Part III. Appendix A. Brief review from: http://en.wikipedia.org/wiki/Vorticity#Vorticity_equation Vorticity is a measure of fluid rotation. Mathematically, vorticity is a vector field and is defined as the curl of the velocity field: B. The vorticity equation describes the evolution of the vorticity of a fluid element as it moves around. In fluid mechanics this equation can be expressed in vector form as follows, where, is the velocity vector, ρ is the density, p is the pressure, is the viscous stress tensor and is the body force term. The equation is valid for compressible fluid in the absence of any concentrated torques and line forces. No assumption is made regarding the relationship between the stress and the rate of strain tensors (c.f. Newtonian fluid ). C. For any flow, you can write the equations of the flow in terms of vorticity rather than velocity by simply taking the curl of the flow equations that are framed in terms of velocity (may have to apply the 2nd Fundamental Theorem of Calculus to do this rigorously). In such a case you get the vorticity transport equation which is as follows in the case of incompressible (i.e. low Mach number ) fluids, with conservative body forces:
Transcript
Part III. Appendix
A. Brief review from: http://en.wikipedia.org/wiki/Vorticity#Vorticity_equation
Vorticity is a measure of fluid rotation. Mathematically, vorticity is a vector field and is defined as the curl of the velocity field:
B. The vorticity equation describes the evolution of the vorticity of a fluid element as it moves around. In fluid mechanics this equation can be expressed in vector form as follows,
where, is the velocity vector, ρ is the density, p is the pressure, is the viscous stress tensor and is the body force term. The equation is valid for compressible fluid in the absence of any concentrated torques and line forces. No assumption is made regarding the relationship between the stress and the rate of strain tensors (c.f. Newtonian fluid).
C. For any flow, you can write the equations of the flow in terms of vorticity rather than velocity by simply taking the curl of the flow equations that are framed in terms of velocity (may have to apply the 2nd Fundamental Theorem of Calculus to do this rigorously). In such a case you get the vorticity transport equation which is as follows in the case of incompressible (i.e. low Mach number) fluids, with conservative body forces:
with the Laplace operator.
D. Vorticity film By Ascher H. Shapiro, MIT, Cambridge, Massachusetts http://web.mit.edu/hml/ncfmf/09VOR.pdf
E. Other linkshttp://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19670028124_1967028124.pdfVorticity meter: http://www.atm.ox.ac.uk/user/read/fluids/fluidsnotes3.pdfVortex ring in the left ventricle of the heart;Vortex ring effect in helicoptersVortex ring formation and structurehttp://en.wikipedia.org/wiki/Vortex_ringSmoke ring: Cigarettes, Vapor Ring, Cooking, Volcanoeshttp://en.wikipedia.org/wiki/Smoke_ringhttp://www.wikihow.com/Blow-Smoke-Ringshttp://www.amasci.com/amateur/vortgen.htmlGround effect (cars)http://en.wikipedia.org/wiki/Ground_effect_in_cars
Figure 1 – Bottle caps with holes drilled between 10% and 90% of the bottle opening diameter. Source: Project report by Janssens, Knerr, and Kretschmer
Figure 2 shows the tank setup over a sink. We chose the sink in the faculty lounge on the second floor of the Engineering building as the location for the performance of this experiment.
Figure 2 – Tank set up over a sink with faucet above.
A rotation rate of three revolutions per second is a good one to start with (15 rev./5 s).
The Vorticity Meter
In order to measure the amount of vorticity that is in the tank at any time, a device called a vorticity meter was needed. The basic design of such a device is the conjoining of four flat rectangular blades in such a way that they all share an edge and are perpendicular to each other. The edge that they share is the axis of rotation of the meter. Figure 3 shows the basic design of the meter.
Figure 3 – Basic design of a vorticity meter.
Based on the concept of this design, six vorticity meters were built and tested for this application. Shown in Figure 4, these meters are designated one though six and will be referred to by these numbers. Table 2 gives a brief description of their construction.
Figure 4 – Vorticity meter designs proposed for the experiment.
Table 2 – Description of the vorticity meter designs.Meter Description
1 Beverage straw core with blades made from soda can aluminum.2 Floor tile spacer with hole drilled in center to allow for rotation. (Version 1)3 Floor tile spacer. (Version 2)4 Lovejoy shaft coupler. (Version 1)5 Lovejoy shaft coupler. (Version 2)6 Four thin steel plates welded together with two steel washers.
All six vorticity meter designs were tested to provide the basis for deciding which one to use in this application. They were placed on a thin rod for placement into the tank. Source: Project report by Janssens, Knerr, and Kretschmer (2011).
Figure 3: The vorticity meter used in this laboratory project.
By Steven Beck, Emily Hauter, Jeremy Hoffman, Fall 2012
Figure 1: The stationary vorticity meter
Figure 2: The suspended meter
By Michael Magsam, Phil Perlich, Jason Trahan, Fall 2012
Figure 8 – The two-liter tank marked with 1 inch Figure 9 – Floating vorticity meter. intervals. The marking on the fins make it
possible to count the number of rotations.
