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
Method of Discrete Vortices
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<m<1.
(I.l3)
This is a particular example of the Riemann problem. Let us find an auxiliary (canonical) solution X(Z) to the problem for the case where g(x) = 0:
(U4) By knowing the properties of functions of type of solution to Equation I.l4 in the form x( z)
= Z 8 (z
1 1 8= - - -Inm
2
Substituting m
=
27Ti
it is easy to obtain a
Z8,
- 1) 1- 8,
(U5)
X(z)
'
lim - z---.x z
=
1.
-X+ /X- into Equation 1.13, we get ig(x)
(I.l6)
6
Method of Discrete Vortices
From physical considerations, it is important to specify the behavior of the solution ncar the ends x = 0 and x = ]. The function cfJ(z) 1 F ( z) = -X-(z-) - 2 7T (A + JL)
I
fa -X-
g(7)d7 7-)-(-7---Z-) ,
t -(
(1.17)
in view of Equation 1.16, will satisfy the condition equation F+(x)
=
o< x
F (x),
< 1,
(1.18)
and hence F( z) is analytic everywhere in the z plane with the possible exception of points z = 0 and z = 1, where F( z) can have poles, dependent on the specified solution behavior at these points. For example, if the only integrable singularities of the solution are physically permitted, these poles are simple; then, according to the Liouville theorem, we obtain
F(z)
C1 =
-
z
Cz
+ --,
(1.19)
z - 1
where C 1 and C z are some constants. Considering Equations 1.17 and 1.19, we obtain the solution in the form
cfJ(z)
=
X(z) 27T(A+JL)
1 X+(7)(7-Z) g(7)d7 1
0
(C
1
+X(z) ----;
Cz )
+ z-1 ' (1.20)
where
The solution of the singular integral Equation 1.11 is determined from Equation 1.20 using Equation 1.5. In this case, the solution depends on the arbitrary constant. The opportunities of the explicit solution were intensively used in thousands of papers and books devoted to numerous applied problems of mathematical physics reduced to the problem in Equations 1.7-1.10. For example, the bibliography of papers on hydrodynamics of jets and cavities in Gurevich (1965) includes nearly 700 main sources. The author also took part in these developments (see Cherepanov 1963, 1964).
Introduction
7
Explicit methods were extremely popular in Russia from 1900 to 1960, and undoubtedly they furthered the success of large-scale engineering projects during that time, despite limited funding and human resources. As a matter of fact, the theory of functions of a complex variable gained more efficient engineering formulas than the rest of mathematics. Joukowski believed that "the task of mechanics was to find out explicit solutions." However, until recently, there were no direct numerical methods in singular integral equations, so that the latter were being quickly forgotten because the yield of explicit solutions was already collected. Numerical method of discrete vortices, treated in detail in this book, enables us to use all the technical advantages of potentials and general solution. Its ease of use is greater than that of general numerical methods (for example, the method of finite elements), so that in cases when the latter require a supercomputer, a calculator sometimes suffices in the former.
1.2. LIFf FORCE OF A THIN AIRFOIL Airfoils are main thrust and lift-generating structural elements of aerodynamical machines, including wings and blades. Let us consider the irrotational steady polytropic flow of an inviscid compressible gas. Following Cherepanov (1987), the governing equations of gas dynamics can be written in the form of f integrals invariant with respect to closed integration surfaces:
i,k=1,2,3,
(1.21 )
(1.22)
(1.23)
Here p and p are the density and pressure of a gas, respectively, V j are the Cartesian components of gas velocity, Px and llx are the pressure and velocity of unperturbed gas flow, n j are outer unit normal components to S, " is the polytr,ope coefficient equal to the ratio of specific heat capacities in the case of adiabatic processes, S is an arbitrary closed surface in gas, and f k are the Cartesian components of the equivalent outer force applied to bodies or singularities inside of S. The f k also can
8
Method of Discrete Vortices
be treated as the dissipation of gas energy inside of S, when the bodies have moved one unit length distance along the X k axis. The rks equal zero if there are no bodies, field singularities, or shock waves inside of S. Repeated indices, as usual, mean summation. Equation 1.21 is the mass conservation law. Equation 1.22 can be considered as both the energy or momentum conservation law. Equation 1.23 is valid for a locally polytropic (e.g., adiabatic) process. When there are only gas particles inside of S, by applying the divergence theorem to Equations 1.21 and 1.22, one easily can derive the traditional differential equation system of gas dynamics. When there is a body inside of S, using Equation 1.22 allows computation of the drag and lift forces acting upon the body. The S may be chosen in any way convenient for calculation. For example, let a body of finite dimensions move with a subsonic speed in a gas that is at rest at infinity. If there are no vortices in the flow, we may move S into infinity and calculate drag and lift force using Equation 1.22. The perturbed flow at infinity, according to the method of small perturbations, has an order dp
O(r 2),
=
dp
=
O(r 2),
(1.24 )
where r is the distance from the body. From Equations 1.22 it follows in this case that f k = 0, which expresses the above-mentioned D'Alembert- Euler paradox. For infinite bodies (e.g., prisms), this paradox is not valid. Now let us consider the two-dimensional problem of a thin airfoil moving in a gas. Engineering applications require the minimum possible perturbations of gas by a moving airfoil, or even maximum lift and minimum drag. The latter is provided by a thin shape profile close to a flat, infinitely thin plate moving with a small angle of attack (Figure 1.2). In this case we designate
p
=
p,J1
+ p),
p=pA1 +p),
( 1.25)
where the dimensionless perturbed quantities are small:
P
I,
P
(1.26)
Here Px, Px, and llx are the density, pressure, and velocity of unperturbed gas flow at infinity with respect to the airfoil (the gas velocity direction at infinity coincides with the direction of the x I axis). Substituting Equations 1.25 into Equations 1.21-1.23 and omitting higher-order terms, in the two-
Introduction
9
i"IGURE 1.2. Flow of a gas past a thin airfoil (a is the angle of attack or incidence, and u., is the unperturbed speed of the gas).
dimensional case one may get
(1.27) (1.28)
(1.29)
(1.30)
Here M~ is the Mach number at infinity, c is the local sound speed, L is an arbitrary closed contour in gas, and [1 and 1'2 are the drag and lift forces acting upon a thin airfoil. Equations 1.27-1.30 form the gas dynamics equation system that holds for an arbitrary body at a sufficient distance from it and for very thin airfoils in the entire flow domain. Using the divergence theorem in the integrals of Equations 1.27 and 1.29 along a contour encompassing only fluid domains, we arrive at the differential equation system
(1.31)
10
Method of Discrete Vortices
Its solution can be written by means of the complex potential as ii]
=
Re W'(z),
( 1.32)
ii 2
=
-VI - M; 1m W'(z),
(1.33)
where (1.34) Here the function W'(z) is analytic in the flow domain. It vanishes at infinity and hence its behavior for z -4 ex; has the form W'(z)
fi =
--,
21TZ~
(1.35)
where f is a constant. It is easy to verify by direct calculation that r is real if there is no mass ejection or suction on the airfoil surface, and the value of f equals the circulation
r
=
f
VI
dx l
+ v 2 dx 2
(1.36)
l.x
consistent with a vortex of strength r at infinity. Using the fact that the integral in Equation 1.29 is invariant with respect to L, we move L into infinity and utilize Equations 1.32-1.35. By calculating the integral, we obtain the lift of the airfoil in the form (1.37)
This formula for M" = 0 was obtained by Joukowski in 1902. Comparison of theoretical and experimental data shows that the linearized aerodynamic theory usually provides good results if the inequality condition ( 1.38)
is met, where Ll and a are the thickness and chord of an airfoil, respectively.
Introduction
II
In the coordinate system of Figure 1.2, it is convenient to write the airfoil contour equation in the form
0<
Xl
< a,
(1.39)
considering the thin airfoil to be close to the straight linear segment of length a, inclined to the Xl axis with angle a, and issuing the coordinate origin. The functions f + (x) and f _(x) describe the upper and lower sides of the airfoil. The condition of nonpenetration on the airfoil contour means that the normal component of the fluid velocity equals zero or VI
tan cp
= V2,
where tan cp
= X~(XI) =
-tan a
+ f~(Xl)'
(1.40)
Using Equations 1.25 and 1.32-1.34, we arrive at
-VI - M! ImW'(z)
=
[1
+ Re W'(z)][f't (Xl)
-
tan a]. (1.41)
In view of a
=
f~ (x I)
a -
VI - M!
X2
=0,0<x l
(1.42)
It is required to find W'(z) that is analytic outside the cut (0, a) and vanishes at infinity, using the boundary value condition in Equation 1.42. Let us introduce the new functions
+ W/(z)],
'I'(z)
=
HW'(z)
O(z)
=
HW'(z) - W'(z)],
(1.43)
where W'(z) = W'(z).
It is evident that z ~ Xl - iO when z ~ Xl + iO. The functions 'I'(z) and O(z) are analytic outside the same cut. In addition, on the Xl axis outside the cut (x 2 = 0 and Xl < 0 or Xl > a), (1.44) (1.45)
Method of Discrete Vortices
12
On the interval 0 < Xl < a (X 2 = 0) of the Xl axis, we obtain according to Equations 1.42 and 1.43, X2
=
0,0 <Xl
< a,
(1.46)
X2
=
0,0 < Xl < a.
(1.47)
Let us represent the functions 'l'(z) and !l( z) using the following Cauchy integrals: 'l'(z) = -
1
la I/J( t) dt ,
O(z)
(1.48)
t - z
27T 0
1 fa w(t) dt = -.
27Tt
0
t - z
(1.49)
.
Here I/J(t) and w(t) are unknown real functions sought in the interval < a. Equations 1.48 and 1.49 satisfy the boundary condition in Equations 1.44 and 1.45 at z = XI and Xl > a or Xl < O. Using the Sokhotsky Equations 1.5 and 1.6, and the representation in Equations 1.48 and 1.49, the boundary condition in Equations 1.46 and 1.47 can be reduced to the
o< t
I/J(X I )
=
-
V1 -1 M",
2
[f~(x) -f'(x)],
(1.50)
~fa w(t) dt 7T
0
t - Xl
o < X I < a.
(1.51 )
Substituting Equation 1.50 into Equation 1.48, we determine the function 'l'(z)
1 =
-
27TV 1 - M!
faf~(t) 0
- f~(t)
t - z
dt.
(1.52)
Expression 1.51 is a singular integral equation coinciding with that of Equation 1.11 at t-t = 0 and A = 1. In this case, m = 1 and {; = t. Let us assume that the thin airfoil has a sharp tail. Such airfoils usual1y applied in
Introduction
13
engineering are called 10ukowski airfoils. They provide a vortex sheet separation locus just where the tail becomes the trailing edge, if the angle of attack is sufficiently small. In this case the fluid velocity and W'( z) are finite at this edge, so that potential o(z) is finite at z = a. At the leading edge, z = 0, the velocity and !l( z) are infinite, but integrable. Hence, the canonical solution to Equation 1.14 will be provided in this case by the function
Vrz=a ----;--z- •
X(z) =
lim X(z) = 1.
(1.53)
z-->x
Similarly to Equation 1.20, the potential O( z) in this case can be found in the form la20'-/~(7)-/'.(7)
X(z)
O(z)
=
2m/I -
M!
dT.
X+(7)(7 - z)
0
(1.54)
According to Equation 1.43, W'(z) = 'I'"(z)
+ O(z),
(1.55)
so that substituting here Equations 1.52 and 1.54, we get the solution in the following shape: W'(z)
l
1 =
2m/l - M! o
+
I: (7) - f'- (7) ------d7
0
7 -
Z
la20'-/:(7)-/~(7)
X(z) 21Tyl
a
-M!
X"(7)(7-Z)
0
d7.
(1.56)
From here, comparing with Equation 1.35 at infinity we find the circulation
f=
~
V1 -
la 2ll' - 1:(7) - 1~(7) M!
IX(7)\
0
~
.
(1.57)
In the limiting particular case of a straight linear airfoil of infinitesimal thickness, when I: (7) = I~ (7) = 0, we get from Equation 1.57,
r-
,20'a~
- ';1 -
M!
~
IriS 1 -x
dx-
1TO'a~
-';1 -
M! .
(1.58)
14
Method of Discrete Vortices
By Equations 1.37 and 1.58 the lift of this airfoil equals 7Taap",u; (1.59)
1'2 = 1 - M! . In the case of relatively small velocity of flight when be considered incompressible, and the lift equals
M~
1, the air can
(1.60) Compare this result with the exact solution for a very thin plate moving in an incompressible fluid with an arbitrary nonsmall angle of attack (see Gurevich 1965): [2 =
7Tap-ycu; sin a.
(1.61 )
Even at a = 30°, the exact solution differs only by 5% from that of the linearized approach. It should be noted that using the Bernoulli integral p + ipv;v; = const in the flow domain of incompressible fluid, the invariant integral in Equation 1.22 can be written in the simple form
( 1.62) Vi
= cp,i,i,k = 1,2,3.
Here cp is the flow potential.
1.3. OPTIMAL AIRFOIL PROBLEM According to Equation 1.59 the lift of a thin airfoil increases when the flight speed grows until the local Mach number achieves unity at a critical point of the upper side of the airfoil. This occurs at a critical Mach Moo = M. < 1. If Moo > M., a local supersonic zone and local shock waves develop near the critical point. At this regime, the drag is sharply increased. If a moving body is "well-streamlined," like a thin plate moving with a very smal1 angle of attack, so that there is no separation of a boundary layer and no formation of drag vortices in the fluid, the drag is defined solely by viscous friction in the boundary layer. For example, in the case of
Introduction
15
a thin plate under study, the drag equals
(1.63)
where JJ is the kinematic viscosity. The dimensionless coefficients of lift and drag, Cr. and CD' are commonly used to describe the forces; as applied to a "well-streamlined" thin plate they are
CL
=
CD
=
r2
27T0'
1 - M;'
tapxU: D
~apxU:
Mx =
=
2.66/ allx JJ
•
u" CX
(1.64)
Thus, when the flight speed grows, the coefficient of lift increases and the coefficient of drag decreases. Hence, for every thin airfoil there exists a certain optimal flight speed before the separation of drag vortices from the airfoil starts or local shock waves foem. It follows that the optimal airfoil shape should provide for a maximum flight speed without drag vortices separation and shock waves. This clear engineering requirement is not easy to formulate in a mathematically unique and strict way. The history of developing aircraft and turbines in this century is the factual record of many events devoted to solving this problem. Some mathematical efforts concerning the sonic barrier are described by Morawetz (1982). Let us treat an approach to this problem formulated by Cherepanov (1973) and based on some engineering observations and numerical experiments. There are three main requirements to an optimal airfoil: 1.
2.
3.
The tail of the airfoil should be a trailing edge, i.e., separation locus of a vortex sheet. To achieve this aim at different regimes of flight, the tail part should be an individually controlled hinged flap with a sharp cusped end. The nose of the airfoil should coincide with the stagnation point and be sharply cusped, which helps eliminate some problems connected with large velocity gradients and separation of drag vortices. (There should be no other stagnation points different from the nose.) To achieve this ailll at different regimes of flight, the nose part should be an individually controlled hinged flap with a sharp cusped spike. The upper side of the middle part of the airfoil should provide for one and the same pressure at all the points at a virtual velocity of
16
Method of Discrete Vortices
FIGURE 1.3. Sketch of an optimal airfoil having a nose flap and a hinged tail flap. Pressure P = Po is uniform along the upper middle part of the airfoil. One streamline doubles at the nose and trails from the tail of the airfoil.
flight, which helps to maximize the critical Mach number at this flight regime. A sketch of the optimal airfoil is shown in Figure 1.3. One streamline doubles at the nose and emerges at the tail of the optimal airfoil. Let us formulate the mathematical problem of an optimal airfoil in this approach for a thin airfoil. Boundary-value condition P = Po = const, according to Equations 1.25 and 1.30, can be written as where
(I.65)
Constant Po or u. should be chosen from additional physical considerations. The boundary condition of nonpenetration holds along the entire surface of the airfoil. It means that on the arc of uniform pressure being sought, two boundary condition equations have to be satisfied. Using Equation 1.65 and a consideration analogous to that used for deriving Equation 1.42, we find the corresponding boundary value problem of an optimal airfoil (see Figure 1.4): Upper side of nose flap: z
= XI
+ io, (1.66)
Upper side of middle part of airfoil: Z = Xl
+ iO, Re W'(z)
=
u •.
( 1.67)
17
Introduction
af'n
a( 1 - E,)
&~~~
o
+
x,
a
FIGURE 1.4. The z plane outside the cut (0, a) along the real axis having the points x I = a En and x, = a(l - E,) at which the mixed boundary value problem changes (0 < En < 1,
o < E,
<
I).
Upper side of tail flap:
+ iO,
Z = XI
a(1 - e,) 0', -
ImW'(z)
=
<XI
1/, (X I)
< a,
VI - M: .
(1.68)
Lower side of nose flap: z
= XI -
iO,
(1.69)
Lower side of middle part of airfoil: Z=X I
-iO,
(1.70)
Lower side of tail flap: Z =
xt
-
iO, (I.71 )
[J~,<X\), f~ (x\), f;"(x\), f/.<x\),
Here unknown constants En'
EI'
and !,'Jx\) are assumed to be given.] 0'/ determine the geometry of
an, am' and
Method of Discrete Vortices
18
a broken chord line of an optimal airfoil consistent with the virtual flight velocity regime. The length and thickness of the nose and tail flaps as well as the form of the lower side of the middle part may be given depending on some free parameters. All these geometrical parameters and u. should be chosen in such a way that all the additional condition equations arc satisfied, including the four conditions of the finiteness of W'(z) at points z = 0, a, a€n + iO, and a(l - €,) + io. Moreover, W'(z) must vanish at infinity. Using the direct method advanced by Cherepanov (1964), the boundary problem of Equations 1.66- I. 71 can be solved in an explicit form. An explicit form solution also can be found using conformal mapping onto a canonical domain and the method treated previously in this chapter. Although it is dependent on several free parameters, after W'( z) is found, the unknown boundary of the upper middle part of the airfoil can be computed from the equation
VI - M; 1m W'(z)
dy
1
dx,
+
ReW'(z)
(1.72)
Here x = y(x I) is the equation sought for the middle part of the optimal airfoil. Equation 1.72 expresses the coincidence of this boundary with a streamline. Because the right-hand side of Equation 1.72 is a defined function of x" y(x I) may be found from this equation by simple integration over XI' We provide the solution result to problem Equations 1.66-1.71 only in one particular case when both the nose and tail flaps are not available, that is, €n = €, = 0:
(1.73) Here, X,(z)
=
Z-a)1 /4 (-z- ,
lim XI(z)
=
1,
z-->x
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 <
< a En' when aE < t < a(1 - E,), when a(l - E,) < t < a, 1
Il
{
fr', (t)
{
f~ (I)
when 0 < t < a En'
[;"(t) f/ (t)
when aEn <
< a(l - E,), when a( 1 - E,) < 1 < a. 1
The unknown functinn y '(x I) determines the upper middle part of the profile and all other functions are known. Then, by Sokhotsky formulas we find Re W'(z) when z -4 XI + iO for aEn < X, < a(l - E) and equate it to u, in view of Equation 1.65. The resulting equation represents a singular
Method of Discrete Vortices
20
integral equation with respect to y'(Xl) and it belongs to the singular integral equations studied numerically in this book.
REFERENCES Chaplygin, S. A. 1902. On Cas Jets. Moscow University Press. Cherepanov, G. P. 1962. The solution to one linear Riemann problem, Appl. Math. Mech., 26(5). Cherepanov, G. P. 1963. The flow of an ideal fluid having a free surface in multiple connected domains, Appl. Math. Mech., 27(4). Cherepanov, G. P. 1964. Riemann-Hilbert problems of a plane with cuts, Dokl. USSR Acad. Sci. (Math.), 156(2). Cherepanov, G. P. 1965. On one case of the Riemann problem for several functions, Dokl. USSR Acad. Sci. (Math.), 161(6). Cherepanov, G. P. 1973. On optimal airfoil in ultimate subsonic regime. Cherepanov, G. P. 1987. Invariant integrals, in Fracture Mechanics of Rocks in Drilling. Nedra Publishers, Moscow, pp. 8-20. Gakhov, F. D. 1952. Riemann problem for systems having n pairs of functions, Adt,. Math. Sci., 7, No. 4(50). Gakhov, F. D. 1966. Boundary Value Problems. Pergamon Press, London. Gilbarg, D. and Shiffman, M. 1954. On bodies achieving extreme values of the critical Mach number, J. Ration. Mech. Anal., 3(2). Guderley, G. 1956. Die Theone der Naneschalgeschwindigkeit Striimungen. Springer-Verlag, Berlin. Gurevich, M. I. 1965. The 'fheory of Jets in an Ideal FluifJ. Academic Press, New York. Khrapkov, A. A. 1971. Problems of clastic equilibrium of infinite wedge with asymmetrical cut at the vertex, solved in explicit form, Appl. Math. Mech., 35(6). Morawetz, C. S. 1982. The mathematical approach to the sonic barrier, Bull. Am. Math. Soc., 6(2). Muskhelishvili, N. I. 1946. Singular Integral Equations. Noordhoff Publishers, Groningen. Sedov, L. I. 1950. Two-Dimensional Problems in Hydrodynamics and Aerodynamics. Wiley, New York.
Introduction
The very essence of the method of discrete vortices considered in this book may be formulated as follows. A continuous vortex sheet, modeling both a lifting surface and a wake downstream of it, is replaced by a system of discrete vortices. Certain points are chosen at the lifting surface (called the reference points) where the nonpenetration condition, according to which the normal flow velocity component is equal to zero, is imposed. The problem of obtaining unknown circulations of discrete vortices reduces to a system of linear algebraic equations. The solution of the problem is not unique and may have singularities at the edges and at angular points of a lifting surface. The required class of solutions is determined by the physical nature of the problem and is singled out by an appropriate choice of the singularities. In the framework of the method of discrete vortices the choice is implemented in the so-called B condition of the method of discrete vortices. According to this condition, at the grid points nearest to where the solution must be unlimited, discrete vortices are placed and where the solution must be limited, the reference points are placed. In addition, the sums that replace singular integrals in the lifting surface theory must correspond to the Cauchy principal values of the integrals. To meet this condition the internal points must be placed midway between the vortices positioned at the surface. In this form, the method of discrete vortices was originally formulated in the doctoral thesis by S. M. Belotserkovsky in 1955 (Belotserkovsky 1955b). Let us illustrate the ideas of the method by considering flow past a thin airfoil modeled by a vortex sheet whose strength is y(x) (see Figure D.O. The problem reduces to the following characteristic singular integral equation of the first kind on segment [ - 1, 1]: 1
27T
JI _I
y(x) dx X
-xu
=f(x o),
x o E(-I,I).
(0.1 )
I
Two grids are introduced: x k = -1 + kh, h = 2/(n + 0, k = 1, ... , n, and X Uk = x k + h/2, k = 0,1, ... , n, for discrete vortices and reference points, respectively. 21
22
Method of Discrete Vortices
-~J",,~
.
:r:
OJ
FIGURE 0.1. of the airfoil.
Vortex sheet simulating a thin slightly curved airfoil, positioned at the chord
:::-o}-o) ?-~ -0},-,
FIGURE 0.2. Mutual positions of discrete vortices and reference points in the case of a problem with nonzero total circulation.
Aerodynamics considers three modes of flow past a body: circulatory (i.e., with nonzero total circulation), noncirculatory (Le., with zero total circulation), and limited everywhere. The first mode is based on the Chaplygin-Joukowski hypothesis about the finiteness of y(x) at the sharptail of the airfoil and is used for analyzing lifting properties of wings. GeneraIly speaking, in this case the solution tends to infinity at the nose of a thin airfoil (see Figure 0.2), and Equation (0.1) is replaced by the foIlowing system of linear algebraic equations:
j
=
1, ... , n.
(0.2)
The second mode is used when considering osciIlations of an airfoil in a fluid at rest. In this case, both edges function under similar conditions, and the total circulation is equal to zero, whereas at the edges y(x) tends to infinity (Figure 0.3). The corresponding system of equations has the form:
j=1, ... ,n-1,
(0.3)
n
L k~t
Yn(xd h
=
0,
j
=
n.
(0.4 )
23
Introduction
-fJ
b
FlGURE 0.3. Mutual positions of discrete vortices and reference points in the case of a problem with zero total circulation.
FIGURE 0.4. Mutual positions of discrete vortices and reference points in the case of a problem with a solution limited at both edges of a thin airfoil.
The latter equation spells out the condition of zero total circulation, and according to the B condition, vortices are placed at the grid points nearest to both edges. The third mode of flow is used when one determines not only the lifting properties of a wing, but also can change the form of the wing in order to avoid flow separation at the nose, which occurs when y(x) tends to infinity there (Figure 0.4). In this case the integral equation is substituted by the following system of linear algebraic equations: j=O,l, ... ,n,
(0.5)
where YOn is an additional unknown related to the deformation of the wing for which the strength y(x) at the nose becomes zero: y( -1) = O. From the mathematical point of view, this unknown may be treated as a regularizing variable, because according to the B condition, the requirement of both y( -1) and y(1) being finite results in the appearance of an extraneous reference point. The method of discrete vortices has the folIowing important peculiarity: There is no need to make any assumptions about the way a solution tends to infinity at both lhe ends of a segment and at angular points. The required class of solution is singled out by choosing relative positions of two grids composed of discrete vortices and reference points.
24
Method of Discrete Vortices
This method is radically different when compared with other numerical methods of aerodynamics developed prior to it. The earliest work close to the direction of this book was the article published in 1932 by M. A. Lavrent'ev that presented a numerical solution to the singular integral equation for flow past a thin airfoil. The solution based on polynomial approximation was a forerunner of many methods in this area. In 1938, Multhopp's work was published, in which the solution to a onedimensional singular integral equation was presented in the form y(x) = w(x)cp(x) where w(x) accounted for the singularities of a solution and cp(x) was a smooth function expanded into a series in the Chebyshev polynomials of the first kind. The right-hand side of the equation was expanded in the Chebyshev polynomials of the second kind. Then, the relationship between the Chebyshev polynomials of the first and second kinds was employed to compose a system of linear algebraic equations for the coefficients of the series. Later the idea was applied to a finite-span wing (Kolesnikov 1957, Falkner 1947). However, this approach was neither sufficiently profound nor applicable to unsteady problems as well as to wings of a complex plan form, to say nothing of an airplane as a whole. On the other hand, the method of discrete vortices was criticized for applying rectangle-type quadrature formulas to singular integrals as well as for ignoring an explicit account of singularities at a wing's edges (Polyakhov, 1973). Because rigorous verification of the basic assumptions of the method was lacking, the authors and their followers concentrated their efforts on solving the problem and developing similar approaches for related areas (Lifanov and Polonskii 1975; Belotserkovsky et al. 1978; Lifanov 1978a, b, 1979a, b, 1980a, b, c, 1981a, b; Belotserkovsky and Lifanov 1981; Lifanov and Saakyan 1982; Lifanov and Matveev 1983; Matveev 1982a,b; Belotserkovsky et al. 1983; Gandel 1982, 1983). This monograph sums up the first results of this work. It consists of four parts. The first part is dedicated to analyzing quadrature formulas for one-dimensional and multidimensional singular integrals. We start by considering the above-mentioned rectangle-rule formulas for onedimensional integrals. In this case, the integration domain is a piecewise Lyapunov curve having angular points only. Also, interpolation-type quadratures for singular integrals on a segment-circle and with the Hilbert kernel, which allow construction of numerical methods for integral equations on the sets, are analyzed. We continue by considering quadrature formulas for Cauchy multiple singular integrals, certain two-dimensional integrals treated in the sense of the Hadamard finite value (Hadamard 1978) one often comes across in aerodynamics, and multidimensional singular integrals. Finally, the Poincare-Bertrand formula for reversing the order of integration in multiple singular integrals is proved by employing quadrature relationships of the type used in the method of discrete vortices.
