Who is faster, the currents or the waves? The Froude number

A very convenient way to describe a flow system is by looking at its Froude number. The Froude number gives the ratio between the speed a fluid is moving at, and the phase velocity of waves travelling on that fluid. And if we want to represent some real world situation at a smaller scale in a tank, we need to have the same Froude numbers in the same regions of the flow.

For a very strong example of where a Froude number helps you to describe a flow, look at the picture below: We use a hose to fill a tank. The water shoots away from the point of impact, flowing so much faster than waves can travel that the surface there is flat. This means that the Froude number, defined as flow velocity devided by phase velocity, is larger than 1 close to the point of impact.

At some point away from the point of impact, you see the flow changing quite drastically: the water level is a lot higher all of a sudden, and you see waves and other disturbances on it. This is where the phase velocity of waves becomes faster than the flow velocity, so disturbances don’t just get flushed away with the flow, but can actually exist and propagate whichever way they want. That’s where the Froude number changes from larger than 1 to smaller than 1, in what is called a hydraulic jump. This line is marked in red below, where waves are trapped and you see a marked jump in surface height. Do you see how useful the Froude number is to describe the two regimes on either side of the hydraulic jump?

Obviously, this is a very extreme example. But you also see them out in nature everywhere. Can you spot some in the picture below?

But still, all those examples are a little more drastic than what we would imagine is happening in the ocean. But there is one little detail that we didn’t talk about yet: Until now we have looked at Froude numbers and waves at the surface of whatever water we looked at. But the same thing can also happen inside the water, if there is a density stratification and we look at waves on the interface between water of different densities. Waves running on a density interface, however, move much more slowly than those on a free surface. If you are interested, you can have a look at that phenomenon here. But with waves running a lot slower, it’s easy to imagine that there are places in the ocean where the currents are actually moving faster than the waves on a density interface, isn’t it?

For an example of the explanatory power of the Froude number, you see a tank experiment we did a couple of years ago with Rolf Käse and Martin Vogt (link). There is actually a little too much going on in that tank for our purposes right now, but the ridge on the right can be interpreted as, for example, the Greenland-Scotland-Ridge, making the blue reservoir the deep waters of the Nordic Seas, and the blue water spilling over the ridge into the clear water the Denmark Strait Overflow. And in the tank you see that there is a laminar flow directly on top of the ridge and a little way down. And then, all of a sudden, the overflow plume starts mixing with the surrounding water in a turbulent flow. And the point in between those is the hydraulic jump, where the Froude number changes from below 1 to above 1.

Nifty thing, this Froude number, isn’t it? And I hope you’ll start spotting hydraulic jumps every time you do the dishes or wash your hands now! 🙂

All pictures in this post are taken from my blog “Adventures in Oceanography and Teaching“. Check it out if you like this kind of stuff — I do! 🙂

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