Improved forcing scheme in pseudopotential lattice Boltzmann methods for multiphase flow at arbitrarily high density ratios

Color-gradient lattice Boltzmann model for immiscible fluids with density contrast

We present a color-gradient-based lattice Boltzmann model for immiscible fluids with a large density contrast. The model employs the velocity-based equilibrium distribution function, initially proposed for the phase-field-based model by Zu and He [Phys. Rev. E 87, 043301 (2013)1539-375510.1103/PhysRevE.87.043301], with a modification necessary to satisfy the kinematic condition at the interface. Different from the existing color-gradient models, the present model allows to specify interface mobility that is independent of the fluid density ratio. Further, we provide a unified framework, which uses the recursive representation of the lattice Boltzmann equation, to derive the governing equations of the system. The emergent color dynamics thus obtained, through an analysis of the segregation operator, is shown to obey the locally conservative Allen-Cahn equation. We use a series of benchmarks, which include a stationary drop, a layered Poiseuille flow, translation of a drop under a forced velocity field, the Rayleigh-Taylor instability, and the capillary intrusion test to demonstrate the model's ability in dealing with complex flow problems.

Unified lattice Boltzmann method with improved schemes for multiphase flow simulation: Application to droplet dynamics under realistic conditions

As a powerful mesoscale approach, the lattice Boltzmann method (LBM) has been widely used for the numerical study of complex multiphase flows. Recently, Luo et al. [Philos. Trans. R. Soc. A: Math. Phys. Eng. Sci. 379, 20200397 (2021)10.1098/rsta.2020.0397] proposed a unified lattice Boltzmann method (ULBM) to integrate the widely used lattice Boltzmann collision operators into a unified framework. In this study, we incorporate additional features into this ULBM in order to simulate multiphase flow under realistic conditions. A nonorthogonal moment set [Fei et al., Phys. Rev. E 97, 053309 (2018)10.1103/PhysRevE.97.053309] and the entropic-multi-relaxation-time (KBC) lattice Boltzmann model are used to construct the collision operator. An extended combined pseudopotential model is proposed to realize multiphase flow simulation at high-density ratio with tunable surface tension over a wide range. The numerical results indicate that the improved ULBM can significantly decrease the spurious velocities and adjust the surface tension without appreciably changing the density ratio. The ULBM is validated through reproducing various droplet dynamics experiments, such as binary droplet collision and droplet impingement on superhydrophobic surfaces. Finally, the extended ULBM is applied to complex droplet dynamics, including droplet pancake bouncing and droplet splashing. The maximum Weber number and Reynolds number in the simulation reach 800 and 7200, respectively, at a density ratio of 1000. The study demonstrates the generality and versatility of ULBM for incorporating schemes to tackle challenging multiphase problems.

Shaping the equation of state to improve numerical accuracy and stability of the pseudopotential lattice Boltzmann method

It has recently been shown that altering the shape of the metastable and unstable branches of an equation of state (EOS) can substantially improve the numerical accuracy of liquid and vapor densities in the pseudopotential lattice Boltzmann method [Peng et al., Phys. Rev. E 101, 063309 (2020)2470-004510.1103/PhysRevE.101.063309]. We found that this approach reduces stability of the method in nonequilibrium conditions and is unstable for bubbles at low reduced temperatures. Here we present an improved method for altering the shape of the metastable and unstable branches of the EOS which remains stable for both equilibrium and nonequilibrium situations and has no issues with bubbles. We also performed a detailed study of the stability of the methods for a droplet impact on a liquid film for reduced temperatures down to 0.35 with Reynolds number of 300. Our approach remained stable for a density ratio of up to 3.38×10^{4}.

A unified lattice Boltzmann model and application to multiphase flows

In this work, we develop a unified lattice Boltzmann model (ULBM) framework that can seamlessly integrate the widely used lattice Boltzmann collision operators, including the Bhatnagar-Gross-Krook or single-relation-time, multiple-relaxation-time, central-moment or cascaded lattice Boltzmann method and multiple entropic operators (KBC). Such a framework clarifies the relations among the existing collision operators and greatly facilitates model comparison and development as well as coding. Importantly, any LB model or treatment constructed for a specific collision operator could be easily adopted by other operators. We demonstrate the flexibility and power of the ULBM framework through three multiphase flow problems: the rheology of an emulsion, splashing of a droplet on a liquid film and dynamics of pool boiling. Further exploration of ULBM for a wide variety of phenomena would be both realistic and beneficial, making the LBM more accessible to non-specialists. This article is part of the theme issue 'Progress in mesoscale methods for fluid dynamics simulation'.

