#### October 2013 Volume 3, Issue 1

# Measurements of the Motion of Polygonal Vortices in Rotating Fluids Showing the Significance of Local Rotational Velocities

**David Weiner1* and E. Jane Wesely2**

Student1, Teacher2: Horace Mann School, 231 West 246th Street, Bronx, New York 10471

*Corresponding author: david_weiner@horacemann.org

### Abstract

Insight into the asymmetries of vortices is important in understanding and predicting weather systems, such as tornados and hurricanes, as well as determining the behavior of vortices in jet engines. In this experiment, the behavior of vortices generated in water by a rotating disc in a cylindrical bucket was studied. Polygonal vortices were observed with two, three, four, five, and six sides, where the number of sides increased with increasing disc rotation speed. The slope of the water surface as a function of distance from the rotation axis was determined at both the sides and at the corners of the polygonal vortices. From this, the rotational water velocity at the surface was calculated at the sides and corners of the polygonal shapes. It was found that the rotational water velocity at the corners of the polygons was greater than the rotational water velocity at the edges of the polygons. This provides new evidence that the polygonal shapes were formed by arrays of subsidiary vortices orbiting a central vortex.

### Introduction

Vortices are fascinating phenomena that occur commonly throughout the world, yet their behavior is still not fully understood. A vortex is a whirling mass in a liquid or gas, and examples of vortices in nature include the winds in tornadoes and whirlpools in bodies of water. Children even create and play with vortices as water drains from the bathtub.

Once vortices form, their surface shapes can alter into a wide variety of configurations. For example, satellite images have demonstrated that the swirling eyes of hurricanes transform into different polygons ranging from triangles to hexagons.^{1} Saturn’s hexagonal structure at its north pole has been attributed to an ongoing raging hurricane.^{2}

It is currently theorized that the observed polygonal shapes are due to stable arrays of multiple vortices that form at equally spaced distances around a central vortex.^{3,4} In 1867, Lord Kelvin observed smoke rings rebounding obliquely from each other and proposed that atoms were vortices similar to smoke rings, proving theoretically that a configuration of three vortices was stable.^{5} In 1883, Thomson predicted that arrays of three to six vortices also were stable.^{6} Multiple vortices seen in rotating superfluid helium were observed to form stable geometrical arrays.^{7 }Researchers have created polygonal vortices in an experimental setup with a disc rapidly rotating at the bottom of a cylindrical container of fluid. The surface shapes of the vortices deviated from circular depending on the rotation speed of the disc, prompting the hypothesis that the polygonal shapes were caused by subsidiary vortices orbiting a primary one in the center of the container.^{3,4}

This experiment extended the study of why different shapes of vortices arise. By determining the rotational velocity of water at the sides and corners of polygonal shapes generated in a swirling bucket of water, the comparison of these measurements was used to show whether subsidiary vortices formed at the corners. If the corners of the polygon shape were due to subsidiary vortices orbiting the main central vortex, then the rotational velocity of the water should be higher at the corners.

The first part of the experiment examined how many lobes were present in polygons produced at various disc rotation speeds and water depths above the disc prior to spinning. The second part of the experiment analyzed the water’s profile as viewed from the side of the container. The slope of the water profile was determined as a function of the radial distance from the center of the disc at both the sides and corners of the shapes. The slope was used to calculate the rotational velocity of the water as a function of distance from the disc’s center at both the polygon’s sides and the corners. The independent variables included the distance from the center of the rotating disc, the depth of water prior to spinning, and the rotational speed of the disc. The dependent variable was the height of the spinning water surface at different distances from the center of the bucket. The rotational velocity of the water at the corners of each polygon was found to be higher than the rotational velocity at the sides of the polygon, indicating the existence of subsidiary spinning vortices.

