A theoretical study of a propeller with skew and rake: An application of the asymptotic matching technique

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A THEORETICAL STUDY OF A PROPELLER WITH SKEW AND RAKE: AN APPLICATION OF THE ASYMPTOTIC MATCHING TECHNIQUE

JAAKKO PYLKKANEN

Thesis for the degree of Doctor of Technology approved after public examination and criticism in the Auditorium Ko 216 at the Helsinki University of Technology on the 19th of September, 1977 at oclock noon.

Initially the author tried to attack some non-linear propeller problems using a vortex lattice approach. In

order to reduce the computing costs the mathematical model was quite simple and crude. However, experience has indicated that it is necessary to systematically determine the most important parameters of each special case.

Asymptotic matching technique seemed to be the most promising method. The off-design problem of a propeller with skew and rake was investigated as the first application.

It

iS

my intention to continue in this line of study. I wish to express my gratitude to Professors V. Kosti-lamen and H. Rikkonen for warm and encouraging support in the course of this work. Similarly I am indebted to Associate Professor S. Laine for niany helpful suggestions and

comments.

I also thank the staff of the Ship Laboratories and the Department of Mechanical Engineering of Helsinki University of Technology for their constructive criticism.

My thanks are also to Mr. J. Beasley for correcting the English text of the manuscript. I wsh to thank Mrs. I. Halenius and Mrs. I. Lauksio for the typing and Mr. J.

Tuovi-nen for drawing the figures.

The financial support of the Academy of Finland has made the present Study possible.

AB STRACT

The equations for determining the performance of a pro-peller with skew and rake are derived using the asymptotic matching technique. The chord-to-diameter ratio is taken as the perturbation parameter. In the present approach velocities calculated from velocity potential connect the Outer and inner regions. A solution is possible only if quite stringent restrictions are made with respect to

pro-peller geometry and flow field. There are three generally feasible flow conditions or parameter domains. In addition, the hub radius in the calculations must be assumed smaller than the propeller hub radius.

At present no numerical results are available. However, the influence of propeller geometry and flow parameters on the effective angle of attack can be intuitively estimated without any computations.

CONTENTS PREFAC E ABSTRACT CONTENTS NOTATION 8 INTRODUCTION 13 PROPELLER GEOMETRY 16 GOVERNING EQUATIONS 20

VELOCITY POTENTIAL AS AN INTEGRAL 23

OUTER FLOW 28

INNER FLOW 32

COMPUTATION OF TWO-DIMENSIONAL VELOCITY 39 7.1 Objective and General Comments 39

7.2 Velocity Components due to Terms

7.3 _{Velocity Components due to} J0 Terms 52

7. _{Velocity Components due to} Terms 54

7.5 _{Velocity Components due to Terms 59} 9G 7.6 Complex Potential 61 INNER POTENTIAL 65 PERFORMANCE 72 DISCUSSION 75 CONCLUSION 78 REFERENCES 80

APPENDIX A Components of Laplace Equation 86

LIST OF FIGURES

FIG. i _{Coordinate Reference Frames for}

Stationary Blade

FIG. 2 Blade Coordinate System (x,r,8)

FIG. 3 _{Blade Coordinate System}

FIG. Blade Coordinate System (010203)

FIG. 5 Profile Geometry

FIG. 6 Schematic View of the Vortex Sheet Leaving the Trailing Edge of a Blade

3 5

NOTATION

vector potential in the expression for velocity in rotating coordinate frame

b integer

C power loading coefficient

thrust loading coefficient

c chord length

E profile shape function

Et profile thickness in (01,02) coordinates

ebi

unit base vector

F force (subscripts denote the direction of force)

F , hypergeometric function

f1 radius of lines of constant circulation in shed

vortex sheet

f2 axial position of lines of constant circulation G function defined by Equation (5.8)

function defined by Equation (8.11)

Gbit abbreviations for mathematical expressions

hsubSCript abbreviations for mathematical expressions

'n Bessel function -t -p

-i,

j, k unit base vectors in the Cartesian reference frame Bessel function

Bessel function

kkm

integers

L lift

leading edge

Î vector along an arc

i integer

m integer

vector normal to surface, pointing into the flow field

unit vector normal to surface, pointing into the flow field

n integer

P(a3) pitch of blade-reference surface

P(a3)

pitch of shed vortex sheet at lifting-line

p pressure

pressure on the + side of the vortex sheet (defined as for _Sv+)

pressure on the - side of the vortex sheet

Q torque

Legendre function velocity vector

velocity vector far upstream

velocity defined in Equation (9.)

R propeller radius

Rh hub radius in the computations

R0 abbreviation for a mathematical expression defined by Equation (7.12)

r _{radial coordinate in cylindrical reference frame} starting point of the trailing vortex line

(x,y,z), position vector of field point

SB SB+ US5_, blade surface

SB; blade surface (+ for pressure side, - for Suction side)

S+

the continuation of _SB+ onto the vortex sheet the continuation of _{SB_} Onto the vortex sheet position vector on blade surface

T _thrust

t time

t integration variable

tangent vector to surface, subscript denotes direction

U magnitude of velocity component far upstream along propeller axis

UA axial velocity induced by propeller

UT circumferential velocity induced by propefler

u variable

Uk element of a series

(v1,v2), 2-dimensional velocity, additional subscripts denote the origin of each term w integration variable

x Cartesian component of position vector along

axis of rotation, pointing downstream

y Cartesian component of position vector

Z number of blades

ZT(03) total rake

z Cartesian component of position vector pointing

along upward vertical

z _or+ii, complex variable

mE effective angle of attack mG geometric angle of attack

r tan , pitch of incoming flow at

lifting-line

hydrodynamic pitch angle

(r) pitch angle of blade-reference surface, measured on a cylinder of radius r

pitch angle of profile section, measured at constant 03

r Gamma function

r circulation enclosed by contour

'r C C o -t e

rB - , jump in potential across blade surface:

local circualtion

defined by Equation (7.53) defined by Equation (8.11) integration variable

a set of linearly independent gage functions defined by Equation (Eq. 5.1)

chord-to-diameter ratio, small parameter in singular perturbation problem

defined by Equation (9.10)

=

f1r+12

position vector of lines of constant circulation in shed vortex sheet

first order approximation to

pe1', 2-dimensional coordinate for conformal mapping

angular coordinate in cylindrical reference frame angular coordinate of propeller blade-reference line (b = _{G,. ..,Z-1)}

oS103) skew angle

A functional defined by Equation (9.10)

A Lagrangian multiplier

arbitrary function defined by Equation (9.10)

p radial coordinate defined by Equation (8.1)

u angular coordinate defined by Equation (8.1)

2-dimensional velocity vector in stretched variables

profile coordinates, subscript denotes direction

p fluid density

S

+ Sv,

notation

ai coordinates based on blade-reference line

=

of/C

, stretched profile coordinate (i 1 2)

"s mathematical abbreviation defined by Equation (7.10)

t dummy variable

r

scalar potential in the expression for velocity, given in inner variables

scalar potential in the expression for velocity, given in outer variables

scalar potential defined by Equation (6.8)

x complex potential

{c +

stream function defined by the complex potential x

r

-wi, rotational velocity of propeller. w in radians per unit time, w> O for right hand rotation

asymptotically equal to (in some given limit) approximately equal to (in any useful sense) /l4 p. 21/

O one writes I4' p. 2'4/ f() r O(â(E)) as

- f(E)

INTRODUCTION

Singular perturbation techniques have _{been used} success-fully in a diverse set of problems in fluid dynamics. Here, a solution is derived to the performance _{evaluation problem} of a propeller with warp and rake _{using asymptotic methods.} The design problem is also discussed. _{The rectilinear and} angular velocities are coaxial and _{constant (i.e. the flow}

is stationary).

The general mathematical and physical foundations of singular perturbation techniques have been presented by Cole /6/, Kaplun /21/, Lighthill /31/, _{Ogilvie /36/, O'Nalley} /37/ and van Dyke /441. _{The references also include}

a number of applications in the field of fluid dynamics. Wings of special ohapes have been investigated by Rotta /39/, Thurber /42/ and Wang /52/. _{The helicopter rotor problem} has been discussed by Johansson /17/, _{/18/, /19/, /20/ and} van Holten /46/, /47/. _{There are two marine propulsion} applications; Brockett /5/ has studied the ordinary screw propeller and James /15/ the cycloidal propeller. The

reference list contains a number of standard propeller texts. They are needed for comparing the results . _{Also some}

compu-tational procedures (mainly integrations containing singu-larities) can be taken from them.

In short, the present approach is _{an extension of earlier} research done by Brockett /5/ and Thurber /42/. The geo-metric notation and general _{formulation has been adopted,} as far as possible, from the first of these two references. The matching technique of Thurber _{has been modified to Suit} the propeller case.

