144 118 63MB
English Pages [534] Year 1955
Table of contents :
COVER
TITLE
COPYRIGHT
PREFACE TO THIRD EDITION
PREFACE TO SECOND EDITION
PREFACE TO FIRST EDITION
CONTENTS
CHAPTER 1. INTRODUCTION
1:1. TYPES OF AIRCRAFT
1:2. TYPES OF AIRPLANES
1:3. TYPES OF HELICOPTERS
1:4. TYPES OF MISSILES
1:5. PERFORMANCE OF AIRCRAFT
1:6. CONTROL OF AIRCRAFT
1:7. CONSTRUCTION OF PRINCIPAL PARTS OF AIRCRAFT
REVIEW QUESTIONS AND PROBLEMS
CHAPTER 2. FLUID STATICS AND NONFLOW THERMODYNAMICS
2:1. PROPERTIES OF LIQUIDS AND GASES: DENSITY, VISCOSITY, CONDUCTIVITY, AND COMPRESSIBILITY
2:2. STATICS OF LIQUIDS: MANOMETERS , BAROMETERS
2:3. STATICS OF GASES: PERFECT GAS LAWS
2:4. SPEED OF SOUND IN AIR
2:5. MOIST AIR
2:6. VARIATION OF AIR WITH ALTITUDE: STANDARD AIR
2:7. PRESSURE AND DENSITY ALTITUDES: USE OF AIR CHART
PROBLEMS
CHAPTER 3. FLUID DYNAMICS AND THERMODYNAMICS OF FLOW
3:1. REALMS OF FLUID FLOW
3:2. STEADYFLOW CONTINUITY EQUATION
3:3. STEADYFLOW ENERGY EQUATION
3:4. STEADYFLOW MOMENTUM EQUATION
3:5. APPLICATION OF STEADYFLOW EQUATIONS
CHAPTER 4. FRICTIONLESS INCOMPRESSIBLE FLOW
4:1. BERNOULLI EQUATION: FLOW IN PIPES AND DUCTS AND BETWEEN STREAMLINES
4:2. MOMENTUM EQUATION: FORCES EXERTED BY JETS
4:3. TWODIMENSIONAL FLOW PATTERNS IN PERFECT FLUIDS
4:4. BODY PRESSURE DISTRIBUTION: PITOT TUBES
PROBLEMS
CHAPTER 5. FRICTIONLESS COMPRESSIBLE FLOW
5:1. RANGES OF COMPRESSIBLE FLOW: SUBSONIC, TRANSONIC, SUPERSONIC
5:2. ONE DIMENSIONAL SUBSONIC COMPRESSIBLE FLOW
5:3. NORMAL SHOCK WAVES
5:4. TWODIMENSIONAL AND AXI SYMMETRIC SUPERSONIC FLOW
PROBLEMS
CHAPTER 6. INCOMPRESSIBLE BOUNDARY LAYER FLOW
6:1. BOUNDARYLAYER FLOW
6:2. LAMINAR INCOMPRESSIBLE BOUNDARY LAYER
6:3. TURBULENT INCOMPRESSIBLE BOUNDARY LAYER
6:4. TRANSITION OF INCOMPRESSIBLE BOUNDARY LAYERS
6:5. SKIN FRICTION IN PIPES AND DUCTS
6:6. SEPARATION OF INCOMPRESSIBLE BOUNDARY LAYERS
6:7. INCOMPRESSIBLE BOUNDARYLAYER FLOW AROUND A CYLINDER
6:8. INCOMPRESSIBLE BOUNDARYLAYER FLOW AROUND A SPHERE
PROBLEMS
CHAPTER 7. COMPRESSIBLE AND HEAT CONDUCTING BOUNDARYLAYER FLOW
7:1. BOUNDARYLAYER TEMPERATURE NEAR A SURFACE: RECOVERY FACTOR .
7:2. HEAT TRANSFER AND SKIN FRICTION
7:3. LAMINAR COMPRESSIBLE BOUNDARY LAYER
7:4. TURBULENT COMPRESSIBLE BOUNDARY LAYER
7:5. TRANSITION OF COMPRESSIBLE BOUNDARY LAYERS FROM LAMINAR TO TURBULENT
7:6. COMPRESSIBLE FLOW AROUND SPHERES AND CYLINDERS
PROBLEMS
CHAPTER 8. AERODYNAMIC TEST FACILITIES
8:1. FLIGHT TESTS
8:2. MOVING MODEL TESTS
8:3. WIND TUNNEL TYPES
8:4. TRANSONIC AND SUPERSONIC WIND TUNNELS
8:5. WIND TUNNEL BALANCES FOR FORCE TESTS
8:6. OTHER WIND TUNNEL TEST EQUIPMENT
CHAPTER 9. AIRFOILS AND ASPECT RATIO EFFECTS AT LOW SPEEDS
9:1. FORCES ON AIRFOILS; AIRFOIL COEFFICIENTS
9:2. STRAIGHTLINE PLOTTING OF LOW SPEED AIRFOIL TEST DATA
9:3. MOMENTUM THEORY OF AIRFOILS
9:4. CIRCULATION THEORY OF AIRFOILS
9:5. ASPECTRATIO CORRECTIONS
9:6. GROUND EFFECT; TUNNEL WALL CORRECTIONS
9:7. EFFECTS OF CHORDWISE SLOT IN WINGS; INTERACTION OF TWO AIRPLANES FLYING SIDE BY SIDE
CHAPTER 10. AIRFOIL COMPRESSIBILITY EFFECTS
10:1. TWODIMENSIONAL AIRFOILS: SUBSONIC COMPRESSIBILITY EFFECTS
10:2. FINITE WINGS: SUBSONIC COMPRESSIBILITY EFFECTS
10:3. TRANSONIC COMPRESSIBILITY EFFECTS
10:4. SUPERSONIC WING CHARACTERISTICS
PROBLEMS
CHAPTER 11. AIRFOIL VISCOSITY EFFECTS
11:1. LOWSPEED SCALE EFFECTS, WING SECTIONS
11:2. COMBINED HIGH SPEED AND SCALE EFFECTS, WING SECTIONS
11:3. SCALE EFFECTS ON FINITE WINGS
11:4. SPANWISE LOAD DISTRIBUTION
11:5. FLIGHT BOUNDARIES
PROBLEMS
CHAPTER 12. HIGHLIFT DEVICES
12:1. NEED FOR HIGHLIFT DEVICES
12:2. TRAILING EDGE FLAPS
12:3. LEADING EDGE SLATS AND SLOTS
12:4. BOUNDARY LAYER CONTROL
PROBLEMS
CHAPTER 13. AIRFOIL SELECTION
13:1. SYSTEMATIC INVESTIGATIONS AND NUMBERING SYSTEMS
13:2. THE NACA 4DIGIT GEOMETRIC SYSTEM
13:3. THE NACA 5 DIGIT GEOMETRIC SYSTEM
13:4. THE NACA 1SERIES AIRFOILS
13:5. THE NACA 6 SERIES AIRFOILS
13:6. OTHER NACA SERIES
13:7. EFFECTS OF AIRFOIL GEOMETRY ON LOWSPEED AERODYNAMIC CHARACTERISTICS
13:8. APPROXIMATE EQUIVALENCE OF MISCELLANEOUS AND NACA SERIES AIRFOILS
13:9. AERODYNAMIC AND STRUCTURAL COMPROMISES
13:10. AIRFOIL SELECTION CRITERIA FOR SUBSONIC AIRPLANES
13:11. AIRFOIL SELECTION CRITERIA FOR SUPERSONIC MISSILES
PROBLEMS
CHAPTER 14. DRAG ESTIMATES AND POWER CALCULATIONS
14:1. METHODS OF ESTIMATING DRAG
14:2. DRAG OF INFINITE CYLINDERS, INCLUDING WINGS
14:3. STREAMLINE BODIES: SUBSONIC
14:4. MISSILE BODIES: SUPERSONIC
14:5. NACELLEWING COMBINATIONS
14:6. FUSELAGES ; COCKPIT ENCLOSURES
14:7. EXPOSED LANDING GEARS AND OTHER PROTUBERANCES
14:8. DRAG ESTIMATE FOR COMPLETE AIRPLANE
14:9. THRUST AND POWER REQUIRED FOR AIRPLANES IN LEVEL FLIGHT
14:10. GENERAL CHARTS OF THRUST AND POWER REQUIRED FOR LOWSPEED AIRPLANES
14:11. DRAG ESTIMATES FOR SUPERSONIC VEHICLES
PROBLEMS
CHAPTER 15. AERONAUTICAL POWER PLANTS
15:1. POWER PLANT TYPES
15:2. PISTON ENGINES
15:3. SEA LEVEL SUPERCHARGERS
15:4. SUPERCHARGED ENGINES
15:5. TURBOPROPS
15:6. TURBOJETS
15:7. RAMJETS
15:8. ROCKETS
15:9. NUCLEAR PROPULSION
PROBLEMS
CHAPTER 16. AIRPLANE PROPELLERS
16:1. PROPELLER CONSTRUCTION AND GEOMETRY
16:2. MOMENTUM THEORY OF PROPELLERS
16:3. SIMPLE BLADE ELEMENT THEORY OF PROPELLERS
16:4. PROPELLER COEFFICIENTS AND PLOTTING OF DATA
16:5. PROPELLER PROBLEM TYPES AND METHODS OF SOLUTION
16:6. CORRECTION FACTORS FOR PROPELLER CHARACTERISTICS
16:7. STATIC THRUST:SLOW VENICLES AND VERTICAL TAKEOFF AIRPLANES
16:8. DETAILDESIGN CONSIDERATIONS
PROBLEMS
CHAPTER 17. HELICOPTER PERFORMANCE
17:1. DEVELOPMENT OF THE HELICOPTER
17:2. LIMITATIONS OF HELICOPTER THEORY
17:3. HOVERING PERFORMANCE ANALYSIS
17:4. SPEED LIMITATIONS
17:5. POWER REQUIRED FOR LEVEL FLIGHT
17:6. HIGH SPEED AND MAXIMUM CLIMB CHARTS
17:7. AUTOROTATIVE DESCENT OF HELICOPTERS
PROBLEMS
CHAPTER 18. AIRPLANE PERFORMANCE
18:1. SPEED AND CLIMB OF PROPELLER DRIVEN AIRPLANES
18:2. CEILINGS OF PROPELLERDRIVEN AIRPLANES
18:3. PERFORMANCE OF PROPELLER DRIVEN AIRPLANES
18:4. PERFORMANCE OF TURBOJET PROPELLED AIRPLANES
18:5. CRUISING RANGE AND ENDURANCE
18:6. TAKE OFF CALCULATIONS
18:7. LANDING DISTANCE CALCULATIONS
18:8. GLIDINGAND DIVING
18:9. LEVEL AND GLIDING TURNS
PROBLEMS
CHAPTER 19. CONVERTIPLANES
19:1. TYPES OF CONVERTIPLANES
19:2. ROTATABLE PROPELLERAXIS AIRPLANES
19:3. UNLOADED ROTOR HELICOPTERS
19:4. OUTLOOK FOR CONVERTI PLANES .
PROBLEMS
CHAPTER 20. MISSILE PERFORMANCE
20:1. OUTLOOK FOR MISSILES
20:2. PERFORMANCE OF SUPERSONIC AIRCRAFT.
CHAPTER 21. STABILITY AND CONTROL
21:1. STABILITY AND CONTROL CONCEPTS
21:2. AXES , ANGLES , AND COEFFICIENTS
21:3. STATIC LONGITUDINAL STABILITY
21:4. STATIC DIRECTIONAL STABILITY
21:5. LATERAL STABILITY
21:6. CONTROL SURFACE CHARACTERISTICS
21:7. CONTROL AT HIGH SPEED
PROBLEMS
APPENDIX 1. NOTATION, ABBREVIATIONS, AND CONVERSION FACTORS
APPENDIX 2. PROPERTIES OF SOME LIQUIDS AND GASES
APPENDIX 3. PROPERTIES OF AIR
APPENDIX 4. COMPRESSIBLE FLOW CHARTS
APPENDIX 5. WING AND TAIL SURFACE DATA
APPENDIX 6. PARASITE DRAG DATA
APPENDIX 7. POWER PLANT AND PROPELLER DATA
APPENDIX 8. AIRCRAFT DATA
ANSWERS TO PROBLEMS
INDEX
UCS
University of
Michigan Libraries 1817 ARTES SCIENTIA
VERITAS
I 1
1
I
TECHNICAL AERODYNAMICS Third Edition
KARL
D.
WOOD , M.E. ,
Ph.D.
Professor and Head , Department of Aeronautical Engineering University of Colorado
PUBLISHED
BY THE AUTHOR
BY ULRICH'S BOOK STORE , ANN ARBOR, MICHIGAN
DISTRIBUTED
East Engin . Library
TL
TECHNICAL AERODYNAMICS
570
Third Edition
.W88
1955 First
and
Second editions
( consisting
of
16,000 copies )
about
1947 , by the McGrawHill Book Company Third edition copyright 1955 by the author .
copyright 1935 ,
All rights thereof
reserved .
, may
This book
not be reproduced
,
in
or
,
Inc.
parts
any form
without permission of the publisher .
The author also serves as Editor of the Prentice Hall Aeronautical Engineering Series of textbooks , which includes " Fluid Mechanics , " by R. C. Binder ; " Thermodynamics , " by Franklin P. Durham ; " Aircraft Jet Powerplants , " by Franklin P. Durham ; " High Speed Aerodynamics , " by H. W. Sibert ; " Elementary Applied Aerodynamics , " by P. E. Hemke ; and These books " Fundamentals of Aircraft Structures , " by M. V. Barton . are referred to frequently in this text , and the format of this text has been intentionally made as similar to Binder's " Fluid Mechanics " as is possible in a lithoprinted book . This edition is published by the author without objection from Prentice  Hall , Inc. , but they are in no way responsible for details . The author is solely responsible and would appreciate having mistakes or errors called to his attention .
Typewritten by Mrs. Erma R. Tucker  Corona Model 88 typewriter with carbon ribbon , elite and " Secretariat Elite " type , and twenty special Greek symbols quickly replaceable on on Smith
keys number 42 and 43 .
Lithoprinted
ratio
with photographic reduction
4 : 3 by Cushing  Malloy , Inc. Ann Arbor , Michigan
in
of
the
8715154
PREFACE TO THIRD EDITION
this edition , like that of the two preceding editions , is to provide a course of study for the engineering student ( or a re fresher course for the practicing engineer ) which will help fit him to The object of
the
make
performance ,
in
current importance With
stability
design calculations of the aircraft manufacturing industry .
an objective ,
such
and
aerodynamic
successive
editions
should
reflect the
manpower , distribution of engineering this basis , this third edition reduces the attention paid to light airplanes and seaplanes and increases the attention paid to jet propelled airplanes , jet propelled missiles , and helicopters and
present , and expected near future
activities
.
helicopter
airplane
On
is almost completely out of propeller the is not , particularly
The biplane
combinations .
picture ,
development
the current but the turbine driven propeller . In the speed spectrum , the techniques of breaking through what was formerly called the " sonic barrier " of " Mach Number
sile
= 1" have been
designers are
now
well established struggling
and
supersonic airplane and
mis
to see how far they can penetrate into
the " thermal barrier " which begins to be formidable at a
Mach
Number
of
about 3 .
With the above increased scope of technical the last edition permissible
ritory
,
is
The development
) .
effort since
principles involved and basic also increased ( in spite of omissions now
the scope of fundamental
data necessary
aerodynamic
aerodynamic
in virgin
engineer continues to work
ter
try to design on the basis of insufficient information , and thus continues to direct the research laboratories , now calling espec and to
ially for
more
in the in the field
information
transonic
field
of mixed subsonic and
in which both the compressibility of air are major factors , and for which no simple scientific analyses are available . It is appropriate here to abstract the definitions of " scientific " and " technical " sometimes attributed to Kettering of General Motors , who supersonic flows and
the viscosity
is
don't understand
it . "
layers
,
and
reported to have said ,
stands
of hot boundary
it
; when
"When
we say a
we say
it is
iii
thing
is
' scientific ' we mean we
' technical ' we mean
nobody
under
PREFACE TO THIRD EDITION
In calling
author accepts Kettering's implication of incomplete understanding as part of the nor the book
" Technical
Aerodynamics , "
the
of the development engineer , but hopes that the areas of approximation reasonable will be properly differentiated from those of mal handicaps
reasonably exact knowledge
the text .
in the design field by " Airplane Design , " 10th Edition ( 1954 ) , distributed by the University Bookstore of Boulder , Colorado ; the 11th edition , scheduled for 1956 , is to be distributed by Ulrich's Book Store of Ann Arbor , Michigan . This text
is
in
supplemented
K. Boulder , Colorado June , 1955 .
iv
D.
WOOD
PREFACE TO SECOND EDITION
Twelve years ago the
of the
cause
this edition
years ,
by the
Not
field
is
45 lessons ,
this
book
The student
is
assumed
instruction in
and mathematics ,
Design , " 8th
in
He may
one
understanding of
This edition Colorado , who
is
dedicated
in
covered
to the
asked the questions and
 especially
,
have taken courses
to Mrs.
Louise
Cornell ,
Purdue
and
the author help them find the
the preliminary H.
the
1.
students at
made
in fluid
necessarily
these subjects needed for Chapter
col
physics
before under
to the University of Colorado V  12 students
fered with the author through thanks are due
of
in
background
good
these courses are not
the fundamentals
this text are
two such courses .
course in calculus
profitably also
and thermodynamics , though
prerequisite , since
for
four sixteen  week terms of
to have completed at least
college
;
first edi
Whereas the
intended to supply material
including
distri
Edition ,
three  credit course of about
a single
engineering and to have a
study .
this
mechanics
answers
" Airplane
has also been extended .
tion was intended to be covered
taking
the last twelve
the University Bookstore of Boulder , Colorado . only has the fund of technical information expanded
instruction in this
lege
in this field in
Be
first edition only in the chapter head text is supplemented in the design field
lithographed text
author's
buted by
This
.
was published .
the
resembles
topics covered
ings and
first edition of this title
important developments
many
second
edition .
Beattie for assistance
in
who
suf
Particular
preparation of
the manuscript .
K. Boulder , Colorado June , 1947
.
1 is due to the high pressures on the front of the body . The base pressure drag , due to the dead  air region seen
tion .
Since skin
in the firing
drag
of such blunt objects
ranges
7 :7.
in Fig . 7:19, makes a minor additional contribu friction and heat transfer are also only minor factors
fairly
are
CALCULATION PROCEDURE
friction for thin
The skin
,
consistent FOR
wedge
if
reasonable assumptions
surface roughness
.
ESTIMATES OF SUPERSONIC
 shaped airfoils
sharp conical noses can be estimated
ter
the data obtained in wind tunnels and
can
SKIN FRICTION .
for cylinders with from the data in the foregoing chap be made as to free  stream turbulence , ,
and
surface temperature , and surface Mach number , so that extent of the the laminar boundary  layer can be estimated and the drag ,
distributed properly between the laminar and turbulent boundary layers . On the other hand , for circular arc airfoils , and for ogival noses involv ing substantial pressure gradients , the estimation is considerably in doubt , chiefly because of the uncertainty of the effect of stream wise pressure gradient
,
for
which
no
data are given
expansion , as on a double  wedge
in this chapter . Beyond a corner airfoil or a cone  cylinder junction , ex
that the corner usually serves as a trip to initiate a turbulent boundary layer one does not already exist . An example of perience has
shown
if
the calculation of skin
friction
on a double
 wedge airfoil follows :
Example . For the 100 semi  angle wedge airfoil of Fig . 5 : 5 , for which surface pressures and velocities were calculated in Chapter 5 , estimate the skin friction and initial rate of heat transfer per foot of span the airfoil chord is 60 in . , M1 = 2.5 , and the air is standard sea  level at point ( ) 1 .
if
Solution . as calculated
(1 ) ,
Assume
in
air
Chapter
Hoerner , Sighard F.
conditions
in
the free stream near the surfaces
5 , namely :
" Aerodynamic
Drag . "
Published by author ,
1951 .
TECHNICAL AERODYNAMICS
718
Front :
M2
=
2.09 ;
T2 = 623 °R ;
M1
M3
M2 Tw
= 519
/
; P2 = 3940 lb ft2 P2 = 0.00369 slug = 1226 a2 sec ; M3 = 2.95 ; T3 = 425 °R ;
ft/
°R
Rear :
/
P3 = 1142 lb ft2 ; = 0.00143 slug P3 = 1013 / sec . a3
30 "
30"
ft
First
/ft3 ;
/ft3 ;
estimate the transition point location : This requires calculation of free  stream Reynolds numbers near the surfaces . For the front faces , with T2 = 623 ° R ( 163 ° F ) , read in Fig . A1 : 1 or A1 : 2 , H2 = 0.43 / 106 lb sec / ft2 ; calculate v2 = H2 / P2 = 0.43 / 106 x 0.00369 = 0.000117 ft2/ sec ; calculate V2 = M2a2 = 2.09 x 1226 =€ 2560 ft / sec ; calculate_Rex /x = V U = 2560 0.000117 = 22 x 106. Hence on the front face Rex / 106 = 22x , where x is the distance in feet from the lead ing edge . In Fig . 7:16 , read for M 2 and Tw T1≈ 1 ( " experimental trend " ) a transition region from Rex 106 = 1.3 to 2.5 . Since this region equals 22x as calculated above , solve for x = ( 1.3 to 2.5 ) 12/22 = ( 7 to 14 ) in . from the leading edge . For the rear faces , with T3 = 425° R ( 35 ° F ) , read in Fig . 41 : 1 or A1 : 2 , = = = 0.32 106 x 0.00143 = из 0.32 106 1b sec / ft2 ; calculate v3 13/03 0.000224 ft2 / sec ; calculate V3 = M3a3 = 2.95 x 1013 = 2990 ft sec ; calcu late Rex x = V U = 2990 0.000224 = 13.3 x 106. Hence Rex 106 = 13.3x on rear faces . Next estimate the skin friction coefficients and skin friction : For the laminar boundary layer on the two front faces , read in Fig . , at Rextr = 1.3x 106 as estimated above 20f 7 : 6 , interpolating for = 0.0023 , and consider that this is the mean skin friction coefficient for the area 7/12 ft2 per foot of span with a value of 92 = P2V22 2 = 0.00369 x 25602/2 = 12,100 lb/ ft2 . Calculate D2 Lam = 0.0023 ( 7/12 ) 12,100 = 16.2 lb. For the turbulent boundary layer on the front faces , the practice is not well established , but a good estimate is obtained it is considered that the turbulent boundary layer starts at the beginning of transition . = Hence x = 30 in . 7 in . 23 in . = 23/12 ft and ReL Turb / 106 = 22 ( 23/12 ) = 42. Read in Fig . 6 : 5 or 7 : 6 for Re = 42 an incompressible value of 2Cf = 0.0048 . In Fig . 7 : 7 , on the line labeled Tw = Too ( T1 ) , read a com pressibility correction factor of about 0.9 for M = 2. With the same value of q as before , calculate D2 Turb = 0.0048 ( 23/12 ) 12,100 x 0.9 =
/
/
/
/
/
/
/
/
/
/
/
M2
/
if
100
lbs . For the rear faces , the
layer is always turbulent , as the to start turbulence it did not already exist . The Reynolds for the turbulent boundary layer may be cal culated as started at the corner ( though the practice is not well established ) . Hence ReL3 106 = 13.3 ( 30/12 ) = 33. For this value of Re in Fig . 6 : 5 or 7 : 6 read an incompressible value of 2Cf = 0.0050 and in Fig . 7 : 7 read for the " cold model " condition [ since Tr3 = 425 ( 1 + 0.2 x 2.952 x 0.88 ) = 10800R compared with Tw = 5190R ] a compressibility cor rection factor of about 0.83 . Calculate 93 = P3732 2 = 0.00143 x 29902/2 = 6400 lb ft2 and D3 Turb = 0.0050 ( 30/12 ) 6400 x 0.83 = 66 lbs . The total skin friction drag per foot of span is thus , to the nearest lb : = = 16 + 100 + 66 = 182 lbs . D2 Lam D2 Turb + D3 Turb Dskin friction This is one of the answers called for .
if it
/
boundary
if
trip "
sharp corner serves as a
" number
/
/
(
COMPRESSIBLE AND HEAT
 CONDUCTING
BOUNDARY
LAYER
719
FLOW
To estimate the heat transfer , use equation ( 7:11 ) and calculate the Stanton number corresponding to each of the three skin friction coeffi cients estimated above and tabulate the results as shown below . Use Pr St Re Pr = hcL ke , and evaluate kc at free stream tem = 0.715 and Nu perature from kc = Hcp / Pr .
/
Surface S
B.L.
No. 2
2 3
St
Cf
.58 Lam .0012 1.92 Turb .0021 2.50 Turb .0021
.00075 .0013 .0013
Re 106
1.3
Nu
Too
103
R
0.7
42
39
32
29.7
L
kc 106
623 .43 623 .43 425 .32
106
ft .
hc 103
4.65 .58 5.6 4.65 1.9 96 3.45 2.5 41
Total rate of heat addition
Tr Tr Tw
R
1090 1110 1080
/
, B sec
Q
2 109 560 57 570
590
168
foregoing heat transfer calculations are based on mean coefficients for the three areas considered ; more accurate estimates are obtainable from local coefficients available in the references cited . Note that heat is being added to various parts of the wing structure at widely varying rates , so that only at the start of a flight can the body surface temper ature be considered uniform . Major structural problems arise from the temperature gradients as well as the high temperatures .
The
PROBLEMS 7: 1. shown
For the wedge
for which
airfoil
surface pres
sures and free  stream tem peratures were calculated in Problem 5: 3 for a = 0 , M1 = 3, and standard sea  level air at point ( ) 1 , assume the wing surface is at a temperature
M1
M2
calculate ( estimating necessary ) the follow 2'0" 2'0" ing : ( a )the transition point location , (b ) the skin tion per foot of span for each surface , and ( c ) the rate of heat transfer from the air to the wing per foot of span for each surface , ( d ) the rate of temperature rise at points 6 in . and 36 in . from the leading edge the wing is hollow with a thin skin of aluminum alloy 24S of thickness 0.065 in . and specific heat 0.23 , neglecting heat transferred from the skin to the internal structure , and neglecting heat transferred between adjacent chordwise stations on the wing . 7 : 2 . Repeat Problem 7 : 1 for the cone  cylinder combination solved in Problem 5 : 4 , for a cone length of 10 ft 0 in . T1 and
where
fric
if
illustrate one step of the series of numer involved in calculating the time  history of the motions is evident that sub and temperatures of a given supersonic missile . stantial progress in missile path and temperature calculation requires a tremendous number of calculations , at present considered feasible only digital computing machines . on automatically programmed NOTE :
The above problems
ical calculations
It
CHAPTER AERODYNAMIC
8
TEST FACILITIES
TESTS . An aircraft (airplane , missile , or helicopter ) is for transporting persons or things rapidly through the air from place to place under the control of a pilot , human or electro  mechanical . The aircraft flies at an altitude or sequence of altitudes determined by FLIGHT
8: 1.
a vehicle
the
pilot .
Measures
its
of
it will travel it can fly , and
merit are the speed at which
chosen altitude , the range of altitudes at which the time required to get from one altitude to another . Quantities
at the
tomarily measured in and
this
connection are :
ceiling (maximum altitude at which the aircraft
can
fly ) .
and landing times , distances , and speeds are also important factors
uting to
the merit of the aircraft .
calculate the mance
in
Manufacturers of aircraft
,
 off contrib
Take
usually
of performance and often guarantee the perfor of a financial penalty for failure to meet the calculated
above items
terms
performance or a
financial
bonus
The purpose of conducting
to establish
cus
level high speed , rate of climb
compliance
for
exceeding
flight tests
with performance
the guaranteed
on performance guarantees , and
is
performance .
normally ( 1 )
( 2 ) to determine
information necessary to permit effective use of the aircraft . test for level high speed at altitudes near the ground can nor
performance
A flight
mally be run by having observers on the ground keeping
continuous
on the location of the aircraft at time intervals and culating speed from the ratio of distance interval to time interval . ground tracking may be done visually ( photographically ) as in Fig . predetermined
Fig .
8: 1 .
Photographer tracking Courtesy NACA . .
aircraft flight
81
tally
cal Such 8 :1,
Fig . 8 : 2 . Ground radar tracking aircraft flight . Courtesy NACA .
AERODYNAMIC
The
of
level
change
flight
of
of altitude condition
.
Fig
is ,
16
14
12
10
130 mph
Observed rate of climb for various constant air speeds From Reed JAS February 1941.
(
indication
unreliable
rate
Time min .
an
17,000
.
a
punctured barometer , and gives
.
indicator " consists of
climb
mph 17,500 120
8
The
,
.
160mph mph 140
)
ditions
6
level con usual " rate of
under exactly
18,000
2
fly
it is difficult to
18,500
3 .
altitudes
,ft
,
bar
At higher
altimeter .
for missiles
flight
are normally measured by ometric
indispensable
19,000
Pressurealtitude
Altitudes of aircraft
is
This procedure
8 : 2.
82
8 :
in Fig .
which carry no observer .
,
or by radar as
TEST FACILITIES
in many cases , determined by plotting full throttle rate of climb against
horizontal
speed
determining the intercept
and
A for zero rate of climb . flight plan for testing climb performance is sketched in Fig . 8: 3.
The
best rate
principle
is
of climb
that the
may be
termined by a series
de
of short ,
steady climbs at predetermined
air speeds . Full throttle climb
is
usually determined
at
sev
eral altitudes and a sufficient number
of air
that the best has
been
speeds to insure
climbing speed
bracketed , and that
.
.
8 : 4 .
.
Recording equipment being Fig accurately formed curve of installed in test airplane by NACA scientists rate of climb against air speed Courtesy NACA is obtained in the region of best climbing speed . Accurate flight testing requires not only good weather but also , as 1 ) (1 ) complete and accurate calibrated instrumentation stated by Allen : (
an
and Stability . " " Flight Testing for Performance ( Allen and his crew were killed a short time later while flight  testing a military aircraft for Boeing Airplane Company ; have since been named in honor of the Boeing Aeronautical Laboratories
( 1 )Allen ,
Edmund
JAS , January , 1943.
test  pilot Allen . )
T.
TECHNICAL AERODYNAMICS
83 so that
all
variables
(2 ) a technique of flying accur
can be measured ,
so that what is measured will be representative of the true optimum performance of the aircraft , and ( 3 ) a mathematically correct method of
ately
interpreting the flight  test
data .
A major step in the development of
flight  test
accuracy comparable with
of techniques for automatic re airplanes helicopters In or this is usually done by
wind tunnel accuracy was the development
cording of
all
data .
collecting the instruments in a panel or by motion picture camera . This device stallation of
is
such equipment
shown
box where they can be photographed
is
as a photorecorder
known
in Fig . which
;
in
In missile tests , in
8: 4 .
photographic records may be landing , instruments are
damaged on
used which
to
radio
send
their readings by
ground based recorders .
This type of instrumentation
is
in
as telemetering . A model strumented with telemetering equip
known
ment
is
shown
in Fig .
8: 5.
Instrumentation for flight test ing usually includes
for
instruments
measuring the pressure and
perature of the
air , the
ten
vertical
horizontal components of velo city of the airplane , and the thrust of the propeller or jet power plant . and
Such instruments must be accurately
Fig .
8: 5 .
Drop test model
instrumentation
.
Courtesy
gravity of the airplane
with NACA .
calibrated to eliminate instrument error . The weight and center of
must also be accurately
recorded since the weight
varies in flight as fuel is consumed . A continuous check on the weight of fuel in the airplane is required for accurate weight calculations . Propeller or jet engine thrust is extremely difficult to measure accur
ately
and
is
The
air  speed
can be done only meter may
air flow , fuel flow , and motor tied in with similar measurements
accordingly often derived from
or rotor rpm measurements , which can be made on a test stand on the ground . meter must be
in flight ,
particularly carefully calibrated
as a wind tunnel calibration of an
differ substantially
from that
of the
same
equipment
.
This
air speed in flight
84
TEST FACILITIES
AERODYNAMIC
effect of the location of the pitot head relative to the airplane wing or body . Air  speed meters are usually calibrated by runs
because of the
in
is
In
which the time
measured
Wind effects must be allowed for by running
estimating the performance of a type of
previously
results
for
a given distance
of in both directions and proper corrections for air density included in the plot of the cali bration . A procedure for planning airplane flight tests so as to check conventional performance calculations is outlined at the end of Chapter 18 . over a speed course
flight .
flown , the most reliable procedure
on the most
aircraft for which
aircraft different from any is to use the flight test
nearly similar
flight  test
data
are available , and to estimate ,
on
the basis of other aerodynamic data , the effect of changes in design on changes
in
performance .
For aircraft differing widely from any previously flown , it is usually necessary to supplement mation
with tests
this infor
on scale models
of
the proposed new aircraft , including an
estimate of the effect of
scale of the model and of the ence between the model
the
differ
test condi
tions and the flight  test conditions
.
Models are constructed as accurately
to scale
as shop conditions
permit ,
ef
with proper consideration of the fect of accuracy of model construc
tion and
on cost of the test program , provision is made for moving the
relative to a body of air or moving the air relative to the model . model
8:2.
MOVING MODEL TESTS .
Models
Fig .
have been tested by dropping from a
great height
facilities .
;
the
Eiffel
Tower
More recently
in Paris is
drop  tests
8:6 .
Research
one
model on rocket
Courtesy
NACA .
of the earliest
drop
launcher
.
 test
from airplanes or balloons . Models have also been mounted on automobiles or airplanes or on a whirling arm. For high speed tests models are sometimes shot from have
been made
TECHNICAL AERODYNAMICS
85 guns or propelled
by rockets and tracked by
instrumented drop  test model model (a
"bird " )
is
shown
is
on a rocket launcher
of forces
The measurement
A typical
radar or camera .
in Fig .
typical free  flight
8 : 5; a
in Fig .
shown
is
on moving models
8:6.
inherently
inaccurate
,
either on account of the unsteadiness of the motion and poorly controlled conditions , as with models mounted on airplanes or automobiles , or because the forces can be only be
inferred from
flight
Most accurate force measurements can be
models .
stationary
model and moving
as a wind tunnel
tions
partially measured directly , and some forces must in flight path , as in the case of drop or free
changes
around
,
air stream ,
but since no
and
this type of
wind tunnel can duplicate the
full  scale , free  flying aircraft
tial error in interpreting the flight characteristics . Hence ,
obtained with a
wind tunnel
there
,
is known
apparatus
is
always
test data in
air condi substan of free
a
terms
both types of measurements
large
may have
errors but of different sorts . The most authentic predictions are made by using the results of both moving  model and moving  air  stream types of tests
when
available .
the order of
Much
time
and
fifty
effort in the last
billions of man hours ) have been devoted aircraft by the world's sovereign
to
years (of
flight  testing
nations as part of a program to develop weapons to fight
and model  testing of
wars , to determine
if
tions , 8 : 3.
any ,
which
shall
WIND
na
of the
endure .
Hun
TYPES .
TUNNEL
dreds of wind tunnels have been
built
air
past
for the a model
of
purpose
while
on the model .
moving
making One
measurements
of the most com
prehensive surveys of the
current
status of the world's wind tunnels is given by Pope , ( 1 ) and abstracts from Pope's summary of available wind tunnel facilities are given in Tables 8 : 1 and 8 : 2 . The exterior view of
Fig .
8:7.
tunnel at
Low Ames
( 1 ) Pope ,
 turbulence
pressure Laboratory of NACA .
Allen .
" Wind Tunnel
is
shown
a
typical
in Fig .
NACA wind
section of a similar
Testing
."
Second
tunnel
8 : 7 , and the cross NACA
tunnel
Edition . Wiley
is
, 1954 .
TEST FACILITIES
AERODYNAMIC
86 splitter vanes
Continuous
Drive motor
1
Propeller
Countervanes
Air

stream direction

Pressure gradient control slots
ANIU
Guide vanes .
Cooling coils
Outer shell of test chamber
Blower

60 mesh screen
Air lock

30 mesh turbulence reducing screens
canopy
Observation
NACA
Test section
Langley two dimensional low turbulence pressure Courtesy NACA

tunnel
.
.
The

Fig . 8 : 8 .

:8 9
a
a
,
though
in
few wind tunnels the return flow is undirected in the surrounding building or neigh
tunnel
is
less efficient
circuit

closed
.
Some
.
in Fig
as those shown
all of
tunnels because
open
circuit
tunnel
An
does not require the cooling
but
:8 8
.
such
as an
known
.
a
is
Such
circuit tunnel
borhood
shown
to the random conditions

left
is
wind tunnel
,
,
another point being
and
a
.
through
a
air flows
of another
are typical of current practice in wind in that closed circuit is provided in which the test section at one point and fan or propeller at
8 : 8
.
: 9 .
.
construction
tunnel
The cross section
Figs
8
in Fig
8 : 8 .
in Fig
shown
open
coils
sort of cooling is necessary on
most
the energy supplied by the motor to
Direction of flow
Sloping ceiling
'
80
Screens
'
12
19
'
cone
Settling chamber
Entrance section
NACA
4
Longitudinal cross section of the foot wind tunnel National Bureau of Standards Courtesy NACA .
.


9 .
8 :
.
12
'
Exit
Test section Fig
25
JOE
.
!
'.
20
'
112
Upper return duct
of the
S.
# Rect
8x10
1.1D
Round
9
70 96
96
70 96
85.2
Rect Rect Rect Rect Rect Rect
7x10 8x12 8x12 7x10 8x12 7.8x11
8
U
..
U.
.
) (
Round
1.09B 1.0D
325
 165
1
est
314
1.25B
111
1+
1+
1 1 1
10
200 160
250
0.835 1.255 2.0D
125
0.25
1
1.0B
250 100
1 4 1 1 1
78.5
59 45
Ellip Octa
.. . . .. 10x7.5
150
250
11
1.5D
0.8D
78.5 63.5
Round
10
1.3B 1.3B
... 80 80
70 200 180
LAAT 1 1 1 1
2.5
1.2
3.6
5.5 .8
.7
4.2 1.2
0.7
0.02
1.12 0.1
3.3 2.5
3.2
5.5 1.6
2.5
1.2
0.15 0.15
0.35 0.7 0.3
0.2
3.0 1.0
1.275 2.25
2.25
1.5
0.25
1.2
0.375 2.0
0.8
2.52 0.2 0.5 1.0 2.5 0.02
6.5 3.8
7.0
1.0
7.0
3.5 6.2
3.7 3.5 1.0
3.56
4.0
0.3 4.0
3.5
1.6 40.0 1.1
6.8 8.5
36.0 0.075
2.84 4.6
6.8 18.0
8.0
1.4
2.3
E.R.
1.6
0.9
0.02
0.05 0.6 1.6
2.0
0.02 1.4 1.1
10.2 8.0
2.5
103
Max
turb_hp_
Min
1.5 10
1.8
1
Acft
2.5B
Rect Rect
6.33 8x8 8x10
300 425 120
1 1
4
24 Consolidated Vultee 25 Grumman Acft 26 Lockheed Acft.Co. 27 North American Acft 28 Northrop Aviation
1.03
64
Round Round
11
21 2
19 Georgia Inst of Tech 20 M.I.T Wright Bros. 21 of Michigan 22 New York 23 of Washington
3.6D
19.6 314 126
Round
  . . . . .  
20
250 100
0.27
111
17 DTMB No.2 Navy and Industries Universities 18 Calif Inst of Tech
Ellip Octa
120 500 300
CARL
Field
4.2D
70
2520 14.4
Rect
7x10 40x80 4.5
Rect
Cleveland
2.3 10
120
1 1 1 1 1 1 1 1
Wright
0.93 2.0D 1.4B
1410 314.2
Ellip Round
30x60 20
7x10
360 300
350 76
Lab
Max
3
13 WADC Wright Field 14 Wash.Navy Yard No.1 15 Wash Navy Yard No.2 16 DTMB No.1 Navy
3.7D 1.5B
36 70
6x6
5
Rect Rect
Lewis 45 260
mph Min
Max
Reft 106
335321 .ya 6
12 WADC
D.C.
(1
U.
) ) )C( ) A( (
A
S.
scale
)L( )
Altitude Ames 7x10 10 Ames full scale 11 Nat Bur Stds Armed Services
full
# 1
High
7x10
Langley
)L(
Stability Langley
1.5D 1.0h
is
0.75B
.. 22.5 24
3x7.5 4x6
284
19.6
1955
Pressure atm
0111
U.
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LTT
,.)L( )L L ( ,. (1 )L( )L(  . .
Press
)A(
Dim
Round Rect
Calif Round
ft
vel
IN
1 1
Vertical
5 19
Lab
,. .
Pressure
Ames
.;
ft
is
:
)C( 4
Two
.;
sq
Length
USE Max
. .. , ..
19
Va
.
Langley Lab density
1 8
is
Shape
Area
IN
% 9
Variable
ft
Size
TUNNELS
.
N.A.C.A.
Tunnel
WIND Section
.
and
Test
SOME LOW SPEED
.
Agency
TABLE
87 TECHNICAL AERODYNAMICS
1 11
.
.U) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )1()2( )3( )4( )5( )6( )7( )8( )9( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
88
Lab
6
Transonic
GDF
PWT
19 AEDC
20 AEDC
18 NACA 6x8
21
Supersonic
BuOrd
Daingerfield
Oaks
Mass Minn
Tex
Md White
Tenn
Md
Conn
8.5x12 8.5x12 8x12
102
Rect
Rect
Rect Rect
1.3x1.7 1.3x1.4 1.6x2.3
,
,
1.5x2 1x1
0.8x1
Rect Rect
Rect
11
5.0
to to
2.5 3.0 4.0
to to to to
3.7 0.75
1
3
8.0
4.3
to
5.0 4.4
1.8 to
to to
5.0 4.5 2.2 2.0
to to
00.95
2.0D
0.25 0.2
2.0 4.0
10.0
4.0
16.0
13.0
100.0 216.0
50.0 100.0
10.0
83.0
54.0 7.0
45.0
0.350 100.0 14.0 01.2 01.2 01.2
01.24 0.81.6
27.0 40.0 2.0
16.0 25.0 11.0 60.0 200.0 0.12
hp 103
00.95 01.22
1.5B 1.16B
1.25B
2.0D
Max
Max
01.5
to
1.8
256 2.1
Rect Rect
6x8 3.3x3.3 16x16
Rect
1x3 6x6
16
36 48
Rect
2x2
102 96
Rect Rect
Rect
4x4
Rect Rect Octa
... . . . . . .. . . . . . . 43
.. ., ,
24 JPL 12 in Pasadena Calif 25 MIT Naval Tunnel Cambridge 26 of Minnesota Rosemount
Navy
Army BuOrd Aberdeen Lab 22 Naval Ordnance
Tullahoma
#
23 Universities
Va
Va
Calif Ames Lab Calif Ames Lab Ohio LFPL Cleveland Tenn Tullahoma
#
Supersonic
ft
ft
Field Field
Calif
256
78.5 0.2
Round Round Rect
121 201
Rect Round
1/6
Min
Pressure atm
TEST FACILITIES
16 NACA 1x3 17 NACA 6x6
Langley Langley
Co
ft ft ft
Buffalo
Pasadena
Y.
8
14 NACA 4x4 15 NACA 2x2
op
8x12
U.
Lab
Wash Seattle 8x12 12 Boeing 8ft East Hartford Aircraft 13 United and Armed Services S. Government
Calif
Aero
N.
Southern
0.5 16x16
10
11x11 16
00.97 01.1
113 201
Round Round 1.5D
01.3
01.2
Range
No.
Mach
64
1.5D
Length
1955
Poly Rect
.
11
10 Cornell
S.
Ohio Tenn
U.
WADC in Transonic Dayton Tullahoma AEDC PWT Transonic and Industries Universities
ft
Ohio
Calif
8x8 12 16
Area
.
Dayton
Ames
Va
., .
Langley Field Va Calif Ames Lab
Va
8
Field
Shape
IN
6
WADC 10
Services
ft
Size
..
NACA 16 NACA 11 NACA 16
Armed
Langley Field Transonic Transonic Pres Langley Calif Ames Lab
and
Location
Test
USE
IN Section
TUNNELS
.
ft ft ft ft ft
ft
ft
and
. . . . , , . , , . , .. , , . , , . ,. . , . . , . , , ,. . ,. . , ,. , ,1 , . ,. ,. , , , , , , , , . , . , , , ,1, , , , , , "A " , .. . . . . . .
Tunnel
WIND
. .
NACA NACA NACA 12
. :
Government
2 8
S.
AND SUPERSONIC
.
U.
SOME TRANSONIC
.
Agency
TABLE
. AERODYNAMIC
88
,,
TECHNICAL AERODYNAMICS
89 the propeller
is
the motor
is
air
started the
is
dissipation
delivered in the form of heat to the
,
coils
either cooling
and
or air
test section is usually the smallest area in the circuit
The
followed
by a gradual
settling
chamber , which
all
expansion
is
the
ual as
circuit
around the
way
and
is
to the
followed by a comparatively short contraction
or entrance section ahead of the throat .
The expansion
is
made as
decreasing velocity
permits because the region of
economy
after
To avoid excessive equilib interchange must be provided .
equal to the energy supplied .
rium temperatures
air stream
continues to rise until the heat
temperature
grad
is
one
of increasing pressure and "unfavorable " pressure gradient as regards the separation and boundary  layer development . Since the pressure inside the throughout
tunnel varies
is at
circuit ,
atmospheric
or
done by leaving a narrow
gap
that
some
This
is
In
point
the
a few low  speed wind tunnels the
the tunnel
must be so constructed other predeterminable pressure , the tunnel circuit at that point .
some
in
test section is
a point
of atmos In tunnels with air  exchange cooling , the atmospheric pressure point is at one of the large sections , resulting in below atmos pheric pressure at the test section . Some tunnels , like Fig . 8 : 8 , have made
pheric pressure .
provision
all
for maintaining
increase the
air
density
and
sections at a high pressure
if
desired to
number of the tests . for maintaining below atmospheric pres been designed for a small fraction of at
therefore
the Reynolds
Such tunnels often have provision
all
sureat
mospheric
points and
some have
pressure at the test section
This permits higher speeds for a
.
given motor power and higher Mach numbers .
for a pressure range from
been designed
providing high Reynolds
number
one
Many of the newer tunnels have
 quarter to four
atmospheres ,
at the high pressure and high
Mach
number
at the low pressure .
principal classification
The
of
wind
tunnels
is
according to
maximum
thirty
section . Table 8 : 1lists about wind tunnels designated as low speed wind tunnels , and nearly all of them have a maximum speed through the test
speed through the test
lists
nearly
termine
of
section of less than
400
miles per hour
an equal number of higher speed wind
compressibility
or
Mach
number
speeds of over 400 miles per hour
miles per hour
effects
Table
8:2
tunnels intended to
de
, and
all of
.
them
are capable
though none are capable of over 2,000
limitations to be discussed later . In fact , there is no expectation of obtaining test speeds over 2,500 miles per hour except in free flight , regardless of the Mach number attained . on account of temperature
TEST FACILITIES
AERODYNAMIC
810
the test section size and the maximum speed obtainable it as well as the maximum and minimum pressure obtainable at the test section for pressurized tunnels . Values are also given of the maxiTable
8 : 1 shows
through
,
mum Reynolds imum
obtainable per foot of reference length
number
turbulence obtainable with existing
free flight
ally in
Reynolds
screen
fast airplane
number of a
the range from ten to twenty million
,
tion of Table 8 : 1 that none of these low  speed of duplicating full  scale free  flight Reynolds pressurized tunnels can closely approach it .
and the
,
installations
.
min
Since the
based on wing chord
is usu
it is
evident from
inspec
wind
tunnels are
capable
numbers , though some
of the that only a very
Note also
few of the wind tunnels provide an air stream of turbulence less than 0.1 % indicated in Fig . 6 : 6 as necessary to get away from major turbulence
fects on skin friction .
Table
8:1
lists
also
ef
the
maximum horsepower
nec
essary to drive the wind tunnel motor expressed in thousands of horsepower and the energy ratio calculated from the equation E. R. =
which

is
in
the ratio of the energy Note
in
Table
the fan
is
mounted as
P AtVt3 1,100
,
(8 : 1 )
Bhp
the jet to the energy supplied by the
that the values of energy ratio given for the typical wind tunnels listed run from 1.0 to 8.5 . In general , a high energy ratio is a measure of the excellence of the wind tunnel design since high energy ratio means low power for a given throat speed and size . drive motor .
In general ,
8 :1
far
from the throat
turbances due to the fan are propogated
installation is seen in Fig shown in . 8 : 9 has the fan too
Such an
The major
the skin
circuit is head loss
with fixed
made up
in
by
each part
area
at
The fan efficiency
well
[
head
the tunnel
it is
/
is
for best results .
wind tunnel
/
=
k
=
APf
=
in
the
that
closed
disc .
The
that part
;
(8 : 2 )
kat throat
velocity at throat , ft/ sec pressure loss coefficient
is defined
to v2
fan
is
also true that
= pv2 2 = dynamic pressure at
Vt
as downstream .
= P + (pV2 2 ) ] around
proportional
dis
The older tunnel
8:8.
rise in pressure APf across the
a
of
ratios
as
near to the test section
in total
loss
upstream
sketch of Fig .
feature of a closed circuit
aerodynamic
friction
the
as possible , as
by
TECHNICAL AERODYNAMICS
811
QAPf 550 Bhp
M₤
of air flowing
where Q = quantity
properly designed value of
8 k (E. R. )
circuit in ft2 /sec .
around the
fan ŉf can usually be
(8 : 3 )
this
about 0.8 and assuming
made
for the wind
can be calculated
nf , pressure drop
With a
coefficient k tunnels listed in Table 8 : 1 from the values of energy ratio there given . Wind tunnels must be expensive to give accurate results . A study of
the cost of
the wind tunnels listed
in Table
8 : 1 shows
that the tunnel
cost per square foot of throat area , including housing , balances
trols , the
in
expressed
tunnel throat
maximum
ing the results to
of the
number
error
tests ,
(mph VAt large
8 :4 .
dollars is
1955
TRANSONIC
Reynolds
to
requires large expense
AND SUPERSONIC
WIND
pressure
rise
The pressure
and replacing and
cities in
be sonic
the expanding
a " bump "
in
surface of the
building
corresponding to
is
small
;
small
Many
in
subsonic wind
tun
parts of the tunnel by
Chapters 3 and 5 show
in
about twice the test section and supersonic
,
Some
.
subsonic wind
velo tun
test section consisting giving local velocities near the a new
Obviously , this can only be
M = 1.
some
the settling chamber . Also obviously
will
is
region beyond the throat
done with tunnels designed to stand
ocity
the Reynolds
TUNNELS .
developed
the subsonic test section , bump
if
velocity at the throat
nels have been run supersonic by
of
large
(mph VAT )
the motor with one adequate to drive the fan .
flow relationships
will
is
correct
the fan with one designed to give greater
that when the settling chamber pressure pressure there
con
.
nels can be redesigned to give supersonic flow the simple expedient of replacing
, and
where mph
The error in
is
number
proportional
mph² ) ,
+ 0.1
$ ( 500
speed in miles per hour .
full  scale which is )
about
appreciable internal pressure
, a smooth shock
 free
supersonic
in
vel
not normally be
obtained in the expanding area downstream of the throat unless it was specially designed for such conditions . Table 8 : 2 lists a number of transonic and supersonic wind tunnels cur
rently in operation
.
The transonic
tunnels are , for the most part
,
over
speeded and redesigned subsonic wind tunnels . The supersonic wind tunnels have
usually been
designed
for
a
particular
Mach number
or with interchange
able or adjustable test section walls for a range of Mach numbers . The design range of Mach numbers is specified in the table , as well as the
size
and shape
of the test section , the
pressures and the
maximum
horsepower
minimum
required
.
and maximum Because
of
test section the tremendous
amount
of
tunnels
, most
812
TEST FACILITIES
AERODYNAMIC
for continuous operation of large supersonic wind supersonic tunnels are either small or operate intermittently .
power needed
The most rudimentary form of intermittent
tank of compressed
air
speeds beyond the hole when the
The
tank pressure
atmospheric pressure ( greater than
30
intermittent supersonic jet
is
supersonic wind tunnel
it .
with a hole in
air
a
at supersonic
flows
is
greater than about twice lbs per sq in . ) . An equally rudi
is
obtained on puncturing a vacuum tank evacuated to less than one  half of atmospheric pressure . Such equip
mentary
useful for small scale teaching demonstrations , but does not per mit accurate measurements of forces , pressures , or temperatures without
is
ment
refinement and expense . The essential elements of two super sonic wind tunnels designed by the NACA for intermittent operation are Quoting from the summary of TN 2189 : "This equipment shown in Fig . 8:10 . consists of an induction tunnel having a 4 in . by 16 in . test section and
considerable
capable of operating at Mach numbers ranging from about 0.4 to 1.4 , and a blowdown tunnel having a 4 in . by 4 in . test section for supersonic Mach The tunnels are actuated by dry compressed air numbers up to about 4.0 . stored at a pressure of
lbs .
300
per sq
in . in
a 2,000
ft
cu
tank by a 150
reciprocating air compressor . This air supply permits inter operation mittent of the tunnels for test periods ranging up to 400 sec
horsepower
onds
(depending
onehalf hour 8: 5. made
on the stagnation pressures maintained ) at approximately
intervals . "
WIND TUNNEL
BALANCES
FOR
FORCE
Force measurements are
TESTS .
by mounting the model on a support connected
draulically
,
or electrically
actuated scales
.
mechanically ,
to
hy
The model may be supported
either by wires through the side of the tunnel , struts through the side of the tunnel , or a single strut from behind the model . A typical wire support is shown in Fig . 8:11 , with provision for measuring all six com of force in
ponents
in Fig . Tail
movement .
8:12 . A model supported
by a
tail
" sting "
is
is
shown
in Fig .
8:13 .
system
shown
sting supports are indispensable for supersonic testing because wires
or struts would generate 8 : 6 . OTHER WIND TUNNEL
the purpose Force
A 3  component strut support
of
measurement
measuring
shock waves which would TEST EQUIPMENT .
forces
equipment
has
,
pressures
affect
the test
Wind tunnels are ,
primarily for
and temperatures
been described
in Art .
results .
8:5.
on models .
Pressures
are commonly measured by manometers or banks of manometers . Equipment for photographic recording of a multiplemanometer bank is shown in Fig . 8:14 .
8:10
Pictorial
. .
Fig
layout
of
small
.
operation intermittent From NACA TN 2189
Schlieren apparatus
AA
11 tunnels
Air dryer
f
Blowdown tunnel
Induction tunnel
developed
Oil filter
Storage tank
by the
. NACA
Air compressor
813 TECHNICAL AERODYNAMICS
AERODYNAMIC
TEST FACILITIES
Balance Room
814
Angleof Attack Indicator
BO Lift
Manometer
W.Yow Balance W. Rings and Balance Static Sections Moment Moment To Entrance Rolling Balance Balance Working E.Yow Balance E.Lift Balance Drog Balance
To Selsyn Generator
Winch
Xx
Model Wind Direction
Wind Tunnel
Room Observation
rig
Original wire balance system of the GALCIT tunnel . This . ging has since been superseded by an improved system . is reproduced here because shows more clearly the essential features of the force resolution than the more complicated system . (From C. B. Millikan , " Aero Reproduced dynamics of the Airplane , " John Wiley & Sons , Inc. , New York . with permission . ) Fig . 8:11
It
it
Counterweight L2 OPivots Screw
Wind Sting '
Fig . 8:12
.
nel balance model .
Simple
Model
type
of wind
tun
using struts to support
Fig . 8:14 .
of
Photographic recording readings . Courtesy
manometer
NACA .
TECHNICAL AERODYNAMICS
815
Fig . 8:13 .
Model
in test
section of 6 x 6 ft supersonic wind tunnel at Laboratory . Courtesy NACA .
Ames
The
air
in
is difficult
density in wind tunnels
tunnel wall has glass sides
to measure directly
in density
differences
,
the test section can be measured optically by
meter shown
in Fig .
means
If
the
points
of the interfero used in photo
is usually
The interferometer
8:15 .
.
between various
graphing supersonic flows but can be used either subsonic or supersonic sufficient care is taken in getting the mirrors truly plane . A simpler
if
A
Testsection
of windtunnel
4 M
Condenser Lens1 lens Spark or monochromatic light
Fig . 8:15 .
Diagram
T
Lens2
Camera

M Mirror
T =Translucent mirror
of Mach  Zehnder interference refractometer to determine density differences .
equipment
TEST FACILITIES
AERODYNAMIC
in Fig . 8:16.
stop
The
provided at that point
is
.
the " schlieren
equipment
"
sketched
if
S1 may be omitted a point source of light is Ordinarily a monochromatic or spark light source
film in the
used with black and white
used
is
than the interferometer
apparatus
816
if
camera , but
light is
a white
with color film in the camera , density gradients appear as color film . An even simpler device is to eliminate the use of
changes on the
51010 Testsection ofwindtunnel
Lens1
Condenser lens
Lens2
Stop Sz
Stop
Spark
S
Camera
Glass port holes
Fig . 8:16 lens
2
in Fig .
sensitized factory
of schlieren
Diagram
.
8:16
for
simply substitute
and
This is
paper .
equipment
sonic flow .
known
getting pictures
for
photographing
of
a sheet
super
photographically
as the " shadow " method and is very of
shock waves .
The
relationship
satis between
these three methods of making photographic records of density changes
explained by
Hilton ( 1 ) by
density changes are large as ing through a shock wave ,
of
use
density , or
its
distance , or
rate of change with
the curvature
of
the
make a
showing
where
record
For the
the shock waves are . dent
who would
field ,
do
laboratory
is particularly recommended of its many helpful detailed
because
( 1 ) Hilton , W. 1951 .
(2)Hilton ,
op .
F.

sensitive
to density
Direct shadow sensitivity proportional to
/
curvature d2p dx2 Schlieren
sensitivity proportional
to
/
do dx
recom
8:17 . Variation of air den and refractive index through ck wave , with notes a typical by Hilton . shock
Fig .
sity
" High speed Aerodynamics
cit .
is
Where
work
Hilton
mendations .
Interferometer
stu
the work of
in this
8:17 .
it is evi
density curve can be used to photographic
in Fig .
in pass
from Fig . 8:17 that either the
dent
the sketch shown
."
Longmans , Green & Co.
CHAPTER
9
AIRFOILS AND ASPECT RATIO EFFECTS AT LOW SPEEDS
9:1.
in
FORCES
AIRFOILS ; AIRFOIL COEFFICIENTS
ON
airfoil is
An
.
defined
general as any body shaped so as to get a useful reaction from an
relative to
stream that moves
it ,
but the term
is
to
most often used
air de
KU727
scribe a body of cross  section similar to Fig . 9 : 1 which is acted on by a large force perpendicular to the
I V
a
tco
D =
planes and
 Chord
air
stream
airfoil in
.
9 : 1 .
.
Forces on
derived from experimental
lift )
data
on
to
voted laws
and
a small
that direction
tail
(1 )
a
( wind
is de
presentation of the
of force  action on
airfoils
air
fuselages
This chapter
.
force
( drag ) .
surfaces of
airplane
some
airfoils
are
Air
an
(
The wings and
Cpc
velocity
Fig
stream
parallel to
c
Zerolift chord chord Geometric of aa Angle attack
air
Drag
tunnel and
airfoils as free flight
tests ) at such low speeds (under 300 miles per hour ) that the compressi bility effect of the air is negligible , and ( 2 ) the use of such laws in form to calculate the forces
mathematical
acting on airplane wing and
tail
surfaces at low speeds .
airfoil result from the distributed pressures over the airfoil . As the speed is changed from low subsonic to super
The forces on an chord
of the
typical pressure distribution pattern undergoes a major by the comparison in Fig . 9 : 2 , suggested by Pope . (1 ) In general , the pressure distributions about the airfoils shown in Fig . 9 : 2 can be calculated from the corresponding velocity distributions and the appropriate Bernoulli equation . It may be noted in Fig . 9 : 2 that for sonic speeds
,
the
change as indicated
the typical subsonic flow pattern most of the
lift
is
due
to negative
; whereas , for a supersonic flow pattern , positive pressures to on the lower surface be
pressures on the upper surface most
of the
lift is
due
surface absolute pressures cannot be less than absolute zero . Special consideration of the effect of high speed , as represented mathe
cause upper
matically
by the Mach
A force ( 1 )Pope
is
number ,
will
be given
in
completely specified by specifying
, op .
cit .
91
Chapter
its
10 .
magnitude ,
direction
,
AIRFOILS
and
line of
2:1
by
(c.p. )
action
AND ASPECT RATIO EFFECTS AT LOW SPEEDS
Forces
.
lift
specifying
airfoils
on
(L) ,
drag
are
( D) ,
commonly
as in Fig .
specified
center  of  pressure location
and
airfoil .
from the leading edge ( LE ) of the
92
Experiments
on
airfoils
0.6
0.6 Upper Surface
0.2
0.2
Lower
0
Surface
0.2
0.2
BTS
20
0
face
100
of chordwise pressure distributions for low flight at small angles of attack .
speed
60 Per cent chord
0
100
80
20
40
60
Per cent chord
and supersonic
lift
air density ( p) , airfoil sur (S ) , and the square of the relative air velocity (v2) for a given velocity of attack ( a ) between the airfoil chord (c ) and the airvec
that
and drag are proportional
tor , provided that other factors surface , and
foil
Surface
80
40
Comparison
. 9:2 .
angle
Lower
o
1.0
1.0
show
Surface
0.6
0.6
Fig
Upper
0
forces
to
, by
specifying
lift
=
from
The
coefficients
and drag
CL
effect of density , specification the of air
constant .
remain
velocity are usually eliminated
(CL , CD )
defined by
L
(9 : 1 )
qS
D CD = qS
/
where q = pV² 2 and
foil .
and D are the
lift
location of the
The
specified
L
by the
total
(9 : 2 )
lift
and drag forces on the
and drag forces on the
coefficient
airfoil is
c.p.
(9 : 3 )
Cp =
where c . p .
as
shown
An
in
is the Fig
alternate
.
air often
distance from the leading edge to the center  of  pressure 9 :1.
means
of specifying
the line of action of the force on the
airfoil is arbitrarily to
locate the L and D forces at some definite point , leading edge such as the or the 25 or 50 per cent chord point and add a
pitching couple or cient
Cm
is
moment
M
to produce
the
same
result .
A moment
coeffi
then defined by the equation Cm =
M
cqs
(9 : 4)
TECHNICAL AERODYNAMICS
93
/
The point
c 4
is
selected as a center of
commonly
location of
and
moments
L and D for subsonic airfoil tests because the pitching moment on most airfoils is approximately constant about this point . With such choice of moments ,
the force system of Fig .
equivalent force system of Fig .
replaced by the
9 : 1 may be
9 :3.
results of tests
The
foils of definite
span
a
/
S )
(A
),
in Fig
Fig
are
.
(
S
=
are
as shown
airfoils
4
9 :
running completely across
airfoil
sections of the aspect ratio
The sec
tion or local characteristics are usually designated by small letters
,
thus
being an
6 .
=
A
center
in Fig
:
6 )
5 )
( 9 :
(9
7 )
9 :
The pitching moment
plotted
is
The aerodynamic
defined as the point about which the pitching angle of attack at which the air strikes the as the
wing
,
.
,
.
,
,
airfoil with c₁ as is shown in Fig 9 : 5
0
abscissa
spanwise
plot of local sec typical
tion coefficients for
"
"
aerodynamic
elementary
distance
28 32
area
the chord and
0.2004
.
a, 8
4
0
9 : 4 .
.
is
the
being
the
9 :
,
0.40.08 dy
Airfoil characteristics of
aspect ratio
moment
respec
a
,
=
cdy
lift
and dM are the
the elementary
on
subsonic
about
,
,
,
tively
10.60.12 dS
4 12 16 20 24 degrees
dD
and pitching
0
4
Co
coefficient
(
.
12 80 8
dL
.
L /D
1.00.20 drag
A
0
Cp
0.80.16
4
dD
qds
1.40.28
60
8
qds
dM
1.20.24 where
20 40
100
dL
cqds
c
S
202
16 22
1.80.36
CL
24 20
Fig
=
Y
:
:
=
2.00.40
1.60.32
Cpl
8
cd
2.20.44
Airfoil Clark Re 3,700,000 5x30: 68.4ft./sec. Pressure21.1 atm
28
C1
=
CL Co 2.4 0.48
5 .
infinite
have
coefficient the moment
center of
a
simulating an
all
is constant airfoil is varied
moment
.
.
made on
tunnel so that
.
,
the same characteristics
To
"
a
wind
airfoil
de
ratio eliminate the effects by aspect
"
.

