Lattice Boltzmann Theory Research Articles

Asymptotic freedom in the lattice Boltzmann theory.

Asymptotic freedom is a feature of quantum chromodynamics that guarantees its well posedness. We derive an analog of asymptotic freedom enabling unconditional linear stability of lattice Boltzmann simulation of hydrodynamics. We further demonstrate the validity of the derived conditions via the special case of the equilibrium based on entropy maximization, which is shown to be uniquely renormalizable.

Метод решёточных уравнений Больцмана: моделирование изотермических низкоскоростных течений

In this work, lattice Boltzmann method on standard lattices was descript as one of the modern method of computation fluid dynamics. The article has main theorems, which prove computational algorithm, different type’s boundary conditions and defect in Galilean invariance. Moreover, the paper has some theoretical background about physical kinetic theory, Hermite polynomials and numeric integration. Here has not any new scientist discoveries, but has explanation of basic lattice Boltzmann theory.

On-node lattices construction using partial Gauss–Hermite quadrature for the lattice Boltzmann method**Project supported by the National Science and Technology Major Project, China (Grant No. 2017ZX06002002).

A concise theoretical framework, the partial Gauss–Hermite quadrature (pGHQ), is established to construct on-node lattices of the lattice Boltzmann (LB) method under a Cartesian coordinate system. Compared with the existing approaches, the pGHQ scheme has the following advantages: extremely concise algorithm, unifies the constructing procedure for symmetric and asymmetric on-node lattices, and covers a full-range quadrature degree of a given discrete velocity set. We employ the pGHQ scheme to search the local optimal and asymmetric lattices for moment degree equilibrium distribution discretization on the range . The search reveals a surprising abundance of available lattices. Through a brief analysis, the discrete velocity set shows a significant influence on the positivity of equilibrium distributions, which is considered as one of the major impacts of the numerical stability of the LB method. Hence, the results of the pGHQ scheme lay a foundation for further investigations to improve the numerical stability of the LB method by modifying the discrete velocity set. It is also worth noting that pGHQ can be extended into the entropic LB model, even though it was proposed for the Hermite polynomial expansion LB theory.

Derivation of lattice Boltzmann equation via analytical characteristic integral**Project supported by the National Science and Technology Major Project, China (Grant No. 2017ZX06002002).

A lattice Boltzmann (LB) theory, the analytical characteristic integral (ACI) LB theory, is proposed in this paper. ACI LB theory takes the Bhatnagar–Gross–Krook (BGK)-Boltzmann equation as the exact kinetic equation behind Navier–Stokes continuum and momentum equations and constructs an LB equation by rigorously integrating the BGK-Boltzmann equation along characteristics. It is a general theory, supporting most existing LB equations including the standard lattice BGK (LBGK) equation inherited from lattice-gas automata, whose theoretical foundation had been questioned. ACI LB theory also indicates that the characteristic parameter of an LB equation is collision number, depicting the particle-interaction intensity in the time span of the LB equation, instead of the traditionally assumed relaxation time, and the over-relaxation time problem is merely a manifestation of the temporal evolution of equilibrium distribution along characteristics under high collision number, irrelevant to particle kinetics. In ACI LB theory, the temporal evolution of equilibrium distribution along characteristics is the determinant of LB method accuracy and we numerically prove this.

Block iterative frequency-based lattice Boltzmann algorithm for microscale oscillatory flow

A recent article (Yong Shi and John E. Sader, Phys. Rev. E 81, 036706 (2010)) developed a linearized frequency-based lattice Boltzmann (LB) model to describe oscillatory Stokes flow in microelectromechanical systems (MEMS). Nonetheless, its numerical algorithm is formulated in the conventional time-marching form with addition of virtual time. In this article, we propose a different algorithm to solve the linearized LB model in the frequency domain using the block iteration scheme integrating the tri-diagonal matrix method, Gaussian Seidel-based alternative direction iteration and over relaxing technique. This change in the LB algorithm leads to direct modeling of microscale oscillatory flow in the frequency domain without mimicking a numerical time evolution. Through simulating the one-dimensional oscillatory Couette flow and two-dimensional flow around an oscillating circular cylinder, we examined numerical accuracy of this block iterative LB algorithm proposed in this article. Importantly, we also explored its computational efficiency in comparison to that of the conventional time-marching LB algorithm and several simplified block-iterative LB algorithms. Our results demonstrate the block iterative LB algorithm in this article is a useful alternative numerical solver exhibiting nearly 2nd order accuracy for simulating frequency-dependent oscillatory flow in MEMS. The simulations also reveal rich possibilities in construction of the block iterative LB algorithm through combining the LB theory in the frequency domain with advanced CFD numerical techniques.

