Introduction to Helicopter Aerodynamics

Citation preview

STS

+

ST36

LIBRARY UNIV

. OF WIS .

The Library

University

of

SIGILLUISINENSIS

RSITATIS REAINS

of the

Wisconsin

SD)

5

Rotary Aircraft Series No.

1

INTRODUCTION TO

HELICOPTER AERODYNAMICS

W. Z. Stepniewski

Chief of Aerodynamics Piasecki Helicopter Corp.

VOLUME

1

PERFORMANCE

Price :

$

2.50

Rotorcraft Publishing Committee Morton , Pa .

:

177

1

SECOND IMPRESSION Copyright 1950 by W. Z. Stepniewski

756957

ago

E ST

6484783

AUTHOR'S PREFACE

1

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weekly The basic material this booklet originated series gave aerodynamics helicopter the Piasecki which lectures on My rough notes Helicopter Corporation during the winter 1947-48 multigraphed leaflets prepared Messrs were the basis Locurto Sloan and Carnese after each lesson These hastily prepared leaflets were finally assembled into one text for which despite numerous shortcomings there Because continuous demand this demand have taken the opportunity review correct and expand the original material make the whole work more order self sustained and thus more useful to those who did not attend the course

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During the preparation appeared that might the revised text original scope beyond be desirable extend the whole the work the problems elementary helicopter aerodynamics include such topics blade loads stability controllability rotorcraft etc. way this the present volume would become only the first part larger project and would serve true introduction more advanced subjects helicopter flight introductory order retain this simplified character the treatment all helicopter flight problems some extent and mathematical complexity avoided much possible hoped that due this approach the reader with the theo retical background an average mechanical engineer will be able grasp the fundamentals helicopter aerodynamics There also another reason for the simplification too much mathematical complex ity introductory book tends particular danger present especially graduates susceptible may develop which young are dealing with long and complicated formulae that much interest students the subject may lose all feeling for the physical significance and true importance various parameters of

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express his indebtedness The author wishes his colleagues especially helped this work and Mr. Sloan for his who him valuable assistance the preparation the text and his assistance together with that Messrs A. Locurto and D. Carnese the in

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is

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The simplification however has been limited the method presentation subject complete quite The matter otherwise that knowledge for the prac this text should provide sufficient amount tically accurate prediction helicopter performance all realms flight power power and off order shorten this volume and thus limit its cost all numerical examples were omitted However special booklet planned which will contain the whole sample compu performance hypothetical helicopter tation sufficient publication people indicate their interest number such writ Publishing ing Rotorcraft Committee Morton Pa will printed very shortly

proof - reading of the plates , Special thanks are due to all members of the Rotorcraft Publishing Committee who made possible the publication of these notes in the improved external form . The author wishes also to thank Mrs. E. Hepler , who very skillfully typed the manuscript and to Mrs. E. Wissler who had to struggle with the Multilith plates . Finally the author wishes to thank the Management of the Piasecki Helicopter Corporation for permission to publish these notes in such a way that they may be made available to readers outside of the Company . W. Z. STEPNIEWSKI Morton , Pa .

March

1,

1950

NOTE TO THE SECOND IMPRESSION In connection with the second impression of this book , the author wishes to express his special indebtedness to Professor G. R. Graetzer of USAF Institute of Technology for his most valuable suggestions regarding the subject matter and his proof reading of the text . Although , for purely technical reasons , not all these suggestions have been introduced in this impression , the author hopes that all of them will be found in the book when the time comes for a new revised edition . W. Z.

Morton August

,

STEPNIEWSKI

Pa .

8 , 1950

PUBLISHER'S

PREFACE

Aerodynamics " is the initial volume of the Rotorcraft " Helicopter Publishing Committee , in what is hoped will be a complete series of works on rotary wing aircraft . The aim of the Committee is to make available for all those interested in the development of rotorcraft , a comprehensive study of the theory and design of this type of aircraft .

In preparation , or under consideration at the moment are texts on Blade Dynamics , Air Loads , Stability and Control , and Helicopter Design . Because of limited interest in some of the proposed subject matter this project can only be made possible by the cooperation of all people having a direct interest in rotorcraft . Comments on the present volume , as well as suggestions for future publications , will be appreciated . The Committee is a non - profit organization , all proceeds from the sale of the present volume will be re - invested in the publication of works aimed at advancing the theory , design and practical development of rotary wing aircraft .

Morton , Pa . March 1 , 1950 A C. Adler

D. J. Carnese R , J. Harris

ROTORCRAFT C. A. Locurto F. R. Mazzitelli L. H. Sloan B. J. Solak

ii

PUBLISHING COMMITTEE W. Z. Stepniewski

T. B. Tarczynski K , W. Ulrich

Table of Contents Chapter

I

REVIEW OF AERODYNAMIC

FUNDAMENTALS

:

Air ,

Streamlines , Bernoulli's Theorem , Two Dimensional Flow , Kutta - Joukowski Theorem , Conformal Transfor mation , Working Formulae , Circulation Around an Air foil , Drag , Induced Drag , Viscosity of Air and Reynolds Number , Boundary Layer , Skin Drag , Stalling , Scale Effect , Compressibility Effects , Critical Mach Number , Drag Coefficient , Lift Slope , Maximum Section Lift Coefficient , a.c. Travel, References pp .

Chapter

II O-

1

-

40

SIMPLE MOMENTUM THEORY : Introduction , Actuator Disc , Thrust in Vertical Climb and Hovering , Power Required in Vertical Climb and Hovering , Thrust and Induced Velocity in Forward Flight , Power Required in Forward Flight , Rotor Tilt in Forward Flight , Horizontal Flight , Power Required Curve , Rate of Climb in Forward Flight , Ceiling in Vertical Ascent , References 58

THE BLADE ELEMENT THEORY

:

Chapter

III -

pp . 41

of

,

72

:

THE VORTEX THEORY

59

Introduction Circulation Along the Blade Downwash Produced by Single Vortex Cylinder Circulation Varying Along the Blade References .

pp

iii

73

-

,

,

,

,

Chapter IV

-

.

pp

-

,

,

,

Introduction Basic Concept Further Development the Blade Element Theory Combined Blade Element and Momentum Theory References

81

Chapter

V.

PRACTICAL METHODS OF CALCULATING ROTOR THRUST AND POWER IN HOVERING AND VERTICAL FLIGHT Introduction , Non - Dimensional Coefficients for Thrust and Torque , Tip Losses , Thrust in Hovering , Maximus Thrust in Hovering , Gust Loads , Induced Power in Hovering , Blade Twist for Uniform Downwash Distribu tion , Induced and Climbing Power in Vertical Ascent , Power Required to Overcome the Profile Drag , Total Power , References pp . 82

Chapter VI

-

-

100

PERFORMANCE IN HOVERING AND VERTICAL

ASCENT

/

:

Introduction , Figure of Merit , Optimum Rotor Radius , Power Loading in Hovering , Average Lift Coefficient , Approximate Pitch Angle , Power Required in Hovering out of Ground Effect , Ground Effect , Vertical Climb , References pp . 101

Chapter

VII -

-

116

PERFORMANCE IN POWERED FORWARD FLIGHT : General , Induced Power , Induced Power of Tandem Rotors , Profile Power in Forward Flight , Contribution of Profile Drag to Parasite Drag , Total Parasite Drag and Rotor Tilt , Power Required Curve , Speeds of Fligh Range

,

Vmax

,

Rates of Climb and Ceilings , Limitations to References pp .

Chapter

VIII

AUTOROTATION

117

-

136

:

Introduction , Basic Concept of Vertical Autorotation , Application of Momentum Theory to Autorotative Verti Descent , Basic Relations of Glauert's Curve , Use of Glauert - Lock Curve , Finding of 1 and V : Hafner's Curve , Rate of Descent in Autorotative Forward Flight References

pp . 137

-

158

LIST OF SYMBOLS

to

whole body whole body

section drag coefficient

friction drag coefficient moment coefficient referred

to

section moment coefficient section moment coefficient

whole body

rotor swept area rotor swept area

)

/

ft -

)

(

,

or

)

ft /

ideal induced power

hovering

lbs / ft

)

in

.)

(

to

actual

lift lbs lift per unit length (

ft

.

)

/

(

of

- -

(

or

of

ratio

,

VI

,

ft -

)

)

(

),

(

-

(

or

,

(

),

length different units Mach number linear momentum lbs sec moment lbs figure merit or abbreviations for Mach number mass slugs general power pressure lbs sq.ft sec power sec lbs climb ideal power i.e. power required by the ideal helicopter lbs sec )

)

)

/

-

hp

/

-

)

ft (

,

(

or ft

hp

,

)

)

/

lbs

-

ft

or

,hp

in

(

)

/

/

-

ft

or

or

hp ,

)

ft -

in

or

or

hp

(

/ (

-

,

/

) ,9

)

(

in

(

Pex

lbs sec

power lbs sec the rotor hp engine horsepower brake power lbs sec excess

V

BR BHP

induced power at

Pi

Pind or

ft

(

Pid

or

PC

(

P

A A

hp

m

,

MN

(

of

M

1

L

k i h H g fa

fi

-

),

(

=

;

-

of

.

of or

)

,

ft -

(

)

drag lbs distance between successive vortex sheets energy per sec energy lbs sec lbs position pressure relative the center Glauert's non dimensional coefficient occasionally force lbs equivalent flat plate area for Cp 1.0 sq.ft Glauert's non dimensional coefficient Hafner's non dimensional induced velocity coefficient Hafner's non dimensional rate of descent coefficient gravity 32.2 sec2 acceleration sq.ft total head lbs height altitude usually (

HOW

FFOAD

,

,

to to

rotor thrust coefficient referred rotor torque coefficient referred

(

,

JFF

Good

or

of

;

to

lift coefficient referred section lift coefficient

drag coefficient referred cd

,

,

)

ft )

or

,

с no

number of blades wing blade chord

CL

sound

R

(

Re

speed

(

tip loss factor

/

b Btop

,

AR

of

Ama

(

);

occasionally aspect ratio area sq.ft abbreviation for aspect ratio occasionally usually slope the lift curve

usually , rotor torque

q

ac

R

Re на

и S S

source ( units of mass , dynamic pressure ( lbs / sq.ft) rotor torque coefficient referred to rotor projected area usually geometric rotor radius ( ft ) ; occasionally Reynolds number effective rotor radius ( ft ) distance of blade element from rotor axis ( feet ) area ( sq.ft ) speed of sound occasionally absolute tempera usually , rotor thrust ( ture time secs usually axial component the total rate flow through occasionally speed fps the outside flow the rotor border the boundary layer fps velocity component along the axis flight fps mph horizontal component the speed great distance flow or speed resultant rate flow through the rotor fps axial component the total rate flow through the rotor fps fpm rate climb fpm rate descent fps flight corresponding speed the minimum power required usually mph air speed along the flight path fps flight corresponding either speed the best range optimum gliding ratio usually mph rotor tip speed fps the rotor axis air flow component parallel parallel component the rotor disc air flow helicopter lbs gross weight speed resultant air blade element fps disc loading sq.ft lbs coordinate along axis usually non dimensional blade stations occasional coordinate along axis the effective blade radius blade station corresponding

F )

(

°

но

,

lbs );

T

exceptionally strength of or volume , per sec )

( ft - lbs );

,

Q

at

of

of

),

or

,

(

or

,

to

)

,

to to

or

),

(

) ,

.

/R

r

(

x

the

occasionally Mach angle ,

;

of

)

attack

2

i Y )

+

effective

in

to

x

or

(

z

,

z

in

y

Y

/

axis axis plane complex variable axis

of

to

Re R )

,

usually angle

vi

the beginning

y

i

+

(

z Z y

Y

blade portion coordinate along coordinate along complex variable coordinate along x

Xi

)

(

at

X

,

хе = blade station corresponding (

хе

)

(

)

(

)

(

of

ly

,

X

х

-

W

)

wear

/

ax

V.

(

Vt

)

(

of

to

)

or

)

or

of ,

,

,

(

Vopt

of

VE

V

)

at

of (

с

va Ve

(

V

of

V ' ах

of of

vi

of

of

V

of

x

u

(

of

(

);

,

,

of

)

(

t U

plan

,

);

specific heat ratio front other symbols means an increment boundary layer thickness ratio actual figure merit ideal figure merit efficiency expressing transmission the ratio rotor

of

/

.)

)

/

ft2

(

of

(

)

,

or

of

)

/

(

,

in in

)

(

/

(

,

,

vii

)

+

/V

=

of

)

to

,

.) of

/

at

or

,

A

of

(

tip

to

to

);

V +

)

/

/

(

a 3

to

)

,

or or

of ,

(

,

=

( м M

(

y

6 al wed

d.orup

)

of

of

)

or

,

,

(

representative pitch angle radians degrees inflow ratio expressing the ratio the axial component speed through the rate flow the rotor v'ax usually tip speed ratio expressing the ratio the flight tip speed speed component parallel the disc viscosity occasionally coefficient Vpar slugs ft.sec viscosity kinematic coefficient sec air density slugs cu.ft air density sea level 0.002378 slugs cu.ft rotor solidity ratio expressing the ratio blade projected area to disc area shearing force per unit area lbs sq.ft induced inflow angle usually radians degrees angle azimuth measured radians from the position blade downwind the direction rotation rotor angular velocity radians sec solid angle

of

box M

(

2

V

to

of

brake horsepower

degrees measured from usually pitch angle radians occasionally special angles zero lift chord degrees total geometric blade twist radians ;

o

,

t

(

n dis

,

8

of

of

of

in

>

occasionally

radians

(

,

of

of

strength circulation usually tilt the thrust vector

CHAPTER I REVIEW OF AERODYNAMIC FUNDAMENTALS

Although it is assumed that the reader is familiar with the basic theories of aerodynamics , it is necessary to review the more generally accepted principles so that the step into the helicopter branch of the subject will be made from common ground , In general discussion will be confined to that subject material having direct application to the study of rotary - wing aerodynamics presented in the succeeding chapters . At the outset , the properties of air are discussed with a view to establishing the regions in which air may be considered as incompress ible and those in which it must be assumed to be compressible . Next , such topics as Bernoulli's Equation and a graphical presentation of the more impor tant two - dimensional flow patterns are recalled and the theory of lift is briefly outlined . Mention is made of viscosity and compressibility . The influence of these parameters on airfoil properties in two - dimensional flow conclude the chapter .

AIR Our understanding of most of the physical phenomena associated with gases is based on the " Kinetic Theory of Gases " . ( see for instance , Ref . 1 ) . According to this theory , any volume of gas is composed of free molecules which can be imagined as being tiny spheres constantly in motion with respect to each other . these molecules is very small , approximately speeds of an inch . The of the individual molecules vary , but the average velocity depends upon the temperature and for room temperature , the 1.5 x

average diameter of The 10

at

of

in

as a

of

at

,

.

an

be as

at

,

valid

to

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to

,

it

of

,

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,

at

of

,

to

,

.

*

of

a

is

dis

average is about 1500 feet per second . For most problems dealt with in normal aerodynamics , the number of molecules included in even very small volumes is extremely large . For instance , a cube having sides of only one thousandth of an inch contains about 4.4 x 1011 molecules at a pressure of one atmosphere and temperature of oºc . Under these conditions the average only about 2.5 tance that molecule can travel before colliding with another 10-6 an inch Thus one can begin understand why cases flow approxi pressures speeds much lower than that the molecular motion and mating that the earth's surface the air can be considered kind continuous incompressible medium dis regarding the molecular motion At speeds approach the same time however starts become apparent that ing that the average molecular velocity the theories which treat air incompressible continuous fluid will begin break down and cease

.

a

)

of

at of

,

of

(

of

of

be

to

as

in

,

a

is

in

of

Any motion body through the air will course produce some dis turbance the molecular motion the immediate neighborhood the body through This disturbance transmitted the air sound waves definite magnitude rate which may be expected the order the mean speed

AIR

1

of the molecules 1

.

Indeed

Theory shows that the speed of

the Kinematic

,

sound

):

:

a

0.7420

(1)

is the mean value of the speed of the molecules

where

.

in

of

of

,

In

is

it

,

body moving through

2 )

T

F .

in °

the absolute temperature

(

)

33,42

$

a

is

I

where

as :

.

in of

(

be

a

is

of

Since , on the other hand , the mean velocity of the molecules depends on the absolute temperature the gas evident that the speed sound temperature only speed air function its absolute fact the sound expressed mph air can

only

is ( Ref .

2

in

Chapter

INSIDE

CONE

SIGNALS BY BODY

of

be

THIS

EMIT

RE

CEIVED

Figure

1

a

v

*

\

a

.o

la

will

2

the moving body

THE NOSE OF CAN

by

a

"

( 3 )

of

that

.

Mach angle OF

the

of

a

at

.

as

"

or

a

.

)

a

V

sound

BE

TED THE

1

(

ONLY

V

,

of

is

the speed called the

is

a

a

sin where Angle

"

be

of

be

"

it .

of

a

is

of

is

as

.

of

in

at

A

air velocities far below that sound creates speed disturbances which are transmitted through the air all directions much higher than that the body itself This means that the air ahead moving body influenced by its motion well that behind When the speed body the disturbance created by its motion cannot sound reached by messages emitted transmitted directly ahead All signals different points the body can received within cone whose angle defined by the expression see Fig

AIR

Chapter I

3

One need not stretch the imagination too greatly to understand that the moving once through the air at a speed much lower than that of body same sound and at the second time at that higher than the speed of sound , may create a completely different flow pattern in each case . It becomes clear that the speed of sound represents a dividing barrier between two types of aerodynamics . It is logical hence to expect that the ratio of speed of flow to the speed of sound ( the so - called Mach Number M ) is one of the most impor tant characteristics in aerodynamics . Since , however , the speed of flow may change at different points of the body , definition of Mach Number requires a

further differentiation

:

-V

1

2

V

V2

2

Figure

2

Imagine a body of the shape illustrated in Figure 2. At a great distance in front of the body ( i.e , at infinity ) the layers of air are moving parallel to each other ( separated by imaginery lines which are usually called streamlines ) .

It can easily be seen from Fig .

that the velocity at infinity ( V ) and the velocities at section 1 ( V1 ) and 2 ( V2 ) are different from each other . The velocity at section 1 will tend to increase since the air is forced to go through a smaller passage exactly like water in a hose , when forced to go through a restricted nozzle , will increase its velocity through that restriction . Therefore , the Mach Number at infinity will be different from the Mach Number at point 1 and 2 , and in fact it may be different at any local station along the periphery of the

airfoil

in Fig .

shape

2

2.

in the most general case not only the speed of flow but also the speed of sound may vary from one section to another , i.e. " a " at infinity may be different from " a " at point l .

It should

be noted that

Consequently the value of Mach Number must be defined for particular

points in the flow : V a

M

(4)

;

V

V 1

ai а

M

;

;

2

a2

M2, etc.

STREAMLINES

Chapter I

BERNOULLI'S THEOREM

4

Summing up this very general discussion of the importance of Mach number the conclusion is reached that as long as both Mach number at infinity 1 ) , the air and local Mach numbers are far below the value of ! 1 ( M may be considered as a continuous , incompressible medium and explanations of the aerodynamic phenomena based on the mechanics of an incompressible fluid should be in agreement with experimental results .

STREAMLINES In describing flow patterns around bodies one can fix the axis of refer ences either in the body itself , or in the fluid at rest far from the body . For practical applications it is usually more convenient to fix the axis of reference in the body and imagine that it is stationary , while the fluid is in motion . In graphical representation of flow patterns the streamline technique is especiall useful . Durand ( p . 185 , Vol . 1 Ref . 2 ) defines the streamline as " a line such that at any one instant of time , its direction at every point is in the line of fluid motion at that point " . This , obviously means that the velocity vectors are tangent to the streamlines , and there is no fluid motion through the streamline Hence , streamlines ( in space ) can enclose a fluid filament thus forming a stream tube . Since the fluid cannot cross the streamline it cannot escape through the walls of the stream tube . This means , of course , that the amount fluid flowing in unit time through any section of a given stream tube is constan

BERNOULLI'S THEOREM Since there is no flow of fluid through the walls of a stream tube then no energy can be carried in or out of the tube by the fluid itself . If, in addition , the fluid is considered inviscous then , of course , no energy can be also dissi . pated through friction . This means that for inviscous fluid the total amount of energy per unit mass of fluid enclosed within the same stream tube remains constant . Through the application of this principle of conservation of the total energy per unit mass of fluid along the stream tube , it is easy to develop Bernoulli's equation for incompressible and compressible fluids ( see for instance Ref . 3 ) . In the case of incompressible fluid Bernoulli's equation reduces its elf to the well known for mula :

H

so H ) -

v2 is

the

const .

2 2 v2

+

as172

of the

P

,

,

2

.

.

1

is

,

,

to

.

,

ut

,

or is

4 ,

of

.

.

,

,

.

,

In

.

(

a

is

,

,

IV ,

,

.

5 )

.

,

3

(

to

1

is

2

at

,

is at

a

is

P

static pressure called dynamic pressure where P = and the sum the two known the total Head This equation states Fig pressure along that the total streamline constant for in velocity section the increases then the static pressure must stance since decrease and lower than the pressure far away from the body The velocity slightly slightly higher section lowered therefore the pressure para 4.1 and 4.2 Ref Chapter para see for instance Chapter pressure along top Ref With decreased the the airfoil and increased along the bottom the airfoil will want move up and thus lift created

TWO DIMENSIONAL FLOW

Chapter I

TWO DIMENSIONAL

5

FLOW

B u

A

z

A

pp

Figure

x

3

Assume that AB , in Figure 3 represents a stream tube in space . Next , assume that the cross section of this stream tube is so small that it can be called a streamline . At any point C along the streamline , the velocity with which elements of air move can be represented by a vector V. This vector in general can be resolved into 3 components along the x , y , and z axis . The corresponding velocity components are u , v , and w . If one component of velocity in the considered flow is always zero ( for instance w = 0 ) then the flow is two - dimensional , and the velocity vector is : V

=

122

V

u

+

v²2

( 6)

Since in further discussion reference to two dimensional flow will often be made , some basic flow patterns belonging to this category will be recalled . It should be mentioned at this point that in graphical representation of the two dimensional flow , the streamlines are usually drawn in such a way that the amount of fluid flowing through a tube formed by two planes parallel to the flow and two surfaces represented on paper by the adjacent streamlines is the same . This way of mapping the flow is especially useful, since looking at the flow pattern we can immediately tell where the fluid flows faster ( crowded streamlines ) and where it slows down ( streamlines further apart ).

Chapter I

TWO DIMENSIONAL FLOW

TWO DIMENSIONAL

.

course

,

(

continued

)

Parallel Flow

a.

motion

FLOW

6

This is the simplest example of the two dimensional fluid Fluid is flowing at a constant speed in one direction and , of all streamlines are parallel straight lines ( Fig . 4 ) .

V

Figure

4

Source

.b .

Imagine that fluid is flowing from a line extending in the direction z and spreading away along the planes parallel to xy in a manner shown in Fig . 5 .

pa

X

Figure

of

( 7 )

r

in

or ,

,

of

other words the the source axis ,

2

in

-

is

(

a

its

r:

=

V

Fig

the so called strength the source flowing fluid out unit time from unit length ) .

in

(

O

amount

of

Q

5

where

TT

of

be

inversely proportional

that the amount of fluid passing in a unit of time origin and given height say surface having O as obviously constant Hence the speed flow will be the radius

to

)

of

through any cylindrical length will unit

1

.

It is obvious

5

Chapter

I

TWO DIMENSIONAL FLOW

TWO DIMENSIONAL C.

FLOW

7

( continued )

Sink

In a two - dimensional sink the opposite takes place , i.e. the fluid flows between two parallel planes into a line . Eq. ( 7 ) holds in this case only with the negative sign . combined into new ones : Since the speed of flow due to each pattern can be represented as a vector , those vectors can be added and thus give the new flow pattern . In Fig . 6 , for instance , is shown how a flow resulting from a combination of a parallel flow and a source

Different basic flow patterns can

can be mapped PARALLEL

be

.

V

Ve

Q

Ve

ZTIT

FLOW

d .

Figure

6

SOURCE

Source and Sink

a

,

or

is

-

so

(

to

7.

)

-

Figure

Figure

8

Х

AY

7

.

.

Fig

8 )

to

in

,

.

in

in

a

of

equal strength are combined then the fluid When source and sink disappears flows out from the first and the second making characteristic Fig pattern flow shown When the distance between the source and sink becomes smaller and smaller and finally they come one point rather one line the two dimensional flow the called doublet obtained

Chapter I

TWO

TWO DIMENSIONAL FLOW

DIMENSIONAL FLOW e.

8

( continued )

Doublet and Parallel Flow

When the doublet is combined with a uniform flow a peculiar flow pattern results with one streamline being a circle . In imagining that this circular streamline is replaced by a physical cylinder , we obtain a flow around a cylinder placed in a uniform stream of ideal fluid . (( Fig . 9 ) .

Х

2 Figure

9

Applying Bernoulli's principle , we see that the pressure forces acting at section 1 and 2 are the same in magnitude but oppositely directed . If the same procedure is repeated for all sections of the cylinder one will find that because of a complete symmetry all pressure forces will be counter - balanced and the cylinder as a whole will produce neither lift nor drag . It must be remembered , however , that this conclusion refers only to the case of ideal ( inviscous ) fluid .

f.

Another important flow pattern is the circulatory flow :

у

vuit X

Figure

10

AERODYNAMIC FUNDAMENTALS

Chapter I

DIMENSIONAL FLOW

TWO

9

( continued )

This flow occurs in concentric circles , where the velocity at any point along any circle is inversely proportional to the radius r of the circular :

Const

r

(8)

val

line

=

Fig .

V

any

streamlines

di

in

2

(

:

)

$

or

circulation

)

(

9

=

of

the intensity

r

(

)

(

const

of V

;

$

10

this

To

.

.

in

of

2

х

Tr

Tr

more commonly

The circulatory flow

eq

.

"

as

to

also referred circulation

is

simply

"

It

V

)

a

is

for

has

,

a

is

the

integral If we take circle where х differential distance along any circle the integration yields dl any radius which the same value constant value val aerodynamics given special name circulation and symbol where

.

*,

10

.

in

or

,

It

" .

is

10

,

,

.

"

in or

of

,

a be as

:

is

it ,

)

It

O.

at

,

At

.

is

)

( 8

O ,

.

is

(

a

,

by

be

a

O

of

be

of

is

of

is

,

it

of be

.

as

defined by irrotational first this may sound paradoxical but should remembered that the requirement for an irrotational flow that elements the fluid should not rotate Indeed Fig except this condition fulfilled for all elements the fluid forming only for the elements the axis rotation where rotational motion expected elements can To this filament fluid extending along the simply vortex special name given the vortex core axis can Fig said that the whole circulatory motion shown created repeated once more that although induced vortex core should the vortex itself vortex core represents rotational motion the circulatory by flow created or induced irrotational

,

to

a

is

(

r)

to

r

) ,

(

In

or .

,

r

at

V

.

r

r

)

r

=

10 )

of

.

(

is

a

be

it it

a

(

as(

V

2ūr measured along any circle Since circulation Øvai having the vortex core its center Fig has the same value strength can be said that the the vortex itself other words may stated that vortex having strength creates induces circu latory motion where the speed any radius proportional the inversely proportional vortex strength and the distance from the core or

.

2 .

Ref

.

Vol

.

,

130

I,

.

"

it

for instance

a

of

."

a

It

be

)

in

it

see

of

"

a

:

,

For mathematical proof

.p

*

at

to

,

a

.

a

(

"

:

to

of

to

may be recalled that accord As the properties the vortex itself general straight Helmholtz The vortex core the case may curved In an indefinite fluid vortex core cannot end in the fluid must either end on solid surface or form closed circuit Kelvin's theorem particles may be added the above All the fluid which are part vortex part any time remain motion for all time ing

AERODYNAMIC FUNDAMENTALS

Chapter I

TWO

DIMENSIONAL FLOW

( continued

)

From the theorem above is evolved the well known principle that : perfect "in a ( inviscous ) fluid no vortex can be generated by an external agency ; and once in existence , it cannot be destroyed " . The definition of circulation which was discussed at length for the case of true circulatory motion can be generalized for a motion about any arbitrary closed path ( Fig . 11 ) :

T

V

=

cos

di

( 10 )

Aalter L

Figure

11

is the angle between the velocity vector path along which the circulation is computed .

where

V and the tangent to the

the velocity distribution is known in any two - dimensional flow the circulation may be found by the use of equation ( 10 ) .

If

tern ,

Doublet and Parallel Flow

pat

plus Circulation

Probably the most important of all two - dimensional flow patterns is obtained by a combination of a doublet with a parallel flow plus circulation .

