Rotational lift – Magnus effect on golf balls

Many science museums have demonstrations in which a ball is levitated by an air jet. Typically the ball is quite light and the air jet quite broad. These setups can produce reasonably stable results with the ball staying supported by the air jet. They even work when the air jet is inclined. There are occasionally explanations associated with the demonstration talking about how the drag from the air jet supports the weight of the ball and as it moves off-center then Bernoulli means that there is a low pressure on the jet side of the ball that draws it back in. This is often unsatisfying as an explanation. It also ignores the fact that, when the air jet is inclined, the ball typically rotates. To get at this in a little more detail I tried it with a golf ball.

Equipment

  1. Shop-vac that can blow air
  2. golf ball
  3. a steady hand or some sort of mount for the hose outlet. I had a spare trolley that I could have a mount built on. See figure 1 below for the setup.

20191017_094929_HDR

Figure 1. image of the shop-vac and adjustable mount for the hose outlet

The mount shown is adjustable so that one can change the outlet flow angle. The shop-vac I used has an outlet nozzle diameter of 2.5 cm and the air speed, measured 25 cm downstream of the outlet, was 36 m/s. The main downsides of this are that it is quite noisy and also it can be tricky getting the ball to sit in the air stream.

Demonstration

  1. turn on the shop-vac and place the ball in the air stream
  2. release the ball such that it remains supported by the flow (This is a lot easier said than done).
  3. observe the behavior
  4. slowly adjust the angle of the air jet (I started at vertical and adjusted from there).

Animated gifs of this demonstration for three different jet angles are shown in figure 2.

verticalmid anglelarger angle_1

Figure 2. Animated gifs of the demonstration for three different jet angles showing the different behavior for each angle. the uninterrupted outlet velocity is the same for each case.

Observations

  1. when the air stream is vertical the ball is quite unstable and, after bouncing around for a while, it falls out of the stream. This differs from many museum exhibits where the ball is lighter and is able to change rotational direction more rapidly.
  2. when the air jet is not vertical the ball is more stable. It rotates rapidly and oscillates backward and forward along the line of the air jet.

Qualitative explanation

The rotation of the ball is key here. The balls rotation deflects the air jet changing the momentum of the flow (see figure 3). To do this the ball must apply a force normal to the direction of the incoming air jet. The reaction to this is a lift force on the ball normal to the direction of the incoming air stream (see figure 4). This is the same process used to throw curve balls (see Lift, Boundary layer separation, and curve balls).

The vertical components of the drag and lift both act upward and together balance the weight of the ball.

golfball c1

Figure 3. Schematic diagram showing the incoming air stream, the ball rotation, and the resulting wake deflection.

golfball c2

Figure 4. (Left) free body diagram of the ball showing the drag, lift, and weight forces. (Right) force triangle showing the force balance when the ball is stable.

The rotation of the ball is established by the air stream. When the ball is placed in the air stream, if it is not rotating, it will fall out. As it does, the air flow across the top of the ball is substantially faster than across the bottom. This drives the ball to rotate which in turn deflects the wake downward and generates the lift force. This is also why it can take a few goes to get the demonstration to work as the rotational inertia of the golf ball limits the rotational acceleration.

Back of the envelope calculation

A standard golf ball has a mass of 46 grams and a diameter of 43 mm. I assumed the density of air to be 1.25 kg/mand the measured air velocity from my set up was 36 m/s and I took the jet angle to be 45 degrees. In this case the sum of the forces in the vertical direction becomes

mg=0.5ρu2A sin45 (CD+CL)                                                                        (1)

Substituting values into (1) leads to CD+CL=0.54. The Reynolds number is approximately 105 which is in the region of the drag crisis for a golf ball (Link) so the coefficient sum would appear to be reasonable.

Application

There are lots of circumstances in which the rotation of a compact object causes a lift force. The most obvious is the in-flight curve of a golf ball when not hit perfectly. One application I am working on is the lift-off and flight of compact debris in severe storms. In particular I am interested in the conditions under which loose-laid roof gravel is removed during hurricanes and tornadoes. On possible mechanism is that the wind shear on the gravel surface pushes a gravel piece to roll over its downwind neighbor. If it is deflected up during this process then the large velocity gradient (wind shear) near the surface could generate the rotation required to generate a lift force and launch the gravel up into the wind field. My student and I will be doing testing on this, and other potential mechanisms, at the Florida International University Wall of Wind Experimental Facility over the next few years.

