i have a project on theory of flight and i need help to do it all the information on the files and keep the project as PDFProf. Dr. Nihad E. Daidzic, ATP, CFII
MSUM, P1 WT Lab, AVIA 201 (Theory of Flight©)
MINNESOTA STATE UNIVERSITY, Mankato
Aviation Department
Theory of Flight© (AVIA 201)
Project 1 Wind-Tunnel Laboratory (Trafton Science Center E 104)
FALL/SPRING
Instructor:
Nihad E. Daidzic, Dr.-Ing., D.Sc., M.S.M.E., B.S.M.E.
Professor of Aviation, Research Graduate Faculty
ATP AMEL, Comm. ASEL/ASES/Glider/Helicopter/LBH, Instrument
FAA “Gold Seal” CFI-IA, ME-IA, CFI-RH, CFI-G, IGI, AGI
Class Meeting:
Project 1 Wind-Tunnel Laboratory (Trafton Science Center E 104)
(During regular class time. Exact date will be announced or syllabus)
PROJECT 1 Assignment
WIND-TUNNEL LAB (AVIA 201)
Name: __________________________________________ Section: AVIA 201-____________
TECH ID: _____________________ Signature: _____________________________________
Semester: ________________________________ Date: _______________________________
Grade/Score: __________________ Comment: _____________________________________
_____________________________________________________________________________
© (March 2020, ver. 1.20), Dr. Nihad E. Daidzic, All rights reserved
1
Prof. Dr. Nihad E. Daidzic, ATP, CFII
MSUM, P1 WT Lab, AVIA 201 (Theory of Flight©)
Purpose: To introduce aviation/aeronautics/aerospace students to wind tunnel(s) and methods
used in experimental identification of various aerodynamic (and stability) coefficients of airfoils
(2D), wings (3D) and scale models. To understand the value and purpose of wind-tunnel and
experimental identification (measurement) of lift, drag, sideforce, and aerodynamic moments
(pitching, rolling, yawing). To review the basic aerodynamic theory and knowledge of wings and
airfoils presented in classroom. To apply and correlate knowledge acquired in the lab to better
understanding of fixed- and rotary-wing flight control, stability and performance. To practice unit
conversions (from SI to English engineering and vice versa). To perform visualization study of the
boundary layer separation and stall patterns at higher AOAs. To visualize wingtip vortices when
3D (finite wing) airfoil is installed.
Grade: Wind-Tunnel lab project is required project in AVIA 201 course and brings 10% of the
overall grade. Two students conduct one measurement, but all project completion is individual.
Measurement report is required with all the graphs constructed.
Project Deliverables, calculations and Graphical presentation: Measure static pressure and
temperature in a lab. Calculate dry air density. Calculate dynamic and kinematic viscosity. Set the
fan speed to achieve desired air dynamic pressure (pdyn = q). Typically, in a range of 600-650 Pa.
Calculate TAS. From the airfoil geometry (2D or 3D) measure chord length and reference surface
area. Calculate the acoustic velocity (speed of sound). Calculate Reynolds and Mach numbers.
Measure lift and drag. Evaluate aerodynamic coefficients. Estimate drag polar, parabolic drag
model and other useful aerodynamic parameters (endurance polar and range polar). Present the
final results in a graphical form for:
1.
2.
3.
4.
5.
6.
7.
C L vs AOA
C D vs AOA
C L vs C D (Polar diagram)
CD vs CL2 (Parabolic drag model CD  CD , 0  K  CL2 )
C L
C
C
32
L
12
L
C D  vs AOA (Aerodynamic efficiency)
C D  vs AOA (Endurance efficiency)
C D  vs AOA (Range efficiency)
in a graphical format (diagrams) by using MS ExcelTM, MatlabTM, Fortran, (Visual) Basic, C++,
MapleTM or any other appropriate calculation/spreadsheet/engineering/mathematical software
(hand drawn graphs on millimeter paper is acceptable) and attach to cover page.
Workload: 2.5 hours of data collection and experiments. 6-10 hours per student for data
processing and plotting/constructing diagrams and results. Overall, this wind-tunnel project should
not exceed 13 effective physical hours workload per student.
Important Safety Note: Discipline and order during lab work will be enforced. Exercise safety
precautions and common sense. Do not touch or come close to the large exhaust fan. Do not touch
any electrical connections and cables. Do not wander around touching and moving things during
the experimentation. Do not stand at the suction or the exhaust end of the wind tunnel.
© (March 2020, ver. 1.20), Dr. Nihad E. Daidzic, All rights reserved
2
Prof. Dr. Nihad E. Daidzic, ATP, CFII
MSUM, P1 WT Lab, AVIA 201 (Theory of Flight©)
Experimental Procedure: Two measurements of lift and drag aerodynamic coefficients will be
conducted at two different wind-tunnel uniform speeds (adjusted by the fan). Reynolds and Mach
numbers are to be evaluated for each experiment. The experimental procedure is:
1. Measure the barometric pressure p and temperature T in the lab. Assume dry air. Convert
from 0F and inches Hg into Celsius (0C), Kelvin (K) and pressure in Pascals (and
hPa=mbar). Call MKT ASOS to verify (KMKT elevation is about 1,000 ft).
2. Calculate the air density assuming that dry air mixture obeys ideal-gas law. The air gas
constant R  287 .053 J/kgK . The isentropic coefficient of expansion for air is   1.4 .
Calculate the speed of sound: a    R  T  20.05  T m/s  .
Calculate the dynamic viscosity of dry air  .
Calculate the kinematic viscosity of the air     .
Chose the airfoil/wing. Measure its chord and span. Calculate its mean chord length (if
applicable), aspect ratio (AR), taper ratio (TR), sweep angle  , planform type. Convert
Imperial units into SI as necessary.
7. Set the airfoil/wing in a wind tunnel under desired AOA. Make sure that the Pitot tube is
parallel with the longitudinal axis of the wind-tunnel test section, i.e., parallel with the
flow. Close and tighten lightly the upper section.
8. Calibrate and re-calibrate the force balance to show about zero in L and D without flow.
9. Set the wind tunnel fan frequency so that the pitot-static system measures desired dynamic
pressure (say 575, 600, 625, 650, etc. Pa). Higher dynamic pressure translates into faster
speeds. At high AOAs too much dynamic pressure could lead to damage of the airfoil
and/or balance. Typical speeds are 60-65 knots (100-110 ft/s or about 32 m/s).
10. Measure the lift and drag force using the dynamic balance. Record lift (L) and drag (D) for
each set AOA. Unit for lift and drag is non-SI “kgf” which is about 9.81 N or 2.2 lbf.
a. Repeat the same measurement of L and D at different AOAs. Maintain constant
measured dynamic pressure (same speed).
b. Check that dynamic pressure and AOA didn’t change during the experiment.
11. Calculate the speed (true) from measured dynamic pressure q and air density  . Convert
measured and all calculated speeds in m/s into km/h, ft/s, mph, and knots.

12. Calculate the Mach number M  u a and the Reynolds number Re    c  u  .
13. After all the measurements are done measure the temperature and barometric pressure
again and find average air mass-density if applicable.
14. Evaluate CL , CL2 , CD , CL CD , CL3 2 CD , CL1 2 CD  at each AOA from measured L and D
and dynamic pressure q and given reference surface area.
15. Construct required diagrams.
3.
4.
5.
6.
Note: Environmental air pressure and temperature (possible multiple) in the lab will be measured
at the time of experiments and dry air density calculated. Wind tunnel corrections, turbulence, and
wall effects are neglected. Use no more than 6 significant digits accuracy when appropriate.
Airfoil tested: ___________________________ (e.g., NACA 2412, 4412, 0012, etc.)
Infinite (2D) airfoil/wing
Finite (3D) wing
© (March 2020, ver. 1.20), Dr. Nihad E. Daidzic, All rights reserved
3
Prof. Dr. Nihad E. Daidzic, ATP, CFII
MSUM, P1 WT Lab, AVIA 201 (Theory of Flight©)
Wing/Airfoil Geometric characteristics
b
[inch/m]
c
[inch/m]
S
[inch2/m2]
AR  b 2 S
[-]
c
[inch/m]
TR    ct c r
[-]

[deg]
Static air data I (beginning of experiment)
p
T
[inch Hg]
[0F]
p
T
[Pa]
[0C/K]

[kg/m3]

[Pa s]
[m2/s]

[kg/m3]

[Pa s]
[m2/s]

Static air data II (end of experiment if conducted)
p
T
[inch Hg]
[0F]
p
T
[Pa]
[0C/K]

Average dimensional and non-dimensional air data (if both static measurements performed)
p
T
[Pa]
[0C/K]

[kg/m3]



[-]
[-]
[-]

[-]
Dynamic air data I (measured, set, and calculated)
q
[Pa]
q  S  [N]
u
2q

[m/s]
a   RT
[m/s]

M u a
[-]
Re 
 c u
[-]

M u a
[-]
Re 
 c u
[-]

Dynamic air data II (measured, set, and calculated if conducted)
q
[Pa]
q  S  [N]
u
2q

[m/s]
a   RT
[m/s]
© (March 2020, ver. 1.20), Dr. Nihad E. Daidzic, All rights reserved

4
Prof. Dr. Nihad E. Daidzic, ATP, CFII
MSUM, P1 WT Lab, AVIA 201 (Theory of Flight©)
Calculate Air Density from the ideal gas equation:

p
[kg/m3 ]
R T
R  287  J/kg K  (DryAir) p [Pa] T [K]
Non-dimensional air data

p
p SL

T
TSL


 SL
    
p SL  101,325 Pa TSL  288.15 K  SL  1.225 kg/m 3
Dynamic viscosity:    SL
or:    SL  
8.14807  10 2  T 3 2
T  110.4
 T K  
 1.78936  10  