If it is imagined as in Para , e that the circular streamline is replaced by a physical cylinder , then , of course , only the flow outside of this cylinder will be of any practical interest . When circulation ( or a vortex ) of a given strength is imposed on a doublet , or a cylinder, placed in a parallel flow the complete symmetry which existed in Fig . 9 ( para . e ) is distorted in a particular way : the speed of flow above the x axis is increased , and below it is decreased . According to the previously established principles of representing the flow with streamlines , these latter will now be more crowded in the top and more spread apart at the bottom ( Fig . 12 ) .

r

Chapter I

AERODYNAMIC FUNDAMENTALS

TWO DIMENSIONAL FLOW

( continued

11

)

у

V

-X

Figure

12

According to Bernoulli's theorem , the pressure on the top will be lower than at the bottom and hence the cylinder will experience an up - force ,

or lift.

axis all у Because of a complete symmetry of flow about the х direction will cancel out and the cylinder will not forces acting in the experience any force along this axis , i.e. , in spite of the presence of lift no It should be drag force is created in this type of two - dimensional flow , remembered , however , that this last statement refers only to the ideal ( inviscous ) fluid .

I

Kutta

Joukowski Theorem

121

Chapter

clear from the preceding discussion that the existence

of

is It is

AERODYNAMIC FUNDAMENTALS

circu

a

of

,

)

.

)

of

Of is

of

,

by

is

a

.

in

in

of

(

V

dę

Apude

VĄ

Figure

13

X

T

de

SA у

.

of

an

be

,

P

,

of

be

)

13

let

a

,

fi

.

(

of

a

of

p of

strength placed the density fluid length then an important relationship expressing the lift per unit terms readily through application and can established the momentum principle and Bernoulli's theorem a

Let cylinder which creates circulation V velocity uniform flow Fig and

be

of

of

"

of

an

-

in

to

it is

"

)

(

an

.

a

a

in

of

(

of

(p

to

is

a

-

.

essential to the creation of lift in two dimensional flow Since the circulatory motion generated vortex the mere presence straight vortex erpendicular the direction uniform flow sufficient for the creation lift from the mathematical point view course the physical picture lift generation would be incomplete without imagining real body which when located uniform flow generates circulation and creating body thus creates lift Since two dimensional flow this lift infinitely long cylinder usually special cross section called air logical cylinder per length compute foil the lift unit the lation

of

a

in

:

,

a

to

,

,

of

of

,

a

) .

13

us

of

of

to

.

Fig

(

.

r

as .

magnitude the force acting between the investigate the change momentum the order do this let examine the varia flows through control boundary radius

it In to

to

is

and body

AV

m

order find the necessary fluid fluid flowing around the body tion the fluid momentum thrown around the cylinder ,

Hence

it in

F

of

If

:

as

of

a

it a

of

be

The principle momentum can formulated follows unit velocity AV undergoes m change time in some direc mass force tion means that during this time the mass m was subjected AV was or forces whose mean value in the direction of

Chapter I Kutta

AERODYNAMIC

FUNDAMENTALS

Joukowski Theorem ( continued

13

)

Over this imaginary boundary the outer fluid will produce an upward force Pc while the body lift L will act on the fluid with an opposite force . This means that the fluid within the control boundary is subjected to a force along the y axis equal to :

F

L

PC

( 11 )

But, according to the momentum principle , this force should be equal to the unit time change of momentum ( in the y direction ) of the fluid within the considered boundary .

found

AM

F

or

L

=

AM

Denoting this change of momentum

Pc

+

L

-

Pc AM

( 12 )

The up - force Pc acting on the fluid along the boundary can easily be from Bernoulli's equation .

ДР

P - Po

=

v2

( v2

1/29

( 13

res )

)

where AP is the difference between the local pressure ( at the boundary ) and that at infinity , and V is the resultant fluid velocity at the point of pressure investigation Vees But the resultant velocity Vres can be obtained as the vectoral sum r of V and the speed induced by the circulation ) V (V

*

or

V

res

V

res

Resolving can be

V

=

y

+

V

+

2T

r

A

V 2

5

1

( 14 )

components along the x and res : into

rewritten as follows

y

axes eq

(

:

v2 res

(

v

+

v2

v2

=+

V + sin o o

or res

|

V

(

)2

r TY

+

( vt Vt

sin

cos

e

2

)? 2

( zon-) 2 TT

( 15 )

14 )

Chapter

I

Kutta

Joukowski Theorem (continued )

AERODYNAMIC FUNDAMENTALS

(

*

]

,

be

(

o

2

TI

do

0

sin

is

the

Pc

)

27

to

0

from

(

17 )

for the whole boundary

yr

(

gur

1/2

PC

18 )

:

Integrating

obtained

(

.eq

(

17 )

I

2

r

t

de

y

?

1/29

)

16 )

sin

PC

.eq (

AP

or substituting for d

orde

sino

AP

Pc

fr

2

|2

of be ;

will d

13 )

.

Fig

at

In

у

axis rde

the control boundary

the element

present any boundary can the normal pressure acting on angle located see

normal

an

the ideal fluid only forces hence the component along the

2

+

sino

to

[

"

g

V

:

16 )

following is obtained

of

1/2

AP

the

( 13 )

,

into

o

( 15 )

for

Substituting

1

of

of it

con

as

)

.

19 )

to

(

,

г

the flow velocity through the con direction and its value the

у

to

in

a

.

ur

)

13

r

present since the being tangent

r

( 20 )

cos

@

r TT

cos

2

V

ty

:

.

(

does not contribute component has

Vt

Fig

radius trol boundary But

Vt r

circle

of

the

is

2

=

,

Vt

of

flow

V

grdo

is

.

is :

time into the boundary through

13 )

Fig

parallel only the speed the circulation

speed component due

of

unit

(

©

a

in

o

at

V

cos

to

)

19

(

.eq

an

of

an

In

-

у

of

a

The amount fluid flowing angle element rde located

dan

.

an

a

,

.

,

its

у

of

:

be

as

(

of

The variation the fluid momentum AM flows through the trol boundary can visualized follows Because the presence circu entering boundary lation the stream lines the left side the control have axis directed up When they leave some velocity component along the equal velocity component but the boundary through left half they have directed downward This means that stream line passing through the con axis trol boundary undergoes change momentum along the

Chapter I

( continued )

The elementary variation of momentum in a unit of time

( 20 ) :

/2 TT

COS

Integrating eq . ( 21 ) for the whole boundary variation of momentum in one second in the у

o

( from 0

T)

2

2

od

COS

2 TT

0

vor

( 22 )

cylinder length becomes according

prv

(

=

the total

direction is obtained : 2TT

vg

( 21 )

to

,

the

L

lift per unit

of

AM

do

2

.

fluid

,

,

)

it

)

23

no in

of

-

to

(

is

,

it

,

of

.

in

.eq

)

(

.

).

(

by

23 )

of

as

(

)

a

an ,

(

a

to be

to

-

as

(

of

of

23 )

(

.

very important relationship expressing the lift infinitely long cylinder wing per unit length the product the speed velocity existing density body uniform flow i.e far from the the fluid Eq and circulation produced the cylinder wing known aero dynamics deducing Kutta Joukowski theorem Since refer ence was made the cross section the cylinder itself means that will true for any physical arrangement rotating cylinder wing etc. produce able circulation when exposed the two dimensional flow represents

)

: ve

Г

23

M

d

dm

1/2

and

( 19 )

Vty

12 ):

or , considering eqs .

Eq

M ) in the

eq

M

d

And the

(d

be :

(

direction will

.

y

Joukowski Theorem

to

Kutta

15

AERODYNAMIC FUNDAMENTALS

AEROD

Chapter I

FUNDAMENTALS

YN AMIC

Conformal Transformation The importance of parallel flow past a circular cylinder having circu . lation about it , is chiefly due to the fact that the circle can be transferred through so - called " conformal transformation " into shapes suitable for practical airfoil sections . In addition the flow around the circle can be trans formed into a flow around the airfoil in such a way that the circulation around the airfoil and the speed at infinity remain the same as for the circle Due to this last condition the lift per unit length of the airfoil section is the same as for the circle ( Kutta - Joukowski law ) .

Conformal transformation was introduced to aerodynamics by Joukowski and further advanced by von Karman , von Mises and others . Basically it consists of " methods whereby a geometrical field 1 , character . ized by an assemblage of points and lines may be transformed into another field 2 >, point by point , and line by line , in such a manner that the indefinitely small element of area in field I shall transform in field 2 into an element of similar geometrical form and proportions , while at the same time the aggregate in field 2 may be quite different from that in field 1 " ( Durand , Vol . 1 , p . 89 , Ref . 2 ) . The conformal transformation can be performed eithet graphically ( see for instance pp 121 - 131 , Ref . 5 ) , or analytically . The analytical transformation is usually done with the help of functions of the complex variable . The use of the complex variable can be considered as a method of representing a vector in a plane , or to give a position of a point in the plane. For instance in plane z the position of point P ( Fig . 14a ) can be determined by the value of z :: z

In another plane Z mined by the value of Z :

=

(

Z

PLANE

Fig .

=

x

+

iy

14b ) the

X

+

position of point P ' can also be deter

iY

Y

Z

PLANE

Z

Р

XT

х Х

ol

Figure

14

lo

х

Chapter

1

AERODYNAMIC FUNDAMENTALS

Conformal Transformation ( continued

If

the

)

relation between Z and z is expressed as

algebraic function

Z

f

17

a

real , continuous

(2 z)

Z plane can be found a correspond then for an assembly of points in the ing assembly of points in the Z plane . It can be further proved ( see for instance Ref . 2 , Vol . 1 , pp 89 -- 104 ) that due to the mere fact that the relation between z and Z is expressed as a real , continuous algebraic function of the complex variable , the condition of conformal transformation ( as defined on p . 16 ) is fulfilled . This indicates that there may be a great variety of functions which can be used in conformal transformation . Indeed , in addition to the well known Joukowski function ,

Z

z

+

b2 12 Z

there are others such as those of von Mises , etc. , which also transform a circle into shapes suitable for airfoil sections ( see von Karman and Burges Ref . 2 , Vol . II , pp 58 - 100 ) .

,

Since in all these transformations the speed at infinity and the strength of circulation is the same in the airfoil plane as it is in the circle plane , both airfoil and circle experience the same lift (Kutta - Joukowski theorem ). In addition , the velocity distribution around the airfoil can be obtained from that existing around the circle , hence for a given lift ( or lift coefficient ) experienced by the airfoil the pressure distribution around it can be predict ed . From pressure distribution such important characteristics as the posi tion of the center of pressure ( c.p. ) , or moment of the airfoil about a given axis can readily be computed . Of special interest to aerodynamicists is a reversed problem : predict the flow pattern , and hence the pressure distribution , around any arbi trarily shaped airfoil section . Possessing such knowledge , it would be possible to study , without recourse to tests , the lift and moment character istics of an airfoil section which seems to look promising on paper . The solution of this problem would be easy if a function transforming the circle into this given airfoil were known . Indeed , some aerodynamicists attack the problem along this line : Methods established by von Karman , Treffetz , Theodors on and others are based on an attempt to transform a given airfoil into a curve of nearly circular shape . Then another attempt is made to distort that nearly circular shape into a circle , or at least a curve very close to a circle . If all those functions of transformation have been estab lished , the flow around the circle can be transformed into that around the airfoil and thus the required pressure distribution obtained ( see for instance Ref . 2 , Vol II , pp 80 -O 83 ) .

Chapter I

AERODYNAMIC FUNDAMENTALS

Conformal Transformation ( continued

)

Of still greater interest is the task of finding the shape of an airfoil section which would produce a given ( desired ) pressure distribution . The first satisfactory solutions to this problem were found in the pre - war years and much progress was made during the war ( see for instance Ref . 6) Here again the desired solutions are obtained through the method of con formal transformation which thus proves once more its usefulness in 1 aerodynamics . Working Formulae

.

;

is

s,

etc.

to

is

,

c

)

cic

(

)

24a

1

C1

1/2

v2

(

a

of

1

as a

is

L

the

)

,

a

of

,

,

)

(

wing axis

the most wing max

usually used express dimensional flow the symbol expressed given length along the cylinder and C. wing times dimension along the flow chord

lift coefficient

its S

two

fuselage

-

In

cross section

a

(

of ,

to

the lift coefficient referred the whole body projected body characteristic area for instance the area

24 )

(

S

va CL

is

where CL

=

s

L

1/2

Experimental aerodynamics paved the way to expressing all aerody . namic forces acting on a body as a function of the velocity squared ( usually the velocity measured far from the body ), a characteristic area of the body, the fluid density , and a non - dimensional coefficient . The usual formula for lift ( force perpendicular to the direction of flow ) is :

): )

is

known

26 )

(

cca

flow

:

in

it

a

v

112

-

r

25

(

c

V of

(

),

wing when the lift coefficient two dimensional then the circulation associated with can be expressed as

or

(

23 )

!

by

(

of

2

ci

.

of

in

c

,

to )

is

.eq (

it

in

to

,

-

Remembering that the Kutta Joukowski theorem gives lift per unit equal length easy express setting terms circulation right unity eqs 24a and equating the sides and 24a

AERODYNAMIC FUNDAMENTALS

Chapter I

19

Working Formulae (continued ) be shown through relatively simple considerations ( see for pp instance . 178 to 180 , Ref . 5 ) that for cylinders with sharp trailing proportional edge ( airfoil sections the section lift coefficient Fig angle the attacka measured from the zero lift line 15 ):

)

(

-

or

is

27

.

so

c

,

is (

called

.

LINE

CHORU

a

,

degrees

2 . 2 )

to

.

II,

48

-

43

)

28 28

(

(

.

in

is

expressed

a

0.11

Ci

«

if

or

TT

a

C1

,

in

2

is

a

(

More elaborate study see for instance Vol Ref shows that for relatively thin airfoil sections should be equal when the angle measured radians i.e.

T

a pp

15

Figure

)

GEOMETRIC

proportionality

C

LIT

the coefficient

"

"

a

Where lift curve slope

ас of

“,

11

ZPRO

the

,

to

of

)

i.e ,

It can

For almost all practical airfoil sections

at

(

)

as

or mean aerodynamic

is )

a

is

mean geometric

,

,

,

about this axis and wing for instance max chord (

of

the

с

(

cm

29

given axis

С

c

pv

ss

?

a

.

of

90

1/2

-

finite span about

the moment coefficient

characteristic chord

).

where Cm

is

M

wing

of

to

M

a

The moment

of

is

a

be

least for low Mach numbers when air may considered an incompressible fluid the lift equal per cent curve slope about its theoretical value

Chapter

I

AERODYNAMIC FUNDAMENTALS

Working Formulae ( continued ) In two - dimensional flow the moment per unit length of the airfoil obviously be ;

M

=

gra с

122

2

wi

( 29a )

cm

The position of the center of pressure , expressed as a fraction e of the chord c and measured from the axis to which the moment coefficient is referred will be : cm

e

( 30 )

cn where chord

4

.

Cn is the coefficient of aerodynamic force perpendicular Using two - dimensional symbols ( Fig . 15 ) :

ci

Cn

hence

'

cos

For most practical cases it may

+

cdo

the

sina and

ci

be assumed that ca

:

cm

e

to

( 30a )

ci Circulation around an Airfoil In finding circulation around an airfoil one may ask whether its valy depends on the type , or shape of the circuit enclosing the airfoil . It can b shown that the circulation has the same value for any closed circuit made around the airfoil . This statement can be proved in the following way :

Li

AL

L2

B

.

mam

(

la )

Figure

6) (C )

16

AERODYNAMIC

Chapter I

Circulation Around An Airfoil

FUNDAMENTALS

21

( continued )

circuits ) enclosing an airfoil ,, Figure 16a and 16b . Introducing a bridge , or barrier AB , we can form one circuit L consisting of L1 and L2 , and two sides barrier indicated Figure 16c The shaded area enclosed by this circuit being entirely free any rotating air such vortex will obviously represent zero circulation different paths

of

in

as

of

.

)

a

(

Lyas

.

Li

the

(

the

We have two

of

.

-

Li

,

,

of

be

if

progres along the part we choose the counter clockwise sense progression along part sion the sense the L2 will clock wise Furthermore the contributions that the two sides of the barrier make to the

Now

to

be

of

L2

,

Li

.

If

.

,

.

)

.

pp

5

.

(

.

is

,

L2

L

plus the circula cancel out The circulation about progres tion about therefore have the sum zero the same sense sion used for both circuits the circulation about and are seen equal Ref 43-45

circulation around

Drag

-

( 31 )

CDS v²

s

D

1/2

:

,

of

to

.

it is is

.

in

in

,

in

of

of

We have seen from the considerations two dimensional flow of the type ideal fluid that this flow there no room for any drag But every practice impossible day experience shows that divorce any motion drag explain physical drag we will the air from Before we the nature expression magnitude recall the mathematical for its

is

S

,

,

,

or

)

of or

,

a

of

in as

(

-

a

is

where Cp non dimensional coefficient the drag coefficient and usually projected area wings such bodies tail surfaces stands fuselages for the maximum cross sectional area the case nacelles

as

312

is

)

wing

usually

31b

(

(

cc

11

of

cd

1

v2

airfoil

)

Co

(

Induced

1/25

=

D

as :

-

In

two dimensional flow the drag per length expressed

pressure

) ,

dynamic

)

С.

q

q

-

V2

=

1/25

D

Calling

(

etc.

Drag

it

,

.

.

in

.

It to is

,

-

a

to

a

,

a

,

is

to

a

in

to

of

drag its existence can be explained As the physical meaning even in the case of an ideal fluid three dimensional flow well According remember that circulation induced by vortex simply Helmholtz's theorems vortex cannot end an ideal fluid infinity must form closed line or extend The simplest picture

Chapter

AERODYNAMIC

1

Induced Drag ( continued

FUNDAMENTALS

22

)

that can be conjured (which explains the physical phenomena of drag in the case of a wing of. finite span ) is the " horseshoe vortex " ( Figure 17 ) .

Z

LR

8

sy

V --X

LIFTINO VORTEX

ZTRATION me (

If

a

)

(b)

votar Figure

we imagine a scheme as in

@

17

Fig .

17

-a ,

air under

the

of the trailing vortices will acquire some downward motion wing itself the flow will resemble Figure 17b .

,

the influence and at the

Flow at the considered section is inclined to the original direction flow V by an angle which for small values can be expressed as :

Q;

;

or

V

this can be called an "induced

"

V

dis

V!

=

V

of

( 32 )

angle .

According to the theory of the flow of an ideal fluid , the aerodynamic force must be perpendicular to the direction of flow , but this new direction of flow is now inclined by an angle of u / v , to that at infinity . This means that force R , Fig . 17 gives a component along V >, or , in other words , induces

drag , whose magnitude

at the considered section will be :

Di or since

R

PL

Di

=

-

R a

ti

Lai

LV

- L

v

/

v

( 33 )

AERODYNAMIC FUNDAMENTALS

Chapter I

23

Induced Drag (continued ) The considered scheme of trailing vortices is a very elementary one . spanwise The downwash distribution produced by it , would be like an arch way with downwash velocities increasing to infinity toward the tips ( Fig . 18a ) .

aa

7‫ןותנ‬

77

Figure

18

-

Ibi

( Horseshoe

Vortex and Vortex sheet )

One can also imagine , that instead of one vortex springing out from wing tips , we have numerous vortices , spaced at different stations the along the wing , and forming the so - called vortex sheet . In this way , different spanwise downwash velocity distributions can be obtained . Of course , as vortices spring out of a wing , the circulation is also being varied , or , the lift distribution is being varied . Of particular interest is an elliptic spanwise lift distribution ( circulation distribution ) which assures a uniform downwash along the span , which in turn makes the induced drag of a given wing a minimum . In this case , the induced drag coefficient is :

CL

TA

CD where

A b S

2

2 b °

/ s ( Aspect Ratio )

Span

Wing Area

( 34 )

Chapter

1

AERODYNAMIC

FUNDAMENTALS

Induced Drag ( continued )

The induced drag is the only kind of subsonic drag whose existence can be explained in the case of an ideal ( frictionless ) fluid , providing , of course , the wing aspect ratio has a finite value , ( or in other words , the flow is three dimensional ). Aerodynamic theories however , based on the concept of ideal flow , cannot explain the existence of drag in two - dimensional flow . In order to do this , a new concept must be considered . Viscosity

of Air and Reynolds Number

In reality air is a viscous fluid , and bodies moving through it are subjected to tangential ( shearing ) forces caused by viscosity . The existence of viscous ( shearing ) forces in gases can be explained by the transfer of momentum by gas particles traveling between layers of gas moving at relative

ly different speeds

.

When the flow is turbulent small masses of air are moving in a direction perpendicular to the main flow . Hence it is obvious that if there is a difference in speed between the adjacent air layers , then those small traveling masses provide a means for exchange of momentum and thus create shearing

forces

.

The flow is called laminar when there is no aerodynamically detectable masses of air traveling between the adjacent layers . In the case of difference in speed between the layers it can be said that they are'sliding " on each other like solids . But even in this last case shearing forces still will be present . The Kinetic Theory of gases explains their existence in considering the transfer of momentum by free molecules of gas which are exchanged between the gas layers moving at different relative speed . ( See Ref . 1 , pp 156-184 ) .

It is clear from all the above considerations that the necessary condition for the existence of shearing forces in gases is the difference in speed of gas layers , or , in other words , the presence of a velocity gradient across the flow . As to the magnitude of shearing forces per unit area ( ! ) , it may be assumed that for both laminar and turbulent flow they are proportional to the velocity gradient au : у

τ Tam 3 у

( 35 )

The coefficient of proportionality M is known as the coefficient of 14,7 psi ; viscosity . For air at standard sea level conditions ( P. T. = 59 ° F ) M becomes , according to the NACA ,

Mм

3.37

х

10-7 slugs / ft.sec

( 36 )

FUNDAMENTALS

so

-

called kinematic coefficient

whosé value under standard sea level conditions

;

is

obtained

)

:

'

y

м

S

Dividing M by the air density

viscosity is

the

Viscosity of Air and Reynolds Number ( continued

25

of

AERODYNAMIC

Chapter I

)

(

.

as

ft

sec

37

/? 2

1

6380

so

Vo

M.

be

it

;

; i

):

pp

5 ,

.

a

of

;

C

of

,

by

in

)

-

of

м

)

( 38

_317

=

R

(

,

in

,

;

be

it is

If

.

be

of

by

(

to

It

drag may has been shown that such aerodynamic forces viscosity present due only drag assumed that not but all aerody viscosity proved namic forces may influenced some way can analysis magnitude aerodynamic means dimensional that the the forces and hence the non dimensional coefficients CL CD special depend etc. will turn on the value ratio called the cdo Reynolds number see for instance Ref 77-78

mle

addition that

V ,

in

Remembering

38b

)

38a

y

(

Vc

(

:

.

concerned

follows

=

c

.

of

M

,

rewritten

asis

flow

as

far

a

is

1

,

,

is

is

can

R

)

38

(

.eq

dimension

,

bodies

as

as ,

§

the body

velocity of air characteristic linear dimension of viscosity For such its coefficient the air density and wings or blades the chord length represents the characteristic the

be

V

where

Reynolds number

)

is

on

c

aerodynamic forces

also

"

" .

The dependence scale effect

known as

of

6380

R.

V

or for sea level conditions

Boundary Layer

U

s

u

Figure

19

).

19

.

Fig

((

U.

),

ТУ

V

at

a

(

velocity reaches its full magnitude air ,

the

that within this layer

of

lar

to

jacent

it , to

is

flowing past body the air layer immediately When the air the body has zero speed adhesion while some distance

ad

perpendicu

This means

Chapter I

AERODYNAMIC FUNDAMENTALS

Boundary Layer ( continued

26

)

where the speed grows from zero to U , there is a velocity gradient dy > O and hence according to eq . ( 35 ) shearing forces exist . presence their limited this layer only since outside and hence the flow may be considered inviscous This layer grows speed air where the from zero its full value and where very thin one and shearing forces acting on body are generated gener ally known boundary layer sometimes Prandtl's layer all

is

,

"

.

a

is

"

"

or

,

a

"

as

.

it

as of

to

,

O is

ay

of

,

du

to

But

Our

be

to

of

A

"

in

2

X

20 )

X ,

X

C

?

NOI937

TRANSITION

!

TURBULENT

Cao carse SC ac cccSc lcci

-se

V

.

LAYER

BOUNDARY

LAMINAR

Fig

considered

(

flat plate will

be

,

example the flow past

a

7 ) .

.

(

Ref

У

As

an

Layer Literature

"

to

,

in

a

In

this course only few topics referring this subject will recalled good boundary layer problems any However reader interested will find Boundary summary and guide the literature Tetervin's Review

Figure

20

,

is

.

çe

x

5.48

( 39 )

:

be

R

VRX the

x

V

R

:

is

х

where the distance measured from the leading edge and Rx Reynolds number local х

is

it

de

as

is

of

,

in

the boundary layer laminar and At the leading edge the flow layer relatively boundary moving thickness the small But downstream expression grows for laminar flow seen from Blausiu's can

x2

and

)

(

40

,

St

can

X

0.377

RX

=

turbulent state

/5 5

St

of

boundary layer Thickness expressed by Von Karman's formula

in

the

.

is

to

x2 ,

,

in xi

At some distance from the leading edge say between point laminar flow starts break down and beyond point the flow fully turbulent boundary layer

the

AERODYNAMIC FUNDAMENTALS

Chapter I

27

Boundary Layer ( continued ) where all symbols are as in eq . ( 39 ) . The mechanics of transition from laminar to turbulent flow and causes producing it were not entirely clear until recent years . But now , according to Dryden , "the circumstances surrounding the breakdown of laminar flow and the beginning at least of the process of transition to turbulent flow are fully understood as a result of work completed during the last decade " ( Ref . 8 ) . Until almost the present time there were two different schools trying to explain the physics of transition . Representatives of the so - called Gottingen school believed that infinites imal disturbances may cause a transition by developing at some Reynolds numbers into unsteady wave motion ( laminar boundary layer oscillations ). Adherents of the British school tried to explain the transition as a result of turbulence already present in the free air stream ,

Experimental results obtained rather recently

reported by Dryden in Ref . 8 , indicate that depending on circumstances , the causes indicated either by the Gottingen school , or that advocated by the British , may start the transition . Of course , such factors as surface roughness , or an adverse pressure gradient may also facilitate the transition . On the other hand , trans sition may be delayed by such action as removing a part of the boundary layer air , favorable pressure gradient , etc. , and

In addition to transition there is another very important phenomenon of boundary layer mechanics : separation . According to Tetervin ( Ref . 7 page 5 ) : " the separation point is the point on the surface of the body at which the surface friction is zero . Upstream of the point the direction of flow in the boundary layer next to the surface is downstream , and downstream of the point the direction of flow in the boundary layer next to the surface is upstream

"

( see

Fig .

21 ) .

V

х

SEPARATION

Figure

21

Chapter

AERODYNAMIC FUNDAMENTALS

1

21

Boundary Layer ( continued )

In the presence of an adverse pressure gradient downstream ) both laminar and turbulent boundary layer separate . However , the turbulent boundary layer would of stability than the laminar one . This is so because in is a greater mixing of air between the layers and hence

( pressure

increasing

will eventually

the

to

of

,

.

an

,

greater degree turbulent flow there more particles of outer layers having higher kinetic energy penetrate the inner layers thus increasing their momentum i.e. the ability air particles move against appreciable slowing down the increasing pressure without show a

.

Skin Drag

velocity gradient within the boundary layer i.e. the so plate and other bodies such called velocity profile the frictional drag

on as

a

,

of

,

Knowing the

in

as or

in

it

is

‫و‬

1/12

(

a

of

In

.

x

2

.

is

q

is be

far from the body and

A

is

Ds

dynamic pressure skin friction drag the wetted area the case flat plate obviously projected wetted area will area

where

the

v2

( 41 )

A s the

the

9

CA

D

.

Cf

" ,

"

to

of

,

.

,

be

wings etc. can general calculated Since the friction drag depends calculating this the contact area between body and fluid seems logical drag projected type use the wetted area rather than the cross sectional area Hence the friction drag coefficient defined

-

at

"

.

22 )

12 ) .

,

7

.

.

(

to

.

.

(

a

in

"

for

/

0

TURB

.

1.0

u

in

of

in

"

so

is

.

of

a

The friction drag coefficient for laminar flow will be different from that the turbulent one This because the velocity profile for the appearance from the velocity turbulent boundary layer differs markedly profile layer Fig boundary the laminar At large boundary layer Reynolds number the turbulent velocity profile shows an extremely rapid very short distance This large slope velocity rise the wall causes the turbulent skin friction coefficient be higher than the laminar skin page friction coefficient Ref

LAMINAR

Yl8 Figure

1o 22

Chapter

Skin

I

AERODYNAMIC FUNDAMENTALS

Drag ( continued

For

)

laminar flow Blausius gives the following expression for the

the

skin drag :

29

1.33

( 42 )

TR

where

is the Reynolds number as given in eq . ( 38a ) .

R

method of computing the friction drag in the case of turbulent boundary layer may be found in Ref . 9 , or it may be approximated by Falkner's ( see for instance Ref . 4 , page 76) :

A

C&A

0.0306

( 43 )

ZR

or Von Karman's formula ( Ref .

5,

-

ost

page 106 ) 0.072

( 44 )

IR

When friction coefficients for laminar and turbulent flow in the boun dary layer are plotted against Reynolds number ( Fig . 23 ) , it becomes ob vious that for the same R.N. the drag coefficient for turbulent flow is much higher than for laminar .

