Method of Discrete Vortices

METHOD OF DISCRETE VORTICES s. M. Belotserkovsky I. K. Lifanov Russian Academy of Sciences Moscow, Russia Translated ...

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METHOD OF DISCRETE VORTICES

s. M. Belotserkovsky I. K. Lifanov Russian Academy of Sciences Moscow, Russia

Translated by

V. A. Khokhryakov English Edition Editor

G. P. Cherepanov Florida International University Miami, Florida

Boca Raton

CRC Press Ann Arbor London

Tokyo

Library of Congress Cataloging-in-Publication Data

Bclotserkovsky, Sergei Mikhallovich. Method of discrete vortices / Sergei M. Belotserkovsky, Ivan K. Lifanov. p. em. Translated from Russian. Includes bibliographical references and index. ISBN 0-8493-9307-8 I. Vortex-motion. 2. Aerodynamics-Mathematical models. /. Lifanov, /. K. (Ivan Kuz'mich) II. Title. QA925.B44 1992 629. 132'3-dc20

92-13460 CIP

This book represents information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Every reasonable effort has been made to give reliable dala and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the puhlisher. Direct all inquiries to CRC Press, Inc., 2000 Corporate Blvd., N.W., Boca Raton, Florida 3343 I. ©1993 by CRC Press, Inc. International Standard Book Numher 0-8493-9307-8 Library of Congress Card Number 92-13460 Printed in the United States of America

I 2 3 4 5 6 7 8 9 0

Printed on acid-free paper

Authors' Preface to English Edition

The advent of high-speed supercomputers resulted in narrowing the gap between fundamental mathematical and applied problems and enhancing their interdependence. The emergence of a new powerful and versatile method of analysis-the numerical experiment-has brought together the physical essence of a problem, its mathematical formulation, and a numerical method of solution taking into account specific features of computers. Most efficient solutions proved to be natural descriptions of physical phenomena embracing the whole of the process of their development. The predominance of discrete and non stationary approaches was established. The ever-growing requirements of practice result in a constantly growing complexity of applied problems for which traditional numerical methods often prove to be inadequate. Quite often applied researchers win the competition in developing a rational approach to solving problems, because their reasoning is based on understanding the essence of a problem, whereas professional mathematicians treat it in a more formal way. A numerical experiment facilitates finding rationale in such an approach by creating favorable conditions for rigorous mathematical verification and generalization. The development of a method encompassing the three major aspects of the problem-physical, mathematical, and computational -is decisive for achieving success. It must be stressed that the role of mathematics is not restricted to verifying or generalizing a method: it and only it makes a method both strict and versatile as well as extendable beyond the limits of a family of problems. This book is the result of 20 years of work by the authors and their pupils, who have traveled together along the aforementioned path. Even before that, the analysis of physical and mathematical peculiarities of aerodynamic problems resulted in elaboration of a rational approach to their solution; the correctness of the method was verified by general analysis and logical reasoning. Then the method was thoroughly verified and perfected by using systematic numerical experiments and tested by comparing exact solutions to special problems of aerodynamics with calculated data. This stage of the development of the method of discrete vortices (MDV) and its application to solving both stationary and nonstationary problems of aerodynamics for a variety of lifting surfaces was summed up by S. M. Belotserkovsky in his doctoral thesis, which was approved in 1955. Here the MDV was treated as a method of solving singular integral equations, iii

iv

Authors' Preface to English Edition

both one dimensional (airfoils, cascades, annular wings) and two dimensional (monoplane wings of arbitrary plan form). Further development of the method called on profound mathematical verification and generalization that was implemented by I. K. Lifanov and permitted spreading the ideas of the method into neighboring areas. The first results of this stage in the development of the MDV were generalized in our monograph published in the Soviet Union in 1985. The present book incorporates all the materials of the monograph, which were revised, corrected, and further developed by the authors. We have also included additional material obtained since 1985. The formation of the new trend, which was developed aggressively during the last two decades, was supported by regular fruitful discussions at the All-Union seminars directed by the authors. While writing this book, we used theoretical and calculated data as well as the useful advice of V. A. Aparinov, N. G. Afendikova, V. A. Bushuev, Yu. V. Gandel, A. V. Dvorak, V. V. Demidov, V. A. Ziberov, A. F. Matveev, A. A. Mikhailov, N. M. Molyakov, M. I. Nisht, L. N. Poltavskii, A. P. Revyakin, E. B. Rodin, A. A. Saakyan, M. M. Soldatov, I. Va. Timofeev, and S. D. Shapilov. To all of them we express our profound gratitude. We also thank V. A. Khokhryakov for translating the book. Sergei M. BeJotserkovsky Ivan K. Lifanov

Contents

Author's Preface to English Edition Contents

iii v

An Introduction to Singular Integral Equations in Aerodynamics ..... 1

Introduction PART I:

21 QUADRATURE FORMULAS FOR SINGULAR INTEGRALS

Chapter 1. Quadrature Formulas of the Method of Discrete Vortices for One-Dimensional Singular Integrals 1.1. Some Definitions of the Theory of One-Dimensional Singular Integrals 1.2. Singular Integral over a Closed Contour 1.3. Singular Integral over a Segment 1.4. Singular Integral over a Piecewise Smooth Curve 1.5. Singular Integrals with Hilbert's Kernel 1.6. Unification of Quadrature and Difference Formulas Chapter 2. 2.1. 2.2. 2.3. Chapter 3. 3. I. 3.2. 3.3. 3.4.

27

29 29 32 39 51 57 58

Interpolation Quadrature Formulas for One-Dimensional Singular Integrals 61 Singular Integral with Hilbert's Kernel 61 Singular Integral on a Circle 66 Singular Integral on a Segment 69 Quadrature Formulas for Multiple and Multidimensional Singular Integrals Multiple Cauchy Integrals About Some Singular Integrals Frequently Used in Aerodynamics Quadrature Formulas for Multiple Singular Integrals Quadrature Formulas for the Singular Integral for a Finite-Span Wing

75 75 83 92 96

Contents

vi

3.5. 3.6.

Quadrature Formulas for Multidimensional Singular Integrals 103 Examples of Calculating Singular Integrals . . . . . . . . . . 110

Chapter 4. 4.1. 4.2.

Poincare-Bertrand Formula ... . . . . . . . . . . . . . . . . . 115 One-Dimensional Singular Integrals 115 Multiple Singular Cauchy Integrals 120

PART II:

NUMERICAL SOLUTION OF SINGULAR INTEGRAL EQUATIONS 125

Chapter 5.

Equation of the First Kind on a Segment and / or a System of Nonintersecting Segments . . . . . . . . . . . . . . . 127 5.1. Characteristic Singular Equation on a Segment: Uniform Division 127 5.2. Characteristic Singular Equation on a Segment: Nonuniform Division 148 5.3. Full Equation on a Segment 154 5.4. Equation on a System of Nonintersecting Segments 162 5.5. Examples of Numerical Solution of the Equation on a Segment 168

Chapter 6.

6.1. 6.2. Chapter 7. 7.1. 7.2. 7.3. 7.4.

7.5.

Equations of the First Kind on a Circle Containing Hilbert's Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Equation on a Circle . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Equations with Hilbert's Kernel 182 Singular Integral Equations of the Second Kind Equation on a Segment Equation on a Circle Equation with Hilbert's Kernel Equation on a Piecewise Smooth Curve with' Variable Coefficients Equation with Hilbert's Kernel and Variable Coefficients

Singular Integral Equations with Multiple Cauchy Integrals 8.1. Analytical Solution to a Class of Characteristic Integral Equations 8.2. Numerical Solution of Singular Integral Equations with Multiple Cauchy Integrals 8.3. About an Integrodifferential Equation

191 191 196 199 203 212

Chapter 8.

217 217 224 239

Contents

vii

PART III:

APPLICATION OF THE METHOD OF DISCRETE VORTICES TO AERODYNAMICS: VERIFICATION OF THE METHOD 243

Chapter 9.

Formulation of Aerodynamic Problems and Discrete Vortex Systems 245 Formulation of Aerodynamic Problems 245 Fundamental Concepts of the Method of Discrete Vortices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 Fundamental Discrete Vortex Systems 251

9.1. 9.2. 9.3.

Chapter 10. Two-Dimensional Problems for Airfoils 10.1. Steady Flow Past a Thin Airfoil 10.2. Airfoil Cascades 10.3. Thin Airfoil with Ejection 10.4. Finite-Thickness Airfoil with a Smooth Contour 10.5. Method of a "Slanting Normal" " 10.6. A Permeable Airfoil

259 259 264 266 271 276 281

Chapter 11. 11.1. 11.2. 11.3.

Three-Dimensional Problems Flow with Circulation Past a Rectangular Wing Flow without Circulation Past a Rectangular Wing A Wing of an Arbitrary Plan Form

283 283 288 293

Chapter 12. 12.1. 12.2. 12.3.

Unsteady Linear and Nonlinear Problems Linear Unsteady Problem for a Thin Airfoil Numerical Solution of the Abel Equation Some Examples of Numerical Solution of the Abel Equation Nonlinear Unsteady Problem for a Thin Airfoil

303 303 309

12.4. Chapter 13. 13.1. 13.2. 13.3. 13.4.

13.5. Chapter 14.

14.1.

Aerodynamic Problems for Blunt Bodies Mathematical Formulation of the Problem System of Integrodifferential Equations Smooth Flow Past a Body: Virtual Inertia Numerical Calculation of Virtual Inertia Coefficients: Some Calculated Data Numerical Scheme for Calculating Separated Flows

315 316 323 323 324 328 331 335

Some Questions of Regularization in the Method of Discrete Vortices and Numerical Solution of Singul~r Integral Equations . . . . . . . . . . . . . . . . . . . . 343 II1-Posedness of Equations Incorporating Singular 343 Integrals

Contents

viii 14.2. 14.3.

Regularization of Singular Integral Calculation 346 Method of Discrete Vortices and Regularization of Numerical Solutions to Singular 350 Integral Equations 14.4. Regularization in the Case of Unsteady Aerodynamic Problems 357 PART IV: SOME PROBLEMS OF THE THEORY OF ELASTICITY, ELECTRODYNAMICS. AND MATHEMATICAL PHYSICS 359

Chapter 15.

Singular Integral Equations of the Theory of Elasticity 361 15.1. Two-Dimensional Problems of the Theory of Elasticity 361 15.2. Contact Problem of Indentation of a Uniformly Moving Punch into an Elastic Half-Plane with Heat Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 15.3. On the Indentation of a Pair of Uniformly Moving Punches into an Elastic Strip 380

Chapter 16.

Numerical Method of Discrete Singularities in Boundary 387 Value Problems of Mathematical Physics 16.1. Dual Equations for Solving Mixed Boundary Value Problems 387 16.2. Method for Solving Dual Equations 396 16.3. Diffraction of a Scalar Wave on a Plane Lattice: Dirichlet and Neumann Problems for the Helmholtz Equation 404 16.4. Application of the Method of Discrete Singularities to Numerical Solution of Problems of Electromagnetic Wave Diffraction on Lattices 414

Chapter 17.

17.1.

17.2. 17.3.

Reduction of Some Boundary Value Problems of Mathematical Physics to Singular Integral Equations Differentiation of Integral Equations of the First Kind with a Logarithmic Singularity Dirichlet and Neumann Problems for the Laplace Equation Mixed Boundary Value Problems

419 419 421 423

Conclusion

425

References

433

INDEX

439

An Introduction to Singular Integral Equations in Aerodynamics t

The present book provides an effective direct method of the numerical solution of singular integral equations for both one and two (or more) dimensions and includes multiple integrals, especiaIly as applied to separated and vortex flows in aerodynamics. The authors of the book are a professional mathematician (Ivan Lifanov) and a numerical experimentalist in aerodynamics (Sergei Belotserkovsky) who have coIlaborated for many years. The book* represents a notable milestone in the brilliant 2SD-year history of the theory of functions of a complex variable and provides a strong impetus for future development of singular integral equations in aerodynamics. With a general aim to popularize singular integral equations in aerodynamics, the purpose of this introduction is to elucidate these issues under the following headings: 1. 2. 3.

A little bit of history Lift force of a thin airfoil Optimal airfoil problem

The latter may be of interest for those seeking future engineering applications of the method presented in the book.

. ~

Prepared by Genady P. Cherepanov, Florida International University, Miami. The content of this bOQk is different from that of the recent book entitled Two-Dimensional Separated Haws (CRC Press, 1993) because the lalter does not cover singular integral equations at all. However, they are connected by common ideas and form an excellent collection of desk references for every aerospace engineer.

1

Method of Discrete Vortices

2

1.1. A LITTLE BIT OF HISTORY Both aerodynamics and the theory of functions of a complex variable originated in the work of the Russian academician Leonhard Euler more than two centuries ago. However, Euler seemed to kill his newly born hydrodynamics when he proved the theorem of zero drag of a body moving in a perfect fluid. Engineers lost interest in this subject, and it took more than a century to revive their interest after works by Helmholtz, Kirchhoff, Kelvin, 10ukowski, Rayleigh, and other great scholars appeared. Their computations of drag and lift forces laid the foundation for future applications of aerodynamics in aviation and turbine construction. The theory of functions of a complex variable was being intensively developed during this time, presumably as a theoretical branch of mathematics until 1900, but it became strongly applied in the twentieth century. 1 will mention here only some results of primary importance for singular integral equations. A Cauchy integral is the complex function c/J( z) of a complex variable z of the following shape:

c/J(z)

1 = -.

f

cp( T) dT

27ft L

T E

,

T -

L,i =

Z

1=1.

(1.1 )

Here L is a closed or unclosed contour in a z plane and cp( T) is a continuous or discontinuous complex function. The function of a complex variable is said to be analytic at a point if it is differentiable at the point any number of times and expandable into a convergent power series at the point. The Cauchy integral in Equation 1.1 provides a function c/J(z) that is analytic at any point z outside Land discontinuous on L, that is, c/J(z) tends to different values c/J'(t) and c/J - (t) when z tends to the same point t of L from the left-hand or right-hand side of L, respectively, if one goes along L. A singular integral is the complex function c/J(t) of a complex variable t defined by the divergent improper integral 1 cp(T) -f-dT 27fi L

T -

t

(1.2)

tEL,

'

understood in the sense of Cauchy principal value as

c/J(t)

1

cp(T)dT

p.v·fL 27ft T - t

= -.

. 1 hm-f

E-->o27fi L.

cp( T) dT T-t

'

tEL,

TEL.

(1.3)

3

Introduction

FIGURE 1.1. The equilateral EO neighborhood of a point t excluded from the integration contour L participating in the definition of the Cauchy principal value. The arrow shows the direction of running L, and + and - show the location of discontinuity points on L relating to this direction.

Here L" is L without its vicinity of t cut off by a circle of a small radius *' centered at t (Figure 1.1). For example, calculating the improper integral

d.x

--= ~x-c b

lim "1-->0 "2 -+ ()

[c-", d.x~ c-X

b - c *', = I n - - + limc - a ",' () €2 ' EZ----+

a

< c < b,

(1.4)

0

we get

P.v.

f

b

d.x

--

=

b - c In--.

aX-C

c-a

Hence, the improper integral does not exist (diverges) because the limit depends on the method of vanishing *'1 and *'2. However, a Cauchy principal value of the singular integral, when *'1 = *'2 = *', exists and equals the first term on the right-hand side of this equation, because the second term vanishes in this case. The same is valid for a Cauchy principal value of an arbitrary singular integral. In 1873, the Russian mathematician Sokhotsky derived the basic equations

cP+(t) - cfJ-(t)

=

cp(t)

(1.5)

cfJt(t) + cfJ"(t)

=

p.v.-.f

1

7Ti

L

cp(T)dT T -

t

.

(1.6)

Sokhotsky's work wa~ unknown in the West, and in 1908 these equations were rederived by the German mathematician Plemeli. Sokhotsky equations allow mutual connection of the following famous boundary-value

4


problems: Riemann Problem:

ep+(t)

=

G(t)cfJ-(t)

+ g(t),

tEL.

(1.7)

Here G(t) and g(t) are some given functions, and cfJ +( z) and cfJ( z) are analytic functions required to be found. In the case of unclosed L,

Hilbert Problem:

+ b(s)v(s)

a(s)u(s)

=

(1.8)

c(s).

Here s is the length of arc on L, a(s) and c(s) are some given real functions, and u(s) and v(s) are the real and imaginary parts of an analytic function required to be found. Singular Integral Equation Problems:

With a Cauchy kernel: d(t)cp(t)

+

1

-.P.v.! 7Tt L

M(t,T) T -

t

cp(T)dT=g(t),

tEL.

(1.9)

Here d(t), M(t, T), and g(t) are some given complex functions and cp(t) is a sought complex function. With a Hilbert kernel:

--1 27T b(s)

a(s)u(s) -

0

a - s u( a )cot-- da

27r

2

=

c(s).

(1.10)

All these problems can be reduced one to another; for example, Equation 1.10 can be reduced to Equation 1.8 in the same designations. Poincare, Hilbert, and Noether studied singular integral equations and established some general theorems by reducing them to Fredholm integral equations. Hilbert, Plemeli, and Carleman found explicit solutions to some important particular cases of these problems. The full explicit solution of the problems as formulated in Equations 1.7-1.10 was given by Gakhov in 1936-1941 and Muskhclishvili in 1941-1946 (for full details, see Gakhov 1966 and Muskhelishvili 1946). Similar problems for systems of singular integral equations reduced to the matrix Riemann problem of Equation 1.7, where g, cfJ+, and cfJ are n columns and G is an m X n matrix, are as yet not solved in explicit form.

Introduction

5

In the general problem, explicit solutions exist only for some particular classes found by Gakhov (1952), Cherepanov (1962, 1965), and Khrapkov (1971). As an illustration of the method, we consider Carle man's singular integral equation A

/LCP(x)

+ -. p.v.l 7Tt

) cp( T) dT

0

=

T - X

ig(x)

(I.ll)

(A and /L are positive constants; 0 < x < 1; g(x) and cp(x) are real). The function 1 l)CP(T)dT (z) = - . 27T"t 0 T - Z

(I.l2)

is introduced, which is analytic outside the unit cut (0, 1) along the real axis in the z plane. Using Sokhotsky Equations 1.5 and 1.6, and the Cauchy integral in Equation I.l2, Equation 1.11 can be written as

cfJt(x) + mcfJ (x)

ig(x) = --,

A+/L

A-/L m=-A + /L'

0< x < 1,

-l<mx

xt(X) is the value of X 1(z) on the upper face of cut (0, a) of the z plane

having argument

/4 and

7T

am

is the angle of attack.

Introduction

19

The function W'(z) is finite at z = 0 and z = a, if equations a

fo x

[;"(x)

O'm -

--------:-1/,..,.-4 3/4

(a - x)

dx =

M _/ 'TrY 2 u, V 1

-

2 M~

,

(1.74) are met. The upper boundary of the optimal airfoil sought in this case is given in the form

1

y=--1 + u,

x,

fc)

(1--

a

v(a -/)X - Vt(a -x) 14

[;"(/) -

14'

27Tfa [t(a - I)] 1 [x(a - x)] 1

1 +-";1 - M;;u, [(a-X)'/4 -ii x ?

-

/ (X - - )1 4]) a-x

1- x

dx.

O'm

dl

(I. 75)

Despite the explicit form of the solution, the problem of optimal airfoil existence, even in such an engineering formulation, has not, as yet, been solved because of difficulties in comprehensive analysis of complicated functions of several free parameters. The methods developed in this book provide a good opportunity for solving the problem of an optimal airfoil using supercomputers. For example, let us show how the problem in Equations 1.66-1.71 can be reduced to a singular integral equation. At first, we notice that the solution to boundary Equations 1.66-1.71 may be written in the form of Equation 1.56, where f'n (t) f' +(t)

f'

y'(/)

=

(I) =

when 0
0 being so small that it inte~ts the curoe at two points t' and t" exactly. Let the arc I be denoted by t' t". Consider the integral

f L\I

cp( t) dt t - to

If for € -+ 0 the integral tends to a finite limit, then the limit is called the Cauchy principal value of the integral, that is, l(to)

=

lim ~--> 0

f L \I

cp(l) dt t - to

~

f cp(t) dt . L

t - to

(1. 1.6)

Muskhelishvili (1952) proved that the class H * of functions on a piecewise smooth curve L is invariant with respect to an integral in the sense of the Cauchy principal value (a singular integral). In other words, if cp(t)€H* on L, then l(to)€H* on L.

1.2. SINGULAR INTEGRAL OVER A CLOSED CONTOUR Let us start by considering the singular integral 1(10)

=

f

L

cp(l)dt t - to

(1.2.1)

around the circle L whose radius is equal to unity and which is centered at the origin of coordinates. Here cp(t) is a class H function on L.

Quadrature Formulas for One-Dimensional Singular Integrals

33

For simplicity, we start by considering the integral Io{to)

=

dt 1. --, I.t -

to

for which it is known that (Muskhelishvili 1952) ( 1.2.2) Let us choose on L two sets of nodes: E = (t k , k = 1, ... , n) and Eo = {t Ok ' k = 1, ... , n}, such that the points t k , k = 1, ... , n, divide the circle into n equal parts, and point t Ok is the middle of the arc t k + I' where t n I = t l · In what follows the sets E and Eo chosen in such a way will be called a canonic division of circle L.

4

+

Lemma 1.2.1.

For any point tOj€Eo the following inequality is fulfilled:

( 1.2.3) where M k = tk; I - t k , k = 1, ... , n. Let O(Iln) or 0li(lln) be a quantity of the order of 1In. Hence, the right-hand side of the inequality equals Bin or Bliln where the constant B or B li is independent of n (and the constant B li depends on the parameter {;).

Because L is a unit circle centered at the origin of coordinates, one may write

where Ok and 00k are polar angles of the points t k and t Ob respectively, andk=1, ... ,n. Taking into account periodicity of the function expU 0) and denoting TIm by 2wmln - win, m = 1, ... , n, we get

~ [

TIm

Ll TIm

. Ll TIm

+ m~1 cotTcos-2- - SIO-2Ll TIm TIm . Ll TIm )]. Ll TIm + i (cos -2- + cot T Sm -2- Sm -2- , (1.2.4)


34

where !iT/m = T/m + 1 - T/m = 27Tln, m = 1, , n. Note that the numbers T/mI2, m = 1, , n, are located symmetrically with respect to 7T/2, and hence, n

" cot Tim '-

l

m=1

(1.2.5)

O.

=

From Equalities 0.2.4) and 0.2.5) it follows that

Ln

m~1

!it k n 27T ( 1 ) . = n sin 2 -7T + i-sin= i7T + 0 - . t k - toJ n 2 n n

(1.2.6)

Together with (1.2.2) this proves the validity of Inequality (1.2.3). Note 1.2.1. The following estimate is true: j = 1, ... , n.

( 1.2.7)

In fact, we note that

[ 6

~

l..J cos m =

I

..

.J. l Sm

2"

1],;,'-+11 sin. !iT/mI 2 . 2

Sin

T/m 1 2

Hence,

(1.2.8) where [xl is the integer part of x. Let us next analyze an analogous quadrature sum for the integral (1.2.1). Let the sets of points E and Eo form a canonic division of the circle L. Then we designate

j and formulate the following theorem.

