格子玻尔兹曼方法入门：理论、工程应用与Fortran代码示例

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“格子玻尔兹曼方法基础工程应用（附计算机代码--Fortran）” 本文档主要关注格子玻尔兹曼方法（Lattice Boltzmann Method, LBM），这是一种用于模拟流体动力学的数值计算方法，特别适合于处理多尺度、复杂流动问题。LBM基于统计力学的玻尔兹曼方程，通过简化和离散化来实现对流体流动的模拟。这种方法在工程应用中表现出色，如在流体力学、传热学、多相流等领域都有广泛应用。作者A.A.Mohamad是机械与制造工程领域的专家，他在Schulich School of Engineering的University of Calgary以及后来的Alfaisal University任职。他的工作提供了LBM的基本理论介绍，同时包含了实际的Fortran编程代码，帮助初学者理解LBM的程序框架并进行实际操作。 LBM的基础理论包括以下几个关键点： 1. **玻尔兹曼方程**：LBM的理论基础是经典的玻尔兹曼方程，这是一个描述非平衡统计系统的微粒动力学模型。在LBM中，流体被看作是由一系列离散的“粒子”组成，这些粒子在格子节点之间移动，并且根据碰撞规则交换动量和能量。 2. **离散速度模型**：LBM中的“格子”指的是空间的离散化，而“速度”则指粒子可以在不同方向上运动的速度模式。常见的离散速度模型有D1Q3、D2Q9、D3Q19等，其中数字代表维度和速度向量的数量。 3. **Boltzmann分布和碰撞过程**：在每个时间步长，粒子按照预设的碰撞规则更新其分布函数，这个过程反映了粒子之间的相互作用。典型的碰撞模型是单 relaxation time (SRT) 和多 relaxation time (MRT) 模型。 4. **流体动力学方程的恢复**：经过足够多的迭代，LBM的分布函数会收敛到满足Navier-Stokes方程的状态，从而模拟真实的流体流动。 5. **边界条件处理**：LBM的边界条件处理也是其独特之处，包括无滑移壁、自由流壁和各种复杂的壁面条件，如温度或速度匹配。 6. **Fortran编程实践**：提供的Fortran代码是初学者学习LBM编程的重要资源，它可以帮助理解算法的实现细节，包括初始化、时间步进、碰撞更新和边界处理等步骤。通过学习和实践这份文档，初学者不仅能理解LBM的基本原理，还能掌握如何使用编程语言（如Fortran）将理论应用于实际问题，从而解决工程中的流体流动计算。此外，对于版权和法律方面的注意事项，文档也提到了必须遵循的相关条款，确保合法使用和复制出版物。

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Note that, the word particle and molecule are used interchangeably in the

following paragraphs.

1.2.1 Particle Dynamics

As far as we know, the main building block of all the materials in nature is the

molecules and sub-molecules. These molecules can be visualized as solid spheres

moving ‘‘randomly’’ in conservatory manner in a free space. The motion satisﬁes

conservation of the mass, momentum, and energy. Hence, Newton’s second law

(momentum conservation) can be applied, which states that the rate of change of

linear momentum is equal to the net applied force.

F ¼

dðmcÞ

ð1:1Þ

where F stands for the inter-molecular and external forces, m is the mass of the

particle, c is the velocity vector of the particle and t is the time. For a constant

mass, the equation can be simpliﬁed as,

F ¼ m

¼ ma ð1:2Þ

where a is the acceleration vector. The position of the particle can be determined

from deﬁnition of velocity,

c ¼

ð1:3Þ

where r is the position vector of the particle relative to the origin, as shown in

Fig. 1.2

In the MD simulation, the above equations are solved provided that F is a

known function.

Fig. 1.2 Position and

velocity vectors

4 1 Introduction and Kinetic of Particles

1.3 Distribution Function

In 1859, Maxwell (1831–1879) recognized that dealing with a huge number of

molecules is difﬁcult to formulate, even though the governing equation (Newton’s

second law) is known. As mentioned before, tracing the trajectory of each mol-

ecule is out of hand for a macroscopic system. Then, the idea of averaging came

into picture. For illustration purposes, in a class of 500 students (extremely small

number, compared with the number of molecules in volume of 1 mm

), if all the

students started asking a question simultaneously, the result is noise and chaos.

However, the question can be addressed through a class representative, which can

be handled easily and the result may be acceptable by the majority.

The idea of Maxwell is that the knowledge of velocity and position of each

molecule at every instant of time is not important. The distribution function is the

important parameter to characterize the effect of the molecules; what percentage of

the molecules in a certain location of a container have velocities within a certain

range, at a given instant of time. The molecules of a gas have a wide range of

velocities colliding with each others, the fast molecules transfer momentum to the

slow molecule. The result of the collision is that the momentum is conserved. For a

gas in thermal equilibrium, the distribution function is not a function of time,

where the gas is distributed uniformly in the container; the only unknown is the

velocity distribution function.

For a gas of N particles, the number of particles having velocities in the x-

direction between c

and c

þ dc

is Nf ðc

Þdc

: The function f ðc

Þ is the fraction

of the particles having velocities in the interval c

and c

þ dc

; in the x-direction.

Similarly, for other directions, the probability distribution function can be deﬁned

as before. Then, the probability for the velocity to lie down between c

and

þ dc

; c

and c

þ dc

; and c

and c

þ dc

will be Nf ðc

Þf ðc

Þdc

It is important to mention that if the above equation is integrated (summed) over

all possible values of the velocities, yields the total number of particles to be N,

i.e.,

ZZZ

f ðc

Þf ðc

Þdc

¼ 1: ð1:13Þ

Since any direction can be x,ory or z, the distribution function should not

depend on the direction, but only on the speed of the particles. Therefore,

f ðc

Þf ðc

Þ¼Uðc

þ c

Þð1:14Þ

where U is another unknown function, that need to be determined. The value of

distribution function should be positive (between zero and unity). Hence, in

Eq. 1.14, velocity is squared to avoid negative magnitude. The possible function

that has property of Eq. 1.14 is logarithmic or exponential function, i.e.,

1.3 Distribution Function 7

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