)

006

TURBULENT

-004

mixed

LAMINAR

ON

Cf

WETTED

AREA

•

.

Figure

23

(

O

BASED

·

002

10

106

RN

it

)

-

of

(

,

in

to

a

,

.

be

of

of

In

of

)

in

of

of as in

it of is

of

as

,

.

of

of

a a

a

,

,

20 ,

(

of

to

be

of

to

is

in

In

of

or

most practical types flow about wings other bodies the flow layer partially boundary the mixed laminar partially turbulent Figure page 26. But similar the example self evident that largest possible part should desirable have the the wetted area covered by the laminar type flow since frictional drag this case will considerably drag be smaller For the reason much because boundary layer on stalling characteristics the influence airfoils this subject will be discussed later the problem boundary layer control has subject been extensive research for the last two decades this course only few topics regarding delay transition will discussed however any reader interested good guide the subject will find literature and summary the present status research on boundary layer control

AERODYNAMIC

Chapter I

Removing of the Boundary Layer

FUNDAMENTALS

Air

One of the long recognized means of delaying transition is the re moving of boundary layer air through suction . There are several ways of doing it . For instance , single or multiple slots running across the flow and located downstream from the anticipated point of transition may be used . A great amount of experimental work on this subject has been done , but , in spite of rather promising experimental results , there have been almost no practical applications of this principle to fixed , or rotary wing aircraft .

Dryden believes that removing a part of the boundary layer air by suction through a homogenous porous material may be a possible method of greatly increasing the stability of laminar flow . He cites German experi ments ( Ref . 8 ) which indicate that the required suction velocity is of the order of 0.0000 14 times the free stream speed , Favorable

Pressure Gradient

The creation of a favorable

pressure gradient as demonstrated by

the

so - called low drag , or laminar , airfoil sections , represents at present prob ably the widest practical application of the principle of boundary layer control . However , in order to produce a better understanding of these airfoils , the mechanics of transition of conventional sections will be briefly discussed

first .

At positive lift coefficients conventional airfoils exhibit a pressure distribution with a high suction area at the leading edge , and a marked drop in suction i.e. increase in pressure in the downstream direction ( Figure 24 ) . This means that the air particles in the boundary layer on the top of the air foil have to move against increasing pressure , or , in other words , against an adverse pressure gradient . This facilitates transition .

1

SUCTION

V PRESSURE

Figure

24

In order to obtain a low drag airfoil section , i.e. an airfoil with lamin ar flow extending as far downstream as possible , it is necessary to create a decreasing pressure downstream , or at least hold it constant for a large part of the chord . This has been achieved through the design of airfoil sections having a desirable pressure distribution , for instance as that shown in Figure 25 .

Chapter

AERODYNAMIC FUNDAMENTALS

I

31

Favorable Pressure Gradient As has been mentioned before

,

this is achieved by means of conformal trans SUCTION

V Figure

25

The reader interested in the subject will find more information and guide to the pertinent literature in Reference 10 , 11 , or 12 .

formations

.

It should be stressed , however , that usually it is impossible to obtain a favorable pressure distribution over an airfoil section for the whole range of practical values of ci · Hence the low drag is limited to a certain range of ci and outside of this region laminar flow airfoil sections may exhibit even higher profile drag coefficients than conventional sections ( Figure 26a ) . It should also be mentioned that low - drag airfoils are extremely sensitive to such factors as surface roughness , deviation of the real shape from the theoretical one (manufacturing tolerances ), waves and notches , etc. In this

SMOOTH

Cars.co cd

ROUGH

ca

Cd LAMINAR

LAMINAR

CONVENTIONAL

CONVENTIONAL

се а

.

Figure

26

b

Cce

respect conventional airfoil sections show more desirable characteristics ( Figure 26b ). However , it should be noted that in both ; low drag , and conven tional airfoils it is possible to obtain a forced transition of the whole boun dary layer by making the leading edge rough as , for instance , by covering it with such material as carborundum . Indeed this procedure is used for obtaining airfoil characteristics at so - called standard roughness .

AERODYNAMIC FUNDAMENTALS

Chapter I

32

Stalling Stalling , or in other words a more or less sudden decrease in lift with increasing angle of attack , is another phenomenon closely associated with the state of the boundary layer . It is caused by separation of the boundary layer when the adverse pressure gradient , ( the pressure difference between the trailing and leading edge ) becomes too high , layer separates , the flow around the airfoil changes from that characterized in Figure 27a to that in Figure 27b . It is obvious from Figure 27b that the circulation around an airfoil in the stalled position is decreased . Between points a - b in Figure 27a there is always some velocity component tangent to the path along which the circulation is computed . But when stalling occurs a large part of the a - b line (Figure 27b ) is occupied by a turbulent wake , which as such would contribute little , if at all , to the circulation . This greatly reduces the whole circulation When the boundary

airfoil

around the stalled page 15 ) .

and consequently decreases the

lift .

( see

eqs

.

( 23 ) ,

b

V

!!!

I Figure Working Formulae

O

Scale Effect

27

at ,

.

)

a

13

96

).

-

)

45

(

atd

°

xx

x 8

. ==

)

(

at

Дcimax +

ci

C1

(

correction term

8

at

-

be

a

by adding

16

were

the NACA the called standard Reynolds number Rstd The maximum section lift coefficient Cimax RN lower than can readily found from those obtained the standard RN cimax std

10

91

-

87

pp

.

airfoil characteristics

so

-

the pre war years most conventional at

given

by

In

(

4 ,

in

of

be

(

by

.

in

to

,

of

,

of

practical application the Since under different conditions Reynolds number usually airfoil elements differs from that which the experimental data were obtained special working formulae were establish ed order evaluate the scale effect on conventional airfoil characteris tics Basic relations regarding the scale effect on conventional airfoil sections were established Jacobs and Sherman Reference but sum mary their work can also found some modern textbooks on applied aerodynamics see for instance Reference and 104

AERODYNAMIC FUNDAMENTALS

I

those

can

page 259

,

on

as

airfoil sections

4 .

13 ,

different types

for different RN such graphs or pp 102 103 Reference

found

of

Acimax -

be

readily

in for

The value of Reference

33

- Working Formulae

Scale Effect

,

Chapter

can

assumed

to

the section drag coefficient

terms

)

the

Camin

(

Acd

+

cd

46

:

of

two

be

The total value

be composed

ca

of

:

,

is

a

of an airfoil at When the section maximum lift coefficient cima given RN drag known the section coefficient cd ifferent ci values can be estimated as follows

,

on

of

ci

profile drag coefficient obtained is

is

at

a

at

c

in

it is

, is

( 47 )

>,

In to

.

function

the

)1

!'

.

,

ci

is

a

at

is

is

7 ( 9

topt max where section lift coefficient for which computed Aca given RN and the maximum lift coefficient the Clopt reaches its minimum value

at

Clopt

)

:

-

=

ca

of

a

can be expressed

)

28

( c

(

The correction factor Acd following ration Figure

as

.

to

xX

8

x

8

R

R

)

be

it

R

,

(

8

X 106 calculated Standard RN and the RN for which cd pointed usrinly although However must out that relation correct greater than 106 and conventional airfoils for not always true for less than 106 even for conventional airfoils and seldom correct given RN finding for laminar sections these two cases always better experimental data obtainedcamin frectly refer the desired RN

)

47

(

(

х

the

*

х

:

)

‫( اه‬

the minimum

with RN

.11

106

R

Cdmin

Cmax which

.

(

std

variation

at

cdmin

is

-

cdmin

std

‫هس‬

as

For many conventional airfoil sections can be expressed

where

Clopt

is

ci

topt

to

cdmin

8

to

corresponding

Cimax

·

of

ci

in

cd

on

component whose value depends directly where cdmin represents RN while profile drag which depends represents the increase the relation the actual the the and Where

Chapter

AERODYNAMIC

1

FUNDAMENTALS

34

Scale Effect - Working Formulae 0

2

1

.

.5

4

.8

.7

.9

10

.O

28

-024

domin

020

Cdo

016

Acdo

012

.

.)

(

/

Fig

004

Clopt

Cemazi

.)

CPopt

-

(

G

·

0.008

28

COMPRESSIBILITY EFFECTS

a

of

,

in

of

.

"

,

.

,

;

"

a

of

of

(

,

M )

,

of

As the speed flow increases or other words as the free stream body Mach number becomes higher the aerodynamic characteristics compressibility This rather undergo changes associated with the effects high speed can be better clarified by the introduction va gue notion such terms as critical Mach number subcritical and overcritical subsonic speed and supersonic speed

,

of

is

a

is

is

It is

of

.

is

at

.

,

it is

at

or

.

,

at

:

of a of

to is

of )

at

(

:

of

to

.

it is

)

,

or

,

of

of

(

of

of

any other body When the speed flow over some part an airfoil equals the speed unity sound the local Mach number becomes said that the speed the free stream reached its critical value The free stream Mach number far from the body corresponding the first occurrence point speed any body the sound the called the critical Mach number Mcr Speeds lower than that corresponding the critical Mach number are supercritical higher called subcritical and When the speed sound ex points ceeded some point the body the flow about mixed charag stage ter subsonic and supersonic As the speed further increases totally supersonic reached which the flow around the entire body be expected that the aerodynamic characteristics the body will undergo change radical first when the critical Mach number exceeded and later wh

.

.

CRITICAL MACH NUMBER

Mr

:

is

be

)

)

(

(

in by

a

The free stream critical Mach number can found from re lations established von Karman and Tsien when the pressure distribu body tion around an incompressible low speed flow known

15

14

as

in

a

.

in

--

in

.

be

be

it

at

of

to

.

,

the flow becomes entirely supersonic However should noted that com pressibility effects start speeds be felt even subcritical The influence these effects on some airfoil characteristics two dimensional flow will briefly discussed below The reader more deeply interested compressi biiity phenomena may find useful guide such textbooks Refs and

AERODYNAMIC

Chapter I

FUNDAMENTALS

CRITICAL MACH NUMBER ( continued

- P.

Pcr

)

7+ ****

2

O

gott [[677 cr Mer

q

35

7

cr

? t)

1

3-1 -

m

1

( 48 )

P. is the static pressure in the free stream , Pcr is the pressure at the point of maximum suction ( highest speed according to Bernoulli's the

is

(

a

in

graphical form see

Bo

the

.

at

at

P

cr Per q

of

is

/q

specific

course the maximum suction coefficient any other body low speeds

or

of

-

.)

P

where Pcr O surface an airfoil section (

z

,

of

Mcr

f

as

be

conveniently presented

13

can page

free stream and

..

1.4

)

.

for instance Ref

dynamic pressure

: =

CV

(

Equation

the

is

q

4, 48 )

heats ratio

/Cp

equation ) .

g

where

-

of

(

)‫ع‬ speed

29 ,

in

as

a

to

at

by

it is

of

)

0

(

.

to

f

=

Mer

a

Figure given airfoil section graph which gives easy read the critical Mach number correspon L.S. any value the section lift coefficient experienced airfoil low

Having for ding

29

Figure

‫ی‬

-Mer

.

C

L.S

)

(

lift coefficient

a

as

,

to

it is

For airfoil sections more convenient for practical purposes represent the critical Mach number function the low speed section

by it .

DRAG COEFFICIENT

of of

of

1

:

at

,

,

of

or

,

of

Drag coefficients any other bodies are influenced airfoils For estimating and there are several theoretical ways compressibility instance the influence on the section drag airfoil sections subcritical speeds can be predicted through an application the Glauert Prandtl correction factor

compressibility

ME

Chapter

I

where

M

AERODYNAMIC FUNDAMENTALS is the Mach number of the free stream

.

30 ,

(

4 )

.

131 ,

its

>

Von Karman has shown that the profile drag coefficient in compressit flow of an airfoil at an angle of attack a 9 having thickness ratio t , and camber m is equivalent to the drag coefficient in incompressible , ( low M.N. ) flow of this airfoil having a t and m multiplied by the practical importance Glauert - Prandtl factor . The of this statement lies in the fact that if the low speed profile drag coefficients are known at differen: angles of attack for a family of airfoils characterized by various thickness ratios and cambers , then the drag coefficient at high M.N. can be found by multiplying t , m and « of the actual airfoil by 1 / ( 1 - M2) 17 , and finding the low speed profile drag of the sections modified in this way . Therefore the low Mach number wind tunnel measurements could be substituted to som extent for the high speed tests . However , it must be noticed that for Mach numbers approaching , and especially exceeding , critical value Karman's gives correction values that are much too low see Figure and page Reference TEST

cao

020

.

KARMAN'S APPROX

.

Figure

30

M

.4

010

2

.

016

.6

to

,

.

is

)

(

at

,

of

is

in

comparison When the free stream Mach number low the critical drag over that found for incompressible flow Mach number the increase obviously negligible obtained low speed However when the ratio

2.0

10 Figure

31

0.0

?

MCR

/

M

10

in

,

of

a

3.0 cdom Icdo

inco

.

is

/

M

,

/

M

Mcr approaches unity the drag usually increases rather rapidly and 1.12 the drag finally when the Mcr ratio exceeds value 1.05 quite crease violent

AERODYNAMIC

Chapter I

FUNDAMENTALS

37

Figure 31 shows a typical variation of Cdcomp / Cdincomp. as a function of the ratio of free stream M to critical Mach number . Any expection that a general curve as shown in Figure 31 can be applied to all airfoil sections is not fully justified since according to Nitzberg and Crandall ( Reference 16 ) " for some types of airfoil sections the drag rises rapidly as soon as the free stream Mach number exceeds the critical , whereas for other types no appreciable drag rise occurs until the Mach number of the free stream is considerably above the critical " . They further suggest in

Reference 16 that a good measure of the free stream Mach number at which abrupt supercritical drag rise begins can be found as that Mach number at which sonic local velocity occurs at the airfoil crest . *

LIFT SLOPE The theory of two - dimensional

,

incompressible flow states that the

section lift coefficient ci is proportional to the angle of attack and that the coefficient of proportionality a ( slope of the lift curve ) is constant ( see page 19 ) . This assumption of proportionality of ci to a can be maintained for compressible flow , but the lift slope coefficient at high speeds ( am ) should be corrected . The simplest correction is given by the Glauert Prandtl formula :

«

ao

ам

( 49 )

VI - M2

where ao is the lift slope for incompressible flow ( low speeds ) . There are also more refined formulae like , for instance Kaplan's correction (Ref erence 17 ) , but for relatively low Mach numbers ( up to M = 0.6 to 0.7 ) eq . ( 49 ) usually shows a good agreement with experimental values . However for higher Mach numbers experiment remains ( at present ) the only reliable source of information regarding the lift slope value ( see for instance

-

/

ao

32 ) . clm

Figure

PRANDTL

TESTS

M

GLAUERT

0.6

.

21

Ref

.

of

and Chapter

9

20

Ref

.

Lecture No.

of

See also

4

*

Figure

32

1.00

,

I

Chapter

AERODYNAMIC

FUNDAMENTALS

MAXIMUM SECTION LIFT COEFFICIENT

.

a

to

)

cimax

(

of

33 ).

Figure

Ma

at

of of

..

(

it

on

ci

of

,

,

be

.

)

of

at

(

18

to

it of

.

(

oi

to

be

can

on

of up

)

goat of

The maximum section lift coefficient of an airfoil at high Mach numbe ( clmaxM ) is usually different from that at low speeds . Normally this is a ' the forma attributed decrease in the cimax ( see Figure 33 ) . , which tion shock waves the upper surface and flow separation breakdown flow the trailing edge But with the Mach number still increasing the may again chiefly due supersonic flow over some the presence portion the airfoil Reference Hence can seen that depending speed distribution the character the airfoil surface different airfoils may show different behaviour regarding the influence the free stream present almost number clmaxM Here again wind tunnel tests provide the only fully reliable source information regarding this matter Howeve can be stated on the basis our present limited theoretical and experima greater tal knowledge that conventional airfoil sections seem show with free stream Mach number than laminar ones see decrease nax

CONVENTIONAL AIRFOILS

M

LAMINAR

To

.

in

the

.

)

(

66

ci

,

For instance the maximum lift coefficient of NACA 230 or NACA series the value decidedly decreases with higher Mach numbers to NACA semnat laminar exhibits much lower sensitivity Cimax free stream Mach number

00

Figure 33

á.c. TRAVEL on

of

has by

)

to 6 %

5

be

to

50

(

),

C1

(

/c

1.9

t

+7 +

t

°

5

5.90

++

-12

:

-

)

)

m² ,

-

L.S.

(ſa

40

)

/

cml

(

M

)4c

/

(

Cmi

It

(

12 %

>

of

.

(

19 )

(

t c

/

a

to

of

.

of

is

.

at it

be

of

to

In

compressibility addition the already discussed effects air position foil section characteristics should mentioned that the airio aerodynamic center varies higher Mach numbers Theoretical and expert mental material presently available not sufficient for establishing expres may only sions for precise quantitative computations those changes very thin airfoil sections stated that the aerodynamic center rather tendency move aft with the increasing Mach number while for something opposite seems moderate and thick sections gives following true Hilton Reference the semi emperical formula quarter for the influence Mach number on the chord moment coefficient

Chapter

a

AERODYNAMIC

I

. c . TRAVEL

FUNDAMENTALS

39

( continued )

the

(

at

,

,,

,

/ at

is

to

,

Flight

,

1945

,

-

&

Prentice Hall

McGraw Hill

" ,

Theory

of

"

Glen H. Peebles

:

5 .

Richard von Mises

6 .

1946

Sons

-

Elementary Applied Aerodynamics

,

John Wiley

,

Elementary Fluid Mechanics

"

,

1943

. ,

.

"

:

"

:

4 .

Paul E. Hemke

.

3

John K. Vennard

Durand Reprinting Committee

"

:

Aerodynamic Theory

Cal Tech

,

,

Gases

"

of

"

An Introduction the Kinetic Theory University Cambridge Press 1940 to

:

2 .

W. F. Durand

and

Chapter

"

1

Sir James Jeans

the airfoil thickness low speed

t c

,

airfoil lift coefficient References

.

at

is

.

/

,

Glauert factor

the

I

.

the

is

)

-V2

is

-

L.s

)

C1

(1 (

M

airfoil moment coefficient about quarter chord where cm1 /4c) M is the same parameter low speed high Mach number Cm14cL.S

A

"

,

,

1948

of

"

.

,

A

,

K ,

and

.:

Present Status Laurence Loftin Jr Paper Research on Boundary Layer Control January presented meeting the 1949

Albert E. von Doenhoff

,

S ,

A ,

I.

at

"

,

of

"

.:

,

4 ,

15 ,

S. ,

A ,

J.

,

"

Low Drag and Suction Airfoils April 1948 Vol No. .

Sydney Goldstein

"

12.

:

,

"

,

at

,

Albert E. von Doenhoff and Laurence K. Loftin Jr Characteris Wing Sections tics Subcritical Speeds NACA University Conference on Aerodynamics 1949 of

.

11

1

.

v

,

,

,

,

"

V.

R.

Boundary Mechanics Applied Mechanics edited Karman Vol Academic of

the

in

Recent Advances

Method for the Rapid Estimation Turbulent Boundary Layer Thickness for Calculating Profile Drag NACA ACR No. L4G14 1944

"

:

9. .

10

NACA TN

1947

Layer Flow Advances by Mises and T. Press Inc. New York Neal Tetervin

,

,

8

14 ,

Boundary Layer Literature

in

"

Dryden

:

Hugh

L.

8.

,

No. 1384

.

S. ,

Review

of

"

A

Neal Tetervin

:

7.

J. A ,

"

,

"

:

Method for Calculating Airfoil Sections from Specification on the Pressure Distribution August 1947 Vol No.

Chapter

AERODYNAMIC

1

References to Chapter

I

FUNDAMENTALS

40

( continued )

13.

Estman N. Jacobs and Albert Sherman : " Airfoil Section Characteris tics as Affected by Variations of the Reynolds Number NACA TR No. 586 , 1937

14.

Hans Wolfgang Liepman and Allen E. Puchett : " Introduction to Aerodynamics of a Compressible Fluid " , John and Sons , 1947

15.

16.

17.

18.

19.

J. Black :

"

Compressibility

An Introduction to Aerodynamic

Publications , London ,

",

Wiley

Bunhill

1947

Gerald E. Nitzberg and Stewart Crandall : " A Study of Flow Changes Associated with Airfoil Section Drag Rise at Supercritical Speeds " , NACA TN No. 1813 H.

E. Murray :

"

Comparison with Experiment of Several Methods of Predicting the Lift of Wings in Subsonic Compressible Flow " , NACA TN No. 1739 , 1948

M. Cooper and P. F. Korycinski

Effect of Compressibility on the Lift , Pressure and Load Characteristics of a Tapered Wing of NACA 66 - Series Airfoil Sections " , NACA TN No. 1697

W.

F. Hilton :

"

:

"

The

Empirical Laws for the Effect of Compressibility Quarter Chord Moment Coefficient etc. " , R & M

on

No. 2195 , 1943 20 . B. H. Goethert

:

"

Experimental Facts on High Speed Aerodynamics

21.

Ira H. Abbott

and

Brief Comparison with Theory

High Speed Aerodynamics Lecture Series given at Air Force Inst . of Technology , Dayton , O. , 1948 and

Albert E. Von Doenhoff : McGraw Hill , 1949

"

"

,

Theory of Wing Sections ",

CHAPTER II

41

SIMPLE MOMENTUM THEORY INTRODUCTION The idea of using an airscrew as a direct lift producing device is not new . Sketches and models made by Leonardo da Vinci indicate that he worked along these lines at the end of the 15th century . Nevertheless , with the devel opment of fixed wing aircraft the airscrew found its principal a pplication as a device producing the thrust required to overcome the drag in forward flight . For this reason the airscrew theories were chiefly developed for its applica tion as a propeller . No wonder , hence , that when the helicopter started to receive more and more practical thought , the already developed propeller theories served as a guide for analysis of the helicopter rotor .

It is obvious that propeller theories were mostly concerned with the movement of the propeller along its axis , or with the development of static thrust . But the most attractive feature of helicopters is their ability to climb vertically , or to hover without any motion of the aircraft as a whole . For both of these regimes of flight the study of propellers provided already developed theories which could be directly applied to the helicopter . One of the simplest approaches to the problem of creating thrust by a propeller , or helicopter rotor , is given by the momentum theory whose develop ment was initiated by Rankine and Froude more than 80 years ago ( 1865 ) . Since this theory is also extremely suitable for the physical interpretation of numerous flight phenomena , it seems to be advisable to begin the study of heli copter aerodynamics by getting acquainted with the principles of the simple mo

mentum theory .

ACTUATOR DISC The simple momentum theory assumes that the rotor works in such a way that the fluid passing through it acquires some additional speed , v , in the axial direction , and this speed v which will be called the induced velocity , is uniform over the whole disc area . It is further assumed that the rotor operates in an ideal fluid , i.e. its rotation imparts only axial velocity to the fluid which passes through it , or , in other words , that there is no rotation of the slipstream . The ideal arrangement as defined above will be called the''actuator disc The increase in the velocity of air passing through the actuator disc forms the basic idea of the whole theory : From Newton's laws of classical mechanics one may recall ( see also page 12 ) that if the linear momentum of a body ( having a mass m ) has been increased in a given period of time t by a value m AV , ( where AV is the difference in linear velocities of the body at the beginning and at the end of the considered increment of time ) , this means that during the time t a force acted on the body in the direction of AV and that the mean value of this force over the period " t" can be expressed as :

F

m

AV

( 1)

".

II

Chapter

SIMPLE MOMENTUM THEORY

ACTUATOR DISC

( continued )

If the time t is chosen as equal to a unit expression for the force F becomes : F

(

sec

1

.

for instance ) then

AV

m

(

the

la )

It is obvious from Newton's third law that if we knew from momentum considerations what force acted on the fluid as it passed through the actuator disc , then a force , equal in magnitude , but oppositely directed must act on the disc . In other words , by knowing the magnitude of the force which has produc ed some given changes in the momentum of fluid passing through the actuator disc , the magnitude of thrust developed by a rotor , or propeller can be evaluated

.

THRUST IN VERTICAL CLIMB AND HOVERING

the

in

air

it

vc с

)

flows past

axial

Vc but directed downward

to

is

far from the rotor equal

(

a

speed

stationary while the

P.

.

(

Fig

rotor

be

1 ).

imagined that direction with

the

The more general case can be considered by assuming that the rotor For convenience it can along its axis with a constant velocity V. up moves

Vc

'

+

V

P.

AP

xe

ve

P P Р

P.

vo

TIT

co

Figure

1

P.

in

as

or

,

as

,

of

is

.

)

4

p

.

(

a

of

,

to

1.

to

be

by

of

The probable shape the stream tube enclosing all air particles which expected are affected the actuator disc may look more less Figure At the actuator disc the air particles acquire some additional axial velocity and since according continuity the mass flow through any the law cross section stream tube constant the stream tube must narrow the velocity increases see also

SIMPLE MOMENTUM THEORY

II

Chapter THRUST

VERTICAL CLIMB

IN

43

AND HOVERING

(

continued

)

It may be assumed that after passing the disc , the speed of flow increases still further (narrowing of the stream tube) and far downstream ( at oo ) it reaches a value of Vc + Vi Since there is a thrust acting on the actuator disc , it is logical to expect that the pressure above it and below it will be different. In other words , some discontinuity in pressure at the actuator disc is assumed . Be cause of this discontinuity in pressure , Bernoulli's equation can only be applied to the upstream part , and to the downstream part of the flow tube separately .

For the whole upstream part of the flow tube , the total head H.

) )?

( 2

)

++

vv

Vc

g

1/2

+

P

2 v

( 3 )

+

с Vc

(

+

§

1/2

AP

+

=

), 2

:

of

the downstream part

v

+ +

V.

(

+

from

momentum

:

( 5 )

vi

v )

R2 )

+

T

vc

(

g

the

A

T

A

where

of

is

la )

(

eq

is

v

is )

.

.

a

vi +

(

9

A

,

(

front and immediately hence the thrust experienced by the disc can also in

AP

pressure immediately

.

R2

6 )

(

4 )

(

can

be

.

from eq

it

),

6

)

( 5

)

AP (

Ti

expressions for expressions and

6

(

of

AP and then

found that 7 )

Vi

---

(

1/2 v

or

V

2

equating the right sides

.

eq

A

ΔΡ

V

Substituting

in

=

T

:

be expressed as

,

The difference behind the disc is

in

of

in

in

is

disc area hence mass nothing else but fluid which flows each second through the disc and velocity the difference this mass Thus full analogy with obtained

,

(4) of the

1

1/2

the increase

T

,

a

of

On the other hand from consideration expressed the total thrust can be as

VilV

Vc

(

S

11

+

AP

Vc

1

H.

()

2

H

H

=

Po

g

true for the total head Hy

P

V

vc

+

ні

Subtracting

2

1/2 ?

P

1/2

is

Ho The same

P

= =

(

should be the same :

SIMPLE MOMENTUM THEORY VERTICAL CLIMB

IN

HOVERING

AND

V1

,

continued

into

(5)

i.e.

)

expression

the

T.

2

?

R2 Rag

v

2

T

?

2

=

T

or

Aşp

2

T

hovering

:

particular case when the speed Vc

in

v

v

))

vtv

vc

(

R

Vc

+

+

2 (

2

T

R

a

In

T

or

AS

2

=

T

Vc

:

.

Substituting this new value of thrust becomes

(

for

THRUST

O ,

II

Chapter

POWER REQUIRED IN VERTICAL CLIMB AND HOVERING

to

the

the

by

it

.

2v ,

+

V. с

to

? ]

v .

-

?

2v ))

++

(

vV

T

(

10 108

) v )

+

(v

I

.

v )

to a

is

)

Vc

(

of a

at

)

(

ideal rotor actuator disc developing thrust steady climb equal speed the sum the rate climb Vc and induced of

thrust and the the disc

(

T

:

,

350

(

v .

)[

v

=

expression giving the power required obtained

in

+

+

?

v )

Vc +

(

A

(

will

vlv

550 an

show that

at of

)

10

(

product

velocity

+

(

T

Vc

for the power required

.

Eqs

Vc

v )

2

in (

Pc

ASIVE

by

E .

Pc )

climb

(

Dividing

AS hp10 )

2

therefore

eq

but

T

=

E

or

be

•

1/29

in an

=

E

2

as :

be of

E

is

to

(

E )

in

be

to

),

1

to to (

V

V. с

a

.

be

at

of

With the rotor imagined stationary and the whole mass air Figure flowing down only performed speed the work rotor will that required cover energy losses due flow through required equal disc The power cover these losses will obviously slipstream away energy Since far the each second carried the below the rotor the speed flow increases while far above equal expressed Vc then can

II

Chapter

SIMPLE MOMENTUM THEORY

45

POWER REQUIRED IN VERTICAL CLIMB AND HOVERING

( continued )

In other words - the power required to climb at a rate V с, is equal to the power required to overcome gravity ( V с x T ) plus the product of the thrust times the induced velocity. ( Tv ) . This product Tv may be called the ideal induced power . The adjective " ideal" being added in order to indicate that as this power corresponds to the minimum energy loss in the slipstream ,it thus represents the lowest theoretical minimum required T with a given induced velocity to produce a given thrust v .