The research and outreach project (including this post) are based upon work supported by the National Science Foundation under Grant No. 1760999. Any opinions, findings, and conclusions or recommendations expressed in the material are those of the author and do not necessarily reflect the views of the NSF.


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 “Solid body rotation – measuring the rotational velocity of a potting wheel”

Here is a link to a video from the “Solid body rotation – measuring the rotational velocity of a potting wheel” demonstration. The video shows the parabolic cavity forming as the water spins up until a steady solid body rotation is achieved. In this video the wheel was slightly wobbly so the cavity was not perfectly parabolic.


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.

Solid body rotation – measuring the rotational velocity of a potting wheel

This demonstration uses the pressure distribution in a fluid undergoing solid body rotation to estimate the rotational speed of the cylinder.

Pre-demonstration calculations

Under solid body rotation about a vertical axis the horizontal variation in pressure in the radial direction is

dP/dr=-ρa=ρω2r

The pressure in the vertical direction is also hydrostatic resulting in the free surface at the top of the water becoming curved concave up. Taking the origin to be at the free surface on the axis of rotation then the pressure along a horizontal radial line from that point will be

P=ρω2r2/2

such that the height to the free surface above that point is

z=P/ρg=ω2r2/2g.

For a cylinder of radius R and depth H when not rotating the water will rise such that the total distance from the new axial water surface to the surface at the edge of the cylinder is given by

h=ω2R2/2g

when fully spun up. The volume of the fluid is initially V=πR2H. This volume is conserved. The volume of the paraboloid of air is given by  V=πR2h/2. which is also the volume of the water above the height of the axial free surface. Denoting the distance that the water surface drops on the axis of rotation by δ we can write that the volume of water above this point must be the same regardless of whether the water is rotating or not. Therefore,

πR2δ=πR2h/2.

or

h=2δ.

That is, the water level at the side of the cylinder rises by the same height that the water level drops on the axis of rotation. Therefore, if the initial depth of water on the cylinder is marked prior to the experiment then one can measure the amount that the water level rises on the cylinder side (δ) using a marked scale and then back calculate the angular velocity as

ω=(4gδ)½/R

The figure below defines the dimensions used above.

Equipment

You will need

  1. a clear cylinder marked with a vertical scale that passes all the way around the cylinder (I bought a large cookie jar and used thin tape to make the scale and mark the initial depth of water I needed)
  2. a method for rotating the cylinder at a constant rate (I was lucky enough to get a repaired potting wheel for the cost of a replacement switch and potentiometer).
  3. A stopwatch to estimate the actual rotation rate

Demonstration

  1. Fill the cylinder to the marked initial depth
  2. start the cylinder rotating at a constant angular velocity such that an air cavity forms as described above. Ensure that the rotation speed is small enough that the cavity does not extend down to the base of the cylinder and that the top of the water surface does not reach the top of the container.
  3. measure the height of the free surface above the marked initial depth line
  4. measure the angular velocity by counting the number of revolutions over some short period of time.
  5. calculate the angular velocity using the equation above and compare.

Comments and difficulties

The main challenge is to get a measurable rise in the water surface at a rotational rate that is also measurable by an observer with a stopwatch. When I tried this the 22.9 cm diameter cookie jar. When rotating at 80 rpm the water rise height should be around 2.3 cm which can be seen. However, that requires the observer to count 20 rotations in 15 seconds. The observed rotational rate has the largest margin for error. One could add a tachometer of something similar but that would obviously add cost. The second major problem is that it is sometimes hard to see the location of the surface at the edge of the cylinder as the water is quite thin there and the cylinder is spinning so the scale is hard to read. I overcame this by marking the height I wanted (2.3 cm in this case) and then adjusting the wheel speed until the water level was at that height. This is admittedly a bit of a cheat but it does still allow for the calculations to be done to confirm the model.


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 “‘Rotational buoyancy’ update – potting wheels and fishing tackle”

Here is a link to a video from the “‘Rotational buoyancy’ update – potting wheels and fishing tackle” demonstration. The video shows the buoyant float moving toward the center of the jar while the weight moves out to the side of the jar.


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.

‘Rotational buoyancy’ update – potting wheels and fishing tackle

This is an update to the “Rotational buoyancy – Hydrostatic pressure in solid body rotation” post but with much better visualization. The analysis is identical to the previous post however, the visualization is different. The demonstration is similar to “a baffling balloon behavior” YouTube video by Smarter Everyday (well worth following on twitter). In this video a heavy ball is suspended from the ceiling of a minivan and swings backward when the van accelerates forward. A Helium balloon is then tethered to the floor and swings forward when the van accelerates. The forward motion is because the horizontal hydrostatic pressure generated by the van accelerating (dp/dx=-ρa) results in a horizontal buoyancy force ρa∀ (where ∀ is the volume of the balloon) that is greater than that required to accelerate the balloon, ma, where m is the mass of the balloon. Therefore, the balloon tilts forward with the horizontal component of the tension in the cord producing the additional force needed to maintain the constant acceleration.

