Original Boltzmann Equation Research Articles

A unified stochastic particle Bhatnagar-Gross-Krook method for multiscale gas flows

The stochastic particle method based on Bhatnagar-Gross-Krook (BGK) or ellipsoidal statistical BGK (ESBGK) model approximates the pairwise collisions in the Boltzmann equation using a relaxation process. Therefore, it is more efficient to simulate gas flows at small Knudsen numbers than the counterparts based on the original Boltzmann equation, such as the Direct Simulation Monte Carlo (DSMC) method. However, the traditional stochastic particle BGK method decouples the molecular motions and collisions in analogy to the DSMC method, and hence its transport properties deviate from physical values as the time step size increases. This defect significantly affects its computational accuracy and efficiency for the simulation of multiscale flows, especially when the transport processes in the continuum regime is important. In the present paper, we propose a unified stochastic particle ESBGK (USP-ESBGK) method by combining the molecular convection and collision effects. In the continuum regime, the proposed method can be applied using large temporal-spatial discretization and approaches to the Navier-Stokes solutions with second-order accuracy. Furthermore, it is capable to simulate both the small scale non-equilibrium flows and large scale continuum flows within a unified framework efficiently and accurately. The applications of USP-ESBGK method to a variety of benchmark problems, including Couette flow, thermal Couette flow, Poiseuille flow, Sod tube flow, cavity flow, Taylor-Green vortex, and flow through a slit, demonstrated that it is a promising tool to simulate multiscale gas flows ranging from rarefied to continuum regime.

Topology optimization using the lattice Boltzmann method incorporating level set boundary expressions

This paper presents a topology optimization method for fluid dynamics problems, based on the level set method and using the lattice Boltzmann method (LBM). In this optimization method, the optimization problems are formulated based on the original Boltzmann equation, and the design sensitivities are precisely obtained without the time-consuming numerical operations encountered when dealing with a large-scale asymmetric matrix, in contrast to previous research in which the LBM uses the lattice Boltzmann equation (LBE) for the formulations of optimization problems and the derivation of their adjoint equations. That is, we newly derive sensitivity formulations from the original Boltzmann equation, not the LBE that can be said to be an approximated equation, and these formulations yield strictly correct sensitivities that are error free. Based on the above formulations, we construct a level set-based topology optimization method incorporating a fictitious interface energy for the design of a fluid channel that minimizes flow friction. Furthermore, two- and three-dimensional numerical examples are provided to confirm the validity and utility of the presented method.

1114 最適流路設計を目的としたレベルセット法に基づくトポロジー最適化

This paper presents a topology optimization method based on the level set method for design problem of flow channel using the lattice Boltzmann method (LBM). In the presented method, the optimization problems and their adjoint equations are formulated based on the original Boltzmann equation, and the design sensitivities are precisely obtained without the time-consuming numerical operations dealing with a large-scale asymmetric matrix, while the previous researches using the LBM use the lattice Boltzmann equation (LBE) for the formulations of optimization problems and the derivation of their adjoint equation. In addition, the Lattice kinetic scheme that has attract attention as a new type of the LBM, enables the reduction of memory usage in computing flow field is employed to reduce the computational cost in the computation of the state field. Based on the above formulations, we construct a level set-based topology optimization method incorporating fictitious interface energy for the design of fluid channel that minimizes pressure drop. Furthermore, two-and three-dimensional numerical examples are provided to confirm the validity and utility of the presented method.

Construction of Normal Discrete Velocity Models of the Boltzmann Equation

Discretization methods have been developed on the idea of replacing the original Boltzmann equation (BE) by a finite set of nonlinear hyperbolic PDEs corresponding to the densities linked to a suitable finite set of velocities. One open problem related to the discrete BE is the construction of normal (fulfilling only physical conservation laws) discrete velocity models (DVMs). In many papers on DVMs, authors postulate from the beginning that a finite velocity space with such properties is given and after that study the discrete BE. Our aim is not to study the equations for DVMs, but to discuss all possible choices of finite sets of velocities satisfying this type of restrictions. Using our previous results, i.e. the general algorithm for the construction of normal discrete kinetic models (DKMs), we develop and implement an algorithm for the particular case of DVMs of the BE and give a complete classification for models with small number n of velocities (n ≤ 10).

