Measuring the Kinetic Energy of a Vortex using Laser Velocimetry

Measuring the Kinetic Energy of a Vortex using Laser Velocimetry

Megan Birch and Brandon Henke

University of Minnesota, Twin Cities

School of Physics and Astronomy

Minneapolis, MN 55455

Introduction

The idea behind this experiment was to explore a technique in measuring the velocity of a fluid called laser velocimetry. Being able to measure the velocity of a fluid is very helpful in the field of fluid mechanics since many subjects in that field are reliant on the velocity field. For this experiment, the velocity of a vortex of water was measured and used to calculate the total kinetic energy of the vortex. Since vortices occur in many situations, such as tornadoes, hurricanes, and accretion disks, it can be helpful to know how much energy is stored in them. For example, knowing the amount of kinetic energy a tornado has can indicate how much damage it may do.

Theory

The method works by creating an interference pattern with light, passing a reflective particle across the pattern, and measuring the amount of light reflected off of the particle. When the properties of the interference pattern are known, namely the distance between two peak intensities (the fringe spacing), one can determine how quickly the particle was moving across the interference pattern. This experiment used a Helium-Neon (HeNe) laser, which had a wavelength of 632.8nm. The laser was split into two parallel beams, 5cm apart, and sent through a converging lens, with a focal length of 25cm. This set up gives the interference pattern shown here:

The fringe spacing found from this was Δx = 3.2μm. As a particle moves across this pattern, the the expected signal is given by letting the position, x, be given by vt, where v is the velocity of the particle, and t is the time. This would give a relationship between frequency of the signal and the speed at which the particle is moving as

v = Δx ν,

where ν is the frequency of the signal.

Calibration

To find this fringe spacing value, the system was calibrated using an optical chopper.

The intersection point was placed on the edge of the rotating chopper and the frequency was measured for different speeds. Plotting the data points for frequency at different speeds and finding the slope of the best linear fit produced the 3.2μm fringe spacing value.

Set Up

After the system was calibrated, the chopper was replaced with 85.09mm diameter cylinder. The cylinder was filled with 500mL of distilled water and two drops of rheoscopic fluid. In more detail, the apparatus is as shown in the following diagram:

The HeNe laser is emitted and reflected off of two mirrors. The mirrors were aligned by adjusting them to go through two iris' that were set to have the same height as the aperture on the optics apparatus, so that the laser was going in parallel to the table. The optics apparatus split the laser into two parallel beams, 5cm apart from each other. These beams were sent into a converging lens, so they intersected at the focal point of the lens. In the region of intersection, the two light waves interfered with each other, which created the interference pattern. A beaker was placed on top of a hot plate, and both were positioned so that the focal point of the lens fell inside the beaker. As a particle crossed the interference pattern, the light would be reflected back into the optics apparatus, which routed it into a photomultiplier tube (PMT). The signal from the PMT was passed through a band pass filter, to reduce noise, and then into an oscilloscope.

Results

When a signal is seen on the oscilloscope, it looks something like the following image.

Looking closely, oscillations can be seen in the signal. This is the interference pattern mentioned above. Taking the Fourier transform can give the frequency of those oscillations. The Fourier transform of this set of data is shown here:

This location of this peak in frequency space is the frequency, ν, of the signal that we're looking for.

The velocity was measured at 10 different radii, and around 10 sets of data were collected at each radii. This allowed for averaging of the measurements, and the standard deviation could be found for each data point, which gave the error for each data point. The "analyze.py" python script was used to analyze all of the data. The script would look for data in a folder named "Data" in the same directory, and then it would look through the folders inside. Each folder was for data of a different radius, so the script would create an info.txt file for each folder that had all of the measured frequencies for each data set at that radius, and it would give the average. Next, the "viewData.py" python script would read each of those text files. It would find the standard deviation from each average frequency at each radius, to use as the error on that frequency measurement. Then it calculated the velocity of the particle using the relation v = Δx ν, and getting the error on v by using error propagation. a scatter plot of the data was created, with error bars, shown here:

Collecting data was pretty tedious since one set of data took 5 minutes to save, and around 100 sets of data were collected total. It also wasn't possible to automatically collect data since we had to wait for a clean signal on the oscilloscope. This meant having to go back to the oscilloscope every 5 minutes to find another clean set of data and start saving the next clean set. Additionally, when moving between radii, both the hot plate and the beaker were moved together, which could have caused sloshing of the water. A better method would be to move the focal lens on a translation stage.

Conclusion

Laser velocimetry was used to determine the velocity of moving particles at different radii inside a vortex. The velocities found were then used in the kinetic energy equation of a fluid to determine the kinetic energy of a vortex. The kinetic energy of a vortex made from a cylindrical beaker containing 500mL of water with a diameter of 85.09mm with a stir rod moving at 6Hz was calculated to be (7.30±3.10)×10^(−2)J. After evaluating the velocity of the vortex at different radii, it was confirmed the velocity increases as you move toward the outside of the cylinder. It was also found the vortex had a slight no slip condition on the edge of the beaker due to the stationary circumference, so the velocity approached zero closer to the outside of the bounded vortex.

To continue this experiment to better understand the behavior of vortices, changing the speed of the vortex using the stirring function would give a better insight on how changing the speed of the stir rod affects the energy. Measuring the kinetic energy of different size cylindrical beakers would also help in discovering the behavior of vortices and how their size affects their kinetic energy.

References

[1] E. J. Hopfinger, “Vortices in rotating fluids”, (1993).

[2] L. E. Landau LD, Fluid mechanics vol 6 (1987).

[3] “Vortex knots dynamics in euler fluids”, (2013).

[4] D. J. Griffiths, Introduction to electrodynamics (2013).

[5] S. Instruments, ed., Optical chopper selection guid (2019).

[6] S. Anthony, “Towards infinite-capacity wireless networks, with twisted vortex radio waves”, ExtremeTech (2014).