As a mathematician, it is rare that I get to speak to the media. My papers don’t always ‘catch the eye’. Yet one did. On 4 January 2023, a random Wednesday in an arbitrary year, my inbox blew up and my phone kept ringing. That day, a piece of my research with Frank Smith, a professor in the department of mathematics at UCL, was published. As a result of this article, 83 news stories appeared around the world. I even had the chance to speak to various radio presenters and a couple of TV reporters to boot. And in all honesty, I was surprised!
What did we do to receive such global interest, I hear you ask? Well, we studied skimming—how objects hit water and bounce back off. Much like the humble pastime we all know and love of stone skimming, also known as rock skimming, ‘ducks and drakes’, skipping (if you are American—the internet made sure we knew this), or ‘piffing a yonnie’ (no idea!? That’s on you, Australia).
Perhaps it was the time of year. A slow news day. A piece of new year’s ‘fluff’ at a time when nobody wants heavy news—‘a scientist redefining rock skimming?’—that’s the stuff people want!
I had expected it to all blow over quickly, but then I made a fatal error. At some point, I mentioned the joy of trying to skim potato-shaped stones. And, well, that was it. They got obsessed. In one small sentence, I soon became globally (but not widely) known as the scientist who suggested potato stones are better for skimming. Their spud-based, mish-mash of interest and enthusiasm boiled over. After that one story, more reporters came, eager to hear about the mysterious potatoes that skim so well.
I learned two lessons that day:
- Be careful what you say when the microphone is on you (lest you become forever known as the ‘expert’ of ‘potato skimming’—ouch), and
- Don’t take the internet too seriously (people are passionate about
skimming… skipping—sorry America).
While it was nice to receive some light-hearted positive recognition, we didn’t really set out to study potatoes… in reality, we were interested in something much more important.
Aircraft icing
All this potato waffle and online roasting began with the study of aircraft, specifically aircraft icing. In the upper reaches of the atmosphere where the air gets cold and clouds get thin, aircraft can meet ice crystals and supercooled droplets (unfrozen water drops that are colder than freezing). Relatively harmless on their own, when lots of these particulates hit an aircraft they can cause ice to form on the wings, engines and other vital components. Without adequate protection, this icing can become a significant hazard. Historically, icing events have led to disaster, with two well-known instances together claiming the lives of almost 300. To prevent future incidents, the aerospace community continues to work to understand the physics of this hazard and protect against it.
This problem was the essence of the motivation for our study. When ice and water particles impact warm aircraft surfaces, such as wings and engines, water layers form. Once formed, other ice crystals may follow up and hit these layers. So, what happens next? Does the ice crystal skim much like a rock on a lake or sink like a potato in a pan? In the case of aircraft icing, the skimming ice isn’t some nice flat rock but potentially all sorts of shapes and sizes. So, the natural questions to ask is what influence does the ice shape and mass have here?
If an ice crystal sinks into a water layer, freezing may ensue and, without protection, an ice block may form. If the crystal skims, we want to know where it goes and whether that changes how we protect an aircraft. So, we studied the relationship between ice crystal mass and shape to show how various types of ice crystal behave—it’s either sink or skim.
Mathematical modelling allows us to peer beyond the constraints of experimentation. Yes, experimental evidence is vital, but it is limited. If you want specific answers to specific questions, experiments are great. But if you are after something a little more general and want to unpick the underlying physics, this is where the beauty of maths reveals itself. With well-defined physics and a set of equations to hand, mathematical modelling enables us to evaluate and understand hard-to-measure problems and, in our case, inform the efficient and effective design of aircraft ice protection systems.
Making it mathematical
To model what happens when the ice meets water, we need some equations.
Modelling fluids is tricky. Most of fluid modelling starts with a classic set of equations (known as the Navier–Stokes equations) but these are a complex beast to solve. To make life easier, we can employ a classic applied maths technique called asymptotic reduction—which is really a fancy way of saying, we compared the big and small factors in our problem and kept what was most influential. To do this, we make some assumptions about different parts of the ice–water interaction, like the water layer being shallow (eg thin and long), the horizontal velocity of the ice being much greater than its vertical velocity (as you would expect when skimming!) and that the fluid has certain physical properties —then our problem simplifies! With some massaging, our set of equations is no longer quite so beastly. We now have a new set called the shallow water equations (included below for awe-inspiring effect):
\begin{align*} \frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}&=-\frac{\partial p}{\partial x},\\ \frac{\partial h}{\partial t}+\frac{\partial}{\partial x}(hu)&=0. \end{align*} If you are unfamiliar with this notation, don’t worry: broadly these two equations together describe how the water layer’s horizontal velocity, $u$, and height, $h$, relate to each other and vary in time and space. With that, we can model the water layer.