=

1, .. . ,n,

(1.2.9)

35


Theorem 1.2.1. Let cp(t) satisJY the condition H( Q') on L. Then the following inequality holds:

j

=

1, ... , n,

(1.2.10)

Proof. For the sake of convenience we put t Oj = 1. Then

-If

I] -

L

12

=

cp(t) - cp(l) d ;, cp(td - cp(l) A t 'u.t k t-l k~] tk - l

Icp(I)1

f -t -dt 1 -

I l.

I

Llt k I Ln -1 . t

k=l

k-

Inequality (1.2.3) gives an estimate for 12 • The expression for I] can be transformed in the following way:

I{"

= I

cp(t n ) tn

-

cp(l) 1

1

27T

n

Because the function cp(t) meets the condition H( Q') on L,

For a unit circle Idtl It - 11 = le iO

=

dO and

+ 11 = I(cos

0 - 1)

+ i sin 01 = 21sin 0/21.

.

36


Hence,

I" < A 2 " I -

O -I +" 1T n sin dO -< / ( ) 0 • 2

l

C

1 dO = 0 (I o n " ).

l 1T/ n0-

It"

For I {" one gets I{".:::;;Alt n

I;

In order to estimate cp(t) - cp(t Oj )

(1) . n"

W l - , ,27T - =0 -

-

n

we use the transform

cp( td - cp( to)

t - t Oj

cp(t) - cp(/d

t Oj

Ik -

(1.2.11) which will be used often in what follows. By (1.2.11) we have

Itk - tl

X

-1--lldtl = t - 1

51

+ 52·

Because cp(t) meets the condition H( 0') on Land t

27T)"n--I j1k

5 O. (1.2.16)

Now let curve L be a set of p non intersecting closed Lyapunov curves L], ... , L p and let the sets Em = {7k, k = n m ] + 1, ... , n m } and E Om = {'TOk' k = n m _ I + 1, , n m } form a canonic Nm = n m - n m .. ] division of the curve L m m = 1, , p; no = O. We denote N

=

min m~],

and suppose that Nm/N ::; R

... ,p

Nm

< +00. We also denote j=I, ... ,n p ,

where 'Tnm ,

m

!i'Tk =

=

'T k +],

k = 1, ... ,n p ' k i=n], ... ,n p , and

!i'Tnm

=

'Tnm

]T]

1, ... ,p.

The following theorem proves to be true.

Theorem 1.2.3. Let cp( 'T) meet condition H on curoe L. Then for any point 'T Oj € U !:, ~ ] E Om , the following inequality holds: A> O.

( 1.2.17)

1.3. SINGULAR INTEGRAL OVER A SEGMENT* Let us assume that in the singular integral (1.2.1), L axis, and the function cp(t)€H* on L, i.e.,

l/J( t) cp(t) = (t - a(b - t)I-" • "Interval" is now commonly used for "segment" (G.Ch.)

=

[a, b] on the real


40

°

where fjJ(t)€H( 0') on [a, b], $ v, /L < 1. Also, let the points to = a, t l' •.• , tn' tn + 1 = b divide the segment [a, b] into n + 1 equal parts h = (b - a)/(n + 1) long, and the point t Oj be the middle of the segment [t j , tj + 1]' j = 0, 1, ... ,n. It will be said that the points of the sets E = {t k , k = 1, ... , n} and Eo = {tOj' j = 0, 1, ... , n} form a canonic division of the segment [a, b] with the subinterval equal to h. The following lemma may be formulated. Lemma 1.3.1.

Inequality

(1.3.1) holds for any point toj€Eo, where B is a certain constant.

Note that the inequality ( 1.3.2) is also true. Lemma 1.3.2. Let the function cp(t)€H( 0') on the segment [a, b]. Then for any point toj€E o, the following inequality holds:

( 1.3.3) Proof. Let us implement the transformation

(1.3.4)


41

Because cp(t) satisfies the condition H( ex), 1?=O(h

IX ),

The use of Formula (1.2.11) gives

For 51 one gets the estimate

For any k and j, the inequality h

- - - s: 2.

(1.3.5)

It; - tojl

holds. Then

;, flh

52 s: Ah '-

k=l k*j

Ik

I

dt

---1-'---,l:---IX It

- t Oj t k - t Oj

The validity of Lemma 1.3.2 is proved by substituting the estimates for 5\ and 52 into the formula for It and the estimates for I.', ... , If into the

formula for II.

•


42

Inequalities (1.3.1) and (1.3.3) allow formulation of the following theorem. Theorem 1.3.1. Let the function cp(t)€H( 0') on segment [a, b] and the sets E and Eo form a canonic division of the segment. Then for any point tOj€E o, one has

(1.3.6) Let us finally prove the following theorem. Theorem 1.3.2. Let cp(t )€H * on the segment [a, b]. Let the sets E and Eo form a canonic division of the segment. Then the inequality j

=

0,1, ... , n, (1.3.7)

holds where the quantity O(tOj) satisfies the inequalities: (a) for all points tOjE[a + 8, b - 8],

( 1.3.8)

°

where 8> is, however, a small number; (b) for all points tOjE[a, b], n

L

O(toj)ILltOjl

j=O

where~ojl = h, j = 0, 1, ... , n. Proof. We may write

:0;

O(h Az ),

( 1.3.9)

Quadrature Fonnulas for One-Dimensional Singular Integrals

43

Here 12 can be estimated with the help of Inequality (1.3.1). To estimate I] we first observe that if r.p(t)€H* on [a, b], then

If I] is represented in the same way as in Inequality (1.3.4), then one gets for If,

because for any j

=

1, ... , n, tj - a -0

=

sin 0'/2

2k

2n + 1

2n + t.

7T, k = 0, 1, ... ,2n,

(2.1.7)

The quadrature formula for the integral (2.1.1) will be constructed in the following way:

S(Oo) =

27r

jo

0 - 00 cot--cpAO) dO

2

0 - 00 2n 1 sin(2n + 1)( 0 - 0d/2 cot-- L cp(Od. dO o 2 k~O 2n + 1 sm( 0 - 0d/ 2 27T

=

1

1 2n + 1

2n

L

k~O

cp( Ok)

j27r 0

0 - 00 sin(2n + 1)(0 - 0d/2 cos-dO. 2 sin(O - 0d/ 2 (2.1.8)


63

Hence, by employing (2.1.2)-(2.1.4), one gets 0 - 00 sin(2n + 1)( 0 - 0d/2 cot-----. dO o 2 sm(O - 0d/2 27r

1

2

n

= 2

L

27T sin m( 00

-

0d

m=l

=

27T cot [

Ok - 00 2

-

cos(2n + l)(Ok -

0 )/2] . 0

sin( Ok - 0 0 )/2

(2.1.9)

At the latter stage we have used the equality 1

- + cos 0 + ... +cosnO + i[sin 0 + ... +sin nO] 2

1 sin(2n + 1)0/2 = _ + e ill + ... e inll = ----------2 2sinO/2

i [ 0 coS(2~ + 1)0/2]. +"2 cos"2 sm 0/2 (2.1.10)

By (2.1.9) the formula for 5(0 0 ) may be rewritten in the form 5(0 0 )

=

2n L

[0 - 0

cp(Od cos

k

0

2

k=O

cos(2n + 1)( Ok - 0 0 )/2 ] 27T . sin( Ok - 0 0 )/2 2n + 1 (2.1.11)

Note that the latter quadrature formula is accurate for any trigonometric polynomial of degree n, because in this case CPn( 0) == cp( 0), and Formulas (2.18) and (2.1.11) provide an accurate value of the integral l( 0 0 ).

Let a function cp( 0) belong to the class H r( Q') on [0, 27T ], Le., cp(r)( 0) E H( Q') and is a periodic function. Let us represent it in the form of Fourier series cp(O)

=

a0 2

oc

+

L k~1

(akcoskO+bksinkO).


64

Then function I( (0) has the following expansion into Fourier series (see Luzin 1951):

I( (0) = 27T

L

(b k cos kO o -

Ok

sin kO o)·

(2.1 .12)

k=l

Let us denote by 0 is a however small number. The latter inequality for

where 0 < v < 1 and cp*(t,7) E H(a, f3) on the set [a,b] limited set of values of 7. Then the function n(to,7)=(to-a)

Vfb

X

T, where Tisa

cp*(t,7)dt

(3.1.10)

v

a

(t - a) (to - t)

belongs to the class H( A, f3 - €) for all points (to, 7) E [a, c] a < c < b, 0 < A < 1, and € > 0 is a however small number.

X

T, where

Proof. Similarly to Theorem 3.1.1, it suffices to consider the difference DUo, 7 + h) - n(t o, 7) presented, in analogy to function k = 1, ... , n, are closed smooth curves. Then, singular integral 0 is a however small number. We shall prove the theorem for the two-dimensional case. By using (3.1.2), 0

fb -b

dx

[I D

y(x,

z) - y(x, zo) - yz/(z - zo) (zo -z)

2

(3.2.6)

Multiple and Multidimensional Singular Integrals

where D

=

D(€, zo)

== [-I, Zo - d U [zo +

€,

87

l],

(3.2.7)

As far as 1K ](x, x o, z, zo)1 ~ 2, we deduce by Inequality (3.2.6) that the first double integral under the limit sign is absolutely convergent. Then

K](x,xo,z,zo) _ 1'-'0' dz - F, ,~ _] + F] D Zo - Z

f

I' l

I

_ +. -

Z

0

21n

IZO Zo

-II + I

(3.2.8) where

In a similar way,

=

2

y(x,zo) (

1

V(x o _X)2 + €2 ) + -'--------

€

xo -

X

K 2 (x,x o,l,zo) +-----(zo -/)(x o - x) K 2 (x, x o , -I, zo) (zo + I)(x o - x) ,

where

(3.2.9)


88 Next we note that

lim

V(x

+

_X)2

€2

-Ix -xl

0

0

0

€(X o

0

=

-x)

X

,

o =f. x.

(3.2.10)

Formulas 0.2.8)-(3.2.10) demonstrate validity of the theorem for the functions y(x, z), such that yx'(x, z) E H on (J. However, from the same considerations it follows that the theorem is also applicable to functions y(x, z) when yx'(x, z) E H* on (J. Note that A(x o, zo) may be written in the form of an iterated integral,

(3.2.11)

where the inner integral may be taken by parts. Thus, we get

_ fb

t

dy(X, z)

-b -I

X

Xo

8z

-x + V(x o _X)2 + (zo

_Z)2

(x o -x)(zo -z)

dxdz,

(3.2.12)

where

In the condition y(x, l) == y(x, -l) == 0 is met (this is the case one has to deal with in aerodynamics), then integral A(x o, zo) may be presented in the form

A(xo,zo)

=

b fl dy ( x, z) f -b -I 8z

X

Xo

-x

+ V(x o _X)2 +

(zo

(x o - x)(x o - z

)

_Z)2

dxdz. (3.2.13)

MuLtipLe and MuLtidimensionaL Singular IntegraLs

89

Let region (J be limited by the straight lines z = -L and z = L as well as by the lines X 2 = xiz) ~ Xl = x](z) belonging to the class H] on [-L, L]. Then integral A(x o, zo) will again be defined by Formula (3.2.5); however, now it has the form

where

We see that A(x o, zo) exists if function dt/Jixo, z, zo)/ dZ E H* on [-L, L] uniformly with respect to X o and zoo However, the criterion is rather inconvenient for checking. • Consider a special case which we shall have to deal with in aerodynamics. Definition 3.2.1. Region (J wiLL be called a canonic trapezoid if it is Limited by the straight Lines z = - L, z = Land the straight Lines X+(

z)

=

aI

+ zb I,

x'(z) >x-(z). (3.2.15)

In what foLLows, [ ( z) wiLL be called the nose and X + (z) wiLL be called the taiL, as is customary in aerodynamics (BeLotserkovsky and Lifanov 1981).

Let us reduce the case of the canonic trapezoid to that of a rectangle. This can be done by considering mapping F of the rectangle D = [0, 1] X [ -l, L] belonging to the plane OX] z, onto region (J, the mapping being defined by the formulas

z

=

z.

(3.2.16)

Point (x~, zo) transforms into point (x o, zo) = (x(xJ, zo), zo), and the Jacobian J of the mapping is given by the formula (3.2.17)

90


For a canonic trapezoid (J, integral A(xo, zo) exists for any function y(x, z), such that function y;(x(x', z), z) E H* on rectangle D. To prove this statement we change the variable appearing in (3.2.14) in the integral for t/J 2, with the help of the first of Equations (3.2.16). Then we get

2

I

xJ(z)dz - -t/J(x(x ,zo)' €

x(xl'2')'2"2,11(2,1].

(3.2.18)

We assume that the function cp(x 1, z) = y(x(x1,z), z)J(z) belongs to the class HI on D. Let us transform 0.2.18) in a manner similar to Formula (3.2.5). To do this, in Equation (3.2.7) we substitute x' for x, cp(x', z) for y(x, z), and K1(x(x', z), x(x~, zo),z, zo) for K1(x, x o, z, zo). Similarly to Formula 0.2.7) one has to calculate the integrals obtained from Integrals (3.2.8) and 0.2.9) by replacing function IK1(x, xu' z, zo)1 ~ 2 by K1(x(X I , z), x(x(L zo), z, zo). To do this we first observe that x( x' , z)

=

a( x I) ""!- zb( x I),

From (3.2.19) one gets

(3.2.20)

Then

/[x(x~,zo) -x(x',z)J + (zo 2

_Z)2

--=----=---------,:----:-1----(zo -z)A(x',xo,zo)

+ C.

(3.2.21)


91

The latter formula may be checked easily by differentiating the right-hand side with respect to z. From 0.2.20) and (3.2.20 we deduce

f

f)

KI(X(XI,Z),X(x~,zo),z,zo)

(zo-z)

2

dz

I I ]2 VI[ A(x I ,xo,zo) + €b(x) +--=---=-------------=--€A

2

+-

€ '

(3.2.22)

where

From the latter formula it follows that the limit

(3.2.23) exists and is a function integrable in the sense of the Cauchy principal value on segment [0,1] of axis OX I • In a similar way it can be shown that the limit

exists and is an absolutely integrable function on segment [0, 1] of axis OX I. Thus, the preceding statement for a canonic trapezoid is proved.

92


Note that if

(J

is a canonic trapezoid, then the following equations hold:

(3.2.24) where

3.3. QUADRATURE FORMULAS FOR MULTIPLE SINGULAR INTEGRALS Let us consider the multiple singular Cauchy integral (sec Equation (3.1.1 a»: cp(t) dt l(to) = {«t - to» .

1.

Similarly to Chapter 1, we start considering quadrature formulas for integral I(t o) beginning with the case when the frame L is a torus, i.e., a product of closed Lyapunov curves. The following theorem is true.

Theorem 3.3.1. Let function cp(t) E H on an m-dimensionaltorus L, and the sets E k = {t ikk , i k = 1, ... , nk} and E Ok = {t~ik' i k = 1, ... , nd fonn a

Multiple and Multidimensional Singular Integrals canonic division of closed Lyapunov curoe L b k

93

=

1, ... , m. Then inequality

(3.3.1) is valid for any point t Oj = (t~jl' ... ' tf:;) belonging to torus L. Tn the latter formula f3 > 0, n = max(n p ••. , n k ), and N Ink $ R < +00 for N -+ oc,

k

=

1, ... ,m,

To prove the theorem we consider the case when m = 2, and L] and L z are circles centered at the origin of the coordinates. One may write

where

By using the results of Section 3.1, we get for singular integrals depending on a parameter,

Because Nlnk $ R < +00, k = 1,2, for N tending to +00, the latter formula terminates the proof of Theorem 3.3.1 for the special case under consideration. The note made in Section 1.4 concerning integrals over Lyapunov closed curves and Inequality (1.2.16) prove the validity of inequalities of the form (3.3.2), and hence, that of Theorem 3.3.1 for the


94

general case. However, 0'], 0'2 entering Inequality (3.3.2) must be replaced taking into account smoothness of the curves L] and L 2 • Note that according to Inequality (1.2.17), Theorem 3.3.1 remains valid in the case when L k , k = 1, ... , m, is a set of nonintersecting Lyapunov curves. 2.

Let now the frame L in multiple integral l(to) be an m-dimensional parallelepiped, i.e., L = L, X ... X L m , where L k , k = 1, ... , m, is an open Lyapunov curve. Then the following theorem is true.

Theorem 3.3.2. Let function q;(t) E H* on m-dimensional parallelepiped L, and the sets E k and E Ok form a canonic division of curve L k with the step h k • Then the inequality

(3.3.3) holds for any point t Oj = (tJ j \, ••• , t(~m quantity O(t Oj ) possesses the properties: 1.

E )Eo =

EO'

X ... X

EOm,

where the

For all points t Oj E L; X ... X L:n, where L~ is the part of curve L k containing no ends of the curve together with their certain neighborhoods, (3.3.4)

2.

For all points n A2 ) "' - O(t O.)h J ] ..... h m -< O(h ,

A2

> 0,

(3.3.5)

j~O

where h

=

max(h], ... , h m ) and h/h k

$

R < +oc for h -40, k

=

1, ... ,m.

Proof. Let us consider the case m = 2. Let L k = [ak> segments of the real axis. Because q;(t) E H * on L,

k

=

bd,

1,2,0
0 is a however small number,

+ 8, 1- 8],

where

(3.4.4) 2.

For all points (X Oj ' zOJ.), n

N

L L

O(X Oj ' zOm)h]h Z

$

O(h A,),

o < Az $

1. (3.4.5)

j=Om=O

Proof. Let us rewrite Formula 0.4.0 as

_t t i~] k~]

y(x;, zod - y(x i ,

ZOk-l)

hz (3.4.6)

Comparing the latter formula with 0.2.12) for integral A(x oj , zorn) and using the results obtained in Sections 1.3 and 3.3, we deduce that Inequality 0.4.3) is true. •


98

Now let the domain of integration of integral A(x o, zo) (see (3.2.1)) be a canonic trapezoid (J. By using the presentation of the integral given by Formula (3.2.24) for the case of a canonic trapezoid, one may construct the required quadrature sum in the following way. I ol I I l' Le tusconsl°d erpolOtsxl, ... ,xnI an d pOlntsxoO,xOI, ... ,xon IOrmlOga canonic division of segment [0,1] with the step hI and points ZI' •. ·' ZN and Zo 0' Zo I' ... ,ZON forming a canonic division of segment [ -I, I] with the step h z . We suppose that h/h k ~ R < +00, k = 1,2; h = max(h p h z ), h -4 O. Denote by B;k(X ik , Zk)' B;Ok(X iOk ' ZOk)' and BOiOk(XOiOk' ZOk) the points of canonic trapezoid (J that are corresponding images of points Bik(xl, Zk)' Bioik(X], ZOk)' and B6iOk(x6;, ZOk) of rectangle D for the mapping F given by Formula (3.2.16). In analogy to sum (3.4.1), let us consider the sum 0

0

(3.4.7) which is a digital analog of integral A(x o, zo) given by (3.2.24). When the sum S(X OjOm ' ZOrn) is written in a form analogous to that of the sum S(X Oj ' zOrn) in Formula (3.4.6) and using the presentation of integral A(x o, zo) in Formula (3.2.24), we come to the conclusion that the following theorem is true.

Theorem 3.4.2. be such that

Let region

(J

be a canonic trapezoid and function y(x, z)

rJ

-(y(x(xl,z),z)J(z» EH* rJz

on rectangle D

=

[0, 1] X [ -I, l]. Then inequality

(3.4.3')

Multiple and Multidimensional Singular Integrals holds for any point (X OjOm ' zOrn), j = 0,1, ... , n, m O,(X6j' zOrn) satisfies Inequalities (3.4.4) and (3.4.5).

=

0,1, ... , N, where

Note that Formula (3.4.7) also may be written in the form N

n S(X OjOm '

zorn) =

L L i~

1

cp(x},zodhl

k~O

The following note is also of interest. Because the equation x = x( x I, z) (see (3.2.16» provides a straight line for any fixed value of x I, the number l X o - x(x , zo) is proportional to the distance between point (x o, zo) and the straight line. Consider integral A(x o, zo) over canonic trapezoid (T at point (X O, zo) lying outside (T but with Zo E ( -I, I). Then Formula (3.2.21) becomes

f

[xo-x(x',z)]dz

(zo - z)\/[ Xo - x(x l , Z)]2

+ (zo - Z)2

V[X O -x(X',Z)]2 + (zo - Z)2 [xo -x(x1,zo)]czo -z)

+ c,

(3.4.9)

where Ix o - x(xl, zo)1 ~ {; > 0. Hence, in the case under consideration, A(xo, zo) reduces to the set of a conventional one-dimensional integral with respect to Xl and a two-dimensional integral with a singularity of the form (zo - Z)-I. Note also that if function y(x, z) entering the integral

A(x o' zo)

=

y(x, z) f bdx fl-t(ZO-Z)

2

-b

X(I+

XO-X

V(xo _X)2

+ (ZO - Z)2

)dxdZ


100

possesses the property described by

fb-b Y (x, z) dx == 0

(3.4.10)

(realized, for instance, in the case of zero-circulation flow past a rectangular wing), then A(x o, zo) may be written in the form

A(x o , zo) =

I

II

b -b

y(x,z)

xo-x

dxdz. (3.4.11)

2

-1(Zo-Z)

V(X O -X)2+(ZO-Z)2

Next we rewrite the latter integral as a repeated one and take the inner integral by parts with respect to x. Then, taking into account Identity (3.4.10) one gets

A(x o , zo) =

f

l fb

-I

Q(x,z) 2

-b [(x o -x)

2 3/2

+ (zo - Z)]

dxdt.

(3.4.12)

Here we neither validated the possibility of integration by parts nor specified the exact sense of the integral entering the latter formula. Let us give the following definition. Definition 3.4.1. Let functions f(x) and g(x) be defined on segment [a, b] and possess the following properties: ['(x) is continuous on [a, b], whereas function g(x) is unintegrable on [a, b], continuous on (a, b], and has the only singularity at point a. However, there exists function G(x) absolutely integrable on [a, b], such that G'(x) = g(x). Then we assume that by definition,

Ic(X)(f(x»

=

tf(x)g(x) dx a

However, if f(x) E H,(a) on [a, b], the function g(x) has a singularity at point X o of the interval (a, b), and the function G(x), G'(x) == g(x), is either absolutely integrable or may be presented in the form cp(x, xo)/(x


- x o) where cp(x, xo) tf(x)g(x) dx

101

H on [a, b], then

E

= -

a

tf'(X)G(x) dx + f(b)G(b) - f(a)G(a), a

(3.4.14) where the right-hand-side integral is assumed to be either an integral of an absolutely integrable function or an integral in the sense of the Cauchy principal value. It can be readily shown that ForfT\'.lla (3.4.14) defines the integral irrespective of the choice of the function's origin. Also, the defined integral operator is linear. Hence, if the point under consideration lies inside the domain, one can define two-dimensional integrals in the manner similar to Formula (3.4.14). Then one gets

A(x o , zo)

=

f

bfl

-b

Q(x, z) 2

-/[(x o -x)

+ (zo

2 3/2

dxdz

-z)]

Q( -b, z)(x o + b)

V(x o - b)2 + (zo + 1)2 + Q( b, -I) --(-x----b-)-(-zo-+-I)-o

+ Q( -

V(x o + b)2 + (zo _/)2 b, I) - ' - - - - - - - - -

(x o + b)(zo - I)

)


102

V(x

o + b)2 + (Zo + 1)2 - Q( - b -I) - - ' - - - - - - - - - - ,

(xo+b)(zo+l)

If G;z(x, z) E H on the rectangle [ - b, b] X [ -I, l], then all the integrals entering this formula are taken in the sense of one-dimensionaltwo-dimensional Cauchy integrals. In the special case of G(x, z) == c = const, one has

(3.4.16)


103

Therefore the quadrature sum for A(x o, zo) will be constructed as follows. Let points Xo = -b,x1, ... ,xn,xn .] =b and xOO,xOI, ... ,xOn form a canonic division of the segment [-b, b] with the step hI' whereas points Zo = -I, z], ... , ZN' ZN+] = I and zoo' ZO], ... , ZON form a canonic division of the segment [-I, I] with the step h 2 • Then we shall designate

n

=

N

L L

fX" 'fZk -

dxdz

I

Q(xoj,zod

2

i=O k=O

x,

_ V(X Oj -X i )2(ZOm

-Zd -zd

(X Oj -X;)(ZOm

[(X Oj -x)

zk

+

(ZOrn - z)

2]3/2

2 ].

