Development and Comparative Studies of Three Non-Free Parameter Lattice Boltzmann Models for Simulation of Compressible Flows

This paper at first shows the details of finite volume-based lattice Boltzmann method (FV-LBM) for simulation of compressible flows with shock waves. In the FV-LBM, the normal convective flux at the interface of a cell is evaluated by using one-dimensional compressible lattice Boltzmann model, while the tangential flux is calculated using the same way as used in the conventional Euler solvers. The paper then presents a platform to construct one-dimensional compressible lattice Boltzmann model for its use in FV-LBM. The platform is formed from the conservation forms of moments. Under the platform, both the equilibrium distribution functions and lattice velocities can be determined, and therefore, non-free parameter model can be developed. The paper particularly presents three typical non-free parameter models, D1Q3, D1Q4 and D1Q5. The performances of these three models for simulation of compressible flows are investigated by a brief analysis and their application to solve some one-dimensional and two-dimensional test problems. Numerical results showed that D1Q3 model costs the least computation time and D1Q4 and D1Q5 models have the wider application range of Mach number. From the results, it seems that D1Q4 model could be the best choice for the FV-LBM simulation of hypersonic flows.     

Nonexistence of H Theorem for Some Lattice Boltzmann Models

In this paper, we provide a set of sufficient conditions under which a lattice Boltzmann model does not admit an H theorem. By verifying the conditions, we prove that a number of existing lattice Boltzmann models does not admit an H theorem. These models include D2Q6, D2Q9 and D3Q15 athermal models, and D2Q16 and D3Q40 thermal (energy-conserving) models. The proof does not require the equilibria to be polynomials.     

Absorption Spectral Study of Mixed Chiral Amino Acid Porphyrin Langmuir-Blodgett Films

Absorption spectra were performed on Langmuir-Blodgett films of a new chiral amino acid porphyrin (Py) and its mixture with stearic acid (SA). Vertical uniformity of the LB films has been demonstrated by the linear relationship between the absorbance and the layer number of films. J-Aggregates were present when the film was transferred at various surface pressures and at different molar ratios of Py and SA. However, no aggregation was present for the molecules in the adjacent monolayers. the observed optical absorption spectra of mixed films were markedly dependent on the combination of Py. When the molar ratio of Py was increased from 0.05 to 0.75, the absorption peaks of Soret and Q bands bathochromically shifted ca. 5.5 and 2.2 nm. When the deposited pressure was increased from 5 to 35 mN/m, the absorption peaks of Soret and Q bands bathochromically shifted ca. 3.7, 3.2 nm and 3.5, 2.2 nm for pure Py and 1:3 Py-SA mixed LB films, respectively. on a hydrophobic substrate, the LB films showed a red shift of ca. 4.5 nm for Soret band compared with that on the hydrophilic substrate owing to their different environments.     

A time-reversal lattice Boltzmann method

In this paper we address the time-reversed simulation of viscous flows by the lattice Boltzmann method (LB). The theoretical derivation of the reversed LB from the Boltzmann equation is detailed, and the method implemented for weakly compressible flows using the D2Q9 scheme. The implementation of boundary conditions is also discussed. The accuracy and stability are illustrated by four test cases, namely the propagation of an acoustic wave in a medium at rest and in an uniform mean flow, the Taylor–Green vortex decay and the vortex pair–wall collision.     

Three-dimensional coupled double-distribution-function lattice Boltzmann models for compressible Navier–Stokes equations

Two three-dimensional (3D) lattice Boltzmann models in the framework of coupled double-distribution-function approach for compressible flows, in which specific-heat ratio and Prandtl number can be adjustable, are developed in this paper. The main differences between the two models are discrete equilibrium density and total energy distribution function. One is the D3Q25 model obtained from spherical function, and the other is the D3Q27 standard lattice model obtained from Hermite expansions of the corresponding continuous equilibrium distribution functions. The two models are tested by numerical simulations of some typical compressible flows, and their numerical stability and precision are also analysed. The results indicate that the two models are capable for supersonic flows, while the one from Hermite expansions is not suitable for compressible flows with shock waves.     

Two-dimensional lattice-Boltzmann simulations of single phase flow in a pseudo two-dimensional micromodel

We compare two-dimensional and three dimensional lattice-Boltzmann (LB) simulations of fluid flow in a pseudo-2D micromodel used in experimental work. We show that 2D LB simulations can be used to compute the average velocity field in 3D systems in which the third dimension is small compared with the other two dimensions, and where the velocity component along the third dimension is zero. Correct results are obtained provided a viscous drag force, representing the effect of the third dimension, is used in the 2D model. The use of 2D simulations allows to significantly reduce the computational cost of the calculations, and hence to increase the size and resolution of the systems that can be studied using the LB method.     

