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Delta-Wing Vortex Lift Enhancement Using Oblique Channeled Distribution
ME 4335: MECHANICAL ENGINEERING LABORATORY
MEAG MCNARY, RUBEN NUÑEZ III, RYAN PARKER, ADAM EAKER, DREW
WAGGONER
ADVISOR: DR. STEPHEN MCCLAIN
DEPARTMENT OF MECHANCIAL ENGINEERING
BAYLOR UNIVERSITY
2 MAY 2011
FINAL REPORT DELTA-WING VORTEX LIFT ENHANCEMENT USING OBLIQUE CHANNELED DISTRIBUTION
I. ABSTRACT
Flow development across a delta wing is very intricate and depends highly on angle of
attack and delta wing geometry. During reasonable angles of attack, vortical formations
allow for an elevated coefficient of lift. However, at high angles of attack, vortical
breakdown and separation occurs, causing decreased wing performance. In order to diminish
the vortical breakdown phenomena, this experiment helps to quantify the steady flow effects
of obliquely aligned channels interacting with vortical structures attached to a delta wing
micro-unmanned air vehicle. The lift and drag coefficients associated with the delta wing
structure are of particular importance to this project. The experiment determines how a delta
wing is affected by the addition of obliquely aligned channels. Two different models were
tested: a control model with a smooth surface, and a model with obliquely aligned channels.
For each model the lift and drag forces were measured at varying angles of attack, ranging
from zero to forty-five degrees (encompassing stall) in five degree increments in order to
calculated lift and drag coefficients. It was found that the addition of the obliquely aligned
channels did not improve the performance of the delta wing. The smooth wing achieved a CL
of 1.44 and CD of 0.69 while the channeled wing had a CL of 1.34 and CD of 0.66 when tested
at a Reynolds number of 179,260. Recommendations for future tests would be to run the
experiment at higher Reynolds numbers and to test delta wings with various distributions of
obliquely aligned elements.
2
II. TABLE OF CONTENTS
I. ABSTRACT ............................................................................................................. 2
II. TABLE OF CONTENTS ......................................................................................... 3
III. LIST OF TABLES AND FIGURES ........................................................................ 4
IV. NOMENCLATURE ................................................................................................. 5
V. INTRODUCTION .................................................................................................... 5
VI. EXPERIMENT METHODS .................................................................................... 9
VII. RESULTS AND DISCUSSION ............................................................................ 17
VIII. CONCLUSIONS AND RECOMMENDATIONS ................................................. 19
IX. ACKNOWLEDGEMENTS ................................................................................... 20
X. REFERENCES ....................................................................................................... 21
XI. APPENDIX ............................................................................................................ 22
3
III. LIST OF TABLES AND FIGURES Table 1: Nomenclature 5 Table 2: Equipment and Support 10 Table 3A: Device Uncertainties 14 Table 3B: Smooth Wing Uncertainties 14 Table 3C: Channeled Wing Uncertainties 15 Figure 1: Experimental Setup 11 Figure 2: Initial Schedule Summary 16 Figure 3: Final Schedule Summary 16 Figure 4: Coefficients of Lift and Drag 18 Figure 5: L/D of Smooth and Channeled Wings 19
4
IV. NOMENCLATURE
Below is a list of variables that are used in the report. This list is made available for quick
reference.
Table 1: Nomenclature
Lift Coefficient Drag Coefficient
c Chord D Drag L Lift Planform Area
s Span V Velocity of Air Λ Sweep Angle
Density of Fluid
V. INTRODUCTION
The delta wing is a wing in the shape of a triangle. It is named after the Greek uppercase
letter delta, ∆, because it is similar in shape. The concept of the delta wing suggested that
there was no need for a tailplane, which was commonly used for equilibrium and stability
prior to delta wing development. At low speeds, it was difficult to ensure continuous flight of
the delta wing, but at higher speeds, it was more efficient than its other designs. This also
significantly simplified plane structures and proved to be extremely easy to manufacture.
Additional benefits of the emerging delta wing were that it allowed the aircraft to travel at
supersonic speeds and have significantly improved maneuverability. This increased
performance is due to the large sweep angle on the rear, which prevents the wings from
contacting the shock wave boundary which forms at the nose (Benson, [1]). Another benefit
that was realized after an immense amount of research and investigation was that the body of
the plane could be combined with the wings to make for a wholly aerodynamic structure that
minimized losses but also has an increase on drag.
5
One benefit of the delta wing design that is of great concern in this research is the wing’s
effectiveness over a high range of attack angles. This is due to the development of vortices
on the wing. Delta wings are used in various applications where performance and
maneuverability are crucial. The constraining mechanism for this effectiveness is governed
by the vortical flow, occurring at the leading edge of the delta wing. At moderate angles of
attack, the vortical flow is robust and promotes a pressure decrease over the wing, creating an
increase in lift (Mitchell and Delery, [2]). At high angles of attack these vortices tend to
break down before reaching the trailing end of the wing. The breakdown location tends to
oscillate from the front of the wing to the rear in an unsteady manner. Thus, these oscillations
cause vibrations and buffeting of the aircraft, resulting in more turbulence, wing fluctuation,
and an increase in drag on the delta wing (Gursul, [3]).
Other than oscillation and the breakdown of vortices, much about vortex behavior is
unpredictable. However, these oblique elements are meant to control the physics of air flow
on micro-air vehicles operating at a low speed and cause complete vortex circulation,
promoting an enhanced overall lift. The research that has already taken place demonstrates
turbulent and transitional flow with increasing Reynolds Numbers, as well as flow
visualization showing vortical substructures and the breakdown that occurs.
The most common justification to instability in vortices credits wave propagation, which
is highly dependent on sweep angle and angle of attack. Lower sweep angles limit this
breakdown, while an increased angle of attack induces vortex breakdown. The best way to
6
quantify these characteristics is to measure the lift and drag, and then calculate the lift and
drag coefficients using the equations below.
(1)
(2)
In the past, research has been done to improve the performance of delta wings by delaying
vortex breakdown through mechanical and pneumatic methods. Mechanical methods attempt
to control vortex breakdown by changing the surface of the wing, while pneumatic methods
attempt to introduce perturbations through forced airflow. However, this report is only
concerned with mechanical methods.
Using jet flaps is one such mechanical device that has been researched in the past. These
flaps can be placed at the leading edge or trailing edge and actively control the vortical flow
around a wing by using actuators to adjust the angles of the flaps. Leading edge flaps
typically improve the lift to drag ratio, while trailing edge flaps typically increase the
stability of the delta wing while only slightly increasing the lift (Mitchell and Delery, [2]). A
disadvantage to jet flaps is that they can be difficult to implement due to the complexity of
adding actuators and altering the wing structure.
Piezoelectric strips are another type of mechanical device that have been researched to
combat the effects of vortical breakdown. These strips bond to a wing and actively control
wing oscillations by countering wing flutter. This is done by sensing oscillations and creating
7
forces that negate the motion of the wing (Shrivastava, [4]). The biggest benefit to these
strips is that they are affordable, lightweight, and easy to manufacture.
Obliquely aligned elements in the form of raised elliptical elements have also been
researched because of their ability to improve the performance of a delta wing by
manipulating air flow. The elements are expected to enhance vortex lift by inducing
secondary flows. The increase in complete circulation and lift are accomplished by
promoting a higher lateral flow in the near-wall region to reduce the resistance to an attached
vortex (McClain, [5]). It is expected that obliquely aligned channels will have a similar effect
as the raised elements and will be the focus of this research.
The objective of the experiment is to determine and quantify any difference between a
delta wing with obliquely aligned channeled elements versus a smooth delta wing. Prior
research has been done proving the existence of an induced secondary flow when
directionally aligned surface elements are mounted to a flat surface, and it is desired to see if
the same is true for channeled elements. Complete vortical circulation could occur due to
these channeled elements.
It is theorized that the channels inset in the delta wing will provoke transverse flow in
turbulent boundary layers and reduce surface resistance for vortical reattachment, causing
increased lift with a limited drag increase. Theoretically, the vortices will have more surface
area allowing manipulation of the air on the surface of the wing to cause the vortices to
reattach, resulting in a significant increase in lift.
8
In order to determine how the performances of delta wings are affected by the added inset
elements, two different models were tested. The tested models were comprised of a control
model with a smooth surface, and one model with a predetermined distribution of obliquely
aligned channels. For each model the lift and drag forces were measured at varying angles of
attack, going from zero until 45 degrees, in five degree increments. By determining these
coefficients at each angle of attack and oblique channel placement, the steady flow effects of
the channeled elements on vortical flow are demonstrated and quantified.