For solid body rotation there is also a buoyancy force toward the center of rotation due to the normal acceleration of the individual fluid particles. See the “Rotational buoyancy – Hydrostatic pressure in solid body rotation” post for more details. Taking inspiration from the balloon in a van video it is possible to do the same demonstration using a rotating jar as described below.

Equipment

  1. large round glass jar with a tightly fitting lid
  2. fishing line, a float and a weight (all easily available in the fishing tackle section of a sporting goods store)
  3. some method for affixing the fishing line to the base and lid of the jar (e.g. self adhesive cable holders)
  4. a potting wheel with a mount for securing the jar (or other type of turntable for mounting the jar.) Fortunately for me the art department was disposing of some old and broken potting wheels that our head technician was able to fix. They then built a mount for the jar I had.

Figures showing the experimental setup (left) full setup and (right) close up showing the suspended weight and tethered float.

Demonstration

  1. attach the float to the fishing wire and the fishing wire to the base of the jar about half way between the center of the jar and the side walls. The wire should be long enough so that it will float at about half the depth of the jar.
  2. attach the weight to another length of wire and the other end to the lid of the so that it will hang about half way between the center of the jar and the side walls. The wire should be long enough so that it will hang at about half the depth of the jar.
  3. fill the jar so that it is about 3/4 full of water and mount the jar on the turntable.
  4. turn on the turntable and slowly increase the speed. Make sure not to go too fast or the water may leak out the top if the lid is not a perfect seal.
  5. After a while the water should spin up completely. At this time the weight should have moved closer to the edge of the jar and the float should have moved closer to the center of the jar.

Image of the demonstration showing the float pushed toward the center of the jar and the weight pushed out to the jar’s side wall.

Analysis

The analysis is the same as for the “Rotational buoyancy – Hydrostatic pressure in solid body rotation” post. There is a hydrostatic pressure gradient with low pressure near the center of rotation increasing radially. The weight moves to the outside because its density is greater than that of the water so that the buoyancy force is not enough to generate the normal acceleration need to maintain the rotation. Conversely, the float moves toward the center as it is less dense than the water and, therefore, the buoyancy force is greater than that required to maintain the normal acceleration. In both cases the horizontal component of the tension in the fishing wire supplies the additional forces required to supply the steady acceleration.

Broader context

While rotational buoyancy forces can produce some cool results, many undergraduate engineering textbooks simply focus on buoyancy forces due to a liquid acting on a submerged solid object (boats floating etc.). However, buoyancy forces occur whenever there are two different density materials next to each other. For example, there is a buoyancy force acting on people due to the surrounding air, though this is a very small force and can reasonably be ignored. Buoyancy forces can also occur on a fluid submerged in another fluid. The obvious example is a gas bubble in a liquid. However, buoyancy forces also occur when there are density differences between two liquids or two gasses.

These buoyancy forces can be significant and have a major influence on flow dynamics in the natural and built environment. For example, a person sitting in a room heats the surrounding air making it less dense. The buoyancy force due to the ambient cooler air drives the warmer air toward the ceiling and can generate a warm upper layer in the room. If the room has vents and floor and ceiling level then the buoyancy force acting on the warm upper layer can drive a flow out the upper vent and induce a flow into the room through the lower vent. The resulting flow is a form of natural ventilation.

Buoyancy forces can also hold a fluid down. For example, the accidental release of Chlorine from tank failures creates a gas cloud that is denser than the surrounding air. It will tend to stay near ground level and spread out horizontally. This is a problem in urban areas as we tend to live near ground level. The density difference also tends to prevent vertical mixing by wind shear as it takes energy to raise the dense gas as the buoyancy force must be overcome. I am currently working on an NSF project to look at the mechanics of dense gas dispersion in urban areas in which we will use salt water to represent the dense pollutant and fresh water to represent the ambient air. The project is ongoing and results will be posted on my work website as they become available.

This material is based upon work supported by the National Science Foundation under Grant No.1703548. Any opinions, findings, and conclusions or recommendations expressed in the material are those of the author and do not necessarily reflect the views of the NSF.


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.