Introduction
25
In the second part, direct numerical methods for solving singular integral equations with one- and multidimensional Cauchy integrals are constructed by using quadrature formulas derived in Part I. The situations encompassed are those for which the right-hand sides of integral equations may incorporate integrable power-law singularities at the end points and discontinuities of the same form or discontinuities of the first kind at inner points. The case when a solution has a singularity of the form (q - t) - 1 at an inner point was also studied. In fact, we succeeded in proving the existence theorems for solutions to characteristic singular integral equations of the first kind on a segment and circle (Le., in developing an analog of the Fredholm theory). It should be noted that the theory of one-dimensional singular integral equations has been thoroughly studied (Gakhov 1977, MuskhelishviIi 1952). However, equations incorporating multiple Cauchy integrals were only considered for toruses (Gakhov 1977, Kakichev 1959). Therefore, we present analytic solutions to a wide class of characteristic singular integral equations of the second kind with multiple Cauchy integrals for the cases when integration domains are products of circles and segments. Then efficient numerical methods are developed to solve the equations. In the third part, mathematical verification of the method of discrete vortices is presented for a wide class of linear steady problems of aerodynamics. The problem of stability of the developed numerical methods for solving singular integral equations is also considered. The actuality of the problem is related to the fact that the equations are unstable in uniform metrics often used in practice. The numerical methods are shown to be regularizing, with the number of reference points being the regularizing parameter. Finally, the fourth part is dedicated to extending the developed numerical methods into alternative areas of interest: the theory of elasticity, electrodynamics, and mathematical physics, i.e., into any area in which the problems may be reduced, either naturally or by special measures, to singular integral equations. Prior to developing reliable stable methods for solving such equations, many attempts were made to reduce the problems to regular Fredholm integral equations of the second kind (in the case of plane problems of the theory of stability in Parton and Perlin 1982, 1984) or to integral equations of the first kind with logarithmic singularity (in the case of electrodynamic problems in Mitra 1977). However, the resulting regular Fredholm equations often have nonsingle-valued solutions, because the equations "set" on the spectrum. This results in significant difficulties while solving the problems numerically. The process of solution becomes unstable, and special methods must be employed for smoothing the resulting fluctuations (Mitra 1977). By reducing the problem to singular integral equations and using the approaches of this book, one may construct unified stable numerical methods for solving a wide class of
26
Method of Discrete Vortices
various problems (Belotserkovsky 1967; Belotserkovsky and Nisht 1978; Belotserkovsky et at. 1983; Gandel 1982, 1983; Lifanov and Saakyan 1982). The fourth part of this book is dedicated to some novel applications of the theory. It also presents methods for deriving singular equations in the mentioned areas (Belotserkovsky et at. 1983; Gandel 1982, 1983; Lifanov and Saakyan 1982; Saakyan 1978). Numerical methods representing a certain physical substance as a set of discrete physical objects (the method of discrete vortices presents a vortex layer as a set of discrete vortices) arc presently called methods of discrete singularities. The studies carried out by the authors after publishing the Russian version of this book required extending the method of discrete singularities to the class of singular integral operators with variable coefficients and Cauchy and Hilbert kernels. It was found that such index x operators transform polynomials of degree n multiplied by the corresponding weight functions into polynomials of the degree n - x. Moreover, the same conclusion is also true for generalized polynomials. By using this property, one may construct numerical methods, similar to the method of discrete singularities, for singular integral equations (of algebraic degree of accuracy) with variable coefficients and the Cauchy kernel on a piecewisecontinuous curve as well as with the Hilbert kernel. These mathematical findings provided the solution to aerodynamic problems for permeable airfoils as well as developed a new approach to calculating finite-thickness airfoils. It was found that if the boundary nonpenetration condition is transferred from the surface of an airfoil onto a certain middle line, then the problem of flow past an airfoil reduces to solving a system of two singular integral equations with variable coefficients. Currently of great practical importance are the problems of flow past blunt (high-drag) bodies with sharp corners, such as buildings, cars, etc., in particular the problems of calculating virtual inertia of the bodies. To solve the problems it was necessary to consider anew the process of shedding of a sheet of free vortices from angular points. In the framework of the new scheme, discrete vortices are positioned at the angular points, also and are assumed to be free. When separated from angular points, the vortices trail along the local flow velocity vector. The preceding problems were solved for both two- and three-dimensional flows. This book is also supplemented with a new chapter on the reduction of certain boundary value problems of mathematical physics to singular integral equations. The chapter follows.
Part I: Quadrature Formulas for Singular Integrals
1 Quadrature* Formulas of the Method of Discrete Vortices for One-Dimensional Singular Integrals
t.t. SOME DEFINITIONS OF THE THEORY OF ONE-DIMENSIONAL SINGULAR INTEGRALS In the following text, only curves in a plane with the right-hand system of coordinates OXY are considered. Sometimes the points of the plane will be considered as complex numbers of the form t = x + iy, where i = !="T. A curve will be called smooth if it is simple, Le., does not intersect itself. A line L will be called smooth and unclosed (an arc) if it can be presented in the following parametric form (Muskhelishvili 1952): x = x(s),
y = y(s),
(1.1.1)
where So and Sb are finite constants; x(s), y(s), x'(s), and y'(s) are continuous functions in the interval [so, Sb]; and the derivatives never vanish simultaneously. Different points of the curve L correspond to different values of the parameter s in [so, Sb]. If points of the curve L are denoted by t = x(s) + iy(s), then a one-to-one correspondence exists between t and s (t E L, S€[so,Sb D and
t; = x'(s) + iy'(s).
• The term "quadrature" is equivalent to the tcrm "numerical integration" that is commonly used now (G.Ch.).
29
30
Melhod of Discrete Vortices
Sometimes the curve L will be denoted by ab, where a b
=
=
I(Sa> and
I(Sb).
A smooth curve will be called a smooth closed contour L if X(Sb) = Y(Sb) and
=
x(sa), y(sa)
X'(Sb - 0) = x'(sa
+ 0).
y'(Sb - 0) = y'(sa
+ 0).
Thus, in this case the functions x(s), y(s) and x'(s), y'(s) can be considered as periodic functions whose period is equal to T = Sb - Sa. A smooth (simple) line is a set of a finite number of closed or unclosed smooth contours having no common points (the end points included). A curve is referred to as piecewise smooth if it consists of a finite number of smooth unclosed curves having no common points (the only possible exception may be the end points). The curve will be said to possess angular nodes only, if at each of the nodes any two smooth curves meet at a nonzero angle, i.e., the nodes are not cusps. In Appendix 1 to Muskhelishvili (1952) the following statement was proved: Let L be a simple piecewise smooth curve that contains angular points only (i.e., L is composed of a finite number of smooth unclosed curves alaZ,aZa3, ... ,an-_lan so that the end point of each preceding smooth curve coincides with the initial point of the next one). Then the following inequality holds for any pair of points II,l z belonging to L: (1.1.2) where: 1.
2.
a(t], t z) is the length of the part of curve L between the points I] and I z. (If L is a closed curve, then a(/],l z ) corresponds to the shorter arc). r{t], t z ) = II] - Izl, Le., the distance between points II' I z in plane
OXY.
3.
The constant K o E(O, 1) is independent of the positions of points t I and t z on the curve L.
Note that if a piecewise smooth curve L contains angular nodes only, then Inequality (1.1.2) is also satisfied for any points of L. Definition l.l.l. A funclion cp(t) (where t is generally a complex variable) meels the condition II( t-t) (the Holder condilion of Ihe degree t-t) on a given set D of the values of the variable if for any Iwo values belonging 10 the set, the
Quadrature Formulas for One-Dimensional Singular Integrals
31
following inequality holds:
(1.1.3 ) where A and /L are positive numbers and 0 < /L $ 1. The constant A is called the coefficient and t-t the exponent of the condition H( /L). If we are not interested in the exponent /L, then the function cp(t) is said to meet the condition H (or belong to the class H) on the set D and will be written in the form cp(t)€H( /L) or cp(t}€H.
Note that if cp(t}€H( /L), then Icp(t)1 €H( /L). The condition H may be generalized onto the case of a function of several variables: A function q;(t" ... , t n ) meets the condition H(/Ll, ... ,/Ln) (or simply H)on the set D of the variables t"t 2 , ••• ,tn, if the following inequality is satisfied for any two points (t{, ... , t~) and (t~, ... , t~) in
where Ai > 0,0 < /Li $ 1, i = 1, ... , n. From (1.1.4) it follows that if cp(t 1, ••• , tn)€H, then cp belongs to the class H( /Lk) with respect to a variable t k , k = 1, ... , n, uniformly t with respect to all the other variables. The opposite statement is also true. Stating in what follows that a function cp(t I' ... , tn ) meets the condition H with respect to each of the variables taken alone, we suppose that the condition is satisfied uniformly with respect to the other variables. A smooth unclosed curve L is cal1ed a Lyapunov curve if the derivative t'(s) meets the condition H(a) on [sa' Sb]' Then, according to Muskhelishvili (1952), the function t - to t(s) - t(so) f(s, so) = - - = - - - S -
So
S -
So
meets the condition H( 0') in both variables s and So and does not vanish on [sa' Sb]' A curve will be called a piecewise Lyapunov curve if all its smooth parts are Lyanpunov curves. t
In other words, 1
,In ) -
Method of Discrete Vortices
32
Definition 1.1.2. A function cp(t) belongs to the class H* on a piecewise smooth curoe L if it has the form p
cp(t) = cp*(t)/P[(t),
PL(t)
=
nit - ckl
vk •
(1.1.5)
k=1
where cp*(t)€H o on L, i.e., cp belongs to the class H on each smooth part of the curoe L, 0 ~ JJk < 1, and Ck' k = 1, ... , p, are nodes of the curoe L. Without loss of generality, it can be assumed that cp*(t )€H on L. Let us recall the definition of a Cauchy singular integral along a piecewise smooth curve.
Definition 1.1.3. Let point to differ from any node of the curoe L. In other words, let it be an internal point. Let us draw a circle centered at this point, with the radius of the circle € > 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)
Method of Discrete Vortices
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
Quadrature Formulas for One-Dimensional Singular Integrals
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
Method of Discrete Vortices
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 <(A - " I -
L.I k~
n
Ik
I
=
exp(i 0),
Idll 1 f1T/2dO (Inn) --
Finally, we get for the sum 52' 1
27T n - I
5
k
~I
It k
-
111 - " {
Ik
Idtl
7T
1T
dO
( 1 )
i' _ _ .:::;;C_ -2- = 0 - . II - 11 3 n {/ n 0 - " n"
Quadrature Formulas for One-Dimensional Singular Integrals
37
By substituting the estimates for SI and S2 into the inequality for I{ and the estimates for I{, I~, and It into the inequality for II' we see that
Thus, Theorem 1.2.1 is proved. Definition 1.2.1. it is of the form
•
A function cp(t) belongs to the class
cp(t)
=
n offunctions on L if
l/J(t ) --t'
q-
where l/J(t)€H on Land q is a certain fixed point on L. Note that one may write
-1-[1 l/J(t)dt - 1 l/J(t)dt]. 1 cp(t)dt t - to q - to t - to t - q =
L
L
L
According to the latter formula, the following theorem is true. Theorem 1.2.2. Let cp(t) belong to the class II on the circle L and the sets E and Eo form a canonic division of the circle L, with q€Eo for j = jq. Then the following inequality holds: j*jq,j= 1, ... ,n.
(1.2.12)
where the quantity O(tOj) has the form O(tOj)
=
It 1- ql 0 (~In n), n Oj
Obviously, the quantity O(tOj) satisfies the inequalities ( 1.2.13) for all toj€L \ I, where I is, however, a small neighborhood of point q, and
j~1 O(to)liltojl O(n~2)' $
(1.2.14)
j*jq
Evidently, the latter two inequalities may be made more accurate by replacing their right-hand sides by O,(n a In n) and O(n- a In n), respectively.
Method of Discrete Vortices
38
Note 1.2.2. If cp(t, to)€H( 0') on L, then Inequality (1.2.10) also remains valid for the integral f(cp(t,/oI{t - to» dl. In other words, ( 1.2.15)
Note 1.2.3. Let L] be a closed Lyapunov contour. Then a one-to-one correspondence T = T(t} exists between points T of the curve and points t of the standard (unit) circle L, such that the derivative T '(1) = dTI dt belongs to the class H( /3) and does not vanish anywhere on L. If cp( T ) meets the condition H( 0') on L" then by the formula of substitution of a variable in singular integrals (Muskhelishvili 1952), one gets
1 cp(T)dT 1 f/J(/,/o)dl, =
I.,
T -
TO
L
I - 10
f/J(t,t o )
I - to =
T(t) -
T'(t)CP(T(t».
T(lO)
Let us next analyze the canonic division of the circle L formed by sets E and Eo. The sets of the points Tk = T{t~k) t k €E, and TOk = T(/ ok ), t Ok €Eo, will be called the canonic division of the curve L I. Let us take the sum
t k~]
cp( Td LlTk Tk -
T Oj
t k=]
cp(T(td)(t k - t Oj ) LlTk T(t k ) -
T(lOj)
Llt k
Llt k t k - t Oj
Because
and T'(t}€H(/3) on L, hence IT/Uk) - T'(tk)I.:s::Altk -Iklf:l, i.e., T'(lk) T'(t k ) + O(ltk - tkl f:l). Thus,
t
cp( Tk) LlTk
k~l
Tk-TOj
n
k~l
cp('T(tk»)(t k - t Oj ) T'(t k ) dt k T( td
- T(lOj)
t k - 10j
=
Quadrature Formulas for One-Dimensional Singular Integrals
39
By using Inequality (1.2.7), it can be readily shown that
IS]I ::; O(n- b
In n).
Hence, in accordance with Inequality (1.2.15) and the preceding estimate for
Sf,
A > 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
Method of Discrete Vortices
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)
Quadrature Formulas for One-Dimensional Singular Integrals
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.
•
Method of Discrete Vortices
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 -<-2 - t Oj - a '
For
t· -
b
-1--.:s:;2. t Oj - b
Ii' one has
X
dt jt) , ----:cv----:P.,-------.I],--v (t - a) (b - t) It - t I
I
j
Oj
(1.3.11)
44
Method of Discrete Vortices
Fikhtengoltz (1959) proved a theorem about the average value of an improper integral; it was formulated as follows. Let functions f(x) and g(x) be integrable on [a, b], with f(x) being limited, i.e., - 00 < m :5. !(x) :5. M < +00, and g(xXhanging its sign. Then the function f(x)g(x) is integrable and ( ... ~)
tf(x)g(x) dx a
=
ILfbg(x) dx,
m :5. IL :5. M.
a
By applying the theorem to all j
=
1, ... , n - 1, one gets
because ~/dtj,tj+l],k!:..I,2,3, and hence, the ratios (~/ - a)/(tOj - a) and (~/ - b) /(tOj - b) are quantities limited for j = 1, ... , n - 1. For j = n in I? we again use the previously formulated theorem about the average value having preliminary divided the interval of integration into two segments: [tn' t n + 3h/4] and [t n + 3h/4, b]. Then we deduce that I? has the same estimate as j = n. Analogously, Ii for j = 0,1, ... , n may be shown to have the same estimate. Finally, we consider Ir Similarly to I? in the proof of Inequality (1.3.3), we write
Quadrature Formulas for One-Dimensional Singular Inlegrals
uSin~~.lO)
By on [a, b], one gets for S],
+h
1 v
(Ij+t-a)
and taking into account that 1l/J(lk)1 is limited
fb
dl I-'
Ij,,(b-t) It-/Ojl
V
+
h I-' (b - I j )
+
hI-' v (lj+]-a)
(
45
1 v (t] - a)
(lbIj~1
h
f
l
I,
2
dl v (I - a) It - tOil
dt (b-/)I-'It-/oji
---2".------
46
Melhod of Discrete Vortices
Next we note that
2
X
[(
I-v ] tj + II ] -2-- a ) -(l]-a)-V
O(lln hi)
j
(/Oj-a)V'
=
1, .. . ,n.
(1.3.12)
Analogously, by considering the integral
f
dl
lj
/2
(I - h - a) 2v II - IOjl
'
one concludes that (1) if 2 JJ $ 1, then the estimate of the form 0.3.12) is valid and (2) if 2 JJ > 1, then by the inequality 1 - 2 JJ < 0, Equation 0.3.12) becomes
f
/!
/, (I -
dl2
h - a) vII -
$ t(ljl
[
hi IO}
~2V
a
+
1
2v]0(lln hi). (1.3.13)
(tOj - a)
For all the other integrals entering the inequality for 51 the estimate of the form 0.3.12) or 0.3.13) is valid if JJ is substituted by t-t and a by b.
Quadrature FormuLas for One-DimensionaL SinguLar IntegraLs
47
Hence, finally we have
hI-I-' v
+
]
(tOj - a) (b - t Oj )
O(lln hi).
Let us consider 52. By applying Formula (1.3.10), one gets
Again using the fact that hilt; - tojl $ 2 for j = 1, ... , n and any i as well as the estimates of the form (1.3.12), one gets
+
vhI-'
1
(tOj - a) (b - 10j)
By denoting T/(tOj' a, f3) may be written in the form 1$ O(tOj)
=
=
21-'
(lIn hi).
(IOj - a)-"(b - t o)-13, the estimate for I
0o(tOj' a,v, IL)O(lln hi),
(1.3.14)
Melhod of Discrete Vortices
48
where
The quantity O(/ O}) entering the latter inequality readily can be shown to satisfy Conditions (1.3.8) and 0.3.9). • Note 1.3.1.
o<
v, /L
that if
JJ
In fact, Inequality 0.3.7) was proved subject to the condition
< 1. However, the preceding considerations allow us to state 0) *- 0, /L
=
0, then the expression for 00
+
T/(tO} , 1 + v,1)h.
0',
v, 0) has the form
(1.3.15)
and (2) that if JJ = /L = 0, then the quantity 0(/ 0 ) has the form of the right-hand side of Inequality 0.3.6).
,
Let us also make the following remark. If a function cp(t) has the form (b - 1)1-' cp(/)
=
(
t - a
)v l/J(/),
where 0 < v < 1, 0 < /L, and l/J(t) belongs to the class H on segment [a, b], then Inequality 0.3.8) is fulfilled for all the points lo}da + D, b]. Note 1.3.2. In aerodynamics (see, for example Belotserkovskii 1967, Lifanov and Polonskii 1975) points t k and IO} are often chosen in the following way. Let us divide the segment [a, b] into n equal parts each h long and denote the parts by Ll k , k = 1, ... , n. The points whose distances from the left end of the segment Ll k are equal to h/4 and 3h/4, respectively, will be denoted by I k and t o7, k = 1,2, ... , n. According to Lifanov and Polonskii (1975), all the statements made for a singular integral on segment [a, b] are also valid for the division. Calculations show that the latter scheme gives better results for model examples than the canonic division (see, for example, Belotserkovskii 1967). A more general
Quadralure Formulas for One-Dimensional Singular Integrals
49
statement is also valid. Inequalities (1.3.1), (1.3.3), (1.3.6), and (1.3.7) are also valid in the case when points belonging to the sets E = {t k , k = 1, ... , n} and Eo = {t Oj ' j = 0,1, ... , n} do not form a canonic division but meet the condition k=I, ... ,n-l, j=O,I, ... ,n-l,
k = 1, .. . ,n, too - a = hq~, b - tOn
where PI and
=
hqt,
ff: are fixed numbers.
Such a situation takes place if a fixed point gda, b) is desired to take a certain position with respect to division points, e.g., if we want to point q to belong to the set E or Eo for an n (see Lifanov 1978a). Let us show how a point q may be made to belong to Eo for any n. To do this we divide segment [a, b 1 into n + 2 parts by points tj" k = 1, ... , n + 1, t~ = a, t~ +2 = b, and l~k (the middle segment [ILl'; + d,k = 1, ... , n + 1). Let point q lie in the segment [t~j . t~jq + .J. Let us displace the set of points {t';, k = 1, ... , n + t} u {tOj' j = 0, 1, ... , n + t} as a rigid whole in such a way that the end of the segment [t~j , t Ojq + 1] nearest to point q would coincide with the point. Let us next discard the end points belonging to the set {t';, k = 1, ... , n + l} and {t~j' j = 0,1, ... , n + l}, from the end to which the displacement was performed. The remaining points of the displaced sets will be denoted by t k , k = 1, ... , n, and t Oj , j = 0, 1, ... , n. If point q coincided originally with one of the points l~j' then the original sets are denoted by E and Eo. If q is required to belong to E, then points lk and t Oj are constructed in a similar way. In the case of flow past an airfoil with a flap, the sets E and Eo are chosen in such a way that point q lies exactly halfway between the nearest points belonging to E and Eo. This can be done in the following way. Let h = (b - a)/(n + 1). Let the point lying at the distance h/4 to the right of point q belong to the set Eo, whereas that lying at the same distance to the left belongs to the set E; the rest of the points of the sets E and Eo are distributed with the step h starting from the chosen ones. The same result can be achieved by displacing the points of the canonic division
Method of Discrete Vortices
50
previously described with the said step, the displacement being no more than h/4. If points belonging to the sets E and Eo take the places of each other, the preceding results for a singular integral remain valid. Hence, an integrand may be taken at the points of the set Eo, whereas the integral is evaluated at the points of the set E. Note 1.3.3. All the results are also valid for the function
cp( t, 7)
=
fjJ(t, 7) (t _ a) v (b - t) I'-
(1.3.16)
'
if the function fjJ(t, 7)€H on segment [a, b] with respect to both variables or if the sum is constructed by using points of the sets E or Eo only. Thus, if we denote (1.3.17)
(1.3.18) then an inequality of the form of 0.3.7) is valid for the absolute value of the difference Ij - 5;. In Section 1.2 we introduced the class II of functions for a circle. A similar class of functions on segment [a, b] is defined as foHows (Lifanov 1978a). Definition 1.3.1. A function cp(t) belongs to the class if it is of the form cp(t)
=
fjJ(t) , q - t
n
on segment [a, b]
(1.3.19)
where function fjJ(t)€H* on segment [a, b], and qda, b) is a fixed point.
The following theorem may be formulated. Theorem 1.3.3. Let cp(t)€ n on segment trt, b] and the sets E and Eo be chosen in such a way that point q belongs to the sec Eo for any n, q = t Oj .. Then, (1.3.20)
Quadrature Formulas for One-Dimensional Singular Integrals
51
where 0q(t Oj ) = Iq - tojl 10(t O)' and the quantity O(t Oj ) possesses the same properties as in Inequality (1.3.7). Therefore, Inequality 0.3.8) holds for the quantity 0q(tOj) for all points t Oj belonging to the set [a + 8, q - 8] U [q + 8, b - 8], and Inequality (1.3.9) also holds in which the sum in the left-hand side is carried out for all j =F j q.
1.4. SINGULAR INTEGRAL OVER A PIECEWISE SMOOTH CURVE 1. Let L be an unclosed Lyapunov CUNeo This means that a one-to-one correspondence T = T(t) exists between points T of L and points t of segment [a, b], such that the derivative T/(t) = dT/dt belongs to the class H( f3) on [a, b] and does not vanish on the segment. Hence, in accordance with what was previously said, the function t - to w(t,t o ) = - - - - T(t) - T(tO)
belongs to the class H( f3) with respect to both variables on [a, b] and docs not vanish on the segment. Let us take on segment [a, b] the sets E = {t b k = 1, ... , n} and Eo = {10k' k = 1, ... , n} forming a canonic division of the segment with the step h. Then one may state that the sets E = {T k = T(tk)' k = 1, ... , n} and Eo = {T Ok = T(tOk)' k = 1, ... , n} form a canonic division of CUNe L with the step h. We denote ii = T(a), Jj = T(b), and Sn(T O) = Lk ~ I «cp(Tk ) Lh k )/ (Tk - T Oj where tJ.Tk = Tk(t k + ,) - T(tk)· The following theorem may be proved.
»'
Theorem 1.4.1. Let cp(T) belong to the class H* on an open Lyapullov curve L and the sets E = {Tk' k = 1, ... , n} and Eo = {T Ok ' k = 1, ... , n} form a canonic division of the curve with the step h. Then the inequality
j
=
0,1, ... , n,
(1.4.1)
holds where the quantity O( TO) satisfies Inequalities (1.3.8) and 0.3.9) and is determined by Equation O.3}4) if T/(t Oj ' v, t-t) is substituted by the function T/(Trv' v, t-t) = IT oj - iirvlb - Tojl-i<, and a by the number A = minta, vf3, t-tf3).
The theorem can be proved by using the formula for substitution of the variable entering the singular integral and superposing functions of the class H (Muskhelishvili 1952).
52
Method of Discrete Vortices
°
If JJ = or /L = 0, then for the quantity O( 'To) entering Inequality (1.4.1) one has to make the changes mentioned in Note 1.3.1 with respect to O(t o) entering Inequality (1.3.14). If only the points 'T Oj lying near, say, the end point ii of the curve L are considered, then Inequality (t.4.1) may be corrected in the following way:
+
l'T oj -
h_ 1 +,,]O(lI nh l).
( 1.4.2)
01
2. Now let curve L be a piecewise Lyapunov curve containing angular nodes only and consisting of 1 Lyapunov curves L 1, ••• , Lt. On each segment [om,b m] mapped onto a curve Lm,m = 1, ... ,1, we choose a canonic division with step h m' formed by sets Em = {t k, k = n m - 1 + 1, ... , n m} and E Om = {t Oj ' j = n m_ 1 + 0, n m_ 1 + 1, ... , n m }, no = O.t We denote
h
=
max h m •
In what follows it is supposed that h/h m .:::;; R < + 00. Then the quantity hp/h m , P = 1, ... ,1 also remains limited for h -40. Next we denote j=0,1, ... ,n 1 ,n 1 +0, ... ,
j t
=
nm -
1
+ 0, ... , n m , (1.4.3)
This meanf that subscript j numbering points of the set E Om on the curve L m acquires the values 0,1, ... , n m - n m _ l "
53
Quadrature Formulas for One-Dimensional Singular Integrals
where 7m(t) is segment [am, bml mapped onto the curve L m = iimbm. Then the following theorem may be formulated. Let function cp(7)€H* on curve L. Then
Theorem 1.4.2.
j
0,1, ... ,n1,n 1 + O, ... ,n/, (1.4.4)
=
where quantity 0(7OJ) satisfies the inequalities: (l) for all points 7 OJ belonging to the curve L' that is a portion of curve L devoid of nodes together with their close neighborhoods,
(1.4.5) (2) for all points
TO j
lying in the vicinity of the node ii,
if function cp( 7) has the form l/J(7) cp(7)
I7-a-Iv'
=
( 1.4.7)
where l/J(7)€H(/3),0 < JJ < 1. If JJ = 0, then the right-hand side of Equation 0.4.6) becomes
Finally we note that Equations (1.4.5) and (1.4.6) imply that n/
L
O(Toj)ILhojl.:::;; O(h AJ ),
j=O
Proof. One can write I
II(7o j ) - S(7oj)l.:::;;
L m=1
where
IIm(7 o) - Sm(7 0j )l,
( 1.4.8)
Method of Discrete Vortices
54
Let us take the point Toj€L p n L'. Then for m = p, Inequality (1.4.1) may be applied to I/m(T oj ) - Sm(Toj)l, and for m *- p the conventional formula of rectangles may be employed. Hence, Inequality (1.4.5) is true. Next we prove the validity of Inequality (1.4.6). Let ii be the common end point of smooth curves L], ... , L/l' I] $ I. For points T Oj in the neighborhood of node ii, one may write /,
I/(T oj ) - S(Toj)1
$
L
I/m(T Oj ) - Sm(Toj)1
+ O(h A ),
O
m~]
Because all the curves L], ... , L/l are Lyapunov curves, Inequality (1.4.2) is true for II/T oj ) - Sp(Toj)1 for any TOj€L p' 1 $ p $ It. Therefore, one has to consider I/m(Toj) - Sm(To)1 for m = 1, ... , I], m *- p. We have
k=nm_l,1
where b is the other end of curve L m and numeration of the points Tk of the curve may be done in such a way that point ii is the starting point on the curve L m (in the direction of counting). Muskhelishvili (1952) showed that if function fjJ( T) is uniquely defined on L and belongs to the class H( /L) on each of the smooth arcs converging at the end point ii, then the function belongs to the class f{( /L) on L throughout the neighborhood of point ii (if node ii is a corner node of the curve 0. Hence, the function Iii - T Iv, 0 < JJ $ 1, belongs to the class H( JJ) on the entire curve under consideration. Taking into account Inequality (1.1.12), one can reduce the estimate for Ilm(Toj) -- Sm(Toj)l, m *- p, to estimates of integrals of the form (1.4.9) for IO} < a < b. Having evaluated the latter integrals, one may return to the variable T with the help of Inequality (1.1.2).
55
Quadrature Formulas for One-Dimensional Singular Integrals
Gakhov (977) showed that for the function
K(x, a)
=
(b - x) 1 -
I'
fb
dt I
a
7T
(b - t) -/L(t - a)/L(t - x)
'
(1.4.10)
the following representation is valid:
K(x,a)
=
(
a - x
)1' .
Sm JL7T
(1.4.11)
,
for x < a and 0 < JL < 1. Hence,
( 1.4.12)
for t Oj < a, 0 < v < 1, and
O(~), a - t
2v
=
1,
Oj
O(~), a - t
(1.4.13) 1 < 2 v < 2.
Oj
Inequalities (1.4.12) and (1.4.13) considered together with what was said when providing Inequality 0.3.7), demonstrate the validity of Inequality (1.4.4). • Note that for a function cp( 7) belong to the class H * on L, i.e., for a function of the form cp(7)
=
fjJ(7)/Pt(7)
(see 0.1.5», one can always assume that fjJ(7) vanishes at the nodes of a curve L and belongs to the class H in the vicinity of the nodes. Also note that all the additional statements formulated upon proving Theorem 1.3.2 dealing with the choice of the grids E and Eo and the function cp(t, 7) depending on a parameter (see (1.3.16» can be generalized onto the case under consideration in the most natural way.
56
Method of Discrete Vortices
Let us now single out two special cases of Theorem 1.4.2 that will be of use for what follows. Theorem 1.4.3.
