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NASA-CR- i7382919840021782

152 PAGES84/08/00

(Dept. of Mechanical EngineeringCSS:

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DISPLAY 01/2/184N29851*# ISSUE 20 PAGE 3125 CATEGORY 2RPT#: NASA-CR-173829 NAS 1.26:173829 CNT#: NCCl-65UNCLASSIFIED DOCUMENT

UTTL: Implementation of uniform perturbation method for potential flow pastaxisymmetric and two-dimensional bodies TLSP: Progress Report, periodending 31 May 1984

AUTH: A/WONG, T. C.; B/TIWARI, S. N.CORP: Old Dominion Univ., Norfolk, VA.

and Mechanics.) AVAIL.CASIAvail: CASI HC A08/MF A02UNITED STATES/*AERODYNAMIC CHARACTERISTICS/*AIRFOIL PROFILES/*AXISYMMETRIC BODIES/*PERTURBATION THEORY/*POTENTIAL FLOW

MINS: / INTEGRAL EQUATIONS/ LINEAR EQUATIONS/ PANEL METHOD (FLUID DYNAMICS)/PRESSURE DISTRIBUTION/ SUPERPOSITION (MATHEMATICS)

ABA: AuthorABS: The aerodynamic characteristics of potential flow past an axisymmetric

slender body·and a thin airfoil are calculated using a uniformperturbation analysis method. The method is based on the superposition ofpotentials of point singularities distributed inside the body. Thestrength distribution satisfies a linear integral equation by enforcingthe flow tangency condition on the surface of the body. The complete

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ABSTRACT

IMPLEMENTATION OF UNIFORM PERTURBATION METHOD FOR POTENTIALFLOW PAST AXISYMMETRIC AND TWO-DIMENSIONAL BODIES

Tin Chee WongOld Dominion University, 1984

Director: Dr. Surendra N. TiwariDr. Chen-Huei Liu

The aerodynamic characteristics of potential flow past an

axisymmetric slender body and a thin airfoil are calculated by using a

uniform perturbation analysis method. The method is based on the

. superposition of potentials of point singularities distributed inside

the body. The strength distribution satisfies a linear integral

equation by enforcing the flow tangency condition on the surface of the

body. The complete uniform asymptotic expansion of its solution is

obtained with respect to the slenderness ratio by modifying and adapting

an existing technique. Results calculated by the perturbation analysis

method are compared with the existing surface-singularity-panel method

and some available analytical solutions for a number of cases under

identical conditions. From these comparisons, it is found that the

perturbation analysis method can provide quite accurate results for

bodies with small slenderness ratio. The present method is much simpler

and requires less memory and computation time than an existing surface-

singularity-panel method of comparable accuracy.

IMPLEMENTATION OF UNIFORM PERTURBATION METHOD FOR POTENTIALFLOW PAST AXISYMMETRIC AND TWO-DIMENSIONAL BODIES

by

Tin Chee WongB.S. (Mechanical Engineering) June 1982, National Taiwan

University, Taiwan, R.O.C.

A Thesis Submitted to the Faculty ofOld Dominion University in Partial Fulfillment

of the Requirements for the Degree of

MASTER OF ENGINEERINGMECHANICAL ENGINEERING

OLD DOMINION UNIVERSITYAugust 1984

Approved by:

~~,(~.Surendra N. Tiwari (Director)

Chen-Huei Liu (Co-Director)

Acknowledgments

I am deeply indebted to the co-directors of my masters program,

Dr. Chen-Huei Liu (NASA Langley Research Center) and Dr. Surendra N.

Tiwari (Old Dominion University) in directing this d;-ss~ertat;bn.Thanks

are also extended to Dr. John M. Kuhlman and Dr. Jeffrey J. Kelly, who

served as the members of the thesis committee.

My sincere thanks are extended to Dr. James F. Geer (State

University of New York, Binghamton) for offering technical assistance

during the course of this research. My sincere appreciation is due to

Mr. Scott O. Kjelgaard, Analytical Methods Branch at the NASA Langley

Research Center for assisting in the solutions of the panel method used

in this study; Mr. Richard J. Margason and his branch staif for their

encouragement and support of this research.

This work was supported by the NASA Langley Research Center under

Cooperative Agreement NCCl-65.

I also wish to thank Ms. Jacqueline Smith for her diligent and

timely typing of the manuscript.

Last but not least, it is an honor and a privilege to extend my

deepest gratitude to my parents and brother for their never-ending moral

and financial support during my entire academic career.

i i

TABLE OF CONTENTS

PAGE

LIST OF TABLES •••••••••••••••••••••••••••••••••••••••••••••••••••••••••v

LIST OF FIGURES •••••••••••••••••••••••••••••••••••••••••••••••••••••••vi

LIST OF SYMBOLS •••••••••••••• ~ •••••••••••••••••••••••••••••••• ••••••••xi

Chapter1. INTRODUCTION ••••••••••••••••••••••••••••••••••••••••••••••••• 1

2. THEORETICAL FORMULATION ••••••••••••••••••••••••••••••••••••••• 6

2.1 Perturbation Analysis Technique ••••••••••••••••••••••••• 7

2.1.1

2.1.2

2.1.3

2.2 Other

2.2.12.2.2

Axially Distributed Singularity Method foran Axisymmetric Body •••••••••••••••••••••••••••••8Line Distribution of Singularities Insidea Two-dimensional Airfoil •••••••••••••••••••••••19Second Order Joukowski Airfoil •••••••••••••••••• 32

Solution Techniques •••••••••••••••••••••••••••••• 38

Some Exact Solutions •••••••••••••••••••••••••••• 38Panel Method •••••••••••••••••••••••••••••••••••• 40

3. APPLICATION OF PERTURBATION METHOD ••••••••••• ~ •••••••••••••• 44

3.1 Perturbation Technique for SlenderBodies of Revolution •••••••••••••••••••••••••••••••••••• 44

3.2 Perturbation Technique for Thin Airfoils •••••• ~ •••••••••46

4. RESULTS AND DISCUSSION ••••••••••••••••••••••••••••••••••••••• 55

4.1 Axisymmetric Bodies •••••••••••••••••••••••••••••••••••• 554.2 Two-dimensional Airfoils •••••••••••••••••••••••••••••••• 56

5. CONCLUDING REMARKS ••••••••••••••••••••••••••••••••••••••••••115

REFERENCES ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• 117

iii

APPENDICES

A. Determination of the Coefficients of a(€) and a(€) forAxisymmetric Bodies •••••.•••••••••••••••••••••••••••••.•••••• 119

B. Boundary Conditons in Complex Plane ••••••••••••••••••••••••••123

C. Expressions for Two-dimensional airfoils ••••••••••••••••••••• 125

1. Linear Operators of the Integral Equation •••••••••••••••• 1252. Determination of the Coefficients of a(€) and a(€) •••••• 125

D. Elliptical Coordinate System ••••••••••••••••••••••••••••••••• 128

E. Simplified Expressions for Axisymmetric Bodies and Two-dimensional Airfoils ••••••••••••••••••••••••••••••••••••••••• 132

1. Axisymmetric Bodies ••••••••••••••••••••••••••••••••••••••132

1.1 Ellipsoidal Body 1321.2 Dumbbell Shaped Body ••••••••••••••••••••••••••••••• 133

2. Two-dimensional Airfoils ••••••••••••••••••••••••••••••••• 133

2.1 Elliptic Airfoil •••••••••••••••••••••••••••••••• ~ •• 1332.2 General Joukowski Airfoil •••••••••••••••••••••••••• 134

iv

TABLE

LIST OF TABLES

PAGE

4.1 Comparison of lift coefficients of the perturbation,panel and exact solutions for the symmetricalJoukowski airfoils ••••••••••••••••••••••••••••••••••••••••••• l01

4.2 Comparison of lift coefficients of the perturbation,panel and exact solutions for the camberedJoukowski airfoi1s ••••••••••••••••••••••••••••••••••••••••••• 114

v

LIST OF FIGURES

FIGURE PAGE

2.1 Axially symmetric body immersed in a uniform stream•••••••••• 9

2.2 Uniform stream past a two-dimensional airfoil withan arbitrary angle of attack •••••••••••••••••••••••••••••••• 20

2.3 Illustrating the derivation of an arbitraryJoukowski airfoil ••••••••••••••••••••••••••••••••••••••••••• 33

2.4 Body immersed in a bounded flowfield •••••••••••••••••••••••• 42

3.1 Ellipsoidal body geometry with € = 0.05 •••••••••••••••••••• 47

3.2 Dumbbell shaped body geometry with e = 0.05; b =3 ••••••••• 48

3.3 Elliptic airfoil of 10% thickness ••••••••••••••••••••••••••• 52

3.4 Symmetrical Joukowski airfoil of 6.5% thickness ••••••••••••• 53

3.5 Cambered Joukowski airfoil of 6.5% thickness •••••••••••••••• 54

4.1 Comparison of pressure distributions from theperturbation analysis method with the analyticalsolution for ellipsoidal body

(a) 0.1; cjI expanded to 2 • •••••.••••••••••.•••••• •59e = up e

(b) 0.1; cjI expanded up to 4 • ••.•••••••••..•••••••.. •60€ = €

(c) 0.1 ; cjI expanded up to 6 •••••..•..••.••..•...•..•61€ = e

4.2 Comparison of pressure distributions from theperturbation analysis method with the analyticalsolution for ellipsoidal body

(a) 0.2; cjI expanded to 2 • •••••••••••••••••••••• •• 62€ = up e

(b) 0.2; cjI expanded to 4 •••..•.••••••.•...•••••• •63e = up e

(c) € = 0.2; 4> expanded up to e.6 ••......•.............. .. 64

4.3 Comparison of the perturbation and exact solutions withthe panel method for ellipsoidal body

vi

(a) e = 0.1; ~ expanded up to e6; 60 panels with cosinespacing •••••••••••••••••••••••••••••••••••••••••••••••• 65

(b) e = 0.1; ~ expanded up to e6; 180 panels with cosinespacing •••••••••••••••••••••••••••••••••••••••••••••••• 66

(c) e = 0.1; ~ expanded up to e6; 300 panels with cosinespacing •••••••••••••••••••••••••••••••••••••••••••••••• 67

4.4 Pressure distributions from the perturbation analysisand the panel methods are compared with the exact solutionfor ellipsoidal body

(a) e = 0.05; ~ expanded up to e6; 300 panels with cosinespacing •••••••••••••••••••••••••••••••••••••••••••••••• 68

(b) e = 0.10; ~ expanded up to e6; 300 panels with cosine

spacing ••••••••••••••••••••••••••••••••••••••••••••••••69

(c) e =0.20; ~ expanded up to e6; 300 panels with cosinespacing •••••••••••••••••••••••••••••••••••••••••••••••• 70

(d) e = 0.25; ~ expanded up to e6; 300 panels with cosinespacing •••••••••••••••••••••••••••••••••••••••••••••••• 71

I

4.5 Comparison of pressure distributions from the .perturbation analysis method with the panel method fordumbbell shaped body

(a) e = 0.03; b = 2; ~ expanded up to e4; 300 panelswith cosine spacing •••••••••••••••••••••••••••••••••••• 72

(b) e = 0.07; b = 2; ~ expanded up to e4 ; 300 panels

with cosine spacing •••••••••••••••••••••••••••••••••••• 73

4.6 Comparison of pressure distributions from theperturbation analysis method with the panel method fordumbbell shaped body

(a)

(b)

4e =0.03; b =3; ~ expand~d up to e ; 300 panels withcosine spacing ••••••••••••••••••••••••••••••••••••••••• 74

e = 0.07; b = 3; ~ expanded up to e4; 300 panels with

cosine spacing ••••••••••••••••••••••••••••••••••••••••• 75

4.7 Comparison of pressure distributions from theperturbation analysis method with the exact solutionfor elliptic airfoil

(a) e = 0.1; ~ expanded up to e2; tIC = 20% •••••••••••••••• 76

(b) e = 0.1; ~ expanded up to e3; tIC = 20% •••••••••••••••• 77

vii

4.8 Comparison of pressure distributions from the perturbationanalysis method with the exact solution for ellipticairfoil

(a)

(b)

g =0.15; ~ expanded up to g2; tiC =30% •••••••••••••• 78

3g = 0.15; ~ expanded up to g ; tiC = 30% ••••••••••••• 79

4.9 Comparison of pressure distributions from theperturbation analysis method with the exact solution forelliptic airfoil

(a) g = 0.05; , expanded up to g3; tiC = 10% ••••••••••••••80

(b) g = 0.10; , expanded up to 3; tiC = 20%••••••••••••••81g

(c) g =0.15; , expanded up to g3; tiC = 30% ••••••••••••••82

4.10 Comparison of pressure distributions from the perturbationanalysis method with the panel method for symmetricalJoukowski airfoil

(a) g = 0.03248; y = 00 ; , expanded up to g2;tiC = 6.5%; 10 panels with cosine spacing ••••••••••••••83

(b) g = 0.03248; y = 00 ; , expanded up to g2;tiC =6.5%; 30 panels with cosine spacing ••••••••••••••84

(c) g = 0.03248; y = 00 ; , expanded up to g2;tiC = 6.5%; 50 panels with cosine spacing •••••••••••••• 85

4.11 Comparison of pressure distributions from theperturbation analysis method with the panel method forsymmetrical Joukowski airfoil

(a) g = 0.03248; y = 60 ; , expanded up to g2;tiC = 6.5%; 10 panels with cosine spacing ••••••••••••••86

2(b) g = 0.03248; y = 60 ; , expanded up to g ;tiC =6.5%; 30 panels with cosine spacing ••••••••••••••87

(c) g = 0.03248; y= 60 ; , expanded up to g2;tiC =6.5%; 50 panels with cosine spacing ••••••••••••••88

4.12 Comparison of pressure distributions from the perturbationanalysis method with the panel method for symmetrical Joukowskiairfoil

(a)

(b)

g = 0.01624;tiC = 3.25%;

€ = 0.01624;tiC = 3.25%;

2y = 00 ; , expanded up to g ;

50 panels with cosine spacing •••••••••••••89

y = 60 ; , expanded up to g2;50 panels with cosine spacing •••••••••••••90

viii

(c) 2£ = 0.01624; y = 120; ~ expanded up to £ ;tIC = 3.25%; 50 panels with cosine spacing ••••••••••••• 91

{a) £ = 0.04872;tIC =9.75%;

(b) £ = 0.04872;tIC = 9.75%;

(c) £ = 0.04872;tIC = 9.75%;

4.13 Comparison of pressure distributions from the perturbationanalysis method with the panel method for symmetrical Joukowskiairfoil

(a) £ = 0.03248; y = 00; ~ expanded up to £2;tIC = 6.5%; 50 panels with cosine spacing •••••••••••••• 92

(b) £ = 0.03248; y = 60; ~ expanded up to £2;tIC = 6.5%; 50 panels with cosine spacing•••••••••••••• 93

(c) £ = 0.03248; y = 120; ~ expanded up to £2;tIC = 6.5%; 50 panels with cosine spacing •••••••••••••• 94

4.14 Comparison of pressure distributions from the perturbationanalysis method with the panel method for symmetrical Joukowskiairfoil

y = 00; • expanded up to £2;50 panels with cosine spacing ••••••••••••• 95

y = 60; ~ expanded up to £2;50 panels with cosine spacing ••••••••••••• 96

y = 120; ~ expanded up to £2;50 panels with cosine spacing ••••••••••••• 97

4.15 Comparison of pressure distributions from the perturbationanalysis method with the panel method for symmetricalJoukowski airfoil

(a) £ = 0.06496; y = 00; ~ expanded up to £2;tIC = 13%; 50 panels with cosine spacing••••••••••••••• 98

2(b) £ = 0.06496; y = 60; ~ expanded up to € ;

tIC = 13%; 50 panels with cosine spacing ••••••••••••••• 99

(c) £ = 0.06496; y = 120; ~ expanded up to £2;tIC = 13%; 50 panels with cosine spacing •••••••••••••• 100

4.16 Comparison of pressure distributions from the perturbationanalysis method with the panel method for camberedJoukowski airfoil

(a) £ =0.02688;tIC = 3.25%;

(b) £ = 0.02688;tIC = 3.25%;

(c) £ = 0.02688;tIC =3.25%;

2y = 00; ~ expanded up to £ ;

50 panels with cosine spacing•••••••••••• l022

y = 60; ~ expanded up to £ ;50 panels with cosine spacing •••••••••••• 103

y = 120; ~ expanded up to £2;50 panels with cosine spacing •••••••••••• l04

ix

(a) £ = 0.08064;tIC = 9.75%;

(b) e = 0.08064;tIC = 9.75%;

(c) e = 0.08064;tIC = 9.75%;

4.17 Comparison of pressure distributions from the perturbationanalysis method with the panel method for camberedJoukowski airfoil

{a} £ = 0.05376; r = 00; ~ expanded up to £2;tiC =6.5%; 50 panels with cosine spacing ••••••••••••• 105

(b) £ = 0.05376; r = 60; $ expanded up to £2;tIC = 6.5%; 50 panels with cosine spacing ••••••••••••• 106

(c) e = 0.05376; r = 120; $ expanded up to e2;tIC = 6.5%; 50 panels with cosine spacing ••••••••••••• 107

4.18 Comparison of pressure distributions from the perturbationanalysis method with the panel method for camberedJoukowski airfoil

r = 00; $ expanded up to e2;50 panels with cosine spacing •••••••••••• 108

r = 60; $ expanded up to e2;50 panels with cosine spacing •••••••••••• 109

r = 120; $ expanded up to e2;50 panels with cosine spacing ••••••••••••110

4.19 Comparison of pressure distributions from the perturbationanalysis method with the panel method for camberedJoukowski airfoil

(a) e = 0.10752; r = 00; $ expanded up to e2; •••••••••••• 111tIC = 13%; 50 panels with cosine spacing

