Thermodynamic Foundations of Kinetic Theory and Lattice Boltzmann Models for Multiphase Flows

This paper demonstrates that thermodynamically consistent lattice Boltzmann models for single-component multiphase flows can be derived from a kinetic equation using both Enskog's theory for dense fluids and mean-field theory for long-range molecular interaction. The lattice Boltzmann models derived this way satisfy the correct mass, momentum, and energy conservation equations. All the thermodynamic variables in these LBM models are consistent. The strengths and weaknesses of previous lattice Boltzmann multiphase models are analyzed.     

A lattice Boltzmann model with an amending function forsimulating nonlinear partial differential equations

This paper proposes a lattice Boltzmann model with an amending function for one-dimensional nonlinear partial differential equations (NPDEs) in the form $u_t+\alpha uu_{xx}+\beta u^n u_x+\gamma u_{xxx}+\xi u_{xxxx}=0$. This model is different from existing models because it lets the time step be equivalent to the square of the space step and derives higher accuracy and nonlinear terms in NPDEs. With the Chapman--Enskog expansion, the governing evolution equation is recovered correctly from the continuous Boltzmann equation. The numerical results agree well with the analytical solutions.     

A multifluid interpenetration mix model

In this Letter, we derive a set of multifluid moment equations with and without internal and external fields from the collisional Boltzmann equation in a self-consistent manner. The new equations are mathematically closed and physically consistent with one free parameter, contained in a phenomenological closure for the collisional frequency and to be determined by experimental data. The new equations provide a theoretical foundation for a large fraction of phenomenological mix models. They contain all the physical terms, particularly the terms associated with the Reynolds stress due to both species interpenetrations and random chaotic motions. Under certain assumptions, the new model equations successfully reduce to the other mix models.     

Comparison between lattice Boltzmann method and Navier–Stokes high order schemes for computational aeroacoustics

Computational aeroacoustic (CAA) simulation requires accurate schemes to capture the dynamics of acoustic fluctuations, which are weak compared with aerodynamic ones. In this paper, two kinds of schemes are studied and compared: the classical approach based on high order schemes for Navier–Stokes-like equations and the lattice Boltzmann method. The reference macroscopic equations are the 3D isothermal and compressible Navier–Stokes equations. A Von Neumann analysis of these linearized equations is carried out to obtain exact plane wave solutions. Three physical modes are recovered and the corresponding theoretical dispersion relations are obtained. Then the same analysis is made on the space and time discretization of the Navier–Stokes equations with the classical high order schemes to quantify the influence of both space and time discretization on the exact solutions. Different orders of discretization are considered, with and without a uniform mean flow. Three different lattice Boltzmann models are then presented and studied with the Von Neumann analysis. The theoretical dispersion relations of these models are obtained and the error terms of the model are identified and studied. It is shown that the dispersion error in the lattice Boltzmann models is only due to the space and time discretization and that the continuous discrete velocity Boltzmann equation yield the same exact dispersion as the Navier–Stokes equations. Finally, dispersion and dissipation errors of the different kind of schemes are quantitatively compared. It is found that the lattice Boltzmann method is less dissipative than high order schemes and less dispersive than a second order scheme in space with a 3-step Runge–Kutta scheme in time. The number of floating point operations at a given error level associated with these two kinds of schemes are then compared.     

Exact solutions of the nonlinear Boltzmann equation

A review is given of research activities since 1976 on the nonlinear Boltzmann equation and related equations of Boltzmann type, in which several rediscoveries have been made and several conjectures have been disproved. Subjects are (i) the BKW solution of the Boltzmann equation for Maxwell molecules, first discovered by Krupp in 1967, and the Krook-Wu conjecture concerning the universal significance of the BKW solution for the large(v, t) behavior of the velocity distribution functionf (v, t); (ii) moment equations and polynomial expansions off (v, t); (iii) model Boltzmann equation for a spatially uniform system of very hard particles, that can be solved in closed form for general initial conditions; (iv) for Maxwell and non-Maxwell-type molecules there exist solutions of the nonlinear Boltzmann equation with algebraic decrease at ; connections with nonuniqueness and violation of conservation laws; (v) conjectured super-H-theorem and the BKW solution; (vi) exactly soluble one-dimensional Boltzmann equation with spatial dependence.Reference due to C. Cercignani.     

A Markov Process Associated with a Boltzmann Equation Without Cutoff and for Non-Maxwell Molecules

Tanaka,⁽¹⁸⁾ showed a way to relate the measure solution {P _t } _t of a spatially homogeneous Boltzmann equation of Maxwellian molecules without angular cutoff to a Poisson-driven stochastic differential equation: {P _t } is the flow of time marginals of the solution of this stochastic equation. In the present paper, we extend this probabilistic interpretation to much more general spatially homogeneous Boltzmann equations. Then we derive from this interpretation a numerical method for the concerned Boltzmann equations, by using easily simulable interacting particle systems.     

