Numerical prediction of propeller performance by vortex lattice method

•

NU~1ERICAL

PREDICTION OF PROPELLER PERFORNANCE

BY VORTEX LATTICE METHOD

by

Richard Shiu Wing Cheung

Submitted August

1987

ACKNOWLEDGEMENT

Grateful thanks is expressed to Prof. Gordon W. Johnston for hi s

gui dance and support duri ng the tenure of thi s y,ork.

Also thanks to Mr. Tony Roberts and Mrs. Winifred Dillon for their

ABSTRACT

The vortex lattice methods of lifting line, and lifting surface with one and two 1 ayers of vortex el ements have been developed to anal yze the perfonnance of propellers. Comparison of the predicted th rust and power coefficients for NACA 109622, NACA 6623-A, NACA 6623-0 straight blade propellers and NACA 4-(4)(06)-057-45A swept Jlade propeller yives close predi c ti on wi th the experimenta 1 resul ts by usi ng the appropri ate vortex 1 atti ce method.

The vortex lattice rnethod of lifting surface with two layers of vortex el enents gives best accuracy among the three methods when it anal yzes the perfonnance of thick blade propellers. For thin blade propellers, the method becomes i1l-conditioned, and the vortex lattice method of lifting surface with one 1 ayer of vortex el ements i s more suitab 1 e. However, when one 1 ayer of vortex el enents is used, the thi ckness effect is excl uded. Both of the above methods account for the camber of the bl ades. For thi n blade propellers with no camber, the lifting line approach would give sufficient accuracy.

Two dimensional airfoil drag data has been included in the deviation of the equati ons of thrust and power coeffi c ients.

improvement of the predicted results is observed.

TABLE OF CONTENTS

Acknowledgement

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Abstract

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L i st of Figures

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Nomencl ature

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1 INTROOUCTION • • • • • • • • • • • • • • • • • • • • • • • • • • • •

1

2 REVIEW OF PROPELLER THEORY •

2.1 MOMENTUM THEORY • • •

2.2 BLADE ELEMENT THEORY •

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• • • e· • • •

2.3 COMBINED MOMENTUM-BLADE ELEMENT THEORY •

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2.4 VORTEX THEORY OF GLAUERT

2.5 GOLDSTEIN'S THEORY •

2.6 THEODORSEN' S THEORY

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2.7 COMPRESSIBLE LIFTING SURFACE THEORY

3 VORTEX LATTICE METHOD

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3.1 LIFTING LINE ANALYSIS

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3.1.1 BLADE GE().1ETRY • • • •

3.1.2 INDUCED VELOCITIES ••

3.1.3 BOUNDARY CONDITION •

3.1.4 FORCES ON THE BLADE ••

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3.1.5 TWO DIMENSIONAL LIFT COEFFICIENTS

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2 4 5 7 10

21

22

33 33

34

35

39

41

46

3.2 LIFTING SURFACE ANALYSIS

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3.2.1 BLADE GEOMETRY •

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3.2.2 INDUCED VELOCITIES •

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3.2.3 BOUNDARY CONDITION •

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3.2.4 FORCES ON THE BLADE

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3.2.5 TWO DIMENSIONAL LIFT COEFFICIENTS AND PRESSURE COEFFICIENTS.

4 NUMERICAL RESULTS •••

4.1 NACA109622

propeller.

4.2 NACA6623-A, 6623-0

straight blade propeller

4.3 NACA 4-(4) (06)-057-45A

propeller • • • • • • •

5 DICUSSION AND CONCLUSION • • • • • • • • • • • • • • • •

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48

48

49

51

53

56

66

66

67

68

83

REFERENCES

90

APPENDIX A ARRANGEMENT OF VORTEX ELEMENTS ON THE BLADE SECTIONS

93

APPENDIX B LIFTING LINE AND LIFTING SURFACE PROGRAM

99

APPENDIX C INPUT DATA FOR NACA 109622 PROPELLER

103

APPENDIX 0 INPUT DATA FOR NACA 6623-A AND 6623-0 STRAIGHT BLADE

104

ffiO~L~R

APPENDIX E INPUT DATA FOR 4-(4) (06)-057-45A SWEPT BLADE PROPELLER

109

"

APPENDIX F APPLICATION OF KUTTA-JOUKOWSKI'S LAW IN THREE-DIMENSIONAL 114 AEROOYNAMIC ANALYSIS

"

LIST OF FIGURES

Figure Page

2.1 ldeal i zed fl ow model of manentum theory 28

2.2 Typical perfonnance curves fr01l manentun theory 28

2.3 Force and velocity diagran of a blade element 29

(induced angle not incl uded)

2.4 Force and velocity di agran of a bl ade el anent 29

(induced angl e incl uded)

2.5 Flow velocity diagran on a propeller blade section in 30

Goldstein's theory

2.6 Hel ix surface of discontinuity 9 - ~ = 0 and hel ical 31

co

surface of constant velocity potenti al 9 - ~

=

ç enclosed

co

inside a constant diameter cyl inderial wake as described in Goldstein's theory

2.7 Coordinate systen used in canpressible hel icoidal surface 32

theory

2.8 Load di screti zat ion in canpressible hel icoidal surface 32

theory

3.1 Coordinate systen of a single lifing line propeller blade 58

3.2 A discrete horseshoe vortex 59

3.3 Flow past the bl ade 60

3.4( a) Locat ion of nodal points and control points 61

3.4(b) A vortex lattice on the blade 61

3.5 Velocity diagram of air flow on a blade section 62

3.6 Oistribution of quadrilateral vortex elanents and horseshoe 63

vortic ies on the mean camber pl ane

3.7 Di stribut ion of quadril ateral vortex el anents and horseshoe 64

vortices on the blade surfaces

3.8 Spanwise and chordwise bound vorticies in a quadrilateral 65

vortex-ring el ement

4.1 Canparsion of efficiency of NACA 109622 propeller with 69

4.2

4.3

4.4

4.5

4.6

Comparsion of thrust coefficient of NACA 109622

propeller

with experimental resul ts

Comparsion of power coefficient of NACA 109622 propeller

with experimental results

Co.nparsion of efficiency of NACA 6623-A propeller with

experimental resul ts

Comparsion of thrust coefficient of NACA 6623-A propeller

with experimental resul ts

Comparsion of power coefficient of NACA 6623-A propeller

wi th experimenta

1

resul ts

70

71 72

73

74

4.7

Thrust coefficient of NACA 6623-A propeller at various blade 75

angles

4.8

Power coefficient of NACA 6623-A propeller at various blade

76

ang 1 es

4.9

Pressure distribution of NACA 6623-A propeller at 0.425

77

radius with J=0.9, blade angle at 3/4 radius = 25 degree

4.10

Pressure distribution of NACA 6623-A propeller at 0.525

77

radius with J=0.9, bl ade angle at 3/4 radius = 25 degree

4.11

Pressure distribution of NACA 6623-A propeller at 0.625

77

radius with J=0.9, blade angle at 3/4 radius = 25 degree

4.12

Pressure distribution of NACA 6623-A propeller at 0.725

77

radius with J=0.9, blade angle at 3/4 radius = 25 degree

4.13

Efficiency of NA CA 6623-0 propeller at various blade angle

78

4.14

Thrust coefficient of NACA 6623-0 propeller at various blade 79

angles

4.15

Power coefficient of NACA 6623-0 propeller at various blade

80

angles

4.16

Thrust coefficient of NA CA 4-(4)(06)-057-45A propeller at

81

various blade angles

4.17

Power coefficient of NACA 4-(4) (06)-057-45A propeller at

82

various blade angles

5.1

Induced velocity field of a hel

ic

al vortex

87

5.2

Comparsion of induced velocity evaluated at the control

88

points and nodal points by

1

ifting surface method with

•

503 Conparsion of induced velocity evaluated at the control 89

points and nodal points by 1 ifting surface method with

Appendix Figure

two vortex 1 ayers

Aol A flat-plate airfoil 97

A02 A flat-plate airfoil divided evenly into panels 98

A03 A panel on the airfoil 98

Bol Flow chart of 1 ift i ng 1 ine program 101

B02 Flow chart of lifting surface program 102

Dol Bl ade fonn curves of NACA 6623-A and NACA 6623-D propell er 107

002 Method of c011bing mean camber lines and basic-thickness fom 108

Eol E02

E03

Blade fonn curves of NACA 4-(4) (06)-057-45A propeller Orientat ion of bl ade sect ion, local bl ade angl e and swept angle