For a particular case of hovering Vc = 0 , the power required for helicopter having the rotor working as an idealized actuator disc would a

be

in bhp .:

Tv

1

Ph

550

But from eq expressed as :

(9 )

( 11 )

the induced

( or

downwash

)

velocity

v

can be

T

7

follows

:

as

be

12 )

(

eq .

?

R

=

rewritten

1

of

rotor

RZ

550

29

)

13

W

(

1

ет

T

:

550

2

-

1

hovering

PhT

)

(

by introducing the Or

and power loading

T P

Ti

P

R2

12a

T

still further rewritten

T

be

12ti5

This last symbol expresses the power required per pound In

.

thrust

can

3/2

of

its reciprocal

(

Frag

V2A

This last formula can disc loading W

symbols

3/2

T

Ph

550

or for the disc area

1350

Ph

:

R

hence

A

ZAS

12 )

T

CHAPTER II

SIMPLE MOMENTUM THEORY

POWER REQUIRED

IN

VERTICAL CLIMB AND HOVERING

( continued )

Eq . ( 13 ) indicates that if the simple momentum theory could be applied to practical helicopters , then there would be no lower limit for the power required to produce a given thrust in hovering flight . It would only be necessary to make the disc loading ( w ) as low as possible . It must be remen bered , however , that the existence of the profile drag , which has been comple tely neglected in the simple momentum theory , will considerably modify the above conclusion . THRUST AND INDUCED VELOCITY IN FORWARD FLIGHT Through an application of the simple momentum theory to the case of vertical flight of the helicopter , formulae were established which express the dependence of thrust on disc area , air density , rate of flow through the disc and magnitude of the induced velocity . These formulae ( 8 and 8a ) can be expressed in words as follows :

the

8

the

.

a

,

i

,

,

.

I

,

as

(

" .

+

.

v, in

in

be

"

of

a

,

.

as

be

is

in

,

of

).

of

or

of

,

is

a

of

),

2

.

.

of

air

,

v )

of

of

(

as

as

+

'(

v ." of

the

" The thrust developed by a rotor moving along its axis with a speed density times of Vc is equal to the rotor disc area T R2 times 2 resultant speed flow through the discV vc and times doubled induced velocity the above relation has been mathemat The accuracy ically proven within the limits validity Unfortunately the theory itself far horizontal flight concerned more generally flight heli translatory velocity rigorous copter with component horizontal the no development yet Nevertheless the formula for thrust can offered relationship proposed by Glauert generally accepted see page 319 Ref and Ref which expressed words will sound exactly for vertical flight There will be however difference mathematical interpretation the notion resultant velocity flow through the disc This last for flight obviously algebraic ward will different from the sum Vc used in the case of vertical climb

it

T

VERT

8

.

Vc

V

V Figure

2

HORIZ

PATH

.

FLIGHT

):

(

).

V )

(

1

:

2

V

+

VE

y2

VF

be

of

of

,

2>

of in

is

fVe

(

is

it in

,

to

In

analyze the translatory flight let order be assumed for con stationary past venience that the rotor while the air flows the manner Figure speed shown and let the air flow far from the rotor flight taken with the opposite sign course the speed The speed Vf can be resolved into two components horizontal and vertical Vc

SIMPLE MOMENTUM THEORY

II

Chapter

THRUST AND INDUCED VELOCITY

IN

47

FORWARD FLIGHT

continued

(

)

It is obvious that the V component will represent the horizontal velocity of flight while V , is equal to the rate of climb ( both will have an opposite direction to that of true flight ) . At the rotor disc the induced velocity v ( directed along the rotor axis ) adds vectorally to the speed of flow V ,f thus making the relative speed of flow at the disc itself ( see Fig . If V ' will denote this relative resultant speed of flow at the disc then the formula for thrust established for vertical flight can now be generalized into the following expression by substituting V ' for ( vc + v ) in eq . ( 8a ) :

T

2

=

V v е viv

TI т R2

2).

( 14 )

'

R

no

is

is

of to

).

at

R2 )

T

(

(

' ,

V

is

.

,

TT

2

.

a

of

of

22

the

It is interesting to make a physical interpretation of eq . ( 14 ) from the point of view of momentum principle : According to its requirements there must be an increase in speed along the line of action of the resultant force ( in this case the thrust T ) . By analogy with eq . ( 8a ) it seems logical to assume that the doubled induced velocity ( i.e. 2v ) represents this increase . By making this assumption TR vi should represent per through mass flow second the stream tube affected by action the obviously rotor This stream tube will be circle radius since its cross section is RS Furthermore since the mass flow per unit time is TR TR2 the stream tube section and there other coefficient smaller than 1.0 This obviously means that the stream tube perpendicular section the rotor disc the resultant speed (

rotor disc

.

the

of

to

,

R )

V '

(

flow

at as

a

is

),

in

be

it

,

.

V '

In

oblique flow which other words can said that the case was substituted for convenience for forward flight the stream tube which affected by the rotor action has at the rotor circular cross section having its radius and being perpendicular the resultant speed

of

.

3 )

a

,

.

(

14 )

R

a

found

from

be

,

forward flight can readily

)

15

1

2

R2 ZTRES

(

V

V

т

14 )

.

(

The induced velocity

eq

become identical

in

R

to

of

.eq

as

by

,

in

,

to

of

a

v,

(

:

is

semis pan equal

of

wing

be

14 )

(

a

it

by

of

An indirect support for the validity this formula .can comparing found with formula giving thrust developed by wing finite span General aerodynamics see for instance Pages 49-50 Ref proves that for wing span equal 2R and downwash velocity uniformly distributed along the span the thrust developed horizontal flight expressed exactly the same formula This means that for horizontal flight both formulae for the rotor radius and

POWER REQUIRED IN FORWARD FLIGHT

t:

of

it

in

(

.

of

as

in

to

is

)

,

of

of

power required Considerations forward flight based on the application the simple momentum theory represent the case flight vertical ascent the most idealized conditions Here again slipstream assumed that there are no energy losses due the rota 11

SIMPLE MOMENTUM THEORY

II

Chapter

FLIGHT ( continued

POWER REQUIRED IN FORWARD

)

whole disc . It is obvious that under such conditions all considerations of power required would lead to the establishment of some ideal minima which , of course , can never be achieved by a real helicopter . Nevertheless knowledge of how to calculate the ideal power will be of considerable use fulness in predicting the performance of a real machine .

)

C '

a

Vf

a

two

(

16 )

is :

to

Vc

-

17 )

(

+

,

of

it ,

in

as

second

V

axico

and

Ves

sliroo-

their values are given by

eqs

.

V2

+

v2

(

!

:V

R

,

R² ?

:

losses per

18 )

(

v

+

2

:

18 )

(

)

to

16

V

+

(

y .

the energy

parsos

2

v²2

clear that

VC

:

becomes

]

v ?

/

in

,

or in

After algebraic simplification the above equation for energy losses per second other words the power required expressed ft.Ibs sec ol

?

+

(

Vecos

2

)

sin q

-

с

12

, v2

V

V

y

-

+

Vsing

2v ) 2

Vcos

(

[

2

T

1

=

T

vi

с

V

V

as

f V2 ;

E

par too

par too

it is

2

)

18

y2

+

v2

(

16 )

(

From Figure

hence

y

Vc to 1

T

sing

slipstream velocity

the

2

=

-

=

slipico

cos

find from eqs in

is

E

to

It

easy the increase

and

V

Vc

16 )

(

be increased by the doubled induced velocity

will

ax100

but

sing

rotor slipstream far behind the rotor the parallel component will remain the same front but the axial compo

V

due

cosy cos

R²2

nent

the

.eq

In

ах Vaxas

sing

is :

V

cosy

V

.

paroo

while the axial component

given by

be

of

of

it .

V

of

is

,

it is

(

V ,

is

an

the

This ideal power will be calculated for the general case of steady flight in which the rotor axis is tilted through an angle ( Fig . 2 ) , while flight helicopter moving along path speed having inclined with component component horizontal and vertical rate climb Vc Agais stationary assumed for convenience that the rotor but the air flows past speed The flow far ahead the rotor can resolved into components the disc The component parallel

II

SIMPLE MOMENTUM THEORY

v )

(

20 )

sing

,

of

:

,

of

is

relative speed

V '

brackets represents the axial component which flow through the rotor disc and the following general theorem can be established the

in

But the expression

( Vc

V ' v is simply the thrust. This means forward flight expressed in Hp will be : &+

T

in

Vy

1

T

550

2

)

RS

+ +

But according to eq . ( 14 ) 2 TT that the ideal power required

Pid

49

FORWARD FLIGHT ( continued

IN

g

POWER REQUIRED

cos

Chapter

a

is

."

It

)

or

(

of

in

of

,

of of

.

be

in

ed

by

.)

/

in

(

to

is

"

any steady flight ideal power lbs.ft sec required helicopter equal the thrust developed the rotor rotors times through speed component the disc flow the relative the axial previously hovering obvious that case vertical ascent or the establish readily developed from the above theorem formulae can The

ROTOR TILT IN FORWARD FLIGHT a

is

of ,

be

of

it

w ,

-T 3

of ) so .

V

T

VERT

.

V

W Figure

Vc )

(

in

,

is

in

FLIGHT PATH

Vf

.

Via

of

2 (

).

(

g

an

,

to

to

.

y 3)

so

.

(

Vf

to

it is

.

may be completely neglected

HORIZ

of

as

in

to

is

a

as

If

rotor be used forward flight not only source obviously propelling be tilted vector must its thrust device lift but also required angle from the vertical position through This rotor tilt overcome the drag due the horizontal component the velocity flight usually small may see Fig The angle that = 1.0 and sin radians Furthermore assumed that cos flight the drag component due the vertical component the speed usually comparison with the helicopter's gross weight small that the

SIMPLE MOMENTUM THEORY

II

Chapter

ROTOR TILT IN FORWARD FLIGHT

50

( continued )

Under the above assumptions , it is justified to write :

T

W

1

cos

V

y

IIS

U

g

cс

vg

sin

( 21 )

V

1

:

the

of

,

2

)

(

tilt i.e.

i

in

be

(

of

when assumptions expressed by power required forward flight can obtaine

550

23 )

(

+

vy

v )

+

VC

(

1

Pid

W

:

20 )

)

.

(

)

22

W

are valid the ideal rewriting eq 21

D

IIS

j

g

For small angles (

V

of

of

D

V ,

Il

sin

eq by

is W ,

to

in

If ,

addition the gross weight the relation between the heli flight copter's drag and its speed known then for any value speed the tilt the rotor axis can readily be obtained

1

or simply V !

flow through

the

relative speed

)

(

the

of

1

the axial component

of

ax

23a

ах

.

disc

W

550

is

where

y

Pid

FLIGHT

HORIZONTAL

:

v

,

W ,

=

T

:

)

(

)

23

(

.

15 )

(

eq .

24 )

(

4

25 )

:

vectorial sum

In horizor see figure

(

v .

depends on

(

it

unknown since the following

as V

V '

)

V ! +

VT

still

expressed

V

V '

(

in

equation But tal flight can

be 24

2 T

R2

W

is S

v

is :

.

be

eq

,

O

a

of

particular case Horizontal flight represents the general forwa flight when the vertical component Vc hence in order to compute power only the ideal from the induced downwash velocity need known From rewritten under the assumption the dow wash velocity

Chapter

SIMPLE MOMENTUM THEORY

II

HORIZONTAL FLIGHT

(

continued

51

)

It is obvious that at high flying speeds the difference between V ' and V becomes negligible as the addition of a relatively low velocity v to the large one V gives a resultant V ' whose absolute value will be only slightly differ ent from its main component ( V ) . Under these conditions , the induced velocity can be expressed by an approximate formula

:

W

V 2

T R2

V

S

( 26 )

flying speeds the assumption V V ' is able , but the rotor tilt ( 3 ) required in steady flight at low It so small that the following approximation can be made : that z velocity O , or , in other words , that the induced lar to the helicopter flying speed V ( see Figure 4 ) , T

For

low

no longer accept

velocities will

be

can be assumed v

is perpendicu

ng

V

V y

V

'

VERTICAL

Figure 4 Under the above assumptions the resultant velocity at the disc ( V ' ) expressed can be as

VE which

,

( V2

introduced into eq .

y2 ) v2

+

( 24 )

will

)(

give

27

W 2 TT

ViV2

R29

y27

+

Solving eq . ( 27 ) for v , the following expression for downwash velocity at low speeds of flight is obtained : V

=

VA

4

*****

14

W

+

2

TT

R2

2 1

2

N

v2

( 28 )

SIMPLE MOMENTUM THEORY

II

Chapter

HORIZONTAL FLIGHT

( continued )

It can be easily seen that for V = 0 , eq . ( 28 ) becomes identical with eq . giving the downwash velocity in hovering . Developing the expression

Reg -* (2kg ) ])"?

v4

[

+v

(12)

1/2

W

R4

+

eq

,

:

g

of

,

be

In finding the downwash velocity , at low flying speeds as well as ones the following graphical method can used which permits taking into consideration the existence the tilt angle

high

according to the binominal theorem , it can readily be shown that for large v values the series quickly becomes convergent , hence only the two first terms of the development can be maintained . In this way eq . ( 28 ) will become identi cal with the high speed expression for downwash velocity , expressed by eq . (4

5

Figure

:

DISC

V

V

-

V '

b?

RELATIVE

the

to

of

) '

'

:

v

INDUCED

VzV

h , .

HOVERING

the

be

VELOCITY

vafiv

V '

v '

=

v

V '

>

be

a

be

; v

-V

a

to

of

AT

is

)

is

V '

(

v

f

v

VELOCITY

of

,

a

24 )

(

of

(

left

as to

pond

,

of

).

scale same

(

.

Using drawn convenient curve giving course the scale the ordinate and the abscissa axes should hyperbola whose point This will obviously will corres the hovering condition and this will the limiting value only the part any practical interest the curve where

SIMPLE MOMENTUM THEORY

II

Chapter

HORIZONTAL FLIGHT

53

( continued )

As previously mentioned , the value of the induced velocity must satisfy equations ( 24 ) and ( 25 ) simultaneously . By referring to figures 6 and one can see the simple graphical method which can be employed in finding the downwash velocity horizontal flight :

vin

aog V

V

+

6

'

=

V

V

VERTICAL

Figure

'

T

v

V

7

)

to

V '

,

v,

V '

of

easily

found

.

can

be of

.

a

V '

.

V '

If

;

,

to

v

be

25 )

(

,

of V ' 24 )

(

and

is

v

of

:

of

of of

6.

to

v

of

(

v

v

of of V ' V '

a

in

5.

.

V ,

The value of and the direction of tilt the rotor are known value the corresponding value found from Figure These values and must also satisfy the vectorial relation Figure ship shown This means that the head vector must lie on ab must be equal line ab parallel the rotor axis while the length corresponding the value the assumed the chosen the first corresponding time and that do not fulfill the above conditions new value should assumed and the whole procedure repeated By cutting and trying the correct pair satisfying both equations values and

Assuming

by

of

to

to

to

at

24 )

.eq )25 .

(

W

.

of

(

v

,

at

of

weight eq relation

the

a

assuming value for the relative velocity possible determine the value produce the thrust necessary balance satisfy and which will the same time the vecto

has been shown that

the slipstream the disc V. required the induced velocity

it is

It

POWER REQUIRED CURVE IN HORIZONTAL FLIGHT

5,

SIMPLE MOMENTUM THEORY

II

Chapter

.

POWER REQUIRED CURVE IN HORIZONTAL FLIGHT

( continued )

29 )

(

+

)

D

:

V

W

(

Pid

"

580

in

is

,

)

22

eq

.

its

(

This value of the induced velocity v , can now be introduced into O , and substituting equation ( 20 ) . Recalling that in horizontal flight Vc for g value from the following expression for the ideal horse power obtained horizontal flight

to

at

,

it is

7

)

Pid (

PR

.

of

in

a

of

flying speeds By choosing easy sufficient number any establish the graph the ideal power required horizontal flight Figure altitude .

MAX

ROTOR

HD

AVAILABLE

REQUIRED POWER

Pemak

V

Ve

VW

-

I

IDEAL

.

-Vopt

7

Vmaxa

of

be

,

.

of f( V )

a

in

at

at

R

P

,

beis

of

When the power available the rotor the ideal helicopter flight given altitude speed known then the maximum horizontal Obviously the maximum found from the graph showing Pid point speed will the abscissa that the curve which has as its :

ordinate PRmax

:

)

7

to

of

(

is

It

evident that the considered ideal helicopter see Figure power when its flying speed will be equal will need the lowest amount Ves

car

Figure

Chapter

SIMPLE MOMENTUM THEORY

II

POWER REQUIRED CURVE IN HORIZONTAL

55

FLIGHT

( continued )

Under zero wind conditions the smallest amount of efficient energy ( measured at the rotor ) will be used for traveling a given distance , when the flying speed will correspond to the abscissa of the point of tangency of a straight line drawn from the origin to the power required curve . ( V opt . in Figure 7 ) . In the case of a head wind of magnitude Vw , the point from which the tangent is drawn should be moved to the right of the origin by this value Vw . While in the case of a tail wind of the same magnitude it should be moved to the left . It is easy to prove that speeds of flight found in the above manner represent the optimum theoretical cruising speeds , as they correspond to the highest possible ratio of distance traveled in unit time to the energy spent at the rotor ( which is obviously proportional to the rotor horse power ).

RATE OF CLIMB IN FORWARD FLIGHT When the total power available at the rotor ( PR of climb can be found in the following manner :

Vc

Vc Horiz

ROTOR

)

is known

// TO

vive

AXIS

.

ROTOR

,

the rate

Axis

Vf

V

'

U

A

Figure

8

be

is

,

U ,

the

UU

.,

+

),

(

550

W

PR

30 )

11

(

U

:

ed

.

eq

7

It is assumed that the rotor inclination , g , remains the same in ascending flight, as it would be in horizontal flight at a speed V equal to the horizontal component of the actual flying speed Vf. Hence , if the drag as a function of forward speed of the helicopter is known , then the rotor tilt can readily be computed . Also , since the power available at the rotor Pr is known , the total rate of axial flow through disc can found from 20a where for the axial.companien Vax substitut

SIMPLE MOMENTUM THEORY

Chapter II

RATE OF CLIMB IN FORWARD FLIGHT ( continued ) climb can

On the other hand the rate of

Vc and substituting

U

=

for U

Vcc

(

eq

vy

expressed

be

+

v)

-

(

2

!

Fig . 8 ) as : ( 31 )

( 30 ) :

.

550 W

PR.

vg

)v

( 32 )

)

+

a

of

.

0O A

. ( 32 )

eq

,

52 )

(

)

(

f

v

.

.

be

,

V '

of

of V '

v

is on

be

A

,

V ' as as .

line

The value of v in eq . ( 32 ) is yet unknown . But , it may be readily From the head found with the help of a simple graph as shown in Figure 8 : of vector V a line parallel to the rotor axis is drawn . Again from the head o vector U a line normal to the rotor axis is drawn , intersecting the first By approximation point may considered the head vector repre senting the relative velocity the slipstream indicates the mag page nitude and from the graph the correspon easily ding can found When this latter value introduced into the rate of climb will be obtained

A

be

,

of V '

(

50 ) .

it

,

at

v,

to

for altitude calculations

remember graph

helpful

.

(

),

v

=

v

of

that

f is

In

finding the induced velocity altitude must be inversely proportional density page the air see altitude while the scale remains constant will

of

the

in

is

the

.

PR )

If of

in

be

it

a

,

finding rate However should mentioned here that climb usually procedure based on the principle power excess accurate enough for all practical purposes the rotor power required horizontal

.))

all

(

,

be

to

is

(

33 )

:

.

)

.

.

in

of

req

av

)

( )PR

.

(

(

550

PR

Vc

by at

of

of

in

it is

to

a

at

.,

(,

req flight given speed and power available rotor PRlav easy assuming that are both known then find the rate climb the rotor power excess that required for horizontal flight used perform work against gravity The rate fps climb will thus

.

,

(

n )

f

=

Vc

RATE OF CLIMB

Ver

.

serv

I

ALTITUDE

max

.

9

.

=

Habs

max

at at

at

)

It is (

of

)

Çmax

V

V

values vs altitude obvious that the altitude ceiling while that represents absolute the zero cmax becomes ceiling yield fpm See Figure the service will 100 Cmax cmax

V

which

which

33

and plotting

.(eq

in

be

Service and absolute ceilings forward flight can obtained finding from maximum rate climb Vcmax several altitudes

by

W

Chapter

SIMPLE MOMENTUM THEORY

II

57

noticed that the method of finding the rate of climb from the excess power can be accepted for higher flying speeds ( ve and higher ),,when climbing would not appreciably change the rate of flow through the disc . For low forward speeds the graphical method previously outlined is more recommend ed , while for purely vertical climb a suitable method is outlined in the next para

It should

be

graph .

CEILING IN VERTICAL ASCENT Rate of climb and ceiling in vertical ascent can be found in the following way : In the case of vertical ascent ( only ) the axial component of the total flow through the disc U ., and the velocity of flow relative to the disc V ' , are identical , this means that

PR

550

U

( 34 )

From the general relationship

( p : 46 ) between thrust T ( in this case equal to the weight W ) , rate of flow through the disc V ' , and induced velocity v , the following expression for the induced velocity in vertical climb ( vv ) can be derived :

SI

W

VV

ZARE

V!

Substituting into this last expression the value of Vi as given by eq .

( 34 ) ,

the following

is obtained

:

1100

R2

we

Vy TT

( 35 )

9

PR

Since the rate of climb in this case is : U

су Ver then, from

equations

Ver

-

-

VV

( 34 )

and

550

PR

( 35 )

:

w2

W

1100TRY

S

( 36 )

PR

As the variation of PR with altitude should be known then 550 Pķ / W can easily be computed for any altitude h . The same applies to W2 / 1100 T R2 S PR , and the vertical rate of climb at any altitude can readily be obtained from equa ( 36 ) .

.

can be

the

( 36 )

to

eq .

the

density can be

relationship between engine power ( hence PR as well ) and air expressed as a simple algebraic function , then by setting VcV = 0 , density corresponding solved absolute ceiling i.e.

the

SH

If

for

tion

II

So

the absolute

Assuming

.

standard atmosphere

ceiling can readily be found from 100 1.67 fps instead Vcy

60

tables zero

for

From this value of

( continued )

,

VERTICAL ASCENT

and

IN

of

CEILING

SIMPLE MOMENTUM THEORY

SH

Chapter

a

is

,

in

.

a

,

be

.

be

in

repeating the above described procedure the service ceiling vertical flight Usually relationship engine power and den can found however the between sity cannot expressed simple way and graphical method more suit able

.Fig

(

the

of

.

,

or

n )

f

(

2² g

PR

TR2

=

1

.

1.66

wa

1100

=

550

Pe

1.67

Vev

SCALE

of fps h

to

,

to

fpm

g PR

is

100

22

IRS

fps

. : C /R

1100

,

of

,

equal

between the ordinates w 550 PS W

in

,

,

at

of

of

of

).

h,

of

,

h

be

36 )

(

)

35

(

10

.

Eqs plotted vs. altitude and should feet see any ure For altitude the difference between the ordinates these two curve will give the rate vertical climb this altitude while the abscissa point intersection the curves indicates the absolute ceiling The service ceiling will correspond where the difference that value course

find )

h

ALTITUDE IANY UNITS

.

Hserv

L :

IV ,

Glauert Durand Reprint ,

H.

Helicopter

,

1943

"

a

of ,

&

Division

1928

.

,

,

A.

by

. ,

"

:

J

"

'

,

.,

Aerodynamic Theory Volume IV Division Applied Airfoil Theory Betz Durand Reprint ing Committee Cal Tech 1943

"

:

F. Durand

R

.

"

On

,

.

,

Horizontal Flight Br A.R.C. M 1157

:

2. 3 .

W.

II "

Airplane Propellers by Cal Tech

ing Committee

H. Glauert

Volume

,

Theory

Aerodynamic

.

"

"

W. F. Durand

:

1 .

References to Chapter

"

Figure

10

.

'

abs

CHAPTER III

59

THE BLADE ELEMENT THEORY INTRODUCTION In the preceding chapter it has been demonstrated that by means of simple the Momentum Theory it is possible to predict the performance of the " idealized " helicopter . However , it will be realized immediately that the concept of a simple actuator disc imparting an axial momentum to the air has several serious limitations , due to such assumptions as the non - existence of profile drag , uniform downwash distribution , etc. The basic assumption of an infinite number of blades in the actuator obviously disc does not permit investigation of the influence of such things as taper , twist , airfoil characteristics and number of blades . In order to study the influence of all these important design factors , it is necessary to make use of a theory which takes into account the phenomena occurring at the blade itself . The Blade Element Theory represents one of the first attempts to solve these problems . In its initial form it was developed almost entirely by S. Drzewiecki , between 1892 and 1920. As in the case of the Momen tum Theory , the Blade Element Theory was first developed in simplified form , sometimes known as the " Primitive " Blade Element Theory ( see page 211 , Ref . 1 ) . 1

dr VE

-R

r

86

CT

108

SIX

'

V

CHORD Woo

*

8

dD

1

27 Figure

dL

Chapter

III

THE BLADE ELEMENT THEORY

BASIC CONCEPT

ro

of

lift

(

3 )

:

of

(

is

),

0

-

©

a

©

the element the angle between the zero angle rotor disc then the attack will be

of

the pitch angle the element and the

If

of

( 1 ) ( 2 )

o

V

:

where tan

line

The

the angle between this resultant velocity and the plane

,

o

be

tation will

,

of

Similarly

+

W

Vc

of

The whole rotor is assumed to be turning at a rotational velocity , while moving in the direction of its axis with a speed Vc : The blade element then experiences a resultant velocity w.dfar from the blade element ) which is Dr. vectorial sum Vc and the speed rotation

the

Consider that the blade of a propeller , or helicopter rotor , is composed of narrow elements ( Figure 1 ) . Each of these elements , of width dr , may now be studied as an airfoil section .

drag

attack Woo referred wind tunnel tests find the elementary lift dL

and for to

possible

in

the angle

of

.

and

usually obtained to

be (

it

2

₂

We

cdr

( 4 ) )

Cog

.

do

dr

( 5

was 2

Ź

Cug

с

2

:

the blade element 1

by

of

),

dL

(

function

the air velocity far from the blade wing finite span would then

drag dD experienced

airfoil iie its lift coefficient CL of

the

as

a

)

coefficient CD were known

of

characteristics as

the

.

in

a

of

to

to

If

at

of

re -

It

be

emphasized that this angle should attack and resultant velo city refer only great the motion air distance from the blade element and not the air velocities its immediate vicinity

Chapter

THE BLADE ELEMENT THEORY

III

BASIC CONCEPT

61

( continued )

From Figure

1

it will be seen that the elementary thrust dT and

torque dQ of the blade element will be : dT

dL cos 0

do

(

dL sin

sin Ø

dD

Ø

+

docos Ø ) r

( 6 )

)

sin

)

CD cos

(

7 )

+

CL sin

CD

Ø

r

g

cos Ø

2

R

).

2

(

be

r

T

A

-

Figure

(

of

dr

of a

points along the span i.e. for number may plotted Figure dT vs.

di dr fir /

( CL

2

bc

1

dr at

the curve

We

dy

/

dT

several values

of r)

/

After finding dT

2

Woo

((

1/2

2

dr

2

/

dQ

bc

/

/

dT dr

bog

Substituting equations (4 ) and ( 5 ) into the above equations and mul tiplying by the number of blades b , the following is obtained :

1

Chapter

THE BLADE ELEMENT THEORY 1

( continued )

.

o

V.

r

.

Q

/

curve of

dr

dQ

)

to

,

)

3

(

( 7

dri

rotor

the vs. obtain the torque

)

/

f(

dr

dQ

=

1

1

/

can the

,

of

s

for

of

exactly the same manner from Equation drawn Figure and graphically integrated In

be

T

The graphical integral of this plot ( the area under the curve ) expresses V с and total thrust the rotor the given conditions

dQ

the

BASIC CONCEPT

=

A

Q

AQ

rotor can readily

found

:

the

be

Now the horsepower required

at

Figure

3

-R

or

8 )

(

PR

20 550

the the disc

.

to

of

i.e. without taking into consideration the fact that the flow pattern element itself may change due the presence induced velocity

at at

,

to

a

of

on

1

It

should be noticed from the above analysis that the Primitive Blade Theory presupposes that the aerodynamic coefficients can be correctly Element referred the air velocity far from the blade estimated the basis

of

of

CD

.