 288.15 
0.76
5
0.76
Kinematic viscosity:  
a SL  340.294 m/s
 SL  1.78936  10 5 Pa  s
Pa s

  SL  0.76  m2 /s 

 SL  1.460702 105 m 2 /s
Speed of sound: a   RT
 RT
 T 
a



 
a SL
 RTSL  TSL 
The “Lift” Equation: L 
1
  u 2  S  CL  q  S  CL
2
The “Drag” Equation: D 
1/ 2
  1/ 2
q
1
  u 2  p dyn
2
1
  u 2  S  CD  q  S  CD
2
Aerodynamic coefficients and efficiencies:
LN 
L  kgf   9.81 N/kgf 
 
q  Pa   S  m 2 
 q  S  N 
DN 
D  kgf   9.81 N/kgf 
CD 

 
q  Pa   S  m 2 
 q  S  N 
CL 
C
L
ED  L 
CD D
C 
EMD   L 
 CD  max

 
 
CL3 2
EP 
CD
 
 C3 2 
EMP   L 
 CD max
CL1 2
ER 
CD
 

 C1 2 
EMRC   L 
 CD max
© (March 2020, ver. 1.20), Dr. Nihad E. Daidzic, All rights reserved

5
Prof. Dr. Nihad E. Daidzic, ATP, CFII
MSUM, P1 WT Lab, AVIA 201 (Theory of Flight©)
1. Tabular presentation of measured and calculated results (Table I measurement)
#
AOA
[deg]
1
-4
2
-2
3
0
4
2
5
4
6
6
7
8
8
10
9
12
10
14
11
16
12
18
13
20
14
22
15
24
16
26
L [kgf]
D
[ kgf]
CL
CD
[-]
[-]
CL2
[-]
CL CD
© (March 2020, ver. 1.20), Dr. Nihad E. Daidzic, All rights reserved
[-]
C L3 2 C D
[-]
C L1 2 C D
[-]
6
Prof. Dr. Nihad E. Daidzic, ATP, CFII
MSUM, P1 WT Lab, AVIA 201 (Theory of Flight©)
2. Tabular presentation of measured and calculated results (Table II measurement)
#
AOA
[deg]
1
-4
2
-2
3
0
4
2
5
4
6
6
7
8
8
10
9
12
10
14
11
16
12
18
13
20
14
22
15
24
16
26
L [kgf]
D
[ kgf]
CL
CD
[-]
[-]
CL2
[-]
CL CD
© (March 2020, ver. 1.20), Dr. Nihad E. Daidzic, All rights reserved
[-]
C L3 2 C D
[-]
C L1 2 C D
[-]
7
Prof. Dr. Nihad E. Daidzic, ATP, CFII
MSUM, P1 WT Lab, AVIA 201 (Theory of Flight©)
3. Tabular presentation of measured and calculated results (Table III measurement)
#
AOA
[deg]
1
-4
2
-2
3
0
4
2
5
4
6
6
7
8
8
10
9
12
10
14
11
16
12
18
13
20
14
22
15
24
16
26
L [kgf]
D
[ kgf]
CL
CD
[-]
[-]
CL2
[-]
CL CD
© (March 2020, ver. 1.20), Dr. Nihad E. Daidzic, All rights reserved
[-]
C L3 2 C D
[-]
C L1 2 C D
[-]
8
Prof. Dr. Nihad E. Daidzic, ATP, CFII
MSUM, P1 WT Lab, AVIA 201 (Theory of Flight©)
Useful equations and definitions:
a    R T  20.05  T
M 
u
a
Re 
uc