(b) e = 0.10752; r = 60; ~ expanded up to e2;tIC = 13%; 50 panels with cosine spacing •••••••••••••• 112

(c) e = 0.10752; r = 120; $ expanded up to e2;tIC = 13%; 50 panels with cosine spacing •••••••••••••• 113

D. Two families of elliptical and hyperbolic curves withdifferent values of ~ and n •••••••••••••••••• ~ •••••••••••• 129

x

a

a .J

A

b

cC(x)

C(n)(x)

e~, en' :13f(x,e), f(x,€)

f n(x), fmn (x)PGj , L

j, L

j

(h,k)

I(x,€), IP(x,€)

1m [F(x)]

LIST OF SYMBOLS

radius of a circle in ~-p1ane; major semi-axislength of an ellipse

coefficients of a binomial expansion

constant determines the curvature of a camberline

minor semi-axis length of an ellipse; profileparameter of a dumbbell shaped body

coefficients of power series expansions

chord

camber distribution

nth derivative of C(x) with respect to x

lift coefficients from the exact, panel andperturbation analysis methods respectively

pressure coefficient

diameter of an axisymmetric body

differential surface area

unit vectors in elliptical coordinate system

singularity strength

functions in singularity strength expansions

linear operators, (p = 0,1)

coordinates of a point in ~-p1ane

scale factors of the elliptical coordinatesystem

integral operators, (p = 0,1)

imaginary part of a function F(x)

xi

k

L

L(e:)

n

p

p00

(r,a,x)

Re [F(x)]

S(x)

s(n)(x)

t

v

(xI,x2,x3)

Z(l;)

a( e:), B( e:)

ak' 13k

r

y

6

€

e: l

II

(~,ThB)

P

ratio of b to a

body 1ength

1i ft force

unit outward normal vector

pressure

static pressure

cylindrical coordinate system

real part of a function F(x)

thickness distribution

nth derivative of S(x) with respect to x

maximum thickness of an airfoil

surface velocity

uniform free-stream velocity

flow velocity

Cartesian coordinate system

transformed point in Z-plane

constants determine the extent of thesingularity distribution

coefficients of expansionsfor a(e:) and B(e:) respectively

strength of circulation

angle of attack

camber line parameter

slenderness ratio

perturbation parameter

doublet strength per unit area

elliptical coordinate system

fl ui d dens ity

xii

1/1

1/1.J

1/10

SUBSCRIPTS

s

T

source strength per unit area

velocity potential

velocity potential in the imaginary internalflowfield

perturbation velocity potential

incident velocity potential

stream functi on

coefficients of 1/10 expansion

incident stream function

boundary surface

trailing edge

free-stream condition

xiii

1

Chapter 1

INTRODUCTION

In recent years, applications of powerful and sophisticated

computers are being made extensively to analyze and solve important

aerodynamic problems. Computer codes are now being developed to

simulate the .flowfield over entire aircraft configurations by

considering the boundary-layer and full-potential formulations. For

realistic physical applications, it would be desirable to work with the

full Navler-Stokes equations. However, this requires large computer

resources (memory and execution time) which makes the computational

costs prohibitive for many applications. For this reason, the former

method is highly desirable for solving many important aerodynamic

problems. Thus, potential solutions become essential in analyzing many

flow problems. Although it would be desirable to obtain potential

solutions for the complete aerodynamic configurations, it is usually

convenient to analyze different basic components of the system

separately. In analyzing the flowfield over an airplane, it is

customary to divide the entire problem into the three separate problems

of fuselage, wing, and tail. Thus, separate programs are developed to

analyze the flow over the fuselage, wing and tail. The analyses of

axisymmetric bodies and two-dimensional airfoils are considered as the

first step for fuselage and wing designs respectively. The two­

dimensional airfoil results are modified usually to account for the

2

effect of three-dimensional wing flow.

The incompressible potential flow over axisymmetric bodies and

wings is essentially governed by the Laplace equation. Although the

Laplace equation is one of the simplest and well known of all partial

differential equations, the number of known closed-form solutions is

quite small. This is due to the non-homogeneous boundary conditions for

the partial differential equation and the requirement for the body

surface to be a coordinate surface of the one of the orthogonal

coordinate system for which the Laplace equation can be solved by

separation of variables. As a consequence, approximate solutions become

more useful in practical applications. One typical example 1s the

slender-body theory where the body boundary condition is simplified for

small slenderness r~tio and angle of attack. In many important fields

successful approximations are found and some of these are discussed in

[1]*. The present study is an application of the slender body th·eory.

Many previous investigators have attempted to solve the potential

flow problem around a slender body. The idea of using a distribution of

singularities inside the body was first treated by von Karman [2]. For

the axisymmetric motion, he used a continuous distribution of sources

along the axis of the given body and gave a method for computing the

distri"bution strength. For the lateral flow past a given body he used a

continuous distribution of doublets along the axis with the doublet axes

normal to the body axis. It is difficult to obtain a closed-form

solution for the exact problem; however, an approximate closed-form

solution may be obtained, if the shape of the body satisfies certain

*The numbers in brackets indicate references.

3

conditions. The body should be slender and should not have any

discontinuities in its surface slope. Zedan and Dalton [3,4] and

Kuhlman and Shu [5,6] extended the method by dividing the singularity

distribution into elements and employing piecewise linear or polynomial

functions to represent the variations of the intensity of the source

distribution over each line element. The effects of the order of the

distribution, the number of elements, the normalization of the body

coordinates, the slenderness ratio and the geometry of the profile on

the performance of the method were studied in detail. It was concluded

that each of these factors should be chosen appropriately in order to

have accurate results.

Another approach to obtain approximate soluti~ns to the slender­

body problem is to use the perturbation method. In 1967 Handelsman land

Keller [7] introduced a method to give a uniformly-valid approximate

solution of axially symmetric potential flow containing a small

parameter, known as slenderness ratio £. The source strength distri­

bution satisfied a linear integral equation. A uniform asymptotic

solution to the integral equation was obtained. The left-hand side of

the integral equation was expanded in the power series of e2 and the

right-hand side in the power series of e2 and a power series in e2

multiplied by e2(log e2) without taking account of the dependence of the

source strength on £. It was assumed also that the source strength

vanishes in an interval near each end of the body and that the profile

curve of the bodY is analytic. It was possible to give a complete

uniformly-valid solution to any order of approximation in this way. The

method used to determine the axially symmetric potential flow about a

slender body of revolution 1s presented in [7].

4

The axial singularity method can be applied to a two-dimensional

airfoil in a similar manner to solve the potential flow problem.

Similarly, one can have an alternative approach that will simplify the

mathematical conditions of the problem and enable the construction of an

approximate closed-form solution. Such an approach is called the

classical thin airfoil theory. The thin airfoil theory breaks down near

stagnation points and near the edges of the airfoil. This is because

the assumption of small perturbation is not valid in the neighborhood of

such points; the perturbation velocity 1s of the same order of magnitude

as the undisturbed velocity. For a survey.of these difficulties and the

methods that are developed to relieve them one should make reference to

Van Dyke [1]. The first approach is the method of matched asymptotic

expansions and the second is the method of strained coordinates [8].

The method of strained coordinates has been modified by Hoogstraten

[9]. However, these methods are so complicated that only the first two

terms of a uniform expansion are generally obtained. Geer and Keller

[10] presented the method to obtain a uniform asymptotic solution for

potential flow around a thin symmetric airfoil by adopting the method of

Handelsman and Keller [7]. This idea was extended and modified by Geer

[11] to obtain the complete uniform asymptotic expansion for the non­

symmetric airfoil. This pertur~ation method 1s presented in Chap. 2.

Since the method to obtain the complete uniform asymptotic expansion for

potential flow past axisymmetric and two-dimensional bodies is known,

the only interest here 1s the assessment of the perturbation method.

Such questions as: how large can the slenderness ratio be or what are

the advantages and disadvantages of this method over other existing

methods are addressed. In the light of these considerations, some

5

programs have been developed to obtain the pressure distributions for.

slender axisymmetric bodies and thin airfoils. In order to make

comparisons, some exact solutions and solutions obtained by using an

existing surface-singularity-panel method have been formulated and

presented in the latter part of Chap. 2. Much effort has been devoted

to obtain the expressions of the asymptotic expansions for the potential

function, with the emphasis on some axisymmetric slender bodies and thin

airfoils, and these are presented in Chap. 3. The pressure distrbutions

on axisymmetric bodies and airfoils obtained by the perturbation

analysis method are compared with the results generated by the panel

method as well as exact solutions whenever available. These results are

presented in Chap. 4. The perturbation analysis method can be applied

to solve the electrostatic and magnetostatic potentials around a slender

conducting body [10-13]. Recently, Homentcovschi eve~ modified this

method to solve two-dimensional elasticity problems [14].

6

Chapter 2

THEORETICAL FORMULATION

Basic potential flow theory is reviewed in this chapter. The

problem under consideration in this study is that of the steady

irrotational flow of .incompressible, inviscid fluid, for which the

Nav1er-Stokes eqaut10ns can be reduced to the classical Laplace

equation. Consider a body immersed 1n an ideal fluid and a space-fixed

reference frame. Then the surface of the body is described by

(2.1)

The variables xl ,x2 and x3 are the Cartesian coordinate system used in

the analysis. Because the fluid is incompressible, the continuity

equation implies that the divergence of velocity must be zero

v.v = 0

Helmholtz's theorem states that the vorticity of an initially

1rrotat1onal, 1nv1scid fluid is zero. Mathematically, one has

vxV = 0

and it follows that

(2.2)

(2.3)

v =v,

7

(2.4)

Here, is the velocity potential. The substitution of Eq. (2.4) into

Eq. (2.2), gives the governing Laplace differential equation for

incompressible potential flow, i.e.,

In order to solve for the potential" Eq. (2.5) should satisfy the

boundary conditions

v+. vF = 0 (2.6)

Also, the components of the velocity should vanish in a certain way with

distance from the body as that distance tends to infinity [15]_, The

Bernoulli equation relat1ng pressure and velocity is given by

(2.7)

Once the velocity on the surface of the body, Us' is determined, the

pressure distribution can be computed through Eq. (2.7).

2.1 Perturbation Analysis Technique

A large number of perturbation techniques to obtain approximate

solutions to physical potential flow problems is available in the

literature [1,7-12]. In this chapter, the special perturbation

8

technique developed by Handelsman and Keller [7,10-12] is reviewed in

detail.

2.1.1 Axially Distributed Singularity Method for Axisymmetric Body

Consider the steady state of incompressible, inviscid and

irrotational fluid past an axially symmetric rigid body which is at zero

angle of attack to the uniform stream, see Fig. 2.1. It is convenient

to analyze the problem by introducing cylindrical coordinates (r,e,x)

with origin at the body·s nose and the x-axis along its line of

synmetry. The surface of the body 1s described by an equation r

= £ R(x), where £ is the slenderness ratio, i.e., the ratio of the

maximum radius of the body to its length. If one uses the length of the

body as a unit of length, then the body intercepts the x-axis at x~O and

x=1. Then, the maximum value of R(x) in absolute value is one. In this

chapter, the slender body of revolution is considered, and e isI

considered to be a small value.

Since it is assumed that the flow is irrotational, the velocity

potential surrounding an axially symmetric body is given as

(2.8)

where .0 is the potential of the incident flow while .b is the

perturbation potential. The perturbation potential .b is due to the

presence of the body, and this is to be determined. This is represented

by a superposition of point sources distributed along the x-axis inside

the body. The boundary conditions for this problem are the no­

penetration (or tangent-flow) condition:

Y/L

u .~

Z/L L: BOuY LEUGTH

r = E Rex) =€~S(X)

/.. _-- ..... - .......... ..... ---­ -- -- .......

{3(E) °1.0 xlL

Fig. 2.1 Axially symmetric body immersed in a uniform stream.

V<I» • ii = 0

and the infinity condition:

on r = eR(x) (2.9)

10

as r + infinity (2.10)

where nis the unit outward normal vector to the surface of the body.. b 0Due to the axial symmetry of the flow, both <I» and <I» are independent

of a and the equation of the profile curve depends on e, so that .b is a

function of x, r2 and e. In this study, our aim is to obtain an

asymptotic expan.sion of .b with respect to e2 around e = O. We define

S(x)=R2(x) and then the cross-sectional area of the body at x is

we2S(x). Assume that Sex) is analytic on 0 < x < 1, with S(O) =0 =S(1) and it can be expanded in power series about endpoints as follows:

•Sex) =r cnxn

n=1

•Sex) =r d (l_x)n

n=1 n

(2.11)

(2.12)

where cn = S(n)(O) }n! and dn = (_l)n S(n)(l) In! • It is assumed that

cl and d1 are non-zero, i.e., the radius of curvature at each end of the

bodY is not zero. The perturbation potential .b is used to represent

the superposit10n of point sources distributed along a segment of the x­

axis 1nslije the body with unknown source strength density f{x,e) per

11

unit length [7]. Thus,

(2.13)

where aCe) and a(e) are constants which determine the extent of the

source distribution and must be found in addition to f(x,e), and they

satisfy the inequalities 0 < a < a < 1.

The potential function +can be related to stream function ~

by 'x = -r'r and 'r =r.x• Thus, one can rewrite Eq. (2.13) as

( 2) o( 2) _ 1 fB(E) (x-t) f(ttE) dt, x,r;e =. x,r ~ 2 2 1/2~T aCe) [(x-~) + r ]

(2.14)

The boundary condition, Eq. (2.9), implies that there is no flow of

flu~d through the surface of the body, i.e.,

a(e)J f(~,e}d~ =0aCe)

(2.15)

Since the body is a continuation of the axial streamline for the total

flow, one can obtain

2,[x,e S(x};e ] =0

Using Eqs. (2.15) and (2.16) 1n Eq. (2.14), it follows that

(2.16)

(2.17)

Equation (2.17) is a linear integral equation from which f(t,e}, aCe)

12

and 8(£) need to be determined. The uniform asymptotic expansion of

the solution of this equation is obtained by adopting the method of

Handelsman and Keller [7]. Since the left side of Eq. (2.17) is

analytic in x for 0 < x < 1, f(x,£) must be analytic in its domain of

definition a , X < 8, and the coefficients in the expansion of f(x,e)

with respect to e2 are assumed to be continuous in the interval

o < x < 1. Both ~ and B are found to be power series in e2 of the form

B(e) =1 - IB e2n

n=1 n

(2.18a)

(2.l8b)

In order to obtain an asymptotic expansion of the source strength:

density f(x,e) of Eq. (2.17) with respect to e2, one first expands each2 .

side of Eq. (2.17) with respect to e without taking account of the

dependency of f on e2• The left side can be expanded in a power series

in e2 because .0 is analytic in r2• The right side can be expanded

asymptotically in powers of e2 and powers of e2 multiplied by

10g(e2) with coefficients which are linear expressions in f. When both

sides are expanded, Eq. (2.17) becomes

aD j . x S(e)4'1 L"'j(x)S (x)e2j = I f d~ - I f d~

j=l (1(£) x

(2.l9a)

where

13

(2.19b)

In Eq. (2.19a) Lj and Gj are l1near operators defined by Eqs. (4.13)­

(4.15) in [7], with e replaced by e2• Upon differentiating Eq. (2.19a)

with respect to x, and by noting that

(2.20)

there is obtained

CD d - 2j4w.L dx [.J-(x)sJ{x)] £ = 2f(x,e)

J=1

CD 2j d 2+ Led (L.+ log e G.)f

j=l x J J(2.21)

To solve Eq. (2.21), the asymptotic solution for f 1s supposed to be of

the form

CD n-1 2f(x,e) = L Len (log e2)m fnm(x)

n=1 m=O(2.22)

where fnm{x) are functions of x and must be determined by substituting

Eq. (2.22) into Eq. (2.21), one obtains

(2.23)

14

For each nand m , the coefficients of &2n(10g &2)m on both sides of Eq.

(2.23) can be obtained as follows:

f 1m = 0 m > 1

n ) 2

(2.24a)

(2.24b)

(2.24c)

(2.24d)

The fnm(x) can be determined recursively by starting with n=O, m=O.

Thus, once the Lj and Gj are evaluated, Eqs. (2.22)-(2.24d) will yield

the desired asymptotic solution for f(x,&) in terms of coefficients 'n.

Inserting Eq. (2.22) into Eq. (2.13), the asymptotic expansion of + is

obtained as

( 2) o( 2·, x,r ;e = + x,r)

1 • n-1 2n 2 S(e) fnm(~) d~- 4;" r r &. (log & )m f

T n=l m=O m(&) [(x-t)2 + r 2]172 (2.25)

The method to obtain the asymptotic expansion with respect to £

about e = a of the integral operator in Eq. (2.17) is available in

[7].

Let the integral operator applied to a function F(x) which is

independent of e be

15

(2.26)

where a(e) and B(e) ,are given in Eqs. (2.18a) and (2.18b). By adding

and subtracting integrals to Eq. (2.26), one finds

I (x, e:) = / F{ t)dt - l F{ t)dt + / F(t) [ 2{x-t~ 1/2 -11dt~ x ~ [(x-t) + e: Sex)] J

(2.27)

On the right-hand side of Eq. (2.27), one sets x-~=v and ~-x=v to the

third and fourth integral respectively. Then, Eq. (2.27) becomes

x a(e) ,I(x,e) = J F(~)d~ - J F(~)d~ + W(x,e) + V(x,e)

a(e) x

Here, W{x,e:) = fX-~ F{x-v) [v{v2 + e:2S)-1/2 - 1] dvo

a-xV{x,e:) = - f F{x+v) [v{v2 + e:2S)-1/2 - 1] dv

o

(2.28)

(2.28a)

(2.28b)

To find the asymptotic expansion of Wand V, the binomial expansion is

considered

~ :(2.28c)

where aj = {-l)j (.l-){l) {~ + j - l)/j! • ThiS expansion is not

valid throughout the domains of integration because these domains extend

to v=O. Therefore, special treatment s~ould be applied. To F(x+v) in

the integrand of Eq. (2.28b), one adds and subtracts the two leading

16

terms in its Taylor series at v=O. To [v(v 2 + e2S)-1/2 - 1], one again

adds and subtracts the leading terms of its binomial expansion. Thus,

Eq. (2.28b) can be written as

B-x 1 B-X-V(x,e) = F(x)! [v(v 2+e2S)- /2 -1]dv + F'(x)! v[v(v 2+e2S)-1/2 -1]dv

o 0

s-x 1 2 j 1 (j ) j+ ! [v (v 2+e2S)-1/2 _ r a. (~) ][F(J(+v) - r F (x ~v /j! ]dv

o . j =0 J v2 j =0 - 1.1 .

s-x 1 (j) j+ a1e2S! [F(x+v)- r F (~)v ]v-2dv (2.29)

o j=O J.