(3.4.17)

It can be shown readily that by rearranging the terms, the latter sum may be presented as a quadrature sum for the integral A(x oj , zOrn) written in the form given in (3.4.15). This can be done by paying attention to the signs of the summands appearing in Formula (3.4.17) and corresponding to the point (Xi' Zk) [see Figure 3.1 where the summands are shown by points and the crosses correspond to points (X Oj ' zorn)]'

3.5. QUADRATURE FORMULAS FOR MULTIDIMENSIONAL SINGULAR INTEGRALS Consider the singular integral (Mikhlin 1948) v(x o )

=

f

f)

f(xo,O) r

2

u(x) dx,

(3.5.1 )

104


z

(-l,b)

.-

- +

"

I
0 is a certain fixed number), i.e.,

A> O.

(4.1.8)

In Muskhelishvili (952) it was shown that

'P,(t) ==

f

b

'P ( t , 'T) 'T -

a

t

d'T E

H*

on [a, b]. Therefore, taking into account that point to is fixed and is one of the points 'Tk' k = 1, ... , n, we get

(4.1.9) where A1 is a fixed number. From (4.1.8) and (4.1.9) it follows that A2 > 0,

(4.1.10)

I.e.,

lim ~n(tO) = A(to)·

(4.1.11)

n-->X

Let us now transform the sum

~

n(tO) in the following way:

(4.1.12) By removing the square brackets in ~ ~(to) and taking into account the remarks made with respect to the sum S; and integral I; in Formulas


118

(I .3.17) and (1.3.18), respectively, one gets

. 11m ~~(to)

n

-->

=

fb

x

d'T fb cp( t , 'T) dt 'T - to a I - 10

--

a

fb

-

d'T fb cp( t , 'T) dt 'T - 10 a I - 'T

--

a

(4.1.13) Next we consider ~~(/o)' We have

tJ.I tJ.'T

n

\83

n

="i..J

}

j=1

k

ll

(tj -/ 0 )

2'

(4.1.14) Let us divide the sum ~~(/o) into two summands, one of which summation is carried out over points I j E [(a + 10/2, (b + 10)/2] = i, and the other over all the rest of I j • In other words \t~~(to)

=

L

+

r/J(l j , to) tJ.t j tJ.'Tk ll

I,c[a,bl\[

L

r/J(t j , 10 ) tJ.l j tJ.'Tk ll

=

D,

+ D2 ,

IjEr

r/J(lj,/ O ) = [cp(lj,/ o) + cp(/(p/o)]/(l j - t o{

(4.1.15)

By (4.1.3) we have C h[f l2

d l1',] = 0 (h) , (4 .1. 1) 6 (b - t)

because 0 ~ VI' ILl < 1. As previously noted, function cp(t,/ o) belongs to the class H( ex.) with respect to I on i Hence,

tJ.ltJ.'T

" } 12kID 2 I -< C 2 '-_ I _ IE=/ I 1 I

j

0

ll

a

< C ha

-

2

n " .'J~ I

1

---~I' _ k _ !1 2 a J

0

2

Poincare-Bertrand Formula

119

Thus, for \B~(to) we have (4.1.17) Note that if Q' = 1, then \B~(to) Hence, for \B~(to) we have

3 \B (to) =

n

L.

1

,

j~1 (j - k o - 2)

=

O(hlln hI).

1

m

2

= 2

L

M

I

k-O

(k

+ 2)

2

L

+

k=m+1

(k

+

tr

1 ,

(4.1.18) where m = min(k o, n - k o - 1) and M = max(k o, n - k o - 1). Because to is a fixed point, n ---) x and m ---) 00 imply that M ---) m. The series '[~ ,= 0< k + 1) 2 converges because the second summand in the latter formula tends to zero for n ---) x. Also, it is shown in Hardy (1949) that

L k-O

1

(2k + 1)

2

8

(4.1.19)

Hence, lim \B~(to)

2

= 7T •

(4.1.20)

n-->x

Then from (4.1.14), (4.1.17), and (4.1.20), it follows that lim \B~(to) = -7T 2cp(tO' to)·

(4.1.21)

n-->x

Thus, Equations (4.1.11)-(4.1.14), (4.1.17), and (4.1.21) demonstrate the validity of the Poincare-Bertrand formula for a singular integral over a segment. It should be noted that the proof relies mainly on the grid points lying in the neighborhood of point to or, more so, on the relative positions of points 'Tk and t j within the said neighborhood. Hence, we deduce that a similar proof is valid in the case when L is a piecewise Lyapunov curve containing angular nodes only, in particular for a circle. In fact, if we choose two arbitrary points a l and a 2 lying on a circle (a, =F to, a 2 =F to) and assume that they are nodes, then we get a curve for which the Poincare-Bertrand formula has been proved already. For a circle the formula may be proved in a straightforward manner by using the quadrature formula presented in Section 5.1.

120


In what follows we will come across the following situation when we have to represent integral A(t o) over L = [a, b], as well as both sides of the Poincare-Bertrand formula, by quadrature sums. Let the sets E = {7k' k = 1, ... , n} and Eo = {t j , j = 0,1, ... , n} form a canonic division of segment [a, b] with the step h. Let integral A(to) be considered at points tOm = 7 m , m = 1, ... , n, i.e., A(7m ) =

f

a

b-dt- fb cp( t, 7) dt . t -

7m

a

t

7 -

(4.1.22)

In a similar way we form the sum

(4.1.23) Then the following theorem is true. Theorem 4.1.1.

Let function cp(t, 7) be of the form (4.1.3) on [a, b]. Then

m

=

1, ... ,n,

(4.1.24)

where quantity O(7m ) satisfies InequaLities (1.3.8) and 0.3.9).

Finally, we observe that the Poincare-Bertrand formula remains valid if function cp(t, 7) is replaced by function cp(t, 7,0 of t.he form cp*(t,7,O cp(t,7,O =

nf,

I _ 1 t

Ck

1{3'1 7 _

Ck

lA,'

(4.1.25)

where cp*(t, 7, 0 E H as a function of three variables within the region under consideration, and the set of values of ~ is limited. Note that point ~ may be a point of an m-dimensional space. In the latter case the Poincare- Bertrand formula becomes

(4.1.26)

4.2. MULTIPLE SINGULAR CAUCHY INTEGRALS Let us start considering the Poincare-Bertrand formula for multiple singular integrals with the two-dimensional case.


121

Let function cp(to, t), where to = (tJ, t(~) and t = (t 1, t z), be defined on L] X L z and have the form (3.1.12). The curves L] and L z are piecewise Lyapunov and contain angular nodes only. Let us show that in this case Formula (4.1.1) is valid. 1. Let L( = [aI' hI] and L z = [a z, h zl Consider the integral

L

=

On segment [a p, hpj we take sets EP = rTf, k p = 1, ... , n p} and Et = {fl, = 0,1, ... , n p}, which form a division of segment [a p, hpj and are such that for k p = k~ point tt is a point of the set EP, p = 1,2. After introducing again the sum

jp

and using Formulas (3.3.3) and (3.3.6) we deduce that

(4.2.1 ) Let us transform the sum

m

(tJ, t~) as

n In 2


122

n, -tiT~\)

nj

L

tiT;,

k,= 1 k,"'kg

j,

L

+tiTfg

k,

~

tiT~,

n2

L L =

1

h~

L L j, = 1 j,

I

1

=

1

k, ",k\'

(4.2.2) where

S",",

~

l, l

)-(-t-l2---t-~-)

[-(-t]-,---t(-\ 1

--------+ (t], - T~,)(tj~ - t~)

(ti,

(4.2.3)

By removing the square brackets in Sn n and using Inequality (3.3.3) as 2 was done for mn 1 n 2(to,1 to), we get for Sn1I n'2 2


123

For Sn, we have

Sn, =

-- SIn, + S2n,·

(4.2.5)

From Section 4.1 it follows that (4.2.6) The sum S,~, will be estimated similarly to the sum ~~(to) (see the preceding section): namely, we split the sum Pn , between the braces in the formula for into the sum over the points t}, -belonging to neighborhood 0«6) whose cfosure does not contain points a, and hI and into the sum over the points t}, lying outside the neighborhood. Remember that

S;;,

cp*( t(L tf", t~, Tf,

(

t ()l -

a I )(3'(b 1 -

]

, . (4.2.7)

t 01 )(3,

Therefore,

f3 >

o.

(4.2.8)


124

Now with the use of the estimates presented in Section 1.3 for a singular integral over a segment, one gets lim

IS;,I.:o;

nl,n2~CC

..

lim

O(h A In 2 h) = 0,

A> O.

( 4.2.9)

nl,nz-+ x

From (4.25), (4.2.6), and (4.2.9) we deduce that

( 4.2.10) In a similar way it may be proved that

(4.2.11)

(4.2.12) The validity of Formula (3.1.16) for the case under consideration follows from (4.2.2)-(4.2.4) and (4.2.10)-(4.2.12). If L] and L 2 are assumed to be arbitrary piecewise Lyapunov curves containing angular points only, then by generalizing the analysis presented in Section 4.1, Formula (3.1.16) may be shown to be valid in this case too. Now it is clear that Formula (3.1.17) may also b' is larger than the sum of the absolute values of all the rest terms of thmw. In fact, we have

_f'k-

1

dt

I

(lOj - t)

Ik

2 '

'.: =

1, ... , n. (5.1.62)

Therefore if t Oj > O. Hence,

et

[t k , t k +

I

J

I]' then a jk > 0, a jj

< 0, and -

(dt/{toj - 1)2)

n

dt

-1(tOj-t)2

L

a jk

+ ajj < O.

(5.1.63)

k~1

k*/

Finally, from the latter equality we deduce that n

-a jj = la j )

>

L

n

a jk =

L

k= I

k= I

k*j

k*j

lajkl.

In a similar way the following theorem may be prove(

(5.1.64)

Equation of the First Kind on a Segment

147

Theorem 5.1.9. Let function f(t) E H on [ -1,1]. Then between a solution to the system of linear algebraic equations

j

1, ... , n,

=

( 5 .1.65 )

and solution 'P(t) of Equation (5.1.54) given by Formula (5.1.55), Relationship (5.1.56) holds. To prove the theorem it suffices to note that System (5.1.65) is equivalent to the system

j=l,oo.,n,

j

where 'Pn(t OO ) = C z and 'Pn(tOn+l) = C] By using Theorem 5.1.1 one gets

=

n + 1, (5.1.66)

+ C z.

kIn 1 f(t)h _ "_/(n'])h"_/(n,.]) OJ 'Pn(tOk ) = '- h Lk ' - h LOj t. - t . i=]

j=]

1

k

+C I "' - _[(n, h I,k i~

I)h

]

I

+ C\. ~

Note that by (5.1.61) and condition 'Pn(t on (5.1.67) for k = n + 1,

I )

=

OJ

(5.1.67)

C1

+ C z one gets from

n +]

L

i~

Ii~ij

I)

=

1.

( 5.1.68)

]

In Matveev (1982) the latter result was generalized onto the equation ] 'P(m)(t) dt

f-]

to - t

=

f(to)'

(5.1.69)

148


5.2. CHARACTERISTIC SINGULAR EQUATION ON A SEGMENT: NONUNIFORM DIVISION* In this section we consider questions of solving numerically Equations (5.1.1') for b = 1,

f

cp(t)dt

l

--- =

to - t

-I

f(to)

by using the representation cp(t) = w(t)l/J(t),

(5.2.1)

where w(t} = (1 - t) + 1/2(1 + t)± 1/2 and l/J(t} is supposed to belong to the class H on [-I, 1]. Therefore, for constructing a system of linear algebraic equations providing a numerical solution to Equation (5.1.1') in a class of functions, we will use the interpolation-type quadrature formulas presented in Section 2.3. Let us start by considering in greater detail the ca.o;e K = 1 when w(t) = (1 - ( 2 )-1/2. Theorem 5.2.1. Let function fystems of linear algebraic equations

10

j

1, ... , n,

=

(5.3.12) where 47T

a

= k

2n + 1

sin 2

k 2n + 1

Ik =

7T,

cos

2k

2n + 1

k

7T,

=

1, ... , n,

2j - 1 IOi

=

cos

2n + 1

j

7T,

1, ... , n,

=

j=1, ... ,n-1, n

L

j

l/Jn(tk)a k = C,

=

(5.3.13)

n,

k=\

where ak

=

7T -,

Ik

n

= cos

j IOi =

cos- 7T, n

2k - 1 2 n 7T ,

1, ... , n,

k

=

=

1, ... , n + 1,

j=1, ... ,n-l,

j

(5.3.14)

where

k

7T

ak

=

IOi

=

--sin2 --7T

n+1

n+I'

2j - 1 cos 2 (n + 1)

7T,

k t k = cos-- 7T, n + 1

j=I, ... ,n+l,

k

=

I, .. . ,n,

160


and the corresponding functions fjJ(t), determining in Equation (5.2.0 the index K solution to Equation (5.3.1). In Relationship (5.2.4) Rn(t k ) is an approximation error for the corresponding integrals in the Fredholm equation of the second kind. (The approximation formulas are applied for points to' j = 0, 1, ... ,n - K.) In a similar way one may write systems of linear algebraic equations for singular integral Equation (5.3.11).

Note 5.3.2. Fredholm's results (Goursat 1934, Privalov 1935) are also valid for a system of integral equations of the second kind if it has a unique solution and the kernels are regular (continuous). Therefore, the previously formulated results are valid for a system of singular integral equations if it has a unique solution of a fixed index and is subjected to corresponding additional conditions; also, a system of linear algebraic equations for the characteristic part may be transposed as was done in the case of a single singular equation.

Note 5.3.3. While calculating steady flow past an infinite-span wing with a flap, one has to consider a characteristic singular integral equation on the segment [0, 1] under the following conditions. If a flap is deflected, then f(t) suffers a discontinuity of the first kind at point q (the point of the flap's deflection). However, if the flap stays undeflected, then f(t) E H on the segment of integration. For concrete calculations it is desirable to have a method for considering both situations from a single point of view. On the other hand, hinge moments at the point of a flap's deflection are calculated by using nodes lying on the flap only. Therefore, if a flap is small (i.e., the segment [q, 1] is short), then it incorporates a small number of nodes, and it is difficult to ensure satisfactory accuracy for calculating the moments. At the same time, it is desirable to construct such a computational procedure for which segments [0, q] and [q, 1] contain the same number of nodes. Here we have succeeded in realizing the following idea. First we choose a mapping of segment [0, 1] onto segment [ -1,1] that is infinitely integrable (or has at least derivatives up to the order r ~ 2 where the rth derivative is limited). The first derivative does not vanish, and point q is mapped onto point 0 on [ -1,1]. Next we choose on segment [ -1,1] either uniform or nonuniform grids subject to the condition that point 0 is a point of the set Eo. The inverse mapping distributes the nodes in the


161

desired manner. One may choose the mapping q( 1

t( T) =

(1 + T)(2q -

+ T) 1) + 2(1 -

q)

,

(5.3.15)

where q E (0,0, t E [0,1], and T E [ -1,1]. We see that t( - 1) = 0, to) = 1, t(O) = q, and t'(T) > for any T E [ -1,1]. Let us substitute the variable in Equation (5.1.1), using Formula (5.3.15) (a = O,b = 1). Then we get

°

f

1

-I

cp(t(T»t'(T)dT t(T) - t(T ) O

=

»'

(5.3.16)

f(t(T O

as seen in Equation (5.3.11). For consideration of the solution limited at the point 1 and unlimited at the point -1, we must examine the linear algebraic system

rL -'-

2n -- I j-]

ti

t Oj

-

=fj,

(5.1.17)

j=1, ... ,2n-1,

where T Oj =

j

=

=

[f(q - 0)

h

=

lin,

1, ... ,2n - 1,

f j =f(t(T Oj »)' fn

-1 + jh,

j

=

1, ... ,2n - 1,

j of- n,

+ f(q + 0)];2.

In applications it is often necessary to construct a direct numerical method, allowing us to have different numbers of grid points on different parts of a segment of integration. To our mind this can be done by finding suitable sets of standard mappings of a segment onto another segment, possessing the following properties: 1.

2.

The first derivative belonging to the class II does not vanish, or there exist higher-order derivatives. Portions of a segment are mapped onto equal or approximately equal segments.

162

Melhod ot Discrete Vortices

Note 5.3.4. By using results of numerical solution of Equation (5.1.54) and the results of this section, we deduce that for the Prandtl equation

f

I

- I

df(t)

dt

.

- d - - - - a(to)f (to) = f(to), 1

10 -

t

t o E(-l,l), (5.3.18)

one has to consider the following system of linear algebraic equations (see (5.1.56)):

j

=

1, ... ,n. (5.3.19)

Relationship (5.1.56) is valid for a solution of the latter system and a solution to the Prandtl equation meeting the condition r< -1) = C z and [(1) = C I + C z .

5.4. EQUATION ON A SYSTEM OF NONINTERSECTING SEGMENTS Let us next consider a singular integral equation of the form

1.

I.

cp(t) dt t - 1 0

=

t(to),

(5.4.1)

where L is a set of I nonintersecting segments [A I' B I]' ... , [A I' BI ]. According to Gakhov (I977) and Muskhelishvili (1952) this equation may have index K = -I, -I + 1, ... , -1,0,1, ... , I determined by the number of segment ends where the sought after solution is limited. Following the procedure proposed by Muskhelishvili (1952), let us renumerate the ends of segments [AI' B)], ... , [AI' B I ] in an arbitrary manner and denote them by c), c z ' ... , C Zl' By 17k), ... , c q) we denote the index K = I - q class of solutions to Equation (5.4.0, which, for t(d E H on L, are limited at the ends C I , •.• , cq and unlimited at the ends c q • I, , CZI' On segment [Am' B m ] we take the sets Em = {t k , k = n m _ I + 1, , n m} and E Om = {t Oj ' j = n m-I + 0, n m_ I + 1, ... , n m} forming a canonic division of the segment with the step h m (m = 1, ... , I; no = 0). Let us denote E = U ~ __ I Em and Eo = U ~ ~ I E om and assume that the sets E and Eo


163

form a canonic division of the curve L if the relationship

h -h . -< R < +x ,

m

=

(5.4.2)

1, ... ,1,

nt

holds, where h = max(h" ... , hI)' h -4 O. Note that according to relationship (5.4.2) the numbers n l and Nm = n m - n m I' m = 1, ... , I, for n -4 x are such that a ratio of any two of them is a limited quantity. In what follows we assume that Relationship (5.4.2) is always valid. By P(q ~ we denote the set of points to} obtained by excluding from the set Eo pOInts cq + p •.. , C ZI nearest to the ends. Similarly to Section 5.1, the following theorems may be proved.

Let function f(t) E H on L. Then, between a solution to the system of linear algebraic equations

Theorem 5.4.1.

to} E

P(q),

n/

L k~

t:CPn,( td !itk

=

C..

€ =

0,1, ... ,

K -

1,1

$

K $

I, (5.4.3)

1

and the index K ~ 0 solution cp(t) belonging to the class 71(C 1, ... , c q ) and meeting the condition

f t'cp(t) dt

=

(5.4.4)

C.,

L

the relationship k=l, ... ,n l ,

(5.4.5)

holds where O(t k ) satisfies the inequalities: 1.

For all points t k ever small,

E

U ~ ,_ ,[Am + 8, Em - 8], where positive 8 is how(5.4.6)

2.

For all points t k

E

L,

n/

L k=l

O(td !it k

$

O(h A,),

0< Az

$

1.

(5.4.7)


164

Theorem 5.4.2. Letfunctionf(t) E H on L. Then Relationship (5.4.5) holds between a solution to the system of linear algebraic equations -K-I "10

'- to/Yo

n, m (t ) tJ,.t ,,"f'n, k k

+ '-

t

k=IOj

10=0

_ t

=

f(tOj) ,

t Oj

E

P(q),

(5.4.8)

k

°

and the index K < solution cp(t) belonging to the class 17(C 1, ••• , c q }, I - q = K < 0, and meeting the conditions €

= 0,1, ... , -

K -

1,

(5.4.9)

where RK(t) is a characteristic function belonging to the class 17(c" ... , c q }. Here the quantities Y.( € = 0, 1, ... , - K - I} are regularizing factors.

Detailed proofs of the latter two theorems are presented by Lifanov (1981). Note 5.4.1. Theorems 5.4.1 and 5.4.2 remain true if function f(t) E H on L and can tend to infinity of the order of Q' E [0,1] at the ends c q + I'.·.' C 21 and to infinity of the order of {3, {3 E [0, ~ at the ends c 1,"... , c q • Here the class 17(C I' ... , c q } corresponds to the class of solutions to Equations (5.4.]) that have a singularity of order less than ~ at the ends C" .•• , c q , and either equal or more than ~ at the ends c q + I'···' C 2 /. If, in addition, function f(t) suffers a discontinuity of the form Ie - tl-", [0,1), on a segment [Am' Em] '3 c, then the sets E Om and Em must be

JJ E

chosen on the segment as was done when considering Theorem 5.1.2. However, if function f(t} suffers a discontinuity of the first kind at point c, then both sets E Om and Em on segment [Am, Em] and the system of linear algebraic equations for the points belonging to the segment should be composed as indicated in the addition to Theorem 5.1.2. In problems of electrodynamics (Gendel 1982, 1983) it is more convenient to require that Equation (5.4.1) meet the conditions k=l, ... ,m,m$l,

(5.4.10)

on alI segments L k from L for which a solution is unlimited at both ends. The system of conditions (5.4.1O) may be readily shown to single out a unique solution to Equation (5.4.0. For the sake of convenience, we rewrite Equality (5.2.1) for the index K solution on segment [ -1,1] in the form: CPK(t) = WK(t}r/JK(t}, w + l(t} = (l - ( 2 )+ 1/2, Wo(t) = ";(1 - t}/(1 + t}, and r/J)t} E H on [-1,1]. Then the system of linear


165

algebraic equations (5.1.3)-(5.1.5) may be written in the form (K

=

0,1, -1):

j=I, ... ,n-K, n

L

nK)CPK,n(td h

=

~(K)C,

(5.4.11)

k=l

where ~(x) = 1 for x> 0 and ~(x) = 0 for x:s:; 0, and points IOj in System (5.1.6) are renumerated from 1 to n + 1. For ~(K) = 0 (i.e, for K = 0 or - 1) the latter equation is an identity and may be ignored. Hence, the systems of linear algebraic equations (5.2.2), (5.2.15), and (5.2.16) may be rewritten in the form j=I, ... ,n-K, n

L

n K)l/!K,n(tdak n K)C,

(5.4.12)

=

k~1

where ak' lk' k = 1, ... , n, and tOj,j = 0,1, ... , n - K, are chosen depending on the index K as indicated in the preceding systems, namely, points lk and t Oj are the roots of polynomials PK, n(t) and QK, n(t), respectively. In what foHows we represent Equation (5.4.1) as a system of singular integral equations on segment [- 1, 1]. Therefore, in accordance with terminology used in the theory of such systems (Muskhelishvili 1952), we say that a solution cp(t) to Equation (5.4.1) has index K = (K" ... , K/), K m = 1,0, -1, and m = 1, ... , I, if it is (1) unlimited at both ends, (2) unlimited at an end, or (3) limited at both ends of segment [Am, Em]' The solution wiII be denoted by cp)t). Let us consider the mapping gm( 'T) of segment [-1,1] onto segment [Am' Em], where

m

=

1, ... , I.