uc 

1
1
  SL  KCAS 2     KTAS 2
2
2
 KTAS 
KCAS


KCAS

 SL 
1
   KTAS 2  S  c L  q  S  c L  S  p  S   pl  pu 
2
1
D     KTAS 2  S  c D  q  S  c D
2
L
Useful units, conversions and constants:
m
N
1 Pa  2
2
s
m
m
1 kgf  1 kg  9.80665 2  9.81 N  2.2 lbf
s
1 lbf  4.448 N  4.448 kg  m/s 2  1slug 1 ft/s 2  32.174 lbm ft/s 2
1 N  1 kg
1 slug  14.59 kg
1kg=2.205 lbm
1 mph  5280 ft/3600 s
1 knot  1NM/h  6080 ft/3600 s  1.689 ft/s
1 km/h  1000 m/h  1000 m/3600 s  3281 ft/3600 s  0.911 ft/s
1 mile  1.61 km
1 m  3.2808 ft
1 NM  1.15 SM  1.852 km
1 inch  25.4 mm  2.54 cm 1 ft  0.3048 m
1 inch 2   2.54  10-4 m 2  6.452 10-4 m 2
2
1m 2  10.764 ft 2  1550inch 2
1 ata = 760 mm Hg  29.92 inch Hg  101,325 Pa  1013.25 hPa  1013.25 mbar  14.69 psi
1bar  14.5 psi 1 inch Hg  3386.53 Pa  33.865 hPa  0.491 psi
t[o F]-32
t[ C] 
1o C  1.8o F
T[K]  273.15  t[o C]
1.8
g 0  9.80665 m/s 2  9.81 m/s 2  32.174 ft/s 2  32.2 ft/s 2
o
© (March 2020, ver. 1.20), Dr. Nihad E. Daidzic, All rights reserved
9
Prof. Dr. Nihad E. Daidzic, ATP, CFII
MSUM, P1 WT Lab, AVIA 201 (Theory of Flight©)
References
Abbot, Ira H. and von Doenhoff, Albert E., Theory of Wing Section, Dover, New York, 1959.
Ackroyd, J. A. D., Axcell, B. P., and Ruban, A. I., Early developments of Modern Aerodynamics,
AIAA, Reston, VA. 2001.
Anderson, J. D. Jr., Fundamentals of Aerodynamics, 2nd edition, McGraw-Hill Book Company,
New York, 1991.
Anderson, J. D. Jr., Aircraft Performance and Design, McGraw-Hill Book Company, New York,
1999.
Anderson, J. D. Jr., Introduction to Flight, McGraw-Hill Book Company, New York, 2005.
Anon, Principles of Flight, JAA ATPL Training, Edition 2, Book 8 (JAR Ref. 080), Atlantic
Flight Training, Ltd., Sanderson Training products, ISBN 0-88487-495-8, Jeppesen GmbH, NeuIsenburg, Germany, 2007.
Ashley, H., and Landahl, M., Aerodynamics of Wings and Bodies, Dover, New York, 1965.
Ashley, H., Engineering Analysis of Flight vehicles, Dover, New York, 1992.
Bertin, J. J., and Cummings, R. M., Aerodynamics for Engineers, 5th edition, Pearson PrenticeHall, Upper Saddle River, NJ, 2009.
Drela, M., Flight Vehicle Aerodynamics. The MIT Press, Cambridge, MA, 2014.
Etkin, B., Dynamics of flight: Stability and control, John Wiley & Sons, New York, 1959.
Etkin, B., Dynamics of atmospheric flight, Dover, Mineola, 2005.
Glauert, H., The Elements of Aerofoil and Airscrew Theory, 2nd edition, Cambridge University
Press, London, 1947.
Hubin, W. N., The Science of Flight: Pilot-oriented Aerodynamics, Iowa State University Press,
Ames, 1992 (1995 second printing).
Hurt, H. H. Jr., Aerodynamics for Naval Aviators, Revised, reprinted with permission by Aviation
Supplies & Academics, Inc., Renton, Washington 98059-3153, 1965.
Kolk, R. W., Modern flight dynamics, Prentice-Hall, Englewood Cliffs, 1961.
Kuethe, A. M., and Schetzer, J. D., Foundations of Aerodynamics, 2nd edition, John Wiley and
Sons, New York, 1959.
© (March 2020, ver. 1.20), Dr. Nihad E. Daidzic, All rights reserved
10
Prof. Dr. Nihad E. Daidzic, ATP, CFII
MSUM, P1 WT Lab, AVIA 201 (Theory of Flight©)
Miele, A., Flight Mechanics: Theory of flight paths. Dover, Minneola, 2016.
Milne-Thomson, L. M., Theoretical Aerodynamics, Dover, New York, 1973.
von Misses, R., Theory of Flight, Dover, New York, 1959.
Moran, J., An introduction to theoretical and computational aerodynamics, Dover, Mineola, 2003.
Nelson, R. C., Flight stability and automatic control, 2nd edition, McGraw-Hill, New York, 1998.
Phillips, W. F., Mechanics of flight, John Wiley & Sons, New York, 2004.
Pope, A., Basic Wing and Airfoil Theory, Dover, Mineola, 2009.
Prandtl, L., and Tietjens, O. G., Applied Hydro- and Aeromechanics, (Translation), Dover, New
York, 1957.
Rae, W. H., and Pope, A., Low-Speed wind tunnel testing, 2nd edition, John Wiley & Sons, New
York, 1984.
Shevell, R. S., Fundamentals of Flight, 2nd edition, Prentice Hall, New Jersey, 1989.
Smith, H. C., The illustrated guide to Aerodynamics, 2nd Edition, ISBN 0-8306-3901-2 (pbk.),
McGraw-Hill Book Company, New York, 1992.
Stinton, D., The anatomy of the airplane, 2nd edition, AIAA, Reston, VA. 1998.
Swatton, P. J., Principles of Flight for Pilots, John Wiley and Sons, Ltd. Chichester, West Sussex,
2011.
© (March 2020, ver. 1.20), Dr. Nihad E. Daidzic, All rights reserved
11
Prof. Dr. Nihad E. Daidzic, ATP, CFII
MSUM, P1 WT Lab, AVIA 201 (Theory of Flight©)
Notes (page 1):
© (March 2020, ver. 1.20), Dr. Nihad E. Daidzic, All rights reserved
12
Prof. Dr. Nihad E. Daidzic, ATP, CFII
MSUM, P1 WT Lab, AVIA 201 (Theory of Flight©)
Notes (page 2):
© (March 2020, ver. 1.20), Dr. Nihad E. Daidzic, All rights reserved
13
Prof. Dr. Nihad E. Daidzic, ATP, CFII
MSUM, P1 WT Lab, AVIA 201 (Theory of Flight©)
Notes (page 3):
© (March 2020, ver. 1.20), Dr. Nihad E. Daidzic, All rights reserved
14
Prof. Dr. Nihad E. Daidzic, ATP, CFII
MSUM, P1 WT Lab, AVIA 201 (Theory of Flight©)
MINNESOTA STATE UNIVERSITY, Mankato
Aviation Department
Theory of Flight© (AVIA 201)
Project 1 Wind-Tunnel Laboratory (Trafton Science Center E 104)
FALL/SPRING
Instructor:
Nihad E. Daidzic, Dr.-Ing., D.Sc., M.S.M.E., B.S.M.E.
Professor of Aviation, Research Graduate Faculty
ATP AMEL, Comm. ASEL/ASES/Glider/Helicopter/LBH, Instrument
FAA “Gold Seal” CFI-IA, ME-IA, CFI-RH, CFI-G, IGI, AGI
Class Meeting:
Project 1 Wind-Tunnel Laboratory (Trafton Science Center E 104)
(During regular class time. Exact date will be announced or syllabus)
PROJECT 1 Assignment
WIND-TUNNEL LAB (AVIA 201)
Name: __________________________________________ Section: AVIA 201-____________
TECH ID: _____________________ Signature: _____________________________________
Semester: ________________________________ Date: _______________________________
Grade/Score: __________________ Comment: _____________________________________
_____________________________________________________________________________
© (March 2020, ver. 1.20), Dr. Nihad E. Daidzic, All rights reserved
1
Prof. Dr. Nihad E. Daidzic, ATP, CFII
MSUM, P1 WT Lab, AVIA 201 (Theory of Flight©)
Purpose: To introduce aviation/aeronautics/aerospace students to wind tunnel(s) and methods
used in experimental identification of various aerodynamic (and stability) coefficients of airfoils
(2D), wings (3D) and scale models. To understand the value and purpose of wind-tunnel and
experimental identification (measurement) of lift, drag, sideforce, and aerodynamic moments
(pitching, rolling, yawing). To review the basic aerodynamic theory and knowledge of wings and
airfoils presented in classroom. To apply and correlate knowledge acquired in the lab to better
understanding of fixed- and rotary-wing flight control, stability and performance. To practice unit
conversions (from SI to English engineering and vice versa). To perform visualization study of the
boundary layer separation and stall patterns at higher AOAs. To visualize wingtip vortices when
3D (finite wing) airfoil is installed.
Grade: Wind-Tunnel lab project is required project in AVIA 201 course and brings 10% of the
overall grade. Two students conduct one measurement, but all project completion is individual.
Measurement report is required with all the graphs constructed.
Project Deliverables, calculations and Graphical presentation: Measure static pressure and
temperature in a lab. Calculate dry air density. Calculate dynamic and kinematic viscosity. Set the
fan speed to achieve desired air dynamic pressure (pdyn = q). Typically, in a range of 600-650 Pa.
Calculate TAS. From the airfoil geometry (2D or 3D) measure chord length and reference surface
area. Calculate the acoustic velocity (speed of sound). Calculate Reynolds and Mach numbers.
Measure lift and drag. Evaluate aerodynamic coefficients. Estimate drag polar, parabolic drag
model and other useful aerodynamic parameters (endurance polar and range polar). Present the
final results in a graphical form for:
1.
2.
3.
4.
5.
6.
7.
C L vs AOA
C D vs AOA
C L vs C D (Polar diagram)
CD vs CL2 (Parabolic drag model CD  CD , 0  K  CL2 )
C L
C
C
32
L
12
L
C D  vs AOA (Aerodynamic efficiency)
C D  vs AOA (Endurance efficiency)
C D  vs AOA (Range efficiency)
in a graphical format (diagrams) by using MS ExcelTM, MatlabTM, Fortran, (Visual) Basic, C++,
MapleTM or any other appropriate calculation/spreadsheet/engineering/mathematical software
(hand drawn graphs on millimeter paper is acceptable) and attach to cover page.
Workload: 2.5 hours of data collection and experiments. 6-10 hours per student for data
processing and plotting/constructing diagrams and results. Overall, this wind-tunnel project should
not exceed 13 effective physical hours workload per student.
Important Safety Note: Discipline and order during lab work will be enforced. Exercise safety
precautions and common sense. Do not touch or come close to the large exhaust fan. Do not touch
any electrical connections and cables. Do not wander around touching and moving things during
the experimentation. Do not stand at the suction or the exhaust end of the wind tunnel.
© (March 2020, ver. 1.20), Dr. Nihad E. Daidzic, All rights reserved
2
Prof. Dr. Nihad E. Daidzic, ATP, CFII
MSUM, P1 WT Lab, AVIA 201 (Theory of Flight©)
Experimental Procedure: Two measurements of lift and drag aerodynamic coefficients will be
conducted at two different wind-tunnel uniform speeds (adjusted by the fan). Reynolds and Mach
numbers are to be evaluated for each experiment. The experimental procedure is:
1. Measure the barometric pressure p and temperature T in the lab. Assume dry air. Convert
from 0F and inches Hg into Celsius (0C), Kelvin (K) and pressure in Pascals (and
hPa=mbar). Call MKT ASOS to verify (KMKT elevation is about 1,000 ft).
2. Calculate the air density assuming that dry air mixture obeys ideal-gas law. The air gas
constant R  287 .053 J/kgK . The isentropic coefficient of expansion for air is   1.4 .
Calculate the speed of sound: a    R  T  20.05  T m/s  .
Calculate the dynamic viscosity of dry air  .
Calculate the kinematic viscosity of the air     .
Chose the airfoil/wing. Measure its chord and span. Calculate its mean chord length (if
applicable), aspect ratio (AR), taper ratio (TR), sweep angle  , planform type. Convert
Imperial units into SI as necessary.
7. Set the airfoil/wing in a wind tunnel under desired AOA. Make sure that the Pitot tube is
parallel with the longitudinal axis of the wind-tunnel test section, i.e., parallel with the
flow. Close and tighten lightly the upper section.
8. Calibrate and re-calibrate the force balance to show about zero in L and D without flow.
9. Set the wind tunnel fan frequency so that the pitot-static system measures desired dynamic
pressure (say 575, 600, 625, 650, etc. Pa). Higher dynamic pressure translates into faster
speeds. At high AOAs too much dynamic pressure could lead to damage of the airfoil
and/or balance. Typical speeds are 60-65 knots (100-110 ft/s or about 32 m/s).
10. Measure the lift and drag force using the dynamic balance. Record lift (L) and drag (D) for
each set AOA. Unit for lift and drag is non-SI “kgf” which is about 9.81 N or 2.2 lbf.
a. Repeat the same measurement of L and D at different AOAs. Maintain constant
measured dynamic pressure (same speed).
b. Check that dynamic pressure and AOA didn’t change during the experiment.
11. Calculate the speed (true) from measured dynamic pressure q and air density  . Convert
measured and all calculated speeds in m/s into km/h, ft/s, mph, and knots.

12. Calculate the Mach number M  u a and the Reynolds number Re    c  u  .
13. After all the measurements are done measure the temperature and barometric pressure
again and find average air mass-density if applicable.
14. Evaluate CL , CL2 , CD , CL CD , CL3 2 CD , CL1 2 CD  at each AOA from measured L and D
and dynamic pressure q and given reference surface area.
15. Construct required diagrams.
3.
4.
5.
6.
Note: Environmental air pressure and temperature (possible multiple) in the lab will be measured
at the time of experiments and dry air density calculated. Wind tunnel corrections, turbulence, and
wall effects are neglected. Use no more than 6 significant digits accuracy when appropriate.
Airfoil tested: ___________________________ (e.g., NACA 2412, 4412, 0012, etc.)
Infinite (2D) airfoil/wing
Finite (3D) wing
© (March 2020, ver. 1.20), Dr. Nihad E. Daidzic, All rights reserved
3
Prof. Dr. Nihad E. Daidzic, ATP, CFII
MSUM, P1 WT Lab, AVIA 201 (Theory of Flight©)
Wing/Airfoil Geometric characteristics
b
[inch/m]
c
[inch/m]
S
[inch2/m2]
AR  b 2 S
[-]
c
[inch/m]
TR    ct c r
[-]

[deg]
Static air data I (beginning of experiment)
p
T
[inch Hg]
[0F]
p
T
[Pa]
[0C/K]

[kg/m3]

[Pa s]
[m2/s]

[kg/m3]

[Pa s]
[m2/s]

Static air data II (end of experiment if conducted)
p
T
[inch Hg]
[0F]
p
T
[Pa]
[0C/K]

Average dimensional and non-dimensional air data (if both static measurements performed)
p
T
[Pa]
[0C/K]

[kg/m3]



[-]
[-]
[-]

[-]
Dynamic air data I (measured, set, and calculated)
q
[Pa]
q  S  [N]
u
2q

[m/s]
a   RT
[m/s]

M u a
[-]
Re 
 c u
[-]