An analysis of the order of accuracy is described in Appendix A of

[7]. It shows that the first two integrals are O(e) and 0(e210g e2),

respectively. The third term is 0(e2), but the fourth integral

is 0(e3), so that it is asymptotically negligible compared to the three

terms. To obtain the asymptotic expansion of the fourth integral in Eq.';

(2.29)~ one can apply the ~ame procedure. To the first factor of the

integrand, one adds and subtracts the second term in the binomial

expansion" given by Eq •. (2.28c). To the second factor one can add and

subtract the next two terms of the Taylor expansion of F{x+v) about

v=O. In this manner, a sum of integrals 1s obtained; some of which can

be evaluated explicitly, others can be expanded as power series

in e2, and the remaining term is asymptotically small compared to the

other terms. The same procedure can be applied to the remainder, and

this can bel done repeatedly. Thus, V{x,e) can be expressed as a series

of integrals which are successively smaller in e2 as e tends to zero.

Furthermore, W(x,e) can be treated in an exactly analogous way, so one

obtains

17

•W(x,e) + V(x,e) = L anSn(x)e2n[Hn(x,e) - H (x,e)]

n=l n

• F(2n)(x)+ n~1 (2n)1 [Pn(x,e) - Pn(x,e)]

• F(2n+l)(x)- n~o (2n+l)1 [Kn(x,e) + Kn(x,e)]

where, Hn 'Pn ,Kn are defined by

(2.30)

X-CI 2n-l (j) j= f v-2n[F(x_y) - L F ~~)Y] dv

o j =1 J ·(2.30a)

X-CI 2 2 2 1/2 n 2S jPn(x,e)=f vn[v(Y+eS)- - L a·(T)]dv

o j=O J v

x n 2 j= f y2n+1[v(y 2+e2S)-1/2 - L a.(e ~)]elv (aO=I)

o j=O J v

(2.30b)

(2.30c)

'V 'V

The ifunctions Hn, Pn, and Kn are defined by Eqs. (2.30a), (2.30b) and

(2.3Oc) respectively, with (x-a) replaced by (B-x). To determine the

asymptotic expansion of I(x,e), one must expand asymptotically ,the

integrals Eqs. (2.3Qa) and (2.3Ob). Hn(x,e) and Hn(x,e) can be expanded

directly as Taylor series in e2 about e = O. The integrals Eqs. (2.30b)

and (2.30c) have been analyzed in detail 1n Appendix B of [7]. Appe;ndi x

A in [7] shows that P = 0(e2n+1), K = 0(e2n+210g e2) and the samen n'V 'V ~

holds true for P and K. However, in the difference P - P , whichn n n n

occurs in Eq. (2.30), the odd powers of £ will be cancelled. By using

the results of Appendix B in [7] to Eq. (2.30), one finds that I(x,e)

has the asymptotic expansion

I{x,e) = W{x,e) + V{x,e) = re2j[L.+{109 e2)6.] F{x)j=1 J J

(2.31)

18

Here, Lj and Gj are linear operators defined by Eqs. (4.13)-{4.15) in. 2 rw rw rw

[7], with e replaced bye. The functions 9k' 9k,t ij , t ij , hk, hk in

Eq. (4.14) are defined by Eqs. (BI0)-(BI7) in Appendix B of [7]

when e is replaced by e2•

We now return to determine the coefficients an and Bn in the

expansions of a(e) and 8(e), respectively. From Eq. (2.24a) it follows

that fl0(x) is analytic on 0 < x < 1, because '1 is analytic and 5(x) is

analytic with 5(0) = 0 = 5(1) and 5 1 (0) # 0 # 5 1 (1). It then follows

that from the recursive nature of Eqs. (2.24a)-(2.24d), the function

fnm{x) will be analytic for 0 < x < 1 if both GjF{x) and LjF{x) are

analytic, provided that F(x) is analytic. This will be true if all of

these functions 9k' gk hk, hk are analytic. The functions 9k(x)

and 9k{x) are defined in Appendix 8 of [7] as

g(x , e)m 2k 2 1/2= r gk{x)e = {[x-~{e)] + e Sex)}

k=O(2.32a)

(2.32b)

where a(e) and 8(£) are defined in Eq. (2.18a) and (2.18b) respectively.

The functions gk{x) depends on the coefficients ~ in the expansion

of ~(e). They are singular at x=O except for certain values of ~, so

the appropriate ak should be chosen such that 9k(x) can be regular at

x=O. To relate gk{x) with ~, one can square both sides of Eq. (2.32a)

and equate the coefficients of the same power of e. Similarly, 9k{x)

can be related to Bk at x=l. Detailed derivation of ak and Bk is

19

available in Appendix A. Using the results Eqs. (A9) and (All), the

expressions for a(e) and S(e) are given by

a( e) (2.33)

(2.34)

The functions hk(x) are defined in terms of the square root of gk(x).

Once the ak are chosen appropriately to make gk(x) analytic, then it#v

implies that hk(x} are also regular. Similarly, the hk(x) are also

regular at x=l, since 9k(x) are made analytic. Therefore, LjF(x); Gj(x)

and fnm(x) are analytic provided that F(x) is analytic on 0 < x , 1.

The asymptotic expansion of +(x,r2;e) can be obtained by using the

appropriate fnm(x), a(e) and S(e) which are determined by Eqs. (2.24a)­

(2.24d), (2.33) and (2.34) respectively in Eq. (2.25). Once the

potential of the flowfield is found, the aerodynamic properties of any

gi"ven axisynmetric slender body can be evaluat"ed.

2.1.2 Line Distribution of Singularities Inside a Two-dimensional

Airfoil

In Sec. 2.1.1, the potential flow past an axially symmetric slender

body has been described and the same technique can be applied to the

two-dimensional airfoil. This idea was presented earlier in [10,11].

In this section, the problem will be analyzed in terms of the functions

of a complex variable. Similar assumptions are made here as in Sec.

2.1.1. Consider the two-dimensional potential motion of an ideal fluid

past an arbitrary airfoil which is shown in Fig. 2.2. The problem is

. j

Y/L-,: Z -plane

.'

.. i

G

~--~-------.......-~ X/Lo 1.0

L:BODY LENGTH

Fig. 2.2 Uniform stream past a two-dimensional airfoil withan arbitrary angle of attack.

No

21

analyzed in Cartesian coordinate (x,y) with the origin at the leading

edge of the airfoil and the trailing edge is some distance intercepts on

the x-axis. Now, one should recall the relationship between the

potential, the stream function and the velocity components in two­

dimensional motion. As before, the potential is , and the stream

function is ,. The combination of +and , into an analytic function as

a complex potential +(z), is given by

+(z) = +(x,y) + i,{x,y) (2.35)

and the x-y plane refers to the z-p1ane (z = x + iy).

For this study, it is convenient to think of +as either the real

or imaginary part of a function

(2.36)

which is analytic in the z = x + iy plane outside the profile curve Gof

the airfoil. The potential ,0 is the potential of the incident flow on

the airfoil, while .b is due to the presence of the body. The

potential .b is represented by the superposition of complex potentials

of point sources and vortices distributed along an appropriate arc

inside the airfoil. The boundary conditions of this problem are.

no penetration condition: v,. n= 0 on G

infinity condition: .b =k tan-1(y/x) + O(1/r) as r + infinity

22

where n is the unit outward normal vector to the surface and r is the

total circulation about the airfoil.

The equation of the profile G of the thin airfoil is given by

y = e[C(x)±/S(x)] on 0 , x , 1, and e is defined as slenderness

ratio. This is assumed to be small for a thin airfoil. The leading and

trailing edges intercept x-axis at x=O and x=l respectively, and the

max.IC(x)±S(x)1 =1. It 1s assumed that both C(x) and Sex) are analytic

on 0 < x , 1, with C{x) and S{x) vanishing at x=O and x=l. These

functions can be expanded 1n a Taylor series at the endpoints as

C{x)•

= L c xn

n=l n

•, C(x) = L C (1_x)n

n=l n

•Sex) = L dnXn

n=1

• IV

, Sex) = L dn(1-x)nn=l

(2.37)

_ C(n){O)where cn - In.

_ S(n){O)dn - In.

__ (_1)nC(n)(1), cn - In.

d = (_1)nS(n}{1), n n! (2.38)

In order to solve for " it is convenient to find a function, Eq.

(2.35), which is analytic in z = x + iy outside G and sati'sfies the

condition

Re {[1-e(C ' tS ' )/2 IS] M-} = 0 on G (2.39)

23

Equation (2.39) is verified and explained in Appendix 8. Here, .b

represents the superposition of the potential due to point sources and

vortices distributed along an arc inside the slender airfoil. Thus,

(2.40)

where f(~,e) is the unknown source strength density, which is a complex

expression. The constants a(e) and B(e) determine the extent of the

source distribution and they can be determined after f(~,e) is found and

they must be inside G.

Using Eqs. (2.39) and (2.40) in Eq. (2.36), one obtains

Re {[i-e(C'tS'/2/S)] ~:o [x+ie(Ct/S)]}

= Re {~w [i-e(C'tS'/2/S)] (2.41)

Equation (2.41) is a pair of equations, corresponding to where the upper

(plus sign) and lower (minus sign) surfaces of G, is obtained. By

adding and subtracting these two equations, a pair of coupled linear

integral equations can be obtained. Upon denoting x+ie(C+/S) and

x+1e(C-/S) by xpu and xpl respectively, one has

2w Re {i(l+ieC')[ d.o(xPU)+ d.O(XPl)] eS' d,o(xpu) d.o(xpl)dz dz - 2/S [dZ - dZ ]}

(2.42)

24

and

2'1 Re {i(1+i C')/S[ d.O(xpu)_ d.O(XP1)] _ ~ S,[d!o(xPU) + ~o(xPl)]}e dz dz 2 dz dz

= Re {e /' 2S{1+i eCI

) - r(X+1eC-t) f("t,e) dtla (x+ieC-~) + e S

(2.43)

In Eq. (2.42) and (2.43), f(~,e), a(e) and B(e) need to be determined.

Both a{e) and a(e) are found to be power series in e of the form [11]

•(I( e) = L (In en

n=l,

00

B(e) =1 L Bnen

n=l(2.44)

Using the results of [10] as an instructive example, the function

f{x,e) shall be of the form

f(x,e) = f{x,e)I[s{e) - x][x - a{e)]

(2.45)

where f{x,e) is singular in x at x =o{e) and x = B{e). The function

f(x,e) is be assumed to be analytic before finding its asymptotic

expansion about e = O. Following the procedures given in [10,11], both

sides of Eqs. (2.42) and (2.43) can be expanded with respect to e,

without taking account of the dependency of f on £. Since ,0 is an

analytic function of x + ie[C{x)±/S{x)], the left-hand sides ofiEqs.

(2.42) and (2.43) can be expanded in a power series in e. Then, Eqs.

(2.42) and (2.43) become

• • 00 • 0L eJ J. (x) = Re {i L eJ L~. [f (x , e:) ]}j=O J j=O

(2.46)

• · • · 1L eJ KJ"(X) = Re {L eJ Lj [f(x,e)]}j=O j=O

Here, Jj{x) and Kj{x) are defined as

25

(2.47)

(2.48a)

(2.48b )\ ",

where (2.48c )

In Eqs. (2.48a) and (2.48b), the notation (~) = nl/(jl){n-j)l has beenJ

used and en] ;s the greatest integer not exceeding n. Applying the same

idea of expanding integral operators asymptotically as in Sec. 2.1.1,

one can expand the right sides of (2.42) and (2.43). To do this,

consider the integral operators IP(x,e), p= 0,1 applied to a function

F(,)/[(B_,)(,_«)]1/2 and F(,) is independent of e. The integral

operators IP are defined as

IO(x,e)=fB 2[x+ieC(x)-,]~1+1eC'(x)] +e2S' (x)F(,) d,o {[x+ieC(x)-~] +e S(x)} I{B - ~)(~ - a)

(2.49a)

I1( )_ f8 2S(x)[I+ieC 1 (x)] - SI(x)[x+ieC(x)-,] F(,) dX,£ - £ 2 2 ~

a {[x+i£C(x)-~] + £ S(x)} 1(8 - ~)(~ - a)

26

(2.49b)

The constants a(£) and 8(£) are defined in Eq. (2.44), where the

coefficients an and 8n need to be determined. The operator IP(x,e) can

be expanded by following the similar method as given in [7] and by using

the results derived in [11], one obtains

(2.50)

The linear operators L~, p = 0,1 are defined in Appendix C. From the

definitions given in Appendix C, one has two simple operators

1 / f(x)LO [F(x)] = 2w 'x I-x' F(x)

(2.51)

(2.52)

To solve Eqs. (2.46) and (2.47), an asymptotic solution for f(x,e) is

assumed as

aD

f(x,e) = L fn(x)enn=O

(2.53)

Here, the functions fn(x) are to be determined. By substituting Eq.

(2.53) into Eqs. (2.46) and (2.47), one obtains

(2.54)

r£j K. (x) = Re [r r £j+n L~ [f (x)]lj=O J j=O n=O J n J

27

(2.55)

By equating coefficients of the like powers of £ in the corresponding

equations, Eqs. (2.54) and (2.55), finds

1 [n 0 ~1m {La [f (x)]} = -In(x) - 1m L L. [fn .(x)]n j=l J -J

Re {Lo1 [f (x)]} = K (x) - Re [I L~ [f . (x)]l

n n j=l J n-J J

Using the results of Eqs.(2.S1) and (2.52), one finds that

0001m {LO[fn(x)]} = 1m { LO [fn(x)]} = LO [ 1m fn(x)]

21m [fn(51n 29)] - 1m [fn(x)]= -4/ de

o (51n29 - x)

(2.56)

(2.S7)

(2.58)

(2.59)

Applying the results of inversion of operators 1n [10] with Eqs. (2.58)

and (2.59) in Eqs. (2.56) and (2.S7) respectively, one has

1m f (x) = f (x) + 1m f (O)n n nwhere

; (x) = x2

{J (x) + 1mr L?[f .(x)] ln 'I n j=l J n-J . j

(2.60)

28

2- 2 [ n ] t = s1 n2e

- L I cos e J (t) + 1m L L~[f oCt)] de (2.61)112 0 (s1 n2e - x) n j=1 J n-J t = x

and

(2.62)

~=a

In Eq. (2.61), the notation [G(~)] = G(a)-G(b). From Eqs. (2.61) and~=b

(2.62), the- funct10n fn(x) can be determi ned recursi vely, once the

constants 1m f n(0) are found. In order to dete·rmi ned 1m f n(0), one has

to use the condition that

B ,.,.1m f f(~,£) d~ = r

(I

where•

r = L r 0 e:jj=O J

(2.63)

Here, each rj 1s a prescribed constant independent of e:. By us1ng Eqs.

(2.45), (2.53) and Eq. (2.61) in Eq. (2.63) can be expressed as

r = i e:j IS f j (~) d~ + 11 f ~ 1m fJ.(O)j=O (I 1(8 - 1;)((; - (I) j=O

(2.64)

By setting ~ = (S-«) sin2e +« and expanding each integrand in Eq.

(2.64) in Taylor series about £ = 0, one obtains

'I .

•. j "2" nr • 2 L e

J 1m L ~I I H:e:) i fJo_n[(s - «) sin2e + «]} de

j=O n=O· 0 £=0

•+ 1r L

j=O1m f. (0)

J(2.65)

29

Equating coefficients of like powers of £ on each side of Eq. (2.63) and

(2.65), yields

11'

rn 2 n 1 '2" a k'" . 21m fn(O)= - - - L IT f {(-a) f n k[(B-a)s1" e + a]} de

11' W k=O • 0 £ - £=0(2.66)

Since fO(x) depends only on JO(x), the 1m fn(O) can be determined

recursively from Eq. (2.66). Thus, functions fn(x) can be found through

Eqs. (2.61), (2.62) and (2.66) in a recursive manner.

Determination of the constants a(£) and B(e) is discussed now.

From Eqs. (2.61) and (2.62), one knows that Re fO(x) and 1m fO(x) depend

on +o(z) which is analytic. Thus, it fa1.lows from the recursive nature

of Eqs. (2.61), (2.62) and (2.66) that all the Re fO(x) and 1m fO(x)

must be analytic for 0 , x , 1 if the LP[F(x)] is analytic. This willq

be true if and only if each gj(x) and hj(x) in Eqs. (C2) and (C3) is

analytic on the interval 0 , x '1. In the manner of Appendix A, one

can determine ~ uniquely by demanding that g(x,e) is regular at x=O.

Similarly, all o·f the Bk can be determined by requiring that the

function h(x,e) is regular at x=l. By squaring both sides of Eqs. (C2)

and (C3) and equating the coefficients of the same power in £, one can

find out all the appropriate ~ and Sk' such that singularities at x=O

and x=l of functions gj(x) and hj(x) are eliminated. The similar

technique for the determination of a(e) and 8(e) is given in Appendix A,

and the general expressions for ~ and Sk are available in Appendix C.