Lattice Boltzmann Model for the Incompressible Navier–Stokes Equation

In this paper a lattice Boltzmann (LB) model to simulate incompressible flow is developed. The main idea is to explicitly eliminate the terms of o(M ²), where M is the Mach number, due to the density fluctuation in the existing LB models. In the proposed incompressible LB model, the pressure p instead of the mass density ρ is the independent dynamic variable. The incompressible Navier–Stokes equations are derived from the incompressible LB model via Chapman–Enskog procedure. Numerical results of simulations of the plane Poiseuille flow driven either by pressure gradient or a fixed velocity profile at entrance as well as of the 2D Womersley flow are presented. The numerical results are found to be in excellent agreement with theory.     

Rotational invariance in the three-dimensional lattice Boltzmann method is dependent on the choice of lattice

The lattice Boltzmann method has recently gained popularity as a tool for simulating complex fluid flows. It uses discrete sets of velocity vectors, or lattices, to create a reduced model of the molecular dynamics of a continuum fluid. While several lattices are believed to behave isotropically, there are reports of qualitatively incorrect results. However, thus far, the reason as to why a lack of isotropy occurs is not known. Based on the hypothesis that lower order lattices may not display rotational invariance, this study tests the isotropy of the D3Q15, D3Q19 and D3Q27 lattices by performing simulations at intermediate Reynolds numbers (50–500) and low Knudsen number (<0.0005) in an axisymmetrical geometry with a nozzle leading to a throat followed by a sudden expansion. The symmetry properties of the results were examined. It was found that at Re ? 250 the D3Q15 and D3Q19 lattices produced different results depending on the plane of the lattice with which the flow was aligned. Lattice planes with fewer than six velocity vectors consistently produced results which were qualitatively different from the planes with six or more velocity vectors. These errors were not observed at Re = 50 or when a D3Q27 lattice was used. They appeared to be independent of grid density, collision operator and Ma. This suggests that the lattices which contain these planes are not fully isotropic and therefore do not properly replicate the behavior of a real fluid in this particular situation, notably downstream from the expansion. Predictions made using these models in more complex geometries may therefore be affected by the orientation of the lattice. When using LBM in CFD simulation (including validation) this study highlights the need for caution to ensure that the solution obtained is independent of the lattice orientation throughout the domain.     

Entropy and Galilean invariance of lattice Boltzmann theories

A theory of lattice Boltzmann (LB) models for hydrodynamic simulation is developed upon a novel relation between entropy construction and roots of Hermite polynomials. A systematic procedure is described for constructing numerically stable and complete Galilean invariant LB models. The stability of the new LB models is illustrated with a shock tube simulation.     

Nonlinear Airy Light Bullets in a 3D Self‐Defocusing Medium

The (3+1)‐dimensional [(3+1)D] nonlinear Schrödinger (NLS) equation is investigated, describing the propagation of nonlinear spatiotemporal wave packets in a self‐defocusing medium, and a new type of Airy spatiotemporal solutions is presented. By using the reductive perturbation method, the (3+1)D NLS equation is reduced to the spherical Kortewegde Vries (SKdV) equation. Based on the Hirota's bilinear method, the bilinear form of the SKdV equation is constructed and Airy light bullet (LB) solutions of different orders are obtained, which depend on the sets of two free constants associated with the amplitude and initial phase. The results show that these Airy LBs can exist in the self‐defocusing medium and their intensities can be controlled by selecting the suitable free parameters along the propagation distance. As examples, three types of low‐order approximate LB solutions are presented and their intensity profiles numerically discussed. The obtained results are helpful in exploring nonlinear phenomena in a self‐defocusing medium and providing a new approach for possible experimental verification of LBs.     

A Consistent Grid Coupling Method for Lattice-Boltzmann Schemes

A method of coupling grids of different mesh size is developed for classical Lattice-Boltzmann (LB) algorithms on uniform grids. The approach is based on an asymptotic analysis revealing suitable quantities equalized along the grid interfaces for exchanging information between the subgrids. In contrast to other couplings the method works without overlap zones. Moreover the grid velocity (Mach number) is not kept constant, as the time step depends not linearly but quadratically on the grid spacing. To illustrate the basic idea we use a simple LB algorithm solving the advection-diffusion equation. The proposed grid coupling is validated by numerical convergence studies indicating, that the coupling does not affect the second-order convergence behavior of the LB algorithm which is observed on uniform grids. In order to demonstrate its principal applicability to other LB models, the coupling is generalized to the standard D2P9 model for (Navier-)Stokes flow and tested numerically. As we use analytic tools different from the Chapman-Enskog expansion, the theoretical background material is given in two appendices. In particular, the results of numerical experiments are confirmed with a consistency analysis.     

