# Bernoulli and collapsing paper tents

This is the simplest Bernoulli demonstration I have seen though there is some subtlety to the explanation. It is also great because it is easy to get 100% class participation. I believe that the demonstration has been around for a while but thanks to Jean Hertzberg for telling me about it.

Equipment

1. A sheet of paper

Demonstration

1. Fold the paper in half along the long axis of the sheet to form a tent
2. Place the tent on a table and blow into one end.
3. The tent should collapse down starting at the end you blow through

Discussion

When you blow into the end of the tent the air jet from your mouth induces a flow of ambient air into the tent. Applying Bernoulli’s equation along a streamline from the stationary ambient into the tent (see the figure below) leads to

0=U22/2g + P2

Therefore, P2 must be negative (sub-atmospheric). Therefore, there is a pressure imbalance across the tent walls near the entrance and the tent collapses.

An index of all the demonstrations posted on this blog can be found here. Don’t forget to follow @nbkaye on twitter for updates to this blog. If you have a demonstration that you use in class that you would like to share on this blog please email me (nbkaye@clemson.edu). I also welcome comments (through the comments section or via email) on improving the demonstrations.

# Take home labs / demonstrations – background

One thing that is often hard to do in class is have full participation in a demonstration except for very simple demonstrations like blowing bubbles or the bendy straw momentum demonstration. It is also often hard to do more involved demonstrations due to a lack of time. One way around this is to have the students do the experiments themselves at home. I have recently started having take-home labs as part of my class (there is also a separate formal laboratory component to the course). The approach is very simple and I will post details about individual experiments later but wanted to post a bit of background and some reflections on my experience of the ups and downs of this approach.

Background

Ben Sill used to do this a lot when he taught Introductory Fluid Mechanics in my Department. The basic approach is really simple.

1. The students are given a task such as measuring the specific gravity of a couple of common fluids. I do not specify the fluids so they can use whatever they have at home
2. The students are required to complete the task using at least two different techniques of their own choosing. No guidance is given on how to do the measurements.
3. They are not allowed to use lab equipment except maybe a scale from our materials lab. Testing is to be done with what they can find around their home or can buy in a store for a few dollars.
4. They are to write a brief report (3 page max) outlining their measurement techniques, analysis, and results. They also have to compare their results from the different measurements and discuss any differences. This includes quantitative error analysis for projects later in the semester.

The assignments are given in groups of 3 to 4 students. I give out 4-5 a semester and I give them about 2-3 weeks to complete the project.

Benefits

1. The students have to think for themselves about how to do the tests.
2. They do research. I have had groups find research articles that they use in their measurements and analysis.
3. They relate the project to their work in this and other courses. I have had groups use construction materials testing techniques and particle dynamics equations as part of their labs.
4. They have to consider sources of error in their measurements and analysis. When they compare their two (or more) different values from their different tests they need to explain why they are different. For example, if you are trying to measure the exit velocity of water from a hose by filling a bucket then the result is very sensitive to your measurement of the hose diameter.
5. They have to work in groups. I require that they provide a photo of themselves doing the tests so that everyone is seen to participate.
6. They relate the analysis that they do in class to physical measurements using things they can find in the kitchen or buy at a grocery store.
7. They can get very creative. I had a group measure their submerged weight by going to a local hotel pool and having someone stand on a scale at the pool edge holding a rope with another group member in the pool holding the other end of the rope.

Problems

1. Some groups really struggle. They just have real difficulty working out how to do the tests and occasionally just end up lost and confused. I  allow groups to self-select so I can get the odd group with a few too many weaker, less motivated, or less organized students.
2. It is time consuming. Grading the reports takes time.
3. The students typically get the measurements and analysis right but have trouble quantifying their errors and uncertainties. I plan to spend more time on that in class in the future.
4. Some of the methods teams come up with don’t work because there is physics or fluid mechanics that they have not covered that means their analysis of their measurements is incorrect.
5. There needs to be some rotation of topics so that students don’t just get ideas form previous semester’s students. I plan to use 4-5 per semester and plan on building a library of around 10 labs to rotate in and out. I will also need to tweak the report requirements to keep things fresh.

Over all I would say that my early experience with this has been positive.  I plan on posting the instructions for each take home lab I use over the coming weeks along with some comments about how they did and did not work.

An index of all the demonstrations posted on this blog can be found here. Don’t forget to follow @nbkaye on twitter for updates to this blog. If you have a demonstration that you use in class that you would like to share on this blog please email me (nbkaye@clemson.edu). I also welcome comments (through the comments section or via email) on improving the demonstrations.

# Video of “Dimensional analysis, conservation of volume and the filling box model” demonstration

The “Dimensional analysis, conservation of volume and the filling box model” demonstration takes quite a while to run (say 10 minutes after setup) so a GIF of the video is too large to post here. The full 8 minute video from a test I ran can be found here. Below is a series of images from that test at various times showing the initial wavy interface which eventually settles down. In the test I ran I had trouble getting air bubbles out of the nozzle so the plume water comes out at a slight angle and from a very small area.

I have also included a GIF (full video here) of the constant head tank being turned on showing the water rising up and flowing over the weir wall.

An index of all the demonstrations posted on this blog can be found here. Don’t forget to follow @nbkaye on twitter for updates to this blog. If you have a demonstration that you use in class that you would like to share on this blog please email me (nbkaye@clemson.edu). I also welcome comments (through the comments section or via email) on improving the demonstrations.