(5.4.13)

Denote

m = 1, ... ,1. We use a uniform division on each of the segments [Am' Em], m By employing the sets Em = {lm,k k = I, ... ,n m} and Em,o =

(5.4.14) =

1, ... , I.

{tmOj,

j

=

Method ofDiscrete Vortices

166

0,1, ... , n m }, m = 1, ... , I, we choose a canonic division of segment [Am' Em] with the step h m. Then the following theorem is true. Theorem 5.4.3. Let function f(t} E H on L. Then, between a solution to the system of linear algebraic equations nm

CPK m.nm< t m,,, )h m

n -Km)YOn m + E

tm,Oj - tm,k

k~1

np

I

+

E E

CPK" ,np( t p . k )h p t mOJ - tl',k

p=1 k=1 p,,-m

m=I, ... ,I,

m=I, ... ,I, (5.4.15)

and a solution cp)t) to Equation (5.4.1) for which the values of the integrals are known for the segments composing L, on which it has the index 1, Relationship (5.4.5) holds. Let us next consider unequally spaced grid points on segments [A m' Em], composed of points tm,k' k = 1, ... , n m, tm,k = gm(T k ), where Tk are the roots of polynomial FKm,n (T) from the system of polynomials orthogonal on [-1,1] with weight wKf~') and points tm,Oj' j = 1, ... , n m - k m and tm,Oj = gm( TO)' where T Oj are the roots of polynomial QK"" n} T) defined by Equality (2.3.5) through P m' n m(T). Then the following theorem is true. K

Theorem 5.4.4. Let function f( t) E H on L. Then, between a solution to the system of linear algebraic equations

j= 1, ... ,n m - Km,m

E

~(Km)t/!Km,nm(tm,damk

=

l( Km)C m,

=

1, ... ,1,

m

=

1, ... ,1, (5.4.16)

k~1

where amk = (Em - Am)ak/2, ak = Qkm,nm(tm,k/PK'""nm(tm,k)' and function t/!K(t} determining the solution cp)O, Relationship (5.2.4) holds on each segment [Am' Em], where Rm,nm(tm,k) is an error of approximating a singular integral on L.


167

Theorems 5.4.3 and 5.4.4 are proved in a similar way. Mappings (5.4.13) allow us to consider Equation (5.4. 1) as a system of I singular integral equations on [-1,1] which has a unique solution, subject to the corresponding additional conditions (the value of an integral of the solution is known on those segments of L on both ends of which the solution is unlimited, i.e., on which the index is equal to 1). Hence, the system (Muskhelishvili 1952) is equivalent to a system of integral Fredholm equations of the second kind, which also has a unique solution. Therefore, by repeating the procedure of passing to a system of integral Fredholm equations of the second kind in discrete form, we conclude that the systems of linear algebraic equations (5.4.15) and (5.4.16) are equivalent to the systems of linear algebraic equations for this system of integral Fredholm equations of the second kind. The passage is possible due to the fact that Systems (5.4.11) and (5.4.12) are solvable for any K = 1,0, -l. Consider the following full singular integral equation of the first kind on a system of nonintersecting segments:

f

L

cp(t) dt

+

to - t

f K(t(), t)cp(t) dt

=

f(t()·

(5.4.17)

L

For this equation analogs of all theorems proved in this section for Equation (5.4.1) arc valid. However, the following sums must be added to the linear algebraic equations: n/

L

K(tOj,tdCPn/(td dt k

k=l

to Systems (5.4.3) and (5.4.8); np

I

L L p~

I

k~

K(tm,()j' tp,dCPKp,n/tp,d h p I

to system (5.4.15); I

np

L1 LI K(tm,Oj' tp,dt/JKp,npap,k

p=

k~

to System (5.4.16). If functions t(t) and K(t o, t) belong to the class H on the corresponding sets or K(to, I) has the form K/t(),I)/lt o - till', 0 ::s:; Q' < 1, where K1(t(l' t) E H on L x L, then the systems of linear algebraic equations are composed on equally spaced uniform grid points by using standard canonic divisions of segments [Am' B m ] forming L. However, if


168

f(O incorporates singularities as indicated in Note 5.4.1, then these grids on the corresponding segment must be chosen in accordance with the note.

5.5. EXAMPLES OF NUMERICAL SOLUTION OF THE EQUATION ON A SEGMENT

00,

Figure 5.1 shows a numerical solution (A A, h = 2/11; X X, h = 2/21; h = 2/41) of the equation

t

y(x) dx =

-]

Xu -

7T,

(5.5.1)

X

whose accurate solution (see the solid line) for a uniform division is given by

/8 -X

y(x)

=

(5.5.2)

--.

1

+x

We see that an increase in the number of points results in the numerical solution converging to the accurate one. The numerical solution is found from the system

j

=

1, ... , n,

(5.5.3)

where Xi = -1 + hi' XUi = Xi + h/2, i = 1, ... , n, and h = 2/(n + 1). Figure 5.2 demonstrates numerical solutions of the equation

r

-]

y(x)dx Xu -

=

-7T,

(5.5.4)

X

whose accurate solution (for the same division) is given by X

y(X)=

~.

vI - x 2

(5.5.5)

The numerical solution was found from the system

f.

Yn( Xi )h

=

-71",

j=], ... ,n-l,

i~] XU} - X i

n

L i~1

Yn(xi)h

=

0,

j = n.

(5.5.6)

169


l'

1

•

11,0

\

\

,.

~o

\;6 ~

':

. 2.r -

~

"~ , •

I"""

-(0 -48 -46 -0.* -0,2

0

0,2

0,*

II...

~ 0.6

0,8

~O

z

FIGURE 5.1. Index K = 0 numerical solution to Equation (5.5.I) For a uniForm division. The solid line corresponds to the exact solution, "" "" to h = 2/11, x X to h = 2.21, and 00 to h = 2/41.

Finally, Figure 5.3 shows numerical solutions of the equation

t

y(x)d.x

-1

Xl) -

(5.5.7)

X

whose accurate solution is given by

y(X)

=

h

-x 2

(5.5.8)

(for the same division). The numerical solution was found from the system

j = 0,1, ... , n. Figures 5.4-5.6, where the solid lines correspond to accurate solutions to n = 10, and 00 to n = 20, show numerical solutions of the same

X X

170


,

21--r-----,,....----,-----,---,----,

fl----l----+--F~-+---l

~.

48

0,6

~o ~

1--+-L-;;.,+---+--4-----4-f

1-+-+--+----l--+--I-2

FIGURE S.2. Index K = I numerical solution to Equation (5.5.4) for a uniform division. The solid line corresponds to the exact solution, "" "" to h = 2/11, X X to h = 2.21, and 00 10 h = 2/41.

equations at unequally spaced grid points. In this case solution y(x) is represented in the form w(x)u(x), and the systems of linear equations are constructed with respect to the values of function (u(x) at the roots of the corresponding polynomials. Thus, numerical solution (5.5.2) is found by considering the system (see Figure 5.4).

t

un(x;)o;

j

= 7T',

=

1, ... , n,

;=1 X Oj - X i

2i

X; =

cos

+ 2n

47T

17T,

o· I

=

i

---sin 2 7T , 2n + 1 2n + 1 2j - 1 X Oj =

cos 2

n + 1

7T ,

171


r f.0

}

~

1.At

~

.... F'"

...."

~

0,L'

r -1,0

~

0,8

~

0,

~

.~

~2

-o,s

-0,6 -0.*

-~2

42

0

44

46

48

f/J z

FIGURE 5.3. Index K = -I numerical solution to Equation (5.5.7) for a uniform division. The solid line corresponds to the exact solution, "" "" to h = 2/11, X X to h = 2/21, and 00 to h = 2/41.

u ..,2 -(0

-0,8 -46 -0,*

as o

-0,2

0,2

0,*

0,6

0,8

f,O z

FIGURE 5.4. Index K = 0 numerical solution to Equation (5.5.1) for a nonuniform division. The solid line corresponds to the exact value of the [unction u(x), where y(x) = w(x)u(x) and w(x) = x)(1 + x); X X corresponds to n = 10 and 00 to n = 20.

J(I -

numerical solution (5.5.5) is found by considering the system (see Figure 5.5)

t i~ I

un(xi)ai X Oj -

j=l, ... ,n-l,

-7T,

Xi

2i - 1

n

Lun(x;)a i

=

0,

j

=

n,

Xi =

COS---7T,

2n

i~l

j

7T

a·I

=

n

X Oj =

COS-7T,

n

172

Method of Discrete Vortices u

/

0,8

1//

0,6 0,4 ~2 -~8

-~6-

~

V·

-~2

~/ o

V·

/

/

~2

~II-

0,6

0,8

(0

z

-~.2

V

-0.11-

-0,6

l/.1

-0,8

,/

-1,0 FIGURE 5.5. Index K = 1 numerical solution to Equation (5.5.4) for a nonuniform division. The solid line corresponds to the exact value of the function u(x), where rex) = w(x)u(x) and w(x) = (I - X 2 )-1/2; X X corresponds to n = 10 and 00 to n = 20.

u ~.2

dB

-

o

FIGURE 5.6. Index K = -1 numerical solution to Equation (5.5.7) for a nonuniForm division. The solid line corresp(lnds to the exact value of the function u(x), where rex) = w(x)u(x) and w(x) = ~; X X corresponds to n = 10 and 00 to n = 20.

and, finally, numerical solution (5.5.8) is found by considering the system (see Figure 5.6) j=l, ... ,n+l, I

x j = cos n + 1 7T,

7T

.

2

i

a· = ---SIO - - - 7 T I

n+1

n+l'

2j - 1

XOj=cos

2(n

+

1)

7T.


173

FIGURE 5.7. Index K = 0 numerical solution to Equation (5.5.9) for a unirorm( x) and nonuniform (e) divisions ror n = 30 and the rderence point coinciding with point q = 0.8.

FIGURE 5.8. Index K = 0 numerical solution to Equation (5.5.9) ror [(xo) = 0,0 < x < 0.8. and [(xo) = -21T, 0.8 <x < 1 for a uniform (x) and nonunirorm (e) divisions ror n = 30 and the reference point coinciding with point q = 0.8.

The calculations were carried out for n = 10,20,30,40. It was found out that lu(x) - un(x)1 $ 5 X 10- 6 . Figure 5.7 compares the results of numerical solution of the equation

1Y (X)d.x - - - =f(x o), o Xo - x

l

(5.5.9)

at equally (x) and unequally (0) spaced grid points and n = 30. It was assumed that f(x o) = -27T, K = 0, and the reference point was placed at the given point q = 0.8 (the hinge point of the flap) over the domain of integration. At point q the right-hand side was put equal to [f(q - 0) + f(q + 0)J!2. Figure 5.8 presents a comparison for the same grid points for

f(x)

=

{a,-27T,

°0.8<
O. The following relationship holds between the accurate solution 'P)t) and an approximate solution 'Pn. K(t) = w~ ~ )(t)t/Jnj _ I; n2(t): j

'.

In analogy to the preceding sections, the numbers 'Yn v will be called reguwrizmguariabks. . Note 7.4.1. From (7.4.28) it follows that an estimate of the difference cp)t) - 'Pn, K(t) in a certain metric of the space of functions on L is equal to an estimate of the function on the right-hand side of (7.4.28) in the same metric. Thus, if w~ )(t) is limited on L (this is so if L is a piecewise smooth curve, but a solution is sought in the class of functions limited at all the nodes), and f(t) is a function of the class Hand L, then, according to Muskhelishvili (952), the estimate may be obtained in a uniform metric. j

Note 7.4.2. Instead of System (7.4.27) one may consider a system of equalities of coefficients before equal degrees of the variable to in generalized polynomials entering Equality (7.4.26). In this case, Formula (7.4.28) preserves its form. Note 7.4.3. A detailed description of the system of linear algebraic equations (7.4.27) corresponding to Equation (7.4.12) in interval [- 1, 1] with real coefficients and a right-hand side is presented in works by Matveev (Matveev and Molyakov 1988, Matveev 1988) together with instructions for calculating all the elements of the system. The results obtained in this section may be transferred in a natural way to the full singular integral equation of the second kind with variable coefficients: b(to) a(to)S(to) + - - . 7Tt

f S(t)dt + f k(to, t)S(t) dt t - to I.

I.

=

f(to)' (7.4.29)


212

7.5. EQUATION WITH HILBERT'S KERNEL AND VARIABLE COEFFICIENTS Considerations similar to those for Equation (7.4.12) may be applied to the equation

b( 0 0 ) a(00)5(00) - - 21T

1 cot--5(O)dO=f(Oo)' 0 - 00 27T

(7.5.1)

2

0

where a(O),b(O),f(O) E H on [O,21T] and are periodic. Therefore, we shall present only the necessary relationships and a corresponding system of linear algebraic equations for a fixed index K. A detailed theory of the equation is presented in Gakhov (1958). Let us introduce the following notation:

where ( t

WK

l( 0) _

exp

-

JL( 00 )

=

-

-

1

21T

(

JL( 0) ) , z(O)

=+=

1277"[arg(a( 0) + ib( 0» 0

Then Equation (7.5.1) for the index written in the form (Gakhov 1958)

K

=

w;+l(Oo)l/J(Oo)

=

2

solution and its solution may be

r(~l(W;+ll/J)(OO) =

5(00 )

0 - 00 - KO]cot-- dO.

f(Oo),

w~+l(Oo)[r( l(w;-lf)(Oo) + b(Oo)PK(Oo)]' (7.5.3)

where for

K ~

0,

A K = -1 1T

1 u( O)sin KO dO, 27T

0

Singular Integral Equations of the Second Kind

1 BK = -

213

2 7T

1 0"( O)COS KO dO,

7T 0

(T( 0) and for

K

< 0,

k = 1,2, ... ,

K -

=

a( O)exp( - /L( (J»/z( 0),

PK ( 0)

1,

O'K'

K > 0 the coefficients 13o, O'k and 13k , are uniquely defined by specifying 2 K relationships

== O. For

k=O,I, ... ,K-I,

k=1, ... ,K-l,

p(O)

=

(7.5.4)

b(O)exp( /L(O»/z(O).

M

7TO"(O) dO *- 0; otherFor K = 0, the number f3 0 is fully determined if wise, 130 has an arbitrary value, and the function cp( 0) is a solution to Equation (7.5.3) only if the condition 27T

1o w;

)f(O) dO

0

=

(7.5.5)

is met. For K < 0 the function cp( 0) as defined by 0.5.3) gives a unique solution to Equation (7.5.3) if the following - 2 K conditions are met: 27T

1o

w~-)(O)f(O)coskOdO=

t7Tw~-)(O)f(O)sin kOdO o

8(0)

=

=

0,

k=0,1, ... ,-K-1,

0,

k

=

1,2, ... ,

-b(O)exp(-/L(O»/r(O).

-K -

1,

(7.5.6)

Similarly to the preceding section, the following theorem may be proved. Theorem 7.5.1. If b(O) is a trigonometric polynomial of degree I, then the operator r< ± )(w~ ±) • ) transforms an arbitrary trigonometric polynomial of


214

degree n

> max(l ±

K; 2 K) into a trigonometric polynomial of degree n

+ K.

Let the full equation

be given, where k( 00 ,0) and f( 0 0 ) are Holder functions. By using the inversion formula (7.5.3) for the characteristic equation, Equation (7.5.7) may be transformed into a Fredholm equation of the second kind in the function w~ - l( 0 )l{!( 0). Because function w~.i l( 0) belongs to the class H, if a(O) and b(O) also do' and the Hilbert operator stays in the class H too, then the kernel and right-hand side of the derived Fredholm equation of the second kind are periodic and also belong to the class H on [0, 27T]. If K < 0, then the Fredholm equation solution is a solution to the original equation, subject to Conditions (7.5.5) and (7.5.6), where f(O) must be replaced by f(O) - ft'k(O, 'r)w; + )(or)l{!('r)dT. Next we suppose that Equation (7.5.7) has a unique solution if Conditions (7.5.4) for the required solution for K > 0 are fulfilled. Then the corresponding Fredholm equation of the second kind also has a unique solution. Suppose that for K $ 0 the Fredholm equation has a unique solution that is a solution to the original equation subject to the preceding conditions. Under the assumptions, a numerical method for solving Equation 0.5.7) is constructed as follows. Let E = {Ok' k = 1,2, ... ,2n + I} and Eo = {OOj, j = 1,2, ... ,2(nK) + l} be a pair of non intersecting systems of points on a unit circle centered at the origin of coordinates. An approximate solution 'Pn . ) 0) will be sought in the form w~' l( 0) l{!n( 0), where I/Jn( 0) is a trigonometric polynomial approximating a function l{!( 0), for example, a truncated sum of Fourier series for function l{!( 0) most often represented by a polynomial of the form 2n + I

l{!n ( 0) =

L

(7.5.8)

l{!n ( Ok) Tn . k( 0),

k~l

where Tn.k(O) is a trigonometric polynomial of degree n such that Tn.k(Ok) = 1, k = 1, ... ,2n + 1. The numbers l{!n(Ok), k = 1, ... ,2n + 1, will be calculated from the following system of linear algebraic equations: 2n + 1

71(-K)b(OOj)YIKI(OOj)

+

L

l{!n(Odqkj

2n + 1

+

L

k~l

k( 00j' Odl/Jn( 0dhk = f( 0 0),

j

=

1, ... , 2( n - K)

Singular Integral Equations of the Second Kind

215

+ 1, 2n

71( K)

i 1

L

r/Jn( 0dCOS(j0k )h k

=

71( K)Cj '

r/Jn( 0dsin(jOdhk

=

71( K )c j

j=O,l, ... ,K-l,

k=1

2n

71( K)

i

1

L

'K

I'

j=l, ... ,K-l,

k = 1

(7.5.9)

where YIK (0) for K < 0 is a trigonometric polynomial of degree IK I and of the same form as in the case of P 0) in (7.5.3) for K > O. The coefficients of the polynomial are regularizing variables chosen under the solvability condition for System 0.5.9). Similarly to the preceding section, they may be shown to be defined uniquely. The coefficients gkj and h k are determined by the relationships K

(

- 1'( +)( W (')Tn,k )( 0OJ' )

gkj -

K

k=1, ... ,2n+l, j = 1, ... , 2( n - K)

k

= 1, ... ,2n + 1.

+ 1,

(7.5.10)

System (7.5.9) corresponding to the characteristic equation (7.5.3) (k == 0) and has a unique solution for any n > max(l + K, 2 K). The foIlowing theorem is valid for System (7.5.9) corresponding to Equation 0.5.7). Theorem 7.5.2. Let Equation (7.5.7) have a unique solution subject to the corresponding assumptions of the form (7.5.4) for K > 0 or of the form 0.5.6) iff(O) isreplacedbYf(O) - f~7fk(O,'T)'P('T)dtfor K < O. Then starting from a certain n, System (7.5.9) has a unique solution, and the rate of comwgence of an approximate solution 'Pn , K( 0) to the accurate solution 'PK( 0) for a given metric of the space ofperiodic functions on [0, 27T ] will be the same as the rate of convergence (in the same metric) of quadrature formulas for a singular integral with Hilbert's kernel and a regular integral, if densities of the latler integrals are approximated by interpolation trigonometric polynomials of the form (7.5.8) on the sets of points E and Eo for the variables 0 and 00 , respectively.

8 Singular Integral Equations with Multiple Cauchy Integrals

8.1. ANALYTICAL SOLUTION TO A CLASS OF CHARACTERISTIC INTEGRAL EQUATIONS Consider a one-dimensional characteristic integral equation of the first kind, depending on a parameter

(8.1.1)

where L I is a curve lying in the plane of complex variable I I, and f(/J, I~) belongs to the class H on the set L I X L 2 , where L 2 is a curve lying in the plane of complex variable 1 2 • In what follows, L I and L 2 are supposed to be limited. Under the constraints imposed on f(tJ, I~), the sought solution cp(tl,/~) must meet the condition H with respect to I~ for any II ELI. Let L, be a smooth closed curve or a system of nonintersecting smooth closed curves. The unique solution to this equation, as well as to Equation (6.1.1), is the function I

2

cp( I ,( 0

_ ) -

1 - -2 7T

f I.,

f(IJ,I~)dtJ I

10 -

2

I

.

(8.1.2)

This is a consequence of the fact that if the function cp(t I ,/6) satisfies the

217


218

identity

(8.1.3) then cp(tl, t~) == o. Now let L] be the integral [a, b]. Then the solution to the above equation has the form (see (5.1.2»

For

K

= 1, the condition

f cp (It ,to2) dt I -b

C( to2)

(8.1.5)

a

holds, and for

K

= - 1, the function cpU 1, t ~) is a solution if the identity b

~

f( t I , tJ) dt l

(8.1.6)

";(11 _ a)(b _ t l ) == 0

is true. If L] is a system of nonintersecting segments, then the index K = m (where m is the number of segments) general solution has the form

(8.1.7) where Pm - l (tl, tJ) = Pl(t~)u])m- I + Plt(~)(t')m -2 + ... +Pm(t(~) and Pk O. (8.2.16)

230


= 0, then Inequality (8.2.15) is fulfilled for all points O(q" 8)] X L 2 • Similarly, if VI = 0, then V2 > o. I 2 Theorem 8.2.1 may be generalized for the case when L i in Equation (8.2.0 is a set of L i nonintersecting circles L 1 , ••• , L~i. Theorem 6.1.3 may also be generalized onto Equation (8.2.1); however, a singularity of the form 1/(q - t) may exist either for one of the coordinates or for two coordinates simultaneously. The cases must be considered separately. Note that if

V2

(11 ,tf ) belonging to [L, \

Definition 8.2.1. A function cp(t', ( 2 ) is said to belong to the class IIi on L, X L 2 where L I and L 2 are unit-radius circles if it has the form

A function cpolving Equation 00.6.2) numerically depends to a large extent on the form of the function F 1( yU o

».

282


»

If F,( ')'(t o == 0 (an airfoil is fully impermeable), then one arrives at a singular integral equation of the first kind for which a sufficiently large number of methods of numerical solution were previously presented. However, if F,( ')'(10» = a(lo)')'(lo), i.e., is a linear function of ')'(t o), then we arrive at a singular integral equation of the second kind with variable coefficients (see Chapter 7) for the methods of numerical solution). Some examples of numerical solutions of such equations may be found in Matveev and Molyakov (1988). Finally, if F 1( ')'(10» is a nonlinear function of ')'(10)' then Equation (l0.6.2) may be solved numerically be employing the method of discrete vortices and an iterative procedure (Belotserkovsky et al. 1987). In this book we present some calculated data; however, the questions of verifying the numerical schemes remain open.

