Lattice BGK Models for Navier-Stokes Equation
LBM算法原理简介
LBM算法原理
• 求解瞬态问题时，计算时间少，精度高 传统方法求解的控制方程对流项是非线性的，每一 个时间步都需要迭代收敛。传统软件计算瞬态问题 时，给定最大迭代次数，计算并未收敛，最终影响 求解精度，而且计算效率低。 LBM算法求解的Lattice-Boltzmann方程对流项是 线性的，不存在这方面的问题。 采用Powerflow计算瞬态问题，精度高，耗费时间 大为减少。
Confidential doc of TS
1 T 2
为运动粘性, T 为温度
Confidential doc of TS
LBM算法原理
• 粒子分布函数 f 满足Lattice Boltzmann方程： 1 fi x ei t , t t fi x, t fi x, t f i eq x, t
Confidential doc of TS
LBM离散方法－2维问题
• D2Q9模型
Confidential doc of TS
LBM离散方法－2维问题
• 平衡分布函数 fi eq
f
eq 0
（根据文献[3]）
4 3 2 1 2 u 9 2c
fi fi
eq
1 3 9 3 2 2 1 2 ei 4 ei 2 u u u 9 c 2c 2c 1 3 9 3 2 2 1 2 ei 4 ei 2 u u u 36 c 2c 2c
Confidential doc of TS
LBM算法原理
• LBM方法将连续介质看作大量位于网格节 点上的离散流体质点粒子. 粒子按碰撞和迁 移规则在网格上运动,通过对各网格流体质 点及运动特征的统计，获得流体宏观运动 规律。 • 粒子分布函数f(r,e,t)drde表示在t时刻，在 空间r处，粒子的速度在e到e+de的粒子数 密度。
格子玻尔兹曼方法(LBM)及其在微通道绕流中的应用
格子玻尔兹曼方法(LBM)及其在微通道绕流中的应用冯俊杰; 孙冰; 姜杰; 徐伟; 石宁【期刊名称】《《安全、健康和环境》》【年(卷),期】2019(019)001【总页数】6页(P7-12)【关键词】格子玻尔兹曼(LBM); 微反应器; 微通道; 绕流【作者】冯俊杰; 孙冰; 姜杰; 徐伟; 石宁【作者单位】中国石化青岛安全工程研究院化学品安全控制国家重点实验室山东青岛266071【正文语种】中文0 前言微反应器在提高反应过程安全性、缩短反应时间、提高转化率、灵活生产等方面具有独特的优势，实现微通道流动的精确测定和控制是微反应器发挥诸多优势的保障和广泛应用的基础[1]。
由于微通道内的流动具有尺度小、多尺度、相界面与边界复杂的特点，传统的计算流体力学(CFD)方法作为宏观模拟方法存在着诸多不足，而格子玻尔兹曼方法(lattice Boltzmann method，LBM)突破了传统计算方法的框架，直接从离散模型出发，通过粒子群的碰撞和迁移代替传统的连续流体模型，更接近流动的微观本质，在微流控领域具有明显的优势[2-3]。
格子玻尔兹曼方法的核心思想是将流体离散为在网格上运动的介观粒子，通过计算粒子的碰撞和迁移规律得到粒子分布函数，进而统计计算得到宏观变量如压力、速度等分布规律，创造性地实现了模拟流体运动的连续介质模型向离散模型的转变[4]。
由于LBM方法基于非平衡统计物理学的Boltzmann方程，因而能成为联系微观分子尺度与宏观尺度之间的纽带[5-6]。
传统的CFD方法主要基于宏观的连续介质假设，而难以计算那些不符合连续介质假设或者难以用宏观方程描述的系统，对于这些体系往往需要借助微观的分子动力学或气体动理论来进行描述[7]。
对于分子动力学来说必须同时跟踪大量粒子的运动，实际求解的计算量非常大。
在这种背景下，基于分子运动论和概率统计力学的LBM方法就成为一种有效方法，其具有更高的计算效率，并且容易实现并行计算[8-10]。
第十讲 不可压缩Navier-Stokes方程的求解
Pij p ij 2ij
连续介质微元体的受力平衡： 应力的概念
热力学压力—— 由分子动力学性质决定 —— 状态方程 完全气体：
p RT
热力学压力
p
p n pn
p
可压缩N-S方程： 动力学与热力学耦合；动力学压力= 热力学压力 不可压缩N-S方程： 动力学与热力学解耦 由不可压缩条件确定压力 （纯动力学概念）
1
p V 0 t V 1 2 V V p V t Re
p n 1 p n )
对于非定常问题，需要内迭代 （效率较低）
V 0 V 1 2 V V p V t Re
p V 0 t V 1 2 V V p V t Re
u13 u 23 u 33
u1n u2n u3 n u nn
a j x j 1 b j x j c j x j 1 d j
x j Aj x j 1 B j
2
Copyright by Li Xinliang
知识回顾
迭代法
2u 2u f ( x, y ) x 2 y 2 u g ( x, y )
a11 a12 0 a 22 A 0 0 0 0
a13 a23 a 33 0
0
a1n a2 n a 3n ann
a11 LU分 a21 解法 a 31 an1
追赶法：
a12 a22 a32 an 2
ui 1, j ui 1, j ui, j 1 ui, j 1 4ui , j f i , j 2
二维Navier-Stokes方程在边界的正则性
独创性声明
本人声明所呈交的学位论文是我个人在导师的指导下进行的研究工作及取得的 研究成果。尽我所知，除文中已标明引用的内容外，本论文不包含任何其他人或集 体已经发表或撰写过的研究成果。对本文的研究做出贡献的个人和集体，均已在文 中以明确方式标明。本人完全意识到本声明的法律结果由本人承担。
Lattice-Boltzmann Simulations of Fluid Flows in MEMS
a r X i v :c o m p -g a s /9806001v 1 11 J u n 1998Lattice-Boltzmann Simulations of Fluid Flows in MEMSXiaobo Nie 1,Gary D.Doolen 1and Shiyi Chen 1,21Center for Nonlinear Studies and Theoretical Division,Los Alamos National Laboratory,Los Alamos,NM 875452IBM Research Division,T.J.Watson Research Center,P.O.Box 218,Yorktown Heights,NY 10598The lattice Boltzmann model is a simplified kinetic method based on the particle distribution function.We use this method to simulate problems in MEMS,in which the velocity slip near the wall plays an important role.It is demonstrated that the lattice Boltzmann method can capture the fundamental behavior in micro-channel flow,including velocity slip and nonlinear pressure drop along the channel.The Knudsen number dependence of the position of the vortex center and the pressure contour in micro-cavity flows is also demonstrated.The development of technologies in Micro-electro-mechanical systems (MEMS)has motivated the study of fluid flows in devices with micro-scale geometries,such as micro-channel and micro-cavity flows [1].In these flows,the molecular mean free path of fluid molecules could be the same order as the typical geometric length of the device;then the continuum hypothesis which is the fun-damental for the Navier-Stokes equation breaks down.An important feature in these flows is the emergence of a slip velocity at the flow boundary,which strongly affects the mass and heat transfer in the system.In micro-channel experiments,it has been observed that the measured mass flow rate is higher than that based on a non-slip boundary condition [2].The Knudsen number,K n =l/L ,can be used to identify the influence of the effects of the mean free path on these flows,where l is the mean free path of molecules and L is the typical length of the flow domain.It has been pointed out that for a sys-tem with K n <0.001,the fluid flow can be treated as con-tinuum.For K n >10the system can be considered as a free-molecular flow.The fluid flow for 0.001<K n <10,which often appears in the MEMS [1],can not be con-sidered as a continuum nor a free-molecular flow.Tradi-tional kinetic methods,such as molecular dynamics sim-ulations [3]and the continuum Boltzmann equation ap-proach,could be used to describe these flows.But these methods are more complicated than schemes usually used for continuum hydrodynamic equations.The solution of the Navier-Stokes equation including the velocity-slip boundary condition with a variable parameter has also been used to simulate micro-channel flows [4].In the past ten years,the lattice Boltzmann method (LBM)[5]has emerged as an alternative numerical tech-nique for simulating fluid flows.This method solves a simplified Boltzmann equation on a discretized lattice.The solution of the lattice Boltzmann equation converges to the Navier-Stokes solution in the continuum limit (small Knudsen number).In addition,since the lattice Boltzmann method is intrinsically kinetic,it can be also used to simulate fluid flows with high Knudsen numbers,including fluid flows in very small MEMS.To demonstrate the utility of the LBM,we use the LBM model with three speeds and nine velocities on a two-dimensional square lattice.The velocities,c i ,in-clude eight moving velocities along the links of the squarelattice and a zero velocity for the rest particle.They are:(±1,0),(0,±1),(±1,±1),(0,0).Let f i (x ,t )be the dis-tribution functions at x ,t with velocity c i .The lattice Boltzmann equation with the BGK collision approxima-tion [6,7]can be written asf i (x +c i δx,t +δt )−f i (x ,t )=−τ−1(f i −f eq i ),(1)where f eq i (i =0,1,···,8)is the equilibrium distribution function and τis the relaxation time.We have assumed that the spatial separation of the lattice is δx and the time step is δt .A suitable equilibrium distribution is [7]:f eqi =t i ρ 1+c iαu α2c 4su αu β.