WL-TR -91 -3073 AD-A240 263
292
WL-TR -91 -3073 AD-A240 263 SUBSONIC WIND TUNNEL TESTING HANDBOOK Captain Michael G. Alexander WL/FIMM Wright-Patterson AFB, Oh 45433-6553 May 1991 P'> 14- 13 19i Interim Report for Period 1 May 1990 - 1 May 1991 Approved for public release; distribution unlimited 91-10437 Flight Dynamic Directorate Wright Laboratory Air Force System Command Wright- Patterson AFB, Ohio 45433-6553
Transcript of WL-TR -91 -3073 AD-A240 263
WL-TR -91 -3073 AD-A240 263
SUBSONIC WIND TUNNEL TESTINGHANDBOOK
Captain Michael G. Alexander
WL/FIMM
Wright-Patterson AFB, Oh 45433-6553
May 1991 P'> 14- 13 19iInterim Report for Period 1 May 1990 - 1 May 1991
Approved for public release; distribution unlimited
91-10437
Flight Dynamic Directorate
Wright Laboratory
Air Force System Command
Wright- Patterson AFB, Ohio 45433-6553
.4
NOTICE
WHEN GOVERNMENT DRAWINGS, SPECIFICATIONS, OR OTHER DATA ARE USED FOR ANYPURPOSE OTHER THAN IN CONNECTION WITH A DEFINITELY GOVERNMENT-RELATEDPROCUREMENT, THE UNITED STATES GOVERNMENT INCURS NO RESPONSIBILITY OR ANYOBLJGATION WHATSOEVER. THE FACT THAT THE GOVERNMENT MAY HAVE FORMULATED OR INANY WAY SUPPUED THE SAID DRAWINGS, SPECIFICAiONS, OR OTHER DATA, IS NOT TOBE REGARDED BY IMPUCATION, OR OTHERWISE IN ANY MANNER CONSTRUED, AS UCENSINGTHE HOLDER, OR ANY OTHER PERSON OR CORPCRATION; OR AS CONVEYING ANY RIGHTS ORPERMISSION TO MANUFACTURE, USE, OR SELL ANY PATENTED INVENTION THAT MAY IN ANYWAY BE RELATED THERETO.
THIS REPORT HAS BEEN REVIEWED BY THE OFFICE OF PUBLIC AFFAIRS (ASDIPA)AND IS RELEASABLE TO THE NATIONAL TECHNICAL INFORMATION SERVICE (NTIS). ATNTIS IT WILL BE AVAILABLE TO THE GENERAL PUBUC INCLUDING FOREIGN NATIONS.
THIS TECHNICAL REPORT HAS BEEN REVIEWED AND IS APPROVED FOR PUBUCATION.
MICHAEL G. ALEXANDER, Captain USAF DENNIS SEDLOCK, Chief
Aerospace Engineer Aerodynamics and Airframe Branch
Airframe Aerodynamics Group Aeromechanics Division
FOR THE COMMANDER
DAVID R. SELEGANActing ChiefAeromechanics Division
IF YOUR ADDRESS HAS CHANGED, IF YOU WISH TO BE REMOVED FROM OUR MAILINGUST, OR IF THE ADDRESSEE IS NO LONGER EMPLOYED BY YOUR ORGANIZATION PLEASENOTIFY W/FIM, . WRIGHT-PATTERSON AFB, OH 45433- 6553 TO HELP MAINTAINA CURRENT MAIUNG UST.
COPIES OF THIS REPORT SHOULD NOT BE RETURNED UNLESS RETURN IS REQUIRED BYSECURITY CONSIDERATIONS, CONTRACTUAL OBLIGATIONS, OR NOTICE ON A SPECIFICDOCUMENT.
form Approved
REPORT DOCUMENTATION PAGE OMB Nppov 0
ccllet C r onqt- -C )Suqgest, ns -a". 'q 0Z "l 0' 5fln -V'0 0e reori So ,'ta.fjl'trstttetletstfl
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1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED
May 1991 INTERIM REPCRT 1 MAY 90 - MAY 914. TITLE AND SUBTITLE 5. FUNDING NUMBERS
Subsonic Wind Tunnel Testing Handbook6. AUTHOR(S)
WU 2404 IOA2
Alexander, Michael G., Captain USAF
7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATIONREPORT NUMBER
WL/FIMMWright-Patterson AFB OH 45433-6553 WL-TR-91-3073
"9. SPONSORING!'MONITORING AGENCY NAME(S) AND ADDRESS(ES1 10. SPONSORING/MONITORINGAGENCY REPORT NUMBER
WL/FIMMWright-Patterson AFB Oh 45433-6553
11. SUPPLEMENTARY NOTES
12a. DISTRIBUTION 'AVAILABILITY STATEMENT 12b. DISTRIBUTION CODE
Approval for Public Release; Distribution Unlimited
13. ABSTRACT (Maximum 200 words)
This handbook is predominantly structured for subsonic (non-compressible flow),
force and moment wind tunnel testing. Its purpose is to provide to the aerodynamic
testing engineer equations, concepts, illustrations, and definitions that wouldaid him or her in wind tunnel testing. The information in this handbook is an
amalgamation of numerous sources and observations. By design, this handbook is a
living document readily expandable to include personal notes and additionalsections. This handbook has not and cannot encompass every wind tunnel testingtechnique or principle.
14,. S1BJECT TERMS 15. NUMBER OF PAGES
306Wind Tunnel, Handbook, Subsonic, Wind Tunnel Testing 16. PRICE CODE
17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION 20. LIMITATION OF ABSTRACTOF REPORT OF THIS OACr OF ABSTRACT
Unclassified Unclassified Unclassified Unlimited
NSN 1540-0' 29O 5500 Stardard t-n, .'98 ý,.v 2 89)
Foreword
This technical report was prepared by Captain Michael G. Alexander from Wright
Laboratory, Flight Dynamics Directorate, Aeromechanics Division, Wright-Patterson AFB,
Ohio, 45433-6553. This work was accomplished under work project 240410A2, Advanced
Tactical Transport. This technical report provides the wind tunnel testing engineer a
handbook with useful equations, definitions, and concepts to aid in wind tunnel testing.
The author wishes to thank the exceptional insights, observations, expertise and help
ii01i, Mr. Tom Tighe, Mr. Jim Grove, Mr. Bob Guyton and Mr. Larry Rogers all from
WL/FIMMC (Wright Laboratory, Airframe Aerodynamics Group).
This report has been reviewed and is approved.
SAcecslon For
iii I
Table of Contents
Section Description Page
I INTRODUCTION ......... ........................... 1-1
11 AERODYNAMIC DEFINITIONS
a) Symbols .......... ........................... 11-1
b) Aerodynamic Center (a.c.) ....... .................. 11-2
c) Center of Pressure (cp) ........ ................... 11-2
d) Mean Aerodynamic Chnrd (M.A.C.) .................... 11-2
e) Neutral Point (N0 or x np) .................... 11-2
f) Static Margin (SM) ....... ...................... 11-2
g) Longitudinal Static Stability ......... ................ 11-2
h) Directional Static Stability ......... ................. 11-3i) Lateral Static Stability .......... ................... 11-3
j) Dynamic Stability ......... ...................... 11-3
k) Trimmed Flight ......... ........................ 11-3
1) Critical Mach Number ....... ..................... 11-3
III FORCE AND MOMENT EQUATIONS
a) General Aerodynamic Symbols and Equations .............. 111-1
b) Aerodynamic Equations ....... .................... 111-51) Trim, Pitching Moments Equations ................ 111-5
Conventional Horizontal Tailed Aircraft ............ 111-5
Tailless ........... ....................... 111-5
Canard .......... ........................ 111-5
2) Stick Fixed Neutral Point ...... ................ 111-9
3) Two Dimensional Lift ...... .................. 111-9
Subsonic ......... ....................... 111-9
Supersonic ......... ...................... 111-9
4) Pressure Coefficient .......... .................. 111-9
Incompressible ...... .................... .. 111-9
Compressible ......... ..................... 111-9
5) Center of Pressure Location ...... ............... 111-10
v
Table of Contents (continued)
Section Description Page
6) Aerodynamic Center (a.c.) Determination ........... . 111-10
7) Mass Flow ........ ........................ I1-10
IV AXIS SYSTEM DEFINITIONS
a) Axis System Definitions ....... ................... IV- I
1) Tunnel Axis (Inerdal Axis) ...... ............... IV-I
2) Body Axis ......... ........................ IV-l
3) Wind Axis ........ ........................ IV- I
4) Stabilit, Axis ........... ................... IV- I
b) Angle Definitions ........... ...................... IV-5
1) Aerodynamic Angles ........................ .. IV-5
2) Orientation Angles ................... IV-5
3) Angle Transformation from Tunnel Axis to Body Axis . IV-6
4) Wing Reference Plane ...... .................. IV-6
5) Wind Reference Plane ....................... IV-6
6) Plane of Symmetry ....... .................... IV-6
c) Coordinate Transformation Equations ..... ............. .IW-8
1) Balance Axis to Body Axis ................... IV-9
2) Body Axis to Wind Axis ...................... I\V-13
3) Body Axis to Stability Axis ..... ............... IV-14
4) Wind Axis to Stability Axis ..... ............... I\V-15
V TRIP STRIPS
a) Boundary Layer Symbols ....... ....................-
b) Boundary Layer Discussion ....... .................. V-2c) Boundary Layer Thickness ....... ................... \--2
1) Laminar ......... ......................... V-2
2) Turbulent ......... ........................ V-2
d) Trip Strips . . . . . . . . . . . . . .. ... V-3
e) Trip Strip Types .......... ....................... V-3
1) Grit. .......... .......................... V-3
vI
Table of Contents (continued)
Section Description Plage
2) Two-Dimensional Tape ....... .................. V-3
3) Epoxy Dots ........ ....................... V-4
4) Thread or String ......... .................... V-4
f) Location of Trip Strips ........ ................... V-4
1) Lifting Surfaces .......... .................... V-4
2) Fuselage ........... ........................ V-4
3) Nacelles ........... ........................ V-4
g) Determination of Trip Strip Height ................... V-5
1) Method 1 (Atmospheric tunnels) ..... ............. V-5
2) Method 2 (Atm. and Pressure tunnels) .............. V-7h) Application of Grit types of Trip Strips ..... ........... V-8
VI PLANFORM CHARACTERISTICS
a) Planform Symbols ....... ....................... VI-1
b) Wing Parameters Definitions ...... ................. VI-3
c) Planform Parameters ....... ..................... VI-5
1) General Planforms ........ ................... VI-5
2) Conventional, Straight-Tapered ...... ............. VI-6
3) Double Delta and Cranked Wing ................... VI-8
d) Planform Example ....... ....................... VI-10
1) Inboard Section ........ ..................... VI-10
2) Outboard Section ....... .................... VI-I 1
3) Total Wing ....... ....................... VI-Il
4) Total Aircraft Aerodynamic Center Location .......... VI-12
Inboard Section ........ .................... VI- 12
Outboard Section ........ .................... VI-i2
Total Aircraft ......... ..................... VI- 12VII DRAG
a) Symbols ............ ........................... VII- I
b) Drag .......... ............................. VI11-2
vii
Table of Contents (continued)
Section Description Page
c) Subsonic Drag ........ ........................ VII-2
1) Minimum Drag ....... ...................... VII-2
2) Profile Drag/Skin Friction Drag ..... ............. VII-2Pressure Drag (Form Drag) .................... V11-3
3) Interference Drag ......... .................... VII-3
4) Miscellaneous Drag ...... ................... VII-3
5) Drag Due to Lift ....... .................... VI1-3
6) Zero Lift Drag ....... ..................... VIl-3
7) Base Drag ........ ........................ VI-4
8) Internal Duct Drag ........ ................... VII-4
9) Wave Drag ........ ........................ VII-4
d) Drag Polar (Subsonic) ....... .................... VII-4
1) (CL vs CD) Polar ....... .................... VII-4
Parabolic .......... ....................... VII-5
Non-Parabolic ....... ..................... VII-5
2) (CD vs CL 2) Polar ....... .................... VII-5
3) Polar Break .......... ...................... VII-5
4) Camber Effects ....... ..................... V\l-5
5) Drag and Performance Equations ................. VII-6
Non-Parabolic Polar ...... .................. .V I-6
Parabolic Polar .......... .................... 1-6
6) Analytically Determined Drag Polar ..... ........... VI1-7Method of Determining a Drag Pola,.. ............. VII-7
Example ........ ........................ V -S
VIII EXPERIMENTAL TESTING AND INTERPRETATION
a) Flaps ............ ............................ V.I-I1) Trailing Edge Flaps ...... ................... VIII-1
2) Leading Edge Flaps ...... ................... VI.l-Ib) Lift Curve (Flaps Up) ......... .................... V111-2
Viii
Table of Contents (continued)
Section Description Page
c) Lift Curve (Flaps Down) ....... ................... VIII-2
d) Drag Curve (Flap Up and Down) ...................... VIII-3
e) Pitching Moment ........ ....................... VIII-3
f) Elevator Stabilizer Power Curve ....... .............. V'IT -3
g) Aileron Power Curve ........ ..................... v 111-4
h) Rudder Power Curves ....... ..................... VIII-5
i) Determine Center of Pressure Shift (cp) ..... ........... VIII-5
j) C.G. Shift .......... .......................... VIII-5
k) Lift Curve Slope ........ ....................... VIII-5
1) Determining Aircraft Parameters from Wind Tunnel Data. . VIII-6
1) Static Margin ........ ...................... VIII-6
2) Aerodynamic Center Location .................... VIII-7
3) Aerodynamic Center Pitching Moment ............... VII-7
4) Center of Gravity Pitching Moment ................. VIII-75) Wing-Body-Tail ....... ..................... VIII-8
6) Longitudinal Static Stability ....... .............. VIII-9
7) Longitudinal Balance ....... .................. VIII-9
8) Trim Angle-of-Attack ...... .................. VIII-9
re)Finding Trimmed Flight Parameters ...... .............. VIII-10
n) Determine any C.G. Location ...... ................. VIII-1I
o) Determining the Average Downwash Angle .............. VIII- II
p) Determining Induced Drag Factor (K) and Oswald's Wing
Efficiency Factor (e) ........... ................... VIII-12
q) Base Pressure ......... ........................ VIII-12
r) Pressure Transducer Selection ........ ................ VIII-13
s) Flow Visualization ........ ...................... VIII-14
1) Surface Flow Visualization ..... ............... VIII-14
Yam .......... .......................... VIII- 14
Fluorescent Mini-Tufts ..... ................. .VIII- 15
Mini-Tuft Installation ....... .............. Vill-15
Surface Plepdic iion Steps. ..... ............ VIII-15
ix
Table of Contents (continued)
Section Description Page
Mini-Tuft Attachment Steps .II .l( ....
Oil Flow. ...... ..... ..... ..... ..... ..... ..........- I7
3) Off-Body Flow Visualization........ ..... . . .. . ... . . . . Ill-Laser Light Sheet (Vapor Screen)....... ... . . .. . .... . . IIl-I
Smoke Seeded Flow................. .. .. .. .. ....... l. V! 1 1
Tufts. ...... .......... ........ .......... ....... VI I I -
4) Deter-mining Aerodynamic Angles from the Model SupptI)(r1(sting) Angles................... . . ..... . . .. . .... . . Ill-1
IX STRESS ANALYSIS
a) Symbols. .. ........ .......... .......... ..... ..... .... I-I
b) Definitions... ..... ........ ......................... . I X-
c) Stress Formulas. .... .................. ..... ..... ...... I-5
d) Ang~e-of-Twist .... .......... .......... ........ . ..... IX-5
e) Polar Moment of Inertia .. .... .......... ..... ..... ...... IN-
f) Radius of Gyration . . . . . . . . . . . . . . . . . . . . ! -
g) Bending, Stress .... .......... .......... ..... ..... ...... I-6
h) Shear Strcess ...... .......... .......... ..... ..... ...... I-6
i) Torsional Formulas .. .. ........ ..... ..... ........... IX-()
jCcmbincd Stress . .. ........ . .. .. .. .. . . . .. I-7
k) Principle Stress..... .......... .......... ..... ..... .... I-S
I) Factor of Safety. .. ........ .......... ..... ..... ...... I-8
m)Calculate Centroid of' a Planform Area .. .... ..... ..... .... IN-
n) Example: Stress Analysis......... ..... . . ... . ..... .. . . . IN-)
X T'RENDS
a) Flap Characteri st ics .......... ... ...... .......... ........-
b) Effect of Vertical Location on C.G. PitchiM2 Moihicii~...... .. . ..-
c0 Typical Longitudinal Stability Breakdown .. .. ........ .......- 2
d) M-,ich Number Trends (Effect). .. .. ........ .......... .....- 3
e) Reynolds Number and Aspect Ratio Effect. .. .. .... .........- 4
x
Table of Contents (continued)
.,ectli ol Description Page
f) Effect of Wing Sweep on CD ................. X-50
g) Aft and Forward Wing Sweep Comparison ................ X-
h) Wing Pressure Distribution in the Presence of a C'ouplCd
Chine Forebody ........ .......................... X-9.)
i) Reynolds Number Effect on C1m .-!.x'
j) Reynolds Number Effect on Drag ...... ................ X-1 I
k) Drag Rise Characteristics (Wing alone) ...... ............ X-12
1) Mach ELfect on a Airfoil Pressure Distribution ..... ........ X-13
XI INTERNAL STRAIN GAGF RALANCES
a) Symbols ...... . ............... Xl-I
b) Strain Gages ......... ........................... XI-2
1) Temperature Effects ...... ................... .. XI-I
2) Deformation Theory and Calculation .............. ... XI-3
3) Measurement of AR/R ....... ................... XI-4
c) Balance Calibration (interaction) Matrix .................. XI-5
1) Example ............ ......................... XI-6
d) Calibration Body . .................. XI-7
e) Check Loading ............. ........................ XI-S
1) Checking Lo ading, Procedure ..................... XI-8
2) Dead-Weight Loading System ....... ............... XI-9
3) Axial Force Check- Loading ........ ................ XI-')
f) Sensitivity Constants ............. .................... XI- I)
g) Obtaining Force and Moment Datai from Raw Balance I)al a.... X. I - I I
h) Balance Calibralion l'qualion L'vxample ................... XI-12
i) Balance Placcenctit in the NModel ....... ................ Xl- 14
. ) Balamce ('alibratitm LItji) X;aiple. ................... X. I - I I
k) 13al'ti c llwc nit , iin l ic the Mlodcl .-.. . . . . . . . . . . . X I-14
RIi'REN.('J." .................................... I
'l
Table of Contents (concluded)
Section Description Page
APPENDIX A .......... ................................ . A-I
Dynamic Pressure Determination ..................... .. A-1
Reynolds Number Determination ..................... A-2
Standard Atmosphere ........... ..................... A-3
Compressibic Flow Pararmeters ...................... ... A-4
Conversion Factors ........... ..................... .. A-8
APPENDIX B ................ ................................. B-I
Tid Bits .......... .......................... B-I
Wind Tunnel First Aid Kit .......................... 13-2
Geometric Equations .......... ...................... 8-3
APPENDIX C ............. ................................ .. C-I
Powered Testing ......... ........................ ... C- 1
Symbols ................ ............................ C I
Power On Aerodynamic Equations ..................... ... C-3
Incremental Coefficients .......... .................. ... C-3
Aerodynamic Coefficients ..... ..................... C-3
Induced Force Coefficients ...... .................. ... C-4
xii
List of Figures
Figure Description Page
111-1 Force and Moment Vectors; Aft Tailed Aircraft .............. 111-6
111-2 Force and Moment Vectors; Tailless Aircraft ..... ........... 111-6
111-3 Force and Moment Vectors; Canard Aircraft ................. 111-7
111-4 Positive Angle, Moment, and Body Axis Definition ............ 111-8
IV-1 Body Axis System ......... ......................... IV-2
IV-2 Wind Axis System ......... ......................... IV-3
IV-3 Stability Axis System ........... ...................... IV-4
IV-4 Angle Definitions ........... ........................ IV-7
IV-5 Rotation Order ............. ......................... IV- I1
IV-6 Balance Axis . ............. .......................... IV- 12V-1 Boundary Layer Thickness ......... .................... V- 10
V-2 Streamwise Grit Location ......... .................... V- I I
V-3 Grit Height ........... ........................... V-12
V-4 Carborundum Grit Number ....... ..................... V- 13
V-5 Minimum Grit Size (method 2) ....... .................. V-14
VI-_1 Airfoil Nomenclature ........... ...................... VI-4
VI-2 General Planform Parameters ....... ................... VI-5
VI-3 Conventional, Straight-Tapered Planform ...... .............. VI-6
VI-4 Double Delta and Cranked Wing Planform ................... VI-8
VI-5 Planform Example ......... ......................... VI- 10
VII-I Drag Tree ........... ............................. Vii-9
VII-2 (CL vs CD) Drag Polar; subsonic ....................... VII- 10
Vll-3 (CL vs C D) Drag Polar, supersonic ...................... VII-10
VII-4 (CL vs CD) Subsonic Drag Polar ........................ VIl-Il
VII-5 (CD vs Co-) Drag Polar ........ ...................... VII- II
VIII- I Downwash Determination ........ ...................... VIII-20
VIII-2 Trim Determination Plot .......... ...................... N111-20
IX- I Stress Analysis Example 1 (centroid determination) ............. IX-9
IX-2 Stress Analysis Example 2 (wing tip missile) ..... ........... IX-15
xii1
List of Figures concluded
Figure Description Page
X- 1 Flap Characteristics ............. ........................ X- 1
X-2 Effect of Vertical Location of C.G. on Pitching Moments ......... X-2
X-3 Typical Component Longitudinal Stability Breakdown ............ X-2
X-4 Mach Number Trends (effects) ........ .................... X-3
X-5 Reynolds Number and Aspect Ratio Trends (effects) .............. X-4
X-6 Effect of Wing Sweep on CDD ..................... X-50
X-7 Aft and Forward Swept Wing-Fuselage Effects ................. X-6
X-8 Wing Pressure Distribution in the presence of a Coupled Chine. ... X-9
X-9 Reynolds Number Effect on C1 (Wing alone) ........... . X- 10max
X-10 Reynolds Number Effect on Drag (Wing alone) ................. X- 11
X- 11 Drag Rise Characteristics (Wing alone) ...... ............... X- 12
X-12 Mach Effect on Airfoil Pressure Distribution ..... ............ X- 13
X- 13 Typical Weapon Separation Data ........................ .. X- 14
XI- 1 Basic Wheatstone Bridge ........... ...................... XI- 15
XI-2 Unbalanced Wheatstone Bridge ....... .................... XI- 15
XI-3 Example Balance Calibration Sheet ...... .................. XI- 16
XI-4 Raw Balance Data ........ .......................... XI- 17
A-1 Dynamic Pressure Determination ....... ................... A- 1
A-2 Reynolds Number Determination ......................... .. A-2
A-3 Standard Atmosphere ......... ......................... A-3
A-4 Compressible Flow Parameters ........ .................... A-4
A-5 Conversion Factors ........... ......................... A-8
B-i Geometric Equations ......... ......................... B-3
C- 1 Powered Effects .......... ........................... C-5
xiv
List of Tables
Table Description Page
III-1 Force and Moment Definitions ....... .................. 111-4
V-1 Nominal Grit Size ........ ....................... .. V-8
IX-1 Stress Constants ........... ........................ IX-7
xv
.ubsonic
Wind
Tunnel
Testing
:Tandbook
By Captain Michael G. Alexander, USAF
xvii
I INTRODUCTION
I Introduction
The aerodynamic engineer who participates in wind tunnel testing often has a need for
reference material to aid him/her in his/her testing. It is nearly impossible for the
engineer to recall every equation, concept, definition, and to carry reference material tothe wind tunnel site. This handbook, Subsonic Wind Tunnel Testing Handbook, attempts to
provide to the testing engineer in one reference some of those equations, concepts, and
definitions that he or she might find helpful during testing. This handbook is a quick
reference for the testing engineer and is an amalgamation of information from numerous
reference materials and personal observations. It is designed to be a living document
readily expandable and to fit in a briefcase to accompany the testing engineer to the test
site. Also by design, this aid has not encompassed every wind tunnel technique or
principle, but offers enough information to facilitate and help ease wind tunnel testing.
This handbook is predominantly structured for subsonic (non-compressible flow), force and
moment, wind tunnel testing. For compressible flow testing, an excellent reference aid
is the NACA 1135 (ref. 1). However, it is no longer in print by the government, but,
reprints of the NACA 1135 can be procured from reference 2.