**Parabolic Surface of Spinning Water:**

In 1689, Isaac Newton observed the curved surface of water spinning in a bucket and became intrigued in the forces that governed its formation.^{8} When a container of liquid is rotated, the surface of the liquid curves to become parabolic in shape due to centripetal force. This effect has been used to make liquid mirrors for telescopes.^{9,10}^{ }Referring to the diagram in Figure 1, the forces on the water at the surface are the downward force F_{g} = mg due to gravity and the normal force N due to buoyancy. The sum of the forces on the water at the surface equals the mass times the radial acceleration given by a_{rad} = rw^{2}_{ }where r is the distance from the center of rotation and w is the rotational velocity.^{10}

**Figure 1. Diagram illustrating the forces on water at the surface.** F_{g} is the force of gravity. N is the normal force due to buoyancy. w is the rotational velocity of the water. The sum of the forces equals the mass times the radial acceleration a_{rad}.

∑F_{i} = ma_{rad}

Equating the x and y components gives:

N_{x} = ma_{rad} = mrw^{2}_{ }

_{y}= F

_{g}= mg

**Slope of the Surface:**

At any point on the surface, the surface of the water will be perpendicular to the normal force.

The slope S of the surface will be the inverse of the slope of N, so S is given by:

S = dy/dr = N_{x}/N_{y} = mrw^{2}/mg

S = dy/dr = rw^{2}/g **Equation 1**

This can be integrated to find y(r)

y(r) = òdy = ò (rw^{2}/g)dr

y(r) = r^{2}w^{2}/(2g) + y_{o} **Equation 2**

which is the shape of a parabola.^{10} y_{0} is the depth at the center.

**Determining Angular Velocity from Slope:**

The derivation of Equation 2 for the parabolic shape assumes that the entire container of fluid is rotating with uniform rotational velocity w, but in the system studied here, w can vary throughout the container, so the surface shape may not be parabolic. In this case Equation 1 can be rearranged to solve for the angular velocity w as a function of the slope S:

w= (gS/r)^{0.5} **Equation ****3**

This equation depends on S^{1/2 }so the percentage errors in w will be half of any errors in S. The experiment measured the slope of the surface at different points and then calculated the rotational velocity at different points.

The vorticity of a fluid is a vector that represents the local angular velocity of the fluid at a given point.^{11}^{ }For a rigidly rotating fluid the vorticity is constant.If the angular velocity varies throughout the fluid then the vorticity will vary.^{11}

### Materials and Methods

The experimental setup shown in Figure 2 was used to study vortices in rotating water. It consisted of a Plexiglas disc immersed in a water-filled clear plastic bucket. The disc was spun rapidly about its axis, which drove the water above it in circular motion and caused it to move outwards from the center. Unlike Newton's bucket^{8} and the rigidly rotating containers used to make parabolic telescopes,^{9} in this system the cylinder walls were stationary.^{2} The water that moved outwards therefore encountered drag due to the cylinder walls.

**Figure 2. Experimental setup used to study vortices in rotating water.**

The Plexiglas disc was attached to a threaded rod that was rotated by a Dewalt DW235G electric drill with a maximum rotation speed of 850 rpm. Although the drill had variable speed capability, in order to set the drill speed accurately an MLCS router speed controller was used. A Cybertech Noncontact Tachometer was aimed at a reflective sticker attached to the drill chuck in order to measure the rotation speed of the electric drill. The accuracy of the tachometer was specified as ±(0.05 %+0.1 rpm) which equals ± 0.4 rpm at 600 rpm.

The Plexiglas discs were cut out of 3/16" thick Plexiglas using a Makita RTO700C router with a circle guide set to the appropriate radius. A 1/2" hole was drilled inthe center of the discs and each disc was attached to a 1/2" diameter stainless steel threaded rod using nuts and washer. A Dico Drill mount was used to clamp the drill vertically to the edge of a table. At high speed the shaft wobbled as much as ±0.5 cm, so a Spyraflo PB1-500-B SAE-840 oil impregnated bronze bearing pillow block attached to a board clamped to the table legs was used to steady the shaft.

