The books once interested me

coolboy

Although V.E. Zakharov and his collaborators in the former Soviet Union derived various Hamiltonian formalisms for different types of waves including three types of fundamental waves in geophysical fluid (surface gravity waves, internal gravity waves, and Rossby waves), only the Zakharov equation for surface gravity waves was extensively investigated, mostly by oceanographers, especially by oceanographers in the United States. The basic conclusion was that the Zakharov equation was better, more accurate than or superior to other types of approaches in describing the nonlinear wave interactions and instabilities. The best or the best known work of using the Zakharov equation to study surface gravity waves was by Yuen and Lake (1982):

Yuen, H. C. (袁子春), and B. M. Lake, 1982: Nonlinear dynamics of deep-water gravity waves. Adv. Appl. Mech., 22, 67-229.

I believe this was probably the last science paper by Henry C. Yuen (袁子春). Afterwards, he pursued a different career and life. He started a high-tech company and became a billionaire. Still, most likely, he will only be remembered for his science contributions to the field of water waves in the years to come.

lwd1981

coolboy 发表于 2014-11-27 15:22
Diaz, J. B. and S. I. Pai (editors), 1962: Fluid dynamics and applied mathematics (Proceedings o ...

I am interested in the two papers included in this book. They are entitled "The Shock Tube and Chemical Kinetics
" and "Some Aspects of Kinetic Theory", respectively.
If possible and convinent, could you send them to me ?
My email is w1li@odu.edu. Thanks.

coolboy

lwd1981 发表于 2015-3-18 09:40
I am interested in the two papers included in this book. They are entitled "The Shock Tube and C ...

“The shock tube and chemical kinetics” described an experimental technique on how to use shocks in a tube to initiate chemical reactions and then to measure the rate coefficients of the gas kinetics.

I have two textbooks on chemical kinetics by the same author on my bookshelves:

Steinfeld, J. I., 1974: Molecules and Radiation. An Introduction to Modern Molecular Spectroscopy. Second Edition. MIT Press, Cambridge, Massachusetts, 493 pp.

Steinfeld, J. I., J. S. Francisco and W. L. Hase, 1989: Chemical Kinetics and Dynamics. Princeton Hall, New Jersey, 548 pp.

The second book also mentioned the shock tube method. I once told a story about my experience in learning chemistry long time ago:

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当年的考试经历(coolboy: 2010/4/6)

有一位叫春阳的网友最近写了一篇回忆说是当年著名“七七级”的大学生的经历和待遇都是很特殊的。不少大学都是安排最强的师资全力精心培养“七七级”的大学生。她是武汉某大学化学系的。化学系给她们那一级主讲物理化学、无机化学、分析化学和有机化学的分别是查全性教授夫妇、赵藻藩教授和刘盛荣老师。尤其讲到当年赵藻藩教授为了考验和挑战由“特殊材料”组成的“七七级”大学生，在大二的一次期中考试中就出了“一道更比一道难”的五道考题直把学生考了个昏天黑地。从下午一点一直考到晚上九点。那次考试，春阳得了七十二分，班里的最高分是九十七分，春阳心里佩服得紧，后来干脆就把他收编成了自己的老公......：

那一年，那一场考。。。 [春阳]
http://blog.creaders.net/jpjliu/user_blog_diary.php?did=58310

coolboy：我来讲一个在美国的考试经历。中学的时候学过简单的有机、无机但大学的时候我们这专业没学化学。来美之后必须学点大气化学，修了一门大气化学课学了个天昏地暗、天花乱坠。期中考试是闭卷，其中的一条大题就是写出大气中甲烷是如何一步步地被氧化成二氧化碳的。期中考试三天之后又是化学课。那教授讲课讲到一半就说：我知道你们在考试之前的记忆力都是惊人的，但是我想看看你们现在到底还记得多少，请你们把这条题现在重新再做一遍。他把投影仪的屏幕拉上之后我们就在黑板上看到他早已写好的那条大题。我记得自己比正式考试时答得差多了，忘了好几个方程。
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coolboy

“Some aspects of kinetic theory” briefly described how to derive the macroscopic fluid equations, i.e., Navier-Stokes equations, from the microscopic/Boltzmann equation. The author also discussed some properties of the Vlasov equation for plasma. I have the following textbooks on this subject on my bookshelves:

Gombosi, T. I., 1994: Gaskinetic Theory. Cambridge Univ. Press, Cambridge, 297 pp.