Because of the conceptual similarities _{to Refs. /5/ and} /42/, the basic equations are presented in a very terse form. Only when there are differences resulting from dissimilar geometry or flow conditions wiJi the expositior _{be more} detailed.

Usually matching must proceed step by step, as explained by van Dyke / p. 93/ and Ogilvie /36 P. 708/. The basic

solution dominates the inner solution, which in turn exerts a secondary influence on the outer expansion and so on.

Sometimes the standard matching order can be short-circui-ted. This can be done, for example, when the nonuniformity is so weak that it does not affect the outer flow.

In the present paper, assumptions will be made in such a way that two outer terms can be computed before one

resorts to matching with the inner expansion. Most of these assumptions are not known and will not be stated beforehand.

They will be made gradually in order to allow the calcula-tions to proceed further. This determination of the domain of validity of the solution forms a major part of the study.

The chord-to-diameter ratio E is taken as the pertur-bation parameter. The parameter is used to magnify the region determined by the intersection of a neighborhood of the airfoil and a plane onto the same plane. The flow near the airfoil with plane sections is matched with the flow at infinity in the new inner coordinate system. The airfoil plane is mapped conformally onto the exterior of a circle. The boundary condition is that the flow is tangent to the airfoil and the Kutta condition is satisfied /2 P. 733/.

By means of an integral representation for the velocity potential, the velocity and the distribution of circulation can be expanded as a function of the chord-diameter ratio. The distribution of circulation can be written as a function of one variable only, which remains fixed under the

stretching. From the form of the integral representation of the velocity potential in terms of the circulation dis-tribution, it follows that the expansion of the velocity potential in the new coordinate system has the same form of expansion in terms of the parameter /2 p. 734/. In this connection it must be assumed that the velocities induced by any blade in the vicinity of any other blade are so

small that their influence upon the inner boundary value problem is negligible ¡'47 p 93/, /18 p. 35/. That is, within a distance O(E) (the distance divided by E is of finite magnitude) from the blade in outer coordinates, there are no perturbations except those generated by the index blade. Correspondingly, it is assumed that the blade number is not very large and the hub radius is not zero. The leading terms in the perturbation velocity potential,

in the original and stretched coordinates, are of orders E, E2inE and E2.

The effect of the transformation of coordinates on the governing equations of the flow is investigated. It is found that the flow in the inner coordinate system is a

collection of two-dimensional problems. The solutions of these two-dimensional problems satisfy Laplace equations for terms of order E and E21nE. _{There are two types of} terms of order E2; one set satisfies the Laplace equation while the other does not. They both satisfy the boundary condition on the airfoil ¡'42 p. 73'4/.

The conformal mapping of the exterior of the airfoil slit onto the exterior of a unit circle is used in conjuction with the Kutta condition to determine the distribution of

In this chapter the propeller geometry is defined, and expressions needed later on are derived. The formulae are very similar to those given in Ref. /5 p. 10/.

Both the Cartesian reference frame (x,y,z) and the cylindrical polar coordinates (x,r,O) are attached to the moving propeller (Fig. 1). Radial distances are measured from the axis of rotation in a (y,z) plane, and angle O

is measured from the z axis in the clockwise direction looking along the positive x axis. The unit base vectors are and

(T,,;0)

respectively, and the con-nection between the unit vectors is

(2.1)

5r -sin O + cosO e0 -cos O - sine k

Usually the blade section offsets are measured from a refe-rence line lying on a cylinder of constant radius. This coordinate system or point of view is adopted in Chapter 3. The second coordinate system is that geometry is defined

for sections cut by a plane at a certain radius. Sets of unit orthogonal reference vectors are defined for each of

these two viewpoints.

For the sections defined on the cylinder, the blade-section reference line on the surface of the cylinder is called the geometric pitch line. It makes an angle

with the (y,z) plane (Fig. 2). The blade-reference line is defined as the line through a datum point at each radius. In Chapter the trailing edge will be the datum point, in Chapter 5 the datum point will be moved up to the leading edge. The blade-reference line is given by x _ZT(r) and o _{- Ob+OS(r); 0b =} 2b/Z (b= 0,1,...,Z-1), Z being the blade number. _ZT(r) is total rake and es(r) is skew angle. On the cylinder surface r constant, a coordinate

system

i

,,r)

is constructed (Fig. 2) with on the

surface of the cylinder and measured from the blade-refe-rence line along a constant pitch line. Positive

values point in the downward direction (i.e. the scalar product of the unit vector in the x-direction and E2-vector is positive). The variable is normal to the C2 axis and points in the upward direction. But the coordinate system

(C1,2,r) is, in fact, not needed when one is interested

in performance only. However, some geometrical features are easier to visualize using this system.

In the second system of blade specification, the sections are cut out by a plane at right angles to the radial compo-nent of the blade reference line (Figs. 3 and 4), the coor-dinates denoted by

_(01,02,03).

The unit reference vector in the ₀₂ direction, e2, forms an angle e0 with the (y,z) plane.

The Cartesian components expressed in the new system are

(2.2) x _{ZT(03) -} + o2sin e0

y - _o3sin(eS(03)+eb)_[clsin e+o2cos $lcos(es(03)+Ob)

z _{a3cos(eS(03)+eb)_{ol sin ß+r2cos e0}

Jsin(OS(03)+Ob)

Corresponding expressions for the cylindrical polar coor-dinates are ) - o cos ß + oSin B (2.3) x

r

_ZT(03 1 o ¿ o 2 2 r

r 03

+ [o sir B + a2cos a e

r

_{+ tan1[(1sin e}

_{+ a2cos 8 )/03]} + 0b where _{80 r 80(03)} and _e r

or

-o100s

+ cr2sin 0 _{X_ZT(0g)}

G

osin 0

+ o cos : r sin(0-0s(03)-Ob)

o 2 o

03 r r cos(0_Bs(o3)_Ob)

denotes the reference blade and it is set ₀₀ 0 for convenience. Later on Eqs. (2.2) and (2.3) are substituted in the position vector defined as

-* + 1

-r xi + yj + zk

Unit base vectors in the ₀₁ and

_°2

directions are +

(2.'-t) e1 = -cos 00+(-sin R0cos 0s)(_sin Osin O

+ e2

2

sin ß+(-cos Ocos 0)+(_cos Osin 0s)

e3 -sin O + cos

The third unit vector is identical to the unit cylindrical polar vector in the r direction.

The blade sections (Fig. 5) are defined as

.5) o = E(a2,a3) for _Rh 03 < R where R is the propeller radius

and Rh is the hub radius.

A point on the surface of the blade is given as a function of two variables

°2 and

03)

by

+ +

(2.6) 5 r i(Z - E cos O + o2sin B)

T o

+ _{(-a3sin O} - [E sin + o2cos 80]cos °s

and two tangent vectors to I-he blade surface by

-4.

(2.7)

r

t2 r

Correspondingly the vector normal to the surface is deter-mined by

(2.8) = X t2

where the sign is positive on the suction or back side and negative on the pressure or face side. In component form the vector product in Eq. (2.8) is

30 (2.9) = (i - (E sin + o2cos o 303 30 3E

+ e - (

2 -i +--- {E303

_{sin O}

0 + o2cos 8}) - 13E 3ZT 30s

+ e I

f+sin

+ E

_{+ a - cos ß}}

02

0303

3303

38 8o 3ZT 3E o + 02 03

_Sifl

8 + cosO

-3. GOVERNING EQUATIONS

The basic flow is exactly the same as in Ref. /5 p. 22/. A Z-bladed propeller of finite radius and finite hub radius

is rotating with constant angular velocity w and advancing at a uniform rate U into an unbounded, inviscid,

incompres-sible fluid.

The flow is steady with respect to a coordinate system rotating with the blade. In the moving coordinate system,

the fundamental equations of the flow are /25 p. 53/

(3.1)

(.V)+1Vp r -2

x - x( x

and

(3.2) 3

where is the velocity vector

p is the pressure

r -w is the angular velocity of the pro-peller

is the position vector

p is the density of the fluid.

The flow field in the inertial reference frame is irrota-tional everywhere except on the boundaries. In the rotating coordinates the vorticity,

Vx,

is given by /5 p. 23/

(3.3) 7x r

Let _SB- represent the pressure side of the blades,

S+ the suction side, the continuation of SB+ onto

the vortex sheet and S_ the other side of the vortex

sheet. _TE(r) denotes the trailing edge. The boundary conditions are

(3.5) r O for on the blade and vortex sheet and

(3.6)

II< +

for E TE,

where is the unit normal, pointing from the bounding surface into the fluid.

The velocity is found to be

(37)

r -j- Vx

where is vector potential and is scalar potential.