I- 1
II AERODYNAMIC DEFINITIONS
II
II Aerodynamic Definitions
Symbols
a.c. Aerodynamic Center
cg Center of Gravity
CL Lift Coefficient
Cn Yawing Moment Coefficient
CM Pitching Moment Coefficient
CM Pitching Moment Coefficient at zero angle-of-attack0
cp Center of Pressure
MAC Mean Aerodynamic Chord
N Neutral Point0
SM Static Margin
Xnp Distance from the MAC leading edge to the aerodynamic center
Xcg Distance from the MAC leading edge to the center of gravity
aC m Pitching Moment Coefficient slope
a ccaCmcg Slope of the cg pitching moment and lift curve
LAC
n Slope of yawing moment coefficient and the yaw anglea p
a Cn Slope of yawing moment coefficient and the sideslip angle
aC1
aC1 Slope of rolling moment coefficient and the sideslip angle
ap3Y Angle-of-Attack
13 Sideslip Angle
(P Yaw Angle
TI-I-
Aerodynamic Center (a.c.) - The point where the pitching moment does not vary with tile
angle of attack.
Center of Pressure (cp)- A point where the pitching moment vanishes. The cp locationvaries with angle of attack and Mach number.
N'Mean Aerodynamic Chord (MAC) - The chord of an imaginary, untwisted, unswept, equal spannon-tapered wing which would have force vectors throughoutthe flight range identical with thobe u," the actual %, ing or
wings.
Neutral Point (xnp or N0 ) - As the cg is moved aft, the slope of aCmcg / aCL becomes
less negative. When there is no change of the pitchingmoment with lift coefficient (aCmcg / aCL = 0), that Xcg
position is the neutral point (No = x cg).
Static Margin - The distance from the center of gravity to the neutral point expressed asa fraction of the mean aerodynamic chord. The 'X' position
of Xnp is measured positive aft from the c.g..
X -X aSM = np cg _ mcg
aCC
aC+SM = mcg < 0 (Stable; N is behind c.g.)
L
aC-SM - C mcg > 0 (Unstable; N 0is ahead of c.g.)
L
Longitudinal Static Stability - Is determined by the sign and the magnitude of the slope
aCof C versus a curve. m must be < 0 and C must bem aam
a~t 0
positive for trimmed, positive longitudinal staticstability. Also, Cm versus CLcan be used to determine
11-2
positive longitudinal stability.
Directional Static Stabiliiy - Is determined by the sign and magnitude of the slope of C
aCversus P3 curve, where 03 is the sideslip angle. n must
a13be positive (> 0) for directional stability. This is also
known as weathercock stability. Also, Cn versus (p can be
dCused where p is the yaw angle. n must be negative (< 0)
for directional stability.
Lateral Static Stability - It is determined by the sign and magnitude of the slope of the8C 1
C1 versus 03 curve, must be negative (< 0) for positivea 0lateral stability. This is also known as the dihedral
effect. An increase in lateral stability causes dutch roll;
too little stability causes spiral instability.
Dynamic Stability - Is the time history of the movements of a body in response to its
static stability tendencies following an initial disturbance
from equilibrium.
Trimmed Flight - For balanced flight, the aircraft flies at a given elevator angle and
angle of attack that produces Cm = 0.
Critical Mach Number - Is the Mach number where there is a sharp increase in drag (drag
divergence, approximately Mach = 1.0).
11-3
This page was intentionally left blank
11-4
NOTES
11-5
NOTES
11-6
NOTES
11-7
NOTES
11-8
III FORCE AND MOMENT EQUATIONS
III
III Force and Moment Equations
General Aerodynamic Symbols and Equations
A Axial Force (lbf) -- (A = Dcosot - Lsinct)
a Speed of Sound (f/s) -- (yRo = 494' = 4(yP/p) ; 7= OR
Speed of Sound (mph) -- 33.42 f--; 9-= °R
a Lift Curve Slope -- CL
CA Axial Force Coefficient (CA = -CLsina + CDcoscc)
CD Drag Coefficient (CD = CNsinoc + CAcosa)
CD. Induced Drag1
CD Zero Lift Drag; Parasite Drag (profile, friction, pressure)0
CL Lift Coefficient (CL = CNCOSot - CAsinax)
/
CL Lift at minimum drag (generally zero)
C1 Rolling Moment
CL Lift Curve Slope -- L8•ot
C Pitching Momentm 8C
C Pitching moment curve slope -- - m
CN Normal Force Coefficient (CN = CLCOSci + CDsinot)
C Yawing Momentn 6C
C Yawing moment curve slope -- n
CT Thrust Coefficient Thrus tq S
x Ccp Center of Pressure cp m
CNc CN
III-1
General Aerodynamic Symbols and Equations (continued)
Cy Y Side Force Coefficient
D Drag Force (lbf) -- (D = Nsinca + Acosot)
Drag Polar-- CD = CD + [CL CL'2o iteAR
e Oswald's wing efficiency factor
g Gravity Constant1
K Induced drag factorireAR
L Lift Force (lbf) -- (L = Ncosa - Asinc)
N Normal Force -- (Lcoscx + Dsinot) (lbf)
8C x NN Neutral point -- m _ cg 0
0 8 CL c
L + T sinatT Ln Load Factor -- - - (for ccT small)
W WM Mach number
PM Pitching Moment (dimensional)
P Pressure
R Gas Constant
T Thrust
ý7 Temperature; (°Rankine)
q Dynamic Pressure -- 0.5pV 2 or 0.7(Ps)M2
RM Rolling Moment (dimensional)
S Surface Area
S Side Force (lbf)
V Tail/Canard Volume Coefficient .....
111-2
General Aerodynamic Symbols and Equations (concluded)
8C-X Neutral point -- m c o
np CLc
Y Yawing Force (side force)
YM Yawing Moment (dimensional)
oxT Thrust vector angle
aX Geometric angle of attack (measured from V. to the wing chord)
oXa Absolute angle of attack (oa - (aZL)
a ZL ; "o Zero lift angle of attack (measured from the wing chord to the wing zero lift
line)
13 Angle of sideslip
Angle of roll
11 Tail efficiency factor (qt/q.)
(P Angle of yaw
C Downwash angle
0 local angle of inclination of surface to V
Subscripts
np -- neutral point w -- wing t -- tail
s -- static pressure T -- Thrust c -- canard
cg -- center of gravity ac -- aerodynamic center
inc -- incompressible o -- freestream conditions
111-3
Table 1
Force and Moment Definitions
Forces
Body Axis Wind Axis Stability Axis
Force Coefficient Force Coefficient Force ICoeffic ientN CN = N/qSw L CL = L/qSw L CL = L/qSw
A CA = A/qSw D CD = D/qSw D Co = D/qSw
S Cs = S/qS Y C = Y/qS Y C = Y/qS
Moments
Body Axis Wind Axis S tabili ty Ax is
Moment Coefficient Moment Coefficient Moment Coefficient
PM Cm = PM/qSwc PM C = PM/qSwc PM CI = PM/qSwc
RM C1 = RM/qSwb RM CI = RM/qSwb RM CI = RM/c1S wb
YM Cn = YM/qS wb YM Cn = YM/qS wb YM Cn = YM/qSw b
Parameter legend
q = Dynamic pressure b Span Sw = Wing area c = MAC
111-4
Aerodynamic Equations
Trim Pitching Moment Equations (ref. 3)
Conventional Horizontal Tailed Aircraft (figure 111-1)
C C w] T ZTVCM =[M1 + C L[ + CD [lz + [ TjJ - CLt)'t L
cg [ac]jw [c C(Vw - Ccgl inlet
For an all flying tail....CLt = at (a + Acx)
a .O8__ )a + ,Aal= at[(l - cx+ a
,ao = a created by control column input
For fixed stabilizer and a movable elevator...
CL = a _O 8_ F )c' - (atZL)t]8t a
Tailless Aircraft (figure 111-2)
CMcg = [CMac] w CL[ 1 + CD [+Lq [ --- j [CMc injlet
Canard Aircraft (figure 111-3)
CMcg = [CMac] CL[ + CD[a] + KSwJ . CLc[ "] C g
111-5
SignConvention
Fumelage Reference line - do
it vact
ZT NT
T ENGINE
w
Aft Tailed Aircraft
Figure III-i Force and Moment Vectors
Aft Tailed Aircraft
a ~ Fgr 11- Force &V Momlaet Vefeectrsn
Meg t 111-6
NWB
Nc LwB
Lc•Dc McDB
ac 'acM • •"C-0.°/ -Zý•ieWB Fuselage.
Meg ca
I c J-7 XW NT
ZT
Canard Aircraft
Figure 111-3 Force and Moment Vectors
Canard Aircraft
111-7
Wind Reference
Win' Refrence + a + CPlane
V Positive Pitching(Moment
VPositive Yawing
+C+U
Symmylng
"÷ CN Up; vertical + C n Nooe rightme"+ C A Aft + CM Nose up + CI
"+ C y Out rt. wingj + C I Right wing down
+ Z
Positive RollingMoment
Figure 111-4 Positive Angles, Moments, and Body Axis System
1l1-8
Stick-Fixed Neutral Point
No = Xcg ;when (8CM / &CL) = 0
N0 = -X (CM /SCL] + (at aw)VT1t[1 - ( 8 /a&x)]
x NSCM/SCL _g 0
Two-Dimensional Lift
Subsonicain (Cxa)
CL - inc a (cx in rads)
inc 1 - M 2
Supersonic
CL2D 4c
CL _ inc CL (flat plate)2D m M2 -1 2D M 2 -1 ((xin rads)
Pressure Coefficient
Incompressible
F 2 p- p Cp.Ci =1 C P C- nc (thin-airfoil
Pinc 7J c q• •- theory)
Compressible
Sp = y2 [ 22 : M2]YCp YMO I + .5(y - 1)M
111-9
C=p 20 0 = >0 compression (0 in rads)
SM 2 _0 = <0 expansion
Center of Pressure Location
CMX = - (units of length)cp3 CN
Aerodynamic Center (a.c.) Determination
Assume the moment reference center is at a distance 'x' from the L.E.
and then take moments about the a.c.
CMacqcS = [CMx qcS] + [CLqS](xac- x)cosCC + [CDqcS1(xac- x)sincU
Solving for Xac...
x C M -CMac x x ac
-c -c CLCosat + CDsina
for a << 1
x CM CMac x x ac
- - CLc cL
Mass Flow (ref. 27)
Mo = pAV
I11-10
NOTES
111- Il
NOTES
III- 12
NOTES
1I1- 13
NOTES
III-14
NOTES
111-15
NOTES
III-16
IV AXIS SYSTEM DEFINITIONS
IV
IV Axis System Definitions
(Ref. 6)
Tunnel Axis - An orthogonal, right-handed axis system which remains fixed with respect to
the tunnel in pitch, roll and yaw.
Body Axis - An orthogonal, right-handed axis system which remains fixed with respect to
the model and rotates with it in pitch, roll, and yaw (figure IV-l). Allforces, moments, and axis systems are referenced from the Body axis.
xb Longitudinal body axis in the plane of symmetry of the aircraft; positive
forward.
Yb- Lateral body axis perpendicular to the plane of symmetry of the aircraft,usually taken in the plane of the wing; positive toward right wing tip.
zb Vertical body axis in the plane of symmetry of aircraft, perpendicular to
the longitudinal and lateral axes; positive down.
Wind Axis - An orthogonal, right-handed axis system which is obtained by rotating through
pitch and yaw, but not roll with respect to the Body axis. This system
defines lift as perpendicular to the relative wind, drag as parallel to therelative wind, and side force as perpendicular to the plane of lift and
drag (figure IV-2).
Stability Axis - An orthogonal, right-handed axis system which remains fixed with respectto the relative wind in pitch, but rotates with the model in yaw and roll.Lift is defined as perpendicular to the relative wind with drag and side
force yawing with the model (figure IV-3). The only difference between the
Stability axis and the Body axis is a.
xs - Longitudinal stability axis, parallel to the projection of the total
velocity vector (V) on the plane of symmetry of the aircraft; positive
forward
Ys- Lateral stability axis, coincident as the lateral body axis; positive out
the right wing tip.
zs Vertical stability axis in the plane of symmetry of the aircraft,perpendicular to the longitudinal and lateral stability axes; positive down.
IV-1
Side View
+xB
CC
Rear View I
+'ýtri+ i urnQ
+zX nun-le
FigreIV1 od AisSyte
RearIV 2iw
Side View
+X • CLw
RearVie clww+ X+-
Fiur I- WidAisSse
Top View CY wI
CL
Rear View CW
+zW~ Cy w -- _+:Yw+Z B
+Y B
Figure IV-2 Wind Axis System
I V-3
Side View
+X B
Sz
Tear View Sy
++ B 8
Fiue"V3StbltyAi Sse
+IV-4
Angle Definitions (figure IV-4)
Aerodynamic Angles
cx - Pitch angle-of-attack, angle between the xb axis and the projection of (V)
on the plane of symmetry. A positive cX is pitch up or rotates the zb axis
into +xb axis. a = tan- (w/u)
-Yaw angle, angle between the projected total velocity vector (V) on the
xbYb plane and the +xb axis. A positive direction rotates the + Xb axis
into the +Yb axis (positive nose right). p( = tan-'(v/u).
- Angle-of-sideslip, angle toetween the total velocity vector (V) and
its projection on the XbZb plane. A positive rotation rotates the +Yb axis
into +xb axis (positive nose left). f3 = sin- I(v/V).
Orientation Angles (looking from origin; figure IV-4)
0 -Pitch angle, positive clockwise about the +Yi axis direction (+ nose up; +zi
into +xi).
p- Yaw angle, positive clockwise about the +z. axis direction (+ nose right;+xi into +yi).
S- Roll angle, positive clockwise about the +x. axis direction (+ rt wing down;+Yi into + z.
(p, 0, and 0 form a system of three angles which defines the orientations of theBody axis with the respect to the Tunnel axis system (inertial reference system). Any
orientation of the Body axis system is obtained by rotationally displacing it from theTunnel axis system through each of the three angles in turn. The order of rotation is
important (figure IV-5) and is defined to be yaw-pitch-roll.
A long discussion ensued over the definition of yaw ((p) and sideslip (13). After many
IV-5
arguments and three dimensional velocity box drawings, yp = -i3 (after yawing and pitching
the model. Roll = 0). When the model is yawed, pi.ched, and rolled, then yaw does not
equal minus sideslip (q * -03). The difference is obviously roll.
Angle Transformation form Tunnel Axis (Inertial Axis) to Body Axis
(figure IV-4; +Z down, +X out nose, +Y out rt wing)
Xb• cosOcos(P cos0sin(p -sinO Xi
-sinsin0cos(p 1 rsinsin0sinp1 sinocos0Y b= -cos~sin IJ [+cOsOcOs(P J Yi
"7 cososin0cosqp I rcososinOsinlp cosOcosOL+sinsin, L-sinocos(p Z
b = body i = inertial
Wing Reference Plane (ref. 4)
It is the plane which passes through the wing tips and is parallel to the longitudinal
axis.
Wind Reference Plane (ref. 4)
It is the plane that passes through the relative wind vector and intersects the wing
reference plane along a line which is perpendicular to the plane of symmetry.
Plane of Symmetry (ref. 4)
It is the plane that passes through the fuselage axis and is perpendicular to both the
wing and wind reference planes.
IV-6
XB
BS
VX
Orenato
-- V-27
Coordinate Transformation Equations
Most external balances measure about the wind axis system and most iniernal balances
measure about the body axis system. Thus it becomes necessary to transfer from one axis
system to another. If the model and balance are fixed to the sting, with no relative
motion between the model and the balance, and the sting is capable of movement in yaw,
pitch, and roll then body axis coincides with the baiaice axis. Therefore, all forces and
moments indicated by the balance are body forces and moments.The axis can pass through
either the balance moment reference center or the desired model c.g.. However, if themodel and balance axes do not coincide, additional transformati',s must be considered.
The coordinate transformations below assume data have been transferred to the desired
c.g. location. Also, at is Cbody and has not been corrected for the wind tunnel upflows
and wall effects.
IV-8
Balance Axis to-Body Axis
(Ref. 28)
Order of rotation: yaw, pitch, roll (See figure IV-6)
Note: a~ and 13are balance to body axis angles and not aerodynamic ang!es.
Forces (roll =0)
= b C NblCosax + C Ablcos~s;a -CYblsnsn
CA U Al cosf~coscL - C Y blsinl~cosa - C asince
C C Yba cosf3 + C A blsinf3
Forces (roll = 0)
C Nb Cosar cosfpsincz sin13sina C Nbal
C A b -sincL coscccos13 coscasin13 C A a
CYb_ L0 -sin13 cos3 -CYbal-
Forces (roll # 0)
C N - cosczcoso sincacos13cos4, sincxsinfpcosap C Nb -sin13sino +cos13sino bal
C A b -sina coscacosp cosa s 1no C A a
CY -coSsczin4 -sincxcoslpsino -sincasinfosino C Yb -sinp3cosp +co s Pco SO bal
IV-9
Moments (roll * 0)
(Note: Body c.g. is located +X, +Y, +Z from balance c.g. See figure IV-6)
Clb coc c (CB)ccasf3 -so 0 (ZB) (YB) CI bal
C -(BC)sacfoso -saspso -(BC)ccaso (ZC) 0 -(XC) Cb -(BC)spco +c cO tubal
Cnb sac Pcp-s Pso (CB)sasp3cO cacO -(YB) -(XB) Cnbao+(CB)cf3so
CAb
CYb
CNb
Legend
(CB) = c (BC) = b (ZC) = Zb c cz Y Xx
(ZB) =- (YB) = - (XC)= (XB) -
b b c b
b = span s = sin c =cos
c = M.A.C X, Y, Z = body transfer di stances
Generally, roll (4) is considered as zero in the balance to body transformation.
IV-10
ROLL-PITCH-YAW PITCH-ROLL-YAW YAW-PITCH-ROLL
xx x
X ROTATE ÷900
ROTATE -900 Y ABOUT Y ROTATE +900ABOUT X 2f (PITCH) ABOUT Z(ROLL) (YAW)
ROTATE *900 X
ABOUT Y TY(P ITCH) ROTATE -900 ROTATE +900
4 ABOUT X ABOUT Y(ROLL) (PITCH)
Nz~zY Y
ROTAE, +90 x xABOUT Z(YAW) ROTATE +90 ROTATE 900
ABOUT Z ABOUT X
z(YAW) (ROLL)
zz
Figure IV-5 Rotation Order
IV-1l
CN
j -- Model Body Axis
bal
-%X
CmXbal CA
Ibal Balance CL
NFMoaci/• +X +Y N
ReferencePo n BalancePoi nt •-SMoment
Reference+Z Center
+X A
+Z
Figure IV-6 Balance Axis
IV-12
Body Axis to Wind Axis (ref. 4 & 8)
Note: a and p are aerodynamic angles
(c/B) or (B/c) is used to add moment coefficients together (apples to apples).
Forces
CL = CN cosa - CA bsina
CD = CA COSaCOS(P + C Ybsin(p + CN sinacos(p
CYw = C ybCosy - CA cosasingp - CN sinasin(p
Forces
CL cosa -sina 0 CN
CD sinacosqp cosacosqP sin(p CA
CYw -sinasinqp -cosasinqp cosq• CS
Moments
C = Cm bcos( + (B/c)CIb cosasinp( + (B/c)C nbsinasinwp
Cl = Cl cosacosp - (c/B)Cm sinp + C nsinacos(p
C =C CnbCosa - Clsinca
Moment
Cm cosqP (B/c) c osasiny (B/c) s inctsiny Cm
C -(c/B)sinp cosacosqp s inccosy C1I
B = wing span C 0 -sina cosa• C_n nbc = M.A.C. w b
IV-13
Body Axis to Stability Axis (ref. 4)
Forces
CL = CNbCOSa - CAbsina
CD = CN sinch + CA Cosot
CYs = CYb
Forces
CL " cosot -sina 0- CN
csD sin. cosoa 0 cAbc y s -0 0 c b
Moments
C =Cm s m bS
C, C1 Cosa + C~ siIII s b nb
Cn C Cn oa - Clsino
Moments
1 1 0 0 m
C 1 0 cosc sinc b C
C 0 -sino. cosaX C
IV-14
Wind Axis to Stability Axis (ref. 4 & 8)
Forces
CL =CLs w
CD =CD CoS(P - Cy singfs W w
Cy -Cy cosq + CD singps w w
Force
I L 1 0 0 CLS
CD = 0 cosq -sing CDS w
Cy 0 sing cosq• Cy"- S" - W
Moments
Cm = Cm cosy - (B/c)C1 singS w w
C1 = C1 cosp + (c/B)Cm sings w W
C =Cn ns w
Mcments
Cm cosp - (B/c)sinp 0- Cms w
C,1 (c/B)sinq cosg 0 Cls w
C 1 0 0 Cn
B = wing span
c = M.A.C
IV-15
Since all of the fore mentioned coordinate transformation equations are orthogonal
matrices (square matrices), one can obtain the 'other' axis transformation by simply
transposing the matrix.
IV-16
NOTES
IV-17
NOTES
IV-18
NOTES
IV-19
NOTES
IV-20
NOTES
IV-21
NOTES
IV-22
V TRIP STRIPS
V
V Trip Strips
Boundary Layer Symbols
Rk Reynolds number based on roughness height, velocity, and kinematic viscosity at
top of roughness (Vkk u k)
R x Reynolds number based on freestream conditions and distance of roughness from
leading edge (VOx / Vc,.)
k Roughness height
x Distance of roughness from leading edge
u Local streamwise velocity component outside the boundary layer
uk Local streamwise velocity component inside the boundary layer at the top of the
roughness particle
"V. Freestream velocity
x Distance from leading edge to roughness strip (trip strip)
8 Boundary layer thickness
"i k Kinematic viscosity at top of the roughness particle
1000 Freestream kinematic viscosity
Tt Total temperature (OF)
VI
Boundary Layer Discussion (ref. 7)
Due to the effects of viscosity, the velocity near a surface is gradually slowed from
the freestream velocity, V., to zero velocity. The region where this velocity changeoccurs is called the boundary layer. A boundary layer in which the velocity varies
approximately linearly from the surface is called laminar and a boundary layer whosevelocity varies approximately exponentially from the surface is called turbulent.
Generally speaking, the upper Reynolds number (Rn) limit for a laminar boundary layer is 1X 106 per ft. However, transition from a laminar to a turbulent boundary layer most often
occurs at a much lower Reynolds number (5 x 105 per ft).
Boundary Layer Thickness
The boundary layer thickness is defined as the distance from the surface to a point
where the velocity in the boundary layer is 99% of the velocity of V\.. The boundary layer
thickness can be approximated by
Laminar
6 = 5.2 (12/Rn) 0 5
Turbulent
6 = 0.37 l/(Rn)02
where I distance from body leading edge
Rn = Reynolds number = (pVl / pt)
p = air density * 32.1741 (ft/sec)
V = freestream velocity
I some reference length1t absolute viscosity
(1.2024 X 10- 5 lbm/ft-sec)
(3.7373 X 107 slugs/ft)
\2
Trip Strips (ref. 8)
A trip strip is an artificial roughness added to the model to fix the location of
transition from laminar to a turbulent boundary layer. The reason a trip strip is added
to a wind tunnel model is to increase the local effective Reynolds number and to
"duplicate" the boundary layer to that of a full scale test article. Some general
guidelines that are applicable to all grit-type trips are listed below (ref. 25):
1) The roughness bands should be narrow (0.125 to 0.25 inches).
2) The roughness should be sparsely distributed. According to reference 8,
approximately one grain per every 2mm (.080 in) along the trip strip is the
closest desirable spacing; grains could possibly be spaced as far as 5mm
(.200 in) apart and still cause transition. However, if the trip strip is
either too high above the surface or is too densely packed with particles, it
can affect the model drag, maximum lift and not fix transition.
3) Two-dimensional trips are unsatisfactory because reasonable heights do not fix
transition at the trip location.
4) Care should be taken no: to build up layers of adhesive which can form spanwise
ridges at the edge of the trip. These ridges also tend to make tile trip act as
a two-dimensional step.
Trip Strip Types (ref. 8)
Grit
The traditional trip strip is a finite width strip of grit. Two commercially
available grit materials that are used are Carborundum and Ballotine micro beads
or balls. Trip strip width is usually 0.100 to 0.25 inches. Clear lacquer or
double sided tape can be used as a gluing agent.
Two-Dimensional Tape
These consist of 0.125 inch height printed circuit drafting tape or chart tape.
Also, cellophane type tape can be used. The trip strip is built up by multiple
layers of tape.
V-3
Epoxy Dots
A vinyl tape of varying thicknesses that has holes of 0.05 inch diameter and
0.10 inch center to center displacement is used. This tape is applied to themodel surface and an epoxy compound is forced into the holes. Once the epoxy
has harden remove the tape.
Thread or String
Thread or string is glued to the model. This technique is not used much any
more.
Location of Trip Strips
Lifting Surfaces
This includes wings, tails, and fins. The trip strip is applied to both sidesof the lifting surface. For four and five digit airfoils and conventional wingconstructions, the full scale transition will occur approximately 10% of tile
chord at cruise conditions.
Fuselage
The trip strip is often located where the local diameter is one-half of themaximum diameter. Care should be taken to ensure that the flow has notreattached or that laminar flow has been reestablished aft of the trip strip.
