Thermodynamic consistency of liquid-gas lattice Boltzmann methods: interfacial property issues

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.

Thermodynamic consistency of a pseudopotential lattice Boltzmann fluid with interface curvature

Thermodynamic consistency of pseudopotential lattice Boltzmann models is a major topic that needs comprehensive evaluations. When interface is flat, pseudopotential models can give density-pressure isotherms in excellent agreement with those from equation of state. When interface is curved, thermodynamic equilibriums are affected by interface curvature, and consistency of pseudopotential models has not been systematically evaluated. In this study, we show that the effect of Laplace pressure on phase equilibrium is quantitatively consistent with Kelvin equation at high reduced temperatures (≥0.7). At low temperatures, inconsistency that can be attributed to the effect of orientation of the interface was noted, and it can be improved by tuning of the pseudopotential. By relating interfacial tension of a simulated fluid to that of a real fluid, the lattice spacing of pseudopotential model is found to be on the order of several molecular diameters, the typical range of intermolecular interactions. Interfacial thickness at different temperatures in pseudopotential model compared well with experiments and molecular dynamics simulations, which confirms that the calculated length scale is reasonable. Evaluation of a free energy lattice Boltzmann model indicate that it is consistent with Kelvin equation at high temperatures. The free energy model, however, is not as accurate as the tested pseudopotential model, and discrepancies may come from the relative inaccuracies in the predictions of vapor densities and the thinner interfaces.

Thermohydrodynamics of an evaporating droplet studied using a multiphase lattice Boltzmann method

In this paper, the thermohydrodynamics of an evaporating droplet is investigated by using a single-component pseudopotential lattice Boltzmann model. The phase change is applied to the model by adding source terms to the thermal lattice Boltzmann equation in such a way that the macroscopic energy equation of multiphase flows is recovered. In order to gain an exhaustive understanding of the complex hydrodynamics during evaporation, a single droplet is selected as a case study. At first, some tests for a stationary (non-)evaporating droplet are carried out to validate the method. Then the model is used to study the thermohydrodynamics of a falling evaporating droplet. The results show that the model is capable of reproducing the flow dynamics and transport phenomena of a stationary evaporating droplet quite well. Of course, a moving droplet evaporates faster than a stationary one due to the convective transport. Our study shows that our single-component model for simulating a moving evaporating droplet is limited to low Reynolds numbers.

Revised Chapman-Enskog analysis for a class of forcing schemes in the lattice Boltzmann method

In the lattice Boltzmann (LB) method, the forcing scheme, which is used to incorporate an external or internal force into the LB equation, plays an important role. It determines whether the force of the system is correctly implemented in an LB model and affects the numerical accuracy. In this paper we aim to clarify a critical issue about the Chapman-Enskog analysis for a class of forcing schemes in the LB method in which the velocity in the equilibrium density distribution function is given by u=∑_{α}e_{α}f_{α}/ρ, while the actual fluid velocity is defined as u[over ̂]=u+δ_{t}F/(2ρ). It is shown that the usual Chapman-Enskog analysis for this class of forcing schemes should be revised so as to derive the actual macroscopic equations recovered from these forcing schemes. Three forcing schemes belonging to the above class are analyzed, among which Wagner's forcing scheme [A. J. Wagner, Phys. Rev. E 74, 056703 (2006)10.1103/PhysRevE.74.056703] is shown to be capable of reproducing the correct macroscopic equations. The theoretical analyses are examined and demonstrated with two numerical tests, including the simulation of Womersley flow and the modeling of flat and circular interfaces by the pseudopotential multiphase LB model.

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.

