Phase-field lattice Boltzmann model with singular mobility for quasi-incompressible two-phase flows.

In this paper, a lattice Boltzmann for quasi-incompressible two-phase flows is proposed based on the Cahn-Hilliard phase-field theory, which can be viewed as an improved model of a previous one [Yang and Guo, Phys. Rev. E 93, 043303 (2016)2470-004510.1103/PhysRevE.93.043303]. The model is composed of two LBE's, one for the Cahn-Hilliard equation (CHE) with a singular mobility, and the other for the quasi-incompressible Navier-Stokes equations (qINSE). Particularly, the LBE for the CHE uses an equilibrium distribution function containing a free parameter associated with the gradient of chemical potential, such that the variable (and even zero) mobility can be handled. In addition, the LBE for the qINSE uses an equilibrium distribution function containing another free parameter associated with the local shear rate, such that the large viscosity ratio problems can be handled. Several tests are first carried out to test the capability of the proposed LBE for the CHE in capturing phase interface, and the results demonstrate that the proposed model outperforms the original LBE model in terms of accuracy and stability. Furthermore, by coupling the hydrodynamic equations, the tests of double-stationary droplets and droplets falling problems indicate that the proposed model can reduce numerical dissipation and produce physically acceptable results at large time scales. The results of droplets falling and phase separation of binary fluid problems show that the present model can handle two-phase flows with large viscosity ratio up to the magnitude of 10^{4}.

Discrete unified gas kinetic scheme for the solution of electron Boltzmann transport equation with Callaway approximation.

Electrons are the carriers of heat and electricity in materials and exhibit abundant transport phenomena such as ballistic, diffusive, and hydrodynamic behaviors in systems with different sizes. The electron Boltzmann transport equation (eBTE) is a reliable model for describing electron transport, but it is a challenging problem to efficiently obtain the numerical solutions of the eBTE within one unified scheme involving ballistic, hydrodynamics, and/or diffusive regimes. In this work, a discrete unified gas kinetic scheme (DUGKS) in the finite-volume framework is developed based on the eBTE with the Callaway relaxation model for electron transport. By reconstructing the distribution function at the cell interface, the processes of electron drift and scattering are coupled together within a single time step. Numerical tests demonstrate that the DUGKS can be adaptively applied to multiscale electron transport, across different regimes.

Implementation of contact line motion based on the phase-field lattice Boltzmann method.

This paper proposes a strategy to implement the free-energy-based wetting boundary condition within the phase-field lattice Boltzmann method. The greatest advantage of the proposed method is that the implementation of contact line motion can be significantly simplified while still maintaining good accuracy. For this purpose, the liquid-solid free energy is treated as a part of the chemical potential instead of the boundary condition, thus avoiding complicated interpolations with irregular geometries. Several numerical testing cases, including droplet spreading processes on the idea flat, inclined, and curved boundaries, are conducted, and the results demonstrate that the proposed method has good ability and satisfactory accuracy to simulate contact line motions.

Multiscale steady discrete unified gas kinetic scheme with macroscopic coarse mesh acceleration using preconditioned Krylov subspace method for multigroup neutron Boltzmann transport equation.

A multiscale steady discrete unified gas kinetic scheme with macroscopic coarse mesh acceleration [accelerated steady discrete unified gas kinetic scheme (SDUGKS)] is proposed to improve the convergence of the original SDUGKS for an optically thick system in solving the multigroup neutron Boltzmann transport equation (NBTE) to analyze the distribution of fission energy in the reactor core. In the accelerated SDUGKS, by solving the coarse mesh macroscopic governing equations (MGEs) derived from the moment equations of the NBTE, the numerical solutions of the NBTE on fine meshes at the mesoscopic level can be rapidly obtained from the prolongation of the coarse mesh solutions of the MGE. Furthermore, the use of the coarse mesh can greatly reduce the computational variables and improve the computational efficiency of the MGE. The biconjugate gradient stabilized Krylov subspace method with the modified incomplete LU preconditioner and the lower-upper symmetric-Gauss-Seidel sweeping method are implemented to solve the discrete systems of the macroscopic coarse mesh acceleration model and mesoscopic SDUGKS to further improve the numerical efficiency. Numerical solutions validate good numerical accuracy and high acceleration efficiency of the proposed accelerated SDUGKS for the complicated multiscale neutron transport problems.

Unified preserving properties of kinetic schemes.