On basis of Eq. (3.2), after some manipulation, the following is easily derived

(3.8)

0

r

_{r(ax_8U)r}

The unknown quantity is now 4 and it must satisfy the boundary conditions

+

(3.9)

V'' O as

x -

-,

r

(3.10)

r

on the blade and the vortex sheet,

(3.11)

Vc < +

for

_{E TE}

and the pressure is continuous across the vortex sheet, that is

(3.12)

_{p+ -}

r

Q

Integrating Eq. (3.1) one obtains

p

22

(3.13) p

r

{constant

+ a r

- qq)

vortex sheet the following can be shown to be valid /5 p.

25/, ¡'42 p. 738, Eq. (12)!

(3.1'4)

r

(2

_o + r +

- (2

o

+V)V-

rE r Q

The above formulae will be needed to establish the equations for the line of constant circulation in Chapter

(Eq. ('4.8)).

Thurber /'42 p. 735/ considers in analogy with two-dimensional flow that imposing the Kutta condition makes the flow unique. This has also been verified by Brockett /5 p. 38/, where he discusses this problem in greater depth.

4. VELOCITY POTENTIAL AS AN INTEGRAL

Using Green's identities and Eq. (3.10) the validity of the following can be shown /5 p. 28/, /42 p. 738/:

z-1

(4.1)

()

r

b0

{(V

° } dS s -* +

.

SB+USB_ jr-sj jr-si

z-1

+ ..L

_j

_{{,+ -}

V

:-.

dS

b0

S

r-s j

where is the unit normal to the positive side of the vortex sheet

is an arbitrary point in the fluid is a point on the boundary.

The nomenclature and basic equations of this chapter are very similar to those in Ref. /5/ and also in Ref. /42 Ch. 4/.

In order to neglect the influence of the blade thick-ness in a formally correct way one must set

-3

(4.2)

ET r

where _ET is the thickness of the airfoil, and

+ ) r O(E3)

for j--j

r O(E5)

That is, the blade must be smooth and is not allowed to contain any sharp kinks.

Correspondingly Eq. (4.1) simplifies to

z-1

1

j }n V_s

_-+

dS+

O(E3).

(4.3)

(r) r

b0 S8US

¡r-s j

The circulation is defined as /5 p. 29/, /25 p. 15/

where dT is the vector length along the closed curve which completely encloses the shed vorticity.

The circulation for a curve, which lies entirely within the irrotational flow except at that point at which it goes through the shed vortex sheet, is given by

V

In the rest of this chapter the continuous vortex sheet consisting of the lines of constant circulation which leave the trailing edge of the blade at the point T5(r) is consi-dered. From a computtion along the curve C2 in Fig. 6 the circulation about the blade at TE(r) is found to be

(14.6) f(TE(r)) r (TE(r))

-The curves 03 and C14 are lying along a line of constant

circulation and after some argument one finds

-3.

('4.7)

Nr

SV) r(TE(rfl

-Conditions are sought that describe the position vector of a line of constant circulation which leaves the trailing edge from the point E TE(r). This point on the blade is denoted by . The position vector of a point along the

line of the constant circulation is given by

(14.8)

Z =

_Z(,e,ob

r f100b)r(0') + f2,8,8b)

where is the parameter describing the starting position

- 2irb

of the line, O is the independent variable,

_{0b = -- is}

the parameter describing the individual vortex sheets. The functions f1(,O,Ob) and

_f2,8,0b)

are the radial and the axial position of the line, respectively. If f1

one obtains

(4.11) VF r Q

After some manipulation it can be concluded that the average velocity is parallel to the curves of constant circulation, and the equation for can be constructed

from two of the three scalar equations which make _up

(4.12i) (2 +

Vflx

O

where t E

+

In the above vector (the vector along the lines In the above formula can be replaced by (the

new coordinates are

defined by Eq. (2.3)).

When referring

to the position vector of constant circulation in the trailing vortex, the following notation is used

(4.9) - &3 - &

Now _{is the parameter describing the starting position} of the line.

The two independent variables (,e)

describe the

trailing vortex sheet /5 pp. 31...32/, /42 p. 7391. The vector normal _{to the trailing vortex sheet is formed} and substituted in the boundary condition (Eq. (3.10)). Observing F _{and the definitions}

V

of constant circulation) is -af2 ae (4.l2ii)

r ---

e + f e + - i r

le

On the shed vortex sheet the following is valid:

+

at.

aq'-(4.l2iii) (2q + Vt). - e + (2af1 +

o

rç

r r

+ (2U + -- i i ax

In component form Eq. (.12 iii) amounts to

i at (4.13) f1(2U + !) -(2f1

+ r_

r Q at (2u + -) r (L .1L ₎

---

ax and (4.15) (2wf + --:-at) at -- f1

-_ r

Q

Formulae ('+.13) and (4.i) make up coupled nonlinear inte-grodifferential equations for the radial and axial position of the lines of constant circulation:

and

i at

af2 U +

(L1ß)

i at

a +

The integral over the shed vortex sheet (4.3) can be simplified using the coordinates

8,e

that describe the vortex sheet.

The vector surface element dS is given by

i at

(4.17) - .

(4.18) dS r X

ddr

= X !i

_ddt

1 2

where T is a dummy variable for O.

Consequently Eq. (4.1) transforms to

z-1 z-1 1

--

_r-s (4.19)

()

_r-

_f

(s)n dS + b,V brû _{SB -s!} brO + + Z-1 q fl o

bO1

SB r5

dS where SB z u SB_ and R dt

f

F()d

f

Rh TTE(o) 8 i-ti3 and (4.3) to z-1

_-+1-

z-1 1 (4.20) i

_f

{+

}n V _dS+ brU 5B brO b,V

5. OUTER FLOW

The position vector of the curves of Constant circula-tion in the shed vortex sheet is expanded as a perturbacircula-tion series /5 p. 55/,

/42

p. 740/

(5.1) r

(0)

_{+ +}

_{22,0) +...}

where in an ordered set of linearly independent gage functions.

From Eqs. (4.16) and (4.17) the zero-order solution is

(5.2) ₊

_{[(e_eb_eS())}

_{+ ZT()]].}

and correspondingly the normal to the vortex sheet is given by +

,

(5.3)

o- U (

-3ZT(8)

u - (ce0

+ - Ùx

\e + 3e w r L o W ₃ U

30s()

_{e +e}

u

-+O(5 =

-ai +

-

w r w + 0(51

It can be chown /42 p. 740/ but is also evident from the general lifting line considerations that

(E)

is of order . For estimating errors one needs to observe the

following:

+ -s

U.

o _-

Je +e

w r w O

r O(E),

is on the blade surface F(8) r O(E).

The last two equations are based on the fact that for E r 3 _{the blade degenerates into a curve and the flow into}

a constant flow. Substituting the right hand sides of the Eq. (5.4) into Eq. (4.19), the error in which results

from replacing by _{is found to be}

_o(E2).

_{In the} following analysis terms of order E2 (angle of attack)2 and E (angle of attack) _{will be neglected in comparison} with terms of order 1. One considers

mE

0(1) /36 p. 675/.

A Taylor expansion of the blade shape defined on a cylinder gives /5 p. 76/

o o

(5.5i) o

- 0LE + O(E2) T

The subscript denotes the position of the datum point, hence

o o o

(5.Sii) ° _°TE °LE

In order to be able to extend the curves of constant circu-lation and the approximation _{up to the leading edge} of the blade one must assume

-+ -r

q n r 0(c)_o for any point on the blade.

Accounting also for Eqs. (4.2) the second expression of Eq. (4.19) is now transformed into

R X . d 1 ¡

r()d

f

o Bt o b,V R

TLE()

ao

Using the fact that the width of the airfoil is of order E _{and Eq. (5.6), the error involved in substituting}

R (5.81)

s =

f r()G(,d

Rh where Z-1 _{_hD}

G() r

(&)+e5 {/[x_ZT_(T_Qç_6b)J2+r242_2cos(_T+O)I brO e (5.6) dT

ZT

ZT(), es =

es(s)

bD r

-

o(x_ZT_(T_0s_eb)) -

2[Z-+ r[Z' - 0]'os(-r+O) + r sin(--r+0)

T u S w

r Eq. (.20) can be shown to be negligible.

After a change of variables

(5.8i1) y

r

hr r /[x-zT-

+ r2

-

2r& cos(0_y_OS(&)_eb) one obtains z-1

(5.8111)

= Z brû +8)

f

b dy. J h

_r

and performing some manipulation this becomes

Z-i

3(1/h

(5.9)

G(,) r

r +

[i

Uw1

3(lIhr)

U By [o w

Uj

0 brU B 3(1/h ) 3(1/h O W

[Zt_U

30

r

ow Zt[ZT_

r

+

uii

T T + (1 )(Z' -3(1 /hr) o }i1 Z-1 _I

_{OC OW}

3(1/h )

3(1/h

J

t°

u

0k])

_CS

r o

brU 0

3e o

e]2'\

3(1/h

r}

U) U T C 1 38 dy o

rG +G +G

y e

The last line is a notation to make some expressions a bit shorter.