11 Three-Dimensional Problems

Il.l. FLOW WITH CIRCULATION PAST A RECTANGULAR WING Consider a rectangular wing, i.e., a plane plate lying in the plane OXZ within rectangle (J = [ - b, b] X [ -I, I]. Let the flow past the wing be steady and characterized by the velocity vector Uo = Uxi + Uyj. According to Belotserkovsky (1967), the wing and the wake occupying the strip [b, + x] X [ -I, l] may be modeled by a continuous vortex sheet; the latter may be modeled by discrete straight horseshoe vortices n;k = n(A;k, A;k + I) of strength I~k = yz(x;, zOk)h" where A;b = A(x;, Zk)' x; = -b+ih h l =2b/(n+1), i=l,oo.,n, x o;=x;+h 1 /2, Zk= -1+ (k - 1)h"2 , h 2 = 21/N, k = 1, ... , N + 1, and ZOk = Zk + h 2 /2. The subscript on y indicates that we seek the components of the vortex sheet of the wing depending on the Z coordinate; in what follows, the subscript Z is omitted. In aerodynamics simulation of continuous vortex sheet of a wing and the vortex sheet downstream of it by discrete horseshoe vortices is justified from the physical point of view. Due to physical reasons, the aerodynamic problem under consideration is described by the function y(x, z), which becomes unlimited as it nears the leading edge - b X [ -I, l] of a wing and vanishes as it nears all the other edges. The fact that the function vanishes at the trailing edge b X [ -I, l] agrees with the Chaplygin-Joukowski condition of the flow shedding smoothly from the trailing edge. The problem thus formulated is called the circulatory problem of flow past a wing. Let us denote by v;{m = I~k wi: the normal velocity component induced by vortex n ik at the reference point Ijm = (x Oi ' ZOrn) and by vim the normal velocity component induced at the same point by the system of 283


284

FlGURE 11.1. Schematic distribution of discrete horseshoe vortices (wavy lines) and reference points (x X X) for a three-dimensional wing of a complicated plan form. (a), (b), and (c) correspond to the circulatory, noncirculatory, and finite-velocity problems, respectively.

straight discrete horseshoe vortices. By the B condition, the discrete vortices and reference points must be located near the leading and the trailing edges, respectively (see Figure 11.1a). Therefore, in Belotserkovsky (1967) it was proposed to find the values of Y(x i , ZOk) corresponding, in the limit, to the preceding problem of aerodynamics, from a system of linear algebraic equations obtained by fulfilling the no-penetration condition at the reference points Ijm, j = 1, ... , n, m = 1, ... , N, i.e., from the system n

N

jm '" - " ' -I 'ikWik

;= 1

k~

--

-

Uy'

j=l, ... ,n,m=l, ... ,N, (11.1.1)

t

or, in a detailed form (see (9.3.17) and (3.4.1), from

Three-Dimensional Problems

285

j

1, ... ,n,m

=

=

1, ... ,N,

By employing the results of Section 3.4, we deduce that the latter system of linear algebraic equations approximates the integral equation _1

Jb JI

47T

-b

_Y_(x_,z----,,-) ( 2 1 + -1(Zo -z) V(x o

xl) - x _X)2

)

+ (zo

_ _

dx dz -

_Z)2

Uv· .

( 11.1.3) System (t 1.1.2) may be obtained from Equation (t 1.1.3) as follows. First, Equation (t 1.1.3) is written for each reference point !jm, j = t, ... , n, m = 1, ... , N, and then the quadrature formula 0.4.0 is applied to the integral entering the equation. Next, upon adjusting System (11.1.2) to straight horseshoe vortices, the system can be shown to approximate: 1. 2.

An integral equation any of whose solutions vanish at the side edges. An integral equation any of whose solutions vanish at the trailing edge and tend to infinity at the leading edge of a wing. We start by writing System (11.1.2) in the form N

n

L L i~

1

k~

y(x;,ZOd h l F (ZOm,Zk,Zk+I)(1

+

sign(x oj

-Xi»)

1

j = 1, ... ,n, m = 1, ... , N,

1 F(zOm, Zk' Zk' I) = - - - zOrn -Zk+1

(11.1.4)

286


Because

sign(x Oj -Xi) =

I,

i s; j, i > j,

{ -1,

( 11.1.5)

System 01.1.4) becomes

j = 1, ... , n, m = 1, ... , N,

( 11.1.6)

where !l(X Oj ' zorn) denotes the right-hand side of System 01.1.4). By using Theorem 5.1.8 one gets

j=I, ... ,n,k=I, ... ,N,

(11.1.7)

where 1a(j, k)1 coincides with the quantity O(x i , ZOk) used in Theorem 8.3.1. Now we see that System 01.1.2) does approximate the integral equation

1 2 X" y(x,z)dx=-z

I

b

7T

1/ 1f!t(Z,ZO) [/ 1/ b

. b

-/

(

Xo

-/

y(x, r) z (zo - r)

- x

(11.1.8)

where the function I/I/(Z, 20) is defined by Formula (8.3.3).


287

From Equation (I 1.1.8) and the properties of the function t/!,(z, zo) we deduce that

jXO XO j - b y(x,/)dx== . b y(x,-/)dx==O, which is equivalent to the identities ( 11.1.9)

y(x, -I) == y(x,/) == O.

Note that Equation 01.1.3) is equivalent to Equation (I 1.1.8) for functions y( x, z) meeting Cond ition (I 1.1.9). On the other hand, System (I 1.1.2) is equivalent to the system

N

11

=

L L

-

v= 1

/L =

y(xv,zO/L)[K(xOj,ZOm,Xv,Zk41) 1

j= l,oo.,n,m

=

K( XO'

ZO,

x,

Z) =

{

Xo -x

=

1, ... ,N,

k i=m, k = 1,

0, -2,

+ V(x o _X)2 +

(ZO

_Z)2

-Iz o -zl

-----------------

(X O - X)( Zo - Z)

( 11.1.10) Hence, by denoting the right-hand side of the latter system by fl1(x Oj ' zOrn) and applying Theorem 5.1.1, one gets

y(x"zom)

=

1 - 22 7T

f2i-x. + L 11

-b I

Xi j~ I

288


i = 1, ... ,n, m = 1, ... ,N,

(11.1.11)

where the quantity IO',(i, m)1 is the same as the quantity O(tk I , t k 2 ) for System (8.2.33)-(8.2.35). System (1Ll.l1) approximates an integral equation of the second kind whose solution (if it exists) meets the condition y(b, z) ==

o.

y( -b, z) ==

00.

(11.1.12)

The integral equation has the form

y(x z ) ,

0

=

1 f2-X b I 1 ---2'IT 2 b + X f_bf_ 1ZO - Z

-

x (_ 'IT +

fb / -b

o 7_)_2_+_(_Z_o Z_)_2 d.x ) y/ _b_+_x_D_V_(_X_ o b-x o (X-X O )(X O -7)

X(7, z) d7dz - 2

-X - - Uv . fb2 +X .

Thus, if System (11.1.2) for straight horseshoe vortices is solvable and the sequence of its solutions is convergent, then in the limit the solutions result in the function y(x, z) possessing the required properties at the edges of the wing. In the following text, the system is shown to be solvable.

11.2. FLOW WITHOUT CIRCULATION PAST A RECTANGULAR WING Consider the problem of noncirculatory flow past a wing. The problem arises, for example, when studying an interaction of flow with a stationary osciIlating wing by employing the so-called virtual inertia. In this case all the edges function under the same conditions, and, hence, the z component Yz(x, z) of the continuous vortex sheet simulating a wing must tend to infinity when approaching either the leading or the trailing edge. Hence, by the B condition, discrete vortices must be located near the leading and the trailing edge (see Figure 11.1b). In this case the problem is modeled as foIlows. Oncoming flow is supposed to have the needed normal velocity

Three- Dimensional Problems

289

component ~.; the vortex layer of the wing is modeled by straight discrete horseshoe vortices lI ik with bounded vortices (A ik , A ik I I) parallel to the OZ axis and the free vortices (Aik,(+x,Zk» and (A ik , 1,(+x,Zk+I» parallel to the OX axis. Because there must be no wake downstream of the wing (j.e., the strength of the vortex wake is equal to zero), the summary circulation due to the discrete vortices along each chord of the wing must be equal to zero: n

L i~

I~k = 0,

k

1, ... ,N.

=

1

Thus, to find the circulations r ik , i = 1, ... , n, k = 1, ... , N, one has to consider the system of linear algebraic equations (Belotserkovsky 1967) N

n

L L i~

1

k~

Aikwlt

=

-~.,

j = 1, ... ,n -I,m

=

1, ... ,N,

I

n

L i~

I~m = 0,

( 11.2.1 )

j=n,m=I, ... ,N.

1

By reasoning in the same way as in the case of circulatory flow past a wing, System (11.2.0 may be easily shown to approximate the integral equation (11.2.3) supplemented by the condition

( 11.2.2)

Similarly, we can show that System (11.2.0 approximates the integral equation (11.2.8) supplemented by the additional condition (11.2.2), and, hence, the solution of this equation again satisfies Conditions (11.1.9). Thus, instead of System (11.1.1 0), one gets 1

- 2"!ll(X Oj ' zOrn), j

1, ... , n - 1, m

=

1, ... , N,

j = n, m = 1, ... ,N.

(11.2.3)

=

n

L i= I

'Yz(x i , zom)h l = 0,


290

From Theorem 5.1.1 it follows that

i= 1, ... ,n,m

=

1, ... ,N,

(11.2.4)

where the quantity 10'20 , m)1 is of the same type as in Equation 01.1.11). The latter system approximates an integral equation of the second kind whose solution (if it exists) tends to infinity when approaching the leading and the trailing edges. In order to find the vortex sheet component parallel to the OX axis Yx(x, z), one can employ the equality (Bisplinghoff, Ashley, and Halfman 1955) dYz(X, z)

ayAx, z)

dZ

dX

( 11.2.5)

or consider the system of equations obtained from 01.2.1) by substituting z and x on the left-hand side for x and z, respectively, Le., by considering straight horseshoe vortices lI ik = I1(A ib Ai. i.k) of strength f ik = Yx(X Oi ' zk)h 2 composed of the vortex segment (A ib A i + I,k) and a pair of semi-infinite vortices (Aib(X i , +oc» and (Ai I i.k,(x i I I' +oc» and putting j = 1, ... , n, m = 1, ... , N - 1. When considering the noncirculatory problem, the following way of modeling a continuous vortex sheet by discrete vortices may also be readily employed. Because the vortex sheet exists only within the area occupied by the wing, it may be conveniently approximated by discrete closed vortices no ik = n(A ik , Ai k t l' A i + I k+ I' Ai t I k) whose circulation is equal to Q ik' = Q(X Oi ' ZOk) and that are compos~d of vortex segments (A ik , Ai, k ~ J), (Ai.klbAi+I,k+I)' (Ai+l,k+I,Ai+l,k)' and (Aitl,k,A ik ), i=O,I,oo.,n, k = 1, ... , N. From Figure 11.1 b it is seen that

(11.2.6 )

because the segment [A ik , Ai,k t , ] is a part of the discrete vortices I1 0,i-l,k' nO,ik characterized, in the former case, by circulation-in the negative direction (with respect to the OZ axis) and in the latter case by


291

circulation in the positive direction. Analogously, we have (11.2.7) Here Q(XOi-I,ZOk) = 0 for i = 0, k = 1, ... ,N and Q(XOi,ZOk I) = 0 for k = 1, i = 0,1, ... , n. From Relationships 01.2.6) and (11.2.7), it follows that m

j

Q(X Oj ' ZOrn) =

L

L

'Yz(x j , zom)h, =

k=

i~O

'Yx(X Oj ' zd h 2'

(11.2.8)

1

where j = 0,1, ... , n, m = 1, ... , N. Because the functions 'Yz(x, z) and 'Yx(x, z) may have at the edges only integrable power singularities of the order of p - 1/2, where p is the distance from an edge, and because the flow is noncirculatory, we deduce that the function Q(x, z) must vanish at all the edges of the wing as pI / 2. The no-penetration boundary condition must be fulfilled at the reference points (X Oj ' ZOrn), j = 0,1, ... , n, m = 1, ... , N, and this results in the system of linear algebraic equations N

n

L L Qjkwlt i~

1

k=

=

-Uy '

j=O,I, ... ,n,m=I, ... ,N, (11.2.9)

1

wit

is the factor before j f in Formula (9.3.23) for x = ZOrn, ZI = Zk, and Z2 = Zk+ I' By using Formula (9.3.24) one gets

where

X2 = Xi + I'

=

X Oj , XI = Xi'

Zo

j=O,I, ... ,n,m=l,oo.,N.

(11.2.10)

From Section 3.4 it follows (see Formula (3.4.17» that System 01.2.1 0) approximates the integral equation

f

bfl

-b

Q(x, z) dxdz

.. / [ (X _X)2 O

+

(ZO

_Z)2 ]

3/2

(11.2.11)

In the class pf functions Q(x, z) vanishing at the edges of a wing, Equation 01.2.11) is equivalent to Equation (I1.1.3) supplemented by Condition


292 (11.2.2), i.e., to the equation

b fl f

'Yz(X,Z)(X\) -x)dxdz

-b - I (ZO - Z) 2 / (X O - X) 2 + (ZO - Z) 2

-Uv

(11.2.12)

where 'Yz(x, z) = :~Q(x, z). Equation 01.2.12) is consistent with Equation (11.2.6). This statement may be easily proved by integrating the left-hand side of Equation (11.2.11) by parts in x and taking into account the condition Q(b, z) == Q( -b, z) == o. Finally, we observe that System (11.2.10) is well-conditioned because its matrix corresponds to the Hadamard criterion. In fact, if(x;, Zk) *- (x j ' x m ), then

.

alZ'

jZk. 'fx",

=

zk

and if i

=

j, k

(X Oj

Xi

-x) +

(ZOm -

2]3/2

> 0, (11.2.13)

't'

< O. (11.2.14)

2)

m, then from (3.4.17) it follows that

=

al= -

dxdz 2

[

t

m

Zm

'f" [(X Xi

Oj -

x)

, +dxd2 (X

Om -

z)

The sum of all elements a{~n of the row (j, m) in the matrix of System (11.2.10) is equal to n

N.

L L al;;'

b

=

dxdz

I

Jf [

;=1 k~l-b -{

(X

0

Oj

- x f + (zOm -z)

l\/2

< O. (11.2.15)

Thus, from Formulas 01.2.13)-(11.2.15) one gets n

N

L L alt ;=0

k~

following identity follows from Identities 01.3.13);

y(xe,O),O) == y(x(x',l),l) == 0,

°on [0, l], the (11.3.14)

298


according to which function y(x, z) vanishes at the side edges of the trapezoid (J. Note 11.3.1. If one of the side edges of the trapezoid (J degenerates into a point (Le., (J is a triangle), then the Jacobian J(z) vanishes at the point, and hence, the preceding analysis docs not allow us to determine the value of y( x, z) at the point. This issue still awaits analysis.

Let us again use Formula 01.3.10) for transforming System (I 1.3.7) as follows:

j = 1, ... , n, m = 1, ... , N,

(11.3.15) where fl'(xil}' zOrn) denotes the terms transferred from the left-hand to the right-hand side. Then, by applying to System (11.3.15) considerations similar to those applied to System (11.3.10) and passing to the limit, one gets

(11.3.16) According to Equation (11.3.16),

(11.3.17) Next we will consider noncirculatory flow past a canonical trapezoid (J. In analogy to the noncirculatory flow problem, simulation of the vortex sheet by slanting horseshoe vortices in the framework of the present problem allows us to conclude that circulations f ik of the vortices may be


299

found by considering the system of linear algebraic equations n

N

" " ''-I 'ikWikjm -i= 1

k~

-

Uy'

j = 1, ... , n - 1, m = 1, ... , N,

I

n

L i~

rim

j =

= 0,

fl,

( 11.3.18)

m = 1, ... , N.

1

Similarly to the circulatory problem, we deduce that in the present case the strength y(x, z) of the continuous vortex sheet on (J satisfies Equation 01.3.8) supplemented by the condition

f

x

x

I( Z)

y(

x,

Z

dx = 0 ,

)

ZE[O,I],

( 11.3.19)

(z)

according to which the vortex sheet circulation along any chord of the wing is zero. Note that by introducing the function Q(X, z) =

fX x

y(x, z) dx,

( 11.3.20)

(z)

one can prove that Equation 01.3.8) supplemented with Condition 01.3.19) is equivalent to the equation l

f-I

fX x

I

(z)

(z)

Q_(_x_,_z_)_ _~ dxdx = :1/'

[(xo _X)2

+ (zo _Z)2]' -

-Uy ,

(11.3.21)

because function Q(x, z) vanishes at all the edges of a wing. In fact, similarly to the circulatory problem, function y(x, z) may be shown to vanish at the side edges because system 01.3.18) approximates an integral equation whose solutions possess the preceding property. Similarly to the previous section, function y(x, z) may be shown to tend to infinity at both the leading and trailing edge of the canonical trapezoid (J. Finally, note that System (I 1.3.18) is well-conditioned. Similarly to the preceding section, we deduce that Q(x i, k' Zk) is the strength of a closed discrete vortex composed of the segments [AC, k> Zk), A(Xi,k+l,Zk+l»)' [A(Xi,klJ,Zk')' A(xi."kt"Zk.'»)' [A(xi+I,ktl,Zktl), A(X i _ 1.k' Zk»)' and [A(x i . I " b Zk), A(x i k' Zk»)'

Note 11.3.2. The results obtained in this section are fully applicable to a wing equipped with a flap, i.e., when the right-hand side of Equation


300

01.3.8) suffers a discontinuity of the first kind on the straight line x = X(ql, z) where x(x l , z) = x'[x. (z) - x (z)] + x (z). However, in this case point q' must be placed midway between the nearest points xl and I

X Oj '

Next we consider steady flow past a finite-span wing of a complicated plan form and a schematic flying vehicle (see Figure 11.2). The lifting surface (I lies in the plane OXZ and its contour is composed of segments. By drawing through the angular points of the contour straight lines parallel to the OX axis, we divide the surface (I into canonical trapezoids 0;" € = 1, ... , p, which can intersect each other along side edges only. Let the side edges 0;, be specified by the equations z = /1, z = 11 < and the leading and the trailing edge by equations

I;,

x~(z) =

a'

+ zb' ,

x: (z)

= a~

+ zb~ ,

I;,

(11.3.22)

respectively, where € = 1, ... , P and a', , a' , b: , and b' are constants. Let us consider p planes OX'Z and choose on each of them a rectangle D, = [0,1] X [11, I;]. Next consider mapping F. of the r~ctangle onto 0;. specified by the formula x(x',z) =x'[x:(z) -x'(z)] +x~(z),

z

=

z. ( 11.3.23)

J.
{Ajk}' where Aik

=

f

ds

ih

(i -

l)h

(kh - s)

1

,,'

Ajm

f

=

ds

jh

(j-I)h

(mh - s)

I

13'

k = 1,2, ... , i = 1, ... , k, m = 1,2, ... , j = 1, ... , m. It is obvious that the sequences satisfy Conditions 02.2.3). Let us construct conjugated sequences {Aid, {A ik } with the conjugation index K = 7T/sin 0'7T and {Ajm}, {Ajm} with K = 7T/sin f37T. Next consider the system of linear algebraic equations N

M

L L i= 1

j~

A;NAjMCPNM( ~i' TJj)hyh~ = f(Mh x , Nh y)'

M, N

=

1,2, ....

1

(12.2.22) This system is solvable, and the estimated difference between its solution and the exact solution 02.2.21) has a form analogous to 02.2.17). Note 12.2.1. equation

An analogous numerical method may be applied to the

x

y

cpU,TJ)d~dTJ f3

f jl/J1O(x Xo

0

(y - TJ)

a

=

f(x,y),

(12.2.23)

Unsteady Linear and Nonlinear Problems

315

where 0 < Q', f3 < 1. It should be mentioned that the case Q' = f3 = [(XII' y) == 0 is of paramount significance for supersonic aerodynamics.

i,

12.3. SOME EXAMPLES OF NUMERICAL SOLUTION OF THE ABEL EQUATION Consider the equation rcp(S)dS =t

(12.3.1)

o~

for 1 E [0,1]. By using Formula 02.2.2) we deduce that the exact solution is given by the function cp(t) = (2/7T)..fi. Equation 02.3.1) was solved numerically with the step h = 0.1 by using Formula 02.2.8). Figure 12.2 compares the exact (solid lines) and approximate (open circles) results. Calculations have shown that the inequality Icp(l k ) - CPn(lk)1 $ 10- 5 , k = 1, ... ,10, is valid. If the right-hand side of Equation 02.3.1) is put equal to 12 , then the exact solution is given by cp(t) = (8/(37T»t 3/ 2 • The exact and approximate solutions obtained by employing the same scheme were compared at the same points; however, the step was put equal to h = 0.0125. Calculations have shown that Icp(l k ) - CPn(tk)1 $ 0.00013 for I k = 0.1,0.2, ... ,1.0 (see Figure 12.3). For the same equation the case of the right-hand side being equal to unity was considered. From 02.1.18) it follows that in this case the results must be less accurate. Figure 12.4 shows that for h = 0.1 the calculated results (denoted by X X ) differ substantially from the exact data in the neighborhood of zero; however, for h = 0.00625 (see open circles) the difference between the exact solution, cP = l/(wIi), and the numerical results docs not exceed 0.02. We have also considered the two-dimensional equation

1 cp(~,TJ)d~dTJ =xy, loo~~ X

Y

(12.3.2)

whose exact solution is given by the formula 4

cp(x, y)

=

-21XY,

X E

[0; 1], Y

E

[0; 1.5].

7T

Numerical solutions were obtained for two cross sections: X = 0.1 and 0.2 (see Figures 12.5 and 12.6, respectively, where open circles correspond to h = 0.1, solid circles correspond to h = 0.05 and X X correspond to h = 0.025). Different steps along the X and y directions were

X =

316


.-

..... ~ ./

l-"""

~

V ~

/

42

/ I

(J

46

0.6

(0

t

FlGURE 12.2. Comparison of the exact (solid line) and approximate (circles) solutions to Equation (12.3.0 at points t k = kh, h = 0.1, k = 1,2, ... ,10. The approximate solution was obtained by using mode (12.2.8).

48

v V

46

1/ / / ,/

42

o

-

,/

"

",.. 42

0.8

f,0

t

FIGURE 12.3. Comparison of the exact (solid line) and approximate (circles) solutions to Equation (12.3.1) at points t j = j x 0.1, j = 1, ... ,10, for h = 0.00125. The approximate solution was obtained by using mode (12.2.8).

used. Let us compare the calculated results with the exact solution at points Yk = k X 0.1, k = 1,2, ... ,10. For x = 0.2, we have Icp(0.2, Yk) CPn(0.2,Yk)I:o; 0.008 for h = 0.05 and Icp(O.2,Yk) - CPn(0.2'Yk)I:o; 0.003 for h = 0.025.

12.4. NONLINEAR UNSTEADY PROBLEM FOR A THIN AIRFOIL In a more accurate formulation the unsteady problem must be considered as a nonlinear one, because free vortices shedding from an airfoil do


"

317

1,0

\ !