Let curve L be a union of segments [a, q] and [q, b]; let < q < b). The sets E = {t k , k = 1, ... , n} and Eo = {t ok , k = 1, ... , n} forming the division of segment [a, b] with step h will be selected in such a way that point q lies halfway between the nearest points belonging to the sets E and Eo. Then cp(t) also belong to the class H* on L (a
bCP(t)dt
n
CP(tdhl
L 1 -t --t-Oj - k=ltk-tOj
1
:0; O(tOj)'
j=O,I, ... ,n,
a
where the quantity O(t Oj ) satisfies the inequalities: (1) O(to):O; 08(h At ),0 < AI:o; 1, for points tojda
+
8,q - 8] u [q
$
1.
+
8,b
- 8]; (2) n
L
O(to)h :0; O(h A2 ), 0
< A2
j=O
Theorem 1.4.4. Let L be either a union of nonintersecting segments [am, b m ], m = 1, ... ,1, or a union of segments [am,qm],[qm,bm],qmdam,bm),m = 1, ... , I. In the former case, the sets Em = ·{tk' k = n m I + 1, ... , n m} and E Om = {t Ok ' k = n m -I + 1, ... , n m} form a canonic division of the segment [am' b m ] with step h m, m = 1, ... , I, no = 0; in the later case, the sets are chosen in such a way that qm EEOm ' Then
where tJ.tk,h m, k = n m -I + 1, ... , n m, m = 1, ... , I, and the quantity O(t o) satisfies the inequalities: (1) 0«0):0; O(hAt),O < A, :0; 1 for all points tOjE U~~I[am + 8,bm - 8], in the former case and tOjE U ~ ~ I([a m + 8, qm - 8] u [qm + 8, bm - 8]) in the latter case; (2) Lj!:oO(toj)ILltojl < O(h A2 ),O < A2 $ 1, where Llt Oj = hm,j = nn ,+ 0, ... , n m, h = max m~ 1, ... ,/ h m, and it is supposed that h/h m :0; R < + 00.
Quadrature Formulas for One-Dimensional Singular Integrals
57
1.5. SINGULAR INTEGRALS WITH HILBERT'S KERNEL Consider an integral of the form 1(00)=
°- °0 dO, 1o cot--cp](O) 2 27T
(1.5.1)
where cp](O) is a function with the period equal to 27T. From Muskhelishvili (1952) it is known that
f
L
cp( t) dl t - to
1 =-1 2
27T
0
°-
00 i 27T cot--cp](0)do+-1 cp](O) dO, (1.5.2) 2 2 0
where L is a unit-radius circle centered at the origin of coordinates, t = exp(iO), to-exp(ioo), cp](O) = cp[exp(iO)]. Let the function cp(t) belong to the class H on L and the sets E = {t k = exp(iOk),k = 1, ... ,n} and Eo = {10k = exp(iOok),k = 1, ... ,n} form a canonic division of L. Then
:t k=]
CP(tk) !J.t k tk
-
10j
:t k~]
cp( e i8 ,) e i8 , (e i !l8, e
i8
, -
e
i8o
-
1)
/
where !J.0k = 27T/n. From Taylor's formula one gets
Hence, we arrive at the equality
( 1.5.3)
58
Method of Discrete Vortices
Because the left-hand side of this equality well approximates the lefthand side of Equality (1.5.2), and L~ 1 'Pl(Ok) !i0k is a good approximation for the integral M"'P,(O) dO, the following is true.
°
Theorem 1.5.1. Let a function 'PI( )€l/ on segment [0, 27T] and its period be equal to 27T. Let the points E = {Ok' k = 1, ... , n} and Eo = {00k- k = 1, ... ,n} be chosen on segment [0, 27T] in the following way: Ok-<- 1 - (fk = 27T jn = h, k = 1, ... , n - 1, 0, + 27T - On = h, 00k = Ok + hj2,k = 1, ... ,n, i.e., points t k = exp(iO k ) and t Ok = expUOok),k = 1, ... , n, form a canonic division of L. Then the inequality
(1.5.4)
Note that functions 'P(t) and 'P'( B) belong to a class H(A), each on its set.
1.6. UNIFICATION OF QUADRATURE AND DIFFERENCE FORMULAS In aerodynamics (Bisplinghoff, Ashley, and Halfman 1955, Ashley and Landal 1967) one often has to deal with the Prandtl integrodifferential equation, which contains the integral
l(to) =
f
b
a
y'(t) dt t - to
,
where the function y'(t) belongs to the class H* on [a, b l, to y(t} belongs to the class Ht on [a, b]. If y'(t}€H* on [a,b], then the integral
(1.6.1)
E
(a, b), i.e.,
(1.6.2) . may be reduced to integral (1.6.0. Here integral 0.6.2) is considered in the sense of Hadamard's finite value, (1.6.3)
Quadrature Formulas for One-Dimensional Singular Inlegrals
59
From 0.6.3) we deduce that _ y( a) 1(2)(/ 0 ) -
-- -
a - to
y( b) --
b -
10
+
fb y' (t) dt a
I - 10
(1.6.4)
Let us next construct a quadrature-difference formula for approximate evaluation of integral (1.6.2). Let the sets E = {tk' k = 1, ... , n} and Eo = {1 0k , k =il, ... , n} form a canonic division of segment [a, b] with the step h. Acco~ding to Section 1.3, it is natural to consider the quadrature formula S(lo)
=
n
y'(/dh
y(b)
k=1
t k - / Oj
b-/Oj
L --- - - -
y(a)
+ -a - / Oj '
j=O,I, ... ,n, (1.6.5)
when considering integral (1.6.2). Convergence of the latter quadrature formula to integral 0.6.1), and hence to integral 0.6.2), was analyzed in Section 1.3. Let us continue by constructing the folIowing quadrature sum for integral (1.6.2): j=O,I, ... ,n,
(1.6.6) where
10 =
a,
In +
I
=
b. The formula may be derived as folIows:
The results obtained in Section 1.3 and Formula 0.6.4) alIow us to formulate the folIowing theorem. Theorem 1.6.1. Lei a funclion ¥(t)EHt on [a, b) and poinls 10 = a, 11"'" tn' In + I = b divide segment [a, b) into n + 1 equal parts; lei t Ok be the middle segment [t k , t k + d, k = 1, ... , n. Then the inequality
( 1.6.7)
Method ofDiscrete Vortices
60
holds for any j = 0,1, ... , n, where O(tOj) has the same properties as in the Theorem 1.3.2. Note that quadrature formulas of the form S{2)(t O) may also be constructed for the integral ( 1.6.8) in the following way. Let
::: t k=1
y(m-I)(tod - y(m
1)(tOk_l)
tk-t Oj
In the latter sum we replace y(m - l)(t ok ) with the help of the difference scheme (y(m - 2)(t Ok ) - y(m - 2)(tOk 1) /h. By continuing the procedure, one gets (1.6.9) where y(m -1)(too), ... , y(O)(t oo ) == y(t oo ) enter w;:j; in other words, are not substituted with the help of the difference scheme (see Matveev 1982).
2 Interpolation Quadrature Formulas for One-Dimensional Singular Integrals
2.1. SINGULAR INTEGRAL WITH HILBERT'S KERNEL In order to construct quadrature formulas for the integral
one has to recall the following facts. In Luzin (1951) the inequalities 27r
0 - 00
mO dO 1o cot--sin 2 27r
=
27T cos mOo,
0 - 00
mO dO = -27T sin mOo' 1o cot--cos 2
m
=
1,2, ... ,
m = 0,1,...
(2.1.2)
(2.1.3)
were proved. In addition, from trigonometry it is known that sin(2n + 1)( 0 - 0d/2 ---::2'---s-in-(-0---0-k-)-/2--
1
=
"2
+ cos(O - 0d + ... +cos n(O - Ok), (2.1.4) 61
Method of Discrete Vortices
62
Hence, for any periodic function cp( 0), the trigonometric polynomial of the power n, 2n
sin(2n + 1)( 0 - 0k)/2 2n + 1 cp( °d sin( 0 _ 0d/ 2 ' 1
kr:O
CPn( 0) =
(2.1.5)
meets the condition
k
=
0, 1, ... ,2n,
(2.1.6)
if 27T O.:s:; f3.:s:; -2-n-+-l '
k=0,1, ... ,2n.
This can be proved by observing that
2n + 1 sin
a =
2
°
for 0'=
sin(2n + 1) 0'/2
lim <>-->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)
Quadrature Formulas for One-Dimensional Singular Integrals
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).
Method of Discrete Vortices
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 1o cot--[cp(O) 2
=
1(00) -lPn(I(Oo»)
27r
CPn(O)]dO
2n
+
L
[lPAOd - cp(Od]
k=O
X
[ cot
Ok - 00 cos(2~ + 1)( Ok - ( 0 2 sm( Ok - (0)/2
)/2]
27T
.
2n + 1 (2.1.13)
In Berezin and Zhidkov (1962) it is shown that in the case under consideration, Icp(O) -
+
In n)En
(2.1.14)
°
for any from [0, 27T], where En is the best possible approximation of. the function cp(O) by polynomials of degree n, i.e., En = inf p .(9) max 9ErO, 21r]1 cp( 0) - Pn ( )1, where Pn ( 0) is an arbitrary polynomial of degree
°
nand
En:O; -n,ta: --,
(2.1.15)
where M is a constant in the Holder condition for the function cp(r l ( 0). Because I( 0o)€Hr ( Q'), an inequality of the form of (2.1.14) holds for 1/(00) - n(I(oo»l.
Quadrature Formulas for One-Dimensional Singular Integrals
65
Thus, from (2.1.10) and (2.1.13)-(2.1.15) one gets
I/( 00) - S( 00)1 :5: 0 (
3
+ In n + n )
(2.1.16)
nr + ex
for any 90 E[0, 27T]. Let us consider Formula (2.1.11) at the points
_ {Ok + 2n: 1 ' °Ok 7T Ok - -2-n-+-l '
k
=
k
=
7T 1, ... , 2n, if 0 :5: f3 :5: - - 2n + 1 7T 27T 1, ... , 2n, if :5: f3 :5: - - 2n + 1 2n + 1
(2.1.17) In this case points 00m' m = 0,1, ... , 2n, are roots of the function cos«2n + l)/2XOk - 00)' because
cos
2n+l 2 (Ok - °Om)
=
cos
[2n+12(k-m)+I 2 2n + 1
7T
]=0.
Also note that . 1 SIO-(
2
for any m, k Thus,
=
Ok - 00m)
. 2( k - m) = SIO
±1
2(2n + 1)
7T *- 0
(2.1.18)
1, ... , 2n.
Ok - 00m
2n
S(OOm)
=
L
cot
k=O
2
27T
cp(Od 2
n +
1
(2.1.19)
at the points 0om. In this case Formula (2.1.16) becomes
(2.1.20)
Method of Discrete Vortices
66
because it can be readily shown that
~lcotOOm-Okl 2
k=O
27T =O(lnn) 2n + 1
(2.1.21 )
for m = 0,1, ... , 2n.
2.2. SINGULAR INTEGRAL ON A CIRCLE Let function cp(t) belong to the class H,( Q') on the unit-radius circle L centered at the origin of the system of coordinates. Let us consider the polynomial (2.2.1 ) where points t k divide the circle L into 2n + 1 equal parts. By dividing a polynomial by a polynomial it can be readily shown that (2.2.2)
k=0,I, ... ,2n,
because t 2n + 1
_
t;n
t 1
--2n + 1 (t - tdtnl%
1, { 0,
1= I k ,
I
=
1m
,
k = 0, 1, ... ,2n,
m
* k, m = 0, 1, ... ,2n (2.2.3)
The latter equality also follows from the relationship sin(2n + 1)0/2 ------ =
sin 0/2 =
1 + 2(cos 0 + ... + cos nO) e- in9
+ e-i(n-
1)9
+ ... +e-i9
+1 + e i9 + ... e inO
Tn +t n+ 1 =
----
1- t
t = e i9 ,
and Formula (2.1.17). From (2.2.4) we deduce that sin(2n + 1)(0 - 0k)/2 sin( 8 - 0d/2
t 2n + I
_
lIn,
1
(I - IdlntJ: '
(2.2.4)
Quadrature Formulas for One-Dimensional Singular Integrals
67
Next we consider the singular integral
_f
1(10) -
I.
cp( I) dl , t - to
where cp(t )€Hr ( 0') on the circle L. We denote
By employing the formula (Gakhov 1977) ~
0, n < 0, n
(2.2.6)
where t = exp(iO), to = exp(iOo), and the evident identity
1 (t - td(t - to)
1 to -
=
Ik
(1 1) t - to - t - t k '
(2.2.7)
we get
-dtI - 10
1- -1tk
-
to
(2.2.8) Thus, the quadrature sum for the integral 1(/ 0 ) has the form
S(lo)
2n
=
L
k=O
cp(ld 7Ti [ -t---t -2-+-1 21 k k
0
n
-
1
t~n + I + tl n+ I --t-n-tn- - . 0 k
(2.2.9)
68
Method of Discrete Vortices
Note that if cp(t)€H,(O') on L, then the function CPI(O) also belongs to the class H,( 0'). Thus, one has 27T
1o
sin(2n + 1)( 0 - 0d /2 . dO = sm(O - 0d/ 2
1 (I + 2 cos( 0 27T
=
cp[exp(iO)]
0d
0
+ ... +2cos n(O - 0d) dO = 27T (2.2.10) and, hence,
(2.2.11)
By employing Formulas (1.5.2), (2.1.16), and (2.2.10, the following inequality may be shown to be true:
1/(t0) - S(to)1 ~ 0 (
In n
n"
+n ) t>
Let us now find the roots of the function (t5 n + 1 + tr' This can be done by observing that
15 n+ 1 + tIn 1 t3 1;:(t0 - Id
. cos(2n + 1)( 00
I
=(
.
sm( 00
-
(2.2.12)
•
-
1)/«31;:«0 -
0d/2
Ok )/2
t k ».
(2.2.13)
Hence, the function has the roots
10m
= eXP{Om +
2n
7T
+
in other words, 10k divides the arc have for the chosen points tOm' 2n
S(tOm)
=
L
k~O
cp(td I - I k
Om
1
m
},
;;h,
1
0, 1, ... ,2n;
into two equal parts. Hence, we
27Til k 2n
=
+ 1'
m=0,1, ... ,2n. (2.2.14)
Quadrature Formulas for One-Dimensional Singular Integrals
69
Then, by using the identity 1
itk ----- =
tk
-
tam
-cot 2
Ok - 00m
+
2
1 -i
(2.2.15)
2 '
one may write 1[ S(tOm) = -2
2n L
k~O
cot
° -° k
2
am 'P,(
0d + i
2n L
k~(J
'PI( 0d
1
27T
2n
+1
.
(2.2.16)
Finally, by employing Formulas (1.5.2), (2.1.20), and (2.2.10, one gets In n ) II(tOm) - S(tom)1 < 0 ( n r t a '
m
=
0,1, ... , 2n.
(2.2.17)
2.3. SINGULAR INTEGRAL ON A SEGMENT In the present section we shall consider integrals of the form _ / ' w(t)l/J(l) dt I(O',{3,t o ), -I t - to
(2.3.1 )
where w(t) = (l-x)a(l +x)l3, 0< 10'1,1{31 < 1, and the function l/J(t) belongs to the class Hr(A) on segment [- 1, 1], i.e., l/J(r)(t)€H(A) on [-1,1]. Because in aerodynamics and some other applications one often comes across the case 0', {3 = ± ~, we shall consider the integrals in greater detail. Let us start by constructing quadrature formulas for the general form of integral (2.3.1) (Korneichuk 1964, Lifanov and Saakyan 1982, Savruk 1981, Stark 1971). Let Pn(t) be a polynomial of the degree n belonging to the system {Pm(x), m = 0,1, ... } of polynomials orthogonal on [- 1, 1] with the weight w(x), where w(x) is a positive function integrable on [-1,1]. In this case integral (2.3.1) will be denoted by I(to). The roots of the polynomial Pn(t) wiII be denoted by ti' i = 1, ... , n, and the polynomial n l/JAt) =
L 1 l/J(td (t
k~
Pn(t) - t )P'(t ) k
n
(2.3.2)
k
by l/J/t). It is evident that the latter polynomial meets the condition k
=
1, . .. ,n,
(2.3.3)
70
Method of Discrete Vortices
because
By 5(t o ) we shall denote the quadrature sum obtained in the following way: I
5(to)=f
-I
W(t)l/Jn(t) t-t dt= 0
n I w(t)Pn(t) dt Ll{!(tdf (t-t)(t-t)P'(t)
k=1
-10k
n
k
(2.3.4)
where _ fl Qn(to) -
w(t)Pn(t) dt t - to
--I
.
(2.3.5)
Let tOm' m = 1, ... , R, be roots of the function Qn(to). Then, the formula for 5(t o) at the points becomes
m
=
1, ... ,R,
(2.3.6)
where k
=
1, . .. ,n.
(2.3.7)
In Korneichuk (1964) inequality
(2.3.8) was proved, where a(t) = f I I w( 'T) d'T and En _ I is the best approximation of the function l{!'(r) obtainable with the help of polynomials of degree n - 1. Hence, if function 1jJ(t)EHr (a) on segment [-1,1], then E n _ 1 ~ O(n r - a + I); accurate values of the constants are presented in Kantorovich (1952). In Stark (1971) it is shown that if ljJ(t) is a polynomial of
Quadralure Formulas for One-Dimensional Singular Integrals
71
degree less or equal to 2n, then S(tOm) = I(tom). Next we present the concrete form of Formula (2.3.6) for the cases when 0', f3 = ± t. 1. Let 0' = f3 = - t, i.e., 2
w( t) = (1 - t)
-1/2
.
(2.3.9)
The polynomials orthogonal with the weight on [-1,1] are Chebyshev polynomials of the first kind, Le., Pn(t) = TAt) = cos(n arccos I).
(2.3.10)
Roots of polynomial Tn(t) are nodes of the quadrature formula (2.3.6) and have the form 2k - I tk=cos
2n
7T,
k = 1, ... , n,
(2.3.1 I)
and the functions Qn(t) are
sin( n arccos I) Un-I(/) = . ) , sm(arccos 1
(2.3.12)
the latter expressions being Chebyshev polynomials of the second kind. In this case, roots of function Q(t) have the form tOm
=
cos m7TIn,
m=1, ... ,n-1.
(2.3.13)
n = 1,2, ... ,
(2.3.14)
Because
it follows from (2.3.7) that k=I,2, ... ,n.
2. Let
0'
=
f3
=
(2.3.15)
t, i.e., w(t) =~.
(2.3.16)
Method of Discrete Vortices
72
Then Pn(t} are Chebyshev polynomials of the second kind sin[( n + 1)arccos I]
PAt)
=
(2.3.17)
------
sin( arccos I)
whose roots are given by k tk
k = 1, ... , n.
= COS--7T,
n
+1
(2.3.18)
For the function Qn
=
7Tsin Osin(n + 1)OdO
1o 7T
cos 0 - cos 00
sin nO o - sin(n + 2)0 0 = sin 0 0
7T
cos(n
+ 1)00 , (2.3.19)
Thus, (2.3.20) and, hence,
10m
=
cos
2m - 1 )
2(n
m=l, ... ,n+1.
7T,
+1
(2.3.21)
Then by (2.3.7) we have Ok =
7T . 2 k --SIO --7T,
n+l
k
n+l
Q'
=
i,
w(t)
=
~
3. Finally, consider the case
f3
=
-
-x
--.
l+x
1, .. . ,n.
=
I
(2.3.22)
.
2' I.e.,
(2.3.23)
In this case polynomials Pn(t} are given by (2.3.24)
Quadrature Formulas for One-Dimensional Singular Integrals
73
and the roots are given by t k = cos
2k 2n + 1
7T,
k=1,2, ... ,n.
(2.3.25)
For the function Qn(t) one gets (2.3.26)
The roots of the function correspond to the points 2m - 1 tOm =
For the coefficients
cos Uk
=
m
7T,
1, ... ,n.
=
(2.3.27)
one has
47T
ak
2n + 1
2n + 1
sin 2
k
2n + 1
k
7T,
=
1, ... , n.
(2.3.28)
Note that inequality
was proved in Sanikidze (1970) for case 1, if function t/J(t)€Hr( 0'). Here TJ = (I + /1 ( 1)/2, and the constant C r is independent of n, I, and t/J. A more convenient estimate, In
n) .~' 1
I/(t o ) - S(/o)1 ~ 0 ( ~ n
was obtained for the case in Sheshko (I 976).
V1 -
Iii
3 Quadrature Formulas for Multiple and Multidimensional Singular Integrals
3.1. MULTIPLE CAUCHY INTEGRALS Let L" ... , L n be piecewise smooth plane CUNes. FoIlowing Gakhov (t 977), their topological product wiII be referred to as L = L 1 X L z X ... X Ln. Let us consider a function cpU) = cp(t', ... ,t n) defined on the frame L. A point t = (t 1, ••• , In) will be caIled an internal point of the
frame L if point t k is not a nodal point of CUNe Lk> k = 1, ... , n. Let point to be an internal point of the frame L. In the plane of CUNe L k we draw a circle of radius €k centered at point t~ and denote by lk the part of CUNe L k lying inside the circle (€k is supposed to be so smaIl that lk is a smooth unclosed arc). The limit
(3.1.1) where
€ I' ... , €n
tend to zero irrespective of each other, will be caIled a
Cauchy integral (a multiple singular integra)) of the function cpU 1, •.. , t n ) over frame L at point to = (t~, ... , to)' 75
Method of Discrete Vortices
76
The limit wiII be denoted by
where dt = dt I
...
dt n. (3.1.1 a)
Similarly to a one-dimensional singular integral, it can be shown that n = cp{t I, ... , t ) meets the condition H within a certain neighborhood of point to = (t6, .. . , tg) belonging to the frame L. On the other hand, Equation 0.1.1) allows us to consider the multiple Cauchy integral as an iterated integral, i.e., one may write
cp(tl, ... ,tn) dt l ... dt n
f
[_IX ...
XL.
(t l
-
t~)
... (tn - tg)
(3.1.2) Let us demonstrate this in the example of two variables. Let function cp(ll, t 2 ) meet the condition H( ILl' IL2) on L] X L 2• We denote (Gakhov
1977) CPI -_ CPl2 -_
cp (I t , to2) - cp (I to, to2) ' cp (I t , t 2) - cp (I / , to2) - cp (I to' t 2) + cp (I to, /02) •
Then the following formulas are valid:
(3.1.3) and hence,
o~ Note also that the
Q'
~
1.
(3.1.4)
Multiple and Multidimensional Singular Integrals
77
holds. Therefore
dt 1
+
CP2
1. fl=ll 1. t 1-,
0
L
2 -
dt 2 t 02
Here all the right-hand-side integrals exist either as conventional improper integrals or as one-dimensional Cauchy integrals. Formula (3.1.2) allows us to transfer many properties of one-dimensional Cauchy integrals onto mUltiple Cauchy integrals. We shall start by considering one-dimensional singular integrals whose density depends on parameter 7,
_1. cp( t , 7) dt ,
L
(3.1.6)
t - to
where cp(t, 7) meets the condition H( /L) with respect to t on L and the condition 1I( JJ) with respect to 7 on a limited set of T. The following theorem is valid. Theorem 3.1.1. Function 0 is a however small number. Proof. Muskhelishvili (1952) has shown that
+ h) -
f
L
cp(t,7+h) - cp(t, 7) t - to
dt.
Method of Discrete Vortices
78
Let us introduce the representation (Gakhov 1977) cp(t,7 + h) - cp(t, 7)
=
CPI 2 =
'PI 2 + CP2'
[cp( t,
7
+ h) - cp(to, 7 + h)]
Then
Because cp(t, 7) meets condition H( IL, v) on L limited quantity on L',
X
T, and fLdt/(t - to) is a
(3.1.7)
On the other hand, according to Gakhov (1977),
From the latter inequalities it follows that Q'
If
Q'
E
[0,1].
= €/v, then
where € > 0 is a however small number. The latter inequality for
Because Q' lies between 0 and 1, 0 < € ~ v. Now let Ihl be such that € = -1/lnlhl ~ v, that is Ihl ~ exp( - v I). Then (3.1.8)
Multiple and Multidimensional Singular Integrals
79
Thus, the validity of our statement concerning function
meets the condition H( /L, v] ever small positive numbers k
- €], = 1,
, Vm -
Em)
on L., where €k are how-
, m.
If L is a segment, then the following theorem is true.
Theorem 3.1.3.
Let function cp(t, 7) be of the form cp*(t,7)
cp(t,7)
=
(t - a)
(3.1.9)
V>
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
n3 =
f
t -a v (0 )
a
b
dt (t-af(to-t)
belongs to th.e ~Iass H on [a,c] (see Muskhelishvili 1952), In~l s: B1hl 13, to E [a,c]. SImilarly, we deduce for nlz that Inlzl s: B]lhl 'e-] for o < € s: f3.
80
Method of Discrete Vortices
Consideration of the latter two inequalities together with the fact that O(t o, 'T) E H with respect to to and uniformly with respect to 'T E T (see Muskhelishvili 1952) completes the proof of the theorem. • Note 3.1.1. Taking into account the results obtained in Muskhelishvili (1952), Theorem 3.1.3 may be generalized to the case of an arbitrary piecewise smooth curve L for which a is a node and v = VI + V Z" o < VI < 1. Let us now return to the multiple Cauchy integrals. The following theorem follows from Theorem 3.1.2. Theorem 3.1.4. Let function cp(t 1, ... , t n ) meet the condition H( ILl' ... , ILn) on an n-dimensional torus L = L I X ... X L n, where Lk> 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),
Using Theorem 3.1.1 about singular integrals depending on a parameter, we deduce that fjJ(t I, t(;) E H( ILl - E, IL;) and
=
,
I n '
p[,(t ). '" 'p[.(tn)
(3.1.11)
en
k C i , i k = 1, ... ,m k , are all where cp* E H on L, Pl':(tk) = n;"~_lltk k k k kk kk k . the nodes of the curve L k , and V = (vI" .. , vm)' 0 $ Vi. < 1, lk = 1, ... , m k , k = 1, ... , n, then it will be called a function of the class H* on the frame L = L 1 X .•. X L n (where L k , k = 1, ... , n, are plane, piecewise smooth curves). Now Theorem 3.1.3 and Note 3.1.1 allow us to formulate the following statement.
Theorem 3.1.5. The class H* of functions on the frame L = L 1 X ... X L n ( where L k' k = 1, ... , n, are piecewise smooth curves) is invariant with respect to the operation of taking a multiple Cauchy integral; in other words, if
MuLtipLe and MuLtidimensionaL SinguLar IntegraLs
81
a function cp(t} beLongs to Ihe class H* on L, Ihen
where L) and L 2 are piecewise smooth curves and cp(t~, t5, tl, t 2 ) has the form
cp*(to' t) cp( to, t) = - m - , - - - - - - - - - - -
n
k,
~
III -
cuvi'l/~ - ck,I
Al
(3.1.12)
,
I
m2
n 11
X
2
clzlvlzl/5 - cll tz
-
k2= I
where cp*(to, t) meets the condition H on L with respect to to and I, I 2) . I· ,I 2 ,2 0 an d t -_ . (t, tare pomts ymg on L ,JJkI I , I\k , JJk ,I\k ~ I 2 2 JJ k + Ai.: < 1, cA:' k; = 1, ... , m; denote all the nodes of curve L;, i = 1,2. 'Let u~ present 1(71, 7 2 ) in the form
I 10 . _- (to' to2), .
I( 7
I
1
2 ,7 ) -
dl5 2
L 2 to - 7
2
(1
dtJ 1
1
1.,1 0 -
1 fjJ(t~,t5,tl'15)dtl)
7
I
1-,
I
I
- to
'
(3.1.13) By the Poincare-Bertrand formula (Muskhelishvili 1952) one has
=
-7T
2
1
2
I
2
fjJ(7 ,/0 ,7 ,to) +
f L,
dl
If
L,
-1..(
1
2
I
2)
d
I
10,/o,t ,/0 10 I I I . (3.1.14) (to - 7 )(t - ( 0 ) 'I'
1
82
Method of Discrete Vortices
Let us substitute the latter expression into Formula (3.1.13) and use the expression for t/J together with the fact that the order of integration over L 1 and L 2 may be changed:
1
2
_
1(7,7)-
f L
dto2 2
to -
2
2
f L
7
d to2
+ / L to2 2
1
2
1
2
/
.1. ( 1 2 1 2) '1'17 ,to ,7 ,I
2
7
2 (
,to' 7 ,t 2 t - to
7T
cp
7
2)
d
t
2
2
I 2 - to2
1. 2
dt 2 '
(3.1.15)
Finally, after using again the Poincare-Bertrand formula for onedimensional Cauchy integrals, one gets
2/ dt 2/ 7,/ I
cp (
-
7T
1'2
1. 2
(to -
2/ dt I/CPt o I
(
-
7T
1
1-,
I.,
2 1 2) 0 ,7 ,I
2
2
7
2
d
)(
2
10
2
2
I - (0 )
1
,7,1,7 1 1
2)
(t - lo)(to -
d
I
to
1 7
)
(3.1.16) Now by applying the method of mathematical induction one can easily derive the formula for changing the order of integration in multiple Cauchy integrals of an arbitrary dimension. Tbeorem 3.1.6. LeI function cp( t) be a function of the class H * on frame L. Then the following Poincare-Bertrand formula for changing the order of
MuLtipLe and MuLtidimensionaL SinguLar IntegraLs
83
integration in muLtipLe Cauchy integraLs is valid: dto
f «(to L
7» n
=
f cp(to,t)dt L
«t - to»
2 (-7T )P
L p=O
L
k,'" .,. ",kp~l
f
dtk" ... ,k"
L k , ...• k p
where dtk" ... ,kp is the product of dt with discarded muLtipLiers co"esponding to k l' ... ,k p ' A simiLar convention is vaLid for L k , , ... ,k I' , and if p = 0, then no terms are discarded, whereas Ttk , ... , k means that the coordinates t k , ... ,t k must be substituted by coordinates ~k"'" 7k I' (a similar conve~tion i~ , assumed for TtOk , , ... , kp ).
