CFD爱好者：我与CFD的故事

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"I do like CFD.pdf" 是一本关于计算流体动力学（CFD）的书籍，由Katate Masatsuka撰写。该书的第二版以PDF形式提供，版权归属于作者。书中涵盖了CFD的基本原理和精确解，强调了源码软件在CFD中的应用。 CFD（Computational Fluid Dynamics）是利用数值方法模拟流体流动的一种科学工具。它在航空航天、汽车工程、环境科学、生物医学、化工等多个领域有着广泛的应用。本书"IdolikeCFD, VOL.1, Second Edition"主要包含以下知识点： 1. **控制方程**：CFD的核心是流体力学的控制方程，包括连续性方程、纳维-斯托克斯方程和能量方程。这些方程描述了流体流动的基本物理现象，如质量守恒、动量守恒和能量守恒。书中可能详细解释了这些方程的数学形式以及它们在不同流动条件下的应用。 2. **精确解**：对于一些简单的流动问题，可能存在解析解，即通过数学方法可以直接求得的解。这些精确解在验证数值方法和理解复杂流动行为方面具有重要价值。作者可能提供了若干经典精确解的例子，如泊肃叶流动、普朗特边界层理论等。 3. **PDF版特点**：此PDF版本具有超链接功能，这意味着读者可以方便地点击书中的内容，如目录、公式、图示和引用，直接跳转到相应位置。此外，还有外部链接，可导向相关网站，增强交互性和学习体验。 4. **源码软件**：书中的内容可能涉及开源或自编的CFD软件，这些软件允许用户深入理解算法并进行定制化计算。这为读者提供了实践和探索CFD技术的机会，尤其适合对编程和软件开发有兴趣的读者。 5. **额外内容**：除了PDF版本，还提到有硬拷贝版本，含有更多额外内容。可能包括更深入的案例研究、练习题、解决方案或者与作者的互动讨论。 6. **版权与许可**：作者明确指出，除非获得书面许可，否则禁止修改、更改、利用、复制或以任何形式重新发布此PDF文件的任何部分。这保护了作者的知识产权，并确保内容的完整性。这本书对CFD初学者和经验丰富的专业人士来说都是一份宝贵的资源，它不仅提供了理论基础，也强调了实践应用，尤其是通过源码软件的学习，有助于读者深入理解和掌握计算流体动力学的精髓。

4 CHAPTER 1. BASICS

It vanishes, of course, because the parallelepiped becomes ﬂat in that case and the volume ( = the scalar triple product)

becomes zero. I now tell you why I really like the scalar triple product. I like it because the volume V of a tetrahedron

formed by the three vectors, a, b, and c, is given by

V =

|a · (b × c)|, (1.2.20)

and a tetrahedron is one of my favorite computational cells in CFD. Note that the vectors a, b, and c are the edge

vectors centered at a common vertex of the tetrahedron. The volume is deﬁned with the absolute value of the scalar

triple product to keep it positive in case any of the vectors points the opposite direction. Of course, then, it is easy to

understand that the scalar triple product does not change at all under rotation of coordinate system:

a · (b × c) = a

′

· (b

′

× c

′

), (1.2.21)

where a

′

, b

′

, and c

′

are the vectors rotated by a rotation matrix R,

′

= Ra, b

′

= Rb, c

′

= Rc. (1.2.22)

This is natural because rotating a parallelepiped (or a tetrahedron) does not change its volume.

1.2.6 Vector Triple Product

Of course, there is a vector triple product. The vector triple product is deﬁned by

a × (b × c). (1.2.23)

The following formula is known as Lagrange’s formula or vector product expansion:

a × (b × c) = b(a · c) − c(a · b). (1.2.24)

It is known also as the BAC-CAB identity. That is useful for memorizing the formula. But that’s not the reason that I

like the vector triple product. I like it because it is useful in CFD. To see how useful it is, consider the dot product of

the vector triple product and a:

[a × (b × c)] · a = 0. (1.2.25)