VI. EXPERIMENT METHODS
Prior to experimentation, the delta wings as well as a connector piece used to attach the
delta wing to the force balance were machined using a Hass VF-1 Vertical Machining Center
in the Baylor University machine shop. A LabVIEW 7 program was also created, and force
balance calibration was performed. The experiment was conducted within the test section of
Baylor University’s subsonic wind tunnel. One delta wing was connected at the wing’s
center of gravity to the force balance at varying attack angles from 0 to 45 degrees.
Adjusting the attack angle was accomplished by employing a set screw and confirming a five
degree incremental difference by using a digital inclinometer. The inclinometer was placed
on the top centerline of the delta wing to measure the attack angle. The calibrated force
balance was then connected to a computer operating LabVIEW via data acquisition cards. A
pitot - static tube was placed on a traverse above and upstream of the delta wing and
connected to the Omega PCL-2A to transmit pressure differences. The Omega PCL-2A was
then connected to the computer via a data acquisition card to transmit and record this
information. Figure 1displays how the apparatus was placed within the test section of the
wind tunnel. Table 2 lists the required equipment for experimentation.
9
Table 2: Equipment and Support
EQUIPMENT NAME EQUIPMENT MODEL AND SERIAL NUMBER
Wind Tunnel Baylor University Subsonic Wind Tunnel Model 406(B), Serial Number: 41403
Inclinometer 7 in. Multi-Function Digital Torpedo Level Serial Number: 320.48295
Omega PCL-2A PCL-2A Serial Number: 50294 Pitot – Static Tube Pitot - Static Tube Force Balance ELD Force Balance Scale Ohaus Scout Pro SP2001
Serial Number: 7123241518 Barometer Oakton Barometer with Digital Thermometer,
Model Number:03316-72 Cold Junction Omega Cold Junction, Model Number: CJ-T Thermocouple Type T Vertical Machining Center Hass VF-1 VMC
The only apparatus that needed to be calibrated was the force balance. The force balance
was calibrated to the range of 0 to 1.5 pound for drag force and 0 to 1.5 pound for lift force.
This range was accomplished by placing known weights on the force balance, receiving the
voltage output via LabVIEW, and creating a trendline graph that allows for a voltage to
pound-force conversion describing the exact relationship between the two. All of the other
apparatus used was calibrated by the manufacturer and does not need additional calibration.
Below is the specific setup of this experiment.
10
Figure 1: Experimental Setup
The following procedure has been used for this project.
1. Fasten the force balance to the calibration stand.
2. Connect the force balance to the computer using the serial port.
3. Calibrate the force balance from 0 to 1.5 lb. for lift and drag. (vertical and horizontal)
Measure the weight of the cup that will hold the lead pellets using the Ohaus Scout
Pro 2000g.
Take LabVIEW reading of force balance when it is unloaded.
Measure 0.1 lb of lead.
Put the lead in the cup and attach it to the force balance.
LabVIEW will give an output voltage that corresponds to the weight of the tin cup.
Continue adding 0.1 lb increments of lead and take readings for each increment until
the force balance is loaded to 1 lb.
PCL-2A
LabView Program
Pitot – Static Tube
Force Balance
11
Produce a calibration curve for the force balance in Microsoft Excel using the
LabVIEW output.
The lift calibration is now complete. Perform the same procedure for drag calibration
by flipping the force balance mechanism.
4. Prepare the wind tunnel for testing.
Hook up pressure probes in the wind tunnel testing section.
Turn on the Velmex Controller, which will allow the traverse to move.
Adjust the pitot - static tube to the desired position in the wind tunnel using the
Velmex Controller.
Attach the force balance to the bottom plate of the wind tunnel test section.
5. Input required preliminary data to LabVIEW.
Ambient Temperature, T.
Turn on the cold junction compensator (Thermocouples).
Record Temperature
Turn off the cold junction compensator.
Pressure, P, is found using the barometer found on the wall of the lab.
R is the specific gas constant for air.
Calculate the density of air using .
6. Attach the smooth delta wing to the force balance by tightening the connector screw.
7. Secure the base to the wind tunnel.
8. Open the small access window to the wind tunnel.
9. Use the inclinometer to set the delta wing at a zero angle of attack, and carefully tighten
the set screw.
12
10. Close the access window.
11. Take an offset for a 0 degree angle of attack in LabVIEW.
12. Turn the wind tunnel on and set the flow to a velocity of 15 m/s.
13. Wait for the air flow to become steady.
14. Take 5 data points consisting of an average of 10,000 individual points taken over 10
seconds using LabVIEW.
15. Turn the wind tunnel off.
16. Open the access window.
17. Loosen the set screw.
18. Increase the angle of attack of the delta wing by 5 degrees using the inclinometer, and
carefully tighten the set screw.
19. Close the access window.
20. Repeat steps 12 through 19 until a 45 degree angle of attack is tested.
21. Remove the wind tunnel base and remove the smooth delta wing from the force balance.
22. Repeat steps 6 through 20 using the channeled delta wing.
23. Remove the wind tunnel base and remove the channeled delta wing from the force
balance.
The lift and drag forces of each delta wing were experimentally determined in this project,
and from this the lift and drag coefficients were calculated (1). These calculations have
sources of error that come from uncertainties associated with instruments and methods used
in the experiment. The overall uncertainty analysis was done using the Kline-McClintock [6]
in Mathcad 15, and the calculations can be found in Appendix iii.
13
Some of the values used in the analysis were assumed based on preliminary specifications
and typical expectations in order to predict the uncertainty before the experiment was carried
out. The delta wing parameters were initially determined in order to be able to be printed
within a Dimension SST 768 Solid printer, and the expected lift and drag coefficients were
based on the delta wing having lift and drag coefficients similar to that of a vertical plate.
The uncertainties of each contributing instrument can be found in Table 6A. The first
source of error comes from the pressure measurements made by the pitot - static tube
connected to the PCL-2A. The uncertainty of the pitot - static tube was stated to be 0.06% of
the full scale. The small pressure differences that are associated with the low free stream
velocity of 15 m/s contributes to the uncertainty in the pressure readings. The uncertainty of
the lengths produced by the Hass VMC was found to be 0.0002 inches. That uncertainty was
stated by the manufacturer and corresponds to the positioning accuracy and repeatability.
The error of the lift and drag forces also comes from the uncertainty of the force balance,
which is 0.001 lbf. It should be noted that the device which had the biggest contribution to
the uncertainty was the force balance, because only 9.72% of the full range of lift and
22.68% of the drag was used. The resulting uncertainty for the lift was on average 1.17%
for the smooth wing and 1.23% for the channeled wing. The resulting uncertainty for the
drag coefficients was on average 1.18% for the smooth wing and 1.23% for the channeled
wing. A detailed listing of the uncertainties by angle of attack can be found in Table 6B a
smooth wing and Table 6C for a channeled wing.
Table 3A: Device Uncertainties
Pitot Static Tube 0.00002168 psi Haas VMC 0.0002 in Force Balance 0.001 lbf
14
Table 3B: Smooth Wing Uncertainties
Uncertainties: Smooth Wing
Attack Angle (degrees)
Lift Coefficient Uncertainty
Drag Coefficient Uncertainty
0 0.39534682 0.00518327 0.05906726 0.00210644
4.9 0.65563342 0.00807623 0.08202781 0.00219825
10.1 0.85811464 0.00856068 0.14412538 0.0024109
15.1 1.08356801 0.01142 0.24158164 0.00319765
20.1 1.22987609 0.01077 0.36603878 0.00371866
25 1.3642587 0.02079 0.51965474 0.00812705
30.1 1.44472537 0.01358 0.68861494 0.00671125
35 1.34143904 0.01946 0.78018633 0.01144
40.5 1.02357147 0.64799467
45.7 0.96679552 0.69848001
Table 3C: Channeled Wing Uncertainties
Uncertainties: Channeled Wing
Attack Angle (degrees)
Lift Coefficient Uncertainty
Drag Coefficient Uncertainty
0 0.410318531 0.00512219 0.060395681 0.0021134
5 0.629921056 0.00714793 0.091825243 0.00222284
10 0.838283826 0.01126 0.150896206 0.0028368
15.1 1.019184535 0.00991963 0.243754295 0.00305657
20 1.220204072 0.01424 0.367174079 0.00467889
24.8 1.328386975 0.01369 0.507141135 0.0054085
30 1.336950184 0.01819 0.657060318 0.00910697
34.9 1.354045736 0.02044 0.743972801 0.01136
39.7 1.09861577 0.645325258
44.8 1.049175114 0.691446995
Upon receiving the research project on the delta wing, a schedule was determined to
project the amount of time that was needed to research the theory behind delta wings and
their development, as well as to begin the testing of prototype delta wings for further analysis
on ways to increase performance. Fig. 2 is the initial timeline that was employed. Fig. 3 is
what actually happened over the course of the semester. Not all of the desired research was
15
accomplished; however, a firm basis for future research was developed as well as results that
will be useful in that research.