M u a
[-]
Re 
 c u
[-]

Dynamic air data II (measured, set, and calculated if conducted)
q
[Pa]
q  S  [N]
u
2q

[m/s]
a   RT
[m/s]
© (March 2020, ver. 1.20), Dr. Nihad E. Daidzic, All rights reserved

4
Prof. Dr. Nihad E. Daidzic, ATP, CFII
MSUM, P1 WT Lab, AVIA 201 (Theory of Flight©)
Calculate Air Density from the ideal gas equation:

p
[kg/m3 ]
R T
R  287  J/kg K  (DryAir) p [Pa] T [K]
Non-dimensional air data

p
p SL

T
TSL


 SL
    
p SL  101,325 Pa TSL  288.15 K  SL  1.225 kg/m 3
Dynamic viscosity:    SL
or:    SL  
8.14807  10 2  T 3 2
T  110.4
 T K  
 1.78936  10  

 288.15 
0.76
5
0.76
Kinematic viscosity:  
a SL  340.294 m/s
 SL  1.78936  10 5 Pa  s
Pa s

  SL  0.76  m2 /s 

 SL  1.460702 105 m 2 /s
Speed of sound: a   RT
 RT
 T 
a



 
a SL
 RTSL  TSL 
The “Lift” Equation: L 
1
  u 2  S  CL  q  S  CL
2
The “Drag” Equation: D 
1/ 2
  1/ 2
q
1
  u 2  p dyn
2
1
  u 2  S  CD  q  S  CD
2
Aerodynamic coefficients and efficiencies:
LN 
L  kgf   9.81 N/kgf 
 
q  Pa   S  m 2 
 q  S  N 
DN 
D  kgf   9.81 N/kgf 
CD 

 
q  Pa   S  m 2 
 q  S  N 
CL 
C
L
ED  L 
CD D
C 
EMD   L 
 CD  max

 
 
CL3 2
EP 
CD
 
 C3 2 
EMP   L 
 CD max
CL1 2
ER 
CD
 

 C1 2 
EMRC   L 
 CD max
© (March 2020, ver. 1.20), Dr. Nihad E. Daidzic, All rights reserved

5
Prof. Dr. Nihad E. Daidzic, ATP, CFII
MSUM, P1 WT Lab, AVIA 201 (Theory of Flight©)
1. Tabular presentation of measured and calculated results (Table I measurement)
#
AOA
[deg]
1
-4
2
-2
3
0
4
2
5
4
6
6
7
8
8
10
9
12
10
14
11
16
12
18
13
20
14
22
15
24
16
26
L [kgf]
D
[ kgf]
CL
CD
[-]
[-]
CL2
[-]
CL CD
© (March 2020, ver. 1.20), Dr. Nihad E. Daidzic, All rights reserved
[-]
C L3 2 C D
[-]
C L1 2 C D
[-]
6
Prof. Dr. Nihad E. Daidzic, ATP, CFII
MSUM, P1 WT Lab, AVIA 201 (Theory of Flight©)
2. Tabular presentation of measured and calculated results (Table II measurement)
#
AOA
[deg]
1
-4
2
-2
3
0
4
2
5
4
6
6
7
8
8
10
9
12
10
14
11
16
12
18
13
20
14
22
15
24
16
26
L [kgf]
D
[ kgf]
CL
CD
[-]
[-]
CL2
[-]
CL CD
© (March 2020, ver. 1.20), Dr. Nihad E. Daidzic, All rights reserved
[-]
C L3 2 C D
[-]
C L1 2 C D
[-]
7
Prof. Dr. Nihad E. Daidzic, ATP, CFII
MSUM, P1 WT Lab, AVIA 201 (Theory of Flight©)
3. Tabular presentation of measured and calculated results (Table III measurement)
#
AOA
[deg]
1
-4
2
-2
3
0
4
2
5
4
6
6
7
8
8
10
9
12
10
14
11
16
12
18
13
20
14
22
15
24
16
26
L [kgf]
D
[ kgf]
CL
CD
[-]
[-]
CL2
[-]
CL CD
© (March 2020, ver. 1.20), Dr. Nihad E. Daidzic, All rights reserved
[-]
C L3 2 C D
[-]
C L1 2 C D
[-]
8
Prof. Dr. Nihad E. Daidzic, ATP, CFII
MSUM, P1 WT Lab, AVIA 201 (Theory of Flight©)
Useful equations and definitions:
a    R T  20.05  T
M 
u
a
Re 
uc