Using the results in Appendix C and the definitions of Eqs. (2.37) and

(2.38), the leading terms for a(e) and S(e) are given by

(2.67)

IV IV IV IV IV IV 2

dl 2 i c1d l £3 + d1(d 2 +164c1 ) .,.4 + O{~5)See) = 1 - ~ e - 4 ~ ~

30

(2.68)

Once f{x,£) has been obtained, the expression for ,(z) is obtained by

using Eqs. (2.45), (2.53) in (2.40) and (2.36) is given by

(2.69)

Equation (2.69) is the desired uniform asymptotic exp.ansion for ,.

Suppose now that the thin airfoil has a sharp trailing edge such

that near x=l, S(x) has the expansion

CD IV •

S(x) = L d.(I-x)Jj=2n+l J

n ) 1 (2.70)

where dj = (-l)j S(j)(l)/j. From Eq. (2.70), it follows that one can

take a{e)=l, the asymptotic expansion obtained above remains valid when

Eq. (2.70) holds. The mathematical proof is available in Sec. 7 of

[10,11]. Physically, the velocity of the fluid near the sharp trailing

edge should be finite. As the velocity of the fluid with the potential

given by Eq. (2.69) near the trailing edge 1s examined, the first term

in Eq. (2.69), .o(z) obviously gives a finite ve~ocity at the trailing

edge, because ,o{z) is analytic at z=l. The remaining terms in Eq.

(2.69) contribute velocity components of the form

1f 1 F(~) d~

a(e) z - ~ I[~ - a(e)] (l-~)

(2.70a)

Equation (2.70a) remains bounded as z+l, if and only if F(l)=O.

31

Therefore, one shall make each fk{x), k > 0 vanishes at x=l by properly

choosing the arbitrary constants of 1m fk{O). The mathematical proof of

choosing 1m fk{O), so that each fk{x), k > 0 vanishes at x=l, is shown

in Sec. 7 of [10], i.e.,

(2.71)

If the flow described by ~o is a uniform stream, the circulation

can be determined by Eq. (2.65), such that the lift L{e) of the airfoil

is gi ven by

L{e) = p U r{e)GO (2.72)

where p is the density of the fluid and U is the speed of the uniformGO

stream at infinity. By using Eqs. (2.65) and (2.71), one finds that Eq.

(2.72) becomes

11'

-. j 1'2" k- 2L(e) = pU r eJ 2 1m r ~ f {(aa~) 1 fJ-_k[{B-a)sin e + a]} de

GOj=O k=O If. 1 0 ~ e=O

(2.73)

The uniform asymptotic expansion for ,(zje) can be determined by finding

fn(x), aCe), and S(e). Then, the velocity, pressure distributions on

the surface, and also the lift coefficient of an arbitrary airfoil can

be determined.

32

2.1.3 Second Order Joukowski Airfoil

Certain assumptions have been made in analyzing the problem in Sec.

2.1.2. The upper and lower surface of the thin airfoil have the

equations y = e[C(x)+/S(x)] and y= e[C(x)-/S(x)] for 0 < x < 1; with

C(O)=S(O)=C(I)=S(l)=O. The equations of the thickness and the camber

line of the airfoil are y = e/S(x) and y = eC(x), respectively.

Here e 1s the slenderness ratio, while S(x) and C(x) give the thickness

and camber distributions. Varying e produces a family of affinely

related profiles. More examples are given by Van Dyke in [1]. Now one

expects to seek an expression for a thin Joukowski airfoil by using the

perturbation method. The parametric form of the Joukowski airfoil is

easily obtained by means of conformal mapping [15]. One employs a

transformation x(~,n), Y(~,n), where the ~-n plane refers to as

the ~-plane which will map the region and its boundaries in

the ~-plane into another desired configuration in the x-y plane, which

is referred to as the z-plane. Fig. 2.3 shows the notation of the

Joukowski transformation of a circle into an airfoil. The center of theio

circle with radius a at any arbitrary point is given by p =me. The

symbol Po denotes the intersection of the vector a e-iS with the y­

axis. A circle with center at Po and radius of magnitude Ipo - tTl

transforms into a circular arc, which is considered as the skeleton of

the airfoil. The arbitrary circle, the points of which are described by

i9~ =a e + p (2.74)

transforms into an airfoil. The angle B determines the mean curvature

33

7]

z-plane

--r-----r~---~----~'----~x

.°L-2C --JT

Fig. 2.3 Illustrating the derivation of an arbitrary Joukowskiairfoil.

34

of the profile and the magnitude Ip - pol determines the thickness of

the profile.

The circle is chosen such that it passes through ~ = ~T = C, where

C is real positive number. The corresponding point on the airfoil is

gi ven by

(2.• 75)

It should recall that the center ~ of the circle, its radius a, and the

angle 8 are connected by

-i 8 i 6C=ae +me (2.76)

Since C 1s a real positive number, one can separate the real and

imaginary parts of Eq. (2.76) as follows:

C =a case + m cos~

a sins = m sin~

(2.77a)

(2.77b)

Using Eq. (2.74) in Eq. (2.75), with the expressions given by Eqs.

(2.77a) and (2.77b), Z(~) can be separated into real and imaginary parts

as x and y respectively. After further simplification, the parametric

form of the airfoil is obtained as follows:

x = a(coss + h) [1 + C2 J (2.78a)a2(1 + h2 + k2 + 2h case + 2k sine)

35

y = a(sine + k) [1 - C2 ] (2.78b)a2(1 + h,2 + k2 + 2h cose + 2k sine)

where h = (m/a) COSo k = (m/a) si~o (2.79)

The trailing-edge point of the airfoil is denoted by ZT with e = -6 in

Eq. (2.75), one finds x = Re(ZT) = 2C and y = Im(ZT) = o. Thus, the

trailing edge of the airfoil intercepts the x-axis at the point T with

OT=2C. In order to have an asymptotic expression for a thin Joukowski

airfoil a small parametric, £1 should be introduced to expand the

available equations in power series. Generally the quantities h,

k, S, and m should be small. If we choose m=elC, then Eqs. (2.77a),

(2.77b) and (2.79) can be expanded to the second order of accuracy as

follows:

2h = (coss cOS6) e1 + O(e1 )

2k = (coss s1n6) el + O(e1 )

(2.80a)

(2.80b)

(2.80c)

Using Eqs. (2.aOa)-(2.aOc) in Eqs. (2.78a) and (2.78b), and retaining

the terms with order of el , one obtains

+ 2 cos2s cose(cos6 - coss cos6 cose - coss sin6 sine)]}

2+ 0(e1 )

y = Csine cos~ e1[2cos2S case - (coss + c~ss)J

36

(2.81a)

(2.81b)

By examining carefully the order of magnitude of sins, cosS, sin2s

and cos2s, one finds that both coss and cos2s are 0(1) and sins and

sin2 S are 0(e1) and O(e~) respectively. Neglecting the terms of

0(e1), Eq. (2.81a) can be further simplified as

x = 2Ccose + 0(e1) (2.82)

Furthermore, Eq. (2.81b) can be simplified by comparing the order of

-t d f - - 2 d 2 t e d T t e tmagnl u e 0 slnB, Sln S, coss an cos S as men lone. 0 re aln erms

of 0(e1), Eq. (2.81b) becomes

(2.83)

Introducing the nondimensional quantities x= x/2C, and y = y/2C, Eqs.

(2.82) and (2.83) become

(2.84a)

(2.84b)

Upon eliminating sine and cose in terms of i, Eq. (2.84b) becomes

37

'" tv / ",2 · (1 1V2 )Y = £1(-COS6)(1 - x) 1 - x + £1 s1n6 - x

-1 < x < 1 (2.85)

Obviously, if ~ = w, Eq. (2.85) will yield a symmetrical second order

Joukowski airfoil, while the second term vanishes. It is consistent to

the second order Joukowski airfoil shown in Sec. 4.6 of [1]. It also

implies that the second term represents the equation of the camber

line. Therefore, Eq. (2.85) can be rewritten as

y = y + ys c (2.86)

h '" ( ) ( tv) / tv2 IV • ( "'2)were Ys = £1 -cos~ 1 - x 1 - x and Yc = &1 Sln~ 1 - x •

To make Eq. (2.86) consistent with the required form of equation, simple

translation and scaling of the coordinates are done, such that Eq.

(2.86) becomes

YC = 2 s1n6 £1 x(1 - x) o < x < 1

(2.87b)

(2.87b)

By def1ning £ = 2(-COS6)£I"' with £ is the slenderness ratio, Eq. (2.87b)

yields Ys = £ IS(x) and Yc = £ C(x)

where

Sex) = x(1 - x)3 (2.88a)

C(x) = Ax(1 - x) , A = -tan6 (2.88b)

38

S(x) and C(x) give the thickness and camber distributions of an airfoil

respectively, and A is related to the airfoil camber.

2.2 Other Solution Techniques

2.2.1 Some Exact Solutions

The analytical solutions of potential flow problems associated with

the motion of a solid body through an ideal fluid are considered here.

The final aim is to determine the pressure distribution over the surface

of the body and the lift force acting on the body whenever available.

The analytical solution for elliposida1 body of revolution and two­

dimensional elliptic airfoil are obtained in this section.

1. Analytical Solution for Ellipsoidal Body in Uniform Stream

Consider a rigid ellipsoidal body moving through a uniform ideal

fluid. The flow is assumed to be steady, incompressible and irrota­

tiona1. To simplify the problem, a body-fixed reference frame and

elliptical coordinate system (~,~,B) are introduced. The derivation is

described in Appendix D. Using the results of Eq. (D5a), one is able to

obtain 0 = U e + U e where U~= ~ :' and U = - h\ *.~ ~ ~ n ~ ~ S ~ n ~ S ~

Here, Ut

and Un are velocity components in t and n directions

respectively, and

2 2a, a, a, a, _~ + -:2 - coth~ - - cotn a - 0at an at n

(2.89)

Consider the Neumann boundary condition on ~ = ~o and the infinity

39

condition as ~ tend to infinity. Making reference to Appendix D, the

boundary conditions can be simplified as follows:

on E: = ~o (2. 90a)

as E: + infinity (2. gOb)

Equation (2.89) along with Eqs. (2.90a)-(2.90b) can be solved by

separation of variables. One finds out that the velocity component

in et direction on the body surface is zero (due to the no penetration

on the surface of the body), and the velocity in e direction is givenTl

by

(2.91)U sinn..

Us= -[ cosht - 1,11[oshto+ sinh2toln{ sin~to )J'COSh2t

o- cos2~

With application of Bernoulli equation, the pressure distribution on the

surface of the elliposidal body is given by

(2.92)

2. Analytical Solution for El11ptic Airfoil in Uniform Stream

In Sec. 2.1.3 the idea of conformal mapping has been introduced to

get the shape of an airfoil. Here, the pressure distribution over the

airfoil, by means of Joukowsk1 transformation is discussed. This method

can be used directly to the actual problem of by means of mapping the

flow past an arbitrary circle, for which the solution is already

known. To define the arbitrary circle, we describe the points by

ie~ = r eo

40

(2.93)

(2.94)

where ro = (a+b)/2 with a and b are the major and minor semi-axis

lengths of the ellipse respectively. Making reference to [15], the

complex velocity on the airfoil surface is given by

-iei 2U sine e•W(z) = dz

~

After simplification,

the velocity distribution along the contour of the ellipse is given by

(2.95)

where k =b/a. The pressure distribution on the elliptic airfoil can be

computed through the Bernoulli equation.

2.2.2 Panel Method

A considerable number of panel methods has been developed in the

past two decades to analyze the steady, inviscid and irrotational flow

fields past a body of arbitrary geometry. The incompressible f10wfield

is governed by the Laplace equation which can be transformed to an

integral equation relating the perturbation potential to the source and

doublet singularities distributed over the surface. There are different

discretization techniques among panel methods for solving the integral

equation. Basically, all the methods approximate the surface by

elemental panels of prescribed ge9metric shape and singularity variation

41

and the integral equation is solved by enforcing a boundary condition at

a control point or points on the panel. There is a summarized table in

[16], showing several useful panel methods with panel geometry,

singularity variation and boundary condition specification. In this

section the governing equation and the basic idea of discretization are

reviewed.

Consider steady, inviscld, irrotational-and incompressible fluid

flow in a bounded domain D, Fig. 2.4, governed by the Laplace equation

(2.96)

where V =0 + v.. The perturbation potential ,(P) at any point P canGO

be expressed as the potential induced by a combination of source (a),

and doublet (~) singularities distribution over a bounded surface S .

1 a 1.(P) = If a( - 41rr) dS + ~f J.l an (41rr) dS (2.97)

where r is the distance from the point P to the bounded surface

and n represents the normal to the surface S directed into the fluid

domain. The boundary condition 1n this study is the flow tangency

condition or

v . n = 0 (2.98)

Since there are two unknowns a and p, while there is one boundary

condition, there exists an infinite number of source and doublet

combinations which satisfy Eq. (2.97) and one can specify the doublet

42

Bounded

Surfaces

Fig. 2.4 Body immersed in a bounded flowfield.

43

strength and solve for the source strength or vice-versa. Using the

idea of real external and imaginary internal fields introduced in [17],

the condition for the perturbation potential interior to the region is

constant and set equal to zero, i.e.,

or 4» = l.l

Thus, it follows that

-a =V; • n (2.100)

For Neumann problems, the source density distribution is given by

Eq. (2.100), the doublet distribution is to be determined. For

Dirichlet problems, the doublet distribution is known from the

prescribed potential on the boundaries, and the source distribution is

to be determined.

In order to introduce lift, wake networks extending from the

trailing edge to downstream infinity are considered. As the free stream

Reynolds number increases, the thickness of the wake region diminishes

and 1s often assumed to be zero. Since fluid properties are continuous

across the wake, the wake region is usually modelled as a stream surface

having zero pressure loading imposed by the ·surrounding flow. The wakes

can be represented by a doublet sheet with zero thickness and streamwise

doublet gradient, 'emanating from the trailing edge and parallel to the

free stream. Determination of the strength of the wake constitutes the

Kutta conditi~n which is available in [16]. Some typical numerical

schemes were already presented in the literature [16-21].

44

Chapter 3

APPLICATION OF PERTURBATION METHOD

3.1 Perturbation Technique for Slender Bodies of Revolution

The formulations of Sec. 2.1.1 are applied to the case of a uniform

ideal fluid flow past an axisymmetric slender body. The coordinate

system is fixed to the body and the velocity of the stream at infinity

is U. The incident stream function is given byCD

(3.1)

The stream:function ,0 can be expanded in powers of e2 and the

coefficients ,.(x) can be obtained by using Eq. (3.1) in Eq. (2.19b);J

this yields

'I (x) = U./2 , ,. (x) = 0J

for j ) 2 (3.2)

By inserting these coefficients into Eqs. (2.24a) - (2.24d) together

with the definition of operators Lj and Gj' the first six leading terms

of fnm(x) are given as follows:

(3. 3a)

45

= _.!. U L { sst _ sst _ SS" + Sus log [4X(t-j}]4 CD dx x I-x 5 x

I-x+ S f [S'(x+v) - S'(x) - vSU(x)] v-2 dv

o

x- S f [S'(x-v) - S'(x) + vSU(x)] v-2 dv}

o

- ~ [fa {Slog [4x(1-x}] - S} + (1 __5_)f ]dx 21 S x I-x 21

2f 32 (x) = - i k (G1f 21) = 10 UCD k [S :x2 (SS")]

(3.3b)

(3.3c)

(3.3d)

(3.3e)

(3.3f)

Equations (3.3a) - (3.3f), together with (2.18a) and (2.18b) depend upon

the profile curve. The expressions for fnm(x), a(e) and e(e) can be

evaluated if S(x) is specified. The complete asymptotic expansion

for t(x,r2 ;e) can be obtained from Eq.· (2.25) with to = U x. The flowCD

46

velocity components over the body surface can be determined by taking

appropriate derivatives of +. Next the Bernoulli equation 1s used and

the pressure coefficient on the body surface is evaluated as

U 2C = 1 - (r!-)p GO

(3.4)

, b ) 2

where Us is the velocity distribution on the surface of the body. The

profile curve is' defined by r = e IS(x) on 0 , x , 1.

In this study, two different axisymmetric bodies are analyzed, and

their profiles are given as follows:

1. El"11psoidal Body

S(x) = 4x (I - x)

2. Dumbbell Shaped Body

S(x) = 4bx{l - x)[1 - bx{l - x)]

With the profile shape r = e IS(x) on 0 , x , 1 mentioned, one can

obtain explicit expressions for fnm(x), a(e) and a(e) from Eqs. (3.3a)­

(3.3f), (2.18a) and (2.18b) respectively for each of the above ~wo cases

(Figs. 3.1 and 3.2). Lists of the required expressions are presented in

Appendix E.