From computations in stretched variables (Chapter 8) one obtains

(5.10)

r(E,)

F(&) + 2log

r2()

+ 2F3(&)

When this is Substituted in Eq. (5.Bi) the outer expansion for is obtained as

(5.11) '' = i() + E2iog

where the functions and t3 are independent of E

One could expect this type of formula on the basis of earlier investigations /2/, /39/.

6. INNER FLOW

The stretched coordinate system is defined by

(6.1) r

02 02

03

r

03

The basic relationships from Chapter 2 with respect to the reference blade are

(6.2) -1cos B0 + 0 (x.'ZT(03)) a1sin B0 + 2c0s B0

1

r sin(OO(o3))

03 r cOS(O_Os(03)) 2 2 -2-- - 2 2 -2 r

= 03 + S

[o1sln ß + o2cos 5]

r 03

+ Q(c )

These formulae are frequently used in this and the following chapter.

Next, this coordinate transformation is applied to the Laplace equation /5 p. 57/, /42 p. 743/

(6.3) r . 1

+ --

o

r

Br r BO 3x

The resulting expression is extremely lengthy, its basic components are given in Appendix A. The Laplace equation now transforms to (6.4) v2

{B2[20

_o+sin2B +(-ZcosB+a3Osin B0)2]_o 2 + _T[s1n B +cos2ß +(-Z'sjnO-o30'cos B )2] a o T S a + B 2

+ 2(Zj. cos 30-o3Osin 80)(-Zsin 80-o38cos

+

o()}

o

If one wants the lowest order term to be a two-dijreiisionai harmonic function in variables

,

and 02, the following condition must be satisfied

(6.5) _{[(Z(o3fl2-(o36(a3fl2J sin2+Z,(3)a1O(a.,)cos280} (6.6) This leads to (6.7) 2 f V r

-(6.8) E -2 ln E E2

oE}-1-

r Q sin 28 = O(E) 0 3Z(o3) - O(s) 303

If Fqs. (6.18) were not taken into account _{at this stage, an}

additional, but temporary, change of variablesuld_{be necessary.}

Next, the coordinate transformation is applied to the expansion of the velocity potential. _{From (5.9) and (5.10)} it follows I sin

r

O(E) BOS(03 ) - O(E) , 303

i.e. there are three possible ways of producing the desired result.

Observing these constraints and the Eqs. (6.18) the Laplace equation is

3E

and

3E 303

where now is expanded in the stretched coordinates. On basis of (6.7) and are seen to be two-dimensional, harmonic functions in the variables and

G2 Consequently, P1 and can be determined using conformal mapping techniques. It will be shown that P3

can be obtained similarly, even though the corresponding potential term does not satisfy the Laplace equation /42 p. 744/

The boundary condition (3.10)

(6.9) r . _{on the blade and the vortex sheet}

is to be expressed in the inner variables. Observing the

not at ions

(6.10) r

and

r

riE

and the fact that /5 p. 59/

(6.11) E(o2,o3;E) r EE(&2,&3) r E

r O(i) 02 rO2

r

O(E)

2re

the basic components, neglecting terms of order E2 of Eq. (6.9) are

-t

(6.12) q

r

+ w/o+E2(E1sin

+a2cos 6)2+

o

3E

r

{-cos 9 (i-h )-sin (i-h1)}

+ {E sin 3 -- _{3+2cos 30)(1_h1) + h3} + (-E)ccs 3

(1sjn 3+2cos

2 (1-h1)} +

_{0sin 3(1-h)-cos}

_2(l-h1) - - i -e (o sin 3 fo COS 3 )h } 1

02

03

= _{(-cas 3) + sin} Boi - h h + 3 - ai + C + Sin [ (

-h8

+ h02 h cos30) °2

1-Eho

03

1-+ 303

1-hO

e + i

( -h

{z

cos 3- 039sin 3}+ 03

rì + 3 1 1

-h+

_{Zsin 3-a3Ocos 3}+03005}

)]

where h1

= E(

sin + &2cos

- 3E o 3E

h3 =

- 3

_.__)+o

ecos30

+ (3E

-_

Zsin S +Zcos 3-a3esin

Z'cos 3_T 33

oOsin 3

0+

0 h01 2 303

Zsin S

33 o3Ocos S h =-01 ₃₀₃ -C C G1sin 3 +

/U2+(wo3)2Sifl(ß0_3)(1_h1)_ /+(o3)2 3E _{cos(ß-3)(1-h1)}

- _{w(a1sin ß} + _{2cos 30)h3} ,

(-u--As in the lifting-line approach, is considered as a parameter only and consequently one must set

(6.131) a (h + o(e)) O(E2)

303

1-EhO

which leads to

3E 3E

(6.l3ij) .- 038cos 6. + Z'sin 6 +Z'cos B -38sin 6

T o T o

O(E)

That is, the radial velocity component in the boundary condition is ignored in a formally correct way.

There are several possible ways of achieving this:

(6.1-t) zj. r O(E)

(i)

= O(e) (i i) r O(E)

Z.

r O(E) sin B = O(E) r _O(E) O(E) Z,1,cos O(E)

The problem defined by the Eqs. (6.1i) can he tackled using the technique given ir Ref. /5/. Observicg the constraint

(6.lttii) (the case (6.lttiii) can be treated similarly) the expression for the boundary condition simplifies drastically to

(5.15) vr (-cos

(i-h )-sin5 --(1-h ))[--(_cos B )+-- sin B

J 1 ₀₃₀₂ 1 U sinL30+0 cosß +

h+

h

_{{E smB0} 1 03 2 °(1-h1) 302 02j

1 -(6.16) 301 sin B cos 8 1 G 2 --- (1_h1)} 03

+ {(sifl

i JZ'cos 8

-G Osin

1G31T

G 3

°

1_EhOs

+ h+

i{z

8 -a O'cos

_{8 })]}

-

1 -Ehe T G

3 S

G 3E

{sin

8(1-h1 )-cos B °2 (1_h1)} D 2

(1-h

2 3E cos 8 o 1 302 ¿ (1-h1) B DE _{(1-h )--- -}

_{sin B0cos}

a (7

+ sin

B i 3J

+ ----

sin28 (i-h ) D 2 -

c-- COS

B G

i

DE 1-h )-!_ +

_{Sin B0coS B(1h}

O G

-

sin 8

cos

8 ( 1

1o38cos 8O)(

_BG2c05BO

3E 03 3o2(l

hi)

3E +

h8cos 80(-cos

8)

(i-h1)

3 ) +

O(E)

-

(1-h1) - -

_(1-h1 302

Setting O(E2) _{the boundary condition}

3E 3

_{- +/u2+32' 3E}

L

is now in agreement with Ref. /5 p. 63/. For the rest of this report it is assumed that r ß +

Q(2)

(6 .18a) (6.61) & (6.lLIii)

This will be denoted as Case (a).

(6.611) &(6.1'11i)

[

Z = O(i)

sin _O(E)

This can be seen to be a subset of Case (a).

(66iii) &(6.1Lii) (6.14i) ( _{Trivial flow}

condition discussed in Reference /5/.)

(6.18b) (6.6i) &(6.l'tiii) Z1, r O(v)

O(E)

COS r

O(/E)

This will be denoted as Case (b). (6.6ii) & (6.1iii) r (6.14i) (6.6iii)& (6.1Ljjj)r (5.14i)

z'

_T

= O(E)

e'

r O(vT) s

sin r

Also, because of Eq. (5.6), the condition that

(-)2

can be neglected must be satisfied in all cases. 2

discussed. In this special case two-dimensional flow is given as

(6.17)

(ai'

+

/u2+03i 3\

where

3E

tana, r--

5

G is the angle of attack

* _{is the two-dimensional velocity vector.}

(7.2)

7. COMPUTATION OF TWO-DIMENSIONAL VELOCITIES

7.1 Objective and General Comments

It would be very difficult to expand the potential function for the lifting-line /5 p. 109/, /'42 p. 7L7/. Instead the expressions of velocity are expanded and on basis of these velocities the two-dimensional potential for the inner flow is constructed. The velocities are computed from. Eqs. (5.8) and (5.9). Cases (a) and (b) of Eq. (6.18) are investigated.

The velocities are needed near the lifting-line in the directions °1'°2' and they are denoted by (v1,v2). In this region, neglecting terms of order

_E2,

one has on basis of Eq. (6.2)

(7.1) r 03 +

O(E2)

03

-Due to the last notation one avoids writing an additional subscript.

