Technical Aerodynamics [Third Edition]

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Technical Aerodynamics [Third Edition]

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 NON-FLOW 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. STEADY-FLOW CONTINUITY EQUATION
3:3. STEADY-FLOW ENERGY EQUATION
3:4. STEADY-FLOW MOMENTUM EQUATION
3:5. APPLICATION OF STEADY-FLOW 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. TWO-DIMENSIONAL 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. TWO-DIMENSIONAL AND AXI -SYMMETRIC SUPERSONIC FLOW
PROBLEMS
CHAPTER 6. INCOMPRESSIBLE BOUNDARY- LAYER FLOW
6:1. BOUNDARY-LAYER 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 BOUNDARY-LAYER FLOW AROUND A CYLINDER
6:8. INCOMPRESSIBLE BOUNDARY-LAYER FLOW AROUND A SPHERE
PROBLEMS
CHAPTER 7. COMPRESSIBLE AND HEAT- CONDUCTING BOUNDARY-LAYER FLOW
7:1. BOUNDARY-LAYER 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. STRAIGHT-LINE PLOTTING OF LOW SPEED AIRFOIL TEST DATA
9:3. MOMENTUM THEORY OF AIRFOILS
9:4. CIRCULATION THEORY OF AIRFOILS
9:5. ASPECT-RATIO 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. TWO-DIMENSIONAL 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. LOW-SPEED 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. HIGH-LIFT DEVICES
12:1. NEED FOR HIGH-LIFT 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 4-DIGIT GEOMETRIC SYSTEM
13:3. THE NACA 5 -DIGIT GEOMETRIC SYSTEM
13:4. THE NACA 1-SERIES AIRFOILS
13:5. THE NACA 6 -SERIES AIRFOILS
13:6. OTHER NACA SERIES
13:7. EFFECTS OF AIRFOIL GEOMETRY ON LOW-SPEED 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. NACELLE-WING 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 LOW-SPEED 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 TAKE-OFF AIRPLANES
16:8. DETAIL-DESIGN 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 PROPELLER-DRIVEN 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 -PROPELLER-AXIS 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

Citation preview

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 McGraw-Hill 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

87-15154

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

7-18

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

7-19

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

8-1

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.

8-2

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

8-3 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

8-4

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

8-5 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

8-6 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

33-5321 .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.

)) . . -. . ( ( -. ,, . . . . . ) U). ) ) ) ) ) )L( ) ) ) ) ) ) ) ) ) ) ) ) )1( )2( )3( )4( )5( )6( )7( )8( )9( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (

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

8-7 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

0-0.95

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 0-1.2 0-1.2 0-1.2

0-1.24 0.8-1.6

27.0 40.0 2.0

16.0 25.0 11.0 60.0 200.0 0.12

hp 103

0-0.95 0-1.22

1.5B 1.16B

1.25B

2.0D

Max

Max

0-1.5

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

0-0.97 0-1.1

113 201

Round Round 1.5D

0-1.3

0-1.2

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

8-8

,,

TECHNICAL AERODYNAMICS

8-9 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

8-10

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

8-11

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

8-12

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 blow-down 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

one-half 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

8-13 TECHNICAL AERODYNAMICS

AERODYNAMIC

TEST FACILITIES

Balance Room

8-14

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 O-Pivots 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

8-15

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

8-16

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 .

9-1

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

9-2

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

9-3

/

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

9-4

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

9-5

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,30-in.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 J-0.4 -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

9-6

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

9-7

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

9-8

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

9-9 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



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

.

9-10

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

9-11

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

9-12

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

9-13

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 .

9-14

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

+



+ ( 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

9-15

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

9-16

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

9-17

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

9-18

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 +





+

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

9-19

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

9-20

AND ASPECT RATIO EFFECTS AT LOW SPEEDS

in

)

9:37

radians

)

8C

9:36

(

-

Δα =

85

(

=

ACD1



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

9-21

/

/

== 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

9-22 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 ) ; 0-0 .