Figure 2: Initial Schedule Summary
Figure 3: Final Schedule Summary
6‐Feb 16‐Feb 26‐Feb 8‐Mar 18‐Mar 28‐Mar 7‐Apr 17‐Apr 27‐Apr 7‐May
Preliminary Data Collection
Test Plan Due
Develop Connector Piece
Develop Rotating Mechanism
Printing of Delta Wings
Data Collection
Final Presentation
Final Project Due
6‐Feb 16‐Feb 26‐Feb 8‐Mar 18‐Mar 28‐Mar 7‐Apr 17‐Apr 27‐Apr 7‐May
Preliminary Data Collection
Research
Test Plan Due
Work on Report
Develop Connector Piece
Develop Rotating Mechanism
Manufacturing of Delta Wings
Data Collection
Final Presentation
Final Project Due
16
VII. RESULTS AND DISCUSSION
The tests were carried out at an average air velocity of 15.09 m/s resulting in an average
Reynolds Number of 179,000. As can be seen in Fig. 4 the lift and drag coefficients of the
smooth and the channeled delta wings follow the same general trend, but the channeled wing
generally has lower coefficients of lift and drag up to stall. Both wings can be seen to stall at
approximately 35˚ and it is interesting to note that the tests show the channeled wing
achieving a moderately higher lift coefficient after stall. From this test it appears to be
conclusive that the oblique striations did not enhance the lift, and instead moderately reduce
it. Furthermore this suggests that the channels did not enhance vortical reattachment as was
theorized. It is possible that this is due to the elements being machined into the surface rather
than being on top, and thus did not create a secondary flow as was seen in prior research.
17
Figure 4: Coefficients of Lift and Drag
Similar to the comparison of lift and drag coefficients, the trend in lift to drag ratio for
both wings is very similar as seen in Fig. 5. Once again the smooth wing generally exhibits a
higher ratio as well as a larger maximum ratio. The maximum difference in lift to drag ratio
is 14.2% at an angle of 5˚. As seen in the trend for lift and drag coefficients, the only time the
channeled wing exhibits a higher ratio is after stall, which occurred at 35˚. This further
supports the conclusion that the channels do not enhance the general performance of the
wing. It is possible that similar structures could be used to enhance performance at angles of
attack greater than stall.
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
1.60
0 10 20 30 40 50
Coefficients of Lift and Drag
Angle of Attack (deg)
Coefficients of Lift and DragRe=179,260 V=15.09m/s
Smooth CL
Smooth CD
Channeled CL
Channeled CD
18
Figure 5: L/D of Smooth and Channeled Wings
VIII. CONCLUSIONS AND RECOMMENDATIONS
The results of the testing conclude that the channeled wing design decreases both lift and
drag coefficients from the values seen with the smooth control model testing. Even though
the channeled wing design did not offer general improvement over this control model, the
results indicate that improvements could be possible. Because the channeled design offers
improved lift and drag after stall, the design could be used to increase performance at
extremely high angles of attack.
In order to determine the validity of this assumption, more testing needs to be done with
the channeled model in the wind tunnel. It is suggested that the channeled design be subject
0
1
2
3
4
5
6
7
8
9
0 10 20 30 40 50
L/D
Angle Of Attack (deg)
L/D: Smooth and Channeled WingsRe=179,260 V= 15.09 m/s
L/D Smooth
L/D Channeled
19
to higher Reynolds numbers in order to see the possible effects of the design. It is possible
that the higher Reynolds numbers will decrease the size of the boundary layer and allow the
same size channels to have a higher impact on improving lift and drag coefficients.
In addition to testing the channeled model at higher Reynolds numbers, a prototype
containing obliquely aligned roughness elements should to be tested. The roughness
elements are theorized to have a greater affect on the breakdown of the vortical structures
associated with the flow across a delta wing aircraft. By delaying or preventing altogether
this breakdown, the oblique roughness elements will increase lift and drag coefficients at
higher angles of attack and prevent other disturbances such as buffeting of the aircraft.
While simply placing the oblique roughness elements on the aircraft should improve the lift
and drag characteristics, multiple arrangements of these elements need to be tested. The
testing of various patterns, sizes, and arrangements of the oblique distributions will determine
the true impact of these structures in regards to its performance. Ideally, the arrangements
could eventually be tailored to specific applications and environments for which the micro-
unmanned aircraft will be used.
IX. ACKNOWLEDGEMENTS
The authors thank Dr. Stephen McClain for his guidance and support in this project. The
authors also thank Ashley Orr, Lance Case, Steven Mart, and Tim Burdett for their help in
making the delta wings, constructing the connector piece for the force balance, setting up the
wind tunnel, and LabVIEW programming.
20
X. REFERENCES
[1] Benson, T. Normal Shock Waves. 9 December 2008. National Aeronautics and Space
Administration, Glenn Research Center. 22 April 2011.<http://www.grc.nasa.gov/WWW/K-
12/airplane/normal.html>
[2] Mitchell, A.M., Delery, J., Research into vortex breakdown control, Progress in
Aerospace Sciences, Volume 37, Issue 4, May 2001, Pages 385-418.