# Dimensional analysis, conservation of volume, and the filling box model

The filling box model has been around for decades and is a simple model for the stratification that develops in enclosures when there is a localized source of buoyancy that forms a turbulent plume. There is a lot to the model but the prediction of the first front requires only knowing the flow rate in a plume and conservation of volume. The plume flow rate can be established from dimensional analysis after which the first front movement can be calculated from conservation of volume.

The filling box

The original filling box model (Baines, W. D. & Turner, J. S. Turbulent buoyant convection from a source in a confined region J. Fluid Mech., 1969, 37, 51-80) looks at a single round turbulent plume ascending or descending in an enclosure of constant cross sectional area. The following discussion focuses on dense plumes as they are easier to create in the lab, but the model works for buoyant plumes as well. When the plume reaches the base of the enclosure it spreads out and forms a dense layer. Over time more dense fluid is added to this layer from the plume and it thickens. Assuming that the ambient fluid in the enclosure is fairly quiescent then the density of the layer inhibits any mixing across the layer. Therefore, the only fluid entering the layer is from the plume. See the figure for a schematic of this flow.

Equipment

1. Fish tank filled with fresh water
2. Salt
3. Food coloring
4. Tubing with some mechanism for holding it steady at the top of the tank.
5. A method for delivering a constant flow rate of dense fluid into the tank (A constant head tank works but it may be hard to bring into the class room.)
6. A valve for turning the flow of dense water on and off.
7. Ruler
8. Stop watch

Image of the experimental setup with the constant head tank filled with dyed salt water, the tubing with valve to control the flow and the glass tank full of fresh water.

Below is an image of a simple constant head tank with a reservoir on the right, a pump to drive water to the left of the weir wall and an outlet at the base of the left hand side. The pump moves more fluid than is required for the experiment so that there is a return flow over the weir. Provided the height from the experiment to the top of the weir wall is large compared to the head over the weir, the total head is very close to constant. The head over the weir will only vary by a millimeter or two over the course of the experiment as the reservoir drains, the pump head increases and the pump flow rate decreases. This setup can be used for hours with the reservoir being topped up to make sure the pump intake remains flooded.

The constant head tank operating with the dyed salt water flowing over the weir wall back into the reservoir.

Demonstration

1. Mix up a dense salt solution in the constant head tank. The denser the better.
2. Fix the tubing to the outlet on the constant head tank and place the other end of the tubing at the water surface in the fish tank
3. Turn on the supply of dense water. A salt plume should form under the outlet that should fall to the base of the fish tank.
4. Measure the thickness of the dense salt layer as a function of time. The time interval will depend on the plume buoyancy flux and the tank dimensions. The larger the tank the larger the time step. The timing starts when the plume first touches the base of the fish tank.
5. Plot the layer depth as a function of time.
6. See analysis for later steps

Here is an image of the demonstration in progress. As you can see the interface can be a bit wavy early on but eventually settles down.

Plume volume flux dimensional analysis

For a fully turbulent plume the volume flux (Q) is independent of the Reynolds number and is only a function of the distance (h) from the source (provided the source is small) and the buoyancy flux of the plume. The buoyancy flux is the source volume flux multiplied by the reduced gravity of the source fluid

F=Q0g0

Where g’=g(ρap)/ρa, ρa is the ambient fluid density, ρp is the plume source fluid density, and the subscript ‘0’ means a source condition. Writing the dimensions of each of the parameters leads to

[h]=L,      [Q]=L3T-1,      and      [F]=L4T-3

There are three parameters, two independent dimensions, and, therefore, one non-dimensional group. This can be written as:

Π =QhaFb

Writing in terms of dimensions leads to

1= L3T-1LaL4bT-3b.

Writing out equations for the powers of L and T results in

0=3+a+4b      and      0=-1-3b

Solving and substituting back into the original expression for Π leads to

Q=CF1/3h5/3

where C is an unknown constant that can be determined experimentally.

Conservation of volume analysis

Returning to the filling box model we can draw a control volume around the dense layer and write conservation of volume

dV/dt+∑ Qout-∑Qin=0

where V is the volume of dense fluid in the lower dense layer. Denoting the tank cross sectional area as AT the volume of fluid in the dense layer is V=AT(H-h). There is no flow rate out of the dense layer and the flow rate in is the flow rate in the plume at the height of the interface between the dense layer and the ambient fluid above. Therefore, conservation of volume can be reduced to

dV/dt=-ATdh/dt= CF1/3h5/3.

Separating variables and integrating leads to

H-2/3-h-2/3=(2CF1/3/3AT)t

Plotting h-2/3 versus t should result in a relatively straight line.

The demonstration is nice for a number of reasons. It is a little more complex than a simple draining flow problem and you can throw in a little dimensional analysis for the plume and talk about data presentation with the final plot of h-2/3 versus t. That said, it is quite time consuming and can be a bit messy. When the plume is first turned on the dense lower layer can be very wavy. I usually use this in my graduate environmental fluid mechanics class.

An index of all the demonstrations posted on this blog can be found here. Don’t forget to follow @nbkaye on twitter for updates to this blog. If you have a demonstration that you use in class that you would like to share on this blog please email me (nbkaye@clemson.edu). I also welcome comments (through the comments section or via email) on improving the demonstrations.