A note on equilibrium boundary conditions in lattice Boltzmann fluid dynamic simulations

It is shown that Lattice Boltzmann (LB) simulations using simple equilibrium boundary conditions at solid walls, provide quantitatively accurate results for backward-facing step flows at moderate Reynolds numbers. The basic reason for such favorable behavior is that well-resolved LB simulations operate in the so-called weak non-equilibrium regime, in which shear effects at the scale of a single lattice spacing are weak, meaning by this that the cell-shear time scale is much longer than the molecular time scales, so that the LB collisional relaxation takes place in a quasi-homogeneous velocity field. Due to their simplicity, it is suggested that equilibrium boundary conditions may represent a viable option for the LB simulation of complex flows with solid boundaries at moderate Reynolds numbers.     

Simulation of blood flow using extended Boltzmann kinetic approach

Lattice Boltzmann (LB) simulations are conducted to obtain the detailed hydrodynamics in a variety of blood vessel setups, including a prototype stented channel and four human coronary artery geometries based on the images obtained from real patients. For a model of stented flow involving an S-shape stent, a pulsatile flow rate is applied as the inlet boundary condition, and the time- and space-dependent flow field is computed. The LB simulation is found to reproduce the analytical solutions for the velocity profiles and wall shear stress distributions for the pulsatile channel flow. For the coronary arteries, the distributions of wall shear stress, which is important for clinical diagnostic purposes, are in good agreement with the conventional CFD predictions.     

带外力项的iDdQ(q-1)多松弛格子Boltzmann模型

以3维14速（iD3Q14）多松弛格子Boltzmann（MRT LB）模型为例，在iDdQ（q-1）MRT LB模型的基础上，采用多尺度展开和反向设计法构造出带外力项的iDdQ（q-1）MRT LB模型，该模型在低Mach数的假设下可恢复到带外力项的不可压Navier-Stokes方程.通过对三维方形管道内泊肃叶流、脉动流以及二维Taylor涡衰减流的模拟发现，数值结果与解析解吻合很好，并且空间精度达到二阶，从而验证了新模型模拟外力驱动的稳态和非稳态不可压流动的有效性.     

Analysis and adaptive synchronization of eight-term 3-D polynomial chaotic systems with three quadratic nonlinearities

This paper proposes a eight-term 3-D polynomial chaotic system with three quadratic nonlinearities and describes its properties. The maximal Lyapunov exponent (MLE) of the proposed 3-D chaotic system is obtained as L ₁ = 6.5294. Next, new results are derived for the global chaos synchronization of the identical eight-term 3-D chaotic systems with unknown system parameters using adaptive control. Lyapunov stability theory has been applied for establishing the adaptive synchronization results. Numerical simulations are shown using MATLAB to describe the main results derived in this paper.     

Simulation of microchannel flow using the lattice Boltzmann method

For microchannel flow simulation, the slip boundary model is very important to guarantee the accuracy of the solution. In this paper, a new slip model, the Langmuir slip model, instead of the popularly used Maxwell slip model, is incorporated into the lattice Boltzmann (LB) method through the non-equilibrium extrapolation scheme to simulate the rarefied gas flow. Its feasibility and accuracy are examined by simulations of microchannel flow. Although, for simplicity, in this paper our recently developed LB model is used to solve the flow field, this does not prevent the present boundary scheme from easily incorporating other LB models, for example the more advanced collision model with multiple relaxation times. In addition, the existing non-equilibrium extrapolation LB boundary scheme for macroscopic flows can be recovered naturally from the present scheme when the Knudsen number .     

基于Lattice Boltzmann方法的圆柱绕流大涡模拟

Lattice Boltzmann(LB)方法是近年来出现的一种流体计算的新方法,在流体力学及相关领域取得了很大的成功。但LB方法在模拟湍流流动时常常引起计算的不稳定。本文基于一种结合大涡模型的LB方法对圆柱绕流问题进行的模拟,并与其他文献的实验结果和计算结果进行对比。计算表明:这种混合LB方法能获得比较满意的结果。     

Preconditioned characteristic boundary conditions for solution of the preconditioned Euler equations at low Mach number flows

Preconditioned characteristic boundary conditions (BCs) are implemented at artificial boundaries for the solution of the two- and three-dimensional preconditioned Euler equations at low Mach number flows. The preconditioned compatibility equations and the corresponding characteristic variables (or the Riemann invariants) based on the characteristic forms of preconditioned Euler equations are mathematically derived for three preconditioners proposed by Eriksson, Choi and Merkle, and Turkel. A cell-centered finite volume Roe’s method is used for the discretization of the preconditioned system of equations on unstructured meshes. The accuracy and performance of the preconditioned characteristic BCs applied at artificial boundaries are evaluated in comparison with the non-preconditioned characteristic BCs and the simplified BCs in computing steady low Mach number flows. The two-dimensional flow over the NACA0012 airfoil and three-dimensional flow over the hemispherical headform are computed and the results are obtained for different conditions and compared with the available numerical and experimental data. The sensitivity of the solution to the size of computational domain and the variation of the angle of attack for each type of BCs is also examined. Indications are that the preconditioned characteristic BCs implemented in the preconditioned system of Euler equations greatly enhance the convergence rate of the solution of low Mach number flows compared to the other two types of BCs.     

