挑战流体高手：伯努利定理中的静压是哪个方向的？

legolasGG

[ADMINOPE=周华|legolasGG|威望由 1 增加至 2|加分！|1149137089]Static pressure is supposed to be used for fluids in still state. Not at all for fluids in motion or in acceleration.
If you want to analyze forcing in the fluid in motion. You have the tensor which can be projected to any direction and on any plane. Then the pressure is simply defined as the average of the diagonal components of the force tensor. So why would we try to tell static pressure from total pressure in that condition?  Also I don';t agree with the original question. Bernoulli equation is valid for unsteady flow, so why do we have to use static pressure in Bernoulli equation? It should be total pressure, period.

coolboy

[这个贴子最后由coolboy在 2006/03/21 04:39am 第 1 次编辑]

legolasGG,
Batchelor (1967) spent one whole page (p. 13) to describe why one needs to introduce the concept of static-fluid pressure: it is spherically isotropic.
+++++++
Batchelor, G. K., 1967: An Introduction to Fluid Dynamics. Cambridge Univ. Press, Cambridge, 615 pp.

legolasGG

Then I simply don';t agree with Batchcelor';s arguments on calling it hydrostatic pressure when the fluid particles are having acceleration.   
In his next page (p14 footnotes) he also said that
    "The term hydrostatic pressure is often used, but the implied association with water has only historical justification and may be misleading. The terms ';hydrodynamics'; and ';aerodynamics'; are likewise unnecessarily restrictive, and are being superseded by the more general term ';fluid dynamics';"
     I think that he tried to introduce the concept of pressure p=-sigma_{ii}/3 in the context of both inviscid and viscous fluid.  The problem is that viscous/compressible fluid has to be at rest for the deviatoric tensor to be zero and the total tensor is only consisted by the pressure component.  On the other hand, for inviscid and incompressible fluid, the deviatoric tensor is always zero regardless whether it is at rest or not. So, what I call pressure is really hydrostatic pressure for viscous/compressible fluid by Batchelor, also total pressure for inviscid and incompressible fluid.
In free surface flow such as water waves, say the surface elevation is defined as eta(x,y,t), the pressure at z is p=rho*g*(eta-z)+ pd, where rho*g*(eta-z) is not related to acceleration of fluid particles, and pd is related to acceleration.  In this context, people call  rho*g*(eta-z) hydrostatic pressure, and pd dynamic pressure.   Obviously, this is not the same ';hydrostatic pressure'; as by Batchelor.
  Finally, in my opinion, pressure is pressure, it is always isotropic in terms of direction, so there there should be no such question of which direction it is acting at or which plane it is acting on unless we are talking about an interface within the fluid or with a boundary in which case pressure should really be called forcing on the interface.

ustcsunl

下面引用由legolasGG在 2006/03/21 10:55am 发表的内容：
Then I simply don';t agree with Batchcelor';s arguments on calling it hydrostatic pressure when the fluid particles are having acceleration.  
In his next page (p14 footnotes) he also said ...

    Batchelor discussed the hydrostatic pressure within the context of "the stress tensor in a fluid at rest" from pg 12 to pg 14. So the arguments is
not valid when the fluid particles are having acceleration. This topic is
addressed in section 3.3 from pg 141 to pg 147. I have not read this part
(totally in English) yet. But some text books in chinese have also addressed the topic (probably after Batchelor';s and Landau';s).
    The discussion about the pressure on free surface is always intresting.
As well known that the pressure associated to acceleration is called fluctent
pressure. To my opinion, this pressure is purely due to the baroclinicity
on the surface, in fact the free surface is the limitation of baroclinic flow. As Batchelor';s and other discussions are mainly valid under the condition of barotropic flow, so that the hydrostatic pressure can be simplified, e.g.,
the gravity is not obvious in the pressure.
    Finally, both legolasGG and coolboy show me some intresting comments. I
agree with legolasGG that pressure is pressure.
   

coolboy

To legolasGG:
Good!
“I think that he tried to introduce the concept of pressure p=-sigma_{ii}/3 in the context of both inviscid and viscous fluid.” – good observation!
Here is what he gained after doing so:
One may note that almost all papers/books (say, Monin’s turbulence monograph) written by authors from former Soviet Union on viscous fluids have two viscous coefficients: first viscosity and second viscosity whereas Western literature has only one coefficient of viscosity. That is because the former follows Landau’s approach whereas the latter follows Batchelor’s approach. Landau did not discuss what exactly p was so people got an impression that “pressure is pressure”. On the other hand, Batchelor introduced “the concept of pressure p=-sigma_{ii}/3 in the context of both inviscid and viscous fluid” so that the effect of the second viscosity can be incorporated into a slightly different definition of p.
+++++++++++++++
Batchelor, G. K., 1967: An Introduction to Fluid Dynamics. Cambridge Univ. Press, Cambridge, 615 pp.
Landau, L. D., and E. M. Lifshitz, 1987: Course of Theoretical Physics. Vol. 6: Fluid Mechanics. Second Edition. Pergamon Press, New York, 539 pp.

legolasGG

Thanks for the quote and the link, I will look into that. My impression, the second viscosity maybe related to the expansion of volume of fluids which is true for compressible flow, yet let me check on that.
legolasGG

海上钢琴师

下面引用由ustcsunl在 2006/03/16 10:49pm 发表的内容：
一个很基础很基础的概念居然都没有搞明白：p究竟是个什么量？

This is also what I want to know. Would you please tell me what it is ? scalar, vector or tensor,

abbbbc

[这个贴子最后由abbbbc在 2006/03/26 10:51pm 第 1 次编辑]