Stability, Convergence to Self-Similarity and Elastic Limit for the Boltzmann Equation for Inelastic Hard Spheres

We consider the spatially homogeneous Boltzmann equation for {\em inelastic hard spheres}, in the framework of so-called {\em constant normal restitution coefficients} $\alpha \in [0,1]$. In the physical regime of a small inelasticity (that is $\alpha \in [\alpha_*,1)$ for some constructive $\alpha_*>0$) we prove uniqueness of the self-similar profile for given values of the restitution coefficient $\alpha \in [\alpha_*,1)$, the mass and the momentum; therefore we deduce the uniqueness of the self-similar solution (up to a time translation). Moreover, if the initial datum lies in $L^1_3$, and under some smallness condition on $(1-\alpha_*)$ depending on the mass, energy and $L^1_3$ norm of this initial datum, we prove time asymptotic convergence (with polynomial rate) of the solution towards the self-similar solution (the so-called {\em homogeneous cooling state}). These uniqueness, stability and convergence results are expressed in the self-similar variables and then translate into corresponding results for the original Boltzmann equation. The proofs are based on the identification of a suitable elastic limit rescaling, and the construction of a smooth path of self-similar profiles connecting to a particular Maxwellian equilibrium in the elastic limit, together with tools from perturbative theory of linear operators. Some universal quantities, such as the "quasi-elastic self-similar temperature" and the rate of convergence towards self-similarity at first order in terms of $(1-\alpha)$, are obtained from our study. These results provide a positive answer and a mathematical proof of the Ernst-Brito conjecture [16] in the case of inelastic hard spheres with small inelasticity.

Model Boltzmann equation for gas mixtures: Construction and numerical comparison

A new model for the nonlinear Boltzmann equation for gas mixtures is constructed by the method employed in the derivation of the McCormack model in the linearized kinetic theory [F.J. McCormack, Phys. Fluids 16 (1973) 2095]. Then it is compared numerically with other existing models proposed in [P. Andries, K. Aoki, B. Perthame, J. Stat. Phys. 106 (2002) 993] and in [L.H. Holway Jr., Phys. Fluids 9 (1966) 1658] (the so-called ES-BGK model) as well as with the original Boltzmann equation. The new model is not restricted to the Maxwell molecule, can fit to general molecular models, and reproduces well solutions of the Boltzmann equation at least in the case of weak nonequilibrium. The numerical comparison is performed in the case of a binary gas mixture consisted of the hard-sphere or pseudo Maxwell molecules, after parameters concerning the molecular interaction are adjusted appropriately.

Database for flows of binary gas mixtures through a plane microchannel

A binary mixture of rarefied gases between two parallel plates is considered. The Poiseuille flow, thermal transpiration (flow caused by a temperature gradient of the plates) and concentration-driven flow (flow caused by a gradient of concentration of the component species) are analyzed on the basis of the linearized model Boltzmann equation with the diffuse reflection boundary condition. The analyses are first performed for mixtures of virtual gases composed of the hard-sphere or Maxwell molecules and the results are compared with those of the original Boltzmann equation. Then, the analyses for noble gases (He–Ne, He–Ar and Ne–Ar) are performed assuming more realistic molecular models (the inverse power-law potential and Lennard-Jones 12,6 models). By use of the results, flux databases covering the entire ranges of the Knudsen number and of the concentration and a wide range of the temperature are constructed. The databases are prepared for the use in the fluid-dynamic model for mixtures in a stationary nonisothermal microchannel derived in [S. Takata, H. Sugimoto, S. Kosuge, Eur. J. Mech. B/Fluids 26 (2007) 155], but can also be incorporated in the generalized Reynolds equation [S. Fukui, R. Kaneko, J. Tribol. 110 (1988) 253] in the gas film lubrication theory. The databases constructed can be downloaded freely from Electronic Annex 2 in the online version of this article.

Mapping of the classical kinetic balance equations onto the Schrödinger equation

Here we find a mapping onto the Sturm–Liouville operator of the first two balance equations derived from Boltzmann's equation. This mapping, which is irreversible and valid only for a subclass of solutions, is achieved by applying a Fourier transform to the momentum coordinate. In light of this irreversibility, it is necessary to develop a set of consistent prescriptions to find the probability of any physical quantity in the p-conjugate space such that it will coincide with the average over the momentum of the true probabilities obtained from the original Boltzmann equation. The one drawback of this prescription is that it is impossible to predict exactly the precise values of the position x and the momentum p at the same time. This uncertainty is limited by the relationship that all conjugate variables in a Fourier transform should obey, namely Δ x Δ p = η/2, where η is a free parameter of the theory. The prescriptions we have found appear to coincide with the postulates of quantum mechanics, when η is set equal to ℏ. This procedure seems to provide a statistical representation of quantum mechanics.