Nacelles
For flow through nacelles, the trip strip is placed inside the nacelle and islocated approximately 5% aft of the inlet L.E. Also, additional trip strip islocated on the outside at 5-10% aft of the inlet L.E.
From reference 8, testing of models through a range of Reynolds and Mach numbers, it isdesirable to eliminate the need for changing grit sizes for each condition. This
elimination is accomplished by using a grit (roughness) size determined for tile
V-4
combination of test Reynolds number and Mach number which require the largest grit size -usually at the smallest Reynolds number and largest Mach number condition. For taperedwings (ref. 10), apply grit at a ccr.aA percer...;: of t,',,e local chord (nominally 5%).In order to permit the use of a single grit size across the span, the grit size iscalculated for the largest chord. For wings with a sharp supersonic leading edge,calculate the grit size using the Mach number and unit Reynolds number based on the flowoutside the boundary layer on the upper surface at the maximum test angle of attack.Apply grit to both upper and lower wing surfaces. For sharp subsonic leading-edge wing orround leading-edge wing, whether or not swept behind the Mach line, the freestream Machnumber and unit Reynolds number are used to determine the grit size. Using the abovecriteria, there will be test conditions when grit particles are larger than the minimumrequired to cause transition. However, careful application of a sparse distribution ofgrit particles on a narrow strip will minimize the drag contribution of the grit itself.
Determination of Trip Strip Height
Grit height determination is a black art. The testing engineer, having an understandingof boundary layer build-up/profile (at low q, the boundary layer height is larger than athigh q), can make a determination of the grit height based on a proven method. Also,don't forget the helpful hints from the the "old guard" that has determined grit heightmany times previously. The correct method in determining the grit height for establishingboundary layer transition is to accomplish a drag study based on grit height. But reality(money and time) dictates the test and the use of reference 25 is generally adequate. Ifthe test has variable dynamic pressure (q) runs, the proper method is to have a new gritheight at each q. However, economics ($) will dictate if a grit study will be
accomplished.
Two methods of determining boundary layer and grit heights based on experimental dataare presented. The first method is for atmospheric tunnels only. The second method can
be used for either atmospheric or pressure tunnels.
Method 1 (Atmospheric tunnels; ref. 11)
This method of determining boundary layer and grit height is for wind tunnels whose testsection is roughly at one (1) atmosphere. In other words, there is no control over thetotal pressure or total temperature and consequently Reynolds number per foot is a
V-5
function of velocity or Mach number. This method is based upon reference II (flIaplate) and each figure (figures V-1 thru V-3) is represented by two total temlperature
curves, Tt= 40OF and Tt= 120 0F.The effect of increasing the temperature is to increase the boundary layer thickness at
each Mach number. Also by increasing the Mach number (Reynolds number per foot), the
boundary layer thickness decreases at any station (on a flat plate).
It is desirable to prevent the transition point from moving on the model during the
test. By applying the correct grit at the correct distance, the boundary layer willtransition from laminar to turbulent near the grit grains at a fixed position. The
conditions which these grains cause transition and remain fixed at subsonic speed, aregenerally dependent upon the length Reynolds number, RX., and the height Reynolds number,
Rk'Reference 25 (using ref. 11 as a reference) points out not to try to cause artificial
transition below a length Reynolds number of approximately 100,000. The first restrictionis line "A" on figure V-1. This restriction has the practical effect of moving the
transition strip aft of the wing leading edge to a fixed dimension rather than a fixed
percentage of the chord. Using the lowest Mach number on figure V-i, the minimum distance
is approximately one (1) inch. Another restriction based upon the length Reynolds number,
R x, is the location on the surface where transition from laminar to turbulent first starts
to occur naturally. Natural transition occurs on a flat plate at an approximate length
Reynolds number of 680,000 and the corresponding distance is represented by line "B" ont
figure V-1. Since transition starts naturally at line "B", it's desirable to place thetrip strip ahead of "B" at a given set of test conditions. This action precludcs the
possibility of the natural transition point moving ahead of the trip strip which could
possibly be attributed to an adverse pressure gradient.
A curve of grit height verse Mach number is shown in figure V-3. Also shown is a curveof boundary layer height at a length Reynolds number Rx = 100,000. The critical grit
height is shown to be well within the boundary layer at each Mach number. Reference 25
has shown that transition may be achieved by the range of grit size above the minimum.
However, when the grit protrudes from the boundary layer, measurable drag is created.Satisfactory artificial transition of boundary layer from laminar to turbulent may be
caused at a given Mach number by grit sizes which lie between the two curves in figureV-3.
Shown in figure V-4 is a curve of Grit height verses Carborundum Grit Number. Thisfigure can be used to determine the use of nominal grit sizes that will provide enough
grains to cause transition provided the spacing of the individual grains is correct.
V-6
In order to satisfy the condition for minimum influence of the particles, some
consideration must be given to particle density and to the width of the grit strip. It
has been demonstrated experimentally in reference 25 that approximately one grit per every
2mm (.080 in) along the transition line is the closest desirable spacing; grains could
possibly be spaced as far as 5mm (.200 in) apart and still cause transition (sublimation
chemicals can help determine the transition point). The width of the trip strip should be
on the order of 1mm (0.04 in).
The maximum forward location of the trip strip required to cause transition is shown in
figure V-2 at two (2) different total temperatures. Figures V-2 and V-3 are used todetermine the trip strip grain size and trip location.
Method 2 (Atmospheric or Pressure tunnels)
This method for determining the height of the trip strip originated from reference 10.
Figure V-5 is derived from reference 10 and is included in this report.
Below is an example using Method 2 to determine the grit height required to starttransition and the corresponding carborundum grit number.
Example
Wing (2-D curves)
X = 0.5 inches (position of grit from LE)
M=3.0
RN/FT = 4 x 106
R /ftX* 6o _X,4 X 106 -4X
I x 106 1 x 106
R / ftX * ___/_ft- (.5)(4) = 2 inches
I x 106
From figure V-5 (2-D, Mach 3.0) at 2.0
R /ft00 -0.0264
1 x 106
V-7
K,4 X 106 -4K=0.02641 x 106
K = 0.0066 inches
From figure V-3 or table V-1 at K = 0.0066 a grit of 90 is to be used.
Table V-I
Nominal Grit Size
Grit Number Nominal Grit Size (in)
10 ...................... 0 .093712 ...................... 0 .078714 .. .............. . ..... 0 .066 116 .. .............. ...... 0 .055520 ...................... 0 .046924 ................ ...... 0 .033130 ...................... 0.028036 ...................... 0.023246 ...................... 0.016554 .. .................... 0 .013860 ...................... 0.011770 ...................... 0.009880 ...................... 0 .008390 ...................... 0 .0070100 ..................... 0.0059120 ..................... 0.0049150 ..................... 0 .0041180 ..................... 0 .0035220 ..................... 0 .0029320..................... 0.0017
Application of Grit types of Trip Strips (ref. 8)
.A,,- -t_ ., ;,, --illy used to lay out the trip strips (at a certain width apart).
Then shellac, lacquer, artist's clear acrylic, or even superhold hair spray is painted or
sprayed over the trip strip area. Once the adhesive material has covered the surface, ihe
grit material is dusted or blown on the wet adhesive. Grit usually is difficult to apply
V-8
to vertical and lower surfaces. To aid in applying grit to those surfaces, a piece of
paper or cardboard can be shaped into a 'V' and with skill can be blown onto the surface.
If the grit is too densely packed, use a tooth-pick to pick off selected grit particles.
V-9
... . . . ...... .. . ..... ... ..
o d -.. -;,. ." -120TT 46-0 f'" T=
Z~ rZ
S .. .............. . .j - . .. .. o.2o
R1=100"000A
0L)120
L
- . . . .T--B4--L no'a Rr~0 MfiCH- NOC0 0.200)
-0 Smn 0.508 '0.95
S'" I' I I F
0.0 0.4 0,8 1.2 1.6 2.0 2,4 2.8 3.2 3.6 1.0OLstonce oLong surFoce, x, Lnches
Figure V-1 Boundary Layer Thickness
V-10
CD
oc
.. ~~ .. .. .. ..... ...
ccc
* . * CD
. . ... . ...
V--c
.T.T. .... I..... .... 120 0 F .. .. .. .. ............ Nom inalT7
. .. .. ... ... ..... .... . ..... .. .... . ..... G r i
.. .. a ............. .... ...... ......
0 0
.. ... ................... .. ...........g ......... ..... . ..0. .. .. .. .. .... ... .. .. ... ..... ... . . ..... ..... ... ......
0.00050.0 .1 0200.5 . 03 0400.5 .5 0550..00.5 ..7 ..7 0.0 080. 0 0.9 .00.~ ~ ~ ~ ~ ~ c ... N....u.... ....... ...... ....... ........ ... .......
V-110
.0 .. . ...
7 . . .. . .
...0. . . . .. . . . . .. . . . . . .. . . . . . . .oD
.. . . . . . ..0. . . . . . . .. ..
C00D
.. .. . ..0. . ... . .
.0 ... ... . . . . . . .
4- C
Co0 ...
CD
CD
.0 . . . .
CD
0000)0 0,0 .1 006 000 .2 .2 002 006 .4 .4
GrtHih,5,i~e
Fiu eV4 Croun u rt N m e
zV 13
MINIMUM GRIT SIZE FOR-U RX = 600
.0 (NACA 'rN-4383)008 - - - - Based on 2I-D Curves. Use for Wing ~1 ~~ and Tail Surfaces 1
Based on 3-D Curves. Use for I0 .24 u slae and Nao a Ilee. et. 'M 1.25
- r fiH
-rr
;7ý1 il7 11.
0. 2 4 8.- 8 0 2741
INC0E5
z .01 0.15(1.t2~~~)(')P !I5si ~ K CJ2
0.026 r
0.0-- ------- -----0..2
0.02
0---- ------ --
0.016
- -- - -- -- ---
'0.0106
* . /ýl
0(H~f3.............
q q
-- -- -- ---
MINIMUM GRIT SIZi: FOR Rx 600r 0.45:NACA TN-4:363
-- -- --- Based on 2-D Curves. use tor 16irig andTail Surfaces, etc...
0.044
0.036
S0.0326
007-~ t.
H120.71 M-.
I.7028__ 0.0 5. 7.-4:
.0.024___. . . . . . . . . .
0.020 .....
0.052
NACA TN-43630.048
0.0404
0.030 - _
0.012 -_.00
0.008 0
77- 77
10O6
X\ F
.. 7b -i~ f--
0.064- .-{---- -<- I~M 3.50
0.050
0.054 - __/
0.05- -~ ------ M 300 5
t -~----
7 L4d
~277V1 A M 25
0 .04 06 -4-~
0.032- -
0.032~~ A a
v -7- MINIMNTM GRITr SIZE FOR Rx 600
___02 NA( A TN -1 16'
---- - --I~~d ''ur'es, Us f. or Wing an d
0.020 4--'t--- t7- -----[- --
0.015-
24 0 -1~-I ''2 14 15
vl III
NOTES
\"-!9~
NOTES
V-20
NOTES
V-21
NOTES
V-22
VI PLANFORM CHARACTERISTICS
VI
VI Planform Characteristics
Planform Symbols
AR Aspect ratio
a Cutout factor
b Full wing span
b/2 Half wing span
b/(21) Wing-slenderness parameter
bi Span of inboard planform formed by two panels
bo 0 Span of outboard planform formed by two panels
c Chord (parallel to axis of symmetry) at any given span station i
c Mean Aerodynamic Chord (MAC)
cB Chord at span break station
c or C Root chordr r
ct or Ct Tip chord
1 Over-all length from wing apex to most aft point on the trailing edge
m,n Non-dimensional chordwise stations in terms of c
p Planform-shape parameter
S Wing area
Si Total area of inboard panelsS 0 Total area of outboard panels
SW Wing urea affected by trailing edge deflection
AS Incremental wing area
x Chordwise location of leading edge at span station y
Xcentroid Chordwise location of centroid of area (chordwise distance from apex to
c/2)
x MAC or x Chordwise location of mean aerodynamic chord
YMAC or y Spanwise location of MAC (equivalent to spanwise location of centroid of
area)
X. Taper ratio
ALE Sweep angle of leading edge
ATE Sweep angle of trailing edge
An ,A Sweep angles of arbitrary non-dimensional chordwise locations
T1 Non-dimensional span stat'lon
VI-I
a• Angle-of-attack
1 i'rio Non-dimensional span stations at inboard and outboard edges of control,
respectively
TIB Non-dimensional spanwise location of leading edge break in wing
G Ratio of chordwise position of leading edge at tip to root chord length
ýB Chordwise location of break in leading edge sweep(s) in terms of chordwise
distance to leading edge at tip = xB/xtSubscripts
B Refers to span station were leading/trailing edge change sweep angle
MAC Mean aerodynamic chord
io Inboard and outboard panels respcctively
ZL Zero lift
VI-2
Wing Parameters Definitions (figure VI-1)
Chord Line A straight line between the leading edge and the trailing edge of the
airfoil in the streamwise direction
Mean Camber Line A line described by the points which are equidistant from the
upper and lower wing surfaces.
Camber Is a measure of curvature of an airfoil and is measured as the maximum
distance between the mean camber line and the chord line and is
measured perpendicular to the chord line. Camber is typically
measured in % chord. Positive camber is when the camber line is above
the chord line and negative when the camber line is below the chord
line.
Angle-of-Attack Angle between the relative wind (V.) and the chord line.
Zero Lift Line A line parallel to the relative wind (V.) and passing through the
trailing edge of the airfoil when the airfoil is at zero lift.
Effective Angle
of Attack The angle of attack of the zero lift line measured from the the
relative relative wind.
otZL Approximately equal to the amount of camber in percent chord for
airfoils and untwisted wings with constant section.
VI-3
y Tangent to Mean Camber Line
~c) maxc
L.E. rI.Ex 1. 0l
X =distance along the chord Yu ordinate of upper surfacemeasured from L.E.U
yl=ordinate lower surfaceY = ordinate at some X =no-iesnazd
c = chord C chord length
l.e.r. = leading-edge radius 0 =slope of i.e.r
X (, a oiino aiu VTh trailing-edge angle
mx camber xt =position of maximum thickness
Ly d maximum ordinate ofmax mean camber line
tmna maximum thickness
(reference 12)
Zero Lift Line
aX eff
Re lat iveloWind
Figure VI-i Airfoil Nomenclaturc
VI-4
Pianform, Parameters
General Planforms
(ref. 12 & 14)
'centroid T xE'/2
YMA4
b
Figure VI-2 General Planform Parameters
S =2 0 c dy c ~fo2 c dy
9 /cx dx ~f/c d
Sx f Tc(x + dXLE 2 centroid So2
(2 yj S~ I tc FT1 p C cRIS dv
r RMSý (-b/2) fC Cr SCJ
Vl-5
Cunventional, Straight-Tapered(ref. 12 & 14)
XEALE
"O cr
C-HA
Figure VI-3 Conventional, Straight Tapered Planform
AR2b b ( 4(1 - X)= C((I - a)(l + 4)tanALE
S = (b/2)C (1 + X) =r W
AS b C2 r [2_(lX)(l 1+T]2)]
2 1 + X + Xc=2Cr I + 4.
-bl [I + 2X
"AI 6
X LE = 2-] rtanA LE = ytanA LE
S2)/ -1 b1 +2),1 b1 tjA ARIT-+-C
Setrid7 I-+ 7,911
tanA 1 C1-0M=X) C tTr r (b/2___]
atanA LB cb2tn r 0 4(l Xfa- A E /2)aALBE AR(1 + 4)Fa-nALE
Tj= (2y)/b
tanA~ = tanA 4 jj(n-m) F I4] n> m
taflA m= tanA LE[l-(l-a)m]
taA I1a 4tanA c 4tALE = ta ATEB -+
tanAE 4 _ anAT (X )
=AR 1 + X)tanALEHi tanAL
%IT ,7
Double Delta
and
Cranked Wings
(ref. 12 & 14)
XLE
XB Cr Xcentroid Xt
ALES
I- b
Figure VI-4 Double Delta and Cranked Wing Planforrm
- b2 A 2b
A =S= - r[i1 X)1lB + ,ki + X.]
b2
AR CS i +o = R (b/2)Cr[(1 - X)rlB + Xi+ ]
C -c + CSo
1 0
VI-8
x = x LE + c/I2
Yis i + YB+ Y]Sy ~S. + S
1 0
- YitanA LE]Sis + [YB tanA LE + y 0 afA LEO]Sox LE Si +so
1 iii+ 0bi+b ljs
b-7 b.. + +s
b Wi =[TIX] [F& 2S
IZB botanAL
1+ 0b. tanA LE.
C C/C = XIXO
x. c B/Cr
0 =C/cB
VI-9
Planform Example
This is an example using the equations for a cranked wing to determine the MAC, c/4, andother items of interest.
Inhadea' t Tre-wition
A 3 5 ° ,040
"L OUUXmrd
Ct
I
C t= 36.51 i
X.=C t~ C ri = 36.5 1/50.45 = 0.724
b /2 =26.96 in
TE (50.45 + 3651)(26.96) = 1172.2 in2 8.14 ft2
C = 50.45)(1.30) 43.86 in
V1-b
i= H [1 + 2Xi =(26.96)(0.473) = 12.76 in
- •2] 1][i 1+2•i
x= [ 1 X jtanALE = (12.76)tan35
= 8.93 in (from LE of C dI
Outboard Section (o)
C = 24.58 inr0
Ct = 7.51 in0
X = 7.51 / 24.58 = 0.306
bo/2 = 33.15 in
s = 24.58 + 7.51 (33.15) 531.89 in =3.69 ft2
- = 2
co 2(24.58)(1.072) = 17.56 ino 3
Measured from LE of Cr
Y= [b I1 [j + 22 0 ] =(33.15)(.4114)= 13.64in
x0= YOtan35 = (13.64)(0.7) = 9.55 in
Measured from LE of Cr.1
Yo = 13.64 + 26.96 = 40.6 in
x = Ytan35 = (40.6)(0.7) = 28.42 in
Total Wing
S = 8.14 + 3.69 = 11.83 ft2
Si/S = 0.688 So/S =0.312
VI-1I
y = 12.76(0.688) + 40.6(0.312) = 21.45 in (1.79 ft)
c = 43.86(0.688) + 17.56(0.312) = 35.65 in (2.97 ft)
Total Aircraft Aerodynamic Center Location
hboa:d Section
xc/4 = xi + Cc/4 = 8.93 + (43.86)/4 = 19.9 in
Yc /4 = 12.76 in (measured from Cr.)1
Outboard Section
x%4 = x + bi tanALE + c0
= 26.96 tan 35 + 9.55 + (17.56)/4 = 32.82 in
Yc/4 = 13.64 + 26.96 = 40.6 in
Total Aircraft
Measured from LE of C1
Xc/4 = 19.9(0.688) + 32.82(0.312) = 23.93 in
Measured from Cr.
Yc!4= 21.146 in
VI-12
NOTES
VI-13
NOTES
VI 14
NOTES
VI-15
V r -T
NOTES
VI-17
VII DRAG
VII
VII Drag
List of Symbols
AR Aspect Ratio
CD Total Drag Coefficient
CD Zero Lift Drag Coefficient (parasite drag)0
CD Minimum Drag Coefficient
ACD Minimum Drag Coefficient change due to camber
CD Drag Coefficient at (L/D)max
CL Total Lift Coefficient
CL Lift Coefficient at minimum drag
CL Lift Coefficient at the Polar Breakpb
CL Lift Coefficient at (L/D) max
e Span efficiency of actual aircraft drag polar
e Span efficiency of a parabolic drag polar
g Gravitational constant
K Induced Drag factor
(L/D) max Maximum lift-to-drag ratio
Rn Reynolds number
V00 Freestream velocity
VII-1
Drag (ref. 13)
Drag is a part of a resultant aerodynamic force produced by the tangential (skinfriction) and normal (pressure) forces acting along the vehicle's surface duc to the
relative fluid motion. This resultant force is resc!ved into the lift and drag components
in the vehicle's plane of symmetry. Drag, the component of the total force that opposes
motion in the equilibrium flight path direction, approximately follows a parabolic
variation with lift and the angle of attack. Total drag can be resolved into various
components (figure VII-1).
Subsonic Drag
The aircraft drag, when taken below the divergent Mach number, is traditionally
decomposed into lift-induced and minimum drag.
Minimum Drag
Minimum drag can be divided into profile (skin friction, pressure, and base drag) andinterference drag (figure VII-l). In general, about two-thirds of subsonic minimum drag
may be attributed to profile (skin friction) drag (figure VII-2).
Profile Drag/Skin Friction Drag (ref. 7)
The skin friction drag is due to the momentum transfer between the fluid particles
adjacent to the vehicle surface and the vehicle. It (in inco" pressibe flow) can he
established for a laminar and turbulent boundary layer. This drag is based upon the
"wetted" surface area of a flat plate, on ONE surface.
Laminar
CDlam =Dlam / S ) (q 1.328 / (Rn)0 5
I X 103 > Rn < I X 106
VII-2
Turbulent
CD turb = Dturb / (q Swet 0.455 / (log Rn) 2,5
I X 106 > Rn < 1 X 109
TurbulentCD _~ *C
cone - 2 CDflat plate
Pressure Drag (Form Drag)
Pressure Drag at subsonic speeds is due to boundary layer displacement effects and
separation effects on aft-facing slopes.
Interference Drag
This drag is the result of mutual interaction of the flow fields developed by the major
configuration components (ie... wing-body).
Miscellaneous Drag
Miscellaneous drag is caused by aircraft protuberances and surface irregularities (ie...
gaps, fasteners).
Drag Due to Lift
Drag due to lift is almost entirely the result of the lift-produced circulation as well
as the vortex shedding from the wing tip.
Zero Lift Drag
Zero lift drag, CD , is the drag at CL = 0 and can be easily found on a drag polar0
curve. The primary contributor to this drag is from skin friction. From figure X-6, the
effect of wing sweep and Mach number on zero lift drag (CD ) can be seen.0
VII-3
Base Drag
Base drag contribution to minimum drag is caused by a rapid expansion of the flow into abase region which causes significantly reduced pressures to act upon a finite base (area).
Internal Duct Drag
This drag is associated with a momentum loss in a flow through nacelle. This drag is
measured by having a nozzle pitot static pressure rake (assume inlet conditions are atfreestream conditions) measuring the momentum loss within the nacelle (duct). Once a
ACD is obtained, subtract it out of the total aircraft CD'Dduct
Wave Drag
Wave drag is primarily due to the lack of pressure recovery on the surface due to a
total pressure loss through the shock wave.
Drag Polar (Subsonic)
A drag polar can be represented by two curve types, CL vs CD or CD vs CL-. Both types
reveal aerodynamic parameters that are important to the performance of an aircraftconfiguration. These two drag polars can be seen in figures VII-2 through VII-5. FiguresVII-2 & VII-3 display some of the major contributors to the overall total drag.
(CL vs CD) Polar
This drag polar, as seen in figures VII-2, VII-3, VII-4, is parabolic (a quadraticfunction) by nature. The eccentricity of the parabola and its origin is affected mainly
by wing camber, twist, and flow separation. A few performance parameters that can beestablished by this curve are CD , CDO, (L/D)max' CL , CL . A drag polar has the
min o pb
characteristic equations listed below.
VII-4
ParabolicC2
CD = CD + L
min rARep
NON-Parabolic (camber, twist, etc.. effects)
CD = CD + (CL - CL)D Dmin ntARe
(CD vs CL 2) Polar
From figure VII-5, the slope of this line is K, the induced drag factor. And 1 4,
Oswald's wing efficiency factor, e, can be obtained. Also, from figure VII-5's cutout,
the intersection of the CD axis and the curve reveals CD0
Polar Break (ref. 14)
(Subsonic)
When the wing leading edge suction has lost its force (separation has occurred), the
polar shape departs drastically from the typical parabolic shape. The drag polar is said
to "break" at this point (figure VII-2). The break point is sensitive to Mach number,
Reynolds number, leading edge radius, and wing geometry.
Camber Effects (ref. 14)
Camber (and twist) essentially shifts the drag polar. The effect of camber (and twist)
can be seen in ACD minresulting in a higher CD and CD The parabolic extent of the
polar is increased and its shape is improved through the use of camber (and twist).
VII-5
Drag and Performance Equations (ref. 14)
'2CD =CD + K(CL - CL)
min
CDmin C Dmin parabolic +ACDmin
CL = (1- e CLpb
AC - [e 1 * CL2 b = CL 2"Dmin [tcARe p2 Lpb AR ep- e)
(CL -CL )2e=
nAR (CD - CD)0
NON-Parabolic Polar
CL K
CD *2(CD -C LY0 0
1(L/D)max =
2(CD - CL TKiP0 0
PRrabolic Polar
CL = K (L/D))max 2 __D
CD = 2 CD0
VII-6
Analytically Determined Drag Polar (ref. 15)
Aerodynamic wind tunnel data (drag) is never exactly parabolic. But, the data can be
approximated by a parabolic relationship. The approach is as follows.