Schunk, R. W., and A. F. Nagy, 2000: Ionospheres. Physics, Plasma Physics, and Chemistry. Cambridge Univ. Press, 554 pp.

Schunk, R. W., and A. F. Nagy, 2009: Ionospheres. Physics, Plasma Physics, and Chemistry. Second Edition. Cambridge Univ. Press, 628 pp.

The following book shows how to derive the Vlasov equation from a more fundamental equation called Klimontovich equation:

Nicholson, D. R., 1983: Introduction to Plasma Theory. John Wiley & Son, New York, 292 pp.

coolboy

So, I mentioned “Boltzmann equation” and “Vlasov equation” above. The Boltzmann equation is for gas kinetics that includes the effects of collisions among different particles whereas the Vlasov equation neglects the effects of collisions but includes the effects of electric and m_a_g_n_e_t_i_c fields when the particles carry electric charges (collisionless plasma). What are the effects of collisions? Collisions among different particles produce viscous frictions and viscous heat in neutral fluid as shown in the Navier-Stokes equations (for neutral fluid). If we neglect the viscous effects, then the Navier-Stokes equations are reduced to the Euler equations, which can be derived from the Boltzmann equation without those collision terms. Since the collisional effects have already been neglected in the Vlasov equation, naturally, the Euler equations can also be derived from the Vlasov equation without the electric and m_a_g_n_e_t_i_c terms for a neutral fluid, which was what I said once in the following post:

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[讨论]量子力学和流体力学有多大联系？   [48楼] [51楼] [56楼]
http://www.cfluid.com/thread-45337-4-5.html

...................

统计力学就不是研究“个体粒子的行为”而是研究大量粒子的平均状态下的行为。其中有一个方程叫Vlasov方程。流体力学中的欧拉方程就可从Vlasov方程通过各种近似简化而推出。从Vlasov方程推出欧拉方程的近似简化程度几乎类似等价于从欧拉方程推出伯努里方程的近似简化。当然，纳维-斯托克斯方程则也可从推广了的Vlasov方程(或即Boltzmann方程)通过各种近似简化而推出。

还没看到有人从数学物理的角度（而非哲学论的角度）研究如何从量子力学原理来推导出Vlasov方程。

...................

我上面引进Vlasov方程的主要目的是想通过比喻说明流体力学的欧拉方程或纳维-斯托克斯方程并非象不少人想象的那样是很基本或很基础的方程：“从Vlasov方程推出欧拉方程的近似简化程度几乎类似等价于从欧拉方程推出伯努里方程的近似简化。”

比如说，有人除了知道一点伯努里方程及其应用之外，从没听说过欧拉方程或纳维-斯托克斯方程却信心十足地老想着如何研究量子力学同伯努里方程的关系。这在知道欧拉方程及知道如何从欧拉方程推出伯努里方程的人的眼里就显得有点浮夸。类似地，我刚好知道Vlasov方程及知道如何从Vlasov方程推出欧拉方程。我当然同样会感到那些不知Vlasov方程为何物的人信心十足地奢谈量子力学同流体力学关系也同样显得浮夸了。

...................

我上面不是已经说过了吗？引进Vlasov方程的主要目的是想通过比喻说明流体力学的欧拉方程或纳维-斯托克斯方程并非象不少人想象的那样是很基本或很基础的方程：“从Vlasov方程推出欧拉方程的近似简化程度几乎类似等价于从欧拉方程推出伯努里方程的近似简化。”

你用流体力学方程解问题，你再怎么把微元划分得如何细致，到一定的时候，流体力学方程本身就已经是很差的近似了。这时要用Vlasov方程或其类似的方程来解决问题。但这时离深入到亚原子结构的量子力学还差着十万八千里呢！还是用伯努里方程来作比喻。有人除了知道一点伯努里方程及其应用之外，从没听说过欧拉方程或纳维-斯托克斯方程却信心十足地老想着如何研究量子力学同伯努里方程的关系。这时，他用测量到的流体压力、密度、速度场等去拟合伯努里方程的解。有的时候拟合得很好，但不少情况下拟合不出来，很差。这位老兄就坚持说这是量子力学的“测不准原理”，需要研究量子力学来解决。人家同他说，你要求解欧拉方程或纳维-斯托克斯方程才会得出更好的拟合，这同量子不量子的没关系。但他就楞楞地坚持着：“伯努里方程研究宏观力学，量子力学研究微观力学，越微观越精确，这有什么错啊？那一闪一闪的像蛇一样前进的火焰波长都隐含着普朗克常数呢，我这测量的压力和速度当然也同量子力学的测不准原理有关啦！”