For computing singular terms a new variable of integ-ration is introduced:

lo

W

-

(o-o) E o -do = c dw

In the neighborhood of a singular point one needs the

e xpans ions

(7.3i) o +

r()

r() + Ew r'()

=

r'(o)

+ Ew r"()

o -2 1 1 -

o

-1_

-

-sln(0_Os(a)) B0 + _{G2CQS BET)} -

w0()

B0+c2cos

B)]

- Ew0()

X _ZT(o) E(-&1cos B0)-EwZ+(o)

The corresponding contributions to the velocity potential and velocities are called here singular terms.

For ordinary points near the vortex line one has

(7.Gii) O = tan1 [E-(c1sn B + o2cos )1

_j

+

x

E(-cl1cos

B +2sin B)

₊ ZT(G)

This is needed when discussing velocities induced by other than index blade.

The first term of Eq. (5.9) can be integrated imme-diately with respect to y. Techniques developed by Morgan and Wrench /35 p. 30g, and also used by Brockett /5 p. 104/ will be adopted here to evaluate the last two terms. One has

-Ix-(&) yjt

(tJ2+r2_2&rcos

(7.4)

_f

e _{[O_Y_Os(&)_ObJ')dt}

rO

The following notations are used

(7.51)

k 1 for rs r 0 rs km 2 for m r 1,2,3,... h0 = -[Z' - eu

.+

(1

. & j ]2\ T w S o w U

U[ T

0j

G

Due to the necessary assumptions (they have been collected -ogether in Chapter ii and are given there as Eqs. (11.3)

(7.51i) h

-[Z' -

O'J + (1 . + ?. _.

G T w S o to w

U)O

o o

The order of integration can be _{switched if one assumes} uniform convergence for the integral. _{This is an additional} constraint on _ZT and _{e. Assuming (x-ZT()) > O the} formulae for velocities contain the expression

z-1 _z-1 (7.6) G ₊ r

f

hG di r hG

bO

dy brû O Z-1 r h5

f

_f

dy

_f

dte

_f

_{k J (-tr)J (t)cos[m(6-'y-6S()-@b)J}

mm

m brOO O

_mu

Z-1 r hG

_f

dt

_f

k J (tr)J (te)

_f

_¡

e cos[m(O-y-OS-Ob) Idi

mm

m O rn=O brO O

Z-1 _[x_z)Jt0

= h5

_f

dt

f

k J ()J (te)

_f

le _{¡ cos[m(O_y_6S_b)]ew} di O rO m brQ _O uy [x_ZT)t + e

_f

cos(m(6_y_OS_Ob))edy}

-(x_ZT)w/U} After a change of variables

(7.7i) t u

Û)

and denoting

(77jj) n

when n _{is an integer Eq. (7.6) becomes}

u) u O w (7.8) GfG8 =h5

_f

¡kJ(U)JuGu)u

n1 + 2Zucos(nZ(x-) )Jcos[nZ(6O3)] + [Z2 u

+ hG

f J(

O w z i{_e_

_ZT)

+ 2}du

ur )J (uo )

U o U U u

For (x-ZT()) < 0 the expression is simpler:

W O U) )WZ +(x_(&))! u (7.9) G _{+ Ge r hG}

_f

J r u)J (a

u U e

du o U o u

ow

wZ i + hG

f

kJz(r u)J(o

u) e nZl O u+(nZY {u cos(nZ(O-O5())) + nZ sin(nz(e-es()))}du

Equation (7.9) will form the basis of velocity computa-tion. Some of the resulting formulae will be lengthy. In

order to avoid repeating them one adopts abbreviations with appropriate subscripts for sorne of the mathematical

expres-s ionexpres-s.

The following equations and notations are needed frequently in Chapters 7.2 to 7.5:

U wo i i

(7.10) -

+

r

tan

tan sin cos

-3 -02 02 0201 2 2 2 -2 2 + _2 2 2 (01+02) 01 +02

°l2

_2. 0201 01 _01 2 2 2 - 2 2 + 2 -2 2

°i°2

01+02 (01f02) r + _[oO(a)} + on page 61, on page '-f8,

= oO(o)[1sin

+ °2° 8]+Z(a)1_1cos60+asin60] on pages 7 and 48.

These velocity components, resulting from the G1 terms of Eq. (5.9), correspond to Eq. (85) of Ref. /142 p. 7521. In the present case they must be understood in the following way:

(7.11)

_e3xV(hOb)

where

5r =

°2

_'

°i

=

The variables and are introduced to facilitate the adoption of complex variables in Ch. 7.6. h0 and hb are defined in the next equation (7.12).

That is, vector 53 _{represents the top of a}

infini-tesirnally narrow horse-shoe vortex. The general formulae transforming a dipole sheet into a vorticity field are given in Ref. /3'4 p. 572!.

The velocity potential

R Z-i _{(1/h )} (7.12) 1o,b

f dr(&)

_{¡{Z+(Z_}

U e)} 1

r

(Ra)/ U

ZrZTwS

+E[r(a)+or'(o)J} 1_ . dw R Z + _dr(&)

10

_{0)} = +} 1 b=1 where 0b 2mb/Z

i

1 1

°

/[x_ZT(&)]2+2+r2_2r8 cos(e_8s())'

E D

_i

(1 )

o,b

₊

i

. dOE

_r

(hh)

/Expre es ion

where

Expression r %2 + r2+(ZT() -

_ZT(G))

_-

_2(ZT()

- z,(o)) (-B1cos B0 + a2sin B0) + E2(-B1cos B0 + B2S1n B0)2

0

-i E

-

2oo[cos @beos{tan (o1sin B0 + 02(:os B0)]

- (Os)_Os(o))}

+ sin

&sinftan [C(BB+

_{B2cos ß0)](es()es(o))}1}

is

differentiated.

Observing

(7.13)

r

_{- -}

i

Sin B [a1sin B0 + 02CO5 B0] sin B0 B0 + O2COS B0] a sin B0 G ax _r -COS B0

ax

_sin R

r

_{B + nos B} o 2 o 2

+ Z()(-a cos B

_i

B sin ß )] + w

1[oe(o))2

o 2 o o

+ [Z4(o)]2)

2 o

hb

r

_a

[1sin

+ &2C0S

ß]

cos

a

and

r

-E2w _a [ sin + cOS ß _I +

o()

i a z a One obtains (7.15) - 3

(i \

-3

(aw[-Zsin

-aOcos

+

w2(-)ecos ß0}

where ( denotes h and

(i'_

-1 (7.16)

'F)

3TT

+ _{[sin(Os(a)_Os(S))c0s} 0b

- cos(65(a)-O5())sin ObJSifl 6+ O(E)}

r

3TT

(7.14)

[r-cos(O-O5(8))J--2 sin

r

G[ a1 a + 2cos aI + -c w a i O ir-a

COSI.-5G))J

r-a

--- 8s(&flsin

Obleas

O

+ o(E)}

where / denotes _hb

The contribution of other, not index, blades simplifies in Case (b) to (the expressions for Case (a) are quite similar)

D(lfhb) _-1 (7.17) O1

_(r)3

{[ZT(o)_ZT()l (-cas )} (1/hb) -1 302 (/-_)3 {[ZT(o)_ZT()] sin

The derivates of the singular term R-o (7.18) - (R-a)/

{o(1+z+2_Z4.

+ Ew[F(a)+oF'(o)]}

() E dw

in inner variables

(R-o)IE

-

1

_{a(1+Z2-Z

9)r

-(R-o)/E

+ Ew[F+0F'J}

--;

()

in inner variables

(the inner variables are substituted after the partial differentiation) are

(7.19) ___i

Dai

(R-a)¡E

+1

j {[ +w[Zjcos

O0-aOsin

O ]}n(1+zf2-Z. e)r(o)

-E (P-a)I-E

+ _{&(r+or'(aflw +}

_{E([F+or'][Z+cos}

_{-08sin BI}

- o(1+Z2-Z'_T e') S F()Osin 80)w2) o and 1 1(R-a)/ (7. 20) + _{+ ([r+or'J[_z+sin B0-oecosB0I} - [1+Z'2-Z'_T

_Tw

8']FO'cos B )w2}_{S S O} 1 dw

The above expressions simplify to some extent when the para-meter domains given by Eq. (6.18) are accounted for. The right hand sides of the equations

dw

V1b

ß a2 + O()

1+(o6+

Q(2)

Z+cos B - oBsin ß0 O(E)

-Zjsin ß - oOcos B -aOcos 8 + O(E)

Z+sin B 02 + O(E)

= 1+(Zj2 Q(-2)

Z+cos - oOsin ß = O(E)

_Z+sin B - oOcos ₀ = - Z+sin B + O(e)

are to be substituted. They are also needed in evaluating the integrals (7.23) to (7.25).