(2)Here c s =1/√2+12).Using the Chapman-Enskog multi-scale expansion tech-nique,we obtain the following Navier-Stokes equations in the limit of long wavelength and low Mach number:∂t ρ+∂α(ρu α)=0,(3)∂t (ρu α)+∂β(ρu αu β)=−∂αP −∂βπαβ,(4)P =c 2s ρ,παβ=ν(∂α(ρu β)+∂β(ρu α)),where ν=c 2s (2τ−1)/(2ρ)is the kinematic viscosity.In classical kinetic theory,the viscosity νfor a hard sphere gas is linearly proportional to the mean free path.Sim-ilarly,we define the mean free path l in the LBM as:a (τ−0.5)/ρ,where a is constant.Our first numerical example is a micro-channel flow [2].The flow is contained between two parallel plates sepa-rated by a distance H and driven by the pressure differ-ence between the inlet pressure,P i ,and exit pressure,P e .The channel length in the longitudinal direction is L .We12take L =1000,H =10(lattice units)in our simulations satisfying L/H >>1.The bounce-back boundary condi-tion is used for the particle distribution functions at the top and bottom plates,i.e.,when a particle distribution hits a wall node,the particle distribution scatters back to the fluid node opposite to its incoming direction.A pressure boundary condition is used at the input and the exit.2468101214161800.10.20.30.40.50.60.70.8V s , M fK nV s M fFIG.1.The slip velocity and the normalized mass flow rate at the exit of a micro-channel flow as functions of K n for P i /P e =2.The ‘+’and ‘×’are LBM numerical results.Thedashedand dottedlinesare Eq.(5)and Eq.(6)respectively.The slip velocity V s at the exit of the micro-channel flow is defined as:u (y )=u 0(Y −Y 2+V s ),where u (y )is the velocity along the x (or flow)direction at the exit and Y =y/H .u 0and V s can be obtained by fitting numerical results using the least squares method.This definition of the slip velocity is consistent with others [2,4].In Fig.1,we plot the slip velocity V s and the normalized mass flow rate M f =M/M 0,as functions of Knudsen number when the pressure ratio η=P i /P e =2.The normalization fac-tor,M 0=h 3P eη2−1.(6)For η=2,the above formula becomes M f =1+24.1K 2n ,which agrees well with the numerical results in Fig 1.In laminar Poiseuille flows,one usually assumes that the density variation along the channel is very small,and the pressure drop along the channel is nearly linear.Inmicro-channel flow,however,the ratio between the length and the width is much larger and the pressure drop is not linear.If there is no velocity slip at the walls,it has been shown [2,4]from the Navier-Stokes equation that the pressure along the channel has the following depen-dence on the dimensionless coordinate,X =x/L :P 2=P 2e [1+(η2−1)(1−X )],(7)If the velocity at the boundaries is allowed to slip,the pressure drop along the channel will depend on the Knud-sen number.In Fig.2we present the LBM simulation re-sults for the normalized pressure deviation from a linear pressure drop,(P −P l )/P e ,as functions of X for several Knudsen numbers,where P l =P e +(P i −P e )(1−X ).It is seen that when K n ≤0.2,(P −P l )/P e is a positive nonlinear function of X .This agrees with the results in [4]using an engineering model.For K n ≥0.2,the LBM simulation shows that (P −P l )/P e becomes neg-ative,which is directly linked to the fact that the slip velocity depends on the square of K n in the LBM.For large K n ,the pressure can be derived from Eq.(6):P =P e [η(1−X )].(8)The negative deviation from a linear pressure drop has not been experimentally observed before and it would be interesting to testify this experimentally.-0.1-0.08-0.06-0.04-0.0200.020.040.060.080.100.20.40.60.81(P -P l )/P eX Eq.(7)K n =0.0194K n =0.134K n =0.194K n =0.388K n =0.776Eq.(8)FIG. 2.The deviations from linear pressure drop for η=P i /P e =2.The top and bottom lines are the analyt-ical results from Eq.(7)for K n =0and Eq.(8)for K n >>1respectively.The other curves are LBM numerical results for the Knudsen numbers indicated.In Fig.3the mass flow rates as functions of the pres-sure ratio ηwhen K n =0.165are shown for our theory,the experimental work [2],the engineering model [4]and the LBM simulation.Our theory and the LBM simula-tion agree well with the experimental measurements.It is noted that for large pressure ratios (η≥1.8),the LBM agrees reasonably well with Beskok et al.[4].But for smaller pressure ratios,the difference increases because of different dependence of the slip velocity on K n .31.41.61.822.22.41 1.2 1.4 1.6 1.82 2.2 2.4 2.6M fηtheory simulation experiment[4]FIG.3.The normalized mass flow rate as a function of the pressure ratio for Kn =0.165.The theory is Eq.(6).010*******102030yx K n =0.0048510203040xK n =0.388FIG.4.Streamlines for two Knudsen numbers.ym a s s f l u xK nFIG.5.The y -coordinate of the vortex center (square sym-bols)and the mass flow(solid circles)as a function of K n .Our second LBM numerical simulation is the two-dimensional micro-cavity flow.The cavity size is L x =L y =40(lattice units).The upper wall moves with a constant velocity,v 0,from left to right.The other three walls are at rest,and bounce-back boundary conditions are used.To see the dependence on the Knudsen num-ber in our simulations,we fixed the Reynolds number,R e =v 0L xc s ≤10−3.In Fig.4,we show the streamlines for two different Knudsen numbers.In Fig.5we show the vertical positions of the vortex center and the mass flux between the bottom and the vortex cen-ter as functions of K n .It can be seen that the center moves upward and the mass flow decreases with increas-ing Knudsen number.This occurs because the slip ve-locity on the upper wall causes momentum transfer to be less efficient.It has been shown [8]that the center of the vortex moves downward when the Reynolds number increases for very small K n .Fig.6shows the pressure contours for the same parameters as in Fig.4.Totally different pressure structures are observed for these two cases.When the Knudsen number is small,the contin-uum assumption is valid and the pressure contours are almost circles with centers at the left or the right corners.On the other hand,due to the slip velocity on the walls,the pressure contours become nearly straight lines at the higher Knudsen number.102030400102030yx K n =0.000485010203040xK n =0.388FIG.6.The contours of pressure for the same two Knudsen numbers shown in Fig.4.。