Because the identify dto
f
------=0, L (t - to)(to - 7)
holds for a smooth unclosed curve, the identity dto
fL«(to -
f -cp(-t) dt- = (-7T )
2 n
7»
cp(T).
(3.1.18)
L«t - to»
holds for a function cp(t', ... ,t n ) smooth closed curves).
E
H on the frame L (L" ... ,L n are
3.2. ABOUT SOME SINGULAR INTEGRALS FREQUENTLY USED IN AERODYNAMICS In the steady problem of subsonic flow of perfect incompressible fluid past a finite-span wing, one has to consider the integral (Belotserkovskii and Lifanov 1981, Bisplinghoff, Ashley, and Halfman 1955) A(xo,zo)
=
y(x,z) (
if (Zo (T
2
z)
1+
...;
Xu
(xu -
X)2
-x
+ (Zo - z) T
)
dxdz, (3.2.1)
where
IT
is a part of the plane OXZ occupied by a plane wing.
Method of Discrete Vortices
84
Before defining integral (3.2.0, it should be noted that integral 0.6.2) may be defined as (see Equation (1.6.3» (3.2.2) Let us show that integral I(2)(x o) exists for any function y(x) In fact, in this case we have
E
H1(a).
(3.2.3) and hence, the following equality holds:
Here the first integral on the right-hand side exists as an absolutely integrable improper integral, whereas the third one exists in the sense of the principal Cauchy value. For the second integral we have lim <,-->0
lim <,-->0
(
~\l(Xo
1 -( Xo -x
dx .
I
Xo - <'
€
1
+--
Xo - a
- b
Xo
2)
b
-€
+ _1_ + ~ _~) Xo -
b
1 Xo
I
X o - x xo+<'
a
lim ( _ _1_ <,-->0
2) --
_X)2
€
€
(3.2.4)
- a
Thus, integral I(2)(x o) is also defined for any function y(x) EO Ht because in this case y(x) E HI in the neighborhood of any point Xo E (a, b).. If y(b) = y(a) = 0, then we have the equality
t
y(x) dx
a (Xo -x)
=
2
_
t a
y'(x) dx . Xo-X
Multiple and Multidimensional Singular Integrals
85
Upon noting that
.. j. one can define the integral
in the following way:
m-2 (_I)k
- L - - y(k)( x k~O
k!
1 _(_I)m
k-Ij
) - - - - ------,---,-0 m - k - 1 €m-k-I '
where y(O)(x o ) == y(x o ). Integral Irm)(x o ) exists for any function y(x) E H:' __ I' because in this case y(x) E H m _ 1 in a certain neighborhood of point X o and the inequality Iy(X) -
y(x o ) -
1\ y'(xo)(x -
xo) - .. ,
1 y (m-I)()( Xo X _ Xo)m'II
holds. The following formula of integration by parts is valid for integral I(m)(x o):
y(a) ( X o - a)
_fb
y' ( x) dx
a (xo-x)m-I
].
m-I
Method of Discrete Vortices
86
Let us return to considering integral A(x o, zo) starting from the simplest case when region IT is a rectangle: IT = [ -b, b] X [-I, I]. Integral A(x o, zo) will be defined by generalizing Formula (3.2.2) in a natural way:
(3.2.5) where Q(€, zo) = {(x, z): Iz - zol < d, I zo = line given by the equation z = zo' and
l/J(x,x(pz,zo)
=
y(X,Z)(l
+ ..;
IT
n L zo ' and L zo is a straight
X02 -x
2 ).
(x o - x) + (zo - z)
Theorem 3.2.1. In the case under consideration integral A(x o, zo) exists for any function y(x, z) and is such that y/(x, z) == dy(X, Z)/dZ belongs to the class 11* on IT. Proof. Suppose that y/(x, z)
E
H( Q') on
IT.
Then by Formula (3.2.3) one
has Iy(x, z) - y(x, zo) - y~(x, zoHz - zo)1 :::; Biz - z oll
+a.
Let us rewrite formula (3.2.5) in the form _
.
A(xo,zo)-hm .-->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)
Method of Discrete Vortices
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
Method of Discrete Vortices
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)
Multiple and Multidimensional Singular Integrals
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
Method of Discrete Vortices
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
Method of Discrete Vortices
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 <
k = 1,2, be
vt, vf < 1.
MuLtipLe and MuLtidimensionaL SinguLar IntegraLs
95
By Theorem 3.1.3 concerning integrals depending on a parameter, function
has the same form as the density of the corresponding singular integral, i.e.,
Hence, one may write
n,
+
h
il~l p::(tiJlt~\ - t(\i,1
Taking into account the comments made in the closing part of Section 1.3 concerning Function (1.3.16), as well as Inequalities (1.3.7) and (1.3.14), one gets
(3.3.6)
By using Formula (1.3.14) for O,(t6j) and OZ
Now let the frame L be a product of piecewise Lyapunov curves containing corner nodes only. With the help of Inequality (1.4.4), Inequality 0.3.3) can be generalized in this case in a natural way.
Method of Discrete Vortices
96
3.4. QUADRATURE FORMULAS FOR THE SINGULAR INTEGRAL FOR A FINITE-SPAN WING Let us start considering quadrature formulas for the integral A(x o, zo) for a finite-span wing beginning with a rectangular wing. In this case
Let points XI' ••• ' x n and x oo , X o 1' ... , X On form a canonic division of segment [-b,b] with the step hI' and points ZI, ••• ,ZN and zoo' Zo I'···' ZON form a canonic division of segment [ -t, l] with the step h z . We suppose that h/h k ~ R < +00, k = 1,2; h = max(h l , h z ). By applying the principle of constructing quadrature formulas for a singular integral along the X coordinate and the principle of constructing the sum 5(2)«0) entering Formula 0.6.6) along the Z axis, one can consider the following quadrature sum for A(x o, zo):
(3.4.1) where
Zo
= -t,
Note that Formula (3.4.0 may be presented in the form
ZN t I
= t.
97
Multiple and Multidimensional Singular Integrals
where
Now we can prove the following theorem. Theorem 3.4.1.
Let region (J in (3.2.1) be a rectangle and the function y(x, z) be such that Yz'(x, z) belongs to the class H* on (J. Then inequality
(3.4.3) holds for any point (X Oj ' zorn), j = 0,1, ... , n, m = 0,1, ... , N, where O(X Oj ' zOrn) is characterized by the following relationships: 1.
For all points (X Oj ' zOrn) E [-b + 8, b - 8] X [-I 8 > 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. •
Method of Discrete Vortices
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
Method of Discrete Vortices
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
Multiple and Multidimensional Singular Integrals
- 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)
)
Method of Discrete Vortices
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)
Multiple and Multidimensional Singular Integrals
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
Method of Discrete Vortices
z
(-l,b)
.-
- +
"
I<
(l,b)
3:'L
-I<
1+-
ai,
1-+ 1
Z,.
Z'"
Z
(I,-b)
(-I,-b)
FIGURE 3.1. Distribution of signs before the summands of the quadrature formula (3.4.17), corresponding to point (x j , Zk)'
where: D is a closed limited region of the plane £2 for whose boundary
1.
Jordan's two-dimensional measure (Fikhtengoltz 1959) is equal to zero. x = (x', x 2 ) and Xu = (xl), x~) are points of region D, r = Ix - yl, and 0 = (x - y)/r is a point of the unit circle u in £2' The density of singular integral u(x) is absolutely integrable on D and meets condition H( Q') on any closed set F lying inside D. Characteristic f(x u, 0) is limited and continuous with respect to 0 for a fixed xu'
2.
3. 4.
If
Xu
lies inside D, then 0.5.0 is an integral in the sense
(3.5.2)
where O(x u, €) is an € neighborhood of point Xo' In Mikhlin (1948) it was shown that to ensure the existence of integral (3.5.1) in the sense of the principal value (3.5.2), it is necessary and sufficient to meet the condition
!f(x u , 0) du IT
=
O.
(3.5.2')
Multiple and Multidimensional Singular Integrals
105
We start considering quadrature formulas for integral (3.5.0 beginning with the case when D is represented by rectangle [al,b]] X [a 2 ,b 2 ] = 1 2 . L ' k k k - bk an d pOInts . k , XU]"",X k k e t pOInts Xuk -- a k,x1,.··'xnk'x x uu n•. , Unk k k form a canonic division of segment [a , b ] when h/h k ~ r < +oc for h ~ 0, when h = max{h], h 2 ], h k = (b k - ak)/(n k + 0, k = 1,2. Let us consider the following quadrature sum for the integral (3.5.0:
(3.5.3)
where
2 r km -_ f(] iij XUk , XUm ' okm) ij , ,
1)2 + (2 2)2 , rijkm -__V/(] Xi - XUk Xj - XOm and
i]
and
i2
are unit vectors of the axes OX] and OX 2 , respectively.
Definition 3.5.1. A characteristic f(x u, 0) possesses property I if it is odd with respect to axis OXJ or OX~, where OX!; is given by the equation x k = x~, k = 1,2. In other words, if points (xl, xf and (xL xp are symmetric with respect to one of the axes, then
(3.5.4) where 0i = (xu - x)/Ix u - xii, i point Xu = (xJ, x~).
=
1,2, or f(x u, 0) is odd with respect to
The following theorem is true. Theorem 3.5.1. Let the characteristic f(x u, 0) of integral (3.5.0 possess property I and be continuously integrable with respect to Cartesian coordinates of points X and 0 E CT. Let there also be u(x) E H(a) on 1 2 • Then the inequality
(3.5.5)
106
Method of Discrete Vortices
holds where the quantity O(X~k' x~m) satisfies the relationship
(3.5.6) for all points belonging to the closed set F lying inside rectangle 1 2 at a nonzero distance from its boundary.
Consider the absolute value of the difference
(3.5.7)
where
uij
=
u( xl, xl),
Multiple and Multidimensional Singular Integrals
107
The expression for I P may be reduced to
where IIi} is a partial rectangle from the division 1 2 whose lower left angular point coincides with point M = (xl, xJ). Let us consider each sum from 1 AP, A = 1,2,3:
n, n,
llP~AfihaLLJl . ().J = ()
1=
n
I}
dx 1 dx2
(rijkm )
~ Afiha[64 + 4127rdCP1~ o
2
dr]
hl{i r
=Afiha[64+87T(lnLl-ln~)] =O(halnh),
(3.5.9)
where lu - uijl ~ A fi ha and Ll is the diameter of the integration domain. Hence, we have
Method of Discrete Vortices
108
< A h . 47T -
I
1 ( vhf
+ h~ a
=
a
3h
in
2 3 h A . 27Th I [ Q' vhf + h~
dr
dr)
1~
-l - + 2 r a hili (r)2-a
4 (-1
+ -- -- + 1-
Q'
ill
a
(3.5.10)
where
-<
(rIJkm
km + h V!f L, )/rIJ -< 3
for any k, m, i, j. Finally, for 13 P one gets
(3.5.11)
In order to analyze the behavior of difference 2 P we note that the position of M(x!, xJ) with respect to each of the points M o = (X6k' xJm) has the following property. If point M o is assumed to be the origin of coordinates and one draws through it axes which are parallel to the original coordinate axes, then for any point Mij from the neighborhood of point M km there exists points Mi,j, symmetric to point M;j with respect to both point M km and each of the axes. Because function f(x o, 0) possesses the property I, its values at points M I and M 2 positioned symmetrically with respect to either axes OXJ, OX~ or point M o are equal in magnitude but opposite in sign. Thus, we have
(3.5.12)
Multiple and Multidimensional Singular Integrals
109
where F/m is a set composed of all the rectangles lying within the 8 neighborhood 0(8, M km ) of point M km , whose left lower angles form a set symmetric with respect to both the point itself and the axes OX6k or OX(~m' Thus,
2P $
L
if
fkm 2
tm (r km )
dx 1 d.x 2
-
(3.5.13) where cfJfm = II \ U s"n s,1' IT"I C n \ 0(8, M km ), max(X',X2)E j2lu(xl, x 2 )1. In view of the properties of function f(x o, 0) we have 2P $
LCl(2..7T4V2h 8 2 7T4V2h 8 (8-hV2)
+ O(h»)
=
and
L
=
O(h). (3.5.14)
because an analysis of the latter sum entering Formula 0.5.13), similar to that done when calculating 12P and 13P' allows one to conclude that the sum is of the order of h. The preceding estimates of liP, i = 1,2,3, and 2 P demonstrate the validity of Inequality 0.5.5). Note that Inequality (3.5.5) remains valid if the rectangle 1 2 is substituted by an arbitrary closed limited plane region D whose boundary has zero two-dimensional Jordan measure. If such is the case, then the quadrature sum must be composed as follows. Consider rectangle II incorporating region D and divided by straight lines parallel to the coordinate axes and separated from each other by the distance hi' The rectangle fully belonging to region D will be numbered and denoted by n i , i = 1, ... , n. By X Oi we denote the center of a rectangle ni , and by Xi we denote the left lower angle of the rectangle. The points thus selected are used for constructing the quadrature. However, if characteristic f(x o, 0) does not possess the property I, then Inequality (3.5.5) may be invalid for an arbitrary relationship between hi and h 2 • For example, function f(x o, 0) = cos 2cp (where cp is the polar angle of point X with respect to point x o) does not possess the property I, and for it Inequality 0.5.5) is valid for hi = h 2 only. In fact, cos 2( cp +
Method of Discrete Vortices
110
7T/2) = cos(2 cp + 7T) = - cos 2 cpo On the other hand, all the points M ij lying within a circle centered at point M km may be divided into pairs in which a point is obtained from another point by means of rotation through angle 7T/2 about point Mkm • Inequality (3.5.5) is also likely to be invalid for the function f(x o, 0) = cos4cp for hi = h 2 = h. In fact,
f
K
cos4cp -dx l dx 2 2
r
=
0
(3.5.15)
'
where K is a circle of radius R centered at the origin of coordinates. Consider the square [2 circumscribing the circle K, whose sides are parallel to the coordinate axes. Next consider a division of [2 into small squares whose sides are equal to h and whose centers coincide with respective origins of coordinates. We assume that
Calculations carried out for hI = R/5 and h 2 = R/l0 have shown that
V,71
=
-
8
+ 0.552,
Vh2
=
-
8 + 0.486.
3.6. EXAMPLES OF CALCULATING SiNGULAR INTEGRALS 1.
The following equality is valid:
x o E(-I,I). (3.6.1) The latter integral was calculated by using the quadrature formula
[Ax oj ) =
n~h
L
i= I
X Oj -
,
j=I, ... ,n+l,
(3.6.2)
Xi
for equally spaced grid points with the step h = 2/(n + 1). The calculated results are presented in Figure 3.2, where the solid line
Multiple and Multidimensional Singular Integrals I(~)
III
t--r--r--.-*",
2,01--+--+--+--t
1,01---h~+--+---1
f/J - 8 - 6 -0,/1. - 2
1---+--it--t---+--t-2,0
FIGURE 3.2. Calculation of singular integral in (3.6.1) with the help of quadrature formula (3.6.2). The solid line corresponds to the exact value, x x to h = 2/11 and 00 to h = 2/17.
2.
corresponds to the accurate value, and h = 2/11 and 2/41, respectively. For an integral of the form
I(xo'zo)
1
=-2 7T
fl f1 -1
- 1
X X
(1 - x)(1 - z)
and
00
correspond to
dxdz
------ ------- =
(1+x)(1+z) (xo-x)(zo-z)
1
,
(3.6.3) calculations were carried out with the help of equally spaced data. The results are shown in Figure 3.3, where the solid curve corresponds to the accurate value X X and correspond to h = 2/9 and 00
Method of Discrete Vortices
112 1(:&0'2.0>
t
z-o,OII1#'
•
~6
•
• •
•
1.2
•
•
0
•
0
0
0
0
0
0
0
•
Q,S
FIGURE 3.3. Calculation of singular integral in (3.6.3) by using equally spaced grid points for xI) = 0.0441. The solid line corresponds to the exact value, x x to h = 2/11, and 00 to h = 2/17.
1(:IfJ,zo)
0
:&-0,6
•
1,6
•
•
0 u
-1,0
-0,8
0
0
•
• 0
0
-0.6 -0.4 -0.2
0
.
•
~2
0
0
0
0
0
o
c
a8 o
FIGURE 3.4. Calculation of singular integral in (3.6.3) by using equally spaced grid points for xI) = 0.6. The solid line corresponds to the exact value, x X to 8 points and 00 to 16 points.
2/17, respectively (for X o = 0.0641), and in Figure 3.4, where the solid curve corresponds to the accurate value and X X and correspond to 8 and 16 points, respectively. For calculating we used the formula
00
1 I nj , n2(x Oj '
zOrn) =
n,
n2
L L
-2 7T j=lk=l
(1 - x j )(1 -
zd
(l+x j )(l+zd
(3.6.4)
j' = 1, ... , n l , m = 1, ... , n 2 • It was supposed that hi = h 2 = h.
3.
Calculations of integral in 0.6.3) carried out with the help of equally spaced grid points corresponding to the weight function under consideration have produced more accurate results as compared with the
113
Multiple and Multidimensional Singular Integrals I Co1w.z,)
(2
I1A
FIGURE 3.5. Calculation of singular integral in 0.6.3) by using unequally spaced grid points for an arbitrary xo' The solid line corresponds to the exact value, x x to 8 points and 00 to 16 points.
equally spaced data. We used the quadrature formula
X; =
z;
=
cos
27Ti 2n + 1
47T
a; = hi =
2n + 1
2i - 1 ,
sin 2
X Ui
7Tl 2n + 1
= ZUi =
cos
2n + 1
for n 1 = n 2 = n.
7T,
(3.6.5)
The results presented in Figure 3.5, where the solid curve corresponds correspond to 8 and 16 points, to the accurate solution, and X X and respectively, have shown that 00
(3.6.6) for any j and m. However, if the weight function is unknown or is of a complex form, unequally spaced grid points are preferable for obtaining preliminary data.
4 Poincare-Bertrand Formula
4.1. ONE-DIMENSIONAL SINGULAR INTEGRALS In Muskhelishvili (1952) the Poincare-Bertrand formula was derived. It regulates the sequence of integration in one-dimensional singular
integrals: dt
7 -
2 7T
7 -
to), (4.1.1)
where L is a piecewise smooth curve, and function
=
(4.1.2)
where
n(t,7)
=
Olt-ck l a '17-ck ll:\
k~1
where c k' k (Xk
+
13k
<
=
1, ... , n, are all the nodes of the curve L,
(Xk' 13k
~
0, and
1.
In this section we prove Formula (4.1.1) for a piecewise Lyapunov curve L containing only angular nodes by means of singular integral quadrature formulas. Let L be a segment [a, b). Then (4.1.2) becomes
(4.1.3) 115
Method of Discrete Vortices
116
where cp*(t,T) E H(a)on [a,b] I, and ILl + ILz < 1. Consider the integral A(to)
=
f
b
a
dt
X [a,b], vk,ILk ~
fb cp( I, T) dl
--
t - to
, a
T -
0, k
=
1,2, VI + Vz <
toE(a,b).
1
(4.1.4)
Let us choose two sets of points, E = {T k , k = 1, ... , n} and Eo = {tj' j = 1, ... , n}, on the segment [a, b], forming such a canonic division of the segment [a, b] with the step h that point to belongs to the set E for = k o. Consider the sums
k
(4.1.5 )
where I::1l j =
tJ.T k
= h, k, j = 1, ... , n, and ( 4.1.6)
Then taking into account (1.3.14) and (4.1.3), one gets
+ T/(t j , 1 +
VI
+ vz, 1 + ILl + t-tz)h ]O(lln hi). (4.1. 7)
After removing the brackets and taking into account that point 10 is fixed within (a, b), we see that all the resulting sums are of the order of h A
Poincare-Bertrand Formula
117
(where A > 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
Method of Discrete Vortices
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
a
dt v (t - a)'
+
1 b
(b+l o>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
Method of Discrete Vortices
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.
Poincare-Bertrand Formula
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
Method of Discrete Vortices
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
Poincare-Bertrand Formula
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)
Method of Discrete Vortices
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<;< shown to be valid although the calculations will, naturally, be more complicated. Note that if function cp(l, T) has the form (3.1.12) and points Tf., k p = 1, ... , n p and 1/', jp = 0,1, ... , n p ' form a canonic division of u~ closed Lyapunov curv~ L p ' P = 1,2 with the step h p , then (4.2.13)
k p = 1, ... ,n~, P = 1,2, where the quantity O(Ti"T;) satisfies inequalities of the form \3.3.4) and (3.3.5). This statement may be proved similarly to Inequality (4.1.24).
Part II· Numerical Solution of Singular Integral Equations
5 Equation of the First Kind on a Segment and / or a System of Nonintersecting Segments
5.1. CHARACTERISTIC SINGULAR EQUATION ON A SEGMENT: UNIFORM nMSION* Consider the characteristic singular equation
f
b
a
cp(t) dt - - =/(to)· to - t
(5.1.1)
According to Gakhov (1977) and Muskhelishvili (1952), the index 0,1, -1 solution to the latter equation is given by the formula
K =
(5.1.2)
where
VI
= 1, V o =
V-I
= 0, and
/2 -t
Ro(t)
=
--,
t - a
1
RI(t)
=
y(b - I)(t - a)
,
'This means equally spaced data or equally spaced grid points (G. Ch.).
127
128
Method of Discrete Vortices
In what follows we do not consider especially solutions with a singularity of the form a)/(b - t) because the alterations to be done in this case will be seen clearly. Let the sets E = {t b k = 1, ... , n} and Eo = {tOj' j = 0, 1, ... , n} form a canonic division of the segment [a, b] with the step h. The following theorem is true.
V(t -
Theorem 5.1.1. Let junction f(t) belong to the class H( A) on [a, b]. Then, between solutions to the :,ystems of linear algebraic equations,
(t )h n 'P k t Oj - t k
n
L k=l
=
f(lo)'
j
= 1, .. . ,n;
(5.1.3)
j=I, ... ,n-l, n
L
'Pn(tdh
=
(5.1.4)
C;
k~l
j=O,I, ... ,n,
and the index K inequality holds:
=
(5.1.5)
0, 1, - 1 solution 'P(t) to Equation (5.1.2), the following
k = 1, ... , n.
(5.1.6)
In (5.1.6), the quantity 0n(t k ) satisfies: 1.
The inequalities (5.1.7)
2.
for all points t k E [a The inequalities
+ 8, b - 8], where 8>
°
is however small;
n
L
0n(tdh
k=!
for all points t k
E
[a, b].
:0;
O(h A2 ),
Az
< 0,
(5.1.8)
Equation of the First Kind on a Segment
129
Proof. Note that due to results presented in Section I.3 for quadrature formulas for a singular integral on a segment, Systems (5.1.3)-(5.1.5) approximate Equation (5.1.1). For the determinants of Systems (5.1.3)-(5.1.5) one gets K =
0,1, - I,
1 /1(n)
tOI-t,
to,-t n
1
1
=
o
.
A(n) _ u] -
(5.1.9)
It can be readily shown (Proskuryakov 1967) that
(5.1.10) whcre ?(x) = I for x > 0 and ?(x) = 0 for x:s O. Let us prove this statement for fj,\~), because for the remaining two determinants it can be proved analogously (with some evident alterations). Subtract the bottom row of determinant t1(~) from all the preceding rows and take out factor Ij(tOn - lk), k = I, ... , n, from all the columns
130
Method of Discrete Vortices
and factor
tOn -
tOm'
m = 1, ... , n - 1, from all the rows. Then we have 1
1 1
Now we subtract the last column of the last determinant from all the preceding columns, take out common factors from each row and column, and expand the resulting determinant in the last row. Then we get
The proof of the formula for Ii(~) is terminated by emplqying the method of mathematical induction. From (5.1.10) it is seen that DLjpfor any n. By employing the Cramer rule for solving a system of linear algebraic equations, one gets
(5.1.11) where V is given by (5.1.2) and t(x) is given by (5.1.10); D~~l and lir;,)k are obtained from D~n) and lir:), respectively, by replacing the kth column by the column of free terms from the right-hand side of the system, and Ii(:(i,k) is obtained from Ii<;) by deleting the jth row and the kth column. Analogously, one gets for lir;?i, k)' K
nn
lirn, KU,k) -
nn
(t m - t p) (top - tOm) 15m
n1:( - n
m = I -
m*j
K) P ~ I
p*k
(tom -
tp )
(5.1.12)
Equation of the First Kind on a Segment
131
Hence, Equation (5.1.10 becomes
k=1, ... ,n,
(5.1.13)
where
l(n) I,k
1 _ t'
= len) O,k t
l(n)
I,Oj
= len) (t - t ) O,Oj On OJ'
k
On
l~n?k = lll:'l(tk -
l~nl,Oj = I(I~Jj-t-_-t-
tOO)'
OJ
00
Next we get 1 h
-l(n)
Pk _
I =
0
k-I (
1+
m=1
t
O,k
-t
Om
1
-p P 2 k-I n-k'
=
)
m,
tm-t k Po 1
1 2
_len) - _p
h
PO,j-
I =
1.
0
I
O,Oj -
(
1
+
m= I
PO,n
I =
n (1
m~j+ I
OJ-I
lOrn l(l} -
+
tm
tOm
P
)
=
1,
(5.1.14)
D,n-j'
,
m
lOrn - 1 IOj - lOrn
),
PO,O
= 1.
(5,1.15)
Method of Discrete Vortices
132
Remember that in the case under consideration, tk
a
=
b-a
+ kh,
h=-n + 1'
(5.1.16) Hence, in accordance with (5.1.14)-(5.1.16),
n
_
Pk
k-I (
1 -
1/2 ) 1- ,
Pn -
m
m=1
II
n- k (
k
=
1
1/2 )
+ ---;;; . (5.1.17)
The work of Hardy (1949) gives the following formula pertaining to the theory of gamma functions:
(I + /3)(2 + /3)
(n
+ /3)
n
1 ·2·
n(3 1'(1 + /3) + O(n(3- I). (5.1.18)
The latter formula may be presented in the form
fI (1 + ~) m
m~1
=
n (3 + O( n (3 I). 1'(1+/3)
Because we need quantities on the order of n in Formula (5.1.18), we can write
n 1/
(
1
m=1
By putting /3
±
=
Pn
/3) +m
(n+1)(3 + O(n + 1)(3 [(1 + /3)
I).
(5.1.18')
+O(n-k+l)-1/2)
(5.1.19)
~, we get
(n - k
-
=
k=
+ 1) 1/2
1'(3/2)
The ~ork of Fikhtengoltz (1959) gives the formulas 1'(1/2)
=
r:;,
1'(a
+ 1)
=
f(a)a,
1'(3/2)
=
r:; /2.
Equation of the First Kind on a Segment
133
Thus,
2/n-k+l + O(k k
Pk - lPn - k-- -
1/2(n -
k + 1)-1/2)
7T
( 5.1.20)
In a similar way,
n
j- I ( PO} - I =
1+
m=1
2 7T
1/2 )
-
m
n
n -} (
,
1/2 ) m
1--,
m=1
1)-1/2( n- j +-1)
j + 1/2 (( +0 j+-2 n + j + 1/2
+0((j+~t2(n-j+~)
1/2)
2
/
2).
3
(5.1.21)
Note that using (5.1.16) we have
~
Vb -
to}
=
Jh(t -
=
Jhfz - j +
k +
1), 1/ 2}
,;t;=a
V
tO} -
=
/hk,
vhf + 1/2)-
a =
(5.1.22)
In accordance with the latter formulas, Equations (5.1.20) and (5.1.21) may be written in the form (5.1.23)
h(b-t dI/2 + 0 ( (t - a) 3/ 2 k
j '
134
Method of Discrete Vortices
BO(n,j) =
o( V
h
a)(b - t Oj )
(tOj -
h(tOj-a) 1/2 +0 (
Similarly, for
A
K =
h
B
([(t
-0
(n)
k
1
_ _ [(
IK,Oj -
7T
. =0
Kln,Oj)
)
3/'
(5.1.24)
.