The contribution of the nonsingular terms amounts to

(7.22a) -I Wç

00

-

E r() cos(O5(o)-O5())sin Obsiri B -- dS Rh

b1

(7.21) Case (a): Case (b):

R Z-1

yb +1 -w

oo y

c j rc ¿ _{&[sin(es(o)_es(&flcos 0b}

Rh bn

i o

-- cos(8(cY)_O(asin 8bJcoS 6 s- do r EV2b

b and (7.22b) O

(r Evib)

+1 Rh bnl _[ZT(a)_ZT()1sinßOdrEv2b

Here _Vib and similar symbols are the mathematical abbre-viations already mentioned. The letters a and b refer to the special cases under investigation.

When computing the contributions of the singular terms, the following evaluations are needed:

R-o E (7.23)

_f

dw 2

{1S

2 2

[]2(]2}

Rh_c

()3

= _2 2 o

o+

G °1°2 E (R-cy)/E -(7.24)

_f

W dw r 2

_2_2

+ E (R_c)(a_Rh) (Rh_o)/E

(/)3

01+02 o os _2 2 + 2

222 -

[(Z)2 + (aO)21 r 2 R +Rh -2o + E (R_o)(o_Rh) + o(E312) and (R-u)/E 2 -2 i F (7.25)

_f

dw _{--+---- Lin (R-o)(o-Rh)} (Rh_o)/E (!/) Gs Gs + ln{i +E ° O(R,Rh) ln('2₀₁₊₀₂₎2

ln{1 1

ii

+ o(E)

7

_22JJ

-- 0S 01+02

These results are only valid for _Z+

_O(/)

_and

O(/)

(see Eqs. (6.18)).

Now one can apply the reasoning presented by Thurber /42 p. 755/. _{The velocity components can be}

decomposed into a sum of two parts . One is harmonic with respect

to variables

l'°2 and the other satisfies the boundary condition by being zero within the _{order of accuracy} adopted (E2 _{order terms neglected) on the blade and on}

the vortex sheet. _{The position of the vortex sheet is} given by ¿ 0+0(E).

These velocity components can be subtracted from y1 and y2 _{that trivially satisfy the}

boundary conditions. _{The remaining flow must satisfy the} boundary condition and must be harmonic.

The fact that the trailing vortex sheet is a helicoidal surface and not a plane calls for an additional comment. An investigation carried out in the outer variables shows that the errors in the induced velocities near the lifting-line are of order E2. Consequently it is correct to use the above procedure in the present case as well.

The velocity component in the _{direction for Case} (a) is 2_2 (7.26a) -(-1) 1 _{W [} u ç (o8cos 02

L0_E ZT

r 2 711+ 1+ _(oe)2} 2o6cos o 02

[Zcos 8-oOsin 80Jo(1 -

_Z+O)F

7- 2 02 - _ _

_2S

-2 1 + em1[I'orl 2 2 + O(e )! 01+02

This must not include any terms that would represent a sourc'e in the two-dimensional flow /36 P. 707/. The follow-ing constraint will accomplish this:

(7.27a) oOcos 6 [Z+cos 60-oOsin

O()

or

oOZ4

(o8)2sin 6

O()

This feasible parameter domain is bounded by the two limits (see Eq. (6.18a))

-5/14

(7.28a) (i) Zj. O(c

_3/14 oO O(e ) sin = O(/E) (ii) Z = O(E312) = O(p/) sin =

Now the velocity components are

1 w -6 (7.29a) - (

1)--IF

2 2+ oF 1 _([08112 - 2r U[ U

Z11 +[aOcos

_TS

]2)1 - b O

j

w G2 2 + 02 2 (_[oe]2 +

[or

2 01+02 - _{+ [0Ocos} 2 + 4n

o8°05 6[2+o)(+2ln + ln(6+6) }

+ E(r+iyr')cyOcos

_{-7-4J oQcos}

_{(2F+o1'T){-2+Th 4(Ro)(oPh)}} R+Rh_20

-u

oFcj8cìos 8

_{(R_a)(cY_Rh)} i

e

Denoe

V

-2ï

2ir

u

(+oi)oQcos B0

1.

_{oß'cos B (2F+aF'){-2+1n}

__U

S o 1 R+Rh_2o

-

_{u ol' oecos B0 (R_o)(o_Rh)}

The sanie steps for Case (b) are:

22

o 2 (7.26b) - _a(1+Z')? 2B

_{(Z+sin B}

) o 2

21+

(B+a)

(Z+)2} ao1 T

2a2Z4sjn

2 2 {Z'cos ßT o -oO'sinS o}o(1 + i

G1 °2

2c + O(E2)] +

à1o?')

2 2 01+02 and

(7.27b) Z4sin

_{80EZcos B}

_-o&sin

r O(E2)

(7.28b) (i)

or

{S2

Z. oO

_O(E2)

Accounting

_{for the constraint (6.6i) & (6.14.iii),}

_one

obtains

the two lintits

Z r O(c )

O(E )

cos ß r o

(jj)

3i)

J o6 -= O(E)

The resulting velocity components are given by

i _U)I -, (7.29b)

(-1).- or

2

jci +

[Zsin B0)2)

2eU[

i '°2

y1

wr

_Lr

2 _2](1 + [Z+sin B0)2) °1 °2

+ q1j

E(?+o?t)Z+sin 90{2lnE+ ln(+)}

R+R- 2e i w +

-r-tr

E +0F')Z+sin -

_.EoTZsinß0

(R_o)(o_Rh) i

w-

-e(r+or')z+sin ßO(_2+ln(R_0)(0_Rh)) -Denote y (r+orT)Zj.sin B0

1w

- -

(r+orT)z+sin8O{_2+ln(R_e)(a_Rh)} i w R+Rh_20 - a Z4sin B0 (R_o)(o-Rh)

Velocity Components due to J Terms

o

The terms containing J0 in Eq. (7.9) will be discussed here.

The potential with its derivates is, assuming r(R) r r(Rh) r (7.33)

w

R wZ 9 J0(ur (x-Z())-u r

1

_{r(&)(-)[Z- 9sT}

-

f

)J (uo -)- e duds o

Uu

e du da

_7

F({Z-

6}1 _J J(ur )J(uS

)1

and

(7.31)

riZ()2

R

O')] [co5 B J (ua W

(A) S a o iyJ0tuaj)

sin B0

W O (A))]

+ s a sin 0 +a cos B )J (uo J ua

a i a 2

al

U o U

[E(-1cos B0+B2sin B01+Z(a)_Z()]ju

e

duda,

au2

zc2f

- O O

[r(a)a{z.- _O}]f[E(-sin B )J(ua )J(u )

R,naa o

-2 cas B0 (A)

+ E (S1sin B0+d2cos B )J1(ua )J ua

_Uo

tJ a

cas B +5 sin B ]+ZT(o)_ZT(&flUu

o

1

a2

a

.e

_duda.

The differentiation of an asymptotic expansion with respect to parameter can be carried Out, both exponential and sin and cos functions being regular near zero /(40 p. 21/.

For the special Case (a) /11 p. 665/ neglecting terms of order 2 one obtains

R / (7 32) v1 cos Z f 1 1 1 _O' o o

-

;) )da for o < a Rh3a R 2

+ - cas B¿4 _a

Z f (r(S

8')

S 021

1

F (-4;i;() )d for

or

can be neglected provided O is of the same order

as

O.

The integrals exist only as Cauchy principal values. Similarly the expressions for the velocity components ¡53 p. 389/ for Case (b) are found to be

(733)

._r

1-cos 8 - Z 3e C k sin 8 o R Rh 2o& 30 vio o

i

(J)

p- sin 8

Z 1+7T 2 R

_{e22+[zTo_z(&1}

'\ o )dO

f

-p---

z4]

_Q_112(

Rh

where

_Q112

is a Legendre function.

7.4 Velocity Components due to Terms

The potential with its derivates is given by the formulae

and

G0

uZ(i

o w

w

u)1

f

2flZJz(rU)J (ru)

U nZ U

_u2+nZ)2

(Z (r)_ZT(8))u

_2e

e T {u _{cos O sinnZ [o-e()]}

U r - (nZ)2 sin

[O-o()1

sin nZ

o

+ (+u)nZ[-coS O -'sin ]cosnZ )du

inner variables

f

2nZJz(ru)Jz(&U)

u2+(nZ)2

U

_{ni U}

(ZT(o)_ZT())u

2

{-u sin O0sjn nZ[OO5()}

-

(nZ)2

sin nZ[O-O3()]

r

+ unZ[sin O-

cos 00]CoS _{nZ[O_O(&)]}dU}

inner variables

The potential function is valid for

Cx-ZT(cY)} O.