uc 

1
1
  SL  KCAS 2     KTAS 2
2
2
 KTAS 
KCAS


KCAS

 SL 
1
   KTAS 2  S  c L  q  S  c L  S  p  S   pl  pu 
2
1
D     KTAS 2  S  c D  q  S  c D
2
L
Useful units, conversions and constants:
m
N
1 Pa  2
2
s
m
m
1 kgf  1 kg  9.80665 2  9.81 N  2.2 lbf
s
1 lbf  4.448 N  4.448 kg  m/s 2  1slug 1 ft/s 2  32.174 lbm ft/s 2
1 N  1 kg
1 slug  14.59 kg
1kg=2.205 lbm
1 mph  5280 ft/3600 s
1 knot  1NM/h  6080 ft/3600 s  1.689 ft/s
1 km/h  1000 m/h  1000 m/3600 s  3281 ft/3600 s  0.911 ft/s
1 mile  1.61 km
1 m  3.2808 ft
1 NM  1.15 SM  1.852 km
1 inch  25.4 mm  2.54 cm 1 ft  0.3048 m
1 inch 2   2.54  10-4 m 2  6.452 10-4 m 2
2
1m 2  10.764 ft 2  1550inch 2
1 ata = 760 mm Hg  29.92 inch Hg  101,325 Pa  1013.25 hPa  1013.25 mbar  14.69 psi
1bar  14.5 psi 1 inch Hg  3386.53 Pa  33.865 hPa  0.491 psi
t[o F]-32
t[ C] 
1o C  1.8o F
T[K]  273.15  t[o C]
1.8
g 0  9.80665 m/s 2  9.81 m/s 2  32.174 ft/s 2  32.2 ft/s 2
o
© (March 2020, ver. 1.20), Dr. Nihad E. Daidzic, All rights reserved
9
Prof. Dr. Nihad E. Daidzic, ATP, CFII
MSUM, P1 WT Lab, AVIA 201 (Theory of Flight©)
References
Abbot, Ira H. and von Doenhoff, Albert E., Theory of Wing Section, Dover, New York, 1959.
Ackroyd, J. A. D., Axcell, B. P., and Ruban, A. I., Early developments of Modern Aerodynamics,
AIAA, Reston, VA. 2001.
Anderson, J. D. Jr., Fundamentals of Aerodynamics, 2nd edition, McGraw-Hill Book Company,
New York, 1991.
Anderson, J. D. Jr., Aircraft Performance and Design, McGraw-Hill Book Company, New York,
1999.
Anderson, J. D. Jr., Introduction to Flight, McGraw-Hill Book Company, New York, 2005.
Anon, Principles of Flight, JAA ATPL Training, Edition 2, Book 8 (JAR Ref. 080), Atlantic
Flight Training, Ltd., Sanderson Training products, ISBN 0-88487-495-8, Jeppesen GmbH, NeuIsenburg, Germany, 2007.
Ashley, H., and Landahl, M., Aerodynamics of Wings and Bodies, Dover, New York, 1965.
Ashley, H., Engineering Analysis of Flight vehicles, Dover, New York, 1992.
Bertin, J. J., and Cummings, R. M., Aerodynamics for Engineers, 5th edition, Pearson PrenticeHall, Upper Saddle River, NJ, 2009.
Drela, M., Flight Vehicle Aerodynamics. The MIT Press, Cambridge, MA, 2014.
Etkin, B., Dynamics of flight: Stability and control, John Wiley & Sons, New York, 1959.
Etkin, B., Dynamics of atmospheric flight, Dover, Mineola, 2005.
Glauert, H., The Elements of Aerofoil and Airscrew Theory, 2nd edition, Cambridge University
Press, London, 1947.
Hubin, W. N., The Science of Flight: Pilot-oriented Aerodynamics, Iowa State University Press,
Ames, 1992 (1995 second printing).
Hurt, H. H. Jr., Aerodynamics for Naval Aviators, Revised, reprinted with permission by Aviation
Supplies & Academics, Inc., Renton, Washington 98059-3153, 1965.
Kolk, R. W., Modern flight dynamics, Prentice-Hall, Englewood Cliffs, 1961.
Kuethe, A. M., and Schetzer, J. D., Foundations of Aerodynamics, 2nd edition, John Wiley and
Sons, New York, 1959.
© (March 2020, ver. 1.20), Dr. Nihad E. Daidzic, All rights reserved
10
Prof. Dr. Nihad E. Daidzic, ATP, CFII
MSUM, P1 WT Lab, AVIA 201 (Theory of Flight©)
Miele, A., Flight Mechanics: Theory of flight paths. Dover, Minneola, 2016.
Milne-Thomson, L. M., Theoretical Aerodynamics, Dover, New York, 1973.
von Misses, R., Theory of Flight, Dover, New York, 1959.
Moran, J., An introduction to theoretical and computational aerodynamics, Dover, Mineola, 2003.
Nelson, R. C., Flight stability and automatic control, 2nd edition, McGraw-Hill, New York, 1998.
Phillips, W. F., Mechanics of flight, John Wiley & Sons, New York, 2004.
Pope, A., Basic Wing and Airfoil Theory, Dover, Mineola, 2009.
Prandtl, L., and Tietjens, O. G., Applied Hydro- and Aeromechanics, (Translation), Dover, New
York, 1957.
Rae, W. H., and Pope, A., Low-Speed wind tunnel testing, 2nd edition, John Wiley & Sons, New
York, 1984.
Shevell, R. S., Fundamentals of Flight, 2nd edition, Prentice Hall, New Jersey, 1989.
Smith, H. C., The illustrated guide to Aerodynamics, 2nd Edition, ISBN 0-8306-3901-2 (pbk.),
McGraw-Hill Book Company, New York, 1992.
Stinton, D., The anatomy of the airplane, 2nd edition, AIAA, Reston, VA. 1998.
Swatton, P. J., Principles of Flight for Pilots, John Wiley and Sons, Ltd. Chichester, West Sussex,
2011.
© (March 2020, ver. 1.20), Dr. Nihad E. Daidzic, All rights reserved
11
Prof. Dr. Nihad E. Daidzic, ATP, CFII
MSUM, P1 WT Lab, AVIA 201 (Theory of Flight©)
Notes (page 1):
© (March 2020, ver. 1.20), Dr. Nihad E. Daidzic, All rights reserved
12
Prof. Dr. Nihad E. Daidzic, ATP, CFII
MSUM, P1 WT Lab, AVIA 201 (Theory of Flight©)
Notes (page 2):
© (March 2020, ver. 1.20), Dr. Nihad E. Daidzic, All rights reserved
13
Prof. Dr. Nihad E. Daidzic, ATP, CFII
MSUM, P1 WT Lab, AVIA 201 (Theory of Flight©)
Notes (page 3):
© (March 2020, ver. 1.20), Dr. Nihad E. Daidzic, All rights reserved
14
AVIA 201 WT lab measurements NACA 0012
Dynamic pressure “q” is costant 625 Pa (2.5 inch H2O column) and is constant for all
measurements. Geometric AOA (Angle of Attack) is changed in increments by 2 degrees from -4
to +16 angular degrees and Lift (L) and Drag (D) forces measured using wind tunnel balance.
NACA 0012 “infinite” wing (2D section) used has the following geometric characteristics:
Chord length (given)
Span (given)
Reference area S
Aspect Ratio AR=
4
12
48.00
3.000
[inch]
[inch]
[inch2]
[-]
0.1016
0.3048
3.09677E-02
3.000
[m]
[m]
[m2]
[-]
Measured aerodynamic data
AOA
[-]
-4
-2
0
2
4
6
8
10
12
14
16
Measured
L
D
[kgf]
[kgf]
-0.6200
-0.3300
0.0550
0.2700
0.8450
1.0800
1.6600
2.0900
2.5500
1.9900
1.8000
0.1170
0.1110
0.1060
0.1050
0.1170
0.1280
0.1650
0.2020
0.2760
0.8000
0.9200
Note: Measured Lift and Drag forces are given in the non-standard force units of “kgf”. The
equation provided in the Lab document contains conversion factor into Newtons or N (force) to
get non-dimensional values of CL and CD for each AOA.
Air density is computed by using ideal gas law and measured data for pressure and temperature.
Make sure you use proper unit conversions from one unit to another (F to C for temperature and
inch Hg to Pa for atmospheric pressure).
Measured lab air data (Wednesday February 12, 2020 at 1:30 PM)
Barometric pressure [inch Hg]
KMKT Altimeter setting [Hg]
Lab Elevation [ft] (approx.)
Temperature [deg F]
29.20
29.85
1020
72.0
measured
measured
known
measured
Based on symmetric NACA 0012 geometric and aerodynamic measurements and air data
measurements (pressure and temperature) find air density, various derived geometric
characteristics, Reynolds and Mach numbers, and various aerodynamic coefficients as required.
We only performed one set of measurements. Accordingly, the average (mean) is the same as the
first measurement. There are no second, third or repeated measurement. Pressure and temperature
were only measured once (Static Air Data I).
Prof. Dr. Nihad Daidzic, ATP, CFII, MEI
WT measurements Tutorial, v 1.3
Minnesota State University, Mankato
Department of Aviation
Wind tunnel measurements and data analysis tutorial
This tutorial gives details on processing of raw aerodynamic data measured directly in the wind
tunnel on airfoils and wings. Also procedures and equations from the lab report are given and
demonstrated how to compute geometric characteristics of wings/airfoils and air data. All numeric
results are typically rounded to five significant digits.
Important Definitions and Descriptions:
A cambered (asymmetric) airfoil is illustrated in Fig. 1. Definitions of various angles is given as
well as the lifting curve CL vs AOA (Angle-of-Attack) or α. At zero geometric AOA, cambered
airfoil still produces some lift as the absolute AOA (αa) is defined not through chord but the ZeroLift-Line (Z.L.L.). The design lift-coefficient CL at the zero geometric AOA (α) is defined as CL0.
The maximum coefficient-of-lift CL occurs at the stalling AOA (αcr) and is designated as CL,max.
Due to production of lift that induces vortices and the net downwash flow (w), the induced AOA
(αi) on finite wing acts to effectively reduces the far-field AOA (α) into the near-field AOA (αE).
Induced drag does not exist in 2D airfoils (infinite wings).
Figure 1: Aerodynamic definitions of cambered airfoils.
Finite (3D) wing projected planform surface area (Sref) is a product of wingspan (b) and mean
geometric chord ( c ). Geometric twist (εg) is given as a difference in incidence angles at the wing
© (2019) Dr. Nihad E. Daidzic, All rights reserved
1
Prof. Dr. Nihad Daidzic, ATP, CFII, MEI
WT measurements Tutorial, v 1.3
root and wing tip. The aerodynamic twist (εa) is the angular difference between ZLLs at the root
and tip. Taper ratio (λ) is the length ratio between wing chords (c) at the tip and the root (see Fig
2). Dihedral angle (Γ) is the angle between the horizontal plane and a plane that goes through root
and tip chords. Sweep angle (Λ) is the angle between LE (at c/4) and the plane perpendicular to
the longitudinal axis.
The resultant aerodynamic reaction force R acting on a moving immersed body in a fluid is a result
of normal (pressure) and shear-stress distributions on the exposed surfaces (see Fig. 3). This
reaction force is determined by computing the surface integral of normal and shear (tangential)
stresses. Normal force to a chord is designated as N, while the axial force is designated as A. Lift
force (L) is by definition perpendicular (orthogonal) to a far-field relative wind (RW). Drag or
fluid resistance (D) force is collinear with the RW but opposite the direction of motion. In the case
of the finite (3D) wing, the variable span-wise and chord-wise circulation generates tip vortices,
which induce downwash, hence resulting in induced (vortex) drag.
Figure 2: Basic finite wing geometry.
Processing of experimental/measurement data
First, one needs to define geometric characteristics of airfoils (wing sections) or finite wings.
Infinite wings or 2D sections (airfoils) span between two walls in the test section, meaning that
apart from wall effects the chord-wise flow (along tunnel’s longitudinal axis) is equal in all
sections. The width of the tunnel test section is 12 inches and that would be the span of the infinite
wing. If the chord length of the chosen rectangular wing (finite or infinite) is, for example, 2.5
inches (6.35 cm or 0.0635 m) then the reference (or projected) area is:
© (2019) Dr. Nihad E. Daidzic, All rights reserved
2
Prof. Dr. Nihad Daidzic, ATP, CFII, MEI
WT measurements Tutorial, v 1.3
Sref  b  c  b  c  12  2.5  30 inch 2
Figure 3: Aerodynamic forces.
Normally we would just use ruler to measure wing geometric data directly. If we have simply
tapered wing than we must first compute the mean chord (Mean Geometric Chord or MGC). We
need to convert surface area in inches squared into meter squared using the fact that one inch is
2.54 cm or 0.0254 meters. One square inch is thus 0.0254 x 0.0254 = 6.452 x 10-4 m2. Hence, 30
inch2 is 1.935 x 10-2 m2.
The aspect ratio (AR) is now:
AR 
b2
b 12
 
 4.8 []
Sref c 2.5
An important note: in infinite wings (airfoils or sections), that span from wall to wall, there is no
induced drag and that aspect ratio is actually infinite in terms of aerodynamic effects.
The taper ratio of any simple wing is:
TR   
ct
cr
In the case of rectangular wing, tip and root chord are identical and TR is one. We used dihedral
angle zero. Sweep angle is the angle between the leading edge at the 25% chord (aerodynamic
© (2019) Dr. Nihad E. Daidzic, All rights reserved
3
Prof. Dr. Nihad Daidzic, ATP, CFII, MEI
WT measurements Tutorial, v 1.3
center) and the vertical plane. For straight rectangular wing with taper-ratio of one, the sweep
angle is also zero. We now have all basic geometric characteristic of the wings and airfoils.
Next, we need to compute (dry) air density in the lab based on the barometric pressure and lab-air
temperature measurements. Let us say that using lab barometer we found out that local atmospheric
pressure is 29.20 inch Hg. Temperature measurements using Mercury thermometer gave us, for
example, 73oF. We need to convert this into Pascals (Pa) and degrees C (oC) and absolute
temperature in Kelvin (K). To convert pressures, we just need to remember that 101,325 Pa is
29.92 inch or 760 mm Hg (column height of mercury). From there we see that 1 inch of Hg is
equivalent to 3,386.53 Pa and 1 mm Hg is equal to 133.3 Pa. Accordingly, 29.20 inch Hg is
98,886.7 Pa. Now convert degrees Fahrenheit into degrees Celsius using:
t[o F]-32 73-32

 22.78o C
1.8
1.8
T[K]  273.15  t[o C]=273.15+22.78o C  295.93K
T[o C] 
All conversion factors and equations are also given in the AVIA 201 Project 1 Wind Tunnel Lab
document. We can now calculate dry-air density (neglecting humidity) from the ideal gas-law:
p
98,886.7