Interface-capturing lattice Boltzmann equation model for two-phase flows

In this work, an interface-capturing lattice Boltzmann equation (LBE) model is proposed for two-phase flows. In the model, a Lax-Wendroff propagation scheme and a properly chosen equilibrium distribution function are employed. The Lax-Wendroff scheme is used to provide an adjustable Courant-Friedrichs-Lewy (CFL) number, and the equilibrium distribution is presented to remove the dependence of the relaxation time on the CFL number. As a result, the interface can be captured accurately by decreasing the CFL number. A theoretical expression is derived for the chemical potential gradient by solving the LBE directly for a two-phase system with a flat interface. The result shows that the gradient of the chemical potential is proportional to the square of the CFL number, which explains why the proposed model is able to capture the interface naturally with a small CFL number, and why large interface error exists in the standard LBE model. Numerical tests, including a one-dimensional flat interface problem, a two-dimensional circular droplet problem, and a three-dimensional spherical droplet problem, demonstrate that the proposed LBE model performs well and can capture a sharp interface with a suitable CFL number.

Consistent lattice Boltzmann equations for phase transitions

Unlike conventional computational fluid dynamics methods, the lattice Boltzmann method (LBM) describes the dynamic behavior of fluids in a mesoscopic scale based on discrete forms of kinetic equations. In this scale, complex macroscopic phenomena like the formation and collapse of interfaces can be naturally described as related to source terms incorporated into the kinetic equations. In this context, a novel athermal lattice Boltzmann scheme for the simulation of phase transition is proposed. The continuous kinetic model obtained from the Liouville equation using the mean-field interaction force approach is shown to be consistent with diffuse interface model using the Helmholtz free energy. Density profiles, interface thickness, and surface tension are analytically derived for a plane liquid-vapor interface. A discrete form of the kinetic equation is then obtained by applying the quadrature method based on prescribed abscissas together with a third-order scheme for the discretization of the streaming or advection term in the Boltzmann equation. Spatial derivatives in the source terms are approximated with high-order schemes. The numerical validation of the method is performed by measuring the speed of sound as well as by retrieving the coexistence curve and the interface density profiles. The appearance of spurious currents near the interface is investigated. The simulations are performed with the equations of state of Van der Waals, Redlich-Kwong, Redlich-Kwong-Soave, Peng-Robinson, and Carnahan-Starling.

Lattice Boltzmann modeling of multiphase flows at large density ratio with an improved pseudopotential model

Owing to its conceptual simplicity and computational efficiency, the pseudopotential multiphase lattice Boltzmann (LB) model has attracted significant attention since its emergence. In this work, we aim to extend the pseudopotential LB model to simulate multiphase flows at large density ratio and relatively high Reynolds number. First, based on our recent work [Q. Li, K. H. Luo, and X. J. Li, Phys. Rev. E 86, 016709 (2012)], an improved forcing scheme is proposed for the multiple-relaxation-time pseudopotential LB model in order to achieve thermodynamic consistency and large density ratio in the model. Next, through investigating the effects of the parameter a in the Carnahan-Starling equation of state, we find that the interface thickness is approximately proportional to 1/√a. Using a smaller a will lead to a wider interface thickness, which can reduce the spurious currents and enhance the numerical stability of the pseudopotential model at large density ratio. Furthermore, it is found that a lower liquid viscosity can be gained in the pseudopotential model by increasing the kinematic viscosity ratio between the vapor and liquid phases. The improved pseudopotential LB model is numerically validated via the simulations of stationary droplet and droplet oscillation. Using the improved model as well as the above treatments, numerical simulations of droplet splashing on a thin liquid film are conducted at a density ratio in excess of 500 with Reynolds numbers ranging from 40 to 1000. The dynamics of droplet splashing is correctly reproduced and the predicted spread radius is found to obey the power law reported in the literature.

Simulation of static critical phenomena in nonideal fluids with the lattice Boltzmann method

A fluctuating nonideal fluid at its critical point is simulated with the lattice Boltzmann method. It is demonstrated that the method, employing a Ginzburg-Landau free energy functional, correctly reproduces the static critical behavior associated with the Ising universality class. A finite-size scaling analysis is applied to determine the critical exponents related to the order parameter, compressibility and specific heat. A particular focus is put on finite-size effects and issues related to the global conservation of the order parameter.