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lwd1981

Coolboy, thanks for sending me that two papers.

coolboy

I recall I started learning professional English around the same time when I had the fluid mechanics course in college. At the time, I learned the Chinese meanings of “kinematic”(运动学的) and “dynamic”(动力学的)  and their clear physical differences in the fluid mechanics. At the same time, I was also aware of another interesting word that was very close to the word “kinematic” but had the meaning and an exactly same Chinese translation as “dynamic”. That word was “kinetic”. I was curious about the difference between “dynamic” and “kinetic” at the time, thinking that the two words were basically the same and people often used it interchangeably.

Later, I learned the differences between the two words. In the field of fluid mechanics, “dynamic” refers to a macroscopic/bulk approach where the fluid is considered as continuum/parcels. “Fluid dynamics” means to use Navier-Stokes equations to study fluid motions. On the other hand, “kinetic” refers to a microscopic/particle approach where the motions of fluid are considered at a molecular basis. “Kinetic theory” means to use Boltzmann equation or Vlasov equation to study fluid motions. Sometimes, people also translate “kinetic” into 动理学的 in fluid mechanics.

The most interesting thing is that the delicate difference in the meaning of “kinetic” versus “dynamic” in fluid mechanics is switched in the field of chemistry. One may already note the title of the above mentioned textbook “Chemical Kinetics and Dynamics”. In the field of chemistry, the macroscopic/empirical approach of studying chemical reactions is referred to as “kinetics” whereas the microscopic/molecular approach of studying chemical reactions is referred to as “dynamics”.

coolboy

周华 发表于 2015-3-3 14:23
是的，这篇文章保留了周培源作为湍流模式理论开创者的地位，其它文献中一般讲到这里就直接讲K41理论了。
...

Here, “K41” refers to the theory of locally isotropic turbulence developed by A. N. Kolmogorov in 1941. The following book made a modern and a detailed exposition of the theory:

Frisch, U., 1995: Turbulence - The Legacy of A. N. Kolmogorov. Cambridge Univ. Press, New York, 296 pp.

I said quite a few words about A. S. Monin and his works above. Monin was a student of Kolmogorov. Kolmogorov’s another famous student was V. I. Arnold whom I will mention later. Here, I like to say a few words about Kolmogorov. To scientists in the field of fluid mechanics, Kolmogorov is known for his contributions by his “K41” theory to turbulence. However, Kolmogorov’s much, much greater contribution is in the field of mathematics because he laid the foundation of the field of probability. To illustrate the significance of his work, let me first refer to an old post about B. A. Cauchy who laid the foundation of the field of calculus:

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[讨论]用非标准分析方法封闭湍流方程（吴峰）
http://www.cfluid.com/thread-45356-3-3.html

coolboy(2012-12-6):

此外，一方面是当时书店多马列的书而无一般的数理化书，另一方面是读马列的数学物理书或为了学习理解马列的书而学习数学物理也不可能犯错误，故当时也买了读了一些诸如马克思的《数学手稿》、恩格斯的《自然辩证法》等书。而马克思的《数学手稿》从后来的观点来看也就是类似于上面所说的“非标准分析”、“全新数学”等的另一个“非主流数学分析”的观点而已。马克思在《数学手稿》中的主要新观点也就是说抛弃“无穷级数、极限”的概念没什么了不起的，传统的“零不可做除数”也不一定就是真理，所谓的导数其实就是“0/0”。当时有些人就认为马克思直接用“0/0”取代“极限和导数”是数学分析中的一种革命性的创新。

