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Hierarchy in wiki:

空气动力学

NS、连续介质力学

The Lattice Boltzmann Method Principles and Practice.pdf - p26 - 连续性方程
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The Lattice Boltzmann Method Principles and Practice.pdf - p27 - the change of net momentum can be due to (i) flow of momentum into or out of the fluid element, (ii) differences in pressure p and (iii) external body forces F.

The Lattice Boltzmann Method Principles and Practice.pdf - p28 - Euler 方程：流体的动量守恒形式
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流体控制方程主要是如下形式：（非守恒形式）

Navier–Stokes equations

\[ \begin{aligned} \frac{\partial \vec u}{ \partial t} + \vec u \cdot \nabla \vec u + \frac 1 {\rho} \nabla p &= \vec g + \nu\nabla ^2 \vec u\\ \nabla \cdot \vec u& = 0 \end{aligned} \]

The Lattice Boltzmann Method Principles and Practice.pdf - p29 - NS方程
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The Lattice Boltzmann Method Principles and Practice.pdf - p29 - 不可压缩流的NS方程（非守恒型）
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The Lattice Boltzmann Method Principles and Practice.pdf - p31 - 理想气体的状态方程
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The Lattice Boltzmann Method Principles and Practice.pdf - p32 - 恒定压强
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动力学

分布函数和它的动量

The Lattice Boltzmann Method Principles and Practice.pdf - p38 - 分布
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The Lattice Boltzmann Method Principles and Practice.pdf - p39 - 动量和质量密度，通过动量来联系
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The Lattice Boltzmann Method Principles and Practice.pdf - p40 - The Lattice Boltzmann Method Principles and Practice-P40-20231014210700
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稳定分布函数

The Lattice Boltzmann Method Principles and Practice.pdf - p42 - The Lattice Boltzmann fMethod Principles and Practice-P42-20231014210732
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Boltzmann eqn and collision operator

The Lattice Boltzmann Method Principles and Practice.pdf - p43 - The Lattice Boltzmann Method Principles and Practice-P43-20231014210927
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The Lattice Boltzmann Method Principles and Practice.pdf - p44 - 守恒律表达
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The Lattice Boltzmann Method Principles and Practice.pdf - p44 - BGK model
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宏观守恒定律

验证细节不展开了。自动满足：

  1. 质量守恒：The Lattice Boltzmann Method Principles and Practice.pdf - p46 - the collision operator’s mass conservation property
  2. 动量守恒
  3. 能量守恒

Boltzmann Theorem

The Lattice Boltzmann Method Principles and Practice.pdf - p49 - The Lattice Boltzmann Method Principles and Practice-P49-20231014211805
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最小化熵。

The lattice boltzmann equation

  • 之前的Boltzmann方程，描述了统计意义下的气体变化性质The Lattice Boltzmann Method Principles and Practice.pdf - p81 - The Boltzmann equation in Sect. 1.3 describes the dynamics of a gas on amesoscopic scale.
  • 尽管BE比NSE更加复杂（七个变量），但如果有好的数值方法，就比nse好算。The Lattice Boltzmann Method Principles and Practice.pdf - p82 - solve the Boltzmann equation numerically,this may also indirectly give us a solution to the NSE

我们还是重新强调一下Incompressible的NSE的形式：

\[ \begin{aligned} \frac{\partial \vec u}{ \partial t} + \vec u \cdot \nabla \vec u + \frac 1 {\rho} \nabla p &= \vec g + \nu\nabla ^2 \vec u\\ \nabla \cdot \vec u& = 0 \end{aligned} \]

一个重要的事实是：

The Lattice Boltzmann Method Principles and Practice.pdf - p82 - The Lattice Boltzmann Method Principles and Practice-P82-20231015101823
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之前的数值方法的主要缺点是，对流项很难处理

The Lattice Boltzmann Method Principles and Practice.pdf - p82 - A major difficulty with these methods is discretising the advection term .u r/u;complicated iterative numerical schemes with approximation errors are introduced to deal with this.

The lattice boltzmann method in a nutshell

A quick intro.