The kinetic theory provides a physical basis for developing multiscale methods for gas flows covering a wide range of flow regimes. A particular challenge for kinetic schemes is whether they can capture the correct hydrodynamic behaviors of the system in the continuum regime (i.e., as the Knudsen number ε≪1) without enforcing kinetic scale resolution. At the current stage, the main approach to analyze such a property is the asymptotic preserving (AP) concept, which aims to show whether a kinetic scheme reduces to a solver for the hydrodynamic equations as ε→0, such as the shock capturing scheme for the Euler equations. However, the detailed asymptotic properties of the kinetic scheme are indistinguishable when ε is small but finite under the AP framework. To distinguish different characteristics of kinetic schemes, in this paper we introduce the concept of unified preserving (UP) aiming at assessing asymptotic orders of a kinetic scheme by employing the modified equation approach and Chapman-Enskon analysis. It is shown that the UP properties of a kinetic scheme generally depend on the spatial and temporal accuracy and closely on the interconnections among three scales (kinetic scale, numerical scale, and hydrodynamic scale) and their corresponding coupled dynamics. Specifically, the numerical resolution and specific discretization of particle transport and collision determine the flow physics of the scheme in different regimes, especially in the near continuum limit. As two examples, the UP methodology is applied to analyze the discrete unified gas-kinetic scheme and a second-order implicit-explicit Runge-Kutta scheme in their asymptotic behaviors in the continuum limit.

Discrete unified gas kinetic scheme for continuum compressible flows.

In this paper, a discrete unified gas kinetic scheme (DUGKS) is proposed for continuum compressible gas flows based on the total energy kinetic model [Guo et al., Phys. Rev. E 75, 036704 (2007)1539-375510.1103/PhysRevE.75.036704]. The proposed DUGKS can be viewed as a special finite-volume lattice Boltzmann method for the compressible Navier-Stokes equations in the double distribution function formulation, in which the mass and momentum transport are described by the kinetic equation for a density distribution function (g), and the energy transport is described by the other one for an energy distribution function (h). To recover the full compressible Navier-Stokes equations exactly, the corresponding equilibrium distribution functions g^{eq} and h^{eq} are expanded as Hermite polynomials up to third and second orders, respectively. The velocity spaces for the kinetic equations are discretized according to the seventh and fifth Gauss-Hermite quadratures. Consequently, the computational efficiency of the present DUGKS can be much improved in comparison with previous versions using more discrete velocities required by the ninth Gauss-Hermite quadrature.

Consistent lifting relations for the initialization of total-energy double-distribution-function kinetic models.

A lifting relation connecting the distribution function explicitly with the hydrodynamic variables is necessary for the Boltzmann equation-based mesoscopic approaches in order to correctly initialize a nonuniform hydrodynamic flow. We derive two lifting relations for Guo et al.'s total-energy double-distribution-function (DDF) kinetic model [Z. L. Guo et al., Phys. Rev. E 75, 036704 (2007)1539-375510.1103/PhysRevE.75.036704], one from the Hermite expansion of the conserved and nonconserved moments, and the second from the O(τ) Chapman-Enskog (CE) approximation of the Maxwellian exponential equilibrium. While both forms are consistent to the compressible Navier-Stokes-Fourier system theoretically, we stress that the latter may introduce numerical oscillations under the recently optimized discrete velocity models [Y. M. Qi et al., Phys. Fluids 34, 116101 (2022)10.1063/5.0120490], namely a 27 discrete velocity model of the seventh-order Gauss-Hermite quadrature (GHQ) accuracy (D3V27A7) for the velocity field combined with a 13 discrete velocity model of the fifth-order GHQ accuracy (D3V13A5) for the total energy. It is shown that the Hermite-expansion-based lifting relation can be alternatively derived from the latter approach using the truncated Hermite-polynomial equilibrium. Additionally, a relationship between the order of CE expansions and the truncated order of Hermite equilibria is developed to determine the minimal order of a Hermite equilibria required to recover any multiple-timescale macroscopic system. Next, three-dimensional compressible Taylor-Green vortex flows with different initial conditions and Ma numbers are simulated to demonstrate the effectiveness and potential issues of these lifting relations. The Hermite-expansion-based lifting relation works well in all cases, while the Chapman-Enskog-expansion-based lifting relation may produce numerical oscillations and a theoretical model is developed to predict such oscillations. Furthermore, the corresponding lifting relations for Qi et al.'s total energy DDF model [Y. M. Qi et al., Phys. Fluids 34, 116101 (2022)10.1063/5.0120490] are derived, and additional simulations are performed to illustrate the generality of our approach.

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.

Perturbation theory of thermal rectification.

