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A SOAP FILM APPARATUS TO STUDY TWODIMESIOAL HYDRODYAMIC PHEOMEA
Laura Borowski*
[email protected]
Mary Lowe**
[email protected]
Robert Bailey***
[email protected]
Loyola University Maryland
4501 N. Charles Street
Baltimore, MD 21210
Abstract
Faculty members and students at Loyola University Maryland are interested in studying
laminar fluid flow phenomena for both instructional and research purposes. A relatively
simple apparatus that has been used to create nearly two-dimensional fluid flows is the
soap film tunnel. There are many variations, but all include a moving film made of soap and
water that flows between guide wires. Objects can be placed within the film, and
interference fringes that appear when the film is properly illuminated (e.g., with a sodium
lamp) enable the resultant flow patterns to be visualized.
In this undergraduate research project, stable soap films were created using a gravity-driven
soap film apparatus with a nearly horizontal test section. The hydrodynamic phenomena
investigated were the wakes behind solid obstructions. The apparatus, which has been
under development for about one year, was first enhanced to produce continuous films with
low flow rates. Tests were then run using single cylindrical obstructions. Other investigators
have extensively studied the wake behind single cylinders, so this allowed us to benchmark
our device against data from the open literature. More specifically, the Reynolds (Re) and
Strouhal (Sr) numbers were calculated for cylinders of different diameters and different flow
rates, and the results were compared with data from other investigators. Our results
generally captured the trends present in the prior work, but our data contained a significant
amount of scatter, and deviations were present.
The wakes behind multiple obstructions, both cylindrical and non-cylindrical, are not as well
understood. For this reason, additional experiments were performed, analyzing one
triangular obstruction and two cylindrical and triangular obstructions in different
orientations. Computer simulations were run using the CFD-ACE+ computational package to
generate data for comparison with the experimental results. The simulation and the
experimental results corresponded reasonably well. Though not all images matched
perfectly as the wakes developed, the comparisons gave us confidence that the simulations
are accurately depicting single obstruction flows.
During the fall of 2009, more work will be done with multiple obstructions and simulations.
Ideally, the information will be used to support research involving the mixing and
interacting of two fluids that would not mix well on their own. Our next step in reducing
uncertainty in our measurements is to use a high speed camera to refine the measurements
of Re and Sr. Ultimately, we also hope to incorporate the soap film apparatus and the
numerical simulations into undergraduate physics and engineering laboratories at Loyola.
Key Words: Soap film apparatus, Reynolds Number, vortex, laminar, CFD
*
**
***
undergraduate student, Department of Engineering
faculty member, Department of Physics
faculty member, Department of Engineering
1
Introduction and Background
Hydrodynamics, a branch of fluid mechanics, deals with flows of fluids with no density change.
The hydrodynamic phenomena investigated in this study were the wakes behind solid
obstructions in liquid flows. While other investigators have extensively studied the wake behind
a single cylindrical obstruction (see, for example references 1,2, and 3), the wakes behind
multiple obstructions in close proximity, both cylindrical and non-cylindrical, are not as well
understood. For this reason, experiments were performed for one triangular obstruction and two
cylindrical and triangular obstructions in different orientations. While investigating the wakes
behind such objects can add to a general understanding of fluid flows, our studies can also be of
help to other engineering applications and physical problems that involve such wakes.
Wake patterns depend on the dimensionless Reynolds number
Re=
ρVd
µ
(1)
determined by the density ρ, free stream velocity V, and dynamic viscosity µ of the fluid, as well
as the characteristic dimension d of the flow geometry. The value of Re is an indicator of
whether a flow is classified as laminar or turbulent. The focus of this study is on wake patterns
in laminar flows, which occur at low Reynolds numbers.
Vortices1 are frequently formed in the wakes of obstacles. Different patterns appear behind
obstacles, and these patterns can be classified according to the Reynolds number associated with
the flow. Two rows of staggered vortices can be created by oscillating eddies formed from the
two sides of a single object when the Reynolds number is in a certain range [1,2]. When 40 < Re
< ~60-100, a Kármán vortex street comprised of laminar vortices is produced behind a single
cylindrical object [2].