Accuracy analysis of high-order lattice Boltzmann models for rarefied gas flows

In this work, we have theoretically analyzed and numerically evaluated the accuracy of high-order lattice Boltzmann (LB) models for capturing non-equilibrium effects in rarefied gas flows. In the incompressible limit, the LB equation is shown to be able to reduce to the linearized Bhatnagar–Gross–Krook (BGK) equation. Therefore, when the same Gauss–Hermite quadrature is used, LB method closely resembles the discrete velocity method (DVM). In addition, the order of Hermite expansion for the equilibrium distribution function is found not to be directly correlated with the approximation order in terms of the Knudsen number to the BGK equation for incompressible flows. Meanwhile, we have numerically evaluated the LB models for a standing-shear-wave problem, which is designed specifically for assessing model accuracy by excluding the influence of gas molecule/surface interactions at wall boundaries. The numerical simulation results confirm that the high-order terms in the discrete equilibrium distribution function play a negligible role in capturing non-equilibrium effect for low-speed flows. By contrast, appropriate Gauss–Hermite quadrature has the most significant effect on whether LB models can describe the essential flow physics of rarefied gas accurately. Our simulation results, where the effect of wall/gas interactions is excluded, can lead to conclusion on the LB modeling capability that the models with higher-order quadratures provide more accurate results. For the same order Gauss–Hermite quadrature, the exact abscissae will also modestly influence numerical accuracy. Using the same Gauss–Hermite quadrature, the numerical results of both LB and DVM methods are in excellent agreement for flows across a broad range of the Knudsen numbers, which confirms that the LB simulation is similar to the DVM process. Therefore, LB method can offer flexible models suitable for simulating continuum flows at the Navier–Stokes level and rarefied gas flows at the linearized Boltzmann model equation level.     

Structural Study on 5-(Dodecyloxy)-1-((6-Trimethylammonio)Hexyl)oxy)Naphthalene—Sodium Dextran Sulfate Langmuir-Blodgett Film Through Fourier Transform Infrared Spectroscopy

The Fourier transform infrared transmission (FTIR-T) and attenuated total reflection (ATR) spectra of the 5-(dodecyloxy)-1-((6-(trimethylammonio)hexyl)oxy)naphthalene — sodium dextran sulfate composite LB films deposited under the surface pressure of 30 mN/m have been recorded and spectral assignments made. By means of polarization techniques and with the application of the uniaxial and biaxial orientation models, the orientation characteristics of the long hydrocarbon chains and the = C-O linkage in the LB films have been investigated, giving rise to the detailed information of the arrangement of molecular fragments in the 1-SDS composite LB films.     

基于格子Boltzmann方法的超声速非平衡流模拟

基于D1Q4可压缩格子Boltzmann模型,按照流通矢量分裂方法的思路,采用坐标旋转技术构造求解三维带化学反应Navier-Stokes方程对流通量求解器.结合有限体积法求解三维化学非平衡流Navier-Stokes方程,采用时间算子分裂算法解决化学反应刚性问题,数值模拟超声速化学非平衡流的三个经典算例.数值结果表明:在高马赫数下,采用D1Q4可压缩格子Boltzmann模型构造的三维对流通量求解器数值模拟中没有出现非物理解,同时在超声速化学非平衡流场中正确分辨激波、燃烧波等物理现象,精度和分辨率均较高,验证了本文构造的三维对流通量求解器的可靠性,拓宽了D1Q4可压缩格子Boltzmann模型的应用范围,为计算超声速化学非平衡流提供一种新方法.     

Quantum phase transition in one-dimensional arrays of resistively shunted small Josephson junctions

We have observed a superconductor-insulator transition in one-dimensional (1D) arrays of small Josephson junctions by changing both the resistance R(S) of normal metal resistors shunting each junction and the ratio of the Josephson coupling energy E(J) to the charging energy E(C). The phase boundary lies at R(S) approximately R(Q) (R(Q) identical with h/4e(2)=6.45 kOmega) when E(J)/E(C) is smaller than about unity. We discuss the obtained phase diagram in terms of theoretical models of the dissipation-driven quantum phase transition, with particular attention to differences from 2D arrays.     

