49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition
4 - 7 January 2011, Orlando, Florida
AIAA 2011-477
Transient Force on a Rotating AR 2 Semi-Span Flat Plate
John Puttmann1 and Aaron Altman2
University of Dayton, Dayton, Ohio, 45469-0238
Transient forces are measured on an aspect ratio 2 semi-span flat plate rotating about its long axis (at
mid-chord) across a variety of tip speed ratios (reduced frequencies). The flat plate is used as an
oversimplified surrogate to Savonius-type vertical axis wind turbines. Little is presently understood about
the process or importance of vortex formation and shedding throughout the rotational cycle of vertical axis
wind turbines. Rotating a thin flat plate reduces ambiguity due to Reynolds number effects on the initial
formation of vortices. Parametrically varying tip speed ratio highlights the influence of vortex convective
time on transient force. This experiment provides an interesting bridge between the flowfields associated with
flapping wings and vertical axis wind turbines. Parallels will be drawn between the two by comparing recent
work in both fields. At the low reduced frequencies tested, static flat plate theory does an excellent job of
describing the lift and drag from 45 degrees to 145 degrees angle of attack. Lifting surface theory described
the lift curve slope well in the linear region up to stall inception.
Nomenclature
AR
c
CD
CL
K
X
V∞
ρ
ω
= Aspect Ratio
Chord
= Coefficient of Drag
= Coefficient of Lift
Reduced Frequency
Tip Speed Ratio (TSR)
Freestream Velocity
Density
Angular Velocity
I. Introduction
Innumerable experiments have been run on Savonius type vertical axis turbines with relatively complex shapes
comprising the “vanes”. None of these experiments are known to be based on the simplest design, a flat plate. It is
thus difficult to determine the most basic of underlying fluid dynamic mechanisms acting on an object spinning
about a vertical axis oriented perpendicularly to the freestream. The purpose of this research was to compare the
forces exerted on a flat plate model in a uniformly imposed freestream as it rotates through 360 degrees
[continuously] in order to understand the baseline fluid dynamic behavior and to subsequently extrapolate this
behavior to Savonius type vertical axis wind turbines and flapping wings. Many characteristics of the flowfields are
expected to be similar and the low Reynolds number at which these tests are performed should make the results
comparable to results already available in the literature, but applied to different problems.
II. Background
There are two primary domains within which complementary supporting literature exists for the present work. The
first that will be addressed is within the domain of flapping wings. The second domain that will be explored is that
of the Savonius Vertical Axis Wind Turbine.
1
Mechanical and Aerospace Engineering, 300 College Park Dr., Dayton, OH 45469-0238,
[email protected], Undergraduate Student, AIAA Student Member
2
Mechanical and Aerospace Engineering, 300 College Park Dr., Dayton, OH 45469-0238,
[email protected], Associate Professor, AIAA Associate Fellow
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Copyright © 2011 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
Recently, much attention has been paid to the aerodynamics of low Reynolds number pitching and plunging
motions. The early work performed in this area typically stopped at pitch excursions under positive and negative
angles of attack less than 25 degrees. Unsteady investigative work subsequently evolved to higher angles of attack,
generally limited to less than 45 degrees angle of attack. For a review, the reader is requested to read Ol et al [1].
Recently these angles of attack have expanded up to angles in excess of 90 degrees, mostly in an effort to accurately
model bird perching. As a result, these investigations rarely evaluate dynamic pitching to angles much in excess of
90 degrees angle of attack. Of great interest in all of these investigations is the crossover point between streamlined,
viscous dominated flow and bluff body, pressure dominated flow.