3.2 Perturbation Technique for Thin Airfoils

The fonmulation given in Sec. 2.1.2 1s applied to the flow problem

when the airfoil is at rest in an ideal fluid which has unit velocity at

infinity. The airfoil 1s at an arbitrary angle of attack y to the x­

axis. Thus, the complex potential for the incident stream is given by

(3.5)

.5-

.3

.2

.1

-.2

-.3

-.~

RC Xl = .OSC-SCRTe S( Xl)

S( Xl =qX( I-XI

LENGTH/DIRMETER= 10.00

47

-.5 ----10 .1 .2 .] .'i .5 .s .1 .e .~ I.~

X

Fig.3.1 Ellioosidal body qeometrv with E = 0.05

48

.s

••

R(X): .030xSORTCSeX)) B= 3.00

S( Xl =8X( J-Xl ( i-8Xe I-XI)

.2

.1

ct: 0

-.1

-.2

-.3

-."1

lENGTH/Ol~EnER= 16.67

--------------_._----~

-.5 '---__1--__.L..-__.L--__"---__.r..-__L.-_-..I~_ ___" __

o .1 .2 .3 .'1 .5 .6 ., .e .~ 1.:)X

Fig. 3.2 Dumbbell shaped body geometrY with E = 0.03, b = 3

49

Introducing Eq. (3.5) in Eq. (2.48c), one obtains

-1y,.(x) =} eJ to

if j = 0

if j ) 1(3.6)

Using Eq. (3.6) in Eqs. (2.48a) and (2.48b) yields

Jo(x) = 41f siny

J1(x) = -4. C1(x) COSy

Jj(x) = 0

K1(x) = -2. SI(X) COSy

Kj (x) = 0

if j ) 2

if j = 0,2, •••

(3.7)

The functions fn(x) are obtained by inserting these coefficients

together with the expressions in Appendix C defining the operator

L~t into Eqs. (2.61) and (2.62). The three leading terms fn(x) are

computed as follows:

Re fO(x) = 0

fO(x) = 2x s1ny

(3.8a)

(3.8b)

(3.8c)

(3.9a)

(3. 9b)

(3.9c)

w . 2~ 2 [ ~~=S1n e- ~ f .C~S e 1m Lr [fO(t)] - 4w C1(t) COSy de

T 0 S1n e - x ~=x

11'

r l 2 "2" - • 21m f 1(O) =- - - I f 1(S1" a) da

11' 'II' 0

50

(3.10a)Re f2(x) = - ~w Ix~t;») [Re L~ [flex)] + Re L~ [fo(X)]]

f 2(x) =~ {1m L~ [flex)] + 1m L~ [fO(X)]} (3.l0b)

f 2 [ ~t=sin2e- ;. f ~o~ 8 1m L~ [fl(t)] + 1m L~ [fO(t)] dey 0 Sln a-x. ~=x

11'

r2 2 ~ - 2 2 21m f 2(O) =.- - y f

o[f2(Sin e)+2 siny(a2 cos e - 62 sin e)]de (3.lOc)

o I 0 I .The operators LI , LI , L2, and L2 are g1ven in Appendix C. When Eqs.

(3.8a)-(3.1Oc) are used in Eqs. (2.53) and (2.45) they give the

asymptotic expansion for f(x,e) up to O(e3). Once f(x,e) has been

found, one can obtain an expression of ,(z;£) by using Eq. (2.69) and

the pressure coefficient can be evaluated from Eq. (3.4). Since Eqs.

(3.8a)-(3.IOc), (2.67) and (2.68) are functions of S(x) and C(x), once

they are specified, the complete expansion for +(z;e) up to O(e3) can be

determi ned.

In this study, elliptic airfoils and second order Joukowski

airfoils have been analyzed using the above method. For a symmetric

airfoil the function C(x) is set to zero; only the thickness solution

has to be solved. If the airfoil 1s symmetric and is set at zero angle

of attack, then there is no circulation about the airfoil and all rn are

set equal to zero in determining the constants 1m fn(O). Moreover, if

51

the airfoi~ has a sharp trailing edge, special treatment is applied to

obtain each rn, which is given in Eq. (2.71). The profile curve of the

thin airfoil is defined by y. = e[C(x)t/S(x)] on 0 < x < 1, with small

slenderness ratio e. Three different airfoils are used as test cases,

Figs. 3.3, 3.4 and 3.5, and their geometries are as follows:

1. Elliptic Airfoil

Sex) = 4x{1 - x)

2. Symmetrical Second Order Joukowski Airfoil

S(x) = x(1 - x)3

3. Cambered Second Order Joukowsk1 Airfoil

C(x) = Ax{l - x)

S(x) = x(1 - x)3

Here, A is related to the airfoil camber and is defined in Eq.

(2.88b). Using the above information, expressions for

fn(x), rn, ~(e) and S(e) can be derived in each case. A brief

derivation and the lists of the resulting expressions are included in

Appendix E.

.3

.2

.1

vex)= .050~SCRT(S(XJ )

T/C= 10.00~

52

Fig. 3.3 Elliptic airfoil of 10% thickness.

53

.5

.3 Y( Xl = .100J'( C( Xl ~/ - SORT( S( XJ ) )

C( Xl = .QOO-X( l-Xl Sl XI =X( i-Xl .-3

.2 T/C= 6.50Z DELTR=lBO.OO

.1

>- a

-.1

-.2

-.3

-.11

.5X

.'t-.S~__",,:-__~__.a.-__...L--__.L--__..L-__..L..-__L-.-_--Jl-_----I

a

Fig. 3.4 Symmetrical Joukowski airfoil of 6.5% thickness.

54

.5

.'1

.2

Y( Xl = .100x( CL Xl ./- SQRT( S( Xl )]

C( Xl =1 .OOO-X( l-Xl S( Xl =X( l-Xl-1I3

T/C= 6.5cr~ DELTA=!35.00

.1

>- a

-.1

-.2

-.3

-.'4

1.0.e.1.aX

.'1.2.1

-.s....-..__~ ~ ~ _a

Fig. 3.5 Cambered Joukowski airfoil of 6.5% thickness.

55

Chapter 4

RESULTS AND DISCUSSION

In this chapter an assessment of the merits of the perturbation

analysis method is made. The pressure distribution on axisymmetric

bodies and two-dimensional airfoils obtained by the above method are

compared with those of panel method as well as exact solutions whenever

avaliable. The results of the panel method are obtained by using the

low-order panel method code (program VSAERO, developed by Maskew

[18]). The analytical expressions for the ellipsoidal body and elliptic

airfoil are avaliable 1n the literature presented in Sec. 2.2.1.

4.1 Axisymmetric Bodies

Since the analytical expression for an ellipsoidal body is

available, this body is selected to demonstrate the accuracy of the

perturbation analysis method with the variations of the slenderness

ratio and also the number of terms in the expansion of ,. Figs. 4.1a­

4.2c illustrate a comparison of the pressure distributions, as calcu­

lated by the perturbation analysis method with different order of

accuracy in the expansion of ., with analytical solutions for the two

values £ = 0.1 and e =0.2, corresponding to LId = 5 and 2.5, respec­

tively. The results show that the perturbation solutions agree well

with the exact solutions to a high order of accuracy, i.e., up to

0(e8). With the same order of accuracy in • up to 0(e8), the pressure

56

distributions obtained from the perturbation analysis method and the

panel method ·are compared with the exact sol uti ons, with di fferent

numbers of panels in the chordwise direction for £ = 0.10. The

calculations are presented in Figs. 4.3a-4.3c. It is seen that the

results using 30x6 panels and 50x6 panels yield better agreement with

the exact solutions than using 10x6 panels. Again for the same order of

accuracy in • up to O{e8), and using 50x6 panels, three sets of pressure

distributions are compared with four different values of £. The results

are presented in Figs. 4.4a-4.4c. The perturbation solutions agree well

with the exact solution with small slenderness ratio up to 0.1, and the

panel solutions compare well with exact solutions for all values of £.

The dumbbell shaped body is also analyzed by the perturbation analysis

method wi th the expansi on of • up to 0(e4), as we11 as by the pane1·

method. A comparsion of pressure distributions obtained by the above

method and the panel method, using SOx6 pane1~, with different values

of e and profile parameter b are presented in Figs. 4.5a-4.6b. For

small £ and b, the perturbation solutions agree well with that generated

by the panel method. In the present examples, each perturbation case

required only 2 seconds computing time versus 5 seconds (30x6 panels) or

14 seconds (50x6 panels) for the panel method. This computation time

comparison is based on the CDC Cyber 173.

4.2 Two-Dimensional Airfoils

The analytical pressure distribution on the surface of the elliptic

airfoil is avaliable 1n the literature discussed in Sec. 2.2.1. It is

convenient to do numerical experiments for the perturbation analysis

method using this airfoil. Perturbation analysis solutions for an

57

elliptic airfoil, set at zero angle of attack, are compared with exact

solutions, with different order of accuracy in , expansion for £ = 0.10

and £ = 0.15, corresponding to the tIC ratio = 20% and 40% respectively,

as presented in Figs. 4.7a-4.8b. It is apparent that the perturbation

analysis solution gives quite good agreement with the analytical

solution for high order of accuracy in • up to 0(e4). The pressure

distributions calculated from the perturbation analysis method are

compared with the exact solutions, with different values of slenderness

ratio and these are shown in Figs. 4.9a-4.9c. The results show that the

perturbation solutions agree closely with exact solutions up to 20

percent thick elliptic airfoil.

Results of the symmetrical second order Joukowski airfoil obtained

by the perturbation analysis method with the order of accuracy of +expansion up to 0(e3) are compared with the panel method in the

following section. Here, 3.25%, 6.5%, 10% and 13% thick airfoils are

chosen as the test cases. For the calculation presented, the thickness

of the airfoil, angle of attack, and the number of panels are given for

each of the calculation. The calculations presented in Figs. 4.10a­

4.1Oc and 4.11a-4.11c are obtained using different numbers of panels in

the chordw1se direction for symmetrical Joukowski airfoil of 6.5%

thickness at 0° and 6° angles of attack, respectively. The results do

not give a smooth curve near some regions of the leading edge until 50

panels are used. A comparsion of the perturbation analysis solutions

for the symmetrical second order Joukowski airfoil at 0°, 6° and 12°

angles of attack with panel method are presented in Figs. 4.12a-4.15c.

The perturbation solutions agree with the panel solutions except in some

region on the leading edge of the airfoil surface. Table 4.1 illus-

58

trates the appreciable difference between the lift coefficients of the

two methods from the exact solutions.

The perturbation analysis method has been applied to the cambered

second order Joukowski airfoil with the order of accuracy in the

expansion for. up to O(e3). Calculations are presented in Figs. 4.16a­

4.19c. The results generated by the panel method with 0°, 6° and 12°

angles of attack are used for comparison. Both pressure distributions

agree closely for small slenderness ratio and angle of attack up to € =

0.03248 and y = 6° respectively, except at some regions near the leading

edge. The lift coefficients obtained from the perturbation analysis and

the panel method compared with the exact solutions are shown in Table

4.2. In the present example, the computation time for each pertur­

bation case required 8 seconds, while 0.5 seconds (10 panels), 2 seconds

(30 panels) and 5 seconds (50 panels) were required for each panel-

method solution.

59

~r Xl = .1 C01f:C:~T( -iX( 1-Xl: c;;I,,

.8 ~LPH~= .10CCE-Ol 8ETH= .~~8CE~aQ

.6 SQU~RE--- PERTUR8~rrCN LINE--- E~T

I;,,

I

Cl.U

.2

a

"t1-~

!

-.2

-.'1

-.8

-.8

1.0.8.1.sX

.2-1.0 ....--.__""--__""---__...a..-.__-....- --'-__-..L.__---J.__--J.__--J

a

(a) € = 0.1; ~ expanded UD to E 2 •

Fig. 4.1 Comparison of oressure distributions from the perturbationanalysis method with the analytical solution for -ellipsoidal body.

.8

.6

.ll

.2

-.2

-.Il

-.8

-.8

L~t\GTH/DlR:1E!ER = 5.GO

~LPh~= .101CE-Ol BET~= .~8~~aJ

saU~RE--- PERTURB~rrON LINE--- E~T

60

-1.00 .1 .2 .3 .'1 .1 •8 .7 .a . .~ 1.0

X

O( b.} E = 0.1 <t> expanded up to e: lt•

Fig. 4.1 Co·nti nued

a.u

.8

.6

.q

.2

o

-.11

-.8

-.8

L~~GTH/OIR~E!ER = 5.GO

~LPh~= .101CE-Cl BET~= .~8~~~OJ

SOU~RE--- PERTUR8~rrON LINE~-- EXACT

61

crI

!\ "

-1.00 .1 .2 .] •• .8 .1 .8 .~ 1.0

X

(C) e: = 0.1 <P expanded up to e: 6•

Fig. 4.1 Concluded.

62

.2

-.2

R( Xl = .20D.SCRT( '-:X( I-Xl)

LENGTH/OIMi~EiER = 2.50

saU~RE--- PERTUR8~r.ICN LINE--- EXI=CT

.J......

1r

-.'.1

-.8

-.8

.e.1X

.2.1-l.O'----__I...-__.L..--__"--__L..-__L-__'--__L.-__.L....--........J~----'

o

(a) € = 0.2; $ expanded uo to €2 •

Fig. 4.2 Comoarison of pressure distributions from the perturbationanalysis method with the analytical solution forellipsoidal body.

0..U

t.cqr-

.J\

.&

.'1

-.2

-.8

-.8

R( x: = .2JQ.S:RT: l1X( l-X~ 1

lE~~7r/QIFr£rE~ = 2·50

saU~RE--- PERTURBhrICN LINE--- E~Ci

63

-1.0a .1 .2 .3 ., .5 .8 .7 .s .~ 1.0

X

(b) E = 0.2 ; <p expanded up to £1+.

Fig. 4.2 Continu~d.

0­U

.9

.6

.2

a

-.'1

-.6

-.8

R( Xl = .20C.S!:RTC 4X~ l-XI 1

LENGTH/DIAMEfER = 2.50

. SaU~RE--- PERTURSHTION LINE--- EXACT

64

-l.aa .1 .2 .\ .1 .e .7 .8 .~ 1.0

X

(c) e: = 0.2 ; <p expanded up to e: 6 •

Fig. 4.2· Concluded.

65

R( Xl = .1 OO:.SCRT( 4X( I-XlI

LENGTH/OIRMEnER= 5.00 NPRN= 60

ALPH~= .JOI0E-Ol BETR= .98~+ao

.6 CROSS----PERTURBRTION LINE----EX~CT SOU~R~----P~~EL

a..u

.2

o

-.z

-...

-.8

-.8

1.0.~.,.1.1X

-1.0~-_Ioo..-_--"'---",":,-------'"'--__..L..-.__...I..-__.L...-_--J~_-Io

(a). E = 0.1 ; ~ expanded up to £6; 60 panels with cosine spacinq.

Fig. 4.3 Comoar;son of the perturbation and exact solutions withthe oanel method for elliosoidal body.

a­u

lorW.8

.6

.'1

.2

a

-.2

-.11

-.8

-.8 .

\

R( Xl = .1OO.SbRTC 4X( I-XI]

LENGTH/OIR~Ensq= 5.00 NPA~ 160

~LPHA= .1010E-at 8ET~= .~89SE+aa

CROSS----PERTUR8~TION L[NE----EX~CT sau~RE----PhNEL

66

-l.aa .1 .2 .] .'1: .5 .8 ., .8 .~ 1.0

X

(b) e: = 0.1 ep expanded up to e: 6 ; 180 panels 'IIi th cosine spacing.

Fig.4.3 Continued.

a.u

I.

.9

.6

.2

o

-.2

-.'1

-.8

-.8

RC Xl = .1 GC.S~T( 4XC I-Xl)

LENGTH/Oln~ErER= 5.00 NPhN= 300

RLPHH= .101CE-al BETR= .9a99E~aa

CROSS----PERTUR8~TION LI~----EXHCT SOJ~RE----P~NEL

67

-1.0a .1 .2 .~ .'1 .8 .e ., .1 .~ 1.0

X

(c) e: = 0.1 <p expanded up to e: 6; 300 pane1s with cosine scacing.

Fig. 4.3 Concluded.

.6

68

R( Xl = .OSC~S;~T( 4X( I-Xl)

lENGTH/OIRMEfER= 10.00 NPAN= 3CO

F1LPH~= .2S0SE~2 8ET~= .9975C·ca

CROSS----PERTURBRTION l[NE----EX~CT SOUHRE----P~NEl

0­U

.q

.2

o

-.2

-.11

-.8

-.&

(a) E = 0.05 ~ expanded up to E 6 ; 300 panels·with cosine spacing.

Fig. 4.4 Pressure distributions from the perturbation analysisand the panel methods are compared with the exactsolution for ellipsoidal body.

a­u

I.

o

-.2

-.'1

-.8

-.8

RC Xl = .1 DC»SeRre 4X( I-X) )

LENGTH/O I.=i:1ETER= 5. 00 ~~PF~~= 300

ALPH~= .101CE-Ot BET~= .9899E~ao

CROSS----PERTURB~TION LIN:----EXHCT SOJhR£----P~~~EL

69

-1.0a .1 .2 ., .'1 . .1 .8 .1 .8 .~ 1.0

X

(b) £ = 0.10 <p expanded up to £6; 300 ·panels with cosine spacin~.

Fig. 4.4 Continued.

70

:·t\

.8

.6

.li

.2

~ ,0

-.2

R( Xl = .2Ca.SCRT( 4X( I-XI)

LE~GTH/OIRrEiER= 2.50 NPR~~ 3eD

CROSS----PERiURSATION L[NE----EX~CT· SOL~R~----P~~EL

-.8

-.8

1.0.~.5X

.2.1-1.O~ --,-__-....__-.-.I'--__-----'__..4- ---'-__~__~

D

(c) E = 0.20 ; ~ expanded up to £6; 300 panels with cosine spac;nq.

Fig. 4.4 Continued.

71

,..1.~

.8

.6

.li

.2

-.2

-.'1

~~ Xl = .2S0.SC.~i( 4XC I-Xl ~

LE~GTH/OJAMETER= 2·00 NPRN= 300

CROSS----P£~TuRa~TION Lrt\E----EX~CT SOJHRE----P~NEL

+...

-.8

-.8

.~.,.1.aISX

.•2-l.O~__...Io.-__""""'__---'--_..Ao-..__-&..-__--L.__--J~__..L--__....L-__--1

o

(d) E = 0.25 ; ~ excanded UP to £6; 300 panels with cosine soacing.

Fig. 4.4 Concluded.

72

.6

.\1

R( Xl: .03C~aX( l-Xl (1-8X( I-Xl) 8= 2.CO

LENGiH/OIF~ErER= 16.67 NPRN= JGO

~LPHA= .1810E-02 BETR= .99~~~CO

saU~RE ---P~NEL LI ~~E----PERTURBrlr[C~

-.2

-.'1

-.6

-.8

-1.O ......__.1--_-1~_---.1-----1_____r._____'_____L____L____L____J

a

(a) e: = 0.• 03 b = 2; $ expanded up to e:~;

300 panels with cosine spaci"ng.