Cole-Hopf Transformation Based Lattice Boltzmann Model for One-dimensional Burgers' Equation

In this paper,we present a Cole-Hopf transformation based lattice Boltzmann(LB) model for solving one-dimensional Burgers' equation,and compared to available LB models,the effect of nonlinear convection term can be eliminated.Through Chapman-Enskog analysis,it can be found that the converted diffusion equation based on the Cole-Hopf transformation can be recovered correctly from present LB model.Some numerical tests are also performed to validate the present LB model,and the numerical results show that,similar to previous LB models,the present model also has a second-order convergence rate in space,but it is more accurate than the previous ones.     

Anomalous temperature dependence of resistivity in quasi-one-dimensional conductors in a strong magnetic field

We present a heuristic, semiphenomenological model of the anomalous temperature (T) dependence of resistivity recently observed experimentally in the quasi-one-dimensional (Q1D) organic conductors of the family in moderately strong magnetic fields. We suggest that a Q1D conductor behaves like an insulator (), when its effective dimensionality is one, and like a metal (), when its effective dimensionality is greater than one. Applying a magnetic field reduces the effective dimensionality of the system and switches the temperature dependence of resistivity between the insulating and metallic laws depending on the magnitude and orientation of the magnetic field. We critically analyze whether various microscopic models suggested in literature can produce such a behavior and find that none of the models is fully satisfactory. In particular, we perform detailed analytical and numerical calculations within the scenario of magnetic-field-induced spin-density-wave precursor effect suggested by Gor'kov and find that the theoretical results do not agree with the experimental observations. Received 20 October 1998     

Optical second-harmonic-generation probe of two-dimensional ferroelectricity

Optical second-harmonic generation (SHG) is used as a noninvasive probe of two-dimensional (2D) ferroelectricity in Langmuir-Blodgett (LB) films of the copolymer vinylidene fluoride with trifluoroethylene. The surface 2D ferroelectric-paraelectric phase transition in the topmost layer of the LB films and a thickness-independent (almost 2D) transition in the bulk of these films are observed in temperature studies of SHG.     

Confinement of polar optical phonons in quasi-one-dimensional Wurtzite GaN-based quantum well wires: propagating and half-space phonon modes

With the aid of the macroscopic dielectric continuum and Loudon’s uniaxial crystal models, the propagating (PR) and half-space (HS) optical phonon modes and corresponding Fröhlich-like electron-phonon interaction Hamiltonians in a quasi-one-dimensionality (Q1D) wurtzite quantum well wire (QWW) structure are derived and studied. Numerical calculations on a wurtzite GaN/A_l0.15Ga_0.85N QWW are performed, and discussion is focused mainly on the dependence of the frequency dispersions of PR and HS modes on the free wave-number k _z in the z-direction and on the azimuthal quantum number m. The calculated results show that, for given k _z and m, there usually exist infinite branches of PR and HS modes in the high-frequency range, and only finite branches of HS modes in the low-frequency range in wurtzite QWW systems. The reducing behaviors of the PR modes to HS modes, and of the HS mode to interface phonon mode have been observed clearly in Q1D wurtzite heterostructures. Moreover, the dispersive properties of the PR and HS modes in Q1D QWWs have been compared with those in Q2D quantum well structures. The underlying physical reasons for these features have also been analyzed in depth.     

On the Three-Dimensional Central Moment Lattice Boltzmann Method

A three-dimensional (3D) lattice Boltzmann method based on central moments is derived. Two main elements are the local attractors in the collision term and the source terms representing the effect of external and/or self-consistent internal forces. For suitable choices of the orthogonal moment basis for the three-dimensional, twenty seven velocity (D3Q27), and, its subset, fifteen velocity (D3Q15) lattice models, attractors are expressed in terms of factorization of lower order moments as suggested in an earlier work; the corresponding source terms are specified to correctly influence lower order hydrodynamic fields, while avoiding aliasing effects for higher order moments. These are achieved by successively matching the corresponding continuous and discrete central moments at various orders, with the final expressions written in terms of raw moments via a transformation based on the binomial theorem. Furthermore, to alleviate the discrete effects with the source terms, they are treated to be temporally semi-implicit and second-order, with the implicitness subsequently removed by means of a transformation. As a result, the approach is frame-invariant by construction and its emergent dynamics describing fully 3D fluid motion in the presence of force fields is Galilean invariant. Numerical experiments for a set of benchmark problems demonstrate its accuracy.