In one example, McGowan et al [2] investigated changes in the lift coefficient at Reynolds Numbers of 10,000 as
an airfoil plunged and/or pitched. They studied pure pitch motions up to 21.5 degrees. They then related the pure
pitch results to pure plunge and devised a relationship. They were able to combine pure pitch motions to get the
same CL. McGowan’s research involved three analytical studies using the matching of the effective angle of attack,
quasi-steady thin airfoil theory, and Theodorsen’s theory. The theory was then tested with experimental flow
visualizations using dye flow in a water tunnel. The computational and experimental data correlated well. The
computational methods all yielded a closed-form solution for plunge height or pitch amplitude depending on which
was initially given. The Theodorsen method had resilience in finding the pitch-to-plunge ratio even when its
underlying mathematical assumptions were violated. This result bears direct relevance in that many lower order
methods have been recently employed in flowfields well beyond the limit where the small angle approximation is
valid. One example can be seen in Hammer [7] where these lower order methods have been successfully applied to
the rapid determination of flowfield characteristics in pitch-ramp-hold cases up to 45 degrees angles of attack. The
point in maximum angle of attack and the conditions where these methods are no longer applicable has yet to be
completely determined. Similar methods have been used to predict the flowfields surrounding Savonius vertical
axis wind turbines [5], [6]; however these methods have not been nearly as rigorously validated as the pitch-ramphold cases. As a result the validity of their applicability is still somewhat ambiguous.
Fujisawa et al [3] researched the flow fields in and around Savonius rotors at various overlap ratios. Using
particle image velocimetry with conditional sampling techniques he compared his data to numerical calculations
obtained using a discrete vortex method. The effects of the overlap rotor blades on the flow and the formation of the
vortices were studied. An increased overlap ratio leads to an increased flow rate through the overlap causing a
decrease in the wake width. The flowfields obtained numerically using a stationary rotor were determined to not be
accurate and for a rotating turbine it was determined to be more accurate but still inaccurate. They suggest the
assumption of flow separation at the tips of the blades was invalid.
Fujisawa et al [4] also experimented with the pressure distribution of a rotating and stationary Savonius rotor.
They measured the pressure distribution to find the power mechanism as well as aerodynamic characteristics such as
torque, power, drag, and side force. Fujisawa determined the torque in a stationary rotor was mainly due to the drag
because of the pressure distribution between the returning and advancing blade and the concave vs. convex side of
the blade. For a rotating rotor the pressure difference of the advancing convex side was low due to the separation of
the moving blade at a high tip speed ratio of 0.9. For the torque distribution the effect of the tip speed ratio was
strong for small angles of the advancing blade and large angles of the returning blade. When the tip speed ratio is
between 0 - 0.4 the torque grows on the advancing blade as the tip speed ratio increases due to the separation control
effect. When the tip speed ratio is beyond 0.4 the torque decreases on the advancing blade due to the reduced
stagnation effect. The returning blade feels a constant negative increase in the torque as the tip speed ratio increases
with the exception of very small tip speed ratios from the increased stagnation effect.
Ross et al [5], [6] investigated the blockage effects of a rotating vertical axis wind turbine. Using static wall
pressures to derive velocities and wake characteristics, the blockage was calculated. The flowfield of a vertical wind
turbine was asymmetrical and, periodic, unsteady, separated, and highly turbulent. Visual observations were also
employed to help find the wall interactions and wake propagation in both static and dynamic rotors as seen in Figure
1. These visualizations can subsequently be used to rationalize the transient force data observed for the low aspect
ratio rotating flat plate.
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Figure 1: Top: Flowfield around a stationary rotor. Bottom: Flowfield around a rotating rotor.
A. University of Dayton Low-Speed Wind Tunnel (LSWT)
The experimental integrated force data was obtained from wind-tunnel tests performed in the University of
Dayton Low-Speed Wind Tunnel (LSWT) seen in Figure 2.
Figure 2: University of Dayton Low Speed Wind Tunnel (LSWT) on the left and rotational stage mount on
the right.