Correspondingly the exponent _{(ZT(cx) - ZT(o))} muSt be inter-presented as

_{_ZT(r)_ ZT()}

. The 0(E)-term at the end

of each equation is due to the differentiation of r with

(7.34)

G

(i

u {

f

(...)du}

O

+ U)30

n1 o

O ( W OW k

r u J

(r-TU) o w U j ,

nZ nZ

U nZ U r n

uL+(nZ)L

-

)u

Cx e U

respect to

o.

After the change into the inner variables they can be neglected.

The equations Simplify when the earlier assumption

BO B is observed. The same formulae for the velocity

components are valid for the two cases under investigation:

i eZ R

f2nzj (cu)J (u)

(7.36)

v10.-.-fr(Th

_u)

_{nZ U nZ U} u2+(nZ)2 ow n'i R {-(nZ)2 sin nZ[Os(G)_Os(8)1 G Zia i

- un - cos B + - sin B ]cos nZ[Os(o)_es()}

U o o o + [(u2+(nZ)2)-(nZ)2)cos

ßRjfl

nZ[6s(G)_Os()]}dud i i aZ

r()(--+)

f2nzJ (GU)Jz(u)

u2+(nZ)2

V2O1T

6h a)

nl

nZ U w

{-[(u2+(nZ)2)-(nZ)2] - sin B sin nZ(es(o)_es(Th]_{u o}

cos B

(nZ)2 ° sin nZ[es(G_es())} ° du do

o

To evaluate the last expressions three types of integra-tions must be performed, the first being

ow

u

(737)

_f

2flZJ(O u)J _u u)

2 2 du

o n u +(nZ)

= 2nZIz(nZo

K(nZ8

) for o <

2nZI

) K(OZ0)

for o > n

For practical purposes this can be computed according to Morgan and Wrench /35 p. 322/.

The second integral

e

ow

(7.38)

_f

J(o - u)J

(a - u)du

U nZ U nZ

r(nz+-)

_{, 2}

-

(o'\ U - \o)

2F1(nZ,nZ+1

o'\

)

a war(nZ+1)r(1/2) o for a

(\nZu

F(nZ+)

2Fi(nZ,nZ1,)

F(nZ+1)F(1/2) for o > is given in Ref. /53 p. 410/.

For

{o/}

i the multiplier

sin (nZ[O(o)_O(&)])

is zero. The convergence of the _{series need be}

n1

investigated. Initially the subseries n 1, consisting of only the first term of the hypergeome-tric series, is shown to be convergent. _{The hypergeometric series itself} is absolutely convergent /33 _{p. 208/, and all other} sub-series, each formed of one element of the hypergeometric expansion, are convergent. _{Then the sum of these subseries} will be shown to be convergent. _{A series converges if}

_/11

p.

5/

(7.39) hrn

°k+l

<1

k-where _{Uk is an element of the series.}

The first subseries (for o < ) and its upper limit are nZ r(nz4) i (7.40) _{nZsin(nZ[O3(o)_05()])j(} \O/ _f(nZ+1Y{

í

nrl o

nZ r(nz4) _{i

.}

nZ (a) F(nZ+1) w n=1 nrl

The recurrence formula /1 P. 256/ is needed to prove the convergence. Applying the convergence test one has

41)

U1

(n+1)Z

(\Z F((n+1)Z4)

r(nz+1) u nZ f((n+1)Z+1) r(nz4) / Z

1+ (nZ+Z 1+

i - ). ..(nZ4) r(nZ4) r(nz+1) n

¼)

9-

1 o (nZ + Z)... (nZ+1) r(nz+1) and un+1

(\Z

hm

n-<1.

Each component of the combined series, denoted by k,

is smaller than { u0) multiplied by

{(/o)2}k and consequently uk+l 2 (7.42)

hm

- (°'\ < i k- Uk

and the sum of the integrals is finite.

The second integral is the upper limit for the third one: U) O w (nZ)2 (7.43)

J(o - u)J

a u u) u2+(nZ)1 du U nZ

<fj(aU)

nZ

fl

U)JzO

,OU) u)du

This last integral can be numerically evaluated in any case. But possibly the following idea could result in a more elegant solution. One of the Besel functions is to

k nZ+2k

(7.44) J (u) (-1) u

2n2kkF(flZ+k+i)

nZ

kzO

Now one might try a method given by Watson /53 p. 43, Ch. 13.6/.

7.5 Velocity Components due to Terms

30

The equations for the potential and the partial derivates are R (7145) _d8 o 4iî ° h R

j{r'()(Z-Y e)+r()(z+_Y

e+z- Y

¡k J (ru)J

al nZ nZ U nz°fl e u2+(nZ)2 (ucos nZ[Q_O()]+nZsinnZ[O_O5(&)1}du}d

md

34, R & aZ j ((+&')(Z' -

o) +

(Z-

8)}

(7.46) T w

I

f

2J(0u)J(&u)

u2+(nZ)2 n=1 O

2w

{-u ECOS

ßcos nZ[O3(o)O3(&)J

sin

+ (nZ)2 COS

_{nZ[O(o)_O(&)]}

o

+ unZ[-cos _{-1-sin]sin nZ[05(o)-05(&)]}du}d&}

+

O()

- R

o _wZ

j {(i+&it)(Z+_

o)+(Z,-

e)}

'41T U

Rh

f

2Jz(où)Jz(u)

U2+(flZ)2

nl O

{+u

_{U51flcC0S nZ[O(o)_B()]}

cos

+ (nZ)2 cos nZ[O(o)_O(8)]

o

+ u(nZ)[ sin - cos ß Jsin nz[os(o)_es()J}du}d

+ E O(E)

inner variables

The same expressions are valid for the two cases under investigation. The evaluation of the integrals was dis-cussed above in Chapter 7»4. The components of the ve-locity are R {(?+&i)(-

8')+F(-

8")} o (7»47) V

jf

_s

_s lo

f 2J (ou)J(

u) u2+(nZ)2

n1

nZ ((-[u2+(nZ)21+(nZ 2 w

) ) cos co5 flZ[Os(G)_Os()1

2 sin

+ (nZ) cos nZ[Bs(o)8s()I

+ (-u)(nZ)(cos +'sjn )sin nZ{65(a)-O5()]}du}d

R - -wZ j 28 _Rh .(

f

2J (o u)J (o u ow 1 nZ nz 2 nrl O u +(nZ)

+ (nZ)2

1

_{con S cos nZ[8(o)_0(&)J}du}d8}

o a

These lengthy formulae can be checked in an elementary way by setting _ZT 0 and 6 0. _{This results in}

(7.48) v = v2o V10 y20 v20r O R o -1 Z 'o -Rh {2(nZ)I (nZo (nZ8 W W _{+'-sjnB )d8} nZ

){cos

_a nZ nr'l for o < Observing (7.49) _{Jm(c&+-i--- )K(c rt)d} =

K'(cm)

one obtains i _wZ V

=+,-(7.50) ie Rh W

+ 1-sin

G nz(0Z0

K'z(nZ)d

(nZ)a(-jcos

n i for a <

which is equal to the corresponding _{parts of Eqs. (23)} and (24) of Ref. /35 p. 310/.

7.6 _{Complex Potential}

Now the velocity components _{from Chapters 7.2, 7.3,} 7.4 and 7.5 are collected _{together to form the basis for} the complex potential.

(7.51) 0r 02 a. -a1 z a + ja. r i y r y + iv. r i

is adopted, i.e. the base flow is in the direction of positive real axis.

Next two-dimensional velocity components are defined as r /U2+wa)21+ (v2ü+v20+v2&) (7.52) - C Vlb i * r (-v10-v1 -v o) + + e i Be 2b De not ing (7.53e) r F and F*' = _F' G r _{(2F*+oF*r)oO cos}

G0r {(o6)2+Z'8'-(Ocos 60)2}F*

_{w TS}

r

o2

and (7.53b) r r F and F*' w _r' U U G r (r*+or*)z+ sin

co

Goz -((Z2 sin

one obtains for Case (a)

a.

/2+2'

+ (v2O+v2O+v2_vlb) - F* (7.5a) -21T .2 _2 r _o.+o i r

a.

_i

+ 2ir O ..2 .2

0. +0

i r

= (+E1nE) ,-_ (2F*+oF* )oOcos

+ f.-.

.(2r*+oF*T)oecos O ln(ä+a2)1+ v2E_{o i}

_2o

+ (_v10v10v1o)E + v2bE: 1 a .-2 .-2 EGe r 2

0.+o

i r i r

G0 is of order E. _{The following term that satisfies the}

boundary condition must be added to

-

,0.