A di screte bl ade sect ion of a swept bl ade

III

112

FU,FV,FW i ,j ,K J K L n R r ro . J 2s u ,v ,w u 0 ,v 0 ,wo J J J V 00 V eff XC, YC,

ze

.

.

NOMENCLATURE number of bl ades chord length d rag eoeffie ient 1 i ft eoeffi e ient

ff ° ° t POWER power eoe lelen pn3 (2R)5

ff ° ° t TORQUE torque eoe _{lelen pn2 (2R)5}

d o 1 f ffo ° t THRUST

ra la oree eoe lClen pn 2(2R)4 Orag

infl uenee eoeffieients of the induced veloeity eomponents u,v,w respeetively

unit veetors in x,y,z direetion respeetively V

advanee ratio,

nU

number of propell er bl ades

1 i ft

rota t i ng speed of prope 11 er, rps radius of propeller

radius of propeller at station r radius of the point j

1 ength of the spanwi se bound vortex el anent

eomponents of indueed velocity in x,y,z direetion respeetively eomponents of induced veloeity at eontrol point j

free stream veloeity relative veloeity

XN,YN,ZN coordinates of nodal points of the vortex element XL,XT,Y

L, coordinates of the bound vortex elenent YT, ZL ,Zr a ~ ~3/4

r

e

p w angl e of attack bl ade angl e

bl ade angle at 3/4 radi us of the bl ade vortex strength

efficiency of the propeller

angl e of turn i ng of the propell er grade difference angle

angle of the point j relative to the x-axis, in the x-y plane radi us of the hel ical vortex 1 ine

angular speed of rotation of the propeller blad~, w=2nn

Note: Non-dimensional lengths are denoted by having a bar over the corresponding variables.

1 INTRODUCTION

In the history of aviation, the propeller played an important role in

the development of powered aircraft. When it cones to moving a

piston-engine airplane throuyh the air, there is no alternative to the propeller.

t~ost conmercial airliners are now driven by propeller in the form of turbofan engines and turboprop engines.

Propellers are the most efficient means of aircraft propulsion. l:3y imparting a small pressure change over a 1 arge area, propell ers achieve much higher efficiency than jet engines. In the past, propellers were highly

efficient at cruise speeds up to approximately Mach 0.6. However, above

thi s speed, 1 arge compressi bil ity losses on the bl ad i ng caused the

efficiency to fa" rapidly. The increase emphasis on fuel conservation has

st imul ated developments on propell er powered ai rcraft. Nowadays, advanced

design concepts make possible the design of high efficiency propeller

capable of Mach 0.8 crusing. Currently, popular nunerical methods in used include the vortex lattice methods, the lifting linejsurface methods, are expected to provide extremely reliable predictions.

In this thesis, various propeller theories are reviewed. Vortex lattice method is then used to predict the performance of a mmber of propell ers. The objective of this thesis is to develop computer codes of vortex lattice method of lifting line analysis, lifting surface analysis with one and two layers of vortex elements. Two dimensional airfoil drag is included in the deviation of the equations of thrust and power coefficients.

2 REVIEW OF PROPELLER THEORY

2.1 MOMENTUM THEORY

The earliest theory of propeller was formulated by Rankine and later by Froude; they regarded the propell er as an actuator d i sc, wi th un i form streaming through it. The propeller is asslJTled to have a large number of blades with no frictional drag, and the thrust is uniformly distributed over the disco Slipstream rotation is ignored. On passing through the actuator disc, the pressure of the fluid increase suddenly by pi, and the axial velocity increase by v as shown in ~g. 2.1.

The increment pressure pi is equal to the thrust on unit area of the disco 8y considering the rate of increase of axial manentuin, th rust is equal to

T = Ap(V",+v)v

1 (2.1)

where A is the area of the actuator disc Pressure increment is

T

PI

-- T

= p(V",+v)v₁ (2.2)

Apply Bernoull i I S equation to the motion before and behind the disc

separately, total pressure heads are: Ho = Po

+~V:= P+~(V",+v)2

1 2 1 2 Hl = Po +

2P

(V", +V_l ) = p + pi + !p (V", +v) respectively; or, 1 pi = Hl - Ho

=

P(V",+2vl)vl (2.3) 1

Compare equation (2.2) and (2.3); v=7v1. That is the axial induced velocity at the far wake is equal to one-hal f of the axial velocity increase

in the sl i pstream.

The work done on the ai r by the thrust of the propell er is E =

~(V~+v)[ (V~+vl)2_V:]

= 2Ap (V +v) 2v ~

=

T(V +v) ~ (2.4)

which is a1so equa1 to oQ, where 0 is the angu1ar velocity and Q is the torque of the propeller.

Consider the actua1 case when the propeller is advancing with velocity V i n st i 11 ai r. The \'K)rk done on the ai r by the thrust is

~

E

=

~(V~+v)vi

= 2Ap (V +v)

i

~

=

Tv (2.5)

and the \'K)rk done on the airscrew is TV • The efficiency of the propeller is

~ TV _~ Tl

=

go-V ~ - V +v ~ Let v=aV ~, _ 1 Tl -

1+a

(2.6) (2.7)

This is called the idea1 efficiency of a propeller, which represents the upper limit of the efficiency that cannot be exceeded whatever the shape of the propeller.

Equate TlP to the work done by the thrust,

TlP

=

TV ~ where

P

=

power

or

1-" _

_(~)(

p ) ~-1t

7"öI

"

p m or

1-" _ (~)(1

)e ~-1t ~ P

"

or T

₌

thrust

=

Ap(V_m+v)v

₁

=

;o2pv:(l+a)a 0

=

diameter of propeller V where J

_=nö

m C

=

P P _{pn 3D5} T where CT

=

~ pn D (2.8) (2.9)

Typical curves of relationship of J,

V~D2

and" is given in Fig. 2.2. The

p m

mOOlentum theory is an one dimensional theory. It bases on the assumption that the only loss of energy is in the kinetic energy of the axial flow in the sl i pstream. The theory cannot be used to pred ict the perfonnance of propellers; however, it gives the ideal efficiency.