To visualize and study such wakes, past researchers have used a number of approaches,
including the soap film tunnel [4,5]. A film made of a soap solution is comprised of surfactant2
soap molecules covering a sheet of water. A soap film apparatus allows the investigator to create
a stable, continuously flowing film of this type. Interference fringes3 enable the flow patterns
present in the film to be visualized. Because of the film’s thickness variation and the additional
factor of air drag on the film, the question arises as to whether or not a soap film is representative
of an incompressible, two-dimensional flow. In general, if the film is thin enough and the film
velocity is slow enough, the effects of air drag become secondary, and the film may be
considered a two-dimensional liquid flow [4]. Our investigation utilizes the assumption that the
film is indeed two-dimensional, making the process of studying flows much simpler, both
computationally and experimentally. A more in-depth explanation as to why the film can be
considered as such is given in Rutgers [4].
1
Vortices represent flows where the streamlines form concentric circles or closed curves [7].
The surfactant lowers the surface tension forces and prevents the water from forming droplets.
3
Interference fringes are caused by constructive and destructive interference of light that enable small changes
in the thickness of the film to be observed.
2
2
Many different types of dynamic soap film devices have been developed, such as the vertical
apparatus driven by gravity, and the water-jet-driven horizontal soap film tunnel [3]. Past work
by Melinda Martino was done at Loyola in the summer of 2008 with a vertical apparatus, but the
velocity was too high to achieve a laminar film, and vortices could not be clearly observed. A
nearly horizontal apparatus was built by Amanda Olmsted at Loyola during Fall 2008 and Spring
2009 semesters. This apparatus could maintain a dynamic (moving) soap film and is based on a
design by Georgiev and Vorobieff [5].
1. Initial films were very unstable, and more control over flow rate and pressure head were
needed to run experiments. With this in mind, the objectives of this study were as
follows: Produce a well-controlled, stable dynamic soap film.
2. Run experimental tests with a single cylinder to investigate the performance of the
apparatus under conditions that have been studied extensively by other investigators.
3. Compare experimental results with results from other investigators.
4. Run experimental tests with non-cylindrical shapes and multiple obstructions to
investigate different wake patterns that are not as well understood.
5. Compare experimental results with results from computer simulations of twodimensional flows.
Materials and Methods
Enhancements of Soap Film Tunnel Apparatus
Enhancing the apparatus was a multi-step process. Before alterations were made, the design of
the nearly horizontal apparatus needed to be understood. An in-depth description can be found in
Georgiev and Vorobieff [5]. A horizontal apparatus was chosen over a vertical apparatus for its
ability to maintain flows at lower velocities. Olmsted built the apparatus frame out of
Unistrut™—rigid steel members that can be easily assembled into a variety of configurations. I
added a low-pressure sodium lamp (SOX) light source to illuminate the film and a peristaltic
pump (McMaster model 4049K77) to recirculate the soap solution. To gain more control over
the flow rates, a high precision needle valve (Ideal Valve V 52-1-11) was added in between the
carboy reservoir and the nozzle. A three-dimensional sketch of the enhanced apparatus is shown
in shown in FIG. 1.
The soap solution, 2-wt% Original Scent Ultra Concentrated Dawn Dishwashing Liquid and 98wt% filtered and deionized water, is dispensed through the upper reservoir (carboy with spigot)
and flows through tubing (silicone, ¼”ID; ⅜”OD) into a needle valve and then into the nozzle (1
ml pipette tip) from which two guide wires (0.80 diameter Ande monofilament fishing line)
come out. A connection piece is located between the valve and nozzle, enabling the tubing to be
separated and re-connected if there is a problem in the system. The soap film is formed between
the two guide wires. The film can be thought of in three sections. The guide wires expand
laterally in the first section at an angle of 30° from the horizontal. At the first set of adjustments,
the angle of the guide wires is changes to 5° from the horizontal. The wires remain parallel in the
second section, at a separation width of 6.0 cm, until the second set of adjustments. At the
second set of adjustments, the angle of the guide wires changes to 30° down from the horizontal.