The dimensions of the clear bucket were measured with calipers. The bucket had slightly sloping sides. The inside diameter of the bucket was 26 cm at the bottom and 27.6 cm near the top. The dimensions of the scale printed on the side of the bucket were measured in order to provide the scale for later analysis. The distance from the bottom of the scale to the 5L mark was found to be 7.2 cm. The Plexiglas disc that was used was 25.2 cm in diameter, slightly less than the diameter of the container. Measurements were also performed in a square bucket. The inside of the bucket was 23.5 cm x 23.5 cm at the bottom and 25.1 cm x 25.1 cm at the top. Experiments in the square container were conducted using a spinning disc 20.6 cm in diameter.

The container was filled to the desired depth with water. The water was tinted with green food coloring to make it easier to see the water flow. The water depth was measured from the top surface of the disc to the top of the water. The drill was set to the desired speed using the motor controller. The rotation speed was monitored with the tachometer in order to make sure that it was steady. Once the desired speed was reached, photographs were taken of the spinning water looking down from the top and also looking straight from the side. The rotation speed was then adjusted to a new value and more photos and video were taken. All photos and video were obtained with a Panasonic ZS-3 camera using a resolution of 3,648 x 2,736 pixels for photographs and 1280 x 720 pixels for videos.

The slope measurement of the water surface is shown schematically in Figure 3. To measure the surface slope as a function of distance from the center at the triangle edge along the blue line in the top view, the container was observed from the side looking past the triangle edge as depicted by the blue curve in the side view. To measure the surface slope as a function of distance from the center along the red line in the top view that passes through the corner of the triangle, then the side of the shape was viewed looking past the corner as drawn as the red curve in the side view.

**Figure 3. Diagram of water profile measurement technique.** The blue and red lines in the top view show the paths along which profiles of the water surface were observed. As shown in the bottom view, by looking past either the side of the polygon or the corner of the polygon then the profile at the side shown in blue or at the corner shown in red could be seen.

To get the needed side views, the videos were stepped in slow motion to the desired orientation using the VLC video player software^{12} and screenshots were taken using the Greenshot screenshot tool.^{13} The screenshots were then rotated using the GIMP image editing tool^{14} to make the bottom of the bucket horizontal. The leveled images were imported into the Dagra digitization program.^{15} This software permitted the images to be overlaid with x- and y-axes so that the coordinates of points could be selected with the mouse in order to digitize them. The horizontal scale was set using the known diameter of the bucket, and the vertical scale was set using the known height of the scale printed on the side of the container. Curves were then digitized by manually selecting points on the images. The data points were exported into Microsoft Excel for calculations and plotting.

**Correction for Refraction Through Side of Container:**

Refraction of light at the curved walls of the container caused objects in the container to be magnified horizontally. Referring to Figure 4, the radius of the bucket is R. If a light ray passes at a distance r_{1} from the center of the bucket, then the angle of incidence q_{1} that the light ray makes with the normal to the bucket is given by:

**Figure 4. The relation between the apparent distance r _{2} from the center C and the actual distance r_{1} can be derived from Snell's Law to be r_{1}=r_{2}/1.33.
**

sinq_{1} = r_{1}/R **Equation 4**

The angle q_{2} that the ray transmitted outside the bucket makes with the normal to the bucket is:

sinq_{2} = r_{2}/R **Equation 5**

where r_{2} is the apparent distance from the center that is observed outside the container.

Snell's Law says that:

n_{1}sinq_{1}=n_{2}sinq_{2} **Equation ****6**

Substituting Equations 4 and 5 into Equation 6 gives:

n_{1}r_{1}/R = n_{2}r_{2}/R

Solving for r_{1 }gives:

r_{1} = r_{2}n_{2}/n_{1}

For air n_{2}=1 and for water n_{1}=1.33.^{16} Substituting these gives:

r

_{1}= r

_{2}/1.33

**Equation 7**

So the container acted as a lens that magnified the profile by 1.33. When the bucket was observed from the side, then the apparent distance r_{2} from the center had to be divided by 1.33 to get the actual distance from the center r_{1} that was being observed.

