Category Archives: Reblogged from E. Darelius & Team

About the influence of viscosity: The Reynolds number

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This blog post was written for Elin Darelius & team’s blog (link). Check it out if you aren’t already following it!

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I read a blog post by Clemens Spensberger over at  a scisnack.com couple of years ago, where he talks about how ice can flow like ketchup. The argument that he makes is that ketchup on your hotdog behaves in many ways similarly to glaciers on for example Greenland: If there is a layer of a certain thickness, it will start sliding off — both the ketchup off your hotdog and the glaciers off Greenland. After most of it has dripped to your shirt or in the ocean, a little bit still remains on the hotdog or the mountain. And so on.

What he is talking about, basically, are effects of viscosity. Water, for example, would behave very differently than ketchup or ice, if you imagine it poured on your hotdog or raining down on Greenland. But also ketchup would behave very differently from ice, if it was put on Greenland in the same quantities as the existing glaciers on Greenland, instead of on a hotdog as a model version of Greenland in a relatively small quantity. And if you used real ice to model the behaviour of Greenland glaciers on a desk, then you would quickly find out that the ice just slides off on a layer of melt water and behaves nothing like you imagined (ask me how I know…).

This shows that it is important to think about what role viscosity plays when you set up a model. And not only when you are thinking about ice — also effects of surface tension in water become very important if your model is small enough, whereas they are negligible for large scale flows in the ocean.

The effects of viscosity can be estimated using the Reynolds number Re. Re compares the effects of the velocity u of the flow, a length scale of an obstacle L, and the viscosity v: Re = uL/v.

Reynolds numbers can be used to separate different flow regimes: laminar flows for very low Reynolds numbers, nice vortex streets for Re > 90, and then flows with a stagnant backwater for high Reynold numbers.

Dependency of a flow field on the Reynolds number. Shown is the top view of a flow field. You see red obstacles, stream lines in blue (so any particle released at any point of a blue line would follow that line exactly, and in the direction shown by the arrow heads)

Dependency of a flow field on the Reynolds number. Shown is the top view of a flow field. You see red obstacles and blue stream lines (so any particle released at any point of a blue line would follow that line exactly, and in the direction shown by the arrow heads)

 

I have thought long and hard about what I could give as a good example for what I am talking about. And then I remembered that I did an experiment on vortex streets on a plate a while back.

Vortices created on a plate

Vortices created on a plate

If you start watching the movie below at min 1:28 (although watching before won’t hurt, either) you see me pulling a paint brush across the plate at different speeds. The slow ones don’t create vortex streets, instead they show a more laminar behaviour (as they should, according to theory).

https://vimeo.com/120239174

Vortex streets, like the one shown in the picture above, also exist in nature. However, scales are a lot larger there: See for example the picture below (Credit: Bob Cahalan, NASA GSFC, via Wikipedia)

Vortex street. Credit: Bob Cahalan, NASA GSFC, via Wikipedia

Vortex street. Credit: Bob Cahalan, NASA GSFC, via Wikipedia

While this is a very distinctive flow that exists at a specific range of Reynolds numbers, you see flows of all different kinds of Reynold numbers in the real world, too, and not only on my plate. Below, for example, the Reynolds number is higher and the flow downstream of the obstacle distinctly more turbulent than in a vortex street. It’s a little difficult to compare it to the drawing of streamlines above, though, because the standing waves disguise the flow.

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One way to manipulate the Reynolds number to achieve similarity between the real world and a model is to manipulate the viscosity. However that is not an easy task: if you wanted to scale down an ocean basin into a normal-sized tank, you would need fluids to replace the water that don’t even exist in nature in liquid form at reasonable temperatures.

All the more reason to use a large tank! :-)

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

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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.

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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?

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

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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.

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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! :-)

 

The ocean is very deep. It’s also very shallow. On the L/H aspect ratio and the size of the tank.

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When we come back from research cruises, one of the things that surprises people back home is how much time it takes to take measurements. And that’s for two reasons: Because the distances we have to travel to reach the area we are interested in are typically very large. And then because the ocean is also very deep.

People usually find it hard to imagine that it can easily take hours for an instrument, hanging on a wire from the ship, to go down all the way to the sea floor and then come back up to the ship again. A typical speed the winch is run at is 1 m/s. That means that for a typical ocean depth of 4 km, it takes 66 minutes for the instrument to go down, and then another hour to be brought back up to the ship. And then we haven’t even stopped the winch on the way up, which we usually do each time we want to take a water sample. So yes, the ocean is very deep!