On the linearized relativistic Boltzmann equation. II. Existence of hydrodynamics

Solutions are analyzed of the linearized relativistic Boltzmann equation for initial data fromL ²(r, p) in long-time and/or small-mean-free-path limits. In both limits solutions of this equation converge to approximate ones constructed with solutions of the set of differential equations called the equations of relativistic hydrodynamics.     

Coarsely grained stochastic Boltzmann equation and its moment equations

The stochastic Boltzmann equation is coarsely grained. The coarsely grained stochastic (CGS) Boltzmann equation has fluctuating terms in its collision term. On the basis of the CGS Boltzmann equation, reduced Grad’s 26 moment equations are derived. Coarsely grained moment equations obtained from the CGS Boltzmann equation show that fluctuating terms remain as nonvanishing terms owing to the nonlinearity in the collision term of the CGS Boltzmann equation. The Navier-Stokes-Fourier law obtained using the CGS Boltzmann equation indicates that the pressure deviator and heat flux include fluctuations of their one-order higher moments.     

Asymptotic analysis of the lattice Boltzmann equation

In this article we analyze the lattice Boltzmann equation (LBE) by using the asymptotic expansion technique. We first relate the LBE to the finite discrete-velocity model (FDVM) of the Boltzmann equation with the diffusive scaling. The analysis of this model directly leads to the incompressible Navier–Stokes equations, as opposed to the compressible Navier–Stokes equations obtained by the Chapman–Enskog analysis with convective scaling. We also apply the asymptotic analysis directly to the fully discrete LBE, as opposed to the usual practice of analyzing a continuous equation obtained through the Taylor-expansion of the LBE. This leads to a consistency analysis which provides order-by-order information about the numerical solution of the LBE. The asymptotic technique enables us to analyze the structure of the leading order errors and the accuracy of numerically derived quantities, such as vorticity. It also justifies the use of Richardson’s extrapolation method. As an example, a two-dimensional Taylor-vortex flow is used to validate our analysis. The numerical results agree very well with our analytic predictions.     

Lattice Kinetic Formulation for Ferrofluids

The lattice Boltzmann approach is used to solve continuum equations describing colloids of ferromagnetic particles (ferrofluids) in a regime, where the particle spins are in equilibrium with magnetic torques. This limit of rapid spin adjustment yields a symmetric total stress tensor that is essential for a kinetic formulation based on the Boltzmann equation. The magnetisation equation is solved using a vector-valued distribution function analogous to the earlier treatment (J. Comput. Phys. 179, 95) of the induction equation in magnetohydrodynamics, but the details are rather more complex because the magnetisation equation is not in conservation form except in a weakly magnetised limit.     

Boundary conditions for regularized 13-moment-equations for micro-channel-flows

Boundary conditions are the major obstacle in simulations based on advanced continuum models of rarefied and micro-flows of gases. In this paper, we present a theory how to combine the regularized 13-moment-equations derived from Boltzmann’s equation with boundary conditions obtained from Maxwell’s kinetic accommodation model. While for the linear case these kinetic boundary conditions suffice, we need additional conditions in the non-linear case. These are provided by the bulk solutions obtained after properly transforming the equations while keeping their asymptotic accuracy with respect to Boltzmann’s equation.After finding a suitable set of boundary conditions and equations, a numerical method for generic shear flow problems is formulated. Several test simulations demonstrate the stable and oscillation-free performance of the new approach.     

A discontinuous Galerkin finite-element method for a 1D prototype of the Boltzmann equation

To develop and analyze new computational techniques for the Boltzmann equation based on model or approximation adaptivity, it is imperative to have disposal of a compliant model problem that displays the essential characteristics of the Boltzmann equation and that admits the extraction of highly accurate reference solutions. For standard collision processes, the Boltzmann equation itself fails to meet the second requirement for d = 2, 3 spatial dimensions, on account of its setting in 2d, while for d = 1 the first requirement is violated because the Boltzmann equation then lacks the convergence-to-equilibrium property that forms the substructure of simplified models. In this article we present a numerical investigation of a new one-dimensional prototype of the Boltzmann equation. The underlying molecular model is endowed with random collisions, which have been fabricated such that the corresponding Boltzmann equation exhibits convergence to Maxwell–Boltzmann equilibrium solutions. The new Boltzmann model is approximated by means of a discontinuous Galerkin (DG) finite-element method. To validate the one-dimensional Boltzmann model, we conduct numerical experiments and compare the results with Monte-Carlo simulations of equivalent molecular-dynamics models.     

