**It has been a difficult summer for me.**

**Putting on the Royal Society Summer Science Exhibition** was utterly exhausting, and even two months on, I have not been able to catch up on all the extra days and hours I worked. And I fell behind on every other project on which I am working.

**So every day** as I enter work I have to catch my breath, staunch my sense of panic, and force myself to stay calm as I begin another day of struggling through tiredness to avoid failure on all the projects on which I am way behind.

**But earlier this week my colleague caught me staring** at the water flowing down the sink in the kitchenette where we prepare tea.

**I was staring at a phenomenon I have been fascinated by** since childhood – the way water falling from the tap onto the bottom of the kitchen sink forms a smooth flat circle for a few centimetres around where the water lands – and then forms a ‘wavy wall’ around this circle.

**My colleague said to me**: “It’s great isn’t it. It’s called a *hydraulic jump*“. Learning that this phenomenon had a name lifted my spirits enormously and made me more curious about what was going on.

**So today** (Saturday) I have wantonly avoided catching up with my weekly tide of failure, stupidly neglected to pack for my week long conference in Belfast starting tomorrow, and spent the afternoon playing at the kitchen sink. I have experienced transitory happiness.

**Hydraulic jump**

**Naming a phenomenon** is stage#1 of the process of understanding it. Knowing this name allowed me to read a number of – frankly confusing – articles on the web.

**But after reading and playing for a while **I think I am now beginning to understand what makes the circle form. There are two parts to my understanding:

**The first insight** arises from comparing:

- the flow speed of the water with,
- the speed at which waves travel on the surface of the water.

**Inside the circle, the flow is faster** than the speed at which waves can travel in the water. So surface disturbances are swept outwards – the waves are not fast enough to travel ‘upstream’, back towards the centre.

**As one moves further away from the centre**, the flow speed falls and at the edge of the circle, the flow speed is just equal to the speed of water waves. So water waves travelling back towards the centre of the circle appear stationary – this what makes the circle appear to be ‘fixed’ even though it is a dynamically created structure.

**Outside the circle**, the flow slows sufficiently that water waves can travel upstream (towards the middle) but they can never travel into ‘the circle’. (*There is actually a scientific paper in which this circle is used as an analogy to the ‘Event Horizon’ in a putative ‘White hole’!)*

**The second insight**, arises from considering turbulence.

**Once waves can travel** in both directions in the water, turbulence builds up which slows the speed of the flowing water dramatically.

**So in the steady state**, the depth of the water builds up suddenly and the ratio of the depth of water inside the circle to the depth outside the circle is simply the ratio of the speeds of water flow just outside and just inside the circle.

**So if the speed of flow is** 10 times slower outside the circle, then the water will be be 10 times deeper outside the circle.

**In the picture above** and the video below, you can see the very strikingly different nature of the liquid surfaces. Shallow and perfectly smooth within the circle, and deeper and turbulent outside the circle.

**Experiments**

**I began playing **by finding a better surface than the bottom of a sink. I used an upside down baking tray and adjusted it to be as level as I could manage.

**Not knowing what to do,** I began by measuring the diameter of the circle formed for different flow rates:

- I measured the diameter roughly with a ruler
- I measured the flow rate by timing how long it took to fill a measuring jug which I weighed before and after filling.

**This produced a pleasing graph**, but no real insight. An increased flow rate meant made the circle larger because it took more time (and distance) for the flowing water to slow down to the speed of water waves.

**Looking at the algebra**, I realised I really needed to know the speed of the water and depth of the water. But how could I measure these things?

**I tried estimating the speed of the water** by injecting food colouring into the flow and making a movie using the slow-motion mode of my iPhone camera.

**Knowing the circle was about 8.8 cm in diameter, **this allowed me to estimate the speed of flow as roughly 1.5 ± 0.5 metres per second in the centre zone. However I couldn’t think how to estimate the thickness (height) of the flowing layer.

**By sticking a needle in I could** see that it was much less than 1 mm and appeared to be less than a tenth of the thickness of the water outside the circle. But I couldn’t make any meaningful measurements.

**Then I realised** that I could I estimate the speed of the water in a different way. If I placed a needle in the moving water, it produced an angular ‘shock wave’.

**This is similar to way an aeroplane **travelling faster than the speed of sound in air produces a ‘sonic boom’.

**For an aeroplane**, the angle of the shock wave with respect to the direction of motion is related to the ratio of the speed of the plane to the speed of the sound.
**For our flowing water**, the angle of the shock wave with respect to the direction of motion is related to the ratio of the speed of the water to the speed of the water waves.

**Unfortunately** the angle changes very rapidly as the ratio of flow speed to wave speed approaches unity and I found this phenomenon difficult to capture photographically.

**But as the photographs below show**, I could convince myself qualitatively that the angle was opening out as I placed the obstacle nearer the edge of the circle.

Observations of the shock wave formed when an obstruction is placed in the water flow. The top row of photographs shows the effect of moving the obstruction from near the centre to near the edge of the circle. The bottom row of photographs are the same as the top row but I have added dotted lines to show how the shock angle opens up nearer the edge of the circle.

**Summary**

**My work remains undone.**
**I still have to pack** in order to leave for the conference at 8:30 a.m. on Sunday morning: less than 8 hours away as I finish this. (*Perhaps I will have a chance to complete some tasks at the airport or on Sunday evening*?)
**I have understood a little something** about one more little thing in this beautiful world, and that has lifted my spirits. For now at least.

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