The adjustments allow the operator to move each guide wire vertically or horizontally by turning
3
screws. A string (invisible thread for hand and machine sewing) is tied around the guide wire
with a special knot [4], goes through a hole drilled into a nylon screw, and then around the thread
of the screw. Using a Unistrut™ square nut in the back of the screw and a conventional nut and
washer enabled the adjustment process to be easier. The solution is collected in a lower reservoir,
drained out with a bulkhead connector, and then pumped (using a peristaltic pump) through more
silicone tubing to the upper reservoir.
The apparatus is 3 m long from the main upstream support beam to the main downstream support
beam. The expanding section is 0.5 m, the parallel test section is 1.3 m, and the contracting
section is 0.5 m. The angles and lengths listed describe the settings used in this study, but the
design of the apparatus allows these lengths and angles to be adjusted.
A weight is attached to the guide wires after the wires pass through a hole at the end of the lower
reservoir to add tension. Depending on the angle of the test section, the weight ranges from 1-2
kg. When the angle is 5°, the weight should be 2 kg. As the angle is lowered to get a more
horizontal test section, weight must be taken off.
Light Source
A light source was needed to illuminate the film. Previous investigators have shown that flow
patterns in the film cause small changes in the film thickness which (i) correspond directly with
the patterns and (ii) can be viewed with proper illumination [5]. Ordinary white light produces
all visible wavelengths of the electromagnetic spectrum, ranging from red to violet. Interference
would cause these colors to be seen in the film, but the presence of so many colors would be
visually confusing. A monochromatic light was needed so that the film was illuminated by one
color and that the interference fringes could be seen clearly. A low-pressure sodium light was
used. Because these lights are not sold as assembled units in the United States, the light was
purchased in three parts: a 35W SOX bulb (GE), a ballast (www.prolighting.com), and a BY22D
socket (www.interlight.biz). Once the light was assembled, a spectrophotometer (VernierSVIS,
SVIS-FIBER) was used to verify that the spectrum had one peak at 589 nm, the accepted
wavelength for sodium. The resolution of the spectrophotometer was such that the sodium
doublet (closely-spaced dual peak) could not be resolved (FIG. 2).
Video Recording
A Canon digital camcorder (ZR100) is held above the film with a ring stand and burette clamp.
The frame rate of the camcorder is the standard 30 fps, and the shutter speed is set at 1/500 s.
The light from the sodium lamp shines upward and bounces off a white poster board situated just
above the camera and oriented parallel to the film. A black board is positioned underneath the
film, resting on the main Unistrut™ piece, providing a black background in the video. The
camcorder is connected to a MacBook Pro computer with a FireWire cable. iMovie is used to
record the video onto the hard drive and do some basic editing. A ruler is placed parallel to the
film to serve as a scale in the videos.
4
Measurement of Flow and Fluid Properties
The level in the upper reservoir was kept constant to maintain a constant head on the flow
system. The flow rate was measured using a “catch and time” approach. A beaker was placed in
the lower reservoir to collect solution while a stopwatch was used to determine the time passed.
Before each obstruction experiment, fluid volumes were collected over three time intervals: 5
minutes, 4 minutes, and 3 minutes. Each volume was divided by its respective time to get an
average fluid flow rate in mL/s. The lowest flow rate we used in our trials was 0.092 mL/s and
the highest was 0.262 mL/s. The apparatus is capable of sustaining higher flow rates, but the
flows were turbulent. Velocity was measured by observing the distance a particle in the flow
moved between two sequential video frames, i.e. ∆t = 1/30 s = 0.033 s. Four trials were averaged
to calculate the mean velocity of the film in m/s. The lowest mean velocity we measured was
0.476 m/s. The highest mean velocity was 1.02 m/s. When the velocity was increased beyond
this point, the flows were too turbulent to identify the tracer particles in the solution. The density
of the soap solution was measured to be 999 kg/m3.
Different seed particles were used in the velocity measurements: dye, tea, and small bits of
paper. These particles produced wakes, indicating that they were not at the same velocity as the
soap film. The best tracking particles were found to be naturally present in the solution. They are
probably tiny bubbles or dust particles, but their composition has not been verified. They were
small enough not to leave wakes, so they appear to be moving with the film.
The angle of the parallel test section was kept at 5° down from the horizontal in all of our
experiments. Changing the angle to approximately 0° lowered the flow rate to 0.007 mL/s, but
for our purposes, the film was easier to stabilize at 5°.