Yes, it is zero because it is a scalar triple product with two identical vectors: a, b ×c, and a. It means that the vector

triple product of a, b, and c is perpendicular to a. That is interesting. In CFD, we sometimes need to ﬁnd vectors

tangent to a face of a computational cell from the face normal vector, or more generally, ﬁnd a vector perpendicular to

a given vector. For example, the situation arises in the rotated-hybrid Riemann solvers [112]. Usually in ﬁnite-volume

methods, a single numerical ﬂux is computed in the direction normal to the face,

n, where the hat indicates that

the vector is a unit vector. The rotated-hybrid method computes the face-normal numerical ﬂux as a combination of

two numerical ﬂuxes by breaking

n into two orthogonal directions and applying two diﬀerent numerical ﬂuxes in the

two directions. The method applies a robust but dissipative numerical ﬂux in a physically meaningful direction,

e.g., normal to a shock wave, and a less robust but much less dissipative numerical ﬂux in the direction

that is

perpendicular to

. The resulting ﬂux has been shown to be very robust for strong shocks without adding too much

dissipation in smooth regions. Since

is determined by a physical consideration, the problem is to ﬁnd

that is

perpendicular to a given vector,

. Then, we can ﬁnd

by the vector triple product [112]:

× (

n ×

)

× (

n ×

, (1.2.26)

where the denominator is the magnitude of the numerator to make

a unit vector. Note that the numerator is not

necessarily a unit vector. Clearly, we have

= 0, i.e.,

is perpendicular to

as desired. That is nice. I like

the vector triple product very much.

1.2.7 Identity Matrix

This is a matrix whose diagonal entries are all 1 and other entries are zero. Such a matrix is called the identity matrix.

For example, the 3 × 3 identity matrix is given by

I =





1 0 0

0 1 0

0 0 1





. (1.2.27)

1.4. INDEX NOTATION 7

where τ

= τ

, τ

= τ

, and τ

= τ

. We deﬁne the following quantities:

= τ

+ τ

, (1.3.31)

= τ

+ τ

, (1.3.32)

= τ

+ τ

, (1.3.33)

= τ

+ τ

, (1.3.34)

nℓ

= τ

ℓ

+ τ

ℓ

+ τ

ℓ

, (1.3.35)

= τ

+ τ

, (1.3.36)

= τ

u + τ

v + τ

w. (1.3.37)

Then, we have the following identities:

+ τ

nℓ

ℓ

= τ

, (1.3.38)

+ τ

nℓ

ℓ

= τ

, (1.3.39)

+ τ

nℓ

ℓ

= τ

, (1.3.40)

dτ

+ τ

dτ

+ τ

nℓ

dτ

nℓ

= τ

dτ

+ τ

dτ

+ τ

dτ

, (1.3.41)

+ τ

nℓ

ℓ

= τ

du + τ

dv + τ

dw, (1.3.42)

dτ

+ dτ

nℓ

ℓ

= dτ

, (1.3.43)

dτ

+ dτ

nℓ

ℓ

= dτ

, (1.3.44)

dτ

+ dτ

nℓ

ℓ

= dτ

. (1.3.45)

I like these identities very much because they can be useful for eliminating tangent vectors from expressions such as

the absolute value of a Jacobian matrix projected along a normal vector that arises, for example, in the upwind viscous

ﬂux [106].

1.4 Index Notation

I like index notation because it is compact and useful especially for proving vector identities. Also, I can save my

time and money because I don’t need to write Σ for summations any more.

1.4.1 Einstein’s Summation Convention

Einstein’s summation convention means that whenever two indices are repeated, summation is taken over that index:

for example,

= a

+ a

, (1.4.1)

which is nice because the dot product a ·b can be simply written as a

without using boldface letters or Σ. The i-th

component of the matrix-vector product Ab can be expressed as

(Ab)

= A

+ A

, (1.4.2)

where (Ab)

denotes the i-th component of Ab. Thus, we can write

Ab =





(Ab)









+ A





. (1.4.3)

Of course, the index used for the summation is a dummy, and so we can use any letter we like, e.g.,

a · b = a

= a

, (1.4.4)

which are all equal to a

+ a

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CFD爱好者：我与CFD的故事

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