 1.1643 kg/m3
R  T 287  295.93
R  287  J/kg K  (DryAir) p [Pa] T [K]

We can also convert that into slug/ft3 by remembering that 1 slug is equal to 14.59 kg and 1 meter
is 3.2808 feet. Hence, 1.1643 kg/m3 is equal to about 0.002259 slug/ft3. Now we can compute nondimensional pressure, temperature and density based on the SL (Sea Level) properties.

p
pSL

T
TSL


 SL
    
 
pSL  101,325 Pa TSL  288.15 K  SL  1.225 kg/m3 aSL  340.294 m/s
Substituting our values, we obtain:

p 98,886.7

 0.9759
pSL 101,325
295.93
 1.026999
288.15
 1.1643


 0.9504
 SL 1.225

     0.9759  0.9504  1.027
   1 2  1.026999  1.01341
© (2019) Dr. Nihad E. Daidzic, All rights reserved
4
Prof. Dr. Nihad Daidzic, ATP, CFII, MEI
WT measurements Tutorial, v 1.3
To estimate Mach and Reynolds numbers we need to find the actual speed of sound in air in the
lab as well as dynamic and kinematic viscosities. The local speed of sound is obtained from the
actual temperature measurements (73oF or 22.78oC or 295.93 K):
a   RT  20.05 T  aSL   340.3    340.3 1.0134  344.86 m/s=1131.43 ft/s  667 knots
The true airspeed (TAS) is computed from Pitot-static measurements. If the dynamic pressure q is
for example 650 Pa, the actual true airspeed (by neglecting measuring errors) is:
q
1 2 1
2q
q
650
 v    SL v 2  v 
 1.278 
 1.278 
 33.42 m/s=64.9 knot
2
2
  SL

0.9504
A student should be able to easily convert between length units (ft, m, NM, etc) and speed units
(knots, ft/s, m/s, kph, etc.) and other imperial and SI units. Important is to always use consistent
units and not mix “apples” and “oranges”. The Mach number is now:
M
vTAS 64.9

 0.0973 []
a
667
A (dimensionless) Reynolds number is defined as:
Re 
vTAS    c


vTAS  c

Here, we used average chord (or MGC) as a characteristic length for chordwise flow to compute
Reynolds number. In our case of rectangular wing, the chord length is constant and equal to 2.5
inch or chord everywhere. Hence, we need to compute first dynamic or kinematic viscosity. We
already computed air density earlier and dynamic viscosity (Sutherland’s equation) is computed
from (see also WT lab document):
8.14807 102   295.93
8.14807 102  T 3 2
5
  SL
 1.78936 10 
T  110.4 
 295.93  110.4 
SL  1.78936 105 Pa  s
32
 1.82665 10 5 Pa  s
1 cP  10-3 Pa  s
Or according to Granger:
   SL 
0.76
 T K  
 1.78936  10  

 288.15 
5
0.76
 1.82595  10 5 Pa  s  1.826  10 2 cP
Kinematic viscosity is now:
© (2019) Dr. Nihad E. Daidzic, All rights reserved
5
Prof. Dr. Nihad Daidzic, ATP, CFII, MEI
WT measurements Tutorial, v 1.3
 1.82665 105
 
 1.5689 105 m 2 /s  15.69 cSt

1.1643
1 St  1 cm 2 /s  10-4 m 2 /s 1 cSt  10-6 m 2 /s
The Reynolds number is now:
Re 
vTAS  c