I t2
C D= C Dmin + K(C L- C L ) 2(1)
CD =CD + K(CL )2
o minI
where CDm, K and CL are unknown; expanding equation 1mi n
CD =CD - 2 KCL CL + KCLC0
CD = a + bCL + cCL2
Method of Determining a Drag Polar
Assume "N" matched points of [CL(i), CD(i)]
N N NaN + bI CL(i) + YCL-i2 C
I= 1 L) c'= i CL i D
N N N NI C ) + YCL(i)2 + CL(i) = Xai L (i)' + b = = L )= ICL(i)CD(i)
N N N NaE C L i 2 +) 3 + Y- C ii)2= L() + CL(i)'+ i CL(i i=CL(i)*CD(
N NOnce F ZCL(i)s & CD(i)'s are determined, use a Gaussian Elimination technique to solve
for the coefficients a, b, c where a = CD , b = -2KC, c = K0
VII-7
ExampleN=5
K CL CD
1 0.0 0.0190
2 0.1 0.0202
3 0.2 0.0234
4 0.3 u.0313
5 0.4 0.0456
Resulting augmented matrix
3 1 3 10.13951
[1 .3 .1 0.03433
1.3 .1 .0354 10.112511
a =0.01945
b = - 0.02398
c = 0.2207
CD = 0.01940
K = 0.2207
CL = 0.0543
VII-8
TOTAL DRAG
Zero Lift Dragý Drag Due to Lift(Induced Drag)
Interference Drag Profile Drag
Skin Friction Pressure Base Drag[Drag EDrag
Figure VII-1 Drag Tree
Vi [-9
Subsonic
'~~ Pressure (Form) pm,: abolic
DragSki ý aircraft
CL DragCLpb \N W ~ \ \ /~
Imsmmam/'
CO N
MCi X11\0
Figu~~~~Re VI- C v D ra\oa
Supersonicn
C MIN
SO CD
Figure VII-3 (C L vs CD) Drag Polar
Supers1-10
CL Hand drawn
Tangent line
CL
CD CD~l
Figure VII-4 (CL vs CD) Subsonic Drag Polar
.08
.06 K - __S7TreAR
.04
.02
0 .2 .4 .6 .8 .9 1.0 1.2
CL 2
CL2
Figure VII-5 (CD VS CL 2) Drag Polar
VII-I
This page -was inientionally left blank
VII- 12
NOTES
VII- 13
NOTES
VII- 14
NOTES
VII- 15
NOTES
Vii- 16
VIII EXPERIMENTAL TESTING AND INTERPRETATION
VIII
VIII Experimental Testing and Interpretation
(Ref. 8)
For each wind tunnel test accomplished, the items of interest for the engineer will
vary. Below is a list of items and facts that are usually of interest to the engineer.
Referenc,• is ani excellent aid for tinding those items and for subsonic wind itui1il
testing. This handbook and reference 8 should be the two reference materials that
accompany the testing engineer to the wind tunnel site.
Aircraft (wing-body-tail)
max C M 0
CD CD CD aomin o
(L/D)max CL0 CMac Cnp
CI CY
Flaps
The purpose of flaps is to reduce the wing area through increasing CL and thus reduce
max
the parasite (skin friction) drag in cruise. Flap systems usually generate large negative
pitching moments and require a large horizontal tail to develop adequate down loads for
trim, which reduce the total (wing-body-tail) CLmax
Trailing Edge Flaps (figure VII-9)
Reduce a for a (a zero lift)0
Increase CLmax
Increases a for stall
Leading Edge Flaps
* Extend lift curve to increase stall
* Extend CLmax
VIII-1
Lift Curve (Flaps Up)
* Used to determine flap-up stalling velocity
* CL range from 0.6 to 1.7 (unpowered)max
0 Wing CL runs 85% to 90% of airfoil values (with no high lift devices)max
0 CL usually increases with Reynolds number (Rn)max
0 L.E. slats are insensitive to Rn, but slats reduce the effect of Rn on
CLmax
0 CL for flaps retracted is less than CL flaps extendedmax max
0 Substantial variations of CM often occur at CL or ((xstall)max
* Model should be as close to trim as possible
* Nacelles usually reduce CLmax
•To be usable, C L must be for trimmed flightmax
oa at 0.9 CL is of interest for landing gear length considerationmax
Stall ac should be taken in very small steps so that its share (mion-linear
portion of CL curve) and CL and astall can be determined accuratelymax
To increase the span of the wing affected by the flaps (increasing CLmax
the ailerons can be drooped
Lift Curve (Flaps Down)
CL flaps down range generally from 1.2 to 3.5 (unpowered)max
* Should have approximately same slope (CLo) as flaps up
* Should have same location for the aerodynamic center as flaps up
* Used to find ACL due to flap deflectionmax
VIII-2
* There is usually little need to take the flap- down lift curve as low as
OI0
Drag Curve (Flaps Up and Down)
Near CD , step a in one degree incrementsmin
CD (clean fighter) -0 0.0120 (120 counts of drag)min
The shape of the drag curve is important for climb and cruise with small
changes in drag due to lift being desired during these portions of the
missions.
The value of CD at CL is needed for takeoff and landing calculations.max
Pitching Moment
Cm must be negative for longitudinal stability.
Generally, the pitching moment curves are used to determine if the aircraft
has static stability through the desired C.G. range at all flight
conditions (trimmed flight).
Elevator or Stabilizer Power Curve
Plotting ACM versus 8 e, elevator deflection, (stab. incidence) is made at
several CL'S. This plot indicates the amount of elevator or
stabilizer deflection is needed to produce a certain moment coefficient.
Stabilizer (elevator) effectiveness (dCM/d~e) is obtained by holding (xwing
constant and varying the tail incidence.
* Plot CM versus CL for several elevator angles. The intersections of thecg
curves with the axis indicate trim conditions (figure VII-7). By
holding a wing constant and varying stabilizer incidence, the pitching
moment about the tail is Mt = -ltqtStCLt. When It, St are known, then
CLt can be found. Use qt = 0.85q.. From CLt and known stabilizer angles,
VIII-3
the slope of the tail lift curve (dCL /datt) can be established.
Elevator power must be sufficient to balance (CM = 0.0) the airplane atcg
maximum lift. The critical condition is gear and landing flaps down and in
ground effects.
On low aspect ratio configurations, with short tails, tail effectiveness
varies with a as the local dynamic pressure changes.
* On swept wing configurations, attention must be paid to pitch-up (reversal
of CM curve) in the CL vs CM curve as it can limit usable CL.
Aileron Power Curves
Aileron criteria are usually determined at (p = 0' and plots of Clvs Cn,
C1 vs Sa, and C1 vs 5a are used.
* Good qualities of ailerons are high rolling moments and low hinge moments.
* Maximum roll rate and maximum helix angles are determined from C1max
Generally, C1 of 0.03 is adequate for one aileron.max
When yaw (yp) equals 0, and there is a slight rolling or yawing moment or
side force when controls are neutral, be sure to subtract them out beforereporting the data. This delta can be attributed to asymmetrical tunnel
flow or model asymmetry.
aC1/ap = 0.0002 is equivalent to 10 of effective dihedral.
If there is enough time and money (which usually there is not), do the aileron
effectiveness test with tail off (ref. 8). The first reason to do this is when the
ailerons are deflected in flight, the aircraft normally rolls and the inboard aileron
trailing vortices are swept away from the horizontal tail by the helix angle. In the wind
tunnel, these vortices stream back close to the horizontal tail and induce a load on (he
tail that does not occur in flight. Secondly, it saves effort in data reduction since
tunnel wall effects on a horizontal tail is then non-existent.
VIII-4
Rudder Power Curves
Rudder power on high-performance multi-engine aircraft must possess
sufficient directional stability to prevent excessive yaw angles (Q).
Rudder power must be able to be balanced directionally at best climb speeds
with asymmetric power (engine out).
* Rudder equilibrium is a plot of rudder deflection, 8r' versus yaw angle
(P.* When doing a rudder study, deflect the rudder and yaw the model only in one
direction. You are allowed to do this due to model symmetry.
Rudder Equilibrium: 8a / 8r ranges from -1.2 to -0.5 (maneuverable to
stable)
Rudder Power: aCn / a8 r= -0.001 is reasonable.
Determine Center of Pressure Shift (C.P.)
To determine the C.P. shift the derivative aCM / aMach is used. The main factor that
contributes to this derivative is the backward shift of the wing center of pressure (C.P.)
which occurs in the transonic range. On two-dimensional symmetrical wings, for example,
the C.P. moves from approximately 0.25c to approximately 0.5c as the Mach number increases
from subsonic to supersonic values.
C.G. Shift
Moving the C.G. forward reduces a trim or CL resulting in an increase in trim speed.
Lift Curve Slope CLot
SCLa ranges approximately
0.110 per degree for thin airfoils }Rn > 1060.115 per degree for thick airfoils
As a rule of thumb; CL is usually < 90% of CL
CCL amakes an important contribution to the dampening of the longitudinal
short period mode.
VIII-5
Determining Aircraft Parameters from Wind Tunnel Data (ref. 29)
Wing-Body wind tunnel data:
(X - 1.5 5.0
CL! 0.0 0.52 a = geometric angle-of-attack
a 1.0 17.88
CM -0.01 0.05 CM - CMcgwb
Static Martin
Lift curve slope
8CL _0,52-0_
a ac L ,5 - - 0.08 per degreewbaa 5-(-1.5)
CMcgwb = CMacwb + awbawb(hhac) (1)
awb " absolute angle-of-attack
at a = 1.00
-0.01 = C Macwb+ 0.08 (1 + 1.5)(h - hac ) (2)Macabwb
at ax = 7.880
0.05 = CMacwb + 0.08(7.88 + 1.5)(h - hacw) (3)
Equations 2 & 3 can be solved simultaneously. Subtracting equation 3 from equation 2...
-0.06 = 0 - 0.55(h - h )aIwb
VIII-6
(h - h a -006aCwb -0.55
(h - hac ) = 0.1"1 (static margin)
Aerodynamic Center Location
h = 0.35c
(h - h ) =O0.11acWb
h = 0.35 - 0.11aCwb
h = 0.24 (a.c. location % c)ac wb
Aerodynamic Center Pitching Moment
Using equation 1 ...
-0.01 = CMacwb + 0.08(1 - 1.5)(0.11)
CM = -0.032
Center-of-Gravity Pitching Moment
For a given wing-body combination, the aerodynamic center lies 0.05 (5%) of a chord
length ahead of the c.g.. The moment coefficient about the aerodynamic center is -0.016.
If the lift coefficient is 0.45, calculate the moment coefficient about the c.g..
CMcg =CMac +CLwb(h - h acwb) (4)
(h - ha ) = 0.05 CwL = 0.45 CMac = -0.016awb wb wb
CMcgwb = -0.016 + 0.45(0.05)
VIII-7
CMcgw = 0.0065
Wing-Body-Tail
Consider the wing-body data above, the area and the M.A.C. of the wing are 1.076 ft2 and
0.328 ft respectively. The distance from the airplane c.g. to the tail a.c. is 0.557 ft,
the tail area is 0.215 ft 2, the tail setting angle is 2.70, the tail lift-slope is 0.1 perdegree, and from experimental measurement, o= 0.0 and a8/cet = -. 35. If x = 7.880,
calculate CMcg
From the information above
C~c C aw+ aeta [(h - hacb V a 1 +Vat~it + Eo) (5)CMcg = Ma c [ - ac -] + VHa( 1 + 5
CMacwb= - 0.032 a a = 7.88 + 1.5 = 9.380
awb = 0.08 (h -hacwb) = 0.11
1 S
VH_ t -t (.557)(.215) = 0.34cSw (.328)(1.076)
at= 0.1/deg a/iact = 0.35
i = 2.70 = 0.0
Using Equation 5
CMcg = -0.032+(.08)(9.38)[0.11-.34 .- 1 (1-.35)] +.34(.1)(2.7+0.0)
CM = -0.032 - 0.125 + 0.092cg
CM = -0065cg
VIII1-8
Longitudinal Static StabiItv
Does the aircraft (wing-body-tail) above have longitudinal static stability and balance ?
aC
(h - H (6)
awb= 0.08 (h - hacwb) = 0.11
VH - 0.34 at - 0.1 per degree
aclaa = 0.35
aCMcg= 0.08[(0.11)- 90.34) 0 (1 - 0.35)]
00.0
aCM cg_ - 0.01331
80:
Since the slope is negative, thus the aircraft is statically stable.
Longitudinal Balance
CMo = CMacwb + VHat(it + e) (7)
CMacwb = -0.032 it = 2.70
CM = -0.032 + (0.34)(0.1)(2.7)0
CM = 0.0
Trim angle-of-attack
8CM
Remember that - ~ C is the slope of a straight line. Therefore by setting CM to zero,R 0¢wb Cc e
VIII-9
and writing the equation of a straight line
y = mx + b (8)
aCMy CM m - cg
cg a
x = trim b =CM0
actrim can be found. From the above example...
0.0 = 0.06 - 0.0133a trim
40(X trim= 4.5
Clearly, this angle-of-attack falls within the reasonable flight range. Therefore, the
aircraft is longitudinally balanced and statically stable.
Finding Trimmed Flight Parameters (ref. 16)
(Unpowered, No thrust components)
Trimmed flight parameters can be easily found from a CM vs CL and CL vs cx plot. One of
the model test parameters an engineer should always consider testing is the elevator.
From this data, trim conditions can be found. When testing the elevator for its
effectiveness, the angle-of-attack for the model configuration should be he!d constant
while only A change of the elevator deflection angle (5 e) is accomplished. Holding the
configuration at a constant angle-cf-attack is not necessary, but it helps when
manipulating the data for presentation. Make as many runs as necessary at different 8 's
while the horizontal tail incident angle is held constant. The resultant plot can be see
,i figure VII-7. On the CM vs CL plot, where each 8 e curve crosses CM = 0, that point is
considered a trim point and the corresponding CL is the trim CL. To find a trini' a line
(this line equates to a constant CL) is drawn from the CM vs CL plot to the corresponding
8e on the CL vs Cc plot. That intersection on the CL vs ac plot a trinican be found. If a
series of trim points are found on the CL vs (x plot a C tL can also be found.
trim
VIII- 10
Determining any C.G. Location
(unpowered, gliding flight)
To find the trim envelope (where CM = 0) for any c.g. location, a line of any slope is
drawn by rotating that line from CM = CL =0 on the CM vs CL curve to any position on theCM vs CL curve. The slope of the newly drawn line is equal to the c.g. shift (refs. 16and 8).
6C M IC M pT
Snew c.g. position -- old c.g. position
where
A = [old c.g. position (%MAC) - new c.g. position (%MAC)]
or
A = [(Xcg- Xac)old- (Xcg- Xac)new]
Example:
Move c.g. from 0.35 to 0.2;
aCMI 0.2
aCMac M .2 = 0.2 - (0.35 - 0.2)
L 0.2
a CT 00S0.2
It appears that to acquire this transfer a subtraction of moment arms (A) wasaccomplished. However, in reference 16, a thorough explanation is discussed and it showsthat it is not just subtraction of moment arms but rather a subtraction of moments.
Determining the Average Downwash Angle (ref. 8)
To determine the average downwash behind the wing ,est the model configuration with the
V III-1 I
tail-off. Then test the configuration with the tail-on at different tail incident angles.
The tail incident angle (it) is the angle between the wing-body zero-lift line (generally
the longitudinal axis) and the tail zero-lift line. If the tail is a symmetric airfoil,
the tail zero lift line and the tail chord line are the same. Also, if the tail is an all
movable tail, the tail incident angle would generally be the same as 8 e. Once the tail-on
and tail-off data are acquired, plot the data similar to figure VIII-1 (a w vs CM) at
different it's and then plot the tail-off data. The intersection of the tai!-on curve
with the tail-off curve are points where, at a given awl the tail-on CM equals the
tail-off CM. Also, at those intersections, the tail is at zero lift.
(For a symmetrical airfoil section)
a t = ccw +it - Ew =0
Ew = wing downwash angle
oxt = Tail angle-of-attack
6 w = (Xw + td
Once L is found then the parameter - can easily be obtained by plotting Lw vs aC.
Determining Induced Drag Factor (K)
and Oswald's Wing Efficiency Factor(e)
Draw a plot of CD vs CL2. The slope of this line is K, the induced drag factor (see
figure VII-5). The intercept is Cd0
Since K =
__ 1Oswald's wing efficiency factor (e) is... e = __ I
Base Pressure (ref. 4)
A base pressure correction is applied to remove the base pressure drag from the total
drag in order to correct the force drag. Such a correction is required because of the
base drag is unknown without the interference or interactions of the sting or a jet. The
measured base pressure is corrected to the reference ambient static pressure which is
VIII-12
considered to act over the entire base of the model. Occasionally more than one static
pressure is measured on the base of the model and averaged to arrive at the base pressure.
This average is multiplied by the area of the base to obtain the base-pressure axial force
correction. IP enýation form:
FA = (Cp ) (AB)(q.)= Cave)
Where
FA = Base pressure axial force to be added to the measured
axial force
CPave = Average pressure coefficient on base,
Poo = Free stream static pressure (in test section)
q* =Free stream dynamic pressure
AB = Base area
Pressure Transducer Selection
To determine the appropriate differential pressure transducer, three items are needed.
The first being the maximum expected pressure coefficient (CP), the second being the
minimum expected CP and finally the dynamic pressure (q). If obtaining the maximum and
minimum CP's are difficult, then use +0.5 and -2.0 for the first iteration (these
suggested CP's are good numbers for wing pressures). Then just "plug and chug" the
equation below.
CP P-_P 0
1 2•pv
Ap = (p - p..) = CP(q)
Differential pressure transducers are usually rated in psi (bf)
1 bf in?2Conversion of 'q' to psf (l-.) may be required.
ft 2
From the maximum or minimum CP determine the greatest magnitude of Ap and then acquire
the appropriate pressure transducer.
VIII- 13
Example:
CPI = +0.5 CP2 = -2.0 q = 2.11 psi
Apl = 1.055 psi Ap2 "-4.22 psiI
Based upon Ap2, a differential pressure transducer of 5 psi will be adequate.
Flow Visualization
Flow visualization offers the testing engineer a unique way to observe the local flow
fields. Flow visualization can be separated into two categories, surface flow
visualization and off-body flow field visualization. Techniques to observe surface flow
fields are tufts and oil flows. Laser light sheet, smoke, and tufts are used for off-body
flow fields.
Surface Flow Visualization
Yam
Tufts do affect the aerodynamic forces.
Yam should be used when the testing engineer is not concerned with the model forces and
moments. Yam has the greatest adverse effect on lift and drag compared to other surface
flow visualization techniques. For yam tufts, use 0.75 inch length of No. 6 floss
crochet yarn (any color). Have plenty of yam available since it does not last a long
time in the wind tunnel. Before applying tufts on the model, clean the model with naphtha
or other solvents to remove any oils. There are two methods used to apply the yarn tufts
to the model. The first method is to use a tuft board. A tuft board is a piece of
scrapwood with two nails in it. The distance between the two nails is the length of the
tuft (generally 0.75 to 1.0 inch). Wrap the tufts around the nails and then cut the tufts
on the backside of the nail to give you the correct length that is needed. To attach the
yarn to the model use cellophane tape or "super glue" and apply the yarn to the model in a
symmetric pattern (0.75 inch x 0.75 inch). Ensure that at least 0.75 inch of the tuft
material is available for the flow to manipulate freely. The second method in applying
tufts to the model is to tape the yarn at two opposite ends of the item of interest on the
model (ie... leading to the trailing edge) and glue/tape the tuft at the desired interval
lengths (0.75 in) then cut the yarn.
VIII- 14
Fluorescent Mini-tufts (ref. 26)
Fluorescent mini-tufts allow a large number of tufts to be applied to a model surface in
a manner that produces negligible interference with model forces and pressures.
Comparisons of tufts-on and tufts-off from Mach 0.5 to 2.4 showed differences of two to
three drag counts. Mini-tufts are an extremely fine nylon mono-filament fiber that has
been treated with a fluorescent dye that renders it visible during fluorescence
(ultraviolet) photography. There are two sizes of mini-tufts generally used during wind
tunnel testing. Those sizes are 3 denier (0.02mm, 0.0007 inch diameter) and 15 denier
(0.04mm, 0.0017 inch diameter). Free moving length (measured from the gluing point) of
the mini-tufts should range from 0.5 to 0.75 inches.
Mini-Tuft Installation
The procedures listed below have proven to be adequate for most applications. Tuft
application utilizing these procedures are able to withstand many hours of testing at
transonic and supersonic Mach numbers without appreciable adhesion failure.
Surface Preparation Steps
1. Wipe model surface with a solvent to remove grease and oil. Use clean paper towelsor tissues rather than shop rags. Preferred solvents are naphtha or any chlorinated
hydrocarbons such as trichloroethylene or trichloroethane. Methyl ethyl ketone or
Freon are less suitable because of rapid evaporation rates. Do not use alcohols.
2. If possible, lightly abrade the surface with a fine grit carborundum pad or aluminum
oxide paper. Do not use silicon carbide paper on aluminum. Care should be taken not
to compromise the aerodynamic smoothness requirements for the model.3. After abrading the surface, thoroughly clean the surface with a solvent using clean
tissues. Then wipe that area with another clean tissue using that tissue only once.
Wiping more than once with the same tissue redeposits the contaminants. Continue to
wipe the surface with solvent until the tissue shows no stains. Avoid letting the
solvent evaporate completely before it is wiped.
4. Apply an Alodine solution to the surface with a swab or a brush. Allow it to remainwet on the surface for three to five minutes. Any areas that resist wetting should
be scrubbed with carborundum pad wet with the Alodine solution (Alodine is a chromateconversion solution that promotes adhesion of the gluing material to the surface. It
VIII-15
is available from aircraft paint suppliers. In this application, add a wetting agent
such as Kodak Photo-Flo® to Alodine to make it a one percent concentration.).
5. Rinse the Alodine from the surface with clean water and then dry the surface with
clean tissues. Be careful to limit skin contact with the Alodine and to wash
thoroughly after use.
6. Cover the cleansed area with paper to avoid recontamination until after the tuft
adhesive is applied. Realize that the above procedures are involved and can be
reduced in scope. Steps one through six are the correct way to prepare the model
surface. However, there is a correct way of doing things and the getting it done way
of doing things. Be flexible in this area.
Mini-Tuft Attachment Steps
1. Lay out alignment marks with a soft pencil (roughly 0.75 inch grid) so that the tufts
can be applied in a symmetrical pattern.
2. Place oversized lengths of tuft material over the extreme ends of the model (i.e...
L.E. to T.E. if a wing) surface by wrapping the tuft around the model and then taping
it to the lower surface.
3. Be careful to use only slight tension t. avoid stretching the filament.
4. Use a "super glue" type of gluing agent to apply the tufts to the surface. Apply a
small drop of glue to the strand of tufting material in intervals of approximately
0.5 to 0.75 inches.
5. Be sure glue drops are dried before cutting the tuft material. Cut tufts just ahead
of each adhesive drop with a new razor blade.
Experience has shown that the visual appearance of these tuft results are greatly
enhanced by carefully keeping the tuft spacing uniform. If the tuft pattern is
asymmetrical, it is still possible to describe the flow, but it is difficult to interpret.
To illuminate the mini-tufts, an ultraviolet light source is required. The ultraviolet
illumination excites the fluorescent material to radiate in the visible spectrum (i.e..
400-600 nanometers). Video tape or black and white still photographs are the normal
surface flow visualization data medium. It also helps if a fluorescent felt tip marker is
used to hi-lite/outline the model. This will allow the model lines to be seen in the
photograph or video tape.
VIII- 16
Oil Flow
Oil flow visualization shows flow separation lines, vortex reattachment lines, shocklines, and complete boundary layer activity on the surface. The oil flow runs should be
held for the last runs of the test since the oil will clog static pressure taps and is
generally a messy procedure. The model is painted a flat color with the oil a color thatwill contrast the model color (generally a black model and white oil). The oil viscosity
is an important parameter to be considered. If the oil is too viscous it will not display
the true surface flow. If the oil is not viscous enough, it will run off the model andnot display the flow on the surface. There are several formulas that can be used for the
oil. The formulas will differ from tunnel-to-tunnel and from test engineer-to-testengineer based on his or her experience. One formula that is used and works well is one
teaspoon of titanium dioxide (a white color), one teaspoon of STP Oil Treatment®, and fivedrops of Oleic acid. Mix this amalgamation thoroughly and apply it on the model using a
syringe with a 18 or 20 gauge needle. When applying the oil to the model it should have alogical and somewhat symmetrical pattern.