The DART Corporation constructed the tunnel in 1992. The LSWT is an Eiffel type with a 16:1 contraction
ratio. The fan was designed and constructed by Hartzell and is driven by a 60 HP motor. The test section measures
30” x 30” x 90” (~0.75 X 0.75 X 2.3 m). The LSWT has six anti-turbulence screens at the tunnel inlet. The highest
flow quality operable speed range of the LSWT is from 20 ft/s (6.7 m/s) to 120 ft/s (36.7 m/s). The tunnel has a
turbulence intensity in the freestream direction of less than 0.1% throughout the test section (measured by hot wire
anemometer), and less than 0.05% throughout the center portion of the test section. For the present test, the models
are mounted on a rotary stage which is located on the top of a force balance. The system balance – rotary stage are
fixed to a support located underneath the tunnel as shown in Figure 2.
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The force sensor and rotational stage arrangement can also be seen in Figure 2. The labels from 1 to 3 are the
force sensor, rotational stage, and mounting bracket, respectively. The mount used and the orientation of plate can
be seen in Figure 3. The plate is slightly offset by one inch due to adapting a previous test setup to this experiment.
Note that the freestream goes from left to right with drag in the same direction and lift perpendicular pointing
towards the side wall. A new mounting system and rotational stage are in the process of being integrated into the test
setup; however this setup was not fully validated in time to produce updated results for this paper.
Flat Plate
Figure 3: Plate mounted in LSWT wind tunnel test section, 25 lb Strain Gage Balance
B. LSWT 25 lb Strain Gage Balance
The force balance, used to determine the forces on the model, was based on an ATI Gamma F/T Transducer as
seen in the setup in Figure 2. The axis system can be seen in Figure 4.
Figure 4: ATI Gamma F/T Transducer (Credit – ATI Industrial Automation)
A list of the maximum allowable forces and moments is listed in Table 1. Sensing ranges and resolution (typical
for a 16- bit data acquisition system) associated with the sensor calibration are also listed in Table 1. A moving
average filter was used to reduce the high frequency noise recorded.
Table 1: Sensing Range and Resolution
Component
Lift – FX
Drag – FY
Side – FZ
Yaw – TX
Roll – TY
Pitch – TZ
Rated Sensing Rates
±7.5 lb
±7.5 lb
±25 lb
±25 in-lb
±25 in-lb
±25 in-lb
Resolution
1/2560 lb
1/2560 lb
1/1280 lb
1/1280 in-lb
1/1280 in-lb
1/1280 in-lb
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C. LSWT 275 lb Rotary Stage
The floor mount orientation of the model in the test section allows for the use of a rotary stage Model RM-5
manufactured by Newmark to change the angle of attack of the models (see Figure 5). The rotary stage is designed
with a maximum load of ±275 lbs and it uses a single axis controller. It provides a resolution of 0.36 arc-secs and an
accuracy of 60 arc-sec. Table 2 shows the specifications for the rotary stage used. As the maximum rotational rate
of this rotary stage is somewhat limiting in the range of tip speed ratios/reduced frequencies it can produce, a more
capable rotary stage has been purchased and is in the process of being integrated into the UD-LSWT. The new
rotary stage will be used to explore higher values of tip speed ratio/reduced frequency that are less likely to produce
the quasi-static results obtained in this paper.
Figure 5: Newmark RM 5 Rotary Stage (Credit -Newmark Systems Inc.)
Table 2: Rotary Stage Specifications
Repeatability
Resolution
Accuracy
Gear Ratio
Max. Load
Moment
Max. Speed
Travel
5 arc – seconds
0.36 arc – seconds
72 arc – seconds
72:1
275 lbs
260 in-lb
1200 RPM
360o Continuous
The flat plate model was mounted to the rotational stage as was seen in Figure 3. The convention for the
freestream velocity and direction of the forces are also labeled in this picture. The plate has a chord of 8” and a span
of 18” giving it an aspect ratio of 2.25. The point of rotation is offset from the mid chord very slightly. The effect of
this slight “wobble” will be investigated in the future with the new experimental setup. This model size represents
greater than normal area blockage for an unsteady flow, however blockage corrections used for Savonius type
vertical axis wind turbines [5] have been used with much success at much larger area blockage ratios. These
techniques were employed to correct the results included in this paper. The resulting corrections appeared to be
unreliable when applied to the rotating flat plate as opposed to a Savonius rotor and a separate investigation will be
performed to determine the source of the inapplicability of the otherwise reliable Savonius rotor blockage
correction.