基于格子波尔兹曼方法的回热器数值模拟
基于格子波尔兹曼方法的回热器数值模拟夏宇栋;陈曦;马诗旻;张华【摘要】Based on Lattice Boltzmann Method ( LBM ) , the flow field in the microstructure of mesh screen and etched foil regenerators were simulated. The velocity and pressure drop were obtained by LBM. The simulation results show that velocity field in etched foil regenerator is better distributed than that of mesh screen. And the etched foil regenerator has less resistance coefficient than that of mesh screen regenerator. The simulation results basically agree with the experiments.%利用格子玻尔兹曼方法,直接对蚀刻薄片和层叠丝网回热器的微观结构流场进行了模拟.得到了两种回热器填料的微观流场和两端的压差.模拟结果显示,当回热器的直径、水力直径和填充率相近情况下,不同流速下蚀刻薄片卷裹式回热器的稳态阻力系数均比层叠丝网回热器小.稳态阻力系数的模拟变化趋势与实验一致.【期刊名称】《低温工程》【年(卷),期】2012(000)005【总页数】5页(P41-45)【关键词】格子波尔兹曼方法;回热器;流阻系数;数值模拟【作者】夏宇栋;陈曦;马诗旻;张华【作者单位】上海理工大学能源与动力学院上海200093;上海理工大学能源与动力学院上海200093;上海理工大学能源与动力学院上海200093;上海理工大学能源与动力学院上海200093【正文语种】中文【中图分类】TB651回热器是低温制冷机的关键部件，也是影响制冷机性能的最重要因素。
A lattice-based public-key cryptosystem
1
1
2 Lattice problems with worst-case/average-case equivalence
cusick@
1 Introduction
Since the origin of the idea of public-key cryptography, there have been many public-key techniques described in the literature. The security of essentially all of these depends on certain widely believed but unproven mathematical hypotheses. For example, the well-known RSA public-key cryptosystem relies on the hypothesis that it is di cult to factor a large integer n which is known to be a product of two large primes. This hypothesis has been extensively studied, but there is still no proof that for a typical such n, the prime factors cannot be found in less than k steps, where k is a very large number. From a computational complexity point of view, we generate a speci c instance of a problem in NP (together with a solution, which is kept secret) and we rely on the belief that the problem is di cult to solve. Apart from the lack of proof that any of these problems is really hard, i.e., there exists no e cient algorithm that will solve the problem in all cases, there is another serious issue. The mathematical hypothesis that these problems are di cult to solve really means di cult to solve in the worst case, but the security of the cryptographic algorithms depends more on the di culty of the average case. For example, even if one day factoring is proved to be unsolvable in probabilistic polynomial time, to the users of the RSA system, there is no guarantee that the key they are actually using is hard to factor. To use these protocols, one must be able to generate speci c instances of the problem which should be hard to solve. But typically there is no way to just generate known hard instances. One way to do this is to generate random instances of the problem, and hope that such instances are as hard on the average as in the worst case. However this property is known to be not true for a number of NP-hard problems. Recently Ajtai 1] proved that certain lattice problems related to SVP have essentially the same average case and worst case complexity, and both are conjectured to be extremely hard. This development raises the possibility of public-key cryptosystems which will have a new level of security. Already Ajtai and Dwork 3] have proposed a public-key cryptosystem which has a provable worst-case/average-case equivalence. Speci cally, the Ajtai-Dwork cryptosystem is secure unless the worst case of a certain lattice problem can be solved in probabilistic polynomial time. Goldreich, Goldwasser and Halevi 11] have also given a public-key cryptosystem which depends on similar lattice problems related to SVP as in 1]. Unlike the work of 3], however, their method uses a trapdoor one-way function and also lacks a proof of worst-case/averagecase equivalence. The cryptosystems of 3] are unfortunately far from being practical. All of them encrypt messages bit-by-bit and involve massive data expansion: the encryption will be at least a hundred times as long as the message. 1 In this paper we propose a public-key cryptosystem, based on the ideas of 1] and 3], which has much less data expansion. Messages are encrypted in blocks instead of bit-by-bit. We o er some statistical analysis of our cryptosystem. We also analyze several attacks on the system and show that the system is secure against these attacks. Whether there is a provable worst-case/average-case equivalence for this system is open.
Navier–Stokes equation
Navier–Stokes equation [Wikipedia]In physics, the Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of fluid substances. These equations arise from applying Newton's second law to fluid motion, together with the assumption that the stress in the fluid is the sum of a diffusing viscous term (proportional to the gradient of velocity) and a pressure term - hence describing viscous flow.The equations are useful because they describe the physics of many things of academic and economic interest. They may be used to model the weather, ocean currents, water flow in a pipe and air flow around a wing. The Navier–Stokes equations in their full and simplified forms help with the design of aircraft and cars, the study of blood flow, the design of power stations, the analysis of pollution, and many other things. Coupled with Maxwell's equations they can be used to model and study magnetohydrodynamics.The Navier–Stokes equations are also of great interest in a purely mathematical sense. Somewhat surprisingly, given their wide range of practical uses, it has not yet been proven that in three dimensions solutions always exist (existence), or that if they do exist, then they do not contain any singularity (they are smooth). These are called the Navier–Stokes existence and smoothness problems. The Clay Mathematics Institute has called this one of the seven most important open problems in mathematics and has offered a US$1,000,000 prize for a solution or a counter-example.Velocity fieldThe solution of the Navier–Stokes equations is a velocity not position. It is called a velocity field or flow field, which is a description of the velocity of the fluid at a given point in space and time. Once the velocity field is solved for, other quantities of interest, such as flow rate or drag force, may be found. This is different from what one normally sees in classical mechanics, where solutions are typically trajectories of position of a particle or deflection of a continuum. Studying velocity instead of position makes more sense for a fluid; however for visualization purposes one can compute various trajectories.PropertiesNonlinearityThe Navier–Stokes equations are nonlinear partial differential equations in almost every real situation.