2.2 SLADE ELEMENT THEORY

The bl ade el ement theory is based upon the idea that the propell er bl ades function basically as wings constrained to rota te about a central point as they advance through the air. In the theory, the propeller blade is viewed as composed of a number of ai rfoil el ements, each el ement acts

independently of its neighbors. The 1 ift and drag forces on each el ement at the radial distance r from the ax;s of rotation is

where

v~ = V~+(2nnr)2

=

n202[

i+(~)

2]

c

=

chord 1 ength

( 2.10a)

(2.10b) The 1 ift and drag forces are perpendicul ar and para" el to the rel ative

inflow velocity respectively (Fig. 2.3). Resolved in the direction of thrust and drag:

dT

=

dLcos$ - dOsin$ dQ = (dLsin$ + dOcos$)r

where $

=

flow angle excluding induced effect

( 2.11a) (2.11b)

The net force acting on the bl ades are the summation of the forces acting upon the individual elements. Consequently, thrust and power coefficients are n 1 2 2 2 . CT

=

"8

f

x h

(J

+n x

)a[

CL COS$-COSl n$ ]dx (2.12a) n 1 2 2 2 . Cp

=

"8

f

x hnx ( J +n x

)a[

CL Sl n$+COCOS$ ]dx (2.12b)

Bc

where a

=

nR r

x=-

_R x

h

=

hub station \tkIere the blade begins CT and Cp are not too sensitive to the value of x

h• The lift coefficient on each individual el ement can be expressed as

CL

=

_aOClO (2.13)

where a

o

=

lift slope for infinite aspect-ratio

a

_o

=

~-~

~

=

b 1 ade angl e

Co can be obtained from airfoil characteristic curves of the blade el ement as a function of Cl. The theory provides no information on the induced field, and neglects interaction between the propeller blade sections and the wa ke •

2.3 COMBINED MOMENTUM-BLADE ELEMENT THEORY

1f induced velocity, and induced angle (ai), is included in the blade element analysis, as shown in Fig. 2.4; we have

and

CT =

~ J~h

(J2+;t2x2)O'[Clcos($+ai )-Cosin($+a;)]dx Cp =

~

J;h nX( J 2+n2x2)

0'[

Clsi n( $+ai )+C oCOS($+a i)] dx

(2.14)

(2.15a) (2.15b)

Assume ai and the drag-to-lift ratio to be small; V_R ",V_R• Equation

o

(2.15a) can be written approximately for B blades as

dT

=

(average mass flow) x (total change in velocity)

=

~v~cao(~-$-ai)cOS$dr

(2.16) Applying momentum principles to the differential annulus of the propeller disc and letting w=<VRa;, we have

Note that according to manenturn theory, the induced velocity at the far wake is twice the induced velocity at the propeller disco

Equate

(2.16)

and

(2.17),

cra V

o:?+o:.(~+

OR) 1 1

x

alv

T V CD where À.

=

wR ' V T

=

uR , V R

=

V TI (x 2 + À. 2 ) <p

=

tan-l~

x

Solve the quadratic equation for the induced angle of attack, cra V cra V cra V

0:.

=

~_(~

+ 0 r) +

I(

(~+

0 R)2+ 0

R(~_<p))]

1

2

x

ax2

v

x

ax 2

v

2x2V

T T T

(2. la)

Thi s theory i ncorporate the resul t of one d imensi onal manentum theory in blade element theory to evaluate the induced angle of attack at the

propeller disco

2.4 VORTEX THEORY OF GlAUERT

The vortex theory of Glauert determines the induced velocity field as a function of the vorticies of the blades and its wake. The vortex theory of Glauert treats the propeller as a bound vortex line with circulation varying along the radial direction of the blades. From the end of each bound vortex 1 ines, trail ing vortic ies spring to form hel ical vortex sheet. Gl auert assumes that the various hel icoidal sheets springing from the various bl ades merge to form one sheet. The propeller is assumed to have infinite mmber of bl ades, and the sl i pstream may be assumed to con si st of a system of cyl indrical vortex sheets, with no sl ipstream contraction. Ti p losses and the effect of multiple vortex sheets are ignored.

L i ft on a bl ade el ement can be expressed by Kutta-Joukows ki 1 aw as (Fig.2.4):

dL

=

pV

R _o rdr where r = vorticity/length (2.19) dr

=

width of the blade element

v

= relative flow velocity

R or in terms of 1 i ft coeffi c ient (CL) as:

dL =

~2

C cdr Z-Ro L Equate (2.19) and (2.20), CLVR c o

r

= --,..---2 o (2.20) (2.21)

The infl uence of the bound vorticies decrease down the wake until it produces no disturbances. The influence of the trailing vorticies, however, still exist in the ultimate wake. The sheet springs from the element is a semi-infinite sheet; the sheet in the ultimate wake is an infinite sheet to either side. Therefore, the induced velocity at the el ement is one-hal f the velocity at a point in the ultimate wake.

The thrust produced by a blade elenent is equal to dT

=

dLcos4>

o

1 2 where dL =

F

V

R

_o

CL cd r

4>0

=

flow angle incl uding induced effect

Thrust is also equal to the rate of change of axial mOOlentum, dT

=

(average mass flow) x (total change in velocity)

=

21tp rdr(V+vcoS4>o) (2vcoslP o)

(2.22)

Equate (2.22) and (2.23), CC L VR2 o v = 81t r(V+vcOS$ )

o

V+veos$ si n$ = V 0 o R

o

Fron Fig. (2.4), Fran eq. (2.21), CL VR

e

o

r

= -,...--2 Therefore,

For number of blades equal to S,

S i nee t he i nd ue ed a ng 1 e, a., i s sm all , 1

sin$ = sin($+a.) = sin$+a.eos$

o

1 1

and

sin$o(tanai) = aisin$o = Solve eq. (2.25) for ai'

a.

=

_--I:ê~-_$~_ 1 1 + 8xs;np <1R

ao

r where x="[ (2.24) (2.25) (2.26)

Eq. (2.26) can then be used with eq. (2.14), (2.15) to calculate the lift eoeffieients, thrust eoeffieient and power eoeffieient. In vortex theory of

Glauert, the induced angle of attack, ai ' is influenced by the wake at that radi us only.

2.5 GOLDSTEINIS THEORY

Goldstein revised the vortex theory by considering the propell er as having a finite number of blades. Each blade on the propeller may be repl aced by a bound vortex system with trail ing vortices spring from every point of the bound vortex. By considering a lightly-loaded propeller, the interference flow of the vortex system is snall compared with the velocity of the bl ades and there is no sl ip stream contraction. Hence, the trail ing vortices approximates a hel ical vortex sheet with constant diameter. Thi s massless mathematical surface, which is impenetrable to fluid particles,

forms a surface of discontinuity.

Gol dstei n further assumes that the di stribut i on of ei rcul ati on along the bl ade is such that the energy requi re to produce the trail ing vortex system per unit time is minimum. According to Betz, this equivalent to the formation of rigid hel ical vortex sheet by the trail ing vortices, and the rearward axial velocity of the rigid hel ical vortex sheet at the far wake is constant. Furthermore, the fl ow outside the rigid hel ic al vortex sheet is considered to be inviscid, continuous and irrotational.

Consider a two-bladed propeller, the geometrie equation of the two rigid hel ical vortex sheets, or hel ix surfaces , of the wake are given by:

or

!