### Results

**Polygonal Rotating Vortices in a Circular Container: **

Measurements were taken using four different water depths: 2.9 cm, 5.1 cm, 7.1 cm, and 9.2 cm. The depths were measured from the top of the disc to the surface of the water while the disc was stationary.The measurements were taken at spin speeds ranging from 60 - 550 rpm.

Figure 5 shows top views and profiles of the shapes that were observed in 7.1 cm deep water at various rotation speeds. It demonstrates the shapes that were formed as the speed of the rotating disc was increased. At a disc rotation rate of 135 rpm the spinning water formed a depression in the center. The side view appeared parabolic near the center and flattened near the sides of the container. At 182 rpm the depression deepened and started to elongate. The shape as seen from the side still seemed parabolic but was starting to touch the disc. At 240 rpm an elongated opening was seen. When viewed from the side, the base of the vortex had developed lobes. At 360 rpm a triangular vortex was observed. At 392 rpm the shape was square and at 417 rpm a pentagonal shape was observed. For all water depths studied, the shape was initially circular and then as the disc rotation speed was increased, the shape developed more lobes, becoming first triangular and then square. At even higher speeds sometimes a pentagonal or hexagonal shape could be seen. The surface of the water showed some faint striations at higher rotation speeds.

**Dependence of Vortex Shapes on Water Depth and Disc Rotation Velocity:**

The disc rotation velocities at which different shapes were produced are plotted as a function of the depth of water used in the container in Figure 6. The rotation rates were accurate to ±2 rpm. The dashed lines are second order polynomial fits to the data, generated with Microsoft Excel. So for example, it shows that at 5.1 cm depth, oval shapes were seen between rotation speeds of 137 - 219 rpm. As the water depth was increased, the shapes appeared at higher and higher rotation speeds. In all cases, the polygonal shapes appeared to rotate, but at much slower rates than the Plexiglas disc was rotating.

**Figure 6. The symbols show the shape type measured at different water depths and spin speeds.** The dashed lines are second order polynomial fits to the data that were generated using Microsoft Excel. More lobes appeared with increasing disc rotation speed, and the disc rotation rates at which more lobes appeared increased with increasing depth.

**Rotating Disc in a Square Container:**

In order to test the effect of the sidewalls on the vortices, measurements were also performed using a square container. Even at disc rotation velocities as high as 680 rpm no opening was seen. The water motion in the square bucket illustrated, however, how the water flowed. The water was driven in a circular motion by the rotating disc and moved outwards due to the centrifugal force. The water piled up at the corners of the container. From the corners, the water flowed back down and towards the center of the disc.

**Determination of Slope of Water Surface and Rotational Velocity of Water at Surface from Surface Profile:**

Next, the profiles of the vortex shapes were measured and digitized. Detailed results that were obtained for three different disc rotation speeds and a water depth of 5.1 cm are shown.

The profiles of the all three vortices were imported into the Dagra software and x- and y-axes were superimposed using the known dimensions of the container. Points along the curve of each vortex profile were then digitized using the Dagra software. In the case of the triangular and square shapes, curves were digitized of the profiles at the polygon sides and the profiles at the polygon corners. All sets of data were then corrected for the horizontal magnification due to the curved container surface using Equation 7.

Figure 7 shows the digitized curves for the 5.1 cm deep water with disc spinning at 106 rpm, together with the parabolic shape calculated for uniformly spinning water using Equation 2. The curves appeared parabolic near the center of the container and then flattened towards the outsides of the container.The curves at the left and right sides appeared similar.

