We’ve been talking about stream lines a lot recently (see for example the flow around a paddle or flow around other stuff). I’ve always heard stories about a neat way of visualizing stream lines that I wanted to show on my blog. So I set out to try it, but it just never worked exactly the way I had imagined it should. Anyway, here you go:
We take paper towels and cut an “obstacle” in it. In this case, it’s a drop-shape. The paper towel is set up such that one end is dunked in water, and that once the water has been sucked up a little, it neatly flows down a slope through the towel. In the picture below you see that the water just came over the edge of the cutting boards.
So once a flow has established (and only then, because I wanted to go for steady state stream lines, not some stuff that happens while things are still adjusting), I started dotting dye in to trace the flow:
As you can see, each dot leaves a streak. In this case, though, the streaks are not nearly clear enough for me, so I decided to “recharge” a little further downstream (making sure I put the dye exactly on one of the stream lines, obviously).
And voila! The flow really goes around the object similarly to what we would have imagined. And this is what the finished drop-shaped obstacle looks like:
stream lines on paper towel
As you see, next time it’s important to make sure there is more paper towel left downstream of the obstacle. We already get interference from the bottom edge of the paper towel where the flow is interrupted.
It’s also important to figure out what kinds of pens work: The picture below is from a test I did at my parents’ which worked a lot better than the pens I tried above.
stream lines on paper towel
And finally I am not sure how the embossed pattern in the paper towel influences the flow. So maybe I should try and find something with either a smaller pattern or something more regular. Plenty to do still!
So, all in all: Interesting visualization which I am definitely going to try again at some point, but there are still a couple of kinks I need to find fixes for!
My sister and I did a little sight-seeing tour in Hamburg the other day, and one of the most fascinating things I saw was — a diving duck. Now, that is not a reflection of how exciting the rest of Hamburg is, but if you don’t see it after watching the movie below, when you read the upcoming blog post on Friday, you’ll understand why this was so exciting to watch :-)
Recently, someone at my university told me about a case of experiments connected to aerodynamics* that they occasionally use for demonstrations and outreach. Obviously, I asked if I might possibly borrow the case, and fast forward: my dad and I spent a whole weekend playing.
I’m gonna go through all the experiments over the next couple of posts, so let’s get started!
The first experiment has a slight ring of the balls balancing on water jets. I’m a little torn on which one I like better. The experiment below looks a little more like magic, because the air jet is invisible. But the balls are balancing on water jets. Water! Tough choice!
Ball balancing on air jet
So this is what happens: The ball sits on the edge of the jet. The jet speeds up where it flows around the ball, and according to Bernoulli the pressure sinks and the ball is being pushed into the jet by the air pressure from outside the jet.
In the movie below you see how the ball can balance quite stably if left alone in the position it finds for itself, and how it reacts to the air flow being disturbed.
A common problem in hydrodynamics is to distinguish between all the different kinds of lines that characterize a flow field: Stream lines, streak lines, path lines, time lines, and probably more that I can’t think of right now.
A common way to think of streak lines is that they are similar to hairs caught in the flow of a blow dryer. So when I saw these long grassy things caught in a flow recently, I thought they would be a nice visualization of streak lines.
Algae showing streak lines in the water
But when you look at them moving, you realize that they are not actually showing streak lines. Streak lines would be visualized if, at the root of each of those blades of grass (or whatever they are, I’m not a biologist), dye was dispersed. The dye streak would be exactly showing the streak line. But looking at the grass move, you see that it is sometimes being jerked one way or another, when the direction of the flow changed and the blade is pulled in the new direction of the flow, even though the downstream end might still be caught up in some old flow.
So yes, there are points in time when a streak line is visualized by hairs in the air or grass in the river, but there are also times when they are not. Right?
I’ve promised a long time ago to write a post on vorticity (Hallo Geli! :-)). So here it comes!
Vorticity is one of the concepts in oceanography that is often taught via its mathematical formulation, and which is therefore pretty difficult to grasp for those of us with less mathematical training. But it’s also a concept that you can have an intuitive grasp of, and I’ll try to show you how.
The easiest way to imagine what “vorticity” is, is to think of a little float in a flow. In a vorticity-free flow, that little float will always keep its orientation (see below). However if there is a shear in the flow, i.e. the flow field carries vorticity, it will start to turn.
Flow fields without vorticity (top) and with vorticity (bottom).
This even holds true for vortices: There are vorticity-free vortices as well as those that carry vorticity (as the name “vortex” would suggest).
Vorticity-laden and vorticity-free vortex. In the left plot, angular velocity of all particles is the same. In the right plot, angular velocity increases the closer you get to the center of the vortex.
If you think back to the discussion on a tank spinning up to reach solid body rotation, you might recognize that only the vortex with vorticity moves like a solid body. To me, a solid body is basically a fluid with so much friction in it, that molecules cannot change their position relative to each other. And that serves as my memory hook for one condition for the formation of vorticity – the flows must have viscous forces and friction in it.