12

of

in

of

to

in

;

.

of

=

,

6,

=

.

to

of

,

A

,

of

.

of

to

as

,

,

in

,

of

.

at

to

,

a

to

,>

at

,

at

at

by

other words

,

In

care should be taken the existence induced vel relationship establishing the blade element the between CL and and coo an equivalent aspect ratio which will assure induced angles identi cal or least similar those existing the blade elements Indeed early attempts analogy working wing span find between finite and rotor blade different wing aspect ratios were proposed best corresponding those requirements proposed AR others For instance some authors AR Actual aspect ratio the blade was also suggested more logical approach analysis would be course base the whole on the flow pattern the imme vicinity diate the blade other words find the true value the angle incidence at the blade element and then to use section coefficients for estimating dL and dD ocity

THE BLADE ELEMENT THEORY

III

Chapter

63

FURTHER DEVELOPMENT OF THE BLADE ELEMENT THEORY It is clear from the concluding discussion of the " Primitive " Blade Element Theory , that if the pattern of flow in the immediate neighborhood of the blade were known , the thrust and torque produced by the blade element could

be accurately

predicted

.

The values thus obtained , would be based on the so - called "section characteristics " obtained in two dimensional flow . In order to distinguish these section characteristics from those referring to a wing as a whole ( three dimen sional flow ) they are usually designated ( i.e. in NACA publications ) by lower case letters ( ci and call) and are shown in separate tables and graphs in NACA reports on airfoil data . re - emphasized that section characteristics are those which refer to "two - dimensional flow " ( infinite aspect ratio ) and therefore the values of etc. are given with respect to the true angle of attack at the airfoil itself. section de?

It should

be

?

ZERO

well

.

speed components

as

с

blade section due to V , and

I

r , but also any other tangential

LIFT CHORD

!

DISC

NOTE

,

THIS FIG

BLADE ELEMENT

Figure

4

AROUND

IT ,

,

STATIONARY WHILE

THE

IS

ROTOR

IN

PARALLEL TO

u

:

To

w

Isr

SHOWN

AIR FLOWS

,

or axial

air

2

с

In order to establish comparable conditions at the blade element, so that section characteristics may be used ALL components of the air velocity at the considered blade section must be known ( in order to determine the true two dimensional angle of attack of the blade element ) . This means that in the case and turning at an angular of a rotor moving along its axis , with a speed V velocity necessary only it is to know not the air speed components at

THE BLADE ELEMENT THEORY

III

Chapter

FURTHER DEVELOPMENT OF THE BLADE ELEMENT THEORY ( continued

)

In the primitive Blade Element Theory , the only resultant air velocity considered at the blade element was the vectorial sum of Vc and 12r . How ever , by analogy with a wing of finite span, as well as from the understanding of the fundamental concept of the Momentum Theory , it should be expected that an additional axial component v will exist whenever lift is produced .* This induced velocity v will create a new resultant w , different from w ' ( see Figu 4 ) . The induced velocity v will provide a further reduction in the angle of attack by an angle O2 , which in most of practical cases can be expressed ( in radians ) as :

,

(9)

V

11

rs ? )

)

(

+

10a

radians

.

all angles are expressed

in

course

,

,

Where

of

(

ve

(

,

su

)

to

in

r comparison

v

v

t

V

are small

10

Q

and

1

Vc

=

a

-

01 Vc

,

usual case where

-

-

-

0

tan

@

a

the

in

or ::

or

=

a

:

of

of

If the pitch angle of the considered blade element is (measured as angle between the zero lift chord and the plane rotation then the final angle of attack the element will obviously be

the

ф

to

it is

a

is

of

,

,

r is

,

be

of

width

:

by the blade element

d

lift experienced

Nr

the

11 )

rewritten

as

2

be

can

12

!

becomes negligib

11 )

) ,

J2r

(

(

Vc

usually true for

(

12 )

to

dr

r)

(

addition

:

12

tangential component

in

(

c

+

;

V

V

,

a

.

in

*

of

Any possibility the existence disregarded this study

of

4

2

1

/

1

dL

а

:

at

)

w

(

in

to

comparison are small which the blade the difference between and equation 10a equation addition for

is

12 r

(

dr

w2

c

1

gad

of v

and

working part Substituting follows

in

Vc

dL When

the

:

c

will

of

The magnitude

and chord

of

.

is

.

of

,

.

ci of

If

at

possible the induced velocity some radius known estimate accurately the lift and drag the blade element using section coef the the slope aa where ficients The section lift coefficient lift curve for the airfoil under consideration which of course must be correcte for the influence Mach number and Reynolds number The effect the usually negligible latter

III

Chapter

THE BLADE ELEMENT THEORY

65

FURTHER DEVELOPMENT OF THE BLADE ELEMENT THEORY The direction of the elementary lift dL will be , of course

ular to the resultant velocity w

,

The elementary drag dD experienced by the blade element 2

į

dD

į

sr

dD

( 12r) 2

perpendic

car will

be :

( 13 )

)(

is practically equal to w : cdo

or again assuming that

cdr

s cdo

( continued )

cdr

14

of

Knowing the magnitude and direction the lift and drag experienced by the blade element the equations from the Primitive Blade Element Theory can now be applied ,

)(

)

16

)

(

r

Ø

v

15

Ø

dD dD

+

cos

+

0 Ø

(

:

v

Vc

123

)

("

17

Vc

dL -dD

(

dT

)

and

"

Ø

Sil

Ø

tan

sin

small Ø

is

When

Ø

tan

sin

sin

relationship

the

8

given

by

is

Where

Ø

do

cos

t

and

aL

dl

:

dT

of

:

(

)

(

be

17a

)

18

(

]

do

+

+

)

v

+

c V

[

dQ

:

simplest formula for the elementary torque will be

(

the

at

,

)

dL

ат Similarly

17

(

)

)

is

+

(

dD

the

Vc usually for the working part expression Since the blade much smaller than dL then equation can further simplified

Chapter

THE BLADE ELEMENT THEORY

III

66

COMBINED BLADE ELEMENT AND MOMENTUM THEORY means of more accurately evaluating the downwash velocity at any blade station , the Blade Element Theory is combined with the Momentum Theory ( see Reference 2 ) . The more general case of a rotor moving up along speed Vc will axis considered first As usual will assumed stationary for convenience that the rotor remains while the whole mass air speed past downward directed V. flows be

it

R

-

th

dr

dr

1

Vc

.

a

is

at

of

a a

.

be

a

at

its

As

ROTOR

PLAN

DISC

VIEW

VIEW

I

1

SIDE

5

Figure

1 ,

by

dr

,

it

the

)

19

(

2v

dr

rotor disc

.

velocity

r

92 at

+

v

(

is

v

the induced

Vc )

:

in

ат where

in

(

to

)

of

5 )

to

(

an

Consider annular ring Figure width and average radius According Theory produced the Momentum the thrust at this elemen tary ring will be equal one second times the the mass flowing through velocity along the rotor axis total increase

in

,

of

)

20

(

dr

be

aq

t)

(

4 1

)

1

2

į

dT

2

:

,

b

in

,

to

On the other hand according the Blade Element Theory and simplifying assumptions accordance with the discussed the preceding para graphs the elementary thrust experienced by number blades can be expressed as

THE BLADE ELEMENT THEORY

III

Chapter

67

COMBINED BLADE ELEMENT AND MOMENTUM THEORY

(

continued

)

The location of a blade element can be defined by non - dimensional

ratio ( Figure

Assigning to this ratio the symbol x , the following relation

5).

ships can be established x

r /R

=

r dr

:

Rx

( 21 )

R dx

=

r

Also to the value of R , which represents a special symbol Vt may be assigned and the speed

Лr

tip speed of the blade , r can be expressed as :

the

R

Vt x

( 21a )

Equating the right sides of equations ( 19 ) and ( 20 ) , introducing the nota tions as given in ( 21 ) and ( 219 ) , and remembering that if the pitch angle at station x is o then ci = a Vc + v ) the following basic equation

zo

can be obtained :

8T R v2

+

(V +

abc

+

8T

Vt x

RVC R Vc )) vv

+

Vt

Vc abc

--

2

V Vé? abcxe

In equation ( 22 ) c is , of course , the blade chord at station x . above equation can be solved for the downwash velocity at station x :

v

=

[

Vt

abc 16

R

TT

abc

V

+

+

2 +

8 TT

0

R

abc 8TT (

In hovering when Vc

=

O , equation

(

23 )

is simplified to the following

( 22 )

The

-

abcx

2Vt

16 | R

2Vt

Vc

= =

V R

23 )

:

2

V

then the

=

abc

Vt

16TI R

If , in addition ,

rotor solidity

*

It should be

blades only .

Vi

abc 16 TT

+

R

abcx 8 TT

(

R

the blade is of rectangular shape ( chord 6 can be expressed as follows : *

6 hence

+

bc

b c R

=

=

6

T

R2

T

R

c

24 )

is constant )

bc

TIR

noticed that this definition of solidity refers to rectangular

с

Vt

Chapter

THE BLADE ELEMENT THEORY

III

COMBINED BLADE ELEMENT AND MOMENTUM obtained

THEORY ( continued )

Substituting the above value for bc into equation ( 24 ) the following is

ar

v

ve{

16

ag

VO

2

+ +

абх

e

( 25 )

8

16

In equation ( 25 ) the symbol o expresses the blade pitch angle at station x . In the case of a blade with linear twist this pitch angle at station can be expressed as :

x

et x where is the pitch angle at zero station while e expressed the total angle of wash out . Substituting the above expression into equation ( 25 ) , the formula for the downwash distribution of a linearly twisted rectangular blade is readil obtained V

=

VV +

a6 16

+

ба 16

2

+

a

6x 8

(

e.

Ot

x

)

( 252

Equations ( 23 ) to ( 25a ) permit the computation of downwash velocity in vertical ascent , or in hovering for a rotor with any number of blades ( b ) of Knowing the true downwash any rectangular form and any pitch distribution . value at any blade station x it is easy to find ( with the help of two dimensiona airfoil characteristics ) the true values of thrust and torque experienced by ev blade element ). ( see equations ( 12 ), ( 14 ) , ( 17 ) and ( 18 )

Forward Flight As in the case of vertical ascent and hovering the Blade Element Theory could provide a proper means for predicting forces acting on the blade element in forward flight. It is obvious that a correct prediction of elementar forces should be based on the two dimensional ( section ) airfoil characteristic and in order to do this it is necessary to know the velocity and direction of the air flow in the immediate vicinity of the investigated element of the blade . Velocity components at the blade element due to the rotation of the rotor and its translational velocities are easy to compute . The most general case of rectilinear helicopter flight assumes that th rotor axis is tilted from the vertical through an angle and is moving at Figure along speed path constant Vf an inclined ( see 6 ) where Vc is the verti cal component ( rate of climb ) and V is the horizontal component .

Chapter

THE BLADE ELEMENT THEORY

III

COMBINED BLADE ELEMENT AND MOMENTUM

69

THEORY ( continued

)

T Vaka

V Vpar

Figure

6

Again it will be assumed for convenience that the rotor is stationary , while the whole mass of air flows past it with a speed Vf inclined down . The speed Ve can be resolved into two components one axial ( perpen dicular to the rotor disc ) and another parallel to the disc . The axial com ponents will obviously be V

Vc

ах

cos y

+

+

V sin

( 26 )

Vc sing

( 27 )

sing

The parallel component will be : V

Since tilt angle fied as follows :

par

J is

ах Vax

V cos

usually small, equations

Vc

=

and

V par

7 -

=

V

+

V7

( 26 )

and ( 27 ) can be simpli

1

(27a )

V ce

As far as the blade elements are concerned , only the component of par perpendicular to the blade axis is important for computing forces acting on the element . Obviously this perpendicular component will vary . The blade azimuth angle is usually with the blade azimuth angle W measured from the blade downwind position ( Figure 7 ) and following this convention the perpendicular component ( Vb ) will obviously be : V

ELEMENT THEORY

THE BLADE

( continued )

V Vpar

parcosy

III

Chapter

V=

V

1

28 )

V of of

sin

)

28

(

of

V

V.

in

is

)

(

(

282

to on

Vb

4

:

)

V

as

In

7

for

(

small

horizontal flight which

most often analyzed the blade element theory par may be considered flight identical with the speed and equation

the case detail along the lines

becomes

sin

par

(

Il

Vb

y

Figure

7

parsın

:

of

a

in

r

.

to

of

,

.

Br

(

sin

292 )

y

+

V

for Vb

s2r

par

Vb

(

+

)

28

W w

or

substituting equation

(

12r

par

29 )

:

be

a

at

is

by

The speed Vb the advancing side adds the speed experienced speed the blade element due the rotational the rotor On the retreating side just the opposite true Hence for blade element situate distance the total speed wpar experienced the plane the disc

is

y

on

.

on

)

of

y

(

of

Sign the sin when measured from the downwind posi adding the translational and rotational tion automatically takes care velocity components the advancing and subtracting them the retreating side

an

at

in

r

*

eur O

as

y

8.

r

in

a

,

at

)

.

(

v

the element along the blade

or

all symbols expressing quantities which may ( r) ,

of

is

*

added change with the location

azimuth angle

)

(

at

ry

to

Subscript

as

be

a

a

of

Y

of

to

of

Assuming that the blade does not change its position with respect the plane the rotor disc no flapping the air flow the immediate neighborhood blade element located distance on the blade Figure Designating the pitch azimuth angle will shown angle blade element radius and azimuth angle

blade

ZERO

Wpar

III

LIFT

THE BLADE ELEMENT THEORY

Axis

71

dL

ROTORJAxis

Chapter

TO

// dD

ur V

ax

-

yr

wr

I

DISC

yr

(

30 )

:

Vaxtot par

the true value of the V. axtot and azimuth angle could be found

If

for

blade element at distance

, a of

$

yr

W

-1

-

tan

ur

obtained

r

Ø

of

be

angle attack ayu will readily angle less the total inflow

a

its

Figure

as

ROTOR

8

TO

/

PLANE

,

at

(w

of

in

sin siny

)?

V

12r

+

Pyr

(

1/2

D

d

dr cdo cr

)

31

(

}

dr

cp

Dyr

)

-

@y

(

)2

y

aa

:

,

is

)

sin

V

+

as

(

of

52r

9(

)

.

of

7

/

d

to

be

the(

by

)

ic

Lrg

)

(

to

)

,

it

30 )

(

0

be

.

of

it

d

(

then

,

y

course knowing the total easy would find the element ri from equation angle attack From the two dimensional section airfoil characteristics easy compute the elementary lift dL and elementary drag would Dpr experienced the element Making usual assumption that magnitude the working part par the blade difference the parallel speed negligible and the resultant flow wr the elementary aerodynam forces can be expressed follows inflow angle

is

d

a

of

by

.

is

,

of

,

.

y

a

be (

y

),

r

at

)

a

,9

of

a

to

in

,

w

to

to

be

it

61

it ,

to

.

in

is

perpendicular Dpr the resultant flow and parallel easy compute would manner similar that pp elementary torque shown and 65 the thrust and blade element Integrating for instance graphic distance and azimuth angle ally the thrust and torque for the whole blade may computed for given azimuth angle By repeating this procedure for several values and finding the average the mean thrust and torque complete one blade for Multiplying the result revolution found the number blades the mean As dL

Chapter

THE BLADE ELEMENT THEORY

III

COMBINED BLADE ELEMENT AND MOMENTUM

THEORY ( continued )

as

is ,

is

0

°

r,

at

)

y

at

is

v

at ,

(

8

no v

,

the

thrust and torque of a rotor in forward flight can be established . However , finding the thrust and torque of a rotor in forward flight , according to practically impossible above described rigorous procedure the down Figure velocity there until now still remains unknown and wash technically satisfactory method for predicting the downwas unfortunately velocity and when the blade different radii blade elements located ** and 360 any azimuth angle between in

is at

it

In

51 con

,

)

)

this paragraph

.

in

of

(

(

of

.

be

is

to

.

of

,

of is

usually small At high flying speeds the induced velocity com parison with the V7 velocity value hence the differencies induced different points the disc area become less important this case legitimate consider that the downwash velocity uniform over the whole disc Under the above assumption the average downwash velocity can equation 26 puted from the Momentum Theory with the help page Having established the downwash velocity from equation 26 further analysis the load on the blade may be carried out along the lines presente

.

Cal

Tech

. ,

of

.

L,

,

"

Durand's

1945

May

in

Forward

of

).

"

,

"

.:

,

,,

in

.

,

to

is

3

an

predict the induced velocity attempt made forward experimental the fore and aft rotor axis on the basis while treats the same problem theoretically .

In

4

.

(p

,

of

,

R. ,

,

4 .

**

Ref

the

.

4,

II , 2

.

,

"

Periodic Aerodynamic Forces on Rotors October 1944 No. J.A.S Vol

Coleman P. Feingold A.M and Stempin C.W Evaluation Velocity Helicopter Induced Fields Idealized Rotor NACA ARR resently unclassified No. L5E10 1945

Ref flight along

1

Aero Digest

,

Rotary Wing Aircraft

"

of

Principles .

1 9,

"

A :

C : "

Flight

,

.

3

Seibel

,

,

2 .

Klemin and June

,

.

1943

Vol IV Division Reprinting Committee Durand

,

,

"

Airplane Propellers

Aerodynamic Theory

"

"

,

1 .

Glauert

H :

REFERENCES TO CHAPTER III

THE VORTEX THEORY

Chapter IV

73

INTRODUCTION In order to complete the list of classical approaches to rotor aerodynamics , a third treatment , based on the Vortex Theory , should be added to the already discussed Simple Momentum and Blade Element

Theories .

In the Vortex Theory the existence of the induced velocity field of a helicopter rotor is explained by means of the lift theory based on cir culation . It has been shown in the review of fundamentals ( see page 15 ) that in a two dimensional flow , the lift per unit length can be expressed as : 1

gwr

(1)

is the strength of circulation around the airfoil section Where is the air density, and w is the air velocity at the considered section .

,

It should also be recalled that if c is the chord at the considered section , then the following relationship exists between the sectional lift coefficient Ci and the circulation ( see page 18 ) . 1

2

ci Wc

(

2

)

In discussing the blade element theory , the reader became fam iliar with the concept that in order to produce thrust at the rotor as a whole , the rotor blade elements must experience lift . In other words , some positive section lift coefficient must exist at least at most of the blade sections . This means that a circulation of magnitude given by equation ( 2 ) is produced at the blade elements experiencing lift . It will be remembered , from the discussion of circulation with respect to a fixed wing of finite span , that the existence of circulation can be identified with the presence of a vortex bound to the lifting surface ; in this case to the rotor blade .

Let it be assumed for the sake of simplicity , that the circulation along the whole blade has a constant value . Since the strength of circulation is expressed by the strength of a vortex bound to the blade ( or it can be said that the blade is replaced by a vortex ) , immediately the question arises as to what happens to this vortex at the ends of the blade , i.e. , at the root and at the tip . From general vortex theorems , it is known that a vortex cannot end in the fluid . This means that the bound vortices ( by which the blade is replaced ) must extend as free vortices beyond the root and the tip of the blade . As to the root vortex , it is apparent that , since at the rotor axis 12r is zero , the only remaining velocity ( in the case of vertical flight, or hovering ) is axial . Hence it is not difficult to imagine that each vortex leaving the blade at the root combines with those of the other blades to form one common vortex line along the rotor axis ( see Figure 1 ) .

THE VORTEX THEORY

Chapter IV

INTRODUCTION

( continued ) 1

898

Figure

1

Vortices springing out from the tip will form approximately a helical line . This line could be compared to the thread of a screw . From the direction of the rotation of the established vortex system , it is easy to anticipate that the fluid in the slip - stream will acquire a double movement The tip vortices will create a flow generally directed along the rotor axis , while the root vortices will impart a rotational movement , in the same direction as the rotation of the rotor .

CIRCULATION ALONG THE BLADE The assumption of constant circulation along the blade is , of course , exactly the same kind of over - simplification of the actual picture lift distribution , as when this assumption was made in explaining the exis tence of induced drag in the case of a wing of finite span ( see page 23 ) . In general 2, circulation varies along the blade span , as in the case of a finite span wing . This variation of circulation is visualized by assuming that at different blade stations , vortices of strengths suitable to account for the change of circulation spring off , as described in the case of an actual wing of finite span , ( see page 23 ) . The vortices springing off the blade forma helical surface analogous to the vortex sheet of a wing .

it

is

of

(

the

to

,

in

,

of

as

to

,

in

by

.

)

V Vc с

a

it

in

,

of

a

as

the

Having outlined the position of the vortices , the whole flow at rotor can be explained fluid motion due the system bound and free vortices plus the axial velocity the rotor its elf usual assut ed that the rotor has no motion the direction its axis but the fluid velocity past m agnitude equal sign moves with but opposite rotor axial velocity

as

.

is

:

is

,

-

in

of

ly

as

by

of

creating lift The picture bound vortices and the induced velocity field very attractive the free vortices the slip stream rigorous problem far treatment the whole concerned Unfortunate this rigorous treatment would bring such mathematical complexity that usually the following assumptions are made

Chapter IV

THE VORTEX THEORY

CIRCULATION ALONG THE BLADE

75

( continued )

The rotor has a very large number of blades . This implies that distances between the consecutive helixes are so small that , instead of having two or three ( depending on the number of blades ) , helical vortex sheets , the whole wake below the rotor may be considered as filled up with

vorticity .

The wake can be sliced up into horizontal circular vortex rings , and longitudinal vortex lines extending along the rotor wake . These vor tices directed along the wake will contribute to the rotation of the slipstream while the ring vortices will produce the downward movement of the air ( downwash ). As to the contribution of the bound vortices to the creation of downwash in the plane of the disc , it is easy to see that (under the assump tion of a large number of blades ) the sum of their contribution will be zero :

At any arbitrarily chosen point on the disc , the tendency to create downwash by the sum of all vortices located to one side of a line through the chosen point and the center of the disc , will be exactly equal in

magnitude and opposite in sign to the sum of those located at the other side of the line . It is obvious , hence that no downwash can be created by the bound

vortices .

The Vortex Theory has found wide application in propeller analy chiefly sis due to the work of Goldstein ( see Reference 1 ) and is also use ful in the study of helicopter rotors ( see for instance Reference 2 ) . As an example of its application to helicopters , an analysis of static thrust will be conducted here following the general lines established by M. Knight and A. Hefner ( Reference 3 ) . ,

DOWNWASH PRODUCED BY SINGLE VORTEX CYLINDER Knight and Hefner also assume an infinite number of blades and thus determine the velocity induced normal to the plane of the rotor by a cylindrical surface of vorticity , bounded on one end by the rotor and extend ing downward to infinity . ( Figure 2 ) у

IZ

P

dz Timire

N = 8 23

Р

-X

THE VORTEX THEORY

Chapter IV

DOWNWASH PRODUCED

BY

VORTEX CYLINDER ( continued )

SINGLE

By application of general theorems of potential flow , it can be shown that if the vortex rings form a cylindrical surface , then velocity induced at any point P of the space by this vortex cylinder can be express ed as :

dr

dz [ 4T W

U =

v

-

w ( 22 )

( 21

]

(3)

Where dr / dz is the intensity of circulation per unit length of the cylinder axis , ( in a steady state condition there is no reason why the intensity of vorticity should change along the same cylindrical surface , or in other words dr dz = constant ), and w is the solid angle * at P subtended by the vortex ring located at distance z along the cylinder axis .

/

*

P ,

a

of

at of

a

far

so

be

A )

)

W

(

P

(

)

.

zero

an

by

of

the

is

be

4īt

to

,

S

or

R2 ,3

the

in

the

P

.

to

of

is

S,

.

,

,

of

is

go

T

4

,

of

),

continues

to

through the surface some other the solid angle then would eventually encompass the sphere the solid angle will

P

+

If

P

P,

at

,

.

of R2

TT

4

it

s,

By

a

S

of

a

of

of

.

(

in

it

S ,

.

as

P

is to

,

point say extreme the total surface

P

point

imaginary unit sphere with its center closer and closer some surface can

If

imagine

a

to

be

space enclosed brought now the point be seen that the surface sphere by unit which cut out conical surface with vertex and having the perimeter for its base will get larger and larger The sphere equals approaching the surface total area angle sphere conical solid cuts out more the until finally when the plane the surface cuts out 1/2 the total surface area sphere angle This means that the solid then equal 2TT

It

helpful

~

Figure the point were very angle the solid would small

if

,

-

S,

from be practically equal may

of

A

to

a

.

can seen then that the surface to (

to

as

away

a

S

be

of

T.

It

4

is

is

S,

sur

In Figure A the solid angle at any point P , subtended by any equal numerically sphere face the portion the surface unit radius which cut out by conical surface with vertex and havi perimeter point angle equals the for base The total solid about

THE VORTEX THEORY

Chapter IV

DOWNWASH PRODUCED

BY SINGLE

Y

/

VORTEX CYLINDER

77

( continued

)

0 A

S Figure

A

rotor which is far from the ground , extends using far below it or notations of Figure 2 ,9 it may be said that it extends . to 22 Hence , if the point P is located in the plane of the , rotor disc but inside the circle of the vortex cylinder then obviously W122 ) , since the supporting ring is infinitely far away from the point P. = 09 since the point z lies in As to the value of W ( 21 ) , it will be equal to 2 TT the plane of the supporting ring , this means that the induced velocity in the plane of the disc ( v ) is ( from substituting the appropriate values in equation ( 3 ) ) : The downwash of a ,

dr

V

dz

22

const .

( 3а)

Since equation ( 3a ) was established for any point on the rotor disc it would indicate that if a rotor produced just one vortex cylinder ( for example , only due to the tip vortices ) then the downwash in the whole rotor plane would be constant . ,

When the point P is located in the plane of the disc , but outside cylinder , the induced velocity is zero . This is obvious since as before the O , as well . The physical interpreta ‫) سب‬22 ( O ; however W ( 21 ) tion of the last statement is that the vortex cylinder does not produce any induced velocity in the plane of the disc outside of the cylinder radius .

,

The downwash velocity far below the rotor can be also found with the help of formula ( 3 ) : If point P is moved far below the rotor , then obviously W ( 22 ) still remains equal to zero , but W ( 21 ) = 4 Tī ( see footnote on page 76 ) . y!

II

dr dz

( 3b )

THE VORTEX THEORY

Chapter IV

DOWNWASH PRODUCED BY SINGLE VORTEX CYLINDER

(

continued

)

Comparing equation ( 3b ) and ( 3a ) one sees immediately that i.e. the downwash velocity far below the rotor disc is equal to twice the downwash velocity at the disc . This coincides with the conclusion arrived at in developing the Simple Momentum Theory .

v'

=

v

2

CIRCULATION VAR YING ALONG THE BLADE Let us now investigate

a case , which is not as purely academic as single the one which considers a vortex cylinder generated by the rotor blades . It shall be assumed therefore , that the circulation along the blade varies from one section to another . By analogy with the case of the wing of finite span ( see page 23 ) , the mechanics of this variation can be visual . ized by imagining a series of vortices springing out of the trailing edge at different stations of the blade . The strength of each vortex is equal in magnitude to the spanwise variation in circulation . KOTOR

AXIS

r

=

f (T)

dr

dr

5 dr

rt

г

CIRCULATION

ruder dr dr

R

-T

Figure

3

RADIUS

de

ti

on

dr

no .

in

of

i

r

of out +

,

,

,

dr dr )

/

at

(

in of

at

dr

.

of

+

r

of

,

,

/

station has influence the downwash blade elements outboard this station The vortex may how affect the inboard elements and this influence can be easily estimat

velocities ever

dr )

at

dr

(

vortex

be

a

,

r

a a

,

It of

at

;

of

3

represent the circulation distribution along the Let Figure blade then the variation circulation for blade element wide and located distance from the rotor axis will dr dr and course vortex this strength should leave the blade station has been proven the preceding paragraph that vortices springing produce any downwash the blade do not the plane the disc outside radius which they separate from the blade This means that the conside

THE VORTEX THEORY

Chapter IV

CIRCULATION

THE BLADE

VAR YING ALONG

79

( continued )

As it has been shown in the previous paragraph , the induced velocity in the plane of the rotor disc is : V

- 12/2

dr /dr

Hence , the downwash produced by the blade element dr will be dr dr + V

-

1

2 2

di "

ar

(4)

dz

be

neglecting the infinitesimals of higher order , this equation becomes , as fore : V

1

- 1/2

į 2

dr

( 4a )

dr ar

The downwash created by the inboard elements V

1

=

d

( r

+

dr. dz

2

/dr

(

o

to

)

will

be

-T )

Neglecting the infinitesimals of higher order , equation

zero .

r

( 5)

( 5)

reduces to

These results give a mathematical proof to the concept that the circulation existing at any blade element influences the downwash at that particular element only . In other words it proves a theorem that the blade elements are reciprocally independent . It should be remembered that the assumption of this independence of blade elements formed the basis of the Combined Momentum and Blade Element Theory .

