Tea Physics
I'm a big tea fan. And a recent edition of Physics of Fluids contained a paper on tea kettle whistles, which reminded me of an older article on tea pot drips. So here are these two papers on the deep physics of tea.
Why do tea kettles whistle?
As of this year, I have a tea kettle that whistles, so it's a good thing that Humanity has finally understood why tea kettles whistle: otherwise my tea kettle would be an object of mystery and wonder (it's not a good enough tea kettle to deserve that). This new understanding comes from a paper titled The aeroacoustics of a steam kettle in the October issue of Physics of Fluids 1 [1 R. Henrywood & A. Agarwal, The aeroacoustics of a steam kettle].
The abstract describes the main results:
The whistle in a steam kettle provides a near-perfect example of a hole tone system, in which two orifice plates are held a short distance apart in a cylindrical duct. This setup leads to distinct audible tones for a large range of flow rates. The main objective of the current paper is to understand the physical mechanism behind the generation of hole tones (whistling of steam kettles). A variety of experiments were undertaken, primarily focusing on how the acoustics of the hole tone system varied depending on the flow rate, whistle geometry, and upstream duct length. These were supplemented by flow visualisation experiments using water. The results show that the whistle's behaviour is divided into two regions of operation. The first, occurring at Reynolds numbers (based on orifice diameter and jet velocity) below Reδ ≈ 2000, exhibits a near-constant frequency behaviour. A mathematical model based on a Helmholtz resonator has been developed for this part of the mechanism. The second, for Reynolds numbers greater than Reδ ≈ 2000, the whistle exhibits a constant Strouhal number behaviour. A physical model has been developed to describe this part of the mechanism where the resonant modes of the upstream duct are coupled with the vortex shedding at the jet exit.
The full paper, which thankfully appears to be open-access, describes things a bit further. Tea kettles whistle due to a particular type of cavity that most tea kettles have; this is a cylindrical cavity, with both of the cylinders closed off except for holes along the axis of the cylinder. When air rushes through the holes, it produces a sound that is proportional to air velocity and inversely proportional to cylinder length. In general, there are three types of whistling:
- Vortex shedding, referred to as constant Strouhal number behavior.
- A feedback mechanism, where the air vibrates slightly, causing sound waves to travel upstream (opposite the flow of air) and vibrate the air.
- Resonance, like a pipe organ.
They use careful experimental analyses to try to figure out which of the above is the right model. As it turns out, the second mechanism is a good model up until a certain Reynolds number (which measures, roughly, speed vs viscosity). Above that Reynolds numbers, the first mechanism becomes a stronger contender (this makes sense, since vortex shedding is a feature of turbulent flow, which is associated with higher Reynolds number). But the sound is mainly produces by the third mechanism, where the duct (the part of the tea kettle which comes before the cavity) begins to resonate.
Using this model, together with some careful calculations of the effect of duct length, Henrywood and Agarwal can compute the pitch of tea kettle whistles to within a few percentage points. That's a very impressive result; I am tempted to go through the math for my tea kettle at home.
Why do tea pots drip?
An earlier, equally fascinating paper is one by J. Keller, published in 1957 in the Journal of Applied Physics 2 [2 J. Keller, Teapot Effect], where Keller describes the mechanism behind tea pot drips. The entire paper makes more sense in the context of some earlier work by M. Reiner 3 [3 M. Reiner, The teapot effect… a problem], as described in an Etsy blog post.
M. Reiner (1956)
Reiner's work was published in a wonderfully understandable article in Physics Today, in 1956. He described the general phenomenon of fluids sticking to surfaces when they are forced (by gravity) to round a corner. The phenomenon had apparently first been noticed for salt crystals, where a layer of dissolved salt water prevents a large salt crystal from being dissolved on all sides, but instead just from the top. You might think this is due to some surface tension or adhesion property, but you can replicate the effect both with salt water piped into fresh water, and with fresh water piped into salt water (one is denser than the other).
This means that there is some other mechanism at work besides adhesion. Reiner describes a complex mechanism based entirely in fluid dynamics. When a fluid flows against a surface, it is sheared: the layers further from the surface flow faster due to friction between the surface and the fluid. Viscosity makes the fluid resist shearing, partly through some internal stresses (I didn't understand this part) and partly by forming vortices. The vortices are in the plane perpendicular to the surface and along the direction of flow. So when the surface turns, the fluid is already turning with it. This causes the fluid to flow by inertia along the surface, even against gravity (for a brief while).