= z

w

V-en ~=z w V-en wZ

where r,e, and z are cylindrical polar coordinates with z-axis as the helix axis coincident with the propeller axis in the direction down the wake, w is the angul ar velocity of the propeller and V is the free stream velocity. e

CD

is measured from z=O • Additional blades YKluld develop similar helix surfaces •

The velocity potential difference across the rigid helical vortex sheet, or hel ix surface , i s constant along the trai 1 ing hel ical vortex 1 ine of gi v en rad i us and is equa 1 to the c i rcul at i on at the correspond i ng poi nt on the propeller where the vortex line is shed. The total velocity potential, <I> of the hel ical wake is a function of rand

r.

where

wZ

r.=e-v-

(2.28)

CD

For given values of

r.

and r, equation (2.28) represent a spi ral line of constant radius. r with constant velocity potential along the line. For a given val ue of

r.,

spi ral 1 ines of different rad i us r add up to form a hel ical surface e - ~z =

r.

with spiral lines of constant velocity

CD

potential. The angle

r.

is measured as the angle between the hel ix surface e - ~z = 0 and the hel ical surface with spi ral 1 ines of constant velocity

CD

potential e - ~z =

r.

at the axis of rotation on any plane cutting across CD

the wake at right angle to the axis of rotation. Note that e and

r.

are defined in the same direction.

Since wake does not exist ahead of the propeller disc, the induced velocity of the fluid at the propeller is half that at the far wake behind the propeller. The fluid velocities normal to the hel ix surface must be the same on both sides of the surface. If wis the rearward axial velocity of the hel ix surface at the far wake, then the velocity component of w normal

v

00 •

to the hel ix surface are equal to \'tCOSa where tana = - 1S the slope of the wr

constant diameter helix at radial location r for

O<a<i.

The slope tana is measured with respect to the propeller disc plane. However, the tangential velocities of the fl uid on each side of the hel ix surface are assumed to be different due to the different in velocity potential s at the surface. The angle a is also equal to the geometric inflow angle on the propeller blade

relative to the propeller disc as shown in Fig. 2.5. In the figure, the induced velocity ~ lies in the free stream direction in the rotating frame on the propeller blade.

The velocity of the fl uid on the hel ix surface can be expressed in terms of velocity potential <1>. The f1 uid velocity nonnal to the hel ix surface is equal to

~cosa

-

r~:sina.

The boundary condition at the helix surface is given as:

or

Nonnal velocity of

the hel ix surface

=

Normal velocity of the fl uid on the hel ix surface

where the nonnal velocity is referred as the velocity nonnal to the hel ix surface at the far wake

WCOSa

=

~osa

_roe

0<1> . _S1

_na

(2.29)

Derivatives of <1>, which are the velocity c~nponents induced by wake singularities, can be detennined from equation (2.28) as

0<1> = _ (~)(o<l»

oZ

V oC

M.

=

0<1>

oe

oC

00 (2.30a) (2.30b)

Substitute (2.30) into (2.29): wV

a<l> _ (~2 )(~)

~- - ~ w at the hel ix surface , (2.31)

h - wr

w ere IJ. -

V-

(2.32)

a>

Note that

e

and

ç

are defined in the same direction. Define non-dimensional velocity potential ~ where

a>

eq. (2.30) becomes

M _ ~2

aç - -

"fi1lT

which must be satisfied in the region r<:R at the helix surface.

(2.33)

The Laplacian equation to be satisfied by ~ at both regions r<:R and r)R is: 'i/2~

=

0

Express the Lapl acian in cyl inderical coordinates:

1

a

M a2~ a2~ 0

'i/2~

=

(rJE"r(

ra

r) + r2ae2 + ~

=

Fran eq. (2.32), M_

_(W

_H

_M» äf-

_r

_a~ a>

a _

(rH

~~)

ä"f-a> V r

=

(wa»~ From eq. (2.30), a2~

=

(W )

2(a2~)

äZT

r

_a>

W

a2~

_

a2~

-W--W

Substitute into (2.34), the Laplacian equation can be written as:

13

(2.35 ) The Lapl acian equation was reduced from three-dimensional in (2.34) to two-dimensional in (2.35). In (2.34), the Laplacian equation is a function of r,e,z. In (2.35), the equation is basically a function of r, ç, Tl ; where Tl is in the direction along a helical line for a given radius r. Since the

velocity potential distribution is independent of Tl; the Laplacian equation

is reduced to a function of rand C, or I.L and ç, as shown in (2.35).

<'fl is a single-val ued function of position, and its derivative must vanish when I.L is infinite. It must also be continuous everywhere except at

the hel ix surface.

If the hel ix surface moves along its axi s with velocity wand rotates with angular velocity ~w about its axis, its displacement is entirely within

co

its own surface. Therefore, the induced fluid motion due to the wake will be the same whether the surface translate along its axis with velocity w without rotation, or rotates about its axis with angul ar velocity w

1' equal to - ~w, without translation. In the latter case, since the induced fluid

co

motion on any plane in the wake perpendicular to the axis is the same, it can be considered as a rotating lamina in a two dimensional pl ane. For a two bladè propeller, the rotating lamina is equivalent to a line of discontinuity of velocity potential with length 2R rotating about its midpoint with angul ar velocity w

1 on a two dimensional plane. The solution for the case of a rotating lamina with angular velocity w₁ without transl ation was determined in Lamb' s Hydrodynamics, section 72. Lamb solved the velocity potential distribution about a rigid ell iptical cyl inder rotating with angul ar velocity w about its center in a two dimensional

fluid by assuming the solution form as:

. 1. ( b) 2 - 2t h t .

$ + 1<\1 = 1+1 a+ woe were =;; + 11)

a, bare the semi-axes of the ell ipse. ;;, 1) are ell iptic coordinates connected with the cartesian coordinates x, y by

x + iy = c cosh(;;+i1))

For a two blade propeller, the solution form of the rotating lamina of a 1 ine of discontinuity of velocity potential rotating about its mid-point in a two dimensional f1 uid is given as in Lamb with a=R, b=O and w =-w . or _{o '}

where t = ;; + i1) ;;,1) are elliptical coordinates given by

reiç = R cosh t

where ,; = re i ç , we have

(2.36)

R2e-2t = 2,;2 - R2 - 2,;Ri { 1 -

~~)

-

~:l(~)

-

~:~:~(~)

-

}

Substitute into (2.36), and by using the identity ,;P= rPe ipç

or

P - rP( cos( lp

t)

+ isin( lp

t))

for

,;

-= rP(cos(lp

t)

isin( lp

t))

for we have for h

I

c; _R

-

...

}

for h l ) R p= 1, 2, 3,

...

p= -1,-2,-3,

$ + i<\l = -

}i~

[ 2r2(cos(2ç)+isin(2ç)) - R2 - 2R2i{

~(cosç+isinç)

-

i-<~)(COS(3ç)+isin(3ç))

-

t~(.~)(COS(5ç)+isin(5ç))

-

~:t~(:)(COS(7ç)+iSin(7ç))

- ••••• } ] = - {iw1 [ 2R2{

~:l(~)(COS(2ç)-iSin(2ç)

-

~:~:~(~)(COS(4ç)-iSin4ç)

- ••••• } ]

for

11: I (

R

for

11: I )

R

Collect the real terms, the solution of the rotating lamina is detennined as:

1 2. 1 2 r 1 r 3 1 (r)5

tI> =

2"

w1 r sln2ç -

2"

R w1 { ~osç -

-2{lr)

cos3ç -

2-4 R

cos5ç

1e₁e_{3(r)7 }}

- Te4-O""'R cos7ç - ••••• for r(R

In the above solution, tI> is a function of rand ç; where ç, as previously defined, is the angle subscripted with the line of discontinuity of tI> on the pl ane of rotation at the hel ix axis. From the above analysi s, we can see that the solution of the potential problem (Laplacian equation)

can be expressed in a series of even multipl es of ç for r)R. Substitute the series into the Laplacian equation, the coefficients of sin2nÇ are found to be a 1 inear function of I2n(2nl-l-) and K2n(2nl-l-), where In and Kn are modified Bessel functions for n=1, 2, 3, ••••• At 1-1- equal to infinite, grad tI> vanish; however, I2n(2nl-l-) intends to infinity. Therefore, only K

2n(2nl-l-) terms should exist in the expression for tI> in the region r)R, or 1-1-)1-1- where

o 1-1-0=

~R.