(b-t(ljY -
1, -1 we have
K(n,k) -
1
_
)
(
t Oj
_
_
h)
a)(b - tdf/2-?1
a
)(b _
t Oj )
] K /'-
K)
+ BKlll,Ojl'
h)
[ (tOj -
a)( b - t Oj )
(5.1.25)
,
]3/2-(lKl
(5.1.26)
•
By substituting (5.1.23)-(5.1.26) into (5.1.13) and acting in a way analogous to the method used when proving Theorem 1.3.2, one gets
k
=
1, ... , n,
(5.1.27)
where the quantity IOK(tk)1 satisfies Inequalities (5.1.7) and (5.1.8). Thus, Theorem 5.1.1 is proved. • Note 5.1.1. Let us consider in greater detail the unknown YOn appearing in System (5.1.5). If the system is devoid of YOn' then the number of equations in it exceeds the number of unknowns and the system, as a rule, has no solution. On the other hand, the system must approximate Equation (5.1.1), which has a unique solution with index K = -1 if the condition
f
a
b
f(to) dt o ..j(t o - a)(b - to) -
-,====.=:=- - 0
(5.1.28)
is met (Gakhov 1977, Muskhelishvili 1952). Therefore, the level of disagreement in System (5.1.5) with zero YOn must decrease. Hence, if the
Equation of the First Kind on a Segment
135
system with YOn has a solution, then lim n -n' YOn = O. However, the statement has to be proved. Thus, the variable YOn makes System (5.1.5) well-posed and, therefore, will be called a regularizing factor. To find YOn we use the Cramer rule again:
YOn
=
n I l "_l(n) .f(t .)h = -
.f.... J~O
h -
I,0J
f( t )h
n
+ O(X I / 2 ).
OJ
"
7T.f....
OJ
J~O
_/(
V
. t OJ
a
)(b - t .) OJ
(5.1.29) Thus limn _~ x YOn = 0 if and only if the index K = - 1 solution for Equation (5.1.1) does exist. Hence, the behavior of YOn can be used as an indicator of the existence of the index K = - 1 solution. Note 5.1.2. In applications (such as aerodynamics, elasticity, etc.), it is often required to calculate not the function cp(t} itself, but the integral fabl/J{t)cp(t) dt, where l/J(t) E H on [a, b]. From Inequalities (5.1.6) and (5.1.8), it follows that the preceding integral may be calculated by using the rectangle rule with respect to points t k , k = 1, ... , n, taking at the points not the function cp(t}, but the values CPn(t k ) for which the inequality A3 > 0, (5.1.30) holds. In fact, one has
It
l/J(t)cp(t) dt -
k~l l/J(tk)CPn(tdhl
$lf+hl/J(t)cP(t) dtl
+
Itt
hl/J(t)cp(t) dt -
k~l l/J(tdcp(tk)h
k~l l/J(tk)( cp( td -
+I
I
CPn(td)h I
n
$ O(h A4 ) + M
L
OAtdh
$
O(h A3 ),
k=l where M
=
max 1l/J(t)l. lEla. b]
Method of Discrete Vortices
136
Note 5.1.3. Theorem 5.1.1 remains valid in the case where function f(t) belongs not to the class H on segme nt [a, b], but to the class H *, i.e., has the form
f(t)
=
l/J( t) (t _ a)v(b - t)I'-'
(5.1.31)
where l/J(t) E H on [a, b]. In this case Theorem 5.1.] remains valid for System (5.1.3)for 0 :5; v < 1,0 :5; /L < t for System (5.1.4)for 0 :5; v, /L < 1 and for System (5.1.5) for 0 .:5: v, /L < ~. The fact that under these restrictions Functions (5.1.2) are solutions to Equation (5.1.1) is a consequence of the Poincare-Bertrand formula. Note 5.1.4. When solving the problem of steady flow past an airfoil with a flap, one has to consider the situation when function f(O E H on segments [a, q] and [q, b] and suffers discontinuity of the first kind at the point q. If the sets E = {t k , k = 1, ... , n} and Eo = {t Oj ' j = 0,1, ... , n} are chosen in such a way that the point q lies in the middle between the nearest points belonging to sets E and Eo (see Note 1.3.2), then Theorem 5.1.1 remains true, but Inequality (5.1.7) holds only for all P9ints t k E [a + 8, q + 8] U [q - 8, b - 8]. In fact, for Theorem 5.1.1, the relative positions of points t k , k = 1, ... , n, and t Oj ' j = 0,1, ... , n are of importance. Also, for these points the quadrature formulas for a singular integral on a segment analyzed in Section 1.3 must be valid.
The results of Section 1.3 and Notes 5.1.3 and 5.1.4 aIlow us to formulate the foIlowing theorem. Theorem 5.1.2.
Let function f(t} be of the form
(5.1.32) where l/J(t) E H on [a, b], a < q < b. Also let sets E and Eo be chosen on segment [a, b] with the step h = (b - a)/(n + i) as indicated in Note 5.1.4. Then Relationship (5.1.6) is valid in the following cases:
1.
2. 3.
For 0 .:5: v, {3 < 1 and 0 :5; /L < t between a solution to the system of linear algebraic equations (5.1.3) and the index K = 0 solution of Equation (5.1.0 in Equation (5.1.2); For 0 .:5: v, {3, t-t < 1 between a solution to the !.ystem of linear algebraic equations (5.1.4) and the index K = 1 solution of Equation (5.1.1); For 0 .:5: v, t-t < ~ and 0 :5; f3 < 1 between a solution to the !.ystem of linear algebraic equations (5.1.5) and the index K = - 1 solution of Equation (5.1.1) in Equation (5.1.2). In Equation (5.1.6) the quantity
Equation of the First Kind on a Segment
137
O(t k ) for all points t k E [a + 8, q - 8] u [q tive 8 is however small) satisfies inequality
+ 8, b - 8] (where posi(5.1.33)
and Inequality (5.1.8) for all points t k
E
[a, b].
Note that if f(t) suffers a discontinuity of the first kind at the point q, then it may be presented in the form (5.1.32), where positive f3 is however small. In this case Solutions (5.1.2) to Equation (5.1.1) have a logarithmic singularity at point q. However, calculations show that good results may be obtained if point q coincides with one of the points t Oj for j = jq, and the right-hand side in Systems (5.1.3)-(5.1.5) is chosen in the following way: fU oj ) for j i= jq and [fU O} - 0) + f{to} + 0)];2 for j = jq, where f(t o} 0) and f{to} + 0) are one"-sided limits ~f the function fUo) at point q: From (5.1.2) the index K = 1 solution cp(t) to Equation (5.1.1) is seen to depend on an arbitrary constant C, which is an integral of the solution over segment [a, b]. However, according to Formula (5.1.2) the constant and hence, solution cp(t) are uniquely defined if cp(q) is fixed at a point q E (a, b). In order to find a numerical value of the solution from System (5.1.4), one has to find C q and substitute it into the last equation of the system. Finding the corresponding constant Cq is a tiresome business, the more so if the equation incorporates a regular part. In the following text, we will show how one can do without the constant C q , by using function cp( q) only. The following theorem is true. Theorem 5.1.3. Let f(t) belong to the class H on [a, b] and the sets E and Eo form a canonic division of the segment [a, b] with the step h. Let us denote by t k the point of the set E that lies at the shortest distance to the left from point q" E (a, b). The Relationship (5.1.6) holds between a solution to the system of linear algebraic equations j
and the index
K =
=
1, ... , n - 1, (5 .1.34)
1 solution cp(t) to Equation (5.1.1):
+ cp(q)
(q - a)(b - q)
(t-a)(b-t)
(5.1.35)
138
Method of Discrete Vortices
Proof. Similarly to Theorem 5.1.1 one may show that the determinant of System (5.1.34) is nonzero and apply the Cramer rule
h
k=l, ... ,n,ki=kq • (5.1.36)
By denoting the first sum in Formula (5.1.13) for write
1 h
+ _len)
Ln
I,k. J= I
K =
1 by S I(t k ), one may
1 cp(q)hh _len) - - - h I,O} ( _ )2 t kq to}
By employing the procedure used when proving the Poincare-Bertrand formula, one gets (q-a)(b-q) (t k - a)(b - td
+0
(
l/2
h V(tk - a)(b - t k )
)
. (5.1.37)
Then taking into account that
_fb V(to a
a)(b - to) dt o
tkq-t O
+
t a
V(to - a)(b - to) dto tk-t O
Equation of the First Kind on a Segment
139
we get
IM,I
$
[
Icp(q)lh +
hlcp(q)1 t k - t kq
V(t
k -
h l/2 a)(b - td
]
O(lln hi). (5.1.38)
The validity of the proof of the theorem follows from Formulas (5.1.37) and (5.1.38) and the procedure used for proving Theorem 5.1.1. •
In the case of the problem of flow past an airfoil with ejection of or sucking air (considered in greater detail later), it is necessary to solve numerically Equation (5.1.1) in the class of functions having the form l/J( t) cp(t) = - - , q - t
where l/J(t) E H in the neighborhood of point q E (a, b). In the subsequent Theorems 5.1.4 and 5.1.5, the sets E and Eo will be taken in such a way that the point q would be located within the set Eo for j = jq. Theorem 5.1.4. Let function f(t} E H on [a, b] and the value of function l/J(t) be known at point q. Then, between a solution to the system of linear algebraic equations
j = 1, ... ,n,j =l=jq,
(5.1.39)
and the solution cp( t} to Equation (5.1.1),
cp(t)
=
-
1
-2 7T
/2- [l b/B;0 t t - a
--
a f(to) dt u b - to t - to
-- --- -
a
2 7T -q---t
18-
a l/J(q) ] , -b-_-q
(5.1.40) Relationship (5.1.6) exists in which the quantity O(t k ) satisfies Inequality
Method of Discrete Vortices
140 (5.1.7) for all t k however small.
E
+ 8, q - 8]
[a
U
[q
+ 8, b - 8], where posilive 8 is
Proof. Similarly to the preceding theorems, we will show that the determinant of System (5.1.39) is nonzero for any n. Therefore, by applying the Cramer rule to the system, one gets
t ) 'Pn( k
=
-
1 h
O,k
"
-
t... h
}~l
+2
_
q
t
t... h
j ~ I
k
(n)
I .
[0, OJ I. _ }q
I
-q
_k__
r/J(q) I - I k
1
"_[(").
q -
1
n -
[n
1
_[(n) _ _
}q
OJ
t.
O,O}
(q-t OJ )f(t OJ )h t - I k
n
f( I(l] )h -
n
1
_[(n)
(
O,Oj
hZ
1
.1,( ) " _len) 'I' qt... h O,Oj-(--_--)-::-z J~l I jq 10j
O}
L h j =1
OJ
1 t - t . O}
k
t jq
1
-
t Oj
)h].
(5,1.41)
Now by using the proofs of Theorem 5.1.1 and the Poincare-Bertrand formula we get
where the quantity 1O(tk)1 satisfies the necessary relationships. In a similar way the following theorem may be proved. Theorem 5.1.5. Let function f(t) E H on [a, b]. Then Relationship (5.1.6) holds just as in the case of Theorem 5.1.4, between solutions to the system of linear algebraic equations
j=O,I, ... ,n,ji=jq,
j
=
1, ... ,n,j i=jq,
j
=
n
(5.1.42)
n
L k=\
'Pn(td h
=
C,
+ 1,
(5,1.43)
Equation of the First Kind on a Segment
141
and solutions to Equation (5.1.0, respectil'ely, 1 J(t - a)(b - t)
cp(t)
=
-
-2 7T
fb
(q - to)f(/o) dt o _/
q-t
a
V{to-a)(b-to)(t-t o )
'
(5.1.44)
(5.1.45) where for the latter case
t
(5.1.46)
cp( t) dt = C.
o
Note that when considering Theorem 5.1.4, one can assume that the required solution is known not at the point t j , but at point t j + I. Calculated results presented in Figure 10.5 confirm this conclusion. '/ Because function fjJ(t) belongs to the class H in the neighborhood of point q, the systems mentioned in Theorems 5.1.4 and 5.1.5 may be constructed as follows. First we choose sets E and Eo on segment [a, b] that form the latter's canonic division. Point t Oj is assumed to coincide with a point from the set Eo located at the shortdst distance from point q. Then the systems have the same form. Estimates between the corresponding accurate solutions to Equation (5.1.0 and solutions to the systems preserve their character. As far as a wing may incorporate a number of air intakes, the previously formulated results may be naturally extended onto the case of finding numerical solutions to Equation (5.1.0 of the form fjJ (t) cp(t)
=
(q]-t)···(qm- t )
,
(5.1.46')
where ql' ... ' qm are fixed points in the interval (a, b) and fjJ(t) is a function belonging to the class H in certain neighborhoods of the points. Then the following theorem is true. Theorem 5.1.6. Let function f(t} E H on [a, b] and the values of the function fjJ(t} be known at the points q], ... , qm. Then, Relationship (5.1.6)
Method of Discrete Vortices
142
holds between a solution to the system of linear algebraic equations
L k I ~
k 1= Jqr .
CPn(tdh
cp(q,)h
t oJ' - t k
(t().J - t Jq , )(t().Jq , - tJq , )
- - - = f(to) - - - - - - , - - - - - : -
. . 14m
j
=
1, ... ,n,j i=jq, ... ,jq""
(5.1.47)
and solution to Equation (5.1.0,
cp( t)
=
_
~2 7T
+
-I b -
t t - a
fb Vto a
a f(to) dt o b - to t - to
-I bt -- at [V bql - - q
a I
t/J (qd q, - t
+ ... +
~
Vb -
t/J( qm) ], q In qm - t (5.1.48)
where the quantity OCt k) satisfies Inequality (5.1.17) for all points t k E [a + 8, ql - 8] U [ql + 8, q2 - 8] U ... U [qm + 8, b - 8], where positil'e 8 is asufficientlysmalinumber,andInequality(5.1.8),whereql
However, it should be noted that points t k , k = 1, ... , n, and t oi ' j = 0,1, ... , n, are chosen as in the case of Theorem 5.1.4 if points ql,"" qm are equally spaced from each other or are as in the case of the note to Theorem 5.1.5. The fact that the Functions (5.1.40), (5.1.44), (5.1.45), and (5.1.48) are solutions to Equation (5.1.1) may be checked straightforwardly. Let us next consider Equation (5.1.1) on segment [-b, b], which is symmetric with respect to the origin of coordinates. In other words, we consider the equation
f
b
-b
cp(t) dt _ - - - -f(to)· to - t
(5.1.1')
Equation of the First Kind on a Segment
143
According to Gakhov (t 977) and Muskhelishvili (t 952), all the solutions to Equation 5.1.1') are given by the formula
cp( t)
=
It is quite evident that if fU o) is an odd function, then a solution to Equation (5.1.1') is an even function, and if fU o) is an even function, then a solution to Equation (5.1.1 /) is an odd function for C = 0 only. This circumstance can be employed for decreasing the order of a system of linear algebraic equations used for solving Equation (5.1.1/) numerically, by a factor of 2. Theorem 5.1.7. Let f(t) E H on [-b, b] and be an even function on the segment. Then, Relationship (5.1.6) holds between a solution to the system of linear algebraic equations,
j
=
1, ... ,m, (5.1.50)
and solution cp(t) of Equation (5.1.1 '),
cp( t)
(5.1.51 )
=
Here the sets E = {t k , k = 1, ... , 2m + l} and Eo = {t OJ ' j = 0,1, ... , 2m + l} form a canonic division of segment [ - b, b] with the step h. Analogously, if fU o) is an odd function and n = 2m, then the system
j=I, ... ,m-l,
1
m
L k~1
CPn(td h
=
2C
(5.1.52)
Melhod of Discrete Vortices
144
must be taken. If one desires to obtain the index the system
K =
-
1 solution, then
j=O,1, ... ,m-1, (5.1.53) must be considered. As an application of Theorem 5.1.1, we consider the issue of solving numerically the equation
f
l
-I
cp'(/) dl I0
-I
(5.1.54)
=f(to),
whose solutions are all given by the formula
C 1[arcsin I + 2 7T] + C 2 , + --:;; cp( -1)
=
C 2 , cp(1)
=
CJ + C2 •
(5.1.55)
Let us present the construction procedure for a numerical method for obtaining a solution to the equation which is zero at the ends of a segment -a situation most often occurring in applications. Let points I J = -1, 12 , ••• , t n In + I = 1 divide segment [ -1,1] into equal parts h long and point IOi be the middle of segment [t i , Ii + J]' j = 1, ... , n. Theorem 5.1.8. Lei function f(to) belong to the class H on [ -1,1]. Then between a solution to the system of linear algebraic equations
n
L CPn(tod k J ~
(1 I
0i
j
- I k
=
1, ... , n, ( 5.1.55
I )
145
Equation of the First Kind on a Segment
and the solution 'P(t) of Equation (5.1.54) vanishing at the ends of segment [ - 1, 1], the relationship
k
1, ... , n,
=
(5.1.56)
holds where A > O.
Proof. System (5.1.55 ') is equivalent to the system
n;"1
'Pn(tod - 'Pn(tOk-
~
dh
=
h
k=1
0
'
j
=
1, ... , n,
j
=
n + 1, (5.1.57)
where 'Pn(t o 0) = 'Pn(t on t I = O. The latter system of equations is seen to coincide with System (5.1.4) for C = O. Therefore,
k=l, ... ,n+l,
or 'Pn(tOk)
=
-
kIn 1 _[(n+')h L _len
L
i=1
h
I,k
j~1
h
'.1) I,OJ
f( t )h OJ
t - t .' k
k
=
1, ... , n, (5.1.58)
OJ
because 'Pn(t o 0) = 'Pn(t On t I) = O. Hence, by Formulas (5.1.25) and (5.1.26) we have
k
=
1, ... , n.
(5.1.59)
Method o:Discrete Vortices
146
This formula proves the theorem because function
(5.1.60)
gives the required solution to Equation (5.1.54).
•
Note that the equality I I n L _lent ')h L . h 1,/ .
n
t
I~
J= I
I
1 h
-I(n~ I) LOj
f( t )h OJ
I· /
t .
=
0
(5.1.61 )
OJ
holds. Also note that from the proof of Theorem 5. Ht follows that (5.1.55') is not an ill-conditioned system, and the mail inverse with respect to the system's matrix is given by Formula (5.1,'!;. This may be demonstrated straightforwardly by using the Hadamard cII:rion according to which the absolute value of a diagonal term in each I'>' 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
Method of Discrete Vortices
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 f
j=I, ... ,n-l, n
L
I/fn ( t k )a k = C,
j=n
(5.2.2)
k=1
[where a k = 1T/n; t k = cos(2k - l)7T/(2n), k = I, ... , n; t Oj = cos j1T/n, j = 0, I, ... , n - I] and the function
l/J(t) =
I __ 1T 2
JI -I
~f(to)dto + -, C
~----
t - to
1T
(5.2.3)
the relationship
(5.2.4) "This means unequally spaced data or unequally spaced grid points (G. Ch.).
Equation of the First Kind on a Segment
149
holds, where Rn(t k ) is the error of approximation of the singular integral appearing in Equation (5.2.3) at point t k by using the interpolation-type quadrature formula constructed for points t Oj , j = 0,1, ... , n - 1, in Section 2.3 (see Equation (2.3.6». Proof. The analysis carried out when proving Theorem 5.1.1 shows that a solution to System (5.2.2) is of the form 1
___
r/J,,(td -
ak
(n)
lu
[n - 1 1 O_j_ _ L 1',Oj f(t) _ C , (n) _ _
j~ 1
ti
k=I,2, ... ,n. (5.2.5)
t(lf
The expressions for l[,n2 and 1i.~lj may be conveniently presented in the form I(n) = I, k
(5.2.6)
By representing the polynomials as a product of linear factors, one gets (see Section 2.3) n
Pn(t) = Tn(t) = En
n
m=1
(t - t m ), n-I
n (t -
Qn(t) = -7TUn _ l (t) = -7TEn
tOm),
m=\
n
P;(tk) = En
L
(t k - t m ),
m=1
m*k n-I
QnCtd
=
-7TEn
n (t k -
m~
tOm),
(5.2.7)
1
where Pn(t) is a polynomial whose roots are used to construct a quadrature formula for the singular integral entering the equation Qn(t) is a polynomial whose roots are collocation points, and En is a coefficient before the senior power of the variable in P/t). Thus, from Equations (2.3.7), (5.2.6),
Method of Discrete Vortices
150
and (5.2.7), one has k
=
1, .. . ,n.
(5.2.8)
Note that the equality of the coefficients before the senior powers of the variable in polynomials Tn(1) and Un _ ,(t) follows from the formulas n inl21 (-1)m(n - m - 1)1
T (t) n
=
" 2m~O
-
(2t)n
m!(n-2m)!
In I 21(-1)m(n-m)1
Un(t) =
L
m~O
m!(n - 2m)!
2m
n=1,2, ... ,
2m
n = 1,2, .... (5.2.9)
(2t)"
Consider integral
(5.2.10)
In this case (see Equations (2.3.17) and (2.3.20) P" _,(t) = Un l(t) and QIl ,(t) = 1TTn (t). Therefore, the quadrature for
j=I, ... ,n-l. (5.2.11)
From Equations (5.2.6), (5.2.7), and (5.2.12), it follows that
j=I, ... ,n-l.
(5.2.12)
By comparing the equality k
=
I, ... ,n, (5.2.13)
which follows from (5.2.5), (5.2.8), and (5.2.12) with (5.2.3), we conclude the proof of the theorem. •
Equation of the First Kind on a Segment
151
From Stark (] 971) it follows that the formula
[n-l
1 L f(t OJ.)bJ 7T2~ j~l t k - t Oj
-
7Te
1
k = 1, ... , n,
,
(5.2.14) gives an accurate value of the solution cp(t) at points t k if f(t} is a polynomial of power less than or equal to 2(n - 1). In the general case we have
Next we consider the solution to Equation (5.1.1 '): b = 1 for K = 1 and 0, i.e., for the cases when w(t} = ~ or w{t) = ";(1 - t)/(1 + t) , respectively. Thus, the following theorem is true. Theorem 5.2.2. Let function f( t) E H on [ - 1, 1]. Then between solutions of the systems of linear algebraic equations
j=I, ... ,n+l,
a
= k
f(lj =
7T k7T --sin 2 n+l n+l
tk
,
(2 - j - 1)7T cos--(-n-+-l-)2
=
k7T cos--, n + 1
k7T sin 2 - - 2n + 1 2n + 1 ' 47T
k -
2j - 1 t Oj
=
cos 2
n
+
1
7T,
k=I, ... ,n+l,
j=I, ... ,n+l;
j
a -
(5.2.15)
j
=
fk =
cos
1, ... , n,
=
1, ... , n,
2k7T 2n + 1
,
(5.2.16)
k = 1, ... , n,
152
Method 01 Discrete Vortices
and the functions fjJ(t},
fjJ(/)
1 =
--2 7T
fjJ(t)
=
_
fl
-1...jl=tf(/-t o)
~2 fl 7T
l(to) dt o
-I
'
/1
+ to l(to) dl o , 1 - to t - to
(5.2.17)
(5.2.18)
Relationship (5.2.4) holds.
Note that if one has to take an approximate solution as a function of I, then it is necessary to employ Formula (2.3.2), which may be presented in the form
where for Theorem 5.2.1, Pn(t) = Tn(t) = cos( n arccos t),
and for Theorem 5.2.2, sin[(n + l)arccos I] Pn(t) = Un(t) = - - - - - sin( arccos 1) for System (5.2.15) and
for System (5.2.16). In the preceding section it was indicated that in some applications (Belotserkovsky 1967) one has to consider a solution to the characteristic equation when its value is known at a point q E ( - 1, 1). In this case the following analog of Theorem 5.1.3 may be formulated: A solution to the system of linear algebraic equations j=l, ... ,n-l (5.2.19)
153
Equalion oj the First Kind on a Segment
(where ak,/ b k = 1, ... , n, and t Oj ' j = 0, 1, ... , n - 1, are defined similarly to Theorem 5.2.1, and k q is the number of point t k located at the shortest distance from point q) converges uniformly to the function
(5.2.20)
In fact, a solution to System (5.2.19) may be presented in the form
k
=
1, ... , n, k
i= k q •
(5.2.21)
Now we introduce the representation of t/lit k ) in the form
where quantity M 2 may be represented as M 2 = t/I(q) + D(n - 1) and IM,I.:<=;; D(n 1). Thus, if function J(t) has a limited second derivative, then It/ln(t k ) - t/I(lk)1 is also a quantity on the order of n 1. If, finally, a solution is sou ht in the class of functions cp(1) = t/I(/)/(q - I), t/I(I) = (1 - I) /( 1 + I) u(I), and the value of u(q) is known at a point q E ( - 1, 1), then the following analog of Theorem 5.1.4 may be formulated. Between a solution to the system of linear algebraic equations
j
=
1, ... , n, j i= jq
(5.2.22)
[where ak' t k , and t Oj are defined similarly to System (5.2.6) and jq is the number of point IOj located at the shortest distance from point q, v(r) =
Method o[ Discrete Vortices
154 u(t}/(q - t}] and the function V(t) =
_
~2 fl 7T
V+
1 to 1 - to
-I
[(10) dtn + t - to
u(q)
q- t'
(5.2.23)
the relationship (5.2.24) holds [if the function [(to) is smooth enough]. This statement can be proved by using Formula (5.1.41) and the procedure for proving the Poincarc- Bertrand formula. One can also formulate analogs of Theorems 5.1.5 and 5.1.6. Similarly to Note 5.1.2, we remark that by employing solutions to the systems of linear algebraic equations (5.2.19) and (5.2.22), one may properly calculate the integrals f I 1 TJ(t)cp(t) dt, where TJ{t) is a function belonging to the class H on [ - 1, 1].
5.3. FULL EQUATION ON A SEGMENT In this section we consider numerical solution of the full equation of the first kind on a segment
f
l cp(t) dt -I
to - t
+
fl -
k(lo,t)cp(t)dt=[(Io)·
(5.3.1)
I
At the start we assume that functions [(0 and k(to, t) belong to the class H on the corresponding domains of definition. By solving Equation (5.3.1) with respect to its characteristic part, we deduce that it is equivalent to the Fredholm equation of the second kind in the sense of obtaining the index K solutions (Gakhov 1977, Muskhelishvili 1952); (5.3.2) where
Equation of the First Kind on a Segment and T, =
7T,
155
To = T _ 1 = O. In addition, the condition
must be met for K = - 1. Note that the kernel NK(t, 7) has the form N (t 7) K'
------
(1-t)"(1+t)(3'
(5.3.4)
where Q' and f3 are equal either to 0 or to {, and function
L
k~l
'Pn(td h + t Oj - t k
n
L
k(toj,td'Pn(td h = f(to) ,
j = 1, ... , n,
k~l
( 5.3.5) n
L
k~l
'Pn(td h + t Oj - t k
n
L
k(toj,td'Pn(td h =f(toj)'
k~l
j=I, ... ,n-l, n
L k~l
'Pn(td h
=
C,
(5.3.6)
Method of Discrete Vortices
156
j
0,1, ... , n,
=
(5.3.7)
and the corresponding solutions to Equation (5.3.0, relationship Equation (5.1.6) holds where the quantity O(tk) satisfies inequalities Equations (5.1.7) and (5.1.8). Here the sets E = {t k , k = 1, ... , n} and Eo = {to, j = 0, 1, ... , n} form a canonic division of the segment [ - 1, 1]. Proof. On the left-hand side of Systems (5.3.5)-(5.3.7), we leave the summands corresponding to a characteristic singular integral equation; all the other terms are transferred to the right-hand side. By employing the results of Theorem 5.1.1, we deduce that the systems under consideration are equivalent to the following systems (K = 0,1, - 0: n
'Pn(td +
L
NAtk,tm)'Pn(tm)h =~ K(td,
k
1" ... , n, (5.3.8)
=
m=l
where
1
N-K ( t k , t m ) = - _len) k K.
f-I,K (t k)
=
n--{;(K)
L
K.
n
j=l-{;(-K)
1
[n-{;(K) " '-
- _len) h K,k
/(n)OJ'
j=l- {;('-K)
k(t t
t) OJ'
k -
t
m
t(t)h I(n). OJ K,OJt - t . k
h,
OJ
-
TKC
]
OJ
[for the definition of ~(x), see (5.1.10)]. We proceed with a more detailed proof for K = 0, because the other two cases are quite similar. From (5.1.27) it follows that by multiplying both sides of System (5.3.8) by the factor (l - t k )I/4(l + t k )3/4, denoting the product of the factor and 'Pn(t k ) by (frn(tk), and considering the resulting system of linear algebraic equations, one deduces that the latter system approximates the integral Fredholm equation of the second kind with a limited kernel: (5.3.9)
Equation of the First Kind on a Segment
157
where
dT
dT
= ---,.--,-----
(1 _ T)I/\1 + T)3/4'
I
d7
I
a
=
J-I (1 _ 7)1/4(1 + 7)3/4·
According to (5.1.27) the order of approximation has the form -
A
O(td$ h(t k +l)
1/4
[
+
(1 -
hl/2(1
+
+ t k )1/4
(1 - td
1/4
3j~(1 + td 3/4]O( lIn hI).
tdo
(5.3.10)
From the theory of numerical methods for integral Fredholm equations of the second kind with a continuous kernel (Kantorovich and Krylov 1952) it follows that the order of approximation (j;n(t k ) of function (j;(t) is the same. By returning to functions cp(t) and CPn(t), we terminate the proof of the theorem for the case K = O. For K = 1 and - 1 the theorem is proved in a similar way. • Note 5.3.1. From the proof of Theorem 5.3.1 it is seen that the theorem stays true if the kernel k(t o, t) is of the form k I(t o, t) lito - t Ia, where o :5: Q' < 1, and k I(t o, t) belong to the class H on a rectangle. In this case kernel NI«t, T) has the same form (this can be shown by using the Poincare- Bertrand formula for changing the order of integration and properties of singular integrals). Therefore, by passing to iterated kernels, one can derive a Fredholm equation with a continuous kernel. Obviously, all the operations may also be implemented in the discrete form. In
Method of Discrete Vortices
158
accordance with the preceding analysis, Theorem 5.3.1 may be reformulated for the equation k(to,t)cp(t) dt _ - - - - - - [(to), -I to - t
f
l
(5.3.11)
where both k(t o, t) and [(t) belong to the class Hand k(to, to) *- 0, to E [ -1,1]. In this case, the construction of systems of linear algebraic equations does not require passing to a singular integral of the form (5.3.1). Thus, to find a numerical index K = 1,0, or -1 solution to Equation (5.3.10 one has to consider, respectively, the systems of linear algebraic equations
j = 1, ... , n,
(5.3.5')
j=I, ... ,n-I,
n
L
CPn(td h
=
C,
j
=
n,
j
=
0, 1, ... , n. (5.3.7')
(5.3.6')
k=1
FinalIy, we note that the preceding method of analysis alIows us to apply alI the notes and theorems made and/or proved for the characteristic singular equation presented in Section 5.1 to Equations (5.3.0 and (5.3.1 1). The analysis may be simplified if one applies the methods of numerical solution developed for the characteristic singular equation in Section 5.2 and based on using unequalIy spaced grid points. In this case the system of linear algebraic equations of the form (5.3.8) converts from the very beginning into a system of linear algebraic equations for an integral Fredholm equation of the second kind with a continuous kernel. Thus, one may formulate the folIowing theorem. Theorem 5.3.2. Let functions f(t) and K(to, t) belong to the class H on [ -1,1] and [ -1,1) X [ -1,1], respectively. Then, Relationship (5.2.4) holds
159
Equation of Ihe First Kind on a Segment belween solulions
the !>ystems 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
Method of Discrete Vortices
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
Equation of the First Kind on a Segment
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
Equation of the First Kind on a Segment
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)
Method of Discrete Vortices
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
Equation of the First Kind on a Segment
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.