Of course, this effect only occurs when the surface turns “away” from the stream of liquid. If it turns “toward” it instead, the rotation and the turn are in opposite directions and the liquid instead is forced to stop. This is used in creating “drip grooves” above windows—check the article for more. Reiner also describes a second teapot effect, more complex than the first, which isn't so important in understanding why teapots drip.
J. Keller (1957)
So now back to Keller. The abstract of Keller paper explains the nub of his work:
When tea is poured from a teapot it often runs along the under side of the spout rather than falling into the cup. Recent experiments have shown that this “teapot effect” is not due to surface tension nor adhesion, as many have supposed. Therefore, a new explanation is presented which is based upon certain exact solutions of the hydrodynamicequations and which seems to account for the effect. In connection with the analysis some new fluid flows with free boundaries are obtained. In addition a method more satisfactory than the usual one is used to study the stability of these flows.
The “recent experiments” refer to Reiner's work. Keller's article is a complex set of hydrodynamics theory which better describes why tea "sticks" to the tea pot. Apparently this is a universal property of ducts that end: the stream turns corners and sticks to the outside. This happens even without gravity. Frankly, I don't have the physics background to explain the effect here, but it suffices to know that the tea pot drips are a completely mechanical artifact and don't require surface tension.
J. Vanden-Broeck and J. Keller (1986)
Keller returned to the problem of drippy tea pots in 1986, together with J. Vanden-Broeck, in Pouring flows, in Physics of Fluids 4 [4 J. Vanden-Broeck & J. Keller Pouring flows] Here's the abstract:
Free surface flows of a liquid poured from a container are calculated numerically for various configurations of the lip. The flow is assumed to be steady, two dimensional, and irrotational; the liquid is treated as inviscid and incompressible; and gravity is taken into account. It is shown that there are jetlike flows with two free surfaces, and other flows with one free surface which follow along the underside of the lip or spout. The latter flows occur in the well‐known “teapot effect”, which was treated previously without including gravity. Some of the results are applicable also to flows over weirs and spillways.
So, the goals of the analysis are the same as in the 1957 paper, but gravity is now included. I again don't have the background to understand the paper fully. In this extremely short paper, Vanden-Broeck and Keller use complex analysis to describe the exact shape of the drip. The paper won an Ig Nobel prize (a parody Nobel prize given for creative and fun, but nonetheless technically challenging and rigorous, scientific work).
Conclusions
It's fascinating that phenomena as simple as tea pot drips and tea kettle whistles have such complex physics. I don't have the physics expertise to really understand these papers, but I think even reading over the abstracts has reminded me how rich the world really is. Feynman once wrote,
I have a friend who's an artist and has sometimes taken a view which I don't agree with very well. He'll hold up a flower and say "look how beautiful it is," and I'll agree. Then he says "I as an artist can see how beautiful this is but you as a scientist take this all apart and it becomes a dull thing," and I think that he's kind of nutty. First of all, the beauty that he sees is available to other people and to me too, I believe. Although I may not be quite as refined aesthetically as he is… I can appreciate the beauty of a flower.
At the same time, I see much more about the flower than he sees. I could imagine the cells in there, the complicated actions inside, which also have a beauty. I mean it's not just beauty at this dimension, at one centimeter; there's also beauty at smaller dimensions, the inner structure, also the processes. The fact that the colors in the flower evolved in order to attract insects to pollinate it is interesting; it means that insects can see the color. It adds a question: does this aesthetic sense also exist in the lower forms? Why is it aesthetic? All kinds of interesting questions which the science knowledge only adds to the excitement, the mystery and the awe of a flower. It only adds. I don't understand how it subtracts.
Edit: You can watch and hear Feynman talk about this in one of the videos recorded of him. It's a real treasure that these are available to all on YouTube. Thank you to Zach Tatlock for finding the video.
Certainly my view of tea pots seems enriched by reading a bit about their drips. And now every time my kettle whistles, I can wonder if its Reynolds number puts it in the resonant or the turbulent regime…
Footnotes:
R. Henrywood & A. Agarwal, The aeroacoustics of a steam kettle
J. Keller, Teapot Effect
M. Reiner, The teapot effect… a problem
J. Vanden-Broeck & J. Keller Pouring flows