[2][3] In some cases, such as one-dimensional flow and Stokes flow (or creeping flow), the equations can be simplified to linear equations.The nonlinearity makes most problems difficult or impossible to solve and is the main contributor to the turbulence that the equations model.The nonlinearity is due to convective acceleration, which is an acceleration associated with the change in velocity over position. Hence, any convective flow, whether turbulent or not, will involve nonlinearity. An example of convective but laminar (nonturbulent) flow would be the passage of a viscous fluid (for example, oil) through a small converging nozzle. Such flows, whether exactly solvable or not, can often be thoroughly studied and understood.[4]T urbulenceTurbulence is the time-dependent chaotic behavior seen in many fluid flows. It is generally believed that it is due to the inertia of the fluid as a whole: the culmination of time-dependent and convective acceleration; hence flows where inertial effects are small tend to be laminar (the Reynolds number quantifies how much the flow is affected by inertia). It is believed, though not known with certainty, that the Navier–Stokes equations describe turbulence properly.[5]The numerical solution of the Navier–Stokes equations for turbulent flow is extremely difficult, and due to the significantly different mixing-length scales that are involved in turbulent flow, the stable solution of this requires such a fine mesh resolution that the computational time becomes significantly infeasible for calculation or direct numerical simulation. Attempts to solve turbulent flow using a laminar solver typically result in a time-unsteady solution, which fails to converge appropriately. To counter this, time-averaged equations such as the Reynolds-averaged Navier–Stokes equations (RANS), supplemented with turbulence models, are used in practical computational fluid dynamics (CFD) applications when modeling turbulent flows. Some models include the Spalart-Allmaras, k-ω (k-omega), k-ε (k-epsilon), and SST models, which add a variety of additional equations to bring closure to the RANS equations. Large eddy simulation (LES) can also be used to solve these equations numerically. This approach is computationally more expensive-in time and in computer memory-than RANS, but produces better results because it explicitly resolves the larger turbulent scales.ApplicabilityFurther information: Discretization of Navier–Stokes equationsTogether with supplemental equations (for example, conservation of mass) and well formulated boundary conditions, the Navier–Stokes equations seem to model fluid motion accurately; even turbulent flows seem (on average) to agree with real world observations.The Navier–Stokes equations assume that the fluid being studied is a continuum (it is infinitely divisible and not composed of particles such as atoms or molecules), and is not moving at relativistic velocities. At very small scales or under extreme conditions, real fluids made out of discrete molecules will produce results different from the continuous fluids modeled by the Navier–Stokes equations.Depending on the Knudsen number of the problem, statistical mechanics or possibly even molecular dynamics may be a more appropriate approach.Another limitation is simply the complicated nature of the equations. Time-tested formulations exist for common fluid families, but the application of the Navier–Stokes equations to less common families tends to result in very complicated formulations and often to open research problems. For this reason, these equations are usually rewritten for Newtonian fluids where the viscosity model is linear; truly general models for the flow of other kinds of fluids (such as blood) do not, as of 2012, exist .Derivation and descriptionMain article: Derivation of the Navier–Stokes equationsThe derivation of the Navier–Stokes equations begins with an application of Newton's second law: conservation of momentum (often alongside mass and energy conservation) being written for an arbitrary portion of the fluid. In an inertial frame of reference, the general form of the equations of fluid motion is:[6]Navier–Stokes equations(general)whereis the flow velocity,is the fluid density,is the pressure,is the (deviatoric) component of the total stress tensor, which has order two,represents body forces (per unit volume) acting on the fluid,is the del operator.This is a statement of the conservation of momentum in a fluid and it is an application of Newton's second law to a continuum; in fact this equation is applicable to any non-relativistic continuum and is known as the Cauchy momentum equation. This equation is often written using the material derivative D v/Dt, making it more apparent that this is a statement of Newton's second law:The left side of the equation describes acceleration, and may be composed of time-dependent or convective effects (also the effects of non-inertial coordinates if present). The right side of the equation is in effect a summation of body forces (such as gravity) and divergence of stress (pressure and shear stress).Convective accelerationAn example of convection. Though the flow may be steady (time-independent), the fluid decelerates as it moves down the diverging duct (assuming incompressible or subsonic compressible flow), hence there is an acceleration happening over