9-23

TECHNICAL AERODYNAMICS

9 : 5 . Repeat problem 4412 - 4412 ; 0-0 .

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 2218-09 ; 0-0 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 23018-09 ; 0-0 . (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 23018-09 ; 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

0-0 ,

ft

it

AIRFOILS

For

Art .

AND ASPECT RATIO

EFFECTS AT LOW SPEEDS

9-24

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 , 10-1

any

particular

to separate .

In this text order :

while

model ; the

It is it is

(1 ) aspect

Chapter

9 has

com

cho

ratio , dealt

10-2

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

10-3

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

10-4

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 ,

10-5

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

10-6

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

:

10-7

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

VI-MO

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

/

10-8

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

10-9

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

10-10

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

10-11

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

10-12

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

10-13

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

10-14

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

10-15

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

=



=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

10-16

.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

.

66,2-215

=

,

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

10-17

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

( 1-1 / 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

10-18

,

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

10-19

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 11-1

11-2

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 662-415

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 662-415 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

11-3

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.

1-5

-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

11-4

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 ( 662-415 ) . 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 ( 662-415 ) 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 0.03-1.38 . 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

11-5

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 633-018 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 64-210 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

.

11-6

the wing chord for sea level flight is 4.3 feet At higher altitudes

-

1.2

.

clmax

.

.

;

flight

For supersonic

--- 369--2-

-369-1-369

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 64-210 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 =

11-7

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

11-8

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

11-9

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

11-10

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

11-11

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

11-12

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 12-1

curva this

Devices of

Some combinations ,

de

Most of these

solely

by the

maxi

the economic merit

TECHNICAL AERODYNAMICS

11-2

661-212

661-212

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

661-212

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

661-212 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

12-3

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 A5-27

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

,

,

A5-25

on C1 max

for

due to Reynolds

some

number

are

for

for

from Re

.

a

.

-

.

a

x

15

are indicated in

106

Reynolds

A5-24

constant

it 1.5

The values

Reynolds

substantial

as indicated

number

few low drag

number at

of airfoils

going

Re

Other

on pages A5-21 Mach

types

in

shown

Approximately the

by

correc

sections with .20c

inclusive

. ,

deflected

corrections

on page A5-20

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 A5-8

,

: 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

12-4

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 A5-25 . mon

as high as 0.3 .

the Mach number goes

that surface condition

A5-25

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

3-0.3

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

12-5

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 A5-26 through A5-29 , 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 A5-30

.

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 A5-27 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 A5-28 and A5-29 , the

an advantage , as an addition

to boundary

slat

may

- layer control

.

12-6

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 X-15 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

12-7

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 A5-28

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

A5-29 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

12-8

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 A5-8 , 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 A5-26 ) . 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

13-1

13-2

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

13-3

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 0020-63 , 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

13-4

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

13-5

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

13-6

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 .

0006-63

0009-33

0012-63

0009-93

0009-63

0009-05

0009-62

0009-35

0009-64

0009-34

0009-65

2209-34

0009-66

2409-34 4409-34

0009-03

b.

C.

NACA

4 - digit

Comparison

airfoils

modified thickness distribution

( From

NACA TR 492.

)

0009-03

0009-62

0009-33

0009-63-0009

0009-63-0009

0009-64

0009-93

0009-66

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

13-7 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

13-8

also be followed by a dash and two additional digits designating other leading - edge radii and maximum thickness locations . The 23012-34 , 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 . 0012-63

0012 68 -60

.