[3] Gursul, I., 2003, “Review of Unsteady Vortex Flows over Slender Delta Wings,” AIAA
21st Applied Aerodynamics Conference. Copyright Clearance Center, Inc., Danvers, MA.
Vol. 42, No. 2.
[4] Shrivastava, S., 1998, “Aerolastic Oscillations of a Delta Wing with Bonded Piezolectric
Strips,” Masters of Engineering thesis, McGill University, Montreal, Quebec, Canada.
[5] McClain, S.T., 2010, “Delta Wing Vortex Control and Lift Enhancement Using Active
Arrays of Oblique Elements,” Technical Report, Baylor University, Waco, TX.
[6] Kline, S.J., and F.A. McClintock. Describing Uncertainties in Single-Sample
Experiments. Mechanical Engineering: the journal of the American Society of Mechanical
Engineers. January 1953, pp.3-8.
21
XI. APPENDIX
i. Preliminary Sizing Data
ii. Experimental Data
iii. Uncertainty Analysis
iv. Executive Summary
22
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27
Appendix ii: Experimental Data
Smooth Wing Angle of Attack Lift(lbf) CL
Lift Uncertainty Std. Dev Drag(lbf) CD Drag
Uncertainty Std. Dev
0 0.2030946 0.40 0.0051833 0.005650 0.03034350 0.06 0.0021064 0.005786 4.9 0.3386498 0.66 0.0080762 0.005653 0.04236931 0.08 0.0021983 0.005626
10.1 0.4416888 0.86 0.0085607 0.005694 0.07418424 0.14 0.0024109 0.005624 15.1 0.5550317 1.08 0.0114200 0.005652 0.12374487 0.24 0.0031977 0.005675 20.1 0.6339781 1.23 0.0107700 0.005686 0.18868736 0.37 0.0037187 0.005779
25 0.7042753 1.36 0.0207900 0.005637 0.26826403 0.52 0.0081271 0.005881 30.1 0.7384211 1.44 0.0135800 0.005676 0.35196268 0.69 0.0067113 0.006677
35 0.6835672 1.34 0.0194600 0.005678 0.39756717 0.78 0.0114400 0.008312 40.5 0.5248721 1.02 0.005712 0.33230145 0.65 0.007926 45.7 0.4939855 0.97 0.005666 0.35688853 0.70 0.006889
Average 0.54 1.05 0.012230 1.169% 0.2166313 0.4227772 0.0049888 1.18%
Channeled Wing Angle of Attack Lift(lbf) CL
Lift Uncertainty Std Dev Drag(lbf) CD Drag
Uncertainty Std. Dev
0 0.2089862 0.41 0.005122 0.005571 0.03076090 0.06 0.002113 0.005722 5 0.3228555 0.63 0.007148 0.005579 0.04706327 0.09 0.002223 0.005724
10 0.4323125 0.84 0.011260 0.005473 0.07781855 0.15 0.002837 0.005562 15.1 0.5220893 1.02 0.009920 0.005475 0.12486636 0.24 0.003057 0.005659
20 0.6286382 1.22 0.014240 0.005497 0.18916378 0.37 0.004679 0.005680 24.8 0.6793303 1.33 0.013690 0.005501 0.25934972 0.51 0.005409 0.005652
30 0.683492 1.34 0.018190 0.005545 0.33591187 0.66 0.009107 0.006319 34.9 0.6897596 1.35 0.020440 0.005543 0.37898809 0.74 0.011360 0.007356 39.7 0.5604469 1.10 0.005536 0.32920541 0.65 0.009086 44.8 0.5376831 1.05 0.005555 0.35435323 0.69 0.007981
Average 0.52 1.02 0.012501 1.23% 0.213 0.416 0.005098 1.226%
28
Lift/ Drag
Angle of Attack Smooth Channeled Percent Difference
0 6.6931838 6.7938899 -1.50% 5 7.9928088 6.8600312 14.17%
10 5.9539438 5.5553917 6.69% 15 4.4852906 4.1811846 6.78% 20 3.3599397 3.3232481 1.09% 25 2.6253065 2.6193602 0.23% 30 2.0980096 2.0347361 3.02% 35 1.7193754 1.8200034 -5.85% 40 1.579506 1.7024231 -7.78% 45 1.3841451 1.5173646 -9.62%
Average 3.7891509 3.6407633 0.72%
y = 125.351578x - 8.437871R² = 0.999910
y = 7.0268863x + 0.0076874R² = 0.9999872
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
0.00 0.05 0.10 0.15 0.20 0.25
Wei
ght [
lbf]
Voltage [V]
Force Balance Calibration Curve:Voltage vs. Force
Lift
Drag
Linear (Lift)
Linear (Drag)
29
Appendix iii Uncertainty Analysis: Smooth Wing
Kline-McClintock Uncertainty Analysis for Drag and Lift Coefficients on Delta Wing:
Air Properties: From Fluids Textbook:
ν 15.68 106
m
2
s Uν 0
m2
s
ρ 1.18517kg
m3
0.07399lbm
ft3
Uρ 0.0006kg
m3
3.74568 105
lbm
ft3
Flow Proerties:
vmax 15.0855m
s so the wind tunnel pressure reading would be:
GivenΔPtest 0psi
ΔPtest1
2ρ vmax
2=
ΔP Find ΔPtest 0.01956 psi UP 0.06%1
27.68psi
2.16763 105
psi
Velocity Uncertainty:Sensitivity Coef:
SCP2
2ΔP
ρ ρ
0.05593s m
2
kg
Fixed Uncertainty for Velocity:
UR_V SCP UP 8.35918 103
m
s
Uv UR_V
30
Delta Wing Properties:
Λ 50deg UΛ 0.005deg Sweep Angle(Set by McClain)
Sspan 6.75in US 0.0002in
CLmax 1.0 Guess for lift coefficients determined, based on a flat plate at90degCDmax 1.0
cwing .625ft UCNC 0.0002in
v vmax Sspan 6.75 in
Rec
v cwing
ν1.83277 10
5
Reynolds Uncertainty:Sensitivity Coef.:
CDv
cwing
ν1.21492 10
4
s
m
CDcwingv
ν9.62085 10
5
1
m
Total Uncertainty:
CDv Uv 101.55767
CDcwing UCNC 4.88739
UR_Re CDv Uv 2 CDcwing UCNC 2
1
23 305.0256
Planform area (S):
Sarea 0.18215ft2
Planform Area of chosen airfoil
Planform Area Uncertainty:
UA1
2Sspan UCNC
21
2cwing UCNC
2
1
2
7.0071 106
ft2
31
Limits of Force Balance Scale:
Lmax 15lbfUFB 0.001lbf
Dmax 8lbf
Drag and Lift Coef Uncertainty:
AOA 0 deg: Smooth:FIXED:
L0s 0.203095lbf D0s 0.030344lbf ΔP0s0.546023psi
27.680.01973 psi
Sensitivity Coef:Lift:
CLP0s1
ΔP0s2
L0s
Sarea 2.88602 10
3
1
Pa
CLS0s1
Sarea2
L0s
ΔP0s 23.19549
1
m2
CLL0s1
ΔP0s Sarea1.9327
1
lbf
Drag:
CDP0s1
ΔP0s2
D0s
Sarea 1.46019
1
in_Hg
CDS0s1
Sarea2
D0s
ΔP0s 0.32196
1
ft2
CDD0s1
ΔP0s Sarea1.9327
1
lbf
Uncertainty Coeff:
UP 0.14945 Pa
32
UA 6.50981 107
m2
UFB 1 103
lbf
CLP0s UP 4.31323 104
CLS0s UA 1.50998 105
CLL0s UFB 1.9327 103
CDP0s UP 6.44431 105
CDS0s UA 2.25603 106
CDD0s UFB 1.9327 103
Total Fixed Uncertainty:
Lift:BR_L0s CLP0s UP 2 CLS0s UA 2 CLL0s UFB 2
1
21.9803 10
3
Drag:BR_D0s CDP0s UP 2 CDS0s UA 2 CDD0s UFB 2
1
21.93377 10
3
RANDOM:
Lift:L0s 0.20309 lbf σL0s 0.00565lbf ntrials 3000 σmL0s
σL0s
ntrials
1
2
Pressure:
ΔP0s 0.01973 psi σP0s .000616psi ntrialsP 30 σmP0s
σP0s
ntrialsP
1
2