It is now possible to establish for each blade station certain relationships existing between the circulation , the geometry of the rotor ( chord , number of blades , etc. ) and the characteristics of the air foil section at this station , without worrying about any possible ramifica tions from any other point along the blade . The lift coefficient of a blade section at a given radius r is :

ci

a (o

0

)

In hovering where e is the pitch angle at this station , and Ø is the total inflow angle ( see for instance Figure 4 , page 63 ) , it may V ) or : usually be assumed that Ø =

Br

ci

a

(

e

)

( 6)

THE VORTEX THEORY

Chapter IV

CIRCULATION

From (

VARYING ALONG THE BLADE ( continued

page 73 , where now the

12r):

r 7

velocity of flow

lu 1

Substituting equation ( 6 ) into blades is readily obtained :

г

src

ci

2

(7),

fo 1

the total

:

le

abc

may be replaced by

w

(7)

circulation for

Ω

V

2

)

)

b

number of

120

(8)

In the case of hovering , as the vortex sheets spring out from each blade and move down with the slipstream , it is clear that the distance measured along the rotor axis ( the z axis ) between successive sheets will

be :

d

d

2

=

T

r

2m

V

b

r

32 7

V

(9)

b

consideration of equations ( 8 ) and ( 9 ) , it can readily be seen that the average change in circulation along the rotor axis in the negative direction is :

From

a

1

dr

clo

dz

TT v2

For

abc

(

2m

S2b

or : 8

abc

2

(e

-I

any value of r , therefore

V

r 12 ,

v

( 10)

b

2

)

equation

327

12 ? r

( 11 )

can be solved for the

induced velocity at that station . It should be emphasized at this point that equation ( 11 ) is identical with equation ( 22 ) page 67 , which was derived by combining the Momentum and Blade Element theories . *

* When Vc

,

o >

NR is

substituted for Vųt , and r / R for x .

Chapter

IV

THE VORTEX THEORY

81

REFERENCES TO CHAPTER IV

1.

S. Goldstein

:

"

On

Vortex Theory of Screw Propellers

Society Proceedings 2.

.

3

R.

P. Coleman

:

(a)

123 , 1929

"

,

Royal

.

A. M. Feingold and C. W. Stempin : " Evaluation of the Induced Velocity Field of an Idealized Helicopter Rotor " , NACA ARR No. L5E10 , 1945 .

M. Knight and R. A. Hefner

Airscrew

"

:

,

Static Thrust Analysis of the Lifting NACA TN 626 , 1937 . "

PRACTICAL METHODS OF CALCULATING ROTOR

Chapter V

THRUST AND POWER

IN

HOVERING AND

VERTICAL FLIGHT

INTRODUCTION As far as practical computation of rotor performance in hovering vertical flight is concerned , the Combined Momentum and Blade Element Theory seems to provide a very suitable tool for solving most of the practical engineering problems associated with this type of flight . However , before app lications of this theory to practical problems can be discussed , the reader should first get acquainted with the phenomenon of tip losses . Since in some tip - loss formulae, and other problems , non - dimensional rotor coefficients are used , this chapter begins with a discussion of this kind of representation of thrust and torque . The procedures associated with practical computation of thrust , torque and other topics of hovering and vertical flight are considered later . and

NON

-

DIMENSIONAL COEFFICIENTS FOR THRUST AND TORQUE

The thrust T , and torque Q of a helicopter rotor can be expresse with the help of non - dimensional coefficients . Following the accepted proce dures of general aerodynamics for expressing forces , it can be expected that

thrust will be proportional to : an area

a) b)

air density

the square of a

c)

Of course

sion .

velocity

torque will be proportional to all these values times a linear

,

dimer

).

to

as

para

2

:

the representative

course

the

of

is

and

9

t

(

V

thrust coefficient

,

non dimensional

?2

Sve

141

?

R )

112

(

,

T

RS ?2

R

Cs

CT

-

is

CT

=

T

where Cr air density

,

or

the

T

T

as

If

disc area and tip speed are chosen meters then thrust can be expressed follows

.

of

is

it

a

,

the

,

Regarding the velocity parameter obvious that the tip speed practical point most suitable from view

is

R

S

=

Vt

(

of

the

As to the area which should be introduced into the thrust expressia there are only two areas which can be logically chosen as representative , i.e. 2 the disc area : TRC , or the projected area of the rotor itself. This will be bd for a rotor with rectangular blades , or in general 6 T R2 , where G is solidity ratio ratio projected blade area disc area

PRACTICAL METHODS

Chapter V

NON

83

-DIMENSIONAL COEFFICIENTS FOR THRUST AND TORQUE

( continued )

In the case of torque , an additional factor representing the linear dimension should be introduced . The most logical choice will obviously be the rotor radius R. The expression for torque hence

( 2a )

vt

dimensional torque coefficient .

(

3 )

( SZR ) ? ve3

(

as :

4 )

(

?

The expression for torque

(

.

)

rotor area

,

the projected

4a )

:

? 2

)

to

referred

blades a

of

the number

12RR

(

S

GTR2

thrust coefficient

is

)

(

arbitrary shape

(

any

SR

of

tc

of

9

t

bcR

rotor with rectangular blades will

( ).

the blade area

5a )

(

5 )

) 2

12R )? 2

(

referred

to

the new torque coefficient

2R

(

§

bc

БпR3

Р

Яс

=

R2

(

where 9c

Q

general is

or

in

Q

qc

:

obviously be

.

be

a

the new

equation

4 )

is

T

rotor having blades

( in tc

b

where

to

)

1

,

to

,

(

for

a

or ,

V

TR

Ce

1

550

T

and

R832

prefer operate Some authors see for instance Reference thrust and torque coefficients referred the projected area The expres having rectangular for thrust rotor blades can written of

with sion

TT

1

,

PR

V

550

RS2

and since

Co

550

PR

are

S20

1

PR or

(

a rotor ( Q ) and rotational speed expression obtain the for rotor horsepower PR

+

to

it is

,

known

easy

(

When the torque of

3a )

-

)

the non

2

V

):

co is

where

Ce Ti R3 s

=

12

Q

(2)

its

or

( 12R)

Co TT R3

Q

can be written as :

,

PRACTICAL METHODS

Chapter V

NON

- DIMENSIONAL

the other

,

COEFFICIENTS FOR THRUST AND TORQUE

By comparing equations it becomes clear that

t and

TIP

( 1)

and

(2)

on one hand and

(continued

( 4 ) and ( 5) on

Ст

16

G

11

ac

calo

LOSSES

.

Re be

R

B

(

R )

one

Theoretical considerations indicate that the lift produced by a wing of finite span or blades , does not disappear suddenly at the tip , but decreases gradually in the tip region ,becoming nil at the tip itself . This phe nomenon of the decrease of lift producing abilities at the spanwise boundaries is known as " tip losses " . In helicopter aerodynamics its amount is usually expressed as a ratio of the effective blade radius ( Re ) to the geometric

-

.

,

r

to

to

of

as

interpreted The effective radius may that radius which will produce thrust equal that the actual rotor with the provision that the lift produce - Re where they suddenly disappear ing abilities are maintained up .

in

of

.

at

a

the

a

,

in

PRESSURE

W

HIGHER PRESSURE

FLOW

Figure

1

INDUCED

1 .

,

at

as

THRUST

LOWER

G

to

to

be

It

at

in is

of

of

it ,

is

as

.

of

on be

tip losses may explained The physical side several ways The simplest picture probably being that based the concept the Simple Theory according theory Momentum will recalled that this the rotor treated an actuator disc imparting downward motion the air flowing pressure difference between through and creating the same time upper and lower surfaces the disc Since the pressure the lower surface higher than that the disc the upper some air will flow around the Figure disc edges an up motion shown

PRACTICAL METHODS

Chapter V

TIP

LOSSES (continued

85

)

This flow , at the disc boundaries in the opposite direction to that of the induced velocity , will indicate that the contribution of the disc edges to the creation of downward momentum is diminished , or in other words , their thrust producing ability is decreased . A more refined explanation of tip losses can be based on princi ples of the Vortex Theory , but the physical picture will remain to some extent similar to that based on the Momentum Theory . Here again the air at the tip will acquire a radial and partially up motion ( due to tip vortices ) and because of that it will not contribute to the increase of downward momentum . to As , the quantitative estimation of tip losses , the most elegant , rigorous , treatment of the problem can be based on the full application of Goldstein's Vortex Theory , but mathematical difficulties connected with this approach are usually considered too great for direct practical applications .

For propellers , Prandtl gives

a simple , but only approximate , tip formula for losses based on the Vortex Theory ( see Reference 2 , page

265 ) : RE

R

1.386 b

1

d (TH 1

+

(8)

T2jV2

where b is the number of blades , d is the propeller's advance ratio , ; V/ V being the ( axial ) velocity of propeller motion , (analo gous to vc с in helicopters ) . It is obvious that if equation ( 8 ) were applied with the above interpretation of a to helicopters then for hovering , when R. Vc = 0 , would be zero too and there would be no tip losses as Re This conclusion , of course , would be incorrect and for this reason , it is sug expres gested to modify the interpretation of equation ( 8 ) by assuming that ses the inflow ratio at the disc .

2R

i

I

dev Vt +

Va

(9)

velocity at the disc , Vc is the rate of = R is the tip speed . According to the momen ( vertical ) climb , and Vt theory given tum for a fixed rotor power PR , and known power required to profile drag overcome the P , the sum v + Vc remains constant and

(

) the gross weight

W

other words

,

T.

(

Por

p

eq.134

.

helicopter

,

the ideal

)

for

(

with

PR

is

10 )

(

identical

support

10 )

’pr

PR

,

.eq

in

Pur

-

Po

rotor thrust

in

rotor

the

,

is

ed by the

I

where

550

or

):

57

*

1

U

page

induced

=

(

equal to

see

where v is the average

PRACTICAL METHODS

Chapter V

TIP

LOSSES ( continued )

Substituting into equation ( 9 ) , equation ( 10 ) for v + Vc , and remembering that for small X's ( as is the case for helicopters , where 2 + usually 1 { 0.10 ) , 1.0 , and equation ( 8 ) can be re

written as follows :

Re R

1.386 b

1

( PR

550

Ppr )

Vt

W

( 11 )

rotor and values of other parameters as encountered in practical design , equation ( 11 ) will give a tip loss factor of Re / R 0.97 which is rather on the higher side of usually accepted figures .

For

a three bladed

There are , of course , numerous empirical and semi - empirical formulae for tip losses . For instance , some authors simply recommend expressing the effective radius as :

Re

=

fo 1

R

с

( 12 )

2

where c is the average blade chord . ( Reference

Re

Cr

-

1

hold R

proposes the following expression :

3 )

( 1

0.7 t

+

1.5

)

( 13 )

R

where cr is the chord length at the root and tt is the blade taper ratio Reference 4 ) recommends another expression

Ст CT

:

be

(

)

T

0.3

R

2

Galth

+

1

,

1

Re

( 14 )

blade begins

For blades with taper starting inboard the half span according Wald the following formula should used to

,

0.5

the

of

R )

rotor thrust coefficient

where taper

( xr ) t V2

( )xp

the

blade station

/

is

0.6

b

of

Ct

xp ) t

СТ уст

-

1

:

.

and

(

is

where

the

Re R

r

(

(

Q. Wald

.

14a

Sissingh

of

.

,

at

.

,

of

in

It

is

of

ed ,

to

of

tip loss problems seems The whole review indicate that until now there no simple and the same time theoretically accurate way estimating tip losses should be noticed however that all the formula spite discussed different approaches and different parameters conside give similar answers usually accurate enough for practice

PRACTICAL METHODS

Chapter V

87

THRUST IN HOVERING In the Combined Momentum and Blade Element Theory it is assum ed that the elementary thrust produced by an annular ring having r as its radius and width dr , is equal to the sum of the elementary thrust produced by all blade elements of the same width dr , and located at the same radius r ( see page 66 ) , hence , the elementary thrust dt may be expressed , for convenience , according to the momentum theory : dT

4

Tr

v²

S

remembering that x r /R; can be rewritten as follows :

rr

( 15 )

Rx , and

T

2

4 TT R2

dT

ar

dr

Rdx

=

x dx

,

the above equation

( 16 )

total thrust T ( in lbs . ) will obviously be : х

T

=

4PTR

Xe

x

va

х

dx

( 17 )

Xi

=

Where R is the rotor radius in feet , p is the air density in slugs / cu . ft ., working part of the blade begins ( usually X ; is the inboard station where the O ; 0.15 assuming 0.2 but even xi Xi O , no great error is introduced .) Xe Re / R , is the station corresponding to the effective blade radius , Re , ( usually Xe 0.95 or 0.96 ) , and v is the downwash velocity in fps .

ointegrationblade

velocity v equation is known as a function of x then the indicated in ( 17 ) can be performed analytically or graphically . Since the analytical relations between v and x are rather complicated , a graphical integration is more suitable for practical purposes . In this last case the procedure is as follows : For a given value of the representative pitch angle e ( when the blade has a built - in twist , it must be stated at which station the pitch angle is considered as representative ) ,9 the downwash velocity v is calculated ( using previously developed formulae ) at several blade stations , say at increments of A x = 0.1 Next , v2 values and vax values for each blade station are calculated . Those values of v2x represent a sufficient number of points to draw a curve vax

If , for some pitch angle

,

of the

,

,

the downwash

.

f ( x) Figure

2.

v²x

vºx

Xit -Xe

.f (

x

)

A Figure

Tх= ATRA AIRS T

2

.

PRACTICAL METHODS

Chapter V

THRUST IN HOVERING ( continued

)

The area under the curve within the limits of x = Xii and represents Xe the desired integral , which , multiplied by 4TTPR yield corresponding will the thrust to the given representative pitch angle

X

V

.

, s

the

to

is

,

rotor having rectangular untwisted blades the relatively simple equation see expressed 2

8

16

( 18 )

+

a

eo x

6 a

o

X

16

ve

|-

+

ба

VE

=

):

(

by

a

and

a

relation 68

page

v

In

the case between and

is

,

pointed out here that in altitude flying when proportional thrust air density

the

remain unchanged

be

x of

It may

@

a

X

=

x

O ;

17 )

3

1

Ga 16

** )

) Jo 231

9

( 19 )

"

ба 16

(

o

(9( )

хе

+

2

1

3

#

:

охе

*

₂ } (( 62 )

*

e

+

16

) ]"

9

5

Ga

o

2

a 2

5

{

vi

Ga

(

+

*

*

5/2

16

96

16

)

)

$15 (

Ga

lei

2

R²

T

a

16

( *)

?

2

mais

4.

(

=

T

:

(

xi

(

(

18 )

(

it

be

lift

of

),

a

be

,

in

If ,

addition the influence the MN on the variation of the slope along the blade radius can neglected i.e. may assumed that introducing constant for all stations then into and performin the indicated integration for simplicity within the limits following the formula can be obtained

much greater

,

.

of

.

of

be

as

19 )

,

in

be seen that even the formula for thrust equation (

It

simplest possible case above the analytical looks rather complicated Wit expressions taper complexity the addition twist and the the thrust will

will

to

of

:

x

to

stalling The pitch angle corresponding the beginning easily blade station can be found by the following procedure

at

.

known

are

,

at

a

be

,

,

it is

In

important some practical problems know which part the blade will stall first what pitch angle and what maximum thrust can given obtained from rotor when the rotor's RPM and the air density

of

MAXIMUM THRUST IN HOVERING

any

PRACTICAL METHODS

Chapter V

MAXIMUM THRUST

IN

89

HOVERING ( continued

)

The section lift coefficient Cimax at different blade stations x should be found ta king into consideration the effect of the Reynolds Number ( see page 32 ) and Mach Number ( see page 38 ) . At the outer portion of increasing , the blade the effect of Mach Number is usually greater than the effect of increasing Reynolds Number . As a result , the maximum lift co efficient , at least for conventional airfoils , usually drops toward the tip , after reaching a maximum somewhere at the mid - span ( see Figure 3 ) . Vs. 5.1T

.

Sonax .

Coman

Cpus

.r

( forgivene

r R

-

Figure

3

*

2

v2

* dx

max

)

2

Ti R ?

20

4

(

dT max

Rag

The elementary thrust dT of a blade element , located at station x , at the beginning of stalling can be expressed according to the Momentum Theory as :

to

)

readily 22 )

(

x

:)

22a

)

Cimax

(

V

GT

х

R

8

where cb

Clmax

R

x

Xmax

blade

8

rectangular

(

a

or

VXmax

for

cb

1

obtained

general expres

:

v

at

,

of

Equating the right sides the above equations sion for the maximum downwash velocity Xmax station

21

(

dx

is

X

x a

Cimax

2

2

Rbc

tV

‫و‬

max

2

dT

1

:

x .

at

be

where Vmax obviously represents the maximum downwash velocity which According can obtained the element the Blade Element Theory elementary the thrust will be

PRACTICAL METHODS

Chapter V

MAXIMUM THRUST IN HOVERING

(

continued

)

The pitch angle of the blade element at station x , corresponding to the beginning of stalling will be :

clmax

Xst

Vxmax

+

for

the

the

.

at

the

-

(

(

the

noof

at

same time

its

As

at

of

washout obvious that lowest value @xst stalling pitch angle begins blade which

is is

the

.

)

to

a

as

several blade exst values have been calculated After fairly clear picture one can have what pitch angle whole the stalling begins When the blade non twisted

), as a it is

,

stations blade

23 )

(

Vt *

ax

:

it

A

.

of

.

In

T is

to

,

fo

),

(

of

of

at

T

of

©

as a

to

the maximum thrust Tmax value for the most prac tical cases can be obtained thrust calculated by the above formulas stalling slightly exceeding the corresponding the beginning calculated for several more thorough study max can be made when stalling values 0. greater than the beginning this case

ca

)

of

.

4 )

(

)

a b ,

(

or

(

)

22

.

ci

(

by

be

,

v

,

must be taken into consideration that at those blade stations where the local stalling pitch angle has been exceeded the downwash velocity computed using instead should tormulae 22a Cimax region Figure the suitable value after stalling

max

.

&

C

b

a

Figure

4

7

a

)

in

to

to

as

as

of

.

of

a

(

in

,

be

it

at

III

Chapter pages 66-67 After the above discussion and that should not difficult for the reader find the thrust vertical flight given rate climb well the pitch angle corresponding the stalling and maximum thrust under those conditions beginning

hovering and vertical flight show som the preceding paragraph and for this ,

in

gust loads

similarity with those considered

in

Problems

of

GUST LOADS

.

reason they are briefly discussed now

PRACTICAL METHODS

Chapter V

GUST LOADS ( continued

91

)

In estimating gust loads the concept of the " sharp edge gust " of determined value 30fps for instance ) is commonly used . In the phys ical picture associated with this concept , the rotor suddenly ( i.e. no grad ient) experiences a vertical air stream ( up " +" , or down " - " direction ), which momentarily changes the rate of flow through the disc by a given gust velocity . It is further assumed that before a new flow pattern is estab lished , the blade elements experience aerodynamic forces simply caused by the increased ( positive gust ) , or decreased ( negative gust ), angles of attack resulting from the additional gust flow through the disc (Figure 5 ) .

RESULTANT

Way

VEL . WITH

BEFORE GUST

Wog

Ser

CUST

ag Wg

=>

drug

Tog

dog

you Figure

For brevity ,

5

the positive gust condition

will only

be

considered

but the reader who thoroughly understands this case should have no in solving load problems in the case of negative gust.

,

difficulty

The momentary increase in angle of attack of the blade elements will cause an appropriate increase in the lift coefficient experienced by them . For the working part of the blade the small angle assumptions can usually be maintained and thus the increase in lift Ac is expressed as :

Da where

a

g

((

:

)

( 24 )

is the gust velocity , a is the lift slope , and air velocity at the considered blade element . U

g

1r is the linear

For the inboard part of flat , and moderately twisted blades , the increase in lift coefficient is usually so great that the resulting + , ( where Clbg is the lift coefficient of the elements ci Clbg value . It should section lift coefficient before gust ) can exceed the ci cimax , , be remembered however that the maximum lift coefficient obtained under dynamic conditions ( sudden increase of the angle of attack ) is usually higher than the ci . max value obtained by normal test methods . Some authors

Ac

believe that an increase of Cimax

Ac

(

as found in

normal wind tunnel data )

PRACTICAL METHODS

Chapter V

92

GUST LOADS ( continued )

will sufficiently approximate

by

or

,

,

6 ).

(

be

)

?

AC Ac

Y

Vs.

©

(

+

vs. I

cing

clmaxa

a

of )

to

, (

a

&

.

a

in

of

is

to

to 20 %

of

d "

"

15

the

where sub Cimaxd script emphasize the condition dynamic increase added Knowing the actual maximum dynamic section lift the angle attack given RN coefficient for MN either from actual dynamic tests graph giving an application static test data suitable coefficients Figure along the blade span can made

by some

Praxd ca

LACTUAL

r

,

vs

Elog

Figure

6

R

T

will

exceed

in

(

be

.

cı

of

(

in

of

In

.

Clmaxd the

sum

not

)

'

s

be

to

).

6

,

is

to be

,

The ci values experienced by the blade elements before the plotted along the span dotted line encountered should also Figure Then the Acı due the gust should calculated for sever blade stations and added the suitable values before the gust The resulting from this addition will greater some places usually obviously that the actual values will means This inboard than Cimaxa be limited to the only other parts the blade the resultant gust

and thus

clmaxa

will represent

the actual

CI the

c

A

in

to

.

,

in

,

.

of

,

it is (h

at

of

6 ).

ci

a

in

be

+

by

In

.

limits with values values combining Cimaxd this way Cibg Figure graph showing actual ci's gust can obtained eavy line Knowing the actual easy each blade station calculate per running plot load foot the blade The load vs. span integrated graphically yields the total thrust experienced by the blade an up gust to

87 .

in

),

v

22

as

(

be

ci

is

17 )) ,

If

be

(

to

compute thrust from the induced veloci the setup made given ties equation the downwash corresponding for hovering can obtained from equation and having the values for the further procedure can page carried out outlined

PRACTICAL METHODS

Chapter V

93

INDUCED POWER IN HOVERING Working formulae for the induced power in hovering can be easily developed from the basic concept of the necessity to supply power to the rotor , in order to cover the losses due to the energy transferred each second to the slipstream . It is clear that for an elementary ring dr wide , and of radius equal to r , the mass flow per second is 2 urp v dr, hence , far below the rotor , where the induced velocity reaches a value of 2 y , the kinetic energy in ft.Ibs / sec. ( dE sec . ) carried away each second be comes

dE

/

į

:

/ sec .

2

1

.2

r v dr (2v)? Ig

Ti

4

Ti

rp v

3

dr

( 25 )

Dividing equation ( 25 ) by 550 , the elementary induced power dPi can be expressed in HP :

550

3

p

26a

)

2 >

Xe

-

and

;

X

a

х

A

Avxx

of

will

( 27 P ): ;

of in

)

..

x

v3

7

,

o

l)

(

is

x )

(

f

=

at

Figure

is

a

the

to

x

v

be

( 13

The most advisable practical procedure for finding integration indicated perform graphically equation the values of have been calculated for number blade stations x say computed and intervals AX curve

drawn

27 )

(

x

13

dx

i*

"

х

хе

:

x

X ; Xi

TRE

.

P.

=

х

II

2

(

)

x

The total induced power for the rotor can be obtained by integration to equation 26a within the limits xe

After

x

TR

r R , it becomes

v2x

4

( 26 )

/

or , using notations for blade station x dP

dr

S

(

550

3

r

dx

I

4

dP

PRACTICAL METHODS

Chapter V

INDUCED POWER

HOVERING

IN

( continued )

T

.

(

,

it in

,

to

It

e o

..

R.

be

to a

of by

the

The area under the curve ( within the integration limits from x to xe ) represents desired integral which multiplied 4/550 gives the induced power corresponding given value the characteris tic blade pitch angle should recalled that for the ideal helicopter flight and the induced power was the only power required sustain because of the uniform downwash distribution its value was

(

remembering that

rewritten

T

3

can be

RZP

as :

)

28

Equation

(

2

T

or ,

550

28 )

Tv

Pid

(

)

28a

T

1 '

2

550

TT

TRS R

Pid

:

k

PM

T3 TR29

the

or

)

.

(

,

of

ta

a

of at k

.

-

,

of

k

,

k

factor For rotors having untapered untwisted blades the usually about by 1.10 1.15 per By proper the blade means twist the downwash least for some thrust values can be made more uniform and value will approach 1.0

is

2

Bija

29 )

550

(

k

Pi

.

is

a

of

,

at

,

is

usually not uniformly the practical rotor the downwash power greater than Pid distributed the disc and the induced To express the ratio symbol these two powers will be used

For

in be

BLADE TWIST FOR UNIFORM DOWNWASH IN HOVERING

of

is a T ,

to

it is

,

,

)

30

be :

(

(

T e 2

2

TREP

,

Re

,

If

,

.

is

it

Let assumed that desired obtain uniform down hovering per wash distribution when the thrust rotor and the tip finding problem speed the blade twist which consists Vt The satisfies these conditions the downwash were uniform the induced velocity within the effective blade radius would

PRACTICAL METHODS

Chapter V

95

BLADE TWIST FOR UNIFORM DOWNWASH IN HOVERING But , from equation ( 22 ) the relation between the downwash velocity Vx ( at any blade station x ) and the corresponding section lift coefficient clx is known . The coefficient which should exist at this sta tion , in to produce the required uniform downwash is :

8

Clx

T

R

box

(

7 V

2 ( 31 )

)

or for a rectangular blade :

(

clx

1

)

( 312 )

required cı ci increases toward the root of the blade ( as x decreases ) . It is obvious hence , that the special condition of a uniform downwash can only be fulfilled down to the value of x where the required cax C1x does not exceed the maximum section lift coefficient, Cimax It is clear from equation

( 31 )

and (( 31a

),

that the

The blade pitch angle ( @x ) at station x , required to produce a downwash v , can be readily obtained from equation ( 23 ) by substituting the corresponding cı value ( from equation 31a ) :

ex

clx ax

V

+

( 32 )

х

INDUCED AND CLIMBING POWER IN VERTICAL ASCENT assumed that for a given rotor geometry , pitch angle , tip speed , air density and vertical rate of climb ( Vc ) , it is necessary to compute the induced power , plus the power required to overcome the force of gravity . Usually , in practice , the opposite question is asked : What will be the rate of climb, under given conditions , when a given power is delivered to the rotor . Nevertheless , the presently considered problem may also find some practical application as for instance , in an investiga tion of engine characteristics required to assure a definite rate of

Let it

be

the downwash velocity computed be Then making the

Chapter

v )

, ) >

(

23

,

(

III ,

From equation

different blade stations can readily

.

at

vertical climb .

PRACTICAL METHODS

Chapter 7

9

INDUCED AND CLIMBING POWER IN VERTICAL ASCENT usual assumption of the stationary rotor and the air flowing at a speed equal to -Vc , it is possible to compute the energy transferred each secon to the slipstream at a disc ring having r as its radius >, and being dr wide the elemin and page 93 By analogy with the considerations on page 44 tary energy per second which should be supplied to the considered ring is obtained as the difference of the energies carried each second in the slip stream far below the rotor , and that present far above the disc :

11

DE / sec

2m r dr s

1

2

( Vc

v)

+

[

(

v.

+

2v ) 2

- v? ]

simplifying , and remembering that dE / sec is the elementary induced plus climbing power (dPic ), equation ( 33 ) can be rewritten as follows : +

( Vc +

v dr

v)

(

as

dr

page 44

.

on

at

v )

++

.

develor

hence

practical computation the combined induced and vertical ascent concerned substitution XR for of

a

,

(

dx

v x

2

v )

+

:

.

(v

is is

of

The induced and

TR2

advisable

S

4

dPic

33a

)

equation

(

in

dr

climbing power and Rdx for

in

As far

climbing power for the whole rotor will obvio xdx

(

v

v ) ?

+

29

( Vc

*пи

4

Pic

==

х

:

ly be

v

agreement with the conclusions arrived as

which

is in

dPic

(

dT

be

as :

r

a

)

(

in

ed

T r s

as

elementary thrust ( vc + v ) v dr is simply ring by having climb its radius and its width equation 32a can written

but 4

dt )

(

Vc ++ vv)) ( Vc

;

Trgç

MT

the

4

dPic

Xi

).

93

page

(

in

be

35 )

in

of

to

(

performed equation indicated can power hovering the case the induced see

The integration

graphically similarly

PRACTICAL METHODS

Chapter V

97

POWER REQUIRED TO OVERCOME THE PROFILE DRAG

coefficient

36 )

(

dx

)3

x

(

V +

ed .

the considered section

as

the drag

1100

х

.

cdo

P Ppr

be

where

is

d

pbc R

The power required to overcome the profile drag of b blade dr wide and located at a station x rr / R , expressed in HP is :

at

elements

,

,

. is

cd

be

)

cdo

4400

)

37

(

tV

3

TR2

:

(

in

6

1

P pr

9

36 )

*,

x

X

O ,

(

)

(

,

,

In

considered can the simplest possible case when constant along the whole blade and when the chord also constant rectangular blade equation integrated between the can easily limits giving hp and 1.0

A

.

a

of

.

