## Similarity requirements of a hydrodynamic model

Why downscaling only works down to a certain limit

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. I’ve talked about surface tension a lot recently. 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.

I’ve noticed that people start glazing over when I talk about this, so in the future, instead of talking about it, I am going to refer them to this post. So here we go:

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 presented on my blog, the Froude number and the Reynolds number are the most important dimensionless numbers.

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.

## Hydraulic jump II

More movies of my kitchen sink.

I am really fascinated by the hydraulic jumps in my kitchen sink. I can’t believe I haven’t used this before when I was teaching! Yes, movies of rivers and rapids are always really impressive, too, but how cool is it to be able to observe hydraulic jumps in your own sink? Let me remind you:

So this is what happens when the water jet hits the (more or less) level bottom of the sink. But what would happen if it instead hit a slope?

Now, if I wasn’t working a full-time job, or if that job wasn’t completely unrelated to anything to do with hydraulic jumps, I would now proudly present movies of all kinds of hydraulic jumps on sloped surfaces. As it is, I can tell you that I have tons of ideas of where to go to make really nice movies, but for now this is all I can offer:

Yes, that is a chopping board in a sink. It shows really nicely how the hydraulic jump occurs closer to the point of impact of the jet as you go uphill (because the water slows down faster going in that direction than going downhill) and again how the radius depends on the flow speed of the jet. Stay tuned for a more elaborate post on this!

## Hydraulic jumps

Water changing its velocity from above to below the critical velocity.

Recently in beautiful Wetzlar: The river Lahn flows through the city below the medieval cathedral at sunset. And I’m showing you this because we can observe a hydraulic jump!

A hydraulic jump occurs when water that was flowing faster than the critical speed suddenly slows down to below the critical speed. Some of its kinetic energy is converted to potential energy (see the higher surface levels of the turbulent part of the fluid {except in this example the water is flowing down a steep slope, so the higher levels are a bit tricky to observe}) and a lot of energy is lost to turbulence. A very nice example can be seen here:

As the water moves away from where the jet hits the sink, it slows down. Can you spot the hydraulic jump? Isn’t it cool to watch how it is pushed away if the flow rate is higher, and how it comes back again when the tap is slowly closed?

P.S.: Yes, I’m being very vague about what that critical speed might be. Stay tuned for a post on that, I’m working on it! Just had to share the Lahn movie :-)