8

\

6

,., .

.........

0,

-

I'-....

~'L

o

0.6

1.0

0,8

t

FIGURE 12.4. Comparison of the exaet (solid line) and approximate solutions to Equation 12.3.0 with the right-hand side equal to unity, at points I j = 0.1 X j, j = 1,2, ... ,10, for h = 0.1 (x X X ) and h = 0.00625 (circles). The approximate solution was obtained by using mode 02.2.8).

9' (:r:.UJ fJ,20 0.15

0,05

o

-

z-4f

o,fO

~~

V

;......- ~

~

0.6

0.2

0.8

f.O

!J

FIGURE 12.5. Comparison of the exact (solid linc) and approximate solutions to Equation 02.3.2) at points I j = 0.1 X j, j = 1, ... ,10, for x = 0.1 and h = 0.1,0.05, and 0.025 {denoted by 0 0 0, • • •, and X X X, respectively.

(e,IJ)

20

.1:-42

0,10

o

Y

.-

_r- .--

-

~

--

/ 42

48

FIGURE 12.6. Comparison of the exact (solid line) and approximate solutions to Equations 12.3.2) at points I j = 0.1 X j, j = 1, ... ,10, for x = 0.2 and h = 0.1 and 0.025 (denoted by o 0 0 and X X X, respectively).

318


not move in its plane and their velocity differs from that of the uniform velocity U o (Belotserkovsky and Nisht 1978). Thus, we suppose that a sheet of free vortices sheds from the trailing edge of an airfoil, the velocities of the vortices coinciding with those of the particles occupying the same places. Let the instantaneous strength of the vortex sheet at point x of the airfoil be equal to y(x, t) and the strength of the free vortex sheet shed at the instant I be equal to 8(t). Suppose that the parametric equation of the CUNe where the sheet of free vortices is located at instant t is given by X=X(I,7),

(12.4.1 )

y=y(I,7),

Because a new location of a free vortex is found by displacing it in the direction of the local velocity, we have the foIlowing system of equations for specifying the curve 02.4. I): dx(t,7) -d-t- = ~(X(t,7),y(t,7»,

dy(t,7) -d-t- = ~(X(t,7),y(t,7»,

( 12.4.2)

7»

where ~ and ~ are the flow velocity components at point (xU, 7), y(t, due to the oncoming flow and the vortex sheet. Therefore, we get (sec (9.3.14»

~

=

Ux

+

~.

=

Uy

+

+

t

y(x,t)Wx<X(t,7),y(t,7),X,O)dx

-1

/ 18(s)Wx<X(t,7),y(t,7),X(t,s),y(t,s» 0

+

t

&

ds ds,

y(x,t)W/X(t,7),y(t,7),X,O)dx

- 1

/8(s)WiX(I,7),y(I,7),X(t,s),y(t'S»)-d& ds, 1o s

wher,e

wAxo,yo,x,y)

=

1 -2 ( 7T

Yo - y )2 + ( )2' Xo - X Yo - Y

(12.4.3)


319

1

Xo

-x

2

2 1T (x o - x) + (Yo - y) 2

dl

=

V[x;(t, S)]2 + [y;(t, s)f ds,

'

0$7$t.

For the system of differential equations (12.4.2) one has to find a special solution subject to the initial conditions x( 7 , 7)

=

(12.4.4)

y( 7,7) = O.

1,

Thus, we have to solve the Cauchy problem at the segment [7, T], 7 $ T, subject to the initial conditions (12.4.4), where T is the instant prior to which the problem must be solved numerically. If unsteady motion starts at a certain instant from the state of steady motion, then to find functions y(x, t) and 8(t) one has to use the no-penetration condition for the airfoil and the value of circulation at the latter. If an airfoil was initially at rest, then the total circulation of the whole of the vortex formation remains zero at any instant t, i.e.,

o$

f

l

-I

y(x,t)dx-+

118(s)-ds=O. dl

( 12.4.5)

~

0

The no-penetration condition generates another equation:

f

y(X,t)dx-

l

---- +

-I

xo-x

1 [Xo -x(t,s)]8(s)(dljds)ds 1

2

o [xo-x(t,s)] +[O-y(t,s)]

_ 2 -

21T~..

(12.4.6)

The system of Equations (12.4.2), (12.4.5), and (12.4.6) and the initial conditions (12.4.4) give a full solution to the nonlinear problem of unsteady flow past an airfoil with the vortex sheet shedding from the trailing edge only. Let us show that the linear unsteady problem considered in the preceding section is a special case of the present problem. To do this one has to require that

Yet, T) == 0,

(12.4.7)

i.e., the vortex sheet must travel along the OX axis. Hence,

Vy

=

u __ 1 fl y

y(x, t) dx- __1_ 277" -IX(t,7) -x 277"

8(s)(dljds) ds 11 X(t,7) == o. -x(t,s) 0

(12.4.8)


320

The latter identity must be fulfilled by the wake no-penetration condition, because in this case ~ is the normal velocity component at a point of the wake. Then, by supposing that the angle of attack is small and the oncoming flow velocity is equal to unity, one gets dx(t,'T) --d-t- = Ux = 1,

dy( t, 'T) - -dt- =0.

( 12.4.9)

Hence, taking into account initial conditions (12.4.4), we arrive at

X( t, 'T)

=

1

+ (t - 'T),

y(t,'T) =0.

(12.4.10)

Thus, the system of Equations (12.4.2) that describes the form of the wake is solved, and the problem reduces to solving the system of Equations (12.4.5) and (12.4.6). Because in this case dl

=

ds,

(12.4.11 )

the system of equations coincides with System (12.1.9). This nonlinear unsteady problem is solved numerically by the method of discrete vortices (Belotserkovsky and Nisht 1978) in the following way. The problem is considered at discrete instants t" t r + 1 - t r = dt, r = 1,2, .... Similarly to the linear problem considered previously, we choose at the airfoil points Xi' i = 1, ... , n, where bound discrete vortices rir as positioned, and reference points x o , j = 1, ... , n. At each instant r = 1,2,... a discrete free vortex A r sheds from the airfoil. At the instant r the free vortex is located at point x n + h of the axis OX, and at the next instant, r + 1, it is displaced (in the direction of the velocity vector) by the distance equal to the product of the flow velocity at the preceding point by dt. In other words, at the instant r the coordinates of a vortex shed at the instant v are given by

Yrn,.v =Yrv

+ [UYr +

t

l~

firwixrv,Yrv,x/,O) I


321

FIGURE 12.7. Computation of the slarting Prandtl vortex.

Xvv=X n +h,Y,'v=O,r= 1,2, ... , p= 1, ... ,r. (12.4.12) For calculating circulations of discrete vortices we use n equations derived by applying the no-penetration condition at points X o ' j = 1, ... , n, as well as the conditions of constancy of the sum of all circulations of discrete vortices: n

L

r

I~rwr<xOj,O,xi'O) +

-Uyn n

i=]

fir

+

A,w(xOj,O,xrs,Yrs)

s= I

i~]

L

L

LA, = 0,

j=1, ... ,n,r=1,2, ... ,

r = 1,2, ....

(12.4.13)

s=]

Because A I' ... , Ar ] are calculated at the preceding steps, only rin j = 1, ... , n, and A r are unknown at the instant r. Thus, a numerical solution to the problem is obtained in the following way: 1.

2.

At the instant r = 1, A] is placed at point Mi I = (x n + h,O) and System 02.4.13) is solved. Then, the flow velocity is calculated at point M] I and Formula 02.4.12) is used to calculate the position of the vorte~ A] at the instant r = 2, i.e., point M 2 ]. At the instant r = 2, A 2 is placed at point M 2 ,'2 = M I ,] and AI is placed at point M 2 I' and System 02.4.l3) is solved for r = 2. Next the flow velocities' are calculated at points M 2,2 and M 2, I and

322


Formulas (12.4.12) are used to calculate the coordinates of vortices Al and A z at the instant r = 3, i.e., points M 3 ,z and M 3 , I are found. Then, the second stage is iterated until the instant T is attained. Figure 12.7 demonstrates some calculated results. However, the problem of convergence of approximate solutions of unsteady problems remains unsolved (in the sense of strictly mathematical verification). Here we note only that due to the choice of the position of vortex A, at the instant r, System 02.4.13) is solvable in the same way as the corresponding system for the linear unsteady problem considered previously.

13 Aerodynamic Problems for Blunt Bodies

Of great practical importance is the problem of separated flow past blunt (high-drag) bodies whose contours contain sharp corners (certain buildings, monuments, towers, etc.). It is also important for designing various vehicles, such as automobiles, ships, aircraft, and so on.

13.1. MATHEMATICAL FORMULATION OF THE PROBLEM The most general problem formulation at the level of selecting an adequate physical model describing flow past a lifting surface was presented in Section 9.1. Through Chapters 10-12 we moved step-by-step from a discrete vortex system approximating a vortex sheet that simulated a surface to the corresponding singular integral equations. However, this was done for models in which (especially in the case of three-dimensional flow past a wing) one can easily single out vortex sheet components affecting the formation of lift and moments of a lifting surface. However, for more complicated separated flow models constructed for finite-span wings of a complex plan form (Belotserkovsky and Nisht 1978), the vortex sheet cannot be modeled by horseshoe vortices (the more so for noncirculatory problems). Therefore, a vortex sheet was modeled by vortex segments augmented by rather complicated relationships between the latter. Simulation of flow past closed surfaces proved to be even more complicated. Although it is very difficult to substitute vortex segments for a vortex sheet modeling a closed surface, it was found that the sheet (both at the surface of a body and in the wake) could be readily modeled by a system of closed rectangular and triangular vortex frames or, in the case of plane flows, by pairs of discrete vortices (Aparinov et al. 1988; Belot323


324

serkovsky, Lifanov, and Mikhailov 1985, 1987). The strength of these vortex formations is equal to the density of a double layer potential distributed at the body and in the wake and generating the same potential as the disturbed flow (Sedov 1971-72). Thus, let a body have the surface-contour u modeled by a vortex sheet. Also, let k vortex sheets up' p = 1,2, ... , k, be shed from the surfacecontour, which move with the fluid particles. The flow velocity induced by the vortex formations at point M at instant t will be denoted by V(M, d and ViM, t), P = 1, ... , k, respectively, and the oncoming flow velocity by Uo(M, d. Then the no-penetration condition at point M o of the surface u may be presented in the form k

V(Mo,t)· OM

(I

+

L

V{,(Mo,t)OM 0

=

-Uo(Mo,t)OM ,1 ' (13.1.1)

p=l

Because the disturbed flow velocities will be presented as gradients of the corresponding double layer potentials in what follows, the condition that circulation along a closed contour embracing a body and the wake be equal to zero is fulfilled automatically. However, if the disturbed flow velocities are presented as flow velocities induced by vortex singularities residing at a lifting surface, then Equation 03.1.1) must be augmented by the zero-circulation condition for any contour embracing the surface and the wake. Let r( M) be the radius-vector of point M. Then, the condition of point Mp ( T, t) of the vortex sheet up that was shed from the surface u at the instant T, moving at the instant t along the fluid particle path, may be written in the form k

r,'(Mp(T,t»)

=

V(Mp(T,t),t) +

L

Vm(MP(T,t),t)

m~l

(13.1.2) subject to the initial condition

where M p is a point of the surface u from which the particle was shed.

13.2. SYSTEM OF INTEGRODIFFERENTlAL EQUATIONS The concrete form of the system of equations depends on the way flow velocities are calculated in the no-penetration condition 03.1.1) and hence, in 03.1.2).

325

Aerodynamic Problems for Blunt Bodies

Previously (see Chapters 9-12) a lifting surface was modeled by a vortex sheet. Let us consider the most general two-dimensional flow when u is a contour containing k angular points Mp(x p , Yp ), p = 1, ... , k, from which vortex sheets are shedding; however, the contour may contain angular points from which no vortex sheets shed. The contour u is assumed to be piecewise Lyapunov (Muskhelishvili 1952) and specified by the parametric equations x

=

x( 0),

Y

=

y(O).

If the contour is unclosed, then 0 E [ -1,1]; if it is closed, then 0 E [0, 27T] and x(O) = X(27T), y(O) = y(27T). Let the strength of the summary vortex layer at point M(x( 0), y( 0» of the contour be equal to y(M, t) = y(O, t) at instant t, and let the strength of a free vortex shed from the corner M p at instant 7 be equal to 8p ( 7), p = 1, ... , k. Then, according to the Biot-Savart formula for a vortex filament, V(M o, t) and V[J(M o, t) entering Equation (13.1.1) arc given by

1 V(Mo,t)=27T

f 'T

r*(M, M o)

zy(M,t)duM ,

Ir( M, M o) I

r* = (M, M o ) = (Yo - y)i - (x o - x)j,

(13.2.1)

where the indices M and M/7, t) for u and ~) signify that the length of the arc is taken care of by the coordinates of these points. If contour u is closed and the parameter u is chosen in such a way that for an increasing 0 the contour is passed counterclockwise, then y'( 0o)i - x'( 0o)j

r*;n( M o )

Ir~n( Mo ) I

vx,Z(Oo)

+ y,z(Oo)

(13.2.2)

is an outward normal. The condition that circulation around a material loop embracing both the body and the wake equal to zero has the form

f y(M,t)du+ IT

k

L p~

I

f Up

8p (7) dup , Mp(T,I) = O.

(13.2.3)

326


Because at instant t the parametric equation of the wake form

o .:5:

7.:5: t,

up

has the

(13.2.4)

then

In this case,

Note that the set of points {X/7,S),Yp (7,S)}, 7.:5: s .:5:p, describes, at instant I, the path of the particle shed from corner M p at instant T. The formulas forV(Mi7, t), t) and Vm (Mp (7, t), t) appearing in Equation (13.12) may be obtained from the corresponding formulas (13.2.I) by substituting r(M, M p ( 7, t)), r(Mm(s, t), M p ( 7, t dUm, Mmls, I) for r( M, M o ), r(M/7,t), M o), dUp,M (T.I)' respectively. The substitution of1ntegral presentations (13.2. I) for velocities entering 03.1.1) and (13.1.2) results in Equation 03.1.1) becoming a singular integral equation and singular integrals appearing in (] 3.1.2). In the special case when U coincides with the segment [ - 1, 1] of thc OX axis, and a free vortex sheet sheds from point x = 1 only and moves linearly in the positive direction of the OX axis with average speed equal to unity, one arrives at Equations 02.4.8)-(12.4.11). An important result was obtained by Poltavsky (1986), who showed that for Equation (12.1.9),

»,

lim y(x, t) X'4

=

8(t).

( 13.2.5)

I

The latter equality confirms the validity of the Chaplygin-Joukowski hypothesis for a bound vortex sheet of an airfoil, according to which the strength of the vortex sheet vanishes when approaching the trailing edge (where the sheet sheds). Hence, at the trailing edge of a thin airfoil the whole of the vortex sheet consists of a free vortex sheet shedding off the edge. This remark allows us to place a free discrete vortex shedding from an airfoil at the trailing edge and to assume that it continues moving along the local flow velocity vector. This is of special importance for analyzing flow past a finite-thickness airfoil containing angular points, when the question arises where should a free discrete vortex shedding from a corner be placed. Now it is clear that, in analogy to the linear unsteady problem (if the Chaplygin-Joukowski

Aerodynamic Problems for Blunt Bodies

327

hypothesis is applied to the bound vortex layer in the framework of the nonlinear problem), the first discrete vortex shedding from a fixed corner must be placed at the corner itself and then allowed to move along the local velocity vector. This principle is used for constructing discrete vortex formations used in Be[otserkovsky, Lifanov, and Mikhai[ov (1985) for obtaining numerical solutions to the problem of separated flow past a contour containing angular points. When considering three-dimensional problems it was found that integra[ equations should be written with respect to the double layer potential jump. The same procedure will also be applied to two-dimensional prob[ems. In this case, one gets the following equation for the velocity V(Mo, t) induced by a body at point M o at instant t: (13.2.6) Here B(Mo, M) = (27T) 1 [n rMlt for the two-dimensional case and B(Mo, M) = (47T) IrMlt II for th~) three-dimensional case; VMf(M) = VMf(x, y, z) = f;i + t;j + f;k. For the velocity V/M o, t) induced at point M o at instant t, one gets

(13.2.7) The relationships for V(M/T, t), t) and Vm(M/T, t), t) entering Equation (13.1.2) may be obtained from the corresponding formulas (13.2.6) and (13.2.7) by using the same substitutions as used before. Note 13.2.1.

Because the oncoming flow is potential, the product UO<Mo, (J is a closed contour-surface, then the following equality is valid:

t) . n Mil is a normal derivative of a harmonic function, and, hence, if

(13.2.8)

Note 13.2.2. If a plane contour (J"" is unclosed, then for M o E (J the integrals entering (13.2.1) have a singularity of the form cot( 0o - 0) - I, and those entering (13.2.6) have the form (0 0 - 0)-2. However, if (J is a closed contour, then the preceding integrals have singularities of the form (0 o - 0)/2 and sin 2 [(0 0 - 0)/2], respectively.


328

13.3. SMOOTH FLOW PAST A BODY: VIRTUAL INERTIA As a rule, a vortex sheet forms both at the surface of a body and in the wake downstream of it. Therefore, hydrodynamic loads must be calculated taking into account vortex formations developing in the wake. However, for unsteady motions of a body, aerodynamic loads must also be known if the vortex wake behind a body may be ignored, as, for example, in the case of the noncirculatory flow mode. The same mode is used for calculating virtual inertia that is, in effect, an augmentation of the inertial properties of a body (such as mass, inertial/moments, etc.) and enter the expressions for forces and moments exerted by outer flow upon a body. According to the mode, outer flow is assumed to be fully potential and it is supposed that no wake forms downstream of a body. In this case differential equation (13.1.2) is ignored, and the problem reduces to finding the flow potential satisfying the Laplace equation outside a body and the nopenetration condition for the body surface. Generally, the no-penetration condition 03.1.1) for a solid body traveling through an incompressible fluid with translational velocity Uo and angular velocity n may be presented in the form .

(13.3.1)

where r Mo is the radius-vector of point Mo at which the no-penetration condition is met and 13k)' composing an approximate solution to Equation (16.2.10, in the following way. By using the values of Uk n (yfn,), i = 1, ... , n k , we construct an interpolation polynomial of the' (n k - l)th degree [denoted by Pk , n , - ](Y), y E (O'k' 13k )] in such a way that Pk.nJ
0,

u ... ,

_

z < 0,

U ,

where the functions u I (y', z') and u (y', z') are 21-periodic with respect to y', u 1 (y'

+ 21,

z')

=

u±(y', z'),

-x

< y'
O

forn>kl/7T.

1m 'Yn

(16.3.6)

From (16.3.4) and (16.3.5) it is seen that the functions u-(y',O) and u'(y',O) coincide for all y' E [-1,1]; hence, n

=

0,1,2, ... ,

n

=

1,2, ....

Thus we have x

u~=Aoe=ikz'

+ " e £...

lYnZ

. (

7Tny'

7Tny' )

A cos-- +B sin--

I

n

n~l

I

n'

'

z'

~

0,

( 16.3.7) and after using boundary condition (16.3.5) at the lattice, we arrive at the equation Ao

L x

+

(

7Tny'

An cos--

I

n= 1

7Tny' )

+ B n sin-l

=

0,

y'

E

CE'. (16.3.8)

Then from Condition (16.3.4), the other equation for determining unknown coefficients in the presentation (16.3.7) of a solution to the problem under consideration may be obtained: •

-ikA o +

x

L n= I

(

7Tny'

'Yn An cos--

I

+

7Tny' )

B n sin-l

=

k,

y' EE'.

( 16.3.9)

Numerical Method of Discrete Singularities

407

Let us introduce the nondimensional quantities

y

7TY'll,

=

K =

lkl7T

211A,

=

(16.3.10) where A is the incident field wavelength. Then, 7Tn 'Yn = -l-

V

1-

(K-;; )2 ,

n EN,

and the branch of the radical is selected in order to meet Conditions 06.3.6). Let us also introduce m

E

=

UE

CE

j ,

= [ -

7T, 7T] \ E,

j=l

-7T< 0'1 < 13, < ...
0, z' < O.

The functions V+(y', z') and V-(y', z') are 21-periodic in y' and satisfy the Helmholtz equation for z' ~ 0, respectively, as well as the matching conditions y'

E

CE',

(16.3.22)

y'

E

CE'.

(16.3.23)


413

At the lattice the normal derivative of the total field vanishes: d V-

I

dZ'

z'~O

(

d V+ d V~ - -) -

o dZ'

I

z'~()

dZ'

V'=ddZ'

-

I

z'~O

y/

-0 -

E'.

E

,

(16.3.24)

In order to ensure uniqueness of a solution to the boundary-value problem under consideration, both the radiation and Meixner conditions must be met (Honl, Maue, and Westphal 1964). A solution to the second boundary-value problem is sought in the form .,

Vt=A:e±,kz o

+

~

'"

"e-Y"z '-

n=1

.(

7T ny /

A: cos-- +B±

I

n

.

7T ny

/) I'

Z' ~

SIO--

n

0,

where 'Yn is defined in Section 16.3.1. From 06.3.22) and 06.3.23), it follows that n

=

0,1,2, ... ,

n

=

1,2, ....

By introducing the nondimensional quantities 06.3.9) and using Conditions (16.3.22) and (1.28), one arrives at the dual equation

L

AO +

(An cos ny + B n sin ny)

=

i,

Y

(1 - EnH nAn cos ny + nBn sin ny)

=

0,

Y EE,

E

CE,

n=1

L

- i KA o +

n~1

or by putting

Ao =

A~

+ i,

( 16.3.25)

one obtains finally a dual equation of the form (16.1.]) and 06.1.2): Ab +

L

(An cos ny + B n sin ny)

=

0,

Y

+ nBn sin ny)

=

0,

y EE,

E

CE,

n~1

bAb

+

L n=l

(l - En)(nA n cos ny


414

V] -

where b = -iK, En = ] (K/n)2, n EN. Thus, the Neumann problem has been reduced to the dual equation previously obtained for the Dirichlet problem. As shown in Section 16.2, unknown coefficients An and En' n EN, appearing in the representations of the fields V- and V+, are defined by Formulas (16.3.7) and (16.3.8) via function F(y), Y E E, which satisfies the singular integral equation of the first kind (16.2.11) with the right-hand

VI -

side equal to -f(x) = K, X E E, b = -iK, and En = 1 (K/n)2 , n E N, and the choice of the branch of the radical is determined by Conditions (16.2.5) and (16.2.12). The coefficient A o entering 06.3.25) is calculated from the formula ]

A o = i - -fF(y)dy,

27T

where

A~

E

was found by using Formula (]6.2.9).