Making a Film
There are two ways of making a film. The guide wires can be pinched together at the nozzle with
two fingers. As the fingers move downstream, the film is made. The downside to making the
film this way is that the wires must be drawn together, so there can be no tension in the
adjustment strings. The other way of making a film allows for the guide wires to stay adjusted.
When the side of one’s hand is wet, it can be dragged down the wires to form a film. Start with
the hand touching the top of both wires. Move over the first set of adjustments and down to the
second set of adjustments. Before reaching them, one’s other hand must be wetted and placed
under the wire to guide the film under the second set of adjustments and into the lower reservoir.
Results
Tests with Single Obstructions
A number of other investigators have studied single obstructions in soap films. The first
experiments run in this project were of this type. Single cylinders, ranging from 0.116 cm to 1.09
cm in diameter, were inserted into the film. The cylinders must be wetted before they are inserted
into the film, or the film will break. The wakes within the film were then recorded by the video
5
camera and transferred onto a computer. An example of a still shot of a Kármán vortex street is
shown in FIG. 3.
The first step in analyzing the wakes and flows is calculating the Reynolds number. Recall from
equation (1) that we need the density, characteristic dimension, velocity, and viscosity. The
density, ρ, of our soap-water solution is very close to that of pure water, 999 kg/m3. The
characteristic dimension, d, is the diameter of the obstruction. The velocity is measured for each
trial using the aforementioned process. Calculating viscosity is a more complicated process than
making a simple measurement. As described by Vorobieff and Ecke [1], film viscosity can be
broken down into contributions from both the bulk film viscosity (µB) and surface viscosity
(µSS), as expressed by the equation
µF = µ B +
2µSS
t
(2)
where t is the average thickness of the film. This equation is modeled after an equation by
Trapeznikov [8]. µB was measured to be 0.95 x 10-3 Ns/m2, using a Gilmont falling ball
viscometer. We were not able to measure µSS. Instead, we used Vorobieff and Ecke’s value for
µSS, 1.5 x 10-5 g/s. Average film thickness t was calculated by dividing the flow rate by the
product of the velocity and the width of the film [1]. Our calculations of Re=ρVd/µF ranged from
77 to 551. Because of the uncertainty in the value of µSS and the film velocity, we sought a way
to test the accuracy and precision of our Re calculations.
While investigating cylindrical obstructions, past researchers have studied the relationship
between Re and Strouhal Number [1, 6]. Strouhal number6 is defined to be
Sr =
fd
V
(3)
where Sr depends on the frequency f of vortex shedding , the diameter d of the obstruction, and
the free stream velocity V of the fluid. Vorobieff and Ecke [1] used an approximation Sr = d/l,
which is valid if the free stream velocity and the vortex advective velocity are equal. According
to Pankanin et al. [9], the vortex advective velocity is less than the free stream velocity in the
wake region near the obstruction. However, the vortex advective velocity approaches the free
stream velocity as the vortex moves away from the obstruction. This implies that Vorobieff and
Ecke’s approximation is reasonable, but we have not verified this in our experiments.
Using Re=ρVd/µF and Sr = d/l, we determined Re and Sr values for nine different trials. The
results are shown in FIG. 4 and Table 1. The results show that our calculations capture the
general trends present in the curve derived from literature data, but significant uncertainty is
present in our calculation of both Re and Sr.
Tests with Multiple Obstructions
Initial experiments were performed with two obstructions, both triangular and circular. Several
different orientations were tested (FIG. 5). The flow rate was kept at a constant 0.092 mL/s. The
6
diameter of the circular obstruction and the side of the equilateral triangle obstruction were both
0.318 cm. The distance between the obstructions from center-to-center is 0.75 cm.
Computer Simulation
The experimental results were compared with simulations run using the CFD-ACE+
computational package. CFD-ACE+ is capable of solving the partial differential equations that
describe fluid motion and uses a finite volume (finite difference) method. The geometry of the
simulations is created in one module (CFD-GEOM), the simulation is run in another module
(CFD-ACE), and the results are viewed in the third and final module (CFD-VIEW). The
boundary conditions and fluid properties in the simulations were modeled as closely to lab
experiments as possible. Ideally the wakes produced in the simulation would match the video of
the wakes formed in the lab.
