重访“ bath-tub vortex”

ustcsunl

因为网站老是更新，链接不好使，所以不得不把这个问题从新找出来。

摘要：
   
     “ bath-tub vortex”经过谢皮罗Shapiro 的宣传，广为人知。但是，真正的“ bath-tub vortex”理论，可以参考下面John Marshall关于地球流体力学讲义部分（内有实验及理论分析）及coolboy所引Pedlosky计算结果可知道，“ bath-tub vortex”和地球自转没有关系。最后给出的是我原来在某个帖子里面的回复。显然地，如果这些涡和地球自转有关系，那么研究航空航天的整个空气动力学就要改写。

Marshall实验要求水要静置一天（为什么），计算说，要求盆子尺寸比漏洞尺寸大200倍（这不是我们通常所看到的家用水槽的尺寸比例）。记得有人曾经贴过上海大学的一篇博士论文，就是做这个实验的，目前帖子找不到，希望有人在回帖中指出。就我个人看法，这篇博士论文的结论是错的（解释他的实验现象不要去找地球自转的麻烦，流体内禀就有那样的流动）。

     最后希望，coolboy能够现身，给出相关文献（我原来看过，但是这两年花了很长时间都没有找到）。

相关帖子内容：

水寿松曾经写到：
“1962年.谢皮罗在《科学》杂志发表论文,认为水漩涡与地球自转有关,如果地球停止自转的话.拔掉澡盆的塞子,水不会产生漩涡.由于地球时自西向东不停地旋转,而美国又处于北半球,所以洗澡水总时逆时针方向旋转。,谢皮罗断言.如果在南半球,则恰好相反.洗澡水将会按顺时针形成漩涡.在赤道则不会形成漩涡。谢皮罗的论文发表后，引起各国科学家的莫大兴趣，纷纷在各地进行试验,结果证明谢皮罗的论断完全正确。这一现象被命名为谢皮罗现象。”
------从下面通流所引用的文章来看，这段话错了，应该是“1962年.谢皮罗在《自然》杂志”，本帖注。

通流更早期提到下面的文章，
麻省理工学院 Shapiro 教授1962年在 nature上的关于 bath-tub vortex的文章
Nature  1962.12.15 vol.196 (p.1080-1081)
Bath-Tub Vortex

PERENNIALLY one hears controversy regarding the direction of the vortex in the drain of the bath-tub or the kitchen sink. Some claim that the direction of swirl is always the same in the northern hemisphere and that in the southern hemisphere it is always opposite to that for the northern hemisphere. Other claim that there is no unique direction of swirl in either hemisphere.
Both schools of thought are in some sense correct. For the everyday observations of the kitchen sink and bath-tub variety. the direction of the vortex seems to vary in an unpredictable manner with the date, the time of day, and the particular household of experimenter. But under well-controlled condition of experimentation, the observer looking downwarl at a drain in the northern hemisphere will always see a counter-clockwise vortex, while one in the southern hemisphere will always see a clockwise vortex.
In a properly designed experiment, the vortex is produced by Coriolis forces. which are counter-clockwise in the northern hemisphere.
There are two reasons for the apparent haphazard results of the everday observations. The effective rim speed of a tank 2 ft. in diameter rotating with the earth at Boston, Massachusetts, is only about 0.04 in./min. This tiny Earth-induced motion may be much less than other motions too small to be perceived by ordinary observation. Unless the water in the vessel is allowed to settle for may hours, for example, the residual motions from the filling of the vessel may far exceed the foregoing figure. Moreover, the very act of pulling the plug from the bottom of the vessel may inadvertently create substantial velocities. Ever if many hours are allowed for settling, are currents in the room or thermal currents associated with nonuniform temperatures around the vessel may easily cause water velocities larger than that due to the Earth's rotation. All these difficulties are quite real;this was amply proved by early experiments with the apparatus desribed later.
The Second source of confusion in the casual everyday observations is that the Coriolis forces due to the Earth's rotation may in fact be much smaller than other forces normally prosent. For a speed of 0.1 in./sec toward the drain, the Coriolis force at latitude 42˚ north is about 3X10-7 times the force of gravity. If the vessel is not perfectly symmetrical about the axis of the drain, the asymmetrical viscous forces at the side walls and at the bottom of the vessel may produce torques larger than that due to the Coriolis force; the direction of the asymmetrical torque would, of course, vary from vessel to vessel. Even the extremely small forces of surface tension at the free surface may, owing to non-uniformities in temperature or chemical impurity, lead to torques.
ASCHER H. SHAPIRO
Massachusetts Institute of Technology, Cambridge, Mass.

Marshall的讲义，内有实验和理论解释。
Theory
We suppose that a particle of water initially on the outer rim of the cylinder at radius r1 moves inwards conserving angular momentum until it reaches the drainhole, at radius ro - see …g.(6.31). The earth’s rotation earth resolvedin the direction of the local vertical is earth sin ' where ' is the latitude. Therefore a particle initially at rest in the laboratory frame at radius r1 of the cylinder, has a speed of v1 = r1earth sin' in the inertial frame. Its angular momentum is A1 = v1r1. At ro what is the rate of rotation of the particle?
If angular momentum is conserved, then Ao = or2 o = A1, and so the rateof rotationearth sin ' Thus if r1/ro>> 1 the earth’s rotation can be ‘ampli…ed’ by a large amount. For example, at a latitude of 42N, appropriate for Cambridge MA, sin ' = 0:67,earth= 7:3£10¡5s¡1; and for our cylinder with r1 = 30cm, ro = :15cm, we…nd that o= 1:96 rad s¡1, or a complete rotation in 3 seconds! This shouldreadily be een.

Marshall的说法应该同我下面的说法，及coolboy所引用的Pedlosky说法一致（Pedlosky也做同样的计算，指示用实际澡盆尺寸带入，结论自然是地球自转没有影响）。因为找不到coolboy的帖子，期望有一天他能光临本帖，从新将Pedlosky的文章贴出。

我的回答：
不过，同样的麻烦也出现在动力论坛上，因为版本的升级，以往的帖子都不能搜索。关于这个问题最好的回答出现在动力论坛上，是coolboy在一个回帖中援引Pedlosky的一个计算来说明，地球自转当然没有关系了（coolboy称其为见到的关于此问题最好的回答）。看来Pedlosky也常为此类问题所不满，所以亲自计算并写了一短文章。
遗憾的是，我今天花了一个小时也没有找到这个帖子。

为啥气体动力学的书和船舶方面的教材中的方程都不包括地球自转的Coriolis 力呢？显然是，根据尺度分析，可以看到对于本问题地球旋转是不重要的。

对于这个问题，其实问问研究飞机、轮船的同志：“你们做气动（水动）外形计算时候要不要考虑地球自转啊？”

[ 本帖最后由 ustcsunl 于 2011-1-5 02:51 编辑 ]

通流

为啥气体动力学的书和船舶方面的教材中的方程都不包括地球自转的Coriolis 力呢？显然是，根据尺度分析，可以看到对于本问题地球旋转是不重要的
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你的流动是什么？没有具体的问题，何来尺度分析？