E: -1 i

(7.55a) - G tan

-2r -

-e r

For Case (a) denote

(7.56a) 'G_r +v o-v

20 2o lb

G. =

-v

-v

O_vi+v2T+v2b

i lo i

The corresponding formulae for Case (b) are

a.

-* -

/U2)2'

_{+ E(v20+v20+v2o) - r} (7.Sb) -2s -.2 -.2 r o.+o i r

a.

_i

22

a +

a

r E(-v

-v10-v1) +

1-. r*

r 10 2s ..2 -.2 0. +0 i r i

E1nEr*+or*z+sin

₈₀ E: i + _{...ç*+oF*)Z+sifl}

80ln(O+a)

a

r -T2_ EG0 -.2 -.2 + V?bE: + V E: 0. +0 i r

2z

and

(7.56b) G = v20+v20+v2

G _{_VlO_V18_Vl&+V21+V2b}

Accounting for the additions the complex potential _x can be written as 2' - -

r*

(7.57a) _x

/2w

z + cG z - cG.

iz

i

in z r 3. + i(-Ein) ,- G_z- i ,- G(zln z-z) e C /2 2! - -

rs

(7.57b) _x ¡U +(eo) z+ CGrZ_ eG.iz - i

rin

Z

+ i .E_

G8inz + (-i)E1nE -. Gz- jE G(zlnz-z).

The constraint

2,To

-(7.58) sin >> e

is based on the fact that the slipstream configuration is fixed. This must be satisfied in ail cases /18 p. 35/.

8. INNER POTENTIAL

The two-dimensional flow problem is solved by mapping conformally the airfoil onto the exterior of a unit circle /42 p. 747/, /5 p. 61/. _{The flat plate airfoil will be} discussed as an example. _{The connection between the old}

(defined by Eq. (7.51)) and new coordinates is

o im0 je _{i T1}

(8.1)

z r e + e G( +

where t ie and c r chord length.

The trailing edge is mapped onto the point -1

The potential being indeterminate within a constant, terms independent of will be dropped. For matching purposes the value of the potential for _{large p is needed /5 p. 63/.} For the two cases the equations (7.57) _{transform to}

im,, ie(.

(8.2a) x (/U2+(WG)2 +

G)(-

e

- iG(4 e

im G 2r 2î

_We

) C im

.1

- i

G{e

G C()[i

iii-

_{ln2+ ln}

E ja,, _im0 In eimO - in + e ' in + 1- e 4 ¿ and im je (8.2b) _x (/U2+(wo)2+ Gr)(_ e

G)

_iG.(-

e

G)

im -

i Ç-

in +

if-

G0ln - i

_{ElnG(-- e}

G)

E im - _{i d- EG{e} G (-)[ic0+ir-ln2 + ln C im im,, _im

Ocr

rcir'

_Oc

+e

-+e

_ln}.

-2

C - in Ç + in Ç} and

/2

2 . c (8.3b)

U(wo)

Ç 005 1

/22'Ç

sin -,.. C c 1 i -s--- G0in Ç - Ç + iEG j-Ç - i in Ç 271

rV

G {a(_Ç)[71+1fl C + j

EinG

1Ç - 1 2r - 4 C s + [-Ç in Ç

+ Ç]--

inÇ + in Ç} The derivates dz C (1

iG

(8.4)

--e

- - e and dx - dX dz dÇ dz _dÇ

are needed when satisfying the Kutta Condition

is finite at Ç = -1

consequentiy

(8 5)

r

O at Ç r -1

dÇ

To deterniine the circuiatiori one assumes the foilowing expansions for x (N0 and M0 being rea i):

i - F* i

-(8.6a) _x

r

1

- CCG[Ç lnÇ -Ç)

+ i[----

cGc]in Ç

C C

-

/U+(wo)

cos

-

i

sin

C i

2' C

(8.3a) _x wo)

Ç COS G_l Ç

/u2+ o)2'Ç

sin

C -

r4

Ç+

iG1

- Ç - 1 in Ç + i

EinG

- Ç i G_{-(-Ç)(ir+in ) + [-Ç in Ç +

-1c

2i C

- _EcG + i a- EÏnEcG C + i

_cGin()C

-s C C

+ lE

. G1 + i EcG in

N+iM

+ n n nrl and

(8.6b) x

r

j

._ EcG

k

in -

+ i[--+

EG8-_ EGcJinC

C _E

/2

2'

c/2

2'

- _{U +(wo) ces cXGC - i U +(wa) sin}

- _{- EcG +} i -. lnEcGC + i

-i.-8ii EcGln(-)C C C C N +1M

EGC+i

+I_ECG'r+

n n 8ir C n r E

fl=l

C

The coefficients _{(N, Mn)} can be found from the fact that on the unit circie (

e) Tm x

s constant.

mia

results in

(8.7) cos cG

c/225.

G+lrC

+ .._ EcG in() + E G. + Mi]

+ sin u[-- /U2()2'

G_.EcGr_ECGE_Nl]

r Q and

(8.8) _{Nk r Mk}

r

O for k > 2

Next the Kutta condition is to be satisfied at the trailing edge so that

E

-EcG1

(8.9) X.

r

j 1

_{EcG in C}

+ [_+

G9 - _{_j} -dC Ss E C

c2

2' _c/u2+( 2'.

r -1

C

-c

-C,-, + j E1nEcG + i _ EcG ln(q-) - G_r+ Ic C C

-

.1.. -

" in N +1M1

- c-G +i

8 - 8ir ccG---7-+--7 2 C C

1-h-

EcG_/u2+(wo)2

cG- E . Gr_E - G E _E + -t--cG - N r Q 8iî E 1 2' r 1 -

_{EcG_IU2+(wo)}

sin '2i -- CG0 + E 87r Gln(-) + E G + EcG + _ E1nEcG + _i.- Ec i Sr -C E E - M1 r Q

From this last equation the circulation is determined, i.e.

rs

-LEG

- 2iî 8 + EcG -/u2+aw2'sin

_G

(8.10) C

+ ElnEcG + EcG in(-) + Eco. o

r .

1 r

E

r

Erc /U2+wo)2' sin

G

E2inE

-cG + E2 [G8 - . cG-i cGln[-] - 1TcG.] Denoting (8.11a)

r5

rc

/ 2

21 _U U +(o) sin e G wo and G (2r5 +

ar)ae

cos S

the components of circulation are given by

where r1 r r2 _{= -} cG, r3 - cG)(l + in()) il

lo

10 l 2y _2bfirzr

+C[V

+

The magnitudes of r2 and r3 are in agreement with earlier experience, that is, terms of order E and ElnE are

invariably of opposite sign /4'4 p. 202/.

The formulae for Case (b) are similar, being

2' U (8.11b) r irc

ÁJ2+()

sin = (r _{+ or,)Z+ sin} and (8.12b) L = r1 + (Eln)r2 + where r1 F2 = - cG r3 (_CG*)(1

+ ln())

+ _{rc[vlQ+vlO+vl_v2y_v2bIFç} i- _{(Z' sin} )2 -- T o L

Of course [v10+v10+v18-v21] is different for the two cases under investigation.

An effective angle of attack can be defined as

/2 p. 750/

(8.13)

/U2+(wa)2cssin

r _{{F1 + lnF2 + r3}}

F1, F2 and F3 are all equal to zero at the propeller tip, agreeing with the assumption of zero circulation at the tip and root. For the hub, Alef /13 p. 21/ has suggested that

Rh should be smaller in the computations than the pro-peiler hub radius. By only adopting this point of view,

the solutions are formally correct. The saíne applies to all calculation schemes based on the trailing vortex sheet.

Assuming O < <1 (ln < 0) the influence of the logarithmic term can be estimated as follows:

At the tip: rr 0, r < 0, 0 > O hence ln r2 < O At the root: F4= O, F, > 0, O > O hence lnE r2 > o r= o, r > o, o o hence clnc r2 < o

Eqs. (8.1ti) and (8.1ii) are in agreement with Ref. /7 p. 100/: Skew induces a positive angle near the blade root and a negative angle toward the blade tip.

A circular arc profile will be discussed as a more dE

general example. The profile is defined by aa

r LE

and the positions of the leading and trailing edges in the

(o o.) coordinates are unchanged. The basic formulae of

r i

the conformal mapping are given in Ref. /25 p. 293/. In Eq. (8.1) new terms of order c appear,and they are disreRarded.

The origin of the unit circle is situated at cot

[J

(1cot[j)2+

i} (denoted as

iT),

and the equation of the unit circle is now

(8.15) j + e i

tan [] + eV

The Kutta condition is to be satisfied at the point _/1_[TI2 Substituting given by Eq. (8.15) in (8.6) and neglecting terms of order and c, only two additional components

appear in the expression for the velocity potential. The first different term in Eq. (8.7) is