Fig. 4.5 Comparison of pressure distributions from the oerturbationanalysis method with the panel method for dumbbell shapedbod.y.

73

a­u

.6

.li •

.2

o

-.2

-.Ii

-.8

-.8

R( Xl = .070.aX( t -X) ( 1-8Xr I-Xl) 8= 2 ·CO

LENGTH/O I R:1E fER= 7•1Lt N?F~~= 300

RLPHA= .100~E-ol BET~= .98~9E.ca

SQU~RE----P~NEL LINE----PERTURSRfION

-1.0~-_........__..L_.___.Ioo...--------_~__..L.--__..J..____.L___ ___I"_____ ___'

a

(b) E = 0.07; b = 2 ~ expanded up to £4;

300 panels with cosine soacing.

Fig. 4.5 Concluded.

74

ALPHA= .2729E-02 BET~= .9973E+OC

RC Xl = .030.8XC L-Xl ( 1-8X( I -Xl ] 8= 3. 00

LE~GTH/OI~~EnER= 16.57 ~PM~~ 3eD

saU~RE~--P~NEL LINE----PERTuR8~rrON

.2

.8

..".6

n.. au

\'\

-.2

-.q

-.8

(a) E = 0.03; b = 3; ~ expanded up to £4;

300 panels with cosine spacing.

Fig. 4.6 Comparison of pressure distributions from the perturbationanalysis method with the panel method for dumbbell shapedbody.

75

a­u

I.@-I

t.e.~

.2

o

-.2

R( Xl = .070)l8XC L-Xl ( 1-8X( I-X) ) 8= 3. 00

LENGTH/DIR~ETE~= 7.1li NPR~J= 300

saU~RE----P~NEl lINE----PERTURB~i[a~

-.8

\ -.8

-l.a.......__':--__~__L--._--JL......-----J-----J--.........J------L----.l.--.-J

o

(b) E = 0.07; b = 3 ~ expanded UD to €~;

300 panels with cosine spacing.

Fig. 4.6 Concluded.

76

.8

TIC = 20.0J%

:3I

Ii

.6 SQUHRE--- PERTURaArrON LINE--- EXQCT

.Ii

.2 ,

a- Du

-.2

-.'6

-.S

-.8

.~.e.,.&X

-1.O~_--':-----';:---~--~-_---L__---1__---Jl.-.__L-__L.._---Jo .,

(a) e: = 0.1; ep expanded up to e: 2 , tiC = 20% .

Fig. 4.7 Comparison of pressure distributions from the perturbationanalysis method with the exact solution for ellipticairfoil.

.'1

Y( Xl = .] DC. S:'~T( 4X( I-Xl)

TIC = 2)·OJ~

SCU~~E--- PERIUR~rra~ LINE--- EX=(i

77

o

T

.2

itLrt;

0~

u

I-.2

-.6

-.8

-1.0a .1 .2 ., ., .& .5 ., .8 ., 1.0

X

(b) e: = 0.1 . <p expanded up to e: 3 , tiC = 20~~,

Fiq.4.7 Concluded.

78

l."~y( x; = .150.S~Tr t:X( I-Xll

TIC = 30.00~

.8

.6 SOU~RE--- PERTUR8~ila~ LINE--- E~T

a­u

.2

o

-.2

-.q

-.8

...l

J

-.8

1.0.a.,.5X

.'t.2.1-1.0 -----. ""'---__..I.--.__~__.L...___L___L.___.L____L-------J

a

(a) E = 0.15; $ expanded up to £2~ tiC = 30% .

Fig. 4.8 Comparison of pressure distributions from the perturbationanalysis method with the exact solution for ellipticairfoil.

a..u

1.

.8

.8

.2

o

-.8

-.8

Y( Xl = •150:aSrRT( 4X( I-Xl J

TIe =30· ocr..!

RLPHA= •2=O:3E-01 BETR= •97XC +00

SCU~E--- PERTU~r[ON LINE--- E~T

79

-1.O ....... ---'----......----'-__--L.__-L-__-'--__.L-_---lL.-..---J

a

(b) E = 0.15; ~ expanded up to £3

Fig. 4.8 Concluded.

tiC = 30%

80

I.~y( x: = .D3G.S:RT~ L1xr 1-x: i

TIC = la.OC~

.8 AL?HA= .2505E-02 8~TA= .9975=-CC

.5 SOU~~E--- PERTUR8~rION LINE--- EX~:i

...

•2

!L.U a

-.2

-.Ii

-.6

-.8

-1.0 L--__..1.-__...L.-__--L..__~___.L.______L_____":......___"'____~_ __...

a .1 .2 •., ,Ii: ,5 .e ,7 ,a .~ 1.0X

(a) E = O~05; ~ expanded up to £3; tIC = 10%

Fig. 4.9 Comparison of pressure distributions from the perturbationanalysis method with the exact solution for elliptica i rfoi 1 •

lot.8

.6

.'1

-.s

-.8

YC Xl = .] DC. 5G~T( 4X( l-XJ J

TIC = 2J.OJZ

SCUq~E--- P£~iU~3rlrrON LINE--- EX=CT

81

(E

I

3I

-1.0a .1 .2 .3 .t .& .5 .1 .e ., 1.0

X

lb) ~ = 0.10 <p expanded up to Eo 3; tiC = 20% .

Fiq.4.9 Continued.

82

y£ X~ = .1S0.SCRT( YX( l-Xl ]

TIC = 30.00~

.8 RLPHA= .2303E-O1 BETR= .977OE+OO

.6 SOU~RE--- PERTURBRrrON LINE--- E~T

a..,u

.2

-.2

-.11

-.6

J

-.8

.5X

.Il.2.1-1 .0 r..-.._----.j"-_---'__--'-__--'-__--'-__~___L___...L_..__...L..__ ___J

o

(cJ e: = 0.15; <p expanded up to E 3 ; tiC = 30% I.

Figure 4.9 - Concluded.

-1.0

-.9

R.3

vex]: .100)l(QX1+1- SCRT(SlXJ)) NPHN= toC(Xl: .OOO:lX(l-Xl S(Xl:X(1-XJlI:s3 ON: 500.0

T/C= 6.S0~ RNGlE OF RTTRCK= 0.00 DEGREE

RLPHR= .2519E-D2. a.

-.6

-.'1

CROSS----PERM8RTION L]NE---P~NEL

-.2

tJ 0

\ \

.8

.8

/f+....

+.,. +

+ .. .. .... ..

1.0-, I 0 .1 .2 ., .'1 .s .8 .7 .e .~ 1.0

X

(a) € = 0.03248 y = o0 ; ep expanded up to e: 2 tiC = 6.5%

10 panels wi th cos i n.e spacing.

Fig. 4.10 Comcar;son of pressure distributions from the perturbationanalysis method with the panel method for symmetricalJoukowski airfoil.

-1.0

-.8

Y( Xl = .10C:J( CC Xl .. /- SCRT( 51 Xl}] NF~~l= 30

C( Xl = •OGO:,X( i-Xl S( Xl =X( L-Xl :- ~3 C~~= SCO.O

TIC: 6.S0~ ~~LE OF RTTRCK= 0.00 CEGREE.

84

~LPH~= .2515E-02. O. - 8ET~=1.0000

a..u

-.6

-.~

-.2

o

•2

.'i

.8

.8

cRass----PERrLR8~TrON LI NE--- -P~~l::L

+ ..

I.a-.1 a .1 .2 .3 .tt .5 .8 .1 .8 .~ 1.0

X

(b)0

expanded to e;2; tIC = 6.5%e: = 0.03248 . y = o ; <P up,

30 panels with cos; ne spac ;.ng.

F; g. 4. 10 Continued.

-1.0

-.8

Y( Xl = .10C:J( C( Xl +/- SeRif Sf Xl)] ~.P~l'~= SO

CC Xl = .OOOj1X( l-Xl S( Xl =XI i-X) ~·.3 c~~= 500.0

T/C= 6.5a~ ~\LLE OF ATT~CK= 0.00 DEGREE

85

RLPH~= .2S19E-Q2. o. EET~=l .0000

-.6

-.\1

-.2

.'1

.8

.8

cRass----PERrLR8~T ION LINE--- -P~~~EL

I.G......__.......-.__.L--__..L.-__""---__.a....-.--~--'___ __.....ll....__ _J____.l____.J

-.1 ax

(c) E = 0.03248 ; y = 0°; $ expanded UD to £2; tiC = 6.5% ;

50 panels with cosine spacing.

Fig. 4.10 Concluded.

86

-9

-8

Y( Xl = .lCC2( C( Xl +/ - SC~T( 51 Xl]) ~,pq'J= toC( Xl = .OCO:sX( l-Xl S( XJ =X( L-Xl ~.3 C~J= 500.0

T/C= 6.50% ~~LE OF ~TTRCK= 6.0J DEGREE

~LPH~= .2519E-02. O. EETH=1.0000

-7 CROSS----PERrLfi8RT ION LINE----?~~~EL

-6

x

+

1L---~:.--_L___-1-__-L..__--'---~:__---':__-~:_----I:__-~:__-__:_-.1

a

-2

·1

-s+

~ -li....

+

-J

(a) e: = 0.03248 y = 6°; ~ expanded up to e: 2 tIC = 6.5%

10 panels with cosine spacing.

Fig. 4.11 Comparison of oressure distributions from the perturbation.analysis method with the panel method for symmetricalJoukowski airfoil.

87

-9

C( Xl = .000-:([ l-X) 51 X} =X( l-X] ~)f3 O,'J= 5eo.o

-9 T/C= S.SOl ~\GLE OF ATT~CK= S.CO OEGR~E

~LPH~= .2519E-02. O. aEr~=1.0CCO

-7 cRass----PERrL.~8~TION LI NE---?H~,EL

-6

-5

x1...-.---..LIIIII---~-_..1..-__..1...---...L.---"'----.L..___.........----"'-_:___-.-...--~-.1

a

-]

-1 .

-2

(b) e: = 0.03248o

; y = 6 ; ~ expanded up to £2; tiC = 6.5%

30 panels with cosine spacinq.

Fig. 4.11 Continued.

88

-9Y( Xl = .100:.w( a Xl .. / - SORTf Sf Xl 1] NPH~I= 50

cex): .OOO:sX(l-X) S[X)=X[l-X)~~3 CN= 500·0

-8 T/C= 6.50% ~NGLE OF RTT~CK=· 6.00 DEGREE

~LPH~= .2519E-Q2. O. 6ET~=l.0000

-7 CROSS----PERfUR8RTION LINE----P~NEL

-8

-5

xl.....-__....._-...l'--- ~_........__-...__-...Io____..____.&..____'_____"____~

-.1

a

-]

-1

-2

(c) E = 0.03248 y = 6°; $ expanded up to £2; tiC = 6.5% ;

50 panels with cosine spacing.

Fig. 4.11 Concluded.

-1.Oi­I

-.8

y(X~ = .CSQ.( a X~ ~/- seRf; S{Xl]l ~?~~= SO

C( x: = ·OJC~:<: t -Xl S( x: =x( t -x: lIlt3 CN= SC-J.O

T/C= 3.25% ~GLE OF ~iT~~{= 0.00 OEG~~E

89

~LPHP= .S262E-03. O. 8ETP=1.CCOO

-.5

-.'1

-.2'--

.8

.8

cRas·s----PERrLR8~TION LJ NE----Ph~:E:L

J1.0L----I----.L..---L.---L.----'----L.----'-:------'::---~:__-__:=__-___;l.Q-.1 a .L .2 .~ ." .5

x

(a) E = 0.01624 ; y = 0° ~ expanded up to 82

; tiC = 3.25%

50 panels with cosine spacing.

Fig. 4.12 Comparison of oressure distributions from the perturbationanalysis method with the panel method for symmetricalJoukowski airfoil.

90

..9V(X]= .OSO~(aX]+/- SORT( S(X))) N?~N= 50

cex): .ooo~xr l-Xl S( Xl =X( t-X)~.3 ON= SCO.O

..8 T/C= 3.2S~ h'~LE OF ATTHCK= 6.00 DEGREE

~LPH~= .62S2E-03. O. 8ET~=1 .0000

..' CROSS----PERn..:R8~TION LINE----PRNEL

-8

-5

-3

-2

-I

xI .......__~_-..I......-.._-"-_-..__-...__-.l._-----_.---...__-L..__~---'

-.1

o

(b) € = 0.01624 ; Y = 6° ~ expanded up to £2 tiC = 3.25% ;

50 panels with cosine spacing.

Fig. 4.12 Continued.

91

-9YC Xl = .OSC:~d cr Xl f-/- SC~T( Sf Xl}) ~?~:'l= 5J

C( Xl = .OOO~X( t -Xl S( Xl =X{ i-Xl • ..3 C,'~= 5oo.J

-8 T/C= 3.25Z ~~LE OF ATTRCK= 12.00 OEGRE~

~LPHR= .6262E-G3. O. EETH=1.0000

-7 CROSS----PERfLR8RT I ON LINE----?~~:EL

-6

-s

-2

-1

a +

.7.1al.....-.-------.......---- ...a.-.__.J--__~___L_____a_____Jo.___....J..___~----'

-.1x

(c) e: = 0.01624 ; y = 12° $ exoanded up to E 2; tiC = 3.25% ;

50 panels with cosine spac.ing.

Fig. 4.12 Concluded.

-1.0

-.8

-.6

-.2

.'1

.8

.8

92

Y( Xl = .1 OC)l~ C( Xl .. / - SeRTf S( Xl ); NP~~= SJ

C( Xl = .oaoJtX( l-X) Si Xl =X( L-Xl .·3 O~= 5CO.0

T/C= 6 .50% ~N;LE Of ATTriCK= a.00 CE~,~EE

~LP~~= .2519E-02. O. 8ET~=1 .cooacRass----PERr~E~T reN LINE----P~~~EL

1.0-,I a .t .2 .3 ... .5 .8 .7 .8 ., 1.Q

X

(a) E = 0.03248 y = 0° cf> exoanded up to £2 tIC =6.5%

~O panels with cosine spacing.

Fig. 4.13 Comparison of pressure distributiqns from the perturbationanalysis method with the panel method for symmetricalJoukowski airfoil.

-9L"--

-8

V( Xl = .1 JC.; :~ Xl ... / - SCR r: S( Xl )) NP~~'~= SJ

CC Xl = ·OCC·"X[ L-Xl S~ Xi =X, l-Xl ~ -3 c~.,= 500.0

T/C= 6.50% ~~~LE Or ~TTrlCK= 6.00 CE~~EE

93

~LPh~= .25I9E-Q2. O. 8ET~= I .CoOO

-7

-5

-5

-3

-2

-I

a

CRass~-- -PERrL~e~T ION LINE--- -P~~~~L

1-.1 a .1 .2 .] .'t .5 .8 .7 .e .~ 1.0

X

(b) e: = 0.03248 ; y = 6° <P expanded up to e: 2 • tiC = 6.5% ., ,

50 panels with cosine spacing.

Fig. 4.13 Continued.

y( X) = .10Q)I( C{ Xl +-/ - SCRT( Sf X] ) 1 N?~N;: SO

C( Xl = ·OOG~:<' L-Xl S{ XI =X( l-X: :..3 OtJ= 5CJ.0

T/C= 6.50% ~GLE Of RTTPC:~ 12.CO OEGR~E

94

~LPH~= .2519E-02. O. 8ET~= 1•DeOD

-7

-6

cRass----PE~ruR8~T ION LI NE----Ph~~EL

x

(c) E = 0.03248 ; Y = 12° ; ~ expanded up to E2 ; tiC = 6.5%

50 panels·with cosine spacing.

Fig. 4.13 Concluded.

-1.0

-.8

YC Xl = .ISQ:s( cr Xl pi - seRif S( Xl }) NP~l'l= sa

C( Xl = .OOC~X( l-XJ S( Xl =X( I-Xl ~ -3 CN= 500.0

T/C= 9.75Z h~~LE Of RTTHCK= 0.00 CEGREE

95

~LPH~= .572CE-a2. o. EETf.l=! .0000

-.6

-.11

-.2

.2

.Ii

.8

.8

cRass----PERM8~T ION LINE----PR~iEL

.. to

I.O'--- """--__-'---_--------....-.-__..A...-__....J--__-L..__--L-__--J.-__~

-,I 0x·

(a) e: = 0.04872 ; y = 0° ; <p expanded up to e: 2; tiC = 9.75% ;

50 panels with cosine spacing.

Fig. 4.14 Comparison of oressure distributions from the perturbationanal.ysis method with the panel method for s.ymmetricalJoukowski airfoil.

96

-3.: Y( Xl = .15Q.( C( Xl ... / - SOR T( SI XJ J) ~~pq'l: SQ

+ C( Xl = .0005X( l-Xl S( Xl =X( l-Xl ·.Jl3 CN= 500.8

8ETR=1.00GO~LPHH= .5720E-02. O.

T/C= ·9.7S~ ~~LE OF ~TTHCK= 6.00 CEGREE

cRass----PERfLRB~TION LINE----?~NEL

I.O"----......---.Ao...---""--__...I---__.........__~___.l...____'____'_____L__ ___'

-.1

-.2

.5

-2.2

-2.&

.-1.8

fj -1.0

x

(b) € = 0.04872 ; y = 6° ~ expanded up to E 2 ; tiC = 9.75% ;

. 50 panels with cosine spacing.

Fig. 4.14 Continued.

97

TIC= 9.75% ~~~LE OF qT7~CK= l2.00 DEGREE

CC Xl = .CCJ-:«( t -XJ S( XI =X~ i-Xl ~ -3 O~~= 5:C·J

~LRH~= .S72CE-02. O.