去看看这个书吧，里面比较详细的讲了静力学和动力学中压强的纯力学定义和热力学定义。当然关于这个问题的讨论最初不是在这个书里，但要找中文版的可能就不太容易，仔细看过这个书我想就应该能够对流体力学中的一些基本概念有比较清楚的理解了。在很多书里，流体（包括静止流体和运动流体）中某点的压强不随方向变化是作为一个基本条件直接引入的，象兰姆的《理论流体动力学》，也都没有对这个问题加以深入讨论。国内各大学自己编的流体书都是根据自己的情况针对自己专业的需要用来入门的，等过了这个入门阶段，就应该自己去找那些真正的经典来看，以求得深入的理解，不要随便看了一点东西就以为发现了什么新大陆，拿来到处显摆。在论坛里看过楼主的好几个帖子，说流体力学教材的这次错误那个错误，好像有多牛X一样，其实那些入门级书籍里面讨论的问题，基本都属于经典、传统的流体力学范畴，在这个范畴内讨论流体力学，不能离开它赖以存在的基本条件，比如连续介质条件，这些条件基本上都是在一开篇就要引入的，然后在接下来的书里面对问题的论述就隐含了这些基本条件。如果抛开这个条件来讲书里面的问题，那就象一个人学过了相对论和量子力学以后，夸夸其谈的讲牛顿力学是错误的那样可笑。
书 名: 流体动力学引论
作 者: G.K.巴切勒著;沈青,贾复译
出版项: 北京:科学出版社/1997.11
ISSN/ISBN: 7-03-004632-3
丛书名: 力学名著译丛
价格/页码等: 58.00元, 707页,32开
内容提要 本书是一本优秀的流体力学教程,作者是近代流体力学方面的权威学者之一。本书系统地介绍了一般流体力学研究取得的基本成果,选材的宗旨是为引导读者熟识流体力学的基本概念、思想及重要应用,着重于阐明流体力学的物理基础。全书共分七章,前三章是研究任何流体力学的前题和基础,讨论了流体的物理特性、流场运动学及基本方程的一般形式。后四章全部讨论均匀不可压缩粘性流体动力学,就其重要性和基础性而言,这部分内容无疑是全部流体力学的核心部分。本书可供应用数学及力学工程系的大学生、研究生,以及从事力学、物理、气象、海洋、航空、水利等方面研究的科技人员阅读、参考。

周华

巴切勒这本书现在有英文影印版，我在北京西单图书大厦买到过，有兴趣的可以去那里找找。

junior

[ADMINOPE=周华|junior|威望由 3 增加至 4|加分！|1149137311]按legolasGG的思路走下去必然要研究动力学中的压强与热力学压强之间的关系，Batchelor在书中引入p=-sigma_{ii}/3时强调这是一个纯粹从力学角度下的定义，目的是为了与人们习惯的从静力学中得到的“压强”概念相对应，而且恰恰p=-sigma_{ii}/3是个不随坐标系旋转而变化的不变量，所以拿这个量出来作为动力学中的“压强”定义是比较恰当的。与legolasGG理解不同的是，在动力学中，压强仍然是与作用面方位角有关的一个量。[br][br][以下内容由 junior 在 2006年04月02日 00:26am 时添加] [br]
legolasGG引用的p=rou*g*(zeta-z)+Pd这个公式是从欧拉静平衡方程推导出来的，因此其中的压强仍然是静力学意义上的压强，由此并不能推出“pressure is pressure”这个概念。在考虑了加速度后，流体仍然可以在达朗贝尔意义上当作静力学问题来处理，这就是平常教科书上所说的“相对平衡”的概念。

coolboy

[这个贴子最后由coolboy在 2006/04/04 00:43am 第 2 次编辑]

Junior’s understanding is essentially correct that the pressure p in Landau’s book is a thermodynamic quantity. In a broad sense, a large part of statistical mechanics is to clarify the concepts of thermal equilibrium and temperature T of a system. Thermodynamic p and (a particular) T are closely related or should be self-consistently defined. The static or dynamic p in Batchelor’s book does not directly involve the definition of T.
In general, T (and thus p) could have different values along different direction even though it is NOT a vector.

junior

是啊，如果考虑分子的内部结构，温度可以有平动温度、转动温度、振动温度之分。在流体力学的基础教材中通常不考虑分子内部结构，即认为流体仅有平动自由度，因而统一使用平动温度作为流体温度的定义。

junior

coolboy, would you explain how the temperature T will vary according to its orientation ? that will make this discussion more complete.

coolboy

[ADMINOPE=周华|coolboy|威望由 1 增加至 2|加分！|1149137355]Random motions of particles define the so-called “thermal speed”. When the thermal speed satisfies a Maxwellian distribution we have a well defined kinetic/translational temperature. If the Maxwellian distribution is isotropic with respect to its three velocity components, then, there is only one value of temperature at a spatial point. Motions of ionized fluid (plasma), including both its bulk/mean velocity and random/thermal velocity, will be influenced by (say, an external) magnetic field. In this case, we may have to use an anisotropic distribution function, the so-called “bi-Maxwellian” distribution, to describe its thermal speed because the velocity distributions parallel and perpendicular to the magnet field are different. This naturally leads to different temperatures parallel and perpendicular to the magnetic field at the given spatial point.

junior

OK, got it, many thanks!
But the problem is when we put a thermometer in the MHD field, how can we judge which temperature we will read, the T parallel to or the T perpendicular to the magnetic field ?