后来考上大学了。我们那一届的同学中有各类不同的学术背景。例如，就有已经从师范大学数学专业毕业在中学任教之后又再考上大学的，自我感觉我的数理化基础还是要远远地高出包括这类同学的其他同学。直到有一天有一位T同学或T老兄给大家介绍这“非标准分析”。这可是我不知或从没听说过的数学内容，我当时就对他升起了一股崇敬之情。他说这“非标准分析”是一种前沿性的新的数学分析方法，在这里大家感到最难理解的所谓“无穷小”就不再是一个抽象的概念而就是一个实在的“数”。在此基础上许多数学分析的问题的求解就都变得非常简单。当时教我们数学分析的程老师是文革前厦门大学数学系毕业的，正讲到实数集的划分什么的。有次在课余时间我就问了他关于马克思的《数学手稿》的“0/0”以及T老兄说的“非标准分析”的“无穷小”是一个“数”的这些理论。这位程老师也知道或十分清楚《数学手稿》和“非标准分析”理论。尽管当时没有“非主流”或“民科”等术语，但他用非常肯定的语气给我解释了这些理论的特性，说是这些都是一些个人或某些人的观点，一般得不到数学学术界的认可，获得大家认可的理论也还是教科书中由柯西建立的以无穷级数和极限为基础的微积分。epsilon-delta语言其实是把极限的抽象概念具体化，这套理论是很完整和严谨的。因为当时众所周知的陈景润也是从那众所周知的厦门大学数学系毕业的，所以我对程老师那段解释坚信不移，后来也从没有再花费任何时间精力于《数学手稿》和“非标准分析”之中去。
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Although I. Newton and G. Leibniz invented the calculus, the calculus they invented was only a tool for people to use or play with it. It did not form a rigorous and self-consistent theory like Euclid geometry that had a well-defined or a logical foundation. Many people tried very hard to lay a rigorous foundation for the calculus and it was Cauchy who finally succeeded.

Likewise, A. N. Kolmogorov in 1933 laid a rigorous foundation for the probability theory based on an axiomatic approach that is similar to Euclid geometry. The classic book in the field of probability was:

Kolmogorov, A. N., 1956: Foundations of the Theory of Probability. Second English Edition. Chelsea Pub. Comp., New York, 84 pp.

coolboy

A. N. Kolmogorov was able to establish a theoretical foundation for probability only because the theories of set, Lebesgue’s measure(测度) and integration became available at the time. The concepts of set, Lebesgue’s measure and integration were briefly described in the following link:

关于“无穷”、“测度”及“长度”的民工式科普   [吴江民工 (2012.01.04)]
http://bbs.lasg.ac.cn/bbs/thread-62615-1-1.html

周华

coolboy 发表于 2015-3-29 00:44
I recall I started learning professional English around the same time when I had the fluid mechanics ...

原来叫“分子动力学”，现在叫“动理学”。最近在讲课的时候我还劝学生们多看一些英文的流体力学教材，主要原因就是中文总是存在各种各样翻译上的问题，直接看英文要简单明了一些。

coolboy

周华 发表于 2015-4-2 21:35
原来叫“分子动力学”，现在叫“动理学”。最近在讲课的时候我还劝学生们多看一些英文的流体力学教材，主 ...

I think you are right. Once getting to the molecular level, it is better to call it “动理学” regardless whether its English term is “kinetics” or “dynamics”. The titles and contents of the following two classic books in the field said just like that:

Bird, G. A., 1976: Molecular Gas Dynamics. Clarendon Press, Oxford, 238 pp.
Bird, G. A., 1994: Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Clarendon Press, Oxford, 458 pp.

Of course, one can also understand the issue in the following way: once getting to the molecular level, there is no difference between the fluid mechanics and chemistry. Both words (“kinetics” from fluid field and “dynamics” from chemistry field) should be understood and called 动理学.

coolboy

The above example of translation in professional English shows that dictionaries will not help much on certain critical terms. In certain cases, one has to really know the field well in order to provide a good translation or one may simply mislead the readers and science community. I think the most interesting example of a bad translation due to misunderstandings in the field of fluid mechanics should be the translation from English to Chinese and its associated explanation of the word “eddy” or “eddies”.

If one looks at the issue from a broad perspective of the fluid “kinematics”, it is quite clear what “eddy” usually means. For a given fluid field variable, such as a temperature or a velocity field, one can decompose it into a mean component and a perturbation component. The mean field can be derived by either a time average, a spatial average, an ensemble average, or some other averages. The perturbation component is simply the difference between the original field and the mean component. At the same time, if the field happens to be a velocity field, then it can also be decomposed into a vorticity component (solenoidal flow) and divergence component (irrotational flow). In general, there is not a relationship between a field being decomposed as “mean” versus “perturbation” components and being decomposed as “vorticity” versus “divergence” components.