**Figure 7. The blue and green curves are the water surface profiles digitized for 5.1 cm water depth and 106 rpm disc rotation speed.** The red curve is the parabolic shape calculated for water spinning uniformly at 100 rpm using Equation 2 as described in the text. The measured profiles were parabolic near the center then flattened at the outsides as the water slowed down due to the drag of the walls.

At a disc rotation velocity of 292 rpm a triangular vortex shape was seen. The rotation speed of the triangular shape was determined from the video to be 78 rpm, much slower than the disc rotation speed. The profiles of the triangular vortex are shown in Figure 8 as the solid blue curve for the side of the triangle and the solid green curve as the corner of the triangle. The slope of the water surface profile at the corner of the triangle was much larger than the slope at the side of the triangle. At a disc rotation speed of 318 rpm a square vortex was observed. The rotation speed of the square vortex was measured from the video to be 93 rpm.The water profiles at the sides and corners of the square vortex are shown in Figure 9. The slope of the surface at the corners was clearly steeper than at the sides.

**Figure 9. The blue curve represents the profile of the water surface at the side of the square vortex observed for 318 rpm disc rotation speed and 5.1 cm water depth.** The green curve represents the surface profile at corner of the square vortex.The surface was steeper at the corners than at the sides.

For all of the profiles, the slopes between pairs of digitized points were calculated from the profile data using the following formula:

S = Dy/Dr = (y_{i} - y_{i-1})/(r_{i} - r_{i-1}) **Equation 8**

where y_{i} is the height of the profile at the i^{th} point and r_{i} is the r-position of the i^{th} point.

The slope curves for the vortex observed at 106 rpm disc rotation rate and 5.1 cm water depth are shown in Figure 10. The slopes at the left and right sides were similar. Equation 1 predicts that for a parabolic shaped surface, the slope increases linearly with distance from the center. The calculated slope of the parabolic shape as a function of distance from the center is shown as the red curve. The slopes match the parabolic shape well up to 5 cm from the center and then decrease farther away. Figure 11 shows the slopes calculated at the corner and side of the triangular vortex and Figure 12 shows the slopes calculated at the corners and sides of the square vortex. In both cases, the slope at the corners was larger than the slope at the sides and then decreased towards the sides of the container as the water slowed down due to the drag of the container walls. The peaks in the slope of the corner at 7.3 cm and 7.7 cm from the center may be caused by the striations that were seen in the surface.

**Figure 10. The blue and green curves are the slopes of the left and right sides of the water surface profiles calculated from the measured data for 106 rpm rotation rate and 5.1 cm deep water using Equation 8. **The red curve is the slope of the parabolic theoretical shape for a rotation speed of 100 rpm. The slopes are parabolic in the range 2 - 5 cm from the disc center.

**Figure 11. The blue curve is the slope of the water surface as a function of radial position at the side of the triangular vortex. The green curve is the slope of the water surface as a function of radial position at the corner of the triangular vortex.** The slope at the corners was clearly greater than the slope at the sides.

**Figure 12. The blue curve is the slope of the water surface as a function of radial position at the side of the square vortex. The green curve is the slope of the water surface as a function of radial position at the corner of the square vortex.** The slope at the corners was clearly greater than the slope at the sides.

The rotational velocity as a function of distance from the center of the container was then calculated from the slopes using Equation 3. The results are shown in Figure 13 for the disc rotation speed of 106 rpm and water depth 5.1 cm.The rotation velocities were close to that of the disc in the range of 2 - 5 cm from the center and then decreased rapidly towards the edges of the container. In the range 0 - 2 cm from the center, the rotation velocity was much slower than that of the disc. The slopes were very small near the center, so small errors in the y-values of the digitized points could cause errors in the calculated rotational velocities.