In this paper, a perturbation theory of thermal rectification is developed for a thermal system where an effective thermal conductivity throughout the system can be identified and changes smoothly and slightly. This theory provides an analytical formula of the thermal rectification ratio with rigorous mathematical derivations and physical assumptions. The physical meanings and limitations of the present theory are discussed in detail. Furthermore, a physical relationship among the thermal rectification, system length, temperature difference, and thermal conductivity is built. It reveals the linear relationship between the thermal rectification ratio and temperature difference. Also, the size dependence of the thermal rectification relies on the specific form of the thermal conductivity. In addition, several previous experimental and numerical observations are well explained by this theory.

Discrete unified gas kinetic scheme for all Knudsen number flows. IV. Strongly inhomogeneous fluids.

This work is an extension of the discrete unified gas kinetic scheme (DUGKS) from rarefied gas dynamics to strongly inhomogeneous dense fluid systems. The fluid molecular size can be ignored for dilute gases, while the nonlocal intermolecular collisions and the competition of solid-fluid and fluid-fluid interactions play an important role for surface-confined fluid flows at the nanometer scale. The nonequilibrium state induces strong fluid structural-confined inhomogeneity and anomalous fluid flow dynamics. According to the previous kinetic model [Guo et al., Phys. Rev. E 71, 035301(R) (2005)10.1103/PhysRevE.71.035301], the long-range intermolecular attraction is modeled by the mean-field approximation, and the volume exclusion effect is considered by the hard-sphere potential in the collision operator. The kinetic model is solved by the DUGKS, which has the characteristics of asymptotic preserving, low dissipation, second-order accuracy, and multidimensional nature. Both static fluid structure and dynamic flow behaviors are calculated and validated with Monte Carlo or molecular dynamics results. It is shown that the flow of dense fluid systems tends to that of rarefied gases as the dense degree decreases or the mean flow path increases. The DUGKS is proved to be applicable to simulate such nonequilibrium dense fluid systems.

Heat vortex in hydrodynamic phonon transport of two-dimensional materials.

We study hydrodynamic phonon heat transport in two-dimensional (2D) materials. Starting from the Peierls-Boltzmann equation with the Callaway model approximation, we derive a 2D Guyer-Krumhansl-like equation describing hydrodynamic phonon transport, taking into account the quadratic dispersion of flexural phonons. In addition to Poiseuille flow, second sound propagation, the equation predicts heat current vortices and negative non-local thermal conductance in 2D materials, which are common in classical fluids but have not yet been considered in phonon transport. Our results also illustrate the universal transport behaviors of hydrodynamics, independent of the type of quasi-particles and their microscopic interactions.

Lattice Boltzmann model for the one-dimensional nonlinear Dirac equation.

In this paper, a lattice Boltzmann model for one-dimensional nonlinear Dirac equation is presented by using double complex-valued distribution functions and carefully selected equilibrium distribution functions. The effects of space and time resolutions and relaxation time on the accuracy and stability of the model are numerically investigated in detail. It is found that the model is of second-order accuracy in both space and time, and the order of accuracy is near 3.0 at lower grid resolution, which shows that the lattice Boltzmann method is an effective numerical scheme for the nonlinear Dirac equation.

Lattice Boltzmann model for nonlinear convection-diffusion equations.

A lattice Boltzmann model for convection-diffusion equation with nonlinear convection and isotropic-diffusion terms is proposed through selecting equilibrium distribution function properly. The model can be applied to the common real and complex-valued nonlinear evolutionary equations, such as the nonlinear Schrödinger equation, complex Ginzburg-Landau equation, Burgers-Fisher equation, nonlinear heat conduction equation, and sine-Gordon equation, by using a real and complex-valued distribution function and relaxation time. Detailed simulations of these equations are performed, and it is found that the numerical results agree well with the analytical solutions and the numerical solutions reported in previous studies.

Lattice Boltzmann equation for microscale gas flows of binary mixtures.

Modeling and simulating gas flows in and around microdevices are a challenging task in both science and engineering. In practical applications, a gas is usually a mixture made of different components. In this paper we propose a lattice Boltzmann equation (LBE) model for microscale flows of a binary mixture based on a recently developed LBE model for continuum mixtures [P. Asinari and L.-S. Luo, J. Comput. Phys. 227, 3878 (2008)]. A consistent boundary condition for gas-solid interactions is proposed and analyzed. The LBE is validated and compared with theoretical results or other reported data. The results show that the model can serve as a potential method for flows of binary mixture in the microscale.

Theory of the lattice Boltzmann equation: Lattice Boltzmann model for axisymmetric flows.