Renormalized dissipation in plasmas with finite collisionality.

A nonlinear truncation procedure for Fourier-Hermite expansion of Boltzmann-type plasma equations is presented which eliminates fine velocity scale, taking into account its effect on coarser scales. The truncated system is then transformed back to $(x,\ensuremath{\upsilon})$ space which results in a renormalized Boltzmann equation. The resulting equation may allow for coarser velocity space resolution in kinetic simulations while reducing to the original Boltzmann equation when fine velocity scales are resolved. To illustrate the procedure, renormalized equations are derived for one dimensional electrostatic plasmas in which collisions are modeled by the Lenard-Bernstein operator.

A generalized Legendre polynomial/sparse matrix approach for determining the distribution function in non-polar semiconductors

Until recently, the Legendre polynomial (LP) expansion method for solving the Boltzmann transport equation was limited to the use of two or three Legendre polynomials. In this work we generalize the method to include an arbitrarily high order LP expansion. The expansion method consists of representing the angular dependence of the distribution function about the field direction in terms of an infinite series of Legendre polynomials with unknown coefficients. The expansion is then substituted into the Boltzmann transport equation. With the use of orthogonality and the LP recurrence relations, an infinite system of equations is then generated from the original Boltzmann equation. This system is then solved numerically, using sparse matrix algebra, for the unknown coefficients of the LP expansion. Once the coefficient are determined, the complete distribution function is readily constructed. In an example calculation the Boltzmann equation is solved to 40th order of the LP expansion. Finally, resulting values for the energy distribution, as well as average energy and average velocity, are shown to agree with Monte Carlo simulation results.

Laguerre polynomial solution of the nonlinear Boltzmann equation for the hard-sphere model

We consider a spatially homogeneous and isotropic gas consisting of hard-sphere molecules. A vector representation of the scattering kernel is used to adapt the original Boltzmann equation to the idealized geometrical situation. By means of an expansion of the distribution function in terms of Laguerre polynomials this scalar Boltzmann equation is transformed to a set of moment equations. All “algebraized” collision integrals can be evaluated analytically. We discuss the truncation of the moment equations necessary for the practical application of this method. The eigenvalues of the linearized relaxation problem show a good convergence with respect to the truncation index.

The discrete model of the Boltzmann equation

Abstract In this paper we study the general model of a gas with a discrete distribution of velocities. The medium is composed of identical particles which can only have velocity vectors belonging to a finite set of p vectors. When the medium is sufficiently rarefied, only binary collisions are considered. The original Boltzmann equation is replaced by a system of p first order partial differential equations. Each equation in the system is linear with respect to the derivatives of the unknown functions and quadratic with respect to the functions themselves. As in classical kinetic theory, by introducing a properly defined H-Boltzmann function, we can prove that for a gas in a uniform state, the distribution of velocities tends to a distribution, called Maxwellian, in which each collision brings no contribution to the evolution of densities. Among all distributions of velocities which correspond to given state variables, one and only one is Maxwellian, and the corresponding H-function is minimal. When the c...

Some nonlinear problems in transport theory

The integro-differential Boltzmann equation governing the diffusion of the particles of a mixture is formulated by incorporating the nonlinear effects due to the collisions of a same kind of particles between themselves. For the case of 1/|; v|; - cross sections the integral version of the original Boltzmann equation is derived, and the general features of the associated nonlinear evolution problem are discussed. A simple application of the theory is developed with regard to the problem of the heating of a plasma.

Solution to the boltzmann equation for monoenergetic neutrons in a slab

This paper presents a physical solution to the linear integro-differential Boltzmann equation, which governs the stationary distribution of the flux of monoenergetic neutrons in the interior and at the boundary of an infinite slab of finite thickness. Results for both the angular and the total fluxes are obtained by properly accounting for the boundary conditions of the problem and by a Fourier transform of the original Boltzmann equation. An integral linear Fredholm equation for the Fourier transform of the total flux is thus derived. The solution of such an equation is given two different representations according to the two decompositions devised for the given Fredholm kernel. The solution obtained in each case is finally inverted termwise to yield the sought total flux in the original space. The result for the angular flux follows, then, directly from the result for the total flux.