Hence

we

have Cl) Cl) K 2n (2nl-l-) tI> -

I

c sin2nç - n=1 n K_{2n (}2n l-l-_o) (2.37)

where Cn and K2n (2nl-l-o) are constants Consider the case for

Put and at ç=O, 1t F or t e reglon h . 0 " (",(1t, ,. - 1t '" -

"2 _

.

4

-

~ co s ( L (2m+1 )2 2 m+ 1 )

ç

1t m=O Cl)

Assume ~1 = fo(ll) +

L

f (ll)cos(2m+1)ç

m=O m with O(Ç(1t ,

(2.38)

(2.39)

(2.40)

(2.41)

differentiate (2.41) term by term, and substitute into (2.39). Note that ~ must be finite at 1l=0, \\e have

The general solution of (2.43) without singularity at ll=O is

(2.42a) (2.42b) (2.43)

~(ll) = T_{1,2m+1( (2m+1)1l) + bm}I_{2m+1( (2m+1)1l)} (2.44)

where b is a constant, and Tl (z) can be determined for given val ues of v

m ,v

z2m

+ ••••• + ""M( 2....,2-_~v2.,...,'I""1(...,,4'""T2 -_ v-2"'-.-. -. (n4r-:::m'""TL-v~2I"'T'", + ••••• 1

+ - (-1) vK (z)

2"

v for v = even integer v = 2m+1 Z

=

(2m+1)

Substitute (2.40), (2.41), (2.42), and (2.44) into (2.38),

we

have 4 DO gm(~d ~ = -

L

2 cos(2m+1)~ 1t m=O (2m+1) DO 4 Tl 2m+ 1 ( (2 m+ 1) IJ. ) =

L {{ -} { ,

}

m=O 1t (2m+1)2 +

a

m where a m = constant to be determined and 0 <; ç <; 1t 1 2m+1( (2m+1)~)

1

_{2m+1( (2m+1} )~o) } cos(2m+l)ç (2.45)

The discontinuity in ~ can be given as the difference in ~ between ç .... O+ and

(2.46 )

The constant a

m can be determined by the conditions of continuity of ~ and

~~ at ~=IJ.. Expand cos(2m+1)ç in a sine series as

u~ 0

cos(2m+1)ç

=

~

nL 4n2-

(~m+1)2

sin2nç

and substitute into (2.45). Since

~

and

~:

must be continuous at

IJ.=~o

equate (2.37) and (2.45), and thei r derivat ives of IJ. at ~=IJ. •

o

El imate c

n' we have the equat i on

which represent an infinite number of 1 inear equations with an infinite number of unknowns, a

m• However, am can be solved mrnerically for finite

val ues of mand n. Substitute the values of a

m into (2.46), we find l1<IJ for

giv~n values of 11. In the above equation, the argument of the functions

T1,2m+1' T{,2m+1' 1_2m+1' 1_{2m+1 is (2m+1)110 ; that of} K_{2n and} K_{2n is 2n110.}

Since the velocity potential difference across the trailing vortex line at a certain radius is equal to the circulation on the propeller blade at that radius, circulation of the bound vortex at radius r can be written as

or r( r)

=

JM.dç

ol::

=

fj,(J> wV CD = ---t,.<IJ w

wr _

l1~ 2nV w - 2n 00

where l1<IJ can be determined from (2.46)

When the number of blades are equal to B, the potential problem

becomes :

with boundary condition

Mi _ 112 .

oç - -

1-i1ï2"

, and grad <IJ van i sh for r=CD.

for r.:;;R, at ç w where <IJ

=

wv-

(j> CD 2n 4n

=

0,

B ' B

,

...

,

2(B-1)n _B

\ fG~l dstein sol ved the potenti al probl em of t:iI! for di fferent mmber of

bl ades B at different non-dimensional radius ~. The sol ution is expressed in

terms of parameter Kwhere

Bwr

21tV w

=

K

co

21tV W

r

=

_&.> a> K (2.47)

The val ue of K for a given number of bl ades Bis expressed in graphi cal form

as a function of r/R for given val ues of

~

[ or 1t/(advance ratio)

J.

co

Force on the blade sections are deteremined by:

Thrust and torque on a propell er

dT ( 1 . ) d r - Bpr w r - 2'v s 1 na

=

Bpr(wr -

iWl~~2)

~

=

Bpr r(Va> + ivcosa) 1 ~2

=

Bprr(Vco

+ 2Wl+~2)

where V is the rel ative fl ow velocity on

the bound vortex line

r

blade section is given as:

where v

=

interference velocity at the

far wake

=

WCOSa

tan ex

=

slope of the hel ix

=

V / _co wr

=

1/~

Substitute (2.47) into the above equations and integrate along the

radial direction of the blade for total thrust (CT) and torque (C_Q)

coefficients: CT

=

~ _{11 -} À. 2 12 C

_Q

=

_{2À.1 1}+ À.2~~13

w

where À.

=

-V co 1 11=

f

oKxdx

and efficiency is: 1 11-

T-

I 2 11+

~~ÀI3

r x = Ir

From the above equations, À can be found in tenns of CT or Tl for any given val ues of !J.

o• Therefore, curves of Tl against CT for different val ues of !J.o ' and curves of Tl against !J.

o for different val ues of CT can be found. Since the above anal ysi s does not incl ude infonnation about the propell er bl ade

shape, only these characteristic curves are obtained.

2.6 THEODORSEN'S THEORY

In Goldstein's theory, the assumption of independence of adjacent blade elements and no slipstream contraction is retained. The latter assLlllption

requires that the blade be lightly loaded, the axial induced velocity in the ultimate wake is snall. The slope of the wake spi ral is considered to be everywhere the same; and is given by the tangent of the tip vortex angle:

1

V

À=(n) (nrr)·

Theodorsen extended Gol dstei n theory to heav i1 y loaded propell er by considering the vortex at the ultimate wake. In his method, the tangent of

1 V+w

behind the propeller, and is not considered to be SInall. The potential flow problem of Theodorsen is o 0 o2cp 0 ~O~(~~) + (1+~2)~

=

where 1.1.

=

~ r 00 (2.48 ) Theodorsen solved the potential problem (2.48) experimentally by using

el ectrical analogy technique instead of Bessel function. The fl ow around a

rigid hel ix moving at a constant velocity down the wake can be represented by electric field around an insulating helix model in a conducting liquid with uniform electric field in the direction of the helix axis. The circul ation along the blade is expressed as

21t(V +w)w

r

= 00 K

&> (2.49)

where Kis the circul ation function determined by el ectric method

Theodorsen function K differs from Goldstein function K in that the 1 atter consider w/V to be infinitely smalle The two functions are the same i f they are cornpared at the same val ue of hel ix angl e in the far wake.

Theodorsen introduce the mass coefficient k, which is physically interpretated as the effective cross section of a propeller in terms of the

propeller disk area, and is given for single-rotation propellers as:

1

k

=

_2JoK(x)dx

2.7 CCJo1PRESSIBLE LIFTING SURF ACE THEORY

Compressibl e hel icoidal surface theory has been developed for propell er

analys is from three-dimensi onal thi n wi ng theory by Hanson; wi ng kernel

multiple blades. Within the restrictions of linearization and thin blade approximation, the kernel function rigorously account for the effects of three dimensionality, compressibility, sweep and blade interference.