Overview

The Lattice Boltzmann Method Principles and Practice.pdf - p83 - 密度和动量的计算从概率密度中恢复。
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单位可以通过一系列转化，变成“相同”的（无量纲数不变）》

The Lattice Boltzmann Method Principles and Practice.pdf - p83 - another possible choice would be Imperial units. The most common choice in the LB literature, however, is lattice units, a simple artificial set of units scaled such that ĩt D 1 and ĩx D 1.

常用的速度集合：The Lattice Boltzmann Method Principles and Practice.pdf - p83 - The most commonly used velocity sets to solve the Navier-Stokes equation are D1Q3, D2Q9, D3Q15, D3Q19 and D3Q27.

The Lattice Boltzmann Method Principles and Practice.pdf - p84 - Lattice Boltzmann Equation：差分
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The Lattice Boltzmann Method Principles and Practice.pdf - p84 - BGK碰撞算子：注意稳定态对速度是非线性的
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其中的 \(w_i\) 的选取和速度集合有关。

这样的\(f_i^{eq}\)能够保证动量和质量守恒。

如果是无粘性流体，\(\Omega_i = - \frac12 (f_i - f_i^{eq})\)

The Lattice Boltzmann Method Principles and Practice.pdf - p85 - 粘性项与等式中的参数的关系
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Time stepping

The Lattice Boltzmann Method Principles and Practice.pdf - p85 - LatticeBoltzmann + BGK Operator ⇒ LBGK eqn
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分成两部分：

  1. Relaxation/Collision
  2. Propagation

The Lattice Boltzmann Method Principles and Practice.pdf - p86 - 不考虑所有其他外界情况下，LBM的两步骤
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The Lattice Boltzmann Method Principles and Practice.pdf - p85 - The Lattice Boltzmann Method Principles and Practice-P85-20231015112638
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需要注意的是：

  1. collision是局部的​

    1. 计算密度 \(\rho\)​
    2. 计算宏观速度 \(u\)​
    3. 从而计算出平衡态的\(f_i^{eq}\) ⇒ \(f_i^*\)
    4. stream，全局更新，但是线性且简单

The Lattice Boltzmann Method Principles and Practice.pdf - p86 - 总结
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Implementation of LBM

The Lattice Boltzmann Method Principles and Practice.pdf - p87 - Full LBM
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初始化：假设流体从平衡态开始，当然也有别的初始化方法

  • 初始条件给出的是\(t=0\)时刻的流体密度和速度场，通过之前的\(f_i^{eq}\)公式给出。
\[ f_i^{eq}(x, t=0) = f_i^{eq}(\rho(x, 0), u(x, 0)) \]

The Lattice Boltzmann Method Principles and Practice.pdf - p87 - 初始化
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时间步：

The Lattice Boltzmann Method Principles and Practice.pdf - p87 - Time step
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很简单吧。为了方便还是将平衡方程放在这里：

The Lattice Boltzmann Method Principles and Practice.pdf - p84 - 平衡态分布函数
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Memory layout and coding hint

速度空间离散化

之前我们没有对于速度空间进行离散化。

当然也可以直接对速度空间用稠密的网格进行离散，但是效率很低。我们可以直接将其限制在一个很稀疏的网格上：

\[ f(v; x, t) = f(v_{i, j} ; x, t) \]

这里的\(v_{i, j}\)经过离散化后只有\(n\)个，所以一般直接写成 \(v_i\) 的形式，避免两重指标。

The Lattice Boltzmann Method Principles and Practice.pdf - p91 - 经过相当粗暴的离散后，这样的方程依然足够求解NSE
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non-dimensionalization（无量纲化？）

The Lattice Boltzmann Method Principles and Practice.pdf - p92 - 无量纲化过程
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The Lattice Boltzmann Method Principles and Practice.pdf - p93 - 维度无关形式
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Conservation laws

我们需要找出在进行速度离散后，积分所对应的离散形式。

The Lattice Boltzmann Method Principles and Practice.pdf - p93 - 守恒定律
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Hermite Polynomials

对于积分离散近似很常用。

The Lattice Boltzmann Method Principles and Practice.pdf - p97 - Hermite展开和积分
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Hermite Series Expansion of the eq distribution