Off Body Flow Visualization
Laser Light Sheet (Vapor Screen)
The vapor screen technique is a simple, yet effective, flow visualization tool to studythe off-body flows about aerodynamic shapes at subsonic, transonic, and supersonic speeds.In recent years, this technique has frequently been employed in wind tunnel experiments toimprove the understanding and control of the vortices shed from slender bodies of
missiles, fuselage forebodies, and wings of fighter aircraft at high angles-of-attack.
The technique features the injection of sufficient water into the tunnel circuit to createa condensation in the test section. An intense sheet of light is generated, usually witha laser (18-watt Argon-ion laser is sufficient), that can be oriented in any selected
plane relative to the test model. The light is scattered as the water particles passthrough the sheet, which enable the off-body flow to be visualized. However, the tunnel
operator/owner might become perturbed at putting water in the tunnel. To alleviatehis/her fear and to remove the water from the tunnel after the vapor screen runs, pump thetunnel down to a Ptotal of 500 psf, start the tunnel(subsonic) and turn the driers on. Dothis approximately twice for 15 minutes each time. This will evaporate a majority of thewater. After doing this have the tunnel technicians enter the tunnel and wipe up wkhat
VIII- 17
little water is remaining.
Smoke Seeded Flow
Flow visualization using smoke is an excellent tool for observing off-body (external)
flow fields that are dominated by vortical flows. If done correctly, superb qualitativedata will result at a Reynolds number that might make it feasible to extrapolate the flow
field to flight conditions.
Typically, smoke is generated by several methods. Such methods include pyrotechnic
smoke devices, chemical (titanium tetrachloride and tin tetrachloride), petroleum products
(kerosene, Type 1962 Fog Juice) and Rosco smoke generating fluid. The technique of
producing smoke will be determined at the wind tunnel site.
A low turbulence wind tunnel is extremely helpful when trying to accomplish a flowvisualization study using smoke. The surrounding air in the test section needs to be as
undisturbed as possible in order that an accurate analysis of the model's flow field can
be made.
Generally, the smoke enters the wind tunnel via a smoke wand that is typically locatedin the stilling chamber of the wind tunnel. The density of the smoke emitting from thewand should range from a mild to moderate fog (qualitatively speaking). The smoke should
then traverse down to the test section and enter as a filament sheet that spans most of
the section. The smoke filament sheet should be set (vertically) in such a manner that
the smoke is "caught up" in the upwash of the wing, chine, or item of interest. It might
be necessary to change the position of the the smoke wand to achieve an optimum smoke
filament sheet position. Once the smoke filament sheet is caught in the upwash, the
external flow field can be observed using a laser light sheet to expose the vortical
systems.
Tufts
To use tufts for off-body flow visualization, a tuft wand or a tufted, framed, wire grid
is used. A tuft wand is a pole with a long tuft on the end of it. The wand is hand held
in the tunnel near the model to observe off-body flow fields. This technique is aqualitative procedure since the person and the pole create a flow disturbance in ihetunnel. A tufted, framed, wire grid gives the testing engineer a planar view of the
off-body flow field. The wire grid contains symmetrical, square (1 inch by 1 inch ) wire
mesh with tufts (usually yam) glued at the comers of each mesh. The wire grid is placed
VIII- 18
downstream of the model and is useful in examining wing tip vortices.Determining Aerodynamic Angles from the Model Support (sting) Angles
Aerodynamic angles a and 13 are calculated using the model support (sting) angles 7, V,
0, andS.
Wherey = prebend angle of sting (only in pitch)
p= yaw angle of the support
0 = pitch angle of the support
= roll angle of the support
Knowingar tan-t w U3 sin -'[Vi
Where
u = longitudinal velocity component
v = lateral velocity componentw = vertical velocity component
V = total freestream velocity
from reference 28 and 38
u = [cos~p (cosycosO - sinysin0coso] - sinysin(Psin(P] V"
V = [sinosin~cos~p - cososirnP] V.
W = [cos~p sinycos0 + cosycososin0] + cosysinosinT] V"
Solving for the model angles a and 03
[cos(PIsinycoso + cosycososin0] + cosysirnosin,(P]tana --
[cosy 4cosycosO - sinysin0coso] - sinysinsinqj
Vill- 19
sino = [sinosinOcosy - cososinwo]
-00
1w
Tail -12 -4 0 +4 +8 ioff
(+) Cm (--)
Figure VIii-i Downwash Determination
C L ( C L _--.
Trimlift curve
a Cmcg
Figure VIII-2 Trim Determination Plot
VIII-20
NOTES
VIII-21
NOTES
VIII-22
NOTES
VIII-23
NOTES
Vlll-24
NOTES
VIII-25
NOTES
VIII-26
NOTES
VIII-27
NOTES
VIII-28
IX STRESS ANALYSIS
Ix
IX Stress Analysis
List of Symbols
A Area
d Diameter
c Distance from Neutral Axis
E Modulus of Elasticity
ftu Ultimate (allowable) Tensile Stress
f su Ultimate (allowable) Stress in pure shear
fty Tensile Yield Stress (point)
f tp Tensile Proportional limit
G Modulus of Rigidity (Shear Modulus of Elasticity)
I Moment of InertiaJ Polar Moment of Inertia
L Total length of element (shaft length etc...)
1 Length of element (not shaft length)
AL Change in length after deformation
M Moment
NF Normal Force
P Load
R Average radius
r Distance to point of interest for torsion
0r Shaft radius
As Change in arc length after deformation
T Torque
t Thickness
y Distance to point of interest for stress
a Constant based on l/t (non dimensional)
03 Constant based on l/t (non dimensional)
E Strain
a Stress
It Shear
IX-I
List of Symbols continued
p. Poisson's ratio
y Angle of shear strain
Subscripts
i-- inside b -- bending n -- normalo -- outside s -- shear
IX-2
Definitions
Stress Stress implies a force per unit area and is a measure of the intensity
of the force acting on a definite plane passing through a given point.The stress distribution may or may not be uniform, depending on thenature of the loading conditions. Tensile load is considered (+) and a
compressive load is considered (-).
Strain It is the change in length per unit length. The strain distributionmay or may not be uniformed depending on the member and loading
conditions.
Normal Stress A unit stress which acts normal to the cross section of the
structural element. These stresses are created by bending
moments and axial forces.
Shear Stress A unit stress which acts parallel and in the plane of the cross
section. These stresses are caused by torsional moments and
shear forces.
Normal Strain Strain •ssociated with a normal stress; it takes place in the
direction in which its associated normal stress acts. Increasein length are (+) strains, decrease in lengths are (-) strains.
Shearing Strains Those strains related to relative changes in angles.
'Y eld Point Where elongation increases with no increase in load. The stress
at this point is known as Tensile Yield Stress.
Proportional Limit Where the stress-strain curve first becomes non-linear.
That point is the maximum stress where the strain remainsdirectly proportional to stress.
Elastic Limit The maximum stress to which a material maybe subjected and still
upon the removal of the load, return to its original dimensions.
Neutral Axis When a beam is deflected, one surface is in compression while theother surface is under tensile stress. There is a plane(neutral) where the stress will be zero. Where this neutral
IX-3
Neutral Axis plane intersects any perpendicular cross section (or plane of
(continued) loading) this location is the neutral axis for that cross
section.
Ultimate Tensile Stress This is the maximum allowable stress of the material.
Factor of Safety Ultimate load / limit load
Modulus of Elasticity Ratio of stress to strain; slope of the straight
portion of the stress-strain diagram.
Shear Modulus of Ratio of shear stress to shear strain at low loads. The
Elasticity initial slope of the stress-strain diagram for shear.
Radius of Gyration The distance from the inertia axis that the entire mass
of an Body would be concentrated in order to give the same moment of
inertia.
Radius of Gyration The distance from the inertia axis to the point where the
of an Area area would be concentrated in order to produce the samemoment of inertia.
Shear Center A point where a load produces no torsion on a asymmetric
beam cross section.
IX-4
STRESS FORMULAS
Normal Stress Normal Strain
P E AL
A L
Modulus of Elasticity Poisson's Ratio
E an lateral deformationC n axial deforma t ion
n
Shear Strain (thin wall, Shear Modulus of Elasticity
circular cylinder: twcc) (twcc)
As "e =- G s_ s
s L C s
ANGLE OF TWIST
Rectangular shaft Circular shaft
TL TL
f1 Gt' JG
Split tube
4, 3TL2irRt 3 G
POLAR MOMENT OF INERTIA
Solid, Circular Shaft Tubular, Circularcross-section
- itd4J d - n (do4 -d. )4
32 32 o
IX-5
Solid. Circular Shaft Tubular, Circularcross-section
J tr- Jr-- 0 ri)S- 'r
2 2
RADIUS OF GYRATION
Body Area
BENDING STRESS
'b = _ Mc 'b = 6M (rectangular)"I bh
SHEAR STRESS
Rectangular cross section (beam) Circular cross section (beam)
f=t -~ f='t -s max = 3s max 3A2A 3
TORSIONAL FORMULAS
Solid, circular shaft Tubular, circular cross section
f=t _ 16T _Tr f=t 6rd0s max s max 4 di 4)
2nrr7tr4 4
7r(r r-r40 1
IX-6
Split tube
3Tfs="s-2rRt'
Rectangular) cross section
(flat plate)
3T Tfs= 'Es =lT>> t f =s -- ltSs i2' s s x•1 t2' 1It
Table IX-1 Stress Constants (ref. 31)
l/t 1.0 1.5 1.75 2.0 2.5 3.0 4.0 6.0 8.0 10.0 00
I a .208 .231 .239 .246 .258 .267 .282 .299 .307 .313 .333
.141 .196 .214 .229 .249 .263 .281 .299 .307 .313 .333
COMBINED STRESS
Shafts under bending and torsional loads (ref. 33)
f b f bl2 +f 2fT= 2_-+ +s
Stresses on an incline plane (ref. 34)
o = y x Ycos0 + t sin202 2 xy
G -aC= 2-Ysin20 + 'r cos20
2
IX-7
Principle Stress
CY Gx + Gy Y x y )2+ 2amax 2 2 r ] (xY)
min 2
Angle on which principle stresses occurs
(measured from x axis)
1;tn20 -- xy
- x - ytan22
Maximum and Minimum Shear stresses
max -2 y)
min
Angle on which max and min Shear stress occurs
(measured from x axis)
CF + ax y
tan20 2
xy
FACTOR OF SAFETY
F ftu ftuSF - stress SF -
T
SF =
f)2 f ( 2
IX-8
Calculate Centroid of a Planform Area
Area y AY Ay2
Elem. Dim ( in) ( in 2 ) (in) (i n3 ) (i n4)
1 1.45 X 4.20 6.1 10.3+2.1= 12.4 75.5 937.9
2 2.3 X 10.3 23.7 10.3/2 = 5.14 122.0 628.6
3 8.98 X 2.0 X 0.5 8.98 8.98/3= 3.0 26.9 80.82
4 8.98 X 2.0 X 0.5 8.98 8.98/3= 3.0 26.9 80.82
5 0.83 X 2.0 1.66 0.83/2= .42 0.7 0.29
6 0.83 X 2.0 1.66 0.83/2= .42 0.7 0.29
7 0.85 X 4.22/2 1.8 4.22/3+10.3= 11.7 21 .1 246.4
8 0.85 X 4.22/2 1.8 4.22/3+10.3= 11.7 21 .1 246.4
' 54.7 in2 ' 294 .9 2221.5
Y _ XAy _ 294.9 - 5.4 inA 54.7
1 = 2221.5 in4
7- -8
2
5t5
Figure IX-1 Stress Analysis Example 1
(centr id determination)
IX-9
Example: Stress Analysis
Shear and bending at section F-F (figure IX-2)
Assuming a rectangular section of average thickness...
f 6M I = baseb t I , t = height (thickness)
( -Mc, I= t = -)(fb= = if C = !
I 12 2
Tf -s (X It2
T = torsion in beam(x = constant dependent upon 1/t
1 = 0.84 in
- 0.044 + 0.030 = 0.037 inave 2
l/t = 22.7 ; cx = 0.333
M = 6.0(0.380) = 2.28 in-lb
T 6 [0.473+0.84 = 5.358 in-lb
L 2
Bending
fb 6(2.28) = 11,896 psi
(0.840)(0.037)2
S.F. - 89100 7.4811896
IX-10
Shear
f = 5.358 = 13,992 psi
(0.333)(0.840)(0.037)2
S.F.= = 3.57
[11896]2 +[[139922[8-+ T5600]j
Shear and bending at first set of attachment holes at section E-E
(figure IX-2)
Assuming a rectangular section of average thickness...
6M
(f -Mc. I= I t, c = )1 12 2
f Ts Xlt2
T = torsions in beama = constant dependent upon l/t
I = 0.840 - 2(0.138) = 0.564 in
- 0.062 + 0.030 0.046 inave 2
l/t = 12.3 ; ax = 0.333
M = 6.0(0.380 + 0.210) = 3.54 in-lb
IX-I1
T= 6[0.473 +0_41 =5.358 in-lb
Bending
fb 6(3.54) = 17,798 psi
(0.564)(0.046)2
S.F. = 89100 = 5.0117798
Shear
f =5.358 13,482 psi
(0.333) (0.564) (0.046)2
S.F.= 1 = 3.23
[1779812 F1348212
Tension in screws at wing tip missile attachment...
Summing moments about A-A
EMAA = 0 = 1.57NF - 0.706R1 - 0.558R2 - 0.154R 3 0.302R 4
Solving screw loads in terms of R,
R2 R 0.558R R3 0.154R 0.302R
0.706 0.706 1 4-0.7061
Substitute into the MAA equation above...
1.57NF = R1 0.706 + 0.5582 + 0.1542 + 0.302 2
0.706
IX- 12
1.57NF = 1.31R 1
RI = 1.199NF
From above...
R2 = 0.948NF R3 = 0.262NF R4 = 0.308NF
From figure IX-2, NF = 6.0 lbs...
IR1= 7.2 lbs1 R2 = 5.69 lbsI R3 = 1.57 lbs 4 = 3.08 lbs
ftu for 0-80 flat-head socket screws is 265 lbs
S.F. - = 36.87.2
Thread pullout in wing tip missile attachment screws...
For screw R1 (0-80 flat-head screws)...
Shear Area = nt(screw pitch dia.)(screw length)(0.5)
(0.5) is and arbitrary fudge factor for conservatism
= 7t(0.519)(0.58)(0.5)
Shear Area = 0.0094(0.5) = 0.0047 in2
Shear f = load / shear area
f = 7.2 / 0.0047
fs= 1532 psi
For flat-head screw 0-80
fsu = 96000 psi
IX-13
S.F. - 96000 62.7
1532
For the wing, 17-4PH stainless steel screws...
f= 120000 psi
S.F. - 120000 = 78.41532
Screw head pullout in wing tip missile attachment...
shear area = 2r(screw head dia.)(t - head depth)2
= ir(0.117)(0.058 - 0.045) = 0.00478 in2
Shear force...(for 4130 steel)
f _ 7.2 - 1506.3 psi
0.00478
S.F. - 89100 -59.21506.3
IX-14
1,1.577
0.706
-0.558
-0.332
0.154
R 2 R
EE
8.0~ .33 2______
0.0 + 4~K 1~ F
Section E-E
0.0.14
0.002 V 0.030
0.840
Figure IX-2 Stress Analysis Example 2
(wing tip missile)
I X-15
This page was intentionally left blank
IX-16
NOTES
IX-17
NOTES
IX-18
NOTES
IX-19
NOTES
IX-20
X TRENDS
X
X Trends
mC maxf
I Flapped L.E.Wing
(L.E. & T.E.)
CL T.E.
f =flap
C L CL max
CL0f
CLa
CL= CLa f a
CL Wing
ZL ÷A astall4aZ
aZf a ZL aStall stall f
(wing alone)
Flaps Extended
Flaps Retracted
CL
CL
CD + CM -
ac
Figure X-1 Flap Characteristics
X-1
E fect of Vertical Location of C.G.on Pitching Moments
0.30 -0.25 -
0.20 -
0.15
0M10
b. 0.05C. 0.0E! -- CL
10.05 1.0 -- Low Wing
--0.3.5 -- -..
SMid Wing
--0.25-0.30 --High Wing
Figure X-2 Effect cf Vertical Location
of C .G. on Pitching Moments
Typical Longitudinal Stebility Breakdown0.25 -
0.20 -
0.15 -
0.10 -
0.05
-0.10
-0.15
-0.20
-0.25
0.0 0.25 0.50 0.75 1.0 1.25 1.50 2.00
CL
Figure X-3 Typical Component Longitudinal
Stability Breýakdown
X-2
Increasing Math number
CL •
0 a
C L ••--•-•
I i I1 2 3
cD
t I 11 2 3
SK = Induced drag
K factor\,
'%
1 ] f
1 2 3
Mact• Nu Irlb•;r
Fiqur• X-4 Ma•;ll Nur, Der Trends (Eff•(:t:.:;)
X-3
... . =• .n• nmmmm•nmnm •mmummn mn nn mnmmmmmmmummnummmmmnmn•
Reynolds Number
(Symmetrical
6 A ir fo il)
CL 3x1
a
9 x 10CL
CD
CL10 AR
4
0(
Figure X-5 Reynolds Number and Aspect Ratio
Trends (Effeccs)
X- 4
Double Wedge() t/c = 0.06
X = 0.0504 A = 4.0
Ater 27°A
.03
CDo A 45 ave 500
.02
.01 0 Data points
Conical Flow Theory
SKIN FRICTION
0 I 1 i I I I I 1
.8 1.0 1.2 1.4 1.6 1.8 2.0
MOD
Figure X-6 Effect of Wing Sweep on CD0
X--5
"" RCH :0.63 flRCN :0.93
.0.1e.g.. I.I
0.I. 0.1
SI.,. 0.7-
0.3, 0.t,
.j 00, .
0.2 0.P 0.
e i -o..
0.21-0. 3 -H "0 9
0.9.. 0-24
0,2.O 0.)9
0.-0.
z01 z046 42 0. t iil?*JF
"0. it -0.4,
00.16. 0046
"0. 08 / 0 6.2
0. 0 M AFTS T 0.04. ( AFT SWET
0:011 ý F'ORdANO &WEF a01S+A
6 , - a•"6 0. i I6 0- '+ -16 -0 -, -2 0 i , "o 12 1, i'
ZLPMA (DEC) 0LP-R 0CEO)
50i
AR = 3.34 A c/4_ = 45 0 S w=257i2
MAC = 5.62 X = 0.4 t/c = 5.5%
hi 0.1? 1.
b/2 = 7.58 C= 6.1 C, = 2.45Cr
Figure X -7 Aft and Forward Swept Wing-Fuselage Ef fects(Re f. IS)
X-6
MACH z0.63 MRCH :0.93
e.g. e.g.0.0, 0.11
0.4 0.4.
0.. 0.1.
.-41 ;4 ýi o 10 12 14 TOAILPHA (DEG) ALPHA (DEG)
-0. -0.
z z6 - 0. & FOatmaO swOP! W.j-O.
7 & FORtAtO lafry
C Co 0"C -0.5" sC -
MACH =O.t. MACH =0.93O.t0 0.29-
0.24 0.24-
0.24 0.24-
0.22 0.22-
"0.'o. 00 l o. -
o. i. a. o..
WO.14. •0. t4.
0. kO 0.I0.
0.64.; a lIft 3EPT 0.04. 0 AFT SOFT
.0 . r apO AN O S N EP/ 0 .0 2 -115f
-o o, i o, . .,
-0 .- . Ii t 'J.. 14 . .Na 1 0 1 .
LIFT COEAFICIENTCL LIFT COEFFICIETN,CL
Figure X-7 continued
X-7
MAOCH 10.63 flRCH -0.9)
0 .S
I....
-4-
3
Is RFI SWEPT 9fFSWP
&£ FORIRO SWEPTA SIWEP T
1- -' ;Z 1, 1" I'sI t 14 1
RLPAR IOEG) ALPHA (O)
0. is1 ZEL=.15(P
-I
K FOhijAlo SM1'
o.11*
0.12
SOil
h.S
0.07
0.1 0.6 0. 7 0.0 0.9 1.0
Figure X-7 concluded
X-8
-2.5 - Wing vortex -...' ti CL suction peaks i. . .0: 9.93 .400 i ; : -- , l I
-2.0 a 12.00 .516 - - - - -•o 14.07 .627 I .zi •
-1.o a 161 .7•2I
-. 18 . 18=• .845 _d / • 1o' 20.!!13 .962• • : i !i . ..
0-.4 .5 .6 .7 .8 .9 1.0y/s
(a) x/c-0.50
-2.0 Wing vortex ....a CL suction peaksLi
o 9.93 .400 AM
-1.5 a 12.00 .516!- -
Cp-1.5
o9 14.7.02
CPU -1.0 " 4 18.18 .845
o 201 .6
•.3 .4 .5 .6 .7 .8 .9 1.0
y/s
(c) x/c.0.62
Figure X-8 Wing Pressure Distribution inthe presence of a Coupled Chine
(ref. 19)
X-9
a CL------ 9.93 .400
-1.5 vor suctione -a--- 12.00 .516S ...... -• J . . • 14.07 .627-" 16.15 .725
S .. .... / \ • 18.18 M8 5
-1.0 )P• '..... - -- 20.13, .9 2/,,Straitsvortex // ' • - L!
C • suction peaks
0.•2 .3 .4 .5 .6 .7 .8 .9 1.0
y/S
(e) x/cxO.75
.0.75 0o1.625210,50 J,
0.4o0.40 lForebody x/c-O.30• !
<A-A
Figure X-8 concluded
X-10
1.6-
1.4
1.2
Cl 1.0
.8
.6
(Q- 14 x 106 M = 0.25
.4 - 6.1 x 106 M = 0.25
t/c = 14-
-5 0 5 10 15 20
-. 2j 1
Figure X-9 Reynolds Number Effect on cImax
(wing alone)
X-11
1.2
1.0
C 1 -8xlOG; M = 0.67
14.5x 10; M = 0.66
t/c = 17%
.2
0.01 .02 .03 .04 .05 .06
Cd
Figure X-10 Reynolds Number Effect on Drag
(wing alone)
X-12
UU-4
I 0\
U CZ
CZ)
il II II t
Figure X-11 Drag Rise Characteristics
(wing alone)
X-13
-2.0-M C 1 Cd
o 0.5 .722 .0082
o 0.6 .746 .0114
6 0.7 .724 .0137
t/c 17%
Shock Effect
-1.0-
Cp
-. 05-
0.5-
1.0
0 20 40 60 80 100
Percent Chord
Figure X-1 2 Mach Effect on Airfoil Pressure
Distribution
X-!4
NOTES
X-15
NOTES
X-16
NOTES
X-17
NOTES
X-18
XI INTERNAL STRAIN GAGE BALANCES
xI
XI Internal Strain Gage Balances
(Ref. 36)
List of Symbols
A Cross-sectional area
E Voltage
F Gage Factor
iFA} Appliea load to balance matrix
IF 0 Output (mv) force matrix[K..]-' Inverted balance calibration matrix
L Length
I Rolling moment
m Pitching moment
mv Milli-volts
n Yawing moment
IDR Resistance
I SC) Sensitivity constant matrixwt Weight
X Axial force
Y Side force
Z Normal force
E Axial (local) strain
A Incremental
11t Poisson's ratio
p Resistivity of the gage material
Subscripts
A - Applied load AF - Axial force NF - Normal force
O - Output (mv) PM - Pitching moment SF - Side Force
RM - Rolling moment YM - Yawing moment
XI-I
Strain Gages
In strain measurement. the simplest device to use is the resistance type strain gage.
Its construction and operation are simple, but it is so precise that strains on the order
of 0.1% may be measured. The gage is about the size of a postage stamp but only slightly
heavier. It consists of a metallic wire or strip of foil whose electrical resistance
varies linearly with strain. The gage is securely bonded to the member to be strained so
that any strain in the member due to a load ie, transmitted to the wire. There areliterally hundreds of types of strain gages available commercially; each gage having been
developed in response to a demand for a gage to meet a specific condition.
The most fundamental part of the gage is the wire itself. The resistance increase of awire, when it is stretched, is due to an increase in the length and a decrease in its
cross section, than to actual change in specific resistance. Typically, an electrical
conductor (wire) is bonded to the specimen (balance) with an insulating cement under a
no-load condition. A load is then applied, which produces a deformation in both the
specimen and resistance element (wire). This deformation is indicated through ameasurement of the change in resistance of the element (wires).
Three common types of resistance strain gages used in internal balances are wire, foil,
and semi-conductor. The bonded wire is most commonly used with the wire diameter varying
between 0.005 and 0.001 inches. The foil gage usually employs a foil less than 0.001
inches thick. The semi-conductor gage employs a silicon base material that is strain
sensitive and has the advantage of a very large gage factor (F-100). (The gage factorrelates the electrical resistance to the physical properties of the gage.) The material
is usually produced in brittle wafers having a thickness of 0.01 inches.