Equation of the First Kind on a Segment
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
Method of Discrete Vortices
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
Equation of the First Kind on a Segment
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
Method of Discrete Vortices
,
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
Equation of the First Kind on a Segment
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.
Equation of the First Kind on a Segment
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< <
X
< 0.8, < 1.
X
(5.5.10)
6 .Equations of the First Kind on a Circle Containing Hilbert's Kernel
6.1. EQUATION ON A CIRCLE Consider the characteristic equation cp(t) dt
1.
L
t - t 0
=
(6.1.1)
f(to)'
where L is a unit-radius circle centered at the origin of coordinates. Let the sets E = {t k , k = 1, ... , n} and Eo = {t Oj ' j = 0,1, . .. , n} form a canonic division of the circle. The following theorem is true. Theorem 6.1.1. Let function f(t) E H on L. Then between a solution to the system of linear algebraic equations
j
where a k = t k
11
-
t k and t n
I ,
cp(t)
=
=
1, ... , n,
(6.1.2)
= t" and the solution to Equation (6.1.1), 1
- -2 7T
f J{to) dt o L
t - to
'
(6.1.3)
175
Method of Discrete Vortices
176
there exists the relationship k
=
1, .. . ,n,
(6.1.4)
where O(tk) satisfies the inequality O
Proof. If h
=
(6.1.5)
a k , then Systems (5.1.3) and (6.1.2) coincide, and hence,
'Pn (
I ) k
-
1 n 1 f(to·)b _len) " _len) } } O. k 1.... b 0, OJ t - t ' ak j~ I j k OJ
(6.1.6)
where bj = t Oj .] - tj , k = 1, ... , n, and t on +] = 10 ], As long as this time L is a circle, multipliers 1(\,n2 and /J,nJj must be treated in a way different from Theorem 5.1.1. Remember that
t Ok = exp(i( Ok + 7Tln» = exp(iO ok ), k
=
1, ... , n.
Therefore one can write
pen) 2. k -tk pen)
(6.1.7)
I, k
Because points t b k = 1, ... , n, divide circle L into equal parts, and t Ok is the center of the arc (tk+]' Ik ), Om - Ok = 27T(m - k)/n and 00m - Ok = 27T(m - k)/n + 7Tln. By relabeling and taking into account periodicity of function expO 0), we can write n-l
pn n (1 =
,
m~1
exp( im ' 27Tln»,
n-]
P~~l =
n (1 -
exp(i(7Tln + m' 27Tln»).
m~O
m
In order to calculate Plnl we note that the numbers exp[im(27TIn)], 0,1, ... , n - 1, are the nth-power roots of the number z = 1. In
=
Equations of the First Kind on a Circle Containing Hilbert's Kernel
177
other words, n-]
zn - 1 =
n (z -
exp(im· 27T/n»
m~O
n-]
=
(z - 1)
n
(z - exp(im' 27T/n»
m=l
or zn-l
n-]
- - - = zn-I + zn2 + '" + 1 = z - 1
n (z -
m~]
exp(im' 27T/n».
The latter equality is, in fact, an identity. Therefore, tending z to unity, one gets in the limit zn - 1
lim - - - = n = pfn1z->I z-1 '
(6.1.8)
In order to calculate p~nl we observe that the numbers exp[i( 7T/n + m . 27T/n)], m = 0,1, ... , n -' 1, are the nth-power roots of the number z = -1, i.e., n-]
zn
+ 1=
n [z -
exp(i(7T/n + m· 27T/n»].
m~O
Because the latter equality is valid for any z, for z
Ptl
=
=
1 we get (6.1.9)
2.
According to (6.1.7)-(6.1.9),
=
.1 (sin 7T + icos 7T) = i~ + nSI07T/n n n 7T
o(~). n
(6.1.10)
178
Method of Discrete Vortices
In a similar way one can show that
~l(n) b 0, OJ
2 en) = t l O.Oj OJ'
n
=
-i~7T
+
O(~) n'
(6.1.11)
J
From (6.1.6), (6.1.10), and (6.1.11) it follows that (6.1.12)
Together with formulas (1.2.7) and (1.2.10), this proves the validity of Theorem 6.1.1. •
In a similar way the following theorem may be proved. Theorem 6.1.2.
Let function f( t} be of the form l/J( t) f(t)
=
(6.1.13)
It _ qlV'
where l/J(t} E H on L, 0 $ JJ < 1. Also let the sets E and Eo forming a canonic division of the circle L be chosen in such a way that the point q is the middle of the arc limited by the two nearest points from E and EO" Then between a solution to the system of linear algebraic equations (6.1.2) and the solution (6.1.3) to Equation (6.1.1), Relationship (6.1.4) holds in which quantity O(tk) satisfies the inequalities: 1.
For all points t k
E
L*, (6.1.14)
2.
For all points t k
E
L, 0< A2
$
1.
(6.1.15)
To prove the theorem one has to employ Formulas (1.4.5) and (1.4.6) for a piecewise Lyapunov curve. Note that L * mentioned in connection with (6.1.14) is the portion of L lying outside the 8 neighborhood of point q:
L*=L\O(q,8).
(6.1.16)
179
Equations of the First Kind on a Circle Containing Hilbert's Kernel
Theorem 6.1.1 and Inequalities (1.2.12)-0.2.14) may be used to prove the following theorem (in analogy to Theorem 5.1.5). Theorem 6.1.3. Let function f(O E H on L and the sets E and Eo be chosen in such a way that point q E Eo for j = jq. Then between a solution to the system of linear algebraic equations
j
=
1, ... , n, j *- jq ,
n
L
CPn(tdak
=
(6.1.17)
C,
k~l
and the class-II solution cp(t) to Equation (6.1.1) on L,
cp(t)
1 =
-
2 7T
(q - t)
j(q-to)f(to)dt o L
t - to
i
C
+ --7T
q - t'
(6.1.18)
jcp(t)dt=C, L
Relationship (6.1.4) holds, in which quantity O(tk) satisfies Inequalities (6.1.14) and (6.1.15).
To prove the theorem it suffices to note that because L is a circle, one can always assume that jq = n. • Note 6.1.1. If function f(O suffers a discontinuity of the first kind at a point q ELand belongs to the class H on the set L \ q, then the sets E and Eo must be chosen in such a way that q E Eo for j = jq' and the system of linear algebraic equations should be composed in the following way:
t k=l
cpn(tda k t Oj
-
tk
=
f( t Oj )' f( q - 0) + f( q + 0) { 2 '
j
=
1, ... , n, j *- jq,
(6.1.19)
Method of Discrete Vortices
180
Then it can be shown theoretically that Icp(t k ) - CPn(tk)l, where cp(t) is a solution of Equation (6.1.0, behaves as in the case of Theorem 6.1.2; however, calculations demonstrate a better convergence. Note 6.1.2. While considering Theorems 6.1.1-6.1.3, we used rectangle rule formulas for a singular integral over a circle (see Section 5.2). Now for constructing systems of linear algebraic equations in (6.1.2) and (6.1.15), we can use interpolation-type quadrature formulas for a singular integral derived in Section 2.2. In other words, we put Ok = -i27Ttd(2n + 0, k = 0,1, ... , 2n. Hence, the sets E and Eo contain an odd number of points. From (6.1.10) and (6.1.11) it follows that 1
_len)
Ok
1 7Ti'
=_
O,k
1
_len) _
b
O,Oj -
}
k=0,1, ... ,2n,
.,
7Tt
j
=
0, 1, ... ,2n,
(6.1.20)
where bj = -i27Tt oJ(2n + 1). According to the latter formula, in the case we have (6.1.21)
Thus, if f(t)
E
H v ( ex) on L, then by Formula (2.2.17) one gets
Icp(td - CPn(tdl
$
In n ) 0 ( n rl a •
(6.1.22)
It must be stressed that in the case under consideration the system of linear algebraic equations possesses the following property: a solution to the system is expressed through the right-hand side with the help of the same quadrature formula that was used for constructing the system. By using the preceding interpolation quadrature formulas in Theorem 6.1.3 we deduce that the right-hand sides of Inequalities (6.1.14) and (6.1.15) must be replaced by O(n- r - a In n) and O(n- r - a In 2 n), respectively. N~t we consider the full equation of the first kind on a circle
cp(t) dt f L to - t
+ fK(to,t)cp(t)dt L
=
f(lo) ,
(6.1.23)
Equations ot the First Kind on a Circle Containing Hilbert's Kernel where function t(1) equation
E
H on Land K(to, I) K(to,/)
f ---cp(t) dl I.
10 -
I
=
E
H on L
X
t(to),
lSI
L, or the
(6.1.24)
where K(to, I) E H on L X Land K(to, (0 ) *- O. Equations (6.1.23) and (6.1.24) are equivalent to the corresponding Fredholm equations of the second kind. If these equations have unique solutions, then the corresponding Fredholm equations of the second kind also have unique solutions. To solve the equations numerically in the class of continuous functions, one has to consider the following systems of linear algebraic equations: j = 1, ... , n,
(6.1.25) or j = 1, ... , n,
(6.1.26)
where points I k , k = 1, ... , n, divide circle L into equal parts, 10k divides arc tktk + I into two halves, and ak = -i27Tt k /(2n + 0, k = 1, ... , n, where n is an odd number (in the general case, a k = t k + 1 - I k ). It is supposed that either t(1) and K(to, I), or K(to, I), K(to, (0 ) *- 0 on L belong to the class H on the corresponding sets. If t(1) suffers a discontinuity of the first kind at point q, then one has to use Note 6.1.1. However, if t(t) suffers an integrable discontinuity at point q, then the points t k and t Ok must be chosen in such a way that point q is the middle of the arc limited by the nearest points from the sets E = {t k , k = 1, ... , n} and Eo = {t Ok ' k = 1, ... , n}. In this case Systems (6.1.25) and (6.1.26) preserve their form. Convergence of solutions of Systems (6.1.25) and (6.1.26) to those of Equations (6.1.23) and (6.1.24), respectively, may be proved in the same way as for the analogous equations on a segment, that is, one has to repeat in the discrete form the transfer to systems of linear algebraic equations for corresponding singular Fredholm equations of the second kind. Naturally, the systems will not be degenerate starting from a certain n I' If one has to find a solution to Equation (6.1.23) having at a fixed point q a singularity of the form l/q - 0, then the points I k and 1 0k must be
182
Method of Discrete Vortices
chosen so that q = t Ok and the following system of linear algebraic equations must be considered:
j
=
1, ... ,n,j i=k q ,
n
L
CPn(tdak
=
C,
(6.1.27)
k~l
Note 6.1.3. All the results obtained in this section can be readily extended onto the case when L dealt with in Equations (6.1.1), (6.1.23), and (6.1.24) is a system of non intersecting circles (Lifanov 1981).
6.2. EQUATIONS WITH HILBERT'S KER.NEL Consider the equation 1 27T
12n-cot--cp(O)dO=f(Oo)' 0 0 0 -
2
0
(6.2.1)
Let us choose on the interval [0, 27T] points 0b k = 1, ... , n, that, being treated as points of unit circle L, divide it into n equal parts. Also let 00k' k = 1, ... , n, divide arc Ok Ok -+ I into two halves. Remember (Muskhelishvili 1952) that Equation (6.2.0 has a solution subject to the condition 27T
1o f(O)dO=O.
(6.2.2)
In order to single out a unique solution, one has either to specify it at a point or to fix the integral of the solution (the latter situation is more frequent in applications). Therefore, if Equation (6.2.0 is solved numerically with the help of the approach used for solving the characteristic equation on a segment (see Chapter 5), then using the quadrature formulas for an integral containing Hilbert's kernel (see Sections 1.5 and 2.1), one must replace Equation (6.2.0 by the following system of linear
Equations of the First Kind on a Circle Containing Hilbert's Kernel
183
algebraic equations:
m
=
1, ... ,n,
(6.2.3)
The number of equations in System (6.2.2) is more than that of unknowns. By the choice of points Ok and (JOk and Formula (1.2.5) we have n
L m-I
cot
80m
-
2
Ok 0,
=
k
=
1, ... , n.
(6.2.4)
Hence, after summing the first n equations entering System (6.2.3) and taking into account Equality (6.2.4), one gets ( 6.2.5) m~
I
Thus the system of the first n equations from System (6.2.3) is iIIconditioned and generally has no solution. The same may be said for the whole System (6.2.3). Naturally, the idea of discarding one of the first equations of System (6.2.3) arises. Then, as shown in the following text, we arrive at a wellconditioned definite system that leads, however, to an unstable process of calculations. Therefore, one can apply the method of regularizing factors, used before for solving a characteristic singular integral equation on a segment for the case of a negative index (everywhere limited flow past an airfoil or flow past an airfoil with a sharp trailing edge). The following theorem is true.
Theorem 6.2.1. Let function f( (J) E H on [0, 27T ] and f(O) = f(27T). Also, let Equality (6.2.2) hold for the function. Then, between a solution to the system of linear algebraic equations
m
=
1, ... ,n,
(6.2.6)
Method of Discrete Vortices
184
and the solution cp( 0)
10
Equation (6.2.1) given by I
cp(O) = --2 7T
1 cot--f(Oo)dO 0 0 o+ C 27T
0 -
(6.2.7)
2
0
(Muskhelishvili 1952) and subject to the condition 1 27T
1 cp(O)dO=C, 27T
(6.2.8)
0
the relationship (6.2.9) holds where A = a E (0, I1, if n is an arbitrary number and f( 0) and 11. = r + a ifn is odd andffr)(O) E H(a).
E
H( a ),
Proof. Let us sum the first n equations of System (6.2.6). Then taking into account (6.2.4) we get
YOn
1 n 27T = -2 L f(Oom)-· 7T m= 1 n
(6.2.10)
Hence, YOn -+ 0 for n -+ 00, if and only if Equation (6.2.1) has a solution. In what follows we will reduce System (6.2.6) to a system of the form of (6.2.2) for an equation on a circle by using Equality (2.2.15). Let us multiply the last equality of System (6.2.6) by ( - i) and sum the result with all the first n equations. Then, taking into account Equality (6.2.10) and multiplying both its sides by 7T, one gets
In
= 7Tf(Oom) - -2 [
L
k~l
27T f(OOk)n
J-
i7TC,
m
=
1, ... ,n,
or
m
=
1, ... ,n,
(6.2.11)
Equations of the First Kind on a Circle Containing Hilbert's Kernel
185
Because System (6.2.11) coincides with System (6.1.2), its solution is given by Equation (6.1.6). By Equalities (6.1.20) we deduce (see (6.1.21)) ~ (
~
4)
1 n _ _"
=
7T
2
~
m=l
f~(t
tk
Om
)b m
(6.2.12)
,
-tOm
where bm = 27Tit om /n. Thus, after using Equalities (2.2.15) and (6.2.4) again, one gets
(6.2.13)
By comparing (6.2.7) with (6.2.13) we terminate the proof of the theorem. •
If instead of considering (6.2.6), one considers the system
Note 6.2.1.
n
1
'YOn
L cot 7T k= I
+ -2
00m - Ok 27T 2 'Pn( 0d n 1
'YOn
+ -
n
=
m
f( 00m)'
=
1, ... ,n,
27T
L=1 'Pn(Ok)n
(6.2.14)
+C,
27T k
then similar considerations result in 'Pn(Od
1
=
° -°
27T 1 exp k 2 Om f(Oom)- - -2 7T m =1 n 7T n
-"2 L
27T f(Oom)n m =1 n
L
+
C.
(6.2.15)
The following theorems may be proved in a similar way.
Method of Discrete Vortices
186
Between a solution to the system of linear algebraic equa-
Theorem 6.2.2. tions
00m - Ok
n
2
L cot 21T k =1
21T CPn ( 0d = f( 00m), n
m = 1, ... , n - I,
n 21T LCPIl(Od-=C 21T k =1 n
(6.2.16)
and the solution cP «(}) of Equation (6.2.1) for any f( 0), cP (0)
=
1
--
21T
1 cot--f(Oo)°- 00 21T 27r
n
2
0
°127Tf( (
- -1c oqt -- 21T 2
0
0)
dO o + C,
(6.2.17)
=
1, ... , n,
(6.2.18)
the relationship k
holds where T/(Ok) satisfies Inequalities (6.1.14) and (6.1.15) in which the numbers A, and A2 are defined by using the properties offunction as was done in (6.2.9). Points Ok and 00k> k = 1, ... , n, are chosen in such a way that eOn = q for and n. Theorem 6.2.3. Let f( 0) E Han [0, 21T] and f(O) = f(21T). Also, let points Ok and 00k, k = 1, ... , n, be chosen so that Ok = q, q E [0, 21T]. Then, between a/olution to the system of linear algebraicqequations ,/
'YOn
+-
I
n
21T
L cot k- 1
00m - Ok 2
21T CP: (Odn
k *k q
1 00m - q
=f(Oom)--cot 21T
2
21T cp(q)-, n
m
=
1, ... , n, (6.2.19)
where cp(q) is known, and the solution cp+(O) of Equation (6.2.1), cp'(O)
=
1
--
21T
177T( 0- 00 cot-0
2
q - 00 ) -cot-f(Oo)dO o + cp(q), 2 (6.2.20)
Equations of the First Kind on a Circle Containing Hilbert's Kernel
187
inequality k
=
1, ...
,n,
(6.2.21)
holds.
Note 6.2.2. the form
Solutions to Systems (6.2.16) and (6.2.19) have, respectively,
lOOn - Ok n . 27T - -cot L j(OOm)21T 2 m~l n
100m - Ok (21T x-cot "cp(q) 27T 2 n
)2
.
+ C,
(6.2.22)
(6.2.23)
From (6.2.22) it follows that the accuracy of calcu[ations deteriorates in the neighborhood of point q whose equation was discarded, because, generally, L;',,~ d(Oom)(27T/n) i= O. In accordance with (6.2.23) we deduce that if cp(q) = 0, then for t<')( 0) E l/( 0') on [0, 27T] and an odd n, the right-hand side of Equation (6.2.21) becomes O(n- r a [n n). Therefore, if the value of cp(q) is known, then it is advantageous to introduce a new function (j;( 0) = cp( 0) - cp(q) for which (j;(q) = O. Note 6.2.3. If function f( 0) entering Equation (6.2.0 suffers an integrab[e discontinuity at point q, then, similarly to the equation on a segment, a system of linear algebraic equations may be taken in the form of (6.2.6). However, points Ok and 00k' k = 1, ... , n, must be chosen in such a way that point q divides the segment into two equal parts limited by the nearest points belonging to the sets E and Eo. If, however, function f( 0) suffers a discontinuity of the first kind at point q E [0, 27T], then the calculations show that the following strategy
188
Method of Discrete Vortices
should be preferred: Points Ok and 00k are chosen so that point q coincides with point 00k , and the right-hand sides of n first equations of system (6.2.6) must be taken as follows: 1.
2.
For equations whose number is equal to m = 1, ... , n, m *- k q , the right-hand sides are given by f( 00m). The right-hand side of the equation numbered m = k q is equal to [f(q - 0) + f(q + 0)];2. Consider the equation 1 -2
7T
127rcot--cp(O)dO+ 00 - ° 1
27T
2
0
K(Oo,O)cp(O)dO=f(Oo)
(6.2.24)
0
or the equation -1
27T
127rK(Oo, O)cot--cp(O) 0 ° dO =f(Oo), 0 -
2
0
(6.2.25)
°
where K(Oo, 00) *and K(Oo, 0) E fl on [0,27T] X [0, 27T], and the functions f(Oo) and K(Oo,O) are periodic with respect to their coordinates. Let us suppose that Equation (6.2.24) has a unique solution subject to the additional condition 27T
1o
(6.2.26)
K](O)cp(O)dO=C,
where K]( 0) is a nonzero function. If kernel K( 00' 0) satisfies the identity 27T
1o
K(Oo, 0) dO o
=
(6.2.27)
0,
and the right-hand side satisfies Equality (6.2.2), then Equality (6.2.6) must be specified subject to some additional conditions demonstrated in succeeding text when considering problems of aerodynamics and elasticity. However, if Equation (6.2.24) is uniquely solvable for any right-hand side, then the additional condition must be taken in the form: 27T
{27T
(27T
0
0
1o dOole
K(Oo,O)cp(O)dO=)"
f(Oo)dO o ·
(6.2.28)
Equations of the First Kind on a Circle Containing Hilbert's Kernel
189
These remarks must be taken into consideration when developing a numerical method for solving Equations (6.2.24) and (6.2.25). Thus, Equation (6.2.24) may be represented by the system of linear algebraic equations:
m
=
1, ... ,n,
(6.2.29) If a unique solution to Equation (6.2.24) is singled out with the help of condition cp(q) = 0, then the following system must be considered:
m
=
1, ... , n, (6.2.30)
where points Ok and OOm are chosen as was done when considering System (6.2.19). The convenience of Systems (6.2.29) and (6.2.30) is associated with solvability of Systems (6.2.6) and (6.2.19), which allows us to transform the former systems into equivalent systems of linear algebraic equations for integral Fredholm equations of the second kind equivalent to Equation (6.2.24) in the class of continuous solutions. Thus, one may prove the convergence of solutions of Systems (6.2.29) and (6.2.30) to the accurate solution of Equation (6.2.24), with the estimates made previously for the characteristic equation. If the right-hand side of Equation (6.2.24) suffers a discontinuity of the first kind or has an integrable discontinuity at point q, then System (6.2.29) must be composed into account Note 6.2.3. The rationale for composing a system of the form of (6.2.29) for Equation (6.2.24) in the case when it has a unique solution for any right-hand side will be explained on the example of the equation 1 127T 27T 0
0 - 00
1 [27T
cot--cp(0) dO + 27T JO cp(O)dO=f(Oo)' (6.2.31) 2
190
Method of Discrete Vortices
which, according to Muskhelishvili (1952), has the solution
cp( 0)
=
-
-
1
27T
127rcot--f( ° °( 0
-
2
0
0)
dO o +
-1
27T
1
27T
f(
(0)
0
dO o' (6.2.32)
Let us, in fact, consider the system
m
=
1, ... ,n,
(6.2.33)
where points Ok and 00k, k = 1, ... , n, are chosen just as in the case of System (6.2.6). When we sum all the equations in (6.2.33), we deduce that the system is equivalent to
m=l, ... ,n-l. m
=
n.
(6.2.34)
By comparing Systems (6.2.34) and (6.2.16), we see that a solution to System (6.2.34) is given by the formula
1 - -cot 27T
° 2-° [ L On
k
n
m~l
(
n f(Oop)27T) -27T] f(OOm) - - 1 L 27T p~l n n
(6.2.35) From Formula (6.2.35) it follows that in the presence of errors in calculilted right-hand sides and the sum L;~ J(00p)27T/n, the error increases by the factor of cot( 00n - Ok) /2, which for Ok ~ 00n has the order of n.
7 Singular Integral Equations of the Second Kind
7.1. EQUATION ON A SEGMENT Consider the equation bj!cp(t)dl acp(t o) + 7T
-
1 -
=
10
f(to),
(7.1.1)
where a and b are real numbers, b *- 0, a 2 + b 2 = 1, and function f(t) belongs to the class H on [- 1, 1]. Let us briefly recaIl some results obtained in Muskhclishvili (1952) for Equation (7.1.1). The index K of the equation is equal to 1,0, -1, and the corresponding solutions have the form cp(/)
=
w(/)l/J(/),
w( I) = (1 - I) a (1 + t) f3 , K=
°< 1 I, 1/31 < a
1,
(7.1.2)
-(0'+/3).
The number a is defined by the equality a + b cot
7T0' =
o.
(7.1.3) 191
192
Method of Discrete Vortices
Let us denote the left-hand side of Equation (7.1.1) by l(to) and call the formula
n
t/lAt) =
L (t k= 1
t/ln(tdP~a,{3)(t)
_ t )p,(a,{3)(t )' (7.1.4) k
n
k
a quadrature-interpolation formula of index K and order n (Lifanov and Saakyan 1982). Here t k , k = 1, ... , n, are the roots of the n-degree Jacobi polynomial p;a, (3)(t) corresponding to function wet), and t/ln(t k ) = t/I(t k ). According to Erdogan, Gupta, and Cook (1973), polynomial p~a,(3)(t) satisfies the relationship
(7.1.5)
Hence, the equality
X
b [2
K
•
SIn C17T
pfa, -(3)(t 0 ) n-K
+
I
w(t)p~a,I3)(t)
-1
t - tk
b -f 7T
dt
1 (7.1.6) '
is valid. Function 'Pn(t) will be called an approximate solution to Equation (7.1.1). It will be found by equating functions InU O) and fn(to), where fnUo) is an interpolation polynomial of the form of (2.3.2) for the function fCt o ) constructed by using roots of the polynomial p~=~, - (3)(t o)' Function 'Pn(t) is denned if the numbers t/ln(t k ), k = 1, ... , n, are known. By equating the functions InU o) and fn(t o ) at the points tOm' m = 1, .. " n - K, where tOm are the roots of the polynomial p~~:, -(3)(t), one arrives at the system of
SinguLar IntegraL Equations of the Second Kind
193
linear algebraic equations m=I, ... ,n-K
(7.1.7) By putting to = t k in (7.1.5), we note that the coefficient a k appearing in (7.1.7) may be written in the form k
=
1, ... , n.
(7.1.8)
Let us consider different values of index K. Let K be equal to O. Then, a unique solution to Equation (7.1.1) may be singled out by specifying the number 0',0 < 10'1 < 1, satisfying Equality (7.1.3). If one has to obtain a solution that is limited at point 1 and unlimited at point -1, then one has to choose a positive value of a. Because polynomials p~a.fJ)(t) and p;=~.-fJ)(t) have thc same number of roots, System (7.1.7) contains n unknowns as well as n equations. Let K be equal to 1. Then -1 < a, f3 < 0 and Equation (7.1.1) has a nonunique solution. A required solution may be singled out by additionally employing Condition (5.1.46). In this case the degree of polynomial p~ _~. fJ)(t) is equal to (n - 1), and hence, System (7.1.7) contains n unknowns and (n - 1) equations. This may be remedied by digitizing Equation (5.1.46), i.e., by passing to the system b _
_
n "
'-
7T k=\
.1. (t )a 'l'n
k
k
tOm -
tk
=
f(
)
m=I, ... ,n-l,
C,
m =n.
tOm'
n
L k
~
t/!n(tk)a k
=
(7.1.9)
1
Finally, let K be equal to -1. In this case polynomial p~ -.~. - fJ )(t) has + 1) roots, and System (7.1.7) has more equations than unknowns and usually has no solutions. Therefore, similarly to the equation of the first kind, we introduce the regularizing factor 'YOn and consider the system
(n
m=I, ... ,n+1. (7.1.10)
194
Method of Discrete Vortices
According to Muskhelishvili (1952) a solution exists only if the equality ]
f(t) dt
{] (1 - t)"(1 +
t)f3
= 0,
0< 0', f3 < 1, 0' + f3 = 1,
holds. Theorem 7.1.1. Let function f(O E H r ( 0') on [ -1,1]. Then, between solutions to the systems of linear algebraic equations (7.1. 7) for K = 0, (7.1.9) and (7.1.1 0) and the co"esponding solutions to Equation (7.1.1), Inequality (5.2.4) holds, where Rn(t k ) is an e"or of the quadrature-interpolation formula of index K and order n - K for the function
l/J(t)
=
f(t) w(t)
which determines the index T] = - 7T[sin 0'7T] - I .
f'
b f(to) dt o 7T -] w(to)(to - t)
a-- - K
+ TkC,
(7.1.11)
solution in Equation (7.1.2); To = T _] = 0,
Proof. As shown when proving Theorem 5.1.1,
.1. (t k )
'l'n
=
1
_/fn)
ak
K,k
[n-K "/f n) '-
m -]
-b- ' 7Tf(t) Om
K,Om
tk -
tOm
+
VK
C
] ,
k = 1, ... , n,
v = 1, V o =
V
I
= O.