This sounds very theoretical, but there are a lot of instances where you can spot vorticity in real life, for example twigs caught up twirling in eddies at the edge of streams are clearly moving in a vorticity-filled environment. Below, for example, the stream is clearly not vorticity-free.
That was quite a teaser on Wednesday, wasn’t it? I said I had the solution to any hydrodynamics problem you might want to illustrate. So here we go:
I recently had the privilege to be given a private demonstration of the “Elbe” flow solver, which is being developed at Hamburg University of Technology. Elbe allows for near real time simulation of non-linear flows, and can be run in an interactive mode.
Look at the Karman vortex street below (their movie, not mine!) – doesn’t it remind you of the vortex street on a plate?
Now. How can we use such an awesome tool in teaching?
There are a couple of scenarios I could imagine.
1) Re-create flow fields.
This is mainly to help students get “a feel” for how a flow reacts to obstacles.
Provide students with a picture of a current field and ask them to recreate it as closely as possible. This is not about creating the exact same field, but about recognizing characteristics of a flow field and what might have caused them. Examples could include a Karman vortex street or a Kelvin Helmholtz instability.
Possible sketches of A) Karman vortex street and B) Kelvin Helmholtz instabilities as examples for flow fields that students could be asked to recreate using Elbe.
In the above examples, students need to recognize, for example, that while a vortex street can be formed in a single-phase flow, a Kelvin Helmholtz instability typically forms on the boundary of layers of different densities in a shear flow (but could also form in a single continuous fluid), and recreate this in the model.
2) Visualize hydrodynamic concepts.
Here we would name a concept and ask students to set up a flow field that visualizes it. They might submit an annotated snapshot, for example. Possible examples are
– difference between stream lines, path lines and streak lines
– hydrodynamic paradox
– dead water
Hydrodynamic paradox. Moored ships are pulled towards each other because the flow is faster between them than upstream of them. (yes, the current in the picture is coming from the left, yet the ships are drawn as if the current was coming from the right. Shit happens.)
3) Test engineering applications.
Here we could imagine giving students different shapes and asking them to find their optimal position in a flow field, for example the pitch of a given wing profile to maximize lift, or the relative placement of a ship’s hull and a submerged ball for maximum canceling of waves.
4) Understanding of limitations of model and/or theory.
In some cases, students might be able to find optimal solutions from theory. In those cases it might be interesting to have them model those solutions and compare results with theoretical values. Can they come up with reasons why the modeled answers are likely different from the theoretical ones?
So far, so good. But how do we make sure that students don’t spend an insane amount of time fiddling with the nitty gritty details of the model, but focus on understanding hydrodynamics?
Combination of individual and group work
One idea might be to have students work individually on defining the important parameters (for example one- versus two-phase flow, obstacle at fixed position or moving, shape of obstacle) and then have them work in groups on putting those parameters into the model. If we were to grade this, we could give individual grades for individual answers to the first part, and then add a group grade in form of bonus points for a good model.
Model as a tool rather than the ultimate goal
Another idea would be to let them use the model as a tool rather than as the final application. As in students could be allowed to play with the model in order to, for example, figure out an approximate shape of an obstacle, and then they sketch their solution and annotate (e.g. “The longer X, the less turbulent region Y”). This would let them experience and explore hydrodynamics.
Whether or not a concept has been visualized well can be judged by the instructor, or it can become a learning activity in itself, for example as peer-review. Figuring out whether a visualization is correct or how it could be improved supports a deeper understanding of the concept as well as all kinds of interpersonal skills. In order to keep this interesting for students, several concepts could be visualized by different students and it can be made sure that the one students work on themselves is not the same as the one they will review later.
I am really excited to really start developing ideas on how to use this model in teaching. How would you use it?
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.
Mediterranean outflow. Mediterranean on the left, Atlantic Ocean on the right. The warm and salty water of the Mediterranean Outflow is dyed red.
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:
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.
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.
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.
The experiment presented in this post was first proposed by Marsigli in 1681. It illustrates how, despite the absence of a difference in the surface height of two fluids, currents can be driven by the density difference between the fluids. A really nice article by Soffientino and Pilson (2005) on the importance of the Bosporus Strait in oceanography describes the conception of the experiment and includes original drawings.
The way we conduct the experiment, we connect two similar tanks with pipes at the top and bottom, but initially close off the pipes to prevent exchange between tanks. One tank is filled with fresh water, the other one with salt water which is dyed pink. At a time zero we open the pipes and watch what happens.
Two tanks, one with clear freshwater and one with pink salt water, before the connection between them has been opened.
As was to be expected, a circulation develops in which the dense salt water flows through the lower pipe into the fresh water tank, compensated by freshwater flowing the opposite way in the upper pipe.
The two tanks equilibrating.
We measure the height of the interface between the pink and the clear water in both tanks over time, and watch as it eventually stops changing and equilibrates.
The two tanks in equilibrium.
Usually this experiment is all about density driven flows, as are the exercises and questions we ask connected to it. But humor me in preparation of a future post: Comparing the height of the two pink volumes and the two clear volumes we find that they do not add up to the original volumes of the pink and clear tanks – the pink volume has increased and the clear volume decreased.