Drag:
D0s 0.03034 lbf σD0s 0.005786lbf σmD0s
σD0s
ntrials
1
2
Student T's:
υ 5 1 4 t95 2.132
33
Total Random Uncertainty:
Lift: SR_L0s CLL0s σmL0s 2 CLP0s σmP0s 2
1
22.24675 10
3
Drag:SR_D0s CDD0s σmD0s 2 CDP0s σmP0s 2
1
23.91763 10
4
Total Uncertainty:
Lift: UR_L0s BR_L0s 2 t95 SR_L0s 2
1
25.18327 10
3
UR_L0s
0.41.296 %
Drag:UR_D0s BR_D0s 2 t95 SR_D0s 2
1
22.10644 10
3
UR_D0s
0.063.511 %
AOA 5 deg: Smooth:FIXED:
L5s 0.33865lbf D5s 0.042369lbf ΔP5s0.549011psi
27.680.01983 psi
Sensitivity Coef:Lift:
CLP5s1
ΔP5s2
L5s
Sarea 4.76004 10
3
1
Pa
CLS5s1
Sarea2
L5s
ΔP5s 38.46674
1
m2
CLL5s1
ΔP5s Sarea1.92218
1
lbf
34
Drag:
CDP5s1
ΔP5s2
D5s
Sarea 2.01671
1
in_Hg
CDS5s1
Sarea2
D5s
ΔP5s 0.44711
1
ft2
CDD5s1
ΔP5s Sarea1.92218
1
lbf
Uncertainty Coeff:
UP 0.14945 Pa
UA 7.0071 106
ft2
UFB 1 103
lbf
CLP5s UP 7.11401 104
CLS5s UA 2.50411 105
CLL5s UFB 1.92218 103
CDP5s UP 8.90045 105
CDS5s UA 3.13293 106
CDD5s UFB 1.92218 103
Total Fixed Uncertainty:
Lift:BR_L5s CLP5s UP 2 CLS5s UA 2 CLL5s UFB 2
1
22.04975 10
3
Drag:BR_D5s CDP5s UP 2 CDS5s UA 2 CDD5s UFB 2
1
21.92424 10
3
RANDOM:
Lift:L5s 0.33865 lbf σL5s 0.005653lbf ntrials 3 10
3 σmL5s
σL5s
ntrials
1
2
35
Pressure:
ΔP5s 0.01983 psi σP5s .0006106psi ntrialsP 30 σmP5s
σP5s
ntrialsP
1
2
Drag:
D5s 0.04237 lbf σD5s 0.005626lbf σmD5s
σD5s
ntrials
1
2
Student T's:
υ 4 t95 2.132
Total Random Uncertainty:
Lift: SR_L5s CLL5s σmL5s 2 CLP5s σmP5s 2
1
23.66407 10
3
Drag:SR_D5s CDD5s σmD5s 2 CDP5s σmP5s 2
1
24.9851 10
4
Total Uncertainty:
Lift: UR_L5s BR_L5s 2 t95 SR_L5s 2
1
28.07623 10
3
UR_L5s
0.661.224 %
Drag:UR_D5s BR_D5s 2 t95 SR_D5s 2
1
22.19825 10
3
UR_D5s
0.082.748 %
AOA 10 deg: Smooth:FIXED:
L10s 0.441689lbf D10s 0.074184lbf ΔP10s0.550281psi
27.680.01988 psi
Sensitivity Coef:Lift:
CLP10s1
ΔP10s2
L10s
Sarea 6.17972 10
3
1
Pa
36
CLS10s1
Sarea2
L10s
ΔP10s 50.05499
1
m2
CLL10s1
ΔP10s Sarea1.91774
1
lbf
Drag:
CDP10s1
ΔP10s2
D10s
Sarea 3.51478
1
in_Hg
CDS10s1
Sarea2
D10s
ΔP10s 0.78104
1
ft2
CDD10s1
ΔP10s Sarea1.91774
1
lbf
Uncertainty Coeff:
UP 0.14945 Pa
UA 6.50981 107
m2
UFB 1 103
lbf
CLP10s UP 9.23577 104
CLS10s UA 3.25848 105
CLL10s UFB 1.91774 103
CDP10s UP 1.5512 104
CDS10s UA 5.47279 106
CDD10s UFB 1.91774 103
Total Fixed Uncertainty:
Lift:BR_L10s CLP10s UP 2 CLS10s UA 2 CLL10s UFB 2
1
22.1288 10
3
37
Drag:BR_D10s CDP10s UP 2 CDS10s UA 2 CDD10s UFB 2
1
21.92401 10
3
RANDOM:
Lift:L10s 0.44169 lbf σL10s 0.005694lbf ntrials 3 10
3 σmL10s
σL10s
ntrials
1
2
Pressure:
ΔP10s 0.01988 psi σP10s .0004993psi ntrialsP 30 σmP10s
σP10s
ntrialsP
1
2
Drag:
D10s 0.07418 lbf σD10s 0.005624lbf σmD10s
σD10s
ntrials
1
2
Student T's:
υ 4 t95 2.132
Total Random Uncertainty:
Lift: SR_L10s CLL10s σmL10s 2 CLP10s σmP10s 2
1
23.8892 10
3
Drag:SR_D10s CDD10s σmD10s 2 CDP10s σmP10s 2
1
26.81424 10
4
Total Uncertainty:
Lift: UR_L10s BR_L10s 2 t95 SR_L10s 2
1
28.56068 10
3 UR_L10s
0.860.995 %
Drag:UR_D10s BR_D10s 2 t95 SR_D10s 2
1
22.4109 10
3
UR_D10s
0.141.722 %
38
AOA 15 deg: Smooth:FIXED:
L15s 0.555032lbf D15s 0.123745lbf ΔP15s0.544446psi
27.680.01967 psi
Sensitivity Coef:Lift:
CLP15s1
ΔP15s2
L15s
Sarea 7.93286 10
3
1
Pa
CLS15s1
Sarea2
L15s
ΔP15s 63.57385
1
m2
CLL15s1
ΔP15s Sarea1.93829
1
lbf
Drag:
CDP15s1
ΔP15s2
D15s
Sarea 5.98929
1
in_Hg
CDS15s1
Sarea2
D15s
ΔP15s 1.31679
1
ft2
CDD15s1
ΔP15s Sarea1.93829
1
lbf
Uncertainty Coeff:
UP 0.14945 Pa
UA 6.50981 107
m2
UFB 1 103
lbf
CLP15s UP 1.18559 103
CLS15s UA 4.13853 105
CLL15s UFB 1.93829 103
39
CDP15s UP 2.64328 104
CDS15s UA 9.22691 106
CDD15s UFB 1.93829 103
Total Fixed Uncertainty:
Lift:BR_L15s CLP15s UP 2 CLS15s UA 2 CLL15s UFB 2
1
22.27251 10
3
Drag:BR_D15s CDP15s UP 2 CDS15s UA 2 CDD15s UFB 2
1
21.95626 10
3
RANDOM:
Lift:L15s 0.55503 lbf σL15s 0.005652lbf ntrials 3 10
3 σmL15s
σL15s
ntrials
1
2
Pressure:
ΔP15s 0.01967 psi σP15s .0005252psi ntrialsP 30 σmP15s
σP15s
ntrialsP
1
2
Drag:
D15s 0.12374 lbf σD15s 0.005675lbf σmD15s
σD15s
ntrials
1
2
Student T's:
υ 4 t95 2.132
Total Random Uncertainty:
Lift: SR_L15s CLL15s σmL15s 2 CLP15s σmP15s 2
1
25.24842 10
3
Drag:SR_D15s CDD15s σmD15s 2 CDP5s σmP15s 2
1
24.41984 10
4
40
Total Uncertainty:
Lift: UR_L15s BR_L15s 2 t95 SR_L15s 2
1
20.01142
UR_L15s
1.081.057 %
Drag:UR_D15s BR_D15s 2 t95 SR_D15s 2
1
22.17138 10
3
UR_D15s
0.240.905 %
AOA 20 deg: Smooth:
FIXED:
L20s 0.633978lbf D20s 0.188687lbf ΔP20s0.5479074psi
27.680.01979 psi
Sensitivity Coef:Lift:
CLP20s1
ΔP20s2
L20s
Sarea 8.94708 10
3
1
Pa
CLS20s1
Sarea2
L20s
ΔP20s 72.15764
1
m2
CLL20s1
ΔP20s Sarea1.92605
1
lbf
Drag:
CDP20s1
ΔP20s2
D20s
Sarea 9.01747
1
in_Hg
CDS20s1
Sarea2
D20s
ΔP20s 1.99517
1
ft2
CDD20s1
ΔP20s Sarea1.92605
1
lbf
Uncertainty Coeff:
UP 0.14945 Pa
41
UA 6.50981 107
m2
UFB 1 103
lbf
CLP20s UP 1.33717 103
CLS20s UA 4.69732 105
CLL20s UFB 1.92605 103
CDP20s UP 3.97973 104
CDS20s UA 1.39804 105
CDD20s UFB 1.92605 103
Total Fixed Uncertainty:
Lift:BR_L20s CLP20s UP 2 CLS20s UA 2 CLL20s UFB 2
1
22.34518 10
3
Drag:BR_D20s CDP20s UP 2 CDS20s UA 2 CDD20s UFB 2
1
21.96678 10
3
RANDOM:
Lift:L20s 0.63398 lbf σL20s 0.005686lbf ntrials 3 10
3 σmL20s
σL20s
ntrials
1
2
Pressure:
ΔP20s 0.01979 psi σP20s .0004374psi ntrialsP 30 σmP20s
σP20s
ntrialsP
1
2
Drag:
D20s 0.18869 lbf σD20s 0.005799lbf σmD20s
σD20s
ntrials
1
2
Student T's:
υ 4 t95 2.132
42
Total Random Uncertainty:
Lift: SR_L20s CLL20s σmL20s 2 CLP20s σmP20s 2
1
24.93033 10
3
Drag:SR_D20s CDD20s σmD20s 2 CDP20s σmP20s 2
1
21.48029 10
3
Total Uncertainty:
Lift: UR_L20s BR_L20s 2 t95 SR_L20s 2