33.42  2.5  0.0254
 1.352648 105  135, 265 []
5
1.5689 10
We see that both Mach number and Reynolds numbers are very low. Mach number of less than
0.1 indicates that the airflow is very slow incompressible subsonic (M < 0.3) and the small Reynold’s number likely affects the maximum coefficient-of-lift (CL) compared to large-scale wings in flight (1-100 million). Now we have all ingredients to compute most important aerodynamic coefficients from experimental data. The coefficient of lift and drag are computed from expressions: L  kgf   9.81 N/kgf  CL  LN   q  Pa   S  m  q  Pa   S  m 2  D  kgf   9.81 N/kgf  DN  CD   2 q  Pa   S  m  q  Pa   S  m 2  2     Various aerodynamic efficiencies (Minimum drag, minimum power or maximum endurance and maximum range cruise) are computed as: EMD C L  L  CD D   EMP CL3 2  CD  EMR CL1 2  CD  We already know that in our example: q  Pa   S  m 2   650 1.935 102  12.578 N This above value is constant and does not change unless different dynamic pressure is used. We will use now an example of measured lift and drag in units of “kgf” at set angle-of-attack (AOA) and set dynamic pressure (q). Let us assume that one particular measurement at AOA of 8o returned measured Lift (L) of 2.500 kgf and Drag (D) of 0.140 kgf. Now we have coefficients: 2.500  9.81  1.95    12.578 0.140  9.81 CD   0.109    12.578 CL  © (2019) Dr. Nihad E. Daidzic, All rights reserved 6 Prof. Dr. Nihad Daidzic, ATP, CFII, MEI WT measurements Tutorial, v 1.3 The aerodynamic efficiency at AOA of 8o is: EMD  CL 1.95   17.86 CD 0.109   The aerodynamic endurance (power required) is: EMP  CL3 2 1.953 2   24.98 CD 0.109  The aerodynamic range performance (range cruise) is: EMR  CL1 2 1.95   12.81 CD 0.109   You will typically find out that for cambered NACA 4412 airfoil, the aerodynamic efficiency peaks around 6 angular degrees AOA, range efficiency around 4 angular degrees AOA, and endurance efficiency around 8o AOA. Stalling (geometric) AOA is at about 16 angular degrees. These computations are then repeated for all measured (geometric) AOAs and associated dynamic pressures (could be different but not recommended as that would change Re and M numbers and possibly affect Boundary Layer structure). When all data are calculated/processed, construct required diagrams as required per WT Lab Project assignment. Also, illustrate Boundary Layer and BL separation. An example of lift-curve and drag-curve diagrams finalized using MS ExcelTM are shown in Figs. 4 and 5 respectively (students need to use their own measured data.). As will be evident from experimental data (see Fig. 4), the lift-curve (CL vs AOA) consists of the linear and nonlinear region. In the linear region, the coefficient-of-lift increases proportionally to increase in AOA. Once the separation (typically at the TE) becomes significant, the lift-curve slope starts decreasing, leveling off, becomes zero at stalling angle, and then becomes negative in poststall region. In linear region (up to about 10o for NACA 4412), the linear region can be described for many practical airfoils with the simple relationship (due to von Mises): CL  CL ,0  k   g dCL 0.1  CL  d 1  2 AR for  in angular degrees More complicated relationships exists for lift-curve. For cambered airfoils and wings CL,0 is typically in the range 0.2 to 0.5, while for symmetric airfoils it is ZERO. For infinite wings (airfoils) the AR is infinite and the slope CL,α is about 0.1/degree for AOA expressed in angular degrees or about 5.7/rad when AOA is expressed in radians. Increasing AR, sweep angle and Mach number all work to decrease the lift-curve slope. The theoretical lift-curve slope from the “thinairfoil” theory is 2π or about 6.28/rad for AOA in radians. All these considerations are valid in the absence of dynamic or vortex-lift and transient effects. Coefficient of drag measured as a function of AOA for given airfoil is given in Fig. 5. © (2019) Dr. Nihad E. Daidzic, All rights reserved 7 Prof. Dr. Nihad Daidzic, ATP, CFII, MEI WT measurements Tutorial, v 1.3 Figure 4: Lift-curve (CL vs AOA) for NACA 4412 airfoil. Figure 5: Drag curve (CD vs AOA) for NACA 4412 airfoil. © (2019) Dr. Nihad E. Daidzic, All rights reserved 8 Prof. Dr. Nihad Daidzic, ATP, CFII, MEI WT measurements Tutorial, v 1.3 Total drag of airfoils or infinite-wing sections (no induced drag) and finite wings (parasitic and induced drag) can be, in general, modeled with a parabolic function in linear lift-curve region: CD  CD ,0  K  CL2 K  k1  k2  k3  k1  k2  1   e  AR Here, k1 is a coefficient of parasitic drag only (form drag and skin-friction drag), k2 is a coefficient of wave drag (only at speeds exceeding critical Mach number and does not exist for slow GA piston-prop airplanes), and k3 is a coefficient of induced- or vortex-drag and only exists for finite wings (not airfoils or infinite wings). The zero-lift coefficient of drag is simply a sum of zero-lift drag coefficient due to parasitic and wave drag (at high speeds only). Accordingly, drag can be modeled as a part that is due to production of lift and the part that is independent of lift production. Close to stalling angle and in nonlinear region many of the coefficients are no longer simple constants and change strongly with AOA. Polar diagram (CL vs CD) for representative WT measurement on example of NACA 4412 is shown in Fig. 6. Parabolic drag model (or CD vs CL2) is shown in Fig. 7. Various aerodynamic efficiencies computed from CL and CD data for given airfoil are shown in Figs. 8, 9 and 10 respectively. Figure 6: Polar diagram for NACA 4412 airfoil. © (2019) Dr. Nihad E. Daidzic, All rights reserved 9 Prof. Dr. Nihad Daidzic, ATP, CFII, MEI WT measurements Tutorial, v 1.3 Figure 7: Parabolic drag model for NACA 4412 airfoil. Figure 8: Aerodynamic efficiency for NACA 4412 airfoil. © (2019) Dr. Nihad E. Daidzic, All rights reserved 10 Prof. Dr. Nihad Daidzic, ATP, CFII, MEI WT measurements Tutorial, v 1.3 Figure 9: Endurance (power required) efficiency for NACA 4412 airfoil. Figure 10: Range (cruise) efficiency for NACA 4412 airfoil. © (2019) Dr. Nihad E. Daidzic, All rights reserved 11 Prof. Dr. Nihad Daidzic, ATP, CFII, MEI WT measurements Tutorial, v 1.3 Normally, all drag and lift coefficients are generally functions of geometry, AOA, sideslip angle, Reynolds and Mach number: CD  CD  ,  , M , Re  CL  CL  ,  , M , Re  For slow-flying GA airplanes cruising at about 100-200 knots and lower altitudes (below 20,000 ft), we can neglect dependence on Reynolds and Mach numbers. Mach and Reynolds number effects only becomes important for high-speed subsonic/transonic (M < 1) high-flying jet transports and, of course, for supersonic airplanes. Regarding an airplane boundary layer (BL) dynamics, we can observe existence and progression of the boundary layer separation and the development of stall patterns on a 3D wing (wing root stalling first). Tufts show flow reversal over parts of the wing (see Fig. 11). Typically, with the trailing edge stall, the BL separation point is located somewhere around the mid chord of the airfoil as the wing is technically (aerodynamically) stalled and the CL starts decreasing with the AOA while CD starts increasing even faster with the AOA. Figure 11: Flow reversal and stall patterns on a GA airplane wing. THE END of Wind tunnel Tutorial December 2019 (ver. 1.3) © Dr. Nihad E. Daidzic © (2019) Dr. Nihad E. Daidzic, All rights reserved 12 Prof. Dr. Nihad Daidzic, ATP, CFII, MEI WT measurements Tutorial, v 1.3 Minnesota State University, Mankato Department of Aviation Wind tunnel measurements and data analysis tutorial This tutorial gives details on processing of raw aerodynamic data measured directly in the wind tunnel on airfoils and wings. Also procedures and equations from the lab report are given and demonstrated how to compute geometric characteristics of wings/airfoils and air data. All numeric results are typically rounded to five significant digits. Important Definitions and Descriptions: A cambered (asymmetric) airfoil is illustrated in Fig. 1. Definitions of various angles is given as well as the lifting curve CL vs AOA (Angle-of-Attack) or α. At zero geometric AOA, cambered airfoil still produces some lift as the absolute AOA (αa) is defined not through chord but the ZeroLift-Line (Z.L.L.). The design lift-coefficient CL at the zero geometric AOA (α) is defined as CL0. The maximum coefficient-of-lift CL occurs at the stalling AOA (αcr) and is designated as CL,max. Due to production of lift that induces vortices and the net downwash flow (w), the induced AOA (αi) on finite wing acts to effectively reduces the far-field AOA (α) into the near-field AOA (αE). Induced drag does not exist in 2D airfoils (infinite wings). Figure 1: Aerodynamic definitions of cambered airfoils. Finite (3D) wing projected planform surface area (Sref) is a product of wingspan (b) and mean geometric chord ( c ). Geometric twist (εg) is given as a difference in incidence angles at the wing © (2019) Dr. Nihad E. Daidzic, All rights reserved 1 Prof. Dr. Nihad Daidzic, ATP, CFII, MEI WT measurements Tutorial, v 1.3 root and wing tip. The aerodynamic twist (εa) is the angular difference between ZLLs at the root and tip. Taper ratio (λ) is the length ratio between wing chords (c) at the tip and the root (see Fig 2). Dihedral angle (Γ) is the angle between the horizontal plane and a plane that goes through root and tip chords. Sweep angle (Λ) is the angle between LE (at c/4) and the plane perpendicular to the longitudinal axis. The resultant aerodynamic reaction force R acting on a moving immersed body in a fluid is a result of normal (pressure) and shear-stress distributions on the exposed surfaces (see Fig. 3). This reaction force is determined by computing the surface integral of normal and shear (tangential) stresses. Normal force to a chord is designated as N, while the axial force is designated as A. Lift force (L) is by definition perpendicular (orthogonal) to a far-field relative wind (RW). Drag or fluid resistance (D) force is collinear with the RW but opposite the direction of motion. In the case of the finite (3D) wing, the variable span-wise and chord-wise circulation generates tip vortices, which induce downwash, hence resulting in induced (vortex) drag. Figure 2: Basic finite wing geometry. Processing of experimental/measurement data First, one needs to define geometric characteristics of airfoils (wing sections) or finite wings. Infinite wings or 2D sections (airfoils) span between two walls in the test section, meaning that apart from wall effects the chord-wise flow (along tunnel’s longitudinal axis) is equal in all sections. The width of the tunnel test section is 12 inches and that would be the span of the infinite wing. If the chord length of the chosen rectangular wing (finite or infinite) is, for example, 2.5 inches (6.35 cm or 0.0635 m) then the reference (or projected) area is: © (2019) Dr. Nihad E. Daidzic, All rights reserved 2 Prof. Dr. Nihad Daidzic, ATP, CFII, MEI WT measurements Tutorial, v 1.3 Sref  b  c  b  c  12  2.5  30 inch 2 Figure 3: Aerodynamic forces. Normally we would just use ruler to measure wing geometric data directly. If we have simply tapered wing than we must first compute the mean chord (Mean Geometric Chord or MGC). We need to convert surface area in inches squared into meter squared using the fact that one inch is 2.54 cm or 0.0254 meters. One square inch is thus 0.0254 x 0.0254 = 6.452 x 10-4 m2. Hence, 30 inch2 is 1.935 x 10-2 m2. The aspect ratio (AR) is now: AR  b2 b 12    4.8 [] Sref c 2.5 An important note: in infinite wings (airfoils or sections), that span from wall to wall, there is no induced drag and that aspect ratio is actually infinite in terms of aerodynamic effects. The taper ratio of any simple wing is: TR    ct cr In the case of rectangular wing, tip and root chord are identical and TR is one. We used dihedral angle zero. Sweep angle is the angle between the leading edge at the 25% chord (aerodynamic © (2019) Dr. Nihad E. Daidzic, All rights reserved 3 Prof. Dr. Nihad Daidzic, ATP, CFII, MEI WT measurements Tutorial, v 1.3 center) and the vertical plane. For straight rectangular wing with taper-ratio of one, the sweep angle is also zero. We now have all basic geometric characteristic of the wings and airfoils. Next, we need to compute (dry) air density in the lab based on the barometric pressure and lab-air temperature measurements. Let us say that using lab barometer we found out that local atmospheric pressure is 29.20 inch Hg. Temperature measurements using Mercury thermometer gave us, for example, 73oF. We need to convert this into Pascals (Pa) and degrees C (oC) and absolute temperature in Kelvin (K). To convert pressures, we just need to remember that 101,325 Pa is 29.92 inch or 760 mm Hg (column height of mercury). From there we see that 1 inch of Hg is equivalent to 3,386.53 Pa and 1 mm Hg is equal to 133.3 Pa. Accordingly, 29.20 inch Hg is 98,886.7 Pa. Now convert degrees Fahrenheit into degrees Celsius using: t[o F]-32 73-32   22.78o C 1.8 1.8 T[K]  273.15  t[o C]=273.15+22.78o C  295.93K T[o C]  All conversion factors and equations are also given in the AVIA 201 Project 1 Wind Tunnel Lab document. We can now calculate dry-air density (neglecting humidity) from the ideal gas-law: p 98,886.7   1.1643 kg/m3 R  T 287  295.93 R  287  J/kg K  (DryAir) p [Pa] T [K]  We can also convert that into slug/ft3 by remembering that 1 slug is equal to 14.59 kg and 1 meter is 3.2808 feet. Hence, 1.1643 kg/m3 is equal to about 0.002259 slug/ft3. Now we can compute nondimensional pressure, temperature and density based on the SL (Sea Level) properties.  