D. Test Matrix
Combinations of different rotational rates were used to create the variation in TSR and reduced frequencies. The
rotational stage was commanded into continuous motion and the resulting transient forces were recorded. The
rotational speeds included 20 deg/sec and 25 deg/sec. In order to obtain a quasi-steady state condition, five full
rotations where allowed and then data was collected for five full rotations to compare differences. The freestream
was measured at 29.4 ft/s creating a TSR of 0.0044 and 0.0054 and reduced frequency of 0.0237 and 0.0300
respectively. Higher TSR’s were studied, however for reasons still under investigation, there was a displacement in
phase as the plate passed through 180 degrees and the lift force would not go to zero until another roughly 5 degrees
had passed. This phase shift did not happen at 0/360 degrees so there is some question as to the source of the error.
It is suspected that aeroelastic effects contributed and as such the new experimental setup has been designed with a
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more robust structure. Additionally, with the reduction in dynamic pressure at speeds less than 20 deg/sec the signal
to noise ratio was too high and thus those results will not be presented.
E. Background Nomenclature
The tip speed ratio (TSR) was measured for each rotational speed and wind velocity using Error! Reference
source not found..
Equation 1
The radius of the turbine was measured from the center point of the rotation to the edge of the plate (or the halfchord in wing profile parlance).
The reduced frequency is also a function of the angular velocity and freestream velocity as seen in Error!
Reference source not found..
Equation 2
The chord, c, is the width of the plate which was measured to be 8”.Static two-dimensional lift and drag flat plate
theory was used as the baseline for comparison in the results. The equations used are as follows:
III. Results and Discussion
The primary baseline used for comparison is the theoretical static flat plate. Figure 6 shows that the lift behavior
is similar for the experimental cases up until initial separation. Agreement with the flat plate theory is quite good
beyond 45 degrees angle of attack where it is assumed the transition between streamline flow and bluff body flow
occurs. It is possible that the overshoot of a CL of 1.0 from 20 degrees to 40 degrees is an artifact of the formation
of a leading edge vortex. If this were the case, it may be increasing in circulation as the tip speed ratio/reduced
frequency are increased which would explain the increase in magnitude in lift coefficient for the higher
TSR/reduced frequency case.
X = 0.0044; k = 0.0237
Static Flat Plate
1.2
Lift Coefficient, CL
X = 0.0054; k = 0.0297
1.0
0.8
0.6
0.4
0.2
0.0
0.0
10.0
20.0
30.0
40.0
50.0
60.0
70.0
80.0
90.0
Angle of Attack (degrees)
Figure 6: Dynamic lift curve from o to 90 degrees shows expected lift behavior through separation and stall
and agreement with flat plate theory beyond 45 degrees.
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In addition, the behavior of the dynamic experimental curves approximates the Prandtl Lifting Surface Theory
aspect ratio corrected lift curve slope quite well in the linear range as seen in Figure 7.
Lift Coefficient, CL
1.0
X = 0.0044; k = 0.0237
AR Corrected ~.076
0.8
X = 0.0054; k = 0.0297
0.6
0.4
0.2
0.0
0
2
4
6
8
10
Angle of Attack (degrees)
Figure 7: Dynamic lift curve highlighting the accuracy of Prandtl Lifting Surface Theory in predicting the
dynamic lift curve slope of the flat plate.