,
of aspect ratio tests are often the test section of
partly
termined
9 : 1
partly by the cross section of the
and
The curves in
Force system equivalent to Fig .
3 .
.
Fig
9 :
9 : 4 .
14
usually plotted
and
defin
surface and hence aspect ratio ite b² Air velocity Mc
air
on
( b )
center of
AIRFOILS
(For some
airfoils airfoil
section
each
The
is
94
EFFECTS AT LOW SPEEDS
such point . )
no
approximately at c
sections and
For
/4
but
is usually
airfoils at low speed the is slightly different for experimentally for
determined
.
results
in Fig .
shown
always some angle at no
is
there
aerodynamic center
different
AND ASPECT RATIO
9 :4
are typical of
many
which the wind can strike an
force perpendicular to the wind direction
(no
airfoils
airfoil
lift ) .
There
.
is
so as to give
Under these con
ditions the resultant air force is parallel to the wind  velocity vector and is drag . The line drawn through the trailing edge of the airfoil in this direction zero 
lift
is
The
chord .
as
known
zero 
tionship to the geometric
is
0.026
lift
The
of
the
airfoil
angles zero 
of attack
lift
chord
9 :4
lift increases lin
beyond which
(CL
as
the
16
changes .
the
stall .
lift The
12 ao
8 0.1Cdo . 14
0
0 CmaC
The
12 16 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 GL
0.4 0.4 0.2
Fig .
does not
4 8
0.2 Cmac 0.3
in
of pressure
limit is known
0.008
0.1
early with the angle , the drag creases at a higher rate , and the center
do 120
0.006
As αa
increased from zero , Fig . that the
0.010
attack and are
usually designated by αa .
28 24
0.002
as
32 Cdo
0.012
ab
solute angles of
shows
Cdo
0.004
known
36
0.014
measured from the
are
40
0.016
with the geometric chord is as the angle of attack (α ) ;
known
is
.
angle that the wind  velocity
makes
144
0018
se
section
c abovechord 0.04 Ja =8,370,000 toinfinite Corrected aspectratio
0.020
lected by the designer for making a drawing
48
aheadc/4 0.011ca.c
0.022
chord ,
arbitrary line
an
152
0.024
no necessary rela
chord bears
which
the
9: 5 .
of
Characteristics
airfoil of infinite
of portion span .
increase with the angle of attack
stalling angle
and
maximum
lift
coefficient
are determined by the separation of the boundary layer , which in as in the flow around a cylinder , depends on the scale (Re ) of the
max )
turn ,
airfoil test coefficient flat plate ) .
and
other factors as discussed elsewhere
(CD min )
is
determined
.
The minimum
chiefly by the skin friction
(
drag
like
a
9 :2 . STRAIGHT  LINE PLOTTING OF LOW SPEED AIRFOIL TEST DATA . The air foil characteristics shown in Fig . 9 : 4 are for a rectangular airfoil of
TECHNICAL AERODYNAMICS
95
Clark
Y wing
of
aspect ratio
6,
with rounded tips , with a
taper
2:1
in
thickness ratio , are shown in Fig . 9 : 6 . wing In each case the characteristics are completely specified by graphs graphs only the of CL, CD , and Cp vs. angle of attack α . Of these three
in
planform , and with a taper
of CL vs. a is a straight equation The of this straight graph
CL Co 2.20.44
Planform ClarkY Airfoil:Tapered sq.in,30in.span Size:150 3,188,000 Re
2.00.40 1.60.32 1.40.28 1.20.24
Cpt 40
1.00.20
60
0.80.16
80
0.60.12
100
0.40.08 40.20.04 0 0
0.2 J0.4 8 4 0 4 8 12 16 20 24 2832 a ,degrees .
Conventional plot of tapered Clark
9 : 6.
characteristics of Y wing of A = 6 .
aLo
lift on
04 0.6 0.8
(9 : 8 )
lift  curve
slope and
the angle of attack of zero the intercept
or
axis
a
of the graph
.
graph
The
of
CD
a
vs.
can be
straightened by replotting CDvs . C} , as shown in Fig . 9 : 7 . The intercept
of this
graph on the Cò axis
is
known
effective minimum drag coef ficient of the wing and is here des as the
the symbol CDoe ; the slope , theoretically equal to 1 A
CL 11.1 1.21.3 1.4 1.5 1.6
reasons
/
to be discussed later
may be designated
is a
1
semi  empirical
/
Aew ,
wing
,
where ew
efficiency
factor , so that the equation of the plot of CD vs. c may be written CD = CDoe + CATAOW
(9 : 9)
The graph of Cp vs. a may be
straight
Co 0.10 0.053 ew = =0.91 0.058 0.05
plotting Cp vs. 1 /CL as shown in Fig . 9 : 8 , for which the equation is
ened by
ACO=0.058 1/6 =0.053
2
Fig . 9 : 7 . of CD for
Straight line plotting Clark Y wing of
tapered
Cp = a.c .
· Cmac /CL
(9:10 )
quantities a.c. and Cmac are the intercept on the Cpaxis and the slope respectively . The quantity a.c. is
The
AC =1
c2
A = 6.
the
is
ignated by
0.15
Срое
is
the
for 0.20
is
where a

αLO)
= a (α
CL
1.80.36
Cp 0
Fig
from analytic geometry
line . line ,
thus the distance back from the
lead
ing
center
edge
to the
aerodynamic
AIRFOILS
expressed as a fraction of the chord
in Art .
The three linear equations
(9 : 8) ,
( 9 : 9 ) , and
completely the airfoil characteristics stall . Three intercepts and three
stalling  lift
plus the
,
efficient , in Fig .
In studying
9 :6.
of
number
co
airfoil
large
sections
Cp
airfoils
than
a number of tapered
0.25
wings
a.c.=0.230.02aheadof 0.25c
con
to deal L.E.
3
is giv
( stalling
max = 1.67
aLO =
)
as
follows :
( Fig .
and slope of
9 : 6)
lift
curve
(Fig .
9 : 6)
= =
Cmac

0.25
0.02 = 0.23
and slope of graph Cp vs.
plotted in Figs
.
from the graphs
from the table and
may be read
ct (Fig .
Intercept
constants of
seven aerodynamic
vs.
for
(Fig .
0.071
CD
ew = 0.91
9 : 6,
a.c.
slope of graph of
9 : 7)
A = 6)
15
(calculate
= 0.058
ΠΑΘΗ
and
9 : 8)
this tapered
wing may also be read
Conversely the con the graphs plotted and
.
intercept
9 : 7,
CDoe = 0.0076
stants
6
0.071 per degree
a =
These
 lift coefficient
intercept
5.20
5
Fig . 9 : 8 . Straight  line plotting of Cp for tapered Clark Y wing of A = 8.
The aerodynamic
wing plotted CL
4
/
1 CL
for the tapered Clark Y in Fig . 9 : 6 are there tabulated
constants
0.07
,
in
Appendix 5.
}Acp =Cmac = +4(1 C ) =1.0
9 : 8 .
en
0.15
/
directly with the plotted graphs sim ilar to Fig . 9 : 6 . Such a tabulation
for
0.50.4 0.30.25 0.2
0.50
or wing
and study these aerodynamic
stants of the
2 18
below the
9:6
CL
plotted a
characteristics it is simpler , brief er , and more illuminating to calcu late
in Fig .
shown
0.75
are seen to spec
( 9:10 )
completely specify
thus
airfoil characteristics
the
center being defined as
aerodynamic
,
9: 1.
ify
slopes
96
RATIO EFFECTS AT LOW SPEEDS
AND ASPECT
lift
infinite
aspect ratio
independent
gives what
is
in the of
few
ar
ratio
A
next
aspect
called the
.
"
is practically
section
"
to
and experimentally
"
correction
9:10
)
Equation

)
,
theoretically
developed (
.
ticles
is
"
,
tion
"
.
(
( a )
( 9 : 9 ) ,
A
appears explicitly in equation The aspect ratio but the curve slope and the wing efficiency ew are also functions of aspect leading to an aspect ratio correc ratio The effect of aspect ratio
TECHNICAL AERODYNAMICS
97
characteristics "
and eliminates
effects of
wing  tip shape and
taper ratio included in the table in Appendix mating or calculating
the
The procedure
5.
particular
characteristics of a is thus to correct
the section characteristics
planform
tapered
the section
for
esti
wing
from
characteristics
for the primary effects of Reynolds number and Mach number and to correct the results to the actual aspect ratio , taper ratio , and wing  tip shape . A rational procedure for doing this requires further study of wing theory developed in the next few articles . analysis of the forces relative to the airfoil involves a major assumption that the airfoil deflects a cylindrical stream of diameter equal to the span of the airfoil . A more rational heading for the article might therefore be the cylindrical air  stream analysis of air 9: 3 .
exerted
foil
MOMENTUM THEORY OF
on
airfoil by
an
action .
AIRFOILS . The following
fluid
a
is
This method
stream that moves
elementary , being
based on the momentum
re
lationships developed in Chapter 4 , but it gives several of the results that are usually ascribed to the more elaborate classical treatment or dinarily described as the circulation theory . The rudiments culation theory of this same problem are given in Art . 9 : 4 . As applied to a deflected as
in equation
fluid ,
stream of
Fy
=
i
fluid
the
ond , Fy
and AVy stream .
is in
is
If
on the
is
the
airfoil in mass
stated
,
the y  direction
fluid
of
deflected
per
the y  component of the vector change of velocity of
AVy
is in feet
per second and
in
is
in slugs per sec
pounds .
Let Fig . section of
9:9.
cir
(9:11 )
mVy
of air past
Fig .
may be
(4 : 5) ,
where Fy is the resultant force exerted by the deflecting stream of fluid ,
unit time ,
law
Newton's
of the
Streamlines of flow past an
9:9
represent the flow
airfoil , which is shown an
the cross .
The
air
velocity relative to the airfoil a short distance in front of the air
foil is represented by the vector the airfoil the velocity is Vs , which is nu to Vo but differs from Vo in direction in that it is de through an angle r ( called the angle of down  wash ) , the
airfoil .
Vo ; a short distance behind
merically
flected
subscript AV
is
equal
downward
r
denoting
that the angle
determined , as shown
in
Fig
.
is in
radians .
The change
9:10 , from the difference
of velocity between the
two
velocity vectors
sin
Ertan r
=
is ,
produce
in Fig . 9:11 .
Vor
=
(9:12 )
laws , the force
y
may be added
air
the
on
stream necessary to
If
this
is called
to F and AV .
The
force ex
is in the direction
change
the y  direction , a subscript erted on the
( for which
from geometry
to Newton's
this velocity
of AV for small angles Er
The magnitude
.
Er)
AV and , according
98
AND ASPECT RATIO EFFECTS AT LOW SPEEDS
AIRFOILS
of AV .
airfoil is in the opposite direction and is designated by Fy In this figure Fy is shown analyzed into lift and drag
Vo
.
Av
JDi
Vs
Change of velocity air stream .
Fig . 9:10 .
in inal velocity
the usual
components
fluid friction
actually
actually
there are how
thin ) .
the
airfoil
(Do) .
The
total on the
eddies
The sum of the
cross  section
total
stream of air . L is perpendicular to the orig indicating is labeled D , the subscript
The drag
Vo .
drag
the
lift .
induced
The mass
by assuming ameter equal
of
air
greater than ( no
Di because there
matter
is
smooth ) and
how
in the air stream behind the airfoil (no matter friction and eddy drag depends on the shape of ( or
profile
is
may be
the =
)
and
Di
is called
:
(9:13)
+ Do
considered to
by the
airfoil deflects a airfoil .
the drag induced
mean
airfoil
cylindrical
to the span of the
the profile drag
of D and Do , thus
sum
per second deflected
that the
i
is airfoil surface
( Dtotal )
drag
lift
drag
Dtotal The term
Resolution of force airfoil on the de
9:11 .
exerted by
flecting
manner ;
induced drag ; the actual
Fig .
of
This
may be
stream
is called
by the
calculated
of air of
di
by Munk ( 1 ) the
" The assumption of a circular cylindrical stream usually of air deflected is stated as a corollary to other assumptions but can just as well be a major assumption . It is justified only on the area
of " apparent
(1)Munk , Ronald
Max .
mass .
"Fundamentals
Press , 1929 .
of Fluid
Dynamics
for Aircraft
Designers , "
TECHNICAL AERODYNAMICS
99 grounds
it
that
correct result .
gives approximately the
The
effective
of air stream for this analysis is usually found experimentally to be 10 to 20 per cent less than that of the circular air stream superim posed on the span of the wing as shown in Fig . 9:12 , and the correction is included in the term ew. For an airfoil of span b , the area of the circular air stream is mb2 4 , area
/
and the mass
air
of
flowing
the area per second
through
m =
Substituting
in
equations =
(9:14 )
Vo
and ( 9:12)
( 9:14 )
,
( 9:11 ) ,
Fy
is
b² VỎE
R™
(9:15 )
For small angles Er , the resultant Fy
is
approximately equal
Front view of air showing area of air stream assumed to be deflected by wing . 9:12 .
plane wing
CL =
,
πAεr/2
=
90S
in Fig .
Note also ,
Dividing by qos ,
L
the
/ 2 , it follows
with L = CLpsv equation ( 9:15 ) that
and ,
Fig .
to
9:11 , that D₁ =
/
Er from equations
( 9:16 ) CDL =
The
total
drag
it
c²
follows
(9:19)
is the " profile  drag coefficient . " If CDo , differentiating equation ( 9:19 ) would give CL then
were independent
where CDO
= ΠΑ
(=
0.053
for
that
is
CD = CDO + CETTA
dcz
L (✯y / 2 ) .
( 9:18 )
ΠΑ
from equation ( 9:13 )
coefficient
from
(9:17)
( 9:17 ) ,
and
lift ;
(9:16 )
CDi = CLEr 2
Eliminating
force
of
(9:20)
A = 6)
Experimental values of dcp / dc for aspect ratio 6 are usually 0.055 to The ra 0.066 , depending on the planform  taper ratio and wing  tip shape .
tio of the ideal to the actual value of dcp / dc of CD vs. c , is the wing efficiency factor ew . ew corresponding to the above mentioned
A mathematical
definition of
ew may
values of
,
determined
from a plot
The range of values
/
dcp dc
thus be written
is
of
0.8 to 0.96
.
910
AND ASPECT RATIO EFFECTS AT LOW SPEEDS
AIRFOILS
1
ew =
/ /
A
( 9:21 )
dCp dcz
lift  curve
slope and angle of attack may also be developed from the foregoing considerations . Note in Figs . 9:10 and 9:11
relation
The
between
direction of the velocity makes an angle Ɛr / 2 with the orig inal velocity . The effective angle of attack of the wing chord may thus that the mean
to be reduced by the angle Ɛr / 2 , which is called the induced angle of attack , designated by the symbol air . From equation ( 9:16 ) calculate
be said
air uation
Since
tio A
= oo ,
aspect
(9:18 ) shows that the induced
profile
the
drag
zero , or aspect ra
drag implies no induced
Since no induced
ratio .
is
drag
of as the drag for infinite
may be spoken
the angle =
αor may be spoken
(9:22 )
= CLATA
angle
of attack
 CLA
αr
of as the angle of attack for
,
(9:23)
infinite
aspect
ratio .
Where
finite aspect ratio and CL is the lift coefficient of either finite or infinite aspect ratio wing , assumed to be the same for either finite or infinite aspect ratio . In some NACA reports α is plotted against CL or c1 in presenting the ap
is
the angle of
results
tips
and
airfoil
of
wind
attack for a
the
effect of rectangular
tunnel walls to be described later .
The above simple an
tests
,
with corrections
for
lift  curve
alysis does not permit arriving at any value of finite aspect ratio . To determine this the more follows 9:4 .
tion
and
under
Circulation
Theory
CIRCULATION THEORY
Art .
of
methods
viscous incompressible
closely duplicated
by
pattern with suitably
OF
Airfoils
of
AIRFOILS .
4 : 3 and assumes
elaborate
appears
in
slope for
that
method
to be necessary
.
This theory follows the assump that the flow patterns of non
fluids about cylinders of various profiles may be the vector combination in space of a uniform flow located sources
fairly
,
sinks
,
and
vortices ,
It
all of
which
this Bernoulli's equation , that the lift L of a portion of length b of an infinite rotating cylinder in a fluid stream of density p flowing with velocity Vo is given in the equation ( 1 ) with
can be
handled
method ,
combined with
simple
L
( 1) The
" Kutta
 Joukowsky
mathematics .
=
pvorb
equation . "
may be shown
by
(9:24 )
TECHNICAL AERODYNAMICS
911
is
where
the circulation of the vortex determined
tion of the cylinder . shown
It
in Fig .
9:13 .
likewise be
may
rota
by the speed of
The flow patterns with and without rotation are
shown
that the flow pattern around an
cyl
elliptic
inder (determined by flattening and extending the circle ) , inclined at an angle a to the direction of free stream velocity , may be approximately
similar flattening of the flow pattern The around a rotating cylinder , the lift being given by equation ( 9:24 ) . elliptic Fig patterns cylinders flow around such are shown in . 9:14 . mathematically by
reproduced
a
(b)
(a)
Fig . 9:13
.
Ideal flow patterns ( neglecting fluid viscosity (a ) stationary and (b ) rotating circular cylinders
If
Ideal flow patterns around elliptic cylinders with cylinder axis at angles to the air stream of zero ( a ) and a ( b ) .
9:14 .
the circulation ♪
flow pattern patterns
the angle of attack a are properly
and
around the
elliptic cylinder closely
The necessary
.
circulation is
relationship
slope dc₁ / d
is MO =
dc1
ness
and of
what
less
Fig .
9 : 14b
25
lift  curve airfoils than
also
approximates
may be shown
= 0.1096
shows
per degree
Application
that the resultant
lift  curve
(9:25)
infinite elliptic cylinders tested in wind tunnels is usually
2π per radian .
the
actual flow
that the
per radian
slope of as
,
ratio of the elliptic cylinder ;
dao
or ao The actual
it
zero , = 2
related
between the angle of attack and the
a function of the thickness
as the thickness ratio approaches
at
around
( b)
( a)
Fig .
) .
lift
of
finite thick
found to be some
of Bernoulli's equation on an
to
elliptical airfoil is
per cent of the chord or ( a.c. ) ' = 0.25
(9:26 )
AIRFOILS
AND ASPECT RATIO
EFFECTS AT LOW SPEEDS
912
of these relationships it is sometimes considered that an infinite elliptic cylinder behaves like a " lifting  line " vortex located at 25 per For an elliptic cent of the chord of the ellipse from the leading edge . Because
cylinder
of finite an end ;
Bust have
length or for a finite airplane wing , the lifting line (1 ) the condi and according to the Prandtl wing theory
with having a finite lifting line by assuming that the three  dimensional flow pattern around an actual wing consists of a horseshoe  shaped vortex system , as shown in Fig . 9:15 . The tion of continuity

down wash
is
made
consistent
pattern behind the horseshoe
system
is
9:16 ;
this
Wing.
Vo
Fig . 9:15
in Fig .
shown
.
Fig . 9:16 .
vortex
Horseshoe
pattern

Down wash system
horseshoe
.
vortex .
for
lift distribution along the span of the wing . To of actual wing  lift distributions , Prandtl conceived of the lift being due to a combination of a group of horseshoe vortices as that shown in Fig . 9:17 , giving a lift distribution as shown in
also corresponds to the take account wing such
P
Fig . 9:17 Fig . 9:18
.
Note
.
in Fig .
Composition
of
horseshoe
vortices .
9:17 that at various points
on the span of the
wing there are various numbers of vortices contributing that point , but there
is
considered to be only
( 1 ) Prandtl , L. , NACA Tech . Rept . 116 .
one
to the
lift
lifting line for
at the
913
TECHNICAL AERODYNAMICS
Using the calculus
wing .
distribution
spanwise
of
this down
lift
9:19 , the necessary
infinite
conception of an
imal vortices combined in
manner , Prandtl
wash
velocity
behind
distribution is elliptic
for
a uniform
wing , as
a
as
,
infinites
number of
shows that
in Fig .
in Fig . 9:20 .
shown
w  Downwashvelocity
y Fig . 9:18 .
Lift distribution
due to
vortices .
several horseshoe
Fig . 9:19 .

lift
сс 9:20 . bution due
infinitesimal vortices .
also that this
He concluded
for
minimum
induced
elliptic
drag and that
planform ( closely approximated These assumptions
were shown
viously
in
developed
Prandtl
found
it
lift
distribution is the condition
occurs with an
by trapezoidal by Prandtl
good
wings
untwisted
of
to result in
the momentum theory of
experimentally
с
Fig . 9:21 . Chordwise distribu tion of lifting lines necessary to account for low  aspect  ratio wing characteristics .
Elliptic lift distri to infinite number of
Fig .
distribution distribution .
Down wash
elliptic
with
airfoils
2 :1
or
equation
3 :1
elliptic taper )
( 9:18 )
.
pre
.
ex later tests on airfoils between theory and ex
agreement between the theory and
airfoils of aspect ratio 1 to 7 , but of aspect ratio 1 to 3 showed major discrepancies periment , particularly in the lift  curve slope , indicating
periments for
that this theory
for low aspect  ratio wings and for tail surfaces . More recent studies (1 ) have shown that a chordwise distribution of lifting
was inadequate
lines ,
as
shown
ratio effect of
in Fig . 9:21 , is necessary low  aspect  ratio wings .
( 1 ) Jones , Robert T. Effect of the Chord . "
" Correction
to
account
of the Lifting  line
NACA TN 817 , 1941 .
for
the
Theory
aspect
for
the
AIRFOILS
ASPECT  RATIO CORRECTIONS
9: 5 .
914
RATIO EFFECTS AT LOW SPEEDS
AND ASPECT
six
Of the
.
aerodynamic
constants (αLO ,
a.c. , Cmac ) needed for straight  line plotting of characteris tics of a particular wing , four of these ( LO , CDo , a.c. , and Cmac ) are substantially independent of aspect ratio . ew ,
a, CDo ,
ratio corrections
Aspect
lift
for the
is
of aspect
also a function
chiefly
of other factors ,
the
in
 curve slope a . To a lesser extent CL
max
ew ,
are needed for
duced drag CDi , and
ratio but
lift
spanwise
it is
which
determines
also a function of a
distribution
number
which are discussed
,
later . theory
The
elliptic
of
in Arts .
wings developed
9 : 3 and
9 : 4 may be
TTA
written
9:28
(
/
πA
)
(
9:27
)
1/2
+ 1
=
1
/m
CD = CDO + CE /
equations are intended to be applicable only to wings of ellip tic planform The lifting line theory has been applied to rectangular )
, ( 1
Lift curve pa slope factor
Induceddragfactor
5 .
equations given in experimental 9:30 and
bi
1.08
120
1.07
t. 8
1.18
1+
1.16 1.14
1.06
,
.
1.05 8 + 1
1.04
1.12
35
1.03
experimental
data
8
6
4
2
0
Fig
A
10
J1.02 12 14
Aspect ratio factors for 9:22 rectangular wings based on lifting line theory From NACA TN 416.
lift
curve
and
)
.
from the theory
efficiency factor
slope
.
for both wing

depart considerably
(
,
the
.
.
.
9:23 and 9:24
.
in Figs
1.10
are
in the wind tunnel
Note that
+ ± 1
1.22
1.09
A
+1
Lift curveslope factor
"
observations
1.10
)
and
)
9:29
,t
theor
between the
1.11
1.26
)
(
)
t
+
( 1
+ d )
.
wings see Appendix
1.28
124
are
for rectangular
values
modifications
)
(
8 ) С
}
/
9:30
TA
9:22
(
etical
the following
) 9:29
πA
and
relation
The
(
9:28
and
/
/
"
tapered
shown
yielding
For trapezoidal or straight
.
wings
Glauert
( 1
factors given in Fig
The
+
(
( 1
+
2π
+ ( 1
CD =
= 1
/
1 m
)
9:27
of equations CDo
wings
(
trapezoidal
+ T )
and
by
.

The above
Corrections
must
data in order to agree well with flight test re sults and accordingly the lines through the experimental data are labeled
London
,
,
Press
1942
and
9:23 that the
Airscrew Theory
"
of Aerofoil
in Fig
.
The Elements
note
.
interesting to
.
"
.
It is
"
Glauert University ,
(
bridge
practice H.
recommended
)1
"
,
be based on experimental
Cam
915
TECHNICAL AERODYNAMICS
1.0
Elliptical 0.9
Prandtl Theory
;
Rectangular : Glauert Theory
I
1 2 and 3  to  1 Taper
0.8 Rectangular Wings
Tail
0.7
Surface Recommended
0.6
Practice
ew
Data : TM 941 , TR 627 ,
0.5
TM 798 , TR 540 , TN 2980
0.4 A = Aspect
0.3
Fig .
4
2
0
8
6
Ratio
10
12
14
Theoretical and experimental variation of
9:23 .
18
20
22
with aspect
ratio .
16 ew
/
dCL da = a .09
Lifting
.085
Line Theory
π2
.08
a =
.075
Lifting
:
ob
A
x A+2
90
Surface Theory by
Points plotted
approximated
.07
a
=
.065
π2
x
90
A A+2.5
Test Data on Surfaces
Tail a =
.06
772
90
X
,
from TR 627 , tests on tapered wings .
A
A+3
=rectangular
tips
• circular tips
DO
=T.M.
941
.055 A
.05
2.5
Fig .
9:24 .
3
Theoretical
Aspect
( Reciprocal
3.5
5
Ratio Scale ) 7 6
10
experimental variation of with aspect ratio .
and
15
20
lift  curve
00
slope
916
AND ASPECT RATIO EFFECTS AT LOW SPEEDS
AIRFOILS
is approximately a constant only for a narrow aspect range of ratios from 3 to 10 , and that it drops off to a very small This fact is a major consideration in se value at large aspect ratios . lecting an aspect ratio for an airplane . The effect of aspect ratio on efficiency factor
wing
of wings ,
induced drag
based on
is
Fig . 9:23 ,
shown
in
Fig
.
9:25 .
.20 .18 ew
60
70
.80
•
.40
d (G2 )
.50
.16
1.00
.14 .12
.10
Lifting
.08
=
)
(
d
C
dC .06
Line Theory
1
TA
.04
)
=
as
a
function
parameter
a
9:25
2
2.5
3
5
Induced drag of wings as
.
.
Fig
10
of aspect ratio with
ew
.
20
00
4
(R 0
Aspect Ratio eciprocal Scale
A
.02

)
(1
.

inFig 9:24 that the experimental data on lift curve slope is sub stantially below that of the lifting surface theory developed by Jones ==
2πA
EA
the form 9:31
the ratio of the semi perimeter to the span of the ellipse
9:31
may be approximated
.

is
)
E
Equation
(
where
in
expressed
(
was
)
which
+ 2
wings
(
elliptical
,
for
)
Note
with
fair
accuracy by the empirical
equation 2πA
given to
tion of this
same
form E =
2πTA
cit
.
.op
,
,
( 1 )
Robert
T.
+3 Jones
show that the
lift
curve

,
,
however
better approximation by the modified empirical equa 9:33
)
is
rectangular wings
(
slope
on
a
The experiments
)
(
9:32
A +2.5
TECHNICAL AERODYNAMICS
917
of aspect ratio
are available , the recommended Aeronautics Administration , ( 1 ) in making aspect ratio
suitable
is
if
( (
if
the
used instead
Also
is
based
of
lift
the
or other test aspect
the correction
ratio
aspect
improved
is
9:23
ew
1.
that
=
9:34
.
shown
improved
is
ratio to on equa
9:35
(
for using published NACA data to estimate the char particular wing is illustrated by the following example
procedure
acteristics of
:
)
instead of
a
A
(
9:33
test
from any
accuracy
in Fig
ratio
from aspect
ratio
an unknown aspect
9:35
) .
,
curve slope correction
tion
equation
by
9:34
A
but the
ratio
ew with aspect
as implied
assuming
ratio
aspect
unknown
due
referred
( 9:28 )
2542
) ,
to any other variation of
and
CETTA
for correcting
can be used
,
Similar equations
/
CL
16
m =
out
drag
6 and then add the induced
61T +

CD = CD6
subtract
)
ratio
Thus equations ( 9:27 )
to the desired aspect ratio . to aspect ratio 6 become
6
the induced
first
use equation ( 9:27 ) and
drag due to aspect
)
corrections
,
of Civil , is essentially to
practice
6
3/8
data
A
test
(
If airfoil

CL ,
a
3 .
Air
Commerce
Manual
=
04.129V2
4,
/
,
=
/
6,
=
A
and
is
.
0
+ +
=
called for in
0.088 CL
( a ) .
0.242
.
, "
)
( 1
CAA
0.063c
ratio correction
aspect
Cp
0.0630
then
0.0073
=
no
Cp
equation requires
These are the answers
is
equation CD
The
блет
, "

The desired drag curve
c
=
CDi
c
of
%
6 .
).
a +
/
a
(
,
.
.
=
A ,
6,
A
=
:
°
=
,
.
=
=
,
a
=
:
5
,
.
,
.
( a )
lift
=
A
c )
.
°,
=
( b )
A
= a 6; ,
CL
.
(
9; a
=
A
( )
CL a
=
5 ,
Given the NACA 4412 wing section data in Appendix find equations for in terms of CD in terms of and Cp in terms of equations for rectangular wing for rectangular wing of equations for an elliptic wing of of Solution Read in Appendix for the 4412 wing the following data 4.0 ao 0.098 Cdo min = 0.0071 a.c. 0.8 ahead alo Cmac = 0.088 0.0073 for 4.28 CD min ть rectangular wing of For use the data uncorrected for and the m6 57.3 4.28 57.3 0.0745 Calculate a6 curve thus For the drag curve 4.0 with aLO = a10 4.0 write CL = 0.074 rectangular wing of In Fig 9:23 for read ew from Fig 9:23 read ew = 0.84 and calculate Example
EFFECTS AT LOW SPEEDS
AND ASPECT RATIO
AIRFOILS
For a rectangular wing of aspect ratio for the  curve slope
(b ) get
9 , use
lift
ag =
9
20.0745
9 + 3
918
equation
( 9:33 )
to
= 0.084
for a rectangular wing of aspect ratio 9 , and read on the line of recommended practice for rec tangular wings ew = 0.77 , for use in equation ( 9 : 9 ) . The estimated equa tions for the rectangular wings of aspect ratio 9 are then To estimate
refer to Fig
the induced drag
. 9:23
CL =
CD
= 0.084
0.0073 +
c²
(α
+
4.0 )
= 0.0073
gnew
+ 0.046c2 0.
Cp = 0.242 + 0.088
CL
These
in
are the answers called for
(b) .
elliptic
wing of aspect ratio 3 use equation ( 9:33 ) for correction from aspect ratio 6 , and equation ( 9 : 9 ) for the induced  drag correction , referring to Fig . 9:23 for ew and assum ing that an actual elliptic wing will approximate the recommended prac tice for a wing of 2 : 1 or 3 : 1 taper in Fig . 9:23 . This procedure gives
( c ) For
an
lift  curve  slope
=
a3
In Fig . 9:23 read
CL = 0.0558 CD = 0.0073
These are the answers 9 :6.
drag than at higher
air
.
ct
=
in
WALL
from the
= 0.0558
4.0 )
+
0.0075 + 0.123c2
(c). CORRECTIONS
ground
.
An airplane
( or water )
this height ,
Below
9:12 cannot
stream from being
(a
эпет
called for
altitudes .
theory sketch in Fig
lar
+
GROUND EFFECT ; TUNNEL
less than one semispan
3 3 + 3
0.86 , and get the equations
=
ew
x 0.0745 x
6
has
the elementary
b
not necessary for high lower surface of the wing , region of increasing velo
are
Ground plane
a
favors
boundary layer .
laminar
flow in
the
(
_"
is
which
The
Prandtl
wing
theory
Fig
9:26
Image wing
.
city ,
circu
Wing
The entire
moreover ,
less
.
.
lift .
flying
momentum
apply , for the ground prevents a
freely deflected
Instead , the wing floats on a layer of com pressed air , and high velocities over the upper surface
wing
appreciably

Vortex system used
for calculating ground effect
.
the
TECHNICAL AERODYNAMICS
919
may be
applied
system
is
so
effects thus calculated are equivalent The equivalent aspect ratio for induced
The
.
ratio
aspect
1.0 0.9
.
1.0
0.8 0.7
Agd
dog da
/
h b
3
0.10
0.2 0.30.40.5
0.10 0.150.2 0.3 0.40.5
/
h b
Fig
Effect of proximity to curve slope or on angle of attack for given lift coefficient a

.
.

lift
.
.
9:28 ground on
.
.
0.8 0.7 0.05
2.5
Fig 9:27 Effective aspect ratio of monoplane wing near the ground for induced drag computation
.
:
.
Fig
,
= €
6
=
=
,
,
0.10
unchanged
,
=
for
is
ew
/
if
h b
9:27
= 0.089
hence
,
0.88
;
a
"
/
ag
=
11.3π
a
x
c ₤
,
and
,
.
.
+ =
6,
a
A
for
9:26
closed throat wind tunnel the same plus similar effect on the
exists
,

a
.
is tested in
in
ft
60
a
as that shown
read
,
effect
= 11.3
0.89
0.075 0.88
span
for
called
When the wing model
.
These are the answers
6
/
=
CL
In Fig
0.10
0.53
= 0.0065
0.10 and
and
are CL = 0.075a CD equations drag and when the wing is
=
=
/
for
h / b
9:28
,
.
In Fig
: =
CD
6/60
Agd =
A
2 : 1
a
h , b .
) 6 ,
Calculate 0.53 Hence
of
wing
in free flight
lift
.
/
A
Agd
/(
ft
Solution read
.
.
lift = .
6
flying
tapered Given and drag equations 0.89 find the from the ground
1.
Example
which the 0.0065+ CL2

The effect on lift curve slope calculation is shown in Fig 9:27 Their use is illustrated by the following example shown in Fig 9:28
drag
is
10
0.9
2
A
0.6
LS
0.5 0.4 0.3 0.05
1.0
A
,
in
an increase
Agd
that the horseshoe vortex
assuming
strength below the ground , that the obvious condition of no flow through the
is fulfilled
ground plane
A
by
matched by an " image " system of equal
as in Fig . 9:26
to
this condition
to
(

.
.


.
a
) ,
out walls
a
tunnel wall For wind tunnel with an open jet test section with an opposite effect occurs because the air stream available is not sufficiently large to provide normal stream deflection as in free flight Wind tunnel tests must therefore always be corrected for the Since the tunnel tunnel wall effect to get free flight characteristics
upper
(
) .

, a
a
,
a
,
,
a
wall effect with closed throat like the ground effect is to increase aspect the effective ratio the correction to free air for such tunnel involves decrease in effective aspect ratio an increase in induced drag Numerical values for such correc decrease in lift curve slope tions are commonly put in the form
920
AND ASPECT RATIO EFFECTS AT LOW SPEEDS
in
)
9:37
radians
)
8C
9:36
(

Δα =
85
(
=
ACD1
c²
AIRFOILS
S is the wing surface and At is the area of the tunnel throat For a circular throat or jet , the values of d are the jet ) . +0.125 and 0.125 , respectively . For rectangular throats and jets of various sorts , Theodorsen ( 1 ) developed the correction factors shown in
in which (or
of
area
Fig . 9:29 . +0.3 Vertical walls only
2
0.2 +0.1
only walls Horizontal
2
0
Closedthroat
8 0.1 02
wall
horizontal One
Openjet
0.4 0.3
1.5
0.5 0.6 0.70.80.91.0 Width Ratio Height
h
0.4
2
0.3

.
.
factors for subsonic flow tunnels and jets
wall correction
Tunnel
9:29
rectangular
in
.
Fig
a
,
"
a
,

drag error due to the static For closed throat tunnels there is also hor along gradient pressure the throat which Diehl aptly describes as
,
it
,
2


"
;
izontal buoyancy since it is proportional to the volume of the model this correction is of the for variable density tunnel tests on airfoils airship may run as high per cent though for tests on models order of
9:37
)
and
(
9:36
for compressibility Fig 9:29 is illustrated
and
.
correction
.
an additional
)
tions
is (
there
,
0.3
)
M >
(
.
as 20 per cent of the drag measured on the tunnel balances For wind tunnel tests in which the Mach number is high say
The use of equa by
the following
:
example
410
.
Rept
.
(
.
NACA Tech
)1
.
=
7
a
=
S
.
,
=
.
Given
a
 2.
by wing of surface 13.5 sq ft tested in on test results wind tunnel The rectangular tunnel 10 ft closed throat and lift and drag are CL 0.080α CD 0.0070+ 0.040С12 Find the drag equations in free air Example
lift
TECHNICAL AERODYNAMICS
921
/
/
== Solution . Calculate b h = 10/7 = 1.43 . In Fig . 9:29 , for b h = 1.43 = 13.5 70 = read for closed  throat tunnels 8 = +0.125 . Calculate S /At 0.193 and 8 ( S At ) = 0.125 x 0.193 = 0.024
/
/
Equations ( 9:36
)
then give
( 9:37 )
and
ACDi = +0.024c2
radians
Δα = 0.024C The corrected
=
when CL = 1.00
= 1.40
1.4CL
results in free air are calculated thus 0.024c2 CD₁ = 0.040C² + 0.024c
CL 0.080
α =
=
=
12.5CL
αι
= a + ▲α =
a'

1.00 =
from the
test :
= 0.064c2
12.5+ 1.4
= 1.00
at CL
12.50
at
= 13.90
CL = 1.00
0.072
13.9
lift
Hence , the corrected drag and CD
equations are :
= 0.0070
+ 0.064c2
CL = 0.072α These are the answers
9: 7. FLYING
EFFECTS OF CHORDWISE
is not
IN
SLOT
WINGS ; INTERACTION
OF TWO AIRPLANES
in panels joined to 9:30 . If the chordwise through the joint , the
Wings are sometimes constructed
SIDE BY SIDE .
joint
gether by a chordwise
joint
called for .
,
in Fig .
as sketched
sealed against air leakage
adequately
two halves of the wing behave somewhat b
dependently , with a large adverse
bs
lift .
on the drag and Plan
in
/
is
a reduction
For
= 0.0021
in
a 40
effective
an increase in induced
k
and
=
0.85
aspect
drag
.
in
in
aspect
of the
(kb ) 2/8
wing
(b = 480
Fig
.
in . ) with
9:31 ) .
(9:38 ) induced
a 1  in .
slot
This corresponds to
ratio in the ratio 0.852 ratio 1 / 0.72 = 1.40 .
the
the
in
ratio of the combination for ( in
on
terms
The reduction
expressed
Aed =
 ft  span
effect
factor " k defined by
.
the effective aspect
calculations .
bg b = 1/480
may be
Munk " span
with chord
wise slot
where Aed drag
Wing
9:30 .
effect
equivalent to a large reduction
aspect ratio
ratio
Front elevation
Fig .
is
drag
The
in
= 0.72
and to
AIRFOILS
AND ASPECT RATIO
flying side
Two airplanes
EFFECTS AT LOW SPEEDS
by side have greatly
922 drag
reduced
for
the
reason , and the same graph (Fig . 9:31 ) can be used to calculate the effect with rectangular wings . For two wings of A = 6 flying separately ,
same
is
CDi
tips at
c
/
б
for
If
each .
a distance apart equal
they are flying side by side with the wing to
1/100
of the
. 9:31 to be 0.82 , and CDi = c to a reduction of induced drag in the ratio
read from Fig
6 0.822
x
=
/
combined span , k
12πk² .
This
can be
corresponds
0.75
12
0.9
k 0.8
0.7 0.001 0.002
Fig .
It is
9:31 .
Effect of
thus seen that two
0.005 0.01
0.02
chordwise 50
ft  span
0.05 0.1
slot in wing
0.2
0.5
1.0
on Munk span
airplanes flying with
wing
factor . tips
if
1
ft
apart have about 25 per cent less induced drag than they keep a large migratory importance may apart This effect be of to birds . distance . PROBLEMS
For Art .
9:1.
A model wing is tested in a wind turmel in which the pressure is the temperature 100 ° F . The wing model is rectangular and has a span of 30 in . and a chord of 5 in . When the air speed is 100 ft per sec , the forces measured on the model are L = 5 lb and D = 1 lb. The pitching moment about the c 4 point is 10 in . lb . Find (a ) CL, (b) CD , 9:1.
24
( c)
in .
Hg and
/
/
Cmc 4 ,
and Cp . 9 : 2 . A wing of 180  sq  ft planform surface S carries a weight of 1,200 coeffi lb in level flight in standard sea  level air . ( a ) Find the cient for speeds of 120 , 100 , 80 , 70 , 60 , and 50 mph . (b ) Find the angle of attack from Fig . 9 : 4 . ( c) Find the stalling speed . 9 : 3 . An airplane wing model extends completely across the throat of a wind turmel in which standard sea level air flows at 100 mph . The wing characteristics are given by Fig . 9 : 5 . The chord of the wing is 18 in . per foot of span of When the angle of attack is 10 degrees , find the the wing model .
lift
lift
For Art .
9 :2 .
9 : 4 . Using the wing constants listed in Appendix 5 , write and plot equations for CL vs. a , CD vs. CL , and Cp vs. CL for the wing of 2 : 1 taper ratio and A = 6 there designated M6 ( 18 )  ( 09 ) ; 00 .
923
TECHNICAL AERODYNAMICS
9 : 5 . Repeat problem 4412  4412 ; 00 .
1
For Art .
for the elliptical
wing
of
A = 6 designated as
9 :3.
9 : 6 . For a particular tapered wing of A = 6 ( wing 221809 ; 00 in Appendix 5 ) , the following test data are reported : a = 12.480 , CL = 1.00 , CD = 0.068 . Calculate CDi , CDoe , and αo . 9:7. Given the following data on the test of a particular wing of A = 10 and taper ratio 3 : 1 , calculate and plot CDoe and ao as ordinates vs. CL as abscissa . (a ) From these graphs , find CDoe min , CL opt , and dCL da o (b ) Also plot CDoe vs. C12 , and find CDoe and ew.
/
α
1.2 0
CL 104Cp
0
2
4
8
0.1
0.27
0.43
0.77
85
82
111
156
16
12
318
1.09
1.38 873
555
18
20
1.42 1.50 1,100 1,640
The above data are from NACA Tech . Rept . 627 , Fig . 13 , without corrections . Check these results with data on the airfoil in Appendix 5 designated 2301809 ; 00 . (The data in Appendix 5 have been corrected as specified in a later report , NACA Tech . Rept . 669. )
For Art .
9 :4 .
lift
per foot of span of an airfoil extending completely 9:8. The across a wind tunnel in which standard sea level air flows at a speed of Using equation ( 9:24 ) , find the circulation necessary 100 mph is 50 lb.
lift
produce this . 9 : 9 . Using equation ( 9:24 ) , find the relationship between the coefficient C1 at any point on the span of the wing of chord c and circulation around the lifting line of the wing at that point .
to
lift
r
For Art .
the
9 : 5.
9:10 . A wing model of rectangular planform , 30  in . span , and 5  in . chord , was tested in a wind tunnel and the results of the tests , when corrected for the effect of the wind tunnel walls , were expressible by
the equations
CL
= 0.070
(a
CD = 0.0077
+ 0.80
)
+ 0.0734c2
Using equations ( 9:34 ) and ( 9:33 ) , write the equations for a rectangular wing of A = 3 , and check with Fig . 9:23 . 9:11 . Using the section characteristics of the NACA M6 wing in Appen dix 5 , follow the method given in the example in Art . 9 : 5 , and write equa tions for the characteristics of a rectangular NACA M6 wing of A = 10 . 9:12 . Using the characteristics of NACA 0009 wing in Appendix 5 , low the method of the example in Art . 9 : 5 , and write equations for the characteristics of an elliptic tail surface of NACA 0009 section of A = 4 .
fol
For Art .
9:6 .
9:13 . A 3 : 1 tapered wing of A = 10 , listed in Appendix 5 as 2301809 ; Using Figs . 9:27 and 9:28 and the data in Ap . has a span of 48 pendix 5 , write equations for CL ( a ) and CD ( CL ) for this wing when is flying 4 from the ground .
ft
00 ,
ft
it
AIRFOILS
For
Art .
AND ASPECT RATIO
EFFECTS AT LOW SPEEDS
924
9:7.
9:14 . A wing model of 5  in . chord and 30  in . span is tested in a wind tunnel of closed throat 39 in . wide and 22 in . high . Equations for and drag characteristics as measured in the tunnel were CL = 0.083α a and
lift
CD= 0.0095
find the 9:15 . of 48
ft .
9:31 and
+ 0.050C12 .
lift and
drag
Using equations
( 9:36 )
and
( 9:37 ) and
characteristic equations in free air .
Fig .
9:29 ,
Each of two rectangular wings of ew = 0.90 and A = 6 has a span They side by side with their tips 4 ft apart . Use Fig . find the percentage saving in induced drag over flying separ
fly
ately a great distance
apart
.
CHAPTER
10
AIRFOIL COMPRESSIBILITY EFFECTS
Airplanes
are of commercial
flight
high speed , and
velocity
ter
is
critical
importance
Mach
must
number "
of their
because
(where
local
some
for
becoming commonplace
indispensable for military
and
signed helicopters
in
beyond the " critical
exceeds the speed of sound )
cial aircraft the
military
and
commer
Efficiently
aircraft .
de
also operate with rotor tip speeds near or beyond
also have satisfactory performance the transonic and subsonic ranges described at the beginning of Chap Supersonic missiles
.
must
5.
of flight is increased beyond M of airfoils begin to be substantially different As the speed
Simple
0.4 , the characteristics
from those at lower speeds . reasonably accurate corrections to low  speed characteristics in the region between M = 0.4 and the critical , which usually
and
can be made comes
≈
between
M =
0.6 and
M =
0.9 .
Transonic corrections
covering
,
in
general the region of partly subsonic and partly supersonic flow , extend ing often up to as high as M = 1.4 , are considerably less accurately known but are discussed
in
in this chapter
the supersonic region
well
as
as
characteristics of
airfoils
.
is
Considerable valuable information
obtainable from a study of two
dimensional airfoil the primary objective of the study predict is to the characteristics of finite aspect ratio wings . This flow even though
chapter
will
consider
first
the
two  dimensional
combined effects of compressibility
and aspect
primarily
Mach number
This chapter concerns
is
not possible to vary the
number .
In the
wind
Mach
tunnel
it
effects ,
and
later the
ratio . effects ,
but
in flight
it
number without also varying the Reynolds is very difficult and seldom done , as a
/
of accurately controlled pressures and or temperatures would provided have to be . For most wind tunnel tests there is a relationship
wide range
between
test
effects
may
mon
practice
Mach
Mach
test
both be large ,
however , to
sen to consider
(2 )
and
number ,
the (3)
Reynolds
number
and they are
for
difficult
try to separate
them .
effects in the following Reynolds
number .
Thus , 101
any
particular
to separate .
In this text order :
while
model ; the
It is it is
(1 ) aspect
Chapter
9 has
com
cho
ratio , dealt
102
EFFECTS
AIRFOIL COMPRESSIBILITY
with aspect ratio effects , this chapter (Chapter 10 ) will aspect deal with ratio and Mach number , and Chapter 11 with aspect ratio ,
principally
Mach number ,
be
and Reynolds
in
insufficient information
to collecting

TWO DIMENSIONAL
10 : 1 .
spite of the billions of
and analyzing data
of
hours
COMPRESSIBILITY
SUBSONIC
infinite
aspect
from 300 to about 600 miles per hour , and develops
devoted
EFFECTS .
ratio in the speed range first the relationship
/4
Mer
Cmc
In
81
man
always
in this field .
AIRFOILS :
airfoils
This article concerns
will
For most new designs there
number .
.05.5 CL .04
.4
.03.3
.2
.02
CD
.01.1
.1
/4
Cmc
0
.2
1
.5
.3
.7
8
creases
(
)
and
pitching
numbers below the
that up to and
slightly
)
moment
,
,
: 1
.
on an NACA 4412
airfoil
critical
is
shown
.
Typi
beyond the
in Fig
critical
Mach
10
number
coefficient increases substantially the moment coefficient in slightly and the drag coefficient either remains constant or rises ,
lift
10
,
the
in Fig
drag
cor
effects
.
with high subsonic Mach Note
second
beyond which the
because of excessive shock wave ,
lift
,
variations of
critical
speed

limit
rections are inapplicable
of speed
,
of
the upper
(
a
study
: 1 .
)M .
compressible and incompressible flow characteristics and
between
Re
.
(a
constant
cal
for
:
.
a
10
1 .
Force and moment coefficient variation with Mach number = 0.250 for NACA 4412 airfoil small angle of attack From NACA TR 646 increasing with
Fig
TECHNICAL AERODYNAMICS
103
Since the test could not conveniently be run at constant Rey nolds number , the effect of increasing Reynolds number would , from con
slightly
.
of boundary  layer skin friction , have resulted in a reduced coefficient . It is evident from Fig . 10 : 1 that there is probably a number effect increasing the drag coefficient to offset the normal
siderations drag Mach
reduction due to increasing
Reynolds
number . M =
0.4
.5 M =
0.6
Cp .5
1.0 25
Fig .
Percent
50
Effect of Mach number on chordwise of a 66 series wing of A = 6.
10 : 2 .
middle
pressure distribution near
From NACA TN 1696 .
insight into the reason for the lift , drag , in Fig . 10 : 1 is obtained by inspection of Fig .
Some shown
effect of
100
75
chord
and moment 10 : 2 ,
variations
which shows the
pressure distribution . Pressures Fig such as those shown in . 10 : 2 can be calculated from the compressible flow equations given in Chapter 5if the velocity distributions are known . In general , it has been shown by Glauert ( 1 ) and Prandtl (2 ) that the Mach
incompressible
number
and
equation
on the chordwise
compressible pressure coefficients are related by the Cp inc Cp
( 1 ) Glauert , Press
,
1942
.
( 2 ) Prandtl , NACA
" Aerofoil
H.
L.
" General
TN 805 ( 1936 ) .
√1

( 10 : 1 )
M2
and Airscrew
Theory . "
Cambridge
Considerations on Flow of Compressible
University Fluids . "
AIRFOIL COMPRESSIBILITY
is
where Moo
tion ( 10 : 1 )
critical
the free stream
is in fair results from Cp
inc
use
=
Cp
It
of
M& √1  M
+
Equa
agreement between the theory and
M2 1
airfoil .
experimental data below the
most
relationship ,
the Karman  Tsien
may be shown ( 1 ) by assuming
104
number remote from the
Mach
agreement with
Mach number , though an improved
experiment
EFFECTS
+ 1/1
 M²
isentropic flow
Cp
( 10 : 2 )
inc
from the
free
stream
to
1.0 0 = NACA 4412 Wing , zero
O=
Elliptic
Cylinders
,
lift
zero
lift
.8
Equation ( 10 : 4 ) modified by equation ( 10 : 2 )
.7
Mcr
.6
Circular cylinder .5
Cp inc
mox
2
1
Fig .
Critical
3
function of maximum incompressible pressure negative coefficient . From Sibert , (2 ) Chart 9 .
10 : 3 .
Mach number as a
( 1 ) e.g . , Sibert , H. W. Prentice  Hall , Inc. , 1948 ( 2) op . cit .
" High .
 speed Aerodynamics
,"
equation 9.8 ,
105
TECHNICAL AERODYNAMICS
Incorporating
number
:

)

))
3
10
8
×
1
÷
2Y=
+
crit
that the
M = 1,
compressible pressure co
(
Mach
maximum
: 1 )
,
=
Y
: 3 )
(

the Prandtl Glauert relationship of equation 10 into gives for air of 1.4 the following relationship be
10
(
equation
YMar
(1
Comas Cpax
local
related to the by the equation + 2
Cpmax
where the
Y F ÷ 7
efficient
airfoil Mer is
[ (
number
☆
Mach
,
ical
*
the point on the
)
:
10
4
·
(
2
[(
0.4M2
0.45 2.4
1 ]
Mar
Mr
3.5
1.431
)
=
incmax

Cp
+ +
tween Mer and Cpincmax
1.0
.90
,
Kaplan calculations Elliptic cylinders
,
.95
TR 624
lift
zero
NACA
digit
CL
.
Moocr
.80
4
.85
symmetrical ,
airfoils
.75
TR
=
592
20
c,
t/
15
is
zero
angle
sweep of sweep
lift
and
on
critical
instead
of Mcr
coefficient
For swept wings read Mer cos
.
: 4 .
Effect of thickness ratio
30
.
where
10 number
,
Mach
A
.
Fig
25
percent
,
10
5
0
.65
0.4
▲
CL
=
.70
It is
elliptic cylinders
with the experimental
data
cal
in Fig
number
shown
.
The .
Mach
The theory
10
is
: 2 ) ,
seen to be
in
good
effect of pressure coefficient
: 3
and
.
tions
: 3,
.
(

a
also possible to derive similar relationship using the more accu rate Karman Tsien relationship of equation 10 and this equation has along with some experimental been plotted in Fig 10 data on wing sec
can be used to calculate
agreement on
criti
the effect of
AIRFOIL COMPRESSIBILITY
critical
.
in Fig
Mach
lift
with
number
coefficient
Mach
number
10
,
critical in Fig
on
data
: 4 .
coefficient
The
and
reduc
seen to be substantial
of the airfoil section and special in developing new airfoils to provide given lift coeffi reduction in critical Mach number for
10
depends on the
efforts have been
shape
NACA
a
the minimum
made by the
,
tion in
lift
are compared with test
.
thickness ratio and
calculations
: 4 ,
such
,
airfoil
106
EFFECTS
.
cient by avoiding large negative pressure peaks
1.0
a
Critical
M
9
)
(
deg
8
12
6
7
10
5
8 6
.
coefficient
CL
2
2
Lift
3
4
Experimental
CL with
8
7
5
6
.
6.
section
=
66 series
number Moo number at various constant angles tapered wing of From NACA TN 1697 Mach
A
of attack for
3 Mach
Variation of a
10
: 5 .
.
Fig
.2
,
0
0
Theoretical
to
and
as
lift
also increases in this ratio as sometimes slightly beyond the critical
the
,
: 1 ) ,
10
,
up
,
10
: 5,
.
in Fig
(
given by equation

1
,
/
.