Kinetic Simulations of Compressible Non-Ideal Fluids: From Supercritical Flows to Phase-Change and Exotic Behavior

We investigate a kinetic model for compressible non-ideal fluids. The model imposes the local thermodynamic pressure through a rescaling of the particle’s velocities, which accounts for both long- and short-range effects and hence full thermodynamic consistency. The model is fully Galilean invariant and treats mass, momentum, and energy as local conservation laws. The analysis and derivation of the hydrodynamic limit is followed by the assessment of accuracy and robustness through benchmark simulations ranging from the Joule–Thompson effect to a phase-change and high-speed flows. In particular, we show the direct simulation of the inversion line of a van der Waals gas followed by simulations of phase-change such as the one-dimensional evaporation of a saturated liquid, nucleate, and film boiling and eventually, we investigate the stability of a perturbed strong shock front in two different fluid mediums. In all of the cases, we find excellent agreement with the corresponding theoretical analysis and experimental correlations. We show that our model can operate in the entire phase diagram, including super- as well as sub-critical regimes and inherently captures phase-change phenomena.

Force methods for the two-relaxation-times lattice Boltzmann

The two-relaxation-times collision benefits the steady lattice Boltzmann method by yielding viscosity-independent numerical errors. We present in an intuitive way how to incorporate popular force methods into the two-relaxation-times collision. We subsequently rewrite force methods into a generic equation to reveal commonalities and differences. We prove that force methods with a second-order velocity moment of the force break the viscosity independence. A force method with only a first-order velocity moment of the force averts this breakage. We validate our proof numerically.

Achieving thermodynamic consistency in a class of free-energy multiphase lattice Boltzmann models

The free-energy lattice Boltzmann (LB) model is one of the major multiphase models in the LB community. The present study is focused on a class of free-energy LB models in which the divergence of thermodynamic pressure tensor or its equivalent form expressed by the chemical potential is incorporated into the LB equation via a forcing term. Although this class of free-energy LB models may be thermodynamically consistent at the continuum level, it suffers from thermodynamic inconsistency at the discrete lattice level owing to numerical errors [Guo et al., Phys. Rev. E 83, 036707 (2010)10.1103/PhysRevE.83.036707]. The numerical error term mainly includes two parts: one comes from the discrete gradient operator and the other can be identified in a high-order Chapman-Enskog analysis. In this paper, we propose an improved scheme to eliminate the thermodynamic inconsistency of the aforementioned class of free-energy LB models. The improved scheme is constructed by modifying the equation of state of the standard LB equation, through which the discretization of ∇(ρc_{s}^{2}) is no longer involved in the force calculation and then the numerical errors can be significantly reduced. Numerical simulations are subsequently performed to validate the proposed scheme. The numerical results show that the improved scheme is capable of eliminating the thermodynamic inconsistency and can significantly reduce the spurious currents in comparison with the standard forcing-based free-energy LB model.

Three-dimensional pseudopotential lattice Boltzmann model for multiphase flows at high density ratio

In this study, we extend the pseudopotential lattice Boltzmann model proposed by Huang and Wu [J. Comput. Phys. 327, 121 (2016)10.1016/j.jcp.2016.09.030] to a three-dimensional model for practical simulations of multiphase flows with high density ratio. In this model, an additional source term is introduced into the evolution function, and the performed high-order Chapman-Enskog analysis demonstrates that the Navier-Stokes equations with accurate pressure tensor are recovered. Also, an alternative geometric formulation is developed to obtain various contact angles and an iteration scheme is involved in the initialization to improve the stability of the model. Theoretical and numerical investigations both validate that the thermodynamic consistency and tuning surface tension independently of density ratio is achieved through varying the two free parameters in the source term. Numerical simulations of droplet wetting indicate that a large degree range of contact angles can be precisely realized with the implementation of the wetting boundary scheme. Further dynamic examinations of droplet impingement on a thin film and a dry surface also verify the stability and capability of the proposed pseudopotential lattice Boltzmann model.

