‘Rotational buoyancy’ – Hydrostatic pressure in solid body rotation

Solid body rotation of a fluid about a vertical axis results in a horizontal pressure variation which provides the centripetal force required to rotate the fluid particles. The pressure gradient is, therefore, given by

dp/dr=ρw2r                                              (1)

Where ρ is the fluid density, w is the angular velocity and r is the distance from the center of rotation.

This is easily demonstrated by spinning a cup of water and showing the paraboloid surface that forms with the low point at the center of pressure. However, the pressure gradient exists regardless of the free surface and, just as with the non-rotating hydrostatic pressure variation, an immersed object will experience a net force in toward the region of lower pressure. This can be demonstrated using a sealed container

Equipment

  1. A small cork
  2. A marble
  3. A Lazy Susan or some other cheap turntable
  4. A mason jar
  5. Some Velcro strips or something else to secure the Mason jar to the turntable.

Photo Apr 13, 3 04 44 PM

Demonstration

  1. Place the cork and marble in the jar and fill it with water
  2. Seal the jar so that there are no bubbles (or at least no bubbles that are large compared to the size of the cork) and attach it to the turntable so that its long axis is horizontal and it is centered on the turntable (see figure above).
  3. Shake the jar until the marble and the cork are near the center of the jar (this is so that when the marble moves it is clearly due to the rotation of the jar).
  4. Rapidly spin the turntable. The marble should be pushed to one end of the jar while the cork should remain centered.

Analysis

The horizontal hydrostatic pressure gradient (equation above) means that any submerged object will experience a net pressure force acting toward the center of rotation. For a rectangular object of width s in the radial direction and area A normal to the radial direction located a distance r  from the center of rotation, the net pressure force toward the center of rotation is given by

FpAw2((r+s/2)2-(r-s/2)2)/2                      (2)

See figure below.

rotation

Expanding leads to

Fp=ρAw2rs= ρ∀w2r                                          (3)

where is the volume of the object. Therefore, the net force toward the center of rotation is the angular acceleration multiplied by the mass of water displaced by the object. This is directly analogous to the buoyancy force in a stationary fluid. Therefore, if the object displacing the fluid has a lower density than the fluid then the centripetal pressure force will exceed that needed to maintain the angular velocity of the object and it will be pushed toward the center of rotation (as in the cork). If the object is denser than the fluid the centripetal pressure force will not be enough to maintain rotation and the object will move radially outward (note that this is a somewhat simplified linearized analysis and the integration constant is left out of the pressure equation used in (2) but it gets the appropriate result).


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 “Spin up, boundary layers, and tracking tea leaves”

Here are the videos of the “Spin up, boundary layers, and tracking tea leaves” demonstration. The video titles link to the full videos.

spinning up

tcup

stopping a fully spun up cup

tcfulldown

stopping a partially spun up cup

tcpartdown

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.

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.

Fire whirl and stretching a vortex

Background

A fire whirlwind, otherwise known as a “fire devil”, “fire whirl” or “fire tornado,” occurs naturally in wild fires (see photos from National Geographic). They occur when a vortex forms around a fire plume. The hot air from the fire plume stretches the vortex vertically narrowing it and intensifying it. In nature the vortex can form due to the ambient wind being deflected around a fire by local topography. In the lab there are a couple of ways to form a fire whirl. The Phaeno Science Center uses an array of air jets blowing tangentially in a circle around the flame to create the vortex. At a smaller scale one can just rotate the flame.

Equipment:

  • Nonflammable Turntable (Lazy Susan)
  • Wire Mesh Trash Can (the finer the mesh, the better)
  • 1 Glass Pyrex Bowl (or any nonflammable container)
  • Cotton Balls
  • Rubbing Alcohol (or any fuel source)
  • Duct Tape
  • Lighter
  • Damp Towel (or any form of extinguisher)
  • fire

Procedure

  1. Using the duct tape, affix the trash can to the center of the turntable.
  2. Place cotton balls, 10-15, into the glass bowl and pour rubbing alcohol over the cotton balls. Pour enough alcohol to lightly dampen the cotton balls, careful not to pour too much; do not completely saturate the cotton balls.
  3. Using the duct tape again, affix the glass bowl inside the trash can, careful to center it in the bottom.
  4. Use the lighter to ignite the cotton balls. Observe the flame.
  5. Begin to rotate the turntable and observe the effect on the flame. Vary the speed (without being reckless) and note how the flame stretches with higher rotational speeds.
  6. Put out the fire using the damp towel.

CAUTION:

  • If not attached well, the trash can will slide off of the turntable when rotated.
  • Also, if not attached, the glass bowl will slide around the inside of the trash can.
  • Be careful when removing the glass container, it becomes very hot during the experiment.

no-whirlwhirl

Flame height (left) before spinning and (right) when spinning

Discussion

This is a good ‘wow that’s cool’ demonstration for any fluid mechanics class. It is particularly useful when discussing vortex dynamics and vortex stretching. Essentially, once the vortex forms around the fire the buoyancy generated by the fire stretches the vortex vertically. This makes the flame and vortex narrower and, by conservation of angular momentum, increases the intensity of vortex. this process is self reinforcing and the flame length can then dramatically increase. While this basic mechanics is fairly well understood, the actual prediction of when such fire whirls will occur in nature is very much an open question.

Thanks to Alex, Meredith, and Ali 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.