(7.1.12) By representing the polynomials by products of linear multipliers and using Equation (7.1.8), one gets k=l, ... ,n,
(7.1.13)
where B~ - <>, - (3) and B~·~' - (3) are coefficients before the senior degrees of the variable in the corresponding Jacobi polynomials.
Singular Integral Equations of the Second Kind
195
As previously noted, function l/J(t), appearing in Equation (7.1.2) for the index K solution, is defined by Equation (7.1.10. Let us consider this equality as an equation in the function ~(t) = [w{tW 'f(t). If cp(t} is the index K solution for Equation (7.1.1), then function ~(t) will be the index - K solution for Equation (7.1.10. Jf, in Equality (7.1.5), I is substituted by 10 , w(t) by l/w{t), and P~lX,(3)(t} by P~::~' -(3)(t), then the number b must be replaced by -b. Therefore, one gets
Let us denote by
k
2
bm
K
1, ... , n,
=
p~a,(3)(tOm)
7T
-= -sincx7Tp,(-a,-(3)(t )' nOm
(7.1.15)
K
By representing the polynomials by products of linear multipliers again, one gets
bm sin
CX7T
B~:::'
2 K 7T
- (J)
B~lX. (3)
m=l, ... ,n-K. (7.1.16)
,
Finally, we observe that the equalities sin
CX7T
sin( - cx) 7T 7T
2
(7.1.17)
are valid. As a result, we get
r/Jn(td
b n-K f(l =
-
L
7T m~1
)b Om
tk
m
-lorn
JJ sin CX7T ----C, K
7T
k = 1, ... , n. (7.1.18)
196
Method of Discrete Vortices
Obviously, sin -
JJK
CX7T
---
=
TK
7T
in Formula (7.1.10. Thus, Formulas (7.1.12) and (7.1.18) actually prove Theorem 7.1.1. The rate of convergence of an approximate solution to the accurate one at points t k , k = 1, ... , n, is determined by the order of approximation attainable by using the quadrature-interpolation formula of the corresponding function t/J(l). • Similarly to the equation of the first kind on a segment, we note that the regularizing factor 'YOn tends to zero for n ---) oc only if the conditions of existence of the index K = -1 solution are fulfilled. The results previously formulated for Equation (7.1.1) are also valid for the equation
acp(t o) +
b -f 7T
1
cp( t) dt
--I
t - to
+
J K(to,t)cp(t) dt 1
-
=
f(to), (7.1.19)
1
where it is supposed that the function K(to, t) E H on [ - 1, 1] X [ -1,1]. We will require that Equation (7.1.19) have a unique index K = 1,0, -1 solution subject to additional conditions that, for K = 1, are imposed on the solution itself, and for K = -1, on the functions K(to, t} and f(to). The systems of linear algebraic equations for Equation (7.1.19) may be obtained from the corresponding systems for the characteristic equation by adding the term n
L
K(tom,tdt/JAtda k ,
m=I, ... ,n-K.
k~1
7.2. EQUATION ON A CIRCLE Consider the equation b
acp(t o) + -
7T
dt f cp(t) t - to
=
f(to),
(7.2.1)
L
where a and b are real numbers, b *- 0, a 2 + b 2 = 1, and the function f(t} E H on a unit-radius circle L centered at the origin of coordinates. As shown in Muskhelishvili (1952), Equation (7.2.1) has index K = 0, i.e., is uniquely solvable for any right-hand side.
Singular Integral Equations of Ihe Second Kind
197
Let us denote the left-hand side of Equation (7.2.1) by 1(/ 0 ) and call
(7.2.2) where points t k = exp(i0k),k = 1, ... ,2n, divide L into 2n + 1 equal parts, 'PnU k ) = 'P(t k ), the quadrature-interpolation formula of the order n for the function l(to)' By using Equality (2.2.6) the function In(to) may be written in the form
'P,,(td(a
+
+ bi)(t(~n+ I + (-a + bi)/
(a + bi)tl n . 1)
2n
L ----------k-O
(2n + 1)(10 - Ik)tgt;
where a k = 27Tilk/(2n + I),k = 1, ... ,2n. Let us search for an approximate solution to Equation 0.2.1) in the class of functions of the form 'Pn(t). To do this it suffices to find the numbers 'Pn(t k ) from the system of 2n + 1 linear algebraic equations
m
=
(7.2.3)
0, 1, ... ,2n,
obtained by equating functions II/(to) and f/l o) at points tom' m 0,1, ... , 2n, where
/'n(/ O )
=
2n t(;1/ - I _ t~::" 1 "f(/() ) n, m (I 0 - t Om )tnt 2 n + 1 m'~0 0 m
=
(7.2.4)
and tOm are the roots of the polynomial t~n + 1 + (-a + bi)/(a + bi)tl1/ t 1. Note that fn(tom) = f(tom) and 10m = t m exp{i(7T + r/J)/(2n + l)}, m = 0,1, ... , 2n, where exp(ir/J) = (-a + bi)/(a + bi). Because b 0, - 7T < r/J < 7T, and hence, lo!!,!-* t k for any m and k. If a = 0, then r/J = 0, and point tOm divides arc Iml m + I into two halves. The folIowing theorem is true.
*
Method of Discrete Vortices
198
Theorem 7.2.1. Let function f(O E lfr ( Q') on L. Then, between a solution to the !.ystem of linear algebraic equations <7.2.3) and the solution cpU) of Equation (7.2.0, the inequality
k=0,1, ... ,2n,
(7.2.5)
holds. Proof. Because System (7.2.3) is similar in structure to System (7.1.7), its solution is given by 7T
CPn ( t k )
_
-
1
bak
2n I(n)
O,k
"
(n)
1.... Io,om,
m~O
f(t) Om -t k
k = 0, 1, ... , 2n.
Om
(7.2.6)
By the choice of points t k and t ok , k proving Theorem 6.1.1, one gets len) 0, k
=
-
tk
.
2n + 1
(1 + e'l/J)
'
=
It(lm
1, ... , 2n, just as in the case of
t =
Om
2n + 1
(1 + e-il/J). (7.2.7)
Because by the definition of exp(il/J) we have [1 exp(-il/J)] = 4b 2 , Formulas <7.2.6) and <7.2.7) result in
k
=
+ exp(il/J)][l +
0,1, '" , 2n,
(7.2.8)
where bm = i(27Tt om /(2n + 0). According to Muskhelishvili (1952), the solution to Equation <7.2.0 has the foqn cp(t)
=
b
af(t) - -
7T
f L
f(lo) dl o 10
-
t
.
(7.2.9)
Singular Integral Equations of the Second Kind
199
Let us denote the right-hand side of Equation (7.2.9) by <1>(1) and take for the function the quadrature-interpolation formula <1>n(1) obtained from <1>(1) by substituting function f(1) by fn(1) with the help of (7.2.4). By using Formula (2.2.6) again, one gets
(7.2.10) From (7.2.8) and (7.2.10), it follows that k=O,1, ... ,2n.
Then, by using (1) the results concerning approximation of periodic functions on a circle by polynomials of the preceding form (Ivanov 1968) and (2) the fact that a singular integral over a circle has the same differential properties that are characteristic of its density, we terminate the proof of Theorem 7.2.1. • Next, we consider the equation acp(t o )
+
b cp( t) dt -f + f K(to, t)cp(t) dt 7T t - to I.
=
f(to)'
(7.2.11)
L
Supposing that the equation has a unique solution, we take for it the system of linear algebraic equations
m=O,1, ... ,2n.
(7.2.12)
If functions f(1) and K(to, I) are such that pr)(1) and Kr(,~)(to, I), Kr)(to, I) belong to the class H( Q') on the sets Land L X L, respectively, then Relationship (7.2.5) is also valid for a solution to the system of linear algebraic equations (7.2.12) and the solution cp(1) of Equation 0.2.11).
7.3. EQUATION WITH HILBERT'S KERNEL Consider the characteristic equation acp(Oo)
+ -b
1
0 - 00 cot--cp(O)dO=f(Oo), 27T 0 2 21T
(7.3.1 )
200
Method of Discrete Vortices
where it is supposed that a*-O and a 2 + b 2 has a unique solution given by
=
1. In this case the equation
127Tcot--f(Oo)dO °0 - ° b o + -2 2
b cp(O) =af(O) - -2
2
7To
2 7T
1 f(Oo)dO o'
7To
Let us denote the left-hand side of (7.3. I) by I( ( 0 ) and call the function In(Oo) obtained from 1(° 0 ) by replacing cp(O) by 1 'Pn( 0)
=
2n
+
2n
I
k~O cpA °d
sin(2n + 1)(0- 0k)/2 sin( 0_ 0d/ 2 '
(7.3.2)
the quadrature-interpolation formula of the order n for function I( ( 0 ), Here points Ok' k = 1, ... , 2n, are the points of a unit-radius circle dividing the latter into 2n + I equal parts. By using Equalities (2.1.2) and (2.1.3), one gets
a sin(2n
+ 1)( 00 - 0d/
2 - bcos(2n + 1)(00 - 0d/2 sin( 00 - Ok) /2
(7.3.3)
Let us look for an approximate solution to Equation (7.3.1) in the class of functions of the form CPn(O). This may be done by finding the numbers CPn( Ok), k = 1, ... , 2n, from the following system of linear algebraic equations: b 2n " 21T k~O cot
° 2-° CPn Om
k
(0)
27T
k 2n
+
1 =
f(
°) Om'
m
=
0, 1, ... ,2n,
(7.3.4) obtained by equating the functions I n( (0) and fn( (0)' where fn( ( 0) is an interpolation polynomial over the points 00m' m = 0, 1, ... , 2n, for the function f( ( 0 ), These points are the roots of the function a sin(2 n + 1) (00 - 0k)/2 - b cos(2n + 1)(°0 - 0k)/2. Note that 00m = Om + (7T2r/J)/(2n + I), where expUr/J) = b + ai, and because b *- 0 and 00m *- Ok for any m and k. If a = 0, then r/J is equal either to zero or 7T, and the
Singular Integral Equations of the Second Kind
201
point 00m divides the arc Om Om ~ 1 into two equal parts, but the systems of linear algebraic equations must be taken as indicated in Section 6.2. The foIlowing is true. Theorem 7.3.1. Let function f( 0) entering Equation (7.3.1) belong to IIr ( ex) on [0, 27T], andf(O) = f(27T). Then ICPn(Ok) - cp(Ok)l, k = 1, ... ,2n, satisfies an inequality of the form (7.2.5), where cp( 0) is a solution to Equation (7.3.1), and CPn(Ok) is a solution of System (7.3.4). Proof. By using the same approach with respect to System (7.3.4) as was used with respect to System (6.2.6), one gets
hi
27T
2n
-- L
27Tp~O
CPn( Op)
Ok - 00m 27T cot----2n+l m =o 2 2n+l 27T
L
(7.3.5)
By the choice of points Ok and 00k we have ~ Ok - 00m 27T '- c o t - - - - - m-O 2 2n + 1
2n
"
'- cot
k~O
27Ta k=0,1, ... ,2n, (7.3.6)
b
27Ta ° -° - 27T -- = -Om
k
2
2n
+1
k
b'
=
0,1, ... ,2n, (7.3.7)
Let us demonstrate the validity of Equality (7.3.7). Let cp( 0) appearing in Equation (7.3.1) be identicaIly equal to unity [cp( 0) == 1]. Then, 1 . a + -b 27T
°
j27Tcot 00- - . 1 dO = 0
2
a.
(7.3.8)
By substituting unity here with the help of formula (7.3.2) and putting 0 0 = 00m, one arrives at Equality (7.3.7). Similarly, by using the formula for cp( 0), one can prove the validity of Equality 0.3.6).
Method of Discrete Vortices
202
Let us sum up all the equations comprising System (7.3.4). Then, taking into account (7.3.6), we get
(7.3.9)
From Equations (7.3.5), (7.3.6), and (7.3.9), we get
k=O,1, ... ,2n. (7.3.10)
Let us next denote the right-hand side of the formula for cp(O) by <1>(0) and take for the function the quadrature-interpolation formula n( 0) of the order n, which is obtained from
(7.3.11)
By comparing Formulas (7.3.10) and (7.3.11), applying the theory of approximation of periodic functions by trigonometric polynomials, and using the fact that a singular integral with Hilbert's kernel has the same differential properties as its density (Luzin 1951), we terminate proving Theorem 7.3.1. • Consider next the equation
acp(Oo')
+-
b j27r 0 - 00 cot--cp(O) dO
27T
0
2
(27r
+ 10 K(Oo,O)cp(O)dO=f(Oo)· 0
(7.3.12)
Singular Integral Equations 0/ the Second Kind
203
We assume that the equation has a unique solution and represent it by the system of linear algebraic equations:
m = 0, 1, ... ,2n.
(7.3.13)
If functions /(00) and K(Oo, 0) have the rth derivatives belonging to the class H(O') on [0, 27T], then Inequality (7.2.5) is valid for Ic,on<0k) - c,00k)l, where c,on( Ok) is a solution to System (7.3.13) and c,o( 0) is a solution to Equation (7.3.12).
7.4. EQUATION ON A PIECEWISE SMOOTH CURVE WITH VARIABLE COEFFICIENTS Consider the equation
(7.4.1 )
where L is a piecewise smooth curve (Muskhelishvili 1952) with nodes c l , c 2 , ••• , C n (see Figure 7.1), and the functions a, b, and / meet the Holder condition on curve L, a 2 - b 2 *- on 1.. In this case the index K of the equation is defined as follows (Muskhelishvili 1952). Let us take the function In G(t), G(t) = (a(t) - b(t»/(a(t) + bet)), keeping in mind its branch varying continuously on each of the smooth curves L 1, L 2 , ••• , L p constituting curve 1.. Denote by In G/c k ) the limit to which the function In G(t) tends as t approaches point C k along the curve L j , and take the function
°
y(z)
1 =
-.
27Tl
f I.
InG(t)dt t -
Then, in the neighborhood of node c k , k representation y(z)
=
(O'k
(7.4.2)
. Z
=
+ if3d l n(z - cd +
1, ... , n, one has the
Yo(z),
(7.4.3)
where Yo(z) is a function analytic within each of the sectors formed by curve L in the neighborhood of point C k and tending to a certain limit for
204
Method of Discrete Vortices
c,
c,
FIGURE 7.1.
Z -+ Ck
A piecewise smooth curve.
along a route staying within a fixed sector. Further,
(7.4.4)
where the sum encompasses al the numbers j of the arcs L j converging to c k • The upper and the lower signs correspond to issuing and incoming arcs, respectively. Hence, near a node C k one has (7.4.5) where O(z) has the same form as 'Yo(z) entering 0.4.3). Node c k will be called singular if the corresponding number Cik is an integer; all the other nodes will be called nonsingular. Let C], ••• , C], 0 ~ 1 ~ n, be nonsingular nodes and C/+ I'.", C n be singular nodes of the line L. A solution to the original Equation (7.4.1) will be sought in the class H* on curve L. In other words, the solutions are either limited at the nodes or have integrable singularities. Therefore, we shall divide all the solutions into classes, assigning to the class h, c" ... , cq , q = 0,1, ... , I, all the solutions cp(t) E H * Oll L that stay limited in the neighborhoods of nonsingular nodes C], ... , C and alIow for integrable singularities at all the rest of the nonsing~lar nodes c q I I"'" c 1• A fixed class h, c], ... , c q , is used to
Singular Integral Equations of the Second Kind
205
specify the canonic function x(z) and the index
K
of a solution
n
cd A; =
x(z) = [y(l). 0 (z -
[y(z). O(z),
(7.4.6)
k=1
(7.4.7) where the integers Ak are chosen, depending on the class h, C 1, ••• , c q , in the following way: 0 < O'k + Ak < 1 for the nodes c l , ••• , c q , -1 < O'k + Ak < 0, for the nodes C q + I' ••• , C I' and O'k + Ak = 0 for the singular nodes
c l + 1'···' Cn· Now the index K solution to characteristic equation (7.4.0 is given by the formula (Muskhelishvili 1952)
cp(t)
=
. b*(t)z(t) a*(t)f(t) 7Ti
1. 2(to)(to f(to) dt o - t) I.
+ b*(t)z(t)P ,(t), K
z(t) = [a(t) + b(t)]x(+)(t) = ";a 2 (t) - b 2 (t) [y(I)II(t), (7.4.8) where for K > 0, PK ,(to) is a polynomial of the degree K K s: 0, PK ,(to) == o. Note that the function z(t} may be presented in the form
-
1, and for
n
z(t)
=
0
w(t)·
cd Y ;,
(t -
(7.4.9)
k~1
where w(t) is a function belonging to the class H o (Muskhelishvili 1952) and nonzero on L, and
k
=
1, .. . ,n.
(7.4.10)
Also, it may readily be shown that lim ZKX(Z)
=
1,
(7.4.11)
z-->c£
i.e., the index
K
canonic function x(z) behaves at infinity as z
K.
206
Method ofDiscrete Vortices
In order to develop numerical methods for solving Equation (7.4.1), let us first point out some of its properties. We start by writing it for the index K solution, in the form
(7.4.12) Formula (7.4.8) will be presented in the form
(7.4.13) where
1 for K > 0, and 71(K) = 0 for K $ O. For K < 0 function cp(t) given by (7.4.13) is a solution to the original characteristic equation (7.4.12) in the case when the conditions 71(K) =
j
=
0, 1, ... ,IKI- 1,
(7.4.14)
hold. It should be noted that (Muskhelishvili 1952) the functions f(t )t j , j = 0, 1, ... , K - 1, are the proper functions of the operator j (+ >( w( + )K • Xt) for K > 0; the same functions are proper functions for the operator j(-)(w(-)'Xt) for K
For.
K ~
1 a solution to the equation (7.4.15)
Singular Integral Equations of the Second Kind
207
is given by the function (7.4.16) subject to the conditions j=O,I, ... ,K-l.
(7.4.17)
°
for K > the question of singling out the unique solution of Equation (7.4.12) arises. Because in this case the solution is not unique due to polynomial P I(t) appearing in (7.4.13), one of the simplest ways of singling out a unique solution is by specifying the values of solution cp(t) at K different points, which do not coincide with nodes. However, more often one has to deal with singling out by means of specification of moments of the solution (Belotserkovskii et at. 1987, Zakharov and Pimenov 1982, Parton and Perlin 1982, Savruk 1981). In this connection we formulate the following result (Nendikova, Lifanov, and Matveev 1987). K
-
Them'em 7.4.1. Let cun:e L be piecewise smooth (in particular, it may be smooth). Then the system of equalities k
=
0,1, ... ,
K -
1,
(7.4.18)
where D k are fixed numbers (either real or complex), singles out a unique solution to the characteristic equation (7.4.12). Proof. Let us assume that the statement is wrong. Then two different polynomials, P 1,1(t) and P I,it}, must exist, which satisfy Equality (7.4.13) for a function cp(t). The difference of the polynomials, P I ,(t) = P I(t} - P I,it), multiplied by + )(t}b(t), satisfies Equalities (7.4.18) for D k = 0, k = 0,1, ... , K - 1. Consider next the equation K
K
-],
K
-
K
w;
K
(7.4.19)
°
If K > for the original characteristic equation, then for Equation (7.4.19) the index is negative (I< = - K), and the solution is unique for any Holder right-hand side subject to the condition j
=
0,1, ... ,
K -
1. (7.4.20)
208
Method of Discrete Vortices
However, the right-hand side of Equation (7.4.19) incorporates a proper function for the operator I l I l(w; + l . Xt), and hence by Formula (7.4.16) one deduces that f(t) == 0 on L, and b(t)P 1 it) == O. Because b(t) is a Holder function on Land b(t} ¥= 0 on L, we'deduce that P 1 it) == o. K
_
K
•
-
Note that the uniqueness conditions for a singular integral equation of the first kind on a system of segments of positive index K > 0 in the form (7.4.18) were originally formulated in Lifanov (I979b); for an equation of the second kind with variable coefficients they were formulated in Lifanov and Matveev (1983). The preceding proof was proposed in Afendikova, Lifanov, and Matveev 1987). For constructing a numerical method we will need a relationship for the operators jl+l(w~+l.). Let us denote ~n;m(z) = Pn(z) + A m(z - c) and ~n'O(z) = Pn(z), where Pn(z) is a polynomial of degree n, A_m(z - c) = C~I(Z - c- 1 + ... +c m(z - c)-m, C $ L. The function ~n'm will be called a generalized polynomial of degree (n; m). The following th~orems are true. Theorem 7.4.2. Let a(t) and b(t) be arbitrary functions belonging to the class H on L. Then the relationship
(7.4.21) holds for any function ~n; m( z).
w;
Proof. Let us prove the theorem for t ). Consider function x(z)· ~n' m(z), where x(z) is the index K canonic function (7.4.6) for Equation (i4.l2). Because x(z) is analytic outside L, when x l j l(t) E H* on L, and at infinity behaves as z - \ there exists such a function ~n _ m( z) that the function
(7.4.22)
t
Taking into account that ~n(:':(t} = ~n(' -,j(t) = ~n' m(t)it~ m(t) = inl _ t~ m(t} = in _ K m(t), and Xl ± )(t) = [a(t) '+ b(t)] . z(t) /(a 2 (t) ~ b 2(t)), one gets from Formula 0.4.22) Relationship (7.4.21) for w~ + l. In order to one has to consider the function get the same relationship for x-I(z)· gn;m(z). Then, the following theorem may be formulated.
w;-)
209
Singular Integral Equations of the Second Kind
Theorem 7.4.3. Let b(t) be a polynomial of degree I. Then the following relationship is valid for operators f (.± >( w~ ± ) • ):
n
+
K ;:::
I - 1. (7.4.23 0
)
For a singular integral operator fJ l )( w~ r ) . ) of the second kind with variable real coefficients of polynomials Pn(t) and L being a segment, Relationship (7.4.23 0 ) was originally derived by Elliott (1980). Elliott also proved that a system of polynomials {Pn(l)} orthogonal with the weight wS ± )(1) transforms into a system of polynomials (Qm{t)} orthogonal with the weight w~ T >(t). Thus, he generalized the known relationships for Chebyshev and Jacobi polynomials. Theorems 7.4.2 and 7.4.3 were originally formulated and proved in Afendikova, Lifanov, and Matveev (1986, 1987). • By using (7.4.23 0 ) one may construct a numerical method for the characteristic equation supposing that b(t) is a polynomial of degree l. Matveev (1988) has also done this for the case b(t) = br(t)· , b,(t), where br,(t) is a polynomial of degree r\ ;::: 0, and b\(1) may vanish at a finite number of points belonging to the line L. For a fixed index K and a natural number n we choose two systems of points Eo = {tol,t02, ... ,ton-J and E = {t l ,t 2, ... ,tn} different in pairs. For K < 0 we get n - K ;::: I + IK I and Eo does not contain any roots of b(to)·
For function f(t} we choose an approximating generalized polynomial I; n,(t) of degree n l + n 2 - K - 1, where n l + n 2 = n. In particular, the polynomial may satisfy the equality fn , I-n (tom) = f(t om ). Then the " index K solution to the characteristic equation (7.4.12) is given by the function w~ + )(t)t/Jn I _ I-,n2(0, where t/Jn I _ I-n (t) is a generalized polynomial of ,2 degree n l + n 2 - 1, and the following equality holds:
fnt-
K
K-
(7.4.23) As an equality of generalized polynomials of degree n l + n 2 - K - 1, the latter equality is equivalent to the following system of n - K linear algebraic equations:
m
=
1,2, ... ,n -
in the coefficients n of the generalized polynomial t/Jn ,_ I-" n .
K,
(7.4.24)
Method of Discrete Vortices
210
If K = 0, then Equation (7.4.23) has a unique solution, and hence, System (7.4.24) also has a unique solution. If K > 0, then by Theorem 7.4.1, Equation (7.4.23) has a unique solution only if the K conditions (7.4.18) are met. Thus, in this case the system of linear algebraic equations (7.4.18) and (7.4.24), where cp(t) is to be replaced by w~' )(t) t/Jn , _ 1, n,(t), has a unique solution. If K < 0, then the function w~ +- )t/Jn, _ I; n,(t) obtained from (7.4.13), where f(t) is replaced by fn , _ K I'' n(t), is a solution to Equation (7.4.23) only if 2 Conditions (7.4.14) for fn, _K I; n.,(t) are met. However, as far as the function fn, _ K - I; n,(t) is an arbitrary function, the conditions will not necessarily be met. Therefore, the function fn, _ K, _ I; n,(t) appearing in Equation (7.4.23) must be replaced by . IKI- 1
fn~-K-I;n,(t) =fn,-KI;n,(t) - b(t)
L
YI1 ,V tV ,
v= 1
where the numbers Yn . v' V = 0, 1, ... , IK I - 1, must be such that all the conditions (7.4.14) are met. Numbers 'Yt" v exist and are uniquely defined by Conditions (7.4.14) for the function fn~-K-I;n,(t), and if Ilf(t)fn, K-I;n,(t)IL. -) 0, then Yn,v -) for n -) 00, JJ = 0,1, ... , IKI - 1. Thus, for K < 0, instead of Equation (7.4.23), one must consider the equation
°
IKI~
J
L Yn,vt(~+ t+)(w~T)t/Jn,
b(tn)
I;n,)(t n ) =fn, K-I;n,(tO)' (7.4.25)
v~()
For the chosen numbers this equation has a unique solution, and, hence, the equivalent system of linear algebraic equations IKI- 1
b(tnm)
L Yn,vt~m
v~o
+ t+)(w~+-)t/Jn, I;n,)(tnm) =!n,-K-I;n,(t nm ), m=I, ... ,n-K,
(7.4.26)
also has a unique solution. From the preceding statements, the next theorem follows.
Theorem 7.4.4. Let K be an index of the solution CPK(t) = w~ +- )(t)t/J(t) to Equa{ion (7.4.12). Then, the system of linear algebraic equations
Singular Integral Equations of the Second Kind
211
(7.4.27) in the coefficients of the generalized interpolation polynomial t/Jn l' n,(t) for K ~ 0 and the set of the coefficients and the coefficients 'Yn . v' V = 0, 1, ... , IK I - 1, for K < 0, is well-conditioned for any n, n - K ~ I .:.... 1. For K < 0 the last K equalities must be discarded, because 71( K) = 0, K ::; 0, and 71( K) = 1, K > 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)
Method of Discrete Vortices
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
Method of Discrete Vortices
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
Method of Discrete Vortices
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
=
l, ... ,m.
(8.1.8)
Singular Integral Equations with Multiple Cauchy Integrals
219
A similar procedure is applicable to a characteristic singular integral equation of the second kind,
(8.1.9)
where L I is a piecewise smooth curvc. It is obvious that arbitrary functions of the variable t~ entering the general solution of a given positive index K must have the same singularities as the function f(t~, t5) or the variable t(f. In a similar way one can consider the procedure of finding solutions to Equation (8.1.1) which have the form cp(t I, ta) = fjJ(t], t 2 ) / (q - t I), with q being an internal point of the curve L] differing from the end points of the curve. By supposing that a(l(\) and b(t/l ) entering Equation (8.1.9) are constants, one can write the equation in the operator form (8.1.10)
where AL"t(\ designates the operators applied to the function cp(ll,t~) entering the left-hand side of (8.1.9). AL"I(\ wiIl be caIled a characteristic singular operator of the second kind. Let us next consider characteristic singular integral equations of the second kind with double Cauchy intcgrals whose operators may be represented by products of the corresponding one-dimensional operators with respect to each of the variables, i.e., equations of the form (8.1.11)
or
meaning
(8.1.12)
220
Method of Discrete Vortices
where a; + b; = 1, ak' and b k are real numbers, k = 1,2, f{tJ, l(~) on L, X L 2 , and L] and L 2 are plane curves. 1.
H
Let L 1 and L 2 be unit-radius circles centered at the origin of coordinates. By using Formula (7.2.9) repeatedly, we conclude that the function
X
2.