Several two-dimensional representations of soap films were created in CFD-GEOM. In each
case, the film was 6.0 cm wide (y-direction), and 50.0 cm long (negative x-direction).
At the inlet (right of FIGS. 6-9), a uniform velocity was specified as a profile in time. A 2
second ramp-up time was assigned for the film velocity to go from 0 m/s to the experimentallymeasured average film velocity for the case in question. At the exit, a fixed pressure boundary
condition was assigned. At both sides of the film, a no-slip boundary condition was applied.
Computational Grid
Initially, an unstructured grid containing approximately 90,000 triangular cells was used for the
entire domain for these simulations; however, it was found that the vortices dissipated much
more quickly than was observed in the physical tests. This is known to be a weakness of the
unstructured grid approach.4 For this reason, we switched to structured grids for the majority of
the domain. Film representations made using mostly structured grids contained approximately
180,000 rectangular cells. Grid points were clustered near the obstructions and at the center of
the film, where the wake would be located (see FIG. 10). A small unstructured region was
retained around the obstructions to accommodate the irregular geometry.
Viewing the Simulations
CFD-VIEW allows the simulation results to be visualized in many different ways depending on
the options selected before the simulation is run. In addition to coloring still shots of the
simulation, the module also animated the simulation.
When deciding how to color the still shots to best visualize the wake, we first looked at the
stream function and associated streamlines. Because the vortices are being carried (advected)
downstream, the streamlines do not reveal the locations of the vortices directly. However, as an
artifact of the plotting routine, “interference” patterns do appear that indicate areas where the
vortices are affecting the streamline trajectories in a periodic pattern (see FIGS. 6a and 7a).
The influence of the vortices on the streamlines can also be seen by plotting the vertical
component of velocity. Depending on the direction of vortex rotation, one side of each vortex
4
Personal communication with R. Bailey, Loyola University Maryland, August 5, 2009.
7
will bend the streamlines up, while the other size will bend them down (see FIGS. 7c and 7d).
When comparing the streamline image to an image of the simulation colored with vorticity (curl
V), it is evident that regions of elevated vorticity correspond to the location of the vortices as
suggested by the streamline interference patterns (see FIGS. 6b and 7b). Therefore, vorticity has
been used as an indicator of vortex position in further comparisons. The program calculates the
magnitude of vorticity, but the direction of rotation can be found when looking at a picture of the
simulation colored with vertical velocity VY (see FIG. 7d).
Comparison with Experimental Results
Images comparing the simulations to the photos taken in the lab are shown in FIGS. 6-9. The
blue lines show a close correspondence between the locations of the vortices and the red lines
show little correspondence. All images show a close relationship in the initial wake.
Conclusion
All of the objectives set out for the summer were accomplished. The horizontal apparatus was
modified so that a stable film could be maintained at low flow rates. Tests were run using single
cylinders as obstructions. Using a low-pressure sodium lamp and camcorder, the wakes were
able to be viewed and recorded. The data collected was analyzed, and computer simulations were
run to provide further analysis and comparison. The simulation and the experimental results did
correspond. Though not all images matched perfectly as the wakes developed, the comparisons
gave us confidence that the simulations are depicting flow patterns behind single obstructions
with reasonable accuracy.
As this project continues in the fall, more work will be done with multiple obstructions and
simulations. Ideally, the information gained will be used to help with the mixing and interacting
of two fluids that would not mix readily on their own. Our next step in reducing uncertainty in
our measurements is to use a high-speed camera and better image processing software. Once
data are taken with the new equipment, we can again compare the Re and Sr relationship.
8
Acknowledgments
The authors gratefully acknowledge the support of the Loyola University Maryland Hauber
Undergraduate Summer Fellowship program, which supplied funding for a portion of this
research. The Loyola Physics Department also purchased much of the equipment used, and their
contributions are greatly appreciated.
References
1.
2.
3.
4.
5.
6.
7.
8.
9.