Now, the word “perturbation” usually contains the meaning of a SMALL deviation from the mean. As a result, the word often occurs in instability studies in fluid mechanics where the deviation from a basic mean state is small and its higher order nonlinear terms can be neglected. To avoid such a drawback in interpretation and consider the decomposition into mean and perturbation to be a general operation to any given field, people consciously replace the word “perturbation” by the word “eddy”. Eddies do not need to be small quantities in comparison to the mean fields. Any field can be decomposed into a mean component and an eddy component. This decomposition has nothing to do with an totally different one of a velocity field being decomposed into a vorticity component and divergence component.

When the word “eddy” was translated into Chinese, it contained a character 涡 that has a Chinese core meaning of vortex. Furthermore, a professor mixed or confused the meanings of “eddy” and “vorticity” and called 涡旋 and 旋涡 most likely in parallel with the two words in a popular textbook in order to distinguish the difference between “eddy” and “vorticity”, which actually confused the entire field. “tingeMM” once described this confusion with a Chinese couplet at the current cfluid bbs, which I can only find now from the lasg bbs:

涡旋是无旋流动，旋涡是有旋流动。尽管两者都有旋转，但是实质是不一样的。
妒忌是无忌行为，忌妒是有忌行为。尽管两者都含忌恨，但是内涵是不一样的。
横批：误人子弟

coolboy

I mentioned above that Chou, Pei-Yuan (周培源) developed “湍流模式理论” to solve the notorious closure problem of turbulence. Specifically, it was cited in Monin and Yaglom’s classic monograph that Chou was the first to use the second order velocity correlation terms such as [u’_i*u’_j] to eliminate the higher order correlation terms such as [u’_i*u’_j*u’_k] or [u’_i*u’_j*u’_k*u’_l].

Where do these nonlinear correlation terms come from? They come from the averaging to the Navier-Stoke equations. Once taking an averaging over a nonlinear equation, one will get the nonlinear correlation terms in the equation describing the mean fields. While trying to establish equations to describe those correlation terms one will get many new higher order correlation terms in those new equations so that there will always be more unknowns than the number of equations. This is the notorious closure problem of turbulence. As long as there exist those nonlinear correlations that play an important role in a fluid system, then it needs to be treated as turbulence.

Note that the so-called the nonlinear correlation terms are actually the products of the perturbation fields. Since “perturbation” often carries the meaning of “smallness”, people thus call those “nonlinear correlation terms” as “eddies” or "eddy terms". Based on my recollection, “eddy” was once translated into Chinese as “涡动” and it should be understood as “perturbation not necessarily being small”. Later, it appeared that the majority or many people translated “eddy” into “涡” and understood it as “vortex”. This is a clear misunderstanding. I have seen many examples on this forum (and also in some Chinese literature). Here is an example:

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[讨论]伯努利方程是能量方程还是动量方程？[227楼]
http://www.cfluid.com/thread-114265-16-1.html

fluent-aero:

我来批驳一下Coolboy的无知。
Coolboy 177楼，225楼：
问：无粘性流体可以产生湍流吗？
答：可以。
-----------------------------------------------------
连什么是湍流都搞不清楚，还在这里谈湍流，还在误人子弟。

湍流是什么，湍流的特征是什么？

湍流的主要特征有（1）所有物理量都是随时间随机变化的量，它们的运动规律是瞬时性不确定的，chaotic。（2）湍流是涡流的集合，有涡的级串，能量从大涡传向小涡。（3）湍流在壁面处有很大的梯度，在壁面处有大的能量耗散，阻力比层流大的多。所以湍流研究的一个主要方面是减租，如航空方面。

无粘流体在壁面处是滑动的。无粘流体是没有粘性的，在壁面处是没有能量耗散的，没有阻力增加。怎么会有湍流呢？
无粘流体在扰动下会有速度脉动，那叫湍流吗？
文献中有人研究过无粘湍流吗？