Figure 8 is quite similar to Figure 6 however the independent axis is limited at 180 degrees angle of attack. It
can be seen in the figure that despite the presence of some hysteresis, there is good symmetry in the curve as the
flow undergoes the transition from bluff body back to attached flow in the last 35 degrees of rotation before
completing 180 degrees of rotation. Once again, the static flat plate theory agrees well in the range from 45 degrees
to 135 degrees.
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1.2
1.0
Lift Coefficient, CL
0.8
Static Flat Plate
0.6
X = 0.0044; k = 0.0237
0.4
X = 0.0054; k = 0.0297
0.2
0.0
-0.2 0
20
40
60
80
100
120
140
160
180
-0.4
-0.6
-0.8
-1.0
-1.2
Angle of Attack (degrees)
Figure 8: Dynamic lift curve from 0 to 180 degrees shows good symmetry in lift behavior with some hysteresis
Figure 9 shows CD vs. angle of attack for the theoretical static flat plate alongside the experimental data. When
comparing these cases the higher angular velocity produces a greater CD. What is not yet understood is the phase
advance in peak drag for the rotating plates relative to the value predicted by the theoretical flat plate however the
lift values in Figure 8 match up well. Also curious in the figure is that the maximum drag does match up with that
expected for a flat plate. It will be interesting to see if this trend remains when higher tip speed ratios/reduced
frequencies are tested with the new test setup. One possibility is that as the rotational rate increases a leading edge
vortex forms more prominently. This remains to be confirmed via flow visualization, however it is conceivable that
the difference in convective time/residence time for a vortex could affect the transient drag force. Additional poststall lift is observable in the lift curves and is of greater magnitude for the higher TSR/reduced frequency case.
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Stationary Flat Plate
2.0
X = 0.0054; k = 0.0297
1.8
Drag Coefficient, CD
X = 0.0044; k = 0.0237
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0
10
20
30
40
50
60
70
80
90
Angle of Attack (degrees)
Figure 9: The dynamic drag curve from 0 to 90 degrees still shows surprisingly close resemblance to the flat
plate theory.
Figure 10 compares static flat plate theory for CD vs. angle of attack from 0 to 180 degrees and it shows similar
behavior from 0 – 90 as it does from 90 – 180 degrees. There is a very slight phase shift between the theoretical
curve and the experimental ones however the agreement is still quite good given that the experimental data result
from a dynamically spinning plate and the flat plate theory should only apply to static changes in alpha.
X = 0.0044; k = 0.0237
X = 0.0054; k = 0.0297
Drag Coefficient, CD
2.0
Static Flat Plate
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0.0
20.0
40.0
60.0
80.0
100.0
120.0
140.0
160.0
180.0
Angle of Attack (degrees)
Figure 10: Drag curve showing reasonably good agreement between the two dynamic experimental cases and
the static flat plate theory with a slight phase shift present between them.
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The dynamic drag polar is presented in Figure 11 from 0 to 90 degrees. The differences in both lift and drag that
appeared small in the previous figures are emphasized when combined in Figure 11.
X = 0.0054; k = 0.0297
Static Flat Plate
2.0
X = 0.0044; k = 0.0237
1.8
Drag Coefficient, CD
1.6
1.4
90 degrees
1.2
1.0
0.8
0.6
0.4
0 degrees
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Lift Coefficient, CL
Figure 11: the drag polar magnifies the differences between the static flat plate theory and the experimental
results in the realm beyond 45 degrees.
The drag polar from 0 to 180 degrees in Figure 12 shows that the second half of the rotation which shows
behavior similar to the first 90 degrees. It is noted that the max CL from 0 to 90 degrees is greater than the max C L
from 90 to 180 degrees. This can also be attributed to stronger leading edge vortex formation influenced by the
rotational directional bias.
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Drag Coefficient, CD
X = 0.0044; k = 0.0237
Static Flat Plate
X = 0.0054; k = 0.0297
2.0
1.6
90 degrees
1.2
180 degrees
0.8
0 degrees
0.4
0.0
-1.2 -1.0 -0.8 -0.6 -0.4 -0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Lift Coefficient, CL
Figure 12: Drag polar from 0 to 180 degrees highlights the asymmetry due to hysteresis effects.