Jens and Arnt recovering the CTD

Jens and Arnt recovering the CTD

And yet, it is not deep. At least not compared to its horizontal extent. The fastest crossing of the Atlantic, some 5000km, took something like 3 days and 10 hours. And according to a quick google search, a container ship typically takes 10 to 20 days these days. So there is a lot of water between continents! And it is really difficult to imagine how large the oceans really are.

One way to describe the extent of the ocean is to use the L/H aspect ratio. It is just the ratio between a typical length (L), and a typical depth (H). A typical east-west length in the Atlantic are our 5000 km used above, and a typical depth are 4 km. This gives us an aspect ratio L/H of 5000/4 which is 1250. That is actually a really large aspect ratio — the horizontal length scales are a lot wider than the vertical ones.

Now think about the kind of tank experiments we typically do. Here is a picture of a very simple Denmark Strait overflow experiment (more on that experiment here). You see the tank in the foreground, and a sketch of the same situation on the wall in the background. What you notice both for the experiment and the depiction is that in both cases the horizontal length scale is only about twice as much as the vertical one, leading to a L/H of 2.

Tank experiments at GFI

Tank experiments at GFI

This L/H of 2, however, is supposed to represent a situation that in the real world has a horizontal scale of maybe 1000 km and a vertical one of maybe 1 km, which leads to a L/H of 1000. So you see that the way we typically depict sections through the ocean is very distorted from what they would look like if they were geometrically similar, meaning that they had the same L/H ratio, which means that they could be transformed into the real world just by uniformly stretching or shrinking.

Below I have sketched a couple of duck ponds. The one on the left (with an aspect ratio L/H of 1) is geometrically shrunk below: Even though L and H become smaller, they do so at the same rate: their ratio stays the same. However going along the top row of duck ponds, the aspect ratio increases: While L stays the same, H shrinks. This means that the different ponds along the top of the picture are not geometrically similar. However, the one on the top right is geometrically similar to the one in the bottom right again (both have an L/H of 6). Does this make sense?

Sketch of different kinds of similarity

Sketch of different kinds of similarity

So in case you were wondering about why we need a tank that has 13 meters diameter — maybe now you see that it allows us to maintain geometric similarity a lot better than a smaller tank would, at least when we want to have water depths that are large enough that allow us to neglect surface tension effects and all that nasty stuff.

More on how Elin actually designed the experiments soon! :-)

Why we actually need a large tank — similarity requirements of a hydrodynamic model

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When talking about oceanographic tank experiments that are designed to show features of the real ocean, many people hope for tiny model oceans in a tank, analogous to the landscapes in model train sets. Except even tinier (and cuter), of course, because the ocean is still pretty big and needs to fit in the tank.

What people hardly ever consider, though, is that purely geometrical downscaling cannot work. Consider, for example, surface tension. Is that an important effect when looking at tides in the North Sea? Probably not. If your North Sea was scaled down to a 1 liter beaker, though, would you be able to see the concave surface? You bet. On the other hand, do you expect to see Meddies when running outflow experiments like this one? And even if you saw double diffusion happening in that experiment, would the scales be on scale to those of the real ocean? Obviously not. So clearly, there is a limit of scalability somewhere, and it is possible to determine where that limit is – with which parameters reality and a model behave similarly.

Similarity is achieved when the model conditions fulfill the three different types of similarity:

Geometrical similarity
Objects are called geometrically similar, if one object can be constructed from the other by uniformly scaling it (either shrinking or enlarging). In case of tank experiments, geometrical similarity has to be met for all parts of the experiment, i.e. the scaling factor from real structures/ships/basins/… to model structures/ships/basins/… has to be the same for all elements involved in a specific experiment. This also holds for other parameters like, for example, the elastic deformation of the model.

Kinematic similarity
Velocities are called similar if x, y and z velocity components in the model have the same ratio to each other as in the real application. This means that streamlines in the model and in the real case must be similar.

Dynamic similarity
If both geometrical similarity and kinematic similarity are given, dynamic similarity is achieved. This means that the ratio between different forces in the model is the same as the ratio between different scales in the real application. Forces that are of importance here are for example gravitational forces, surface forces, elastic forces, viscous forces and inertia forces.

Dimensionless numbers can be used to describe systems and check if the three similarities described above are met. In the case of the experiments we talk about here, the Froude number and the Reynolds number are the most important dimensionless numbers. We will talk about each of those individually in future posts, but in a nutshell:

The Froude number is the ratio between inertia and gravity. If model and real world application have the same Froude number, it is ensured that gravitational forces are correctly scaled.

The Reynolds number is the ratio between inertia and viscous forces. If model and real world application have the same Reynolds number, it is ensured that viscous forces are correctly scaled.