position.A significant feature of the Navier–Stokes equations is the presence of convective acceleration: the effect of time-independent acceleration of a fluid with respect to space. While individual fluid particles indeed experience time dependent-acceleration, the convective acceleration of the flow field is a spatial effect, one example being fluid speeding up in a nozzle.Regardless of what kind of fluid is being dealt with, convective acceleration is a nonlinear effect. Convective acceleration is present in most flows (exceptions include one-dimensional incompressible flow), but its dynamic effect is disregarded in creeping flow (also called Stokes flow). Convective acceleration is represented by the nonlinear quantity:which may be interpreted either as or as with the tensor derivative of the velocity vector Both interpretations give the same result, independent of the coordinate system — provided is interpreted as the covariant derivative.[7]Interpretation as (v·∇)vThe convection term is often written aswhere the advection operator is used. Usually this representation is preferred as it is simpler than the one in terms of the tensor derivative [7]Interpretation as v·(∇v)Here is the tensor derivative of the velocity vector, equal in Cartesian coordinates to the component-by-component gradient.In irrotational flowThe convection term may, by a vector calculus identity, be expressed without a tensor derivative:[8][9]The form has use in irrotational flow, where the curl of the velocity (called vorticity) is equal to zero. Therefore, this reduces to onlyThe effect of stress in the fluid is represented by the and terms; these are gradients of surface forces, analogous to stresses in a solid. Here is called the pressure gradient and arises from the isotropic part of the , which has order two. This part is given by normal stresses that turn up in almost all situations, dynamic or not. The anisotropic part of the stress tensor gives rise to , which conventionally describes viscous forces; for incompressible flow, this is only a shear effect. Thus, is thewhere is the 3×3 identity matrixpressure.and are yet unknown, so the general form of the equations of motion is not usable to solve problems. Besides the equations of motion—Newton's second law—a force model is needed relating the stresses to the fluid For this reason, assumptions based on natural observations are often applied to specify the stresses in terms of the other flow variables, such as velocity and density.The Navier–Stokes equations result from the following assumptions on the deviatoric stress tensor :[12], also does not depend directly on the flow velocity, but only on spatial derivatives of the flow velocityof the tensor gradient of the flow velocity with, a fourth-order viscosity tensor , i.e.,the fluid is assumed to be isotropic, as with gases and simple liquids, and consequently is an isotropic tensor; furthermore, since the deviatoric stress tensor is symmetric, it can be expressed in terms of two scalar": where is the rate-of-strain tensor and is the rate of expansion of the flowthe deviatoric stress tensor has zero trace, so for a three-dimensional flow 2with the quantity between brackets being the non-isotropic part of the rate-of-strain tensor The dynamic viscosityThe pressure p is modeled by an equation of state.[14]The vector field represents body forces. Typically, these consist of only gravity forces, but may include others, such as electromagnetic forces. In non-inertial coordinate frames, other "forces" associated with rotating coordinates may arise. Often, these forces may be represented as the gradient of some scalar quantity , with in which case they are called conservative forces. Gravity in the direction, for example, is the gradient of . Because pressure from such gravitation arises only as a gradient, we may include it in the pressure term as a body force The pressure and force terms on the right-hand side of the Navier–Stokes equation becomeOther equationsor, using the substantive derivative:Incompressible flow of Newtonian fluidsNavier–Stokes equations(Incompressible flow)in tensor notation:Navier–Stokes equations(Incompressible flow)f represents "other" body forces (per unit volume), such as gravity or centrifugal force. The shear stress term becomes , where is the vector Laplacian.It's well worth observing the meaning of each term (compare to the Cauchy momentum equationOnly the convective terms are nonlinear for incompressible Newtonian flow. Convective acceleration is caused by a (possibly steady) change in velocity overindividual fluid particles are accelerated and thus under unsteady motion, the flow field, a velocity distribution, will not necessarily be time-dependent.Another important observation is that viscosity is represented by the vector Laplacian of the velocity field, interpreted here as the difference between the velocity at a point and the mean velocity in a small surrounding volume. This implies that – for a Newtonian fluid – viscosity operates as a diffusion of momentum, in much the same way as the diffusion of heat in theThis is more specifically a statement of the conservation of volume (see divergence andWith the velocity vector expanded as , we may write the vector equation explicitly,Note that gravity has been accounted for as a body force, and the values of g x, g y, g z will depend on the orientation of gravity with respect to the chosen set of coordinates.The continuity equation reads:When the flow is incompressible, does not change for any fluid parcel, and its material derivative vanishes: . The continuity equation is reduced to:The velocity components (the dependent variables to be solved for) are typically named u, v, w. This system of four equations comprises the most commonly used and studied form. Though comparatively more compact than other representations, this is still a nonlinear system of partial differential equations for which solutions are difficult to obtain.Cylindrical coordinatesA change of variables on the Cartesian equations will yield[15] the following momentum equations for r, , and z:The gravity components will generally not be