The rotational velocities calculated for the triangular vortex shape are shown in Figure 14. The dashed green line shows the rotation speed of the Plexiglas disc, which was 292 rpm. The purple dashed line shows the rotation speed of the triangular vortex shape, determined from the video to be 78 rpm. The solid blue curve shows the rotational speed of the water at the side of the triangular vortex shape as a function of radial distance. The solid red curve shows the rotational speed of the water at the corner of the triangular vortex as a function of radial distance. The maximum speed at the edge of the triangular vortex was 150 rpm. The speed at the corner of the triangular vortex was much higher, 200 rpm or more. Even near the center the water speeds never reached the 292 rpm speed of the rotating disc, which shows that there was some slippage between the disc and the water. Also, the rotation speeds became much lower near the sides of the container. This indicates that the water near the sides was slowed by the drag produced by the walls of the container.

**Figure 14. The dashed green line shows the rotation speed of the Plexiglas disc, which was 292 rpm. The purple dashed line shows the rotation speed of the triangular vortex shape, which was 78 rpm.** The blue curve shows the rotational speed of the water at the side of the triangular vortex shape as a function of radial distance. The red curve shows the rotational speed of the water at the corner of the triangular vortex as a function of radial distance. The rotational velocity at the corners was clearly greater than that at the sides.

The rotational velocities calculated for the square vortex are shown in Figure 15. The dashed red line shows the rotation speed of the Plexiglas disc, which was 318 rpm. The orange dashed line shows the rotation speed of the square vortex shape, determined from the video to be 93 rpm. The solid blue curve shows the rotational speed of the water at the side of the square vortex shape as a function of radial distance. The solid green curve shows the rotational speed of the water at the corner of the square vortex as a function of radial distance. The maximum speed at the edge of the square vortex was 144 rpm. The speed at the corner of the square vortex was much higher, 190 rpm or more. Even near the center, the water speeds never reached the 318 rpm speed of the rotating disc. Additionally, the rotation speeds became much lower near the sides of the container. This indicates that the water near the sides was slowed by the drag due to the sides of the container.

**Figure 15. The dashed red line shows the rotation speed of the Plexiglas disc, which was 318 rpm. The orange dashed line shows the rotation speed of the square vortex shape, which was 93 rpm.** The blue curve shows the rotational speed of the water at the side of the square vortex shape as a function of radial distance. The green curve shows the rotational speed of the water at the corners of the square vortex as a function of radial distance. The rotational velocity at the corners of the square vortex was clearly higher than at the sides of the square vortex.

In addition to the disc rotation speeds shown, the water profiles were measured for polygonal shapes in initial water depths of 2.9 cm, 7.1 cm, and 9.1 cm. The water surface profile was digitized for each water depth and disc rotation speed. The rotation rates as a function of radial distance were calculated from the measured slopes. If a polygonal vortex was observed then separate measurements and calculations were made for the edge and corner of the shape. Where possible, the polygon rotation speeds were measured from the videos. The peak velocities calculated for the edges and the corners of each polygonal shape were determined. The results are plotted in Figure 16 for the four different water depths. The green curves are the peak rotational velocities calculated for the water surface at the sides of the polygons and the purple curves are the peak rotational velocities calculated for water surface atthe corners of the polygons. The blue curves are the rotational velocities of the polygons themselves that were observed. It is seen that in all cases, the rotational velocities of the polygons themselves were much slower than the rotational velocities of the water at the sides and corners of the shapes. The water rotational velocities in turn were slower than the rotational speed of the Plexiglas disc.

**Figure 16. (a) - (d) show rotation velocities determined from the slopes of the water surface for stationary water depths of 2.9 cm, 5.1 cm, 7.1 cm, and 9.1 cm respectively.** The green curves represent the peak water rotation velocity at the edges of the polygonal vortices. The purple curves represent the water velocities measured at the corners of the polygonal shapes. The blue curves represent the velocities at which the polygonal shapes themselves rotate. The rotational velocity was higher at the corners than at the edges at most points, providing evidence for the existence of subsidiary vortices at the corners.

