Study of phase-field lattice Boltzmann models based on the conservative Allen-Cahn equation.

Conservative phase-field (CPF) equations based on the Allen-Cahn model for interface tracking in multiphase flows have become more popular in recent years, especially in the lattice-Boltzmann (LB) community. This is largely due to their simplicity and improved efficiency and accuracy over their Cahn-Hilliard-based counterparts. Additionally, the improved locality of the resulting LB equation (LBE) for the CPF models makes them more ideal candidates for LB simulation of multiphase flows on nonuniform grids, particularly within an adaptive-mesh refinement framework and massively parallel implementation. In this regard, some modifications-intended as improvements-have been made to the original CPF-LBE proposed by Geier et al. [Phys. Rev. E 91, 063309 (2015)PLEEE81539-375510.1103/PhysRevE.91.063309] which require further examination. The goal of the present study is to conduct a comparative investigation into the differences between the original CPF model proposed by Geier et al. [Phys. Rev. E 91, 063309 (2015)PLEEE81539-375510.1103/PhysRevE.91.063309] and the so-called improvements proposed by Ren et al. [Phys. Rev. E 94, 023311 (2016)2470-004510.1103/PhysRevE.94.023311] and Wang et al. [Phys. Rev. E 94, 033304 (2016)2470-004510.1103/PhysRevE.94.033304]. Using the Chapman-Enskog analysis, we provide a detailed derivation of the governing equations in each model and then examine the efficacy of the above-mentioned models for some benchmark problems. Several test cases have been designed to study different configurations ranging from basic yet informative flows to more complex flow fields, and the results are compared with finite-difference simulations. Furthermore, as a development of the previously proposed CPF-LBE model, axisymmetric formulations for the proposed model by Geier et al. [Phys. Rev. E 91, 063309 (2015)PLEEE81539-375510.1103/PhysRevE.91.063309] are derived and presented. Finally, two benchmark problems are designed to compare the proposed axisymmetric model with the analytical solution and previous work. We find that the accuracy of the model for interface tracking is roughly similar for different models at high viscosity ratios, high density ratios, and relatively high Reynolds numbers, while the original CFP-LBE without the additional time-dependent terms outperforms the so-called improved models in terms of efficiency, particularly on distributed parallel machines.

Comprehensive comparison of pore-scale models for multiphase flow in porous media.

Multiphase flows in porous media are important in many natural and industrial processes. Pore-scale models for multiphase flows have seen rapid development in recent years and are becoming increasingly useful as predictive tools in both academic and industrial applications. However, quantitative comparisons between different pore-scale models, and between these models and experimental data, are lacking. Here, we perform an objective comparison of a variety of state-of-the-art pore-scale models, including lattice Boltzmann, stochastic rotation dynamics, volume-of-fluid, level-set, phase-field, and pore-network models. As the basis for this comparison, we use a dataset from recent microfluidic experiments with precisely controlled pore geometry and wettability conditions, which offers an unprecedented benchmarking opportunity. We compare the results of the 14 participating teams both qualitatively and quantitatively using several standard metrics, such as fractal dimension, finger width, and displacement efficiency. We find that no single method excels across all conditions and that thin films and corner flow present substantial modeling and computational challenges.

Numerical simulation of three-component multiphase flows at high density and viscosity ratios using lattice Boltzmann methods.

In this paper, we propose a multiphase lattice Boltzmann model for numerical simulation of ternary flows at high density and viscosity ratios free from spurious velocities. The proposed scheme, which is based on the phase-field modeling, employs the Cahn-Hilliard theory to track the interfaces among three different fluid components. Several benchmarks, such as the spreading of a liquid lens, binary droplets, and head-on collision of two droplets in binary- and ternary-fluid systems, are conducted to assess the reliability and accuracy of the model. The proposed model can successfully simulate both partial and total spreadings while reducing the parasitic currents to the machine precision.

Extended lattice Boltzmann scheme for droplet combustion.

The available lattice Boltzmann (LB) models for combustion or phase change are focused on either single-phase flow combustion or two-phase flow with evaporation assuming a constant density for both liquid and gas phases. To pave the way towards simulation of spray combustion, we propose a two-phase LB method for modeling combustion of liquid fuel droplets. We develop an LB scheme to model phase change and combustion by taking into account the density variation in the gas phase and accounting for the chemical reaction based on the Cahn-Hilliard free-energy approach. Evaporation of liquid fuel is modeled by adding a source term, which is due to the divergence of the velocity field being nontrivial, in the continuity equation. The low-Mach-number approximation in the governing Navier-Stokes and energy equations is used to incorporate source terms due to heat release from chemical reactions, density variation, and nonluminous radiative heat loss. Additionally, the conservation equation for chemical species is formulated by including a source term due to chemical reaction. To validate the model, we consider the combustion of n-heptane and n-butanol droplets in stagnant air using overall single-step reactions. The diameter history and flame standoff ratio obtained from the proposed LB method are found to be in good agreement with available numerical and experimental data. The present LB scheme is believed to be a promising approach for modeling spray combustion.