Lift Curve Slope Effect At constant angle of attack on an air foil since all local pressures increase in proportion to √1 M² shown Mach
TECHNICAL AERODYNAMICS
inc
is
in Fig .
plotted
infinite
=
√1
lift
relating compressible
and
is
ratio
aspect
10
130
Figure
10 : 6 .
also shows that coefficient as
which
)
 curve  slope for 80
this
10 : 5 ,
increased
The corresponding equation
ao
and
with
reduced
(
lift
incompressible
is
number

critical Mach noted in Fig . 10 : 4 . the
5
line in Fig .
number , designated by the broken
:
107
be
10 : 6 shows good agreement
Prandtl Glauert theory and some experimental values , though slopes accurately  curve are difficult to measure and must be cor
tween the
lift
for
rected
wall
a tunnel
effect of uncertain accuracy , so that
some
of
the experimental data do not agree with the Prandtl Glauert theory as well as the samples
in Fig .
shown
VIMO
0.8
0.9
1
10 : 6 .
0.6
0.7
0.6
1.6 1.5
10.7
10
.
: 5
Eq
1.4
4412
TR 646
o a
alo
0.8
NACA
Wing
,
1.2
inc
inc
1.3
0.9
airfoils
,
4A series NACA
c1
= 0
1.1 TN 3162
11.0
0.2
0.3 0.4
0.6
0.5
0.8
0.7
Moo
.
,
often designed with sweep ,
or sweepforward
is aeroelastically
but which
rigidity
A
,
aerodynamically advantageous
the wings have low flexural
major
and
important
number
a
(
,
A
▲ ,
/
beneficial effect of sweep is to increase the critical Mach portion to cos where is the angle of sweep for thickness ratio when viewed normal to the wing axis not 1
aspect
,

are
sometimes with negative sweepback
) .
if
4a
flight
(
is also
.
which
,
as
1 :
in Fig
back
speed


lift
for high
Wings intended
unstable
infinite
compressible flow correction to ratio curve slope zero sweep
Subsonic
10
: 6 .
.
Fig
given
in pro airfoil
as seen normal
AIRFOIL COMPRESSIBILITY
to the lateral axis of the airplane ) .
is
This
/
108
EFFECTS
is
because Mcos
line , where effects of sweep ,
the
com
is usually
ponent of Mach number
normal to the wing c 4
sweep
measured .
several adverse
however , and the
There are
selection of the best more
in detail later .
if
sweep ,
of the
One
This effect exists even for
is
any ,
adverse
infinite
a complicated problem discussed effects is on lift  curve  slope .
aspect ratio
40°
.11
:
(
▲
/57.3
1.0
27 a=
.08
cos
:a
=
10 : 7 ,
20°
) cos
Theory
.07
in Fig .
angle
Sweep
.10
a√ M²
as shown
30 °
A
.09
,
.9
A
аодес
.8 Moo0
Theory
.06
.7
FA
Mo50.6
.6 Data NACA TN 1739
.05
.5
cos A .04
1
0.8
0.7
1.0
0.9
lift
Fig . 10 : 7 . Effect of sweep on infinite  aspect  ratio  curve  slope , including Prandtl  Glauert correction for high  subsonic sub  critical Mach number .
which includes the Prandtl
 Glauert
compressibility
correction
on the Mach
Note that the elementary theoretical correction based on COSA very good . An empirical correction factor FA is better . This can is not number .
the experimental right of the chart .
be read on
Drag and
in thickness ratio
Pitching on
curves in Fig Moment
critical
Mach
.
10 : 7 ,
Effects . number
with the FA scale at the
Since the effect of changes in Fig . 10 : 4 to
have been seen
similar to those of change of maximum negative pressure coefficient (1) shown in Fig . 10 : 3 , it is convenient to consider , as suggested by Stack ,
be
( 1 ) Stack , John . " Compressible Flows nautical Sciences , April 1945 .
in Aeronautics
."
Journal of Aero
TECHNICAL AERODYNAMICS
109
airfoil of
that the low  speed
airfoil is
speed
in Fig .
shown
mean line curvature coefficient than the actual ,
in
reduced
airfoil will
Such an
a greater
lift
characteristics as
same
of shorter chord
one
10 : 8 .
the
 M&
Cmac
and
this relationship
is
seen
some
experimental observations .
to be consistent
airfoil .
resulting relationship
The
sible flows
is
tios
The
The
.
Mach number analogous to the
( 10 : 5 )
and
( 10 : 6 )
is ,
lift
however , not
be based on empirical
must
and
repre data
coefficient with thickness ratio for incompres relation between the effective and actual thickness ra
of drag
given by
variation of
moment
Fig . 10 : 8 . Geometric interpretation of subsonic compressibility effects .
similar simple equation but
variation
be
coefficients sug
с
with
variation of equations
on the
,
Lc√1_M²
coefficient variation with
sentable by a
ratio
and a greater angle of attack for a given
in
10 : 1
moment
Mã as
(10 : 6 )
Fig .
The drag

have a greater thickness
tween incompressible and compressible pitching gested by Dwinnell (1 ) is
CacincV1
ratio V1
the
high
a given
(t/c ) effective minimum
 (t / c) actual / √1  M drag
section
NACA NACA
(10 : 7)
coefficient with thickness ratio
digit
4 and 5 66 series
series
.012 Rough
Surface
.008
1
Cd min
Smooth.
.004
Surface
/
t 4
Fig .
10 : 9 .
percent 12
Effect of thickness ratio for use with equation
(1)Dwinnell 1949 .
8
, James
H.
20
16
on minimum ( 10 : 7 ) .
" Principles
of
24
section drag coefficient
Re = 6
x
106 .
Aerodynamics . "
Mc Graw
,
Hill ,
AIRFOIL COMPRESSIBILITY
for two types of smooth and others , (1 ) is shown in Fig .
airfoils
rough
1010
EFFECTS
as summarized by Abbott and
,
with equation
10 : 9 and can be used
( 10 : 7 )
to
estimate the effect of compressibility on drag coefficient with zero sweep . Sweep angle also has an effect on infinite  aspect  ratio or section
coefficient
drag
if it is
ing (as
in Fig .
10 : 10b
considered that the
rotation
from the unswept wing by
in Fig .
( as
was
generated
rather than by shear
ratio of the
The thickness
).
swept wing
10 : 10c )
section
streamwise
with Fig . 10 : 9 to estimate the effect of sweep on section drag coefficient , but on airplanes the aspect ratio is always finite and some
may be used
times very small swept
( as
in
" delta wing " airplanes ) ; the
compressibility correction
wing
is
finite
article .
discussed in the next
V
ratio
aspect
V
Normal
Section
Streamwise
Section
(a )
(b)
Unswept
Fig . 10:10 10 : 2 .
Illustration of
.
This
FINITE WINGS
:
Effect
ainc a
CDi
for
.
the
=
effect of high subsonic
1.8
given
infinite  aspect  ratio
VI 
MZ ,
/TA is
independent
of
(10 : 8 )
+
Mach
number ,
but neither
drag and
of these
Ira H. , von Doenhoff , Albert E. , and Stivers , Lewis of Airfoil Data . " NACA Wartime Report L  560 . othert , B. " Plane and Three  dimensional Flow at High
( 1 ) Abbott ,
lift
resulting
often assumed that the relationship between induced
C
in
Mach number .
 M200
1.8+ A
a swept
EFFECTS .
The aspect ratio corrections
equation (2 )
in Göthert's
lift
COMPRESSIBILITY
of generating
in the lifting  line theory by the factor
curve  slope

( c ) Swept by rotating section thinner )
( streamwise
methods
modifying only the
commonly done by
It is also
different
SUBSONIC
9 need to be modified
is
two
section
wing from an unswept wing .
Lift Curve  Slope Chapter
Swept by shearing
( streamwise same )
S. ,
Jr.
" Summary
()
2 Speeds . "
NACA TM 1105
( 1946) .
Subsonic
1011
TECHNICAL AERODYNAMICS
is well verified
results on which
by more
and the assumptions
recent experiments
they are based are admitted
validity
by Göthert ( 1 ) to have poor
at low aspect ratio . A desirable
ratio
aspect
for high subsonic
correction
numbers
Mach
should be applicable to a wide range of aspect
ratios (0 to only on
1 ) and
ratios ( 1 to 10 ) and taper major should also cover the effect of sweepback , not
lift  curve  slope
drag , but also on
and induced
critical
Mach
num
information of this sort is ( in 1955 ) , having by by passed this need been recent NACA research because of the even more urgent need for supersonic test data . Murray (2 ) found in a study of wing test data in range aspect
ber .
not yet available
Complete
of
the
3 to 6 , and including
ratios from that Jones
(3 )
lifting  surface
sweepback
theory
up
discussed
to 45° on the c / 4
in
Chapter
9
line ,
could be
modified for high Mach number effect , including a sweep correction factor FA ≈ cos▲ , but requiring additional semi  empirical corrections as yet un determined , by the equation a where ẞ = √1

/
AE
1
=
ao inc
57.3 2π 2A
/
AFA B
( 10 : 9 )
small wing  thickness effect
M2 and a
of
Kaplan
(4 )
was
neglected , and E is the same factor used in Jones ' lifting  surface theory in equation ( 9:41 ) , and is very closely approximated for A = (1 to 10 ) by the equation
=
E Combining
equations
and ( 10:10 ) gives
( 10 : 9 )
1 57.3 2π
Equation (10:11 )
becomes ,
for ainc
( 1 ) Gothert ,
, (2 )Murray,
of Predicting 1739 ( 1948 ) .
(3 )JJones ,
B. , op .
Harry E. the Robert
/
/
( 10:10 )
0.94 + 1 A ≈ 1 + 1 A
2+ ( B/FÄ ) ß = 1 and
=
2 πT
for
(1 + A)
ao inc = 2
/57.3 ( 10:11 )
FA = 1 ,
A
57.3 3 +
( 10:12 )
cit .
of Several Methods " Comparison with Experiment Wings in Subsonic Compressible Flow . NACA TN
Lift of
T. , op .
cit .
( 4 ) Kaplan , Carl . " Effect of Compressibility at High Subsonic Veloci ties on the Lifting Force Acting on an Elliptic Cylinder . NACA TN 1834
( 1946) .
AIRFOIL COMPRESSIBILITY
is identical
which
with equation
particular
tests on a
( 9:33 ) and
available , present information ( 1955 ) sug for thin swept wings at high subsonic speeds .
( 10:11 )
Equation
also be written
( 10:11 ) may
A
a ☐ ao inc
( 10:13 )
and
2 + (1 + A)
( 10:12 ) can
be
ainc a
Effect .
effect of
equation
the
form of equation
 M² /FA
√1
3+ A
The
in
combined
= 2 + ( 1 + A)
( 10:13 )
 M² /FA
√1
( 10 : 8 ) thus
Drag
low speed
at
unless special high  speed
,
wing are
gests use of equation
Equations
well verified
Accordingly
ratios .
for a wide range of aspect
1012
EFFECTS
(10:14 )
on the low  speed induced
Moo
/
(10:15 )
CD₁ = CZ πAеW has been inadequately as are
available
explored , but such spanwise load distribution data
suggest
9:23 should be reduced The the
profile
effect
" rotating
drag
depending
is
that
ew
somewhat
for
as
Moo
also affected
on
whether
approaches
by
the wing .
was
tive thickness ratio sweep reduces the effective thickness ratio
sweep ,
the amount of
swept by " shearing " or by
Increased
of the wing as given by
Fig .
Mor
and by
Moo
A , as given in
ratio
any aspect
as described in Fig . 10:10
"
drag
Moo
increases the
equation
( 10 : 7) ;
effec
increased
factor cos▲ , but in creases the chord and hence the Reynolds number . The net effect of sweep on profile dragis hence partly a viscosity phenomenon ; viscosity effects are considered 10 : 3.
lift ,
in
TRANSONIC
Chapter
by the
11 .
COMPRESSIBILITY
EFFECTS
.. Rules
for the variation of
airfoils covering the range of speeds from the high subsonic super critical , as in Fig . 5 : 1b , through the low supersonic range involving subsonic flow areas , as in Fig . 5 : 1c , have not drag , and pitching
been , and quite possibly
stall
moment
of
can not be , simply formulated .
and separation as well as
Mach
number effects
This
is
because
are involved
.
How
general rules can be stated for a limited high subsonic super region . This is well worth while because it is the region in
ever , some
critical
competing military aircraft designs are currently for superiority for obvious military reasons .
which
The
airfoil
air
( 1955 )
striving
flow patterns in this regime of speeds and the corresponding pressure distributions are shown in Fig . 10:11 . Figure 10 : 11a
TECHNICAL AERODYNAMICS
1013
shows the
shock waves which have formed
point ) of
a 23015
airfoil
exceeded .
Figure
slightly
sure distribution
sufficiently
exceeded so that the upper
layer have interacted This
surface .
is
number
Mach
critical
known
of the flow
as
the
number
Mach
24
,
, 1.0
c
,
b .
.
a
=
in
10:12 that no marked change when Mer
is
reached
corres in
are shown
either the drag
lift
and increase in Glauert rule can be


a
The
and note also that when
major drop off in there has been that the Prandtl curve the from evident
reached
slightly
coefficient
due to exceeding has started
airfoil
coefficients of this
drag
coefficient occurs
It is
0.73
the symbol Ms.
to
beyond Mer
for the prediction of variation of section and that the onset of serious trouble
with Mach number the
rise
critical
Mach number does not appear
appreciably
This point
,
applied
Moo
,
.
drag
and
in Fig
designated by
.
is
Note
.
.
.
10:12
lift
lift
section
burble
)
.
.
= 0.60
"
a .
,
x
Chordwise station
x
/c
/
.8
Simultaneously obtained schlieren photographs and pressure 20. From NACA TN 1813 for NACA 23015 airfoil section at
"
(
Moo
compressibility
lift
1.0
6
.6
=
4
.2
,
Pressure coefficient
M
ponding
Mg
stall
M 1.0.
Fig 10:11 distributions
or
shock
1.6
Chordwise station
Fig
of
Crest
Crest
160
or
from the upper
∞
,P
has been
Mach number
surface shock waves and boundary
separation
produce
to
number has
Mach
(highest been only
: 11b shows the shock wave pattern and pres
0.73 when the
Moo =
at
beyond the crest
critical
when the 10
just
until
where the slope
the drag
of the
drag
AIRFOIL COMPRESSIBILITY
1014
EFFECTS
.
a
,
,
,
a
curve vs. Mach number has attained value of 0.10 is labeled Md At some slightly higher value peak in the but smaller than Mg there is .7 .14
c,
assuming lower Variation of surface lift contribution constant above M 0.65
ratio also
1
1
Aspect
12
.

.6
,
See
.10
3
.06
coefficient drag
7 .
coefficient
.08 O
lift
, cd
a
= A = . 2
.
,
a
large has TN 2720 which shows increase in Md from 0.68 at A = to 0.80 at for NACA 0012 wing at 20
effect
cd ./
Mcr
.
Md
02
.6 .7 Free stream Mach number Section lift and drag coefficients 20.
of
Moo
for
sub is evi
beyond the
0.95
indicated by Fig 10:13 which shows that the shocks that form at
a
small loss
pattern
.
small disturbance
of
,
0.85 1.0 1.1 1.2 1.3 1.4 1.5 1.6 Machnumberi front shock
numbers involve
energy
and
hence
of the general flow
n
Mach
0.90
Fig
10:13 Ratio of total pres sure across shock as function of Mach number in front of shock .
critical

low super
is
.
Mach number
1.00
.
corrections
sub critical
, M ,
critical
The reason
of subsonic
of
applicability
a
for the
1.05
.
dently no longer applicable
.
correction
Totalheadlossacross shock wave

curve and the Prandtl Glauert
sonic compressibility
only
M
as functions From NACA TN 1813 .
a
at
=
airfoil
NACA 23015
lift
,

10:12
.
.
Fig
O
8
.5
.4
Section
.04
2
Section
Ms
TECHNICAL AERODYNAMICS
1015
below which no serious
The Mach number
by the Prandtl
other than those covered
effects
 Glauert
compressibility
due to
or
 Tsien correc
Karman
, is thus seen to be not only greater than Mer but greater usually , , Mg also than Md . It is however less than insofar as compressibility effects are concerned , but the effect is greatly depen Fig dent on angle of attack and airfoil section as shown in Fig . 10:14 .
tions are applicable
ure 10:14 gives calculated
and experimental
values of Mer ,
Md , and Mg ,
as
well as calculated and experimental values of Mg which is defined as the free  stream Mach number giving M = 1 at the crest ( highest point ) of the
airfoil .
The calculated
excellent
agreement and reasonably close to Md .
the newer
airfoils
by
Fig .
Figure
10 : 14a .
larger than
typified
,
values of
are seen to be in also be noted that
Mg may
10 : 14b
than the older
also shows that
Md
is
airfoils
,
critical typified
very substantially
Mer at the higher angles of attack .
For determination of the actual flyable plane , the additional
adverse
Junction
into
must be taken
SUPERSONIC
this is
flight
tests
CHARACTERISTICS .
WING
pressure or pitching
moment
Mach
compressibility
account ;
without special wind tunnel or 10 : 4 .
It
by Fig . 10 : 14b , have a much higher
a given angle of attack
for
Mach number
and experimental
number
for
a given
air
effect due to wing  fuselage almost
impossible to predict
.
The
lift , drag ,
and center
infinite  aspect
coefficient for
 of
ratio super
sonic airfoils can be calculated from shock and expansion wave theory as in Chapter 5. For thin airfoils at small angles of attack the results , as simplified by Ackeret and presented
by
Sibert
( 1 ) may
be written as
follows : 4α
CL
/2
Cmc
/
where Zc = 2 ( t c ) ² a symmetrical
calculated
=
C²
=0
√M²

1 +
( infinite A )
2Zc
√M20

(infinite A)
( 10:17 )
( 10:18 )
for a symmetrical double  wedge airfoil and Z = ( 8/ 3Xt /c ) ² bi  convex airfoil . For other airfoil shapes , Zc can be
as outlined
( 1 ) Sibert , op .
(2) Ibid .
( 10:16 )
VM20
CDwave
for
(infinite A )
cit . ,
in
Sibert
. (2)
chapter 19 .
AIRFOIL COMPRESSIBILITY
EFFECTS
1016
.8
)
(
Mg calc
Ms
(
Melexp
יו
1
Md
,
Mcr
6
,
section
8
4415
NACA
8
.
deg
.
Airfoil
a
2
of attack
Md
Ms
Mplexp
(
Free

Angle
7
stream
Mach
2
a,
4.6
,
number
M.
.5
.6
MB'calc
Mc
of attack
.
B
0.6
.
stall

drag divergence and shock two
different
NACA
airfoil
Mach
sections
.
66,2215
=
,
a,
NACA
Variation of critical of attack for
numbers with angle From NACA TN 1813
deg
a
section
,
.
.
10:14
Airfoil
,
b .
Angle
Fig
K
2
2
,
4
.5
TECHNICAL AERODYNAMICS
1017
Finite
Rectangular Wings .
Except
for
boundary
 layer effects ,
finite
rectangular wings behave like infinite aspect  ratio wings except for the region included in the tip Mach cones , as shown in the sketches in Fig . 10:15 . It is seen in Fig . 10:15 that the difference between
nite
and
fi
infinite
 ratio characteristics is for small flying at low supersonic speeds as in Fig . 10 : 15b .
aspect
aspect  ratio wings
small except
a . High M , Low A
Fig .
10:15 .
Sketch
in tip
conical flow A wings . The
showing
region
.
b . Low
M , High
A
areas of finite rectangular wings affected by Unshaded areas behave nearly like infinite
analysis of Schlichting ( 1 ) as presented by Sibert
, (2)
results in the
equations 4α
CL
√μ20 CDwave
/2
CTRC
( 11 / 2A / M2 1 )
·
=
180
1
A α
3A (M²
/ /M2

1 2
in
has the same meaning as
be added to the wave drag
lift
( Delta ) Wings .
drag , and pitching
( 10:20 )
1
( 10:21 )
drag must
,
2Zc
VM20
 1)
where Zc
Triangular
·
( 10:19 )
equation of equation
( 10:17 ) ,
and skin friction
( 10:20 ) as
The general problem
moment of tapered
before .
of
calculating
and swept trapezoidal
wings
is
too complicated for presentation here , but has been effectively presented by Cohen (3 ) and others . However , the case of triangular wings with straight ( 1 ) Schlichting ,
1939 .
H, "
Airfoil
Theory at Supersonic Speed , "
NACA TM 897 ,
( 2 ) Sibert , op . cit . , p . 136 . ( 3 ) Cohen , Doris , " Formulas and Charts for the Transonic Lift of Flat Sweptback Wings with Interacting Leading and Trailing
NACA
TN 2093 , 1950 .
and Drag Edges , "
AIRFOIL COMPRESSIBILITY
a
very commonly has been
angle
of the delta leading
TN 1955
are
.
,
"
edge
10:16
The
Ellis
,
aerodynamic
to the
m
well
pointed out
in
characteris
tangent
of the
results of the tests reported in and Hasel draw the following con
:
tan E tan m Jones theory
8/1
(
1000
Y
1.43 1.71 1.75 reference
)1
)
'
(
00
angle
Mach
(
(

6
Measuredlift curve slope Theoretical two dimensional ft curve alope
1.0
in Fig
of
it is
wings
their experimental study
clusions from

( 1 )
.
tangent
.
,
is the ratio of the shown
For such
factor in determining the
major
a
NACA TN 1955
and Hasel
.
Ellis
by
presented
"
,
known as delta wings which constitute special case of the tapered and swept trapezoidal wing
used
tics
1018
,
trailing edges
EFFECTS
.2
00
2
.8
.4
1.0 tan E tan m
1.2
1.8
1.6
1.4
2.0
.
0
(
% )4
T
max
с ☐
0
M
1.43 1.71
.08
6
.09
min
O
O
2
4
8
O
O
NACA
1.0
tanE
1.2
1.4
1.6
1.8
2.0
1949
.
October
.
.
From NACA TN 1955
Preliminary Hasel and Sweptback Wings
Investiga
, "
C. ,
,
at
( 1 ) 1955
E.
Macon Jr. and Lowell Supersonic Speeds of Triangular
Ellis
tion ,
series of delta wings ,
Test results on
"
10:16
.
.
Fig
a
tan m
NACA
TN
TECHNICAL AERODYNAMICS
1019
lift of
thin , triangular plan  form wings may be calcu slender wing theory up to values of tan Ɛ / tan m ≈ 0.3 , where is the wing vertex half  angle and m is the Mach angle . For values of tan ɛ / tan m above 1.0 , the is essentially the same as that obtained theoretically for a two  dimensional wing . 2. The center of pressure of thin , triangular plan  form wings is coincident with the center of area . 3. For low drag coefficients approaching those due to skin friction alone and for the highest values of maximum  drag ratio , both triangular and sweptback wings should be operated with their lead ing edges well behind the Mach cone . " " 1.
The
lated by Jones
'
lift
lift
PROBLEMS
Using the Clark Y ( CY ) airfoil data in Appendix 5 , estimate a 10 : 1 . and cd0 min for a Clark Y wing of 7  ft chord and 35  ft span , flying at 450 mph in standard sea  level air , neglecting Re corrections . 10 : 2 . An airfoil is tested at low speed and found to have a maximum negative pressure coefficient on the upper surface of 1.0 . Using Fig . 10 : 3 , find the critical Mach number and critical speed at 40,000 stand
ft
ard altitude . Using equations ( 10:19 ) , ( 10:20 ) , and ( 10:21 ) , write equations 10 : 3 . , drag , and pitching moment coefficients of a rectangular wing for the of aspect ratio 4 , flying at a Mach number of 2.0 . Assume t c = 0.05 . 10 : 4 . Using Fig . 10:16 , calculate the , drag , and pitching moment about the center of area , for a wing consisting of an isoscles triangle 10 ft on each side , flying at a Mach number of 1.5 in standard sea  level
lift
lift
air .
/
CHAPTER
11
AIRFOIL VISCOSITY EFFECTS 
11 : 1 .
SECTIONS . The low speed Reynolds
SCALE EFFECTS , WING
LOW SPEED
number affects chiefly the boundary layer and hence the skin friction drag and , through effect on separation , the stalling or maximum lift coeffi cient . Since boundary  layer transition (from laminar to turbulent ) and
separation
also affected
are
by
surface roughness
surface
,
curvature
,
surface pressure gradient , and other factors , as discussed in Chapters 6 and 7 , there is a complicated interaction between the boundary layer and the outside air stream , resulting in a number of minor secondary effects which could not be anticipated from any simple theoretical considerations . Some
are evident from Fig .
such interactions
speed scale effects
attack
for
lift
zero
( Fig .
stall
11 : la ) ,
section drag and
maximum
section
results
shown
in Fig .
dimensional low  turbulence tunnel .
" rough
L.E.
"
coefficient
moment
the major
( Fig .
and angle of
effects
ft
10
to
lie
near
11 : 1c ) .
obtained in the Langley two The smooth models were of polished the " smooth on
"
curves
the smoothness
could be maintained in service
on airplanes
and
(c ) on
11 : 1 were
curves , depending
wing construction
ft to
lift
low
minor
shows
free  flight characteristics of an actual airplane
reasonably be expected
tual
pitching
(b )
in addition to
center
mahogany wood ;
from 4
11 : 1 , which
lift  curve  slope
11 : 1b ) ,
The test
of the
on (a )
(Fig .
aerodynamic minimum
below the
with stalling
.
in
might
direction
the
to which the
ac
For wings of chords
speeds of 50 mph to 100 mph ,
is cross  hatched on the c1 max chart of Fig . sizes and level high speeds of 100 mph to 300 mph ,
the range of values of Re 11 : 1c .
For the
same
the lower part of the range of values of chart of Fig . cause of Mach
11 : 1c . number
The most abundant
Re
is
cross  hatched
Beyond 300 mph , the charts
effects discussed in the
next
section test data available
are
on the cd min not applicable be
article
and
are either
elsewhere .
at
Re = 3.2
million (from the NACA variable density tunnel , with turbulence of about 2.5%, as given in Tables A5 : 1 through A5 : 3 ) or Re = 6 million (from NACA Wartime Report L  560 , as given in Figures A5 : 1 through A5 : 5 ) . Correc tions of these data to other values of Re from 3 million to 9 million 111
112
TECHNICAL AERODYNAMICS
.11
.10
TH
1
Smooth
ao per deg
alo ,
deg
Rough
2
B
L. E. A
L. E.
Rough
.09
/
3
Re 106
.08
1.0
.5
2
345
10
4
Smooth
/
Re 106 1.0
.5
2
345
10
Section 662415
Lift
a.
curve effects (minor ) .
.05 .20
C mac
.06
Smooth
.07
or
ac ,
Rough
of
/
1.0
2
Pitching
b.
3
/
.25
Re 106 1
.08 0.5
fraction chord
4 5
moment
Re 106
1.0
0.5
10
and aerodynamic
center effects
Rough
10
( minor
).
Stall
range
L.E.
Smooth
10
1.6 Smooth
High speed
7
range
of
1.4
airplanes
of airplanes clmax
5 1000
1.2
Cdmin
s
Rough
L.E.
1.0
3
/
Re 106
2
0.5
1.0 C.
Fig .
345
1.8
15
4
2
2
345
Minimum drag
/
Re 106
0.8 10
0.5
and maximum
lift
1.0
effects
2
3
45
10
( major ) .
Minor and Major effects , at low speed , of Reynolds number on section characteristics of NACA 662415 section . For comparison with other sections see Chapter 13 and Appendix 5 , Fig . A5 : 15 . From NACA TN 1945 . 11 : 1 .
AIRFOIL VISCOSITY are given in Figures
cl
and
max up
data
The
to
113
EFFECTS
A5 : 6 to A5 : 8 and A5 : 11
to A5 : 14 ; corrections of cd min given in Figures A5 : 9 and A5 : 10 ,
million are conflicting and nearly
Re = 25
are often
This
incomplete .
always
is
necessarily true because the task of maintaining even a nearly complete catalog of airfoil section characteristics as affected by most of the variables involved is prohibitively expensive . It is moreover unneces of the desired accuracy can usually be estimated , flight condition , by a judicious combination particular specified for a existing of data such as are given in Appendix 5 .
sary
,
as information
For example , suppose it is desired to estimate the low  speed section characteristics of a smooth wing section designated by the four digits for a range of Reynolds numbers from 8 million ( for cl max ' at stall ) to 18 million ( for cd min , at high speed ) , with other characteristics es timated at 13 million . The data here available ( other data may be avail 2418 ,
able ; consult at
Re
3.2 million
first
the
indexes ) are those given
NACA
Item
Data
Table
C1 max
A5 : 1
/
ao deg cd min
of Table
Figs
Data .
A5 :
a.c.
0.239
0.241
3.2
6.0
first ,
1.28 ±0.10
2.0 ±0.1 0.105 ±0.03
0.0068
0.045
Re 106
involved
Estimated Values ( see text )
0.098
0.0076
Estimate
listed in
below.
15
2.1
0.094
Cmac
is
test data
A5 : 1
OF PROCEDURE
1.35
0.038
/
11 : 1
(VDT
These data are
FOR ESTIMATING AIRFOIL SECTION CHARACTERISTICS
EXAMPLE
1.53
1.9
α10
)
two columns
TABLE 11 : 1 .
in Table
and Figures A5 : 1 to A5 : 5 .
0.00761.001
0.048 ±0.003 0.243 ±0.002
the minor effects
since the graphs for
8
13
Re
to
Re = 13
18
million
million . Since the effects are unlikely to be very far off ; extrapolation should possible , but here , as often , it cannot be avoided tend beyond Re = 9
An extrapolation
.
corrections of these items minor , always .
the be
do
not
ex
results are avoided
if
The extrapolations
are explained item  by  item below : a10 :
Data in columns 1 and 2 are 1.90 and 2.10 respectively . Refer to Fig . A5 : 6 and note that while data for the NACA 2418 are not given , the effect of Re on alo for other 4  digit air foils between 3 million and 9 million is less than 0.20 . For may reasonably be judged from inspection of the the 2418 , graphs that no greater change is involved and that a10 changes
it
TECHNICAL AERODYNAMICS
114
less than 0.20 up to is 2.00 ± 0.10 .
Re = 13
million
.
Hence , the estimated
alo
in columns
Re 1 and 2 are 0.094 and 0.098 respectively . A5 : 7 and note that ao for other 4  digit airfoils is in the range from 0.102 to 0.110 with small changes between Re = These data are inconsistent with 3 million and Re = 9 million . Table A5 : 1 for the airfoils given in both Fig . A5 : 7 and Table A5 : 1 ( 0012 , 4412 , and 4415 ) being about 0.005 to 0.010 higher in Fig . A5 : 7 . is recommended that Fig . A5 : 7 be accepted as more accurate because is a more recent study , and that the value 0.094 be thrown out as probably improperly measured or corrected . Correcting the value 0.098 at Re = 6 million to 0.104 , is 0.003 at timated by inspection of Fig . A5 : 7 that a = 0.105
ao :
Data
fer to Fig .
It
it
it
million
Re = 13
.
in
columns 1 and 2 are 0.038 and 0.045 respectively . No general Re corrections are given in Appendix 5 , but Fig . 11 : 1 3 shows the order of magnitude of the correction between Re million and Re  9 million to be 0.004 for another airfoil ( 662415 ) . It is judged by inspection of Fig . 11 : 1b that another increase of about 0.003 would be involved in going to Re = 13 Data
cmac

million a.c
es
.
0.045
.:
Hence , the estimated value 0.003 = 0.048 ± 0.003 .
of
Cmac
for
Table
11 : 1
is
No Data in columns 1 and 2 are 0.239 and 0.241 respectively . general Re corrections are given in Appendix 5 , but Fig . 11 : 1 shows the correction on another airfoil ( 662415 ) to be about 0.002 between Re = 6 million and Re = 9 million . Hence , the es timated value of a.c. is 0.243 ± 0.002 at Re = 13 million .
effects of with similar
The major comparison
Re on
cl
max and
data on other
cd min may also be estimated
airfoils in
Appendix
5 as
by
follows :
Data in columns 1 and 2 are 1.53 at Re = 3.2 million ( and high free  stream turbulence ) and 1.35 at Re = 6 million ( low turbu lence ) . A procedure for taking account of turbulence ( given in second edition of the text ) by calculating an " effective Reynolds number " higher than the actual Reynolds number has been largely abandoned as basically incorrect . The preferred procedure now ( 1955 ) is to use only low turbulence data . Hence , the figure = C1 max 1.53 from Table A5 : 1 will be disregarded as a basis for estimating the free  flight value . No low turbulence data for Re
C1 max :
for
of C1 max NACA 2418 sections are available in , though some might be found by a thorough study of the indexes . Data for Re corrections to c1 max of some other
corrections
this text
NACA
4  digit
airfoils ( 4412 and 4415 ) are given in Fig . A5 : 14 , and these data are consistent with Fig . A5 : 2 which also shows the 2400 series to be about 0.15 less than the 23000 series at Re = 6 million . the same sort of variation of C1 max with Re is assumed for the 2418 as is shown in Fig . A5 : 14 for the 23015 , the increase in C1 max between Re = 6 million and Re = 8 million should be about 0.03 . Hence , the estimated C1 max for a smooth 2418 airfoil is 1.35 0.031.38 . Note , however , that " stand ard " roughness produces a drop of 0.3 . Hence , moderate surface irregularities might well be assumed to produce a drop of 0.1 , and the best estimate for an actual wing is c1 max = 1.28 ± 0.10 . Data listed in columns 1 and 2 are 0.0076 and 0.0068 at Re = min 3.2 million ( high turbulence ) and Re == 6 million respectively .
If
Cd
AIRFOIL VISCOSITY
115
EFFECTS
airfoil section are not given , but from other airfoil data as follows : Refer to Fig . A5 : 8 and note that cd min for 4 digit and 5  digit = 6 million 15 percent thick airfoils drops 0.0002 between Re and Re = 9 million . Further change to Re = 18 million may be inferred from inspection of Fig . A5 : 9 to be small . In fact , Fig . A5 : 9 shows the drag of 6  series airfoils to rise between Re = 9 million and Re = 18 million , but this is presumably due to a larg is here judged that on er area of turbulent boundary layer . the 2418 airfoil the same conflicting factors are acting ( reduced cd min with higher Re due to normal skin friction changes for a given ratio of turbulent to laminar areas ; increased cd min with Re due to change of transition point resulting in larger area of turbulent skin friction ) . The net change is estimated as zero but uncertain to the extent of the rise seen in Fig . A5 : 9 for the 633018 airfoil ( about 0.001 ) . Hence , the estimated cd min for a smooth 2318 airfoil section at Re 18 million is 0.0066 ± 0.001 . Note in Fig . A5 : 9b that standard leading edge rough ness can add 0.004 ; hence , for actual wings in free flight , part of the difference may be added ( say 0.001 ) . The net result estimated at cd min = 0.0076 ± 0.001 . Re corrections may be estimated
for the 2418 or inferred
It
is
For 6  series
airfoils
which are coming
,
considerably
as the above can be made
airfoil are
discussed above , because
wide use , estimates such
accurately
more
( e.g. ,
data
than
Figs
.
for the
2418
A5 : 10 )
A5 : 9 and
available .
11 : 2 .
COMBINED
HIGH
SPEED
AND
SCALE EFFECTS ,
variations of
results with independent dom
more
into
Mach
WING
and Reynolds
SECTIONS .
Test
numbers are
sel
obtained .
The combined
for high
effects on
minimum
drag
coefficient are usually estimated
subsonic Mach numbers from tests at the proper Reynolds
number
For supersonic Mach numbers , drag estimates usually are based on tests at the proper Mach number , and skin friction
by the methods of Chapter
variation
with
10.
Reynolds
number
is
estimated by the methods of Chapter
7
(which are notably incomplete and often inadequate ) . A few test results of the independent available .
airfoil
Some such
section
( see
effects of
flight of airplanes at high Fig Note in . 11 : 2 that the Mach number effect on factor for values of M as low as 0.2 or even 0.15 independent effects in Fig . 11 : 2 must be combined calculation of stalled
conditions
.
results are
For the shown
M and
Re on c1 max
are
results are shown in Fig . 11 : 2 for an NACA 64210 also Fig . A5 : 15 ) . Such data are indispensable to
three conditions
in Fig .
11 : 4 .
shown
altitudes or in turns .
cl for
max becomes
according to the
in Fig .
a major
smooth wings .
11 : 3 ,
The
flight
the combined
TECHNICAL AERODYNAMICS

number of
The Reynolds
Fig
11
1.65 feet
wing
chord
For condition
.
=
c
of
con
corresponds to
level flight with
sea
mph
2 ,
lof
dition
1.3
M =
standard sea
a
69.
200
200/761
0.26
=
1.4
in
designated by numbers plotted
6.
flying at level air
For an airplane
: 3
/
Re 106
.
1.5
.
116
the wing chord for sea level flight is 4.3 feet At higher altitudes

1.2
.
clmax
.
.
;
flight
For supersonic
 3692
3691369
sort should be lating maneuverability
TN 2824 .
is classified for reasons of military security
available
.
NACA
when
,
11 : 2 .
current
most
information of this sort
,
(
for
inde pendent variations of M and Re on From NACA 64210 airfoil section . Test results
Fig .
1955
)
.5
.4
.3
this
available for calcu
M
.2
.1
0
of
data
;
0.9
at 50,000 ft times as great
,
1.0
the chords are about
3
Rough
5
the chords are larger
1.1
Condition
Low speed
/
8
4
2
: 2
.
After the effects of as
Re
.
11 : 3 ,
with
M and
in Articles
Re 11
: 1
0 .
: 4 .
C1 max
wing have been estimated
wing can be calculated
,
.
a
a
the
gration over
of wing
variation of
characteristics of finite inte If dS is an element series of chordwise elementary areas area equal to cdy the relationship between the section coefficients
: 2 ,
and 11
finite
10
Combinations of Figs 11 flight showing and
,
.
on each section of
11
smooth
EFFECTS ON FINITE WINGS a
:
3 .
SCALE
Fig
condi
)
8
flight
Mand Re effects
(
Possible
10
.
: 3 .
.
6
0
2
4
combining From NACA TN 2824
Re 106
1.0
6
/
Re 106
11
2
1.1
3
Condition
1
Condition
1.2
1
High airplane Condition Large
.2
Fig 11 tions for
Condition
1.3
altitude
.
High
.3
Clmax
1.4
2
Small altitude Condition
M
airplane
1
.4
3
1.5
.5
by
AIRFOIL VISCOSITY
coefficients specified
and the wing
equations
by the
CLS =
117
EFFECTS
( 11 : 1 )
cids
( 11 : 2 )
= Scads CDS
Cmaccavs
is
where cav
when multiplied
lie is
.
is
( 11 : 3 )
/
( 11 : 3 ) ,
the line
about
if
only
the aerodynamic
straight line .
on the same
wings and
jcmaccds
wing
Equation
coefficient
applicable
(about a.c. line )
chord S b . Equations ( 11 : 1 ) and by the dynamic pressure q , give the lift and drag
the average
wing respectively moment
=
in
as noted
of
aerodynamic
therefore
applicable to tapered wings only
,
the
on
gives the
is
centers and
hence
all
centers of
It is ,
parentheses
( 11 : 2 ) ,
if
sections of the wing not applicable to swept
,
the geometry of the taper
in a straight line . flight approximately For subsonic this line is the wing quarter chord ; for supersonic flight it is approximately the fifty percent chord line . A more general statement than equation ( 11 : 3 ) would require writing an centers of the wing
such as to keep the aerodynamic
other integral expression for
the pitching
the root section , of the forces acting
applied
( 11 : 1 ) can be
can be related
only
.
distribution of air
The spanwise
critical
the
distribution of
Mach
of a
number
lift
structed with zero
of construction
chords
, however ,
have , however , a
values of
of
finite
all
is to
have
special
also wing .
spanwise
twist .
α10
known as a " wash ;
positive .
The
A twisted
wing
wing .
to es
the
of
factor deter
Wings are often con
stations
parallel
;
such
An equally common method
geometric
chords
parallel
,
in
said
between the root and
tip sections
built
negative sign
importance
a major
with an intentional twist , the most involving a reduced angle of incidence near the tip . ever , often
twisted
coefficient cl considered in
to have zero geometric twist ; it will then aerodynamic definite twist corresponding to the differ
is
the wing
which case
is
is
This
.
in
Equation
.
loads for structural analysis
lift is
wings are said to have zero aerodynamic
in
lift
local
the
It is of
LOAD DISTRIBUTION .
SPANWISE
a wing .
ence
if
to the angle of attack of the wing
timate the spanwise mining
practice
sections
article .
the next 11 : 4
in
about some point
moments ,
on the outboard
the
 out " of incidence reverse twist
,
and
.
of
of twist
This kind of twist
is arbitrarily
a " wash  in "
Wings are , how
common type
designated by a
incidence
,
is
twist in is has a different spanwise load distribution from Negative twist is often built into a tapered wing to degrees
commonly designated by the
called
symbol € . an un
reduce
TECHNICAL AERODYNAMICS
118
the tendency
twist
on
When
of such
wings
stall first
to
stall distribution is
at the
tips .
The
effect of
discussed later in this article . a wing with a positive angle of twist has zero total , the not zero at all points ; zero negative is the result of a
lift is lift near
the middle and a positive
lift lift near
lift
the tips , as sketched in A wing without twist has a spanwise distribution similar Fig , to that sketched in . 11 : 6 with a maximum at the middle and a reduced
Fig .
11 : 5 .
lift
near the tips , the
form
of the
is
wing .
lift
distribution
The Prandtl
obtained with an
elliptic
wing
lift
depending
chiefly
theory concludes
elliptic
planform and an
that
lift
on the minimum
plan drag
distribution
.
Wing Wing
Fig .
11 : 5 .
due The
lift
sidered additive
tion
in Cla
as sketched in Fig
c1c1b
which
is
Fig .
components due to twist ,
+
cla
lift
Additional distri 11 : 6 . bution due to angle of attack .
Basic lift distribution to positive twist .
and .
angle
11 : 7 and
Clb
+
of attack are usually con as represented by the equa
ClaiCL
( 11 : 4 )
clb is the basic lift coefficient at any spanwise station additional lift coefficient . Since cla is proportional
the
and
to
Wing alone
&Wing Wing +2 nacelles
lift
distri Resultant bution on wing with positive twist at positive angle of attack . Fig .
11 : 7 .
Fig .
Effect of nacelles on distribution . ( From Vol . IV , p . 161. )
11 : 8 .
spanwise Durand ,
lift
to the average lift coefficient proportionality is designated by clal on the wing CL . The constant of and may be considered to represent the local additional lift coefficient
angle
of attack
,
it is
also proportional
lift coefficient CL is 1.00 . wings with rounded tips and various taper ratios tapered straight For theoretically calculated values of clal and clb /a € and aspect ratios , widely used in are given in tabular form in NACA TR 631 and have been
when
the average
AIRFOIL VISCOSITY
calculating
lift
spanwise
There is is justified , because
sis .
some doubt ,
distribution for however , as to
the effect of
EFFECTS
119
of structural analy
purposes
such a refined
whether
method
fuselage or nacelle interference
is
neglected by this method , and such
neglect results in wide departures from theoretical calculations as shown in Fig . 11 : 8 . Accordingly , an equivalent approximate method , developed by Schrenk ( 1 ) which is explained in detail in textbooks on aircraft structures such as that of Peery , ( 2 ) has been widely adopted . The assumptions of the Schrenk are as follows :
method
1
)
of twist 2
ematic
)
any point
is
equal to the angle
lift chord multiplied by one half lift  curve  slope . lift distribution is proportional to the arith
aspect ratio
additional
The
the actual wing chord and the chord of an ellipse area as the wing .
mean between
having the
same
These assumptions work
distribution at
measured from the mean zero
infinite
the
lift
The basic
fairly
well
are
difficult
to
justify theoretically
but are found
to
in practice .
1.7
Point of
first stall
C1 max
1.6
CL =1.52 =CL Imax
CLE 1.45
CL
1.5
CL
1.40 1.35
1.4
1.3
0.1
Fig .
If ted
in
11 : 9 .
11 : 9 ,
Semi
and
0.4
0.5
Fraction of  log plot of C1
0.6
0.7
semi  span max and
c
to
0.8
determine
0.9
1.0
CL max ·
if the
lines in Fig .
solution
can be made
(1 ) sSchrenk wise
0.3
the distribution of c1 max along the span of a wing can be estima terms of M and Re , resulting in a plot such as the solid line shown
in Fig . dotted
0.2
distribution of c₁ is also known as shown by the for each of a number of values of CL , a trial for the spanwise location of the point of first stall 11 : 9 ,
" A Simple Approximation Method NACA TN 948 , 1940 .
, O. ,
Lift Distribution , "
( 2) Peery , David
,
J.
" Aircraft
Structures
."
for Obtaining the Span
Mc Graw
 Hill ,
1950
.
TECHNICAL AERODYNAMICS
1110
for the value of CL
and
if
in Fig .
shown
point of heaviest shading
Fig .
b.
11:10 ,
BOUNDARIES
lift . the lift
of speed and times
These
lift ,
The
tion
is
are constant
In flight ,
.
= nW =
flight
the
if
L
is
speed
constant .
equation
number
Data on the
relationship
that of Fig .
10 : 5 may be
some
Clas
= CLSamph²
,
tapered , and
as shown .
8
, a wing may
stall ,
other combinations by the
following
/391
in miles per hour . In
1481
rewritten
/
p Po
condi (11 : 4 )
Note that
terms of the pressure
qCL
/
and nW S
/
ratio
= p Po '
as
MCL
( 11 : 5)
for a particular wing such as replotted logarithmically as shown in Fig . 11:11 , between CL and
M
this logarithmic replot lines of constant
pressure ratio
lines
airplane
but also under
,
( 11 : 4 ) may be nw
on
on an
relationships are indicated
ds
and
rectangular
load factor , and speed are related by the
L
and Mach
and sweptback
Tapered
in turning or other accelerated flight may be several of the airplane or , in general , L = nw, where n is the
the weight
load factor .
where mph
C.
Tapered
not only at the landing condition argument :
as shown
of rectangular , tapered , and the initial stall occurring at the
sweptback wings .
FLIGHT
11 : 5 .
an outflow on swept wings
Initial stall distributions for
11:10 .
particularly
.
Rectangular
a.
max '
stall distributions
Typical
swept wings are
indication of CL
is
sweptback , as there
11 : 10c .
larger
done by using increasingly
the local c❘ distribution is tan as shown by the heavy dashed line . This
,
however , only a rough
is
the wing
in Fig .
is
until
( 11 : 4 )
distribution
gent to the c1 max
calculation is ,
This
max .
values of CL in equation
are also lines of constant
For level
flight
(n
= 1)
/
wing
loading
nw S and
MCL ,
and are
straight
at sea
level (8
= 1) ,
a wing
100 lb / ft corresponds to the line labeled 100 in Fig . 11:11 . Fig . 11:11 that when the same airplane goes to about 30,000 ft , Note in
loading of
sq
AIRFOIL VISCOSITY where the pressure
1111
EFFECTS
/Po≈
1/3 , or at sea level in a turn with a load factor the condition is specified by the line nw /ds = 300 , and that under these conditions there has been not only a substan tial reduction in CL max but also the airplane must operate in the re
ratio
gion
d = P
flight
n = 3,
of the peaks of the constant
1.5
100
200
a
300
lines
which , as noted
400
_nw
= 1481 M
8S
.
in Fig .
CL
CLmax
1.0
CImax
140
.9
t
CImax at each const a ( High speed
100
.7
CL
.6
at
each const M
120.
.8
11:11 ,
buffet )
80 .
.5 .4 40.
.3 α
.2
20
.4
.3
.2
3
5
·6
.5 6
7
Moo
/
.7
.9 1.0
.8
Re 106 as tested
Fig . 11:11 . CL variation with M and Re at constant a for a tapered wing of NACA 66  series airfoil sections and A = 6 (from NACA TN 1697 ) , with Mer and several flight conditions shown . represent a condition of possible high speed buffet ( a condition of un satisfactory handling conditions in flight ) . For the wing whose charac teristics are shown in Fig . 11:11 , it is judged on this basis that values
/
300 do not represent satisfactory flight conditions . of nw 88 In using a graph such as Fig . 11:11 , it is important to know that the
combinations of
Moo
tions in free flight on CL max
and Re as tested
may
not correspond to the combina
of a particular airplane
should be applied
,
if
,
so that a scale
data are available
,
to Fig
.
correction
11:11 .
This
1112
will result in
TECHNICAL AERODYNAMICS
a
new
Reynolds
number
scale
representing the conditions
as flown or contemplated . PROBLEMS
ft
11 : 1 . A small airplane has a rectangular wing of S = 180 sq and A = 7 and a NACA 4412 section ( see Table A5 : 1 for data ) . The stalling speed at sea level ( determined by CL max ) is about 40 mph with a particular gross weight , and the level high speed at sea level ( determined chiefly by CD min ) is 100 mph with a particular engine and propeller . Using the
methods of Article 11 : 1 , estimate the proper values of CL max and CD min to use for stalling speed and high speed flight calculations . 11 : 2 . The root section of a large airplane has a chord of 15 and a 23021 section . Using the data in Table A5 : 2 , for flight in sea level standard air , estimate c1 max at 60 mph and cd min at 240 mph . 11 : 3 . An airplane of wing loading W, S = 50 lb/ sq ft flies at an tude where the pressure ratio d = P Po = 1/2 and a load factor of n = 2 and the wing characteristics are those given by Fig . 11:11 . From inspec tion of Fig . 11:11 , find the maximum Mach number at which satisfactory flight without high speed buffet is permissible under these conditions .
correction
ft
/
alti
CHAPTER
12
HIGH  LIFT DEVICES
12 : 1 .
ing
FOR HIGH LIFT DEVICES .
NEED
airplane
are high
speed
speed requires
quirements
are
lift
lift
a high
conflicting
wings without high 
devices , requires
12 : 1 .
Airfoil
conflicting
boundary
 layer
turbulent
flow
boundary
sections
requirements , whereas
layer
in Fig .
high
which
the
section and low Low drag ,
12 : 1 .
lift
devices
,
is
resists
= C1 max Re = 6 x
satisfying the
good low drag and good high
lift
for
extent of laminar
maximum
lift
high 
stall
general , these re
b . Good high section ; 1.7 , but cd min = 0.006 at 106 ( smooth surface ) .
without
of
low landing or
In
section .
illustrated
as
and
drag wing
a low
wing
a . Good low drag section ; cd min = 0.003 , but c1 max = 1.0 at Re = 6 x 106 ( smooth surface ) .
Fig .
Desirable characteristics of an
in level flight
High maximum speed requires
speed .
stalling
maximum
lift .
favored by early development of separation . These requirements
could be simply compromised by the use of a retractible turbulence gener ator near the leading edge . The airfoil section shapes most suitable for the two purposes are also conflicting and velocity gradients involved . An obvious
solution to this
dilemma
lift
vices for
airfoil
sections are
ie chordwise
pressure
is to use a low drag section for auxiliary retractible or enclosed
the high speed condition and provide an high Various device to get high cl max low  drag
of
on account
shown
used high 
commonly
in
Fig
.
lift
12 : 2 .
devices provide either for change in effective section centerline ture
,
or for delay of boundary  layer separation
sort are currently
( 1955 ) under
development .
ary layer removal and flaps , such as Fig
in cl mum
max ; but
lift
.
12 : 2f
the merit of a device cannot
coefficient obtained ,
as
,
or both .
show
of bound
tremendous increases
be judged
in the last analysis 121
curva this
Devices of
Some combinations ,
de
Most of these
solely
by the
maxi
the economic merit
TECHNICAL AERODYNAMICS
112
661212
661212
Thicker low drag section ; C1 max 1.2 with ca min = 0.0035 . Simula ted .2c split flap shown dotted .
b. Low drag section with plain (sealed ) flap ; C1 max = 2. From
64A010
661212
a. =
C. Low drag section with leading edge slots ; c1 max = 2 , but at high∞ . From NACA TN 3129 .
NACA TN 2502 .
d.
Low drag
flap : 3007 .
C1 max =
section with slotted
2.5 .
From NACA
TN
661212 e . Low drag section with nose flap and slotted flap ; cl max = 3. From NACA TN 3007 .
f.
Low drag section with flap and vacuum boundary  layer removal ; C1 max = 4. From NACA TN 2149 , 3093 .
= 0.8 no flaps C1 max = 1.9 opt . flaps C1 max
g.
Fig .
trail
Low drag supersonic section with leading edge flap as well as ing edge flap for high subsonic From NACA TN 2149 , 3093 . .
lift
12 : 2 .
Values of
Various high 
lift devices
applied to low drag airfoil sections
.
values for low Mach number , optimum C1 max shown are approximate any , optimum flap angle any , optimum boundary  layer slot location any , and suction 106. For specific arrangements and test data ,
if
see Appendix
if
5.
Re6x
,
if
HIGH
LIFT
123
DEVICES
of a high  lift device must be judged from economic considerations of the over all installation , including the effect of high stalling angles on landing
suitability
design as well as the
gear
devices associated with full  span high 
trol
of possible
lift
devices
lateral con
.
effects of various high  lift devices on wing lift curves are shown in Fig . 12 : 3 . Note in Fig . 12 : 3 that the effect of flaps is to displace The
3.0
the effect of
either flaps , is alone or in combination with to increase substantially the angle of attack at which the wing stalls . This provision
signer as layout of
be made
must
the landing
CL•max . 1.5
0.5 0
for very
gear
+
1.0
de in
a major handicap to the airplane
+
2.0
Fig
high wing angles of attack on landing .
15
Effects of high
12
lift
: 3 .
is
or
2.5
boundary layer removal ,
lift

, whereas
Flap Wing and slot slot Suction
3.5
.
slots or
with
devices on so page A527
See
al
.
curves
.
of the airplane
lift
in the stalling angle
change
Wing
out major
flaps Wing alone
the angle of attack of zero
have
flaps of
edge
or another
sort
one
since 1945
is
Their purpose
not
in
high CL max for slow landing but also to provide an during steeper glide drag crease in the landing approach which permits landing and smaller fields Such flaps are usually deflected about 600
where
,
,
A525
on C1 max
for
due to Reynolds
some
number
are
for
for
from Re
.
a
.