The upwash angle ai induced by the pressure distribution is given as a function of the kernel function as bel ow:

r

.

ai(r,z) '" tana i

=

J!

_{h L}

JrT~cp(ro,zo)KL(r,z;ro,zo)drodzo

(2.50) where ~cp

=

distribution of pressure difference on the thin blade

surface

KL

=

kernel function (or influence coefficient) rT' r

L

=

trailing and leading edge value of ro respectively zh

=

normal i zed radi us at centerbody

The kernel function KL (r ,z;r o,zo) gives the induced upwash at a field point (r,z) due to unit load at a source point (Fo'Zo). The coordinate system is given as in Fig. 2.7. The ordinate r is the distance in streamwise direction measured at constant radius from the axis of rotation. The function ~cp(ro'zo) is defined to be zero outside the blade surface. By applying the boundary condition that the flow be tangent to the blade surfaces, pressure difference distribution, ~c , and forces on the blade can

p

be determined. In wing theory, the kernel function is given as:

K . wlng

~ing

1 r-r 0

=

81t{z-z )2[1+/({r_r

)2+~2{z-z

)2)] 0 0 0

1

r-r

0

=

41t{z-zo)2[/(r-ro)2+~L(Z-zo)2)]

for M<l for M> 1

While the kernel function of propeller is derived by Hanson as:

K

_L

= -

~(z-z

_{41tZ- 0}

)H(r-r )

₀

(2.5Ia)

(2.5Ib)

where

a

=

j-B

=

number of bl ades

~

=

I (l-M2)

H(r-r 0)

=

Heavyside function

I , K _{n n}

=

modified Bessel functions

J

=

advance ratio

J_n, Y_n = Bessel functions of first kind k

=

wave number variable

m = summation index M

=

forward Mach number

Z<, z>

=

lesser, greater of Zo and z

cr ,cr

=

1{l+a2_{z2 ) and 1{l+a}2_{z2 ) respectively}

o

0

Equation (2.52) is of the same form in either subsonic or supersonic regions ; therefore, high speed propell ers, with subsonic rel ative speed at the roots and supersonic speed at the ti p, can be anal yzed accuratel y.

The kernel function in eq.(2.52) consist of five terms. The first term represent the increase in axial and tangential velocity components associate with the gross thrust and power of the propeller. The term H(r-r_o) is zero upstream of the leading edge, increases between the leading and trailing edges, and equal to a constant aft the trailing edge to the ultimate wake. The second term ;s odd in

(r/a-rolao)

and decays upstream and downstream of the bl ade. The thi rd term does not depend on r or ro and does not include compressibility. The leading singularity in this term is 8n(z:z )2 ,

o

as compare with the wing kernel function. The fourth term is odd in (r/a-ro/ao), and is identical to the third term for r+-a>. The fifth term is even in

(r/a-rolao)

and decays far upstream and downstream. For incompressible flow, the integration range l/(l+M)<k<l/(l-M) vanishes and the term \'«)uld equal to zero.

Due to the complexity of the kernel function, the integration equation (2.52) cannot be solved analytically. However, it can be solved nlMllerically by discretize the load in terms of load elements on the blade surface, and enforcing the flow tangency boundary condition at control points. The blade surface is divided into spanwi se panel s with widths at a constant percentage of the chord. Load 1 ines are di stributed on the quarter chord, and control poi nts at the three quarter chord of each panel, as shown in Fi g. 2.8.

Hanson rewrite eq. (2.50) in discrete form as follow: W

=

L

L

K

p. v v p.v

where Wis the downwash angl e at the control points

K

=f

1 R.(Z)K (z.,z)dz

~v zh J 0 ïTr

_,

1f 1 0 0

Z. ,R. (Z ) are the radial modes of the control points and load

1 J 0

points respectively

~, vare control point and load point index respectively

m,n are the subscripts of the chordwise locations of the control points and load points respectively

K _

J

z i ' zo) =

IJ

K L (r , z ;r 0 ' Zo ) G _ (r , z i ) C _ (r 0 ' zo) dr 0 dr ( 2. 54 )

m,n !TI n

G (r,z) =

2~;~2exp[

-

4~~2

(r-r_)2] (2.55)

m

m

(chordwise distribution factor for control point)

2;1 n2 41 n2 2

=

flo/n exp[ -

~r

o-r o-n-) ] (2.56)

(shape factor for chordwise loading)

fI, flo are the width at the half amplitude points of the

Gaussian functions

G ,

C respectively.

iiï l'f

The section lift coefficient is

1 NCP_

CL

=

NCP

J

_{(flC p),;}

n=l

Thrust and power coefficients dC

C "=

f1--1.

dz

T zhdz

are

where NCP

=

nlJTlber of chordwi se stations

dCT 1 Co

where

az-

=

jSaJ2BoCL(az - CL)

BO

=

chord-to-diameter ratio

CD

r.

=

friction drag + wave drag

L

+ vortex d rag

dCp 1 CD

where ëf'Z

=

iBaJ3BoCL aZ(1+az

Friction drag can be determined from 20 ai rfoil tabl es as a function of section lift coefficient, camber, thickness and relative Mach nLlTlber. Wave _, drag can be predicted fr om radiated acoustic energy. Vortex drag is the assoc i ate drag force produced by the trai 1 ing vortex system. It can be determined as

C crBBO CL B3 cr 3 ex>

ac t ua tor d

i

sc

_Va:> V +v₁

- - GD

Po

FIG. 2.1 Idealized flow model of momentum theory

1.Or---~ _{1. 0 r---:::::;:;~} ~ _~ u _u c: c: Q) _Q) .r- .r-U _U .r- .r-4- 4-4- 4-Q) _Q) .-_/ti .-/ti Q) _Q) -=:7 _-=:7 ... _... O~---~-J _p

~

P GD J

Or

=

21tnr

plan e

0 f ra ta t ; on

FIG. 2.3 Force and velocity diagram of a blade element (induced

angle not included) in blade element theory

pl ane of rotat ion

FIG. 2.4 Force and velocity diagram of a blade element (induced

angle included) in combined momentum-blade element

theory and vortex theory of Glauert

i nterference velocity

wr

w

"2"

FIG. 2.5 Flow velocity diagram on a propeller blade section in Goldstein's theory

LV ...

I

I

I

I

\

/

/

/

r

---~

z

Rigid helical vortex sheet (or helix surface of discontinuity of velocity potential) e-~=o

=

\

\

\

/

"'~~---Hel ical surface with spiral 1 ines

of constant velocity potential

e-~=c

=

FIG. 2.6 Rigid hel ical vortex sheet

e -

!f-

= 0 and hel kal surface with

=

spiral 1ines of constant velocity potential e - ~

=

C enc10sed

=

inside a constant diameter cy1inderial wake as described in

z

FIG. 2.7 Coordinate system used in compressible helicoidal surface theory

•

spanwi se panel control points at 3/4 chord of each panel laad 1 ines at 1/4 chord of each panel

FIG. 2.8 Load discretization in compressible helicoidal surface theory

3 VORTEX LATTICE METHOD

The vortex lattice method approximates a lifting surface and its wake by a discrete vortex lattice system. The propeller is considered to be bl ades of arbitrary pl anefonn, rotating with a constant angul ar velocity about a common axis in an unbounded fluid. Interaction between blades is considered. The presence of the hub is ignored. The vortex wake of the

propell er is assumed to be a hel ix of constant pitch and diameter. Thi s asslIl1ption is coincident with Goldstein ' S hel ical vortex model. In both

cases, the propeller is considered to be lightly loaded, and to have no slipstream contraction.