16.4. APPLICATION OF THE METHOD OF DISCRETE SINGULARITIES TO NUMERICAL SOLUTION OF PROBLEMS OF ELECTROMAGNETIC WAVE DIFFRACTION ON LATTICES Much research has been done on wave diffraction on lattices. However, of special importance is the work by Agranovich, Marchenko, and Shestopalov (1962), where a method for solving the problem of diffraction of a plane monochromatic electromagnetic wave on a plane ideally conducting lattice was proposed. The method, based on reduction to the Riemann-Hilbert problem, was widely used for solving various problems of wave diffraction on periodic structures. The results are presented in monographs (Shestopalov 197'1, Shestopalov et at. 1973) where the interested reader will find extensive lists of references. The problems of diffraction of plane monochromatic electromagnetic waves on a plane ideally conducting lattice are reduced to the Dirichlet and Neumann problems considered in detail in the preceding section. Some problems were solved by the method of discrete singularities for the purpose of comparison with calculated results obtained by the method of reduction to the Riemann-Hilbert problem (Shestopalov ]971). The approach proposed herewith ensures a simple and uniform method for solving'the problems of wave diffraction on lattices irrespective of the number of strips per period. Let a lattice be placed in the plane OXY, which is composed of periodically repeated (with period 2l) groups of infinitely thin, ideally


415

conducting strips parallel to the OX axis. Let, further, a plane monochromatic wave fall onto the lattice from the upper half-space (z > 0):

where k = w/c is the wave number and the time dependence is determined by the factor exp( - i wt). First we will consider the problem of reflection of such a plane wave from an infinitely thin, ideally conducting screen placed in the plane OXY. The field in the upper half-space will be presented in the form of a sum of an incident and a reflected wave:

A solution to the Maxwell equations will be sought in the form of a sum of the initial wave (zero for z < 0 and a superposition of the incident and the reflected waves for z > 0) and the scattered wave: E

=

E(inc )

+

E(scal),

"

= "(inc)

+

"(scat).

The field must be continuous outside the strips; the components of the electric vector parallel to the plane OXY must vanish at the strips, and hence, the magnetic vector components normal to the plane the strips lie in must vanish too. Additionally, both the radiation conditions at infinity and the Meixner conditions at the ribs must be met. It is obvious that in the framework of the problem under consideration, all the planes perpendicular to the OX axis are physically equivalent, and hence, the scattered field is independent of x. The MaxweII equations split into a pair of independent systems of equations for the components Ex, flv' Hz and H x' E y , E z (Honl, Maue, and Westphal 1964). The requirement that the electric vector components parallel to the plane be equal to zero results in the boundary conditions Ex = 0 and aHx / az = 0 (at the strips). Thus, one may consider separately the cases of E and H polarization for which the corresponding vector is polarized paralIel to the OX axis (Belotserkovsky and Nisht 1978). Because the problem is linear, the amplitudes of the components Ei incj and Hync) will be assumed to be equal to i. Then, from the boundary conditions for z = 0 one can easily obtain the amplitudes of the components Eyef) and Hyef) of the wave reflected from the screen:


416

Hence, E(ine)

=

H(ine)

=

z > 0, z < 0,

{2 sin kz,

0,

x

z > 0,

{2i cos kz,

z < 0.

0,

x

a. Let us start by considering the case of E polarization. The problem reduces to finding the function u :;: Ex presentable in the upper and lower half-spaces in the form u =

{2~in kz +

u+,

U ,

z> 0, z < 0,

and satisfying the Helmholtz equation outside the lattice, the first boundary condition at the lattice, the radiation conditions at infinity, and the Meixner conditions at the ribs. This is the Dirichlet problem. A'i shown in the preceding section (see Section 16.3.1), it reduces to solving singular integral equation (16.2.11) subject to additional conditions (16.2.5) and 06.2.12). A numerical solution is found by the method of discrete singularities discussed in Section 16.2. After finding u, all the other components of the total field are determined by the formulas (Honl, Maue, and Westphal 1964) 1 au H = --Z

ik ay .

b. In the case of the H polarization, we arrive at the Neumann problem for the function V = H x sought in the upper and the lower half-spaces in the form

V = {2i cos kz + V+-, V- ,

z > 0, z < 0.

The problem was considered in the preceding section (see Section 16.3.3), where it was shown to be solvable by the method of discrete singularities. The remaining components of the total field are defined by the formulas (Honl, Maue, and Westphal 1964)

Ey

=

-

] av --a;'

ik


417

TABLE 16.4 K

IAol

K

IAol

lAd

0.1 0.2 0.3 0.4 0.5 0.6 n.7 0.8 0.9

0.0694 0.1394 0.2106 0.2838 0.3598 0.4401 0.5267 0.6232 0.7383 0.9476

1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

0.6086 0.5538 0.5256 0.5060 0.4769 0.4642 0.4696 0.4826 0.5050 0.6506

0.9108 0.8320 0.7942 0.7721 0.7576 0.7009 0.6801 0.6665 0.6589 0.6506

I

0.3527 0.3426 0.3361 0.3314 0.3264

TABLE 16.5 n

IAol

lAd

4 6 8

0.6056 0.6085 0.6085

0.9123 0.9110 0.9108

The problem of diffraction of an E-polarized plane monochromatic wave falling downward onto a simple (one strip I wide per period) lattice was solved for 20 values of the parameter K E (0; 3]. The number of grid points used in the method of discrete singularities was chosen equal to 10. The values of A o and A q, and then of IAol and IAql, q = 1,2, were calculated for nonattenuating diffraction harmonics. The results are presented in Table 16.4. The calculated values of B 1 and B 2 were equal to zero. A comparison of the numerical results with those presented in the monograph (Shestopalov 1971) shows that calculation of amplitudes of diffraction spectra by the method of discrete singularities produces quite satisfactory results and is very economical. For K = 1,2 the problem was solved with the number of grid points equal to 4, 6, and 8 (see Table 16.5).

17 Reduction of Some Boundary Value Problems of Mathematical Physics to Singular Integral Equations

In this chapter it is shown how some boundary value problems of mathematical physics may be reduced to singular integral equations by employing the theory of potential.

17.1. DIFFERENTIATION OF INTEGRAL EQUATIONS OF THE FIRST KIND WITH A LOGARITHMIC SINGULARITY Let a problem under consideration be reduced somehow to the solution of one of the following integral equations:

1 InsinI to - - t Iy (t)dt+ 1 K(to,t)y(t)dt=[(to)' 2

1 0 7T

27T

27T

0

(17.1.1) 1

-

7T

fl -1

Inlt o - tly(t) dt +

f

I

. 1

K(to, t)y(t) dt

=

[(to),

t oE(-l,l),

(17.12)

where the functions K(to, t) and [(to) are such that their derivatives with respect to to and t belong to the class H( a) on the corresponding 419


420

segment, and both the functions and their derivatives are periodic (with the period equal to 27T) in Equation 07.1.1). As a rule, the equations have unique solutions: the former, in the class H on [0, 27T]; the latter, in the class of functions of the form

y( t)

r/J( t) r;--:z' vI - t-

=

( 17.1.3)

where function r/J(t) belongs to the class H on [ - 1, 1]. Let us differentiate each of Equations 07.1.1) and 07.1.2) with respect to to' As shown in Gakhov (1977) and Muskhelishvili (1952), in the presence of a logarithmic singularity, the signs before a derivative and an integral may be changed to 1 127T to - t cot--y(t) dt + 27T 0 2

1 27TK;(to,l)y(t) dt 0

f

=

I'(to), (17.1.4)

II

fl'

1 I y( t) dt + K;(to,t)y(t)dl =I'{to)' (17.1.5) 7T - 1 to - t _ 1 II

-

A" a rule, the latter two equations have, in the class of functions, solutions to an accuracy of a constant, whereas for the former one the condition

(17.1.6) must also be met. The latter requirement is fulfilled if K(to, t) == O. In order to single out a unique solution of Equations (17.1.4) and (17.1.5) it suffices to specify an integral characteristic of the solution. Therefore, instead of Equation (2.1.1), one has to consider the system 1 127T to - t -2 cot--Y(l) dl + 7To 2

-1 7T

1 27TK; (to , t)y(t) dt = I'{to), 0

II

127TIn I-a 2--t Iy(t)dl + 127TK(a,t)y(l)dt =f(a), 0

a

€[O, 27T],

0

(17.1.7)

Reduction 0/ Some Boundary Value Problems

421

where a is a fixed point of the segment [0, 27T]. Equation (17.1.2) must be substituted by the system

-7T1 f-) y(to t)- dtt + f- )K;,,(to, t)y(t) dt I

-1

fl

7T -]

Inla - tly(t) dt +

I

fl

K(a, t)y(t) dt

=

!'(to),

t o€( -1,1),

=

/(a),

a€(-l,l).

-I

(17.1.8)

In a similar way one may reduce the following equation with a logarithmic singularity

tod -b,b] (17.1.9) (where 0 ~ b ~ 7T) to a singular integral equation of the form (17.1.7) or (17.1.8). Equation (17.1.9) is derived when solving various plane problems of radio wave diffraction (Zakharov and Pimenov 1982, Nazarchuk 1989).

17.2. DIRICHLET AND NEUMANN PROBLEMS FOR THE LAPLACE EQUATION It is well known that numerous applied problems of fluid mechanics reduce to the Dirichlet or Neumann problems for the Laplace equation (Bitsadze 1981, Vladimirov 1976, Zakharov and Pimenov 1982, Abramov and Matveev 1987, Nazarchuk 1989, Tikhonov and Samarskii 1966). Thus, the problems of aerodynamics considered in the preceding chapters of this book are variants of the Neumann problem for the Laplace equation. Let the Laplace equation (17.2.1)

be given on the plane OXY within a limited closed domain D with a piecewise smooth boundary (J (Muskhelishvili 1952). It is required to find a solution to the Dirichlet problem subject to the condition

Mo E

(J,

(17.2.2)

where /(M o), /;(M o), and /;(M o ), M o E (J, belong to the class H o on L, i.e., belong to the class H on all smooth portions of u.

422


The problem thus formulated has a unique solution (Tikhonov and Samarskii 1966) that will be sought in the form of a simple layer potential on (J, i.e., in the form (17.2.3)

where M o is an arbitrary point of the domain D. Because the simple layer potential is continuous on D, the boundary condition (I7.2.2) may be written in the form ( 17.2.4)

As long as (T is a closed contour, differentiation permits us to reduce Equation 07.2.4) to a singular integral equation of the first kind with Hilbert kernel (see Section 17.0. Let us now find a solution to the Neumann problem, i.e., consider the case when the condition (17.2.5)

is specified, where n M o is a unit vector of the outward normal of the curve (J at point M o, and function f(M o ) may have the same form as in Equation 07.2.2). A solution to the problem will be sought in the form of a double layer potential (17.2.6)

though traditionally it is sought in the form of a simple layer potential (Bitsadze 1981, Vladimirov 1976, Tikhonov and Samarskii 1966). Because the normal derivative of a double layer potential is continuous in the domain D, boundary condition 07.2.5) acquires the form Mo E

or the form (I 3.3.8).

(J,

(17.2.7)

Reduction

01 Some Boundary Value Problems

423

Note that Equation (13.3.8) furnishes simultaneously a solution to the external Neumann problem [subject to the condition that circulation due the gradient of function .......:;~

.!.·::l:-!- -:•• ,

. ,, ' ..... -.. .,...\t::i

~ .. ..... ..• - . .... .. - . ..!!::.::..- -:.:--' II

...

~...

••~~. :~. .-. '.....

.~.~~~

V·- ....·~1· ~:-

.A::c-

:--,..-:-.: :.: ~

..... ~ .

FIGURE

c.1.

:,:'.

;, .-;. :..e::-. : fl.....-. - . -, ....

. .

··.~t -.,.-.t" ".-:• •~,~ •

.-r

.• .

. .. . .... .-::,-:.r\• ...·r".· --..,-.:=,.::: ... -....... ,.. ..-.- .

,

Construction of the Karman vortcx street hy the method of diseretc vortices.

Chaplygin-Joukowski condition that the flow velocity finite at both edges is used. Because circulations of bound vortices vary in time, the flow past the plate is accompanied by shedding free vortices. This ensures conservation of circulation in accordance with the theorems of hydrodynamics and allows us to meet the Chaplygin-Joukowski conditions at both edges of the plate. The shape of the vortex wake is determined in the process of calculation subject to the conditions that the free vortices be frozen into the fluid and their circulation be independent of time. As a result, we managed to simulate not only integral but local effects, including the loss of stability of vortex surfaces and the formation of clusters composing the Karman vortex street. The method also proved to be most efficient for solving three-dimensional problems of aerohydrodynamics. The analysis of the problems called upon somewhat different mathematical formulations as well as required us to extend notions related to both the solution procedure and peculiarities of organization of numerical calculations. The interested reader will find a detailed description of the approach in Belotserkovsky and Nisht (1978). In connection with the foregoing comments, let us point out the following problems whose actuality becomes ever more evident. First, the mentioned ideas must be actively transferred into other areas of mathematical physics and applied science. The first steps in this direction, made in elastodynamics, have already brought promising results. Second, rigorous verification of new approaches to solving boundary problems should be undertaken; the approaches should be fully based on discrete presentations. Third, it is necessary to analyze and put into practice the possibilities of creating optimal software for high-speed supercomputers. The problem acquires even greater importance in view of the development of a unified mathematical methodology that permits us to solve various problems of

Conclusion

431

different physical nature by organizing conveyerized and co-current calculations. In conclusion, we would like to draw the reader's attention to some new processes under way in science. Numerical experiment is a qualitatively new method of study possessing a number of most unusual features; moreover, it incorporates what is presently called "artificial intelligence," because after being fully developed, a model finds its way from contradictory situations without employing some special algorithms. It is common knowledge that no discrete distribution of point vortices allows simulation of a stable Karman vortex street. However, we were able to do this in the framework of our model of separated flow past a plate by forming finite vortex clusters (Belotserkovsky and Nisht 1978). As known, flow velocities induced at the ends of a thin vortex surface tend to infinity. Nevertheless, a model describing the formation of an initial vortex wake downstream of a plate resulted in constructing a vortex spiral-a model of the initial Prandtl vortex. The so-called "effect of beneficial separation" is widely used in modern aviation. It is realized by creating favorable conditions for flow separation at the leading edges of small-aspect-ratio triangular wings or at a "bulb" of a swept-back wing. To induce flow separation, it suffices to sharpen the leading edges. Subsonic flow separates from a sharp edge, because otherwise its velocity would tend to infinity. Under real conditions a separation zone forms, which may be modeled by a vortex surface. The method of discrete vortices permits development of mathematical models of such flows (Belotserkovsky and Nisht 1978), which incorporated a special algorithm for describing the roll-up of vortex sheets into vortex cores. However, in our calculations bow vortex cores were formed automatically (Belotserkovsky and Nisht 1978). Note that the presence of the cores results in increasing lift of a wing and extending the working range of the angles of attack. The selection of stable vortex structures takes place in the process of either analyzing flow development (unsteady problems) or carrying out iterations (steady problems). Sometimes the results are quite startling, as in the case of a steady jet outflowing through a square orifice into a space filled with a fluid at rest. The pressure at the boundary of the jet must be equal to that in the space. High flow velocities developing in the corner points of the square-like orifice result in a local decrease of pressure and indentation of the jet surface. Because the outflowing fluid is supposed to be incompressible, the continuity equation results in conservation of the cross-sectional area of the jet, which acquires a star-like configuration. The accumulated experience of mathematical simulation permits us to state that without numerical experiments one cannot comprehend either mathematics or physics of a complicated phenomenon. Only after carrying out a numerical experiment can a mathematician be

432


sure that the very essence of a problem and its solution are understood. Applied scientists may be sure that they thoroughly understand a more or less complicated phenomenon only if they are able to construct its mathematical model for calculation on a computer. One of the realistic ways of increasing research efficiency in a number of novel areas, which was prepared by the long period of development of the method of discrete vortices, is the use of the latter's achievements on the basis of the method of discrete singularities. Along with the development and practical use of the two methods, it is worthwhile to develop specialized software and architecture of multiprocessor computers oriented onto the methods. The authors hope that this book will help to solve these problems.

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Index

INDEX

A Abel equation, 308 numerical solution of, 309-316 two-dimensional, 310, 315 Abel integrals, 309 Adiabatic processes, 7, 8 Aerodynamics, 48, 245-258, see also Airfoils for blunt bodies, see Blunt body aerodynamics formulation of problems in, 245-248 fundamental concepts of discrete vortices method and, 248--250 fundamental discrete vortex systems and, 251-258 linearized theory of, 10 linear steady problems of, 25 regularization in unsteady problems of, 357-358 singular integral equalions in, see Singular integral equations three-dimensional problems in, see Three-dimensional airfoil problems two-dimensional problems in, see Twodimensional airfoil problems unsteady problems of, 357358 Aerohydrodynamics, 430 Airfoils, see also Aerodynamics ca~cades and, 264-266 chord of, 10 contour of, II critical point of, 14 curved, 22 defined, 7 finite-thickness, 271"-275 finite-velocity, 260 flow past, 49, 139, 183 free vortex jumping across surface of, 358 Joukowski, 13 lift of, see Lift mass ejection on, 10 mass suction on, 10 moving, 8 nose of, 15, 16, 17, 18

optimal, I, 14-20 permeable, 281-282 with sink, 267, 269 straight linear, I3 tail of, 15, 16 thickness of, 10, 13 thin, see Thin airfoils three-dimensional problems in, see Three-dimensional airfoil problems two-dimensional problems for, see Twodimensional airfoil problems Analytic at a point, 2 Analytic functions, 4 Angle of attack, 9, 13, 14, 18,249 Angle of incidence, 9 Angular nodes, 30, 52 Angular points, 30 Angular velocity, 328 Artificial intelligence, 431 Asymmetric unsteady separaled flow, 339

B

B-condition, 250, 262, 301 Bernoulli integral, 14 Bessel equation, 391, 392 Bessel function, 408 Biot-Savart law, 249 Blades, 7 Blunt body aerodynamics, 323- 342 integrodifferential equations in, 324-327 mathematical formulation of problems in, 323-324 numerical calculalions in, 331-335 separated flows and, 335-342 smooth flow and, 328-331 three-dimensional problems in, 333 virtual inertia and, 328-331,336,337 numerical calculations and, 331-335 Boundary conditions blunt body aerodynamics and, 330

441


442 in boundary value problems, 388, 391, 392, 393, 422, 423, 424 at contour, 361 elasticity theory and, 371 no-penetration, 259, 291 uniformly moving punches and, 371, 372 Boundary layers, 14 Boundary point displacements, 375 Boundary value problems, 16, 17, 248, 387-417 Dirichlet and Neumann problems and, 404-414 formulation of, 404-408 Laplace equation and, 421-423 model problems in, 408-411 discrete singularities and, 402, 414--417 dual equations in, 387-395, 413 method for solving, 396-404 mixed, 387-395, 423-424 reduction of to singular integral equations, 419-424 three-dimensional, 390 Bound vortices, 247, 248, 250, 304 Buildings, 340

c Canonic division, 229 of circles, 33, 34, 37, 38 of closed Lyapunov curve, 93 of curves, 38, 51, 52, 93, 124 of segments, 40, 42, 49-50, 51 finite-span wings and, 96 full equation and, 156 for nonintersecting segments, 166 singular integral equatins and, 137, 143 of unclosed Lyapunov curve, 124 Canonic domain, 18 Canonic function, 205, 208 Canonic solution,S, I3 Canonic trapezoids, 89, 90, 98, 99, 294, 299, 300 leading edge of, 299 noncirculatory flow past, 298 steady flow past, 293 trailing edge of, 299 Carleman's singular integral equation,S Cartesian coordinates, 105, 252, 393, 404 Cauchy integrals, 2, 3, 5, 12, 24, 25, 32, sec also specific types

defined,2 double, 219, 237 multiple, see Multiple singular Cauchy integrals one-dimensional, 77, 82, 102 two-dimensional, 102 Cauchy kernel, 4, 26, 274 Cauchy-Lagrange integrals, 247 Cauchy principal value, 2, 3, 21, 32, 84, 250 Cauchy problem, 319 Cavity hydrodynamics, 6 Chaplygin-Joukowski condition, 22, 248, 250, 262,430 blunt body aerodynamics and, 326-327 three-dimensional airfoil problems and, 283 Chebyshev polynomials, 24, 71, 72, 209 Circles, 352, 354, 357 eanonic division of, 33, 34, 37, 38 elasticity theory and, 365 equations of first kind on, 175-182 Hilbert's kernel in equations on, 182-190 quadrature formulas for singulaf integrals on, 180 singular integral equations on, 196-199, 233 singular integrals on, 66-69, 180, 345 unit-radius, 224 Circulation, 10, 13, 249, 423, see also Flow change in, 247 constancy of, 303, 304 of discrete vortices, 21 of free vortices, 304 around material loop, 325 nonzero total, 22 problems in, 246, 268 past rectangular wings, 283-288 summary, 289 time-independent, 303 total, 22, 23 two-dimensional airfoil problems and, 260 unknown, 339 zero total, 23 Circulatory flow, 22, 248, 295 Closed contours, 2, 327, 423, sec also specific types blunt body aerodynamics and, 331 Lyapunov, 38 singular integrals over, 32-39 Closed curves, 39, 79, 92, 93, 217, 368 Closed rectangular vortices, 255, 323 Closed triangular vortices, 323

Index

443

Closed vortex polygons, 249 Close neighborhoods, 53 Collocation points, 250 Common multipliers, 227 Complex function, 2 Complex potential, 10 Complex variables, 1, 2, 7 Condition 11, 30, 31 Conformal mapping, 18 Conjugated sequences, 311 Conservation of cnergy, 8 Conservation of mass, 8 Conservation of momentum, 8 Constructing theory of Fredholm, 155 Contact pressure, 369, 380, 386 Contact stress, 379 Continuous curves, 273 Continuous kernels, 155, 157, 273 Continuous vortex surface, 248 Contours, see also specific types boundary conditions at, 361 closed, see Closed contours smooth, 30, 271 -275 unclosed, 2 Coordinates, 105, 252, 322, 325, 393, 404, see also specific types Coulomb law, 381 Cramer rule, 130, 138, 226 Cross-flow velocity, 246 Cubes, 337, 341, 342 Curves canonic division of, 38, 51, 52, 93, 124 closed, 39, 79, 92, 93, 217, 368 continuous, 273 Lyapunov, see Lyapunov curves node of, 32, 204 nonintersecting smooth closed, 217 open, 51 piecewise Lyapunov, 52, 95, US, 325 piecewise smooth, see Piecewise smooth curves plane, 224 smooth, see Smooth curves unclosed, 31, 51, 83, 333 Curvilinear free vortices, 250 Cusps, 30

D D'Alembert-Euler paradox, 8 Delta wings, 281

Density of gases, 7, 8 Difference formulas, 58-60 Dirac function, 371 Dirichlet and Neumann problems for Helmholtz equation, 404-414, 416 formulation of, 404-408 model problems in, 408- 411 for Laplace equation, 421-423 Discarded multipliers, 83 Discontinuities, 3, 25, 427, see also specific types of first kind, 179, 181, 187, 189, 300, 426 infinite, 343 power-law integrable, 426 Dissipation of gas energy, 8 Divergence theorem, 8, 9 Divergent improper integral, 2 Double Cauchy integrals, 219, 237 Double layer potential, 422 Drag, 2, 8, 9, 14, 15, 342 Drag vortex separation, 15 Duhamel-Neumann relationships, 372

E Eigenfunctions, 364, 365 Ejection, 10, 139, 266-270 Elasticity, 135, 188 theory of, 361-386 contact problem of indentation and, 368-380 Duhamel- Neumann relationships and, 372 Euler constant and, 374 Fourier transform and, 371, 373, 374 Jacobi polynomial and, 385 Lame equations and, 372 Poisson coefficient and, 385 two-dimensional problems of, 361-368 uniformly moving pun'ches and, sec Uniformly moving punches Elastodynamics, 370, 372, 374, 430 Electrodynamics, 402 Electromagnetic wave difFraction, 395 Electrostatics, 402 Energy conservation, 8 Equally spaced data, 111, 112, 127 Equally spaced grid points, 112, 127, 173, 346, 355, 356