.
a
x
15
are indicated in
106
Reynolds
A524
constant
it 1.5
The values
Reynolds
substantial
as indicated
number
few low drag
number at
of airfoils
going
Re
Other
on pages A521 Mach
types
in
shown
Approximately the
by
correc
sections with .20c
inclusive
. ,
deflected
corrections
on page A520
106
to
up
12 2a
higher drag than
split flap
,
than Re
60 °
to less split flaps
=
the test data with flaps down
these data
to
all
a
on
,
number correction
there
but
106
=
for
page A58
,
: 2 ,
.
obtainable
have
on low drag sections
.
max
A5
for split flaps x
Fig
line in Fig
off
take
obtainable either with the plain or Re = 6
cl
of
is
max
,
cl
increase
likely to
CL max for landing but is
plain flap at smaller flap deflections for
same
without substantial
:
max
such as that shown by the dotted
Re
are
may be noted
106 to
shown
Typical on page
that the gain
Re =
x
CL
for take
9
in
increase
used
commonly
is
A
good
a
.
a
provides
moderate
split flap
is
=
provide
in drag
smaller flap deflection angle
x
off to
A
.
for landing
a
.
,
a
a
only to provide
built
Most subsonic airplanes
EDGE FLAPS
.
TRAILING
trailing
.
12 : 2 .
106
is
TECHNICAL AERODYNAMICS
124
if
entirely lost
almost on page
well as
C1 max as
and
Mach
usually intermediate types of flaps
Reynolds
is
factor in
a major
Actual
numbers .
Note also determining
airfoils will
be
between the " smooth condition " and " rough condition "
data shown on page A525 . mon
as high as 0.3 .
the Mach number goes
that surface condition
A525
Comprehensive
as a function
of
data
on
airfoil
cl
shape ,
for various com flap angle , surface
max
condition Mach number should be available to permit a wise selection of subsonic flaps but are not now ( 1955 ) available ; the necessary information for design decisions must usually be pieced together ,
and Reynolds
number ,
information . Occasionally tables have been prepared given ( such as those in NACA TR 664 ) for comparing various types of flaps fragmentary
from
flap  slot
and
at a particular value of Reynolds
combinations
number , Mach
surface condition and airfoil thickness but such tables are helpful more misleading than used to select a high device for flight conditions other than the test conditions . Few airplanes use full span flaps because of the necessity of reserving
number ,
likely
,
,
if
to be
lift
the outer 30 to 50 percent of the span of each wing for aileron or other lateral control devices . The effect of using partial span flaps as com pared with full span flaps is a great reduction tainable as shown in Figs . 12 : 4 and 12 : 5 .
. D /L
6
at 0.2
S
0 .
) .
12
60 40 80 20 Flaplengthpercent wing span
Effect of flap
split
1000
span on
wing characteristics 0.15c flaps deflected 60 from NACA ,
tapered TR 611
(
a
8
0.4
Fig span on 60 on
from NACA TR 611
(
3
0.4
16 20
°
.
0.15c tapered wing
5 :
, 4
Effect of flap split flaps deflected
12
: 4
.
4 Angle attackdegrees 0
0.4 16 12 8
of
30.3
12
50% 100%
0.2
0.8
°
0.1
.
0.6
CL max
Wing
1.2
Co
.
50%
0
Wing
1.6
5 .
0.2
100%
:
0.4
).
0.6
2.0 1.8
CL max
100 50%
%
0.8
alone
C₁ 510
Span Span flap flap
1.2
Wing
and Co
1.4
∞
at
1.6
Fig
Tapered Tapered
CL
1.8
ob
5 : 5 : 13
2.0
max
12
max O
2.2
in the value of CL
HIGH
If
full
tain
span high 
a value
of CL
lift
max
LIFT
DEVICES
devices are used
of
125
it is
usually
possible to
about 0.9 to 0.95 c1 max ' but special
ob
devices
for lateral control with flaps fully deflected . Some such devices for use with plain , split , and slotted flaps are shown in Fig . 12 : 6 . In general , the highest lifts are obtained with combinations of trailing edge flaps , leading edge slats , and boundary  layer control . must
be developed
The development of the necessary
the determination
of optimum
lift
to say at the present time what tory lateral control , as many
in
lateral control devices configuration
maximum
such
lift
is
,
and
has lagged
it is
behind
not possible
satisfac
obtainable with
devices are under active
development
lateral control devices sketched in Fig . 12 : 6 do not include devices suitable for control at maximum lift coefficients in the range from 3 to 4 indicated in Fig . 12 : 2 and on pages A526 through A529 , though 1955.
it is
The
not considered
difficult
to develop such devices because minor
/ lift
de
partures of the slat and or slotted flap from their optimum locations pro always involves a re , and control at high duce major losses of duction of
lift
on one of the wings .
It
lift
should also be possible to provide
lateral control by varying the boundary  layer duct pressure or boundary layer duct flow volume in devices where boundary  layer suction is used . For supersonic wings
very thin wing sections
with sharp leading and trailing edges , are needed from considerations of optimum supersonic flight configuration . It is extremely difficult to get a high maximum coef landing wings ficient for slow with such . The best device developed to ,
,
lift
date
(1955 ) appears
to be a combination of leading and trailing
resulting in a thin wing shown in Fig . 12 : 2g , for possible to
make such
structural thin wing ,
design
12 : 3 .
with a which
highly
data
flaps slotted
cambered ,
edge
are given on page A530
.
flaps
line
broken , median
It
as
may be
in the deflected position , but the is difficult for such flaps because of the very flaps of this sort appear not to have been reported . when
problem
and tests on
LEADING EDGE SLATS AND SLOTS . An extensible
leading edge slat
,
leaving a slot between the slat and airfoil , as sketched in Fig . 12 : 2c , a simple and effective means of delaying stall and increasing the max
is
lift
coefficient , as shown in Fig . 12 : 3 , either alone or in combina tion with other high  lift devices . Note on page A527 that a leading edge slot adds a substantial increment of maximum lift even when double  slotted flaps and boundary  layer control have already produced a very high maximum imum
lift .
For thicker wings
not be so
clearly
,
as shown on pages A528 and A529 , the
an advantage , as an addition
to boundary
slat
may
 layer control
.
126
TECHNICAL AERODYNAMICS
Slot lip aileron
Flap Plain aileron
WY
NATIONAL ADVISORY COMMITTEE FORAERONAUTICS
A

Q15c plain sealedaileronona ClarkY15 wing with020c split flap 62c
Fake
A retractableaileronon a Clark X15 wing with 6/6c
020c split flap
JAFF⋅
046c
A retractable
Ú
aileron onanMACA23012wing with a 02566cslottedflap
A 0.10c spoiler
hinged at Q50con an NACA23012wing with a 02566c slatted flap
hinged
A QIOCdeflector at Q50c onan NACA23012wing with 0.2566cslotted flap
AQ.10cspoiler hinged at0.50cand010cdefledor hinged at Q60c onan NACA23012wing with a Q2566 slotted flap
I
AQUOCdeflector hinged at0.50cand a retractable aileron on an NACA23012wing with a 02566slotted flap
J
hinged
at
A Q10cspoiler and ac deflector
050c on an NACA23012 wing with a 02566 slotted flap NATI ALADV COMMITTEE FOR AERONAUTICS
A QOC spoiler hinged at 0.50c and a Q10cdeflector hinged at 0.60c with a slot on an NACA 23012 wing with a 0.2566 slotted flap.
Fig .
12 : 6 .
Some
split ,
lateral control devices for use with full span , plain , flaps , reported in NACA TN 1404 .
and slotted
HIGH  LIFT DEVICES
flaps , since
as double  slotted
are obtained without leading
is
fulness on thick wings
maximum
lift
slats ,
edge
127
coefficients in
of
excess
but the question of
4
their use
pre
not answered conclusively by the data here
sented . 12 : 4 .
the
BOUNDARY
lift
maximum
in
pores
is
drag ,
minimum
 LAYER
A most effective device for increasing
CONTROL .
of subsonic
wing sections , which
blowing or sucking of
the
air
also decrease
may
the
slots , holes ,
through
or
chordwise locations on the upper surface controlling purpose for the of the boundary layer , which in turn controls skin friction and separation . Suction has been found more effective , and the surface at suitable
more economical
for
been found
of blower power , than blowing . Substantial effects have suction removal of boundary layer anywhere between the
leading edge of the wing and the leading edge of the flap . through porous materials near 4  digit
symmetrical
to raise the
found
the
section
NACA
lift
maximum
small quantities of suction
leading
air
,
edge
as sketched
12 : 2f ,
.
coefficient from 1.3 to
"
thick
percent
of a 10.5
in Fig
suction
"Area
has
been
1.8 with very
Various types of porous surface were used , including sintered steel as well as perforated plates backed by felt or filter paper . For a quantitative measurement of
air
the suction
flow
it is
flow
customary
( NACA
/
is
3093 ) .
to use a flow coefficient
,
9
CQ =
where Q b
TN
( 12 : 1 )
bc
the volume rate of flow per unit span in
ft3/ft  sec .
For the
leading edge experiment , values of CQ less than 0.001 were found to give nearly the maximum increase in cl max . This flow coefficient repre
porous
sents the ratio of the stream which NACA
tions
suction
would flow through
studies a spanwise
slot
pages A528
and
0.1
percent
has been used ;
have been found to be 0.45c
as shown on
flow to the
duct
(NACA
of
amount
of the free
In
the wing area .
air
other
effective suction slot loca
TN 2149 ) and
A529 respectively .
0.75c The
(NACA TN
1631 ) ,
flow coefficients
necessary for effective use of such spanwise slots are , however , ten times as great as for a porous leading edge , and the size , weight , and power re quirements
in
of the associated " vacuum cleaner " device
the design of
It is
will
possible
an
airplane
that
" porous
major
factors
.
area suction " at more rearward
locations
installations in connection with slotted flaps and lateral control devices , but no general rules for the design of
permit economical
adequate
become
using such suction slots
128
TECHNICAL AERODYNAMICS
such equipment
are
boundary
that economical
slight reduction in crease
in
currently available
 layer
minimum
removal
drag
tice within
Current developments by "area suction , "
coefficient
indicate
providing
a
well as an enormous in = max' 4.0 , possibly higher ) ,
lift coefficient ( up to c1 lateral control , is feasible and
maximum
with adequate
.
as
may become
accepted
prac
a few years . PROBLEMS
12 : 1 .
Using
data from Table
A5 : 2 on an NACA
0012
wing with 20 percent
split flaps deflected 60 ° , write equations for CL vs. a , CD vs. CL , and Cp vs. 1/CL for a rectangular wing of aspect ratio 6 with full span flaps chord
.
Using the graphs on page A58 , estimate c1 max at Re = 6 x 106 an NACA 4412 section with 20 percent chord split flap deflected 60º . Referring to Fig . 12 : 5 , find the angle with the horizontal of a 12 : 3 . steady stalled glide of the 5 : 3 tapered wing of characteristics there shown with 50 per cent span flaps . 12 :4 . An airplane weighing 1400 pounds and having a rectangular wing span and 180 sq ft area , is equipped with full span slotted flaps of 35 and with a leading edge slat in the optimum position (data as given on page A526 ) . Assuming CL max = 0.93 c1 max , calculate the stalling speed of the airplane in standard sea level air ; also calculate the Reynolds number at stall , and compare with the Reynolds number specified for the data . 12 : 2 .
for
ft
CHAPTER
13
AIRFOIL SELECTION 13 : 1 .
SYSTEMATIC INVESTIGATIONS AND NUMBERING
have been systematically
investigated
Airfoil
SYSTEMS .
for at least
shapes
a hundred years .
The
primary objectives have usually been to determine a shape which would give high maximum Only within , low minimum drag , and low pitching moment .
lift
the last twenty years , however , have the systematic investigations been reasonably enlightened as to the nature of the air flow in the immediate
vicinity of
an
as noted in
Chapters
so
far
tions Navy ,
airfoil , particularly in 6 and
more an
Air
art than
Force ,
a science .
of
NACA ) , many
many
airplane
educational research organizations
a number
of private
with the
Catholic
investigators
University ,
 layer flow
is
airfoil sec
agencies (Army ,
agencies ( England
,
France ,
( e.g. , Boeing ) ,
manufacturers
Goettingen University )
( e.g. ,
( e.g.
Even today ,
.
optimum
Many U. S. governmental
foreign governmental
, Japan , U.S. S. R. ) ,
layer
on boundary
from complete as to leave the development
Germany many
the boundary
the information
7,
, and
in
1955
Washington , D. C. ) , have studied and
pub
,
Max Munk ,
Dr.
connected
lished results of investigations . The NACA has attempted to serve as a clearing house for information of this sort and is also responsible on its own
for
systematic investigations
the most comprehensive Each investigator and investigating account
of classifying airfoil
Tests of the
shapes .
wind tunnels have appeared
in
many
but most of these discrepancies ences
in
turbulence
,
Reynolds
Prior to
about
wing sections mary
and
variable .
same
have been tracked down as due to number ,
for effects of
most investigations
/
and or surface
wind
differ condi
tunnel wall or open
were made on
relatively thin
the shape of the center line of the section was the
With the development
during World War
I it
seemed
to
many
of structurally feasible investigators
pri
monoplanes
that the study should
attempting to isolate the effects of the upper and lower on which pressure measurements could be made independently . The
be extended surfaces
1910
all .
cases to give widely different results ,
number , Mach
tion , when the proper corrections jet test section have been made .
of
its own system airfoils in different
agency has devised
by
131
132
TECHNICAL AERODYNAMICS
conclusion was that the lower surface was relatively unimpor systematic variations of the upper surface could lead to
tentative
tant and that
airfoil ,
the best
in flight
out
but
the
tunnel test results were often not borne
wind
of imperfectly
because
differences
understood
tunnel and free  flight flow conditions
between wind
In the alphabetical designation system of Colonel V. E. Clark , with airfoils designated alphabetically from A through Z , a number of very satisfactory airfoils were found to be feasible using a lower surface that was a plain flat surface from trail .
ing edge forward to nearly the leading
edge . For the "Clark Y" , the upper thickness of 11.7 percent at a station 30 per cent of the chord from the leading edge . This Clark Y became one of the
surface reached a
maximum
airfoils
most widely used
Later studies by exploration
the
first
by
ever investigated .
provided for a
NACA
investigating
edge radius , and maximum thickness
ing
usually requires
shapes , including
in
resulted 4
digit
some
series .
airfoils
" 4  digit "
numbering
the
) which
This inves system which ,
lead
.
a 5  digit system
( it did not provided an additional set of center
 digit series
4
and
the thickness .
digits to designate
thickness location
airfoils
include symmetrical
line
NACA
for
symmetrical
a dash and two more
edge radius and maximum
A variant of the
and shape
leading
,
airfoils ,
symmetrical
of
subsequently varying the center line curvature
however ,
geometric
thickness distribution
the
location
distributions found most suitable for tigation involved the so  called NACA
systematic
more
was
trailing
reflexed
edge .
The 5  digit
series
airfoils
considered to be an improvement over any of the For a number of years the 5  digit airfoils of the NACA
were considered to be the best .
Later theoretical studies of effect of free  stream pressures
airfoil
pressure distribution layer , resulted
boundary
on
,
and
in
the
another
pressure on the up per surface , and later also on the lower surface , were specified in the numbering system . Such airfoils were found to permit a considerably ex
series of
NACA
tended range system
is
values of of a
maximum
number
which the point
of laminar boundary  layer
currently
provided by pages .
airfoils in
very
lift ,
widely used since
it is
lift
,
of
flow
minimum
and the
even though
considered that
it
" 6  series " of
this
does not give high maximum
lift
can be
of the high devices described in Chapter 12. Details designation systems are given in the following few of airfoil
some
AIRFOIL SELECTION 13 : 2 .
airfoil shown 13 : 2 ,
Mean
THE NACA
4
 DIGIT
GEOMETRIC
designation involves a
in Fig . 13 : 1 , and laid off at right
The NACA 4  digit
.
SYSTEM
line consisting
mean
angles to the parabolic
system of
parabolas
two
in
shown
,
as
Fig
.
arcs .
Horizontal tangent to both parabolas here Parabola
Parabola Chord
Location of max . ordinate of
mean line SECOND DIGIT
=
of
thickness distribution as
a basic
line of airfoil ·
Xyc max 4 digit
133
airfoils Fig .
=
max
for
13 : 1 .
Mean
. ordinate of
mean
line
" camber " = Ус max = FIRST DIGIT for 4  digit airfoils
line
designation for
4
digit airfoils
=
.
с
0.3c X
y FLE
= 110 (t
ty
t / c ) 2 in
=
0.2969
per cent c
√x 

.1260x
.3516x²
+ .2843x3
 .1015x4
air
Basic thickness distribution for NACA 4  digit and 5  digit Above airfoil is designated 0020 or 002063 , where 6 denotes normal leading edge radius and 3 denotes maximum thickness at 0.3c . See Fig . 13 : 6 for other L.E. radius and maximum thickness location designations . LAST 2 any ) are t c , per cent . DIGITS of 4 and 5  digit airfoils ( before dash ,
Fig .
13 : 2 .
foils .
if
The two parabolas
of
maximum
in Fig .
ordinate of the
of the 4  digit designation DIGIT .
horizontal
13 : 1 have a
mean
line
system .
.
The
This ordinate
/
tangent
is
at the point
the FIRST
location of this ordinate
DIGIT
is
the
are the thickness ratio in per cent . The equation for the basic thickness distribution is given below the sketch SECOND
in Fig .
/
13 : 2 .
110 ( t c ) 2.
The LAST TWO
DIGITS
The leading edge
radius
The maximum thickness
Other leading edge radii and other
is
specified by the equation гLE
=
for the basic distribution is at 0.3c . maximum
thickness locations
are
indi
cated by adding a dash and two more digits after the 4  digit airfoil desig nation . A code table describing the meaning of the supplementary digits
after the
dash
is
given
in Fig .
13 : 6 .
TECHNICAL AERODYNAMICS
134
A sample
airfoil for
4
 digit
which
x
is
layout
in Fig .
shown
and y  coordinates
of the
13 : 3 .
sured from the chord line through the leading
in Fig .
13 : 4 .
designations
34
omission because " low drag
in
range
A wide
as shown in Fig . cambers from 0 to 20 to 70 per cent ,
and 35 ,
this
and
is
developed .
the equipment available
2415
airfoils
have been tested ,
them were tested with the dash  number now recognized to have been a major
airfoils
of these
some
NACA
ratios from 6 to 25 percent , mean line ordinate locations of
maximum
all of
though not
the
mea edges are given
trailing
and
4  digit
NACA
6 per cent , and
airfoils later
"
of
covering thickness
13 : 5 ,
is
This
upper and lower surface
correspond
Even
at the time
,
if such
closely to
very
airfoils
the
had been tested
the low drag would not have been
detected becauseno low  turbulence wind tunnel providing for high Reynolds
available
and Mach numbers was
at the time the tests were run .
0.3c %c
"LE=2.48
t
=
.15c
yc
=
2%c
measurements
of
B = 0.1 Xyc max
=
Fig .
.4c
it
bolas
shown
Principal
13 : 3 .
While the cause
max
4
 digit
NACA 2415
airfoil .
is usually considered obsolete in 1955 be for other mean lines than the two tangent para 13 : 1 , some of the 4  digit airfoils involving such system
does not provide
in Fig .
parabolas are very close approximations to the Upon re  testing
discovered .
in
" low  drag "
airfoils later
low  turbulence wind
tunnels , they have also been found to have low drag with the proper leading  edge radius and thickness distribution designated by a dash  number after the four digits . Another range of
mean
THE
was
NACA 5
 DIGIT
GEOMETRIC
ond
to the
with
mean
ordinate
.
two
digits
,
NACA
5 digit
series de
of
The NACA 5  digit system can system by replacing the
which designate not only the
line but also the
The range
SYSTEM .
from the 4  digit
be considered to be evolved
digit
provided by the
article .
scribed in the next 13 : 3 .
lines
mean
shape of the mean
lines considered is
ordinate
maximum
line aft of the shown
in
Fig
.
sec
maximum
13 : 7 ,
and
135
cent of chord
per cent
ratio
Position of yc max at tenths of chord 4
is
thickness 15 per
is
88STRIN
of chord
2
Among
Maximum
is
NACA 2415 per given Stations and ordinates cent ofairfoil or in Apper Surface Lower Burtes StationOrdinate Station Ordinate
29 28 28 enge
Camber , yc max
FEEBETRERAR
15
no97872393282588
2
NACA
4
AIRFOIL SELECTION
2512
6206
6409
.
Dalala
6418 6421
:
2606
2706
2609
2709
2612
2712
2615
2715
2618
2718
2621
2721
4606
4509
4609
4512
4612
4515
4615
4518
4618
4521
4621
6506
6606
6509
6609
6512 6515 6518 6521
+ 0 +
6612 6615 6618
6621
reported in Technical Report 350 See Airplane Design D. Tenth Edition
,
Wood
1 0 0
! 0=0€ 4506
NACA
.
or
K.
560
,
digit airfoils

L 
: 5 .
.
for
4
6221
6412 6415
"
6218
NACA
4421
6209 6215
Report ordinates
4418
6406
6212
Wartime
4415
of
560
FFFFFFFFFFFF
FEIFSC FFGGES
4221
2518 2521
DDDDD
)
4215
4218
,
0025
DDDDD
0021
2515
FFFFFFFFO
FFM FFFFFF
O
FFFFEN
0012
Frrrrr
2509
0018
13
2506
0009 0015
and ordinates
Report
.
.
2406
0006
Fig
designation
From NACA Wartime
L
d4
airfoil
, "
igit Example of 2415 airfoil
13
: 4 .
.
Fig
,
2 .
L. through radius 2.48 Slope ofradius L.B.1 0.10
136
a.
TECHNICAL AERODYNAMICS
First digit
L.E. radius
Seconddigit
Max.thickness location, tenths
0 3 6 (standard) 9
Sharp 4 normal Normal 3 X normal
2 3 (standard) 4 5 6
2 3 4 5 6
Meaning of two digits after dash indicating nonstandard leading edge radius and nonstandard maximum  thickness location .
000663
000933
001263
000993
000963
000905
000962
000935
000964
000934
000965
220934
000966
240934 440934
000903
b.
C.
NACA
4  digit
Comparison
airfoils
modified thickness distribution
( From
NACA TR 492.
)
000903
000962
000933
0009630009
0009630009
000964
000993
000966
of airfoils with
modified nose radii
Fig .
with
nose shape .
13 : 6 .
.
Modified
and
Comparison of airfoils with modified location of point of maximum thickness .
d.
NACA 4  digit
airfoils
.
AIRFOIL SELECTION a sample
airfoil
first
is
designation
in Fig . 5 digit
shown
137 13 : 8 .
in Fig .
As noted
13 : 8 ,
system have the same meaning last  two digits of the digit system except , that the first digit is only the approx as for the 4 imate maximum ordinate of the mean line and is actually intended to be a measure of the lift coefficient at which the airfoil is designed to oper ate , usually designated by cli , and actually equal to ( 20/3 ) x Cli ′ The
the
and
Cubic
Straight
yc
line : third digit Inverted cubic : third digit =
max
0
=
1
Approx . FIRST DIGIT , per cent chord
Xyc
max
One
half of the per cent expressed
(Actually 20/3 of design c₁ )
by SECOND AND THIRD DIGITS
Fig .
13 : 7 .
Mean
line
designation
for 5  digit airfoils
.
2 30 15
thickness is 15 per cent of chord
ordinate of line is about per cent of chord
Max .
Max . mean
2
ordinate of
Max .
line is at Fig .
second
and
13 : 8 .
Aft portion of mean line is straight
mean
of
3/20
chord
Example
of
5
third digits constitute
digit airfoil
third digit tells
The
ter the maximum ordinate designating straight and of the
5 digit
along with
some
The mean
ly
was
mean
in
reported
,
NACA
TR 610 are shown
line of
in Fig .
some 13 : 9 ,
also there reported . system were found
the 4  digit
to provide just as low
system , with a reduction
in pitching
thought at the time to be important , but with the
adoption of high 
af
the symbol o
or an inverted cubic designating inverted cubic . Sketches
the 5  digit as
twice the
line ordinate
mean
straight
 digit airfoils
lift
which
widespread
4
lines of
drag and high moments ,
airfoils
is 1
maximum
the portion of the
whether
.
is
a per cent of chord which
per cent of chord from the leading edge to the
location .
designation
lift
devices as
fair
described in Chapter
12 ,
the quest for low pitching The basic
moments had to be abandoned . thickness distribution and leading  edge radii for the
system are the same as
for the
4
digit
system .
Hence , 5  digit
5  digit airfoils
TECHNICAL AERODYNAMICS
138
also be followed by a dash and two additional digits designating other leading  edge radii and maximum thickness locations . The 2301234 , shown as one of the collection in Fig . 13 : 9 , is a very close approximation to
can
airfoils later
discovered to have very low drag
not detected at the
time
,
of the
this low
though
tests reported in
drag was
of a
TR 610 because
in the wind tunnel . It is now considered slightly low  drag airfoils only on account of a flat or concave
trace of turbulence
inferior to area near
such
the
structurally
leading edge on the
disadvantageous
is
lower surface which
judged
aerodynamically unimportant
though
to be
Aero
.
characteristics of 4 and 5  digit series of airfoils as determined high in the  turbulence variable density wind tunnel (VDT ) are given in
dynamic
Tables A5 : 1 and A5 : 2 . 001263
0012 68 60
.
23006
°
0006
23009
43009
63009
001264
23009
0012
23012
43012
63012
001265
23012
0015
23015
43015
63015
2301233
23015
0018
23018
43018
63018
2301234
0021
23021
43021
63021
2301264
16066
0009
23021 43009
A
43012
22012
32012
42012
23012
33012
43012
24012
34012
44012
43012
000
21012
60°
62021
63009
63021
64021
airfoil
testing
involving
Mach
the
NACA
to develop
undertook
systematic variations of the
mean
a
an attempt
.
,
)
:
(
.
In
.
number
Note see NACA Wartime Report Tenth Edition for ordinates
series
of line in con
a
of airfoils of high critical
, "

SERIES AIRFOILS
,
"
D. ,
THE NACA
digit airfoils
Airplane Design

,
.
4
:
13
4 and
Some
Wood
1
or
L 
.
: 9 .
13
560
K.
Fig
5
25012
program
junction with optimization of leading edge radius and thickness distribu tion based on velocity and pressure calculations assuming theoretical
it is
in
to add algebraically the velocity patterns for flow around various mean lines to the velocity pat terns for flow around various thickness distributions The basic symet closely igit correspond trical airfoils tested to those of the NACA
in
which
permissible
d
4
.
,
compressible flow

1
.
,
,

In the series series followed by dash numbers 34 35 or 36 second digit designates the minimum pressure location in tenths of
the the
139
AIRFOIL SELECTION leading edge , for the basic symmetrical
chord from the
lift .
For
this
and
a minimum
is
group
pressure at 0.6 chord
often
as the
known
16
,
 series
two
The
.
section at zero
digits are 16 , third digit is the
first
the
lift coefficient cliin tenths , being zero for symmetrical airfoils last two digits are again the maximum thickness in percent of chord A typical 1  series airfoil designation is shown in Fig . 13:10 . design
.
The
.
1 62 15
1
 series
Maximum
Design
Minimum pressure at 0.6 chord for basic symmet
rical
lift
Fig . For most
it
at
section
1
Typical
13:10 .
 series airfoils
1  series
the
airfoil line is
mean
of
chord
lift coeffi
cient Cli
zero
is
thickness
15 per cent
=
0.2
designation curved
in
.
such a manner
pressure difference that an approximately uniform chordwise between the upper and lower surfaces at the design coefficient , and the designation of such airfoils is often indicated by a = 1.0 following produces
lift
the numerical series designation worthwhile to provide uniform
.
lift
In
some cases ,
however ,
it was
thought
over only part of the chord with a
lin
lift at the trailing edge . In such cases the fraction of the chord designed for uniform lift coefficient is designated as shown in Fig . 13:11 . ear taper to zero
16215 Same
as
in Fig .
a
=
0.5 Mean line designed to give uniform loading to 0.5c , then linear decrease to
13 : 10.1
trailing
Fig . A
Modified 1  series airfoil designation with region of unspecified , a = 1.0 ) . uniform loading specified (
13:11 .
more
flexible
was designed
designed
edge
later
sections
if
and
and
is
set of aerodynamically  designed series the most widely used of these aerodynamically
improved
the
6  series
namic data on some 1  series
described in the next article
airfoils
are given in
NACA
TN 976 .
.
Aerody
TECHNICAL AERODYNAMICS
1310
13 : 5 .

rently , the

most widely used of the NACA The designations
6 series .
The most widely explored and ,
AIRFOILS .
6 SERIES
THE NACA
is
tional information
series
are similar to those of
supplied by
a
cur
airfoils is the NACA the 1  series but addi
of
third digit in front of
dash as
the
to cli where an extensive region very of laminar flow exists and the drag is low . A typical NACA 6  series designation is shown in Fig . 13:12 , and the meaning of the third and fourth the range of values of c1 above and below
digits in
terms of a graph of ca against 6 5, 3

2 15
is
c₁
line designation , indi cating resultant of upper and lower surface pressure is un iform for 0.5 chord from L.E. Mean
location of mini pressure position is 5 tenths of chord for basic symmetrical section at zero mum
g
thickness is 5 per cent of chord
Maximum
drag is 3 C1 range for low tenths above and below Cli
Fig .
13:13 .
L.
Chordwise
( see Fig .
.
a = 0.5
Series 6
lift
in Fig
shown
Design
13:13 . )
Sample designation
13:12 .
lift
Cli is
of
coefficient
2 tenths
6  series
airfoils
.
.008
.006
са Low drag range ± 0.3
°1700
of extensive laminar flow Region
.002
.2 .1
.3
Fig .
Cli
L
u
0
.1
1
C1
L
.2
.3
1 .4
lift
Typical Note
surface pressure distributions for symmetrical are compared in Fig . 13:14 with those of an NACA
that
the peak
farther aft troduced
in
each value
.7
Typical " bucket " in graph of section drag coefficient vs. coefficient specified by designation in Fig . 13:12 .
13:13 .
section
foils
.6
.5
on the
the of
negative pressure
airfoil
6  series ,
cli
to get
NACA 6  series 4  digit
air
airfoil .
is
considerably reduced and moved in the 6  series . The greatest improvement
however ,
is in
in
adjustment
a nearly uniform resultant
of the
mean
line for
pressure over
the
AIRFOIL SELECTION
airfoil
chord of the
is
in Fig .
shown
.8
and the comparison
,
1311
with
a
similar
4
 digit airfoil
13:15 .
(77/v )2
1
.4 .8 .2 a.
NACA
.8
.2
0 x c
1.0
/
b.
0012
.6
.4
1.0
65,2012
NACA
=
a
=
a
.5 theoretical line
=
0.2 a
0.2
line
Mean
a
0.6 0.8 Fraction of chord

0.2 0.4 0.6 0.8 1.0 Fraction of chord NACA 1 and series loadings for given Cli ·

6
NACA 64
1.0 b .
a.
+1.0 =
.06 0.4
1.0
Resultant Pressure
a = 0
Mean
1.0
)
Resultant Pressure
0.5
(
Difference and lower
1.0.
.
a = 0
of upper surface
2.0
0.2
.8
Comparison of basic thickness forms and surface pressure distribution of 4 digit and 6  series airfoils .
13:14 .
coeff
Fig .
.6
.4
pressure
0
.
with the third digit recently been superseded
more
by
a
is designated
in
which
subscript
as
separated by
sim
the half width shown
in Fig
.
13:12
a

6
d "
pressure distributions

has
and resultant
series airfoils ,
.
in Fig
comma
bucket
"

.
and
with improved thickness distribution
of the low drag 13:16
shown
igit
,
The designation
4
for
from the second by
ilar series
of mean lines
Comparison
a
13:15
.
.
Fig
TECHNICAL AERODYNAMICS
1312
as
Same
in Fig .
13 :
a = 0.5
215
653
12
Same
as before
as
in Fig .
13:12
0.3 from cli c1 range for " bucket " of ca graphs , with thickness distribution improved from theoretically derived value specified in Fig . 13:12
Fig .
13:16 .
all airfoil series
Unlike
previously
 series airfoils is a of 6  series airfoils may ,
bution of the
A
6  series airfoil section similar to that designated in Fig . 13:12 .
Improved
sub  group
6
discussed , the thickness
function of the therefore
maximum
be obtained by
,
distri
thickness
.
linearly in Fig .
or decreasing the ordinates of an airfoil such as shown 13:12 in a constant ratio . Such a linearly increased airfoil of this sort is designated as shown in Fig . 13:17 . increasing
65 ( 215 ) Same as
in Fig .

218
as before 13 :
12
Cli and thickness ratio obtained by linearly increas ing ordinates of Fig . 13:12
new
and thickness from Fig . 13 : 12
Cli ratio Fig . 13:17 .
a = 0.5
Modified
6
 series airfoil obtained by linearly increasing ordinates of Fig . 13:12 .
in
A similar sub  group of 6  series airfoils is obtained by linearly creasing or decreasing the ordinates of an airfoil such as designated in Fig . 13:16 , and such an airfoil designated in Fig . 13:18 . 65 (315 ) Same
,
with linearly
2 18
Same as
Fig . 13:18
.
,
is
as before
thickness ratio obtained by linearly increasing ordi
ratio
in Fig .
Modified
ordinates
a = 0.5
as in Fig . 13:16
Original thickness
increased
6
New
nates
13 : 16
 series airfoil
ordinates of Fig
in Fig .
13:16
obtained by linearly increasing .
13:16 .
Characteristics of a wide variety of 6  series airfoils as a function of the digits designating the airfoils at various Reynolds numbers (and
1
AIRFOIL SELECTION a few high subsonic 5 and
will
13 : 6.
2,
While there are
NACA
series designated by
4 , and 5 , they are not widely known
3,
Appendix
later .
SERIES .
OTHER NACA
in
presented graphically
Mach numbers ) are
be discussed
first digits
the
1313
7  digit series is , however , fairly widely is shown in Fig . 13:19 . In the 7  series
used
and
The
a sample designation
is
attempt
an
or used .
the range of favorable pressure gradients over both
made
to specify
the upper and lower
surfaces and the thickness distribution and mean line are specified by a serial letter in the middle . The last three digits designate c11 and the thickness ratio as usual .
2 4 2 A 4 15 7  series
airfoil
Thickness ratio
Favorable pressure grad ient for 0,4c on upper surface at design q
Design
1S
Series no . wedge ,
(2

( 70 ) ( 03 )

13:20 .
For supersonic and
inar sonic
boundary
0.4
7
 series airfoil .
L
(70 ) ( 03 )
Max . thickness of lower surface = 0.03
Max . thickness of lower surface at 70 per cent chord
Max . thickness of upper surface at 70 per cent chord
trailing
of
arc )
Supersonic
Fig .
=
Favorable pressure gradient at design c1
1 denotes
circ .
=
Cli
on lower surface
Sample designation
13:19 .
=
per cent
Serial letter designating thick ness distribution and mean line
for 0.7c Fig .
C1
,
Max .
Sample designation
flight , airfoils
with
thickness of lower surface = 0.03
of supersonic
airfoil .
sharp , or nearly sharp , leading
are favored lam  layer is also quite different from that favorable for sub edges
airfoils
.
;
It is
the geometry conducive to extensive
customary
to designate supersonic airfoils
by
1314
TECHNICAL AERODYNAMICS
separate groups
ilar
digits for
of
thickness
maximum
would have a
airfoil using circular first digit of 2 instead of
tics of
airfoils
supersonic
such
13 : 7 .
ISTICS
In
airfoil
an
is
profitably
on wing
This
chord .
number
on minimum
million
.
will
a,
of
( 2 ) graphs
of ca vs.
six
in
in
mil
,
data
location for
(a )
practically
,
Angle of attack on zero independent
the 4  digit series
of thickness ratio
and with
considered unimportant
lift
Cli for
in the choice of
It is
simply necessary that order to set the wing , incidence properly
(b ) importance .
of c vs.
c,
5,
beginning with Fig .
in
terms of thickness ,
the 4 and 5  digit
series ,
negative pressure location
is
, but
in
Fig
. A5 : 1 to be to vary with camber for
seen
It
is airfoil because it deter on the airplane to get low fuse alo be known to the designer in
each of the 6  series
mines the angle of incidence of the wing .
alo
on ( 1 )
with reference to graphical
Appendix
terms of thickness cli and maximum (second digit ) for the 6  series airfoils . 1. Lift  curve Effects . ,
effects separately
and ( 3 ) graphs
C₁ ,
considered below
low turbulence data given
camber , and maximum mean  ordinate
airfoils
.
simply
.
lift
The section  curve  slope per degree a is also of little is seen in Fig . A5 : 1 to be almost independent of thickness ,
It
decreasing slightly with thickness for the
slightly
about
to speeds near the stall The effect of higher sub  critical
These graphs present the aerodynamic
lage drag
of
number
character
for by considering only the effect of coefficients up to a Reynolds number of
drag
of c₁ vs. and these are systematically
and
Low speed
.
correspond
convenient to consider the aerodynamic
graphs
summaries
tur
to have
then adequately accounted
about 20
A5 : 1 .
characteristics
compared at a Reynolds
Reynolds
It is
often convenient or desirable
therefore highly desirable
scale airplanes in free flight .
full
speeds
be
CHARACTER
AERODYNAMIC
ideas as to the effects of minor changes
geometry on aerodynamic
istics can lion based for
it is
, and
reasonably accurate general
airfoil
lines
characteris
the chapter .

it is
A sim
section slightly different from any for which low
bulence tests are available some
in
the
behind
13:20 .
The aerodynamic
1.
ON LOW SPEED
the design of an airplane
and
of straight
instead
arcs
are discussed later
EFFECTS OF AIRFOIL GEOMETRY
.
to use
in
in front of is shown in Fig .
portions
the
A sample designation
.
with
value of 0.11
thickness for the per degree
or for
6
4  digit
series and increasing
 series airfoils from the theoretical very thin airfoils . Small changes in ,
AIRFOIL SELECTION aspect ratio can far
indicated in Fig . (c )
over
 shadow
1315
the small changes
in
with geometry
a
A5 : 1 .
lift coefficient
Maximum
ness and camber on
c1
max
without flaps .
without flaps ,
ulated split flaps deflected out flaps nearly all reach a
60 ° ,
is
20 per
with
and
effects of thick
The
shown in Fig . A5 : 2 .
cent chord
sim
with for thick
The graphs
peak at c₁ max between 1.5 and 1.6
nesses in the region from 12 to 15 per cent of chord , with slightly higher values for high camber within the range investigated . For airfoils with
flaps , thickness ratios in the region and give values
Much higher values
flaps
and
noted
in
are , however
with the
Chapter
from 18 to 20 per cent are
addition
12 ,
,
obtainable
of slats
significance for
optimum double
with
/
and or boundary
so none of the graphs
favored
full  span split flaps .
for
C1 max around 2.6 to 2.8
of
in Fig .
 slotted
 layer control
of an airfoil , since high lift can best be obtained with auxiliary high  lift devices . Drag  curve Effects . 2.
much
(a)
as
A5 : 2 are considered of
the selection
maximum
Minimum drag
coefficient cd min comparative tests run at Re = for 4  digit and 6  series airfoils in Fig . A5 : 3 , but there are major effects due to Reynolds number shown in Figs . A5 : 8 and A5 : 9 and these must also be considered . Low minimum drag at high Rey
6 x
106 are plotted
nolds
number
teristic
is
of an
here
to be the most important aerodynamic charac Note that all the airfoils for which data are
judged
airfoil .
A5 : 3 show an increase of cd min with thickness for 4  digit , digit , and 6  series airfoils . Camber and maximum mean ordinate location 5
plotted in Fig .
are seen to be unimportant for the the 6  series airfoils portant but the position of
the design
range
lift
minimum
of cambers there plotted
cli is
coefficient
pressure
is
.
seen to be
of major importance
coefficients are obtained with
minimum
drag
position of
minimum
pressure , though the gain from the 66  series to
67  series
is
very small .
Minimum drag
or possibly
choice of the 66  series of all the extensively tested ,
considerations
the 67  series
airfoil
,
series
.
results are , however , believed to be obtainable series airfoils followed by the dash numbers 34
,
as the NACA has largely
data on these are available
series ,
however modified ,
tion
boundary
and the
hence lead to the
suitable equally good
as the most Almost
from 35 ,
modified or
36 ,
4
 digit
but few
abandoned the 4  digit
favor of the wider variety of specifications Major reductions in cd min with suitable suc
in
6  series .  layer removal
covered by the
NACA
im
farthest aft
the lowest
the
For
have been found possible in preliminary
tests
,
TECHNICAL AERODYNAMICS
1316
field is
but this
evident , however ,
in Fig .
such as that shown
coefficients could
drag
lowest of the
 layer
boundary
friction ,
in skin
from the basic studies
that the
It is
as yet inadequately explored .
be
divided by
6:5
if the
5
held completely laminar , and tremendous amounts
could be
power would be economically justifiably in the design of an airplane the boundary layer could be kept laminar over the fuselage and tail surfaces as well as wing .
of
boundary
(b )
 layer suction
if
The induced drag
of aspect ratio ,
as
shown
efficiency factor in Fig . 9:23 , has
ew ,
while chiefly a function
been found
4  digit
for
air
to be substantially higher for
foils
thin airfoils (with proper camber ) , than for thick airfoils ( 1 ) but these effects , like those on lift  curve slope , are far overshadowed by small changes in aspect ratio . 3 . Pitching Moment Curve Effects . (a )
center a.c. The effects
Aerodynamic
on aerodynamic
center location
in
4  digit
general the
are seen in Fig
airfoils
series
a
show
a.c. with increased thickness movement with increase in thickness
, whereas the 6
aft
little
.
of thickness and
. A5 : 5
slight forward
 series airfoils This
is
camber
to be small , though
in slight
movement
show a
considered to be of
center simply serves as a refer the center of gravity , the distance between aero
importance , since the aerodynamic
ence point
for locating
dynamic center
center of gravity being a factor in the longitudinal
and
stability calculations . Pitching
(b )
airfoil always
moment
coefficient
/
in
geometry on Cmc 4 are shown
small without flaps and
thickness
is negligible .
objectionable
Cm ac
Fig
large with
A high value of
Cm
or
. A5 : 4 .
flaps
c/ 4 is
/
The effects of c 4 The pitching moment is
Cm
;
the effect of
airfoil
considered structurally
to the wing to provide adequate torsional strength and rigidity , but this objection is usually outweighed by the advantage of flaps in producing high stalling speed . Reynolds A5 : 6 through Mach
because weight
number A5 : 15 ,
number effects
13 : 8 .
FOILS .
must be added
effects for inclusive
,
not covered
most
the
in
of the above items are
last
named
Chapter
figure
Some
airfoils
many
in
Figs
.
also
some
SERIES
AIR
10 .
APPROXIMATE EQUIVALENCE OF MISCELLANEOUS developed
shown
showing
years ago are
AND
NACA
known from
flight
tests
to have been very satisfactory , but their characteristics at low turbulence
( 1 ) Dwinnell , James H. " Principles of Fig . 9.8 . Mc Graw  Hill , 1939 .
Aerodynamics , "
First Edition ,
AIRFOIL SELECTION and high
Reynolds
coefficient
drag
not been determined
have
number
1317 Often the
.
minimum
with better accuracy than from the mea
can be estimated
surements
reported in a high  turbulence wind tunnel by inspection
ordinates
and determination of the equivalent
example , consider
the Goettingen 593
airfoil ,
in Fig .
sketched
specified
listed in
by the ordinates
Table 13 : 1 .
airfoil .
NACA
For
Medianline
Referencechord
is to find
The problem
the equivalent
L.E.T.E.Chord
13:21 , and
of the
airfoil .
series
NACA
Fig . 13:21
The
Goettingen
.
593
airfoil .
solution follows . TABLE 13 : 1 .
5
%C
ORDINATES
20
10
Find the
1.
is
and
means
13 : 1
same
as a
of the equivalent Find the
Goettingen second
70
80
95 95 100
90
10.85 9.45 7.655.50 3.00
11.70 0
0
0
shows that the
60
0
maximum
maximum
0
10
1.65 0 0
O
 thickness location .
thickness
is
In
11.9 percent
located at 30 per cent of the chord from the leading edge . This that the Goettingen 593 airfoil has a thickness distribution approx
imately the 2.
0.10
50
thickness and the
maximum
spection of Table
12.00
0.15
Lower 3.00 0.85 0.40
40
30
Upper 3.00 7.85 9.75 11.50
593 AIRFOIL .
OF GOETTINGEN
593
NACA
 digit NACA airfoil , and 4  digit airfoil are 12 .
maximum mean  line
airfoil ,
line of Table
used as a reference
these
items
ordinate
that the last
and
are calculated
its as
location
in Table
two
.
digits For the The
13 : 2 .
represents the ordinates of a line through
13 : 2
leading and trailing TABLE 13 : 2 .
4
the
similar triangles . This line is line of the NACA 4  digit system .
edges calculated by
line for
the mean
CALCULATION OF MAXIMUM MEDIAN CAMBER AND MAXIMUM LOCATION L OF GOETTINGEN 593 AIRFOIL
CAMBER
Location
,
%c
20
L. E.T. E. , chord ordinate Mean Mean
 line line ,
ordinate from L.
,
, %c
c
E.T.
E. chord , %
Inspection
30
40
50
60
2.40
2.10
1.80
1.50
1.20
5.82
6.05
5.85
5.42
4.72
3.42
4.05
3.95
3.92
3.52
of Table 13 : 2 shows that the maximum ordinate of the mean line 4.05 per cent and that it is located between 30 and 40 per cent of chord the mean line is plotted and is judged to be from the leading edge .
is
If
TECHNICAL AERODYNAMICS
1318
representable with reasonable accuracy by imum mean  line ordinate to the Goettingen 593
Low turbulence be read
Figs
from
closely to
the
A5 : 5 , and
may
by
4312
or
NACA
4412 may
airfoil in flight
University
Goettingen
4412 .
to correspond
be assumed
the Goettingen 593
results published
AERODYNAMIC
digits
4
characteristics of the
number
characteristics of
turbulence and low Reynolds 13 : 9 .
NACA
parabolas tangent at the max airfoil approximately equivalent 2
designated by the
A5 : 1 through
.
the test
than would
then the may be
high Reynolds
,
more
,
at high
number .
AND
STRUCTURAL
It is
COMPROMISES .
important for
the aerodynamicist to realize that airplane wings are not designed and built exclusively from aerodynamic considerations . There can be no com promise should
it is
structural safety
with
;
light
line operation involves
factors
many
Low drag
.
is
It if
operation at high speed .
economical
labor costs
volved is a
in
minimum
the last analysis
is
of wings
the cost
other than the drag of the wings of
important because
and maintain and may also weigh more
in
advantageous .
also
also be remembered that commercial airplanes will only be flown financially profitable to do so , and the financial profit of an air
the airplanes
a
is
weight
it
permits high
Low drag wings
build
All
costs
than wings of higher drag . , ( 1 ) and
only
if the
total labor
a low drag wing economically desirable . ,
as components
speed and
are expensive to
in
Major items
operation rather than
of an airline
as components of an airplane , are maintenance costs and costs of inspec tion and rebuilding to meet certification requirements for safety . Thin wings that are aerodynamically desirable may not only have to be heavier the structural requirements , but also may be so compact in equip design as to be less accessible for repair and rebuilding than thicker Boundary  layer control devices , including porous lead and simpler wings .
to
meet
ment
suction flaps ,
ing edges and
reliable
,
and
more
troublesome
line operation using
them
is
may be
sufficiently
more complicated ,
to repair , that the total cost of the greater rather
than
less
,
less
air
so that apparent
gains may turn out to be economic losses in flight operations . supersonic airplane and missile field , aerodynamic considera In the tions may dictate sharp leading and trailing edges but , at high accelera
aerodynamic
tions
and
high
Mach
numbers , a
knife  like leading
edge
is structurally
( 1 )Wood , K. D. " Airplane Design , " Tenth Edition ( 1954 ) , Chapter 7 , distributed by University Bookstore , Boulder , Colorado . (Eleventh edi tion to be distributed by Ulrich's Book Store , Ann Arbor , Michigan . )
AIRFOIL SELECTION
impossible stresses
which
13:10.
AIRFOIL SELECTION CRITERIA FOR
The forces
AIRPLANES .
SUBSONIC
on an airplane in a glide are shown in Fig . 13:22 . For as flat a as possible , the ratio D L should be a minimum or L D should be a
/
/
flattest
be one of the chief factors determining the
lon of fuel ) of a conventional airplane per gallon of fuel ) of a jet .10 airplane .
.06
Airplane نم
Complete
•04
Wing
Alone
Dp .02 KCDP min
/
( L D) max
for for
/)DL(
a
.
1.2
(
13
13
(
divide equation
CL equal to zero and
solve for (
is 13
(
/
, ( L D ) max
by CL
13
2CDp min
)
a
complete
A
is
Effect of Flight
,
and Configuration Variables on shaped Supersonic Wings JAS November 1951

in Diamond
1.0
coefficients of
, "
The
"
,
F.
)1
P.
and the corresponding
Durham Thermal Stresses
D ) ,
coefficient for CL (L/ D)
.8
™
lift
=
max
√
The corresponding
.6
.
(L/D),
CD
to
with respect
CDp
CD CL
max =
derivative of
maximum
min
the condition of
for
max
cz πAе
CDpmin
( L /
CD and put
for
wing
Graph showing conditions complete airplane and
and drag
written
/
To solve
for
wing alone
lift
between the
=
may be
. D )
13:23
max
a
relationship
.4
Fig
(
13:22 . Forces acting on airplane in glide . The
.2
0
Fig .
airplane
CL
CDo min
L /
.
D
1
13:23 .
:
in Fig .
complete
airplane
)
CL for a wing alone and for a complete airplane is shown
, for (L/D ) max
.08
(hours
2
vs.
CD
maximum
and the maximum endurance
)
A typical graph of
,
also be shown to range ( miles per gal may
.
of
:
The condition
, ( L / D) max
glide
: 3
maximum.
(
thermal
. (1)
)
glide
failure
cause
involved produce
: 1
acting
will
gradients
temperature
the
because
1319
TECHNICAL AERODYNAMICS
1320
/D) max
(L
(13 : 4 )
√πAe /4CDp min
=
, Equation ( 13 : 4 ) states that the highest values of ( L / D ) max are obtained with the highest values of the product Ae and the lowest values of CDp min This requires high aspect ratio , as shown in Fig . 13:24 , though aspect ratios over
are seen to be unprofitable , and low values of
15
CDp
min '
10
8 6
Aew
5
3
2 A
1.5
Fig .
2
By an analysis
sinking speed endurance
4
3
Plot of
13:24 .
Ae
5 6
similar to the
above ,
shown
maximum
is
that for
minimum
=
(13 : 5 )
4CDp min
jet  propelled airplane
,
it
can also be
similarly
that
and (CL1
/2 /CD )
is
jet
also be
range =
(4/3 )CDp min
(13 : 6 )
a maximum .
For each of the conditions should
can be shown
a maximum .
range of a
CD max
may
it
rate of descent ) Vs min . corresponding to maximum ( time aloft per gallon of fuel ) of a conventional airplane
(C₁3 /2 /CD )
For
20
15
10
( vertical
CDys min and
7 8
vs. A , based on data in Fig . 9:23 .
shown
be a minimum
presented
by equations ( 13 : 5 ) and ( 13 : 6 )
it
that the product Ae should be a maximum and CDp min . The foregoing studies indicate that a low value of
CD min is the most important aerodynamic
characteristic of an airfoil sec ratio is also highly important to good airplane perfor mance . The relative importance of low CD min and high aspect ratio is discussed later . tion ; high
aspect
AIRFOIL SELECTION Where the
in
utmost
high
1321
is
subsonic speed
objective
a major
,
as
in
military fighter aircraft , high critical Mach number is also of major importance . The means for obtaining high critical Mach numbers have been Chapter discussed in 11 and are usually not in conflict with the require ment
of low
13:11 .
drag .
minimum
AIRFOIL SELECTION CRITERIA
sonic missiles
to be supported
by
also a compromise between aerodynamic practices of supersonic missile wing lished
of
adequate
structural
and
is
of solid metal
made
high hot  strength , though a given
if
,
requirements
but the
are less well estab
construction
so that the nature of the compromises
sonic wings are
MISSILES . For super strength , there is
FOR SUPERSONIC
wings
less clear
Many super
.
its
often stainless steel because of strength can
bending
be obtained with
hollow with fairly thick plates for the wing surfaces . wings , for a given strength , will have higher Such hollow drag and require more fuel , which may more than neutralize the weight sav less weight
is
the wing
made
ing due to making the wing hollow , depending on the mission or specifica tion of the missile . Studies reported in TN 2754 provide means for cal culating the " optimum hollowness " of a supersonic wing , but no simple general rules
AIRFOIL SELECTION CRITERIA
13:12 .
as currently
of the blades tabs
operated
( 1955 ) are
through near
the tips ) .
the
importance
to evaluate , copter .
of at least as
much
FOR HELICOPTER
controlled by
In either
a
case
be selected
Helicopters
ROTORS .
cyclic variation of pitch
it
in
( or ,
is
a few cases , by
important
that
for negligible pitching
the
moment
center . The blade section should preferably also drag and high maximum without flaps , though the
aerodynamic
have low minimum
ative
is
.
forces applied at the hubs
helicopter rotor blade section about
profile
as the cross  section
importance
control
be formulated , as the planform
can
lift
of low
minimum
drag
and high maximum
lift is
and depends on the design operating conditions
Inspection
of the pitching
moment
data
in Fig .
rel
difficult
of the
heli
A5 : 4 shows that
only the symmetrical airfoils , among those there reported , have zero pitch ing moment . Recalling from Article 13 : 7 the effect of airfoil geometry on aerodynamic characteristics , it should be noted that only thin airfoils have low minimum
for the
drag , and
66  series .
It
that the lowest drag reported in Fig . A5 : 3
airfoils
should also be noted that only
per cent thickness have high the most favorable for high
maximum maximum
lift ,
lift .
of over
and that the 66  series To
find
a
is
is 12
not
better compromise
TECHNICAL AERODYNAMICS
1322
airfoil for rotor
blades than any reported in Fig . A5 : 3 , several special trailing series of reflex  edge airfoils have been tested by the NACA , ( 1 ) but criteria for selection from those tested are not well established .
is some doubt as to whether a high value of the ratio c1max /cd min proper a measure of the merit of a helicopter blade section , as only
There
is
is
Cd min
involved
if
so
new
or
some
that
it is
weighted
not
The art
possible to say whether
must be maintained .
airfoil
series
not only
of construction
is
drag or maximum
minimum
agreement that the low pitching
It is entirely
with some sort of boundary
minimum
limited by
not
of such blades
also
lift
selec require
combination of the two should be a criterion for
tion , but there is general ment
rotor is
the forward speed of the
retreating blade stall .
drag and high maximum
the complicated mechanical
moment
possible that a symmetrical
 layer control
lift ,
device
but also
cyclic pitch control
( often
eer's nightmare " ) , possibly in conjunction with lieu of a conventional gear  driven rotor .
a
may
66
provide
a substitute for called " an engin
rotor tip jet burner in
PROBLEMS
Describe the airfoil designated 2311535 . Coordinates of the upper and lower surfaces of a Clark X airfoil are given in Table 13 : 3 ( below ) . Find the constants for straight  line plotting of the variable  density  tunnel characteristics of a rectangular wing of A = 6 using this airfoil . 13 : 1 .
13 : 2 .
05
с
TABLE 13 : 3 10
Upper
4.00
7.96 9.68
Lower
4.00
1.14 0.50
20
40 4
30
11.28 11.70 11.40 0 0.03 0
13 : 3 . An airplane wing mensions : span 40 ft , chord
is
60
50
70
80
90
10.529.157.355.222.80 0
0
10
10
95
100
1.490.12
10
following di in . , location of flat aft of 25 per portion of lower mean line to be lo
measured and found to have the
in . ,
maximum thickness 6 maximum thickness at 30 per cent chord , lower surface cent chord , height of leading edge above plane of rear 50
surface 2.0 in . Assuming the maximum ordinate of the cated at 0.3c , what is the approximately equivalent NACA 4  digit airfoil ? Using data presented graphically in Figs . A5 : 1 through A5 : 5 , 13 : 4 . write equations at Re = 6 x 106 for CL (a ) , CD ( CL ) , and Cp ( CL ) for a 2 : 1 tapered wing of A = 10 ( without sweep ) the airfoil is a smooth 662315 Also estimate CD min at section . Neglect " bucket " in the drag curve .
if
Re = 25
x
106 .
( 1) Tetervin , Neal , " Tests in the NACA Two  dimensional Low Turbulence Tunnel of Airfoil Sections Designed to Have Small Pitching Moments and High Lift / Drag Ratios . " NACA Wartime Report L 452 .
1
CHAPTER
14
DRAG ESTIMATES AND POWER CALCULATIONS 14 : 1 .
lift The
OF ESTIMATING
METHODS
coefficients for data in Fig . 14 : 1
a
typical
DRAG .
The
may be noted
where CD
/
1 TAе
is
is the
is
to be approximated
by the equation CD =
relationship
airplane
complete
between drag and
in Fig .
shown
14 : 1 .
with good accuracy
c
CDL
( 14 : 1 )
ПAе
the intercept on the CD axis of a graph of CD vs. cz and slope , theoretically ( for lifting  line theory ) equal to 1 /πA ,
.12
Line
.10
Test
.08
Data Straight
Typical
Approximating
CD
.06
°170
.02
CDE
CDp
c²
min
.2
Fig .
.6
Comparison
14 : 1 .
between
.8
1.0
typical airplane approximation .
1.2
1.4
test data
and parabolic
but modified by the factor e to take account of discrepancies lifting  line theory for wings and experimental data for airplanes
tion
( 14 : 1 )
parabola
intercept
if
is
called
on the
CD
axis
is
used as coordinates , as sometimes referred 141
between .
it
plots
in Fig .
13:23 .
a parabolic approximation because
CD and CL are
1.6
to ( as in
Chapter
Equa as
a
The 13 )
as
TECHNICAL AERODYNAMICS
142
min , but is here called CD which small amount shown in Fig . 14 : 1 .
CDp
may
differ
from CDp min by the
very
of estimating CDp min or CDf is to designate the product = Df q , called the " equivalent parasite area of the CDS as equal to airplane , and to estimate this area by adding numerically the values of f for the component parts of the airplane . Each component parasite area One
is
method
/
f
designated by the product CDA , where CD
the component part of the
airplane
A
and
is
is
the drag coefficient
that drag coefficient is based , usually the area Ac of the maximum section . These relationships are summarized in the equation
f
CD S and
typical values of
which they The
= Df
for airplane
CD
are usually associated are
total
f
for
an airplane
component parts plus
5
/9
or
10
=
cross ( 14 : 2 )
ZСDAT
components , and the areas A shown
of
the " proper " area on which
in Table
with
14 : 1 .
is approximately the sum of CD of the per cent to allow for mutual interference
per cent allowance may well be made for small protuberances such as handles , hinges , and cover plates , unless they are specifically included on the basis of data in Appendix 6 . between the components ; an
The drag
additional
coefficients given in Table
5 or 10
14 : 1
are not specific but simply
dicate the range of values usually encountered
flying at
critical
speeds below the
.
for
More detailed
in
full
scale airplanes suggestions for es
the specified range ( sometimes slightly beyond the specified range ) are given in later articles of this chapter and in Ap pendix 6. Their application to a specific airplane is illustrated by timating
within
CD
an example
later in this chapter . efficiency factor "
The " airplane
"wing efficiency factor "
ew
fuselage and other airplane cient
,
as well
An approximate
e
it
differs substantially
in that includes increments parts with angle of attack and
from the
of
drag of
lift coeffi
effects due to wing planform and thickness ratio . rule for estimating e for a complete airplane is given by
as the
the equation е
=