\11-

-8

-7- '"

CROSS-- --PERflR8RT rG~ L1~E-- - -P~~~EL

-6

-5

-3

-2

-1

a

I ......__...-..-........_~__...a...-__.........--.,...---~__---'-__~__---'--__-.A..-__-

-.1x

(c) e: = 0.04872 ; y = 12° ; <p expanded up to E 2 ; tiC = 9.75%

50 panels with cosine spacing.

Fig. 4.14 Concluded.

98

Y( Xl = .200:a( a Xl +/ - SORi( S{ Xl ]] ~,?~N= 50

CC Xl = .OOO~X( t -Xl S( Xl =X( L-X) »~3 DN= 5eo.o

T/C= 13.00% ~~LE OF RTTPCK= o.co OEGR~E

~LPH~= .1030E-O l. O. 8ET~.: 1.0000

-.6

-.2

~ 0

.2

.8

.8

cRass----PERnR8~TION LI NE---?h~~EL

+

1.0'---__.....__..L.-__-&-.__..L-__-'------'-------------L----lo-----"------.1 G

x

(a) E = 0.06496 ; y = 0°; ~ expanded up to E 2 ; tiC = 13% ;

50 panels with cosine spacing.

Fig. 4.15 Comparison of pressure distributions from the perturbationanalysis method with the panel method for symmetricalJoukowski airfoil.

99

-3.0Y( Xl = .200:.d a Xl .. /- SORT( S{ XJ)) ~l?~N= 50

C( Xl = .QOO:Jxr L-Xl S( XI =X( l-XJ ~ -3 ON= seQ.o

-2.8 T/C= 13.ao~ ~GLE OF RTT~CK= 6.00 DEGREE

~LPH~= 11 030E-Q l. aI BET~=l .occa

CROSS----PERfCHBHTION LJNE---P~NEL

.i .2 ., .'1 .5 .8 .7 .8 ., 1.0X

(b) e: = 0.06496 ; y = 6° 4> expanded up to e: 2 tiC = 13%

50 panels with cosine spacing.

Fig. 4.15 Continued.

.8

-.6

-.2

1.0~-_........__........__--a...-__-.a.._-~-_---.l_-_.L-..__.l-__.J..-__...L-----'

-.1

-1.8

-2.2

~-l.O

100

-9-

C( Xl = .COO.:«( l-X~ 5{ XI ::X( l-X1 :. - 3 O~j= 5:0.0

-9 TIC=" 13.00% ~'~L~ OF qTT~CK~ L2.CO OcGRE~

~LPHM= .103CE-Ol. O. 8ETH=1.ODeo

-7 cRass----PERrLR8~iIC~ LINE----?A~EL

+

x.,a

l~__~:11111'~....a...-__----__~_--.....----.....-_._..___....___'_____...____J

-.1

o

-8

-s .

-1

-2

" -3

(c) e; = 0.06496 , <P expanded up to e;2 tiC = 13%

50 panels with cosine spacing.

Fig. 4.15 Concluded.

Table 4.1 Comparison of 11ft coefficients of the perturbation, panel and exactsolutions for the symmetrical Joukowskl airfoils

. i

Slenderness Exact Perturbat1on Panelratio C - C C' - C

C.. ,ex C.. ,pt C.. ,P"!,pt !,ex x 1001 "'8" ",ex x 100%

£ C.. ,ex t,ex

y = 6°0.06496 0.71588 0.72245 0.67684 0.92 -5.45

0.04872 0.70234 0.70603 0.66472 0.53 -5.360.03248 0.68797 0.68961 0.65377 0.24 -4.970.01624 0.67278 0.67319 0.63080 0.06 -6.24

y = 12°0.06496 1.42391 1.43698 1.34485 0.92 -5.550.04872 1.39698 1.40432 1.32128 0.53 -5.420.03248 1.36840 0.37167 1.29823 0.24 -5.13

0.01624 1.33819 1.33901 1.23750 0.06 -7.52

-.8

-.5

r

-'2~I

:J D~II

I

.2~

.8

.8

102

Y( Xl = .050.' a Xl ~/ - SORT( Sf Xl )) 'L:)~~;.: SC

C( Xl =l·OOC:tX[ L-X) S( Xl =X( l-Xl "»3 ON= 5CJ.O

T/C= 3.25Z A~LE OF RrT~CK= O.CO QEGR~~

~LP~~= .6216E-Q3. -.3137E-04 3cT~=1 .OCCO

cRass----PERrLJ~8~ r ION II NE --- -P~~E'-

1.0-,t .L .2 .] .'1 .5 .8 .7 .8 .~ 1.0

X

(a) e: = 0.02688 y = 0° <p expanded up to e: 2 tIC = 3 .. 257~

50 panels with cosine spacing.

Fig. 4.16 Comparison of pressure distributions from the perturbationanalysis method with the panel method for camberedJoukowskf airfoil.

-9

-8

-7

-8

-s

-3

-2

-I

Y( Xl = .OSO:t( a Xl +/- SDRTI Sf Xl)) N?~N= 50

C( Xl =1•OOOllX( L-Xl S( Xl =X( l-XJ ll1.3 O~J= 5eo.o

T/C= 3.25% ~LE OF RTT~CK=6.CO DEGREE

~LPH~= .62~6E-03. -.3137E-a! 8ETR=1 .OGeo

CROSS----PERrLRSHT ION II NE----?~~~EL

103

o

1~__......-.__~__"",,--_~ ......... -L..__....L---~-_--L-._-""

-.1x

(b) e: = 0.02688 ~ exoanded up to £2; tiC = 3.25% ;

50 panels with cosine soacinq.

Fig. 4.16 Continued.

-9-

-8

-7

-s

-s

veXl= .05Q~raX] ~/- SCRT( S(Xl I 1 ~?~~= SO

C( Xl =1·OOC.X! L-Xl Sf Xl =X{ ~ -X; ~»3 ON= SCJ.O

T/C= 3.25% R\GlE QF AiT~CK= l2.00 DEG~EE

~LPHR= .62~6E-a3. -.3137E-04 3E i h:l .CCOO

CROSS----PERfLREfiT ION LI NE--- -P~~~EL

104

-3

·2

-I

a

1-,I a .1 .2 .] .i .8 .8 .7 .e .s 1·0

X

(c) e: = 0.02688 y = 12° <P expanded UD to £2 . tiC = 3.25%,

50 panels with cosine spacing.

Fig. 4.16 Conc 1uded.

-l.C-

-.8

-.6

vex}: .loO-(aX} ~/- SO~T(S{Xl J) ~?qN= SO

C( Xl =1·000]1 X( l -X1 S( X) =x(. l -Xl ~ t 3 O~~= 500 •a

T/C= 6.50Z ~GLE OF RTT~CK= 0.00 OEGRE~

~LPH~= .249~E-02, -.2538E-03 8ET~=1 .ODeD

cRass----PERfLqS~T ION LI NE----?~NEL

105

-.'1

-.2

.8

.8

x

1.a........__~ -"-__--L.__-..I. L...-__..L.-.__..L-__-l-__-I-__-J

-.1 a

(a) e: = 0.05376: '; Y = 0° $ expanded up to E 2 ; tiC = 6.5% ;

50 panels with cosine spacihg.

Fig. 4.17 Comparison of pressure distributions from the perturbationanalysis method with the panel method for camberedJoukowski airfoil.

106

YC Xl = .1 QO:s( CC Xl +-/ - SORT( Si Xl )) ~P:~~'J= 50-9

CeX)=1.0CO:sX(L-X) S(XJ=X(l-X)"I .. 3 ON= 5CC.O

-8 T/C= 6.SQZ h~LE Of RTTRCK= 6.CO DEGREE

~LPH~= .2~9qE-02. -.2538E-03 8ET~=1 .ODCO

-7 CROSS----PERrLR8~TICN LINE----P~NEL

-6

-s

-3

-2

-I

a

1.0.~.1x

,tt.2a1......--__.c;p---....----~ --.a.....---.&....---"""'---.a--_ _.JL...__._ ___J~_~

-.1

(b) e: = 0.05376 ; Y = 6° ~ expanded up to £2; tiC = 6.5% ;

50 pa ne1s wi th cos i ne spac.i ng.

Fig. 4.17 Continued.

107

Y( Xl = .1 OO~( a Xl +/- seRTf Sl Xl )] ,'J?~~~:: SO-9

C( Xl =1·OOC·sX{ l-Xl S( XI =X( t -Xj 11-3 ON= 5CJ.C

-8

-7

-6

-5

-3

-2

·1

a

T/C= 6.50% ~GLE OF RTT~C~= l2.CO OEGR:E

~LPH~= .2~9~E-a2. -.2538E-03 8ET~=1 .0000

CROSS----PERrL.~8~T ION lINE----P~~~~L

+

1~-~--.-~-------'_---'--__..L...----J.--....L---L-----1---..J...--J-.1 .2,3.II .5 .8 •1 .8 •3 I •0

x

(c) e:"= 0.053761 ; y = 12° ~ exoanded UD to £2 tiC = 6.5%

50 panels with cosine spacinq.

Fig. 4.17 Concluded.

0­U

-I.!)

-.8

-.6

-.q

-.2

o

.2

.Ii

.8

.8

108

.,

+ f' .~

Y( Xl = .150.{ C( Xl +/- SORTf 51 Xl )) NP~N= 50

CeX)=1.000-X{1-Xl SlXl=X(1-X) ••3 ON= SCO.O

T/C= 9.75% ~~LE OF ATTRCK= 0.00 DEGREE

RLPHA: .5593E-C2. -.8722E-Q3 BET~=l.coao

CROSS----PER~~8~TION LINE----PRNEL

+

1.0-.1 a .2 .3 ., .8 .e .7 .8 ., 1.0

X

(a) € = 0.08064 y = 0° <f> expanded up to £2. tiC = 9.75%,

50 panels with cosine spacing.

Fig. 4.18 Comparison of pressure distributions from the perturbationanalysis method with the panel method for camberedJoukowski airfoil.

109

-~.aY( Xl = .1SCJl( CC Xl +/ - SC~ 'i( 51 Xll] ~~pq.'~= 50

C( Xl =1.OCO~X( l-Xl S( Xl =X[ l-XJ :It -3 C~~= 500·0

-3.5

l'

-3.0~

++...

-2.5

-2.0

~-l.S

-1.0

-.5

a

T/C= 9.7S~ ~'~LE OF HTTHCK= 6.00 DEGREE

RLPHH= .5593E-02. -.8722E-03 EET~=l .0000

CROSS----PERM8~TION LINE----PRNEL

...+

x

.5 . , .

I.a-....-.__.... .---~__.........__......_.____'_ ____"___.llo...o__~

-.1

(b) e: = 0.08064 ; Y = 60~ expanded UD to E

2; tiC = 9.75%

50 panels with cosine spacinq.

Fig. 4.18 Continued.

110

-~­

I

II. ·9...-.

·7J

I~,I

: If --:

Y( Xl = .1 sa.( a Xl "./ - s:.~ ~( Sf Xi ~; t\P.~'·~= 50

TIC= 9.75Z ~~LE CF ~Tr~CK= t2.00 C:GRE~

~LPH~= .SS93E-C2. -.8722E-03 eET~=!.OOGO

cRass----PERfL~a~i I eN LINE-- --?~~~EL

-6

-3

-2

-1

o•

.~

x••.1a1~-_.....-.-~~......-.--....a--__....L.-___I_____'______L..__........____""____..___~

-.1

(c) e: = 0.08064 ., $ expanded up to £2 tiC =9.75%

50 panels with cosine spacing.

Fig. 4.18 Concluded.

111

-1.0

..

Y( Xl = .200.( a Xl +/- SQRT( SI Xl )1 NPf1N= 50

C(X)=1.000-X( l-Xl SI XJ=X[ 1-~ ••3 ON= 500.0

T/C= 13.0rrl ~GLE OF RTTRCK= 0.00 DEGREE.8

••

-.6

-.8

-.'1

+-+

,.-.2

+..

+

a.. au

.1

~LPHR= •9900E-{]2. -.2 I 20E-(2 BETR:1.0000

CRCSS----PERTURBATION LINE----PRNEL

xo

I.OL-__..L__--L.__--1 L-__~--~--___'::__-~::_---'=_--_:_-__:_

-.1

(a) e: = 0.107521 Y = 0° ; ~ expanded up to £2 tiC = 13%

50 panels with cosine sp~c;ng.

Fig. 4.19 Comparison of pressure distributions from the perturbationanalysis method with the panel method for camberedJoukowski airfoil.

112

-3.C.-·'(( x: = .2JJlI( a x: .. / - S~R ~( S~ Xi 1) ~.;)~n= 50

:( X1 =1 .CJD.:<! l-Xl S(:<: =X( l-X: '-.3 O~L: 5CG.O

-2.5- TIe = ! 3 . JO% ~,~~= CF Mi i~C.<':: 5.CO OEGR::::

-2.2- CRass ----PE.~ fl,R3~T [ O~l LI~~E.- - - -?~~Ef_

..

-1.9

-l.ll

~ -1.0

-.6

-.2

1.0

... + .. • .. • ~ .,.

x••.2.tal.a'--__--'-..----..I~--~__.._._I'___ __' __'_ ...._.__.....---

-.1

.8

(b) e: = 0.10752" ~ expanded up to £2 tiC = 13%

50 panels with cosine spacing.

Fig. 4.19 Continued.

113

-9Y( Xl = .20(h( a Xl .;./- SORTr S( Xl }) N?~N= 50

(eX)=1 .OOO~X{ l-Xl S( XJ =X( L-~~~3 ON= SCO.O

-8 T/C= 13.00% ~~LE OF RTT~CK= 12.00 DEGREE

-7 +

~LPr~= .9900E-Q2. -.2120£-02 8ET~=1 .oeoacRass----PER~~8RTION LINE----P~NEl

-6 +

•

x1.......__..........-....~.....-.__..-...-__.....-.__---.__--..._--"'"--_-.lI.-.-__L....-_-"-------.&

-s

a

-]

-I

-2

(c) E = 0.10752 ; Y = 12° $ expanded up to £2 tiC = 13%

50 oanels with cosine spaci"nq.

Fig. 4.19 Concluded.

Table 4.2 Compar1son of 11ft coeff1cfents of the perturbat1on. panel and exactsolutions for the cambered Joukowski airfoils

Slenderness Exact Perturbation Panel C - C C - Cratio C C.. ,pt C t.pt t.ex x 100% t,pn t.ex x 100%i,ex -t,pn C.. ,ex C

£ t,ex

y = 00

0.10752 0.61721 0.69115 0.63346 11.98 2.630.08064 0.46641 0.50658 0.46610 8.61 -0.070.05376 0.31268 0.32987 0.30390 5.50 -2.810.02688 0.15689 0.16101 0.14812 2.62 -5.59

y = 6°0.10752 1.32350 1.40981 1.29947 6.52 -1.820.08064 1.16264 1.20984 . 1.12378 4.06 -3.340.05376 1.99734 1.01767 0.95359 2.04 -4.390.02688 1.82841 0.83332 0.77878 0.59 -5.99

y = 120

0.10752 2.01528 2.11303 1.94816 4.85 -3.330.08064 1.84614 1.89984 1.76670 2.91 -4.300.05376 1.67107 1.69432 1.58984 1.39 -4.860.02688 1.49084 1.49649 1.38619 0.38 -7.02

115

Chapter 5

CONCLUDING REMARKS

The perturbation analysis method has been used to analyze the

potential flow, due to a uniform stream, past an axisymmetric body or a .

two-dimensional airfoil. The theoretical formulations have been

described in Chap. 2. Results obtained from the above method for

certain classes of either axisymmetric slender bodies or two-dimensional

thin airfoils have been compared with those generated by the panel

method, as well as with exact solutions whenever available.

Numerical experiments for ellipsoidal bodies and elliptic airfoils

have shown that the higher order expansion of +in £ gives better

results, as compared with exact solutions than lower order repre­

sentations. The perturbation analysis method for axisymmetric slender

bodies give pressure distributions· which compare well with the panel and

exact solutions for small slenderness ratio up to £ = 0.1, corresponding

to a LId ratio of 5. The perturbation analysis method is quite

inexpensive, because little computer memory is required; but it 1s only

applicable to slender bodies. It is suggested to be an applicable

method as a preliminary step in modelling some simple fuselages as part

of the analysis of a wing mounted on'a slender axisymmetric fuselage.

The utility of the perturbation analysis method for the two­

dimensional airfoils is also confined to the thin airfoils with small

slenderness ratios up to £ = 0.03248, corresponding to tIC = 6.5%. The

116

perturbation analysis method seems to relieve the numerical instability

of the panel method at some regions near the leading edge of the airfoil

surface. The perturbation analysis method can also provide more

accurate results for thin airfoils (i.e., £ tends to a small value) as

compared with the panel method. Moreover, the computing time and

computer memory required are relatively small as compared with an

existing panel method. The perturbation analysis method is recommended

as a method which will provide reliable solutions for thin airfoils at

low cost. Using the thin airfoil analysis as an illustrative example,

the perturbation analysis method can be extended to analyze the three­

dimensional wing as the next task. If the extension to the three­

dimensional wing can be completed, the next step will be the development

of a program for the wing design problem with the axisymmetric fuselage

effects on a pair of wings. This study has been only a preliminary step

for the wing-fuselage design problem.

In addition, the perturbation analysis method can be applied to

solve the electrostatic and magnetostatic potentials around a slender

conducting body in a similar manner.

117

REFERENCES

1. Van Dyke, M., Perturbation Methods in Fluid Mechanics, TheParabolic Press, 1975.

2. von Karman, T., ·Calculation of the Flowfield Around Airships,"NASA TM -574, July 1930.

3. Zedan, M. F. and Dalton, e., ·Potential Flow Around AxisymmetricBodies: Direct and Inverse Problems,· AlAA Journal, Vol. 16, No.3, March 1978, pp. 242-250.