Wire and foil gages may be manufactured in various ways, but the important point is that
the resistance element (wire/foil) be securely bonded to its mounting. Most wire strain
gages employ either a nitrocellulose cement or a phenolic resin for the bonding agent with
a thin paper backing to maintain the wire's configuration. Such gages may be used up to
300°F. A Bakelite mounting is usually employed for temperatures up to 5000F. Foil gages
are manufactured by an etching process and use base material of paper, Bakelite, and epoxy
film. Epoxy cement is also used on wire and foil gages.
XI-2
Temperature Effects
The major soure of strain gage error is the fact that the resistance of most wires
changes with temperature. This variation is not only a function of the change intemperature but, may also be a function of the number of heating cycles to which the gageshas been subjected and of the time elapsed between cycles. Temperature compensation mayeasily be accomplished by installing a second strain gage, often known as a dummy gage, onan unstrained piece of the same metal that the active gage is bonded to. There are twochanges in gage resistance which are caused by temperature changes. The first effect is
the temperature coefficient of resistivity which is a random effect that may b, consideredas independent of the gage factor. The second effect is the difference in linear
expansion between the gage and the member (material) on which it is mounted. The lattereffect is one in which the error in the strain gage output is significant. Semi-conductorgages offer the advantage that they have a lower expansion coefficient then either wire orfoil gages. It's best to mak , sure the strain gages are temperature compensated.
Deformation Theory and Calculation
Basic relations for the resistance strain gage are
R = L)A
L = length
A = cross-sectional area
p = resistivity of the materialDifferentiating (1)
dR + dL dA (2)
R p L A
The area may be related to the square of some transverse dimension, such as the diameter
(D) of the resistance wire.
dA _ dD (3)
A D
XI-3
The unit axial strain E is defined as
_ dL (4)
LPoisson's ratio is defined as
= dD/D (5)
dL/L
Substituting equations (3), (4) and (5) into (2)...
dR = e(1 + 2gt) + dP (6)
R p
The gage factor 'F' is defined as
F dR/R (7)
Substituting (7) into (6)
F =1 + 2g. + (!)(p) (8)
Rearranging (7)
Local Strain
= (AR (9)
FR
The value of the gage factor 'F' and the resistance (R) are usually specified by the
manufacturer so that the user need only measure the change of resistance (AR) in order to
determine the local strain due to a load.
Measurement of AR/R
Consider the basic Wheatstone bridge in figure XI-1. The voltage at the detector is
given by
XI-4
R !PED E 1 2 (10)
IR I +R 4 R 2 R 3
If the bridge is balanced then E0 = 0.0 volts.
Let the strain gage represent R in the circuit in figure XI-2 and a voltage readout is
used such that the br'idge operates as a voltage sensitive circuit. We assume that the
bridge is balanced at zero strain conditions and that a strain , e, on the gage results in
the change in the resistance ARI and a change of voltage AE on the bridge. R will beused to represent the resistance of the page at zero strain conditions.
The voltage due to strain is...
A 2 R +AR RD _ I 1 2 (11)
E R +AR +R R +R1 1 4 2 3
Solving for the resistance change...
AR, (R 4/R I ) [AE D/E + R 2/(R R2+R3)] 1(3- - 1(13)
R1 I - AED/E - R2/(R2+R3
Balance Calibration (interaction) Matrix
(ref. 8 and 37)
No internal balance is able to measure the pure loads that it was intended to measure.
This is due to errors within the balance. There are generally two types of errors. The
first type of 'balance' errors are the linear (first degree) errors. This type of errors
are primarily due to construction errors, improperly positioned gages, variation in gage
factors, and electrical circuits. The second type of error is the non-linear (2nd degree)
error. This error is primarily attributed to elastic deformation (deflections) of variousbalance parts. Both types of errors create interaction of forces and moments.
To account for these interactions (errors) in the balance, the balance is calibrated.
The purpose of this calibration is to acquire a set of interaction equations that can be
used to determine the loads applied by the model through the balance output (voltage)
signals. This type of calibration will not (generally) be accomplished by the testing
XI-5
engineer. All the testing engineer will receive will be a sheet(s) of paper with the
linear and non-linear interaction coefficients on it (figure XI-3). For a six component
(3 forces and 3 moments) balance, the linear coefficients will be uscd as a 6x6 matrix
with near unity (1) on the diagonal. The non-linear coefficients will be used as a 6x21.
For a six component balance, there is a total of 27 interaction terms. Generally
speaking, the 6x6 matrix will need to be inverted to be used in the data acquisition
computer. This can be seen below in the theory of how the computer acquires force/moment
data from a balance.
(F 0 [Kij]IF A
(F A = [K.i] '(SC) (Fo} (14)
Example:
Original Balance Calibration Matrix
(non-dimensional)
Fz] 1.0000 .0336 -. 1764 -. 0549 .0059-.00041 ZAX j -.0169 1.0000 0.uftO .0131 .0000 .0000 XA
m -.0059 -.0022 1.0000 -.0032 .0000 .00021
1 -. 0004 -. 0006 .0078 1.0000 .1012-.0044'
0000 .0002 .0000 -. 0412 1.0000-.0064 n
.0012 .0049 .0294 -. 15). -. 0.384;i.u . I A
Xj9A]
XI-6
Inverted Original Balance Cal ibration Matrix
ZA 1.0000 -. 0332 -. 176 .0555 -. 01 15 .00051 ZX A .0169 .9994 .0031 -. 0121 .001 1 -.0000 Xm .006 .002 1.001 .0034 -. 0004 -.0002 m
A -W .0004 .0006 -. 0078 .9964 -. 1007 .0037 1
n A .0000 - .0002 -.0005 .0421 .9959 .0065 n IY -.0013 -. 0048 -.0309 .1564 .0225 1.0008 YI
The interactions of the forces and moments can be seen in the inverted matrix. For anexample, thete is a -17.6% interaction on the normal force due to the the pitching momentand a 3.7% interaction on the rolling moment due to the side force.
The testing engineer needs to know if the balance coefficients are indeed
loadnon-dimensional or dimensional (ie... (vTl-)) If the interaction matrix is
non-dimensional, then the testing engineer will have to acquire a sensitivity constant
I oadVol-) for each force and moment. The acquisition of the sensitivity constants is
acquired during the check loading of the balance (see page XI-10).
Calii.aticn Podv
In order to load a balance accurately, it is first necessary to construct a fixture.This fixture is geiteraily called a calibration body (cal. body). The --al. body On-,liatesthe model to be tested which will allow loads to be accurately transmitted to the balancein the same manner as the model will during testiiig. It should be attached in the samemanner as the model is attached to the balance (generally a single pin) and should havereference surfaces which will remain fixed with respect to the balalnce reference center
(usually the balance electrical moment center)The cal. body is an accurately machined 'rectangle box' with precisely located V-shaped
grooves in which the loads are applied during calibration or check loadings. It isimportant that both the cal. body and the model be aligned with the balance in exacily thesame majiner. Also, the balance should fit tightly (no relative motion between the two
items) in the cal. body and in the modei. All fixtures (moment arms, weight pans etc...
XI-7
should be manufactured as light as possible a.rnd as stiff as possible.
Loads applied to the balance (via the cal. body) components are considered absolute
loads. Therefore, the weight of the cal. body and fixtures and even parts of the balance
must be considered as applied loads. However, the weight of the cal. body and fixtures
are taken out in the 'zero' reading. The zero reading is the reference voltage to be
subtracted out from the load reading to acquire a 'true' reading.
Check Loading
The purpose of check loading a balance serves several purposes.
1) Acquire sensitivity constants (if needed)
2) Proof (check) load the balance
3) Check gaging and wiring
4) Determine hysteresis and repeatability
5) Determine component sensitivity
6) Determine deflections when a load is applied
7) Determine accuracy
It is important .o check load tile balance over the entire anticipated load and
center-of-pressure range and that sufficient points are taken to assure accurate and
repeatable loadings. The minimum check load to hang would be 10e7- of tile balance limit.
Check Loading Procedure
This procedure will assume that a dead weight (ie... disks) will be used to acquire the
loadings required. However, a load is a load whether it is generated by a disk or
hydraulic actuator or sonie other source. Just make sure it's a point load and not a
distributed load on the calibration body.
Hang the weights on the calibration body at CONSTANT INCREMENTS, as an example,
A50 lbf or A100 lbf or whatever weight increment you choos-, to use. Be sure to maintain
that increment at all times. Don't forget to take a 'zero' voltage reading prior to
hanging weights. A zero reading includes the weighis of the cal. body, ",eight pan or any
other weight that's considered a *zero' weight. After loading a weight on the weight pan,
stop the weight pan from swaying. Place an inclinometer on the cal. body and raise or
lower the cal. body (via the sting) until it is level (or < 4± 3 mins of ,, degree). This
XI-8
takes out the weight component due to the deflection of the sting. Also, sting deflection
measurements can be taken at this time. After the weight has been added and the swaying
has been stopped, have the control room take a data point and record the output (normally
in milli-volts [mv], and then add the next incremental weight and repeat the above
procedure. Do this until the maximum load is achieved. Once the maximum load is applied
and voltage taken, remove the weights at the same increment as when you added ihe weights
and take a data point at each weight going down. Do this until all the weights haýe been
removed and then take another zero. This up and down procedure of check loading will show
accuracy, iepeatabiliy and a zero shift of the balance. Do this for all force and moment
check loadings.
Dead-Weight Loading System
When the dead-weight system is employed, the weights are generally in the forni of cast
iron disks (accurate to 0.1%) hanging on a weight pan supported by the cal. body or moment
arm fixture. For a highly accurate check loading, one can weigh the cast iron di,;ks prior
to check loading.
When hanging weights for the normal force and side force, al a minimum, hang the weight,
at the balance electrical moment center and use this check loading to determine the normal
and side forces sensitivity constant. Taking the sensitivity constant for the forces at
the balance electrical moment center, allows the contribution (interactions) of the
principle moments (pitching and yawing) to become negligible.
Hanging the weights for the pitching and yawing moments, at a minimum, at one moment arm
ahead and behind the electrical moment center but at different moment arms. This will
result in a positive and a negative moment. When check loading for the rolling moment,
place the rolling moment fixture at the balance electrical moment center and at a minimum,
apply loads at two different moment arn on the rolling moment fixture (expect to see a
normal force reading).
Axial Force Check Loading
Axial force check loading is generally accomplished with a pulley rig. A pulley rig has
two wires that are connected to the cal. body that run down the side of the sling and over
the pulleys down to a 'T' weight pan where the weights are placed. The 'T' weight pan
must be absolutely level. Any angle in the 'T' weight pan will produce undesirable
XI-9
interactions and an erroneous axial force will be acquired. Also, it might be desirable
to check load m the negative axial force direction. If this is desirable, set the pulleyrig in front of the cal. body and hang weights. Be sure to check the balance limits to
see if a negative axial force limit exists. Normally, a negative axial force calibration
is not dor:e.
Sensitivity Constants
Sensitivity constants are required when the balance calibration matrices arenon-dimensional. They are acqiired during check loading and are nothing but slopes of a
line [fAload and are required for each force and moment. The sensitivity matrix is a
diagonal matrix.
T acquire sensitivity constants...
1) Put balance calibration matrix in the data acquisition computer.2) Initially set sensitivity constants to unity (1).
3) Accomplish the check loading in all planes and at multiple positions within that
plane.4) When the check loadings are complete, obtain the 'Raw Balance Data' (figure XI-4). Be
sure and account for the zero reading. Usually it needs to be subtracted out of the
output readings. There should be 'n' forces and moments usually titled X, Y, Z, 1,m, n. Where X = axial force, Y = side force, Z = normal force, I = rolling moment,
m = pitching moment, and n = yawing moment.
5) At this point use a linear regression program to acquire the slope of the line with
the x-axis as the voltage and the y-axis as the load. The slope of this line is the
sensitivity constant for that force or moment. Do this for each position and foreach force and moment. The force and moment sensitivity constants should not varymuch from one another when taken at different positions on that plane. For the force
sensitivity constants, use the sensitivity constants at the balance electrical moment
center. This eliminates, theoretically, the influence of the moment of that plane.6) If a linear regression program is not available at the test site, take the first
weight reading (mv) and subtract it from second weight reading (mv). Then take the
second weight reading and subtract it from the third weight reading and so on. The
XI-10
Amilli-volts (Amv) should be close to one another since the weight increment (Awt) is
held constant.
7) Obtain an average Amy.
8) Divide the Awt by the average Amv. This is the sensitivity Zonstant for that position
on the plane.
9) Repeat steps 6 3 for each position on the plane.
10) Steps 6-8 are not as accurate as using a linear regression program on the computer.
11) For the moments, add your sensitivities together and divide by the number of
sensitivities to obtain and average. This average sensitivity constant for the
moment is the one given to the software/tunnel personnel.
12) After the sensitivity constants have been acquired, re-hang selected loads on the
cal. body once again to verify the validity of the constants. The applied load to
the output load should be within 0.5% of one another.
Example Sensitivity Constant Matrix
Z 2.3184 0 0 0 0 0 Z
X 0 - .7844 0 0 0 0 X
mA 0 0 -. 1867 0 0 0 mi
1 A 0 0 0 .1324 0 0 1
nA 0 0 0 0 -. 07780 n
Y 0 0 0 0 0 -1.188 Y IA 0J
Obtaining Force and Moments from Raw Balance Data
1) Take the balance output (mv) and subtract out the zero reading (mv).
2) Take that increment (Amv) and multiply it by that position on the balance calibration
coefficient (ie... if you are checking the normal force, on the nxn matrix (which is
usually inverted) go to the Z,Z position), and multiply it by the its sensitivity
constant.However, don't forget the interaction terms. They'll need to be added or
subtracted to the force or moment you are looking at for an accurate reading.
XI-II
F or M = Amv(balance calibration)(sensitivity constant) + interaction terms
F = force; M = moment
Don't forget that each interaction term also has a sensitivity factor associated with
it. Keep a close eye on the units. Don't mix apples and oranges.
Balance Calibration Equation Example (ref. 8)
The example below uses a 6 component balance calibration matrix. The calibration matrix
below is considered non-dimensional. If the calibration matrix was dimensional, then
consider the sensitivity constants on the diagonal of the sensitivity constant matrix as
unity (1).
From equation 14
(F0 )(SC) = [Kiji]FA}
(F ) = [K.i.]'SCIHFo}
Example Original Ba I ance Calibration Matrix
(non-dimensional)
Z 1.0000 .0336 -. 1764 -. 0549 .0059 -. 0004 Zo A
X -.0169 1.0000 .0000 .0131 .0000 .0000 X0 A
m -. 0059 -. 0022 1.0000 -. 0032 .0000 .0002 mA
1 -.0004 -. 0006 .0078 1.0000 .1012 -. 0044 10 A
n .0000 .0002 .0000 -. 0412 t.0000 -. 0064 nA
Y .0012 .0049 .0294 - .1555 -. 0384 1.0000 YA
XI-12
Example Inverted Balance Calibration Matrix
Z 1.0000 -. 0332 -. 176 .0555 -. 0115 .0005 ZA0
X .0169 .9994 .0031 -. 0121 .001 1 -.0000 XA 0
m .006 .02 1.001 .0034 -. 0004 -.0002 m
1A .0.00 W0006 -. 0078 .9964 -. 1007 .0037 !
n .0000 -. 0002 -. 0005 .0421 .9959 .0065 n
Y -.0013 -. 0048 -. 0309 .1564 .0225 1.0008 YA o
Example Sensitivity Constant Matrix
Z 2.3184 0 0 0 0 0 ZA 0
X 0 .7844 0 0 0 0 XA 0
m 0 0 -. 1867 0 0 0 mA_ 0
1 0 0 0 .1324 0 0 1A 0
n 0 0 0 0 -.0778 0 nA o
YA 0 0 0 0 0 -1.188 Y
A typical load (normal force) equation can be seen below.
ZA = ZoK K 1 SCNF + XoK ISCAF + YK 13SCSF + 1oK 14SCLM + moK 15 SC M +"'
... + n K SC (15)0 15 YM
Using the inverted and sensitive constant matrix above, the total normal force (Z) and
pitching moment (m) equations are listed below.
Z = Z(1)(2.3184) + X(-.0332)(-.7844) + m(-.176)(-.1867) +
... + 1(.0555)(.1324) + n(-.01 15)(-.0778) + Y(.0005)(-1.188)
m = Z(.006)(2.3184) + X(.002)(-.7844) + m(l.001)(-.1867) + ...
... + 1(.0034)(.1324) + n(-.0004)(-.0778) + Y(-.002)(-1.188)
The rest of the forces and moments equations are derived in the same manner.
XI-13
Balance Placement in the Model
The physical placement of the balance in a wind tunnel model can vary. Usually, the
balance electrical moment center is placed at the center of pressure of the configuration
at the maximum aerodynamic load. Since the center of pressure position varies with
angle-of-attack, make re the aerodynamic load is not outside the load of the balance.
XI- 14
B
AE /l Ci
S A Cx,
' D
St
Figure XI-1 Basic Wheatstone Bridge
RbB
E b E A CT
Figure XI-2 Unbalanced Wheatstone Bridge
XI-15
N ~ ~ ~ ~ ~ ~ % -Il Nw N WJI 1I
Q ini
0z X-tx ný Lm- ---- m
('D .: 'F). "
-4 ..-"4 "4"4 .,34A q
-' 220 b %A 4 r "
N upNa zrb 4ý t 4w V
a)
I• I I I # II I
IN. I i 0 I S I I//
CD 0 000 0 0 0 0 0001a , 3 Soo -.g -,4 WN 4- 4 M I
WV -
tn Mi w I I 2 I
-) e" 'A'")* ) 04 "• n1
CD 0 4.d Ci31 '-"kw% %MZ - jo' s CA 0 4Z.l x m O
(DW0 V ~ 0 ' CA -4.VCt 0 V C go
I- m I. I MI I If I IlIl I
H 0 000 0 0 0 0 000W
X1-16
Normal Force (Z) Calibration(Balance Electrical Moment Center)
Wt Inc. fData Pt X Y Z I m n0 1 -1329.8 960.6 0.0 -1112.7 705.4 536.3
100 2 -1308.8 962.6 191.1 -1110.7 710.4 536.3200 3 -1289.8 964.6 384.2 -1108.7 714.4 537.3300 4 -1268.8 965.6 575.3 -1105.7 718.4 536.3400 5 -1250.8 966.6 766.5 -1104.7 722.4 539.3500 6 -1230.7 966.6 958.6 -1103.7 726.4 536.3600 7 -1208.0 966.8 1150.9 -1101.9 731.6 534.4700 8 -1186.7 967.6 1342.8 -1100.7 735.4 534.3600 9 -1211.0 967.8 1149.9 -1102.9 730.6 536.4500 10 -1227.7 964.6 957.6 -1103.7 725.4 536.3400 11 -1250.8 966.6 765.5 -1106.7 721.4 536.3300 12 -1268.8 965.6 573.3 -1107.7 727.4 536.3200 13 -1287.8 965.6 381.2 -1110.7 713.4 538.3100 14 -1306.8 964.6 189.1 -1112.7 709.4 540.30 15 -1326.8 962.6 0.0 -11-2 2,1 4, 541.3
Pitching Moment (m) Calibration(2 inch moment arm)
WtInc. DataPt X Y Z I m n0 16 -1324.8 962.6 -2.0 -1113.7 688.4 543.3
100 17 -1310.8 963.6 182.1 -1110.7 567.3 544.3200 18 -1298.8 964.6 369.2 -1108.7 446.3 545.3300 19 -1286.8 964.6 555.3 -1105.7 326.2 545.3400 20 -1271.8 965.6 742.4 -1102.7 206.1 545.3500 21 -1260.8 965.6 928.6 -1100.7 85.1 546.3600 22 -1248.7 966.6 1114.7 -1099.7 -36.0 545.3700 23 -1233.7 966.6 1301.8 -1097.7 -157.1 545.3600 24 -1247.7 965.6 1114.7 -1098.7 -35.0 547.3500 25 -1256.8 966.6 928.6 -1100.7 86.1 545.3400 26 -1273.8 965.6 742.4 -1102.7 208.1 545.3300 27 -1287.0 965.8 556.4 -1103.9 329.3 545.4200 28 -1298.0 964.8 370.3 -1106.9 450.4 545.4100 29 -1309.8 964.6 183.1 -1108.7 571.3 547.30 30 1-1322.8 963.6 0.0 -1110.7 692.4 547.3
All readings are in milli-volts (my)Wt Inc. is in ibf
Figure XI-4 Raw Balance Data
X 1-17
This page was intentionally left blank
XI-18
NOTES
XI-19
NOTES
XI-20
NOTES
XI-21
NOTES
XI-22
LIST OF REFERENCES
R
R List of References
1) Equations, Tables, and Charts for Compressible Flow, NACA 1135, 1953.
2) Ametec Engineering, Inc., 11820 Nor'hup Way Suite 200, Bellevue, Wa 98005,
(206)827-3304.
3) Nicolai, L., Design of Aircraft Vehicles, United States Air Force Academy,
Colorado Springs, Co, July 1972.
4) Lesko, James S., Reduction of Forces and Moments Taken on Internal Balances and
the Effect of Axes Orientation, Wright Air Development Center, Wright-Patterson
AFB, Ohio, Technical Note 9, April 1952.
5) Stewert, V.R.,Low Speed Wind Tunnel Test of Ground Proximity and Deck EdgeEffects on a Lift Cruise Fan V/STOL Configuration, NASA-CR-152247, May 1979.
6) Henderson, C., Clark, J., Walters, M., V/STOL Aerodynamics, Stability &
Control Manual, Naval Air Development Center, Warminster, Pa., NADC-80017-60,
January 1983.
7) Hoerner, Sighard, F., Fluid Dynamic Drag, Published by Author, 1965.
8) Rae, William, H., and Pope, Alan, Low Speed Wind Tunnel Testing; WileyInterscience Publication, New York, New York, 1984.
9) Lan, Edward, C. and Roskam, J., Airplane Aerodynamics and Performance; Roskam
A iaition and Engineering Corporation, Ottawa Kansas, 1981.
10) Braslow, A.L., Knox, E.C., Simplified Method Determination of Critical lleight of
Distributed Roughness Particles for Boundary Layer Transition at Mach Number 0
to 5, NACA-TN-4363, Sept 1958.
11) Taylor, Robert, Boundary Layer Transition Strips in Atmospheric Tunnels, Memo,
March 7, 1967, NASA-Langley Research Center.
12) Hoak, P.E. et al., USAF Stability and Control Datcom, Air Force Flight Dynamics
Laboratory, Wright-Patterson Air Force Base, Ohio, 1978.
13) Jobe, Charles E., Prediction of Aerodynamic Drag, AFWAL-TM-84-203, Flight
Dynamics Laboratory, Wright -Patterson AFB, Ohio, July 1984.
R-1
List of References continued
14) Smith, C.W., Aerospace Handbook, General Dynamics, Convar Aerospace Division,
Fort Worth, Tx, Rev B, March 1976.
15) Alexander, Michael G., Personal class notes, AEE 502 Aircraft Performance, 1988.
16) Roskam, Jan, Airplane Flight Dynamics & Automatic Flight Controls, Roskam
Aviation & Engineering Corporation, Ottawa, Kansas, 1982.
17) Perkins & Hage, Airplane Performance Stability and Control, John Wiley & Sons
Inc., Jan 1967.
18) Simms, Kenneth L., An Aerodynamic Investigation of a Forward Swept Wing, Thesis,
AFIT/GAE/AA/77D-14, Air Force Institute of Technology, Wright-Patterson AFB,
Ohio.
19) Erickson, G.E., Rogers, L.W., Schreiner, J.A., Lee, D.G., "Further Studies of
Subsonic and Transonic Vortex Flow Aerodynamics of a Close-Coupled Forebody
Slender Wing Fighter", AIAA paper AIAA-88-4369, Aug 1988.
20) Alexander, Michael G., Cavity/Separation Characteristics of an Axisymmetric Air
-to-Air Missile from Mach 2.0 to Mach 5.0, Wright Research and Development
Center, Wright-Patterson AFB, Ohio, WRDC-TR-89-3041, April 1989.
21) Gieck, K., Engineering Formulas 5th Edition, McGraw- Hill Inc., 1986.
22) Bartel, H.W., McAvoy, J.M., Cavity Oscillation in Cruise Missile Carrier
Aircraft, AFWAL-TR-81-3036, June 1981.
23) Whitford, Ray, Design for Air Combat, Janes Publishing Company Inc., New York,
New York, 1987.
24) Abbot and Doenhoff, Theory of Wing Sections, Dover Publications, Inc., NY, NY,
1958.
25) Braslow, A.L., Hicks, R.M., and Harris, R.V., Use of Grit Type Boundary-Layer
-Transition Trips on Wind Tunnel Models, NASA-TN-D3579, Sept 1966.
26) Crowder, J.P., Fluorescent Mini-Tufts for Non-Intrusive Flow Visualization,
Douglas Aircraft Company, McDonnell Douglas Corporation, MDC J7374, 1980.