Improved locality of the phase-field lattice-Boltzmann model for immiscible fluids at high density ratios.

Based on phase-field theory, we introduce a robust lattice-Boltzmann equation for modeling immiscible multiphase flows at large density and viscosity contrasts. Our approach is built by modifying the method proposed by Zu and He [Phys. Rev. E 87, 043301 (2013)PLEEE81539-375510.1103/PhysRevE.87.043301] in such a way as to improve efficiency and numerical stability. In particular, we employ a different interface-tracking equation based on the so-called conservative phase-field model, a simplified equilibrium distribution that decouples pressure and velocity calculations, and a local scheme based on the hydrodynamic distribution functions for calculation of the stress tensor. In addition to two distribution functions for interface tracking and recovery of hydrodynamic properties, the only nonlocal variable in the proposed model is the phase field. Moreover, within our framework there is no need to use biased or mixed difference stencils for numerical stability and accuracy at high density ratios. This not only simplifies the implementation and efficiency of the model, but also leads to a model that is better suited to parallel implementation on distributed-memory machines. Several benchmark cases are considered to assess the efficacy of the proposed model, including the layered Poiseuille flow in a rectangular channel, Rayleigh-Taylor instability, and the rise of a Taylor bubble in a duct. The numerical results are in good agreement with available numerical and experimental data.

Conservative phase-field lattice Boltzmann model for interface tracking equation.

Based on the phase-field theory, we propose a conservative lattice Boltzmann method to track the interface between two different fluids. The presented model recovers the conservative phase-field equation and conserves mass locally and globally. Two entirely different approaches are used to calculate the gradient of the phase field, which is needed in computation of the normal to the interface. One approach uses finite-difference stencils similar to many existing lattice Boltzmann models for tracking the two-phase interface, while the other one invokes central moments to calculate the gradient of the phase field without any finite differences involved. The former approach suffers from the nonlocality of the collision operator while the latter is entirely local making it highly suitable for massive parallel implementation. Several benchmark problems are carried out to assess the accuracy and stability of the proposed model.

Finite-difference lattice Boltzmann method with a block-structured adaptive-mesh-refinement technique.

An adaptive-mesh-refinement (AMR) algorithm for the finite-difference lattice Boltzmann method (FDLBM) is presented in this study. The idea behind the proposed AMR is to remove the need for a tree-type data structure. Instead, pointer attributes are used to determine the neighbors of a certain block via appropriate adjustment of its children identifications. As a result, the memory and time required for tree traversal are completely eliminated, leaving us with an efficient algorithm that is easier to implement and use on parallel machines. To allow different mesh sizes at separate parts of the computational domain, the Eulerian formulation of the streaming process is invoked. As a result, there is no need for rescaling the distribution functions or using a temporal interpolation at the fine-coarse grid boundaries. The accuracy and efficiency of the proposed FDLBM AMR are extensively assessed by investigating a variety of vorticity-dominated flow fields, including Taylor-Green vortex flow, lid-driven cavity flow, thin shear layer flow, and the flow past a square cylinder.

Multiple-relaxation-time lattice Boltzmann method for immiscible fluids at high Reynolds numbers.

The lattice Boltzmann method for immiscible multiphase flows with large density ratio is extended to high Reynolds number flows using a multiple-relaxation-time (MRT) collision operator, and its stability and accuracy are assessed by simulating the Kelvin-Helmholtz instability. The MRT model is successful at damping high-frequency oscillations in the kinetic energy emerging from traveling waves generated by the inclusion of curvature. Numerical results are shown to be in good agreement with prior studies using adaptive mesh refinement techniques applied to the Navier-Stokes equations. Effects of viscosity and surface tension, as well as density ratio, are investigated in terms of the Reynolds and Weber numbers. It is shown that increasing the Reynolds number results in a more chaotic interface evolution and eventually shattering of the interface, while surface tension is shown to have a stabilizing effect.

Phase-field modeling by the method of lattice Boltzmann equations.

In this paper, at first, a lattice Boltzmann method for binary fluids, which is applicable at low viscosity values, is developed. The presented scheme is extension of the free-energy-based approach to a multi-relaxation-time collision model. Various benchmark problems such as the well-known Laplace law for stationary bubbles and capillary-wave test are conducted for validation. As an appealing application, instability of a rising bubble in an enclosed duct is studied and irregular behavior of the bubble is observed at very high Reynolds numbers. In order to highlight its capability to simulate high Reynolds number flows, which is a challenge for many other models, a typical wobbling bubble in the turbulent regime is simulated successfully. Then, in the context of phase-field modeling, a lattice Boltzmann method is proposed for multiphase flows with a density contrast. Unlike most of the previous models based on the phase-field theory, the proposed scheme not only tolerates very low viscosity values but also emerges as a promising method for investigation of two-phase flow problems with moderate density ratios. In addition to comparison to the kinetic-based model, the proposed approach is further verified by judging against the theoretical solutions and experimental data. Various case studies including the rising bubble, droplet splashing on a wet surface, and falling droplet are conducted to show the versatility of the presented lattice Boltzmann model.