-

of

,

,

,

,

of

is ,

,

cd

by

,

In

reality course rarely constant along the blade judiciously selecting the constant value reasonably accur nevertheless ate predictions over all rotor profile power can be established satisfactory value obtaining method of this cdo will be discussed later ,

c,

,

(

as :

.

a

a

it

is

In

).

to

to

ci ,

of

,

,

to

cd

of as

In

may vary along the span and the blade chord changes mentioned above also from one station another because usually the influence the local RN and MN this case necessary graphical integration take recourse For blade with variable chord the profile power can be expressed general

dx

it

,

of

to

of

.

to

is

,

to

.

a

In

*

)

(

37b

the calculation for the thrust and induced power the integration here carried out the very tip Although the tip end directly the blade does not contribute the creation lift does profile drag have

contrast

)

(

dx

v

ve

cdo

x3

T RP

is

1

1100

372

1.0 2

P pr

one

cdo

3

rectangular

c

3

bg Rvi :

1

1100

6

while for

a

Ppr

13

1.0

PRACTICAL METHODS

Chapter V

POWER REQUIRED TO OVERCOME THE PROFILE DRAG

(

continued )

In both cases above , the cdo should be calculated first for a given char acteristic pitch angle e 9, at several blade stations x ( say at intervals X of x3 should be drawn as , or 0.1 ), then the curve c cd x3 cdo Obviously , the area under the curve ( within a function of x . ( Figure 8 ) . the limits x 0 and x 1.0 ) will represent the required integral.

Ax

cca.x

,

,

or

coox ?

A X

=

to

Figure

8

Multiplying this integral by the value of the constants in front integral sign , the power required to overcome the profile drag can of the

be

obtained .

recalled at this point that in calculating the pro file drag coefficient of a practical rotor , some allowance should be made for the surface roughness of real blades , by multiplying the cdomin values from airfoil wind - tunnel data by a roughness coefficient . It is obvious that depending on the blade construction and state of the blade sur face , the value of the roughness coefficient may vary within rather wide limits . For conventional airfoil sections a roughness coefficient is sug . gested from 1.15 ( for smooth metal blades ) to 1.5 ( for fabric covered It should

be

ones ) .

As should be remembered from Chapter I , laminar airfoil sections are very sensitive to surface roughness and for this reason when ever there is no absolute assurance that the blade will be kept in perfect condition , roughness correction factors in excess of 1.5 are recommend ed . As a practical short - cut in calculating power required to overcome the profile drag of a rectangular , untwisted blade the following procedure is recommended : 1.

For given conditions of pitch angle , tip speed , and air density compute the section lift coeffi

cient at x

0.75

.

PRACTICAL METHODS

Chapter V

99

POWER REQUIRED TO OVERCOME THE PROFILE DRAG

( continued )

,

the

Assume that this lift coefficient ( obtained in = 0.8 . Then compute the 1 ) exists also at x at x

cde value taking into account and Mach Numbers existing

ci

, the

3.

Correct the

0.8 and the

thus obtained for roughness

cd

above

Reynold's

the

2.

of

*

37 )

(

,

,

)

a

(

profile multiplying cdomin component drag coefficient by suitable correction factor and further assuming that this cdo value exists along the whole blade compute the profile power from equation

( PR )

for

TOTAL POWER

the

(

Vc

a

,

).

38 )

(

P;

)

(

+

39 )

(

VC

W

+

pr

P

+

:

P ; a

Pi

=

term representing the work

helicopter gross weight

.

is

W

where

Ppr

speed

against gravity should be added

PR

,

,

the

climbing

at

In

PR R

of

as a

of

hovering The total rotor horsepower required under given conditions pitch angle tip speed and air density can readily be obtained sum induced and profile power Ppr

.

to

)

(

to

by

*

P

Numerous comparisons made between pr obtained in the above out leading procedures equation 37b way computed lined with that the seem indicate that additional accuracy usually does not warrant the work involved

PRACTICAL METHODS

REFERENCES TO CHAPTER V 1.

H. B. Squire

2.

H.

Glauert :

3

.

G.

Sissingh :

4.

Q. Wald :

:

The

"

Flight of

a

Helicopter

",

R

&

Propellers " , Durand's " Airplane Theory , Volume IV , Division L "

M 1730 "

Aerodynamic

Contribution to the Aerodynamics of Rotary No. 921 , 1939

Aircraft " , NACA TM

- Wing

I The Effect of Planform on Static Thrust Performance Sikorsky Aircraft , SER 442 , November 1944 .

CHAPTER VI 101

PERFORMANCE IN HOVERING AND VERTICAL ASCENT INTRODUCTION Several theories of the lifting airscrew in vertical flight have been reviewed thus far and at this point the reader should have acquired a sufficient amount of general knowledge for discussing the importance of the various design parameters . The main parameters which a designer must establish before a final concept of the rotor is fully developed are :

1.

Tip speed

2.

Disc loading

3.

( V +) (w )

Number of blades ( b ) and the geometry of the blade ( planform , solidity , 6 ,) )

4.

Built - in twist

5.

Airfoil section at different blade stations

(

et )

,

The combination of these parameters determines whether or not the rotor will provide the desired performance in hovering or vertical flight . * No matter how fully developed is the theoretical background in some engineering branch , the process of designing always remains an art . And it may be added , that this is chiefly the art of finding a judicious compromise among the usually conflicting influences of the different design parameters . The opening part of this chapter is devoted to the considerations which may help the reader in choosing rotor design parameters best suited for the purpose , while in the concluding portion practical methods of predicting perfor mance in hovering and vertical flight are presented .

* In this chapter only these particular types of flights are considered

.

102

PERFORMANCE IN HOVERING AND VERTICAL ASCENT

Chapter VI

FIGURE OF MERIT a

of

,

of

T

or in

.

in

P,

.

/P

W

,

/

T

a

of

of

)

and

( 1 ) 1 )

equations

(

expressed with the help

of

be

follows

)

( 2

Vt

SZR

/

C

/C

ca

CO

:

as

=

P page 82 and 83

P T

( 2 ),

power ratio can

to

thrust

the

W is , :

in

ft

(

in

to

is to

be

of

,

,

in

be

,

There are course many aspects successful rotor design which cannot translated easily into precise technical criteria expressed numbers but some can and these will be considered this chapter The helicopter rotor hovering seems most logical criterion the success flight power loading type the this i.e. the since thrust equal the supported weight the ratio Remembering that power expressed lbs./sec

/P

T

. :

called the figure

merit

( 3 )

S

co of

defined

3/2

the helicopter

,

rotor

Ст

of

and the parameter

ĪT R2 is

so

P T th

M

proposed by writing

T

-

suitable non dimensional form

is

t

a

a

2 )

(

A

)

(

1

page 311 consider that Some authors see for instance Reference equation does not provide suitable criterion since the ratio varies inversely with the tip speed V. for given rotor design

(

(

)

.

be

the thrust

(4)

2

‫م‬

T

2

T

Theory see page 44

( 5 )

3

RSS

,

,

la )

82

the thrust

Vt

6 )

2

V

(

2

Ст

page

:

is

of T

Substituting the above value into equation readily found as coefficient value

(

g

TT

Tv

2

=

P

is :

R2

as :

the Simple Momentum

hovering can be expressed

while power

of

of

)

According

in

be

highest merit will course for the ideal rotor uniform no profile drag and can easily calculated for this case to

downwash

,

The figure

PERFORMANCE IN HOVERING AND VERTICAL ASCENT

Chapter VI

FIGURE OF MERIT

( continued )

Then , introducing into equation ( 1 ) the P value from equation ( 5 ) and Q value from equation ( 2a ) , page 83 , the torque coefficient is easily found as :

10

co

2

(lu )

3

(7 )

The ideal figure of merit is obtained by substituting Cş and Ce from equations ( 6 ) and ( 7 ) into equation ( 3 ) . Then it becomes :

M.

2 V2

Mid

1.414

(8)

merit which can theoretically be obtained by any helicopter rotor . All practical rotors will obviously have a much lower figure of merit . Since Mid = 1.414 is the highest possible value , then for comparative reasons , it is possible to establish another criterion of the efficiency of the design of a rotor in hovering , taking the ratio of the figure of merit of the actual rotor to the ideal one : This is , of course

,

the highest figure of

M M id

n des n

des

.

M

1

1.414

can be expressed as a

(9)

fraction , or in percent .

Some authors believe that a great advantage of the figure of merit is its independence of tip speed . This concept , if improperly interpreted , may misleading be rather . For instance , some students have a tendency to assume that the figure of merit obtained for a given rotor under one set of conditions ( pitch angle , tip speed , density ) will remain the same , if , for instance , the tip speed changes . Such an assumption will , in general , be erroneous , as it should be remembered that the tip speed , even simply because of its influence through the MN and R.N., plays a decisive role as far as performance of a rotor , hence its figure of merit , is concerned .

OPTIMUM ROTOR RADIUS

in 10 )

Total power required

(

pr Рpx

).

density

3

99 ): +

P

Р

Pi

(

some known see page

P

(ie, at is

altitude hovering

air

In discussing the influence of different design parameters it may be interesting to know what will be the theoretically optimum rotor radius R for supporting a given gross weight W in hovering at some known density

103

PERFORMANCE IN HOVERING AND VERTICAL

Chapter VI

OPTIMUM ROTOR RADIUS

ASCENT

( continued )

where P : is the induced power and P pr is the power required to overcome the profile drag . The induced power can be expressed as the ideal power ( Wv ) times the coefficient k ( to take into account the tip losses and non uniform downwash distribution ). As to the Ppr , let it be assumed that it can be expressed by equation ( 37 ) , page 97 ( constant value of cdo' rectangular blades ) . Equation ( 10 ) expressed in ft. lbs./sec . can be

rewritten as :

P

=

k

W3

AZT R

2

*

тр

,, в стеча v'я*

( 11 )

Vt

12 )

(

6k

W

3

0.8 out on

Ropt

Assuming that all parameters except R are constant , it is easy to find the minimum power required by differentiating equation ( 11 ) with respect Then it becomes clear that equation to R , and equating the result to zero . ( 11 ) is a minimum when the first term of the right side of equation ( 11 ) ( i.e. the induced power ) is equal to twice the value of the second term of the right side of equation ( 11 )) . That is , when the profile power is equal to one half of the induced . The optimum blade radius in hovering (Ropt) can be expressed therefore as :

cdo

and

.

in

W

11 )

(

to

a

)

12

.

be

as

is is

Vt ,

of

of

(

as

as

a

at

of

as to of

to

k .

,

as

,

to W

a

shows quite

is

12 )

(

clearly that for given weight and air den inversely proportional optimum this radius the tip speed inversely proportional profile drag the cubic root the coefficient solidity directly proportional the while the cubic foot the induced power coefficient implies long This that cdo can considered practically independent the most important Vť the tip speed parameter far the optimum blade radius concerned The Ropt Equation value found from and introduced into will give the power required given weight hovering minimum value sustain flight given density altitude Equation

sitys

RHP LBS

Figure

1

?

LOADING POWER ROTOR

-

14

15

16

17

18

3

|

2 1

101

2.0

2.5 DISC

25 = V LBS

4.0

AT

T

9

3.5

So

/ : SQ.FT

DISC

4.5

OF

LOADING

3.0

L

VALUES

LOADING

Poco

LOADING

5.0

5.5

/V

1.5

600

400

300

200

100

50

FOR

:

2

/

23 DIFFERENT

-

/S

20

POWER

vs.

21

ROTOR

Chapter VI

PERFORMANCE IN HOVERING AND VERTICAL ASCENT

.

60.6

105

PERFORMANCE IN HOVERING AND VERTICAL ASCENT

Chapter VI

!

POWER LOADING IN HOVERING Equation ( 11 ) can be transformed into a more useful form for practical designing . By expressing power in hp and dividing weight W by power , relationship between the rotor power loading ( w P ) R in lbs /hp and disc loading ( w ) in lbs / sq.ft. in hovering , at sea level for instance , is readily

/

obtained

:

1( *P )R ll W

550

2

so

Vt

5

8

.

.‫و‬

+

( 13 )

1

Cd

W

-1B

k

,

k

/L

S

at

,

of

,

.,

Vt is ,

is

), ),

94

is

So

,

6

,

as

for

Cd .

So

it of is of of Vť3

the

,

be

13 )

conveniently represented graphically for different values the characteristic parameter

can

w )

so

(

f

)

P

R

Vť

=

PR

/

W

(

Equation

.

tip speed

(

the

is

(

w

is

where the disc loading the induced the air density power correction factor see page profile average cdo the drag coefficient the geometric solidity ratio and course

a

'

of

,

t

.

is

V

6 G

бо

of

,

of

'

/P

)R

(W

,

as

I

Cdo

is ,

it

)

there

and

see

63.

Figuren is

3

t

V

an optimum

in

a

.

of

,

to

,

,

disc loading Wopt which will assure the highest rotor power loading hovering i.e. the lowest power required given gross weight sustain flight this type

in

clear that for each value

of

a

higher disc loading becomes moredesirablet

(

aa

of

as

far

as

be

:

,

,

,,

of

to

is

of

at

A

inthe

,

is

.

is

W

1

in

as

or

6

value Vt86 some existing contemplated design easy graph assumed find by means Figure the corresponding value rotor power loading hovering when the gross and then calculate the rotor power required weight known This graph also very useful for an estimation the influence such design parameters the tip speed Vt the average w profile drag coefficient cdo the solidity ratio and the disc loading Figure glance product will show that when the cdo relatively small then course the real helicopter will more closely resemble the ideal and conclusions reached on the basis the simple momentum theory can be applied i.e. that decreasing the disc loading will obtaįning high power loading beneficial the possibilities When

cdo

AVERAGE LIFT COEFFICIENT to

,

oth

of

.

to

a

,

to

in In

c

it is

a

In

performing maneuver necessary vary the blade section lift coefficients from the values corresponding the steady flight condition necessary margin order assure the for the possible increase high rotor efficiency on the maneuvers on one hand while retaining

PERFORMANCE IN HOVERING AND VERTICAL ASCENT 107

AVERAGE LIFT COEFFICIENT

of

a

in

,

a

.

be

of

as

c

,

of

to

guide

.

some kind

a

as

it

may still serve

of

,

be

c

or

blades although ,

other types

of

,

,

In

a

ta

be

"

. is

..

lie

it is recommended by some authors that the average blade lift coefficient ( Clay ) be kept within specified limits . The reader may find , for instance , a Establishing such suggestion that clay should between 0.4 and 0.6 desirable limits of having quite justified similar for blades Clav generalizing this value for all distributions along the span However applied by different combinations rotors some caution should twist span along per great variety obtained can the distributions and rectan criterion established for instance for this case the average fully applicable gular untwisted moderately twisted blade may not

:

Chapter VI

14 )

(

dx

R

c

х

vt

bo v

ở

of

of a

,

x2

xV22

(

TT

ta

RS2

:

1

Clavs

Tx3

),

2

)

15

and

)

(

the

15a

rotor blade

)

(

)

{

156

)

(

tc

6 @

(

x

,

)

8 3

lar

8

(

x2

Using notations for the thrust coefficient referred equation 15a becomes area see page tc ,

see page

to

СТ

1

is

CT

:

as

follows

6

clav

(

(

)

x

x

3

T

9 (

v

xi

2

the thrust coefficient

rewritten

6

can

be v

TT

15 )

/

T

(

SRP

TR2

tV.Z

11

Clar

But equation

6

:

hence

CR

14 )

(

Xe

T

Xi

blade stations

Sày Clar

Taking for example the case rectangular blade equation integrated within the limits working yields and

)

TR2

5

=

R

(

b c

can be integrated

.

2

ở

1

dT

xỏ2

2

,

.

is

ci

be

to

of

Analytical relations leading the computation the average blade lift along the blade coefficient can established by assuming that constant assumption expression elementary Under this the for the thrust

PERFORMANCE IN HOVERING AND VERTICAL ASCENT

Chapter VI

AVERAGE LIFT COEFFICIENT

11

( continued )

solo

char

tc

6

or ,

(

Lav

6

16 )

Since the average lift coefficient is chiefly used in practice for a relative comparison of different rotors , or the same rotor working under different conditions of air density and RPM , it is rather customary to assume 1.0 . Under the above assumptions equations ( 152 ) Xi x = 0 and Xe and ( 15b ) become

APPROXIMATE PITCH ANGLE At the initial stage

in

hovering and blade pitch 15a )

:

in

)

, W )

as

be

.

.:

An approximate relationship between thrust angle can obtained the following way be

or

(

T

a

of

to

,

to

it of is

,

in

solving such performance problems power required hovering approximately desirable know even the pitch per angle given thrust weight the blade corresponding Starting with this approximate value an accurate relationship rotor between thrust and pitch angle can later established

(

(

.

as

R

x

,

a

or at

16 ).

)

a

or

,

)

(

(

Compute the average blade lift coefficient Clay av from equations 15b for still rougher approximation from equation Assume representative blade station Exper that this lift coefficient exists also moderately twisted blades ience indicates that for untwisted and

:

)

17

(

a

)

(

For

.75

v

find

a

89

to

page

).

it is

22a

easy

(

V,

V

and

)

'

equation

18

as

)

(

in

Vt vį

0.75

Clar

to

64

of

to

(ci

But knowing

1.75 equation analogous

0.75

(

Y

be

of

is

course equal the angle attack plus the induced angle expressed and this last can radians see page .75

,

)

IP

be

.

clav

2.75 Pitch angle

at

( a ),

of

be

considered this representative station Knowing the average lift curve slope the angle attack the 0.75 station may found 0.75 can

from an

rectangular blade

Chapter VI

PERFORMANCE IN HOVERING AND VERTICAL ASCENT

APPROXIMATE PITCH ANGLE 0.306

0.75 and the approximate

equations

( 17 )

,

( 18 )

0.75

( continued )

Vt

representative and

(

19 )

will

clav a

109

( 19 )

Clay

pitch angle ( in radians ) obtained from

be :

+

0.408

Claro

( 20 )

POWER REQUIRED IN HOVERING OUT OF GROUND EFFECT The outline given in this paragraph will probably constitute the simplest procedure for establishing an exact relationship between thrust ( gross weight ) and power required in hovering out of ground effect ( under given conditions of air density and tip speed ). This procedure can be broken down into the following three steps : 1 .

Establishing of the exact relationship

between representative blade pitch angle and thrust deyeloped by the rotor . 2.

3.

Computation of the induced and profile powers for the pitch angles for which thrust has been computed .

Establishing of the desired direct relation ship between thrust and power through these common pitch angles . The practical execu

tion of all these steps can be achieved as follows :

Knowing the range of gross weights ( thrust per rotor ) of interest , one may compute approximate pitch angles corresponding to the limiting weights (max . and min . ) from formulae given in the preceding paragraph . For pitch angles slightly higher and slightly lower than the limiting ones , as well as for some intermediate values , the downwash velocity v should be computed at a number of blade stations using equation ( 24 ) , or ( 25 Chapter III . Then the corresponding thrust can be found by methods out Chapter lined in V ( page 87 )

),

Knowing the downwash distribution for the selected pitch angles and fol lowing procedure outlined in Chapter V , the induced power corresponding to these angles may be computed . Also applying the procedure outlined in the same chapter , the profile power corresponding to the chosen pitch angles can be predicted .

PERFORMANCE IN HOVERING AND VERTICAL ASCENT

Chapter VI

POWER REQUIRED IN HOVERING OUT OF GROUND EFFECT ( continued ) Summing up the induced and profile powers computed for the representa tive blade angles the total rotor power (P. ) equation ( 38 ) , page 99 ) is obtained . Knowing or assuming transmission efficiency which usually takes into account not only the actual transmission losses but also such items hydraulic units etc. cooling running electric mechanical losses due readily obtained engines the brake horsepower BHP the engine as

(

,

:

Pв

2 .

in

as

.

is

in

a

of

pitch angles Since thrust and BHP were calculated for number relationship developed hovering desired direct and exact between thrust Figure power required plotted obtained This can be BHP vs. THRUST FOR ALT.ing ha

BHP

hi

Figure

2

THRUST

3 )

.

.

vs. ALT

.

VS. Prequy

ALTITUDE ALT

Figure ,

-

Including the anti torque rotor

3

.

HOVERING

ALT

,

Pav

.

BH

BHP

*

a

-

a

2. be

in

BHPav ,

&

.

BH Preg

(

)

is

(

,

an

as

repeated for several altitudes and the The above procedure can Figure Making results represented cross plot for given thrust relationship Weight important Gross between the BHP required and Figure readily obtained altitude

and the

t

.

(

To

21 )

)

is

,

or

,

or

(

)

of

,

to

(

BHP

*)

,

the

,

CHAPTER VI

PERFORMANCE IN HOVERING AND VERTICAL ASCENT

POWER REQUIRED

IN

HOVERING OUT OF GROUND EFFECT ( continued

)

Knowing how the BHP available from the engine , or engines , changes VS. with altitude , and plotting this variation on the same graph as BHPreq altitude , it is easy to find the hovering ceiling ( for given G.W. ) . This will be the abscissa of the point of intersection of the two curves . It may be added here that the above procedure is probably the most accurate way of predicting the hovering ceiling . GROUND

EFFECT

When a helicopter is hovering with the then the power required to produce a given

rotor disc close to the ground thrust is less than when it is away far . This reduction in the BHP ( PR ) required results from the fact that because of the proximity of the ground the downwash velocity " has no 27. Since in addition thrust time" to develop to its full value of Voo while the induced power to is proportional to the first power of voo square , , the of it hence it becomes obvious that any decrease in the va value will result in an improved ratio of thrust to induced power . The analytical approach to the ground effect problem is usually based "mirror concept " . It is assumed that the actual vortex system is ground " reflected " in the surface as in a mirror . Through a mathematical analysis of the above concept , formulae expressing the ground effect can be developed ( see for instance Reference 2 ) . on the

ill

Chapter VI

PERFORMANCE IN HOVERING AND VERTICAL

50

2.0

CT

.76 A POWER

ASCENT

=

-025 .55

'

D

1.8

CONSTANT

.60

.70

15

TIT

•

•

1.4

75

)

-20

CONSTANT

.65

10

WITH

•

1.6

WITH

2

.80 1 :

85 .90

.95 1.0 RATIO

Figure

4

D

/

h

•

6

.2

.8

1.0

sala

.

THRUST

·

05

PERFORMANCE IN HOVERING AND VERTICAL ASCENT

Chapter VI

GROUND

EFFECT ( continued

)

There are also numerous graphs , based on empirical data , which may be used to determine the ground effect . These usually express the decrease of the power required to produce a given thrust , or the increase in thrust for a given power , as a function of the ratio of rotor - disc distance from the ground to the rotor radius . From a theoretical point of view these empirical corrections usually have one common shortcoming : they refer to the total power , while strictly speaking all these corrections should be directly applied to the induced power only . Since , however , the induced power represents a larger percentage ( 60-70 % ) of the total power in hovering and since for all.practical helicopters this percentage does not vary too much from one type to another , the corrections to total BHP established for one helicopter can be applied in practice to other types too . Figure 4 gives an example of ground effect corrections based on British tests ( Reference 3 ) . The advantage of this particular empirical correction lies in the fact that at least it considers the influence of the altitude through the Ct parameter .

VERTICAL RATE OF CLIMB

22 )

(

be

Veid

V

UU

(

( v ):

U )

of

the

Concepts of the Momentum Theory become quite useful in establishing working formulae for computation of the vertical rate of climb . It has been shown in Chapter II that for an ideal helicopter the vertical rate of climb (Vcid can be expressed ( see page 57 ) as difference between the rate total flow through the rotor and the induced velocity

a

of

)

(

be of

by

multiplying

found .

nt

)

at

Rotor power PR available the rotor should BHP available by the transmission efficiency (

the

be

:

.

to

to

a

of

a

,

a

.

be

,

W ,

a

of

(

)

The above equation true for the ideal helicopter can worked out practical procedure computing into for rate climb Vc real helicopter when its gross weight and BHP delivered by the engine are known This will done through substitution for real helicopter equal an ideal one which will have vertical rate climb that the practical machine The reasoning leading this substitution can represented as follows

.

be

)

to

)

Ppr

)

PR

23

Pid

(

:

)

(

, (

.

is

(

)

(

ly

in

The ideal helicopter vertical ascent uses the rotor power exclusive for covering induced losses induced power and work against gravity Hence the power available for these two functions Pid should com puted easily obtained by subtracting from the rotor power This available PR the rotor power required overcome the profile drag Ppr

113

1

PERFORMANCE IN HOVERING AND VERTICAL ASCENT

Chapter VI

VERTICAL RATE OF CLIMB The rate of flow

(

continued

)

for the ideal helicopter is

( U)

57

page

( see

Pid

550

U

)

W

( 24 )

where w is the gross weight of helicopter

*

The induced velocity in vertical ascent for the ideal helicopter is

57

page

)

u

( see

:

v

W

AS

11

2

U

( 25 )

The numerator and denominator of the above relationship can be multi plied by the downwash velocity in hovering ( Vhov ) without changing the validity of the equation : V

W

Vhov

),

hovering

(

downwash follows

hence

for the ideal helicopter can

be

fps

)

in

of

climb

represent

(

?

Vhov

)

Pid

fp

W

28a

)

Vhov

(

Pid

W

)

(

cid

550

60

U

V

11

?

?

in

or

Cid

U

550

V

V

28

:

and the rate ed as

(

(

U

hov

27 )

V

V

Vhov

2

be

)

26

(

But W / 2 A 9 Vhov is simply equation rewritten can

( 26 )

U

in

AP

2

hov

:

A

as the

V

of

.

It

of

,

-

multi rotor machine power and gross weights per rotor may

be

;

in

.

a

)

(

*

For

used

.

to

be

it

a

is

It

to

,

)

28

(

in

of

With the proper interpretation the physical meaning the symbol readily equation applied Vhov can the real helicopter too equivalent induced velocity only necessary substitute for Vhov the interpretation may proper by which be obtained the relationship power has been shown hovering between the induced and downwash page 45 that

PERFORMANCE IN HOVERING AND VERTICAL ASCENT

Chapter VI

VERTICAL RATE OF CLIMB W

Pin hence

( continued )

V ,

hov

550

:

550

Vhov

Pin

( 29 )

W

where Pin is the induced power . For the ideal helicopter the Vhov com puted from equation ( 29 ) represents the downwash velocity at any point of the disc . All that is to be done for the real helicopter , is to go to the actual relationship between the induced power and gross weight ( as out lined on page 93 ) , find Pin corresponding to the weight w , and compute from equation ( 29 ) the equivalent Vhov for the real helicopter . Having Vhov it is easy to calculate the vertical rate of climb Vc from equation ( 28 ) which in fpm will be ( 30 )

Pilihan

by

Pid

to

is VS

,

(

is

R

/

C

/C

R

VERT

G.W.

Figure

5

OPERATIONAL

G.W

F

GROSS WEIGHT

C

REQ'D

/

.

VERT

MIN

R

RIC

.

a

S

/L

at

Repeating this calculation for several gross weights tionship giving vertical vs. Gross Weight obtained Figure

rela ).

Chapter

5

in

found by the exact method outlined

as

equation the power defined the induced power corresponding the weight Was V.

Pindh

gross weight Pid ,

the

33000 ( W

is

and

P is

( 23 ),

where

W

Vc

Pid

2

115

PERFORMANCE IN HOVERING AND VERTICAL

Chapter VI

VERTICAL RATE OF CLIMB

ASCENT

( continued )

graph as in Figure 5 can be used for establishing an operational Gross Weight at which a given amount of vertical R C is desired .

A

ALT

.

CEILING

VS.

6

Durand's

"

"

"

,

Chapter VI

Airplane Propellers

Glauert

:

H.

1.

References

to

Figure

R / C

,

7

RIC MIN REQ'R'D

Aerodynamic

1948

,

,

"

"

February

and

1948

,

,

,

12 ,

February

Flight

January 5

,

paper delivered

Royal Aeronautical Society

and summarized

1941

29

of

NACA TN 835

"

" ,

"

of

Ground Effect

Helicopters

in

the

Analysis

Lifting Airscrew

Flight Testing

to

Stewart

"

W.

:

.3

on the

Hefner

:

A.

R.

2 .

"

Theory

M. Knight and

an

/C

R

an

.

6 )

acceptable minimum

ALTITUDE

HOV

.

. .

RIC

SERV

Figure

CEILING

(

See

which helicopter will still show

VERT

altitude

at

.

as

,

V

, cal

Fixing the Gross Weight and repeating the vertical rate of climb culations for several altitudes another graph can be obtained giving altitude From this graph service hovering ceiling can be obtained

vs

/

CHAPTER VII

117

PERFORMANCE IN POWERED FORWARD FLIGHT GENERAL In forward flight of present day helicopters ( almost all of which use articulated blades ) aerodynamic and dynamic problems are closely inter woven . These inter - related phenomena peculiar to forward flight are generated to a large extent by the asymmetry of air speed acting on the blade elements : As every blade goes through one full cycle of azimuth angles from y = 0 to V 360 ° ( see Figure 1 ) , it undergoes periodical changes speed in the resultant air . Because of this periodicity ( cyclic character ) of air speed changes , it may be anticipated that the dynamic , as well as aerody namic phenomena , associated with forward flight will be of a periodic charac ter also . And indeed they can be best described and analyzed with the help of geometric ( periodic ) functions .

