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

Lattice Boltzmann method for interface capturing

Accurately solving phase interface plays a great role in modeling an immiscible multiphase flow system. In this paper, we propose an accurate interface-capturing lattice Boltzmann method from the perspective of the modified Allen-Cahn equation (ACE). The modified ACE is built based on the commonly used conservative formulation via the relation between the signed-distance function and the order parameter also maintaining the mass-conserved characteristic. A suitable forcing term is carefully incorporated into the lattice Boltzmann equation for correctly recovering the target equation. We then test the proposed method by simulating some typical interface-tracking problems of Zalesaks disk rotation, single vortex, deformation field and demonstrate that the present model can be more numerically accurate than the existing lattice Boltzmann models for the conservative ACE, especially at a small interface-thickness scale.

Short-range repulsive force model for near-contact interactions of bubbles

We introduce a short-range repulsive force model to tackle near-contact interactions when the collision occurs between bubbles. In contrast to the previous numerical method adopting the adaptive mesh refinement technique, such a mesoscale model can be applied to a relatively coarse mesh, which can prevent the nonphysical coalescence between bubbles without excessive mesh refinement. We assume that the repulsive force is inversely proportional to the third power of distance, as a reasonable approximation to the short-range phase interactions. The model is validated against two different experiments. In both experiments, two identical bubbles rising side by side were considered. First, the experiment performed in a water-glycerol mixture helps determine the model parameter K. Second, three typical combinations of bubble size and initial separation distance are simulated, presenting different types of interactions, i.e., coalescence, bouncing coalescence, and bouncing separation, agreeing well with the second experiment, which was performed in pure water. Owing to its simplicity, this model can be easily implemented into existing codes, and it can be extended to the case with multiple bubbles or droplets.

Modified radius-weighted lattice Boltzmann model to address singularities in axisymmetric multiphase flows

The radius-weighted lattice Boltzmann model has achieved great success in the simulation of axisymmetric flows. However, severe spurious currents near the axis are observed when this model is extended to simulate axisymmetric multiphase flows. In this study, to determine the origin of this singularity, we conducted a truncation error analysis based on high-order Taylor series expansion and identified the leading error terms through dimensionless analysis. By neglecting the error terms in proportion to the radius, we obtained the final forms of the singular terms in the axisymmetric lattice Boltzmann model. We proposed a modified model by including an additional correction term, to remove the singularity at the third order. We validated the proposed model using numerical tests for flat and spherical interfaces. Results showed that the present modified model reduced the spurious currents near the axis by two orders of magnitude compared with the original model. This modified model also has been successfully applied to predict bubble dynamics in an air-water system. Our numerical results are in excellent agreement with available experimental observations in terms of bubble shapes and terminal velocities.

Discrete unified gas-kinetic scheme for the conservative Allen-Cahn equation

In this paper, two discrete unified gas-kinetic scheme (DUGKS) methods with piecewise-parabolic flux reconstruction are presented for the conservative Allen-Cahn equation (CACE). One includes a temporal derivative of the order parameter in the force term while the other does not include temporal derivative in the force term but results in a modified CACE with additional terms. In the context of DUGKS, the continuum equations recovered from the piecewise-linear and piecewise-parabolic reconstructions for the fluxes at cell faces are subsequently derived. It is proved that the resulting equation with the piecewise-linear reconstruction is a first-order approximation to the discrete velocity kinetic equation due to the presence of the force term and the nonconservation property of the momentum of the collision model. To guarantee second-order accuracy of DUGKS, the piecewise-parabolic reconstruction for numerical flux is proposed. To validate the accuracy of the present DUGKS with the proposed flux evaluation, several benchmark problems, including the diagonal translation of a circular interface, the rotation of a Zalesak disk and the deformation of a circular interface, have been simulated. Numerical results show that the accuracy of both proposed DUGKS methods is almost comparable and improved compared with the DUGKS with linear flux reconstruction scheme.

