Solution to Feynman’s reverse sprinkler puzzle also applies to “silly sprinklers”

费曼“反向喷水器”难题的解决方案同样适用于“趣味喷水器”

Watering your lawn in the summer can be both pragmatic and fun with so-called “silly sprinklers,” designed to create amusing loops and spirals of water jets. And there’s some fascinating physics at work to boot. Researchers at New York University’s Courant Institute conducted a series of experiments with different silly sprinkler designs to find the answer to a longstanding problem in fluid dynamics, according to a new paper published in the Proceedings of the National Academy of Sciences.

夏天给草坪浇水时，使用所谓的“趣味喷水器”（silly sprinklers）既实用又有趣，它们被设计成能喷出各种好玩的环形和螺旋状水流。而且，这背后还蕴含着迷人的物理学原理。根据发表在《美国国家科学院院刊》（PNAS）上的一篇新论文，纽约大学柯朗数学科学研究所的研究人员针对不同的趣味喷水器设计进行了一系列实验，旨在解决流体力学中一个长期存在的难题。

As previously reported, the reverse sprinkler problem is associated with physicist Richard Feynman because he popularized the concept, but it actually dates back to a chapter in Ernst Mach’s 1883 textbook The Science of Mechanics (Die Mechanik in Ihrer Entwicklung Historisch-Kritisch Dargerstellt). Mach’s thought experiment languished in relative obscurity until a group of Princeton University physicists began debating the issue in the 1940s. Feynman was a graduate student there at the time and threw himself into the debate with gusto, even devising an experiment in the cyclotron laboratory to test his hypothesis.

正如之前报道的那样，反向喷水器问题与物理学家理查德·费曼（Richard Feynman）有关，因为他普及了这一概念，但它实际上可以追溯到恩斯特·马赫（Ernst Mach）1883年的教科书《力学及其发展的历史批判》（The Science of Mechanics）中的一个章节。马赫的这一思想实验在很长一段时间里鲜为人知，直到20世纪40年代，一群普林斯顿大学的物理学家开始讨论这个问题。当时还是研究生的费曼兴致勃勃地投入了这场辩论，甚至在回旋加速器实验室设计了一个实验来验证他的假设。

One might intuit that a reverse sprinkler would work just like a regular sprinkler, merely played backward, so to speak. But the physics turns out to be more complicated. “The answer is perfectly clear at first sight,” Feynman wrote in Surely You’re Joking, Mr. Feynman (1985). “The trouble was, some guy would think it was perfectly clear [that the rotation would be] one way, and another guy would think it was perfectly clear the other way.”

人们可能会直觉地认为，反向喷水器的工作原理就像普通的喷水器一样，只不过是“倒着放”而已。但事实证明，其中的物理原理要复杂得多。费曼在1985年的著作《别闹了，费曼先生！》（Surely You’re Joking, Mr. Feynman）中写道：“答案乍一看非常清楚。问题在于，有人会觉得（旋转方向）是向一个方向，而另一个人会觉得向另一个方向才完全清楚。”

Mach proposed that there would be no rotation with a reverse sprinkler: the reaction force on the nozzle as it sucks in water pulls the nozzle counter-clockwise, while the water flowing into the inside of the nozzle pushes it clockwise. The two forces cancel each other out in this steady-state scenario. Feynman’s own experiment showed a slight tremor when pressure was first applied to pump water through the nozzle, and then the sprinkler returned to its original position and remained still.

马赫提出，反向喷水器不会发生旋转：喷嘴吸水时的反作用力将喷嘴向逆时针方向拉，而流入喷嘴内部的水流则将其向顺时针方向推。在这种稳态情况下，这两种力相互抵消。费曼自己的实验显示，当最初施加压力通过喷嘴抽水时，喷水器会有轻微的震颤，随后便回到初始位置并保持静止。

But others suggested that if the friction was low enough and the inflow rate high enough, a reverse sprinkler would start to turn in the opposite direction of an ordinary sprinkler, thanks to the formation of a vortex inside. Since Feynman’s efforts, experiments have been all over the place: some showed steady reverse rotation, some showed only transient rotation, and some produced unsteady rotation that changed direction or flowed in a direction determined by the contraption’s geometry.