constants, however for most applications either the coordinates are chosen so that the gravity components are constant or else it is assumed that gravity is counteracted by a pressure field (for example, flow in horizontal pipe is treated normally without gravity and without a vertical pressure gradient). The continuity equation is:This cylindrical representation of the incompressible Navier–Stokes equations is the second most commonly seen (the first being Cartesian above). Cylindrical coordinates are chosen to take advantage of symmetry, so that a velocity component candisappear. A very common case is axisymmetric flow with the assumption of no tangential velocity (), and the remaining quantities are independent of :Spherical coordinatesIn spherical coordinates, the r, ϕ, and θ momentum equations are[15] (note the convention used: θ is polar angle, or colatitude,[17] 0 ≤ θ ≤ π):Mass continuity will read:These equations could be (slightly) compacted by, for example, factoring from the viscous terms. However, doing so would undesirably alter the structure of the Laplacian and other quantities.Stream function formulationTaking the curl of the Navier–Stokes equation results in the elimination of pressure. This is especially easy to see if 2D Cartesian flow is assumed ( and no dependence of anything on z), where the equations reduce to:Differentiating the first with respect to y, the second with respect to x and subtracting the resulting equations will eliminate pressure and any conservative force. Defining the stream function throughresults in mass continuity being unconditionally satisfied (given the stream function is continuous), and then incompressible Newtonian 2D momentum and mass conservation degrade into one equation:where is the (2D) biharmonic operator and is the kinematic viscosity, . We can also express this compactly using the Jacobian determinant:This single equation together with appropriate boundary conditions describes 2D fluid flow, taking only kinematic viscosity as a parameter. Note that the equation for creeping flow results when the left side is assumed zero.In axisymmetric flow another stream function formulation, called the Stokes stream function, can be used to describe the velocity components of an incompressible flow with one scalar function.Pressure-free velocity formulationThe incompressible Navier–Stokes equation is a differential algebraic equation, having the inconvenient feature that there is no explicit mechanism for advancing the pressure in time. Consequently, much effort has been expended to eliminate the pressure from all or part of the computational process. The stream function formulation above eliminates the pressure (in 2D) at the expense of introducing higher derivatives and elimination of the velocity, which is the primary variable of interest.The incompressible Navier–Stokes equation is composite, the sum of two orthogonal equations,where and are solenoidal and irrotational projection operators satisfying and and are the non-conservative and conservative parts of the body force. This result follows from the Helmholtz Theorem (also known as the fundamental theorem of vector calculus). The first equation is a pressureless governing equation for the velocity, while the second equation for the pressure is a functional of the velocity and is related to the pressure Poisson equation.The explicit functional form of the projection operator in 3D is found from the Helmholtz Theoremwith a similar structure in 2D. Thus the governing equation is an integro-differential equation and not convenient for numerical computation.An equivalent weak or variational form of the equation, proved to produce the same velocity solution as the Navier–Stokes equation,[18] is given by,for divergence-free test functions satisfying appropriate boundary conditions. Here, the projections are accomplished by the orthogonality of the solenoidal and irrotational function spaces. The discrete form of this is imminently suited to finite element computation of divergence-free flow, as we shall see in the next section. There we will be able to address the question, "How does one specify pressure-driven (Poiseuille) problems with a pressureless governing equation?"The absence of pressure forces from the governing velocity equation demonstrates that the equation is not a dynamic one, but rather a kinematic equation where the divergence-free condition serves the role of a conservation law. This all would seem to refute the frequent statements that the incompressible pressure enforces the divergence-free condition.Discrete velocityWith partitioning of the problem domain and defining basis functions on the partitioned domain, the discrete form of the governing equation is,It is desirable to choose basis functions which reflect the essential feature of incompressible flow – the elements must be divergence-free. While the velocity is the variable of interest, the existence of the stream function or vector potential isnecessary by the Helmholtz Theorem . Further, to determine fluid flow in the absence of a pressure gradient, one can specify the difference of stream function values across a 2D channel, or the line integral of the tangential component of the vector potential around the channel in 3D, the flow being given by Stokes' Theorem . Discussion will be restricted to 2D in the following.We further restrict discussion to continuous Hermite finite elements which have at least first-derivative degrees-of-freedom. With this, one can draw a large number of candidate triangular and rectangular elements from theplate-bending literature. These elements have derivatives as components of the gradient. In 2D, the gradient and curl of a scalar are clearly orthogonal, given by the expressions,Adopting continuous plate-bending elements, interchanging the derivative degrees-of-freedom and changing the sign of the appropriate one gives many families of stream function elements.Taking the curl of the scalar stream function elements gives divergence-free velocity elements.