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! :-)
Joe Buchanan says:
I am really enjoying this blog. I am trying to understand hydraulic jumps. Can you explain why faster flowing water slows down? Friction? ( I thought water flows had negligible friction). Collision with slow water?
But how is this maintained? And how does fast flowing water entrain slower water? Is turbulence required? And finally, are hydraulic jumps really any different from standing waves? Do the former just have enough energy to break upstream?
Joe Buchanan says:
Thanks Mirjam, that makes sense and the billiard ball analogy is a good one. I think I was thinking that if I have a large number of billiard balls, and I am shooting at a group of stationery balls at the other end of the table, then eventually I would be able to knock all the stationery balls out of the way and create a path that my fast moving balls could move through without collisions. So in a river I would expect to have a fast current down the middle of the river and slow or still pools on either side. But instead rivers at normal flow tend to speed up (where the gradient is steeper, forming rapids) then slow down (at a hydraulic jump or the pools between rapids).
Swollen or flooded rivers often do move at fairly constant speed – the speed is similar bank to bank and for long stretches of river. I have also seen this in a kind of “river without banks”. I was kayaking across Cook Strait (between the North and South Islands of New Zealand). It was a still day and the sea was flat and glassy. But Cook Strait is funnel shaped in three dimensions – there is a wide deep area to the south-east, and a narrow shallow area to the north-west. This makes for some impressive tidal currents. What was strange was that the tidal currents formed a distinct “river” in the sea. In a few kayak lengths I went from flat, glassy, apparently still water into a river of fast moving (at least 5 knots) water with large (1-2 metre) standing waves). The “river” was maybe 100 metres wide, and after several minutes of hard paddling I crossed the “river” to the other side – more still glassy water. I don’t know how far this current extended, probably 10s of kilometres.
So can I try and answer my own question: If there is sufficient energy – in a flood state river or a tidal current (the one I am describing must have been several thousand cumecs) there is enough energy for most of the water to gain speed and maintain it. In a normal river turbulent interactions with slow moving water in eddies and pools transfers energy from fast to slow water decreasing the speed of the main current. Does this make sense?
But now I am wondering about the interactions between the fast and slow water in a normal river. Even if the velocity of the water at the steep part of a rapid is more or less linear – directed straight downstream, the water will slow down and push sideways into an eddy or become turbulent at the eddy-line. It seems almost non-newtonian for the linear water to start pushing or swirling towards still water at the sides of the river – as if the billiard balls not only hit the stationary balls directly in their way, but curve towards balls lying to the side and entrap them. This would make sense if there was friction between the fast moving and stationery balls, but I am used to thinking of water as having very little friction. So why does waterin a current apparently curve to the sides?
Thanks, I love your blog.
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