Force approach for the pseudopotential lattice Boltzmann method

One attractive feature of the original pseudopotential method consists on its simplicity of adding a force dependent on a nearest-neighbor potential function. In order to improve the method, regarding thermodynamic consistency and control of surface tension, different approaches were developed in the literature, such as multirange interactions potential and modified forcing schemes. In this work, a strategy to combine these enhancements with an appropriate interaction force field using only nearest-neighbor interactions is devised, starting from the desired pressure tensor. The final step of our procedure is implementing this external force by using the classical Guo forcing scheme. Numerical tests regarding static and dynamic flow conditions were performed. Static tests showed that current procedure is suitable to control the surface tension and phase densities. Based on thermodynamic principles, it is devised a solution for phase densities in a droplet, which states explicitly dependence on the surface tension and interface curvature. A comparison with numerical results suggest a physical inconsistency in the pseudopotential method. This fact is not commonly discussed in the literature, since most of studies are limited to the Maxwell equal area rule. However, this inconsistency is shown to be dependent on the equation of state (EOS), and its effects can be mitigated by an appropriate choice of Carnahan-Starling EOS parameters. Also, a droplet oscillation test was performed, and the most divergent solution under certain flow conditions deviated 7.5% from the expected analytical result. At the end, a droplet impact test against a solid wall was performed to verify the method stability, and it was possible to reach stable simulation results with density ratio of almost 2400 and Reynolds number of Re=373. The observed results corroborate that the proposed method is able to replicate the desired macroscopic multiphase behavior.

Interface tracking characteristics of color-gradient lattice Boltzmann model for immiscible fluids

We study the interface tracking characteristics of a color-gradient-based lattice Boltzmann model for immiscible flows. Investigation of the local density change in one of the fluid phases, via a Taylor series expansion of the recursive lattice Boltzmann equation, leads to the evolution equation of the order parameter that differentiates the fluids. It turns out that this interface evolution follows a conservative Allen-Cahn equation with a mobility which is independent of the fluid viscosities and surface tension. The mobility of the interface, which solely depends upon lattice speed of sound, can have a crucial effect on the physical dynamics of the interface. Further, we find that, when the equivalent lattice weights inside the segregation operator are modified, the resulting differential operators have a discretization error that is anisotropic to the leading order. As a consequence, the discretization errors in the segregation operator, which ensures a finite interface width, can act as a source of the spurious currents. These findings are supported with the help of numerical simulations.

Density gradient calculation in a class of multiphase lattice Boltzmann models

The multiphase lattice Boltzmann (LB) models based on pairwise interactions show great potential for simulating multiphase flows due to the conceptual and computational simplicity. Although the dynamics of multiphase flows are reproduced by the pairwise interaction force, the gradient of density (or effective density, i.e., pseudopotential) is implicitly involved in these models via the specialized forcing scheme or the consistent scheme for ɛ^{3}-order term. This work focuses on the calculation of density gradient in this class of multiphase LB models. Theoretical analyses are first carried out to reveal the involvement and calculation of density gradient. On the basis of a low Mach number approximation, an improved scheme is then proposed to calculate the density gradient for the recent LB model with self-tuning equation of state. Analytical and numerical calculations show that the improved scheme is more accurate and can help to reduce the numerical error when the reduced temperature is relatively low.

Regimes of Head-On Collisions of Equal-Sized Binary Droplets

Through molecular dynamics simulations, head-on collision processes of two identical droplets with a diameter of 10.9 nm are elaborately scrutinized over a wide range of impact Weber numbers (from 6.7 to 1307) both in vacuum and in an ambient of nitrogen gas. As the impact Weber number exceeds a certain critical value, a hole or multiple holes in apparently random locations are observed in the disklike structure formed by two colliding droplets. We name this a new "hole regime" of droplet collisions, which has not yet been reported in previous studies. As the impact Weber number increases, the number of holes increases. The hole or holes may disappear unless a second critical impact Weber number is exceeded, when the merged droplet is likely to experience dramatic shattering. It is also found that the existence of ambient gas provides a "cushion effect" which resists droplet deformation, thus delaying or even preventing the appearance of hole formation and shattering regimes. Moreover, increasing ambient pressure suppresses hole formation. A model based on energy balance is proposed to predict droplet behaviors, which provides a more accurate estimate of the maximum spreading factor compared to previous models. Finally, we further extend the current nanoscale droplet collision regime map and analyze the similarities and dissimilarities between nano- and macroscale droplet collision. Our study extends the current understanding on nanodroplet collisions.