[19][20] The requirement that the stream function elements be continuous assures that the normal component of the velocity is continuous across element interfaces, all that is necessary for vanishing divergence on these interfaces.Boundary conditions are simple to apply. The stream function is constant on no-flow surfaces, with no-slip velocity conditions on surfaces. Stream function differences across open channels determine the flow. No boundary conditions are necessary on open boundaries, though consistent values may be used with some problems. These are all Dirichlet conditions.The algebraic equations to be solved are simple to set up, but of course are non-linear , requiring iteration of the linearized equations.Similar considerations apply to three-dimensions, but extension from 2D is not immediate because of the vector nature of the potential, and there exists no simple relation between the gradient and the curl as was the case in 2D.Pressure recovery [edit ]Recovering pressure from the velocity field is easy. The discrete weak equation for the pressure gradient is,where the test/weight functions are irrotational. Any conforming scalar finite element may be used. However, the pressure gradient field may also be of interest. In this case one can use scalar Hermite elements for the pressure. For the test/weight functions one would choose the irrotational vector elements obtained from the gradient of the pressure element.Compressible flow of Newtonian fluids [edit ]There are some phenomena that are closely linked with fluid compressibility. One of the obvious examples is sound. Description of such phenomena requires more general presentation of the Navier–Stokes equation that takes into account fluid compressibility. If viscosity is assumed a constant, one additional term appears, as shown here:[21][22]where is the second viscosity .Application to specific problemsThe Navier–Stokes equations, even when written explicitly for specific fluids, are rather generic in nature and their proper application to specific problems can be very diverse. This is partly because there is an enormous variety of problems that may be modeled, ranging from as simple as the distribution of static pressure to as complicated as multiphase flow driven by surface tension .Generally, application to specific problems begins with some flow assumptions and initial/boundary condition formulation, this may be followed by scale analysis to further simplify the problem.a) Assume steady, parallel, one dimensional, non-convective pressure-driven flow between parallel plates, the resulting scaled (dimensionless) boundary value problem is:The boundary condition is the no slip condition . This problem is easily solved for the flow field:From this point onward more quantities of interest can be easily obtained, such as viscous drag force or net flow rate.b) Difficulties may arise when the problem becomes slightly more complicated. A seemingly modest twist on the parallel flow above would be the radial flow between parallel plates; this involves convection and thus non-linearity. The velocity fieldmay be represented by a function that must satisfy:This ordinary differential equation is what is obtained when the Navier–Stokes equations are written and the flow assumptions applied (additionally, the pressure gradient is solved for). The nonlinear term makes this a very difficult problem to solve analytically (a lengthy implicit solution may be found which involves elliptic integrals and roots of cubic polynomials ). Issues with the actual existence of solutions arise for R > 1.41 (approximately; this is not the square root of 2), the parameter R being the Reynolds number with appropriately chosen scales.[23] This is an example of flow assumptions losing their applicability, and an example of the difficulty in "high" Reynolds number flows.[23]Exact solutions of the Navier–Stokes equationsSome exact solutions to the Navier–Stokes equations exist. Examples of degenerate cases — with the non-linear terms in the Navier–Stokes equations equal to zero — are Poiseuille flow , Couette flow and the oscillatory Stokes boundary layer . But also more interesting examples, solutions to the full non-linear equations, exist; for example the Taylor–Green vortex .[24][25][26] Note that the existence of these exact solutions does not imply they are stable: turbulence may develop at higher Reynolds numbers .For example, in the case of an unbounded planar domain with two-dimensional — incompressible and stationary — flow in polar coordinatesthe velocitycomponents and pressure p are:[27]where A and B are arbitrary constants. This solution is valid in the domain r≥ 1 and for.In Cartesian coordinates, when the viscosity is zero (), this is:For example, in the case of an unbounded Euclidean domain with three-dimensional — incompressible, stationary and with zero viscosity () — radial flow inCartesian coordinates the velocity vector and pressure p are:[citation needed ]A two dimensional example[hide]A three-dimensional example[hide]There is a singularity at .A nice steady-state example with no singularities comes from considering the flow along the lines of afor arbitrary constants A and B. This is a solution in a non-viscous gas (compressible fluid) whose density, velocities and pressure goes to zero far from the origin. (Note this is not a solution to the Clay Millennium problem because that refers to incompressible fluids where is a constant, neither does it deal with the uniqueness of the Navier–Stokes equations with respect to anyfrom the Pythagorean quadruple parametrization. Other choices of density and pressure are possible with the same velocity field:Other choices of density and pressureWyld diagramsWyld diagrams are bookkeeping graphs that correspond to the Navier–Stokes equations via aextension of Keldysh's technique for nonequilibrium processes in fluid dynamics. In other words, these diagrams assign。
- 1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
- 2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
- 3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。
1 February bis 1992 published in January 1992
ቤተ መጻሕፍቲ ባይዱ
Europhys. Lett., 17 (6), pp. 479-484 (1992)
Lattice BGK Models for Navier-Stokes Equation.