23006

°

0006

23009

43009

63009

0012-64

23009

0012

23012

43012

63012

0012-65

23012

0015

23015

43015

63015

23012-33

23015

0018

23018

43018

63018

23012-34

0021

23021

43021

63021

23012-64

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

13-9

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 6-2 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

16-215 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

13-10

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

,

13-11

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

65,2-012

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

13-12

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

13-13

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

13-14

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

13-15

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

13-16

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

13-17 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

13-18

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

13-19

TECHNICAL AERODYNAMICS

13-20

/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

13-21

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

13-22

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 23115-35 . 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 662-315 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



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 14-1

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

14-2

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

14-3

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

14-4

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

14-5

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

14-6

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

164-66 64-66 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

/

14-7

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

64-66

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

14-8

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

14-9

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

14-10

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

14-11

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 : 0012-0.55 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

14-12

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

14-13

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

14-14

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

14-15

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

14-16

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

14-17

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

14-18

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

14-19

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



x 7.8 x 0.83

lift

The desired equation relating the drag and

answer called

14 : 9.

AND

THRUST

coefficients is then

0.0490-2

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

14-20

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

14-21

,

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

14-22

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

14-23

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

14-24

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

14-25

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

14-26 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

)

.

.

480-483

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

14-27

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

14-28

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

15-1

,

TECHNICAL AERODYNAMICS

15-2

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

15-3

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

15-4

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

15-5

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

15-6

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 A7-1

.

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

15-7

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

15-8

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

15-10

,

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

15-9

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

15-10 AERODYNAMICS

.

)

POWER PLANTS

AERONAUTICAL

15-11

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

.

15-12

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

15-13

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.

15-14

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

15-15

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

15-16

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 TG-180

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

15-17

( )

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

15-18

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

15-19

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

.

15-20

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

280-284

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

15-22 .

as a function

of the are shown in Fig .

liquid

propellant

for

system

rocket

From

Dur

Injection plate

15:27 , page 15-22 . 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

15-21

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

White-fuming

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

(

15-22

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

15-23

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 15-24 . 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 .

15-24

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 Life-Time , Inc. , from Life Magazine of February 7 , 1955. )

AERONAUTICAL

15-25

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

16-18

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

16-19

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 .

16-20

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

16-21

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 .

16-22

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

16-23

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 A7-13 through A7-16

tractor

A7-17 and

plane and cedure

they

, and composite design charts for propellers of two to eight blades are given on pages A7-18 . 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

16-24

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 . 1-86 .

TECHNICAL AERODYNAMICS

16-25

(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

16-26

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 5868-9 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

16-27

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 5868-9 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

16-28

.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

16-29

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



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

=

=

/

/



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

16-30

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

16-31

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

16-32

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

16-33

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 37-3647 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



/

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

16-34

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

( 1452-1519 )

,

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

1-6 and 1-7 . 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. 24-27 , 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 . 17-1

of Aircraft Can

Do , The

, "

Mc Graw

Story of the

HELICOPTER PERFORMANCE

17-2

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

17-3

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)

17-4

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

17-5 cient of the form Cq =

in

a

+

CT

+

/

c CT3 2

d

+



(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

17-6

YAAR

σ 19 819

۲۲ 89to

C

B

O

188-8

JIG

D

B

( 17 : 4 ) and ( 17 : 5 ) .

10

O

C

TECHNICAL AERODYNAMICS

17-7

#

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

17-8

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

17-9 400

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 )

17-10

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 = 0.475-0.63μis =

)

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 = 0.425-0.63μLS

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

17-11

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

17-12

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

17-13

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 17-14

37.

.)2 =

rh

(

TECHNICAL AERODYNAMICS

where R

is

P

= Rp

rn

the ratio of power required

Ri

17:25

)

=

(

*

R

1 + +

17-15

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

17-16 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

.

.

1500-1000

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

17-17

/

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

17-18

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

17-19

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

18-1

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

18-2

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

18-3

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

18-4

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

18-5

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

18-6

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 " " Csm-0.9

-

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

18-7

)

(

.

)

18-8

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

=

18-9

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.

18-10

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

18-11

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

18-12

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

"

.

/

"

0.45-1.89 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

18-13

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

18-14

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

18-15

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

18-16

(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

18-17

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 Tc-Dc

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

18-18

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

18-19

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

(

18-20

18

assumed

) )

be given

.

18:47

K2

"

Reverse Pitch pp 441-447 .

,

,

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

18-21

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

(

18-22

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