Phase-field lattice Boltzmann model for two-phase flows with large density ratio

In this work, a lattice Boltzmann (LB) model based on the phase-field method is proposed for simulating large density ratio two-phase flows. An improved multiple-relaxation-time (MRT) LB equation is first developed to solve the conserved Allen-Cahn (AC) equation. By utilizing a nondiagonal relaxation matrix and modifying the equilibrium distribution function and discrete source term, the conserved AC equation can be correctly recovered by the proposed MRT LB equation with no deviation term. Therefore, the calculations of the temporal derivative term in the previous LB models are successfully avoided. Numerical tests demonstrate that satisfactory accuracy can be achieved by the present model to solve the conserved AC equation. What is more, the discrete force term of the MRT LB equation for the incompressible Navier-Stokes equations is also simplified and modified in the present work. An alternative scheme to calculate the gradient terms of the order parameter involved in the discrete force term through the nonequilibrium part of the distribution function is also developed. To validate the ability of the present LB model for simulating large density ratio two-phase flows, series of benchmarks, including two-phase Poiseuille flow, droplet impacting on thin liquid film, and planar Taylor bubble are simulated. It is found that the results predicted by the present LB model agree well with the analytical, numerical, and experimental results.

Instability of a liquid sheet with viscosity contrast in inertial microfluidics

Flows at moderate Reynolds numbers in inertial microfluidics enable high throughput and inertial focusing of particles and cells with relevance in biomedical applications. In the present work, we consider a viscosity-stratified three-layer flow in the inertial regime. We investigate the interfacial instability of a liquid sheet surrounded by a density-matched but more viscous fluid in a channel flow. We use linear stability analysis based on the Orr-Sommerfeld equation and direct numerical simulations with the lattice Boltzmann method (LBM) to perform an extensive parameter study. Our aim is to contribute to a controlled droplet production in inertial microfluidics. In the first part, on the linear stability analysis we show that the growth rate of the fastest growing mode [Formula: see text] increases with the Reynolds number [Formula: see text] and that its wavelength [Formula: see text] is always smaller than the channel width w for sufficiently small interfacial tension [Formula: see text]. For thin sheets we find the scaling relation [Formula: see text], where m is viscosity ratio and [Formula: see text] the sheet thickness. In contrast, for thicker sheets [Formula: see text] decreases with increasing [Formula: see text] or m due to the nearby channel walls. Examining the eigenvalue spectra, we identify Yih modes at the interface. In the second part on the LBM simulations, the thin liquid sheet develops two distinct dynamic states: waves traveling along the interface and breakup into droplets with bullet shape. For smaller flow rates and larger sheet thicknesses, we also observe ligament formation and the sheet eventually evolves irregularly. Our work gives some indication how droplet formation can be controlled with a suitable parameter set [Formula: see text].

Coupled lattice Boltzmann-large eddy simulation model for three-dimensional multiphase flows at large density ratio and high Reynolds number

A coupled lattice Boltzmann-large eddy simulation model is developed for modeling three-dimensional multiphase flows at large density ratios and high Reynolds numbers. In the framework of the lattice Boltzmann method, the model is proposed based on the standard Smagorinsky subgrid-scale approach, and a reconstructed multiple-relaxation-time collision operator is adopted. The conservative Allen-Cahn equation and Navier-Stokes equations are solved through the lattice Boltzmann discretization scheme for the interface tracking and velocity field evolution, respectively. Relevant benchmark cases are carried out to validate the performance of this model in simulating multiphase flows at a large density ratio and a high Reynolds number, including a stationary droplet, the process of spinodal decomposition, the Rayleigh-Taylor instability, the phenomenon of a droplet splashing on a thin liquid film, and the liquid jet breakup process. The maximum values of density ratio and Re number are 1000 and 10 240, respectively. The capability and reliability of the proposed model have been demonstrated by the good agreement between simulation results and the analytical solutions or the previously available results.

Phase-field lattice Boltzmann model for interface tracking of a binary fluid system based on the Allen-Cahn equation

A lattice Boltzmann (LB) model is proposed to track the interface of binary fluid system based on the conservative-form Allen-Cahn (A-C) equation for phase field. Utilizing an equilibrium distribution function and a modified LB equation, this model is able to correctly recover the conservative A-C equation through the Chapman-Enskog analysis. A series of two-dimensional (2D) and three-dimensional (3D) phase-capturing benchmark tests have been conducted for validation, which include the diagonal translation of a circular interface, the rigid-body rotation of a Zalesak disk, and the deformation of 2D circular interface and 3D spherical interface in shear flows, all illustrating better accuracy and stability of the proposed model than the previous models tested. By coupling the incompressible hydrodynamic equation, a stationary droplet, a spinodal decomposition, and the Rayleigh-Taylor instability are simulated as well, showing the satisfying performance of the model in dealing with complex interfaces of binary fluid systems.