P. Vorobieff, R. E. Ecke, Cylinder wakes in flowing soap films, American Physical Society Physical Review
E, volume 60, issue 3, 2009, pp. 2953-2956
R. L. Panton, Incompressible Flow, John Wiley & Sons, 1933.
M. Beizaie, M. Gharib, Fundamentals of a liquid (soap) film tunnel, Experiments in Fluids, volume 23, 1997,
pp. 130-140
M. A. Rutgers, X. L. Wu, W. B. Daniel, Conducting fluid dynamics experiments with vertically falling soap
films,” Review of Scientific Instruments, volume 72, issue 7, 2001, pp. 3025-3037
D. Georgiev, P. Vorobieff, The slowest soap-film tunnel in the southwest, Review of Scientific Instruments,
volume 73, issue 3, 2002, pp. 1177-1184
U. Fey, M. König, H. Eckelmann, A new Strouhal-Reynolds-number relationship for the circular cylinder in
the range 47 < Re < 2 x 105, Physics of Fluids, volume 10, issue 7, 1998, pp. 1547-1549
B. R. Munson, D. F. Young, T. H. Ohiishi, W.W. Huebsch, Fundamentals of Fluid Mechanics, John Wiley &
Sons, 2009.
A. A. Trapeznikov, Proceedings of the Second International Congress on Surface Activity, 1957, p. 242
G. L. Pankanin, A. Kulińczak, J. Berliński, Investigations of Karman vortex street using flow visualization
and image processing, Sensors and Actuators, A 138, 2007, pp. 366-375
9
Tables and Figures
FIG. 1. Sketch of apparatus (based on [5])
[5]).
10
FIG. 2. Spectrum of low-pressure
pressure sodium lamp.
treet; d =0 .660, V = 60.6 cm/s, Re = 55
551 (See Results for a description
FIG. 3. Kármán vortex street;
of the measurement of Re.)
0.25
0.2
Sr
0.15
Relationship
between 1/√Re
and Sr [6]
0.1
0.05
0
0
0.05
0.1
0.15
0.2
1/√Re
FIG. 4. Comparing the Re-Sr
Sr data for a cylinder with the relationship from the literature.
11
TABLE 1. Data, from nine experiments, that was used in Re
Re-Sr comparison.
Velocity
(cm/s)
Length
Between
Vortices
(cm)
Sr
(d/l)
Film
Thickness
(µm)
µF [µss = 1.5 ×
10-8 kg/((m⋅s)]
Re
0.140
60.61
0.936
0.150
4.76
0.0073
116.8
0.173
0.660
60.61
3.427
0.193
4.76
0.0073
550.7
3
0.262
0.660
101.51
3.167
0.208
4.30
0.0079
844.7
4
0.195
0.116
67.68
0.592
0.137
4.80
0.0072
108.6
5
0.195
0.180
67.68
0.825
0.231
4.80
0.0072
169.4
6
0.234
0.116
80.30
0.592
0.195
4.86
0.0071
130.1
7
0.234
0.180
80.30
0.825
0.219
4.86
0.0071
203.0
8
0.136
0.116
53.30
0.860
0.134
4.25
0.0080
76.9
9
0.136
0.180
53.30
0.844
0.214
4.25
0.0080
120.0
8umber
Flow
Rate
(mL/s)
Diameter
(cm)
1
0.173
2
FIG. 5. Orientations of obstructions (flow is downward).
12
a.
b.
c.
FIG. 6. Different views of simulation. (a) Streamlines; (b) streamlines superimposed over the
magnitude of vorticity (curl V); (c) streamlines superimposed over the vertical velocity VY ,
where red indicates positive, upward motion and blue indicates downward, negative motion.
13
FIG. 7. Alignment of the vortices. (a) Streamlines; (b) streamlines superimposed over the
magnitude of vorticity; (c) streamlines superimposed over the vertical velocity VY. (d) arrows
showing the direction streamlines are bent by the alternating vortices.
FIG. 8. Comparison of two 0.318 cm side length triangle obstructions parallel to flow.
Simulation uses a mostly structured grid.
14
FIG. 9. Comparison of one 0.116 cm diameter circular obstruction. Simulation uses an
unstructured grid.
FIG. 10. View of two-dimensional simulation grid. The white regions show a greater number of
cells.
15