Experimental investigation and numerical simulation on liquid phase separation of ternary Fe-Sn-Si/Ge monotectic alloy

The liquid phase separation of small Fe-Sn-Si/Ge alloy droplets under reduced-gravity condition is investigated experimentally by free fall technique and theoretically by lattice Boltzmann method. In the drop tube experiments, the Fe-Sn-Si/Ge monotectic alloys are heated by induction heating in an ultrahigh vacuum chamber and further overheated to 200 K above their liquid temperatures for a few seconds. Finally, the molten alloy melt is ejected out from the small orifice of a quartz tube by high pressure jetting gas of He and dispersed into numerous tiny droplets, which are rapidly solidified during free fall in a protecting He gas environment. These droplets benefit from the combined advantages of high undercooling, containerless state and rapid cooling, which can provide an efficient way to study the liquid phase separation of high-temperature alloys in microgravity. In order to efficiently reproduce the dynamic process of phase separation inside drop tube equipment, the effects of surface segregation and Marangoni convection are introduced into the interaction potential of different liquids within lattice Boltzmann theory. Based on this modified model, the dynamic mechanism of phase separation can be sufficiently analyzed and the phase separation patterns can be realistically simulated. Experimental results demonstrate that conspicuous liquid phase separations have taken place for both Fe-Sn-Si and Fe-Sn-Ge alloy droplets and the corresponding morphologies are mainly characterized by core-shell and dispersed structures. The phase separation process can be modulated by the third-element addition. As the Si element of Fe-Sn-Si alloy is replaced by the Ge element with the same fraction, the distribution order of Fe-rich and Sn-rich zones is reversed within core-shell structure. A core-shell structure composed of a Fe-rich core and a Sn-rich shell is frequently observed in Fe-Sn-Si alloy droplets whereas the Fe-Sn-Ge alloy droplets tend to form a core-shell structure consisting of a Sn-rich core and a Fe-rich shell. Theoretical calculations show that the droplet cooling rate is closely related to droplet size: a smaller alloy droplet has a higher cooling rate. The liquid L2(Sn) phase always nucleates preferentially and forms tiny globules prior to solid Fe phase. Stokes motion can be greatly weakened in this experiment and the Marangoni migration dominates the globule movement in the process of liquid phase separation. Furthermore, the intensity of Marangoni convection within Fe-Sn-Ge alloy droplets is significantly stronger than that inside Fe-Sn-Si alloy droplets. Numerical simulations reveal that the cooling rate, Marangoni convection and surface segregation play the important roles in determining the selection of core-shell configurations and the formation of dispersed structures. Ultrahigh cooling rate contributes to forming the dispersed structures. When the Marangoni convection proceeds more drastically than the surface segregation, the minor liquid phase with a smaller surface free energy migrates to droplet center and occupies the interior of droplet, otherwise most of the minor phases appear around the periphery of droplet.

Consistent two-population lattice Boltzmann model for thermal flows

Theory of two-population lattice Boltzmann equations for thermal flow simulations is revisited. The present approach makes use of a consistent division of the conservation laws between the two lattices, where mass and the momentum are conserved quantities on the first lattice, and the energy is conserved quantity of the second lattice. The theory of such a division is developed, and the advantage of energy conservation in the model construction is demonstrated in detail. The present fully local lattice Boltzmann theory is specified on the standard lattices for the simulation of thermal flows. Extension to the subgrid entropic lattice Boltzmann formulation is also given. The theory is validated with a set of standard two-dimensional simulations including planar Couette flow and natural convection in two dimensions.

Computer simulation techniques for food science and engineering: Simulating atomic scale and coarse-grained models

Abstract We describe computer simulation techniques that have been, or can be, used in Food Science and Engineering. We describe models which cannot be utilized without employing computer simulation but do not explicitly address models such as self consistent field approaches or DLVO theory. We describe techniques which either continue to play fundamental roles in computer simulation of soft matter and fluids, or newer developments which have shown increased use in the last decade. Here we outline the background to statistical mechanics followed by descriptions with selected examples of Molecular Dynamics, Coarse-Grained modeling, Monte Carlo techniques, Dissipative Particle Dynamics and Lattice–Boltzmann theory.

Multidimensional on-lattice higher-order models in the thermal lattice Boltzmann theory

We present a set of uniform polynomial equations that provides multidimensional on-lattice higher-order models of the lattice Boltzmann theory while keeping compact the number of discrete velocities. As examples, we explicitly derive one-, two-, and three-dimensional on-lattice models applicable to describing thermal compressible flows of the accuracy levels of the Navier-Stokes equations with smaller numbers of discrete velocities in comparison to the existing models. We demonstrate the accuracy and stability of the one-, two-, and three-dimensional models by using the Riemann problem.

Univariate polynomial equation providing on-lattice higher-order models of thermal lattice Boltzmann theory

A univariate polynomial equation is presented. It provides on-lattice higher-order models of the thermal lattice Boltzmann equation. The models can be accurate up to any required level and can be applied to regular lattices, which allow efficient and accurate approximate solutions of the Boltzmann equation. We derive models approaching the complete Galilean invariant and providing accuracy of the fourth-order moment and beyond. We simulate one-dimensional thermal shock tube problems to illustrate the accuracy of our models. Moreover, we show the remarkably enhanced stability obtained by our models and our discretized equilibrium distributions.

How to obtain higher-order multivariate Hermite expansion of Maxwell–Boltzmann distribution by using Taylor expansion?

We obtain the higher-order multivariate Hermite expansion of the Maxwell–Boltzmann distribution by using a new, compact tensorial notation and present a method to obtain the nth order multivariate Taylor expansion, which is identical to the nth order multivariate Hermite expansion of the Maxwell–Boltzmann distribution. This study enables us to find higher-order models of discrete kinetic theories such as the lattice Boltzmann theory.

Uniform Polynomial Equations Providing Higher-order Multi-dimensional Models in Lattice Boltzmann Theory

We present a set of polynomial equations that provides models of the lattice Boltzmann theory for any required level of accuracy and for any dimensional space in a general form. We explicitly derive two- and three-dimensional models applicable to describe thermal compressible flows of the level of the Navier-Stokes equations.