Now, what about the ice (which we will refer to as the ‘body’ from now on)? When the body first descends into the water’s surface, pressure $p(x,t)$ builds up underneath as the water resists the body’s entry. This pressure along the body produces a force, hence we can turn to Newton’s second law, the classic $F=ma$ (force equals mass times acceleration) to understand how the body reacts, lifting (changing its vertical position $y$) and rotating (changing its angle $\theta$) as follows:
\begin{align*} \int_{x_1}^{x_0}p(x,t)\,\mathrm{d}x&=m\frac{\mathrm{d}^2y}{\mathrm{d}t^2},\\ \int_{x_1}^{x_0}xp(x,t)\,\mathrm{d}x&=I\frac{\mathrm{d}^2\theta}{\mathrm{d}t^2}. \end{align*} Note that $m$ and $I$ (the body’s moment of inertia) here are multiplied by vertical acceleration ($\mathrm{d}^2y/\mathrm{d}t^2$) and the angular acceleration ($\mathrm{d}^2\theta/\mathrm{d}t^2$), just as in Newton’s second law.
Now let’s think through what happens while the body skims. Before entering the water, the body is ‘dry’. Then, as soon as the body first touches the water, only a single point is wet. Let’s call it $x_0$. As the body plunges further still into the water, more water contacts the body—as shown above right. Underneath the body then, there is a contact region (the ‘wet bit’) where the body meets water, which we denote as being between two points $x_0$ and $x_1$, the trailing and leading edges of this wet region, respectively.
A stone skimming the surface of a liquid layer. If $x_1$ reaches the body’s leading edge, we deem it to have sunk, while if it reaches the trailing edge, this is a skim.
As the contact region grows larger, the pressure between fluid and body increases also, pushing back against the body and potentially lifts it back out of the water. If the force is sufficient, the body skims! Yet, depending on how the body initially makes contact with the water, it may well sink! If the body is too heavy or descends too quickly, that the water will stretch past the end of the body and it will flood, claiming the body to the water’s murky depths.
What does this mean for the avid amateur skimmer?
Through our study we uncovered a relationship between a body’s shape and mass that determines its ability to skim. Greater mass can lead to a ‘super-elastic’ response (a most enticing name)—an almighty leap from the water, with the body leaving at a greater height than it entered. Have we broken physics? Did we find a way to overcome the pesky constraints of the conservation of energy? Fortunately, not!
Bodies with larger masses sink deeper into the water and for longer; this increases the pressure under the body and its contact time with the water. In doing so, the large horizontal velocity of the body (much like your thrown stone) is converted into vertical velocity. Hence the almighty leap. But, interestingly, if the same body becomes more curved—this effect is inhibited. The body rotates more and leaves the liquid layer faster. A shorter, sharper skim. Altogether then, curvature enables heavier bodies to skim that would otherwise sink if they were flatter. For our aircraft icing problem (and your holiday by the beach), this means a broader range of bodies may skim than expected with some possibly rather dramatic results.
The body might sink…
…or skim.
This graph shows the vertical position during a skimming motion of three bodies with different amounts of curvature ($c$):
All three bodies enter the water layer at the same height. The flat body ($c=0$) sinks and so the blue curve stops mid-skim. For the largest curvature case ($c=10$, the orange curve), the body undergoes a successful skim, descending into the liquid layer and rising out again, leaving the liquid layer at a lower height than it entered (this is due, in part, to how the body rotates!). The most interesting case here is $c=1$ (the green curve) in which the so-called super-elastic response is seen: the body is ejected from the liquid layer at a far greater height than it entered!
So, while rock skimming is fun—aircraft icing is a more significant application our findings. Thankfully, our paper is only part of the varied research on skimmers and not the single authority (though one Redditor did question whether we had any authority given ‘they’d never seen a scientist throw anything further than 5 feet…’—it’s the ones closest to the truth that hurt the most).
Still, skimming is as complex as it is fun. So, when you next stand by that body of water, glistening in the sun, ready to skim—you can leave your King Edwards, Maris Pipers and Russets behind. Pick up a weighty stone and see if you can get that almighty super-elastic leap. And when you do, remember, that the phenomenon of skimming goes far beyond the lake.
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