—
Σ + = [A
(14 : 3 )
( ÷ ) parts ]
The values of 1 / ew implied by Fig . 9:23 include
(a ) a planform
18
shape cor
theoretical factor in Fig . 9:22 , (b ) equal an airfoil thickness correction factor to 0.005 to 0.010 for usual thickness ratios , and ( c ) a correction to practical wing construction for rection factor ,
given by the
143
DRAG ESTIMATES
TABLE 14 : 1 .
APPROXIMATE DRAG OF
AIRPLANE COMPONENTS AT LOW SPEED (M < SPEED AT SEA LEVEL FOR MEDIUM
0.4 ) AND Re CORRESPONDING TO LEVEL HIGH SIZE AIRPLANES . (1 ) SEE APPENDIX 6 FOR Re Length
Part
Description
Wing
Re
roughness ,
Usual
t /c
=
AND M CORRECTIONS
for
Calc .
.
Area for Drag Calc .
Range
CD
Chord
S
.004

.1 to .2
.010
Flaps
60% span , deflected 300
Chord
S
.02
 .03
Tail
Usual roughness , t c = .08 to .12
Chord
St
.006

Fuselage
Smooth
Length
Ac*
.03
·..08
"
"
.07
.10
:11
:"1 :
/
streamline
body
transport
11
Large
"
Bomber
:00
Small plane , nose
"

.08
hull "
"1
Very low drag
Nacelle
performance
Above wing , small
.04
Ac
.08

:
:
.07
·
.12
1 =9
"= 1
.04

.07
:
.04
Ac
.05
Ac
.07
"
.15
:"1
"
.20
airplane
wing , large airplane
"
In
"
For turbojet engine
:
tip
"
Ext . tanks
Centered on wing
tip
"
"
Below wing
11
"
Inboard below wing ( incl . support ) ( incl .
·
Ac
Usual for bestwa
ter
.12
.20 ,
.09
engine
Boat
.008
":1 : 3
·
.08 2
.07 .07 .10 .30
Bomb
Below wing
Float
Best
streamlining
"
"
.05
"
Usual
for best wa
"
"1
.12

.25
.5
·
.8 .30
support )
ter Landing gear
"
"1
strut
well faired
dwheel
** bdwheels
"
"
.15
·
"
"
.3
 .5
wheels with " pants"
"
11
 .08
performance
Nose wheel and Two
.30
Wheels and exposed
struts
area of cross  section ; ** b = width , d = diameter , for wheels 11 of these data are based on Hoerner , S. F. , " Aerodynamic Drag , .; this published in 1951 by author at 148 Busteed , Midland Park , N. reference should be consulted for more detailed information than that given here or in Appendix 6 . Ac
( 1 ) Most
J
144
TECHNICAL AERODYNAMICS
high aspect ratio
involving non  optimum spanwise lift distribution and resulting in the " recommended practice " lines of Fig . 9:23 . The principal incremental value ▲ ( 1 / e ) parts , is due to the fuselage (and nacelles , if any ) . Data for estimating still
/
A ( 1 e ) fuselage are rather incomplete but may reasonably be assumed to be proportional to the ratio of fuselage frontal area to wing area as implied by the plot of Fig . 14 : 2 , which should be noted as limited to the rather unusual case of zero wing incidence . The use of Fig . 14 : 2 is illustrated by the fol lowing example :
Example . Estimate the airplane efficiency factor e for a combination a rectangular wing of aspect ratio 7 and 180 sq ft area , with a rec tangular fuselage of S₁ = 15 sq with zero wing incidence .
of
/
1
attack
An
2 .
.
:
: 3 .
In
and
7.
ficient
can
,
e )
(
/
▲ ( 1
a
of the ,
: 2
.
profitably the
NACA TR 540
check
be
line
of
.
in
the
on
components
made by
cal
total area of
the
parts exposed to the air stream the wetted area and esti
mating 6
.
.
Chart for estimating effect of fuselage on airplane efficiency fac tor for zero wing incidence See also from Chapters
value out
additional
culating
in
" )
14
is
"
(
) 15
8 9
7
5
6
20
A
4
3
10
drag
the case
drag of the airplane
(
0886
2
Art
in
with the test data
0.5
14
little
Constellation fuselage indicated in Fig 14
as having
Constellation Est
Fig
loss
minimum
increased angle of
a
,
which
Round fuselage
0.6
as
Lockheed
Points plottedfrom TR540
10 09 0.8 07
how
.
)

very
2.4
fus ST fus
(Al/
is
crease with
1.5
kept to
fus can be
the fuselage so that there
shaping
positive
an additional
of
(
,
.
)
e
is
there
Increments
Rectangularfuselageor 2.0 round fuselagewith cowledengine
)e /S
1.37
a
.
/
to nacelles
by properly
15/180
0.73 With few degrees of wing incidence due to fuselage could be much less
nacelles are involved due
1.75
/e1
If
of
=
= 1
,
,
ever
e
1.37 the loss of
e
Hence
=
1.22
/e
4( 1
x
/
read
14 : 2
culate
9:23 read ew = 0.82 and calculate 1/ ew = 1.22 . In e ) fus ( S fus / S ) = 1.75 . Using equation ( 14 : 3 ) cal +
Fig .
ft
In Fig .
Solution .
the skin
friction
coef
general
be
a
will

area
fis
)
4
14
:
CD wetSwet
(
CDS
=
=
f
: 3 ,
.
wetted
.
on
,
the drag coefficient CDf wet substantially higher than that of flat plate of the same Reynolds number because certain pressure drags are in volved as well as skin friction An over all chart for checking given in Fig 14 where based
145
DRAG ESTIMATES
is
indicated
the "wetted " area of the airplane
on the chart .
and values of CD wet are estimates based on wetted area are
More detailed
given later in this chapter for the various components
wet
..
CD .015 100
.
.010 .008 .006
300
.
;
and Swet
.005 .004 .003
70

B 29
50 40 30
f
20
C 47
10 8
L 5
P  38
6
4
F  80
2
flat
plate as airplanes
30,000
8000
of
area of
wetted
)
function
function of Reynolds
ary layer transition
a
of cir to be
on bound
a
)
(
1949
.
"
Airplane
"
Hage
,
Robert E. .
and
Wiley
,
in Perkins Stability and Control
suggested
pressure drag
"
rear of the cylinder and its related ,
( )1
in the
D.
area
Plot
Performance
their effects
numbers and
C.
turbulent
and Mach
in Art
from laminar to turbulent and the effects of tran separation which results in the generation of low pressure
on
,
sition
been pointed out
has
The drag .
,
cylinders
INCLUDING WINGS
6 : 7
and
complex
INFINITE CYLINDERS
elliptic
,
cular
DRAG OF
, "
: 2 .
14
.
.
Parasite
a
1000
4000
(1
14
:
.
Fig
400 600 3 .
200
100
2000
ft
sq
Swet
1
TECHNICAL AERODYNAMICS
146
For thin
elliptic cylinders ,
thickness
,
is
coefficient location pends
is
the drag
effects are
almost
accordingly approximately given by
transition
of the
chiefly
for wings of less than entirely skin friction
and
The
in Fig .
on the skin
around an
elliptic
cylinder
The
layer de
if
friction
thermal
in Fig .
is
drag
of the theoretical given
drag
6 : 5.
boundary
turbulence and surface roughness
effect of thickness
taken account of by Hoerner ( 1 ) from consideration
ocity distribution
and the wing
2C
from laminar to turbulent
on free  stream
absent .
or 15 per cent
10
vel
4 : 8 , where
.015
CDo min
Pressure Drag
4 digit airfoils and streamlined

.010
=
Cf
struts
0.003
J
W
Skin friction
Skin
0.005
\\\\\\\\\
11
.005
/c)
(1 + 2t
friction
Pressure
LLL
(1
+
Drag
/
1.5t c) 0.0035
16466 6466 series
airfoils
t/c .10
.20
.30
.40
wings in Low speed minimum drag coefficient of cylinders and ratio . of thickness range as a function 8 million 3 to from Re sub critical
Fig .
it is
14 : 4 .
noted that
very close to
1,
/
Vmax Vo = 1 +
it is
t/c .
From the
approximately true that
/
rule for squaring amax
/40 = 1
+ 2t
/c ,
numbers though
of course the average ratio amax o must be less than the maximum . Hoerner finds , as shown in Fig . 14 : 4 , that the minimum drag coefficient of 4  digit airfoils and the skin friction coefficient and thickness ratio are re lated by the equation
/
/
CDo = t 1 + 2t c + 60 ( t c ) 4 , pres . drag 2C1 skin fric .
(1)Hoerner
,
S.
F. , op .
cit . ,
p . 61 .
( 4  digit
airfoils )
( 14 : 5 )
DRAG ESTIMATES
first
The
right
two terms of the
hand member
the skin friction effect ; the third
is
term
/
147
of equation
represent
( 14 : 5 )
an empirical
additional
term
for the pressure drag based on values of t c up to 0.5 and obviously not intended for extrapolation to the circular cylinder case . For airfoils and struts similar to the
series airfoils with the
6466
maximum
thick
ness near the 50 percent chord point , Hoerner finds that the increase of the skin friction portion of drag coefficient with thickness is slower , as given by
/c, fric .
1 + 1.5t
CDO =
skin
2Cf
125 ( t / c ) 4 pres . drag
+
(65  series
airfoils
( 14 : 6 )
)
but the pressure drag term increased more rapidly with thickness in this case as noted . This analysis intends to imply that streamline struts of a given thickness ratio will have about the same minimum drag coefficient as
airfoils
if
tion of Fig .
the Reynolds 14 : 4
it
for
effect
predominant
may
is in
and
Mach
noted
be
From inspec
numbers are the same .
that the skin
friction effect is
thickness ratios of less than 15 per cent
if
the
the
 critical range . For very low Reynolds numbers , of coefficient will be high , partly because of the typical skin friction variation with Reynolds number and partly be cause the flow is more likely to be laminar in the region where there is Reynolds
number
the
super
course , the drag
danger
of separation
14 : 3.
, and a
large pressure drag
STREAMLINE BODIES :
same
build
up .
The drag of spheres
SUBSONIC .
pattern of variation with Reynolds
may
follows
and Mach numbers as that of
the
cir
cular cylinders , as may be noted in Fig . 6:16 . Likewise , the drag of ellipsoids follows the pattern for elliptic cylinders noted in Fig . 6:15 . For streamline bers
bodies at low Mach number and super
gives the semi  empirical
, Hoerner ( 1 )
/ +6 (d/ 1 ) 4
CD wet =
1 + 0.5d 1, skin fric .
Cf
/ is the
where d 1
diameter
experimental data on
in Fig .
frontal area is CDTI
Cf
( 1 ) Hoerner
=
/ length 14 : 5 .
given
1.5+
pres . drag
ratio .
/
, S. F. , op .
num
equation (streamline bodies )
Equation
( 14 : 7 )
is
( 14 : 7)
compared with
as
/
+ 18 (d 1 ) 3
cit . ,
Reynolds
The corresponding drag coefficient based
by Hoerner
3 (1 d)
,
critical
p . 70 .
( streamline bodies )
( 14 : 8 )
TECHNICAL AERODYNAMICS
148
this
and
is
compared with
first
based on a
Equation ( 14 : 8 ) data in Fig . 14 : 6 . relationship in between wetted area and
experimental
approximation
frontal area of to 0.8 )
Swet = (0.7
smaller constant applies to conventional streamline bodies one to ellipsoids and to fuller bodies such as are used
which the
larger
rela
value of 0.75 this results in the
average
With an
.
Swet
/S
/
= 3 (1 d)
.02
14:10
)
for airships tionship
(
and the
.10 CDπ
.04
Drag
0.5 d
/
/1 )
Fig
6
a
d
l/
.
2 ,
.

a
surround
given
airplane
(b
.
a

6 ) .
69 .
,
F.
cit
distance of laminar
maximum
ratio to
in Fig
14
: 6 .
/
by the
Later low turbulence
varied series of streamline bodies
might be obtained than any shown S.
Hoerner
,
(1
values of CD
body
Reynolds
in Appendix If boundary layer vel transpiration lowing through porous analysis in Fig suggests that lower
data
were reduced
rear portion of the
critical
super
the
incorrect
to maintain
optimum length diameter
the region of
ocity gradient
)
airfoils
in
ratio
and that
but tests on actual fuselages cited by
.p
in
0.03
systematically
6 (
cockpit
low drag

"
show an
number
,
flow
like
optimum
)
tunnel tests on
shaped
about
conclusion to be a
this
"
show
.op
)
(1
number
Hoerner wind
is
CD
range at low Mach
.
value of
minimum
inferred that the
circular fuselage is approximately ,
diameter of
. ,
of length to
erroneously be
might
a
: 6
.
it
14
10
CD for streamline bodies 14 at low and super critical Re as function of ratio
/
Fig
8
d
CD wet for streamline 14 : 5 . bodies at low M and super  critical Re as a function of d 1 ratio .
4
1.0

.8
2
.6
Fig .
From
31
l/
d 1
.4
1.5+
0.003
.02
0.003
.
.2
+
7 : 8
(1
(
.005
/
Pressure
.06
)d
= 0.003
dra
Cf
.01
:
CD wet
Pressure
.08
.015
the
(14 : 9)
circumference )
max .
M 6 .
in
( length x
149
DRAG ESTIMATES
A streamline fers
airplane
body such as an
air
an angle a to an
in
an increase
The relationships
stream , has a
lift
drag nearly proportional
shown
in Fig .
If ,
in
equation
of
for circular fuselages
equation ( 9:33 )
( 14:11 ) , a
is
is
/
A(1 e) fus S fus /S
put equal
=
it
shown
a,
by the
suf 14 : 7 .
equation ( 14:11 )
for rectangular fuselages
/
.
lift
to CL a and the  curve  slope then for zero angle of incidence
may be shown
that
11.6 CD minfus K (A
and equation ( 14:12 ) corresponds
in Fig .
adeg 2 15
and 4 to 6
used to determine
the wing on the fuselage ,
to a² , as
to a and
)
=1 + K
CDπ
inclined at
a wing , when
14 : 7 may be approximated
СDπ min where K = 1
like
,
nearly proportional
+ 3)
²/A
( 14:12 )
closely to the line labeled
" round
fuse
if
lage " in Fig . 14 : 2 CD minfus equals 0.05 (as shown in Fig . 14 : 7 ) . A small and appropriately chosen angle of incidence of the wing on .25
.20 CDπ
Fuselage
.15
Rectangular
.10
Circular CDπ
Fuselage
= CDT
min
(1 + α
/ )
deg 225
.05 a 2 , degrees2 200
100
0
Fig .
+ 5
14 : 7 .
10
α
Variation of fuselage
from data by Hoerner ,
(1 ) Hoerner
, S. F. , op .
cit . ,
.
300 15
400 20
deg .
coefficient with angle of attack and 1 /d = 6.9 . at Rei =
drag
p . 72 .
,
TECHNICAL AERODYNAMICS
1410
the fuselage can
,
completely eliminate
however , almost
/
the unfavorable
A ( 1 e ) fus indicated in equation ( 14:12 ) for the cruising and climbing range of lift coefficients of an actual airplane . With the zero lift
line of the cruising conditions , equation (14:12 ) greatly over  state the adverse effect of the fuselage .
wing set at 3 to 5 degrees
chord of the fuselage
,
Fig .
and
is
as 14 : 2
common
for
to the
drag
minimum
optimum
The effect of subsonic compressibility on the drag of streamline bodies of revolution as analyzed by Hoerner ( 1 ) is shown in Fig . 14 : 8 , and the
effect of fineness ratio on critical Mach number from the same source is shown in Fig . 14 : 9 . Hoerner develops from rational considerations what 1.0
3
CD wet
2
.8
Mer
comp
/
Mer
0.5
d 1
.6 0.1
f
O
0.2
.4 d
0.3
Effect of subsonic com pressibility on drag of streamline bodies of revolution From Hoerner
.6
209
.
From Hoerner
,
.
.4
.
.
: 8 .
14
.
.2
203
.
.p
1/2
.8 Fig 14 Critical Mach number of streamline bodies of revolution
1.0
.p
.8
,
.6
: 9 .
.4
0
0
Fig
x
M
.2
+
d
.2
p "
203
)
)
(
4
/
d 1
(
p "
"
the
.
p
.
,
cit
(
for several typical values of
show good
numbers up through
.
.op
,
14:14
experimental
the
critical
critical
Mach
.
1
"
.
: 8
to
presented
empirical study that F.
Hoerner
,
(d
14

semi
S.
from
(1 )
finds
a
Mach
,
)
is
of this rule for
/1
and other evidence
14:13
VI  M2
this equation is plotted in Fig
cation
¹.5
/
=
p "
where
+ 6
/1
0.5
)
comp
d
= 1
CD wet
+
giving
Cf
and
resulting in the modification of factor l.5 in the last term of "
"
insertion of
by the
a
)
,
that equation
rule
Prandtl
"
14
: 7
equation
the extended
(
calls
he
number
verifi Hoerner Mcr
for
DRAG ESTIMATES
streamline bodies of revolution
is
1411
single  valued
a
function of
" effec
an
tive " diameter / length ratio defined by the equation ( d / 1 ) eff where
x
thickness
= d
is
the distance from the
.
This ratio
verification
is
is
supersonic missiles
is
older
in
subsonic
+
0.5
( 14:15 )
1)
leading edge to the
plotted vs. Mer in Fig .
14 : 9
point of
SUPERSONIC
based on
.
Much
ballistic
information
research and
on
the drag of
this information
in many respects more extensive than information on flight , but basic studies on shock waves and boundary
and
and their interaction have only recently been extensively comparison
for
the optimum bodies
of
in Fig .
indicated
maximum
with experimental
.
MISSILE BODIES :
14 : 4 .
/( x
14:10 .
Subsonic
made .
bodies layers
A rough
subsonic and supersonic flight is and fuselages have often been
wings
1
с
} t
d
low drag wing section : 00120.55 50 ; CD wet = 0.0015 , CD min 0.003 Good NACA
low drag subsonic body : 0.9 ; at Re / 106 .15 ; Mer = CD wet 0.0015 , CD≈ 0.03 1 Good
Mcr.75
/
d 1
= 4
+
d = diameter
T
/
noses
Comparison
14:10 .
of the
been found
NACA
digit airfoils
better than those
fly
required to
sible
4
X2
at M = 2 but d 10.07 M as in Fig with Re . of near  optimum subsonic subsonic and supersonic bodies . From Hoerner .
low drag missile body : varies considerably with
Good
Fig .
+
2x
X
shown
. CDT = 0.18 . A6 : 36 . also
without subsequent dash numbers have Fig . 14:10 only because they were
in
at wide range of angles of attack . For the lowest pos relatively small leading  edge radii shown in Fig .
drag the
minimum
Good supersonic missiles
14:10 are necessary . a nearly
sharp
curvature
.
point
;
have been found to require
any point , however sharp , has some small radius of
stresses considered , a perfectly sharp point can not highly accelerated supersonic flight ; a nose bluntness of
Thermal
exist long in
the order of a few per cent of the diameter
(1 ) ,Hoerner
,
S. F. , op .
cit . ,
has
little effect
pp . 198 , 210 , and 226 .
on the drag
1412
TECHNICAL AERODYNAMICS
but permits a large reduction in thermal
sile
in
shown
Fig
.
14:10
coefficient against
(d 1)
is
/
stresses
mis
The near  optimum
.
very long and slender but the graph of drag
ratio is very flat
con
near the optimum , and
in
siderably
shorter missiles have nearly as low drag as the optimum , as dicated in Fig . 14:11 . Note in Fig . 14:11 that for a given nose length
is
the ogive shape
as good as a conical
not
,
nose , but on the other hand
for
a given nose angle the ogive has less drag , as may be seen by compar ing the " proper " drag coefficients for Figs . 14 : 11a and 14 : 11j . The sharp junction between a cone and cylinder , while it shows no disadvantage from shock wave theory , is actually a substantial handicap in practice , pre
h
.5d
2.5d
°1
FTTTIALL
j
i
It
.
1 .
5d
2.5d
junction
the
A.A
3.5d
layer behind g
d
junction in assisting the
sharp
boundary
a turbulent с
b
a
effect of the
of the
of
new development
f
sumably on account
to provide
tail is

,
still
substantial function
exploration
under
the supersonic drag of
"
Mach
in
drag at
number
simple cone
some
in
semi
at
of
a
vertex angle
and boundary
also of nose shape
Data are provided
used
degrees
of 5.7
reduction
of the
quite possibly
and
tail
,
timum boat
bulence
boat
are
but the
op
layer
tur

=

one diameter
missile
to provide For the tests reported in
noses
+
seen
conical
the drag
,
5
if
between the cone and cylinder
a
,
a
.
is
14:11
.27

fairing
smooth
Fig
good
nose and tail shape on 2. From and M
practice
1 d
accordingly
a
is
=
this matter
though
Appendix
cylinder
6
Effects of

.
.25
1.64
"
14:11
115.7
.23
.26
M = 2 ,
.29
.24
1.7
.94
.
.25
.45 .47
,
.28
/
Fig .
.21
.
CDπ
is
for estimating
combinations
at
Mach
Flight at speeds beyond is currently being con sidered only at very high altitudes of the order of 100 miles because missiles flying in the lower stratosphere become red hot even at very quickly Such missiles can get out of the lower atmosphere without but the problem of cooling
or of delaying the melting
,
,
overheating
,
.

M = 4
)
(
M = 4
numbers up to 4.
upon
225
.
.p
cit
. ,
.op
,
F.
,
S.
( 1 )
Hoerner
;
)
(
reentering the lower atmosphere is as yet 1955 unsatisfactorily solved is some question as to whether the solution will be judged worth
there
1413
DRAG ESTIMATES
its
if
cost
It
is
obtainable . to analyze the drag of a supersonic missile into three
customary
parts , ( 1 ) nose Figs . A6 : 36 and
drag ,
The pressure
A6 : 37 .
can be calculated
friction ,
skin
(2)
from the conical
lations of this sort are seen to Figs . A6 : 34 and A6 : 35 . Pressure boat tail angle are shown in Figs .
sonic
skin
Turbulent
.
relative
( or " wave " ) drag
in
flow charts be well
noses
Appendix
4 , and
calcu in
and the
Skin
A6 : 38 and A6 : 39 .
effects of
friction
may be
includes a chart for laminar
7 , which
well
and 4 , as
is still
on the
the missile wall and the recovered temperature
of
of an unsolved problem
somewhat
sub
as
depend greatly
these factors affect transition of laminar to turbulent boundary
How
in
shown
conical
corroborated experimentally
friction coefficients at Mach 2 skin friction is there seen to
temperature
, as
of
coefficients
drag
estimated by the methods of Chapter and turbulent
(3 ) base drag
and
.
layer
indicated by the charts show
, as
ing current status of NACA information on this subject in Chapter 7. There is some evidence to indicate that missiles that are cold relative to the recovered temperature theories
some
Figs .
A6 : 36
)
of the boundary
transition
and
Reynolds
A6 : 37 must
 layer
have very high ( infinite by The drag
number .
analyses given
be considered as only a rough guide
to
in
esti
if
the drag of proposed new missile configurations , particularly they are long and slender and the skin friction is a major factor in
mating
if
the drag
,
because
transition varies with
Mach
and
Reynolds
numbers
in
ways as yet inadequately explored . NACELLE WING COMBINATIONS
14 : 5 .
largely laminar
have
boundary
.
A smooth wing
 layer flow .
designed
to
A fuselage can also be so
de
can be
signed ; good subsonic designs of
this sort Fig . 14:10 . ever ,
in
have been sketched Combine the
any
way
two ,
in
is
junction .
Usually a substantial
velops are
generated
, no matter
made
turbulent
imaginable and
turbulence
region of separated
laminar
how
at
flow also what
the de
attempts
Fig .
14:12 . Generation of turbulent flow region by wing  nacelle or wing fuselage junction on laminar flow streamline body .
to provide " fillets " at the junction
.
The
effect of
wing  fuselage
junction
on transition in some combined wing  fuselage tests of the NACA on an otherwise completely laminar flow fuselage are shown in Fig . 14:12 . The adverse
junction
at the
effects of
the junction can be minimized by having the rear of the fuselage , as shown in Fig . 14:13 , but this
TECHNICAL AERODYNAMICS
1414
is flyable
combination
only with a large
the center of gravity of the airplane wing chord .
sweep ,
Forward
turally objectionable
 forward
is
to put
shown ,
as
at the proper location on the main
while aerodynamically advantageous
This
.
sweep
,
is struc
because wing deflections produce increased
angle of attack
, which
further increases
increases
lift
and
This
the deflections .
 elastically critical speed and
combination is said to be aero unstable beyond
Wind
the
c.g. +
critical
less the wing
some
speed may be rather
is
merous combinations
Mean wing
chord
have been
.
un Nu
of wings and nacelles
. Some of Figs in . A6 : 11
tested by the
the test results are through A6 : 18 .
low
rigid
exceptionally
NACA
shown
are seen to show an incremental drag coefficient Some combinations
Fig . 14:13 . Advantageous com bination for maximum laminar due to nacelle of only about ACD = 0.035 , flow (but structurally disad which is very little above the skin fric vantageous ) . drag Reynolds tion coefficient at the numbers tested . Many of the studies are applicable either to fuselages or nacelles . If used for a guide to wing fuselage combinations
, the drag
of other fuselage parts
must be added ,
particularly the cockpit canopy or enclosure . If used as a guide to con struction of wing  nacelle combinations , consideration must be given to the amount of air flowing into and out of the nacelle for cooling of the power plant or , in the case of turbo  jet nacelles , for intake and dis charge of combustion air and combustion products respectively . Turbo  jet power plants
are often most advantageously located in nacelle pods slung diameter below the wing ( as in Fig . 1 : 4a ) , though nearly
about one nacelle
equally good performance
has been obtained with power plants
built inte
grally with the wings as in the British Comet airplane . In general , it is most efficient to take air in through the nose of the nacelle and to discharge
it
pressure and the tail . 14 : 6.
a fuselage Some
.
is
the
tail
discharged with
FUSELAGES
drag
; COCKPIT
estimate
reported values of
tails , Fig
through
is
CD wet
canopies , protuberances
14:14
, as
air is
the
minimum
loss into a turbulent
ENCLOSURES .
the drag and CD ,
" rammed in " by the impact
coefficient of
a
for
streamline body .
without wings , are shown in Fig . 14:14 . flat plate skin friction data
or slipstream
includes laminar and turbulent
through
starting point
The usual
for fuselage
wake
bodies
DRAG ESTIMATES
for
comparison
with
based on equation ( 14:10 )
scale for CD ratio of about 7. Fig .
CD wet and a
and a length / diameter range of Reynolds
1415
full
for
numbers
scale
14:14 also
fuselages in
the usual
shows
flight ,
which
is
to be substantially beyond that of all data there reported . Drag coefficients corresponding to less than that of turbulent skin friction on a flat plate are almost never reported . Flow in the slipstream of a propeller is , of course , always turbulent . Minor protuberances generate seen
turbulence ; flow aft of the leading
1.20 o
.005
.10
CD
CD wet
.002
=
/
1 d
Plate
Flat
.08
,0
Data

.001
=
7
 Flat
Cf Laminar
for
X
d
Turbulent
.04
body
.02
Transition
on
Plate
Large
A6 13
mph
.01
400
mph
Airplane . ,
Fig
, Small
Int
Wing
Fus
part lam . turb . :
. A6 : 15 , A6 : 15 ,
Fig .
.
Fig
O = NACA , ♂ = NACA , e = .
.0002
100
Fuselage data D = Hoerner p . 122 X = Hoerner p . 123
Airplane
Data
.0005
.004
Re1
in
.
,
,
109
the form of
be
NACA
tests
on
to put
laminar ring of sandpaper
fuselage bodies
in
an
where
a
.
,
,
in
,
,
ingly common practice part of the flow might erator
108
turbulent and any cockpit or canopy to provide vision for usually an additional source of turbulence It is accord
,
pilot is
a
nearly always
3.2
107
Comparison of fuselage drag coefficients without wings tails canopies protuberances or slipstream ,
14:14 .
,
Fig .
106
105
104
a
.0001
the
is
of the wing  fuselage junction
edge
.01
large
artificial turbulence gen
few per cent from the leading
of the fuselage so as to permit more accurate comparisons The effect of fuselage wing junctions may be inferred from data dis cussed in the preceding article to contribute little to the drag coeffi 
.
edge
:
,
.

.
:
A6 14
a
fuselage in which the flow is already turbulent if the optimum fuselage wing is selected from the data in Figs A6 13 and combination
cient of
TECHNICAL AERODYNAMICS
1416
effect of cockpit canopy for Mach numbers less than 0.4 may be in ferred from the NACA tests reported in Fig . A6 : 19 . For a typical ratio The
Scan / S
1/10 , the worst canopies had an incremental drag coeffi cient ACD fus = 0.03 , while the best of the canopies there studied ( X  1 and X  2 ) had only one  tenth of this amount or about 0.003 , which may be fus
=
only about ten per cent of the fuselage drag . Inspection of data in Fig . A6 : 19 shows that the canopies must be carefully selected in order to avoid Tests on
critical Mach numbers . tail surfaces reported by /Dprofile of tail = 4% to 8% for most
effects
extreme adverse
on
fuselages with
ratio Dinterference
arrangements , with a very a 2  surface or V  type The lowest drag
slight
improvement
EXPOSED
conventional tail (to 3 %) for
in interference
tail .
flying boat hulls are
seen
in Fig .
A6 : 23 to have
as low proper drag coefficients as the best fuselages ence and cockpit enclosure are included in each case 14 : 7 .
Hoerner ( 1 ) show a
LANDING
GEARS AND OTHER
landing gears are all but obsolete
wing
when
nearly
interfer
.
PROTUBERANCES
.
Non
 retractable
being used only for airplanes
in which considered relatively unimportant . A wheel has a high drag coefficient even streamlined as well as possible ; a combination of
high speed
is
,
if
or
dinary wheels and round struts would be incongruous wing and fuselage , as the landing gear drag
of all the rest of the airplane . ing gear drag data in Fig . A6 landing gears
shown
in Fig .
This : 26 ,
A6 : 25 .
well streamlined could easily exceed the drag on a
may be seen by
which Drag
consulting
the land
refers to the typical exposed is given in terms of f = D q
/ in
stead of proper drag coefficients to save estimating time . Data on various types of protuberances , including antenna masts and wire and air scoops and outlets , are given in Appendix 6. The proper drag
coefficients of are small
,
these
where otherwise
it
are very high , though
items
effect
the drag
may
did not
"aerodynamic atrocities
, " but
if
the items themselves
is
be small unless turbulence
exist .
Such items such
as
generated
radio masts are
the aerodynamicist should be quick to admit
are not designed for drag alone and that an airplane is a designed to transport persons and things quickly and comfortably
that airplanes vehicle
,
from place to place , and that
it
is
even more
important
for
a
pilot
to be
comfortable , to see where he is going , and to know where he is , than for the vehicle to proceed as rapidly as possible . It may also be true that ( 1) Hoerner , S.
F.
,
op .
cit . ,
p.
112 .
DRAG ESTIMATES
1417
faulty landing  gear retraction mechanism would be more of a handicap to the sale of a light airplane than its relatively slow speed without retractable landing gear . Making good use of published information on a
the drag of airplane components , it is possible to arrive at an accur ately estimated value of f = D/q for a complete airplane as has been done by Hoerner ( 1 ) in Fig . 14:15 . In Fig . 14:15 the effect of surface fric tion , irregularities , and exposed parts on the drag of each of the major
irreg
rivet
heads ,
ft
and
+
.
L
1.4 1.6 1.8 2.0 2.2
.
Interf
N26
Skin Friction Turbulence Paint Smooth
(
Fuselage
bolt
sq
1.2
NYL 3
Wing
with good
can be estimated
drag estimate , wing
.
include sheet metal
0.2 0.4 0.6 0.8
In Hoerner's
.
edges , surface gaps ,
1.0
0
tail )
power plant ,
,
practice
)
ularities
fuselage
,
little
/
(wing accuracy after a
D q ,
components
Irregularities Additional Parts
537 Radiator
Parts
Engine
Tail
Induced Drag
DRAG ESTIMATE
: 8 .
14
.
installation exhaust
,

,
parts
mast and
,
.
antenna
drag estimate includes
stacks
oil
cooler
FOR
AIRPLANE
COMPLETE
and
air
ventilating
For preliminary
considerably
simpler answer can be obtained by considering major parts without such detailed analysis
due to the analysis of this sort is illustrated
.
a
loss
and wing radiators
,
openings
His
momentum
wheel
by the
following
studies only the drag
An approximate
example
:
intake
and
engine
.
scoop
tail
canopy
His fuselage drag
landing gear ,
pilot
other irregularities
,
estimate includes
aileron gaps aileron edges ailerons and flaps air speed
as
ends of
holes around retracted
and
as presented
,
,
,
head
.
pitot
well
as
gaps at
airplane
,
blisters
metal
aileron balance weights
109
,
sheet
by Hoerner
,
.
.
and several
of the Messerschmitt ,
Drag analysis
14:15
.
Fig
.
Example
.
a
.
151
.
.p
cit
. ,
.op
,
F.
S.
(
(
)2
Ibid
,
Hoerner
)
1
.
.
( )
= a
( b )
"
"
,
( a )
Estimate the flat plate area equal to the parasite drag of assuming the data given below Lodestar airplane Use the result calculated in to write an equation for CD in terms of CL2 for the airplane Use VL 270 mph for calculation of Re at high speed Lockheed
TECHNICAL AERODYNAMICS
1418
Sπ "
Part Wing , assumed span 65.5
ft
equivalent to
sq
Total .
.
Solution
.
35
0.0054 0.080 0.10
200
0.006
551 40
Fuselage , length = 50 ft Nacelles . total for two , including Tail surfaces , total
cooling drag
.
ΔΙ ,
ACDI
section ,
NACA 23012
.
ft
...
sq
ft
3.0
3.2 3.5 1.2 10.9
Wing Drag Estimate . To estimate the minimum profile drag coeffi cient of the wing , calculate first the mean chord c = S b = 551 65.5 = For V = 270 x 1.467 = 396 ft sec , calculate Re = 396 x 8.4 x 8.4 . 6380 21 x 106. For a 23012 airfoil at Re 106 = 6 , read in Fig . A5 : 3a cd min = 0.006 . To extrapolate to higher values of Re , consider Fig . A5 : 8 , which shows a slight reduction up to Re 100 = 9 , and consider also Fig . A5 : 9 showing a slight increase in cd min from Re 1069 up to Re 106 = 25 . These 6  series airfoil data may not be applicable as they refer to bottoms of " buckets . " Plotting this point on a skin  friction graph such
ft
/
/
/
/
/
/
/
as Fig . 14:14 and extrapolating from 6 to 20 million , gives a net reduc tion of about 10 per cent or CD min = 0.0054 . Insert this value as the proper drag coefficient in the above table and calculate A fwing and
= 0.0054
insert this value in the table
x
551
=
3.0
.
/
Fuselage . For a fuselage length of 50 ft at 396 ft sec , calculate = 396 x 50 x 6380≈ 1.2 x 108. From streamline body data in Fig . 14:14 , estimate CD = 0.05 , allowing about 20 per cent for cockpit enclo sure , another 20 per cent for interferences , and another 20 per cent for protuberances such as antenna mast , etc. A value of CD = 0.08 is judged
Rel
to be reasonable .
With
this
Affus and
value of
= 40
x
insert this value in the table
calculate
ACD
0.08
=
3.2 sq
ft
.
Nacelles . For the nacelles , refer to Figs . A6 : 15 through A6 : 17 and note that while the best cowlings for air  cooled engines with substantial cooling air flow show proper drag coefficients in the region 0.06 to 0.08 , is conservative to assume additional drag due to non  optimum intake and discharge of air resulting in CD = 0.10 . Using this value , calculate
it
Afnacelle and
= 0.10
insert this value in the table
x 35
=
3.5 sq
ft
.
Surfaces . Calculate Retail 6 x 106 and in Fig . A5 : 3 , assum section consisting of an 0009 airfoil , read cd min = 0.0055 and increase this about 10 per cent for interference with fuselage as discussed in text , and calculate ing
Tail
a
tail
Aftail and
insert this value in
= 200 x 0.006 = 1.2 sq
the table .
ft
DRAG ESTIMATES
1419
Adding the values of Af in the above table gives a total equivalent Using this value , calculate the = Af = 10.9 sq ft . area of = total parasite drag coefficient as CDf f/S 10.9 551 = 0.0196 . To get the induced drag , calculate the wing aspect ratio as
f
flat plate
/
A
=
/551
65.52
7.8
=
/
in Fig . 9:23 for tapered wings ew = 0.87 For the contribution of the fuselage to the
and calculate 1 ew = 1.15 . induced drag , read in Fig .
Read
14 : 2
for
[A( 1/ e )fus ]/ ST round fuselages without
/
This is
as
ances
0.75
0.75 x 40/550
=
0.055
value and can allow for effects of protuber variation with angle of attack and optim
maximum
as normal drag
well
wing incidence
=
nose engine , and calculate
▲ ( 1 e )fus =
a probable
fus / S )
.
Calculate
/
1.15
1 e
and
+
0.055
=
1.205
/
e = 1 1.205 := 0.83
/π Ae
Calculate
=
1
1
/π
x 7.8 x 0.83
lift
The desired equation relating the drag and
answer called
14 : 9.
AND
THRUST
coefficients is then
0.04902
CD = 0.0196
This is the
= 0.049
for
POWER REQUIRED FOR AIRPLANES
IN LEVEL FLIGHT
principal forces acting on an airplane in level flight are 14:16 to be the lift L , the weight W , the drag
the power plant thrust T ; there
D,
is also
usually
to balance
lift
on the
clude Ft . velocity
if
Ft necessary L is considered to be
entire airplane
flight
For level
,
it
follows that the
lift
L
= CLPSV2 =
2
The drag
and
lift
D
Fig .
14:16 . an airplane
Forces acting on
in level flight .
( 14:16 )
Clas
coefficient necessary for level flight CL
The
in
at constant
W=
It
can
.
in Fig .
Ft
a tail force
, but
shown
is
/
(14:17 )
= W Sq
coefficients are usually related
by an equation of
the form CD = k1
as shown earlier
in
the chapter , and
+
(14:18 )
k₂CL2
this equation
is
true at
all
speeds
TECHNICAL AERODYNAMICS
1420
if
altitudes
and
compressibility
of the airplane
The drag
effects are neglected . calculated
may be
from


D = DW = CDW
Since power force x velocity and required for level flight is
hpr
1 hp = 550
= DV
ft  lb / sec , the horsepower
= D mph
( 14:20 )
375
550
calculation of the
An example
( 14:19 )
power required
for level flight at
sea
level follows : Example . For a Lockheed Lodestar airplane , assume S = 551 ft2 and that and drag coefficients are related by the equation CD = 0.0196 + 0.04901 , as found in Art . 14 : 8 , and calculate the power required for level flight at sea level ( ) with a gross weight of 17,500 lbs at speeds (mph ) of 300 , 250 , 200 , 150 , 125 , 100 , and at stalling speed ( flaps up ) , assuming CL max = 1.55 for a power  on stall with flaps up . Plot hpro vs.
lift
the
mph .
Solution .
Calculate
/
W S = 17.500
/ 551
=
31.7 lbs per sq
ft
Using q = 0.00256 ( mph ) 2 at sea level and equations ( 14:17 ) through the power required for level flight at sea level is calculated 14 : 2 and plotted in Fig . 14:17 . TABLE 14 : 2 .
CALCULATION OF POWER REQUIRED OF A LOCKHEED LODESTAR q = 0.00256 (mph ) 2 , CT CL = 31.7 lb per sq
mph
ft
300 250 200 150 125
160
102.4
57.8 40.0 25.6 20.4
100
89.1 min
1.55
max
AT SEA LEVEL
AIRPLANE =
CDi 0.049C12
0.138 0.198 0.310 0.548 0.792 1.24
230
FOR LEVEL FLIGHT
( 14:20 ) ,
in Table
0.0009 0.0019 0.0047 0.0147 0.0307 0.0752 0.118
CD =
0.0196 + CDi 0.0205 0.0215 0.0243 0.0343 0.0503 0.0948 0.138
D= WCD
CL
lb
hPro
D mph
375
1
2,080
2,600 1,900 1,375
1,268 733 438 370 358 370
1,095 1,110 1,340 1,560
When the horsepower available from the engine and propeller is plotted on the same sheet , the sea  level performance of the airplane can be de termined . Methods of calculating the power available are discussed and illustrated in Chapters 15 and 16 .
Equations (14:16
)
or altitude
conditions
(14:16 ) .
the
If
( 14:20 ) are applicable
through
same
if
the proper
speeds
air
density
are assumed for the
to either
is
used
sea
in
level
equation
calculation of
power
required at sea level , new values of CL and Cp and hp , may be calculated using the proper air density in equation ( 14:16 ) for the specified alti
,
DRAG ESTIMATES
It is
tude .
CL
convenient , however
much more
and CD as were found
sea  level
for the
1421
,
to
assume
the
calculation .
values of
same
It is
then neces
sary to calculate only the values of mph and hpy for the altitude condi The equations for calculation of power required at altitude are then , for constant CL :
tions .
alt
k₂c²
mphsL √1
=
)
14:21
CL
(
)
14:22
hPr SL 11/0
14:23
)
hpr
/0
k1 +
(
=
mphalt
 CDS CD
(
Dalt = DSL
The calculation of power required at sea level in Table 14 : 2 is continued in Table 14 : 3 for the speed and power at 10,000 ft standard altitude , at
the
same values of CL as in Table 14 : 2 , and the results from both figures plotted are in Fig . 14:18 . Fig . 14:18 differs from Fig . 14:17 only in that 2500 2000
2000
hpa. Chap .
1500
f
1500
leve
1000
1000 900 800
Sea
hpr
700 600 500
500
400
Stall 100
300L 70 80 90 100 125 150 200 250 300350 mph
300
200 mph
Replot of Fig 14:17 Power required for level Fig 14:18 Lockheed Lode with logarithmic scales and with airplane as calculated in Table power required at 10,000 tude added from Table 14 .
.
.
3
:
Note
A3
: 1 ,
,
chart
.
.
air
standard altitude the or from the standard
10
/
0.7384 from which calculate Table 14:18 that each sea level point plotted is one may
=
σ
is
Fig
√
in
A3
ft
At 10,000

.
read from the
Table
used
=
,
table
1.164
scales have been
ratio
: 2 ,
density
.
.
logarithmic
air
ft alti
.
:
2
14
.
14:17
flight at sea level for star
.
ft
10,000
hpro
Fig
16
10,000
moved
a
.
to the right and up the same distance on the logarithmic chart The equality of distances over and up is peculiar feature of logarithmic
TECHNICAL AERODYNAMICS
1422
scales :
a given
TABLE 14 : 3 .
is
ratio
represented by a given distance on the scale
CALCULATION OF POWER REQUIRED FOR LEVEL FLIGHT AT 10,000 ALTITUDE FOR A LOCKHEED LODESTAR AIRPLANE
ft ,
10,000 Sea Level , from Table 14 : 2 σ mph
CL
hpro
300
0.138 0.198
2,080 1,268
250
0.310 0.548
200 150 125
0.792 1.24
100
1.55
89
FT
σ = 0.738 ,
= 1.164
hpr
mph
2,420
349
1,475
291
733 438 370 358 370
.
233
854
174.7 145.5
431
510
116.4 103.6
416 431
Fig . 14:18 also shows a typical graph of power available from cating engine and propeller at 10,000 ft altitude , as developed
recipro in Chap
of graphs of power required and power calculating available on the same chart permit the performance of the plane in level and nearly level flight at various altitudes , as explained in Chapter 18 . ters
15
and
14:10 . PLANES .
16.
The combination
GENERAL
Fig .
CHARTS OF THRUST
plotted as ratios to
trary standard of maximum L/D
for
an
( 14:18 )
in
14:18 can be put
commonly
selected
(minimum
D
/ L) .
is
condition
of
of this minimum
CL
/
CD CL
CL₁ and
for level flight at
maximum
L/D
given
/
+
The
may be
are
arbi found
by equation
( 14:24 )
K₂CL
with respect to CL =
0 and
D L ( or maximum L/ D ) by ( ) 1 ,
k₂CL1² so that
of
characteristics are
H
Setting the derivative
speed and power .
AIR
first
CL
ing the that
if
the speed and power
the speed and power at the condition
The condition
calculate
follows :

AND POWER REQUIRED FOR LOW SPEED
general form
arbitrary standard
some
airplane whose aerodynamic as
air
/
=√k1
/ k2 ;
it is
found
( 14:25)
= k1
CD1
designat
= 2k1
a given gross weight
W,
(14:26)
DRAG ESTIMATES
CLI CL
14:27
is ,
The ratio of the drag to the drag at the reference condition
coefficients
D
CDV2
=
D1
14:28
CD1V12
which
ratio )
14:31
(
1
)
(
)² speed
V1
in Figs
)
(
)
+
~ +
1
2
= 2
(
14:30
V
14:19 and 14:20 and
and power required
for low
speed

constitute general charts of thrust
(1
)²1
+
÷

of the
are plotted
ratio as
2
V1
.
14:31
14:29
the speed
terms
V3
=
+
in
+
D1 V1
)
and
(
)
(
14:30
V
= D
2
2
/
P1 Equations
1
can be calculated
of
terms
=
)=
(1 ·+
2
2
ratio
√ )
11/1 Vi

23/12
and the power
v2
=
k2
in
so that the drag ratio can be calculated
D1
1
CL2
+
24
=
V /
k₂cz
2k1
/
=
k1
k₂CL2
1
+
k1 + k₂CL12
CD1
2
 k1
CD
k1 +
in
from the
,
)
drag
(
definition of
)
=
(
V V1
1423
air
.
planes
For any particular airplane
these charts
5.0
specific
charts
6.0
4.0
5.0 D
4.0
D1
3.0
3.0 ala
2.5
2.0
2.0 1.8 1.6
1.5
1.4
1.0 0.9 0.8 0.6 0.70.80.91.0
3.0
2.5 3.0
Fig
General power required 14:20 chart plotted from equation 14:31
) .
) .
(
.
.
,
chart
2.0
(
General thrust required plotted from equation 14:30
14:19
1.5
.
Fig
1.5
.
.5
2.0
,
1.0
>>
SA
1.2 1.0
may be made
TECHNICAL AERODYNAMICS
1424
by
calculating specific values of
going equations .
V1 ,
to these charts and plained later . An example of the application of Fig . airplane follows : be added
Example .
Given :
CD = 0.0196
fore
In
V1 , D1 , tude .
equation
and P1 , at sea
/
and W S = 17.500 / 551 = 31.7
level
and at 10,000
ft
alti
from the given data ,
( 14:24 ) ,
k₁ k1
14:20 to the Lockheed Lodestar
0.049C12
lb/ft2
Find :
Solution .
as given by the
D1 , and P1 ,
available from the power plant can also used for performance calculations as ex
Power or thrust
k₂
= 0.0196 ,
= 0.049
From equation ( 14:26 ) calculate CL1
/
Vk kz = 10.0196 / 0.049
=
CD1 = 2k1 = 2
definition of
From the
91
For standard V₁ or at
10,000 V1
alt
sea level =
√91
ft =
=
/
/
W S CL1 = 31.7 0.632 = 50.1
air
/ ( p /2)
/
with
SL V170
,
calculate
/ 0.00119
D1V1
,
with
= CD1 W = 0.0392
0.032
for
calculate
sea
= 205
ft / sec
/
= 239
ft/ sec
17,500 = 1,085
level
( 140 mph )
= 0.738 σ = p Po
x 1.164
= 205
CL1
calculate
p 2 = 0.00119 ,
= 150.1
V1
D1
0.0196 = 0.0392
CL calculate
standard altitude
For any altitude
Since P1
=
x
= 0.632
and 10,000
/
= 1 1.164
( 163 mph )
lb
ft
standard altitude
ft lb/ sec = 403 hp 259,000 ft lb / sec = 470 hp
P1 SL = 1,085 x 205 = 222,000 P1
alt
=
1,085 x 239
=
To check Fig . 14 ; 18 against Fig . 14:20 , read a point on Fig . 14:18 , such = altitude . Calculate for this point as mph 349 , hpr = 2,420 , at 10,000
ft
V = 349 = 2.14 V₁ 163 Above
this value of
/
V V₁
in Fig . P
= 5.1 x 470 = 2,390 hp
This value checks the value 2,420 hp calculated
in Fig .
14:18 .
/
= 5.1 , and calculate 14:20 , read P P1
in Table
14 : 3 and
plotted
DRAG ESTIMATES
.
14:11
DRAG ESTIMATES
FOR
1425
Inhabited
SUPERSONIC VEHICLES .
vehicles
( supersonic airplanes
) are limited by the " thermal barrier " to M = 1.5 to troposphere 2 in the and lower stratosphere from considerations of cockpit
refrigeration
lent
( tr = 215 ° F in the lower stratosphere at M = 2 with turbu layer ) and from considerations of hot  strength of transpar
boundary
suitable for windshields
ent materials
siles )
limited in
are likewise
flight
speed of prolonged
mis in the
( guided
Uninhabited vehicles
.
to
M = 3
troposphere and lower stratosphere from considerations of hot  strength of
available structural materials ( tr = stratosphere at M = 3 with turbulent
flights
at
M = 4 and
1030
°F , or
" red  hot , "
boundary
layer
lift of some
wings
in
the
lower
though
short  time higher are possible without structural disintegra ) ,
tion . Data
for estimating the
able for supersonic timating the drag of
vehicles
have been
drag and
flight
have been presented
in Art .
data for es suitable for supersonic
bodies (cone + cylinder ) given earlier in this chapter .
some
wing + body combinations can
, however , NOT be
suit
(rectangular )
The
10 : 4 ;
lift
and drag
of
estimated with useful accur
ef
acy from knowledge about the separate components , as the interference fects for typical supersonic configurations , unlike subsonic , are likely to be larger than the separate effects . Bonney (1 ) portrays this graphi
cally ,
in Fig . 14:21 . The wings are usually not greatly affected but their effect on the body is often large .
as shown
XXX
by the body ,
missile
have enough
lift
cit
at
weight without wings but this in Fig 14:22 and the resultant
Engineering Supersonic Aerodynamics .
.op
,
can
characteristics .
ahead of the usual center of gravity
"
)
is
E. Arthur . A.
Bonney
,
)
(
2
(1 )BBonney , Hill , 1950 .
(
to the axis
E.
normal
N
,
angle of attack to support its mostly on the nose as sketched
body
is
body with wing
not
From
small
lift is force
location
,
sum of the separate
a
,
speeds
a
At supersonic
characteristics of
aerodynamic
. "
of
,
Sum
even approximately equal to the
,
14:21 .
.
Fig .
McGraw
TECHNICAL AERODYNAMICS
1426 a wingless
and such
missile
attack unless stabilized
not maintain a constant angle of
rifle
in
shells )
and gun
(as in Fig . 14:23 ) . Resultant normal force
=
fins
N
or
will
body
by spin ( gyroscopic , as
Noody N
tail
80
Net pressures normal to axis
=
Wind
Wind
Axial force
W
Forces on wingless
.
14:23
fin stabilized missile
usual c.g. location
.
.
on
when of semi vertex angle eg to the inclined at an angle
Behind the cone usual junction the net normal ,
.
as
/
3
pressure drops to
a
deg
tity in
M
.01
vertex angle
,

Semi
calibers
so that the
is not far from the centroid of the platform area
center of pressure
deg
20
15
of the nose
.
5
10
few
negligible quan ,
α
q8
cylinder
a
d
dCN
=
N
2
.
1.5
.03
,
,
wind
1.0
.02
)
may be estimated from the re sults of theoretical calculations shown in Fig 14:24 where CN =
83
,
T
t..
a (d
,

nose
.05 .04
conical eg ,
N
The normal force
cap
body
flight
able of steady level
a
tail for
.
.
.
out
Fig
Sketch showing unbal on missile body with
14:22
anced forces