4. Zedan, M. F. and Dalton, e., -Higher-Order Axial SingularityDistributions for Potential Flow About Bodies of Revolution,"Com uter Methods 1n lied Mechanics and En ineerin , Vol. 21, No., arc , pp. •

5. Shu, J.-Y. and Kuhlman, J. M., ··Calculation of Potential Flow PastNon-Lifting Bodies at Angle of Attack Using Axial and SurfaceSingularity Method,U NASA CR-166058, February 1983.

6. Kuhlman, J. M. and Shu, J.-Y., ·Potential Flow Past AxisymmetricBodies at Angle of Attack," Journal of Aircraft, Vol. 21, No.3,March 1984, pp. 218-220.

7. Handelsman, R. A. and Keller, J. B., -Axially Symmetric PotentialFlow Around a Slender Body,· Journal of Fluid Mechanics, Vol. 28,part I, Apr1l 1967, pp. 131-147.

8. L1ghth1ll, M. J., "A New Approach to Thin Airfoil Theory," T:leAeronautical Quarterly, Vol. 13, November 1951, pp. 193-210--.--

9. Hoogstraten, H. W., ·Uniform Valid Approximations in Two­dimensional Subsonic Thin Airfoil Theory,· .Journal of EngineeringMathematics, Vol. 1, No.1, January 1967, pp. 51-66.

10. Geer, J. F. and Keller, J. B., "Uniform Asymptotic Solutions forPotential Flow Around a Thin Airfoil and the ElectrostaticPotential About a Thin Conductor,I' SIAM Journal on AppliedMathematics, Vol 16, No.1, January 1968, pp. 15-101.

11. Geer, J. F., "Uniform Asymptotic Solutions for the Two-dimensionalPotent1 a1 F1 e1d About a 51 ender Body, II SIAM Journal on Appli edMathematics, Vol 26, No.3, May 1974, pp. 539-553.

12. Handelsman, R. A. and Keller, J. B., "The Electrostatic FieldAround a Slender Conducting Body of Revolution," SIAM Journal on

118

Applied Mathematics, Vol. 15, No.4, July 1967, pp. 824-841.

13. Homentcovschi, D., "Uniform Asymptotic Solutions for the Two­dimensional Potential Field Problem with Joining Relations on theSurface of a Slender Body," International Journal of EngineeringScience, Vol. 20, No.6, 1982, pp. 153-767.

14. Homentcovschi, D., '·Uni form Asymptoti c Sol uti ons of Two-dimensi analProblems of Elasticity for the Domain Exterior to a Thin Region,"SIAM Journal on Applied Mathematics, Vol. 44, No.1, February 1984,pp. 1-10.

15. Karamcheti, K., Principles of Ideal-Fluid Aerodynamics, Wiley,August 1966.

16. Thomas, J. L., Luckring, J. M•. , and Sellers, W. L., III,"Evaluation of Factors Determining the Accuracy of LinearizedSubsonic Panel Methods," Presented at the 1st AIAA AppliedAerodynamics Conference, Paper No. AIAA-83-1826, July 1983.

17. Bristow, D. R. and "Grose, G. G., "Modification of the DouglasNeumman Program to Improve the Efficiency of Predicting ComponentInterference and High Lift Characteristics," NASA CR 3020, August1978.

18. Haskew, B., "Predicting of Subsonic Aerodynamic Characteristics--aCase for Low-order Panel Methods, II Presented at the AIM 19thAerospace Sciences Meeting, Paper No. AIAA 81-0252, January 1981.

19. Bristow, D. R. and Hawk, J. D., "Subsonic Panel Method for theEfficient Analysis of Multiple Geometry Perturbations," NASA CR3528, March 1982.

20. Haskew, B., Rao, B. M., and Dvorak, F. A., ·Prediction ofAerodynamic Characteristics for Wings with Extensive Separations, ..Paper No. 31 in Computation of Vlscous-inv1scid Interactions,AGARD-CPP-291, September 1980.

21. Maskew, B., ·Predicting Aerodynamics of Vortical Flows on Three­dimensional Configurations Using a Surface-Singularity-PanelMethod,· Paper No. 13 1n Aerodynamics of Vortical Type Flows inThree Dimensions, AGARD-CP-342, April 1983.

22. Glauert, H., IIA Generalized Type of Jouko~ski Aerofoil," Reportsand Memoranda of the Aeronautical Research Committee No. 911,January 1924.

23. Currie, I. G., Fundamental Mechanics of Fluids, McGraw-Hill BookCompany, February 1979.

APPENDICES

119

APPENDIX A

DETERMINATION OF THE COEFFICIENTS OF aCe) and e(e) FORAXISYMMETRIC BODIES

Appropriate constants ak need to be determined such that 9k(x) is

regular at x=O. One can proceed in the following way. The

functions 9k(x) are defined by

g(x,e) • 2k 2 2 1/2= L 9 (x)e = {[x - a(e)] + e S(x)}k=O k

(AI)

• 2where «(e) = r «n en. Since S(O) =0 is assumed, as x is set equaln=l

to zero 1n Eq. (AI), one obtains

, k ) 1 (A2)

To determine «t explicitly, one squares both sides of Eq. (AI) and has

2 2 · "[k ] 2k[x - «(e)] + e S(x) = r .r gJ.(x) gk_j(x) ek=O J=O

Here, «0 =- x is defined, such that «(e) can be written as

•a( e) = L

n=O

Using Eq. (A4) in Eq. (A3), the left side of Eq. (A3) becomes

(A3)

(A4)

120

aD[k ]2k 2r .r ClJe ~_Je e: + e: S(x)k=O J=O

Equating the coefficients of the same power of £2 on both sides of the

above equation, one obtains

kr Cl.~_. + ~k 1 Sex) =

j =0 J J ,

k.r gJe (x )9k_Je(x)J=O

(A5)

where 6mn is the Kronecker delta, i.e., 6mn = 1 if m=n, 6mn =0 if m ~ n.

For k = 0

For k = 1

(A6)

. (A7)

Using Eq. (A2) and with the Taylor series expansion of Sex), Eq. (A7)

yields a1= 51 (0)/4 •

For k > 2, Eq. (A5) becomes

(ABa)

Using Eq. (A2) in Eq. (ASa), one has

(ABb)

One can determine ot recursively by using Eqs. (ABa) and (A8b) and has

the result as follows:

121

_ 51 (0) 2 51 (0)5 11 (0) 4a(e) - 4 e - . 32 e

1 [[SI(O)]2SIII(O) SI(O)[SII(O)]2] 6 + 0(£8)+02r 6 + 2 £ (A9)

Similiarly, 9k(x) can be made regular at x=1 and the coefficients Bk are

determined by following the same procedures as above. The resulting

expressions are as follows:

g(x,e)• #v 2k 2 2 1/2= I gk(x)e ={[x - B(e)] + e S(x)}

k=O(AIO)

• 2n .where B(e) =1 - I Bne with BO = 0 and Bl = - SI(I)/4 •

n=l

-(All)

k kI BJo B + ~k 1 S(x) = I 9J

o(X)9k_J- (x)j=O k-j' j=O

90(x) =1 - x

For k ) 2

k ) 0 (AI2)

(A13)

(A14)

(A15)

(A16)

122

See) =1 + SO(l) e2 _ SO(l)SOO(l) e4-,- 32

(All)1 [[so(l)]2sooo(l) SO(l)[SOO(1)]2] 6 8

+ 10 6 + 2 e + O(e )

By using the above resulting equations, a(e) and B(.e) can be evaluated.

123

APPENDIX B

BOUNDARY CONDITIONS IN COMPLEX PLANE

The complex potential ,(z) is defined in Eq. (2.35) as

+(z)= +(x,y) + i,(x,y) which is analytic in the z-plane outside G. The

derivative of the complex potential with respect to z is denoted by W{z)

and is known as the complex velocity, i.e.,

w(z) =~ =.!t + ..!tdz ax ' ax

=!! ,.!tay - ay

=u{x,y) - iv{x,y)

where u and v are the velocity components in x and y direction

respectively.

From Fig. 2.2, on the surface of the airfoil, one finds that

~ =!.ax u

The equation of the profile G is given by

(Bl)

(B2)

y = e [C(x) t IS(x)] on a < x < 1

*= € [C'(X) ±~]

Equating Eqs. (82) and (83), yields

v - € U [C' (x) ± ~~~{~)] = 0

Using Eq. (B1) in Eq. (2.39), and considering the real part of the

124

(83)

(84)

resulting expression, one has the same expression as Eq. (84).

To satisfy the boundary condition at infinity in Eq. (2.39), one

needs

8 SRe f f(~,£)d~ =0 and 1m f f(~,£)d~ = r

« B

where r is the total circulation about the body. Physically, the first

condition says that there is no fluid flow through the surface of the

body.

125

APPENDIX C

EXPRESSIONS FOR TWO-DIMENSIONAL AIRFOILS

1. Linear Operators of the Integral Equation

Linear operators L:,p=O, 1 in Eq. (2.50) are defined 1n Sec. 4 of

[11], and some of the related operators are given by

g(x,e:)

nbn(x) = &n,o - j~l aj(x)bn_j(x)

• n 2 2 1/2= L 9n(x)e ={[m - x - 1eC(x)] + e Sex)}n=O

(Cl)

(C2)

h(x,e) = r hn(x)en = {[s - x - 1C(x)]2+ e2S(x)}1/2 (C3)n=O

hn,j (x,F) = tr I~t~) j [ (T-x-i eC) -( 2k+l) [ F(T) (C4)

- 2~ F(P) (x+i eC) (T-x-i eC) p]l} deP=O pI e=O

where T = [m(e)-S(e)] cos2e + S(e) and C =C(x).

In particular, with n = j = 0 in Eq. (C4), yields

2. Determination of the Coeffi.cients of a(e) and 8(e)

-The functions g(x,e) and h(x,e:) are defined in Eqs. (C2) and (C3)

126

~ .where aCe) = L am em and B{e) =1 - L amem •

m=2 m=2

Following the similar procedures mentioned in Appendix A, the general

expressions for the determination of <1(£) and a{£) can be found as

below:

9 (O) = a , m) 2m m

91 (x) = iC{x)

For m ) 3

1 m-2 I

«- = - iC'(O) ~ - ~ ~ (I 9 (0) (C6)m in-I c. j~2 j m-j

9 (x: ) = -(I _ 1C(x) [ ()] 1 m~2[ () ()] (C7)m m x· am-I+ 90.-1 x + 2xj~2 aj a.n-j. - gj x gm-j x

Applying Eqs. (C6) and (C7) with m=3 and m=4, one obtains

8 = 0 = 8o 1

h (x) = 1 - xo

h (1) = Bm m

, m ) 2

h (x) = 8 + S(x)2 - 2 2(1-x)

For m ) 3

m-2 .8m = - ; Bm.,_,'1,. CI ( I) +~ LB. hi. ( I) .

, j =2 J rn-J

Using Eqs. (C8) and (C9) with m=3 and m=4, this yields

= C(x}S'(x)- 63 ~ i. 2

2(1-x)

h (x) = - B + B S(x) _ [C(x)]2S(x) _ TS(x)J~4 4 2 2(1-x)2 2(1-x}3 8(l-x)

(C8)

(C9)

By using the resulting expressions, a(E) and 8(£) "can be determined.

128

APPENDIX 0

ELLIPTICAL COORDINATE SYSTEM

Consider the equations

x =Acosh~ COSn

y =A s1nh~ sinn cose

z =A sinh~ sinn sine

(01)

With appropriate combination of x,y,z in Eq. (01), one has two families

of elliptical (~ is constant) and hyperbolic (n is constant) curves as

shown in figure D. The fineness ratio is defined by

cotht =length of major axislength of minor axis

The system of the reference unit vectors at point P are denoted

by e t , en' and es' corresponding to the coordinate t, n, s. Making

reference to [15], the scale factors can be determined

(D2)

2 2 1/2ht =hn =A(cosh t - cos n)

hS =A sinht sinn

(D3a)

(D3b)

129

! RIA

~o

X/A----t---t---t------+---+--~~-~--.~---I---\---+----~

Fig. 0 Two families of elliptical and hvoerbolic curveswith different values of ~ and n .

l::SU

The unit vectors of the desired coordinate system in terms of the

~artesian coordinate system are given by

et

=~ (f sinht CoSn + J cosht sinn coss + ~ cosht sinn sins)~

e = -}- (i cosht sinn - J sinht COSn coss - I( sinht COSn sins) (04)n n

es = - J sins + k coss

Making reference to Chap. 2 of [15], one obtains

(D5a)

ht

et

h e hs

esn n

vxv 1 a a a (D5b)= hthnhs at all as

htVth v 0

Tl n

Using Eqs. (03a) and (D3b) in Eqs. (05a) and (D5b), yields

2 2a • + a • _ coth t .!! - cotn .!! = 0~ ~ at an

To find out the boundary conditions, one uses the requirements of no

penetration and infinity conditions on Eq. -(D5a), and has

(06)

,(~,n) = constant = C· on (07)

as ~ + infinity

131

(08)

By using Eqs. (D7) and "(08), Eq. (06) can be solv"ed by separation of

variables method.

132

APPENDIX E

SIMPLIFIED EXPRESSIONS FOR AXISYMMETRIC BODIES ANDTWO-DIMENSIONAL AIRFOILS

1. AXISYMMETRIC BODIES

Using the Eqs. (3.3a)-(3.3f) with the specified Sex), one can find

out some leading terms of fnm(x), «(e) and See) as follows:

1.1 Ellipsoidal Body

Sex) = 4x(1 - x)i

246 8«(£) = £ + £ + 2£ + D{e )

246 8B(e) = 1 - e - e - 2e + O(e )

1.2 Dumbbell Shaped Body

133

S(x) = 4bx(1 - x)[l - bx(l - x)] , b ) 2

f10(x) = 4bw U~(2x - 1)[2bx(1 - x) - 1]

3 2 2 432+ b(28x -42x +16x-1)+2x-1] + b x(168x -42Ox +356x -114x+10)

224 3! 2f 21 (x) = 8b • U.[b x(36x - 90x + 76x -24x+2)

3 2+ b(28x -42x +16x-l)+2x-l]

2 2 4 6B(e) =1 - be - b (1. + b)e + O(e )

2. TWO-DIMENSIONAL AIRFOILS

2.1 Elliptic Airfoil

If the airfoil is ~ymmetric and is set at zero angle of attack,

134

then y =0, f n =0 and C(x) =O. Notice that the imaginary parts of the

functions fn(x) all vanish, which simplifies the derivation. With all

these assumptions, one can find out f(x,e) up to 0(e3) as shown below:

f 1(x) = - 2{1 - 2x)

f 2(x) = - 4(1 - 2x)

f 3(x) =- 8(1 - 2x)

B{e) = 1 2 4- e - e -

2.2 General Joukowski Airfoil

With the specified S(x) and C(x), all the operators can be

evaluated, through Eqs. (3.8a)-(3.10c) can be evaluated and the

constants 1m fn(O) can be computed from Eq. (2.71). With this

information, the following three fn(x) are obtained as follows:

fO(x) = i(2 siny)(x - 1)

Re f1(x) = (cosy + Asiny)(x - 1)(1 - 4x)

1m f 1(x) = 2{x - 1)[2(Acosy - siny)x + s1ny]

135

Re f 2(x) = O.S(x - 1){3 (COSy + Asiny)(8x2+ ax + 1)

+ 2A [6(Acosy - siny)x(1 - 2x) + siny(1 - 4x)]}

2O.S(x - 1)[A(cosy + Asiny)(24x - 16x + 1)

+ (Acosy - siny)(24x2 - 20x + 1) + 4siny(2x - 1)]

Notice that fn(x) are complex-valued functions, so each fn(x) is

represented by

, n = 0, 1, 2

Using the above results in Eq. (2.73), with the definition of lift

coeff~c1ent Ct,pt one obtains

Ct,pt =2. [s1ny + O.S(Acosy + s1ny)e + 0.2SA(cosy)e2] + 0(e3)

The leading terms for a(e) is given in Eq. (2.67). Since this airfoil

has a sharp trailing edge, S(e) should be set to one. Thus, by using

the resulting expre~sions, the asymptotic expansion for ,(z;e) is given

by

· 1 2 k 1 In(z-~) fk(~) dl; 3,(z;£) = e-1y z - -- ~ ~ fL'- -----.-.0lIl..-..-. + O( £ )

2. k=O «(e) [(1 _ ~)(~ _ «)]1/2

In particular, if A = 0 or C(x) = 0, the real part of the results are

136

suitable for the symmetrical Joukowski airfoil. For this case, the

first two terms of +(z;e) above agree exactly with Eq. (12.9) of [10],

when his results are expanded up to 0{e3).

The third order lift coefficient for the general Joukowski airfoil

is formulated to compare with the results obtained from the perturbation

analy~ls and the panel methods. Making reference to [15, 22, 23], the

exact ] i. ft coeffi ci ent. ·for the Joukowski ai rfoi 1 is gi yen by

Ct ex= 8. a si~{y~J/t, where y represents the angle of attack, .-,

and! is the chord of the airfoil. The radius a, the angles Band 6 are

connected by the Eq. (2.76). By applying the similar technique in Sec.

2.1.3, a perturbation parameter £1 1s chosen, ~hen

a = C{1-e1cos~)/{1-0.5 e~Sin2~) and t = 4C{1-e1cos~)2/{1-2e1cos~).

By using the resulting expressions and retaining terms up to order··

of e12, the lift coefficient can be.expressed as

Ct,ex =.21rSi,n{Y+S){1 - e1cos~ - e12cos2~ - 0.5 e12sin2~) + 0{e13)

where S =t~n-1{e1sin&/{1-e1cos~» and the lift coefficient for the

Joukowski airfoil can be computed up to 0{e13) if the angle of

attack y, the parameters ~and e1 are specified.