R-2
List of References concluded
27) Shapiro, Ascher H., The Dynamics and Thermodynamics of Compressible Fluid Flow,volume 1, The Ronald Press Company, 1953.
28) Tighe, Thomas, Personal notes, 1989.
29) Anderson, J.D., Introduction to Flight, McGraw-Hill Book Company, 1978
30) Strength Analysis 4 Percent Scale Model Aero/RCS Wind Tunnel Model", McDonnell
Aircraft Company, Report # SA-151, 4 Aug 1981.
31) Peery, David J., Aircraft Structures, McGraw Hill Book Co, 1950.
32) MIL-Handbook 5
33) Cerrnica, John N.,Strengths of Materials, Holt, Rinehart, & Winston. 1977
34) Nash, William A., Strength of Materials Schaums's Outline Series inEngineering, McGraw-Hill Book Company, 1972
35) Unknown; "Definition & Measurement of Net Lift and Drag", 707/CFM56 AerodynamicStaff, June 1978
36) Holman, J.P., Experimental Methods for Engineers, Second Edition, McGraw-Hill
Book Company, 1966, 1971.37) Volluz, R.J., Handbook of Supersonic Aerodyanmics, Section 20, Wind Tunnel
Instrumentation and Operation, John Hopkins University Applied PhysicsLaboratory, Silver Springs, Maryland, NAVORD Report 1488 (Vol. 6), DTIC
AD261682, January 1962.
38) Thelander, J.A.; Aircraft Motion Analysis, FDL-TDR-64-70, March 1965, DTICAD617354, Air Force Flight Dynamics Laboratory, Wright-Patterson AFB, Ohio.
R-3
APPENDIX A
A
APPENDIX A
20.0 -
15.0
10.0
7.005.0 A I _:
DYNAMICPRESSURE (q) 4.0 -
(psi)
3.0
2.0 / / /
MCNU/
i.u -i Dcse
(a dA se
.- i!
-~ -- r1-
Reynolds number per unit lengthx (RA ft ) 14.7 pot
N X, u4 1 j
I N
4-
EIN
-TIT
N NN U " 0 ~ ~ INo
-t# . .. .. ..
Fiur A- Renod Nube Determ ff I t 1LInaVtio I rf.
I il .. 1 tA-21 1
Speed of Absolute Kinematic
Altitude Temperature Pressure Density Sound Viscosity Viscosityft F R psf psi lbm/ft^3 ro/ro0 ft/sec mi/hr Ibm/ft-s ftf2/sec
0 58.7 518.4 2116.0 14.700 0.002378 1.0000 1107.64 760.9 3.725x10^-7 1.566x1O^-42000 51.5 511.2 1968.0 13.672 0.002242 0.9428 1100.07 755.7 3.685 1.6444000 44.4 504.1 1828.0 12.699 0.002112 0.8881 1092.36 750.4 3.644 1.7256000 37.3 497.0 1696.0 11.782 0.001988 0.8358 1084.64 745.1 3.602 1.812
8000 30.2 489.9 1572.0 10.921 0.001869 0.7859 1076.78 739.7 3.561x10"-7 1.905x10-410000 23.0 482.7 1455.0 10.108 0.001756 0.7384 1068.92 734.3 3.519 2.00412000 15.9 475.6 1346.0 9.351 0.001648 0.6931 1060.91 728.8 3.476 2.10914000 8.8 468.5 1243.0 8.635 0.001545 0.6499 1053.05 723.4 3.434 2.223
16000 1.6 461.3 1146.0 7.961 0.001448 0.6088 1046.21 718.7 3.391x10^-7 2.342x10"-418000 -5.5 454.2 1056.0 7.336 0.001355 0.5698 1036.75 712.2 3.348 2.47120000 -12.6 447.1 972.1 6.753 0.001267 0.5327 1028.60 706.6 3.305 2.60822000 -19.8 439.9 893.3 6.206 0.001183 0.4974 1020.59 701.1 3.261 2.756
24000 -26.9 432.8 819.8 5.695 0.001103 0.4640 1012.15 695.3 3,217x10"-7 2.916x10-426000 -34.0 425.7 751.2 5.219 0.001028 0.4323 1003.71 689.5 3.173 3.08628000 -41.2 418.5 687.4 4.775 0.000957 0.4023 995.26 683.7 3.128 3.26830000 48.3 411.4 628.0 4.363 0.000889 0.3740 986.82 677.9 3.083 3.468
32000 -55.4 404.3 572.9 3.980 0.000826 0.3472 978.23 672.0 3.038x100-7 3.678x10^-434000 -62.5 397.2 521.7 3.624 0.000765 0.3218 969.50 666.0 2.992 3.91136000 -67.3 392.4 474.4 3.296 0.000705 0.2963 963.67 662.0 2.962 420438000 -67.3 392.4 431.1 2.995 0.000640 0.2692 963.67 662.0 2.962 4.625
40000 -67.3 392.4 391.9 2.723 0.000582 0.2448 963.67 662.0 2.962x10^-7 5.089x10"-442000 -67.3 392.4 356.2 2.475 0.000529 0.2225 963.67 662.0 2.962 5.59944000 -67.3 392.4 323.7 2.249 0.000480 0.2021 963.67 662.0 2.962 6.16146000 -67.3 392.4 294.2 2.044 0.000437 0.1838 963.67 662.0 2.962 6.778
48000 -67.3 392.4 267.4 1.858 0.000397 0.1670 963.67 6620 2.962x10°-7 7.459x10"-450000 -67.3 392.4 243.1 1.689 0.000361 0.1518 963.67 662.0 2.962 8.20652000 -67.3 392.4 220.9 1.535 0.000328 0.1379 963.67 662.0 ,2.952 9.02854000 -67.3 392.4 200.8 1.395 0.000298 0.1254 963.67 662.0 2.962 9.933
56000 -67.3 392.4 182.5 1.268 0.000271 0.1140 963.67 662.0 2.962x10-7 10.93x10-458000 -67.3 392.4 165.9 1.153 0.000246 0.1036 963.67 662.0 2.962 12.0260000 -67.3 392.4 150.8 1.048 0.000224 0.09415 963.67 6620 2.962 13.2362000 -67.3 392.4 137.1 0.952 0.000203 0.08557 963.67 662.0 2.962 14.56
64000 -67.3 392.4 124.6 0.866 0.000185 0.07777 963.67 662.0 2.962x100-7 16.02x10'-465000 -67.3 392.4 118.7 0.825 0.000176 0.07414 963.67 662.0 2.962 16,870000 -67.3 392.4 93.53 0650 0.000139 0.05839 963,67 662.0 2.962 21.3375000 -67.3 392.4 73.66 0.512 0.000109 004599 93.67 6620 2.962 27.09
Figure A-3 Standard Atmosphere
A-3
CQUATIONS, TABLES•, AND CHIARTSP ORl COMPRESIBLE FLWW
SUBSONIC FLOW
PI A... .P .
ow G• 0.50O 0."30)0 0 9524 O0S 011475 1 •33" 0 &.345
. .02 ln 294m .gin + 021 1 .s52 .m .81,6 0187 1,642 1574 1 3034
.0 M.7 n 99 2 1 13 .32 5 391 8• 3 ., 7 21469 &40 ]N2W 564%3
.0 .181) om o M .19 '.lll 46.15 0 I3816 .5 .W40 .8067 9446 97 2 1 67 13 7703 57501
09 m wol 9967 .1709 - 1 1, 51114 .03476 o1 .'142 - I9 , . -u : -•. .• 47 -, 1;~ .:
,WA WIG+ '59 :3' 3+ ', .7" 1 24W,: .,+ +. ..
oh 90+ Low 9867 9061 .4460 it 42?+ý 2M 75 58+ 910 :1 175 23 1W
0. .7m6 .Wiggll 2711 VO WI 094 .0 71•'0
14 '211 .94,5 9714 •I4 747 - !822 .• 62 .7 93109 .79m &.25 6%2 48T •l
I S I 91 M• 911 /'61 . '1"M .7 71353 4 7•+ .... i" .64 .5 1 .7136 .9243 21., 1,3 1 142O 67402
345 W144] ow" 91163 7'+~ 3••' V: l 3 :&1.+95 8 1 7 7 0• '71 '.':5 t1 IL .56• .W74,7 9" 09473 9740 ,9o 1'5A .... 74 2 .. . 6 6 9 74'. . .. 277 1. .IM., . I42
.. , MO, 11113 .9130 lJ# 957 " M 3,4614 14tB 8 PI" 62 • 2 46 11
,a VAM7 :V0 Wit,' A712 .17'40ý+7 6lI • 1+ .10661 .11"9 1146 3311 I 0,071 1 1.
41 9 7 176 o1'w 9 ,1 9 W I2 1 7• i 12171ý3 13. 72 7 1 9 2• + 3 T 3 0 l • + . 4 Z +9-.51 O M . 9 M.4 I • :98119 l7464 31 234 2U72•39 .09 7271 .7966l ,14.9 41 0131 -,Z392/
47 %-. •9783 9195 977.87 .1294 2I 0829 2M4 77• 1 .14. •• 19 1 1 .1 WM 25•21 1 0 I .9412
TA2L 7IW-14UPERSONC7FL633 91LIA M2 3569 2 ý,WA -4A 7 3/ 06 475 1
3r --.0 --1 1 1 1 1, -- -- -. 161. 7-(15-1 3 7 C 2 4 1
' I 01,7K9WA III 9W1 0&56 :41 02 3 6W21 7W 619 , 05, MZ.0
7 40 96129 984 P• 9f, 1, 2 & 01 " 81 M 1 41 1I 709 $ 62 93 60 & 99 3 3(M -'9 6, , 5 .. 5]05 , 613 029,4 1 - bl.W
:1 g4 s 9,, W2 I Z1131 Mý7 6"9 i 3V 1 9 1 11 10 &I 78 2, 19 3 +2 • 4 4O
I2 oil .9iT 0 9 94 4•lt •• 177 +3 1.11 72 21",Y 0 1 . I M .4 333
I13 4 •1 .ni • 9 4 ;069. 1 . 9 0 .3 U2 l 9i W Q 7 8 1, 71 + .• I +2 1 3Z 542 ] 31 2 IX ofi m 4 •P
0, = .:9 SO+ 371 ,= 8-,, I,.. ... ,.+ +, , ,6 VI , W .• 5' .9 .. . . .. . 62o 3 , . . •S1 4]2 •4 1 '•2 • 42. V,". 1112 3 17 . 7K3•4 9711 I lI$ q1+ 2'
.I7 g . ... 4IB4 7 9117 M'•1 P,• `5 . 70 , l31 1 41 .
1 om+_ 9113 91 712W 9 4V 1 , ($ Il• 431 1 .77 :7501 327 81 012 8g718 I 14
.5 : SIM - 11i Ia a e t r (r e f .:3 407 4). 6 .3 59 M b7 2
,A , 916a g 2K 'I 9C, 6 W2 6 I 844A -0436
4- MN 867 937 It 1 .1110.mmm •IIIIIIII I
REPORT I 13b-NATIONAL ADVISORY COMMIrErr MOR AERONAUTICS
SUPERSONIC FLOW-Coltioiued
p LT A_ , Et ý ýM . 17 V .J P. A. o.p . T, P,~ P.,
386.10 .0117 70360 75BM .4291 3047 1.30522 4. AW 33131 .8135 1.05W 1.429 1 1511 .007) .3911:?w0 .00 724 70 .423 3 05 0 1 .3 312 5 (W3 82.53 .071 1006 1 406 1.109 0037 .3W
1.27 .2375 .41071 .7361 7020 .. 420 064 1 3D77 5.3W" 53.00 40011 1 715 1 461 1. 172 .0042 .30191.3 .3m0 .4021 7332 70010 .421U I06 21410 5 027 31.301 .7 l 1 745 1.091 1.179 .002T 377,1
23 .20560 47 0 04 00 3062 1 2240f b.RON 30.02 .-,Oil 1.775 1 49N 1.103 .11111 3720
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EQ•UATIONS, TABLU•, A." CRAMSI FOR COMPR BLZ FLOW
SUPERS8ONIC FLOW
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A-9
To Convert Into Multiply By
A
Atmospheres Ton/sq inch 0007348
Atmospheres cm of mercury 76Atmospheres dynes/cm ^2 1.013xl106Atmospheres Ft of water (@ 4 degs C) 33.9Atmospheres In. of Mercury @ 0 degs C 29.92Atmospheres kgs/ sq cm 1.0333Atmospheres kgs/sq meter 10332Atmospheres Ibf/sq ft (psf) 2116.4Atmospheres Ibf/sq inch (psi) 14.7Atmospheres tons/ sq ft 1.058
B
Bars atmospheres 0.9869Bars dynes/sq cm 10^6Bars kgs/sq meter 1.020x10"4Bars Ibf/sq ft (psf) 2089Bars Ibf/sq in (psi) 14.5
C
Centigrade fahrenheit (C - 9/5) - 32Centimeters (cms) feet 3.281x 10-2Centimeters inches 0.3937Centmneters kilometers 10'-6
Centimeters meters 0.01Centimeters miles 6.214xl0^-6Centimeters millimeters 10Centimeters mils 393.7Centimeters yards 1.094x10-2
Centimeters of Mercury atmospheres 0.01316Centimeters of Mercury feet of water 0.4461Centimeters of Mercury kgs/sq meter 136
Centimeters of Merwry Ibf/sq ft (psf) 27.85Centimeters of Mercury Ibf/sq inch (psi) 0.1934Centimeters isec feet/min 1.9685
Centimeters /sec feet/sec 0.03281Centimeters /sec kilometers/hr 0.036
Centimeters /sec knots 0 1943
Figure A-5 Continued
A-10
To Convert Into Multiply By
Centimeters/sec meter/min 0.6Centimeters/sec miles/hr 0.02237Centimeters/sec miles/min 3.728x10^-4Centimeters/sec/sec feeVsec/sec 0.03281Centimeters/sec/sec kms/seqcsec 0.036Centimeters/sec/sec meters/sec/sec 0.01Centimeters/sec/sec miles/hr/sec 0.02237Cubic centimeters cubic feet (ft^3) 3.531x 10^-5Cubic centimeters cubic inches (in^3) 0.06102Cubic centimeters cubic meters (m'3) 10^-6Cubic centimeters cubic yards (yd^3) 1.308x10'^-6Cubic centimeters gallons (gal; US Iiq.) 2.6242x10"-4Cubic centimeters liters 0.001Cubic centimeters pints (US liq.) 2.113xl0^-3Cubic centimeters quarts (US liq.) 1.057xl 0^-3Cubic feet cubic centimeters 2832Cubic feet cubic inches 1.728Cubic feet cubic meters 0.02832Cubic feet cubic yards 0.03704Cubic feet gallon (US liq.) 7.48052Cubic feet liters 28.32Cubic feet pints (US liq.) 59.84Cubic feet quarts (US liq.) 29.92Cubic inches cubic centimeters 16.39Cubic inches cubic f get 5.787xl 0"-4Cubic inches cubic meters 1.639x10'-5Cubic inches cubic yards 2.143x10"-5Cubic inches grdlons 4.329xl0"-6Cubic inches liters 0.01639Cubic inches pints (US liq.) 0.03463Cubic inches quarts (US liq.) 0.01732Cubic meters cubic certimeter 10^6Cubic meters cubic feet 35.31Cubic meters cubic inches 61023Cubic meters cubic yards 1.308Cubic meters gallons (US liq.) 264.2Cubic meters liters 1000Cubic meters pints (US liq.) 2113Cubic meters quarts (US liq.) 1057
Figure A-5 Continued
A-11
To Convert Into Multiply By
Centimeters/sec meter/min 0.6Centimeters/sec miles/hr 0.02237Centimeters/sec miles/min 3.728x10"-4Centimeters/sec/sec feet/Vcc/sec 0.03281Centimeters/sec/sec kms/sec/sec 0.036Centimeters/sec/sec meters/secisec 0.01Centimeters/sec/sec miles/hr/sec 0.02237Cubic centimeters cubic feet (ft^3) 3.531xl 0"- 5Cubic centimeters cubic inches (in^3) 0.06102Cubic centimeters cubic meters (m^3) 10^-6Cubic centimeters cubic yards (yd^3) 1.308x1 0^-6Cubic centimeters gallons (gal; US liq.j 2.6242x104Cubic centimeters liters 0.001Cubic centimeters pints (US liq.) 2.! 13x10^-3Cubic centimeters quarts (US liq.) 1.057x10"-3Cubic feet cubic centimeters 2832Cubic feet cubic inches 1.728Cubic feet cubic meters 0.02832Cubic feet cubic yards 0.03704Cubic feet gallon (US liq.) 7.48052Cubic feet liters 28.32Cubic fet pints (US liq.) 59.84Cubic feet quarts (US liq.) 29.92Cubic inches cubic centimeters 16.39Cubic inches cubic feet 5.787x10^-4Cubic inches cubic meters 1.639x10^-5Cubic inches cubic yards 2.143x 10^-5Cubic inches gallons 4.329x 10-6Cubic inches liters 0.01639Cubic inches pints (US liq.) 0.03463Cubic inches quarts (US liq.) 0.01732Cubic meters cubic centimeter 10^6Cubic meters cubic feet 35.31Cubic meters cubic inches 61023Cubic meters cubic yards 1.308Cubic meters gallons (US liq.) 264.2Cubic meters liters 1000Cubic meters pints (US liq.) 2113Cubic meters quarts (US liq.) 1057
Figure A-5 Continued
A-12
To Convert Into Multiply By
Cubic yard cubic centimeters 7.646x1 05Cubic yard cubic feet 27Cubic yard cubic inches 46656Cubic yard cubic meters 0.7646Cubic yard gallons (US liq.) 202Cubic yard liters 764.6Cubic yard pints (US liq.) 1615.9Cubic yard quart (US liq.) 807.9
D
Days seconds 86400Degree (angle) quadrants 0.01111Degree (angle) radian 0.01745Degree (angle) revolutions 2.7778x1 0^-3Degree (angle) seconds 3600Degree/sec radian/sec 0.01745Degree/sec revolutions/min 0.1667Degree/sec revolutions/sec 2.778x10"3Dyne sq cm atmosphere 9.869x10^-7Dyne sq cm inches of Mercury at 0 degs C 2.953x10^-5Dyne sq cm inches of water at 4 degs C 4.015x10^-4Dynes grams 1.00x10^-3Dynes joules centimenter 10^-7Dynes joules meter (newton) 10"-5Dynes kilogram 1.02x10^-6Dynes poundals 7.233x10"-5Dynes pounds 2.248xl 0-6Dynes bar 10^-6
E
Ergs Btu 9.84x10^-11Ergs dyne-centimeters IErgs foot-pounds 7.367x10-8Ergs gram-calories 0.2389x10^-7Ergs gram-centimeters 1.020x1 0-3Ergs horsepower-hrs 3.725xl0"-14Ergs joules 10^-7
Figure A-5 Continued
A-13
To Convert Into Multiply By
Ergs kg-calories 2.389x10^-11Ergs kg-meter 1.02x10^-8Ergs kilowatt-hrs 0.2778x1O^-13Ergs watt-hrs 0.2778x10'-10Ergs/sec Btu/min 5.688x10^-9Ergs/sec ft-lbf/min 4.427x10"-6Ergs/sec ft-lbf/sec 7.3756x1 0^ -8Ergs/sec horsepower 1.341xl0^-10Ergs/sec kg-calories/min 1.433x10"-9Ergs/sec kilowatts 10^-10
F
Feet centimeters 30.48Feet kilometers 3.048x10^-4Feet meters 0.3048Feet miles (nautical) 1.645x10"-4Feet miles (statute) 1.894x10^-4Feet millimeters 304.8Feet mils 1 .2x104Feet of water atmosphere 0.0295Feet of water inches of Mercury 0.8826Feet of water kgs/sq centimeters 0.03048Feet of water kgs/ sq meter 304.8Feet of water lbf/sq feet (psf) 62.43Feet of water Ibf/sq inch (psi) 0.4335Feet/min centimeters/sec 0.508Feet/min feet/sec 0.01667Feet/min kilometer/hr 0.01829Feet/min meters/min 0.3048Feet/min miles/hr 0.01136Feet/sec centimeters/sec 30.48Feet/sec kilometers/hr 1.097Feet/sec knots 0.5921Feet/sec meters/min 18.29Feet/sec miles/hr 0.6818Feet/sec miles/min 0.01136Feet/sec/sec cms/sec/sec 30.48Feet/secqsec kms/hr/sec 1.097
Figure A-5 Continued
A-14
To Convert Into Multiply By
FeeVseq/sec meters/secrsec 0.3048FeeVsec/sec miles/hr/sec 0.6818Foot-pounds Btu 1286x10^-3Foot-pounds ergs 1.356x10"7Foot-pounds gram-calories 0.3238Foot-pounds hp-hrs 5.05x10^-7Foot-pounds joules 1.356Foot-pounds kg-calories 3.24x10"-4Foot-pounds kg-meters 0.1383Foot-pounds kilowatt-hrs 3.766x10^-7Foot-pounds newton-meters 1.35582Foot-pounds/min foot-pound sec 0.01667Foot-pounds/min horsepower 3.03x10"-5Foot-pounds/min kg-calories/min 324x10^-4Foot-pounds/min kilowatts 2.26x10"-5Foot-pounds/sec Btu/hr 4.6263Foot-pounds/sec Btu/min 0.07717Foot-pounds/sec horsepower 1.818xl0^-3Ft ot-pounds/sec kg-calories/min 0.01945Foot-pounds/sec kilowatts 1.356x10"-3
G
Gallons cubic centimeters 3785Gallons cubic feet 0.1337Gallons cubic inches 231Gallons cubic meters 3.785xl0"-3Gallons cubic yards 4.951xl0-3Gallons liters 3.785Gallons (liq. Br Imp) gallons (US liq) 1.20095Gallons (US) gallons (imp) 0.83267Gallons of water pounds of water 8.3453Grams dynes 980.7Grams joules centimeter 9.807x10^-5Grams joules meter (newtons) 9.807x1 0^-3Grams kilogram 0.001Grams milligrams 1000Grams ounces ounce (troy) 0.03215Grams poundals 0.07093
Figure A-5 Continued
A-15
To Convert Into Multiply By
Grams pounds 2.205x10"-3Grams/cm pounds/inch 5.6xl 0'-3
H
Horsepower Btu/min 42.44Horsepower foot-lbf/min 33000Horsepower foot-lbf/sec 550Horsepower (metric) hp (550 ft Ilbflsec) 0.9863Horsepower hp metric (542.5 ft Ibf/sec) 1.014Horsepower kg-calores/min 10.68Horsepower kilowatts 0.7457Horsepower watts 745.7Horsepower (boiler) Btu/hr 33.479Horsepower (boiler) kilowatts 9.803
Inches centimeters 2.45Inches meters 2.54x10^-2Inches miles 1.578xl 0^-5Inches millimeters 25.4Inches mils 1000Inches yards 2.778x10^-2Inches of Mercury atmospheres 0.033422Inches of Mercury feet of water 1.133Inches of Mercury kgs/sq cm 0.03453Inches of Mercury kgs/sq meter 345.3Inches of Mercury pounds/sq ft (psf) 70.73Inches of Mercury pounds/sq in (psi) 0.4912Inches of Water @ 4 degs C atmospheres 2.458x10^-3Inches of Water @ 4 degs C inches of Mercury 0.07355Inches of Water @ 4 degs C kgs/sq centimeter 2.54x10^-3Inches of Water @ 4 degs C ounces/sq inches 0.5781Inches of Water @ 4 degs C Ibf/sq ft (psf) 5.204Inches of Water @ 4 degs C Ibf/sq in (psi) 0.03613
Figure A-5 Continued
A-16
To Convert Into Mulitiply By
J
Joules Btu 9.840x10"-4Joules eogs 10^7Joules foot-pounds 0.7376Joules kg-calories 2.389x10^-4Joules kg-meters 0.102Joules watts-hrs 2.778x10^-4Joules/cm grams 1 .0210^4Joules/cm dynes 10^7Joules/cm joules/meter (newtons) 100Joules/cm poundals 723.3Joules/cm pounds 22.48
K
Kilograms (kgs) dynes 980665Kilograms grams 1000Kilograms joules/cm 0.09807Kilograms joules/meter (newtons) 9.807Kilograms poundals 70.93Kilograms pounds 2.205Kilograms ton (long) 9.842x10^-4Kilograms ton (short) 1.102x10^-3Kilogram/cubic meter grams/cu cm 0.001Kilogram/cubic meter pounds/cu ft 0.06243Kilogram/cubic meter pounds/cu inch 3.613x1 0^5Kilograms/meter pounds/ft 0.672Kilogram/sq cm dynes 980665Kilogram/sq cm atmosphere 0.9678Kilogram/sq cm feet of water 32.81Kilogram/sq cm inches of Mercury 28.96Kilogram/sq cm Ibf/sq feet (psf) 2048Kilogran/sq cm lbf/sq inch (psi) 14.22Kilogram/sq meter atmospheres 9.678xl0^-5Kilogram/sq meter bars 98.07xl0^-6Kilogram/sq meter feet of water 3.281xl0^-3Kilogram/sq meter inches of Mercury 2.896x10'-3Kilogram/sq meter lbf/sq feet (psf) 0.2048Kilogram/sq meter lbf/sq inch (psi) 1.422x10'-3Kilometers (kms) centimeters 10^-6
Figure A-5 Continued
A-17
To Convert Into Multiply By
Kilometers feet 3281.0Kilometers inches 3.937x1 04Kilometers meers 1000.0Kilometers miles 0.6214Kilometers millimeters 10"6Kilometers yards 1094.0Kilometers/hr centimeters/sec 27.78Kilometers/`hr feet/min 54.68Kilometers/hr feet/sec 0.9113Kilometers/hr knots 0.5396Kilometers/hr meters/min 16.67Kilometers/hr miles/hr 0.6214Kilometers/hr/sec cms/sec/sec 27.78Kilometers/hr/sec ft/sec/sec 0.9113Kilometers/hr/sec meters/sec/sec 0.2778Kilometers/hr/sec miles/hr/sec 0.6214Kilowatts Btu/min 56.92Kilowatts horsepower 1.341Knots feet/hr 6080Knots kilometers/hr 1.08532Knots mautcal miles/hr 1.0Knots statute miles/hr 1.151Knots yards/hr 2027.0
feet/sec 1.689
L
Liters bushels (US dry) 0.02838Liters cubic cm 1000Liters cubic feet 0.03531Liters cubic inches 61.02Liters cubic meters 0.001Liters cubic yards 1.308xl0^-3Liters gallons (US liq) 0.2642Liters pints (US liq) 2.113Liters quarts (US liq) 1.057Lumen spherical candle power 0.07958Lumen watt 0.001496
Figure A-5 Continued
A-18
To Convert Into Muitiply By
M