A lattice Boltzmann equation (LBE) for axisymmetric flows is proposed. The model has some distinct features that distinguish it from existing axisymmetric LBE models. First, it is derived from the Boltzmann equation so that it has a solid physics base and is easy for generalization; second, the model can describe the axial, radial, and azimuthal velocity components in the same fashion; and third, the source terms of the model contain no velocity gradients and are much simpler than other LBE models. Numerical tests, including steady and unsteady internal and external flows, demonstrate that the proposed LBE can serve as a viable and efficient method for low speed axisymmetric flows.

Spontaneous shrinkage of droplet on a wetting surface in the phase-field model.

Phase-field theory is widely used to model multiphase flow. The fact that a drop can shrink or grow spontaneously due to the redistribution of interface and bulk energies to minimize the system energy may produce ill effects on the simulation. In this Rapid Communication, the spontaneous behavior of a drop on a partially wetting surface is investigated. It is found that there exists a critical radius dependent on the contact angle, the domain size, and the interface width, below which the drop will eventually disappear. In particular, the critical radius can be very large when the surface becomes very hydrophilic. The theoretical prediction of the critical radius is verified numerically by simulating a drop on a surface with various contact angles, the domain sizes, and the interface widths.

High-order lattice-Boltzmann model for the Cahn-Hilliard equation.

The Cahn-Hilliard equation (CHE) is widely used in modeling two-phase fluid flows, and it is critical to solve this equation accurately to track the interface between the two phases. In this paper, a high-order lattice Boltzmann equation model is developed for the CHE via the fourth-order Chapman-Enskog expansion. A truncation error analysis is performed, and the leading error term proportional to the Peclet number is identified. The results are further confirmed by the Maxwell iteration. With the inclusion of a correction term for eliminating the main error term, the proposed model is able to recover the CHE up to third order. The proposed model is tested by several benchmark problems. The results show that the present model is capable of tracking the interface with improved accuracy and stability in comparison with the second-order one.

Lattice Boltzmann equation with multiple effective relaxation times for gaseous microscale flow.

The standard lattice Boltzmann equation (LBE) is inadequate for simulating gas flows with a large Knudsen number. In this paper we propose a generalized lattice Boltzmann equation with effective relaxation times based on a recently developed generalized Navier-Stokes constitution [Guo, Europhys Lett. 80, 24001 (2007)] for nonequilibrium flows. A kinetic boundary condition corresponding to a generalized second-order slip scheme is also designed for the model. The LBE model and the boundary condition are analyzed for a unidirectional flow, and it is found that in order to obtain the generalized Navier-Stokes equations, the relaxation times must be properly chosen and are related to the boundary condition. Numerical results show that the proposed method is able to capture the Knudsen layer phenomenon and can yield improved predictions in comparison with the standard lattice Boltzmann equation.

Multiple-relaxation-time model for the correct thermohydrodynamic equations.

A coupling lattice Boltzmann equation (LBE) model with multiple relaxation times is proposed for thermal flows with viscous heat dissipation and compression work. In this model the fixed Prandtl number and the viscous dissipation problems in the energy equation, which exist in most of the LBE models, are successfully overcome. The model is validated by simulating the two-dimensional Couette flow, thermal Poiseuille flow, and the natural convection flow in a square cavity. It is found that the numerical results agree well with the analytical solutions and/or other numerical results.

Lattice Boltzmann model for high-order nonlinear partial differential equations.

In this paper, a general lattice Boltzmann (LB) model is proposed for the high-order nonlinear partial differential equation with the form ∂_{t}ϕ+∑_{k=1}^{m}α_{k}∂_{x}^{k}Π_{k}(ϕ)=0 (1≤k≤m≤6), α_{k} are constant coefficients, Π_{k}(ϕ) are some known differential functions of ϕ. As some special cases of the high-order nonlinear partial differential equation, the classical (m)KdV equation, KdV-Burgers equation, K(n,n)-Burgers equation, Kuramoto-Sivashinsky equation, and Kawahara equation can be solved by the present LB model. Compared to the available LB models, the most distinct characteristic of the present model is to introduce some suitable auxiliary moments such that the correct moments of equilibrium distribution function can be achieved. In addition, we also conducted a detailed Chapman-Enskog analysis, and found that the high-order nonlinear partial differential equation can be correctly recovered from the proposed LB model. Finally, a large number of simulations are performed, and it is found that the numerical results agree with the analytical solutions, and usually the present model is also more accurate than the existing LB models [H. Lai and C. Ma, Sci. China Ser. G 52, 1053 (2009)1672-179910.1007/s11433-009-0149-3; H. Lai and C. Ma, Phys. A (Amsterdam) 388, 1405 (2009)PHYADX0378-437110.1016/j.physa.2009.01.005] for high-order nonlinear partial differential equations.