Spin up, boundary layers, and tracking tea leaves

Background:

Boundary layers play an important role in many fluid mechanics applications including drag, lift, and flow in conduits. This demonstration illustrates the role of boundary layers as part of the classic spin-up problem. The demonstration is a cheap and easy version of one written up by Nicholas Rott in the book ‘Experiments in fluid mechanics’ (out of print but worth getting a second hand copy of). The demonstration uses tea leaves to visualize the secondary vortex that forms during spin-up. See here for more on tea cup fluid dynamics.

Equipment:

  • Turntable (Lazy Susan)
  • Bag of tea
  • Scissors
  • Half-filled glass of water
  • Tape (to secure the glass to the turntable)

Procedure:

  1. Place the tea bag in a glass of hot water to wet the tea leaves.
  2. Fasten the half-filled glass of water to the turntable with tape.
  3. Using the scissors, cut open the used tea bag and dump roughly half of the tea leaves into the glass of water fastened to the turntable.
  4. Spin the turntable quickly, so that the tea leaves move to the outer edge of the glass. Keep spinning until the water is fully spun up (at least thirty seconds for the glass we used you will need to test this out prior to using the demonstration).
  5. After the elapsed thirty seconds, stop the turntable abruptly.
  6. The tea leaves should move from the outer edge and settle in a heap in the center of the bottom of the glass.
  7. Alternatively, if you do not allow the water in the cup to fully spin up, when you stop it the tea leaves will form a circle at the edge of the secondary vortex (see analysis below).

CAUTION: If not attached well, the glass of water can slide off of the turntable when rotated.

The images below show the tea leaves location when, from left to right, the cup is being spun up, the cup is stopped having been fully spun up, and the cup has been stopped after partial spin up.

spinupspindownfullspindownpart

Analysis (qualitative)

When, starting from rest, the cup is initially spun, a boundary layer forms along the base of the cup. This drives the fluid in a circumferential direction. However, in the absence of any force to balance the resulting normal acceleration, the water in the boundary layer is driven radially outward. This drives the tea leaves to the edge of the cup. The radial outflow is then forced up the side of the cup, though the tea leaves stay in the corner at the base as they are denser than the water.

The vertical flow then turns back in toward the cup center and then down when it reaches the water surface. This creates a cylindrical vortex around the edge of the cup (see figure below). Inside the cylindrical vortex is a non-rotating core with a flat water surface.

partialup

Over time, the cylindrical vortex grows toward the center of the cup until there is no longer a non-rotating core and the water surface is curved all the way across (see figure below). At this point the flow is fully spun up and the tea leaves should still be at the corner of the cup.

fullup

When the cup is abruptly stopped, the water in contact with the base also stops moving. There is, therefore, no longer anything driving the flow radially outward. Instead, there is a hydrostatic pressure gradient toward the center of the cup due to the curved water surface (the water surface remains curved as all the fluid outside the boundary layer does not know the cup has stopped and is still rotating). Therefore, the flow in the bottom boundary layer reverses and the tea leaves are driven into the center of the cup (see figure below).

spindown

In the event that the cup is not fully spun up (step 7 in the procedure section), the hydrostatic pressure gradient only extends from the side of the cup to the edge of the cylindrical vortex (recall that the water surface in the non-rotating core is horizontal). Therefore, the lower boundary layer only flows radially inward to the edge of the cylindrical vortex. The tea leaves thus accumulate at the inner edge of the cylindrical vortex (see figure below).

partialdown

This is a remarkably robust experiment. It is almost impossible for it not to work (provided that the cup is secured to the center of the turntable). Thanks to Alex, and Meredith for putting together this write up and demonstration. Videos to follow soon.

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.

Videos of “Fire whirl and stretching a vortex”

Here are the videos from the “Fire whirl and stretching a vortex” demonstration. The full videos are linked from the GIF titles (the entire demo video is here).

Setup

setup

Ignition (with low flame height)

ignition

fire whirl (with much larger flame height)

whirl

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.

Videos of Vorticity and walking in circles

Here are the GIFs of the demonstration “Vorticity and walking in circles“. The headings link to the full videos.

(1)    Rotational flow around a circle

CR

(2)    Irrotational flow around a circle

CI

(3)    Rotational flow along a straight line

SR

(4)    Irrotational flow along a straight line

SI

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.

Vorticity and walking in circles

This is a really simple demonstration that requires no equipment at all, though it is slightly enhanced if you have a stool that you can place at the front of the room. The demonstration shows the difference between rotational and irrotational flow for both straight and circular streamlines.

Equipment

Ideally you would have a stool or chair to walk around for the first two parts of the demonstration. If you do not, then you just need a space on the floor to walk around. The demonstration is exactly the same with or without the stool. You just don’t have the visual reference to walk around if you don’t have a stool.

Demonstration

The demonstration has four parts covering rotational and irrotational flows for both circular and straight streamlines.

(1)    Rotational flow around a circle

Place the stool in the front of the room with space all around it. Start off facing the class with the stool in front of you. Simply walk around the stool while facing the stool the whole time. Stop after one trip around the stool and point to the back of the room. Point out to the class that your body (the model fluid particle) is facing the back of the room. Start walking around the stool again and stop half way around, you should have your back to the class. Point to the front of the room and point out to the class that you are now facing in the opposite direction and have, therefore, rotated. Hence, this is a rotational flow with circular streamlines.

(2)    Irrotational flow around a circle

Repeat the first demonstration, only this time always face the back of the room as you walk in a circle around the stool. Again, stop after one trip around the stool and point to the back of the room. Continue for another half circle and stop. The stool should be behind you and you should be facing the back of the class room. Point to the back wall and point out that, while you were walking in a circle you were not rotating (you always faced the same direction). Hence, this is an irrotational flow with circular streamlines.

(3)    Rotational flow along a straight line

Start at one side of the room with a clear path across the front of the room. Roll yourself across the front wall of the room. That is, walk across the front of the room while rotating as if you were a wheel rolling along the wall. Stop a third of the way along while you are facing the back of the room and point to the back wall telling the class which way you are pointing. Continue rolling and stop a bit further on when you are facing the front wall. Point out to the class that you are now facing the front wall and must therefore have rotated. This is a rotational flow with straight streamlines.

(4)    Irrotational flow along a straight line

This is the easiest part. Simply walk across the front of the room in a straight line while always facing in the same direction. The degree of difficulty can be raised (though only slightly) by facing the students while walking sideways. This is an irrotational flow with straight streamlines.

Discussion

For each of the components of the demonstration you can give an example of such a flow. Examples might include (1) solid body rotation, (2) the bath tub vortex, (3) laminar flow in a pipe, and (4) Wind above the atmospheric boundary layer where there is negligible shear. You can also use examples from outside of fluid mechanics. For example, (1) is analogous to how the moon rotates around the earth such that we only ever see one side of the moon. Example (3) is analogous to a car tire as it drives along a flat road.