但其他人认为，如果摩擦力足够小且流入速度足够快，由于内部形成了涡流，反向喷水器会开始向普通喷水器的相反方向旋转。自费曼的研究以来，各种实验结果各不相同：有的显示出稳定的反向旋转，有的仅显示出短暂的旋转，还有的产生了不稳定的旋转，其方向会发生改变，或者由装置的几何形状决定。

In 2024, New York University applied mathematician Leif Ristroph and several colleagues built their own custom sprinkler that incorporated ultra-low-friction rotary bearings so their device could spin freely. They immersed their sprinkler in water and used a special apparatus to either pump water in or pull it out at carefully controlled flow rates. This let the team observe how water flowed inside, outside, and through the device. Adding dyes and microparticles to the water and illuminating them with lasers helped capture the flows on high-speed video. They ran their experiments for several hours at a time, the better to precisely map the fluid-flow patterns.

2024年，纽约大学应用数学家莱夫·里斯托夫（Leif Ristroph）和几位同事制造了他们自己的定制喷水器，其中采用了超低摩擦旋转轴承，使装置能够自由旋转。他们将喷水器浸入水中，并使用特殊装置以精确控制的流速进行抽水或排水。这使研究团队能够观察水流在装置内部、外部以及穿过装置时的流动情况。通过在水中添加染料和微粒，并用激光照射，他们成功地用高速摄像机捕捉到了水流轨迹。他们每次进行数小时的实验，以便精确绘制流体流动模式。

The team found that the reverse sprinkler rotates 50 times slower than a regular sprinkler, but it operates along similar mechanisms, which surprised them. Ristroph described the behavior as an “inside-out rocket,” where the internal jets shoot inside the chamber where the arms meet and collide—but they don’t collide head-on, which results in the forces that rotate the sprinkler in reverse. By contrast, a forward sprinkler is more like a rotating rocket, with jets shooting out of its arms.

研究小组发现，反向喷水器的旋转速度比普通喷水器慢50倍，但其运行机制相似，这令他们感到惊讶。里斯托夫将这种行为描述为“由内而外的火箭”，内部喷射的水流在喷臂汇合的腔室内发生碰撞——但它们并非正面碰撞，从而产生了使喷水器反向旋转的力。相比之下，正向喷水器更像是一枚旋转的火箭，水流从喷臂中喷射而出。

The 2024 experimentally observed flow patterns were in excellent agreement with the group’s mathematical models—which they dubbed the momentum flux theory. However, it didn’t definitively rule out competing theories. Also, the group only looked at sprinklers with S-shaped arms. So this latest paper builds on that earlier work by extending the experiments to silly sprinklers the team created themselves. Ristroph et al. tested them in both forward mode (where water sprays out) and reverse mode (where water is sucked in). Their observations strongly supported Ristroph et al.’s momentum flux theory and were inconsistent with both Mach’s and Feynman’s hypotheses.

2024年实验观察到的流动模式与该小组的数学模型（他们称之为“动量通量理论”）高度吻合。然而，这并没有彻底排除其他竞争理论。此外，该小组之前仅研究了S形喷臂的喷水器。因此，这篇最新的论文在之前工作的基础上，将实验扩展到了团队自己制造的趣味喷水器上。里斯托夫等人分别在正向模式（喷水）和反向模式（吸水）下进行了测试。他们的观察结果有力地支持了里斯托夫等人的动量通量理论，且与马赫和费曼的假设均不一致。

They also found that the arm shape of a given sprinkler can control the jet flow, and the team devised specific guidelines for designing structures to control flow to produce torque and rotation. “Our findings provide a firmer understanding of how components respond to fluid flows—knowledge that can guide future engineering and technological advances for devices, such as turbines, that co

他们还发现，特定喷水器的喷臂形状可以控制喷射水流，团队为此制定了设计结构以控制水流从而产生扭矩和旋转的具体指南。“我们的研究结果为理解组件如何响应流体流动提供了更坚实的认识——这些知识可以指导未来工程和技术的发展，例如涡轮机等设备……”