves, an unconstrained rising bubble, a Taylor bubble in a tube, and vertical and oblique drop impacts.
• Simulations were cross-validated using independent codebases and compared against analytical models and experimental data from the literature.
• Non-dimensionalized metrics such as splash crown diameter, cavity depth, and interface position were used to quantify accuracy across different resolutions and time steps.

Experimental results

Research questions

• RQ1 How accurately can the PFLBM simulate surface tension-dominated flows, such as capillary waves, compared to analytical solutions?
• RQ2 Does the FSLBM achieve comparable accuracy to the PFLBM with significantly lower computational resolution due to its sharp interface formulation?
• RQ3 How do the PFLBM’s simulation results depend on the choice of mobility and interface width parameters in complex interfacial flows?
• RQ4 To what extent do both models accurately reproduce experimental behaviors in drop impact scenarios, including vertical and oblique impacts?
• RQ5 In which flow regimes does the FSLBM outperform the PFLBM in terms of computational efficiency and resolution requirements?

Key findings

• The PFLBM successfully simulated surface tension-dominated flows, such as capillary waves, with reasonable accuracy, demonstrating its capability for interfacial dynamics modeling.
• The FSLBM required lower resolution than the PFLBM to achieve comparable accuracy in most cases, although this advantage did not hold for the standing gravity wave case.
• For rising bubble simulations, the FSLBM accurately predicted both the bubble’s shape and rise velocity at low computational resolution.
• The PFLBM’s accuracy was found to be highly sensitive to the mobility parameter and interface width, with deviations observed when these were not carefully tuned.
• In vertical drop impact simulations, both models captured the splash crown evolution, but the PFLBM overestimated the crown diameter at higher times, especially at lower resolutions.
• For oblique drop impact, the PFLBM predicted deeper cavities and larger crown diameters than the FSLBM, with both models showing convergence toward experimental trends as resolution increased.