Y. H. QIAN, D. D'HUMIERES and P. LALLEMAND Laboratoire de Physique Statistique (*), Ecole Normale Suphrieure 24, rue Lhomond, 7'5231 Paris 05, France
Home
Search
Collections
Journals
About
Contact us
My IOPscience
Lattice BGK Models for Navier-Stokes Equation
This article has been downloaded from IOPscience. Please scroll down to see the full text article. 1992 Europhys. Lett. 17 479 (/0295-5075/17/6/001) View the table of contents for this issue, or go to the journal homepage for more
Recently, lattice gases have attracted much attention since models were proposed to simulate fluid flows: the FHP for 2D [l]and the FCHC [2] for 3D have been shown to satisfy the incompressible Navier-Stokes equation. Other models have been designed to allow the numerical simulation of a variety of physical situations and their properties have been studied in great details [3]. Typically, all lattice gas models are built with the following ingredients: a regular lattice, on each node a set of variables {ni}, E { 1, ..., b } , describes the i population of b given velocities ci. The dynamics is divided into two steps: streaming from node to node according to the velocity ci and redistribution of the populations at each node during the collision step. To represent some physical situation, such a model must satisfy the following constraints: the lattice must be sufficiently symmetric, basic physical quantities must be conserved (mass, momentum, energy), local equilibrium exists and depends only upon the conserved quantities. By now the main advantages of lattice gas techniques are well established: intrinsic stability, easy introduction of boundary conditions, simple numerical codes allowing efficient parallel implementation.
(*> Associb au CNRS et aux Universites Paris 6 et Paris 7.
480
EUROPHYSICS LETTERS
However, lattice gases suffer from some drawbacks. A fundamental one is statistical noise which results in the need for spatial (or time) averaging (loss in efficiency) and to some interesting physical behaviour like the coupling of statistical fluctuations, due t o the nonlinearity of the governing equations, that leads to a divergence of transport coefficients in low dimensions [4].In addition lattice gases usually are not Galilean invariant (presence of a so-called g(p) density-dependent factor in the nonlinear advection term of the NavierStokes equation), they have a velocity-dependent pressure and they present spurious invariants that correspond to unphysical hydrodynamical quantities. These difficulties can be cured as their origins are well understood. The Fermi-Dirac (or alternatively Bose-Einstein) statistics causes g(p) to depend upon the density. However, in the discrete kinetic theory [5] where the gas is Maxwellian, g(p) does not depend upon the density. The pressure dependence upon the velocity is related to g when g is different from unity. The regular lattice structure on which the particles reside produces the nonphysical invariants. Such quantities are scalar and they influence the dynamics through the pressure term, but their production by shock waves and boundary conditions is not yet clear. In order to solve some of these difficulties, there exist three similar more or less efficient ways using modifications of the collision step. The first one was proposed by McNamara and Zanetti [6], who used the mean population of particles instead of the Boolean variables of lattice gases. (ni is now a real variable between 0 and 1instead of 0 or 1.)The expression for the collision step uses the same expression as in the Boolean counterpart which leads to the use of polynomials in the ni that have as many terms as there are effective collisions. This is unpractical for anything but extremely simple cases and in particular rules out the consideration of the 3D FCHC model. The second one was proposed by Higuera, Succi and Benzi[7], who used a linearized expression for the collision term, the so-called enhanced collision, with a matrix whose elements depend only on the angle of two interacting particles. This method was generalized to the 3D FCHC model. These two versions of lattice Boltzmann equations suppress completely the statistical noise and control the initial spurious invariants easily. However the problems related to the density-dependent g and to the velocity-dependent pressure are unsolved because the preceding authors dealt with Fermi gases. The third way was studied by Qian, d’Humieres and Lallemand [8,91, who used Maxwellian particles that violate the semi-detailed balance of binary collisions and introduced more populated rest particles. Thus they obtained noise-free models with Galileian invariance and a velocity-independent pressure. Some universal expressions for the speed of sound and viscosities were obtained for all dimensions. One-dimensional shock waves were studied and compared to the results of the classic characteristic method [91. The detailed choice of collision terms for 2D model is complicated and 3D collision rules are not yet worked out [8]. The last two ways of modifying lattice gases for simulating fluid flows may cause numerical instability. The question is how to keep some of the advantages of lattice gases and of the lattice Boltzmann equation while overcoming their drawbacks. To answer this question, we propose a new kind of model which can be used to simulate the Navier-Stokes equation in a simple and efficient way. From computational fluid dynamics, it is known that the method of relaxation has good stability property if the relaxation parameter w is between 0 and 2. This method can be described simply as follows: if N(t) is a quantity at time t and N e is its predicted value, then N(t + 1) is given by