Additionally, in almost all cases, the peak rotational velocity of the water at the corners of the polygons was higher than the peak velocity of the water at the sides of the polygons.

### Discussion

The graph of vortex shapes versus water depth and rotation velocity in Figure 6 is qualitatively similar to what was seen by Jansson et al,^{3} but the rotation rates were higher here. For example, at 7.1 cm depth triangular shapes were observed at rates between 288 and 378 rpm, but Jansson et al observed triangular shapes at 210 - 290 rpm.^{3} Jansson et al^{3} used a container with radius 19.4 cm whereas the results here were obtained using a container with a smaller 13.3 cm radius. This suggests that the rotation rate at which the shapes form has an inverse dependence on the container radius. Since w = v/r also depends inversely on radius, then it may be that the rate at which given shapes formed depended on the linear velocity of the water. The striations in the surface at higher speeds were attributed by Jansson et al to Gortler vortices^{3} which tend to form on fluid surfaces near concave walls.^{17} The error in the measured rotation rates due to variation in the drill rotation rate was ±2 rpm, which is much smaller than the range over which the shapes were stable.

As shown in Figure 7 and Figure 10, at slow rotation speeds, such as 106 rpm with starting water depth of 5.1 cm, the profile and slope of the water surface were well-fitted within 6 cm from the center of the container by the parabolic shape calculated for water spinning uniformly at 100 rpm using Equation 2. The profile then deviated at larger distances from the center. Figure 13 shows that the calculated water velocities at the surface were in the range of 85 - 106 rpm at distances of 2 - 6 cm from the center and decreased at smaller distances or larger distances. This shows that near the center of the container the water was spinning uniformly with slight slippage if any between the water and the rotating disc. Between 0 and 2 cm distance from the center the slopes were very small, so the uncertainty in the calculated velocities was larger. At distances greater than 6 cm from the center the water rotational velocity decreased. The surface slope approached 0 at the walls of the container. The boundary condition for fluid flowing past a surface is that the tangential velocity of the fluid at the surface is zero.^{18} It is therefore believed that the decrease in velocity approaching the stationary container walls was due to the drag caused by the walls of the container due to the viscosity of the water.

At higher rotation speeds the shape of the water surface lost its cylindrical symmetry and geometric shapes were observed. To understand these exciting results it is necessary to discuss the multiple vortex model that has been previously proposed.^{3,4} This theory is schematically illustrated in Figure 17. The theory is that there is a main vortex centered on the rotating disc. Due to the effects of the stationary walls of the container, subsidiary vortices form in a geometric pattern that orbit around the main vortex. This is shown schematically in the left half of Figure 17. As the rotation velocities of the subsidiary vortices increased then eventually centrifugal force caused the center of each subsidiary vortex to become clear of water. The vortices merged to form the geometric shape such as the triangle shown in Figure 17. The effect that asymmetries of the cylinder walls can have was illustrated for the extreme case of a square container in which the water welled up at the container walls and then flowed back in towards the center along the surface, as was postulated by Jansson et al.^{3}

**Figure 17. Left diagram shows model in which multiple subsidiary vortices orbit around a central vortex. ^{3,4}** At high enough rotation speed dry areas will open up at the locations of the subsidiary vortices to form the polygonal vortex shown at right. The rotational velocity of the water at the corners of the polygonal shape should be greater than at the edges of polygonal shape.

The rotation speeds of the water at the edges of the polygonal shapes were set by the rotation speed of the main vortex, while the rotation speeds at the corners of the polygonal shapes were due to the rotation speed of the subsidiary vortices. The rotation rate of the polygonal shapes was just the rate at which the subsidiary vortices orbit the main vortex.

As shown in Figures 14, 15, and 16, the rotational velocities of the water at the corners of the polygonal shapes were generally higher than the rotational velocities at the edges of the same polygonal shapes. The vorticity of the fluid was therefore higher at the corners. This supports the theory that the corners are formed by subsidiary vortices.