3.1 LIFTING LINE ANALYSIS

In lifting line analysis, the propeller blade is assumed sufficiently thin to be represented by distribution of horseshoe vortices lying in the mean canber surface of each blade. Steady flow conditions are assumed; the circulation is only a function of location along the radius. The induced flow at any radial location due to the horseshoe vortex is given by the Biot-Savart equation:

-+-

_v

_r

_f~

t

_xds

-+-V CD

=

41tRV CD 0 I_t31

(3.1 )

where

l'

Jare non-dimensi onal

t

=

vec to r from the poi nt where the induced f1 ow is sorted to the location of the vortex element ds

R

=

radi us of the propell er

By imposi ng the boundary cond it i on that the local velocity component nonnal to the mean camber surface at the control point must be zero,

circulation at each vortex filament can be determined. Once the circulation is known, the force on each element is calculated by the equation:

F.

=

p ~ . x dr . ( 2 s) . ( 3 • 2 )

1 1 1 1

where

V

i

=

rel ative inflow velocity at the midpoint of bound vortex element i

dr.

=

vortex strength per unit length of element i

1

(2s) i = length of vortex el ement i

3.1.1 BLADE GEll-1ETRY

(a) Coordinate of nodal points and control points

In lifting line analysis, the coordinate system is defined as shown in Fig.1 • lhe propeller is modelled by a single bound vortex line joining the nodal points (XNi, YNi ,ZNi) at the quarter chord of the bl ade sections and a set of control points (XCj ,YCj ,ZCj) along the three quarter chord line.

The coordinate of the control points at the 3/4 chord can be cal cul ated by

YC.

=

(YN.+YN. 1)/2 + (ë./2)cos~.

J J J+ J J

ZC.

=

(ZN.+ZN. 1)/2 + (ë./2)sin~.

J J J+ J J

XC.

=

I (

r .

2 -YC .2 )

J J J

where ~j

=

angl e of pitch of the propell er chord 1 ine

C

_j

=

mean chord of the vortex el ement r.

=

radial distance of the control points

J

-XC.

=

r.cos</l. _{where </Ij} = a n 9 1 e subscripted by the

J J J

control point and the X-Z pl ane at the axis of rotation

YC. =

r.

si n</l.

J J J

ZC j =

Z.

_J

(b) Helical vortex lines

Assume rigid wake, the helical vortex lines are determined by the following equations:

x.

= p-' cose 1 1 where

ë

=

e

+

e

p (3.3a)

Vi

=

Pi

si

na

V

Zi

=

~

3.1.2 INDUCED VELOCITIES

e

= angle of rotation

e

_p= q(P-l) , p=1,2, ••• ,k k = number of bl ades (principle blade at p=l)

p. = radius of the helical vortex lines

1

(3.3b)

R = radius of the blade (3.3c)

Induced velocity at (XCj,YCj,ZC j ) due to circulation

r

_i about vortex element i can be divided into two contributions.

(a) Bound vortex u. _{J _}

r.

₁

V -

4nRV FUB ij

<XI <XI

v· _{J _}

r·

₁

V -

~VBij CD CD w· _{J _}

r·

₁

V -

4n R V FW Bij CD CD wi th F UB ij = [ (V L _VI ) (Ir -I L ) - (Z L _TI ) (Y r -Y L) ] 1 FVB_{ij =[ (rL _fl ) (Xr} -X L)- (XL -'XI

)(lr

-ZL)] 1 FWB_ij=[ (XL _XI) (Yr-Y L )-(YL -yl) (Xr -XL)]1 where and 1 a+b b

l=ac-WI{ a+2b+c) IC]

a= (Xr-X

L )2+ (V

r

-VL )2+( Zr -ZL)2 b=(Xr-X_L) (XL-XI )+(Yr-Y

L) (YL _yl )+(Tr-IL) (ZL -II) c= (XL -Xl )2+ (Y L -VI )2+C

f

L _Zl )2 I X = XC. J I V =

YC.

J I

7.

=

Të.

J

and for multiple blade propeller: XL=p. cos(e+e )=XN. 1 P 1 Xr=Pi+l cos(e+ep )=XN_{i +1}

V

_L =Pi sin(e+ep)=YN_i Vr=Pi+l sin(e+ep)=YNi +1 (3.4b) (3.4c)

ZL=!N'i

I

_T=ZN

_i+

₁

The induced velocity at a point on a bound vortex line or the extension of the line due to the bound vortex is indetenninant, according to the above equat ions; and consi dered to be zero.

(b) Helical vortex

Induced velocity due to the ith hel ical vortex 1 ine on the jth control point can be obtained by Biot-Savart law:

+ -t +

V

r

Jco

Ixds

v:

=

4nRV

_co

0

W

where

1',s

are non-dimensi onal . and

l'

=

CX.-XC.)t + (V.-VC.)j +

(Z.-zc.)t

1 J 1 J 1 J V - - - + - - - + = (p.cose-r.COS~.)l + (p.sine-r.sin~.)J + 1 J J 1 J J (~R-w

-Z.)t

J V

S

= p.cosë1 + p.sinëj + ~t 1 1 wK-V

dS

=

[-Pisinêt + Picosëj + ~Jde

+{

(p

icosë-

r

/os~ j) (p i cosë) - (-p i si në) (p i si në -

r

j si n~ j )} ~J de

v

Z.

=

[(w~){;;;sinë-rjSin~j-(e-v /~R)P;

co se}

t

v

Z.

+(w'R){

-P

i cosë+ r jCOS<\l

r

(a-_V

7

~R)p

i si nel

j

CD +{ P~

-

r . p . cos ( ë -<\I • ) } ~ J de 1 J 1 J V 7. ( CDR) [ -p.cosë+r.cos<\l.-(a-~R-)p.sinë W 1 J J v IW 1 + CD ~

Induced velocities in x,y,z directions on control point j due to helical

vortex line i of k blades can be written as:

r·

u 1 (3.5a)

r=

4nRV FUH_ij CD CD v

₌

r·

_{1 FVH .. (3.5b)}

r

_CD 4nKV _CD lJ

r·

w 1 (3.5c)

V-=

4nRV FWH_ij CD CD where FUH ij

v

T.

k ( w CDR)[p .sinë-r.sin<\l 1 J J .-(a-

v

/R)P·coseJda W 1

=

f~

L

CD

- - - -FVH .. 1J FWH .. 1J 9=9+9 P

The induced velocity at each control points are contributed by both the helical vortex lines and the bound vortex lines: (Fig. 3.2)

u· M

r·

-v

J

=

L

4

R

V

FU .. co i=1 1t co 1J ( 3.6a) v· _{J _ \'}

Mr·

₁

V -

'~141tRV

FV ij co 1- co ( 3.6b) w· M

r·

-v

J

=

L

4

R~

FW .. co i=1 1t co 1J ( 3.6c)

where FU .. =FUH. 1 . +FUH .. +FUB ..

1J 1+,J 1J 1J

FV .. =FVH. 1 .+FVH .. +FVB ..

1J 1+,J 1J 1J

FW .. =FWH. 1 .+FWH .. +FWB ..