Corrosion of Ultrathin Iron Layers and Titanium Nitride Coated Iron

In this paper results are presented on the corrosion resistance against SO₂-polluted atmospheres of ultrathin Fe layers (<2.5 nm) deposited on SiO₂/Si wafers by two methods: vacuum evaporation and Langmuir–Blodgett (LB) technique. It is shown that the corrosion resistance of the evaporated films is similar to that shown by massive iron while the corrosion resistance of the LB films is remarkably higher. We also show that the corrosion resistance of iron evaporated films can be greatly enhanced by using TiN coatings. Results on the influence of the SO₂ concentration, the atmospheric relative humidity and the exposure time on the corrosion behaviour of thin (30 nm) TiN layers deposited onto evaporated Fe films on SiO₂/Si wafers are discussed. This revised version was published online in July 2006 with corrections to the Cover Date.     

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.     

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 Simulations of Blood Flow: Non-Newtonian Rheology and Clotting Processes

The numerical simulation of thrombosis in stented aneurysms is an important issue to estimate the efficiency of a stent. In this paper, we consider a Lattice Boltzmann (LB) approach to bloodflow modeling and we implement a non-Newtonian correction in order to reproduce more realistic flow profiles. We obtain a good agreement between simulations and Casson’s model of blood rheology in a simple geometry. Finally we discuss how, by using a passive scalar suspension model with aggregation on top of the LB dynamics, we can describe the clotting processes in the aneurysm     

Finite volume TVD formulation of lattice Boltzmann simulation on unstructured mesh

A numerical scheme is presented for accurate simulation of fluid flow using the lattice Boltzmann equation (LBE) on unstructured mesh. A finite volume approach is adopted to discretize the LBE on a cell-centered, arbitrary shaped, triangular tessellation. The formulation includes a formal, second order discretization using a Total Variation Diminishing (TVD) scheme for the terms representing advection of the distribution function in physical space, due to microscopic particle motion. The advantage of the LBE approach is exploited by implementing the scheme in a new computer code to run on a parallel computing system. Performance of the new formulation is systematically investigated by simulating four benchmark flows of increasing complexity, namely (1) flow in a plane channel, (2) unsteady Couette flow, (3) flow caused by a moving lid over a 2D square cavity and (4) flow over a circular cylinder. For each of these flows, the present scheme is validated with the results from Navier–Stokes computations as well as lattice Boltzmann simulations on regular mesh. It is shown that the scheme is robust and accurate for the different test problems studied.     

A numerical study on pattern selection in crystal growth by using anisotropic lattice Boltzmann-phase field method

Pattern selection during crystal growth is studied by using the anisotropic lattice Boltzmann-phase field model.In the model,the phase transition,melt flows,and heat transfer are coupled and mathematically described by using the lattice Boltzmann(LB) scheme.The anisotropic streaming-relaxation operation fitting into the LB framework is implemented to model interface advancing with various preferred orientations.Crystal pattern evolutions are then numerically investigated in the conditions of with and without melt flows.It is found that melt flows can significantly influence heat transfer,crystal growth behavior,and phase distributions.The crystal morphological transition from dendrite,seaweed to cauliflower-like patterns occurs with the increase of undercoolings.The interface normal angles and curvature distributions are proposed to quantitatively characterize crystal patterns.The results demonstrate that the distributions are corresponding to crystal morphological features,and they can be therefore used to describe the evolution of crystal patterns in a quantitative way.     

Lattice Boltzmann modeling and simulation of compressible flows

In this mini-review we summarize the progress of Lattice Boltzmann (LB) modeling and simulating compressible flows in our group in recent years. Main contents include (i) Single-Relaxation-Time (SRT) LB model supplemented by additional viscosity, (ii) Multiple-Relaxation-Time (MRT) LB model, and (iii) LB study on hydrodynamic instabilities. The former two belong to improvements of physical modeling and the third belongs to simulation or application. The SRT-LB model supplemented by additional viscosity keeps the original framework of Lattice Bhatnagar-Gross-Krook (LBGK). So, it is easier and more convenient for previous SRT-LB users. The MRT-LB is a completely new framework for physical modeling. It significantly extends the range of LB applications. The cost is longer computational time. The developed SRT-LB and MRT-LB are complementary from the sides of convenience and applicability.     

Monte-Carlo algorithm for a two-dimensional Hubbard model of high-T c superconductivity

A Monte-Carlo procedure is given for the two-dimensional (2-D) Hubbard model using the Suzuki-Trotter transformation. The resulting three-dimensional (3-D) classical model does not have the usual problems with negative transition probabilities in the large-U limit (U-repulsive interactions). Numerical simulations based on the algorithm described are expected to be of importance for the theory of high-T _c superconductivity.     