Fig
W
Triangular fins are usually pre
more
movable
,
"
all
because add
,
(
)
1 :
must be
A
)
.
.
480483
information
Inclined
Body
of this sort Revolution
, "
little
of
very
an
,
(
"
but
Supersonic Flow Over
pp
.
,
1938
"
tunnel tests S.
H.
,
Tsien October
( 1 )
fins
they
in the
large flaps are very ineffective at supersonic speeds interference lift data is currently 1955 being compiled from
supersonic wind
JAS
Guidance
given area
as
7a
edge
"
body of
"
"
trailing
to the body
.
lift
interference
a
for
Fig
.
inical
ferred over rectangular Nike missile
"
.
Tsien
as
Normal force coefficient cal noses as calculated
,
co
(
for
,
14:24
by
slope
.
.
Fig
1427
DRAG ESTIMATES
has been published
It ents
.
customary to analyse
has become
, one independent
which
is called
lift
of
two compon
) and one nearly proportional
(CD
CDi by analogy .
missile force data into
Hence , for a
it is
missile
write
to
customary
this
analysis
cients are largely
is
not very useful because
for
body
not yet
stability
wing +
fin
been formulated
, but
of such missiles
flight test
lift ,
students
if it is
The
coefficients are not ,
drag , and pitching
may
moment
supersonic
and
wind
 tunnel
between wind
 tunnel
data will be due to the large difference temperature
have
the performance
calculate
assumed that
coefficients
M and dimensions
principal discrepancy
conditions as the missile surface path ( see Chapter 7 for details ) .
to
while the subsonic coeffi
combinations as a function of
test data are available . and
,
independent of M , the supersonic
Simple rules for estimating
,
(14:32 )
CD = CDo + CDi
but
CL2
in
boundary
changes along
layer
its flight
PROBLEMS 14 : 1 . For a particular wing  fuselage combination , the drag added by the fuselage per sq ft of fuselage frontal area at 100 mph at sea level is 10 lb. Find ACD for the fuselage . 14 : 2 . For a streamline body of revolution of / d = 5 , read CD in Fig . A6 : 15 and compare with general data of Fig . 14 : 6 .
l
Using Figs . A6 : 34 through A6 : 37 , estimate CD at M = 1.5 and cylinder missile body without boat  tail for flight Tr for a cone in standard sea level air . Cylinder diameter = 10 in . , cone , length = 25 in . , cylinder length = 25 in . Assume transition at Re = 106 . 14 :4 . A " streamlined " wire of nominal 1/4 in . size ( diameter of threaded end ) has a cross  section 0.087 in . wide and 0.348 in . long . Using the drag data on elliptic cylinders ( Fig . A6 : 2 ) , estimate the drag per ft of length of this wire at 100 mph in standard sea  level air . 14 : 5 . For the Ercoupe airplane sketched in Fig . 1 : 3 , assume the data given below . ( a ) Estimate the equivalent parasite flat plate , and ( b ) write the equation for CD vs. CL2 . Use mph = 120 to calculate Re for the high  speed wing drag estimate . Tw
14 : 3 . =
Part section , b = 30 ft · Fuselage , length 20 ft 9 in . Tail surfaces • Nose wheel ( low pressure ) and support Main wheels and supports . ·
Wing , 4412
Total .
Sπ , sq
ft
Af, ACDπ
142.6 12
30
0.60 1.50
0.12 0.006 0.6 0.4
sq
ft
TECHNICAL AERODYNAMICS
1428
For the Lockheed Constellation airplane , a photograph of which 1 : 4d , assume the data given below . ( a ) Estimate the equivalent parasite flat plate , and ( b ) write the equation for CD vs. CL² . Use mph = 350 to calculate Re for the high  speed wing  drag estimate .
is
14 : 6 . shown
in Fig .
effective section 23015 , span 123 ft Fuselage , length 95 ft . Nacelles , total for four , with cooling air flow Tail surfaces ·
Wing , mean
Total
tion
ft
1,650 100 70 700
ACDП
sq
ft
0.070
0.080 0.006
•
14 : 7 . 14 : 5 ,
Af,
Sπ ,
sq
Part
For an Ercoupe airplane assume that the drag and
for which drag was estimated in problem coefficients are related by the equa
lift
CD = 0.030
0.066012
the power required for level flight at sea level with a gross weight of 1,260 lb at speeds ( mph ) of 120 , 100 , 80 , 70 , 60 , 50 , and minimum speed , assuming CL max = 1.50 at minimum speed , and plot hpr vs. and calculate
in Fig .
mph as 14 : 8 .
14:17 . a Lockheed Constellation airplane , Fig . 14 : 6 , assume that the drag and
For
for
lift
timated in lated by the equation
CD = 0.0154
which
drag was es
coefficients are
re
+ 0.0426C12
calculate the power required for level flight at sea level with a gross weight of 86,250 lb at speeds ( mph ) of 400 , 350 , 300 , 250 , 200 , 150 , = 1.60 at 125 , 100 , and minimum speed ( flaps retracted ) , assuming CL max flaps plot Fig hpr speed , mph , minimum ( retracted ) and vs. as in . 14:17 . 14 : 9 . For the Ercoupe airplane , calculate the speed and power quired for level flight at 12,000 ft standard altitude at the same values of CL as in problem 14 : 7 and plot on logarithmic ruled graph paper . Check and
re
by
Fig .
14:20 . 14:10 . For the and power required
Lockheed
Constellation
airplane
,
calculate
the speed
for level flight at 20,000 ft standard altitude at the same values of CL as in problem 14 : 8 , and plot on logarithmic ruled graph paper . Check by Fig . 14:20 .
CHAPTER
15
AERONAUTICAL POWER PLANTS
15 : 1 .
siles )
POWER PLANT TYPES .
are
to
combustion
rearward
titative terms
the
( airplanes ,
aircraft
energy of discharge of
mechanical
helicopters
plants which convert chemical
air
energy
/
of
gases
It is
forward .
summary of typical quan the student or engineer can
chapter to present a brief
of this
on such power plants
data
mis
,
and or exhaust
generating a thrust which propels the aircraft
,
the purpose make
Most
propelled by power
so
that
rapid and reasonably accurate estimates of the propulsive thrust in of fuel consumption , as a basis for estimating the performance of
aircraft . Typical
15 : 1 ,
and
power plants
aircraft some
sketches of Fig . as the British
are
shown
in the
photographs
of their engineering characteristics are The engine
15 : 2 .
more
 driven
screw propeller
descriptively characterize
it , is
of Fig .
shown
in the
or " airscrew , "
the most efficient
of aircraft propulsion yet devised for flight speeds under 400 mph . The efficiency of screw propellers is considered in Chapter 16 in some detail . At speeds around 450 mph , with the usual optimum propeller blade settings in the region of 450 , the Mach number along the helical blade
means
unity
path approaches
Under these circumstances ,
near the propeller efficiency . propulsive losses of thrust and propulsion by means of high  speed jets gener shock waves begin to
and
blade tips , with substantial
ated by internal combustion more
efficient
.
At
still
turbines
For flight at very high
optimum ramjet be by rockets as
designs (Fig .
( " turbojets , " Fig .
15 : 1c ) become
higher speeds the turbines are unnecessary
the impact or " ram " pressure of the air plant operation and the " ramjet " (Fig .
ical .
form
become
is sufficient for
, as
power
economical
reasonably econom 15 : 1d ) becomes altitudes where air density is very low ,
prohibitively large
15 : 1e ) , which
,
carry along their
and own
propulsion
oxidizer
must
as well
fuel . For these various types of power plants the thrust
propulsive of speed
efficiency
in Fig .
,
15 : 2 .
efficiency
all efficiency are summarized as functions civil aircraft in use or under construction
and over Most
, thermal
151
,
TECHNICAL AERODYNAMICS
152
a.
Gasoline engines
, unsupercharged supercharged .
b.
Turbine
and
propeller ( turboprop ) .
c. Turbojet ( Pratt and Whitney
J57.
40 Ramjet ( Marquardt ) . d. ( Courtesy Aviation Week ,
April
HYDROGENPEROXIDELINE
e.
Fig .
STEAMTURBINE PUMP
18 ,
STEAM EXHAUST
1953 .
ROCKET MOTOR
Rocket ( liquid propellent , using gasoline and H202) . 15 : 1 . Current common types of aircraft power plants . ( See Fig . 15 : 2 for characteristics . )
AERONAUTICAL POWER
Accelerating Thermal force Thrust efficiency ( T) (n₁₂)
mph
mph
0
.8
4000
.4
.1
mph
AL 500
500
8
2
Prop
° F
1
.2
Jet
.1 mph
moh 1000
1000
mph
500
500
8 1500
F
O
.2
Turbojet
mph
500
.3
mph
Jet
mph
500
.1
°
moh
500
.2
mph
500
1500
Prop
Turboprop
(16)
mph
500
0
500
( p)
L
propeller
efficiency
0
.1
Overall
efficiency
L
engine and
Propulsive
0
Piston
.3 .2
1
Prop
2
D
° F
m
153
O
Mass per second
PLANTS
.4
.2
moh
moh 1000
1000
mph 1000
12
Ik .2
mph
Rocket
possible
placement of
internal
O
0
2000
engines
that the next few decades engines
are
will
by turbines
driving
.
)
screw pro
under construction
,
combustion
propelled airplanes
combustion
mph
types of aircraft power plants Westinghouse Engineer March 1945.
internal "
it is
by
turboprop
"
.
and
few
2000
, "
: 2 .
1955 are propelled
mph
2000
common "
0
Characteristics of Presentation suggested in
15
A
in
2000
.8
5000
mph
2000
(
.
mph
2000
0
mph
pellers
mph
.2
0
Fig
L
2000
k
.5
mph
2000
0
mph 2000
.2
°F
2000
0
mph
0
1
Ramjet
0
8
.8
see the gradual re
for propeller drives
TECHNICAL AERODYNAMICS
154
under
of the inherent lightness
mph because
400
simplicity of the
and
turbine compared with the internal combustion engine (wt /hp for turbo props is about half that of piston engines ) . Such developments , however , take many years , but most
the gas turbine
experts agree that there
is
where weight
The supersonic missile field in . always be dominated by turbojets , ramjets , and rockets ;
course ,
transonic military
field of
the
for
a good future
airplanes
at even more of a premium than
will , of
is
particularly in the field of helicopters
,
aircraft , is
speed commercial
aircraft
expected
and , to a
to be
lesser extent , high by the
dominated
turbojet
power plant . 15 : 2 .
PISTON ENGINES .
first
The
power driven airplanes
were powered
by steam engines . Consideration has also been given to the powering of large airplanes by steam turbines from steam generated by nuclear reac
tor boilers
.
The
principal
steam
ternal
engine of equal
combustion
turbines
are
to be used for
to
condensers
to the use of steam or other
handicap
fluid in an engine is that turbine is usually about ten
mediate
inter
the "water rate " of a steam engine times
power .
airplane
the fuel consumption Thus ,
if
of the
steam engines
or
in
or steam
power , they must be equipped
with
for recirculation of the water in order to have Suitable steam condensing equipment has thus far proved
provide
reasonable range
.
excessively
it is quite possible that wing surface condens at a reasonable weight similar to the wing surface radi
ers can be
heavy , though
built
ators used with
liquid cooled gasoline
some
Internal
Intake manifolds ThrottleCarbureter Float E

Gasoline
15
: 3 .
a .
Fig of
Air
Stroke jacket Water
Cylinder head Intake valve ... port Intake
k
Exhaust valve Exhaust port Spark plug Cylinder Piston rings Piston pin
Bore Piston
ConnectCrankcase ingrod Crank shaft Crank Crankpin
Principal
elements
(
gasoline engine From Chatfield and Taylor The plane and Its Engine the study
Fig .
15 : 3
.
. "
)
,
"
.
Air
which burn
inders
engines . combustion
air
and
" piston " engines ,
and gasoline
drive
a
in
metal
of pistons and connecting rods ,
by means
are the principal current type of plane power plants . The principal ments
of
cyl
rotating crankshaft
one
combustion
cylinder of engine
are
air ele
internal Fig in . 15 : 3 .
such an
shown
While the construction of such engines assumed
to
be
familiar to
is
most students
it
of Technical Aerodynamics , is consid ered worthwhile to review here the basic operating principles
in
connection with
of the characteristic limitations of such engines . Note in that the cylinder surrounds a piston which is connected by
155
AERONAUTICAL POWER PLANTS
of
means
peller
.
a connecting rod to a crankshaft
is
There
also a carburetor
air ,
and a manifold that conducts
also
a system of valves
in
the proper time open
manifold to form a combustion
burning .
pipe after
starting the
The valves
.
to permit the products of combustion
exhaust
combustion
in
is
There
the
the fuel in There
the mixture into the cylinder
the stroke of the piston
the cylinder from the
atomize
the mixture to the cylinder .
for admitting
pro
that delivers power to the
with jets that
,
first
at
close
chamber , and
off
later
discharged through
to be
is
an
also a timed ignition system for
cylinder .
Measurements of the pressure inside of one of the cylinders while the engine is running yield an indicator diagram similar to that shown in
Fig .
of
15 : 4 shows
operation consists
is
the mixture
approximately adiabatical
ly
and at the end of which
the mixture
is
ignited
a
rise in
and burns ,
with
ucts of combustion piston
on the
;
bustion flow out
;
(3 )
prod
the
expand and do work
(4 )
an exhaust
four  stroke cycle and engines .
atmosphere .
is
Fig
15 one engine
Bottom dead center
for
Indicator diagram cylinder of an airplane
stroke during which the
of the exhaust
discharged into the
Power stroke Compression stroke 15 stroke Exhaust s troke Intake O 4 0 Top dead center PistonPositionIn
resulting
pressure and temperature
a power stroke during which
1000
100
, .
compressed
( 2 ) a compression
;
which
4,400
.
stroke during
500 .I.n8 400 , 300 200
.
into the cylinder
600
Lb.per Sq Pressure $
of four strokes of ( 1 ) an intake the piston as follows : during stroke which the mixture flows
cycle of
that the
°F
diagram
°F
indicator
The
.
15 : 4 .
: 4
Fig .
valve and through
an
products of com exhaust pipe to be
is known as the in most airplane
The cycle of operation
the cycle
of
operation used
A lighter engine
can be built , with some sacrifice in fuel economy , slight positive if a intake pressure is maintained by means of a blower and the exhaust and intake occur simultaneously at the end of the power
stroke cycle
,
.
in
which case
Such engines
the cycle of operation are
" outboard" motorboat engines . They have also gliders , and some small helicopters . The
at the center "
ratio of the
is
cheap and simple and
volume contained
known
as the two  stroke
are most widely
been used
in the cylinder
for target when
" bottom dead center " to the volume when the piston
is
known
as
the compression
ratio
.
With a
is
known
as
drones ,
the piston
is
at " top dead sufficiently high
TECHNICAL AERODYNAMICS
156
ratio , the isentropic rise in temperature on sufficient to ignite the fuel without an ignition system .
compression
in this
operates
occasionally
is
manner
fuel
where
that
engine ; such engines have
known as a Diesel
aircraft
on
been used
is
compression An engine
economy
is
a primary con
sideration . indicator
The net area of an
is
diagram
a measure of the work per cycle done
is
what
delivered by the crankshaft
,
is
( Bhp ) ,
known as the mechanical
efficiency
the torque
( Ihp )
15 : 4
and determines
of the engine
The power
.
sometimes measured by a brake and therefore
called brake horsepower power . The ratio of the
If Q is
inside the cylinder
horsepower
as the indicated
known
in Fig .
such as that shown
always
than the indicated
less
brake horsepower
horse
is
horsepower
to the indicated
.
is
delivered to the crankshaft and rpm rpm , then
the rate of
3500
Characteristics of typ for light airplanes
engine Note
in Fig
15
: 5
a
)
.
a
the mixture flowing
through
manifold and valves
The
is
being
that
if
somewhat
termined
arbitrary
primarily
by
the
rating
de
considera
tions of durability of the engine
propeller the re rpmis determined by the propeller
the engine
drives
between propeller power and for fixed pitch propeller the horsepower varies approximately as rpm3 Specific fuel consumption based on brake horsepower for propeller load Ratings of many and full throttle conditions are also shown in Fig 15 airplane engines currently manufactured in the United States are shown in page A71
.
Appendix
7,
: 5 .
.
.
a
;
lationship
to increased friction power loss of pressure in
and increased
2000 2500
rpm
.
.
parts
1500
due
a
5 .
:
15
ical
are
,
Full
1000
engines
.
0
throttle
800
proportional
,
8
bhp
20
is
brake horse power and rpm where the torque has not yet begun to drop off with rpm
throttle
,
.
usually rated at
load
x
airplane
full throttle
Airplane
a
50
:
(
A
=

.
.
full
small
that the
rpm
rpm shown
Propeller BSFC
Approximate Propeller fixed
Bhp also BSFC 100 lb.fuel perBhp
15
This corresponds to constant torque over the range of
BSFC
60
.
with rpm for
in Fig
to the
RatedBhp and rpm
70
Fig
15
brake horsepower
100
hr 80
,
Note
rpm
5,250
horsepower
.
.
rpm 2π 33,000
: 5 .
typical variation of brake is shown in Fig 15
engine
: 5
A
Bhp
1
rotation of the crankshaft in
15 : 3 .
SEA
 LEVEL
built lighter for
is
cylinders
SUPERCHARGERS
such as that
shown
in Fig .
15 : 6 ,
are
supercharged
percharger
only
.
horsepower
into the
"Scroll"
Diffuservanes Air fuel mixture tomanifolds
Geardrive
(Powerfromshaft engine ) "Scroll "orcollector
gases for compressed
Cutaway view of gear driven centrifugal supercharger .
15 : 6 .
but
70
about 300 horsepower
Fig
be
admitted to the cylinders .
Rotatorvane and impeller
The supercharger would add ,
air
Fuel injector nozzle
su
about 700 horsepower .
about 370 horsepower would require about
the
usually
can
flowing
blower , or " supercharger , "
it is
before
deliver
might
a rotary
of
engine
Fuelline from carburetor
15 : 7 .
Such an engine without
if
power
Carburetor Air inlet
engine
in Fig .
shown
airplane
An
by means
The relationship between indicated and brake horse power for a typical sea
level
.
a given delivered
compressed
157
POWER PLANTS
AERONAUTICAL
to run
it ,
resulting in
with only a slight increase
in
a net gain of
weight
(necessitated
by the requirement for designing the engine to withstand the extra stresses and higher
signed
resulting from
temperatures
for use
on
the higher pressures ) .
land and water vehicles
less important , rarely
,
in
which weight
which is
ButnotALLof thispoweris available for "outside work" ..... Some ofitisused toovercome friction engine within the . andsome ofitisrequired to things pumps , drive such asfuel oilpumpsmagnetos . on endasexplained , some poge ofit 18 isused torunthe SUPERCHARGER
BRAKE HORSEPOWER
ical
15 : 7 .
Engines
is what'sleft for drivingthepropeller
1150 H P.
minus
15.0 H.P ●quals
1000 H.P
Relationship
horsepower Motors Corporation . ) 1000
de
considerably
use sea  level superchargers or " ground boosting . "
INDICATED HORSEPOWER power developed means within thecylinders
Fig .
is
between indicated and brake horsepower sea level  supercharged engine . (Courtesy
for
typ
General
Without supercharger , or with a sea  level supercharger , the indicated
is
horsepower which
the
density of the air in flying and hence drops off markedly with increasing
approximately proportional
engine
is
to the
altitude . Since the friction and blower power does not drop off in pro portion to the air density , the reduction in power with altitude is accen tuated .
For an airplane
engine
in
which the
friction
torque
Qf
is
13
per
158
TECHNICAL AERODYNAMICS
)
15
15
)
2
is
and
a
)
: 8,
a
a
.
4 .
,
as shown
various
,

been used
altitude Fig in 15
power with
Note
: 9 .
7
have
ENGINES
SUPERCHARGED
In order to avoid excessive loss of
.
of blowers
arrangements
uni
shown
.
: 8 .
.
engine
also
:
15
variation of torque with altitude for sea level Typical
15
is
which
.5
with
resulting in uniform scale of altitude of
.

)
(
.6
plotted a,
(
)
non
Uniform scale
.8
.9
.
.
,
is
form scale σ
like
15
altitude relations for

many power
: 1
15
Fig
,
(
0/00
Equation
1.0
is available Fig
to the contrary
.5
in
that superchargers may operate in either one or two stages and either gear driven or driven by an exhaust turbine sometimes with inter cooler between stages or an after cooler after the second stage
Fig
.
in
Such
the
right
is

,
.
.
page
(
15:10
considered independent
in the left
given
hand graph
involving
in Fig
shown
hand graph and
al
ratings
Several sea level
"
performance
15:10
ratings
is
limitations of both manifold pressure
read from the chart are shown
in
Table
15
.
.
and rpm

.
are shown
such engines
"
level titude performance in Sea
A useful
the intake manifold
and manifold pressure are
rpm
,
)
variables
for
in
: 1
graph of performance
in which
power
can be set by The constant rpm
predetermined
.
to
.
governor

of
1510
,
form
a
operate at any throttle then determines the pressure of
means
level
sea
.
are
of 30,000

Such engines
superchargers
on the
altitudes ft or more ordinarily used with propellers that
up to
can be maintained
.

,


With suitable pressure regulators

.
:
9
15
may be
an
represented
assumed to represent the
unless information
engines
Fig
15
loss of power with altitude at given rpm e.g. in NACA TN 579
.6
.4
is
in Fig
graphically commonly
.8
0.13
: 8 ,
Equation

:

ft

(
.9
1.130
(
Qalt
Non uniform scale
variation
the
given by
)
1000
.
.
is
.
Alt
Std
is typical ) ,
( which
Q
20
15
10
5
1.0
torque
with altitude
torque
: 2
full  throttle
of
(
sea  level brake
cent of the
altitude chart is plotted on the following basis at constant rpm very nearly proportional horsepower and full throttle the brake is to the density ratio which is the uniform scale abscissa of the right hand chart ).
hence non uniform

(
altitude scale is
Accordingly
,
,

:
The
lines of constant
rpm
AERONAUTICAL POWER PLANTS
159
SUPERCHARGER TYPE 1ALTITUDE SINGLE STAGE WITH ONLY ONE SPEED ho Below 1000 High gear Air 4000 feet Carburetor Throttle ratio engine engine Critical altitude Sea level has Power from advantage TYPE I Manifold She Latitude Fuel mixture Versuspressure engine TYPE feet &000 Above toengine SEA Type Supercharger provides LEVEL Isubstantial Control Supercharger gain Altitude output 2 ALTITUDE SUPERCHARGER hp inpower TYPE 500 ; TWO engine SINGLE ,MECHANICAL STAGE SPEED CLUTCH Two speed drive Sea dutch mechanical AirThrottle level Carburetor engine engine from Power Manifold pressure mixture Fuel Sea control Supercharger level toengine 10,000 20,000 30,000 infeet Altitude above sea level SUPERCHARGER TYPE, 3 ALTITUDE SINGLE STAGE WITH VARIABLE SPEED CLUTCH speed Variable hydraulic Carburetor AirThrottle clutch 1000 ho engine Power from CriticalTYPE 4 Manifold Type Caltitude mixture Wet Fuel 3 TYPE Versuspressure 4 atas good 3 control Superchargertoengine TYPE lower as stage TYPE 4 No.3Auxiliary but altitudes inoperation TYPE 4 SUPERCHARGER bethrottled must TWO STAGE WITH MECHANICAL CLUTCH hp 500 Mechanical clutch Air engine Power from Auxiliary stage Supercharger toengine Micure Carburetor Engine stage supercharger 10,000 20,000 30.000 ALTITUDE Altitude infeet above sea level TYPE 6 SUPERCHARGER ,VARIABLE AFTERCOOLER WITH SPEED STAGE TWO speed Variable Critical jaltitude clutch (hydraulic hp 1000 Air auxiliary stage Since TYPE engine Power from 7 Auxiliary byexhaust , stage isdriven supercharger to Mixture the total increase in engine power gain Engine isanet stage Carburetor Aftercooler supercharger hp ALTITUDE 500 TYPE 7 SUPERCHARGER ,WITH TYPE7 EXHAUST DRIVEN INTERCOOLER TURBO byexhaust engine Versus Turbine run from TYPE 6 driving Turbo unit runs Turbo Sea auxiliary stage supercharger regulator level engine from Power 10,000 20,000 30,000 Auxiliary Mixture toengine Altitude infeet above sea level stage Air Engine stage Intercooler Supercharger

Fig .
I
Types of supercharger arrangement and their effect on engine 15 : 9 . power at altitude . ( Courtesy General Motors Corporation . ) TABLE 15 : 1 .
Point
on S. L.
throttle
Take  off Max . continuous
2,100
36.5
1,800
34 26
on the
will
pres
rpm
Cruising
are plotted
Man .
Rating
chart
KB
SEA  LEVEL RATINGS
graph as
sure ,
1,900
straight .
in
Hg .
Point
Bhp
alt .
760
on
chart
L
675 430
For each altitude
give a determinable manifold pressure
;
and
therefore
rpm , ,
full
lines of
constant full  throttle manifold pressure may be drawn and these are the arcs sloping diagonally upward from left to right . The other diagonal lines XY , LE , VW , and CA are graphical constructions to determine various power
fold
throttle limitations at other altitudes with pressure
.
For a given
increases with altitude because of reduced
tion should
be a
function
a
given
rpm and
mani
rpm and manifold pressure , brake horsepower exhaust
of pressure ratio
,
pressure and
interpola
but the density ratio scale
2100 rpm 1100
&
TO 1.
.
.. , , .C +7 ,A B.A .T: ) °
on on
to
of Ts
by
hpat hp
Ts +
2. 3. 4. By
[
off
Take
SR
000L
5000 4000
C
s $
horsepowerBrake
in
hp
pressuremanifoldRated
atD
%
10Ts
,.
airplane
0009
. .
1820F
53.
engine
000'11 0000
(
Wright
12,000 altitude
feet
13,000
supercharged
8000
Standard
14,000
,
No ram
2100rpm
performance
data
25,000 24,000 23,000 22,000 21,000 20,000 19,000 18,000 17,000 16,000 15,000
Typical
34
0 50
15:10
+42

Hg
Hg
S0006
Standard altitude temperature Ts
11,300
Absolute manifold pressure
in
..
150
1800 1700 1600 1500
50
200
N
P
300
D 51
250
R Normal horsepower 2100rpm
33
350
H
400
450
G
.,
F °
temp altitudeStandard
Fig
pressureand temperaturestandard
600 HORSEPOWER VS 550 MANIFOLDPRESSURE at SEA LEVEL PERFORMANCE 500
. . 650
..
Absolutemanifold pressure in Hg
,
222
. Y 32
11
700
TECHNICAL
B.
C D 750
:1
850
WRIGHT AERO ENGINE NORMAL PERFORMANCE Engine SR 1820F Propeller gear ratio 6.40 Compressionratio 8.31 Blower gear ratio 11 Impellerdiam ins Carburetion Fuel Date 53
800
HORSEPOWERAND MANIFOLD PRESSUREALTITUDEPERFORMANCE
:: 1
900
950
1000
1050
.
FINDACTUALHP GIVENDATAFOR 2000 ALTITUDERPM MAN.PRESSCARB.AIR TEMP Locate fullthrottlealt.curve for givenrpmandman pressure 1900 Locate sealevelcurvefor rpm andtransfer andmanpress Connect and straightline 1800 andread givenaltitude Modify forvariation carb air temp fromstd.alttemp 1700 formula 460 Actual Hp.at DxV460 1600 correction Approx.1 for Full throttle each F.variation horsepower from 1500
1510 AERODYNAMICS
.
)
POWER PLANTS
AERONAUTICAL
1511
if
use is restricted to standard altitudes , for which P / Po definitely and p Po are related and if corrections are made for departures from standard temperature . A graph of standard temperature is given on
/
facilitate
the chart to
such corrections , which are noted on the chart to
a function of √Tsta /T .
is
in Table 15 : 2 .
shown
is
15 : 2 , which
PART  THROTTLE
TABLE
for
The procedure
OPERATING LIMITATIONS
S.L. ratings pres sure
Max . continuous 1,900 Cruising 1,800
34 26
AT ALTITUDE .
Alt . ratings ( full throttle Man .
Man .
rpm
altitude interpolation
an extension of Table 15 : 1 .
(part throttle ) Ratings
such
pres sure
Bhp
rpm
675
1,000 1,800
443
Alt
hp
22243
be
. ,
can be used
ft
Point E A
8,000 14,000
750 550
26
)
If
.
it
?
delivering Solution
Hg
,
a
.
Example the above engine is cruising at 2,050 rpm and 29.2 in manifold pressure at standard altitude of 11,300 ft what power is
shaft
chart
.
TUVW on
given horsepower to be supplied to
combustion
gas
turbine can be
built
a
a
.
For
internal
an
,
:
ler
.
.
TURBOPROPS
.5
15
See points
Read the answer as 650 hp
propel
for approxi
etic
energy
,
ler
this
is
though
is left
(
AB
,
.
,
shown
tend to either are provided
in the sketch of a in Fig 15:11
.
substantial
amount
of
kin
after the gases pass through the turbine wheels jet which augments the thrust of the propel discharged in propulsive power due to the jet is usually the additional over
a
and
,
through reduction gears the propeller
,
and
controls
they
which drives the compressor A
CD
)
turbine
(
and
a
BC
)
chamber
(
.
.
15:11
combustion
typical turboprop power plant are compressor The principal parts are a
Parts of Fig
with minor in unless complicated electronic
they
) ,
,
,
or stop
speed
a
"
"
run away
piston engines
unlike
changes
;

are not self regulating
.
have an inherent disadvantage in that
,
light
,
.
,
.
a
a
mately half the weight of piston engine and with small fraction of the currently number of moving parts Such turbines are under intensive de velopment by power plant manufacturers Such power plants while very
,
.
h s
.
from
to
B ,
compressor
A
.
P
v
.
less than 10 per cent of that supplied by the propeller The principle of operation of the turboprop power plant is shown in diagrams of Fig 15:12 the and which is labeled to correspond 15:11 Air is compressed approximately isentropically in the to Fig where
fuel is injected
and combustion
takes
place
TECHNICAL AERODYNAMICS
Fuel line Fuel pump
Jet
D
Propeller
JUL
hub
chamber
and Whitney T57 turboprop power plant on manufacturer's photographs
,
of Pratt
Sketch
based
°
t≈ 1500
F
.
.
.
15:11
Turbine
Combustion
Compressor
Fig
nozzle
HI
0
0
0
0 0 0 0
Reduction gears
.
1512
Combustion
D
Intake Pressure
diagrams
for
gas turbine
power plant
.
and
h s
v
S
P
.
.
15:12
exh
and
V
Fig
Turbine
Combustion Pressure
.
B Compression
h
P
Turbine
Compressor
B
a
.
C
D ,
B
at approximately constant pressure between and C. The products of com bustion expand approximately isentropically in the turbine from to where they are discharged at nearly atmospheric pressure The efficiency cycle is limited by the maximum temperatures which are permis of such .
feasibility
of various turbine blade cooling
Efficient fuel air ratios are far
/
.
vices
now used
above the stoichometric
de
ideal
(
1500
temperatures

the
blade material and the
Maximum
limited by the hot strength of the turbine 
in
row of turbine blades
F,
are
first
region of
°
sible at the
POWER PLANTS
AERONAUTICAL
ratio
for
needed
all
burning of
complete
1513
in
oxygen
air ) .
the
Since the
exhaust gases have a great surplus of oxygen , they may be "burned
again "
by feeding
called
fuel in
more
an " afterburner . "
a tremendous
behind
is
increase
in
is
This arrangement
the turbines .
Such combustion
very inefficient
It is
thrust of the jet .
result in
but can
seldom used with tur
on military turbojets for short bursts of high or for getting through the " sonic barrier . " turbine engines , given in Table A7 : 2 , show that ( except
boprops , but often used
for
speed ,
take
off ,
Data on gas
for
small turboprop power plants being developed by Boeing ) the only United States turboprops now under development are the T56 by the Allison some
Division of
General
Corporation
Motors
Pratt
and the T34 and T57 of the
Aircraft
Typical 15:13
thrust
as under development
full  throttle is
in
England .
turboprop data are shown
a generalized
plot of shaft
in Figs
horsepower
. 15:13 and 15:14
( Shp )
.
gross jet
and
against rpm in terms of the rated values ( ) r , as analyzed by 15:14 is a similar plot of fuel and air flow rates . The
( Fg )
(1 ) Fig .
parameters
turbine
(rated at 5,500 and 15,000 shaft horsepower respec eight large turboprops are listed in recent aviation
At least
statistics
Durham .
Aircraft Division of United
Corporation
tively ) .
Fig .
at 3,750 shaft horsepower )
( rated
and Whitney
used
inlet
the analysis
in
by
Durham
(which
is
based on a
limiting
temperature ) are :
8t1 =
0 +1 =
Pti Tti
inlet total pressure standard sea  level pressure
(15 : 3 )
inlet total
( 15 : 4)
compressor
Po
= compressor
To
temperature
standard sea  level pressure
In
equation ( 15 : 3 ) the compressor inlet total pressure is to be calcu lated from the isentropic pressure rise ( designated by a prime ) and the
ram
efficiency
Пram
defined by Pt1 Пram =
/P  1
(15 : 5)
/Pa · 1
Pt1
/
where Pt1 Pa = ( 1 + 0.2M² ) 3.5 as
( )a
total
denotes
ambient
temperature
condition .
is to be
( 1 ) Durham , Franklin P. 1951 .
given
In
calculated
" Aircraft
in
Chapter
equation
( 15 : 4 )
5 and
subscript
the compressor
assuming Tt1 / Ta
Jet
the
Til / Ta
inlet
1 + 0.2M² .
Powerplants , " Prentice  Hall , Inc.
1514
TECHNICAL AERODYNAMICS
1.50
/
SHPr
1.0
SHP
1.00
10 t1
Pt1
8t1
.90
2.00
Pa
.80 HP
.70
,
Shaft
.60
1.40.8 1.2 =
ram
1.1
mph
0.8
=
400
.50
1.50
,n ram
1.0
1.00
mph
200
t
1.2 1.11
Jet
1.0
.30 .25 75
80
Fig .
15:13 .
Gross
Fg Fgr
Thrust
/
.40
811
/
( rpm /rpmr ) √0t1
85
90
.50 100
95
105
110
General plot of turboprop power and thrust data .
1.5
/
Wa War
/
St1 10t1 and
1.0
Pti t1 /Pa
/
.9
we Wfr
.8
St110t1
1.4
.7
1.2 1.0
.6
Air Fuel
1.2
/
rpm rpmr
14
60
Durham
90
flow data
)(
General plot of turboprop fuel and air 15:13 and 15:14 replotted from Durham
, F. P. , op .
110
100
.
igs
.
(1 ) .
15:14 .
(F
Fig .
Vet1 80
70
cit . ,
p . 81 .
)
.3
1
.4
.
1.0
/
.5
AERONAUTICAL POWER PLANTS
The use of Fig . 15:13 for calculating boprops
is
best illustrated
off rated
by the following
1515
items of tur
performance
example taken from Durham's
work . Example . Given a turboprop power plant , whose performance is given 1513 , which has a sea  level rated shaft horsepower and jet thrust of Shpr 3,000 and Fgr = 1,000 lb at rpmr = 9,000 . The rated air flow = = 0.6 lbm sec ( air fuel 50 lbm sec and the rated fuel flow war ratio = 83.3 lb air 1bm fuel ) . Find the shaft horsepower , gross thrust , and fuel and air flow rates , at 35,000 standard altitude at an air speed of 400 mph and at 8,500 rpm , the ram efficiency is 85 per cent . Solution . In Appendix 3 , at an altitude of 35,000 , read the am
by Fig .
/
is
/
fr
/
ft
if
ft
bient conditions Pa
= 498
lb/ft ?
Ta
= 394 °R
To get total pressure and temperature
Ma
Fig .
and from
/
Pt1
/Pa
and calculate
/
Tt1
0t1
Tt1 /Ta
=
(1
=
0.2
x
0.602
= 1.072
+ 0.2M2 ) 3.5 = 1.275
ram = 0.85
10.85
x 0.275
= 1.234
394 = 422 °R
x
= 422 =
0.815
15:13 ,
as calculated
/
519 rpm
t1 = Voti
Pti
rpm
1
rpmr
10 t1 Noti
=
8,500
=
8 +1
1
lb/ft2
498 = 615
=
615
2,116
= 0.290
=
1.05
9.000 V0.815 V0.815 rpm
Shp Shpr = 1.50
/
x
ratio is then
for the above corrected above , read
Fg Fgr t1
= 1.234
0.905
8t1 Vet1
the
calculate
= 0.60
1 + 0.2M2 = 1 + = 1
= 1.072
The corrected
From
inlet ,
at compressor
= 400/665
665 mph
next
and
In Fig .
=
from
,
Pt1 Pa
Calculate
=
aa
A4 : 1 read
Tt1 Ta and
/
1.60
given values of Shpr , Fgr , Wfr , above , calculate
ratio
for
and
/
wf wer
/
Pt1 Pa
= 1.234
= 1.20
8t√t1
/
Wa War
8+ 10+1 and war and
=
1.05
the values of 8t1
and 0 t1 calculated
Shp = 3,000 x 1.50 x 0.290 x 0.905 = 1,180 hp
TECHNICAL AERODYNAMICS
1516
lb thrust
Fg
= 1,000
x 1.60 x 0.290
= 464
wf
= 0.60
x 1.20 x 0.290
= 0.209
x
50
wa =
/ 0.905
1.05 x 0.390
=
/
lbm sec
16.8 lbm/ sec
are the answers called for . Application of the above data to and airplane is considered later .
These
a
propeller
is
While the foregoing analysis
general , and valid for altitude as well
as sea  level performance , some modification
is
power plant control
inlet
desirable
to have the
re
sults directly applicable to flight performance of turbine  powered air planes at altitude , as pointed out by Domasch . (1 ) The usual turboprop system involves
a turbine
(semi automatic ) .
( automatic ) and an rpm control
limitation effects of altitude
temperature The
inlet temperature are shown in the British turboprop calculations plotted in Figs . 15:15 and 15:16 . Note on performance at constant rpm and turbine
as 8
.7
for variation
.6
)
power
.4
400 200 200
δ = p Po 8
.4 .5 .3 Fig 15:16 British calculations 81.5 of Fig 15:15 plotted vs.
.4 .5 .3 British calculations on .6
Fig . 15:15 . Bristol Theseus turboprop performance 1944 .
From " Aviation , " December
.8 .7
.
450mph
cessive shock losses at the blade tips
it is
,
ex the
jet alone
diagram
.
a
4,
Flight Test
Manual Courtland Perkins Performance of Turbojet Airplane
"
Agard
Chap
.
0.
Daniel
Editor Vol
I
,
turbojet
.
General
)
(
Typical
Dommasch
)
1
.
15:17
for
"
and some design data
have
economical to eliminate
where propellers
A flow means of typical axial flow turbojet are shown in Fig thrust and fuel consumption data are shown in
and propel the airplane by
,
propeller
a
.
: 6 .
TURBOJETS
,
For speeds beyond
15
.6
/
.7
.
.8
/
8 01.5 λ
.
1.0
1
.3 1.0
8
/
ft
mph
SHP O
.5
/0.8
1000
SHP
1000
.
HP
Std .
40
30
Alt . ,
.
Shaft
(jet
7
mph
1.5 .
,
Shaft HP
/
D.
2000
SHP
.9 .8
graphs
1500
ft
1000
Necessary slope of
8
20
. "
Alt
10
.
Std . 2500
.,
oe or
to
30 1 .
, " (
3000
proportional
400
20 T
10
Shp
200
that these calculations show
POWER PLANTS
FundamentalData GE TG180
EXHAUST SYSTEM

,
15:17 .
(

Flow diagram for design data for General Electric ( Courtesy " Aviation Week , " July 7 , 1947. )
TG 180
turbojet .
24
20
20
Thrust
,
24
. )
)
)
(
Fig .

(
TG 180 Turbojet
Schematic Flow Diagram of GE
(
)

(
AND ACCESSORIES GEAR DRIVE ACCESSORY
.
. )( (
Maximumdia. 36% in. length 166in. Maximum Weight ) 2,380 (includingall accessories lb. (av.) ; 2,450lb. (guaranteed max .) military rat (15 min. takeoffand Thrust ing)4,125 ; 3,750 lb. (av.) lb. (guar anteedmin.) 7,700 Rpm takeoffand military deg F. max Exhaustgastemp 1,250 rating 3,420 lb. Trust max continuous av. Sfc. lb./hr./lb. thrust) 1.026 av. cruise) Fuel Gasoline ANF28 or Kerosene ANF34 Lubricant 3606 hydraulic fluid or 1065 engineoil .
IGNITOR PLUG SYSTEM COMBUSTION FUIL HOZZLE
1517
( )
AERONAUTICAL
lb
/
100
16
Thrust
10,000
12.0
8
1.5
4
1.0
0
Fig . 15:18 . level , Jumo
800
600
TSFC
lb
Fig
400
mph
800
600
Thrust at altitude with 15:19 Jumo 004B turbo TSFC as parameter jet at 8700 rpm .
,
.
Thrust and TSFC at sea 004B turbojet at 8700
rpm .
True Airspeed 200
,
mph
400
40,000
,
True Airspeed ,
30,000
.
TSFC
200

1.5
TSFC
20,000
lb
8
ft
1.9
, /
12
1.7
/100
lb
1.6
Thrust ,
the
which
in
.
shown
is
the
by 15
One

.
two values of rpm need be considered
be a
for calculating the sea Usually only ratio is the rated rpm for full throttle
maybe used
function of rpm and pressure
.
15
: 5
.
in Art
to
.
e2
Fig 15:21 that found factor at constant rpm and turbine inlet temperature
)
VeVa
in
as given
is
net thrust
two
Fig
: 6

level gross thrust as
a
The methods outlined
is
(
ram drag
in
.Vj

= where Fg Wa V .. Note the net thrust parameter
inlet
15:21
"
in Fig
temperature
"
and turbine
Fg
minus the
ratio on gross jet the thrust ef
.
,
15:20
Fn
"
"
gross thrust
plotted
which duplicates

The thrust
rpm and ram pressure
altitude effect studied separately for
The
"
.
15:21
.
.
fect shown in Fig 15:13 turbojets at constant rpm
.
shown
in Fig
.
are
effects of
The
"
15:18 and 15:19
.
.
Figs
jet thrust
is
12
hr
16
TECHNICAL AERODYNAMICS
1518
1.8 1.5
/
Fg Fgr
8t1 1.0
.9 .8
Pt1
Pa
.7
1.4
1.27 1.0
.6
.5
/
.4
rpm rpmy
I
Vet1
.3
60
80
70
90
90
Fig .
100
15:20 . Effect of rpm and ram pressure ratio on turbojet gross thrust (General Electric Co. ) . Note that both 8 and 8 must be varied for tude performance , and that turbine inlet temperature is proportional to 0 .
alti
1.0 1.0K
.8 .7 .6
.9f
•.5 5
0/0
.8
.25
4
 8/02
I
.7
x = G.E.  16 + = Jumo 004 B
.6
/
Fn Fno
.5 .4
X
.3 Std .
.25 0
Fig .
15:21 .
10
20
Alt
. 1,000
30
Effect of altitude
of two turbojets inlet temperature
at
.
ft 40
on net thrust constant rpm and turbine (SAE Journal , Sept. 1946. )
AERONAUTICAL POWER PLANTS
1519
operation and the other is a reduced rpm of about 80 or 85 per cent of rated rpm for cruising . Also , only two values of ram efficiency are com monly involved
in
is
and 100 per cent
installations in aircraft . A value between 95 usually reasonable for nacelle or " pod " installations
turbojet

value of between inlets at the side of a fuselage
as on the
for
B 52 ; a
 layer
selage boundary
Fig .
is
reduced
ature at altitude
.
inlet
with
temperature
preferable
and for
in
cent
is
is usually
which a substantial
assumed
amount
of fu
For altitude effects ,
if it
is desired to assume that the turbine inlet in proportion to the reduction in absolute temper
The more reasonable assumption
this
given by Fig . 15:21
,
diminishes the ram recovery .
15:20 may be used
temperature
80 and 85 per
altitude
Inlet cowling
constant turbine
is
,
reasonable .
more
(
of
giving better altitude performance
the variation of net thrust with altitude
purpose
Supersonic spike Diffuser cone )
,
Subsonic diffuser Flameholder
Fuel injection manifold
Exit
Fuel inlet
nozzle
line
Total head probe
Combustion chamber
Inner body rear cone
air turbine driven fuel pump
Fuel control
Fig .
15:22 .
15 : 7 .
ramjet
ical
is
Ram
Cutaway sketch
(Courtesy
RAMJETS . A cutaway shown
in Fig .
15:22 .
of typical supersonic ramjet Marquardt Aircraft Co. )
power plant .
sketch showing typical parts of a supersonic Ramjets will also operate subsonic , a typ
subsonic ramjet having been
in Fig .
shown
is
subsonic speeds the ram pressure ratio ciency , as noted in Fig . 15 : 2 . With a fixed inlet configuration
15 : 1d , but even at high
such as to give very poor
the shock
wave
from the
effi
conical
nose
is
" captured " by the inlet cowling with good efficiency only at a partic ular design Mach number , as indicated in Fig . 15:23 . With a variable
position inlet
cone , as shown
in Fig .
tained over a wide range of
Mach
iable , satisfactory control
can
15:24 , good
numbers .
also
be
efficiency
can be
ob
With the outlet area also var obtained to provide a thrust to
TECHNICAL AERODYNAMICS
or airplane
match the missile Capture area
steady
level
at any desired
within
the
Mach number
Designcondition
c .
Swallowedshock

for useful configurations
,
(
security
classified
.
15:24 are those of an example worked out by
of course
,
(2 )
and are
)
)
1
in Fig
The values given ham
1955
) (
(
.
From Durham
ap
pear to be at the present time
.
,
.
.
Fig 15:23 Effect of off design condi tions on inlet flow for supersonic ramjet
Numerical
values of thrust coefficients
not typical of the best
Dur
of current practice
but are judged to be feasible and usable for preliminary
design studies
.
b .
a .
ramjet and vehicle Spillover
flight
limitations of the .
Capture area
Capture area
for
drag
.
1520
1234 .
EXIT AREA FIXED AT MAX IMUM PRACTICAL VALUE
NORMAL
COEFFICIE ENT
1
1
SHOCK
Fn =
.

yob
.
/
FUEL GASOLINE STOICHIOMETRIC FUEL AIR RATIO
INFINITELY VARIABLE INLET
1.0
THRUST
THRUST
SUPER
CRITICAL
MAXIMUM SUBCRITICAL
NET
1
TYPICAL
£ 0.5
1.0
AIRPLANE
1.5
0.5 FIXED INLET AREAS DRAG
CHARACTERISTICS
2.5
2.0
3.0
.
FLIGHT MACH NO
280284
266
.
.p
cit
. ,
.
,
op
.
,
Ibid
pp
.
(2 .)
,
Durham
P.
F.
).
(
1
(
)
)
( 2
.
)
,
.
15:24
(
.
inlet
.
Effect of inlet geometry on ramjet with infinitely variable Courtesy Marquardt Aircraft Co. Bulletin MP 520. Typical nu merical values of thrust coefficients added from Durham Fig
POWER PLANTS
AERONAUTICAL
15 : 8 .
differ
Rockets
ROCKETS .
all
from
Gas
other power plants pre
viously considered in this chapter that they carry along their
oxidizer
Fuel
in
dependently of the surrounding
at
mosphere or beyond the earth's
at
Gas
Fuel
typical liquid propel lant rocket systemis shown in Fig . 15:25 and a typical rocket combus tion chamber in Fig . 15:26 . Typi shown
Table
page
15 : 3 ,

Valve
exhaust
are
Rocket motor
1522 .
as a function
of the are shown in Fig .
liquid
propellant
for
system
rocket
From
Dur
Injection plate
15:27 , page 1522 . feed Propellant
Typical fuel
Fig 15:25
area ratios
.
pressure and
pump
Valve
ideal thrust coefficients ob
tainable from rockets
Oxidizer
Turbine
.
The
in
Shaft turbine
pump
mosphere . A
combinations
Oxidizer tank
tank
own
and hence can operate
cal propellant
generator
. (
in
1521
Throat
Ab
Combustion chamber
.
)
2
,
"
.
a
The
specific thrust
/
1
: 3 .
. ,
Ibid
.p
( 2 ).
290
cit
. ,
.op
,
( 1 )
Durham
reactors are
288

,
of quantitative F.
the publishing
on
using uranium
security classified basis that prohibits technical data but some non classified .
under development
plants
Power
.p
.
PROPULSION
a
NUCLEAR
,
: 9 .
15
currently
P.
some
tremendous rate
in the form of in lbs for lbm sec
often stated
obtainable
.
.
for
rocket
of fuel and oxidizer specific optimum Values of thrust for mixture ratios are given of the fuel combinations in Table 15
meaning the thrust consumed
is
rocket at
"
consumption
in
consumed
a
rate of fuel
are
a
(
Fuel and oxidizer
liquid propellant
(
)
combustion chamber for From Durham
Typical
.
15:26
.
.
Fig
Nozzle
TECHNICAL AERODYNAMICS
gasoline
)
hydrazone hydrogen
242 255 239 243 238 255 259 335
Red  fuming
aniline
220
Whitefuming
furfural alcohol
214
Hydrogen
hydrazine ethanol
methane
ethanol
,
%
25
%
75 ethanol methanol
Oxygen
water
ammonia
nitric
)
/
lb in.2
2
)
300
(
and Pp about
/
Fuel
Oxidizer
/(
Specific thrust lb lbm sec with optimum mixture ratio ,
.
1
BIPROPELLANT COMBINATIONS
LIQUID
: 3
TABLE 15
(
1522
ethanol
acid
nitric acid
peroxide
methanol 2.0 200 Ptb Po
1000
*
A
Pb
/
Fn
=
CT
1.8
500
,
333
200
1.6
100 50
33.3
Thrust coefficientCT
1.4
20.0 10 Lineof maximum thrustcoefficient 5.0
LO
+3.5
/
08
2.5 2.0 30
/
.
From
)
y
=
,
306
101.
Wiley
,
Elements
.p
Propulsion
, "
Rocket
100
.
p .
. ,
cit "
.
op
P.
,
)
,
George
.
J.,
F.
Sutton
,
Durham
F.
( )1,
(2 .
1949
80
60
1.2 Ideal thrust coefficient for rockets with Jour Franklin Institute October 1940.
,
Malina
P.
15:27
.
.
Fig
40
(
TO Arearatio
A
8
0.6
AERONAUTICAL
POWER PLANTS
1523
published which are worth discussing here to show the
studies
have been
sort of
development
is
that
being given
consideration .
A hypothetical
airplane driven by uranium  powered turbojets , as studied for Life Maga zine by physicist Lyle Borst and aeronautical engineer Frederick Teichmann of New York University and presented in Life Magazine for February 7 , 1955 ,
is
shown
in Fig .
15:28 .
Possible details of the
power plant considered
helf klep
Fig .
15:28 . Hypothetical nuclear  powered turbojet  propelled supersonic long  range bomber . ( Reproduced with permission of artist Rolf Klep and Life Time , Inc. , from Life Magazine of February 7 , 1955. ) for such an airplane are shown in Fig . 15:29 , page 1524 . While the power
plant goes
is
into
feasible by its designers , the article in Life Magazine detail as to the extensive precautions which would be neces
judged some
sary for servicing this type of power plant and the limitations of crews due to radiation absorption in spite of the elaborate and heavy radiation shielding . was the judgment of the designers that crews for the
It
plane might safely
make
only a few long  range
flights
air
per lifetime
,
so
that while the power supply might last practically indefinitely the useful range of the airplane would be limited by radiation absorption by the crews .
1524
TECHNICAL AERODYNAMICS
TAIL ET ENGINE CUTAWAY
JET ENGIN VARIABLE EXHAUSTCONE
COMPRESSOR TURBINE SUPERHEATED COMPRESSED AIR STEAMBOILER
CONTROLRODS
TURBINESHAFT HEATEXCHANGER
LIQUIDMETAL PUMP
CCC
STEAM TURBINE
COP
AUTOMATIC UNCOUPLE
LIQUIDMETAL
TAT
COMPRESSED AIR
COMPRESSOR
REACTOR
URANIUM FUEL
POWERDRIVE FOR PLANE'S ELECTRICAL SYSTEM
LAYERSOF SHIELDING
AIR INTAKES
WING
Fig . 15:29 . Hypothetical nuclear turbojet power plant for supersonic long  range bomber . (Reproduced with permission of artist Rolf Klep and LifeTime , Inc. , from Life Magazine of February 7 , 1955. )
AERONAUTICAL
1525
POWER PLANTS
PROBLEMS
15 : 1 . For the engine data given in Fig . 15 : 5 , read values from the graphs and calculate , ( a ) the full  throttle torque , ( b ) the fuel consump tion rate in lbs hr at 2,550 rpm, full throttle , and ( c ) the fuel con sumption rate in lbs hr at 1,550 rpm on propeller load . 15 : 2 . The engine shown in Fig . 15 : 5 is rated 75 hp at 2,550 rpm at sea level . Using equation ( 15 : 2 ) , find the power developed at 12,000 ft standard altitude at ( a ) 2,550 rpm and ( b ) 2,000 rpm . 15 : 3 . For the engine data in Fig . 15:10 , at 14,000 ft standard tude and 1,800 rpm , read Bhp = 550 with 26 in . Hg manifold pressure ( point A on the graph ) . For the same rpm and manifold pressure at sea level read Bhp = 430. Using the method of Art . 15 : 4 , ( a ) find the brake horsepower available at this rpm and manifold pressure at 8,000 ft standard altitude , and ( b ) find the brake horsepower available at this rpm and manifold pres sure at 8,000 ft pressure altitude and a temperature of 10 ° F . 15 : 4 . A turboprop is rated as follows : Shpr = 3.750 , Fgr = 500 , rpmr  50 lbm/ sec . Find the Shp , Fg , wf , and = = 10,000 , wfr 0.56 1bm sec , war 9,000 rpm wa at . Using the turbojet data in Figs . 15:14 and 15:20 , ( a ) calculate 15 : 5 . Fn Fno for standard air altitudes of 10,000 ft , 20,000 ft , 30,000 ft , 40,000 ft , and 50,000 ft for constant rpm ( rpm /Ve variable ) ; ( b ) calcu late also turbine inlet temperature ( Tti ) if Ttio = 2,000 R. Assume a constant Mach number of 0.5 ; ( c ) calculate the thrust specific fuel con estimate the effect of keeping Tti constant as sumption ( TSFC ) ; ( g ) tude is changed . Using the ramjet data in Fig . 15:24 , (a ) calculate the net thrust 15 : 6 . of a ramjet of 24 in . burner diameter at M = 2 at 30,000 ft standard titude , assuming a capture area ( Fig . 15:23 ) corresponding to 18 in . ameter and a fuel air ratio of 0.06 ; ( b ) estimate the fuel consumption 925 . Using the rocket data in Table 15 : 3 and Fig . 15:27 , ( a ) esti 15 : 7 . mate the ideal thrust at sea level of a rocket which burns gasoline and oxygen at a combustion chamber pressure of 300 lb/ in.gage pressure with air area ratio Aj A** = 4 and a throat area of 7 sq in .; ( b ) estimate the 1,000 lbs of ( fuel + oxidizer ) with optimum mixture burning time for
/
/
alti
/
/
alti
al
/
/
ra
tio .
(1 )Durham, F. P. 1951.
Chapters
" Aircraft 4 and 5 .
( 2 ) Ibid . , Chapter
( 3 ) Ibid . , Chapter
12 .
13 .
Jet
Powerplants , "
Prentice
 Hall
,
Inc. ,
CHAPTER
16
AIRPLANE PROPELLERS
propeller , or British call it two or more rotating airfoil  shaped blades driven by a piston engine or turbine . A propeller blade element and the forces acting on it are shown in Fig . 16 : 1 . The lift 16 : 1 .
PROPELLER CONSTRUCTION
airscrew as the
and drag
AND GEOMETRY .
,
An airplane
usually consists of
forces on the blade element can also be analyzed into thrust and tangential force components dT and dF , dr respectively , as shown in Fig . 16 : 1 . The thrust force propels the airplane tangential
dD
known as the " effective pitch ' of the pro
and
Propeller blade el forces acting on .
it
is the
same at all radial sta effective pitch angle must therefore decrease in going radially out ward from the hub to the tip of the blade .
tions .
and
The
chord chord direction thrust wind Zero Geometric
25 20
(
is
fifth
calculation of the
).
The
AIRPLANE PROPELLERS
PROPELLER PROBLEM
16 : 5 .
the selection lems
is
and use
TYPES AND
1618
A guide
METHODS OF SOLUTION .
to
of the proper equations for various types of prob 16 : 2 , which is illustrated by examples follow
provided in Table
ing the table . TABLE 16 : 2 .
OUTLINE OF PROCEDURE FOR SOLVING OF VARIOUS TYPES
Given
Problem type
1. Selection of diameter
2.
Thrust
power calcu
V, n, P, P
To
if
fixed pitch )
V , n , P , P,
T , 1 , Thp
Thrust
lation , const .
v,
Q,
P, D,
B
T , Thp , 1 , n
torque .
4. Best cruis ing fuel con sumption .
Calc . Cg , equation or (16:38 ) .
2.
Read V nD and on " max " line , as in Fig . 16:18 .
3.
Calc
1. 2.
Calc . Cp , equation or ( 16:39 ) . Calc . V/ nD .
4.
Calc . Thp
3.
rpm .
3.
1.
D
lation , const .
power calcu
Procedure : Calc . calculate Read = read on chart
find
T, n , (also ẞ
D,
1. 2.
3.
V, C=
T,
p , D; n) ,
f (P ,
lb/ (hp ) ( hr )
n, P , n, best fuel lb mile
/
PROPELLER PROBLEMS
/
.
V/D ; T = TP.
D =
V/nD
( 16:25
)
in Fig . A7 : 9 . Bhp ; P = 550 = Bhp ; and T = np v . ,
Read
as
/
, equation ( 16:33 ) . Read CT CQ and V nD , as in Fig . 16:17.
Calc . Cos
/
Calc . T
/
=
CT& D са
V/D V nD
n =
/
4.
Calc . Thp = TV/ 550
1. 2.
3.
Calc . Pc , equation ( 16:37 ) . Assume several V / nD values . Read , as in Fig . A7 : 10 .
4.
Calc . P
5. 6. 7.
/
= TV n ; n =
Select best of
of
fuel .
V / nD
/
; n = TV P.
V /D
/
V nD or Fig.
Read C , Fig . 16:20 A7 : 2 . Calc . fuel lb / mile = PC / mph . ues
550
val
assumed minimum
for
to are typical only ; the most recent chart of the type specified should be used . Charts referred
( 16:32 )
available
TECHNICAL AERODYNAMICS
1619
Example 1. Propeller  selection problem , Lockheed Lodestar airplane . in Tables 14 : 2 and 14 : 3 , Given the airplane power  required calculations the engine data in Fig . 15:10 , and the propeller data in Figs . A7 : 5 through A7 : 9 . Find the propeller diameter for maximum efficiency at a speed of 240 mph at 10,000 standard altitude and 2,100 constant rpm ; also find n , Thp , and T. Solution . Following the procedure outlined in Table 16 : 2 , calculate first Cs . In the specified engine data read Bhp = 780 per engine , and for this brake horsepower and the specified =rpm and altitude read in Fig . 21.3 ; and calculate 16:19 : σ 1/5 = 0.941 ; Bhp1 5 = 3.79 ; rpm² 5
ft
/
= 0.638
Cs
In Fig .
for this
A7 : 8 ,
24.5 degrees
/
/
x
3.79
240 x 0.941 = 1.79
x 21.3
value of Cs , read on the " maximum " line Bo.75R At the same Cs , read = 0.88 .
=
and V nD = 1.01 . D =
/
V n
/
V nD
= 240
x ( 88 / 2,100 ) 1.01
=
9.95
ft
Calculate also 0.88 x 780 = 687 per engine
Thp
and
T
= 687
x
375/240 = 1,073
lb
called for .
These are the answers