根据Holmholtz涡定理，没有粘性，涡是守恒的。若来流为均匀流，永远也不会产生涡，而涡是湍流的本质。没有涡，流体之间就没有能量传递，何来湍流？
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In the above, when fluent-aero said “湍流是涡流的集合，有涡的级串，能量从大涡传向小涡”, it was technically ok as long as one understands that“涡” means “eddies”. However, the last statement showed that he/she thought that “涡” meant “vortex”, which was a misunderstanding. I once gave an interesting example to show how people often make mistakes by misunderstanding or mistranslating between English and Chinese words:

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菜鸟请教特征线问题 [25楼]
http://www.cfluid.com/thread-143194-2-1.html

Coolboy:
那“云雨”的例子刚好也可照搬到这里来。中文的“云雨”其实可以对应于英文中的两个不同的含意。含意之一是 “make love”，含意之二是“cloud and rain”。英文的两个词语“make love”同“cloud and rain”的含意以及它们的数学物理解释之间并不存在任何联系。但只因为“make love”同“cloud and rain”刚好都可以对应中文的“云雨”，而有人偏偏只理解了其中的一个含意，结果也可能就会产生如下的瞎扯：

问：云雨如何产生雷电？
答：云雨的时候若男女之间具有共同的理想和人生目标，则云雨就能产生爱情。爱情的力量是无穷的。在“天人合一”的中国古代哲学思想以及“人体科学”这一现代创新思想的的启发下，这无穷的爱情之魅力就能转化为巨大的自然力，这雷电也就随之而生了。
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coolboy

Here is another example on this bbs that misunderstands (or did not understand) “eddies” in turbulence as “涡”，“漩涡”，“漩涡是指旋转量大于剪切量”:

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请问湍流一定会产生漩涡吗？ [1楼] [4楼] [6楼]
http://www.cfluid.com/thread-74630-1-1.html

Talentliam:
用fluent模拟流动问题，如果一个管道内是湍流的话，那么在速度矢量图中，一定会看到涡吗？
谢谢各位大侠！

zery:
漩涡理论是湍流理论的核心，
湍流运动的特征主要是存在大量尺度不一的漩涡

Dswayb:
个人认为取决与你是说漩涡还是涡量，层流只要有平均速度就有涡量，但是有涡量不代表有漩涡，漩涡是指旋转量大于剪切量，也就是一般大家都看的Q-D图
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coolboy

I once also told another interesting story of “The universe is a big bag” due to a bad translation of “Big Bang” into “Big bag”:

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<地球流体中的涡和波>之广告   [4楼]
http://bbs.lasg.ac.cn/bbs/thread-63199-1-10.html

联想起了曾听到的两个故事，都与术语的混淆而影响讲座效果有关。一个是“宇宙是个大口袋”，另一个是“这恩索到底是什么东西”。前一个故事（好像是）发生在复旦大学，而后一个故事发生在科学院大气所。

故事一：才改革开放时来中国访问的外国学者作学术报告时，都有翻译帮忙。但当时翻译一般多不是专业人员，故听报告的专业人员常常需要根据自己的专业知识来猜测一些报告内容。有一世界知名天体物理学家来华访问，有一个关于“宇宙的结构”的报告是在复旦大学作的。报告中展示了不少数据和数学公式，但可能是专业背景的差别和差距，台下的人都听得有点云里雾里的。这报告完了之后也总得提几个问题才显出和谐的气氛吧！并不是很懂大师的报告，但大师在报告中多次提到“大口袋”了，不如就这“大口袋”来问一个比较得体的问题。有人就问了：现在多数人认为这宇宙应该是无限的，你现在的理论认为宇宙是个大口袋，这大口袋的边界怎么同无限的宇宙协调起来呢？另一个人又问：关于这大口袋的形状及结构，其本质上是否应该由最小作用量原理来确定？不知那大师是怎么回答那些问题的。不过到最后发现，其实大师在报告中说的是“Big Bang”，而那个翻译人员则误听、误翻成了“Big Bag”。“Big Bang”是“大爆炸”，“Big Bag”是“大口袋”。
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The above story also tells us some truth that one may still conduct an interesting scientific research along the line of an initial misunderstanding. One may propose a scientifically reasonable theory/hypothesis describing the structure of the universe based on an idea of a “Big bag”. Still, a misunderstanding has occurred or a mistake has been made. This is also true on the investigation of turbulence. One may carefully study an important problem of how a big vortex becomes unstable, breaks and is converted into many smaller vortices. However, the result may not be so directly related to the general problem in turbulence of what roles the eddies play in driving the mean fields.