Water moving around a cylindrical pendulum makes the cylinder swing at the vortex shedding frequency.

### What it Shows:

When fluid flows around a cylindrical object, there is a range of flow velocities for which a von Karman vortex street is formed. The shedding of these vortices imparts a periodic force on the object. The force is quite small and not enough to accelerate the object to any significant amount, especially if the object is relatively massive. If the situation is such that the object can vibrate about a fixed position, we have the possibility of simple harmonic motion; and if the frequency of the periodic driving force matches the natural frequency of the oscillation, then resonance obtains and the amplitude of the oscillations can be significant.

The object is a cylinder attached to a physical pendulum whose frequency of oscillation is adjustable. The end of the cylinder is submerged in flowing water. When the frequency of the pendulum is adjusted to match the frequency of vortex shedding, the cylinder swings transverse to the direction of flow with a peak-to-peak amplitude of 2 to 3 centimeters.

### How it Works:

Consider the flow of fluid around a smooth cylinder. For velocities exceeding laminar flow, the inertia of the fluid starts to become significant and, as the fluid stream passes the topmost part of the cylinder, it is unable to negotiate the rear half of the cylinder. Hence the fluid tends to separate from the top surface and peel off in a clockwise motion as it approaches the rear end of the cylinder, ending up as a shed vortex (it will peel off in a CCW motion from the bottom surface). For a given velocity of flow, this model suggests the vortex formation time will be proportional to the distance around the cylinder (or its diameter) and thus the frequency of vortex formation will be inversely proportional to the diameter. Furthermore, if the flow velocity increases, the frequency of vortex formation will likewise increase, leading to a direct relation between the two. This is what Strouhal found empirically in 1878 (ref 1).

The proportionality constant is called the Strouhal number and turns out to be a function of the Reynolds number. For that reason it is now known as the Strouhal-Reynolds number. It is very nearly equal to 0.2 for a large range of Reynolds numbers.

Strouhal's empirical formula for the frequency of vortex shedding is f = (1/5)(v/d)

where v is the velocity of flow and d is the diameter of the cylinder (a full and modern treatment of vortex shedding can be found in references 2-4). Thus, for water flowing past a 2.54 cm diameter cylinder at a velocity of 10 cm/s, one can expect a frequency of 0.8 s^{-1}.

The cylinder is only partially immersed in the water and does not rest on the bottom of the channel. Rather, it is supported by a threaded rod and is part of a physical pendulum. The pendulum's frequency of oscillation is adjustable and can be made to closely match the vortex shedding frequency.

One can see the vortex shedding by shadow projection. Similar to a ripple tank, if a point light source is set up over the tank, then shadow patterns of the water's motion can be seen on the bottom of the water channel. Thus, not only can one demonstrate resonance oscillations of the cylinder, but one can also see the relative motion of the vortex eddies and the cylinder—they move in opposite directions.

As with most fluid dynamics phenomena, the physics of vortex-induced vibrations is quite rich and very complicated—even in two dimensions. For example, various vortex wake modes are possible with fluid flow jumping from one mode to another. Additionally, the sideways motion of the object in the fluid affects the formation and/or shedding of vortices and can have a positive or negative feedback effect. Also, depending on the phase between object and fluid motion as well as their frequency difference, a lock-in or synchronization effect may or may not occur. Furthermore, the ratio of the object-to-fluid mass as well as damping forces have a significant effect, leading to parameters described in the literature as effective mass, critical mass, high-mass ratio, etc. Reference 4 is an excellent resource for those that wish to go deeper into the subject matter—and it can be quite deep indeed!

### Setting it Up:

A 55-gallon aquarium has been converted into a flow tank. A barrier divides the aquarium into two halves. Water is pumped from one half of the aquarium to the other. When the water spills over the barrier, it returns to the other end via a wide flow channel. The flow rate in the channel can be adjusted by enlisting one or two pumps, as well as by changing the amount of water in the tank. A convenient flow rate for this demonstration is about 10 cm/second. The flow can be readily measured by placing a plastic meter stick (with decimeter markings) in the channel and letting a ping-pong ball float downstream, noting how long it takes to pass the markings.

The object is a 1" dia. × 2" long aluminum cylinder constrained to move transverse to the flow. The cylinder is attached to a 15" long, 10-32 threaded rod which, in turn, is suspended from a knife-edge. The combination is a physical pendulum whose period of oscillation can be changed by its length, or screwing machine nuts onto the threaded rod. The pendulum can be suspended over any part of the water flow channel, allowing one to choose a location where the flow rate is optimal for what one wants to demonstrate.

### Comments:

Vortex-induced vibrations are important in that they can have a strong influence in countless situations ranging from tethered structures in the ocean, pipes bringing oil from the ocean floor to the surface, aeolian harps, tall buildings, and chimneys, to name but a few. For example, the tallest building in the world, the Burj Khalifa in Dubai, UAE, incorporates a variation in cross section with height to help ensure that vortices are not shed coherently along the entire height of the building (reference 5). The Tacoma Narrows Bridge collapse is discussed in practically every introductory physics course as a dramatic example of resonance. Although vortex shedding is often cited as being the culprit, Billah & Scanlan say that this is oversimplified physics and posit that the real culprit was *flutter*—a non-linear phenomenon in which the motion of the bridge was the source of self induced periodic impulses (reference 6).

### References:

1. V. Strouhal, "Ueber eine besondere Art der Tonerregung," *Ann. Physik. Chem* (Leipzig) **5**(10), 216-251 (1878).

2. C.H.K. Williamson, "Vortex Dynamics in the Cylinder Wake," *Annual Review of Fluid Mechanics* **28**, 477-539 (1996).

3. U. Fey, M. König, and H. Eckelman, "A New Strouhal-Reynolds-number relationship for the circular cylinder in the range 47<Re<2×10^{5}," *Physics Fluids* **10**(7), 1547-1549 (1998).

4. C.H.K. Williamson and R. Govardhan, "Vortex-Induced Vibrations," *Annual Review of Fluid Mechanics***36**, 413-445 (2004).

5. P.A. Irwin, "Vortices and tall buildings: A recipe for resonance," *Physics Today* **63**(9), 68-69 (2010).

6. K.Y. Billah and R.H. Scanlan, "Resonance, Tacoma Narrows bridge failure, and undergraduate physics textbooks," Am. J. Phys. **59**(2), 118-124 (1991).