1
20.01077
UR_L20s
1.230.876 %
Drag:UR_D20s BR_D20s 2 t95 SR_D20s 2
1
23.71866 10
3
UR_D20s
0.371.005 %
AOA 25 deg:
Smooth:FIXED:
L25s 0.704275lbf D25s 0.268264lbf ΔP25s0.5487046psi
27.680.01982 psi
Sensitivity Coef:Lift:
CLP25s1
ΔP25s2
L25s
Sarea 9.9103 10
3
1
Pa
CLS25s1
Sarea2
L25s
ΔP25s 80.04219
1
m2
CLL25s1
ΔP25s Sarea1.92325
1
lbf
Drag:
CDP25s1
ΔP25s2
D25s
Sarea 12.78328
1
in_Hg
43
CDS25s1
Sarea2
D25s
ΔP25s 2.83249
1
ft2
CDD25s1
ΔP25s Sarea1.92325
1
lbf
Uncertainty Coeff:
UP 0.14945 Pa
UA 6.50981 107
m2
UFB 1 103
lbf
CLP25s UP 1.48112 103
CLS25s UA 5.21059 105
CLL25s UFB 1.92325 103
CDP25s UP 5.64171 104
CDS25s UA 1.98476 105
CDD25s UFB 1.92325 103
Total Fixed Uncertainty:
Lift:BR_L25s CLP25s UP 2 CLS25s UA 2 CLL25s UFB 2
1
22.42803 10
3
Drag:BR_D25s CDP25s UP 2 CDS25s UA 2 CDD25s UFB 2
1
22.00439 10
3
RANDOM:
Lift:L25s 0.70427 lbf σL25s 0.005637lbf ntrials 3 10
3 σmL25s
σL25s
ntrials
1
2
44
Pressure:
ΔP25s 0.01982 psi σP25s .0007762psi ntrialsP 30 σmP25s
σP25s
ntrialsP
1
2
Drag:
D25s 0.26826 lbf σD25s 0.005881lbf σmD25s
σD25s
ntrials
1
2
Student T's:
υ 4 t95 2.132
Total Random Uncertainty:
Lift: SR_L25s CLL25s σmL25s 2 CLP25s σmP25s 2
1
29.68522 10
3
Drag:SR_D25s CDD25s σmD25s 2 CDP25s σmP25s 2
1
23.69418 10
3
Total Uncertainty:
Lift: UR_L25s BR_L25s 2 t95 SR_L25s 2
1
20.02079
UR_L25s
1.361.529 %
Drag:UR_D25s BR_D25s 2 t95 SR_D25s 2
1
28.12705 10
3
UR_D25s
0.521.563 %
AOA 30 deg: Smooth:
FIXED:L30s 0.738421lbf D30s 0.351963lbf ΔP30s
0.543264psi
27.680.01963 psi
Sensitivity Coef:Lift:
CLP30s1
ΔP30s2
L30s
Sarea 0.0106
1
Pa
CLS30s1
Sarea2
L30s
ΔP30s 84.76341
1
m2
45
CLL30s1
ΔP30s Sarea1.94251
1
lbf
Drag:
CDP30s1
ΔP30s2
D30s
Sarea 17.10931
1
in_Hg
CDS30s1
Sarea2
D30s
ΔP30s 3.75346
1
ft2
CDD30s1
ΔP30s Sarea1.94251
1
lbf
Uncertainty Coeff:
UP 0.14945 Pa
UA 6.50981 107
m2
UFB 1 103
lbf
CLP30s UP 1.58419 103
CLS30s UA 5.51793 105
CLL30s UFB 1.94251 103
CDP30s UP 7.55094 104
CDS30s UA 2.63008 105
CDD30s UFB 1.94251 103
Total Fixed Uncertainty:
Lift:BR_L30s CLP30s UP 2 CLS30s UA 2 CLL30s UFB 2
1
22.5072 10
3
Drag:BR_D30s CDP30s UP 2 CDS30s UA 2 CDD30s UFB 2
1
22.08428 10
3
46
RANDOM:
Lift:L30s 0.73842 lbf σL30s 0.005676lbf ntrials 3 10
3 σmL30s
σL30s
ntrials
1
2
Pressure:
ΔP30s 0.01963 psi σP30s .000469psi ntrialsP 30 σmP30s
σP30s
ntrialsP
1
2
Drag:
D30s 0.35196 lbf σD30s 0.006677lbf σmD30s
σD30s
ntrials
1
2
Student T's:
υ 4 t95 2.132
Total Random Uncertainty:
Lift: SR_L30s CLL30s σmL30s 2 CLP30s σmP30s 2
1
26.26123 10
3
Drag:SR_D30s CDD30s σmD30s 2 CDP30s σmP30s 2
1
22.99221 10
3
Total Uncertainty:
Lift: UR_L30s BR_L30s 2 t95 SR_L30s 2
1
20.01358
UR_L30s
1.440.943 %
Drag:UR_D30s BR_D30s 2 t95 SR_D30s 2
1
26.71125 10
3
UR_D30s
0.690.973 %
47
AOA 35 deg: Smooth:
FIXED:
L35s 0.683567lbf D35s 0.397567lbf ΔP35s0.5416308psi
27.680.01957 psi
Sensitivity Coef:Lift:
CLP35s1
ΔP35s2
L35s
Sarea 9.87179 10
3
1
Pa
CLS35s1
Sarea2
L35s
ΔP35s 78.70332
1
m2
CLL35s1
ΔP35s Sarea1.94837
1
lbf
Drag:
CDP35s1
ΔP35s2
D35s
Sarea 19.44289
1
in_Hg
CDS35s1
Sarea2
D35s
ΔP35s 4.25258
1
ft2
CDD35s1
ΔP35s Sarea1.94837
1
lbf
Uncertainty Coeff:
UP 0.14945 Pa
UA 6.50981 107
m2
UFB 1 103
lbf
CLP35s UP 1.47537 103
CLS35s UA 5.12343 105
CLL35s UFB 1.94837 103
48
CDP35s UP 8.58083 104
CDS35s UA 2.97982 105
CDD35s UFB 1.94837 103
Total Fixed Uncertainty:
Lift:BR_L35s CLP35s UP 2 CLS35s UA 2 CLL35s UFB 2
1
22.44448 10
3
Drag:BR_D35s CDP35s UP 2 CDS35s UA 2 CDD35s UFB 2
1
22.12916 10
3
RANDOM:
Lift:L35s 0.68357 lbf σL35s 0.005678lbf ntrials 3 10
3 σmL35s
σL35s
ntrials
1
2
Pressure:
ΔP35s 0.01957 psi σP35s .0007286psi ntrialsP 30 σmP35s
σP35s
ntrialsP
1
2
Drag:
D35s 0.39757 lbf σD35s 0.008312lbf σmD35s
σD35s
ntrials
1
2
Student T's:
υ 4 t95 2.132
Total Random Uncertainty:
Lift: SR_L35s CLL35s σmL35s 2 CLP35s σmP35s 2
1
29.05631 10
3
Drag:SR_D35s CDD35s σmD35s 2 CDP35s σmP35s 2
1
25.2742 10
3
49
Total Uncertainty:
Lift: UR_L35s BR_L35s 2 t95 SR_L35s 2
1
20.01946
UR_L35s
1.341.452 %
Drag:UR_D35s BR_D35s 2 t95 SR_D35s 2
1
20.01144
UR_D35s
0.781.467 %
50
Appendix iii Uncertainty Analysis: Channeled Wing
Kline-McClintock Uncertainty Analysis for Drag and Lift Coefficients on Delta Wing:
Air Properties: From Fluids Textbook:
ν 15.68 106
m
2
s Uν 0
m2
s
ρ 1.18517kg
m3
0.07399lbm
ft3
Uρ 0.0006kg
m3
3.74568 105
lbm
ft3
Flow Proerties:
vmax 15.0855m
s so the wind tunnel pressure reading would be:
GivenΔPtest 0psi
ΔPtest1
2ρ vmax
2=
ΔP Find ΔPtest 0.01956 psi UP 0.06%1
27.68psi
2.16763 105
psi
Velocity Uncertainty:Sensitivity Coef:
SCP2
2ΔP
ρ ρ
0.05593s m
2
kg
Fixed Uncertainty for Velocity:
UR_V SCP UP 8.35918 103
m
s
Uv UR_V
51
Delta Wing Properties:
Λ 50deg UΛ 0.005deg Sweep Angle(Set by McClain)
Sspan 6.75in US 0.0002in
cwing .625ft UCNC 0.0002in
v vmax Sspan 6.75 in
Rec
v cwing
ν1.83277 10
5
Reynolds Uncertainty:Sensitivity Coef.:
CDv
cwing
ν1.21492 10
4
s
m
CDcwingv
ν9.62085 10
5
1
m
Total Uncertainty:
CDv Uv 101.55767
CDcwing UCNC 4.88739
UR_Re CDv Uv 2 CDcwing UCNC 2
1
23 305.0256
Planform area (S):
Sarea 0.18215ft2
Planform Area of chosen airfoil
Planform Area Uncertainty:
UA1
2Sspan UCNC
21
2cwing UCNC
2
1
2
7.0071 106
ft2
52
Limits of Force Balance Scale:
Lmax 15lbfUFB 0.001lbf
Dmax 8lbf
Drag and Lift Coef Uncertainty:
AOA 0 deg: Channeled:FIXED:
L0c 0.208986lbf D0c 0.030761lbf ΔP0c0.5413574psi
27.680.01956 psi
Sensitivity Coef:Lift:
CLP0c1
ΔP0c2
L0c
Sarea 3.02114 10
3
1
Pa
CLS0c1
Sarea2
L0c
ΔP0c 24.07401
1
m2
CLL0c1
ΔP0c Sarea1.94935