Meters centimeters 100Metrs feet 3.281Meters inches 39.37Meters kilometers 0.001Meters miles (nautical) 5.396x10"-4Meters miles (statute) 6.214x10^-4Meters millimeters 1000Meters yards 1.094Meters/min cms/sec 1.667Meters/min feet/min 3.281Meters/min feet/sec 0.05468Meters/min kms/hr 0.06Meters/min knots 0.03238Meters/min miles/hr 0.03728Meters/sec feet/min 196.8Meters/sec feet/sec 3.281Meters/sec kilometers/hr 3.6Meters/sec kilometers/min 0.06Meters/sec miles/hr 2.237Meters/sec miles/min 0.03728Meters/sec/sec cms/sec/sec 100Meters/sec/sec ff/sec/sec 3.281Meters/sec/sec kms/hr/sec 3.6Meters/sec/sec miles/hr/sec 2.237Microns meters 1.Ox1 0^-6Miles (nautical) feet 6080.27Miles (nautical) kilometers 1.852Miles (nautical) meters 1853Miles (nautical) miles (statute) 1.1516Miles (nautical) yards 2027Miles (statute) centimeters 1.609x1 0^5Miles (statute) feet 5280Miles (statute) inches 6.336x1 0^4Miles (statute) kilometers 1.609Miles (statute) meters 1609Miles (statute) miles (nautical) 0.8684Miles (statute) yards 1760Miles/hr cms/sec 44.7Miles/hr feet/min 88
Figure A-5 Continued
A-19
To Convert Into Multiply By
MilaqIhr feet/sec 1.467Miles/hr kms/hr 1.609
Mile4hr kms/min 0.02682
Miles/hr knots 0.8684
Miles/hr meters/min 26.82Miles/hr miles/min 0.1667
Miles/hr/sec cmstjmvsec 44.7
MilesWhr/sec feet/sec/sec 1.467
Miles/hr/sec kms/hr/sec 1.609
Miles/hr/sec meters/sec/sec 0.447Miles/min cms/sec 2682Miles/min feet/sec 88
Miles/min kms/min 1.609
Miles/min knot-/min 0.8684Miles/min miles/hr 60
Millimeters centimeters 0.1
Millimeters feet 3.281x10^-3Millimeters inches 0.03937Millimeters kilometers 10^-6Millimeters meters 0.001
Millimeters miles 6.214x10^-7
Millimeters mils 39.37
Millimeters yards 1.094x10^-3MilS centimeters 2.54x10-3Mils feet 8.333x10^-5
Mils inches 0.001
Mils kilometers 2.54x10^-8Mils yards 2.778x10^-5Minutes (angles) degrees 0.01667
Minutes (angles) quadrants 1.852x1 0^-4
Minutes (angles) radians 2.909x10^-4
Minutes (angles) seconds 60
N
Newtons Dynes 1.0x10^5Newtons pounds (Ibf) 0.2248
Figure A-5 Continued
A-20
To Convert Into MultiF* By
0
Ounce drams 16
Ounce grains 437.5
Ounce grams 28.349527
Ounce pounds 0.0625
Ounce ounces (troy) 0.9115Ounce tons (long) 2.79x10-5
Ounce tons (metric) 2.835xl0"-5Ounce (fluid) cubic inches 1.805
Ounce (fluid) liters 0.02957
Ounces (troy) grains 480
Ounces (troy) grams 31.103481Ounces (toy) ounce (avdp.) 1.09714
Ounces (boy) pennyweights (troy) 20
Ounces (troy) pounds (troy) 0.08333
Ounce/sq inch Dynes/sq cm 4309
Ounce/,c inch Ibf/sq in (psi) 0.0625
P
Parsec miles 190x!0^12Parsec kilometers 3.084x1 0^ 13
Pennyweight (troy) grams 1.55517
Pennyweight (troy) pounds (troy) 4.1667x10^-3Pints (dry) cubic inches 33.6
Pints (liq) cubic centimeters 473.2
Pints (liq) cubic feet 0.01671
Pints (liq) cubic inches 28,87
Pints (liq) cubic meters 4.73210"-4Pints (liq) cubic yards 6.189x10"-4
Pints (liq) gallons 0.125Pints (liq) liters 0.4732
Pints (liq) quarts (liq.) 0.5
Pounds (avoirdupois) ounces (boy) 14.5833Poundals dynes 13826Poundals grams 14.1
Poundals joules/crn 1.383xl0^-3Poundals joules/meter (newtons) 0.1383Poundals kilograms 0.0141
Poundals pounds 0.03108
Figure A-5 Continued
A-21
To Convert Into Multiply By
Pc,'nds drams 256Pounds dynes 44.4823xl 0^4Pounds grains 7000Pounds grams 453.5924Pounds joules/cm 0.04448Pounds joules/meter (newtons) 4.448Pounds kilograms 0.4536Pounds ounces 16Pounds ounces (troy) 14.5833Pounds poundals 32.17Pounds pounds (troy) 1.21528Pounds (lbf) slugs 3.1081xl0-2Pounds tons (short) 0.0005Pounds (troy) grains 5760Pounds (troy) grams 373.24177Pounds (troy) ounces (avdp.) 13.1657Pounds (troy) ounces (troy) 12Pounds (troy) pennyweights (troy) 240Pounds (troy) pounds (avdp) 0.822857Pounds (troy) tons (long) 3.6735 x 10^-4Poends (troy) tons (metric) 3.7324x10^-4Pounds (troy) tons (short) 4.1143xl0^-4Pounds of water cubic feet 0.01602Pounds of water cubic inches 27.68Pounds of water gallons 0.1198Pounds of water/min cubic feet/sec 2.670x10"-4Pounds-feet cm-dynes 1.356xi 07Pounds-feet cm-grams 13825Pounds-feet meter-kgs 0.1383Pounds/cubic feet grams/cubic cm 0.01602Pounds/cubic feet kgs/cubic meter 16.01847Pounds/cubic feet poundsjcubic inch 5.787x10^-4Pounds/cubic inch grams/cubic centimeter 27.68Pounds/cubic inch kgs/cubic meter 2.768x10^4Pounds/cubic inch pounds/cubic feet 1728Pounds/sq foot (psf) atmospheres 4.725x 10' 4Pounds/sq foot (psf) bars 4.788x10"4Pounds/sq foot (psf) dyne/sq cm 4.788xi 0'2Pounds/sq foot (psf) feet of water (4 degs C) 0.01602
Figure A-5 Continued
A-22
To Convert Into Multiply By
Pounds/sq foot (psi) inches o Mercury @ 0 degs C 0.01414Pounds/sq foot (psf) kgs/sq meter 4.882Pounds/sq foot (psf) lbf/sq inch (psi) 6.944xl O"-3Poundsisq foot (psi) newton sq. meter 47.88Pounds/sq inch (psi) atmospheres 0.06804Pounds/sq inch (psi) dynes/sq cm 6.8948x10"4Pounds/sq inch (psi) feet ot water @ 4 degs C, 2.307Pounds/sq inch (psi) inches of water @ 4 degs C 27.681Pounds/sq inch (psi) inches d Mercury @ 0 degs C 2.036Pounds/sq inch (psi) newton sq meter 6.8948x10^3Pounds/sq inch (psi) kgs/sq meter 703.1Pounds/sq inch (psi) pounds/sq inch (psi) 144
Q
Quadrants (angle) degree 90Quadrants (angle) minutes 5400Quadrants (Ingle) radians 1.571Quadrants (angle) seconds 3.24x10^5Quarts (dry) cubic inches 67.2Quarts (liq.) cubic centimeters 946.4Quarts (liq.) cubic feet 0.03342Quarts (liq.) cubic inches 57.75Quarts (liq.) cubic meters 9.464x10^-4Quarts (liq.) cubic yards 1.238x10"-3Quarts (liq.) gallons 0.25Quarts (liq.) liters 0.9463
R
Radians degrees 57.3Radians minutes 3438Radians quadrants 0.6366Radians seconds 2.063xA 05Radians/sec degrees/sec 57.3Radians/sec revolutions/min 9.549Radians/sec revolutions/sec 0.1592Radians/sec/sec revs/min/min 573Radians/sec/sec revs/min/sec 9.549
Figure A-5 Continued
A-23
To Convert Into Multiply By
Radians/sec4sec revs/sec/sec 0.1592Revolutions degrees 360Revolutions quadrants 4Revolutions radians 6.283Revolutions/min degree/sec 6Revolutions/min radians/sec 0.1047Revolutions/min revs/sec 0.01667Revolutions/min/min radians/sec/sec 1.745xl 0^-3Revolutions/min/min revs/min/sec 0.01667Revolutions/min/min revs/sacjsec 2.778xl 10-4
S
Seconds (angle) degress 2.778xl0^-4Seconds (angle) minutes 001667Seconds (angle) quadrants 3.087x10"-6Seconds (angle) radians 4.848x10^-6Slug ibm 32.2Slug kgs 14.594Square centimeters circular mils 1 .973x1 0^5Square centimeters square feet 1.076xl 0^-2Square centimeters square inches 0.155Square centimeters square meters 0.0001Square centimeters square miles 3.861 x1 0^ -11Square centimeters square millimeters 100Square centimeters square yards 1.196xl0'-4Square feet acres 2296x 10-5Square feet circular mils 1.833x10"8Square feet square centimeters 929Square feet square inches 144Square feet square meters 0.0929Square feet square miles 3.587xl0"-8Square feet square millimeters 9.29x104Square feet square yards 0,1111Square inches circular mils 1.273x10"6Square inches square centimeters 6.452Square inches square feet 6.944x10"-3Square inches square millimeters 645.2Square inches square mils 10^6
Figure A-5 Continued
A-24
To Convert Into Multip*y By
Square inches square yards 7.716x1 0-4Square kilometers acres 247.1Square kilometers square centimeters 10O10Square kilometers squarefeet 10.76x10^6Square kilometers square inches 1.55x10"9Square kilometers square meters 10^4Square kilometers square miles 0.3861Square kilometers square yards 1.196xl 06Square meters acres 2.471x1O^-4Square meters square centimeters 10^4Square meters square feet 10.76Square meters square inches 1550Square meters square miles 3.861xl 0"-7Square meters square millimeters 10^6Square meters square yards 1.196Square miles acres 640Square miles square feet 27.88x10"6Square miles square kilometers 2.59Square miles square meters 2.59x10^6Square miles square yards 3.098xi0"6Square milimeters circular mils 1973Square milimeters square centimeters 0.01Square milimeters square feet 1.076x1 0"-5Square milimeters square inches 1.55x10^-3Square yards acres 2.066x10"-4Square yards square centimeters 8361Square yards square feet 9Square yards square inches 1296Square yards square meters 0.8361Square yards square miles 3.288xl0"-7Square yards square millimeters 8.361 x 0^5
T
Tons (long) kilograms 1016Tons (long lbf 2240Tons (long) tons (short) 1.12Tons (metric) kilogram 1000Tons (metric) lbf 2205
Figure A-5 Continued
A-25
To Convert Into Multiply By
Tons (short) kilogram 907.1848
Tons (short) ounce 32000Tons (short) ounce (troy) 29166.66Tons (short) lbf 2000
Tons (short) pounds (troy) 2430.56Tons (short) tons(long) 0.89287
Tons (short) tons (metric) 0.9078
W
Watts Btuthr 3.4129Watts Btu/min 0.05688
Watts ergs/sec 107Watts foot-lbf/min 44.27
Watts foot-lbf/sec 0.7378
Watts horsepower 1.341x10"-3Watts horsepower (metic) 1.36x10'^-3Watts kg-calodes/min 0.01433Watts kilowatts 0.001
Y
Yards centimeters 91.44Yards kilometers 9.144x10"-4Yards meters 0.9144Yards miles (nautical) 4.934x10^-4Yards miles (statute) 5.682x10"-4Yards millimeters 914.4
Figure A-5 Concluded
A-26
NOTES
A-27
NOTES
A-28
APPENDIX B
13
APPENDIX B
Tid Bits
This section offers tid bits of information that can be useful. The information listed
below is not in any particular order.
1) One count of drag is 0.0001.
2) A A drag count of 30 counts is a significant increase.
3) Wing sweep decreases CLa and CLmax
4) Critical Mach is increased by wing sweep.
5) In a powered test (V/STOL), vary q, dynamic pressure, to obtain the appropriate
thrust coefficient.
6) (xZL does not vary with aspect ratio.
7) Principal contributor to C, (dihedral effect) is wing geometric dihedral, 'Z'
position of wings and sweep angle.
8) Supercritical airfoils (wings) delay the drag rise due to shock formation.
9) When the Mach number is > 0.3, be sure to correct the dynamic pressure
(density) for compressibility effects.
B-1
Wind Tunnel First Aid Kit
This section is a collection of suggested items that would be helpful to the testing
engineer if they accompanied him/her to the testing site. It keeps the testing engineer
from loosing an 'atta boy'.
1) Extra money for testing
2) Aspirin
3) Coffee mug
4) Triangle, ruler, symbol template, and other graphing equipment
6) Pencils with extra lead and erasers
7) Binders and dividers to put plots in
8) Labels & gummed tabs for binders and dividers
9) Paper clips
10) Calculator
11) Notebook for personal notes
12) Notebook for wind tunnel log. It is most important that the testing engineer
write down everything that happens during the test. By the time the reduced data
arrives back 'home', you will have forgotten what happened during the test.
13) Test Plan
14) Run Schedule
15) Stress report
16) Previous data for comparison
17) Model drawings
18) Reference material
19) Raucous reading material for the wee hours in the morning
20) Telephone numbers from back home and at the test site
21) Don't forget extra money for a post test party.
B-2
Square Circular Are Sector
S= !LI• h = z fp,
* 22
b -T
dý,- 2.in -9li
2b d2- 2.b.... T
Rootanillo
Acb Iab3 Quarter Circular
22 b
Pazallelolraan
dCi n A = b 1---- D--- -i = • -'D•
.i A [a±.- h -l:h
T n r. LICuba
h 2a + b-=
-+ v
A =GA
xquilatrialTrian
T
PeIrallelpiped ml n-.q,
_A 2(.1
Ln 2
a Parallelpiped
A
Circle A nr2
! 4 d2 0.785d2 , - -- '; :;: " "
4icu 27-7-r b h V =hA 1 1 ,1 1
CI. = T 4
x4-= V = 0 1
1.( .
CylinderSemi-Cirole A E 1 2 V d:-
A TM 4 1 1'
Sx y 4, -- rh=-- £--- 3,-r AS urfe 2nrjr + h) I v :
I.= ly 0 .10913r4
I2-a--PIrbollob ~ ~ 3 Co -
h A
iP2Žsr: obol1b.l~ 4
Parabelio
Aa A ab sphere
Sb Aeurf 4.ýr2
Figure B-i Geometric Equations
B-3
NOTES
B-4
NOTES
B-5
NOTES
B-6
APPENDIX C
C
APPENDIX C
Power On Symbols
Moment Arms
I = Horizontal Ram Drag Arm
12 = Vertical Thrust Arm
13 = Vertical Ram Drag Arm
14 = Horizontal Thrust Arm
15 = Lateral Thrust Arm
Thrust Induced Aerodynamic Forces (Pwr ON - Pwr OFF)
AL Lift
AD Drag
AM Pitching moment
ARM Rolling moment
Total Coefficients
CL Lift coefficient, L/qS
CD Drag coefficient, D/qS
CM Pitching moment, M/qSc
CRM Rolling moment, RM/qS
Incremental Thrust Coefficients
CT Thrust coefficient, T/qS
ACLT Lift coefficient due to Thrust
ACD Drag coefficient due to thrust
ACMT Pitching moment coefficient due to Thrust
C-1
Symbols continued
ACRMT RolliAg iionoent coefficient due to Thrust
AC yMT Yawing moment coefficient due to Thrust
ACM Pitching moment due to Ram drag
M. VCD Ram drag coefficient, 1
Aerodynamic Coefficients (direct thrust effect removed)
CL or CL Lift coefficientCAero LA
CD or CD Drag coefficientCAero CA
CM or CM Pitching moment coefficientCAero CA
C R or C R olling moment coefficientCRAero CM
Miscellaneous
x = Fan Number
0 = Total Thrust Vector (measured from longitudinal axis)
0 = Fan Thrust Vectorx
c = M.A.C.
b - Wing Span
h/D Non-dimensional he ight with respectto the nozzle diameter
C-2
Power On Aerodynamic Equations (ref. 5)
Incremental Aerodynamic Coefficient (x fan number)
OT = sin' IETX sinOx / T (ITx cosOx) 2 + (ETx sinOx) 2
ACLT = CT sin(OT + (x)
ACDT =CT cos(OT + (a)
ACMT Tx(14x sin0Tx + 1 2T x cOSoTx qSc
x
ACMT = 1.ACMT (Fan 1,2,3,...x)
ACM D= DR( 3 cosa + 11 sina) / qSc
x
AC M D XACM (Fan 1, 2,3,....x)
ACRM = Txl5 [sinOx cosa + cosO sina] / qSc
ACYM = Txl5 x[cosOxcosz - sinOxsina] / qSc
Aerodynamic Coeffi,;ients (total - thrust effects)
(figure 111-5)=CLL AL
CLAero L
CD = CD ACD TCDCAero _ CR
CM = CM ACD T ACMDCAero°
CRM = CRM - EACRMx
CyYM = CYM - YACYMx
C-3
DR = MiV
_DR
DR qS
Induced Force Coefficients
AL CLA -CL
AD pwr on pwr offT CT
AD~ CA " CD
AM pwr on pwr off
T CT
CRMA - CRMAM - pwr on pwr off
T • CT
C-4
LIFT C~rFnCIENTDUE TO THRUST
CL At
CLA
INDUCED LIFT COEFFICIENT O)
4 L Puwcr oft
(Acro Lift)
24 it6 6
(Total Drug minusT con 8)
(+) IND RAM DRAG COEFFICIENT
CDA
Power OffCD 0 (Aero Drug)
2 4 68INCREMENTAL
H ~~DRAG COEFFICIENT h1DUE TO THRUST
ACA
CM
PITCHING MOMENT (Total Pitch minusCOEFMCIENT DUE Thrust X Arm)
cmA
Power OnINCREMENTAL PITC11ING (Aero Fitch)
ACM T
CM o Power On
2 4 68h/) MtDX~ED P'ITCHING MOMENT
COEMVCIE NT
CM
Power Off
(Total Pitch)
Figure c-i Powderedt Effects
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NOTES
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NOTES
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NOTES
C-9
NOTES
C-10
NOTES
C-II
NOTES
C-12
NOTES
C-13
NOTES
C-14
NOTES
C-15
NOTES
C-16
NOTES
C. 17
NOTES
C.- I,
NOTES
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NOTES
C-20
NOTES
C-21
NOTES
C-22
NOTES
C-23
NOTES
C-24
NOTES
C-25
NOTES
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NOTES
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NOTES
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NOTES
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NOTES
/ /,.,
SUPPLEMENTARY
INFORMATION
DEPARTMENT OF THE AIR FORCEWRIGHT LABORATORY (AFMC)
WRIGHT-PATrERSON AIR FORCE BASE, OHIO
FROM: WL/DOA 7 January 1994
Wright-Patterson AFB OH 45433-6523
SUBJ: Notice of Changes in Technical Report(s)#WL-TR-91-3073/ADA240263
TO: Defense Technical Information CenterAttn: DTIC-OCCCameron StationAlexandria VA 22304-6145
Please change subject report as follows:
Include the attached 2 corrected pages in subject Tech Reportas an ERRATA SHEET.
WM F. WHALEN Cy to: WL/DOL (M. Kline)Chief, STINFO and Technical Editing Branch
Operations and Support Directorate
Corrections to version 1.0 of Subsonic Wind Tunnel Testing Handbook
WL-TR-91-3073; DTIC AD A240263
1) Page 11-2; Mean Aerodynamic Chord (MAC); Add a comma after 'span'.
2) Page IV-7; Figure IV-4, Top figure 'Aerodynamic Angles'; -3 is on the wrong plane.It should be on the +VI plane.
3) Page IV-9; Balance to Body A."is , force equations (roll=O) [not matrix]; on CN term
C sino3sina should be positive +C sinosinox.-- Cbal +Cba1
4) Page IV-9; Balance to Body Axis , force equations (roll=0) [not matrix]; on CA term
.C sino3cosca should be positive +Cy sino3cosoc.-- Cbal +Cba1
5) Page IV-9; Balance to Body Axis , force equations (roll=O) [not matrix]; on CY
term ... +CAbal sino3 should be negative -CAbal sinp.
6) Page IV-13; Bod,- Axis to Wind Axis, Force matrix; Change C toCb Cb
7) Page IV-14; Body Axis to Stability Axis, Force matrix; Change C to C
8) Page IV-15; Wind Axis to Stability Axis, Add a 'w' sub-subscript to CL in the vector
column of the force matrix.
9) Page V-2; change the units on 32.1741 to (lbm-ft/lbf-sA2) from (ft/secA2).
10) Page VI-6; Figure VI-3; Tli is measured on this figure from the centerline to
the inboard trailing edge flap chord.
11) Page VI-10; Inboard Section, b /2=26.96in; Change to bi/2=26.96 in.
N12) Page VII-7, Method of Determining a Drag Polar; Change X CL(i)*CD(i)2 to
N i=l
XCL(i) 2*CD(i).izi
13) Page VII-7, Last line; change b=-2kC to b=-2KCL
14) Page VII-8; Resulting augmented matrix; Change first line from '3 1 3' to'5 1 0.3'; Change 0.11251 to 0.0 11251
15) Page XI-9; 4th line; Change 'in milli-volts [mv]' to 'in milli-volts [my])'
D-1
16) Page XI-12; Example Original Balance Calibration Matrix, no action is required. Thisis a point of clarification. Part of this matrix came from Figure XI-3. The original
matrix was inadvertently entered and subsequently inverted in COLUMN order when in fact itshould have been entered in ROW order. All the preceding calculations use the column
order inverted matrix, but their ORDER of calculations and calculations is correct. If
you "pretend" they were entered in row order everything works out fine. This was not
corrected in this report.
17) Page A-3; Standard Atmosphere; The values used in this figure were obtained from adated U.S. Standard Atmosphere source ie., speed of sound @ S.L. = 1107.64 when it should
be 1116.1.
18) Delete Page A-12.
19) Page A-10...; Conversion Factors;
To Convert Into Multiply By Change to
Centigrade fahrenheit (C*9/5)-32 (C*9/5)+32
Centimeters kilometers 10A-6 10A-5Cubic feet cubic centimeters 2832 28,320.0
Cubic feet Cubic inches 1.728 1728.0
Cubic inches gallons 4.329x 10^-6 4.329x I0A-3Degree/sec revolutions/sec 2.778x 10A3 2.778x 0^A-3
Ergs/sec BTU/min 5.688x 10A-9 5,688x 10^A9
Hp (boiler) BTU/hr 33.479 33,479.0Knots kilometers/hr 1.08532 1.8532
Knots mautical mi/hr L.0 nautical mi/hr
20) Page B-3; Figure B-I Geometric Equations, Parallelogram; tile length of 'a' is the
base of the parallelogram.
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