Example 2. Power available calculation , Lockheed Lodestar airplane . Given the airplane , engine , and propeller data in Example 1 , and assume that a propeller diameter of 10 ft 6 in . was selected on the basis of ad Find the full  throttle ditional considerations to those in Example 1. power available at 10,000 ft standard altitude 2,100 constant rpm , and airplane speeds ( mph ) of 291 , 233 , 175.5 , 145.5 . 116.4 , and 103.6 ( same speeds as in Table 14 : 2 ) . Solution . Following the procedure outlined in Table 16 : 2 , calculate first Cp (which applies to all airplane speeds , because P and n are known and V is not involved in the definition Cp = P / pn3D5 ) . A more useful form of equation ( 16:25 ) defining Cp for practical calculations is Cp =
Using
Bhp
= 780 ,
σ
= 0.738 ,
Cp =
Calculate
0.5
/
Bhp 1,000
/
( rpm/ 1,000
) 3 ( D 10
( 16:39 )
)5
rpm = 2,100 , and D = 10.5 , calculate
0.5
0.780
0.738 2.13 x 1.055
also = mph
= 0.045
88 = x 2,100 x 10.5
mph 251
list this value opposite each value of mph as in Table 16 : 3 below . For each V/ nD , with Cp = 0.045, read ẞ0.75R in Fig . A7 : 6 and ŉ in Fig . = 780 x 2 for two engines , giving the results A7 : 7 , and calculate Thp and
shown .
Table 16 : 3 gives the answers called
for in this
example .
1620
AIRPLANE PROPELLERS
CALCULATION OF FULL  THROTTLE THRUST HORSEPOWER AVAILABLE FOR LOCKHEED LODESTAR AIRPLANE AT 10,000 FT ALTITUDE
TABLE 16 : 3 .
/
V nD
mph
1.16 0.93 0.70
291 233
175.5 145.5 116.4
Thp 1,360 1,340 1,280 1,230 1,120 1,060
0.87 0.86 0.82 0.79 0.72 0.68
260
22° 180 160
0.58 0.464 0.412
103.6
n
Bo.75R
150
14.50
Power  available calculation , Piper Cub airplane with Given a Piper Cub airplane for which
Example 3.
pitch propeller . W = 1,400
lb
ft
S = 180 sq
fixed
CD = 0.046 + 0.060 C12
powered by a Lycoming six  cylinder opposed  type engine rated 120 hp at 2,000 rpm and equipped with a fixed  pitch propeller of diameter D = 6.5 set at 0.75R = 180 of characteristics shown in Fig . 16:17 . Find the full  throttle power available at sea level at speeds ( mph ) of 120 , 100 , 80 , 70 , 60 , and 50. Calculate also the thrust at mph = 0 . Solution . Following the procedure outlined in Table 16 : 2 , calculate first CQs = V√√pD3 Q , equation ( 16:37 ) , using Po = 0.00238 = 1/420 , D3 = 6.53 = 275 , Q = 120 x ( 5,250 / 2,600 ) = 242 lb  ft , and V = mph x 1.467 .
ft
/
Cos
1.467 mph
=
V420 x
For each speed ( mph ) , calculate CQs
in Fig .
16:17 ,
for
Bo.75R
242/275
For each Cos and list as in Table 16 : 4 . Calculate 18 ° , read CT CQ and V nD .
/
/
=
/
/
= moh 13.1
/
/
T = ( Cr Cq ) ( Q D ) = ( Câ / Cq ) ( 242 6.5 ) = 37.2 ( GT CQ )
/
Thp = (T x mph ) 375 , and rpm = 60n =
giving the results shown TABLE 16 : 4 .
in
Cos
100
9.16 7.64
80 70 60
5.35 4.58
120
50
0
/
16 : 4 .
CALCULATION OF FULL  THROTTLE FOR 120
mph
Table
88 mph
6.5 (V nD)
6.11
3.82 0
Cr
CQ
6.3
6.9
7.45
7.9
8.15
8.5 9.8
HORSEPOWER
/
PIPER
V nD
T
0.74 0.65 0.57
234 257 277 294 303 316
0.51
0.45 0.38 0
364
=
13.5 mph V nD
/
THRUST HORSEPOWER CUB AIRPLANE
Thp
75.0 68.5 59.1 54.9 48.5 42.1 0
AVAILABLE
rpm
2,190 2.010 1,830 1,800 1,740 1,720
Indeterminate
TECHNICAL AERODYNAMICS
1621
Since the values of rpm in Table 16 : 4 come out well under the rated is evident that the propeller is not properly selected or set ; proper diameter and setting could have been determined by the method the of Example 1 , the data in Fig . A7 : 12 being used . Example 4. Calculation of best cruising fuel economy for Lockheed Lodestar airplane . Given the same airplane , engine , and propeller data as in Example 2 , plus the specific fuel  consumption data in Fig . 16:20 . Find the engine rpm and brake horsepower for most economical cruising at 233  mph true air speed at 10,000 ft standard altitude , and find the cor responding number of miles the airplane will go on 6 lb ≈ 1 gal ) of fuel .
2,600 ,
it
(
0.55
perbhp
,
0.50
500
hp
600
350 hp
hp hp
400 he 450
lb. BSFC
hp
hp 300 hp
hr
700
800
0.45
0.40
800
1400 1600 1200 Crankshaft rpm
1800
for
engine
2000
.
data in Fig
15:10
: 2
)
,
:
=
0.078
2
x
,
=
)
(
on the
, S
/4
X
π
=
/4
x
=
interpolation
551
86.5
cruising propeller chart in Fig
.
by
0.0243
.
line
(
с
=
,
2, .
=
про
Draw this A7 10
data
=
:
ft
consumption
Following the procedure outlined in Table 16 taking the value of CD Calculate Pe from equation 16:37 14 as CD 0.0243 for 200 mph at sea level or 233 mph at = 10.52 86.5 ft2 пD2 Then with Ad 551 ft2 2
Step from Table
10,000 and ne =
fuel
1. .
Solution
:
Assumed

16:20
.
.
Fig
.
,
1000
.
. 
,
/
V
,
C = .
.
=
=
)
V /
,
(
= .
/ C
/
.
: 2, =
/
/
V
/(
,
x
=
/
=
V
,
(
=
) V /
,
/ x
, = a
/ 6
(
$
).
/
,
6
: 3 .
/
7.
).
=
/
lb /
.
.
/D ) /(
V
,
:
.
,
(
V
V / =
C
.
6.  5.
/
,
/
,
/n
n
3.
.
: 2
:
,
5
,
/
V
.
.
,
;
V /
a
,
a
V /
Step 2. As guide to the range of values of nD to assume for the specified trial calculation range to the right of the peak of propeller efficiency is suggested for high nD values will give lower values of rpm and Fig 16:20 shows that low rpm favors low fuel consumption Ac cordingly the range of values of nD from 1.4 to 2.2 is chosen for tab ulation as in Table 16 below for each value of nD assumed as out lined in Table 16 Step Read from Fig A7 10 e.g. at nD 1.4 read 0.889 Step 4. Using Thp = 854/2 = 427 per engine from Table 14 calcu = = late Bhp Thp for each nD e.g. at nD 1.4 Bhp 427 0.889 480 1,400 rpm For this condition 60 nD 88 233 10.5 1.4 Step Read in Fig 16:20 For the same nD as above read 0.495 lb hp hr Calculate fuel mile = Bhp mph Step For the same nD as above calculate fuel lb mile 480 0.495 233 1.02 per engine 2.04 lb mile for two engines Step Select the lowest fuel lb mile in Table 16 In this case the best rpm is 980 corresponding to Bhp = 500 per engine and giving 1.82 fuel consumption of 1.82 lb mile for the airplane gas mileage = 3.30 miles per lb gallon
AIRPLANE PROPELLERS TABLE 16 : 5 . LODESTAR
V /nD
CALCULATION OF BEST CRUISING FUEL ECONOMY FOR LOCKHEED AIRPLANE WHEN CRUISING AT 233 MPH TRUE AIR SPEED AT 10,000 FEET STANDARD ALTITUDE Bhp , 1 engine
η
1.4 1.6 1.8 2.0 2.2
0.870 0.855 0.815
500
CORRECTION
0.467 0.444
0.425 0.425
890
, 2 engines
2.04 1.94 1.87
0.495
1,400 1,220 1,080 980
523
fuel lb / mile
с
rpm
480 485 490
0.889
0.880
16 : 6 .
1622
1.82 , optimum 1.91
Wind  tunnel
FOR PROPELLER CHARACTERISTICS .
FACTORS
cruising
tests on propellers are seldom available for the exact arrangement contem plated in a proposed new design . In making performance estimates it is accordingly necessary to use such test data as are available and make cor rections for conditions .
at
the difference
principal
The
1.
Number
items
for which corrections
are
are :
made
of blades .
2.
Blade width and planform .
3.
Blade thickness
4.
Blade
5.
Body interference
6.
Tip
airfoil
ratio .
section
.
between propeller
(compressibility
speed
the blade tips )
and nacelle
correction
due
or fuselage
to high Mach
.
number
.
Each of these items has an appreciable
the
and the proposed
between the test conditions
optimum diameter , on
,
the efficiency
though sometimes minor ,
for
, and
effect
on
fixed  pitch propellers
on the blade angle necessary
(a ) tests
Number
to absorb a given power at a given rpm . of Blades . The results of a large number of NACA
have been summarized by Weick
is usually desirable propellers . " This
(1 )
as follows : "Two blade propellers are used in all ordinary cases , for the fewer the blades the lighter , cheaper , simpler , and more efficient will be the propeller ; and two is the smallest number of blades with which proper balance of mass and air forces can be obtained . ..· Vibrations are however set up in two  bladed propellers when the air plane is turning , due to the varying gyroscopic moment of the two bladed arrangement and , when the airplane is sideslipping , due to the uneven air loading . . . . Vibration difficulties considered , it on propellers
statement
is still
to have three or
good
develop a one  bladed propeller ( 1 )Weick
,
F. E.
" Aircraft
in ,
It
1955 .
using
more
may
blades
in
have inspired
large geared attempts
a blade  stump counterweight
Propeller
Design , " McGraw
 Hill ,
for
to
mass
1929 , p. 252 .
TECHNICAL AERODYNAMICS
1623
a rubber mounted
balance and
pivot to permit the centrifugal couple to balance the
air
forces and give
of pitch . usually This is not considered satisfactory , for the simultaneous adjustment development
drag of the blade  stump balance more than offsets the gain in
ef
ficiency blades
.
due to reduced number Moreover , as
designs have developed
forward speeds and slower
tive
speeds , the
ofblades
of
airplane to larger
effect of
rota number
on efficiency has become
small to negligible and while three blade designs predominate
simplicity and smooth Fig . 16:21 . Eight blade counter  rotating because of propeller . ( General Motors " Aeromatic . " ) ness , many careful studies have yielded four , five , six , and even ten blades as optimum design . An eight blade "counter rotating " design , with four blades rotating in each
is
rection ,
shown
in Fig .
when a large amount
of
16:21 .
is
Counter rotating
di
designs are often used
installed in a small airplane to eliminate in efficiency due to inter sets of blades rotating in parallel planes a small
power
the " torque reaction . "
While there are losses
ference between the two
distance apart , there are compensating gains due to elimination of the energy of slipstream rotation ; in general , counter  rotating designs will
efficiency as high as , or higher than the best single plane propellers . Test data on counter  rotating designs of four , six , and eight blades
have
are given on pages A713 through A716
tractor
A717 and
plane and cedure
they
, and composite design charts for propellers of two to eight blades are given on pages A718 . Special test data on each number of blades , both single counter  rotating , appear to be necessary because a general pro
and pusher
for
will
different
correcting be
the results of tests on one number of blades so that
applicable
number
to
of blades ,
calculations not available .
performance
is
on
a
propeller of
(b )
Blade Width and Planform . The effects on power absorption of blade width , and blade width distribution along the radius , are usual
ly
properly accounted
and calculated
for in
in Art .
16 : 1 .
terms The
of
the activity factor , as defined
effects of activity factor
on maximum
AIRPLANE PROPELLERS
propulsive efficiency
recent 70 =
Fig .
gible loss of efficiency . In fact , the highest efficiencies thus far reported
(c ) Blade is usually
Thickness Ratio taken as a
blade thickness ratio ports tests on
16:22
1
.
8
n
/bh n
per cent
thickness on maximum propulsive efficien cy is shown in Fig . 16:23 , the efficiency being used as
thick propeller
0.97
Airfoil
blade
Fig .
Section .
0.6
0.4 0.94 0.070.080.090.10 0.11 0.120.13 at 0.75
Fig
high values of Jm the effect of thickness on maximum propulsive Blade
0.8
20.96 0.95
16:23 shows that at (d )
1.0 Jm
= 0.98
.
a reference value .
1.00 0.99
/
cent
propeller
the
h b
of a 7.5 per
thickness ratio
measure of
re
effect of blade
The
max
The blade  section
typical single
NACA TR 1126
per cent , though values around are more typical .
.
Effect of A.F.
on
blades as thin as 5
some
80 90 100 110 120 100,000 16 ffx³dx
=
Fig
0.075
at 0.75R
70
between 150 and 200 per blade .
range
max
factor
cent are the activity
.
in
with very thin blades
0.96 60
93 per
of about
1126 )
0.97
AF
maxat
( NACA TR
Jm 2.0 1.0 0.6 0.4
12100 9004
negli
FAF
with
1.00
at 0.99 0.98
16:22 that relatively wide blades can be used under these conditions
1.01
b
evident from
(
is
J
.
it
and
more
are
as
R
2.5 or 3.0
In
Jm may be as high as
.
shown
as a parameter .
propeller designs ,
of V/ nD , designated
which the design value
in
16:22 ,
Recent
16:23
.
is
in Fig .
(1 )
as studied by Thomas , Caldwell , and Rhines ,
max Пmax AF
shown
1624
on
efficiency
Effect of max
is
/
h b
very small .
designs of propellers with
air
high efficiency at high tip speeds have favored low  drag , high  speed foil sections , such as those shown in Fig . 16 : 7 . For conditions where
tip is
not operating near sonic velocity tion has been proved relatively unimportant . This
the blade
at the usual Reynolds
numbers
,
the blade
is
airfoil sec
presumably
for propeller blade  tip operation
because ,
, the
bound
is mostly turbulent with service leading  edge roughness . Re gardless of the airfoil section , for a given thickness ratio , the maximum
ary layer
propulsive
No systematic correction
it is
best to use data
tip
about the same
for
available the effects of blade section
data are not
at low
all good airfoil sections . factors for blade airfoil sections are available ; based on the proper airfoil section , but if such
efficiencies are
may be
safely neglected
speeds .
( 1 ) Thomas
,
Caldwell
,
and Rhines , JRAS , January 1938 , pp . 186 .
TECHNICAL AERODYNAMICS
1625
(e)
This is partly because the propulsive efficiency , as defined , includes
Diehlp.334
stream . Most propeller data are obtained with a propeller mounted in front of a na celle ; often the nacelle is mounted on a wing . A propulsive efficiency correction
0.90
D B
0.85 0.1 0.2 0.3 0.4 0.5 0.6 0.7 = Body diameter Propellerdiameter
max Speed
critical
plot of maximum propulsive particular configuration number for
for cruising
speeds

propellers operate at speeds beyond the high speed of the airplane some even oper
Many
flight
.
.
efficiency against helical tip Mach shown in Fig 16:25 These tests involved
is
thin blade section oper angle of attack Maximum ,
.
a
relative low lift coefficient and efficiency corrections to other thicknesses and angles of attack
ating at
in
,
ate at super
given
;
the level
is
A
critical for
body interference
.
Tip
.
)
(f
n
/D B
factor due to Fig . 16:24 .
Effect of
.
.
16:24 on
slip
the effect of the drag added by the
Tractor
0.95
a
= 1.00 D
Fo
in front propulsive
on the
efficiency .
1.06
Fig
effect
adverse
a
/B
has serious
or fuselage

max 0.42 maxfor
of or behind the propeller
η
A large nacelle
Body Interference .
as
re
31HARM
(
( 3 )
062
045
)
10
NACA
efficiency
,
Maximum
.8
.6
stream Mach number

Air
!

Air
stream
Mach
)
number
08 045
3 )(
(
NACA 10
,
.8
.7
.6
.
.
Fig 16:25
for
Effect of tip
Leading edge
.9 Mach
1.0
1.1
NACA
1.3
1.2
number on maximum propulsive efficiency reported in NACA TN 2881
the particular configuration
.
H4T
Helical tip Mach number
M
………………………… ++++++
.2
Rhines ,
and
1626
are
in
shown
Figs
16:26
.
Caldwell
Thomas ,
)
ported by
(1
AIRPLANE PROPELLERS
and
16:27 . 140
.
Yor
sections
.
EHTS= HTS Fcxfhx fa 1.20
fa 1.10
Fig
0.75R
deg
of
.
16:27 Effect of angle attack on limiting Mt.
AND VERTICAL
TAKE OFF AIRPLANES
.
:
: 7 .
SLOW VENICLES
Bo.75R

Effect of thickness limiting Mt.
STATIC THRUST
16
=
α
.
16:26 on
1.5
.
.
Fig
1.0 at 0.75
.
0
/
h b
0.5
10
5
1.00
R
LOOL
,
speed
High
15
tipsections

Clark
1.10
1.30
RAF
6
fn 1.20
tip
130
propeller at zero forward speed called the particular importance in the calculation of the static thrust is of airplane take off of an and is basic factor in the design of verti
power coefficient increases
a "
.
= 0,
nD
/
V
.

.
rotating
rapidly
the static thrust
wing
to the blade angle
of total
As would be expected
16:28
setting
;
in Fig
,
of the blade as
cient is nearly proportional
a
.
shown
three blade 58689 propeller .
are
227
a
=
activity factor
data on
stalled
.
of
,
static thrust
usually
is
greater than Bo.75R150 most the blade may be seen by inspection of Fig 16:13
as
For blade angles at
are seldom available
0
nD
=
at
/
Wind tunnel tests
V
,
static thrust considerations
consideration
,
.
,
V /
,
primarily
Typical
Air
5b
Since such propellers operate at very airplanes they may usually be designed compared nD with
small values of from
in Fig
for the propulsion of slow vehicles
used
sleds and bicycles
meaning
sometimes VTOL
such as that shown
" ) ,

"
vertical propellers are also occasionally such as boats
VTO
1 :
take off and landing
.
"
usually designated
,
airplane
off

take
(
cal
.

a
,
"
"
,
airplane
The thrust of an
from
coeffi
the corresponding
relationship
.
between static static thrust coefficient is found to be approxi mately parabolic as shown in Fig 16:29 More careful and more extensive hovering required helicopters power Chapter 17 studies of for show that usually 3/2 For the three blade pro better approximation The
)

.
cit
.
Rhines
.op
and
,
)
a
Caldwell
,
(
,
Thomas
)
1
(
CPS CTS
is
(
.
,
and
.
coefficient
power
more
TECHNICAL AERODYNAMICS
1627
peller data in Fig .
16:28 ,
it is
seen
CPS
=
0.006
in Fig .
16:29 that the
static
power
by the equation
thrust coefficients are related
and
( Fig .
+ 3CT3
16:29
)
( 16:40 )
.16 .14 .12
.10
CTS
CTS
and CPS
.08 .06
.04
CPS .02
$0.75 R
0
5 Fig . 16:28 .
0
20
10
Static thrust data on three  blade 58689 propeller of total activity factor = 227. NACA data .
Studies of power for
the static condition
of
other propellers
indicate
or zero thrust value of Cpg is a function of the total activity factor of the propeller , and accordingly the data in Fig . 16:28 generalized by writing in the form may be somewhat that the
minimum
If , mum
as
thrust
angle
in
may be
For a
it is
the design of a slow vehicle ,
for
a given
(16:41 )
= 0.006 (A.F.total/227 ) + 3CT3
CPS
amount
selected from the following
specified tip
Cps
maximum
InD
:
/
maximum
T
( 16:42 )
1,730 Bhp
/
propeller blade
considerations
speed , maximum T Bhp requires
стя For a specified rpm ,
desired to get the maxi
of horsepower , the proper
T Bhp requires
maximum
AIRPLANE PROPELLERS
/
CT3 2 205 13/2
/2
=
)
(
requires
BT Bhp
PDp1
CPS
/Bhp
T
maximum
maximum
16:44
720
)
diameter ,
16:43
Tn2
(
For a specified
T
P
5/2
R2
1/4
=
75/403
=
CPS
/
4
/
CT5
1628
.151
Test Data approximation 2 3
+
+
/
CT
CP
/
of
3/2 CT
max =
CTS
/
5/4 CT CP = max
.
0.006
CP = max
.
.05
CPS
.
CPS
=
.10
:
Parabolic
2
→
104 CTS O
.
:
ample
/
.
16:30
/2
C3
In Fig ,
rpm
Bhp
/
Cp ,
,
speed
.
.
The known values
factors in the design
thrust for and
is
or diam
the optimum blade angle has a
or
(
read from Fig 16:28 from similar the propeller of desired activity fac
for
and number
of blades
).
blade section profile
of
,
blade thickness
When
Cpg and CTs can be used calculate the unknown This procedure is illustrated by the following ex to
tor
,
,
CPg and CT can been selected graph experimentally determined
design
tip
.
in the
,
specified
on whether .
assumed
depending
16:28
the design
the optimum blade angle for maximum
be
,
is
eter
degrees
,
3,5 or 7.5
about
16:28
in
T
"
.
are plotted against blade angle for the data of Fig note that for this propeller
maximum
CT5
.
static
/
desired to have
/4
.
also
Values
,Cp
it is
figure of merit
/
where
given power and diameter
called the
CT
,
sometimes
"
is
16:44
of helicopters
of
Equation
(
.
.
16:29
225
Parabolic approximation to static thrust data of Fig )
Fig
200
150
100
50
Cp a
0
TECHNICAL AERODYNAMICS
1629
2.0
10
₹9.0
1.77
CTS
= 0.0775
CPS
= 0.024
( Specified
8
1.8 D)
1.6 5/4 CTS
7 6
3/2
10 CTS
CPS
5
3
CTS
CTS
= 0.0575
CPS
=
/1
514
0.016
( Specified
3.70
CTS
rpm
)
1.2
+ 1.0
CPS
0.045
CTS
=
CPS
= 0.012
0.8
(Specified
nD )
0.6
CPS
2
0.4
1
0
Fig .
1.4
CPS
0.2
B0.75 R 10 16:30 .
/
0
0
20
10
30
Thrust power ratios for optimum static thrust designs with various criteria , based on data in Fig . 16:28 .
Example . Given a bicycle , to be propelled by an air  cooled outboard motorboat engine rated 5 hp at 4,000 rpm, ( a ) for a direct drive at 4,000 rpm, find the maximum thrust and corresponding propeller diameter ; ( b ) for
diameter propeller with a belt drive of ratio to be determined , maximum thrust and the corresponding rpm ; ( c ) for drives at con stant tip speeds of 900 , 700 , 500 , and 300 ft/ sec , find the maximum thrust and corresponding diameter and rpm . Solution . (a ) For this propeller may be noted that CT5 4 Cp = 1.77 at Bo ..75R = 50 and Cp = 0.016 . For the example above , using Bhp 5 and = rpm = 4,000 , use Cp 0.016 to calculate the optimum diameter of a three blade propeller thus : a 24
in .
find the
//
it
Cp = 0.016 =
D = 10
0.0025 0.016 x 64
and , from equation ( 16:43
(0.5 x Bhp)/ 1,000 (rpm/ 1,000 ) 3(D/ 10 ) 5 D =
=
5/410 5/1410
),
10
3.33
=
/
2.5 1,000 x ( D/ 10 ) 5
64
= 3.00
ft
= 36
in .
AIRPLANE PROPELLERS
39
8.16
/ /
27
(
) x x x
= (
,
=
=
)
,
2
(π
πD
nD
)
(
rpm = 60
)
=
900
ND 3/2 = 1,000
3
πnD 100
)3
(
2,980
and
nD
that gives at sea level
Cp
/
=
54,500
107
of
=
рCp πnD
read CT = 0.045 5x 1,730 3.70 nD
50
300
64.0
550,3 Bhp
lb
)
45.7
3/2
/ /
(
.
T
,
==
700
35.6
ND
38.8
60
6,880
calculate
and
;
x
6,88012 x 24
the diameter from the definition
D =
32
16:30
500
900
or
1,000
=
=
/
/
D²
rpm
rpm
( c ) For CT CP max 3.70 and 0.75R and Cp = 0.012 . For the specified Bhp calculate 32,000 nD thus : πnD T
105
6.88
in Fig
ẞ0.75R
1,000
=
لوسو
0.0775
=
3/326
For the ex
at
2.5
(
/
=
0.00238
x
0.0775
=
) 3 ( D 10 ) 5
=
read CT
,
thrust Ts
( rpm/ 1,000
x
32
=
To calculate
Bhp 1,000
σ
250 0.024
rom
1,000
ft
/
0.5
= 0.024 = Cp =
for
132
lb.
(b) Note that Cr3 2 Cp max = 0.90 at ẞ0.75R = 7.50 . , use Cp = 0.024 ( read ample above , using Bhp = 5 and D = 2 7.50 ) to calculate the necessary rpm , thus :
Solve
0.221
=
2,750
x
1.77
=
) =
=
0.00238
√66.7
/
or Ts
5 x 550 x
= 1.77
In
x
/4 Po 1/4
/
= C5 Ts5 4 Cp
1630
12.2
5.2
2.02
2.95
4.47
7.65
8,500
4,530
2,140
750
.
18.5
speed is limited by noise or by compressibility of limit on thrust and diameter is seen above
the effect of the
.
tip If choice
300
,
rpm
500
=
ft
700
=
D
) "
(
which gives
( b ).
a )
.
(
;
a
,
a
These answers permit wide choice of diameter and rpm based on conven ience but it should be noted that they do not give the maximum thrust for given diameter or rpm these items have been found in parts and These are the answers called for
1631
TECHNICAL AERODYNAMICS
22 22 DETAIL  DESIGN CONSIDERATIONS .
16 : 8 .
structed of
dural
wood ,
or steel in the
,
manner
Brass tipping
of propeller blade construction
is
hollow steel construction 8
feet . Propeller
tion of
and
forces , as shown
carefully
be
tensile stresses
in Fig .
Curtiss Hollowsteel
(From Nelson . ) ( 1 )
.
usually lighter for propeller
hubs and blades must
bending
16:32 .
If
analyzed
set up by the thrust
Tilt
T
diameters over
for the combina and centrifugal
fails
the blade or hub
Centrifugalforces.
16:31 . The
Weld
Aeroproducts Hollowsteel
Solid Dural
Types
16:31 .
in Fig .
shown
Brazed. joints
Solid wood
Fig .
are usually con
Propellerblades
it is
,
thrown
In
out with
enormous
such
event , the unbalance
an
of the
blade usually
remaining
tears the airplane
energy .
engine
tilt
being
approximately 1/2 degree
.
The
this
out
of the
of disaster Fig . 16:32 . Forces acting on propeller has occurred on several occa blades . sions in experimental testing of airplanes . The centrifugal and thrust forces can be made to offset each other partly by having a small amount of forward tilt in the blade , as shown in Fig . 16:32 , the usual amount of ;
type
propeller in . at the tips for
deflection of
blades under load , which sometimes amounts to
2
or 3
the
gives an additional effective tilt maximum thrust that must be considered in making an accurate stress analysis . A system (1 ) Additional gy atic procedure for doing this is outlined by Nelson . large propellers under
,
stresses are added to the propeller blade
roscopic bending
plane has a rapid rate of pitch or yaw ; a procedure for
when
the
air
calculating the
stresses is also outlined by Nelson . A rotating propeller blade that is free to turn about the blade axis
gyroscopic bending
is
acted
on
zero pitch . 16:33 ,
in
by a powerful centrifugal couple tending to set the blade at The cause of the
which
spection of Fig
blade
the
equivalent of two .
weights
16:33
( 1 ) Nelson , Wilbur Inc. , 1944 .
Sons ,
is assumed to be represented by its dynamic displaced slightly from the blade axis .
In
shows
C.
centrifugal couple is illustrated in Fig .
that the centrifugal forces tend to
" Aircraft
Propeller Principles
,"
rotate
John Wiley
&
AIRPLANE PROPELLERS
its
the blade about
axis
own
in
as to reduce the pitch .
such a way
propellers the centrifugal couple
some
is
1632
balanced
pair of
out by a
In
weights
located in a plane perpendicular to the mean plane of the blade , but in most controllable  pitch propellers this centrifugal couple represents sim
ply
load to be taken by the pitch  changing
an additional
pitch  changing be either
mechanism , which may
electrical
mechanical ,
or hydraulic
,
Bladeaxis
( as shown in previous
,
Shaftaxis Shaft axis
photographs ) must not only overcome
friction of
the
The
mechanism .
but also
must
trifugal
couple
the blade bearings
Fig . 16:33 . Sketch showing source of centrifugal couple tending to reduce pitch of controllable blades .
counteract the cen
pitch
the
when
Blade axis
is
being increased and the aerodynamic
couple
pitching
due to the
moment on
the blade , which usually acts in the same direction as the centrifugal couple . The efficiency of the pitch  changing mechanism is usually made somewhat
than 50 per cent
less
to
make
it irreversible
.
in pitch ( for
that do not have to be feathered or reversed
namic brakes ) a rate of pitch change of 5 or 6 degrees
For propellers use as
aerody
per second has been
found satisfactory , but rates of pitch change as high as 45 degrees per (2 ) to be necessary for quick feathering or second are reported by McCoy aerodynamic
braking by
sary to avoid
ing by
means
damage
means
in
of reverse pitch
case of engine
of reverse pitch
is
Quick feathering
.
failure in flight ;
is neces brak
aerodynamic
incorporated in most large propeller
signs because aerodynamic braking has been found to be considerably effective than braking by means of wheel brakes . On
airplanes
for the pellers be
pilot
with three or
it
propellers
to synchronize the propeller
This
is
usually accomplished
in the propeller  governing
propeller
governors
synchronization
by means of synchronous
made
speed
is
approached
to avoid hunting .
and
( 2 ) McCoy ,
, W.
C. , op .
H. M. ,
"
liquid
must
motors
propellers
in that
governor
they
.
in
speed as the
mechanism " dead  beat "
are also equipped
for spraying the blades with antifreeze
( 1 ) Nelson
the
makes the governing
Most propellers
operated
" anticipatory
lude an accelerometer element that decreases
iesired
of constant  rpm pro
governors
circuit for electrically are being
more
is practically impossible
manually ; therefore , some means of automatic
provided .
Most
more
de
with
slinger  rings
to minimize the detrimental
cit .
Jour . Aeronautical Sciences , July
1944 .
TECHNICAL AERODYNAMICS
1633
effects of ice formation
Rubber de  icers , such
on the blades .
times have been used on the leading edges of wings
useful on the inboard portions of
some
propeller
, have
also
as
Some
been found
blades .
PROBLEMS
16 : 1 . Using equation ( 16 : 4 ) and the method outlined in Table 16 : 1 , calculate the activity factor for the 373647 propeller for which blade form data are given in Fig . A7 : 4 . 16 : 2 . Using equation ( 16:13 ) , calculate the slipstream velocity ratio of 90 per cent . Vs Vo necessary to get an ideal efficiency 16 : 3 . An airplane traveling at 500 mph is propelled by a jet that has a discharge velocity ( Vs  Vo ) of 1,000 ft/ sec . Using equation ( 16:13 ) ,
/
calculate
i·
/
A propeller operates at a disc loading T Ad = flying 200 mph in standard sea  level air . Using equation 16 : 4 .
late ni .
10
lb/ ft²
( 16:14 ) ,
16 : 5 . An Ercoupe airplane is to be powered by an engine 2,550 rpm . Using the design chart in Fig . A7 : 12 , select a
rated
while
calcu
75 hp
at
propeller diam at sea level , and
eter and blade angle for maximum efficiency at 120 mph find the propulsive efficiency at this speed . 16 : 6 . Assume that a two  blade propeller of diameter 6ft set at Bo.75R = 16 ° is selected for the Ercoupe airplane powered by the engine of char acteristics shown in Fig . 15 : 5 ( direct drive , no reduction gear ) and with the propeller characteristics shown in Fig . 16:17 . Assume the same air plane speeds as in problem 14 : 1 . calculate the maximum full throttle thrust horsepower available for level flight at sea level and the engine rpm at each flight speed . Using Fig . 16:17 , calculate the static thrust ( V nD = 0 ) for the 16 : 7 . airplane  engine  propeller combination in problem 16 : 6 . 16 : 8 . Assume the Lockheed Constellation airplane to be powered by four engines of characteristics shown in Fig . A7 : 2 , with a propeller reduction gear ratio of 0.4375 . Assume the propeller characteristics shown in Figs .
/
A7 : 5 through A7 : 8 , and select a propeller diameter for maximum efficiency when delivering the maximum rated power ( 1,800 hp at 2,400 rpm ) at 10,000 standard altitude at a true air speed of 375 mph . Find also max . 16 : 9 . are selec Assume three  blade propellers of diameter D = 15 ted for the Lockheed Constellation airplane powered by the engine of char
ft
ft
acteristics given in Fig . A7 : 2 and with given in Fig . A7 : 5 through A7 : 8 . Assume
' problem
14 : 2 and
calculate
the
maximum
the
propeller of characteristics
the
same
airplane speeds as in thrust horsepower 10,000 ft altitude at
full throttle
from four engines available for level flight at 2,400 rpm . 16:10 . For the Lockheed Constellation airplane with engines and pro pellers as in problem 16 : 5 and power  required calculations as in problem 14 : 2 , use the cruising  propeller chart in Fig . A7 : 10 and find the mini per mile for sea  level cruising at 250 mph . Find also mum fuel consumed the brake horsepower and rpm for most economical cruising . 16:11 . A four  blade hollow  steel propeller blade of 12 ft diameter weighs 65 lb and has a center of gravity 40 per cent of the radius from the shaft axis . is acted on by a centrifugal force of 150,000 lb and has a maximum centrifugal blade torque of 10,000 in . lb . Using a blade friction torque coefficient of 0.014 in . , pitch  changing  mechanism ciency of 50 per cent , and a rate of pitch change of 45 deg / sec , calculate the maximum power required to change the pitch .
It
effi
AIRPLANE PROPELLERS
1634
Using the methods and data of Art . 16 : 7 , find the blade angle available from a 2 ft propeller of three blades driven at 2,200 hp gasoline engine . For an engine developing 1 hp driving at 2,200 rpm a three blade propeller at the static  thrust characteristics shown in Fig . 16:30 , find the maximum static thrust that can be developed and the corresponding propeller diameter and blade angle . 16:14 . For an engine developing 1 hp driving a three  blade propeller of the static thrust characteristics shown in Fig . 16:30 and a propeller diameter of 36 in . , find the maximum static thrust that can be developed and the corresponding rpm and blade angle . 16:15 . A sled used for servicing high  tension lines in the mountains is powered by a small airplane engine rated 75 hp at 2,550 rpm at sea level . The sled , carrying power plant , equipment , and two servicemen , weighs 800 lb. At 10,000 ft pressure altitude and a temperature of 0 ° F (σ = 0.77 ) , the engine develops 55 hp at 2,550 rpm . With this rpm specified , find the optimum blade angle and propeller diameter , using the three  blade  propeller data in Fig . 16:30 . Also , find the static thrust and the steepest grade climbable . 16:12 . and thrust rpm by a 1 16:13 .
CHAPTER
17
HELICOPTER PERFORMANCE ORMA
( 1)
is a vehicle of military become of considerable and commer cial importance with the satisfactory solution of the control problem . The invention of the helicopter is often attributed to Leonardo da Vinci 17 : 1 .
DEVELOPMENT
ancient lineage
( 14521519 )
,
HELICOPTER .
though he appears
,
several centuries
well in Fig .
OF THE
The
helicopter
only recently
(2 ) .
1 : 6 , pages
The helicopter
(3 )
the Chinese by
history of the helicopter
The more recent
handled by Gregory .
by
to have been anteceded
Several current
has been
types of helicopters are shown
16 and 17 . consists
essentially of
a body which encloses the pay
rotor or rotors which provide lift by action sim airplane propeller , and a control system . The dominant
load and power plant , a
ilar to type of
that of an
helicopter
has
become
that with a single main
lifting rotor
and
an anti  torque rotor in a vertical plane in the rear , as shown in Figs . 1 : 6a and 1 : 6e . The control system for a helicopter of this type consists of a controllable pitch anti  torque rotor for control about the vertical
axis ,
and a " cyclic pitch
about the other two axes .
pitch control
" for the rotor blades
control The
main
rotor blades for control
must also have a " collective
either vertical climb , hovering , or safe descent without power . The collective pitch control is very similar to that of a controllable pitch propeller . The cyclic pitch control changes the pitch " to permit
(1)Much of
the material on this chapter was presented by the author at the 23rd annual meeting of the Institute of the Aeronautical Sciences , 11 Jan. 2427 , 1955 , as a paper entitled " Aerodynamic Design of Helicopters , which constituted an abstract of a Ph.D. dissertation of the same title at the University of Michigan . Microfilm copies of the original 95  page dis sertation are obtainable from the Graduate School of the University of Michigan , at about 3 cents per page . The IAS supply of preprints of the paper was exhausted early in 1955. As of June 1955 no arrangements have been made for publication . , ( 2 )Magoun
,
Hill , 1931 , p . (3 ) cGregory ,
F. A. 8.
and
Eric
Hodgins .
"A
History
Colonel H. F. " Anything a Horse Helicopter , 11 Reynal and Hitchcock , N. Y. , 1944 . 171
of Aircraft Can
Do , The
, "
Mc Graw
Story of the
HELICOPTER PERFORMANCE
172
C Fore and aft
cyclic
Lateral cyclic
Collective
(a )
(b)
( Above ) Bell Two blade Mechanism .
( Right ) Sikorsky
Three blade Mechanism .
Figure 2.
Pitch
Two Types
of
Control Mechanism .
Fig . 17 : 1 . Types of pitch  control mechanism . ( Courtesy Bell and Sikorsky advertising . )
TECHNICAL AERODYNAMICS
173
of each blade each time the rotor goes around , providing small pitch  angle The as the blade goes forward and large pitch angle as the blade retreats . pitch control
mechanisms , which have made safe
helicopter flight possible
,
are complicated and expensive ; they are also heavy because they must be designed for fatigue limits much lower than yield stresses ; they are also troublesome because they introduce vibration problems . Common types of
pitch control
in Fig . 17 : 1 . The principal com cyclic pitch mechanism . On small is due to the helicopters great control simplification is obtainable by substituting weight shifting for cyclic control , as shown in Fig . 17 : 2 , though only very mechanisms
are
sketched
plication of the control
inher
slow forward speeds are obtainable with this device because of the ent limitations of center  of  gravity from
lift
rolling of the
over sidewise as advancing
and
it
movement
moves
necessary to keep the machine
forward , due to the difference
in
retreating blades .

Fig .
Helicopter ( de Lackner , DH 4 ) without cyclic pitch  control , 17 : 2 . steered by shifting of the weight of the pilot . ( Courtesy " Aviation Week , " April 4 , 1955. ) The problem ment of
with which this chapter
simplified
methods
is chiefly
for calculation , rapidly
concerned and
is
a develop
accurately
,
of the
nearly level , flight performance of single  rotor helicopters It is found convenient to use such as those shown in Figs . 1 : 6a and 1 : 6e . the hovering performance as a point of reference .
level ,
and
17 : 2 . ment
LIMITATIONS
of helicopter
OF HELICOPTER
performance
is an
theoretical treat extension of airplane propeller theory THEORY .
The usual
HELICOPTER PERFORMANCE
and has been
well presented
by Stepniewski , (3)
174
by Gessow and Myers ,
(1 )
by
Dommasch ,
(2 )
and
of the theory are only very the resulting equations require empirical coeffi cients to be applicable with good accuracy to the helicopter performance but since the assumptions
fulfilled
approximately problem .
In helicopter based on
tip
propellers .
work
it is
customary
to
speed (Vt
= ПnD )
The usual
coefficients are
rather
CT
use torque and thrust
than forward speed
coefficients (V ) as used for
given by the equations
T
A
PAdVt
( 17 : 1 )
2
P
СО
( 17 : 2 )
PAdVt3 The momentum theory
in Fig .
16:10 ,
is
of propellers , involving the flow pattern shown to climbing helicopters ; and , in the
commonly applied
case of zero climb , to hovering helicopters , even though the airflow pat tern through a hovering helicopter , as typified by Fig . 17 : 3 , bears little
similarity to that of a climbing helicopter or an advancing propeller . On this basis it is shown in the references previously cited that CQ for a helicopter is proportional to C3 /2 .
Fig .
Streamlines of flow through a hovering rotor .
17 : 3 .
(Official
NACA
photograph . )
A careful integration of a blade element analysis
including
the effect of " inflow " at the disc , for uniform chord blades of optimum twist is shown by Gessow and Myers (4 ) to yield an expression for hovering torque coeffi
(1 0Gessow , Alfred and Garry C. Myers copter , " Macmillan , 1952 .
)
( 2 )Dommasch , Daniel 0. 11 namics , Pitman , 1953 .
I,
( 3 )Stepniewski , " Performance
."
W.
Z.
" Elements
Jr.
"Aerodynamics
of Propeller
" Introduction
Rotorcraft Publishing
( 4 ) Gessow , A. and G. C. Myers ,
,
Jr. ,
of the Heli
and Helicopter
Aerody 11
Aerodynamics , Vol . Committee , Morton , Pa . , 1950 .
to Helicopter op .
cit . ,
p . 83 .
TECHNICAL AERODYNAMICS
175 cient of the form Cq =
in
a
+
CT
+
/
c CT3 2
d
+
c²
(17 : 3)
which a , b , c , and d are constants for a particular rotor . The quantity
is the zero thrust torque coefficient usually designated by Coo The quantity b is usually a small negative quantity depending on the shape of a
the blade
profile
drag curve near the zero
and d are nearly the
same
for
all
lift
region .
helicopter rotors
The
quantities
and depend
chiefly
c on
the " tip loss , " or on the extent of the blade tip (Fig . 17 : 3 ) , on which there is upflow rather than downflow . For a simplified analysis of the
it
hovering performance of helicopters is desirable , convenient , and rea sonably accurate to replace equation ( 17 : 3) by either of the following two approximations
Cqk1
+
k₂C+2
(17 : 4 )
= k3
+
k₁G3 / 2
( 17 : 5 )
CQ
A
of the accuracy of
comparison
the basis of
rotor test
all
tower
these
two approximations , as judged
the full  scale hovering test data available (Fig . 17 : 4 )
is
in Fig .
shown
C3/2 is
mation involving
within
on the NACA
Note that the
17 : 5 .
on
approx
nearly always
a few per cent of the
test data .
The approximation involving CT2 is some times better at high values of CT , such as are often involved
but
Fig .
17 : 4 .
mounted on
tower .
of
Helicopter rotor NACA hovering test
is usually CT , more
in propeller tests ,
poorer at the
low values
typical of current helicopter
hovering condition has been used
.
in
The CT2 approximation
Chapter
16
in
handling
static thrust data on airplane propellers , and has also been used by the 1) author ( for a simplified helicopter design procedure . The C3 / 2 approx
will be used here in the interests of slightly for lightly loaded helicopters . imation
17 : 3 .
HOVERING
PERFORMANCE
ANALYSIS .
( 17 : 5 ) is the torque coefficient at zero theory (2 ) to be given by the equation
(1 )Wood,
K. D.
( 2 ) Gessow ,
" Airplane
Design , " Tenth
A. and G. C. Myers ,
Jr.
, op .
The quantity
lift
Co
Edition cit . , p .
and
improved
accuracy
in
equation
k3
is
indicated
( 1954 ) , Chapter 60 .
5.
by
55
Fig .
ce
/
te
.
.
50
17 : 5 .
,0
T
5
.076
2 11 12
090 Scele
5 59

T
Comparison
40
30
25
40 19 8 20 я10
( (
Scale .042 20
8758
105c 60
8
0
O
P
.
of approximations in
20 6
4
.038
ga
la
00
)G2 (
equations
15 2
Scale
30
20
9 to
4
)A
=
VLLE
5
Scale
.027
+++++
20
270.5
1050Q Scale 60 40 40
5.5
6.5
4.5
3.5
2.5
1.5
60 30
)A (
Scale
T
C)
1
( 40
Also
.σ6σ
teg 20
1000
GT
80
(
σ
2
1.5
Scale 2086 TN .027 .023
Scale 70
5
e =
x
)
=
Scale .038
2.5
(
.(
10000
3.5
)G (
4.5
.
)A)
=
A
30 10
Scale 1000CT
6 10
3/2

5.5 12
T
1000 .060 Scale
T
) (
4 3 3 7
x
(
A)
, .
(
.076
3
+
Scale
, 14
( (
4
20
( , .
5
10 Scale
? )
)
(
k₂
)
=
=
7
)
C
X
)
. 1698 .042
)
A
k₂
)
B
.
T.4
7
1000
k₂
te
.20 .15
■
16
co
/
18
of
3 *
46.5
105
2318 .027 2277 .038
3/2
0.5 80 twist 1.0 0.5 .15 1.0 .12 1698 .060 T.0 0.5 Fabric not smooth 1.0 .20 11.5 X 2318 .076 8 ARRL5F25b Full scale tunnel test check points
Sym
,
x
20
K4
.20 5.5 80 twist .15 .15 .12 Fabric not smooth .20 14 scale tunnel test check points
Helicopter tower test data fitted by family of straight 2.7 lines form CK3 1000GT with FE of Notes Ref TN 105kg 105
) (
2318 .027 2277 .038 1698 .042 1698 .060 2318 .076 ARR15F256 Full
22
k
T.A
Sym
Helicopter tower test data fitted by family of straight lines of form 105ca 3 1000GT with 1.15 FE of Ref TN Notes 105K1 105
HELICOPTER PERFORMANCE
176
YAAR
σ 19 819
۲۲ 89to
C
B
O
1888
JIG
D
B
( 17 : 4 ) and ( 17 : 5 ) .
10
O
C
TECHNICAL AERODYNAMICS
177
#
K3
is
where σ
solidity defined
the
Cao
= Ocde
(17 : 6)
8
by
Bce
D =
( 17 : 7)
πR
in
which B
is
ce Equation ( 17 : 8 ) integrates
= 3
blade ce
is
f₂ cx²dx
is
ratio of the blade area to
the
weighted
( 17 : 8 )
thus ,
a blade of uniform chord c .
in
the case of a uniform chord
For
the disc area .
a
tapered blade
weighting each chord in proportion to square of and hence , for constant angle of attack , in proportion
rational
A more
.
mean
be CeQ comparison
which would involve torque
The quantity
·/·
in
purpose
would
( 17 : 9)
cx3dx
O
blade
on the
elements
is
thrust difference usually neglected .
is
Cde
effective chord for this
= 4
of
than equal
rather taper ratios , and
, but the
equation
is
basis of equal
small for the usual
effective minimum drag Studies of the relationship between
( 17 : 6 )
coefficient for the blade airfoil .
a mean
CQ and a show a range of values of Cde from 0.008 for rotors with 3 drag blades to 0.012 for rotors with 2 relatively high  drag blades .
to
1
low
( 1 ) to be indicated by theory The slopes of the curves plotted in Fig . 17 : 5 differ from
The quantity equal
,
mean chord ,
the local velocity , to the thrust
defined by
cc for
to give
The meaning of the term " solidity" ,
is
the number of blades and ce
in
k
/√2 .
equation
this theoretical value by
( 17 : 5 )
is
a factor of about 1.2 because of blade
tip losses
non  uniform downwash .
This factor is called kå and is the correction factor from ideal to actual hovering induced torque coefficient . and
and k in equation ( 17 : 5 ) as modified above with the definitions of thrust and torque coefficients for a helicopter given in equations ( 17 : 1 ) and ( 17 : 2 ) , permits writing an ex
Using the coefficients k3
and combining
pression for hovering rotor horsepower which
is
here called Ph
( 1 ) Gessow
=
Bhphr
/
W 1,000 , A.
per thousand
pounds gross weight ,
Ph , as = 1,000
550
Poocde V+3 + 1,000 8w
and G. C. Myers ,
550
Jr. ,
op .
cit . ,
Kn √w
1200
p . 30 .
( 17:10 )
HELICOPTER PERFORMANCE
178
where W Po
W =
(17:11 )
Ad P
The appearance of equation
simplified , and its plotting fa specific case of helicopters with three
( 17:10 )
cilitated , by restricting to low  drag blades , and writing still fairly general
can be
the
Vt / 100
Ph P₁
=
=
For this special case
v.
kg K5 Ov³
+
, which
is
(17:12 )
kg √w
where k5 = 4.32 for three low  drag blades ( 6.48 for two high  drag blades ) and k6 = 31.9 for the usual tip loss and downwash correction kn = 1.2 . Since equation ( 17:12 ) has too many variables for a simple network chart
,
it is
dividing by
considered desirable
30
to reduce
giving
/ /3
Ph = v3 15 To w 02 Equation
( 17:13 ) may be
abscissa
respectively
the
number
of variables
/ / /3
by
(17:13 )
+ x6 V
w 02
plotted as a network chart using as ordinate Ph
and
(17:14 )
Зго
/ /
= w 02 3 X =
and
this
maxima
has been done
in Fig .
17 : 6
may be shown
are characterized
ratio =
Th
where
It
17 : 6 .
and minima that the bottoms of the " valleys
speed curves power
in Fig .
(17:15 )
Phi
=
k√w
( 17:12 ) and Php tion (17:12 ) . Note that

Fig .
power per 1,000
and
is
/ is
of the constant of the hovering
( 17:16 )
= 2
Php power "
term
in equation
the " hovering profile power " term in
is , for
of
tip
by a value
the " hovering induced
k50v³ w k5 °v³ /w 17 : 6
Phi
"
by the calculus
a given
lbs of gross weight
solidity
, a
equa
plot of hovering horse
against disc loading
( corrected to sea
and that the " optimum line " of rh = 2 gives minimum power for a given diameter and tip speed , though it does not necessarily represent the best design . In fact , Fig . 17 : 6 is not very useful for design , as it pro
level )
vides no guidance for selection of
ity . It is
a
tip
speed , diameter , power , or
simply a chart which satisfies hovering requirements
.
solid To be
TECHNICAL AERODYNAMICS
179 400
300
"Optimus line" 3√0
200
N
Pap
const
Vt
250

1600 500 400 300
Vft /bec
150
100 90 80 6
5
Fig .
7
8
v12213 20
9 10
First draft of
17 : 6 .
useful , it must consider the special copter operates . 17 : 4 . may
be limited by
tips of the
itation ,
it is
Cb
cient "
stall
advancing
customary
To develop
to think
/
When
ition
is
in
defined by W B =
where W
speed
of the retreating
blades .
the gross weight
equation
( 17:17 )
is
CLb b
and
B
35 40
limitations
R
50
.
under which
, the speed
of
a
helicopter the
retreating blade stall
lim
the
lift coeffi
( pV2 / 2 ) cdr
the number
it
may be shown
CLb = 6CT / 0
(17:17)
of blades .
integrated and combined with the usual
of thrust coefficient CT ,
heli
a
blades orby shock waves on
terms of a " mean blade
[S*0 is
30
hovering power chart
In forward flight
LIMITATIONS .
SPEED
25
15
defin
that ( 17:18 )
1710
HELICOPTER PERFORMANCE
It in
is
customary
terms
where V
to think of the speed of forward
of the ratio
is
a helicopter
of
movement
/
the forward speed
με V QR and R is the tip
(17:19 ) speed Vt .
Elementary
the
that with a given stalling angle of at degrees , , tack such as 12 or 16 an increase in forward speed requires a reduction in the blade lift coefficient to avoid stall of the retreating blade and that the reduction should be proportional to the square root of
oretical
considerations
indicate
the mean blade lift coefficient . Flight tests and wind tunnel tests on rotors show this to be the case , as shown in Fig . 17 : 7 , where the " rotor limit " data of Brown (1 ) (based on Bell Helicopter flight tests )
0.6
are
com
0.35
g
0.30
G
11/2
( Squared scale )
0.5 Equation :
α = ((d290
0.25
· 0 . V = 0.4750.63μis =
)
16°
0.20
"Rotor Limit "
0.4
/
PL Gessow 0.3
0.16 0.15 0.14 0.13 0.12 0.11 0.10 0.09 0.08 0.07
0.2
0.06
0.3
0.05
0.4
0.2 Equation :
Va )
(α 270=12°
9.2 = 0.4250.63μLS
0.3 0.4
0.04 0.03 0.02 0.01
0.1
M15 0.1
Fig .
17 : 7 .
R
0.3
0.4
of retreating blade stall limitations . Solid lines and Myers . ( 2 ) " Rotor limit " line from Brown . ( 1 )
Summary
from Gessow
(1 ),
0.2
VIS
Brown , Eugene "Helicopter Performance and Aerodynamics , " craft Corp .. Helicopter Div . Document , 1954 . ( 2 ) Gessow , A. and G. C. Myers , Jr. , op . cit . , p . 266 .
Bell Air
TECHNICAL AERODYNAMICS
1711
flight
pared with a summary of
(1 ) ,
test and wind tunnel test data presented by
in which the total drag / lift ratio P/ L is considered to be a minor parameter . Equations for the limiting speed ratio μLS in terms of CT o are given in Fig . 17 : 7 .
Gessow and Myers
/
flying at the speed of retreating blade stall , the advancing blades may be limited by shock waves at the tips . For a retreating blade stall limitation of When
mphLS
if
critical
the
tion
on
stall is
parameters
itations Two
speed due
(QR)LSC
Equations .