To obtain equality of Froude number and Reynolds number for a model with the scale 1:10, the kinematic viscosity of the fluid used to simulate water in the model has to be 3.5×10-8m2/s, several orders of magnitude less than that of water, which is on the order of 1×10-6m2/s.

There are a couple of other dimensionless numbers that can be relevant in other contexts than the kind of tank experiments we are doing here, like for example the Mach number (Ratio between inertia and elastic fluid forces; in our case not very important because the elasticity of water is very small) or the Weber number (the ration between inertia and surface tension forces). In hydrodynamic modeling in shipbuilding, the inclusion of cavitation is also important: The production and immediate destruction of small bubbles when water is subjected to rapid pressure changes, like for example at the propeller of a ship.

It is often impossible to achieve similarity in the strict sense in a model experiment. The further away from similarity the model is relative to the real worlds, the more difficult model results are to interpret with respect to what can be expected in the real world, and the more caution is needed when similar behavior is assumed despite the conditions for it not being met.

This is however not a problem: Tank experiments are still a great way of gaining insights into the physics of the ocean. One just has to design an experiment specifically for the one process one wants to observe, and keep in mind the limitations of each experimental setup as to not draw conclusions about other processes that might not be adequately represented.

So much for today — we will talk about some of the dimensionless numbers mentioned in this post over the next weeks, but I have tried to come up with good examples and keep the theory to a minimum! :-)

Of swirls, eddies and fairy dust

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Similarly to last Friday’s Kelvin-Helmholtz instabilities, observing swirls and eddies made from green fairy dust is not really what we are in Grenoble for. But are they pretty!

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And it is actually very interesting to observe the formation of eddies. If you look at the picture above and focus on the sharp edge “downstream” of the canyon, you see that there are some small instabilities forming there that detach as eddies. And in the picture below you see that there are more, and larger ones, a little while later.

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And below you see how they have grown into larger eddies.

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And in the gif below you see that the structures of those eddies inside the canyon are actually coherent throughout the uppermost three layers (which are the only ones in which the shelf is lit, for the lower three layers we can just observe what’s going on deeper than the depth of the canyon). So a nice and barotropic flow, just like we had hoped!

eddies_scan

Don’t those eddies look just like phytoplankton patches observed from a satellite?

Why are we rotating a huge tent with our tank?

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When watching the images or movies that show the rotating tank from the outside, you may have been wondering about why the whole structure — tank, office above the tank, everything — is inside a rotating tent, which itself is inside a large room.

Tank, office, everything in fast rotation

Tank, office, everything in fast rotation

Remember the last time you were on a merry-go-round? Remember the wind on your face and in your hair? Yes, that’s exactly what we don’t want. Neither for us sitting in the office, nor, more importantly, for our tank.

If there wasn’t a tent around the whole structure, rotating with it, we would always have “wind” on the tank’s free water surface, because the water would be in motion relative to the room in which the tank is located. The friction between air and water would then cause wind-driven surface currents, which might disturb our experiments. Now, however, the air inside the tent is rotating with the tank, hence there is no motion of the air relative to the water, no wind, no wind-driven currents, perfect conditions for our experiments!

And believe me, when you step out of the tent on your way off the rotating platform, or from the stationary room onto the platform on your way in, you definitely feel the wind!

Totally not the focus of our experiments, but so beautiful! Kelvin-Helmholtz instabilities

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This is really not the focus of our experiments here in Grenoble, but they are too nice not to show: Kelvin-Helmholtz instabilities!

Sheer instabilities in the flow

Sheer instabilities in the flow

They showed up really nicely in our first experiment, when we only had neutrally-buoyant particles in our source water (and not yet in the ambient water). The water that shows up as the lighter green here is thus water that originally came from the source (and at this point has recirculated out of the canyon again).

Sheer instabilities in the flow

Sheer instabilities in the flow

I get so fascinated with this kind of things. How can anyone possibly not be interested in fluid dynamics? :-)

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Watch the movie below to see them in motion! The scanning works as explained here.

 

About neutrally buoyant particles, popcorn, and more bubbles

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When you see all our pretty images of currents and swirling eddies and everything, what you actually see are the neutrally buoyant particles that get lit by the laser in a thin sheet of light. And those particles move around with the water, but in order to show the exact movement of the water and not something they are doing themselves, they need to be of the exact same density as the water, or neutrally buoyant.

But have you ever tried creating something that just stays at the same depth in water and does neither sink to the bottom or float up to the surface? I have, and I can tell you: It is not easy! In fact, I have never managed to do something like that, unless there was a very strong stratification, a very dense lower layer in which stuff would float that fell through a less dense upper layer. And in a non-stratified fluid even the smallest density differences will make particles sink or float up, since they are almost neutral everywhere… One really needs stratification to have them float nicely at the same depth for extended periods of time.