Stabilized lattice Boltzmann-Enskog method for compressible flows and its application to one- and two-component fluids in nanochannels

A numerically stable method to solve the discretized Boltzmann-Enskog equation describing the behavior of nonideal fluids under inhomogeneous conditions is presented. The algorithm employed uses a Lagrangian finite-difference scheme for the treatment of the convective term and a forcing term to account for the molecular repulsion together with a Bhatnagar-Gross-Krook relaxation term. In order to eliminate the spurious currents induced by the numerical discretization procedure, we use a trapezoidal rule for the time integration together with a version of the two-distribution method of He et al. [J. Comput. Phys. 152, 642 (1999)]. Numerical tests show that, in the case of a one-component fluid in the presence of a spherical potential well, the proposed method reduces the numerical error by several orders of magnitude. We conduct another test by considering the flow of a two-component fluid in a channel with a bottleneck and provide information about the density and velocity field in this structured geometry.

Forcing term in single-phase and Shan-Chen-type multiphase lattice Boltzmann models

Numerous schemes have been proposed to incorporate a bulk forcing term into the lattice Boltzmann equation. In this paper we present a simple and straightforward comparative analysis of five popular schemes [Shan and Chen, Phys. Rev. E 47, 1815 (1993); Phys Rev Lett. 81, 1618 (1998); He et al., Phys. Rev. E 57, R13 (1998); Guo et al., Phys. Rev. E 65, 046308 (2002); Kupershtokh et al., Comput. Math. Appl. 58, 965 (2009)] in which their differences and similarities are identified. From the analysis we classify the schemes into two groups; the behaviors of the schemes in each group are proven to be identical up to second order. Numerical test simulating the two-dimensional unsteady Taylor-Green vortex flow problem demonstrate that all five schemes are of comparable accuracy for single-phase flow. However, for two-phase flow the situation is different, which is demonstrated by incorporating these schemes into different Shan-Chen-type multiphase models. The forcing scheme in the original Shan-Chen (SC) multiphase model turns out to be inaccurate in terms of the resulting surface tension for different density ratios and relaxation times. In the numerical tests, a typical equation of state and interparticle interactions including next-nearest neighbors were incorporated into the SC model. Our results confirm that the surface-tension values obtained from the original SC lattice Boltzmann method (LBM) simulation depend on the value of the relaxation time τ. For τ<0.7Δt, the surface tension agree well with the analytical solutions. However, when τ>0.7Δt, the surface tension turns out to be systematically larger than the analytical one, exceeding it by more than a factor of 2 for τ=2Δt. In contrast, with the application of the scheme proposed by He et al., the SC LBM produces very accurate surface tensions independent of the value of τ. We also found that the densities of the coexisting liquid and gas can be adjusted to match those at thermodynamic equilibrium if the particle interaction term includes next-nearest-neighbor contributions. The obtained results will be useful for further studies of two-phase flow with high density ratios using the SC LBM approach.

Phase separation in thermal systems: a lattice Boltzmann study and morphological characterization

We investigate thermal and isothermal symmetric liquid-vapor separations via a fast Fourier transform thermal lattice Boltzmann (FFT-TLB) model. Structure factor, domain size, and Minkowski functionals are employed to characterize the density and velocity fields, as well as to understand the configurations and the kinetic processes. Compared with the isothermal phase separation, the freedom in temperature prolongs the spinodal decomposition (SD) stage and induces different rheological and morphological behaviors in the thermal system. After the transient procedure, both the thermal and isothermal separations show power-law scalings in domain growth, while the exponent for thermal system is lower than that for isothermal system. With respect to the density field, the isothermal system presents more likely bicontinuous configurations with narrower interfaces, while the thermal system presents more likely configurations with scattered bubbles. Heat creation, conduction, and lower interfacial stresses are the main reasons for the differences in thermal system. Different from the isothermal case, the release of latent heat causes the changing of local temperature, which results in new local mechanical balance. When the Prandtl number becomes smaller, the system approaches thermodynamical equilibrium much more quickly. The increasing of mean temperature makes the interfacial stress lower in the following way: σ=σ(0)[(T(c)-T)/(T(c)-T(0))](3/2), where T(c) is the critical temperature and σ(0) is the interfacial stress at a reference temperature T(0), which is the main reason for the prolonged SD stage and the lower growth exponent in the thermal case. Besides thermodynamics, we probe how the local viscosities influence the morphology of the phase separating system. We find that, for both the isothermal and thermal cases, the growth exponents and local flow velocities are inversely proportional to the corresponding viscosities. Compared with the isothermal case, the local flow velocity depends not only on viscosity but also on temperature.