p pSL  T TSL    SL        pSL  101,325 Pa TSL  288.15 K  SL  1.225 kg/m3 aSL  340.294 m/s Substituting our values, we obtain:  p 98,886.7   0.9759 pSL 101,325 295.93  1.026999 288.15  1.1643    0.9504  SL 1.225       0.9759  0.9504  1.027    1 2  1.026999  1.01341 © (2019) Dr. Nihad E. Daidzic, All rights reserved 4 Prof. Dr. Nihad Daidzic, ATP, CFII, MEI WT measurements Tutorial, v 1.3 To estimate Mach and Reynolds numbers we need to find the actual speed of sound in air in the lab as well as dynamic and kinematic viscosities. The local speed of sound is obtained from the actual temperature measurements (73oF or 22.78oC or 295.93 K): a   RT  20.05 T  aSL   340.3    340.3 1.0134  344.86 m/s=1131.43 ft/s  667 knots The true airspeed (TAS) is computed from Pitot-static measurements. If the dynamic pressure q is for example 650 Pa, the actual true airspeed (by neglecting measuring errors) is: q 1 2 1 2q q 650  v    SL v 2  v   1.278   1.278   33.42 m/s=64.9 knot 2 2   SL  0.9504 A student should be able to easily convert between length units (ft, m, NM, etc) and speed units (knots, ft/s, m/s, kph, etc.) and other imperial and SI units. Important is to always use consistent units and not mix “apples” and “oranges”. The Mach number is now: M vTAS 64.9   0.0973 [] a 667 A (dimensionless) Reynolds number is defined as: Re  vTAS    c   vTAS  c  Here, we used average chord (or MGC) as a characteristic length for chordwise flow to compute Reynolds number. In our case of rectangular wing, the chord length is constant and equal to 2.5 inch or chord everywhere. Hence, we need to compute first dynamic or kinematic viscosity. We already computed air density earlier and dynamic viscosity (Sutherland’s equation) is computed from (see also WT lab document): 8.14807 102   295.93 8.14807 102  T 3 2 5   SL  1.78936 10  T  110.4   295.93  110.4  SL  1.78936 105 Pa  s 32  1.82665 10 5 Pa  s 1 cP  10-3 Pa  s Or according to Granger:    SL  0.76  T K    1.78936  10     288.15  5 0.76  1.82595  10 5 Pa  s  1.826  10 2 cP Kinematic viscosity is now: © (2019) Dr. Nihad E. Daidzic, All rights reserved 5 Prof. Dr. Nihad Daidzic, ATP, CFII, MEI WT measurements Tutorial, v 1.3  1.82665 105    1.5689 105 m 2 /s  15.69 cSt  1.1643 1 St  1 cm 2 /s  10-4 m 2 /s 1 cSt  10-6 m 2 /s The Reynolds number is now: Re  vTAS  c   33.42  2.5  0.0254  1.352648 105  135, 265 [] 5 1.5689 10 We see that both Mach number and Reynolds numbers are very low. Mach number of less than 0.1 indicates that the airflow is very slow incompressible subsonic (M < 0.3) and the small Reynold’s number likely affects the maximum coefficient-of-lift (CL) compared to large-scale wings in flight (1-100 million). Now we have all ingredients to compute most important aerodynamic coefficients from experimental data. The coefficient of lift and drag are computed from expressions: L  kgf   9.81 N/kgf  CL  LN   q  Pa   S  m  q  Pa   S  m 2  D  kgf   9.81 N/kgf  DN  CD   2 q  Pa   S  m  q  Pa   S  m 2  2     Various aerodynamic efficiencies (Minimum drag, minimum power or maximum endurance and maximum range cruise) are computed as: EMD C L  L  CD D   EMP CL3 2  CD  EMR CL1 2  CD  We already know that in our example: q  Pa   S  m 2   650 1.935 102  12.578 N This above value is constant and does not change unless different dynamic pressure is used. We will use now an example of measured lift and drag in units of “kgf” at set angle-of-attack (AOA) and set dynamic pressure (q). Let us assume that one particular measurement at AOA of 8o returned measured Lift (L) of 2.500 kgf and Drag (D) of 0.140 kgf. Now we have coefficients: 2.500  9.81  1.95    12.578 0.140  9.81 CD   0.109    12.578 CL  © (2019) Dr. Nihad E. Daidzic, All rights reserved 6 Prof. Dr. Nihad Daidzic, ATP, CFII, MEI WT measurements Tutorial, v 1.3 The aerodynamic efficiency at AOA of 8o is: EMD  CL 1.95   17.86 CD 0.109   The aerodynamic endurance (power required) is: EMP  CL3 2 1.953 2   24.98 CD 0.109  The aerodynamic range performance (range cruise) is: EMR  CL1 2 1.95   12.81 CD 0.109   You will typically find out that for cambered NACA 4412 airfoil, the aerodynamic efficiency peaks around 6 angular degrees AOA, range efficiency around 4 angular degrees AOA, and endurance efficiency around 8o AOA. Stalling (geometric) AOA is at about 16 angular degrees. These computations are then repeated for all measured (geometric) AOAs and associated dynamic pressures (could be different but not recommended as that would change Re and M numbers and possibly affect Boundary Layer structure). When all data are calculated/processed, construct required diagrams as required per WT Lab Project assignment. Also, illustrate Boundary Layer and BL separation. An example of lift-curve and drag-curve diagrams finalized using MS ExcelTM are shown in Figs. 4 and 5 respectively (students need to use their own measured data.). As will be evident from experimental data (see Fig. 4), the lift-curve (CL vs AOA) consists of the linear and nonlinear region. In the linear region, the coefficient-of-lift increases proportionally to increase in AOA. Once the separation (typically at the TE) becomes significant, the lift-curve slope starts decreasing, leveling off, becomes zero at stalling angle, and then becomes negative in poststall region. In linear region (up to about 10o for NACA 4412), the linear region can be described for many practical airfoils with the simple relationship (due to von Mises): CL  CL ,0  k   g dCL 0.1  CL  d 1  2 AR for  in angular degrees More complicated relationships exists for lift-curve. For cambered airfoils and wings CL,0 is typically in the range 0.2 to 0.5, while for symmetric airfoils it is ZERO. For infinite wings (airfoils) the AR is infinite and the slope CL,α is about 0.1/degree for AOA expressed in angular degrees or about 5.7/rad when AOA is expressed in radians. Increasing AR, sweep angle and Mach number all work to decrease the lift-curve slope. The theoretical lift-curve slope from the “thinairfoil” theory is 2π or about 6.28/rad for AOA in radians. All these considerations are valid in the absence of dynamic or vortex-lift and transient effects. Coefficient of drag measured as a function of AOA for given airfoil is given in Fig. 5. © (2019) Dr. Nihad E. Daidzic, All rights reserved 7 Prof. Dr. Nihad Daidzic, ATP, CFII, MEI WT measurements Tutorial, v 1.3 Figure 4: Lift-curve (CL vs AOA) for NACA 4412 airfoil. Figure 5: Drag curve (CD vs AOA) for NACA 4412 airfoil. © (2019) Dr. Nihad E. Daidzic, All rights reserved 8 Prof. Dr. Nihad Daidzic, ATP, CFII, MEI WT measurements Tutorial, v 1.3 Total drag of airfoils or infinite-wing sections (no induced drag) and finite wings (parasitic and induced drag) can be, in general, modeled with a parabolic function in linear lift-curve region: CD  CD ,0  K  CL2 K  k1  k2  k3  k1  k2  1   e  AR Here, k1 is a coefficient of parasitic drag only (form drag and skin-friction drag), k2 is a coefficient of wave drag (only at speeds exceeding critical Mach number and does not exist for slow GA piston-prop airplanes), and k3 is a coefficient of induced- or vortex-drag and only exists for finite wings (not airfoils or infinite wings). The zero-lift coefficient of drag is simply a sum of zero-lift drag coefficient due to parasitic and wave drag (at high speeds only). Accordingly, drag can be modeled as a part that is due to production of lift and the part that is independent of lift production. Close to stalling angle and in nonlinear region many of the coefficients are no longer simple constants and change strongly with AOA. Polar diagram (CL vs CD) for representative WT measurement on example of NACA 4412 is shown in Fig. 6. Parabolic drag model (or CD vs CL2) is shown in Fig. 7. Various aerodynamic efficiencies computed from CL and CD data for given airfoil are shown in Figs. 8, 9 and 10 respectively. Figure 6: Polar diagram for NACA 4412 airfoil. © (2019) Dr. Nihad E. Daidzic, All rights reserved 9 Prof. Dr. Nihad Daidzic, ATP, CFII, MEI WT measurements Tutorial, v 1.3 Figure 7: Parabolic drag model for NACA 4412 airfoil. Figure 8: Aerodynamic efficiency for NACA 4412 airfoil. © (2019) Dr. Nihad E. Daidzic, All rights reserved 10 Prof. Dr. Nihad Daidzic, ATP, CFII, MEI WT measurements Tutorial, v 1.3 Figure 9: Endurance (power required) efficiency for NACA 4412 airfoil. Figure 10: Range (cruise) efficiency for NACA 4412 airfoil. © (2019) Dr. Nihad E. Daidzic, All rights reserved 11 Prof. Dr. Nihad Daidzic, ATP, CFII, MEI WT measurements Tutorial, v 1.3 Normally, all drag and lift coefficients are generally functions of geometry, AOA, sideslip angle, Reynolds and Mach number: CD  CD  ,  , M , Re  CL  CL  ,  , M , Re  For slow-flying GA airplanes cruising at about 100-200 knots and lower altitudes (below 20,000 ft), we can neglect dependence on Reynolds and Mach numbers. Mach and Reynolds number effects only becomes important for high-speed subsonic/transonic (M < 1) high-flying jet transports and, of course, for supersonic airplanes. Regarding an airplane boundary layer (BL) dynamics, we can observe existence and progression of the boundary layer separation and the development of stall patterns on a 3D wing (wing root stalling first). Tufts show flow reversal over parts of the wing (see Fig. 11). Typically, with the trailing edge stall, the BL separation point is located somewhere around the mid chord of the airfoil as the wing is technically (aerodynamically) stalled and the CL starts decreasing with the AOA while CD starts increasing even faster with the AOA. Figure 11: Flow reversal and stall patterns on a GA airplane wing. THE END of Wind tunnel Tutorial December 2019 (ver. 1.3) © Dr. Nihad E. Daidzic © (2019) Dr. Nihad E. Daidzic, All rights reserved 12 Purchase answer to see full attachment




Why Choose Us

  • 100% non-plagiarized Papers
  • 24/7 /365 Service Available
  • Affordable Prices
  • Any Paper, Urgency, and Subject
  • Will complete your papers in 6 hours
  • On-time Delivery
  • Money-back and Privacy guarantees
  • Unlimited Amendments upon request
  • Satisfaction guarantee

How it Works

  • Click on the “Place Order” tab at the top menu or “Order Now” icon at the bottom and a new page will appear with an order form to be filled.
  • Fill in your paper’s requirements in the "PAPER DETAILS" section.
  • Fill in your paper’s academic level, deadline, and the required number of pages from the drop-down menus.
  • Click “CREATE ACCOUNT & SIGN IN” to enter your registration details and get an account with us for record-keeping and then, click on “PROCEED TO CHECKOUT” at the bottom of the page.
  • From there, the payment sections will show, follow the guided payment process and your order will be available for our writing team to work on it.