Towards a quantitative kinetic theory of polar active matter

A recent kinetic approach for Vicsek-like models of active particles is reviewed. The theory is based on an exact Chapman- Kolmogorov equation in phase space. It can handle discrete time dynamics and “exotic” multi-particle interactions. A nonlocal mean-field theory for the one-particle distribution function is obtained by assuming molecular chaos. The Boltzmann approach of Bertin, et al., Phys. Rev. E 74, 022101 (2006) and J. Phys. A 42, 445001 (2009), is critically assessed and compared to the current approach. In Boltzmann theory, a collision starts when two particles enter each others action spheres and is finished when their distance exceeds the interaction radius. The average duration of such a collision, τ₀, is measured for the Vicsek model with continuous time-evolution. If the noise is chosen to be close to the flocking threshold, the average time between collisions is found to be roughly equal to τ₀ at low densities. Thus, the continuous-time Vicsek-model near the flocking threshold cannot be accurately described by a Boltzmann equation, even at very small density because collisions take so long that typically other particles join in, rendering Boltzmann’s binary collision assumption invalid. Hydrodynamic equations for the phase space approach are derived by means of a Chapman-Enskog expansion. The equations are compared to the Toner-Tu theory of polar active matter. New terms, absent in the Toner-Tu theory, are highlighted. Convergence problems of Chapman-Enskog and similar gradient expansions are discussed.     

Selection rules,causality, and unitarity in statistical and quantum physics

The integrodifferential equations satisfied by the statistical frequency functions for physical systems undergoing stochastic transitions are derived by application of a causality principle and selection rules to the Markov chain equations. The result equations can be viewed as generalizations of the diffusion equation, but, unlike the latter, they have a direct bearing onactive transport problems in biophysics andcondensation aggregation problems of astrophysics and phase transition theory. Simple specific examples of the effects of severe selection rules, such as the relaxational Boltzmann transport equation and the diffusion equation, are also given. Finally, partial differential equations for the probability amplitudes of quantum mechanics are derived, usingunitarity instead of causality, and a selection rule is applied directly to obtain ageneralization of the Dirac equation in which infinite transitions between states arenot allowed.     

Relaxation and transport in FCHC lattice gases

FCHC lattice gases are the basic models for studying flow problems in three-dimensional systems. This paper presents a self-contained theoretical analysis and some computer simulations of such lattice gases, extended to include an arbitrary number of rest particles, with special emphasis on non-semi-detailed balance (NSDB) models. The special FCHC lattice symmetry guarantees isotropy of the Navier-Stokes equations, and enumerates the 12 spurious conservation laws (staggered momenta). The kinetic theory is based on the mean field approximation or the nonlinear Boltzmann equation. It is shown how calculation of the eigenvalues of the linearized Boltzmann equation offers a simple alternative to the Chapman-Enskog method or the multi-time-scale methods for calculating transport coefficients and relaxation rates. The simulated values for the speed of sound in NSDB models slightly disagree with the Boltzmann prediction.     

Generalized Boltzmann equation for lattice gas automata

In this paper a theory is formulated that predicts velocity and spatial correlations between occupation numbers that occur in lattice gas automata violating semi-detailed balance. Starting from a coupled BBGKY hierarchy for then-particle distribution functions, cluster expansion techniques are used to derive approximate kinetic equations. In zeroth approximation the standard nonlnear Boltzmann equation is obtained; the next approximation yields the ring kinetic equation, similar to that for hard-sphere systems, describing the time evolution of pair correlations. The ring equation is solved to determine the (nonvanishing) pair correlation functions in equilibrium for two models that violate semidetailed balance. One is a model of interacting random walkers on a line, the other one is a two-dimensional fluid-type model on a triangular lattice. The numerical predictions agree very well with computer simulations.     

Numerical method based on the lattice Boltzmann model for the Fisher equation

In this paper, a lattice Boltzmann model for the Fisher equation is proposed. First, the Chapman-Enskog expansion and the multiscale time expansion are used to describe higher-order moment of equilibrium distribution functions and a series of partial differential equations in different time scales. Second, the modified partial differential equation of the Fisher equation with the higher-order truncation error is obtained. Third, comparison between numerical results of the lattice Boltzmann models and exact solution is given. The numerical results agree well with the classical ones.     

Hydrodynamic Limit of the Boltzmann Equation with Contact Discontinuities

The hydrodynamic limit for the Boltzmann equation is studied in the case when the limit system, that is, the system of Euler equations contains contact discontinuities. When suitable initial data is chosen to avoid the initial layer, we prove that there exist a family of solutions to the Boltzmann equation globally in time for small Knudsen number. And this family of solutions converge to the local Maxwellian defined by the contact discontinuity of the Euler equations uniformly away from the discontinuity as the Knudsen number ε tends to zero. The proof is obtained by an appropriately chosen scaling and the energy method through the micro-macro decomposition.