1
lbf
Drag:
CDP0c1
ΔP0c2
D0c
Sarea 1.50588
1
in_Hg
CDS0c1
Sarea2
D0c
ΔP0c 0.3292
1
ft2
CDD0c1
ΔP0c Sarea1.94935
1
lbf
53
Uncertainty Coeff:
UP 0.14945 Pa
UA 6.50981 107
m2
UFB 1 103
lbf
CLP0c UP 4.51518 104
CLS0c UA 1.56717 105
CLL0c UFB 1.94935 103
CDP0c UP 6.64596 105
CDS0c UA 2.30675 106
CDD0c UFB 1.94935 103
Total Fixed Uncertainty:
Lift:BR_L0s CLP0c UP 2 CLS0c UA 2 CLL0c UFB 2
1
22.00102 10
3
Drag:BR_D0s CDP0c UP 2 CDS0c UA 2 CDD0c UFB 2
1
21.95049 10
3
RANDOM:
Lift:L0c 0.20899 lbf σL0c 0.005571lbf ntrials 3000 σmL0c
σL0c
ntrials
1
2
Pressure:
ΔP0c 0.01956 psi σP0c .0005792psi ntrialsP 30 σmP0c
σP0c
ntrialsP
1
2
Drag:
54
D0c 0.03076 lbf σD0c 0.005658lbf σmD0c
σD0c
ntrials
1
2
Student T's:
υ 5 1 4 t95 2.132
Total Random Uncertainty:
Lift: SR_L0c CLL0c σmL0c 2 CLP0c σmP0c 2
1
22.21162 10
3
Drag:SR_D0c CDD0c σmD0c 2 CDP0c σmP0c 2
1
23.81665 10
4
Total Uncertainty:
Lift: UR_L0c BR_L0s 2 t95 SR_L0c 2
1
25.12219 10
3
UR_L0c
0.411.249 %
Drag:UR_D0c BR_D0s 2 t95 SR_D0c 2
1
22.11341 10
3
UR_D0c
0.063.522 %
AOA 5 deg: Channeled:FIXED:
L5c 0.3228555lbf D5c 0.0470633lbf ΔP5c0.5447688psi
27.680.01968 psi
Sensitivity Coef:Lift:
CLP5c1
ΔP5c2
L5c
Sarea 4.60898 10
3
1
Pa
55
CLS5c1
Sarea2
L5c
ΔP5c 36.95824
1
m2
CLL5c1
ΔP5c Sarea1.93715
1
lbf
Drag:
CDP5c1
ΔP5c2
D5c
Sarea 2.27518
1
in_Hg
CDS5c1
Sarea2
D5c
ΔP5c 0.50051
1
ft2
CDD5c1
ΔP5c Sarea1.93715
1
lbf
Uncertainty Coeff:
UP 0.14945 Pa
UA 7.0071 106
ft2
UFB 1 103
lbf
CLP5c UP 6.88826 104
CLS5c UA 2.40591 105
CLL5c UFB 1.93715 103
CDP5c UP 1.00412 104
CDS5c UA 3.50714 106
CDD5c UFB 1.93715 103
Total Fixed Uncertainty:
Lift:BR_L5c CLP5c UP 2 CLS5c UA 2 CLL5c UFB 2
1
22.05611 10
3
56
Drag:
BR_D5c CDP5c UP 2 CDS5c UA 2 CDD5c UFB 2
1
21.93975 10
3
RANDOM:
Lift:L5c 0.32286 lbf σL5c 0.005579lbf ntrials 3 10
3 σmL5c
σL5c
ntrials
1
2
Pressure:
ΔP5c 0.01968 psi σP5c .0005524psi ntrialsP 30 σmP5c
σP5c
ntrialsP
1
2
Drag:
D5c 0.04706 lbf σD5c 0.005724lbf σmD5s
σD5c
ntrials
1
2
Student T's:
υ 4 t95 2.132
Total Random Uncertainty:
Lift: SR_L5c CLL5c σmL5c 2 CLP5c σmP5c 2
1
23.21099 10
3
Drag:SR_D5c CDD5c σmD5s 2 CDP5c σmP5c 2
1
25.09163 10
4
Total Uncertainty:
Lift: UR_L5c BR_L5c 2 t95 SR_L5c 2
1
27.14793 10
3
UR_L5c
0.631.135 %
Drag:UR_D5c BR_D5c 2 t95 SR_D5c 2
1
22.22284 10
3
UR_D5c
0.90.247 %
AOA 10 deg:
57
Channeled:FIXED:
L10c 0.4323125lbf D10c 0.0778185lbf ΔP10c0.5481454psi
27.680.0198 psi
Sensitivity Coef:Lift:
CLP10c1
ΔP10c2
L10c
Sarea 6.09576 10
3
1
Pa
CLS10c1
Sarea2
L10c
ΔP10c 49.18326
1
m2
CLL10c1
ΔP10c Sarea1.92521
1
lbf
Drag:
CDP10c1
ΔP10c2
D10c
Sarea 3.71577
1
in_Hg
CDS10c1
Sarea2
D10c
ΔP10c 0.82249
1
ft2
CDD10c1
ΔP10c Sarea1.92521
1
lbf
Uncertainty Coeff:
UP 0.14945 Pa
UA 6.50981 107
m2
UFB 1 103
lbf
CLP10c UP 9.11028 104
CLS10c UA 3.20173 105
CLL10c UFB 1.92521 103
58
CDP10c UP 1.6399 104
CDS10c UA 5.76329 106
CDD10c UFB 1.92521 103
Total Fixed Uncertainty:
Lift:BR_L10c CLP10c UP 2 CLS10c UA 2 CLL10c UFB 2
1
22.13013 10
3
Drag:BR_D10c CDP10c UP 2 CDS10c UA 2 CDD10c UFB 2
1
21.93219 10
3
RANDOM:
Lift:L10c 0.43231 lbf σL10c 0.005473lbf ntrials 3 10
3 σmL10c
σL10c
ntrials
1
2
Pressure:
ΔP10c 0.0198 psi σP10c .0006752psi ntrialsP 30 σmP10c
σP10c
ntrialsP
1
2
Drag:
D10c 0.07782 lbf σD10c 0.005562lbf σmD10c
σD10c
ntrials
1
2
Student T's:
υ 4 t95 2.132
Total Random Uncertainty:
Lift: SR_L10c CLL10c σmL10c 2 CLP10c σmP10c 2
1
25.18463 10
3
59
Drag:SR_D10c CDD10c σmD10c 2 CDP10c σmP10c 2
1
29.52888 10
4
Total Uncertainty:
Lift: UR_L10c BR_L10c 2 t95 SR_L10c 2
1
20.01126 UR_L10c
0.841.34 %
Drag:UR_D10c BR_D10c 2 t95 SR_D10c 2
1
22.80368 10
3
UR_D10c
0.151.869 %
AOA 15 deg: Channeled:FIXED:
L15c 0.5220893lbf D15c 0.12486636lbf ΔP15c0.5444826psi
27.680.01967 psi
Sensitivity Coef:Lift:
CLP15c1
ΔP15c2
L15c
Sarea 7.46102 10
3
1
Pa
CLS15c1
Sarea2
L15c
ΔP15c 59.79655
1
m2
CLL15c1
ΔP15c Sarea1.93816
1
lbf
Drag:
CDP15c1
ΔP15c2
D15c
Sarea 6.04275
1
in_Hg
CDS15c1
Sarea2
D15c
ΔP15c 1.32864
1
ft2
CDD15c1
ΔP15c Sarea1.93816
1
lbf
60
Uncertainty Coeff:
UP 0.14945 Pa
UA 6.50981 107
m2
UFB 1 103
lbf
CLP15c UP 1.11507 103
CLS15c UA 3.89264 105
CLL15c UFB 1.93816 103
CDP15c UP 2.66688 104
CDS15c UA 9.30989 106
CDD15c UFB 1.93816 103
Total Fixed Uncertainty:
Lift:BR_L15c CLP15c UP 2 CLS15c UA 2 CLL15c UFB 2
1
22.23638 10
3
Drag:BR_D15c CDP15c UP 2 CDS15c UA 2 CDD15c UFB 2
1
21.95645 10
3
RANDOM:
Lift:L15c 0.52209 lbf σL15c 0.005475lbf ntrials 3 10
3 σmL15c
σL15c
ntrials
1
2
Pressure:
ΔP15c 0.01967 psi σP15c .0004822psi ntrialsP 30 σmP15c
σP15c
ntrialsP
1
2
Drag:
61
D15c 0.12487 lbf σD15c 0.005659lbf σmD15c
σD15c
ntrials
1
2
Student T's:
υ 4 t95 2.132
Total Random Uncertainty:
Lift: SR_L15c CLL15c σmL15c 2 CLP15c σmP15c 2
1
24.53295 10
3
Drag:SR_D15c CDD15c σmD15c 2 CDP15c σmP15c 2
1
21.1015 10
3
Total Uncertainty:
Lift: UR_L15c BR_L15c 2 t95 SR_L15c 2
1
29.91963 10
3
UR_L15c
1.020.973 %
Drag:UR_D15c BR_D15c 2 t95 SR_D15c 2
1
23.05657 10
3
UR_D15c
0.241.274 %
AOA 20 deg: Channeled:
FIXED:
L20c 0.6286382lbf D20c 0.189164lbf ΔP20c0.5475928psi
27.680.01978 psi
Sensitivity Coef:Lift:
CLP20c1
ΔP20c2
L20c
Sarea 8.88192 10
3
1
Pa
CLS20c1
Sarea2
L20c
ΔP20c 71.59099
1
m2
62
CLL20c1
ΔP20c Sarea1.92716
1
lbf
Drag:
CDP20c1
ΔP20c2
D20c
Sarea 9.05066
1
in_Hg
CDS20c1
Sarea2
D20c
ΔP20c 2.00136
1
ft2
CDD20c1
ΔP20c Sarea1.92716
1
lbf
Uncertainty Coeff:
UP 0.14945 Pa
UA 6.50981 107
m2
UFB 1 103
lbf
CLP20c UP 1.32743 103
CLS20c UA 4.66043 105
CLL20c UFB 1.92716 103
CDP20c UP 3.99437 104
CDS20c UA 1.40237 105