Chequerboard effects on spurious currents in the lattice Boltzmann equation for two-phase flows

Spurious currents near an interface between different phases are a common undesirable feature of the lattice Boltzmann equation (LBE) method for two-phase systems. In this paper, we show that the spurious currents of a kinetic theory-based LBE have a significant dependence on the parity of the grid number of the underlying lattice, which can be attributed to the chequerboard effect. A technique that uses a Lax-Wendroff streaming is proposed to overcome this anomaly, and its performance is verified numerically.

Shear stress in nonideal fluid lattice Boltzmann simulations

The viscous stress in nonideal fluid lattice Boltzmann methods is investigated theoretically and by simulations. Three representative liquid-gas models are compared in a steady-state situation, where an analytical expression for the viscous stress is available. It is shown that, in the presence of nonideal fluid interactions or strong body forces, the accuracy of the computed viscous stress depends strongly on the underlying model.

Thermodynamically Constrained Averaging Theory Approach for Modeling Flow and Transport Phenomena in Porous Medium Systems: 8. Interface and Common Curve Dynamics

This work is the eighth in a series that develops the fundamental aspects of the thermodynamically constrained averaging theory (TCAT) that allows for a systematic increase in the scale at which multiphase transport phenomena is modeled in porous medium systems. In these systems, the explicit locations of interfaces between phases and common curves, where three or more interfaces meet, are not considered at scales above the microscale. Rather, the densities of these quantities arise as areas per volume or length per volume. Modeling of the dynamics of these measures is an important challenge for robust models of flow and transport phenomena in porous medium systems, as the extent of these regions can have important implications for mass, momentum, and energy transport between and among phases, and formulation of a capillary pressure relation with minimal hysteresis. These densities do not exist at the microscale, where the interfaces and common curves correspond to particular locations. Therefore, it is necessary for a well-developed macroscale theory to provide evolution equations that describe the dynamics of interface and common curve densities. Here we point out the challenges and pitfalls in producing such evolution equations, develop a set of such equations based on averaging theorems, and identify the terms that require particular attention in experimental and computational efforts to parameterize the equations. We use the evolution equations developed to specify a closed two-fluid-phase flow model.

Lattice Boltzmann method for Lennard-Jones fluids based on the gradient theory of interfaces

In the present study we propose a lattice Boltzmann equation (LBE) model derived from density gradient expansions of the discrete BBGKY evolution equations. The model is based on the mechanical approach of the gradient theory of interfaces. The basic input is the radial distribution function, which is related exclusively to the molecular interaction potential, rather than semiempirical equations of state used in previous LBE models. This function can be provided from independent molecular simulations or from approximate theories. Evidently the accuracy of the interaction potential, and thus the radial distribution function, reflects on the accuracy of the thermodynamic properties and consistency of the derived LBE model. We have applied the proposed model to obtain equilibrium bulk and interfacial properties of a Lennard-Jones fluid at different temperatures, T, close to critical, T(c). The results demonstrate that the LBE model is in excellent agreement with gradient theory as well as with independent literature results based on different molecular simulation approaches. Hence the proposed LBE model can recover accurately bulk and interfacial thermodynamics for a Lennard Jones fluid at T/T(c)>0.9.