1J 1+,J 1J 1J

M=number of radi al bl ade el enents •

3.1.3 BOUNDARY CONDITION

The boundary condition that the velocity component normal to the chord plane is always zero is satisfied at the control points (Xë,YC,ZC). The resultant velocity at an arbitrary point on the blade is given by:

(Fig. 3.3)

(3.7) where r j

=

radial distance of the point from axis of rotation

Yj

=

angle subscripted by the point and the X-Z plane at the axis of rotation

The surface nonnalof each vortex el ement j is determined as the cross product of the two vectors from the control point j to the two adj acent nodal points j and j+l :

n

.

=

i\

x ~ J where

i\ =

~

=

(XN j +1-XC j

)i

+ (YN j +1-YC j

)!

+ (ZN_j+ 1-ZCj

)t

(X'N.-XC.)t + (YN.-YC.)j + (ZN.-ZC.)t J J J J J J

and can be written as

n.

= n .t+n .!+n .~

J XJ YJ ZJ (3.8)

where n

.=(YN'.

l-YC.)(ZN.-ZC.)-(ZN.+1-ZC.)(YN.-YC.) XJ J + J J J J J J J n . =( ZN ·+l-ZC.) (XN . -XC. )- (XN '+1-XC .)( ZN. -ZC . )

YJ J J J J J J J J

n .

= (

XN· 1 - XC . ) ( YN . - YC . ) - ( YN. 1 - YC . ) (XN . - XC . ) ZJ J+ J J J J+ J J J

Apply the boundary condition, ~.n=O

,

at the control points; we have

u;

v.

w.

R

J + J ( J

(W )[-.

-

]

(-V ) n . (-V ) n . + -V ) n .

=

-V r . s 1 ny .• n . - r . co sY . • n. - n .

(J) XJ (J) Y J (J) ZJ Q) J J XJ J J YJ ZJ

Substitute the expression for

~

,

1-,

V- ;

the boundary condition at control

(J) (J) Q)

point j can be written as:

M

r.

L 4 R

~

[( FU .. ) n . + ( FV .. ) n . + ( FW .. ) n .]

=

i =1 1t (J) lJ XJ lJ YJ lJ ZJ

(wVR)U.)[siny .• n .-cosy .• n.] - n .

ex> J J XJ J YJ ZJ

By applying the boundary condition on M control points, we have M

r·

₁

simultaneous equations. The circulation 41tRV can then be found by solving ex>

the equat ions.

3.1.4 FORCES ON THE BLADE

Force on each vortex el ement can be cal cul ated by the Kutta-Joukowski I s

1 aw:

F.

=

p

V .

x dr . ( 2 s) .

1 1 1 1 (3. 10)

Coordinate of the mid-point i can be written as

Y mi

=

ris i ny i Zmi

=

zi

where r

i

=

radial distance of mid-point axis of rotation

from

Yi

=

angle subscripted by the mid-point and X-Z plane at the axis of rotation

Consider the ith vortex elanent,

F. = p[(u.-wr.siny.)i+(v.+wr.cosy.)j+(w.+v )kJ

1 1 1 1 1 1 1 1co

x [(XNi+CXNirt+(YNi+l-YNi)!+(ZNi+l-ZN;

)~J

Ir; I = p[{(v.+wr.cosy.)(ZN·+1-ZN.)-(w.+v )(YN. l-YN.)}1

1 1 1 1 1 1 co 1+ 1

+{ (w . +V ) ( XN . + 1 - XN. ) - ( u . -wr. s; ny. )( ZN . 1 - ZN. ) } j

1 co 1 1 1 1 1 1+ 1

+ { (u ;-wr i si ny ;) (YN i + l-YN i )- ( V ;+w r iCOSy ;)( XN i + l-XN i )} tJ Ir; I

Let Ir; I = r· , ₁

XNi+1=X T XN; =X_L YN;+l=Y T YN; =Y L ZN;+l=ZT ZN; =ZL

Li ft force on el ement ; can be wr; tten as:

t

_{L = (F x) Lt+(FY)L1+(Fz)Lk}

where (F_x \=pr;[ (vi+wr; COSy; )(ZT-ZL)-(V_co +w; )(YT-YL)J (F )L=pr.[(w. +V ) (XT-XL)-(u.-wr. s;ny. )(ZT-ZL)J

y 1 1 co 1 1 1

( F z \ = pr; [ (u; -w r; si ny; ) (Y T - Y L ) - ( v; +w r; COSy i ) (X T -X L )]

Orag force on blade element; is g;ven by:

or o

Moreover ~

Orag: O=Cd

(~VfCi

)(2S)i

1 Li ft: L = Cl (ml V? Co) (2 s) 0 =p I V 0 Ir 0 (2 S ) 0 c 1 1 1 1 1 1 where C i (2s) i = area of wi ng el enent

o

Cd L

=

c,-Cd

o

=

i,{

p I~ i I r i (2s) i) 1 Cd 0

(Fx)O = G,1'r i (ui-wri s1nyi )(2s)i (3.12a) Cd 0 (F ) 0 = c"pr 0 (v 0 +w ros 1

ny

0 ) ( 2 S ) 0 ( 3. 12 b) Y 1 1 1 1 1 1 Cd (F )0 = c"pro (wo +V )(2s) 0 (3.12c)

z

1 1 leD 1

Total force on the bl ade el ement i is

(3.13) In x~y~z directions, force on an element is:

· Cd

(F ).= pr.((u.-wr. Slny. )(YT-YL)-(v.+wr.cosy. ) (XT-XL)+-C (w.+V )(2s).)

Z 1 1 1 1 1 1 1 1 1 1 CD 1

(a) Thrust coefficient:

Sin eet het h rus t i sin th ene gat i v e z - dir eet ion, T = - F • Th rus t

z

coefficient is defined as

w

where n=z:; , D=2R

For a propeller with B blades, thrust loading coefficient of element sect ion i is:

Thrust coeffic ient of the propell er is:

M C

=

L

r_

T i=1 -ri where

M

= number of vortex _{el anents}

(b) Power coefficient

Aerodynamic torque on a bl ade el ement is given as

Q.

=

Fe r. 1 . 1 1 where Fe. 1 = F cosy.-F siny. Yi 1 Xi 1

(3.14)

•

Power coefficient of the propeller is:

M

C

_P

=

_{. 1}

I

Cp.

1 = 1

(c) Radi~ force coefficient

Radial force coefficient is given as: FR.

1

CR.

=

pnLD4

1

where FR

=

F cosy.+F siny.

i x 1 y 1

Radial force loading coefficient is:

Radi al force coeffic i ent is: t~ CR =

.I

CR. , =1 , (d) Efficiency: _ TV Tl - 2nnQ

CT

=

- ( J ) 2nC Q where J-

n~

n= rotation/ sec.

3.1.5 TWO OIMENSIONAL LIFT COEFFICIENTS

W· Y-Y

(1+')( T l) V _a>

25.

_,

(3.16)

(3.17)

Fran previous calculations, lifting force on blade elenent is given by:

t

_{L =}

(Fx)Lt+(FY)lj+(FZ)L~

where (Fx\ = pr;[(vi+wricosYi)(ZT-ZL)-(Va>+wi)(YT-Yl)] (Fy)L = pr i [ (wi +Va» (XT-XL)-(ui-wri sinYi )(ZT-ZL)]

(Fz)l = pri[ (ui-wri sinYi )(YrYL)-(vi+wricosYi) (XrX L)] lift force on elenent i can be expressed as: (FIG.3.5)

Li

=lt

_L

I

=/[(Fx)t+(Fy)t+(Fz)t]

=pr i [{ (vi+wri COSy i )(ZT-ZL)- (Va> +wi ) (Y T-Y l )}2

+ {(w.+V )(XT-XL)-(u.-wr.siny. )(ZT-ZL)}2