But luckily, here in Grenoble, they know how to do this right! And it’s apparently almost like making popcorn.

You take tiny beads and heat them up so they expand. The beads are made from some plastic like styrofoam or similar, so there are lots of tiny tiny air bubbles inside. The more you heat them up, the more they expand and the lower the density of the beads gets.

But! That doesn’t mean that they all end up having the same density, so you need to sort them by density! This sounds like a very painful process which we luckily didn’t have to witness, since Samuel and Thomas had lots of particles ready before we arrived.

Once the particles are sorted by density, one “only” needs to pick the correct ones for a specific purpose. Since freshwater and salt water have different densities, they also require different densities in their neutrally buoyant particles, if those are to really be neutrally buoyant…

Below you see Elin mixing some of those particles with water from the tank so we can observe how long they actually stay suspended and when they start to settle to either the top or the bottom…

Elin experimenting with the buoyancy of our particles

Elin experimenting with the buoyancy of our particles

Turns out that they are actually very close to the density of the water in the tank, so we can do the next experiment as soon as the disturbances from a previous one have settled down and don’t have to go into the tank in between experiments to stir up particles and then wait for the tank to reach solid body rotation again. This only needs to be done in the mornings, and below you see Samuel sweeping the tank to stir up particles:

Samuel sweeping particles from the topography that sank to the bottom over night

Samuel sweeping particles from the topography that sank to the bottom over night

Also note how you now see lots of reflections on the water surface that you didn’t see before? That’s for two reasons: one is because in that picture there are surface waves in the tank due to all the stirring and they reflect light in more interesting pattern than a flat surface does. And the other reason is that now the tank is actually lit — while we run experiments, the whole room is actually dark except for the lasers, some flashing warning signs and emergency exit signs close to the doors and some small lamps in our “office” up above the rotating tank.

But now to the “more bubbles” part of the title: Do you see the dark stripes in the green laser sheet below? That’s because there are air bubbles on the mirror which is used to reflect the laser into the exact position for the laser sheet. Samuel is sweeping them away, but they keep coming back, nasty little things…

Samuel sweeping particles from the topography that sank to the bottom over night

Samuel sweeping particles from the topography that sank to the bottom over night

I actually just heard about experiments with a different kind of neutrally buoyant particles the other day, using algae instead of plastic. I find this super intriguing and will keep you posted as I find out more about it!

No, the edge of our tank is not “the equator”

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A very common idea of what goes on in our tank is that we have a tiny Antarctica in the center and that the edge of our tank then represents the equator. We are rotating in Southern Hemisphere direction, clockwise when looked from above the pole. And when looking at the Earth that way, where the Earth seems to end is at the equator. It makes sense to intuitively assume that the edge of the tank then also represents “the end of the world”, i.e. the equator.

But then it is confusing that our Antarctica is so big relative to a whole hemisphere and that we don’t have any other continents in our tank. And it’s confusing because the idea that the edge of our tank represents the equator is actually wrong.

Let’s look at the Coriolis parameter. The Coriolis parameter is defined as f=2 ω sin(φ). ω is the rotation of the Earth, which is  so constant everywhere. φ, however, is the latitude. So φ is 90 at the North Pole, -90 at the South Pole, and 0 at the equator. And this is where the problem arises: The Coriolis parameter depends on the latitude, which means that it changes with latitude! From being highest at the poles (technically: Being highest at the North Pole and the same value but opposite sign at the South Pole) to being zero at the equator. And with the latitude φ obviously changes also sin(φ), and f with both of those.
Sketch of f as a function of latitude

Sketch of f as a function of latitude

In our tank, however, we don’t have a changing latitude, it’s constant everywhere. You can imagine it a little like sketched below: As if the top of the Earth was cut off at any latitude we chose, and then we just put our tank on the new flat surface on top of the Earth: the latitude is constant everywhere (at least everywhere on the shaded surface where we are putting our tank)!

How we simulate f in a tank

How we simulate f in a tank

Since the latitude is constant throughout our tank, so is the Coriolis parameter. That means that if we want to simulate Antarctica, we will match our f to match the real Antarctica’s, except scaled to match our tank. And if we wanted to simulate the Mediterranean*, we would match our f to the one corresponding the Mediterranean’s latitude.

This means that we actually cannot simulate anything in our tank that requires a change in f, much less half the Earth! So currently no equator in our tank (although that would be so much easier: No need to rotate anything since f=0 there! :-)

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*which, in contrast to my sketch above, is well in the Northern Hemisphere and not at the equator, but I am currently sitting at Lisbon Airport and this sketch is the best I can do right now… Hope you appreciate the dedication to blogging ;-)