A Brinkman penalization method for compressible flows in complex geometries

To simulate flows around solid obstacles of complex geometries, various immersed boundary methods had been developed. Their main advantage is the efficient implementation for stationary or moving solid boundaries of arbitrary complexity on fixed non-body conformal Cartesian grids. The Brinkman penalization method was proposed for incompressible viscous flows by penalizing the momentum equations. Its main idea is to model solid obstacles as porous media with porosity, ϕ, and viscous permeability approaching zero. It has the pronounced advantages of mathematical proof of error bound, strong convergence, and ease of numerical implementation with the volume penalization technique. In this paper, it is extended to compressible flows. The straightforward extension of penalizing momentum and energy equations using Brinkman penalization with respective normalized viscous, η, and thermal, η_T, permeabilities produces unsatisfactory results, mostly due to nonphysical wave transmissions into obstacles, resulting in considerable energy and mass losses in reflected waves. The objective of this paper is to extend the Brinkman penalization technique to compressible flows based on a physically sound mathematical model for compressible flows through porous media. In addition to penalizing momentum and energy equations, the continuity equation for porous media is considered inside obstacles. In this model, the penalized porous region acts as a high impedance medium, resulting in negligible wave transmissions. The asymptotic analysis reveals that the proposed Brinkman penalization technique results in the amplitude and phase errors of order O((ηϕ)^1/2) and O((η/η_T)^1/4ϕ^3/4), when the boundary layer within the porous media is respectively resolved or unresolved. The proposed method is tested using 1- and 2-D benchmark problems. The results of direct numerical simulation are in excellent agreement with the analytical solutions. The numerical simulations verify the accuracy and convergence rates.     

Double MRT thermal lattice Boltzmann method for simulating convective flows

A two-dimensional double Multiple Relaxation Time-Thermal Lattice Boltzmann Equation (2-MRT-TLBE) method is developed for predicting convective flows in a square differentially heated cavity filled with air (Pr=0.71). In this Letter, we propose a numerical scheme to solve the flow and the temperature fields using the MRT-D2Q9 model and the MRT-D2Q5 model, respectively. Thus, the main objective of this study is to show the effectiveness of such model to predict thermodynamics for heat transfer. This model is validated by the numerical simulations of the 2-D convective square cavity flow. Excellent agreements are obtained between numerical predictions. These results demonstrate the accuracy and the effectiveness of the proposed methodology.     

Flux Limiter Lattice Boltzmann Scheme Approach to Compressible Flows with Flexible Specific-Heat Ratio and Prandtl Number

We further develop the lattice Boltzmann (LB) model [Physica A 382 (2007) 502] for compressible flows from two aspects. Firstly, we modify the Bhatnagar--Gross-Krook (BGK) collision term in the LB equation, which makes the model suitable for simulating flows with different Prandtl numbers. Secondly, the flux limiter finite difference (FLFD) scheme is employed to calculate the convection term of the LB equation, which makes the unphysical oscillations at
discontinuities be effectively suppressed and the numerical dissipations be significantly diminished. The proposed model is validated by recovering results of some well-known benchmarks, including (i) The thermal Couette flow; (ii) One- and two-dimensional Riemann problems. Good agreements are obtained
between LB results and the exact ones or previously reported solutions. The flexibility, together with the high accuracy of the new model, endows the proposed model considerable potential for tracking some long-standing problems and for investigating nonlinear nonequilibrium complex systems.     

Assemblies, characterization, and luminescent enhancement of organized molecular films based on rare earth complexes

Langmuir-Blodgett (LB) films of different molar percentages of Eu(TTA)₃Phen (TTA=2-thenoyltrifluoroacetone; Phen=1,10-phenanthroline) with Gd(TTA)₃Phen coexisting with arachidic acid (AA) (complexes:AA=1:l, in molar ratio) were fabricated and the luminescence enhancement of Eu(III) in the films was studied in this investigation. The monolayers and LB films were characterized by π-A isotherms, fluorescence microscopy, UV-vis spectroscopy and low-angle X-ray diffraction. High-quality LB films and strongly luminescent films were obtained. It was learned from the present study that an efficient intermolecular energy transfer occurred from Gd(TTA)₃Phen to Eu(TTA)₃Phen in the films, which resulted in the luminescence enhancement effect. According to the proposed model of the “active enhancement circle” the distance of energy transfer from Gd-, Tb-, La-, and Y-complex to Eu-complex were calculated to be 1.2, 1.2, 0.7 and 1.0 nm, respectively.     

Surface-grafted block copolymer gradients: Effect of block length on solvent response

We outline a method to fabricate gradient combinatorial libraries that explore architectural parameters of surface-grafted block copolymers (BCs). In addition, we demonstrate the utility of such libraries for the rapid, thorough assessment of the response of grafted BCs to solvent exposure. Our fabrication route uses surface-initiated controlled radical polymerization to produce a tethered polymer block with uniform length (in this case, poly(n-butyl methacrylate), PBMA), followed by a graded synthesis that adds a second block that varies in its length over the library (here, poly(2-(N,N′-dimethylamino)ethyl methacrylate), PDMAEMA). Our demonstration study maps the response of PBMA and PDMAEMA blocks to hexane and water, and defines regimes of behavior to this respect. Moreover, our study illuminates a narrow BC composition window that exhibits the strongest possible response to water and hexane treatment.