Transition point prediction in a multicomponent lattice Boltzmann model: Forcing scheme dependencies

Pseudopotential-based lattice Boltzmann models are widely used for numerical simulations of multiphase flows. In the special case of multicomponent systems, the overall dynamics are characterized by the conservation equations for mass and momentum as well as an additional advection diffusion equation for each component. In the present study, we investigate how the latter is affected by the forcing scheme, i.e., by the way the underlying interparticle forces are incorporated into the lattice Boltzmann equation. By comparing two model formulations for pure multicomponent systems, namely the standard model [X. Shan and G. D. Doolen, J. Stat. Phys. 81, 379 (1995)JSTPBS0022-471510.1007/BF02179985] and the explicit forcing model [M. L. Porter et al., Phys. Rev. E 86, 036701 (2012)PLEEE81539-375510.1103/PhysRevE.86.036701], we reveal that the diffusion characteristics drastically change. We derive a generalized, potential function-dependent expression for the transition point from the miscible to the immiscible regime and demonstrate that it is shifted between the models. The theoretical predictions for both the transition point and the mutual diffusion coefficient are validated in simulations of static droplets and decaying sinusoidal concentration waves, respectively. To show the universality of our analysis, two common and one new potential function are investigated. As the shift in the diffusion characteristics directly affects the interfacial properties, we additionally show that phenomena related to the interfacial tension such as the modeling of contact angles are influenced as well.

Consistent forcing scheme in the cascaded lattice Boltzmann method

In this paper, we give an alternative derivation for the cascaded lattice Boltzmann method (CLBM) within a general multiple-relaxation-time (MRT) framework by introducing a shift matrix. When the shift matrix is a unit matrix, the CLBM degrades into an MRT LBM. Based on this, a consistent forcing scheme is developed for the CLBM. The consistency of the nonslip rule, the second-order convergence rate in space, and the property of isotropy for the consistent forcing scheme is demonstrated through numerical simulations of several canonical problems. Several existing forcing schemes previously used in the CLBM are also examined. The study clarifies the relation between MRT LBM and CLBM under a general framework.

Modeling mass transfer and reaction of dilute solutes in a ternary phase system by the lattice Boltzmann method

In this work, we propose a general approach for modeling mass transfer and reaction of dilute solute(s) in incompressible three-phase flows by introducing a collision operator in lattice Boltzmann (LB) method. An LB equation was used to simulate the solute dynamics among three different fluids, in which the newly expanded collision operator was used to depict the interface behavior of dilute solute(s). The multiscale analysis showed that the presented model can recover the macroscopic transport equations derived from the Maxwell-Stefan equation for dilute solutes in three-phase systems. Compared with the analytical equation of state of solute and dynamic behavior, these results are proven to constitute a generalized framework to simulate solute distributions in three-phase flows, including compound soluble in one phase, compound adsorbed on single-interface, compound in two phases, and solute soluble in three phases. Moreover, numerical simulations of benchmark cases, such as phase decomposition, multilayered planar interfaces, and liquid lens, were performed to test the stability and efficiency of the model. Finally, the multiphase mass transfer and reaction in Janus droplet transport in a straight microchannel were well reproduced.

Analysis of force treatment in the pseudopotential lattice Boltzmann equation method

In this paper, different force treatments are analyzed in detail for a pseudopotential lattice Boltzmann equation (LBE), and the contribution of third-order error terms to pressure tensor with a force scheme is analyzed by a higher-order Chapman-Enskog expansion technique. From the theoretical analysis, the performance of the original force treatment of Shan-Chen (SC), Ladd, Guo et al., and the exact difference method (EDM) are ɛ_{Ladd}<ɛ_{Guo}<ɛ_{EDM}≤ɛ_{SC} with the relaxation time τ≥1, while ɛ_{Ladd}<ɛ_{Guo}<ɛ_{SC}<ɛ_{EDM} with τ<1; here ɛ is a parameter related to the mechanical stability and the subscripts are the corresponding force scheme. To be consistent with the thermodynamic theory, a force term is introduced to modify the coefficients in the pressure tensor. Some numerical simulations are conducted to show that the predictions of modified force treatment of the pseudopotential LBE are all in good agreement with the analytical solution and other predictions.

Pseudopotential lattice Boltzmann equation method for two-phase flow: A higher-order Chapmann-Enskog expansion

In this paper, a higher order Chapmann-Enskog expansion technique is applied to pseudopotential lattice Boltzmann equation (LBE), and the contribution of third order error terms to pressure tensor is analyzed in detail. To be consistent with the thermodynamic theory, a force term is introduced to modify the coefficients in the pressure tensor. Some numerical simulations are conducted to validate the LBE, and the results show that the predictions of the present LBE agree well with the analytical solution and other predictions.