IV. Discussion
Problems still exist with plate deflections, transient inertial contributions, and precise phase alignment between
the force transducer and the rotary stage. The data still show the dynamic effects of the rotation and that information
can be reliably used as verification of the correct phenomenological behavior. Future experiments will have a
MicroLYNX and LabVIEW common trigger across the two interfaces in order to produce the exact alignment of the
plate and force data acquisition.
Since the plate was kept as thin as possible to remove ambiguity with respect to vortex formation location, this led
to slight aeroelastic deflection of the top of the plate. Associated noise was subsequently observed in the lift and
drag measurements. This problem can be overcome by working at lower air speeds or having a thicker plate.
V. Conclusions
The transient forces on the rotating flat plate in this experiment are compared to the behavior of two dimensional
static flat plate theory. The behavior closely mimics the theoretical result with a variation only in the initial 45
degrees. Of the variance from theoretical behavior, only the CL varies greatly. Lifting surface theory predicts the lift
curve slope well for the linear portion of the lift curve. Once the rotating plate reaches 45 degrees, the behavior
suddenly becomes similar to the static flat plate. This behavior is very likely the result of the formation and shedding
of a leading edge vortex and has serious implications for aerodynamic prediction techniques for use in real-time
vehicle control of perching MAVs. The CD graph is almost identical to the static flat plate theory. The slope of the
drag increase is nearly the same but initial increase in slope occurs earlier. In general, the static flat plate theory
described the dynamically rotating flat plate well. The next logical step is to repeat these tests at higher tip speed
ratios/reduced frequencies to see how well the flat plate theory applies outside of the quasi-steady regime. This
result does seem to imply that at least at such low reduced frequencies, lower order vortex methods and quasi-steady
methods could be used to predict lift and drag at angles of attack in excess of 45 degrees since the flat plate theory
results match up so well with the experiments.
VI. References
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[1] Ol, M., Altman, A., Eldredge, J., Garmann, D., Lian, Y., Résumé of the AIAA FDTC Low Reynolds Number
Discussion Group’s Canonical Cases, AIAA 10-1085, 48th AIAA Aerospace Sciences Meeting and Exhibit, January
2010, Orlando, Florida.
[2] McGowan, G. A., Gopalarathnam, A, Ol, M, Edwards, J, (2009). Analytical, Computational, and Experimental
Investigations of Equivalence Between Pitch and Plunge Motions for Airfoils at Low Reynolds Numbers, AIAA 09535, 47th AIAA Aerospace Sciences Meeting and Exhibit, January 2009, Orlando, Florida
[3] Fujisawa, N., Experimental Study on the Aerodynamic Performance of a Savonius Rotor, ASME Journal of Solar
Energy Engineering August 1994, Vol 116
[4] Fujisawa, N., Velocity Measurement and Numerical Calculations of Flow Fields in and Journal of Wind
Engineering and Industrial Aerodynamics, Journal of Wind Wngineering and Industrial Aerodynamics 59 (1996)
39-50
[5] Ross, I., Altman, A., Bowman, D., Mooney, T., & Bogart, D., Aerodynamics of Vertical-Axis Wind Turbines:
Assessment of Accepted Wind Tunnel Blockage Practice, AIAA 10-397, 48th AIAA Aerospace Sciences Meeting
and Exhibit, January 2010, Orlando, Florida.
[6] Ross, I., Wind Tunnel Blockage Correction an Application to Vertical-Axis Wind Turbines, Master’s Thesis,
University of Dayton, May, 2010.
[7] Hammer, P, Altman, A, Eastep, F, Discrete Vortex Method Investigation of Canonical Pitch Ramp Hold Case,
49th AIAA Aerospace Sciences Meeting and Exhibit, January 2011, Orlando, Florida.
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