444 Errors, 353, 399, see also specific types Euler constant, 374 Euler equations of motion, 247 Explicit methods, 7, see also specific types

F Field singularities, 8 Fikhtengoltz formulas, 44, 132 Finite elements method, 7 Finite-span wings, 96-103 of complex plan form, 323 rectangular, 239 schematic representation of, 246 separated flow models for, 323 in stationary flow, 239 steady flow past, 300 subsonic flow past, 83 Finite-thickness airfoils, 271-275 Finite velocity, 269, 276, 284 Finite-velocity airfoil problems, 260 Flight, 14, 15-16, 18 Flow, see also Circulation past airfoils, 49, 139, 183 past blunt bodies, 323 past canonic trapezoids, 295 circulatory, 22, 248, 295 gas, 7, 8 of incompressible fluid, 14 past infinite-span wings, 160 irrotational, 7, 246 modes of, 22 no-circulation, 261, 270 noncirculatory, see Noncirculatory flow perturbed, 8 polytropic, 7 potential of, 14 past rectangular wings, 283-288 separated, see Separated flow smooth,328-331 stationary, 239 steady, see Steady flow strength of, 275 subsonic, 83 past thin airfoils, 21 past thin wings, 249 time-independent, 303 two-dimensional, 325, 342, 357 unbounded, 262 unperturbed gas, 7, 8

unsteady, 340, 341, 342 velocity of, 21, 254, 258, 324, 342, 358, 430 past wings, 248 Huid, see also specific types ideal, 303, 357 incompressible, 83, 245, 303, 357 inviscid incompressible, 245 perfect incompressible, 83 veloci ty of, 13 Fourier heat conduction law, 370 Fourier series, 214, 344, 391, 397 Fourier temperature transformant, 371, 374 Fourier transform, 371, 373, 374, 404 Fredholm equations, 4, 214 elasticity theory and, 362· 363 integrodifferential equations and, 239, 241 of second kind, 181, 189, 224, 239, 241, 362-363 segments and, 154, 155, 156, 157, 160 Fredholm's theory of constructing, 155 Free vortices, 247, 248, 250, 303, 318 blunt body aerodynamics and, 340 circulation of, 304 discrete, 304 jumping across airfoil surface by, 358 strengths of, 304 velocity of, 318 Friction, 14,369,370,381

G Gamma functions theory, 132 Gases, see also specific types density of, 7, 8 dynamics of, 8, 9 energy dissipation of, 8 flow of, 7, 8, 9 moving in, 8 pressure of, 7 unperturbed flow of, 7, 8 unperturbed speed of, 9 velocity of, 7 Gaussian quadrature formulas, 410, 411 Generalized functions theory, 374 Generalized polynomials, 208, 209 Grid points, 21, 23, 112,357, see also specific types equally spaced, 112, 127, 173, 346, 355, 356 unequally spaced, 113, 148, 173, 276, 353, 356

Index

445

Grids, 23, 55, 233, 250, 350, see also specific types

H

Hadamard criterion, 146, 292, 296 Hadamard finite value, 24, 58 Harmonic function, 327, 388, 393 Heat conduction law, 370 Heat conductivity, 371 Heat generation, 370 Helmholtz equation, 388, 390, 416 Dirichlet and Neumann problems for, 404-414 formulation of, 404-408 model problems in, 408-411 Hilbert's kernel, 4, 26, 348, 352, 357 boundary value problems and, 422 elasticity theory and, 363 equations on circles with, 182-190 singular integral equations with, 199-203 of first kind, 422 variable coefficients and, 212-215 singular integrals with, 57-58, 61-66, 215 singular integrals over circles with, 345 Hilbert's problem, 4 Hilbert's transform, 397, 403, 408 History of singular integral equations, 2-7 Holder conditions, 30, 64, 203 Bolder-continuous functions, 398, 403 Holder functions, 208, 214 Horseshoe vortices blunt body aerodynamics and, 323 discrete, 284, 289 oblique, 249, 254, 255 slanting, 293, 294, 295, 296, 300 straight, 255, 284, 289, 382 usual,254 Hydrodynamics, 6, 430

I Improper integral, 2, 3 Incompressible fluids, 14,83,245,271,303, 357 Inertia, see Virtual inertia Infinite discontinuities, 343 Infinite-span vortex filament, 249

Infinite-span wings, 160 Influence functions, 374 Integrable singularities, 6, 25 Integral equations, see Singular integral equations Integrals, 10, see also specific types Abel, 309 Bernoulli, 14 Cauchy, see Cauchy integrals Cauchy-Lagrange, 247 Cauchy principal value of, 3, 21, 32 divergent improper, 2 improper, 2, 3 invariant, 14 multiple Cauchy, see Multiple singular Cauchy integrals singular, see Singular integrals two-dimensional, 24, 101 Integrodifferential equations, 239-241 in blunt body aerodynamics, 324-327 Prandtl,58 two-dimensional, 239 Internal point of the frame, 75 Interpolation polynomial, 399 Invariant integral, 14 Inversion formulas, 224, 309 Inviscid compressible gas, 7 Inviscid incompressible fluid, 245 Irrotational flow, 7, 246 Iterated kernels, 157

J Jacobian, 89 Jacobi polynomials, 194,209,385 Jet hydrodynamics, 6 Jordan's two-dimensional measure, 104, 109 Joukowski airfoils, 13 Joukowski theorem, 247

K Karman vortex street, 430, 431 Kernels, see also specific types Cauchy, 4, 26, 274 continuous, 155, 157, 273 Hilbert's, see lIilberl's kernel iterated, 157


446 Kinematic viscosity, 15 Kolosov-Muskhelishvili formulas, 361 Komeichuk inequality, 70

L Lame equations, 372 Laplace equation, 246, 328, 329, 423 in boundary value problems, 388, 390, 393 Dirichlet and Neumann problems for, 421-423 Leading edge, 13, 248, 304 of canonic trapezoids, 299 equations for, 300 of trapezoids, 296, 301 two-dimensional airfoil problems and, 260 Left-hand direct product, 233 Lift, 8, 9, 10, 14 coefficients of, 15 maximum, 8 of thin airfoils, 1,7-14 Lift-generating structural elements, 7 Lifting surface, 21 Linear multipliers, 194, 195 Linear steady problems of aerodynamics, 25 Linear unsteady problems, 303--309, 322 Lines, 29, 30 Liouville theorem, 6 Local flow velocity, 342 Locally polytropic processes, 8 Local shock waves, 15 Local sound speed, 9 Local supersonic zone, 14 Logarithmic singularities, 419 -421 Lord Kelvin condition, 337 Lyapunov contour, 38 Lyapunov curves, 24, 31, 52, 54, see also specific types closed, 39, 92, 93, 368 elasticity theory and, 363 nonintersecting, 39, 94 open, 51 piecewise, 52, 95, 115, 325 unclosed, 51, 124

M Mach number, 9, 14, 16 Mapping, 18 Mass conservation law, 8

Mass ejection, 10 Maximum flight speed, 15 Maximum lift, 8 Maxwell equations, 415 Meixner conditions, 405, 408, 415 Metric of the space of functions, 211, 215 Minimum drag, 8 Modified Bessel equation, 391, 392 Momentum conservation, 8 Multi-dimensional singular integrals, 24 quadrature formulas for, 103-110 Multiple singular Cauchy integrals, 24, 25, 92,217-241 analytical solutions to, 217-224 defined, 32 integrodifferential equations and, 239- 241 inversion formulas for, 224 numerical solutions for, 224-239 canonic division and, 229 common multipliers and, 227 Cramer rule and, 226 uniform grids and, 233 unit-radius circles and, 224 one-dimensional, 82, 238 Poincare-Bertrand formula and, 120-124 on products of plane curves, 224 quadrature formulas and, 75-83 of second kind, 234 Multiple singular integrals, 24, 75-113 in aerodynamics, 83-92 Cauchy, see Multiple singular Cauchy integrals for finite-span wings, 96-103 quadrature formulas for, 92-95 Multipliers, 83, 194, 195, 227, see also specific types Muskhelishvili equations, 363

N Natural regularizing factors, 345, 346 N-dimensional torus, 80 Neumann problem, 412-417, 421, 423, see also Dirichlet and Neumann problems No-circulation flow, 261, 270 Node of curves, 32, 204 Node vicinity, 53 Noncirculatory flow, 22, 247, 261, 266, 275, 290,293 blunt body aerodynamics and, 338

Index

447

three-dimensional airfoil problems and, 298 Nonintersecting Lyapunov curves, 39, 94 Nonintersccting segments, 56, 162-168, 218 Nonintersccting smooth closed curves, 217 Nonlinear unsteady problems, 316-322 Nonpenetration condition, 11, 16, 21, 26 Nonsingular nodes, 204 Nonzero total circulation, 22 No-penetration boundary conditions, 259, 291 No-penetration condition, 245, 261, 265, 271, 304,319 blunt body aerodynamics and, 324, 328 three-dimensional airfoil problems and, 284 two-dimensional airfoil problems and, 279 wake, 320 No-slip condition, 245 Numerical integration, see Quadrature

o Oblique horseshoe vortices, 249, 254, 255 One-dimensional Cauchy integrals, 102 One-dimensional singular integrals, 29 60, 61-66, 238 Cauchy, 77, 82, 102 on circles, 66-69 over closed contours, 32-39 defined, 29-32 of first kind, 217 with Hilbert's kernel, 57-58, 61-66 over piecewise smooth curves, 51-56 Poincare- Bertrand formula and, 115-120 on segments, 39-51, 69-73 theorems for, 42-A8 theory of, 29-32 Open curves, 51 Optimal airfoil, 1, 14-20 Optimal flight speed, 15 Oscillating wings, 288 Outer force, 7

p Parachutes, 281 Perfect incompressible fluid, 83 Permeable airfoils, 281-282

Perturbed flow, 8 Piecewise Lyapunov curves, 52, 95, 115, 325 Piecewise smooth curves, 30, 32 singular integral equations on, 203 -- 211 singular integrals over, 51-56 Plane curves, 224 Plane strain, 361 Plane stress, 361 Poincare-Bertrand formula, 24, 81, 82, 115--124 full equation on segments and, 157 multiple singular Cauchy integrals and, 120-124 one-dimensional singular integrals and, 115-120 singular integral equations on segments and, 136, 138, 140, 154 validity of, 119, 120 Poisson coefficient, 385 Polygons, 249 Polynomials, 429, see also specific types Chebyshev, 24, 71, 72, 209 generalized, 208, 209 imerpolation, 399 Jacobi, 194, 209, 385 trigonometric, 202, 213, 214, 215 Polytropic processes, 8, see also specific types Potential, 13, 14, 422 Power-law integrable discontinuities, 426 Prandtl equation, 58, 162 Prandtl vortex, 321 Pressure, 15, 16 contact, 369, 380, 386 decrease in, 379 distribution of, 341 gas, 7 gas 11ow, 8

Q Quadrangular-triangular vortices, 330, 333 Quadrature formulas, 24, 29-60, 61-73, 75-113, 426, see also· specific types coefficients of, 429 construction of, 426, 427 Gaussian, 410, 411 for multi-dimensional singular integrals, 103-110 for multiple Cauchy integrals, 75-83 for multiple singular integrals, 92-95 regularizing factors and, 350


448 for singular integrals in aerodynamics, 83-92 on circles, 66-69, 180, 197, 199 over closed contours, 32-39 estimation of, 346 for finite-span wings, 96-103 with Hilbert's kernel, 57-58,61-66, 182 multiple Cauchy integrals and, 75-83 over piecewise smooth curves, 51-56 on segments, 39-51, 69-73, 148, 194, 348 theorems for, 42-48 theory of, 29--32 unification of difference formulas and, 58-60

R Radiation, 405, 416 Rectangle rule formulas, 180 Rectangles, 54 Rectangular vortices, 255, 323 Rectangular wings finite-span, 239 flow with circulation past, 283-288 flow without circulation past, 288--293 Rectilinear vortex segments, 249 Reference points, 21, 22, 23, 250 elasticity theory and, 368 three-dimensional airfoil problems and, 284,296 two-dimensional airfoil problems and, 76, 260, 262, 264, 267, 276 velocity component at, 304 Regularizing factors, 135, 183, 193, 196,345 natural, 345, 346 self-, 345 singular integral calculations and, 346-350 for singular integral equations, 350-357 two-dimensional airfoil problems and, 262 in unsteady aerodynamic problems, 357-358 Regularizing parameters, 25, 350 Regularizing variables, 23, 211, 275, 332, 334, 337 Relative velocity, 247 Restriction of function, 402 Rhombus, 341 Riemann-Hilbert problem, 395, 414 Riemann problem, 4, 5

s Segments canonic division of, 40, 42, 49-50, 51 finite-span wings and, 96 for nonintersecting segments, 166 singular integral equations and, 137, 143 full equation on, 154-162 canonic division and, 156 Fredholm equations and, 154, ISS, 156, 157,160 Poincare-Bertrand formula and, 157 Prandtl equation and, 162 non intersecting, 56, 162-168, 218 rectilinear vortex, 249 singular integral equations on, 127-147, 156,233 approximate solution to, 192 canonic division and, 137, 143 Cramer rule and, 130, 138 gamma functions and, 132 Hadamard criterion and, 146 nonuniform division and, 148-154 numerical solution of, 168-173 Poincare-Bertrand formula and, 136, 138, 140, 154 of second kind, 191 -196 singular integrals on, 39-51, 69-73, 348 theorems for, 4248 union of, 56 Self-regularization, 345 Separated flow, 23, 323, 429 asymmetric unsteady, 339 numerical calculations for, 335-342 symmetric unsteady, 339 unsteady, 342 Separation locus of vortex sheet, 15 Sharp tail, 12 Shedding, 26, 283, 319 Sherman-Lauricella equations, 363 Shock waves, 8, 15 Simple layer potential, 422 Singular integral equations, 1-20, 21, 127, 191-215,427, see also Singular integrals in boundary value problems, 387, 397, 398, 400, 401, 403 boundary value problems reduced to, 419-424

Index Carleman's, 5 with Cauchy kernel, 274 on circlcs, 196-199, 233 degenerate, 363 differentiation of, 419-421 direct numerical methods in, 7 of elasticity theory, see Elasticity, theory of of first kind, 21, 274, 419-421, 422 future development of, I general theory of, 401 with Hilbert's kernel, 199-203 of first kind, 422 variable coefficients and, 212-215 history of, 2-7 with logarithmic singularity, 419-421 with multiple Cauchy integrals, see Multiple singular Cauchy integrals numerical solutions for, 350-357 optimal airfoil problem and, 14-20 on piecewise smooth curves, 203-211 problems of, 4 regularizing factors for, 350-357 of second kind, 219, 363 on segmcnts, see under Scgments systems of, 4 two-dimensional airfoil problems and, 280, 282 with variable coefficients, 203-211 Singular integrals, 2, 3, 21, 426, see also Singular integral equations in aerodynamics, 83-92 calculations for, 110-113 Cauchy principal value of, 3 on circles, 66-69, 180,345 over closed contours, 32-39 defined, 29-32 diagonal, 280 equations for, see Singular inte-gral equations estimation of, 346 for finite-span wings, 96-103 with Hilbert's kernel, 57-58, 61-66, 215 ill-posedness of equations with, 343-346 multi-dimensional, 24 quadrature formulas for, 103-110 multiple, see Multiple singular integrals multiple Cauchy, see Multiple singular Cauchy integrals one-dimensional, see One-dimensional singular integrals over piecewise smooth curves, 51-56

449 Poincare-Bertrand formula and, 115-120 quadrature formulas for, 24, 346 regularizing factors and, 346- 350 on segments, 39-51, 69-73 theorems for, 42-48 theory of, 29-32 two-dimensional, 101 Singularities, 21, see also specific types discrete, 26,402,414-417 at end points, 25 field, 8 integrable, 6, 25 of lifting surface, 21 logarithmic, 419-421 Singular nodes, 204 Singular operator of second kind, 219 Sink, 267, 269 Slanting horseshoe vortices, 293, 294, 295, 296,300 Slanting normal, 276-281 Smooth contours, 30, 271-275 Smooth curves, 29, 32 closed, 79,217 piecewise, see Piecewise smooth curves unclosed, 31, 83 Smooth flow, 328--331 Smooth line, 29, 30 Sokhotsky equations, 3, 5, 12, 19 Sokhotsky-Pleneli formulas, 208 Sonic barrier, 15 Specific heat capacity, 371 Speed, 8, 9, 14, 15, see also Velocity Squares, 336, 341, 342 Stability prOblems, 351, 354 Stagnation points, 15 Stationary flow, 239 Stationary wings, 288 Steady flow, 160 past canonical trapezoid, 293 past finite-span wings, 300 ideal incompressible, 271 incompressible, 271 polytropic, 7 past thin airfoils, 259-264 two-dimensional, 259 unbounded, 262, 264 Strain, 371, see also specific types Streamlining, 14 Strength, 21, 249, 251 constant, 254, 257 distribution of, 269 flow, 275


450 moving concentrated source of, 372 of summary vortices, 325 of vortex, 10,260, 261, 325 of vortex layer, 248, 275 of vortex sheets, 259, 269, 304 Stress, 364, see also specific types concentration of, 386 contact, 379 plane, 361 tangential, 368-369, 375 Structural elements, 7 Subsonic flow, 83 Subsonic speed, 8 Suction, 10 Summary circulation, 289 Summary vortices, 250, 325 Supercomputers, 425, 430, 432 Superposition principle, 375 Supersonic zone, 14 Symmetric unsteady separated flow, 339

T Tail, 12, 13, 15, 16, 17, 18,89 Tangential stress, 368-369, 375 Tangential velocities, 267 Taylor's formula, 57 Theory of functions of complex variables, I, 2,7 Theory of gamma functions, 132 Thin airfoils, 11, 12, 15 with ejection, 266··270 with flaps, 262, 263 flow past, 21 gas flow past, 9 lift of, 1, 7-14 linear unsteady problems for, 303-309 nonlinear unsteady problem for, 316-322 steady flow past, 259-264 two-dimensional problem of, 8 Thin plate, 15 Thomason condition, 337 Three-dimensional airfoil problems, 283-301 arbitrary plan form wings and, 293-301 rectangular wings and flow with circulation past, 283-288 flow witkout circulation past, 288-293 Three-dimensional blunt body aerodynamic problems, 333 Three-dimensional boundary value problems, 390

Three-dimensional Cartesian coordinates, 252 Three-dimensional Dirichlet and Neumann problems, 404 Thrust, 7 Total circulation, 22, 23 Trailing edges, 13, 15, 183,248,304,319 of canonic trapezoids, 299 equations for, 300 flow shedding smoothly from, 283 two-dimensional airfoil problems and, 260 Translational velocity, 247, 303, 328 Trapezoids, 296, 298, 299, 300, 301, see also specific types canonic, see Canonic trapezoids Triangles, 340, 342 Triangular vortices, 323 Trigonometric polynomials, 202, 213, 214, 215 Two-dimensional Abel equation, 310,315 Two-dimensional airfoil problems, 259-282 cascades and, 264-266 ejection and, 266-270 finite-thickness airfoils and, 271- 275 finite-velocity, 260 permeable airfoils and, 281-282 slanting normal and, 276 -281 steady flow and, 259-264 Two-dimensional Cauchy integrals, 102 Two-dimensional elasticity problems, 361-368 Two-dimensional elastodynamic problems, 372 Two-dimensional flow, 325, 342, 357 Two-dimensional integrodifferential equations, 239 Two-dimensional problems, 8, see also specific types Two-dimensional singular integrals, 101 Two-dimensional steady flow, 259

u Unclosed contours, 2 Unclosed curves, 31, 51, 83, 333 Unclosed line, 29 Unclosed Lyapunov curves, 51, 124 Unequally spaced data, 148 Unequally spaced grid points, 113, 148, 173, 276, 353, 356

Index

451

Uniform grids, 233 Uniformly moving punches, 368- 386 boundary conditions and, 371, 372 contact pressure and, 369, 380, 386 contact stress and, 379 Duhamel-- Neumann relalionships and, 372 Euler constant and, 374 Fourier transform and, 371, 373,374 friction coertieients and, 369, 370, 381 heat conduction and, 370 heat conductivity and, 371 heat generation and, 370 Jacobi polynomial and, 385 Lame equations and, 372 Poisson coefficient and, 385 tangential stress and, 368-369, 375 Uniform metric, 345, 351,357 Unit-radius circles, 224 Unperturbed gas flow, 7, 8 Unsteady flow, 340, 341, 342 Unsteady problems, see also specific types of aerodynamics, 357-358 linear, 303-309, 322 nonlinear, 316-322 regularization in, 357-358

v Vector now velocity, 342 Vekua's approach, 224 Velocity, see also Speed angular, 328 average translational, 303 cross-flow, 246 finite, 269, 276, 284 finiteness of, 248 of flight, 14, 15-16, 18 flow, 21, 254, 258, 324, 342, 358, 430 fluid, 13 of free vortices, 318 gas, 7 gas flow, 8 normal, 295 relative, 247 tangential, 267

translational, 247, 303, 328 virlual, 15-16, 18 of vortices, 318 Virtual inertia, 26, 288 blunt body aerodynamics and, 328-331, 336, 337 numerical calculations and, 331 335 coefficients of, 330, 331335, 336 of cubes, 337 of squares, 336 Virtual velocity, 15-16, 18 Viscosity, 15 Viscous frieri on, 14 Volterra equation, 309 Volumetric expansion, 372 Vortex grids, 250 Vortex sheet separation locus, 13

w Wake, 21 Wake no-penetration condition, 320 Wings, 7 of arbitrary plan form, 293-30 I delta, 281 finite-span, see Finite-span wings flow past, 248, 249 infinite-span, 160 oscillating, 288 rectangular, see Rcctangular wings stationary, 288 steady motion of, 245 thin, 249 unsteady motion of, 245

z Zero-circulation condition, 324 Zero drag theorem, 2 Zero total circulation, 23 Zero two-dimensional Jordan measure, 109

Method of Discrete Vortices

Method of discrete vortices

Theory of Concentrated Vortices

The combined finite-discrete element method

Vortices in Bose-Einstein Condensates

Ginzburg-Landau vortices

Vortices in high-temperature superconductors

On planar selfdual electroweak vortices

A shadowing lemma for abelian Higgs vortices

Vortices in Bose-Einstein Condensates

Discrete Variational Derivative Method: A Structure-Preserving Numerical Method for Partial Differential Equations

Pearls of discrete mathematics

Method of Madness

Adaptive method of lines

Adaptive Method of Lines

Renaissance Concepts of Method

Generalized Method of Moments

Descartes's Method of Doubt

The Method of Coordinates

Descartes's Method of Doubt

Descartes's Method of Doubt

The Method of Nature

Vortices in the magnetic Ginzburg-Landau model

Method of Madness

Rules of Sociological Method

Method of Madness

A Method of Programming

Pearls of Discrete Mathematics (Discrete Mathematics and Its Applications)

Discrete Mathematics

Discrete mathematics

Schaum's Outline of Discrete Maths

Method of Discrete Vortices