Cascaded lattice Boltzmann method with improved forcing scheme for large-density-ratio multiphase flow at high Reynolds and Weber numbers

A recently developed forcing scheme has allowed the pseudopotential multiphase lattice Boltzmann method to correctly reproduce coexistence curves, while expanding its range to lower surface tensions and arbitrarily high density ratios [Lycett-Brown and Luo, Phys. Rev. E 91, 023305 (2015)PLEEE81539-375510.1103/PhysRevE.91.023305]. Here, a third-order Chapman-Enskog analysis is used to extend this result from the single-relaxation-time collision operator, to a multiple-relaxation-time cascaded collision operator, whose additional relaxation rates allow a significant increase in stability. Numerical results confirm that the proposed scheme enables almost independent control of density ratio, surface tension, interface width, viscosity, and the additional relaxation rates of the cascaded collision operator. This allows simulation of large density ratio flows at simultaneously high Reynolds and Weber numbers, which is demonstrated through binary collisions of water droplets in air (with density ratio up to 1000, Reynolds number 6200 and Weber number 440). This model represents a significant improvement in multiphase flow simulation by the pseudopotential lattice Boltzmann method in which real-world parameters are finally achievable.

Multipseudopotential interaction: A consistent study of cubic equations of state in lattice Boltzmann models

A method is developed to analytically and consistently implement cubic equations of state into the recently proposed multipseudopotential interaction (MPI) scheme in the class of two-phase lattice Boltzmann (LB) models [S. Khajepor, J. Wen, and B. Chen, Phys. Rev. E 91, 023301 (2015)]10.1103/PhysRevE.91.023301. An MPI forcing term is applied to reduce the constraints on the mathematical shape of the thermodynamically consistent pseudopotentials; this allows the parameters of the MPI forces to be determined analytically without the need of curve fitting or trial and error methods. Attraction and repulsion parts of equations of state (EOSs), representing underlying molecular interactions, are modeled by individual pseudopotentials. Four EOSs, van der Waals, Carnahan-Starling, Peng-Robinson, and Soave-Redlich-Kwong, are investigated and the results show that the developed MPI-LB system can satisfactorily recover the thermodynamic states of interest. The phase interface is predicted analytically and controlled via EOS parameters independently and its effect on the vapor-liquid equilibrium system is studied. The scheme is highly stable to very high density ratios and the accuracy of the results can be enhanced by increasing the interface resolution. The MPI drop is evaluated with regard to surface tension, spurious velocities, isotropy, dynamic behavior, and the stability dependence on the relaxation time.

Assessment of interaction potential in simulating nonisothermal multiphase systems by means of lattice Boltzmann modeling

The pseudopotential lattice Boltzmann model (PP-LBM) is a very popular model for simulating multiphase systems. In this model, phase separation occurs via a short-range attraction between different phases when the interaction potential term is properly chosen. Therefore, the potential term is expected to play a significant role in the model and to affect the accuracy and the stability of the computations. The original PP-LBM suffers from some drawbacks such as being capable of dealing with low density ratios only, thermodynamic inconsistency, and spurious velocities. In this paper, we aim to analyze the PP-LBM with the view to simulate single-component (non-)isothermal multiphase systems at large density ratios and in spite of the presence of spurious velocities. For this purpose, the performance of two popular potential terms and of various implementation schemes for these potential terms is examined. Furthermore, the effects of different parameters (i.e., equation of state, viscosity, etc.) on the simulations are evaluated, and, finally, recommendations for a proper simulation of (non-)isothermal multiphase systems are presented.

Entropic lattice Boltzmann method for multiphase flows: Fluid-solid interfaces

The recently introduced entropic lattice Boltzmann model (ELBM) for multiphase flows [A. Mazloomi M., S. S. Chikatamarla, and I. V. Karlin, Phys. Rev. Lett. 114, 174502 (2015)] is extended to the simulation of dynamic fluid-solid interface problems. The thermodynamically consistent, nonlinearly stable ELBM together with a polynomial representation of the equation of state enables us to investigate the dynamics of the contact line in a wide range of applications, from capillary filling to liquid drop impact onto a flat surfaces with different wettability. The static interface behavior is tested by means of the liquid column in a channel to verify the Young-Laplace law. The numerical results of a capillary filling problem in a channel with wettability gradient show an excellent match with the existing analytical solution. Simulations of drop impact onto both wettable and nonwettable surfaces show that the ELBM reproduces the experimentally observed drop behavior in a quantitative manner. Results reported herein demonstrate that the present model is a promising alternative for studying the vapor-liquid-solid interface dynamics.