CDD20c UFB 1.92716 103
Total Fixed Uncertainty:
Lift:BR_L20c CLP20c UP 2 CLS20c UA 2 CLL20c UFB 2
1
22.34055 10
3
Drag:BR_D20c CDP20c UP 2 CDS20c UA 2 CDD20c UFB 2
1
21.96817 10
3
63
RANDOM:
Lift:L20c 0.62864 lbf σL20c 0.005497lbf ntrials 3 10
3 σmL20c
σL20c
ntrials
1
2
Pressure:
ΔP20c 0.01978 psi σP20c .0005888psi ntrialsP 30 σmP20c
σP20c
ntrialsP
1
2
Drag:
D20c 0.18916 lbf σD20c 0.005680lbf σmD20c
σD20c
ntrials
1
2
Student T's:
υ 4 t95 2.132
Total Random Uncertainty:
Lift: SR_L20c CLL20c σmL20c 2 CLP20c σmP20c 2
1
26.58598 10
3
Drag:SR_D20c CDD20c σmD20c 2 CDP20c σmP20c 2
1
21.99099 10
3
Total Uncertainty:
Lift: UR_L20c BR_L20c 2 t95 SR_L20c 2
1
20.01424
UR_L20c
1.221.167 %
Drag:UR_D20c BR_D20c 2 t95 SR_D20c 2
1
24.67889 10
3
UR_D20c
0.371.265 %
AOA 25 deg:
Channeled:
64
FIXED:
L25c 0.6793303lbf D25c 0.2593497lbf ΔP25c0.5435612psi
27.680.01964 psi
Sensitivity Coef:Lift:
CLP25c1
ΔP25c2
L25c
Sarea 9.74105 10
3
1
Pa
CLS25c1
Sarea2
L25c
ΔP25c 77.93775
1
m2
CLL25c1
ΔP25c Sarea1.94145
1
lbf
Drag:
CDP25c1
ΔP25c2
D25c
Sarea 12.59349
1
in_Hg
CDS25c1
Sarea2
D25c
ΔP25c 2.76428
1
ft2
CDD25c1
ΔP25c Sarea1.94145
1
lbf
Uncertainty Coeff:
UP 0.14945 Pa
UA 6.50981 107
m2
UFB 1 103
lbf
CLP25c UP 1.45583 103
CLS25c UA 5.0736 105
CLL25c UFB 1.94145 103
CDP25c UP 5.55795 104
65
CDS25c UA 1.93696 105
CDD25c UFB 1.94145 103
Total Fixed Uncertainty:
Lift:BR_L25c CLP25c UP 2 CLS25c UA 2 CLL25c UFB 2
1
22.42719 10
3
Drag:BR_D25c CDP25c UP 2 CDS25c UA 2 CDD25c UFB 2
1
22.01953 10
3
RANDOM:
Lift:L25c 0.67933 lbf σL25c 0.005501lbf ntrials 3 10
3 σmL25c
σL25c
ntrials
1
2
Pressure:
ΔP25c 0.01964 psi σP25c .0005152psi ntrialsP 30 σmP25c
σP25c
ntrialsP
1
2
Drag:
D25c 0.25935 lbf σD25c 0.005652lbf σmD25c
σD25c
ntrials
1
2
Student T's:
υ 4 t95 2.132
Total Random Uncertainty:
Lift: SR_L25c CLL25c σmL25c 2 CLP25c σmP25c 2
1
26.32043 10
3
Drag:SR_D25c CDD25c σmD25c 2 CDP25c σmP25c 2
1
22.42012 10
3
Total Uncertainty:
66
Lift: UR_L25c BR_L25c 2 t95 SR_L25c 2
1
20.01369
UR_L25c
1.331.029 %
Drag:UR_D25c BR_D25c 2 t95 SR_D25c 2
1
25.54085 10
3
UR_D25c
0.511.086 %
AOA 30 deg: Channeled:
FIXED:
L30c 0.683492lbf D30c 0.33591187lbf ΔP30c0.5433894psi
27.680.01963 psi
Sensitivity Coef:Lift:
CLP30c1
ΔP30c2
L30c
Sarea 9.80692 10
3
1
Pa
CLS30c1
Sarea2
L30c
ΔP30c 78.44
1
m2
CLL30c1
ΔP30c Sarea1.94206
1
lbf
Drag:
CDP30c1
ΔP30c2
D30c
Sarea 16.32151
1
in_Hg
CDS30c1
Sarea2
D30c
ΔP30c 3.58145
1
ft2
CDD30c1
ΔP30c Sarea1.94206
1
lbf
Uncertainty Coeff:
UP 0.14945 Pa
67
UA 6.50981 107
m2
UFB 1 103
lbf
CLP30c UP 1.46567 103
CLS30c UA 5.10629 105
CLL30c UFB 1.94206 103
CDP30c UP 7.20325 104
CDS30c UA 2.50956 105
CDD30c UFB 1.94206 103
Total Fixed Uncertainty:
Lift:BR_L30c CLP30c UP 2 CLS30c UA 2 CLL30c UFB 2
1
22.4336 10
3
Drag:BR_D30c CDP30c UP 2 CDS30c UA 2 CDD30c UFB 2
1
22.0715 10
3
RANDOM:
Lift:L30c 0.68349 lbf σL30c 0.005545lbf ntrials 3 10
3 σmL30c
σL30c
ntrials
1
2
Pressure:
ΔP30c 0.01963 psi σP30c 0.0006846psi ntrialsP 30 σmP30c
σP30c
ntrialsP
1
2
Drag:
68
D30c 0.33591 lbf σD30c 0.006319lbf σmD30c
σD30c
ntrials
1
2
Student T's:
υ 4 t95 2.132
Total Random Uncertainty:
Lift: SR_L30c CLL30c σmL30c 2 CLP30c σmP30c 2
1
28.45367 10
3
Drag:SR_D30c CDD30c σmD30c 2 CDP30c σmP30c 2
1
24.15959 10
3
Total Uncertainty:
Lift: UR_L30c BR_L30c 2 t95 SR_L30c 2
1
20.01819
UR_L30c
1.341.357 %
Drag:UR_D30c BR_D30c 2 t95 SR_D30c 2
1
29.10697 10
3
UR_D30c
0.661.38 %
AOA 35 deg: Channeled:
FIXED:
L35c 0.6897596lbf D35c 0.37898809lbf ΔP35c0.5414506psi
27.680.01956 psi
Sensitivity Coef:Lift:
CLP35c1
ΔP35c2
L35c
Sarea 9.96785 10
3
1
Pa
CLS35c1
Sarea2
L35c
ΔP35c 79.44274
1
m2
69
CLL35c1
ΔP35c Sarea1.94902
1
lbf
Drag:
CDP35c1
ΔP35c2
D35c
Sarea 18.54663
1
in_Hg
CDS35c1
Sarea2
D35c
ΔP35c 4.0552
1
ft2
CDD35c1
ΔP35c Sarea1.94902
1
lbf
Uncertainty Coeff:
UP 0.14945 Pa
UA 6.50981 107
m2
UFB 1 103
lbf
CLP35c UP 1.48972 103
CLS35c UA 5.17157 105
CLL35c UFB 1.94902 103
CDP35c UP 8.18528 104
CDS35c UA 2.84152 105
CDD35c UFB 1.94902 103
Total Fixed Uncertainty:
Lift:BR_L35c CLP35c UP 2 CLS35c UA 2 CLL35c UFB 2
1
22.45369 10
3
Drag:BR_D35c CDP35c UP 2 CDS35c UA 2 CDD35c UFB 2
1
22.11411 10
3
70
RANDOM:
Lift:L35c 0.68976 lbf σL35c 0.005543lbf ntrials 3 10
3 σmL35c
σL35c
ntrials
1
2
Pressure:
ΔP35c 0.01956 psi σP35c .0007582psi ntrialsP 30 σmP35c
σP35c
ntrialsP
1
2
Drag:D35c 0.37899 lbf σD35c 0.007356lbf σmD35c
σD35c
ntrials
1
2
Student T's:
υ 4 t95 2.132
Total Random Uncertainty:
Lift: SR_L35c CLL35c σmL35c 2 CLP35c σmP35c 2
1
29.51562 10
3
Drag:SR_D35c CDD35c σmD35c 2 CDP35c σmP35c 2
1
25.23378 10
3
Total Uncertainty:
Lift: UR_L35c BR_L35c 2 t95 SR_L35c 2
1
20.02044
UR_L35c
1.351.514 %
Drag:UR_D35c BR_D35c 2 t95 SR_D35c 2
1
20.01136 UR_D35c
0.741.535 %
71
Appendix iv- Executive Summary
Executive Summary It is important to keep in mind that the entirety of the report was collectively done with the entire group. Each member is able to give a full explanation of any part of the Test Plan Report. Below is a breakdown of the Test Plan Report, and who in the Delta Wing group completed the majority of each portion. Abstract- Ruben Nuñez and Ryan Parker Nomenclature- Ryan Parker Introduction- Meag McNary Experimental Methods- Meag McNary Experimental Procedure- Ruben Nuñez Uncertainty- Adam Eaker Results and Discussion- Ryan Parker Conclusion and Recommendations- Drew Waggoner Acknowledgements- Ruben Nuñez References- Ryan Parker and Ruben Nuñez General Formatting- Meag McNary
72