=
( 17:20 )
about 0.8
, the combined
limita
blade shock and retreating

( 840 to 900 )
( 17:21 )
1.467 mphLS
to Fig .
blade
since the
same
to provide a hovering power chart with speed Such a chart is Fig . 17 : 8 .
lim
( 17:20 )
( 17:21 ) can be added
and
are involved
17 : 6 ,
,
shown
in Fig .
17 : 8
relative to the One of these
of shock  free landing from a descent without power .
limitations is that of
a mean blade
corresponds approximately to CL Another limitation added speed of
involving
/ /o
√w
22
to advancing
additional limitations are also
problem
cal
0.50R
number of the tips is
Mach
rotor tip
=
gliding
unusual
the vertical
is
skill for
conditions of forward
a
that of approximately , which
30
is
ft / sec
descent
will
have been more
also includes scales for rotor diameter
minimum
verti
judged to be a region
a safe power  off landing .
gliding
flight
for
coefficient of about 1.4 , which typical helicopter blade section .
without power
descent
speed of
max
lift
The estimation
of
be discussed later after the carefully studied . Fig . 17 : 8
and hovering rotor horsepower
per
1,000 lbs of gross weight to facilitate performance calculations and de sign selections . The use of Fig . 17 : 8 is illustrated by the following example : Example . Given a helicopter rotor with a solidity of σ = 0.04 and a rotor diameter of 27 ft . The helicopter weighs 1,000 lbs and operates at a rotor tip speed of 600 ft / sec . Find the power required to hover at sea level out of ground effect ( at an altitude greater than 1 or 2 rotor dia meters ).
Solution . In Fig . 17 : 8 , for a sea level diameter of 27 ft and a solid of σ = 0.4 , read w /o2 / 3 = 15 ; at a tip speed of 600 ft / sec read on the horsepower scale for σ = 0.04 a value of Bhphr = 63. An engine of greater power than this would be required to provide also the usual tail rotor Since these losses usu power , cooling losses , and gear friction losses . ally total 15%, a 75 hp engine would provide satisfactory hovering . Ad ditional information provided by this chart is , ( 1 ) that the advancing blades would not be shock  limited , since the upper shaded area is avoided ,
ity
( 1 )Gessow ,
A. and G.
C.
Myers ,
Jr. ,
op .
cit . ,
P. 266 .
HELICOPTER PERFORMANCE 30

0.04 0.0
30
350
25 2000 and
25
1
.
S
120
for
W
.. Dam
13015
1712
σ
400
0.06
150 0.01
110 140 19
1000 and
18
17
16
0.09
15
τσ

for W
locating
.
20
Diam
σ
SL
.
35
.
130 110
1525
100
0.08 300
diameter
cal
20 90 100 110
250
Tip speed
80 90
1
ΩR
Advancing blade shock in at phys
100+
1
Zod
wm 108
106
200
165
phys
70
60
= 0.04 //////////
70
600
80+
σ
60 50
J
70
gliding
.
Approximate 30 ft./sec
=.04
descent
150
ph
Bat
is 
55
65+
150
for σ
specified
3.09
60
MO 100
50
QR 35
40
45
)
(
Shock free landing not possible if Laax 14 =
phs
00
(
GLb1.4
400
55

50
10 )
ایک
σ
300
100 NIT
Lo
9
10
15
20
Hovering power chart with speed
25
30
limitations
35 .
17
: 8 .
Fig
8
8/02/3 7
specified
80
6
0
.
σ
/
W
0
.
σ
5
= 60
for
.04
BuPhr 1000
=
90
'
= 290
30
40
50
TECHNICAL AERODYNAMICS
1713
and ( 2 ) that the forward speed would about 160 mph .
POWER REQUIRED FOR LEVEL FLIGHT .
17 : 5.
stall  limited until it
not be
The
rotor blade profile power is to increase the turn the rotor . While the velocity increase
effect of forward
torque on
reached
speed
on
and power required
to
the advancing
is
blade
the
as the decrease on the retreating blade , the forces vary as the square of the velocity and the net effect is an increase in the ratio
same
Pp Php
= 1 +
also a body drag which forward which the
it
flat
must
well
may
allow for rotor hub
to
( 17:22 )
(17:22 )
higher values than
though various authors have suggested
cient of μ² in equation
3μ²
be overcome by
additional
the
3 as
drag .
tilt
coeffi is
There
rotor
of the
Designating by
be considered at the same time .
f
plate area of unity drag coefficient equivalent to the body drag that
may be shown
f
Ppb = 1 + 3µ² + Rp = Php Cde Ao
43
,
( 17:23 )
effect of forward speed on induced power is quite the opposite of the effect on profile and body power , since the rotor acts on a much larger volume of air when a component of forward velocity is added to the inflow . The
The usual assumption would
flow
normal
through
to the
flight test
a
flight
data
is that the stream of air affected is the stream circle of diameter equal to the rotor diameter path .
show
it
curacy of this assumption
prising
if it power
is
Equations ( 17:23 and
to be of the
)
glance
order of On
and
=
Pi Phi
=
1 8μ
( 17:24 ) are
thrust coefficient respectively
power
in
1
/
W =
ν
20 per
cent would not be
the basis of this assumption
ratio R₁ of induced given by the equation Ri
and
this seems optimistic , but substantially correct , though an inac
At
could be detected .
be shown that the induced
first
that
52μ
plotted
forward
flight
as parameters .
.
17 : 9
sur may
to hovering
(17:24 )
1,000 CT
in Fig
it
with drag
ratio
Since the variation of
each of the two components of hovering power with forward speed has been determined
and
plotted
in Fig .
17 : 9 ,
if
the ratio of these two
components
= when hovering is known ( e.g. , гh Phi / Php 2 ) a proper weighting of these plotting permits general factors a chart of power required for level flight .
This has been done in Fig . 17:10 for гh = 2. A similar chart for any other can be plotted from the equation
value of г
0
=
Fn
Right
+ 72 15
R
9
flight
of
required =
for =
/ Php
chart
profile flight
(
Phi
and
level
body
level
.
power
in
induced
+
ratio
0.1
)
helicopters
optimum
17:10
: . .
Fig
) ) ( .( . of
power with speed Above General
Drag
17
P7 0.3
By
hi
0.1
0.2
1000CT
defo Cdokamra
3424
except where shownbroken

+
Fig
0.2
0.2
0.3
3.4
0.5
0.7
Rp
Variation
0.1
μ
1000 CT
High
0.5
гороха 1.0 0.9 0.8
3
0.6
4
0.7
5
0.8
,
1000 CT
 Ppb1 Php dray
Xbox
0.3
Ασ
f
0.9
.
=
1.0
2R
10
98 7 6
1.5
2.0
1
HELICOPTER PERFORMANCE 1714
37.
.)2 =
rh
(
TECHNICAL AERODYNAMICS
where R
is
P
= Rp
rn
the ratio of power required
Ri
17:25
)
=
(
*
R
1 + +
1715
rh
in
level
flight
to hovering power
. The use of Fig . 17:10 is illustrated by the following example : Example . Given the helicopter for which hovering calculations were made in Art . 17 : 4 . This helicopter has a body of " low drag " specified by the ratio f/ cdeAσ = 12.5 , and it flies at a thrust coefficient of CT = 0.02 . Find the minimum power necessary for level flight and the forward
required
speed at which the power is a minimum . Solution . In Fig . 17:10 , for " low drag " and 1,000 CT = 3 , read in Fig . Since 17:10 a minimum value of R* of 0.56 at an advance ratio µ = 0.14 . 63 hp was required for level flight , the minimum power for forward flight is 0.56 x 63 = 35 hp . Since the tip speed of the rotor was 600 ft sec and the value of u at minimum power was 0.14 , the forward speed of the copter under minimum power conditions is 0.14 x 600 = 84 sec = 57 mph .
/
17: 6 .
HIGH SPEED AND MAXIMUM CLIMB CHARTS .
ft/
If more
power
heli
is available
at the rotor than the minimum necessary to hover , the ratio of the avail able power to the hovering power may be plotted on Fig . 17:10 ( or similar may be read ( which chart for the appropriate value of г½ ) , and a value of is designated HL) . This is the ratio for the level high speed of the helicopter . A chart of this sort is plotted in Fig . 17:11 . For each of
a number of values of tip speed (Vt = R ) such as 500 , 600 and 700 scales of level high speed can be added to Fig . 17:11 as shown.
ft / sec ,
Level high speeds
in mph for R shownbelow ( IF NOT LIMITED BY VIBRATION) .60
=
12.5 1.50
t,/
cdeσ
150 200 190 180 170 160
170 160
150
130
110
140
120
100
130
11c
,
120
High
/
30
25 80
speed chart
2.0
for helicopters
15
2.5
designed
with
гh
2 .
level high

Sea
Pa ph
1.5
=
NR
& C
/
=
20
= BOOK
80
70
600
700 ↑ .
35
M
1.0 .
37.5
90
= R
17:11
drag
000
90
100
100
Fig
40
Looger
7
1405
110
drew
130
150
120
Low
140
HELICOPTER PERFORMANCE
A
sea  level rate of climb chart
maximum
for
1716 any given
ratio of
avail
able to hovering power can be plotted in a similar manner , measuring the net available power from the bottoms of the " valleys " of the curve in Fig . 17:10 . This has been done in Fig . 17:12 . Since the climb ratio calculated
in this added
manner
for
is
Ch max / 33Ph , scales
of rate of climb in
This has been done
value of Ph .
= 40 , 60 , and 80 ;
values of Ph
tion
is
any specified
for
other values of Ph .
ft / min
in Fig .
can be
for linear interpola 17:12
satisfactory . MAX. CLIMB, FT./MIN . FOR VALUES OF Ph AT HEADOF COLUI .
·P
80
6000
values
2.5 1000CT
1.5

4500r 3000
2.0
100077 5000
2500 3500+ 3000 2000
drag
2000 2000
High
Approx
1000 1000
.
.
15001000
climb
Low
1500
vertical
2500+ 3000
1.5
10000 drag
max
4000
1.0 Comax
0.5
500
sea
2.5
level rate of climb chart designed with rh =
Maximum
2.0
1.5
.
The use of Figs
level rate of climb of in Art
17
for
helicopters
the level high speed and
may be
illustrated
by con
: 4 .
example started
a
tinuing the
.
sea

maximum
17:11 and 17:12 to calculate
helicopter
3.0
2 .
17:12
.
.
Fig
1.0

0
0
500
helicopter
,
.
.
=
"
it
.

"
a
is
.
,
,
with
: 4
The
.
Example equipped
.
on which calculations were made in Art 17 125 hp engine of which 20 hp is used in the cooling blower tail rotor and gear losses so the net available horsepower per 1,000 lbs gross weight Pa 105 hp Assume the body of the helicopter may be and its rotor hub are exceptionally well streamlined so that classified as low drag helicopter Find the level high speed and maximum sea level rate of climb of this helicopter
is
TECHNICAL AERODYNAMICS
1717
/
Solution . Calculate w = 1,000 x 13.521.75 ; v = 6 ; 1,000CT = 42w v2 x 1.75/ 36 = 2.04 . Calculate also Pa / Ph = 1.67 . In Fig . 17:11 above = 1.67 , read L = 0.41 and above QR = 600 read a level high speed Pa Ph of about 165 mph . Note , however , that the hovering power chart showed a retreating blade stall limit of 160 mph . The top speed would be , there fore , 160 mph at which point roughness would develop and the helicopter could not safely be flown at the level high speed corresponding to wide
= 42
/
throttle . In Fig . 17:12 , above Pa / Ph = 1.67 , interpolation between the two low drag lines given permits reading Ch max / 33 Ph = 1.1 . The maximum sea  level rate of climb for Ph  60 is read as 2,100 ft / min ; at Ph = 80 read Ch = 2,900 ft / min . Interpolate linearly to the design value of Ph = 63 , and get Ch = 2,220 ft/ min . These are the answers called for . By calculating rate of climb at each of a number of altitudes , plot ting rate of climb against altitude , and reading the intercept where rate open
of climb equals
zero
solute and " service
ceilings are
"
forward
some practical value such as 100 ft / min , the ab ceilings of the helicopter can be determined . These flight ceilings ; they are substantially higher than
or
the hovering ceiling determined 17 : 7 .
AUTOROTATIVE DESCENT
equipped
with a " free wheeling
tion
from the hovering chart OF HELICOPTERS .
minimum
rate
descent
the pilot
rotor
stops .
the engine
is
are
usually
and the
tail
rotor
will con
Rotation at a favorable speed for
greatly enhanced by prompt action on the part of
the engine
, when
Helicopters
device between the engine and the reduc
"
gear box , so that both the main
tinue to rotate after
.
stops ,
in setting the collective pitch
down
to
position . In most helicopters the " rate of stable auto windmill is not far from that of the design level flight
a good windmilling
rotation "
as a
, though
condition
flow conditions
flight . since
trol .
air
there may be a substantial difference because the windmilling are quite different from those of level
when
is
The condition of windmilling
all
the main rotor
Under windmilling
must
the middle of the blade radius
it
blade drives
is
drive the
lift
, and the
blade
a large
of
there
,
forward , overcoming
the inner third of the the presence
do
conditions
approximately that of zero torque ,
is .
If
the
minimum
, the rate of descent power required for level гh
of the " valleys
"
in Fig
.
17:10 )
rotor to maintain
stalled
Approximately
under these conditions ;
a handicap
to the
calculation of
the rate of stable autorotation
may
flight
be
flight
is supplied
con
force on this portion of the
be assumed to correspond to the design level
sign value of
tail
a high angle of attack near
the drag of the tips .
usually
stalled area is
the rate of stable autorotation
is
condition at the
can
de
calculated by assuming that corresponding ( to the bottoms
by gravity .
This assumption
HELICOPTER PERFORMANCE
is
admittedly poor
17:10
,
and
(approximately
imum rate of descent
the horizontal .
In
violates the conditions
level flight )
, because the
in
are usually
1718
in
assumed
flight
the neighborhood
path angles at
of
a few cases , however , nearly correct
ues have been obtained by
this
method and the
deriving
results of
45 degrees
flight
these
Fig .
min with
test val
assumptions
800 700
QR, feet per second
600 500 400 v 300
60 50
/02/3.1
9 10
20
MinimumSinking Speed, feet per second
:/ Nort
40
041
10004
Lines
for
h=2
/
0.006 CT 0.004 CT 0.0015 S01e7
30
Lines
for
Drag gh Drag Yow Drag
20
h
=3
cois
HighDra Low
15
Glide test check points on Bell H 13 B plotted x = Calculated = Flight Test
Drag Low 10
Drag High Drág Low
o
w, pounds per square foot
8
3
Fig .
ڈن 4
5
6
7
8
Simplified approximate calculated min 17:13 . imum sinking speed of helicopters in autorotative descent at the autorotative speeds specified above . are therefore tionary words
presented
in
graphical form
" simplified approximate "
in Fig .
17:13 , with the precau
title . A analysis of this maneuver is sorely needed as a design cri as the providing of shock  free landing without power is considered in front of the figure
more adequate
terion , to be one of the principal sales advantages of helicopters compared with other forms of transportation . A rough check on the general validity of
TECHNICAL AERODYNAMICS
1719
Fig .
rates calculated by this well established rate of vertical descent given by
17:13 may be made by comparing
method with the
fairly
the descent
the equation ( 17:26 )
Vy = 29 VW where Vy Note
is
clined glide descent
the vertical descent
in Fig .
is
speed
17:13 that the minimum found to be
in ft /sec . sinking speed for the optimum
between 40 and
60
in
per cent of the vertical
speed . PROBLEMS
17 : 1 . A helicopter weighs 2,000 lbs and has a rotor diameter of 34 The solidity is 0.0625 . The normal rated tip speed is 640 sec . Using Fig . 17 : 8 , calculate ( a ) the power required to hover out of ground effect at sea level ; ( b ) the retreating blade stall  limit speed ; and ( c ) the ratio of hovering induced power to hovering profile power . (d ) Will the advancing blades be shock  limited at the stall  limit speed ? 17 : 2 . Consider that the helicopter of problem 17 : 1 has a body drag corresponding to the " low drag " line in Fig . 17:10 ; calculate ( a ) the power required for level flight , and ( b ) the forward speed at minimum which the power is a minimum . 17 : 3 . The helicopter of problem 17 : 1 is driven by an engine rated 240 hp at sea level . Assuming 15% losses in gears , cooling blower , and tail rotor , use Figs . 17:11 and 17:12 to calculate ( a ) the level high speed at sea level , and ( b ) the maximum rate of climb at sea level . 17 : 4 . Calculate the rate of climb of the above helicopter at 10,000 standard altitude , assuming the engine power varies with altitude as specified by equation ( 15 : 1 ) . 17 : 5 . Using Fig . 17:13 , estimate ( a ) the minimum speed of vertical descent at sea level , and ( b ) the minimum speed of gliding descent at sea
ft/
feet .
r
ft
level .
CHAPTER
18
AIRPLANE PERFORMANCE
18 : 1 .
SPEED
CLIMB
AND
OF
PROPELLER  DRIVEN AIRPLANES .
in
such as have been presented
Chapters
From tabular
of power calculations power airplane flight required for level available from an en and of an gine propeller unit , it is customary to plot power required and available ,
speed and solve graphically
against
designated by the subscript
in
the manner outlined
( ) L,
for the
speed of
maximum
and the maximum rate
in Tables 14 : 2 and Figure Fig . 18 : 1 . in
requirements were calculated
available
are reproduced from the engine
titude the power  required
level flight
of climb (Ch
,
max )
below .
For a particular airplane , the Lockheed Lodestar 14:18 , and
14 and 16 ,
 propeller units in
as a parameter , as determined
level flight
14 : 3 and 18 : 1
power
plotted in Fig .
also shows the power
as a function of speed with
Chapter
power available graphs
and
,
15.
The
any given
of
al
intersection of
maximum speed of level flight at that altitude . Fig = 250 at sea in . 18 : 1 , read mph 2500 level and 280 at 10,000 ft . For
resents the
altitude rep For example ,
full
Ch The rate
being given by
= 33,000
W
(18 : 1 )
500
alti
400 300 70 80 90100 125 150 Mph
a given
tude varies with the
airplane
speed
in the
in Fig .
18 : 2 ,
which
manner shown
also
maximum
climbing
shows
the
definitions of max and
best
Since Fig .
18 : 2
rate of climb Ch speed mph .
Max.X710hp at sealevel
700 600
X
of climb at
the
1000 900 800
181
Fig
200 250 300
18 Power required and available at sea level and 10,000 ft for Lockheed Lodestar airplane
.
ft /min
climb in equation
climb , the rate of
1 .
will
:
the airplane
ft
10000 level Sea 10,000
18 : 1 , and
.
in Fig .
as indicated
1500

for level
.
flight
over that required
,
available
2000
Horsepower
throttle flight at speeds lower than mph there is an excess horsepower X
TECHNICAL AFRODYNAMICS
182
is
15
$
a
max.
I
18
is
point
a measure of the angle of climb , and a line drawn through the origin tangent to the graph is the maximum angle of climb , as any
10
5
plot of vertical vs. horizontal speed , line drawn from the origin to the graph at a
Best climbing Max.angle speed ofclimb 100 200 Mph
shown
in Fig .
18
: 2.
Choice of the
Lodestar as an airplane 300
example
in this text
in
sonably appropriate
Fig .
18 : 2 . Rate of climb at sea level as a func tion of speed , for Lockheed Lodestar airplane .
availability in Fig .
for
is
an
Lockheed
illustrative
considered to be view of
its
rea
current
modified form , as shown
in
18 : 3 .
0 Leerste
Fig . 18 : 3 . Modified Lockheed Lodestar ( " Learstar " ) , distributed in 1955 by Lear , Inc. of Santa Monica , Calif . ( Courtesy Aero . Digest , March 1955. ) 18 : 2 .
formance
full  throttle per plotting the maximum Fig . 18 : 4 . For airplanes
CEILINGS OF PROPELLER  DRIVEN AIRPLANES .
of
an
airplane
is
commonly
The
summarized by
rate of climb Ch max vs. altitude , as shown in with gear driven superchargers , the graph of Ch max vs. altitude is very nearly a straight line above the critical altitude ; for airplanes without superchargers , it is a straight line from sea level up . For any airplane
is
of climb is zero , ceiling airplane Since the rate and this is known as the absolute of the . of climb is zero , the absolute ceiling is attainable theoretically only in infinite time . A ceiling can be attained only if there is a positive small rate of climb ; it is customary to define the service ceiling as the there
always
altitude at
in
some
which the
altitude at
which the maximum rate
rate of climb
is
100
unfavorable weather or over mountainous
craft a true
in
maximum
terrain
and
commercial
flight
for military
100 ft /min rate of climb feasible operating altitude of
formation , this arbitrary
picture of the
ft /min . For
air
does not give the airplane ,
AIRPLANE PERFORMANCE
it is
and
ceiling
accordingly
for
customary
some
183
purposes to speak of an operating
higher rate of climb , usually set at ceilings 300 to 500 ft /min . These three are shown in Fig . 18 : 4 for the Lockheed Lodestar airplane . corresponding to a
35 30
somewhat
35
Absoluteceiling, 29,800 ft ft Serviceceiling, 28,200
30
25
ceiling ,26,000 Operating ft.
20
Critical Typical altitudeof climb variationwith engine unsupercharged engine (130hp CubCruiser )
h 100015 10
25 20 1000 15
5
2.4
6
8
/
10 12 14 16
0
Ch max 100
18 : 5 . Variation of stalling speed , best climbing speed , and level high speed with altitude for
Lodestar airplane . The speed
of
 shaped
is
and
it
and outside
in Fig .
shown
area of speed
Lodestar airplane
Figure
18 : 5 .
altitude within
fly .
.
its relation
of climb in
rate
maximum
and level high speeds
YAYL 50 100 150 200 250 300 V
Fig .
Fig . 18 : 4 . Graphical determin ation of ceilings for Lockheed
dome
LVC
10
5 0 0
Vs
to the stalling
18 : 5 shows a
which an airplane
typical can
fly
exists be
Such a dome  shaped
of which cannot area cause the stalling speed increases with altitude and , above the critical altitude of the engine , the level high speed decreases with altitude .
Below the critical altitude of the engine , there is seen to be an increase in level high speed with altitude , and this increase is usually represented by a straight line of slope such that the level high speed increases almost
alti
exactly
1 per cent for each 1,000 ft of altitude up to the critical This feature of the performance of supercharged airplanes is what made possible a new order of high  speed performance unattainable with un engine airplanes . supercharged For example , an airplane that will reach
tude .
a
level high
speed of 300 mph at sea
of nearly 400
mph
To determine and
h₂ ,
18 : 6 .
it is Since Ch
if
the
its
engine had a
time At required
level
critical altitude to climb between
convenient to replot Fig . max
= dh
At
/dt , it =
իշ
18 : 4
in the
high speed
of 33,000 two
form
ft .
altitudes hi shown in Fig .
follows that
Sh² at
ել
would have a level
dh =
իշ
Sh1
1 Ch max
dh
( 18 : 2 )
TECHNICAL AERODYNAMICS
184
If Ch
30
titude 20
arbitrary varia tion of rate of climb with altitude , the time for
10
/
Fig .
in
0.3
0.1 0.2 1000Ch max
termining time to climb between
altitudes
two
CL , as
tween CD and
in Art .
quired

Assuming a parabolic
the development
charts for
of general
power
from the engine
opofv3 + 550 Thp = 2
in
which may be put
Equation =
2 w2
( 18 : 3 )
поρ еb² (V)
the form
550 =
v3
840
840 (Thp / of )
π
W
1
eb
σThp / V
( 18 : 4 )
permits solving for the level high speed of the airplane
( 18 : 4 )
VL/ 1.467 in terms of the parameters
/1
and Lt
is called is called
( 18 : 4 )
has
where Lp
abscissa
plot
re
for level flight to propeller unit , it may be
that
shown
mph
AIR relationship be
the power required
equating
available
in
may
Fig .
PERFORMANCE OF PROPELLER DRIVEN
PLANES .
14:10 , and
the thrust horsepower
general , for any
to climb between two altitudes
minutes
18 : 3 .
.
in
t ; in
be represented as an area , as shown 18 : 6 , and integrated graphically .
Plot for de
18 : 6 .
as
straight line can be substituted in equation ( 18 : 2 ) resulting in a logarithmic expression
Min time(minutes to climb from 10,000 ft.to20,000ff
1000
,
plots as a straight line vs. al in Fig . 18 : 4 , the equation for 8
max
,
( 18 : 5)
= W eb2
( 18 : 6 )
Lt
= w Thp
( 18 : 7 )
/ /
the " parasite loading , " Lg is called the " span loading , " (1) Equation horsepower loading " by Oswald .
the " thrust
been plotted and
↳p = W
Lg
LL /
in Fig .
18 : 7 with mph
o as parameter .
(2 ) suggested by Perkins and Hage .
for the level high
This
is
Fig .
speed of an airplane
in
as
18 : 7
terms
/
ordinate , Lp oLt as of the Oswald
a modification
permits a quick solution
of the
maximum horsepower
available without calculating the complete charts of power required and available , as in Art . 18 : 1 . The equation on which Fig . 18 : 7 is based ( 1 )Oswald
, W.
B.
Airplane Performance
(2)pe Perkins ,
C.
" General ."
Formulas
NACA TR 408 ,
D. and R. E.
Hage ,
and Charts
for the Calculation
1932 .
op .
cit . ,
p . 168 .
of
185
AIRPLANE PERFORMANCE
500
400 mphL
( Uncorr
for
.
compr . )
300 250
LsLt 200 100 200
150 Lp
It
120 20
Fig .
40
30
propeller driven airplanes , neglecting Level high speed of Prppell compressibility corrections . ( 1 ) See Fig . 18 : 8 for corrections .
tions
no compressibility
and
correcnecessary to take
it is
separate account
of the effect
high
on
Mach
the
mph .
number
This
if
fect
on wing drag
is
and
.
the major
500
critical
Mach
wing , additional
in Fig .
.18 10
.12
Amph
O
ef
10
com
35000
20
AmphL
numbers than the
are involved
.
18
20000
20
15
ft .
t/c=
0
10
ft .
t/c =
,
, have
corrections to those
18 : 8
(1 ) ,Replotted
of
If other items
nacelles or cockpits
0
speed
the compressibility
pressibility effect
shown
400
of
can be done by the use
18 : 8
such as
level high
region between
Fig .
lower
1000
500
300
18 : 7 .
involves
in
200
100
50
Amphi
 CL =0.2 C₂ =0.4
SEA
0.18
LEVEL
Uncorr .
mph
from Oswald , W. B. 200 450 TR 408) as modified by Perkins , 500 C. D. and R. E. Hage , " Airplane Per Fig . 18 : 8 . Effect of wing thickness formance , Stability , and Control . " ( unswept ) on compressibility correc Wiley , 1949 . Chapter 4. tion for propeller driven airplanes ( 1 )
( NACA
TECHNICAL AERODYNAMICS
186
Similar charts are available for maximum sea  level rate of climb and for absolute ceiling . The Oswald charts for climb and ceiling of airplanes with unsupercharged engines are shown in Figs . 18 : 9 and 18:10 . Figures airplanes in 18 : 9 and 18:10 show the climb and ceiling for unsupercharged terms of the ceiling parameter A = LgLt4 3 / Lp1 /3 . The climb chart uses
/
altitude as a parameter as well as the design Cg (CSm) of the propeller . The ceiling chart has a single family of absolute ceiling charts for var ious values of Csm of the propeller . The absolute ceiling corresponds , by definition , to the service ceiling with L = 0. For other values of Lt designated
on the
responding to
100
18 : 4 .
chart
ft / min
of
ting drag and
rate of climb .
turbojet
a
cor
service ceiling
is the
shown
OF TURBOJET PROPELLED
PERFORMANCE
performance
ceiling
, the
AIRPLANES .
The full
is
analyzed by
propelled airplane
thrust against speed .
commonly
is
high speed
The level
of the graphs of airplane
throttle
plot
determined
thrust available , just as speed for a propeller  driven airplane was deter mined from thrust horsepower required and available . A maximum angle of from the
climb
is
intersection the level high likewise
drag
thrust available to
from the maximum excess
determined
of gravity along the flight
overcome a component
and
path .
Charts for level high speed , maximum rate of climb , and absolute ceiling of jet  propelled airplanes may be developed in a manner similar to those
It is
customary to
/
to
make
drag gives (
18
98
W2
8292
solved graphically
subsequently
18
(
q 1
L
Ls
1,000
as
made by means
in Fig
18:11
.
and
1,000 π
of Fig
.
can be plotted
with compressibility correction
=
+
W
feb²
q 1
W
,
: 9 )
(
18
= q +
1
T Equation
(
thrust
to
written π
which may also be
= D =
+
T
equating
,
)
speed calculation
1/8
level high
For the
, and
)
.
8
for propeller  driven airplanes
:
18 : 3
assume the parabolic approximation CD (CL2 ) for this purpose subsequent correction for compressibility .
18:12
.
in Art .
: 9 )
developed
usually cover the range from 400 to 600 mph instead of the 400 to 500 mph range common for propeller driven airplanes jet propelled airplane assuming The maximum rate of climb of con corrections
stant thrust Ta
and
.
a
,
√πT
,
is 18:10
)
eb2
path
(

/

D1 = Ta
flight √1
forces along the
2
balanced

a

Compressibility
17
6 16 5 15
4 14
3 13 2 12
F11
10
.
9
Ch +
L
8
7
6
5 сл
4
3
3 3 4 +/
15
=
10,000
117
9 8 7 6 5
2
1
0
1 airplanes TR
/
NACA
with 408.
Fig
:.9
.
Oswald unsupercharged
. . with
18:10
0
for
A 9 8 7 6 5
(
. From
T
chart
Absolute engines
ceiling
20
25 30
40
0
LT =
50 60 70
chart for airplanes From NACA TR 408.
28474 Lp13
=
1
climb
ceiling
10
15
engines
== =
50 60 70
/
Oswald
40
.. ..
30
3
unsupercharged
20
Service
284 Lp
5000
10,000
15,000
ceiling
18
ft
10
feet Ceiling = H
Peak eff prop All Csm Best perf.prop Csm 1.6 " Csm 1.2 " " Csm0.9

Fig
level
1000
20,000
25,000
30,000
dP 1.00 aN
..
5000
Sea
10
35,000
=
dN
dP
Peak eff prop All Csm Best perf prop Csm 1.6 Csm 1.2 Csm 0.9
AIRPLANE PERFORMANCE
187
)
(
.
)
188
TECHNICAL AERODYNAMICS
.
600
ft
9
500
at
10.
35000
8 7
mphy
6 5
( Uncorr .
91 100
for
comp .
400
700
3
600
776
2.5
2
300 500
1.5
250
Ipa 100g
T100
10
Fig tion
400
2.5
1.5
200
3
Level high speed of turbojet propelled airplanes from equa neglecting compressibility corrections See Fig 18:12 for compressibility corrections ,
.
.
.
: 9 ) ,
18
(
.
.

18:11
where
/
18:11
(
ZW SCL1 Po
)
V1 = and
obtained by neglecting 18:10
from
,
sea
(
/0
18:13
level
The condition
)
.
0
at
emp
.
/(T
)
.
=
the
.
(
can be determined

a
,
Temp
(
and
=
Po
/
=
P
where
σ
Ta To
)
(
V1 may be

term
)
first
approximation to Ch
of the right hand member of equation jet propelled airplane The absolute ceiling of the consideration that as shown in Fig 15:21
second
)
(
/s
hand member
/
a
,
first
of equation 18:10 is often small jet airplanes for with adequate thrust for take
of the right
compared with the
off so
18:12
).
term
VAеf

The second
=
CL1
for
AIRPLANE PERFORMANCE
t/c
10 Amphi
35,000
ft
O
t/c
10
17
20
20,000
10 Amphi
·CL
=
0.2
CL
=
0.4
Uncorr .
ft .15
t/c
=
. 12
18
mphL
400
Fig .
= .12
.15
18
Δmphi
20
.12
.15
18
20
=
189
SEA
LEVEL 600
500
Effect of wing thickness ( unswept ) on compressibility correc tion to level high speed of jet  propelled airplanes . ( 1)
18:12 .
absolute
ceiling
( altitude
for zero climb)
(0/0 )ch= 0
/
is
/
2 √£ ob2 √π To W
=
/
( 18:14 )
To W in the range from 0.2 to 0.3 , which are necessary for reasonable sea  level take off , absolute ceilings of 50,000 ft to 60,000 ft are quite common , but are rarely used in civilian jet aircraft because of
For values of
the high pressures necessary for cabin supercharging which impose a se vere structural penalty on the fuselage design . The use of the foregoing methods in calculation of the full throttle performance of the civil jet airplane ( Morane  Saulnier 760 , shown in Fig . 18:13 , is illustrated by the following example . Example . For the airplane shown in Fig . 18:13 , assume the following data and calculate ( a ) the level high speed at sea level , ( b ) the maximum rate of climb at sea level , ( c ) the level high speed at 20,000 ft standard altitude , and ( d ) the absolute ceiling .
( 1 )Replotted from Perkins
, C. D. and Hage , R. E. ,
op . cit ..
Chapter
4.
1810
TECHNICAL AERODYNAMICS
lb
W = 8,480 e = 0.85
ft2 b²/s = 5.6
S
= 194
A
=
Rated thrust at sea level To Thrust variation with altitude as =
Solution . ( 1 ) Calculate
VAеf /S
=
2f /S
=
CD1
=
D1
in Fig .
=
√ x
33.3
ft
( 2 Turbomeca Marbore 15:21
/
5.6 x 0.85 x 4.4 194
.
engines ) .
= 0.584
= 0.0453
= 582 CD1 W = 0.0453 7,480
0.534
/
12W/ SCL1 ° Po =
V10
=
at (L/D) max from equations ( 18:11 )
the conditions
and ( 18:12 ) thus CL1
= 1,760
b
f = 4.4 ft²
1/840
x
7,480
lbs
/194
x
0.584 = 235
ft/ sec
( 160 mph
)
WGVOE
F
Fig .

Morane Saulnier 760 four  place civil turbojet airplane , in in 1955 at Beech Aircraft Corp. plant in Wichita , Kansas .
18:13 .
production
(2)
Plot
(3 )
W
V1 =
(4 )
on
mph
Ch = T
Ch

0.157
=
D1 =
1,760
L1
7,480
x
At 20,000
ft
D1
14:19
/
0.235.078
ft/min
=
0.62 x 1,760
=
1,909 582
=
= 0.157
( checks published 2,260
standard altitude , read in standard 0.86 , 0/0 = 0.62 , and calculate
Ia =
Ta a
In Fig .
0.0453 = 0.584
235 x 60 = 2,230
0.533 , 0 = 4470/5180
mphL =
/
/
/
Fig .
14:19 : Ta D1 = 1,760 582 = 3.03 and read VI V1 2.45 x 160 = 394 mph at S.L. From equation ( 18:10 ) calculate at sea level
2.45 , hence
ft /min )
air table
σ =
= 1,090
1.88
/
read V V1 = 1.85 . Calculate V₁ = 160 10.533 = 219 mph , and = 405 mph (checks published value ) .
1.85 x 219
1811
AIRPLANE PERFORMANCE
( 5 ) For the absolute ceiling
//
2 Vf
=
(0/0 ) ch = 0
V1
eb2
, =
To W
from equation ( 18:14 ) calculate
/
2 14.4 0.85 x 33.32 1,760 7,480 Υπ
=
/
0.33
ft
From standard air tables , for o / 8 = 0.33 , read absolute ceiling = 39,000 ( published value 36,000 ft , presumably based on a slightly less favorable variation of thrust with altitude ) . (6 ) As a check on the level high speeds , use Fig . 18:11 . Calculate
first
Lp
7,480
=
L's
7,480
=
=
1,000 x 4.4
1,000
LpLs
7.95 x 1.70
=
1,000
7.95
=
0.85 x 33.32
/f
/
1.70
=
13.5
= 1,760 sea level , with T 4.4 read in Fig . 18:11 mph = 390 mph . 20,000 altitude , with T / = 1,090 44 = 248 and the same value of = 0.533 calculate LpLs , read qL = 225 , and with o
For For
ft
/
are the answers called for
to
x 25.6
is
" Range "
endurance .
Airplanes are
are of
endurance
it is
customary
and endurance
for
is
to
make
can
corresponding time
the
for both
importance
proposed
commonly cruised at the interests of long range
in
the distance an airplane
running out of fuel ; " endurance " major
= 405
.
75 per cent of rated power or thrust
or long
and
/
CRUISING RANGE AND ENDURANCE .
18 : 5 .
50
/
= 1001225 0.533
mph
These
f
civil
and
.
fly
without
Range
and
military operations
,
extensive and careful calculation of range designs . The basic equations for these
new
calculations are Range
R =
= E =
Endurance
is
where AW
the weight of
AWf
fuel
AWf
calculations in Chapter Propeller  driven Airplanes .
equations
,
usually attributed
consumed
,
and the rates
/
of fuel
consump
and or engine data as
in the
16 .
The basic incremental range and endurance
to Brequet
dR = 375 dE =
( 18:16 )
(lb fuel /hr) av
tion per mile or hour are based on propeller sample
( 18:15 )
(lb  fuel /mi ) av
L
CD
,
are
dw
( 18:17 )
/2
W
18.91 C13 CD
( 18:18 ) 33
1812
in
TECHNICAL AERODYNAMICS
which
dW
consumption
is
the change in airplane
if it
weight
fuel
differential
due to a
Both of these equations can be conveniently integrated
dwp .
is permissible to assume CL , CD , n (propulsive efficiency ) , C (brake specific fuel consumption ) , and σ ( altitude density factor ) constant be tween the limits of integration . While the optimum values of these fac tors are not truly constant , it is customary to assume reasonable mean
).
)
( W₁
(
18:20
1 )
√W
/
08
18:19
(
D
 37.9
=
Exax
giving
,
10910 Wo
HIA
 863.5
)/2
Rax
the equations
)
order to integrate
023
in
them
(
values of
Wo
80
)
70
hours
10
for
60
1.89
60
.
80
= 1.89
100
.85 for 7 = .45
12
Endurance
,
J
in
mi .
1000
=
Range
1
(a)
14
(b
16
50
HA
20 8
40
6
30
S
20
4 =
5

).
in Fig 18:14 for typical val is plotted for the no wind
Figure 18:14
"
.
/
"
0.451.89 no effect on the endurance but it is customary to as function of altitude for various values of winds ex
18 0.85
Wind has
at each altitude in order to determine the operational using in general the optimum cruising speed for the airplane ,
which
is different
from the optimum
cruising
speed
in
utility
of
each wind
still
air
.
range
pected
velocity
18:20
(
and
.
are plotted
18:19
)
) ,
(
C
/. ¶
)
(
)
(
18:20
.4 .5 propeller d riven for
.6
a
calculate
.
n
/c
of
condition
and
root scale
)
(
.7
.8
1.0
.
.
ues
18:19
.
/ .4
Cruising range and endurance no wind 18:14 See equations airplanes with typical values of Equations
Sq
0
)
WO
(
Fig
.5
.6
.8 .7
1.0 .9
W1
T
10
Log scale
0
Wo
(
/
W1
2
1813
AIRPLANE PERFORMANCE
pounds
08
/ L
C11
MP
CD
W
18:21
dw
18:22
(
D W
10910
(
/
)

(
WO
W1
1
18:23
Wo
18:24
)
)
max
to
(
2
(
)
Wo
CD
for typical
val
W1
in Fig
are plotted
18:15
.
(1) 18:24
:2
and
18:23
(
(

2.3
Bmax
Equations
1
/

39.6
Rmax
GL1 2 )
assuming constant value of C ' , CL , CD , and σ , integrate
,
equa
endurance
and
)
1391W
=
dE
=
as
range
)
to write the basic incremental dR
which
Since turbojets are characterized by a fuel per hour per pound of thrust ,
.
in
C'
)
Airplanes
consumption
customary
tions
of equations
(
propelled
specific fuel
is
by means
and ( 18:20 ) .
Turbojet
it
/C
to other values of
18:14 can be corrected
( 18:19 )
/2
Figure
8
)
.
(
(
)
.
,
,
C '
Graphs for other values of these parameters may ues of CL and CD be constructed by use of equations 18:23 and 18:24 As for propeller
'
C
CL CD
Endurance
,
14
1.10
0.4 0.025
'
for
=
hours
1.10
L D
10
100
5
= = =
for
6
(
in mi
1247
1000
C
Range
.
7
b )
(a )
16
8
4
80 60
3
6
:
40
.7
.6
.5
.4
.8 .7
Log scale
.6
)
/
W1
1.0.9
Wo
(
J
0
1
1.0 .9
Fig
.8
root scale
)
Sq
(
W1 WO
O
.
/
1
2
2
4
.
Wg
.4
.5
).
18:24
(
and
)
(

)
.
(
,
C ' ,
.
.
Cruising range and endurance no wind of turbojet propelled 18:15 airplanes with typical values of CL and CD See equations 18:23
TECHNICAL AERODYNAMICS
1814
calculations are customarily made at various altitudes with various head and tail winds at each altitude to determine opera
driven airplanes and
,
utility .
tional
in general the comparative range and and 18:14 . For a given ratio of empty
Note 18:13
endurance weight
to
in Figs .
scales
original weight
(W1 /Wo) , the range and endurance of turbojet  propelled airplanes are only to 40 per cent of the corresponding values for propeller driven air planes for the no  wind condition . With the usual adverse winds , the jet 30
airplanes
show up
In
more favorably .
" long
general , however , the terms
range " and " jet propelled " are contradictory . Elaborate techniques for refueling in flight are necessary for " long  range jet bombers . " For long range commercial
transoceanic hops
order of 3,000 miles ) there
( of the
is
considerable question as to whether the small pay load remaining ( after the necessary fuel is pumped aboard ) will willingly pay the premium for the
mph
100
 is
DC 8
18 : 6 .
cause
extra
speed .
TAKE
 OFF
primarily
a take off runway length take
is
CALCULATIONS .
take off distances
plane designed
if
(This
to say
the financial success
of the
not assured . )
off
off calculations are
often limiting factors
from
flight
in excess of
occasionally
must
Take
are
is likely
considerations commonly
those
be made from an
important
in design .
An
be air
to require
available , especially a high altitude
airport at
level .
above sea
Take off calculations
difficult
accurately because the take piloting technique relative to off distance varies considerably with the the ground and wind conditions , and because the principal forces involved cannot
are
be very accurately
distance
velocity ,
,
tions of velocity equations ,
it
and
and
to
estimated .
acceleration
acceleration
follows that take
As
off
in
any simple problem involving
basic equations are the
, the
V
:
make
= ds
/dt
gration
VTO
/
and a = dv dt .
distance can
be
defini
From these
obtained by the
inte
V
( 18:25 )
a = Net Force Mass
(18:26 )
xto
=
√
V dv
where
and VT
is
the take  off speed in
of the stalling partly down at
It is
speed
in the
a predetermined
customary to make take
ft / sec ,
which may be any speed
take off configuration optimum
( usually
in
excess
with flaps
position ) .
 off calculations
from a standing start to
AIRPLANE PERFORMANCE
a
point
18:16 .
xtı ,
shown
,
and a climbing
in Fig .
, as shown
as
distance xc
= xg + Xt1
(18:27 )
+
general , the lower the
take  off speed
will
high could be cleared
into which it is customary to analyze the take  off in Fig . 18:16 to be a ground run xg , a transition dis Xto
In
ft
50
The components
distance are tance
obstacle
an
where
1815
the
,
shorter
total horizontal distance to clear an obstacle . An airplane can actually be be the
pulled off the
round
at
a
speed somewhat lower than the
stalling
calculated
speed
Ground run
ob
tained by using a wind  tunnel
test value of CL a higher under
value
again .
To avoid " mushing
practice to specify take 20
conditions
involves
 off stalling
per cent
, hence
a take
a
in"  off
Components
of
Xc
take
 off
that
speed at some
arbitrary
This margin of safety
common
per cent above the
is
commonly set at
to assume ( 18:28 )
VTO = 1.20V STO A smaller value
of
dis
the airplane will settle down after a too early take  off , it is a common
hazard
speed VSTO
it is
18:16 .
50ft.
h₁
tance over a 50 ft obstacle . of increasing angle of attack . This pro
obtainable
the transient
cedure , however ,
Fig .
climb 10,
·Xti
Xg¨¨
max because
is
Transition V=VTQ
VÃO may be assumed
if it
is
desired to get a shorter
take off distance with a correspondingly higher risk . Forces acting on the airplane during the three phases of the assumed take off are shown in Fig . 18:17 . An analysis of the horizontal distance
for these three
phases
is
made
separately in the following three para
graphs .
L=1.44W O R=μW
Fig
a. .
N=W
вер
W

Tc Dc Vw
C. Climb b . Transition Ground run phases of assumed 18:17 . Forces acting on airplane during three take off .
TECHNICAL AERODYNAMICS
1816
(a )
Ground Run .
tion of the three forces T ,
Equation ( 18:29 ) assumes that
is
tion for
most favorable
very high
( as
R = μW ,
D , and
T ually the
accelerates horizontally under the
The airplane
·
D
lift
· is
IEa
HW =
(18:29)
zero during ground run
condition unless the coefficient of
field )
on a muddy
any wing as low as the
ac
so that
is
since there
is us
, which
fric
rolling
/
of L
no value
coefficient of rolling resistance of wheels
D
on
surface . Typical values of rolling resistance coeffi cient for braked landings as well as for rolling take  off , are shown in Table 18 : 1 . is often not possible to take off at zero because of a reasonably hard
It
lift
TABLE 18 : 1 Brakes off , average ground
Type of surface
Concrete or Hard
turf
Firm and dry . · Soft turf · Wet concrete . · Wet grass Snow or ice  covered
zero
lift .
placed by ground
may
effect )
is
0.07 to 0.10
set on the airplane
prevent take off with
in
as calculated
( 18:29 )
the
Chapter
nose
, as
landing gear
the
sufficiently
a take
The acceleration
9.
in equation
should be inserted
 off distance
.
( 18:25 )
to get
down
the ground run based net thrust
is
on
=
(T

D

)
and
determined
integrated
To avoid a graphical
quicker but less accurate calculation may be the average net thrust during acceleration by
, a
TNav
age
0.30 0.20
0.10 0.02
field
graphically to determine defining
0.33
In this event , the term µW in equation ( 18:29 ) should be re L) and D must include the induced drag of the wing ( with μ (W
by equation gration
0.4 to 0.6 0.4
0.07 0.05
the angle at which the wing arrangement
coefficient
μ
0.03 to 0.05 0.05
macadam
•
, average
wheel braking Hb
resistance
coefficient
fully
Brakes
applied
made as
corresponding to the aver
1 vto xg = 29 TNav W
( 18:31 )
/
Studies of the relationship between average
:
(18:30 )
HW av
a constant acceleration
inte
follows
and
initial
net thrust
during
1817
AIRPLANE PERFORMANCE
in
by Diehl , (1 ) show that the
equation
is
18:18 ) , and
( 18:31 )
if a
initial
correction factor
used instead of 1 / 2g
Ig
in equation Kg vto
net
/
=
For
18:32
TNI W
accurate calculations
more
should be integrated
in Fig .
giving
( 18:31 )
Note in Fig . 18:18 that Kg is a function of the ratio of to initial net thrust , thesenet thrusts being determined 18:19 .
thrust TNI can
Kg (as plotted
)
be used
made
(
take off , as
graphically
, the
net thrust
final
net thrust
in Fig .
as shown
shown
in Fig .
18:19
. Tor D
0.040
Thrust
0.035
available
0.030 TcDc
Ks 0.025
TNI
flight level Drag
D+μW
/
Fig
Distance .
.

Variation of forces dur showing the values of necessary for use in Fig
.
TNI and TNF 18:18
transition
The
from the level
ground run to
given by the equation 01 =
Tc
Dc
18:33
W
take place at constant normal
to
)
81 ,
may be considered
off
(
(b) Transition climb at an angle
18:19
ing take
Fig . 18:18 . Diehl's ground  run coefficient for use in equation (18:32 ) .
.
0.4 0.5 0.6 0.7 0.80.91.0 TNF TNI
,
0.3
μW
.
0.015 0.2
2
Takeoff
0.020
acceleration
due
to
lift
a
,
a
.
)
(
18:35
W
revised edition
)
vio 0.44g
(
=
Engineering Aerodynamics
"
W. S.
18:34
approximately
Xt1r91 ( 1 ) Diehl ,
vio
0.44g
.
is
r=
Dc
the horizontal distance
is
, "
path
To
flight

dius of such
a
a
at
operating CL max while traveling speed VTO 1.2VSTO giving equal to 1.44w and The ra normal acceleration of approximately 0.44g
1818
TECHNICAL AERODYNAMICS
and the corresponding
vertical distance
is
x+19
b1 =
In
the distance hi
cases
some
which
will
be calculated
to reach
50
50
ft ,
ft altitude
in
. may
relationship
from the approximate
Climbing Distance .
ter than
turn out to be
Itxt1 /50/ (c )
,
(18:36)
the transition distance necessary
case
the geometry of Fig . 18:16
, from
Equilibrium
(18:37)
h1
of
in Fig .
the forces
18 : 17c
for
steady climb at an angle 81 requires that W , Te , and De be related as in equation ( 18:33 ) . The horizontal distance necessary to climb a vertical distance of 50  hy is accordingly xc =
50
(Tc

 hi
(18:38 )
Dc) /W
10
Typical
8
long runways
Take off
7
distance
6
1000
ft
,
Typical short
5
runways
Propeller driven
Jet
4
ft
3
50 Over
2
run
Ground
1 i
W W
1
5
CLTO
100
Bhp 200
or
W W
2
1
I
STOCLTO
300
400
500 600 700
Fig . 18:20 . Approximate ground run (no wind ) and total take  off distance for either propeller or jet airplanes , replotted from Perkins and Hage . ( 1) An over  all picture of the effects of wing loading , power or thrust loading
,
air
density
, and
as studied by Perkins
and
take  off
(1 ) Hage ,
18:20 that high wing loadings
lift
coefficient is shown in Fig .
and high
on take 18:20 .
power or thrust
off
distance
, C. D. and R. E. Hage , op .
cit . ,
p . 197 .
in Fig .
Note
loadings require
take off distances nearly to the limits of current typical runways
( 1 )Perkins
,
.
Figure
AIRPLANE PERFORMANCE
is
18:20
plotted for no headwind or typical small values of headwind
there is
a substantial
shortened
, as
18 : 7 .
headwind , the take
in Fig .
shown
DISTANCE
LANDING
18:21 .
der development ) .
Reasonable landing
tance for airplanes
on ice requires
,
 off
distance
,
0.6 Xq go
dis
,
like
cannot
Diehldata for landplanes
0.4 Xa·=(1 Vw2 X90 SK Forseaplanes fromR&M 1593 )
(
0.2 0
pro
.
Calculations for landing distance
08
either
reverse  thrust jets or reverse  thrust
take
1.0
CALCULATIONS .
sign , particularly with jet airplanes which have no provision for reversing the thrust of the jets (such items are currently un
calculations for
If
.
off distance is very appreciably
Landing distance , like take  off distance is important as a limiting factor in de
pellers
1819
0
Fig .
0.2
18:21
0.4 0.6 VW VTO
/
.
1.0
0.8
Effect of wind
on
take off ground run xg , replot ted from Diehl . ( 1)
be made with very good accuracy because they are capable of major reduc tion by proper piloting technique , particularly the " sideslip " on landing ,
For commercial airliners the sideslip is usually not considered a permis sible maneuver except in an emergency , because it tends to frighten the passengers ; high  drag flaps are often chosen for the primary purpose of avoiding the necessity of a sideslip landing .
of landing distance over
50
foot obstacle
.
Components
18:22
.
.
Fig
Xt
a
Xg
50ft.
,
a
,
rolling distance before
revised edition
.
Engineering Aerodynamics
a
,
, "
S.
,
W.
Diehl
transition distance
, "
ground run
of the
( 1 )
sum
a
.
.
.
a
,

are commonly made for landing as for take off over 50 Typical phases of the landing over foot obstacle 50 foot obstacle are 18:22 shown in Fig The total landing distance is considered to be the Calculations
TECHNICAL AERODYNAMICS
reverse pitch propellers
braking distance which
calculation
is the stalling
based
+
X
( 1 ) that the glide is at a speed of 1.3VSL , speed in the landing configuration ; ( 2 ) that the
a deceleration
if reversed  pitch
propeller braking until the airplane stops . The relationships these are applied
CLA
1.3V SL at
where
Dg
= Dg
)
18:42
(
2
/)
at
18:41
(
1
VSL
Xt =
18:40
CDg
)
50
)
=
Xg
)
(
( and ,
is
used , the reversed thrust
(
are applied
equations that express
from Vg = 1.3VSL to Vg = VSL ; ( 3 ) that required before the brakes are applied ;
is
time of about 3 seconds
( 4 ) that the brakes
is
(1 ) is
V
a
by Crocket and Bonney
recommended
assumptions :
transition involves
rolling
18:39
Xp
V &
where VSL
procedure
Xt
by
as indicated
W + g
on the following
B Xg
+
XL The
jets ,
or reverse thrust
include the use of
may
)
, and a
+
brakes are applied
(
1820
18
assumed
) )
be given
.
18:47
K2
"
Reverse Pitch pp 441447 .
,
,
E.
,
B.
, "
,
to
+
K₁
=
T
.
.
.
)
( (
( F
,
is
Crockett Harold and Arthur Bonney JAS October 1945 as Airplane Landing Brakes )
retard
Subscripts
of the form
Fig 18:23 Forces acting on K₂V² airplane during braked ground run with the coefficients K1 and
(1
the net

by an equation
N =W L
is
18:23
refer to glide and stall respec If the variation of reverse pitch
thrust with speed
μ(W L)
o ) ,
in Fig
(
.
18:46
.
and
tively
shown
.
equation
S
g
18:46
(
In
ing force nT
)
Fg
a W
18:45
)
where
a
)
VSL
(
=
1
Xx
18:44
3VSL V
=
Xr
18:43
CDS ground
V &
1.69C
Dg +
=
(
Xt
W1S
giving
determined
Propellers
AIRPLANE PERFORMANCE
)
(
a
18:46
is
and
applied
integrat
complicated expression for
ground run
braked
propeller
)
( CLS
CD
)/

18:45
integrated
may be
graph
.
)
)
18:49
HDCL
(
equation
Но
(
/
calculation
CD
,
more accurate
ically
loge
CLS
Hb
( a
For
)

HCL
mph
(
0.0334
Xb
2
,
.
For the simpler case in which the terms involving thrust are removed the result of the integration is
)
(
18:48
reversed thrust
into equation
rather
CLOP
+ Hb
on which the
18:48
)
(
K₂V² )
becomes
.
of engines
Putting the force of equation ing equation 18:45 gives
+
force
,
is the number
where ner
ner (K₁
+

)
v2s
= спорот 2
F
the net retarding
,
2
characteristics
(
from the propeller
1821
The forces acting on an airplane
= 0 ,
ΣFx
,
= 0
glide
steady
and 2Fy = 0
(
)
18:50
8
18:51
(
)
sin
W
in
equa
these
e
=
D
shown
,
as
= W COs
L
x
y
.
Since acceleration axes
(
D
)
.
propeller
with and give tions
.
W ,
D ,
L,
.
.
:
8 .
GLIDINGAND DIVING
a
in steady glide without power are shown in Fig 18:24 The tail force is negli gible compared with or but its moment about the center of gravity is not negligible and must be considered when calculations of balance in glide are made The drag force must include the drag of the idling 18
and
.
)
Fig
18:24 Forces acting on airplane in glide
elevator
.
.
,
)
)
D ]) D )
/
/( L /
(
/
sin
e
and
L
(
18:53
8
+ (
e
1.0
SCL
²
0.002560 mph (
=
L
)
(
and
/S cos
sin
are negligible
then 18:54
)
W
0.002560CL
(
mphg =
/S
;
compared with 1.0
glide
,
For small angles of
18:50
W
√0.00256 0.002560CL cos e
mphg
from equation
mph
,
glide
in
of
The speed
,
is
stabilizer
.
settings
by the
is de
The angle
W
.
with the horizontal termined
(
the angle of the path of glide
.
is
√
where
18:52
8
HIP =
D
cot
TECHNICAL AERODYNAMICS
a
vertical dive ,
sin
= 0,
cos
w /s
mphg max
For any
assumed
18:55
0.002560CD min
of glide
angle
9 = 1.0 , and
)
For
(
1822
8,
L / D can be calculated
from equation
corresponding angle of attack , CL , and Cp can be read from a graph of the airplane characteristics ; thus equation ( 18:53 ) can speed of glide is plotted be used to calculate the speed of glide . ( 18:52 ) and the
If
against angle of glide as in Fig
gliding  velocity
10
Stall
18:25 , the
.
resulting
graph
diagram , a
is called
special
a
form of
" hodograph . " The
limiting
speed of
monly calculated
Pancake
in connection with the
lem of determining design loads on the
300
Vertical dive
is
vertical dive
com
prob
wings
of the airplane . Under these conditions drag of the idling propeller is a major
60°
tor
90°
and
can not be neglected .
which the propeller
will
rotate
the
fac at
The speed
idling
when
Fig . 18 : 5 . Polar diagram depends on the friction horsepower of the en of gliding speed vs. angle gine glide if " dead " and on the idling  jet adjust of . Terminal dive calculations for de ment of the carburetor if throttled . termination of design loads for wings are
commonly
made
on the assumption
is or limited to some arbitrary value . Since the propeller drag coefficient depends on V/nD , the propeller drag can not be estimated until the terminal dive velocity is known , hence , the solution for terminal dive velocity must be by trial . governed
on airplane
r is
( 18:56 ) and
cos
L sin
in horizontal
tions
lift
is
= W
18:56
)
L
where
must
flight ,
available for supporting the weight . In Fig . 18:26 , 2Fy = 0 and ΣFx = w gry2. hence
Forces acting
turn without sideslip
a
gr
of bank .
Solving equa
=
18:57
.
is the angle the radius of turn and ( 18:57 ) together gives tan
❤ =
v2 gr
18:58
)
18:26 .
W
L
in
(
Ľx
Fig .
lift
turn are shown in Fig . 18:26 . The be greater in a turn than in horizontal for only the vertical component of the
.;
W
on an airplane
)
R=
The forces acting
,
properly banked horizontal
TURNS .
(
LEVEL AND GLIDING
22/225
18 : 9 .
rpm
(
that the propeller
AIRPLANE PERFORMANCE