The Lattice Boltzmann Method Principles and Practice.pdf - p98 - Hermite多项式展开形式
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The Lattice Boltzmann Method Principles and Practice.pdf - p98 - 这边说明了，展开的前三项就能够计算出完整的质量动量能量
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因为这样的性质，我们可以值计算前三项

The Lattice Boltzmann Method Principles and Practice.pdf - p99 - 足以能够恢复宏观空气动力学守恒方程。
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The Lattice Boltzmann Method Principles and Practice.pdf - p99 - 因此近似时，可以只需要保留前三项
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Discretization of the eq dist function

The Lattice Boltzmann Method Principles and Practice.pdf - p101 - 对于连续方程，选取有限个速度分量并不影响守恒律。
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The Lattice Boltzmann Method Principles and Practice.pdf - p102 - 最终的分布函数离散
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Discretization of the particle distribution function

The Lattice Boltzmann Method Principles and Practice.pdf - p103 - 对于求解变量用Hermite展开，取三项，然后离散化
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速度集合

Construction and requirements.

The Lattice Boltzmann Method Principles and Practice.pdf - p105 - 从旋转无关等角度考虑
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一些常用的速度集

The Lattice Boltzmann Method Principles and Practice.pdf - p106 - D1Q3,D2Q9
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The Lattice Boltzmann Method Principles and Practice.pdf - p107 - D3Q15 19 27
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Table for implementation

The Lattice Boltzmann Method Principles and Practice.pdf - p108 - The Lattice Boltzmann Method Principles and Practice-P108-20231015150757
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The Lattice Boltzmann Method Principles and Practice.pdf - p109 - d3q19
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时间和空间离散化

终于是正常一点的东西。

一般就是等距网格，以及可以选择如下的Refinement。

The Lattice Boltzmann Method Principles and Practice.pdf - p114 - local grid refinement of the LBM on structured grids is possible ([31] gives an overview of this), the most common form of space discretisation is a uniform and structured grid.

原始的LB算法假设其每个格点都迁移到其邻居（或自身）。

The Lattice Boltzmann Method Principles and Practice.pdf - p114 - Overall, the original LB algorithm assumes that populations fi move with velocity ci from one lattice site to another. After one time step ĩt, each population should exactly reach a neighbouring site.

这对于原始问题有一定的要求。

  1. 空间网格是等距规则的
  2. 速度分量是\(dx/dt\)的倍数

特征线方程

The Lattice Boltzmann Method Principles and Practice.pdf - p114 - 类似于传输线方程，但含一个系数
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The Lattice Boltzmann Method Principles and Practice.pdf - p116 - The Lattice Boltzmann Method Principles and Practice-P116-20231015163732
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First and second order discretisation

The Lattice Boltzmann Method Principles and Practice.pdf - p117 - LBE
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The Lattice Boltzmann Method Principles and Practice.pdf - p118 - Second Order method
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BGK Operator

The Lattice Boltzmann Method Principles and Practice.pdf - p119 - This collision operator is only suitable for gas simulations, as it only accounts for binary collisions between molecules.

The Lattice Boltzmann Method Principles and Practice.pdf - p119 - 碰撞算子的一大性质是保持质量和动量守恒
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The Lattice Boltzmann Method Principles and Practice.pdf - p120 - LBGK方程，将BGK算子离散化后代入
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Over Relaxation

The Lattice Boltzmann Method Principles and Practice.pdf - p120 - 如果选取的参数错误，将导致问题，\(\tau\)决定了 \(\Delta t\)的取值。
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The Lattice Boltzmann Method Principles and Practice.pdf - p121 - 这里直接给出了Stability 的必要条件。
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Streaming and collision

The Lattice Boltzmann Method Principles and Practice.pdf - p122 - LBGK算法，类似Operator Splitting
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The Lattice Boltzmann Method Principles and Practice.pdf - p122 - Now we have derived everything required to write a first LB simulation code, except boundary conditions and forces.