Possible sources of error included errors in measurement of the water surface profiles. The horizontal dimension of the screenshots was about 250 mm. The horizontal resolution of the video was 1280 pixels. This means that each pixel represented 250 mm/1280 = 0.2 mm. Errors of 2 – 5 pixels would make an error of 0.4 – 1 mm.The points of the digitized curves were about 5 mm apart so this would make an error of from 0.4 mm/5 mm to 1 mm/5 mm or 8 % - 20 % in the slopes. From Equation 3, w depends on S^{1/2} so the error in the calculated rotational velocities would be about 4 % - 10 %. Where possible, still photos were used, which had a higher horizontal resolution of 3,648 pixels and therefore smaller errors in digitization. Additionally, there were sometimes slight variations in the disc rotation speed of about ±2 rpm or less.

A more important factor is that in the calculation of the rotational velocity from the slope of the surface in the experiment, the radial distance from the center of the rotating disc was used. This may underestimate the rotational velocity at the corners since the radii of the corners were smaller than the distance to the center of the rotating disc. In the case of points where the calculated rotational velocity at the corners was slightly smaller than that at the edges, then using the radius of the corner in the calculations would produce a rotational velocity or vorticity at the corners that was higher than at the edges.

In conclusion, in this paper it has been demonstrated that polygonal vortices can be produced in a system in which water in a circular container was driven by a rotating disc. Shapes with two, three, four, five, and six sides were seen, similar to those seen by previous researchers.^{3,4} The shapes appeared at disc rotation speeds that were higher than recorded earlier,^{3} but the container and disc diameters used here were smaller, suggesting that the rotation velocity for polygon formation depends inversely on the diameter of the container. A novel technique for determining the rotational velocity of the water at the surfaces from the measured slopes was developed. With this technique it was determined that the rotational velocity and therefore the vorticity of the water at the corners of the polygons was higher than at the sides of the polygons. This supports the theory that the polygons were formed from subsidiary vortices that were orbiting a central vortex.^{3,4}

Knowledge of asymmetries in vortices is important to understanding weather systems such as tornados and hurricanes. Predicting the patterns of vortices and their formation can also be useful in other systems where water or gas is spinning rapidly in a confined space, such as in jet engines^{19} or water pumps.

Future work could include testing liquids with different viscosities than water and testing different sizes of containers to see their effect. Oval containers with varying eccentricities could also be used in order to see what effect the asymmetry of the container has on the vortex shapes. It would be interesting to find out what determines the rate at which the polygonal shapes rotate. Vatistas et al found evidence that the rotation rate was related to the diameter of the rotating disc relative to the container size.^{4} The Navier-Stokes equations describe the behavior of fluids, including effects of viscosity.^{20} Fluid flow due to an infinite rotating disc has been solved analytically with the Navier-Stokes equations.^{21} Future work could also include solving the Navier-Stokes equations for the system described here. This would have to be done numerically, perhaps with computational fluid dynamics software such as the open source program OpenFoam.^{22}

### References

1. Lewis, B. M., H. F. Hawkins, 1982: Polygonal Eye Walls and Rainbands in Hurricanes. Bull. Amer. Meteor. Soc., 63, 1294–1301.

2. Johnston, Hamish, "Polar Vortex Replicated in a Bucket" http://physicsworld.com/cws/article/news/2008/may/08/polar-vortex-replicated-in-a-bucket

3. Jansson, T. R. N., M. P. Haspang, K. H. Jenson, P. Hersen, and T. Bohr, Polygons on a Rotating Fluid Surface," Phys. Rev. Lett. 96, 174502 (2006).

4. Vatistas, G. H., H. A. Abderrahmane, and M. H. Kamran Siddiqui, "Experimental Confirmation of Kelvin's Equilibria," Phys. Rev. Lett. 100, 174503 (2008).

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