By Scott Gaskill, Kyle Hartman, and Jennifer Heck, Fall 2012
Figure 2
The vorticity meters had fins 0.5” tall. The length of the fins for the surface meter was 0.75” and the bottom meter had 0.625” fins. By David Hiser, Nick Jewson, and Josh Kestner
The materials could be just about anything thin and flat that can be cut into a rectangular shape.
The vorticity will need to be measured at the surface as well as at a certain depth in the tank.
Other characteristics to look for include some material that floats and another that sinks or can be
held under the surface. The vorticity meter also needs to spin freely to show the vorticity.
Starting with materials that are on hand, a metal rod was chosen. It was discovered that the shell
of a pen worked as a tube to hold the blades of the vorticity meter and would allow fairly free
rotation about the metal rod. Two types of slats from vinyl blinds, dulled razor blades and egg
carton foam were turned into blade prototypes (Figure 3). The 1/8 inch thick vinyl blind
material worked very well for a free floating vorticity meter until the vorticity began to increase
with large tank openings and began to suck the vorticity meter under the surface and even pulling
it into the tank exit. The meter design used for testing consisted of the rod for a shaft and had
one meter that sank to the bottom of the shaft while the other floated on the surface (Figure 4).
The vinyl blades, when attached to the plastic from the pen, sank as the bottom meter and the egg
carton foam connected to a section of a plastic straw floated at the surface.
Figure 3
Six vorticity meter prototypes were considered for testing.
Figure 4
Two vorticity meters were chosen to measure the magnitude of the vorticity at the free surface (right) and the magnitude of the submerged vorticity (left). The meters were colored to give more visibility during testing. By David Hiser, Nick Jewson, and Josh Kestner
A.3. Design of Vorticity Meters
By Vanessa Ray, Afrid Sarker, Eduardo SztrajtmanVanessa Ray
A device was needed to assist in measuring the vorticity within the tank. Such a device was
needed to be able to rotate around its center. The device needed to have four rectangular vanes
that were 90 apart. The previous group that ran this experiment tested six different vorticity
meters. In class we learned that the lightest design, one made out of a straw and aluminum,
worked the best. Figure 3 below shows the basic design of a sample vorticity meter. This is the
design that worked best for the previous group.
Figure 3: Basic design of a vorticity meter
In order to build off of previous designs, the group decided to also try different designs in order
to find the vorticity meter that worked the best. Floor tile spacers were purchased and tested.
Aluminum roof flashing, plastic tubing, and epoxy glue were used to create the rest of the
designs. A hole was drilled into the two floor spacers and a very thin wire was found that could
fit though the hole. The end of the wire was bent and a bead was threaded before the vorticity
meter could be put on. Before cutting the aluminum into different shapes, templates were made
out of paper to get ideas. The vorticity meters are shown in Figure 4 and Figure 5 below.
Figure 4: This picture shows the top view of all the vorticity meters that were tested.
Figure 5: This picture shows the side view of all the vorticity meters that were tested.
The group tested a total of nine different vorticity meter designs. These designs are evaluated
below. The number of the vorticity meter correlates to the number shown in the pictures. The
dimensions, weight and materials for each vorticity meter are summarized in Table 1.
Vorticity Meter 1:
This vorticity meter didn’t spin well at times. Since the height of the vanes wasn’t very large,
the sides of the vanes do not have much surface area. This floor spacer didn’t work well.
Vorticity Meter 2:
This vorticity meter worked better than the previous design. The aluminum that is wrapped
around the vanes created a larger surface area for the meter to flow with the water, but it was not
large enough to catch the vorticity for larger exit diameters.
Vorticity Meter 3:
This vorticity meter had better surface area for the water to catch, but it wobbled as it spun
making the data inconsistent.
Vorticity Meter 4:
This vorticity meter worked very well when placed at the free surface. When it was placed with
nothing connected to it, the meter would spin freely with the water as it followed the free surface
perfectly. Problems occurred with this meter when we tried to get submerged meter readings.
Vorticity Meter 5:
This meter didn’t have long enough vanes to capture the data.
Vorticity Meter 6:
Due to limitations for this design, the vanes were not exactly perpendicular to each other.
Hence, the rotation of the vorticity meter was not acceptable.
Vorticity Meter 7:
This design worked very well for the free surface and the submerged depth. The vanes were
perpendicular and the lengths of the vanes were sufficient to represent the vorticity in the water.
Vorticity Meter 8:
This vorticity meter didn’t have a large enough surface area to capture the vorticity at all exit
diameters. This design inspired the design of meter 9.
Vorticity Meter 9:
In this design, we made a larger vane that was directed vertically rather than horizontally. This
design worked well for all exit diameters. It was made of 1 piece of aluminum that was cut using
a template that we created. Since it was one piece of aluminum it was durable and was easier to
attach to the tubing.
Vorticity meter 9 was used for all experiments since, in the group’s opinion it gave a better
representation of vorticity present in the tank for all exit diameters. If two vorticity meters were
used, one for free surface and one for submerged depth, our group would suggest further testing
vorticity meter 4. The group decided to be more consistent and use only one vorticity meter for
all experiments. Our vorticity meter rod design is shown in Figure 6 below.
Table 1: Dimension, Weight and Materials of Different Vorticity Meters Tested