Consider the forward flight of a rigid rotor ( i.e. with blades fixed in plane the of the rotor axis ) with pitch angle constant with azimuth . As the blades advance , the forward speed is added to the rotational , causing increased resulting in decreas lift . As the blades retreat the forward speed is ysubtracted 90 ° VE ed lift . ; AIR SPEEDS SHOWN OPPOSITE TO THE MOTION

NOTE

V

AS

180 °

--09

DIRECTION OF F'W'O FLIGHT

Vq

V

2700 Figure

1

This means that every blade would experience periodical increase and decrease in lift . For a rigid rotor that would introduce a more or less steady ( depending on the number of blades ) rolling moment acting on the rotor as a whole . It is obvious that the existence of this rolling moment would be an objectionable feature , and indeed special means are applied to eliminate it . The most widely accepted ways of eliminating rotor rolling moment in forward flight are : 1 .

2.

Permit the blades to flap in the vertical direction , Vary the blade pitch angle with the azimuth in such a way that the rolling moments experienced by the blade on the advancing and the retreating side cancel out .

PERFORMANCE IN POWERED FORWARD FLIGHT

Chapter VII

118

GENERAL (continued ) of both methods is also used , chiefly through such an arrangement of flapping axes that flapping - up motion is automatically accom panied by a decrease in the pitch angle , and vice versa . As to the aerodynamic

A combination

importance of methods 1 and 2 in eliminating rolling tendencies , it can be assumed with Lock ( Reference 1 ) that aerodynamically they are equivalent . It will be assumed hence in this study that rolling tendencies are completely eliminated through proper cyclic variation of the blade pitch angle . This means that in the present performance study no reference will be made to flapping and other dynamic phenomena encountered in an articulated rotor . * Using the convention for measuring blade azimuth angle as shown in Figure 1 ( 4 = O at downwind position ; angle increases in the direction of rotation ) , the variation of blade pitch angle ( @ ) required to eliminate rolling moments acting on the rotor can be expressed as :

sin y

ei

e

(1)

e is the average pitch angle ( measured from the zero lift line ) and , e is the maximum value of the angle required to eliminate the rolling

where

moments .

,

Under the simplifying assumptions of uniform downwash distribution , , and rectangular untwisted blades , both e o and e 1 can be expressed in a relatively simple manner . Solving for the average pitch angle the expressions given by Glauert and Squire ( Ref . 2 pp 5 and 6 ) , 2. becomes ( in radians ) : ** M

2.25 ,

T

(

6

tc

(

CT

is

,

V )

a

for most practical purposes may flight the average

:

the inflow ratio

ах

.

rotor

.

of

the many

in

,

be

obvious that because approximate value an

only

in

thoroughly investigated further volumes con junction with aerodynamic blade loading and helicopter stability and controll ability

** It beis

Blade dynamics will

total flow through

the

the

axial component

of

the

ах

is

,

V

=

Vt

*

where

M

2 )

+

2

'

1.5

M2

v !

is

I

,

and

+

the blade area Vpar being the speed

of

,

to M

:

/ .

to

as

to

the thrust coefficient

lift curve slope

1

M

L*

tip

is

tc

is

M

+

referred V speed ratio the par V+ flight component parallel the disc which equal assumed the horizontal speed

where be of

ал ^

( H = 10:52 ] G *(* )(

te

a

(

6

,

simplifying assumptions this will

PERFORMANCE IN POWERED FORWARD FLIGHT

Chapter VII

GENERAL

( continued )

The amplitude of the periodic variation of the pitch angle ( also in radians ) is :

8

.

0,

+ 3

3

eo

4

M

+

Z

3

33 2

where all symbols are the same as in equation

(3)

M M

(2) .

As to the inflow ratio , it should be recalled ( from Chapter II , page 49 ) that the axial component of flow through the rotor in the general case of forward flight can be expressed ( for small tilt angles ) as :

viах

( vc

+

( vg

+

vg

+

vv )

(4)

while for horizontal flight : ах vax

v)

( 5)

In both equations ( 4 ) and ( 5 ) g is the tilt angle of the thrust vector , practical methods of finding this value are given below in the paragraph entitled " Total Parasite Drag and Rotor Tilt " .

and

INDUCED POWER It has been mentioned before ( see page 72 ) that at present there is no satisfactory method for accurate prediction of the downwash distribution of a rotor in forward flight . Fortunately , from the stand point of performance pre diction , this lack of precise knowledge regarding the downwash is not too important. This is chiefly due to the fact that the induced power in the region of forward speeds of particular interest (from that of best climb to Vmax ) represents only a relatively small fraction ( 1/3 to 1/5 ) of the total power required ; hence possible errors made in its computation have little influence on the final results . Because of all this , a method based on the Momentum Theory is sufficiently accurate for practical prediction of the induced power . It should be recalled from Chapter II that the ideal induced rotor power Pinid can be expressed in hp as :

Pinid

W v

(6)

550

where W is the weight supported by the rotor and v is the induced velocity , which in horizontal flight at speed V ( see page 51 ) is :

119

Chapter VII

INDUCED POWER

PERFORMANCE IN POWERED FORWARD FLIGHT ( continued )

)(

7

W

2TR²g RegV where R is the blade geometric radius , and

3

is the air density ..

In calculating the induced power of practical helicopters , values obtain ed from equation ( 6 ) should be slightly corrected for single rotor machines , while for tandem helicopters a modified approach is suggested in order to consider the influence of this particular configuration . The increase of the induced power of a real single rotor helicopter ove the ideal value is caused , as in hovering , by : uniform downwash distribution , and

1 .

Non

2.

Reduction in the effective disc area due to tip losses .

-

Since at our present stage of knowledge neither of these two factors can be accurately estimated in forward flight , it is suggested that the real induced power be approximated by multiplying the ideal , expressed by equa tion ( 6 ) , by the correction factor k computed for hovering ( see page 94 ) :

Pin

=

W

550

v

(8 )

k

INDUCED POWER OF TANDEM ROTORS As has been shown in Chapter II ( page 47 ) the Momentum Theory implies that in forward flight the air stream affected by the rotor action has a circular cross section of radius equal to that of the rotor . When the rotors are arranged in tandem ( Figure 2 ) , the simplest thing would be to assume that the cross section of the affected air stream remains the same as for the single rotor . Under this assumption , AIR STREAM AFFECTED BY ROTORS

R C

| --RY 1

Figure

2

1

-R

Chapter

VII

PERFORMANCE IN POWERED FORWARD FLIGHT

INDUCED POWER OF TANDEM ROTORS ( continued

121

)

the induced power would obviously be equal to twice the value computed for two isolated rotors supporting together the same total gross weight W. This increase in power would be caused by the fact that the average induced velocity ( vt ) computed for a tandem helicopter having rotors of radius R and supporting total gross weight W would be : W

Vt

2 TT R2

while for two isolated rotors average downwash is :

Vis

,

V

9

( 9)

each rotor supports the weight

1/2 W

=

RS

2T

1

/ 2W

,

and the

( 10)

v

A comparison with the theory of tandem biplanes rather supports equation ( 9 ) . Denoting semi - span of wing by b , the following relationship can be developed for the induced velocity of a tandem biplane . W

Vtb

2

TT

(Kb ) 2

‫م‬

V

( 11 )

where W is the total gross weight , V is the speed of flight , b is the semi span , and K is Munk's span factor . Glauert gives K values ranging for the practically possible biplane tandem arrangements from K = 0.92 ta 0.99 . ( see for instance page 50 , Ref . 3 ) . If the analogy between tandem helicopter and tandem biplane could be fully extended , it would mean that the average downwash velocity , because of the presence of K < 1.0 , higher given equation by , would be even than ( 9) . However because of the inflow velocity through both rotors , the actual amount of air affected each , second by the rotors will be greater than T R2 R2 V , where VV is the horizontal component of flight speed . Introducing instead of v the total resultant flow through the rotor Vi and assuming that because of the usual ly small angle of rotor tilt , V ' in horizontal flight can be approximated by

:

),

V

+

vi

+

vectoral sum

V

the

V2

the

following relationship

a

is

it

(

9

as

.eq

v

,

,

is

)

in

neglected entirely for low flying speeds The component VƏ comparison with where all axial while for high ones components are omitted provides sufficient accuracy for practical pur poses

small

.

*

(

instead

of

V!

=

3)

Vez

( Figure

‫م‬

T

PERFORMANCE IN POWERED FORWARD FLIGHT

Chapter VII

INDUCED POWER OF TANDEM ROTORS ( continued

)

ya V

6400

Sivi

Figure

w

2

where Re

is

=

етті

?

Vt

vt

V

2

V2

)

vt ?

+

Vt

1

v2

vt

( 12 )

2

radius

the effective

.

w

gross weight W ) and downwash of tandem

+

rotors ( vt) is proposed :

( total

ra

thrust

between the total

3

(

V

in of

it

9 ),

(

.

equa

( 1

+

2

)v2

Re2

1

2

2

' ((

Re 4

112

TT

W

14 )

w

11

13 )

:

(

S

e

v2 )

2

.?

terms under the radical sign

,

of

TR

:

as

-

+

W

(

Re ?2

T

4

Vt

8g of

can

arrangement rewritten follows re

a

simple be

)

13

(

tion

by

W

Vt

By

solved for the induced speed

vt

can easily

be

)

12

(

Equation

a

to

it

a

V ,

.

vt

,

,

it is

12 ),

(

Inspecting equation becomes iden clear that for hovering supporting tical with that for two isolated rotors each one half the total weight While for high forward speeds when compare becomes small approaches the relationship expressed equation ison with which bears close resemblance the formula for tandem biplane

Chapter VII INDUCED

PERFORMANCE IN POWERED FORWARD FLIGHT ROTORS

POWER OF TANDEM

(

continued

123

)

It is clear that the first term in the brackets represents the effective loading disc of a tandem helicopter , while the second one is the dynamic pressure of the horizontal speed . It can be stated hence that in these cases when the dynamic pressure of horizontal speed is large in comparison with the disc loading * , then the first term in the bracket may be neglected and the induced velocity of the tandem helicopter can be computed from equation ( 9 ) . In hovering , or in flight at low speeds when the dynamic pressure in forward flight is small in comparison with disc loading , equation ( 10 ) , which is true for isolated rotors , becomes correct . For intermediate cases , as for instance flights at best endurance , or low cruising speeds , formula ( 13 ) , or ( 14 ) should be used for the computation of vt . The induced power of the tandem helicopter expressed as for the single rotored machine as : W

Pint

( in

hp )

will ,

of course

vt

,

be

( 15 )

550

In this case no additional correction factor ( similar to k , equation 8 ) is applied , as tip losses were allowed for in equation ( 13 ) by introducing Re , and the whole approach seems to be too approximate to warrant any further refinements .

PROFILE POWER IN FORWARD FLIGHT In the study of the real power required in hovering , a basic knowledge for computing the power required to overcome the profile drag has been acquired . This study is now extended to the case of forward flight . The elementary profile drag experienced by an element of the blade , dr wide and located at a distance r from the rotor axis will be : dD

Ź

sco

cdo

wW?

dr ar

where w is the component of the resultant air velocity perpendicular the blade at the considered blade element .

* Since Re

0.95 to 0.97R the difference between geometric and loading effective disc is not appreciable .

( 16)

to

PERFORMANCE IN POWERED FORWARD FLIGHT

Chapter VII

PROFILE

POWER

FLIGHT ( continued

FORWARD

IN

)

Assuming that the rotor inclination 7 is small enough to allow V , and further neglecting the influence of induced velocity , and of the resultant velocity w at the blade element , V8 on the magnitudeexpressed solely as a function of the peripheral speed the value of w can be

V cos 7

,

r , forward speed V , and azimuth angle

90 °

/

lv

:

( 17 )

be

velocity component parallel

the blade

is

sin

to

of

expressed

usually

18

19 )

)

( (

2

y

sin

V

so

+

(

b

c

dr jar

,

of

dr 20 )

(

) usa

sin

rev

S2

2

bogado

(

+ El

2

T

1

ST

.

O

R

2

:

(

)

T

as in

as

V )

a

(

Cd.fr

the general case the torque dQ varies with the azimuth angle well with the location blade element the average torque full revolution will obviously be

Since

for

=

dQ

The elementary torque offered by all these elements will

be :

dr

12

y

Set

sin

:

of

yields the following expression the rotor

v

cdo

(

bc

9

1

2

dD

b

(

)

17

of

(

Introducing equation into elementary drag for the all blades

16 )

.

the

velocity component per

that the resultant

y

sr

W

The influence neglected

4

the blade span can +

to

clear from Figure

4

Figure

. 4

V

O

It is pendicular

r Jomo

w

dr

·

sin

A

°

y

V

V

270

1180

as :

S2

PERFORMANCE IN POWERED FORWARD FLIGHT

Chapter VII

PROFILE POWER In general

FLIGHT ( continued

FORWARD

IN

125

)

equation ( 20 ) can be solved either graphically , or semi graphically by computing the torque Q at several azimuth angies , plotting , and finding the average for one complete revolution . Q vs. 4 But it can be easily integrated with a few simplifying assumptions : ,

The drag coefficient is constant along the blade ; The drag does not vary as the blade changes in azimuth angle , and That the blade chord is constant .

( 1)

(2) (3 )

,

bc

for

/

t

/2

( 21 )

of

will

be

hp

)

+

in

(

1

R

, v

RS

O

overcome the profile drag

M2

:

to

The power required course

TT

cdo

we obtain

:

TT

8 £

6

R ,

Integrating equation ( 20 ) within the indicated limits , remembering that R V = the tip speed radio M V V + , and substituting

1

in

hovering

,

profile drag

in

,

( 22 )

M2

+

(

1

VAS3

the

power required

the

R2

v

Ti TT

5

)

forward

)

23

hovering

.

overcome

(

)

M2

in

to

power required

+

Ppro

( 1

=

the

Ppr

.

where

pr

is

P

flight

simply

overcome the profile drag

is :

which means that the power required

tois

) :

21

6

ve3

d .

cdo

R2

TT

. 6

(

1/4400

9

1

4400

(

550

equation

do

Ppr But ,

Q

substituting for

cd

or

Ppr

PERFORMANCE IN POWERED FORWARD FLIGHT

Chapter VII

CONTRIBUTION OF PROFILE DRAG TO PARASITE DRAG be

.

of

the

In forward flight , the profile drag of the rotor contributes also to easily com parasite drag the whole machine This contribution can

located the distance the drag direction will

i

in

,

b

blades component

at

of

Its

).

18

equation

(

of

by

Figure

the elements

24 )

(

FLIGHT

°

180

y

sin

D

=

a

Dpr

d

):

see

all

5

be

The drag expressed (

is

.

puted




ryo

(4)

( 4a )

case will occur when the horizontal components of dL and dD will be equal in magnitude and opposite in direction . Then , of course , the element dr will have no tendency to either accelerate , or slow down , and obviously its moment contribution to the acceleration , or slowing down of the rotor , will be zero : The second

dM

a

3)

dD dd cos

dL sin Ø

is small :

dM

2)

(

.

=

(

dL 0

-

r

dD )

=

O

(5 )

The third case will occur when the absolute magnitude of the horizontal component of the elementary drag dD is greater than the horizontal component of the lift element dl , In this case , the element dr will contribute to the deceleration of the rotor , since dM

( dL

a

0

dD )

r

so

(6)

It is clear that for the rotor as a whole some elements may exert accelerating , and some decelerating force . Therefore , the sign of the integral of all elemen tary moments : ( ) will determine the tendency either to increase ,

R

d Ma or to decrease the RPM . In the particular case when R

S* the

a Ma dM

=

0

(7)

rotor will maintain constant RPM .

Of course , in steady autorotative descent the RPM should be constant i.e. condition ( 7 ) should be fulfilled . Any continued tendency to slow down the rotor RPM may , of course , be dangerous . In other words , the condition expressed by equation ( 6 ) cannot be tolerated over any large part of the blade . This means that dD should not exceed drag offered by the element dr ,

de 0 O.

But dD is the profile

VIII

AUTOROTATION 14

2

-

1

dD

w2 c cd ,

9

с

or ,

1

$

(

12 =

)

dr

)?

dr

( 8 )

dD

( continued

de

įį

BASIC CONCEPT OF VERTICAL AUTOROTATION

como

Chapter

is

it

,

of

,

«

,

cd of

on

,

( 8 )

In

equation the only parameter the section profile drag coefficient cdo depending directly the blade element and attack the angle

a so

to

to a

3

in ),

be of

of

( 5 )

by

.

ø 0

of

,

of

of

e

of

is V

)

( 9

V

magnitude from because s2r increases towards the tip that autorotation the local angle attack see that

decreases

in

to

C1

of

in

)

),

(

,

cdo

.

(

It

be

to

of

it is

V '

easy constant tip the the blades can seen hence equation decreases towards the tip

3

With the root

rs

=

Ø

hence

Ø

C

0

cdo

o

but

O

:

on a

of

it to is

·

, (

to

it is

,

be

usually increases with emphasized here that must Remember ing this easy why understand see equation essential bring the rotor blades order rotor into decrease the pitch angle autorotation from powered flight when the angle attack tends increase equilibrium sign the change The condition horizon because equation tal forces acting blade element expressed can rewrit ten as follows

of

,

of

2 )

.

(

a

0

cio

0

>

,

.

is

a

is

is

,

of

a

in

.

of

-

of

a

,

to

,

is

at

it

in

Now becomes clear why autorotative flight stalling the blade usually occurs first the root end section the blade Farther outboard usually negative has there section the blade where cdo ci accelerating torque being supplied Then still value i.e where some tip portion closer the the blade there where cdo decelerating torque generated which counteracts the acceler and where ating torque from the preceding part the blade thus maintaining autorota equilibrium see Figure tion

Chapter VIII

AUTOROTATION

141

BASIC CONCEPT OF VERTICAL AUTOROTATION

STALLED

ACCELERATING

( continued )

OLCELERATING

dL

dL

dD W

W

Figure

2

APPLICATION OF MOMENTUM THEORY TO AUTOROTATIVE VERTICAL DESCENT Since the Simple Momentum Theory was so useful in understanding dif ferent flight phenomena in powered flight , it is only natural to try to adapt it to the case of autorotative descent as well . Unfortunately , in this latter case , serious limitations to the application of this theory are encountered . By assuming , as usual , that the rotor as a whole does not move , but the air flows upward past it , at a speed equal to the rate of descent Vd , one can imagine the following shape of the air stream effected by the rotor action ( Figure 3 ) : Far below the rotor the air stream velocity is Vd . As it approaches the rotor this velocity is reduced ( the air stream widens ) , since it encounters the downwash produced by the rotor . * 1

V

'-

V

-VV V 1 1 1

V *

Figure

3

The presence of the downwash is , of course , necessary because the case where an upward thrust exists is being considered .

AUTOROTATION

Chapter VIII

14

APPLICATION OF MOMENTUM THEORY TO AUTOROTATIVE VERTICAL DESCENT (continued ) velocity at the rotor itself is v , then the rate of flow will obviously be : V ' v . According to the Va simple momentum theory the downwa.sh downstream of the rotor ( in this case far above the rotor ) can reach a maximum value equal to twice the induced velocity at the disc . This means , that higher up the rate of flow in the stream tube affected by the rotor action will still decrease and it will still widen as in Figure 3 . It is generally accepted that in the limiting case it can be ima ed that the air above the rotor comes to rest i.e.

If

Va

=

2

v

,

,

gir

the downwash through the rotor disc

,

or

Va 2

11

-1

)

(

10

2 1

)

(

11

:

as

W ,

to be

V V

wles

rate

)

to

2 VE d

-

va

(

??

R

2

d va be

the weight the found follows

29

TR2

disc loading

by

the

3

W

Vd

.

is

w

where

)

(

of

at

T

in

steady flight the thrust equal Since must descent sea level for the linting case can

Va

the highest downwash

YŽ

as(

I

R²

T

fo

e

finally

corresponding

TT

2

v

T

11

Vi

S

TT

R2

:

and

2

T

)

be

The thrust for this limiting condition velocities can easily calculated

is

11 )

(

to

3

to

.

,

of of

,

.

to to

it is

it

to is

is

a

4,9

in

of

,

of

a

a

of

be

.

to

(

of

)

(

a

)

at

,

to

a

as

in to

is

3 )

(

It

.

of in

of

as

given equation higher than usually The velocity descent observed actual flight tests should be added here also that the assumed condition the downwash speed increasing far above the rotor the Va imagine value rather difficult final state Under this assumption Figure the air flow the stream tube Figure would come rest with respect the rotor disc while the whole mass air outside the upward equal speed tube would still flow Va The existence such situation would obviously mean that some kind vacuum would created brought over that layer the air where rest Of course this picture illogical would be and more reasonable assume that before this ulti mate state air coming rest reached the air will flow into the stream Figure tube as thus creating new flow pattern

AUTOROTATION

Chapter VIII

143

APPLICATION OF MOMENTUM THEORY TO AUTOROTATIVE VERTICAL DESCENT

(continued )

REST

BROUGHT TO

AIA

Va

Va

V

Figure

4

The presence of the downward component in the flow affected by the rotor ( above it ) may include the answer as to why the real rate of descent can be lower than that given by equation ( 11 ) .

BASIC RELATIONS OF GLAUERT'S CURVE In order to find a practical solution to the difficulties illustrated by the above considered case , Glauert , Lock and other British authors ( Reference 1 and 2 ) tried to establish semi - empirical relations ( based on tests with low pitch airscrews ) , which would permit calculation of the actual rate of descent in vertical autorotation . In order to introduce parameters important for practical application two sets of relations were proposed . One between Thrust T and rate of descent , Vas ( speed at ) , the other between thrust and the rate of flow through the disc , v ' . The first of these formulas is expressed as :

o

(

FOT

R2

-

another non dimensional

12 )

dimensional thrust coefficient .

coefficient

13 )

2

2

some kind of a non

-

is

where

F

T

,

R2 va

(

where f is , of course The second is :

f

T

.

2

S

on

=

vi2

T

F

f

of

5

)

2

.

of

(

1

,

be

of

it

f

F

depends The relation between and the working conditions analytically while for expressed the rotor and for some states can early others the works Lock and Glauert Reference and still remain the chief source information regarding this matter Figure represents this relationship between the coefficients with the and

Chapter

VIII

AUTOROTATION

BASIC RELATIONS OF GLAUERT'S CURVE ( continued

)

noticeable peculiarity that instead of the normal values the reciprocals of these coefficients are plotted . This is done in order to avoid infinite values for some conditions .

As can be seen from Figure Glauert's curve . They are :

on

four basic working states are

Windmill Brake

1.

All

5,

2.

Turbulent Brake

3.

Vortex Ring

4.

Propeller

the above states

will

briefly discussed

be

.

7

D

6 MOMENTUM

foF / (

5

THEORY

)

1+ F 2

WINDMIL LOCK - GLAUERT

)

/F

(F

V5

MOMENTUM

OR

,

41-2

=

B

f)

3

TURBULENT

4

)

F

(

/F

E

-

10

F12

D

PROPELLER

Figure

5

I

/F

5

2

4

-

RING

WORTEX

1

=

11-2f1 -73

/

2

= F

f

VORTEX THEORY

marked

Chapter VII

AUTOROTATION

145

BASIC RELATIONS OF GLAUERT'S CURVE 1.

( continued )

Windmill Brake State

If

rotor works in a way corresponding to that considered in the simple momentum theory ( 0 < Va v < 1/2 Vd) relation between and can easily be established theoretically and this condition may be called the Windmill Brake State The word brake used because some thrust present The term windmill braking force applied since generally gravity than necessary more power delivered overcome the blade profile and induced drag perform power The excess hence available ordinary some extra work the case the windmill is

to

is

to

is

.

.

of

in

as

is

"

"

"

"

" .

by

.

is

is

)

(

"

F

f

the

the

(

14 )

(

15 )

(

16 )

(

17 )

:

)

13

(

)

(

12

F

)2

(

:

v

)

v

-

F

Va +

F

obtained

14 ):

1

(

2

be

)

1

(

F

F +

16 )

into

is

(

Iva

the following

f

(

Substituting

Va

:

R2

TT

2 11

)

13

(

and v

15 )

(

Equating

v

F

2

V

F

the momentum theory

T

According

to

2

fyVd

Va

:

of

f

In

this state the relationship between and can be established theoretically Equating the right sides equations and one obtains

in

it

F

.

is

in

,

Vd1/

/f 1

as

of v

,

(

,

is

(

in

,

1

.

is

F

to /

5 )

17 )

/F 1

/ f

1

plotted When vs. obtained from equation can seen good agreement with test region AB that the curve thus obtained Figure approach the for higher values However and values corresponding the limiting case the theoretical 1/2 radically departs proves curve from the test values which that this region the theoretical approach based on the Simple Momentum Theory no longer valid

Chapter VIII

AUTOROTATION 1

BASIC RELATIONS OF GLAUERT'S CURVE 2.

Turbulent Brake State

с a cal

r

(2

Vd

Figure

6

In this state the air does not flow smoothly through the rotor . It can be expected from a scheme shown in Figure 6 , that the rotor will offer more resistance than when it works as a windmill . Therefore , the rate of descent should be lower . As to the torque distribution along the blade , it is difficult ( because of a rather erratic character of the flow ) to predict exactly which elements of the rotor are experiencing accelerating , or decelerating forces , or what the character of the downwash distribution will be . In general , however , it can be said that enough accelerating torque is present to cause the rotor to autorotate , and that the power supplied by gravity is entirely absorbed in creating the turbulence , overcoming the profile drag , and creating a downwash . The part of Glauert's experimental curve which corresponds to this state is the curved line extending between points B and C. This curve can be approximated for this state and for the next one ( vortex ring state ) by the following equation :

(1

Bennett

( Reference

(1

-C

2 3) 2

f€

)2

=

3

f ( f/F

)

Figure

5).

18 )

suggests an even simpler expression :

f )

3

(

f /F

)2

which graphically amounts to two straight lines , on

(

( 19 )

( Shown

by dotted lines

AUTOROTATION

Chapter VIII

147

BASIC RELATIONS OF GLAUERT'S CURVE 2.

Turbulent Brake State

( continued )

These lines connect points B , C and D in Figure 5.

In solving equa

tion ( 19 ) we get :

(1

-

V3 ( f /

25 )

F

( 20 )

)

The part referring to the turbulent brake state is obtained by ascribing the plus sign to the square root of 3 , or :

(

1

-

2f ) 28

/

(( f F )

13

=

( 21 )

This equation represents the portion of the line between points B and C.

V3 :

By ascribing a minus sign to (

1

-

25 )

- 13

=

(f

/F )

( 22 )

straight line portion between points C and D is obtained . This last line approximates that curved portion of Glauert's curve which represents the relation between f and F in the vortex ring state .

the

3.

Vortex Ring State

Vd

Figure

7

and

+

)

,

VH

;

downwards

( u'-

flow

is

the

At

tips

;

rotorvortices (v + Vat ). ring are formed

.

,

the

In this state , the flow is upward through the middle portion of the

AUTOROTATION

Chapter VIII

BASIC RELATIONS OF GLAUERT'S CURVE ( continued 3.

148 )

Vortex Ring State (cont'd )

It is possible to associate the transition from one state into another with the variation of the ratio of the average induced velocity to the rate of descent . At the windmill state , the ratio of v / Va is small , it increases in the turbulent state , and still more in the vortex ring state . Finally it can be imagined that the vortex ring is so large that the flow through the disk is down only ( see Figure 8 ) , and thus the rotor will start to work as a propeller . 1

-Figure

4 .

Propeller

8

State

In the fully developed propeller state a definite flow pattern exists , and the action of the rotor as a propeller has already been studied in Chapters II , III and IV , with the help of the Simple Momentum , the Blade Element and the Vortex Theories .

This state is depicted in Figure

as the portion of the curves between points D and E. The equation relating f and F can easily be established from the Momentum or Vortex Theory as :

f

=

I

F F

T

It can

from Figure

be seen

experimental curve

)

5

5

( 23 )

that the theoretical line is parallel to the

.

USE OF GLAUERT LOCK CURVE Since there are curves available that establish the relation between the flow through the disc and the rate of descent of a rotor the method of attack for the problem of autorotative descent presents itself : First a means of calculating the " inflow ratio " will be developed ,where :

,

Chapter

VIII

AUTOROTATION

USE OF GLAUERT LOCK CURVE

i

149

( continued )

va

vi

( 24 )

Vt

Vt

Secondly , a means of finding the value of Vt will be established . Then knowing the values of and V to it will be possible to find the value of V ' , the flow through the rotor disc , by means of equation ( 24 ) .

Y'

X

Vt

( 24a)

Knowing V ' , the Glauert dimensionless equation ( 13 ) :

. 28

( 25 )

2

2

TT

from

can be found

ve )?

(

RZ

TT

29 TTRZ ( vije

F

W (

viz

W

R

F

coefficient

(( 19 )

by

" f"

"

be

F

),

(

18

or it

or

,

1

),.




.

State

If

mill Brake

,

be

(

(

(

17 )

" "f

of

(

5 ),

"

Having the empirical data giving the relation between and Figure can read directly from this curve can found equation equations use Windmill Brake State Turbulent and Vortex Ring State use the equation for Wind

,

.

of

at

,

of

.

,

used

or

)

1

< be

1

F

the higher

")

i.e ,

(

is

recommended that the more conservative least an average both

of

with values

(

of

descent

of

on

in

1 ,

1