Stability

Analysis

The Lattice Boltzmann Method Principles and Practice.pdf - p148 - The Lattice Boltzmann Method Principles and Practice-P148-20231015170322
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BGK model

The Lattice Boltzmann Method Principles and Practice.pdf - p150 - BGK model：一定条件下，所有的eq非负，但eq是由速度决定的，可以通过速度来推导
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The Lattice Boltzmann Method Principles and Practice.pdf - p151 - 在参数满足的条件下，对流速有如下的要求可以保证稳定性
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The Lattice Boltzmann Method Principles and Practice.pdf - p153 - 可以看出，粘性对于LBM的稳定有很大的意义
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Boundaries

Periodic

The Lattice Boltzmann Method Principles and Practice.pdf - p191 - The Lattice Boltzmann Method Principles and Practice-P191-20231015170953
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The Lattice Boltzmann Method Principles and Practice.pdf - p192 - 虚拟节点，直接按周期传递。
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Solid boundaries：Bounce back approach

The Lattice Boltzmann Method Principles and Practice.pdf - p195 - Despite its age, it is still the most popular wall boundary scheme in the LB community, largely due to its simplicity of implementation.

Principle

The Lattice Boltzmann Method Principles and Practice.pdf - p196 - 图示
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The Lattice Boltzmann Method Principles and Practice.pdf - p196 - 原理是，碰到墙反弹到它原来的地方。
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满足no-slip条件！

Fullway v.s. Halfway bounce-back method.

两种实现方式：

  1. Fullway：下一个时间步反弹回来
  2. Halfway：当前时间步骤直接反弹

The Lattice Boltzmann Method Principles and Practice.pdf - p197 - The Lattice Boltzmann Method Principles and Practice-P197-20231015191535
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The Lattice Boltzmann Method Principles and Practice.pdf - p198 - 无论是哪种，都假设了碰撞发生在Midway。
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  • 实现上：fullway简单
  • 准确性上：Halfway

The Lattice Boltzmann Method Principles and Practice.pdf - p198 - The question that arises is: which strategy to implement? Fullway or halfway? There is no definitive answer. If simplicity is our main criterion, then fullway bounce-back wins. Here, the boundary treatment is independent of the direction of fi and the execution time is shorter, cf. Chap. 13. 13 Yet, halfway bounce-back is more accurate for unsteady flows as explained below.

Resting walls

The Lattice Boltzmann Method Principles and Practice.pdf - p199 - 固定边界
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Moving walls

The Lattice Boltzmann Method Principles and Practice.pdf - p200 - 移动边界
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第二项的作用是使其能够满足质量守恒。

The Lattice Boltzmann Method Principles and Practice.pdf - p209 - The Lattice Boltzmann Method Principles and Practice-P209-20231015192559
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Solid bc: Wet-node approach

idea：在边界上赋予合适的值，期望LBM能有合理的输出。

The Lattice Boltzmann Method Principles and Practice.pdf - p209 - The idea of the wet-node approach is to assign suitable values for the unknown boundary populations such that the known and constructed populations reproduce the intended hydrodynamics at the boundary.

通常而言，会有导致欠定

The Lattice Boltzmann Method Principles and Practice.pdf - p209 - there are typically more unknown boundary populations than macroscopic conditions

变成法向相对速度为零

The Lattice Boltzmann Method Principles and Practice.pdf - p210 - The continuity of the fluid at the boundary is described by the well-known impermeable wall condition, i.e. zero relative normal velocity of the fluid

Eq state

用\(f_i^{eq}\)来设置边界处的BC

The Lattice Boltzmann Method Principles and Practice.pdf - p211 - The Lattice Boltzmann Method Principles and Practice-P211-20231015193430
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好用，但问题也很大。

Non-eq Extrapolation

The Lattice Boltzmann Method Principles and Practice.pdf - p214 - The Lattice Boltzmann Method Principles and Practice-P214-20231015193824
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用一部分流体项修正。

NEBB

The Lattice Boltzmann Method Principles and Practice.pdf - p216 - The Lattice Boltzmann Method Principles and Practice-P216-20231015193904
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很麻烦，很精确。

Open bc

The Lattice Boltzmann Method Principles and Practice.pdf - p219 - The Lattice Boltzmann Method Principles and Practice-P219-20231015194010
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