E
fUlL t~) dlJ dl~ (I L,XL z 10 -1 1)( 102 -1 2)
if
(8.1.13)
is a unique solution of Equation (8.1.12). Hereinafter A /.,1,/, denotes operations performed on the function f{t(\, l~) with respect to a corresponding variable either in Formula (7.2.9) or Formulas (5.1.2) and (7.1.2) for the corresponding indices of the operators under consideration. Let now L 1 be a circle and L z be the interval [ -1,1]. We start by seeking a solution whose index with respect to the coordinate 1 2 is equal to 1. In this case operator A L" rA is uniquely invertible for any right-hand side. To invert the operator ALz,r.j uniquely one has to know the function fL l CP{tI,12)dt 2, which will be defined uniquely if the function (8.1.14) supposed to belong to the class H on L I' is specified. Thus, to find a unique solution to Equation (8.1.12) or Equation (8.1.11), one has to consider the system
(8.1.15)
Singular Integral Equations with Multiple Cauchy Integrals
By applying Formulas (7.2.9) or (7.1.2) for
K
221
= 1, one gets
where (see 0.4.35» T,(0'2) = -sin 0'7T)/7T, -1 < 0'2 < 0, a 2 + b 2 cot 7T0'2 = 0, w;X,(t 2) = (1-t 2 )"2(1 +t 2 )t\ and -(0'2 + f32) = 1. If for a solution to Equation (8.1.10, K = or -1 in the variable t 2 , then it is given by
°
(8.1.17) where Wf2(t2) must be replaced by w ",(t2), K = 0, -1. If the index is equal to - 1 in the variable t 2, then the function cp(/', t 2) given by (8.1.17) is a solution if the condition K
(8.1.18)
°
3.
is met. Note that the possibility of obtaining the general solution to Equation (8.1.1 from System (8.1.15) allows us to change the order of operations performed with respect to the variables t I and 1 2 • The form of the result will be the same. However, if the result is found by inverting the operators in Equation (8.1.10, then it does depend on the order in which the operators are applied [because the arbitrary function Cl(tb) enters in different ways]. The convenience of System (8.1.15) will become clear when we construct direct numerical methods for solving Equation (8.1.11). Let L) and L 2 be parts of the interval [-1, 1]. We will seek the index 1 solution to both variables, i.e., K = 0,1). In this case opera-
222
Method of Discrete Vortices tor
AI."t~
is uniquely solvable if the function
is known. For operator A L 2· ,20 to be uniquely solvable one has to know the function II. 2 cp(t l , t 2) dt 2, which may be determined uniquely with the help of the operator AL"t,', if the function ClUJ) = AL"r,', l (IL,CP(t ,t 2 )dt 2 ) and the number C = Il.,(JL,CP(tl,t 2 )dt 2 )dt' are known. Thus, in the case under consideration, a unique solution will be found from the system
AI""li
({,cp(t" t
2 )
dt
2 )
= Cl(tJ),
~.' dt (~.,cp(t', t l
2 )
dt
2 )
= C. (8.1.19)
By applying Formula 0.1.2) for the index 1, one gets
(8.1.20) Let the solution now have index 0 in the variable t l and index 1 in the variable t 2 , i.e., K = (0, 1). Then the operator A L " r,', is uniquely invertible for any right-hand side; to make the same true for the operator A L20 r"0 one has to know the function IL 2 cp(t',t 2 )dt 2 , and hence, the index K = (0, 1) solution may be found uniquely from the system
(8.1.21)
Singular Integral Equations with Multiple Cauchy Integrals
223
By again using Formula (7.1.2) for K = 0 with respect to the variable t I and K = 1 with respect to the variable t z, one gets
Let us stress once again the functions C1(t(\) and CitJ) meet condition H on the corresponding curves. Let us indicate the solutions and the systems they can be obtained from, for all the other combinations of indices. If K = ( -1,0, then subject to the condition
(8.1.23)
the system has the form (8.1.21), and the solution has the form (8.1.22). Unique solutions of the indices K = (0,0), (0, -0, (-1,0), (-1, -1) may be found from Equation (8.1.20. However, for the variable corresponding to a negative index, the right-hand side must satisfy identity (8.1.23). Solutions of the indices K = 0,0),0, -0 are found in a similar way. Thus, if L] and L z are portions of the segment [ -1,1], then the
Method of Discrete Vortices
224
(8.1.24) Note that for the index K = 0, - 1) the function cp(t 1, t 2 ), given by Formula (8.1.24), is a solution to Equation (8.1.1 1) if the condition (8.1.25)
4.
5.
is fulfilled. 01 = 02 = 0 in Equation (8.1.11), then the preceding formulas for solutions of the equation provide inversion formulas for multiple singular Cauchy integrals on products of plane curves (Lifanov 1978a, 1978b, 1979). . If a full singular integral equation with multiple Cauchy integrals, that is, an equation of the form If
=
f(tJ, t~),
(8.1.26)
is considered, then by using Vekua's approach (Muskhelishvili 1952) (i.e., by solving the equation with respect to the characteristic part for a given index) it may be reduced to an equivalent (in thc sense of finding solutions) integral Fredholm equation of the second kind.
8.2. NUMERICAL SOLUTION OF SINGULAR INTEGRAL EQUATIONS WITH MULTIPLE CAUCHY INTEGRALS Consider equation (8.2.1) where L, and L 2 are unit-radius circles. Let the sets E i
=
{t£"
k;
=
Singular Integral Equations with Multiple Cauchy Integrals
225
1, ... , nil and Eb = {tb j ;, ji = 1, ... , nil form a canonic division of the circle L i , i = 1,2. The following theorem is true (Lifanov 1978a). Theorem 8.2.1. Let the function f(t], t Z ) E H on the torus L] X L z. Then between a solution to the system of linear algebraic equations,
ji = 1, ... ,n i , i = 1,2,
(8.2.2)
and a solution to Equation (8.2.1),
(8.2.3)
the relationship
ki
=
1, ... ,n i , i
=
1,2,
(8.2.4)
holds where the quantity O(tl I , t; 2 ) satisfies the inequality
(8.2.5) and ni/N ::s:; R < +oc. For the number A we have:
1.
A= a -
af
A=
1)],
n"
p
2.
€
tf r+
=
p
(where
+]
-
€
is however small), iff E H(a) on L]
p
if f,f() E H(a), k = 1,2 and = 2m" + 1, P = 1,2. a -
X
L z and
tf· €,
af
I'
= i[27Ttfp /(2m"
+
Proof. Let MJn,',l be a matrix obtained from the matrix whose determinant is ti\~) in Syst~m (5.1.3), by substituting tJ and t l for to and t, respectively. Then the determinant of System (8.2.2) may be written in the form
(8.2.6)
226
Method of Discrete Vortices
where a~, is in degree n i , i
n' n2)
( il' o
=
1,2,
= 1 M- o(n') ,--:0----;:,r t2 _ 12 On 2
1
By calculating determinant il~" n ) with the help of the approach 2 used for calculating determinant il(;), one arrives at il(;" n I) Continuing the process, we get 2
(8.2.7)
where il(;') is obtained from il(;) by substituting n i for n and for I k , lOb tiAk respectively, i = 1,2. I I 2 It is evident that D~)n , n ) *- 0 for any n 1 and n 2 • Hence, by applying the Cramer rule to System (8.2.2), one gets I
k;=1, ... ,n i ,i=1,2, (n', n 2 )
(8.2.8)
(n' n 2 )
where D o,(k"k 2 ) is obtained from Do' by substitutinR the free terms* column number (k il(;'('kn)k) (J" J") is obcolumn for the l , k 2 ); I 2 ' I' 2 I' 2 tained from il(; ,n ) by striking out the column number (k l , k 2 ) and the 1
o
*(k" kJ.) > k 2 = k 2•
(k" k 2 ) if
k2 >
k 2 ; or
k2 ~
k 2 , but
k, >
k,;
(k" k 2 ) =
(1<" k 2 )
if k, =
k"
Singular Integral Equations with Multiple Cauchy Integrals •
•
- (n')
227 - (n')
row number (iI, jz). Let us denote by MO,/I,(k,.},) (Mo,r1,(k,,0); Mrr"~(o,}) the matrix obtained from matrix M(~,~:) by striking out the column number k l and the row number j, (the column number k, only; the row number jl only, respectively). Then one gets
(8.2.9) Next we move the matrix column with Mcin,',l (k 0) and the matrix row - (n') , , " with Mo,r',ro,},) to the first positions in the columns and rows, respectively. The rearrangement does not result in a change in the sign of determinant (8.2.9). In fact, the movement of a column of determinant (8.2.9) from the said matrix column to the first position requires n l (k 2 - 1) rearrangements, whereas the movement of the entire matrix column requires (n l On l(k 2 - 1) such rearrangements, because no rearrangements are performed in the matrix column itself. The number thus obtained is evidently even. The transformed determinant (8.2.9) will be denoted by Ii(;'(kn,~~,),(j"h)' It can be calculated by using the following approach. First, the last matrix row is subtracted from all the preceding matrix rows save the first one. Next the last matrix row from which the j, th row is preliminarily struck out is subtracted from the first matrix row, and common multipliers appearing in the matrix rows and columns are taken out of the determinant sign. Then the last matrix row becomes
A similar operation will be performed in the last determinant with respect to the matrix columns. The resulting common multipliers will be taken out of the determinant sign again. Then the last row in the last determinant will have the form
(0 0
228
Method of Discrete Vortices
By applying the rule of expanding a determinant in a matrix row, we come to the block determinant whose order is smaller by 1 as compared with the original. By repeating the preceding operations n 2 times, one gets
nn
l~m
(t5p
-
t5m)
nn
l~m
h- I
(t~
- tn
n (t5m - t(~h)
m=1
p*k 2
p*h
Thus, we have
(8.2.10) By substituting this formula into (8.2.8), we get
(8.2.11) The use of Formula (5.1.13) for 'Pn l
1 k (tkI ,tk2) -_ -1-[0 Uk (n l
n2 ,
1
2
'
)
1
K =
0 results in
(n 2 )
-2-[0 k 1 Uk, • 2
l
(8.2.12)
Singular Integral Equations with Multiple Cauchy Integrals
229
where b~j, = t~j, +] - t~j, or b~j, = 27Tit(~jJ(2mk + 1) (if afp have the corresponding form). Hence, by using Formulas (6.1.10), (6.1.11), and (6.1.20), Formula (8.2.12) may be reduced to
+ O( N ' In 2 N),
(8.2.13)
because if one uses Formulas (6.1.10) and (6.1.11), then
and if Formulas (6.1.20) are used, then the quantity O( N-' In 2 N) must be discarded in Equation (8.2.13). The use of the quadrature formula for multiple singular integrals terminates the proof of Theorem 8.2.1. • Theorem 8.2.2.
Let function f(t6, t~) have the form
(8.2.14) where l/J(t6, t~) E H on L] X L 2, qi ELi' Vi E [0,1), i = 1,2, and the sets E i and Eb fomz a canonic division L i such as in the case of Theorem 6.1.2. Then between a solution to the system of linear algebraic equations (8.2.2) and the solution (8.2.3) to Equation (8.2.1), Relationship (8.2.4) holds where 0(11 1, t1), k i = 1, ... , n i, i = 1,2 satisfies the inequalities:
1.
for all points (11 1,t12 ) ever small:
E
[L] \ O(q" 8)]
0(t11,t1,) 2.
$
O(~AI)'
X
[L 2 \ O(q2' 8)], 8 is how-
(8.2.15)
For all points (t k'I ,t12 ),
A2 > O. (8.2.16)
230
Method of Discrete Vortices
= 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 cp
n, 2
on L I X L 2 if, other conditions
The following theorem is true. Theorem 8.2.3 (Lifanov 1978a). Let function f( t I , t 2) E H on L I X L 2' where L I and L 2 are unit-radius circles. Then between solutions to the systems of linear algebraic equations,
jp = 1, ... ,n P , p = 1,2,j, i=jq"
(8.2.17)
Singular Integral Equations with Multiple Cauchy Integrals
231
n2
n'
L L
CPn'nz(ti" t;,)ak,az, = C,
k,~lk2~1
where the sets E P and EC form a canonic division L P' P = 1, 2 (for (8.2.17), ql = t(\j' and for (8.2.18), q p = tCj , P = 1, 2), and the solutions ~(t 1, t 2) to Equation (8.2.1), qp
I I_[ff 1
12 __ cp( t , t ) 4 7T
q -t
(q,-tl)f(tJ,t(~)dtl)dtJ
L,xL,
(I 1 (2 2) t -to) t -to
(8.2.19)
1
X
[if.
L,XL 2
(ql - t(\)(q2 - tJ)f(tJ, tJ) dt~ dtJ (t 1-to 1 ) (t 2-to 2)
Method of Discrete Vortices
232
(8.2.20)
respectively, Relationship (8.2.4) holds in which the quantity O(t; , t1 ) satisfies Inequalities (8.2.16) and (8.2.15) at all points (t k , t1) E L2 X [L 1 \ O(ql' 8)] in the former case and at all points (tk" tl) ~ [[,1 \ O(q" 8)] X [L 2 \ O(q2' 8)] in the laller case. Note that the numbers AI and A2 depend on the differential properties of the functions f(t', t 2), C1(t(\), and Cit~) as in 7heorem 8.2.1 (if a1 , ,a; 2 are chosen in the same manner). Note 8.2.1. equation
Theorems 8.2.1-8.2.3 may be readily generalized onto the
f···J
L,
cp(t) dt x '"
XL,
«(to -
t»
=f(to)'
(8.2.21)
where L I' ... , L{ are circles whose solution in the class of functions H on L 1 X ... X L{ [if f(to) belongs to the same class] is given by the function (see (2.1.12» (-1){
cp(t)
= -2-{ 7T
f··· f
L,X .. ' xL,
f(to)dt o
«t - to» .
(8.2.22)
In order to write the subsequent results in a compact form and to emphasize the special structure of the matrices of Systems (8.2.2), (8.2.17), and (8.2.18) of linear algebraic equations, one should be reminded of the following definitions (Proskuryakov 1967). The right-hand direct product of matri~ A of the order of m X m and matrix B of the order n X n is an array C = A X· B whose element posed
233
Singular Integral Equations wilh Multiple Cauchy Integrals
at the intersection of the jth row and the ith column is defined by
Cji =
aj;B.
The lefl-hand direct product of the same matrices is an array D = Ax B, where Dji = Abji' The products possess the properties Ax' B
=
B'XA,
A X· (B X·
C)
= (A
X·
B) X· C,
Similar properties are also valid for the left-hand product. For any direct product,
Let us write in matrix form the systems of linear algebraic equations for characteristic equations on a segment and a circle for the case of uniform grids: 1.
M(n)
2.
M(n)
3.
M(n)
.0
O,tCP, = Jlo'
(fl~l) C'
=
l,tCP,
1,'
(A on ) ='J,o CPt
I
'
where
M(n) O,t
=
M(n)
l,t =
(aO)n J,k
J,k~
1
,
aO
j, k -
ak ---
It'
OJ -
(l)n aj,k
j,k~ l '
j,k = 1, ... ,n, j *- n, a~ k
(n) M -1, I
( =
j,k=l, ... ,n,
k
=
ak,j
=
n; k = 1, .. "n,
,)n"I,n
a j, k
j
~ I, k _ 0'
k = 1, ... , n, j = 1" .. , n + 1, aj~J = 1, k = 0, j = 1, ... , n + L
234
Method of Discrete Vortices
~
] = ( fUo) )nj~l' +]
Here the points numbered earlier in j starting from zero are numbered starting from 1. If the grids are nonuniform, then CPt must be replaced by t/Jt [cpt = w{t )t/Jt l. In light of the latter definitions, the systems of linear algebraic equations (8.2.2), (8.2.17), and (8.2.18) for a torus may be presented in the form (8.2.2')
(8.2.17')
(8.2.18')
where jq, = n i , i = 1,2, and the pairs (k], k z ) are arranged in lexicographic order. In a similar sense, we treat the right-hand side matrix columns appearing in Systems (8.2.2'), (8.2.17'), and (8.2.18'). Let us next consider a characteristic integral equation of the second kind with multiple Cauchy singular integrals (see Equation (8.1.12) for the case when both L] and L z are circles). The following theorem is true. Theorem 8.2.4. Let the function f(t6,t5) E Hr(a) on the torus L] Then between a solution to the system of linear algebraic equations,
X
L z·
(8.2.23) where M6,nt\l and Mci~t2! are matrices obtained from the matrix of System (7.2.3)
Singular Integral Equations with Multiple Cauchy Integrals
235
by subscribing the letters or, in other words, tP
Om p
(-a p + bpi)
7T + t/Jp } = t P exp { i----'-m" 2nP + 1 '
(a p + bpi) ,
divide the circle L p, p = 1,2, into equal parts, and a k = 27Titf / (2n p + 0, and the solution cp(t l , t 2 ) (see Equation (8.2.13) to Equati~n (8.2.12), an inequality of the form (8.2.4) holds where the quantity 0(11" t1) satisfies Inequality (8.2.5) with A = r + a - € and € > 0 being however small.
Now let L I and L 2 associated with Equation (8.1.12) be a circle and a segment, respectively. Then the following theorem is true. Theorem 8.2.5. Let f(l I , t 2) appearing in Equation (8.1.12) belong to H on L I X L 2 , and a l = a 2 = 0, b l = b 2 = 1. Then Relationship (8.2.4) holds, on the one hand, between solutions to the systems of linear algebraic equations, (n').
(
(II')) CPIJ
M(l , r J X M o• I'
,
_ lr' rO,O (',
I' -
0.
(8.2.24)
lJ
(8.2.25)
M(n') ( M(n'rX (l,1 -1,1
1)( 'Yr',on') = CPr',r'
FC?, -, I JIII·l o
(8.2.26)
'
where Mr,'? is the matrix of System (7.2.3), and the matrices M~:I;'!, K = 1, -1, are the matrices of Systems (5.1.3)-(5.1.5) multiplied by 1/ 7T, and the solutions cpU I, t 2) of an equation under consideration, 1
+ t~
--2
f(tJ, t~) dt6 dt(; I
I
2
2'
I-to (t -to)(t -to)
(8.2.27)
236
Method of Discrete Vortices
(8.2.28)
(8.2.29) where in the case of Equation (8.2.28) function cp{tl, t 2) meets the condition
(8.2.30) and in the case of Equation (8.2.29) function cp(tl, t 2) is a solution to the original equation subject to condition (8.1.18). In Equation (8.2.4) the quantity o(ti l , tf) satisfies both Inequality (8.2.15) for the points (tll' tf) E L] X [-1 + 8,1 - 8] and Inequality (8.2.16).
Note that System (8.2.25) has the form
j] = 1, ... ,n',j2 = 1, ... ,n 2 - 1,
j]=I, ... ,n',j2=n 2. (8.2.31)
In order to ensure a higher-order convergence of an approximate solution to the accurate one, one has to account for the singularity. For example, in the case under consideration with L] and L 2 being a circle and a segment respectively, function cp(t], t 2) may be represented in the form (8.2.32) where w)t 2 ) is of the same form as in (5.2.1).
Singular Integral Equations with Multiple Cauchy Integrals
237
n'" . Th en, t/Jt',t' = «t/Jn'.n' Ik]" t k2 , »n' k,:k,~ I' lor 'Pt',t' cn!enng Systcms (8.2.24)-(8.2.26) must be substituted and thc matrices m~n:L K = 0,1, - 1, should be replaccd by thc matriccs of the systems of' lincar algebraic equations (5.2.16), (5.2.2), and (5.2.15) multipicd by 1/7T. The rate of convergence of an approximate solution to the accurate onc, i.c., the numbers A] and A2 in formulas of thc form (8.2.15) and (8.2.16) for all points
,I; )
,1, "
J lu.'O
C.'n l , l(~
(8.2.33)
C'll,n 2
(8.2.34)
(
M(n')
,'X
'YOn' on,) 'Yt',On' =[,1'1]
M(n'),)
1,1
-1,1
(
'YOnl,t2
to,f o
'
(8.2.35)
'Pt'.t'
where MK,r p , K = 0,1,- 1, P = 1,2, are thc Systems (5.1.3)-(5.1.5) matrices multiplied by 1/ 7T. Bctwccn solutions to these systcms and thc corresponding solutions of the characteristic singular integral equation of the first kind with a double Cauchy integral, Rclationship (8.2.4) holds where the quantity o
tn
238
Method ofDiscrete Vortices
jp
= O,I, ... ,n p ,p = 1,2,
(8.2.36)
where YOn' • On', Yrk' l , and YOn' • 12k ... are the regularizing factors. Similar systems may be constructed for indices K = 0,0), (-1,0), and (-1,1). If nonuniform grids are used, then the solution must be presented in
the form (8.2.37) where
K], K Z =
0,1, -1. In Systems (8.2.24)-(8.2.26) and (8.2.33)-(8.2.35),
t/Jr', I ' must be substituted for 'Pr',r', and the matrices of Syste,!TIs (5.2.16),
(5.2.2), and (5.2.15) must be substituted for the matrices M~~), p = 1,2; K = 0,1, - 1. Now let 01' and hI' entering Equation (8.1.12) be real, o~ + hi; = 1, P = 1,2, and L] and L z be either a segment or a circle. Because the equation was constructed by using a product of one-dimensional singular operators, the corresponding systems of linear algebraic equations must be constructed with the help of the left-hand-side direct product of matrices of corresponding systems of linear algebraic equations for one-dimensional characteristic singular integral equations. Thus, the systems of linear algebraic equations will have the form of Systems (8.2.24)-(8.2.26) and (8.2.33)-(8.2.35), where t/JI' I' is substituted for 'Pr' r2, and the matrices (n P ) , " M K • rP , P = 1,2; K = 0,1,- 1 are the matrIces of Systems (7.1.7), (7.1.9), (7.1.10), or (7.2.3). Next we consider full equation (8.1.26) with a regular kernel, for which the systems of linear algebraic equations may be obtained from the corresponding systems for characteristic equation (8.1.12) by adding the terms nl
n2
LI L
k,~
k2~]
K(tl,m" t~m2' ti" tl,)t/Jn'n (ti" tl,)ok,ok" 2
We will require that Equation (8.1.26) have a unique solution subject to corresponding conditions imposed onto the characteristic part. In this case, the system of linear algebraic equations for the equation is equivalent to a system of linear algebraic equations approximating the corresponding
Singular Integral Equations with Multiple Cauchy Integrals
239
Fredholm integral equation of the second kind, to which the preceding equation is equivalent and that has a unique solution. Thus, an approximate solution to Equation (8.1.26) is seen to converge to the corresponding accurate solution.
8.3. ABOUT AN INTEGRODIFFERENTIAL EQUATION The following two-dimensional integrodifferential equation is used in aerodynamics to calculate approximately aerodynamic characteristics of a thin rectangular finite-span wing in stationary flow (Bisplinghoff et at. 1955):
f
b
f' y/(x, z)
Jb
-b -{
+
2
+ (zo (x o - x)(zo - z)
Xo - x
Z)2
dxdz
=
f(x o , zo), (8.3.1)
where X o E (-b,b), Zo E (-1,1), and f(x,z) E H on [-b,b] In the class of functions y(x, z) meeting the conditions
X
[-1,1].
y(x,/) == y(x, -I) == y(b, z) == 0, yz' ( x, I)
=
00,
yz'(x, -I) =
y( -b, z)
oc,
=
oc,
(8.3.2)
characteristic of aerodynamics, the equation is equivalent to the following Fredholm equation of the second kind:
y(t,O + - 1 'TTl
12- fb -b-t
+t
-b
y(x,
n dx -
1
-2 'TT
f'
y(t,
-{
Note 5.3.4 implies that the following theorem is true.
z)K(~, z)
dz
Method of Discrete Vortices
240
Theorem 8.3.1. If Equation (8.3.3), and hence, Equation (8.3.1), has a unique solution within the required class offunctions, then between a solution to the system of linear algebraic equations
j=I, ... ,n,m=I, ... ,N,
(8.3.4)
where Xi = -b + ih lo hI = 2b/(n + 0, x o; = Xi + h 1/2, i = 1, ... , n, Zk = -I + (k - Oh z , h z = 21/N, ZOk = Zk + h z/2, k = 1, ... , N, ZN+I = I, and the solution y(x, z) to Equation (8.3.0 subject to Conditions (8.3.2), the relationship
i holds where quantity O(x;, 1.
For all points
ZOk)
(Xi' ZOk) E
=
I, ... ,n, k
=
1, ... ,N,
(8.3.5)
meets the following conditions: [-b + 8, b - 8]
X
[-I, l], (8.3.6)
2.
For all points (x;,
ZOk) E
where h
=
X
[-I, l],
N
n
L L i~
[-b, b]
I
k~
O(x i , zodhJhZ
=
O(h A,),
(8.3.7)
I
max(h l , h z ), h/h;.:o;; R
<
+oc, i
=
1,2 for h ---)
o.
Proof. It suffices to prove that System (8.3.4) is equivalent to the system of linear algebraic equations approximating Equation (8.3.3). To do this we
Singular Integral Equations with Multiple Cauchy Integrals
241
write System (8.3.4) in the form
N
n
L L
=f(xoJ,zOm) -
v~
x [X Oj - Xv + (X Oj
I I'
~
'Yn,N(X",Zo,J I
~b2 + (ZO~- Z/L)2 Xv )( ZOm
- b
zd
j = l,oo.,n, m = 1,oo.,N.
(8.3.8)
The matrix of System (8.3.8) with respect to the unknowns on the left-hand side is a left-hand side direct product of the well-conditioned matrices M6,n; and Mf.~) of Systems (5.1.3) and (5.1.55), respectively. By solving the system with respect to the left-hand-side unknowns and performing appropriate transformations, one arrives at the system of linear algebraic equations approximating Equation (8.3.3). • This is a Fredholm equation of the second kind. Because it is physically clear that Equation (8.3.1) has a unique solution, Equation (8.3.3) also has this feature. Thus, starting from certain sufficiently large values of nand N, the system of linear algebraic equations (8.3.4) becomes solvable, the solution converging to the corresponding solution of Equation (8.3.0.
Part III: Application of the Method of Discrete Vortices to Aerodynamics: Verification of the Method
9 Formulation of Aerodynamic Problems and Discrete Vortex Systems
9.1. FORMULATION OF AERODYNAMIC PROBLEMS Here we consider steady and unsteady motion of a wing of an arbitrary plan form in an inviscid incompressible fluid (see Figure 9. O. In aerodynamic problems both the form of a body and the law of its motion are supposed to be known. If a body under study is elastic, then the law of deformation is also supposed to be specified. * The conditions under which a motion (flight) proceeds are also assumed to be known. As a rule, the medium through which a body travels is supposed to be boundless and initially undisturbed. However, one may consider the motion of a body through a medium disturbed by wind, currents, atmospheric turbulence, etc. In this case, one has to determine disturbed flow velocity V(x, y, z, t) = {I/", ~,~} and pressure p(x, y, Z, 0. For calculating the four required functions 1/", ~, ~, and p, one has three Euler equations as well as the continuity equation (Golubev 1949). Of paramount importance are boundary conditions at the surface of a body. If a medium is viscous, then the "no-slip" condition must be employed, i.e., at the surface of a body V = O. For an inviscid fluid the only no-penetration condition is used according to which the normal component of the flow velocity is equal to zero at the surface of a body: Vrel
•
n = 0,
(x, y, z)
E
(T,
(9.1.1)
• Acroclasticity studies problems where both the deformations of a body and the flow of a fluid have to be determined (G. Ch.).
245
246
Method of Discrete Vortices
/---($,--
FIGURE 9.1.
Schematic representation of a finite-span wing. The circulatory problem.
where n is a unit vector of the normal to the body surface
(T
at a point
(x,Y,z).
It should be pointed out that there exists a class of problems associated with flows past parachutes and paragliders whose surfaces are formed by thin fabrics. In this case, the surface is permeable, but the law of cross flow is known. It is determined experimentally and is actually a relationship characterizing a given material. Usually, the relationship connects the cross-flow velocity and the pressure drop across a fabric. Further simplification of the formulation of the general problem is closely related to the following experimental evidence. Flo~ past a body is accompanied by formation of an aerodynamic vortex wake within which the fluid participates not only in translational and deformational motions, but in rotational motion as well. Outside the region flow is irrotational. In this case the perturbed flow outside a body and the wake is potential, i.e., may be characterized not by the three unknown functions 1/", ~., and ~, but by only one function
v"
=
~=a
a
~ =
8
(9.1.2)
and the continuity equation becomes the Laplace equation (9.1.3) outside ( I and (II' where (II is the wake formed behind a moving body. Because it is natural to suppose that far from the surface of a body ( I and ~ wake (II flow disturbances die away, a solution
=
(9.1.4)
0,
r-->X
for points (x, y, z) located infinitely far from the body
(I
and its wake
(II'
FormuLation of Aerodynamic ProbLems and Discrete Vortex Systems
247
The Euler equations of motion may be integrated, resulting in the well-known relationship between pressure p and derivatives of the velocity potential
(9.1.4')
where Vrel and V* are the relative and translational velocity components, respectively, p is the density of the fluid (assumed to be constant), and Px is the known pressure at infinity. Thus, the unknown function p(x,y,z,t) may also be excluded from the general problem formulation. The following major peculiarity of the general problem formulation should be mentiorled. When constructing the vortex wake behind a body, one has to take into account general theorems of hydrodynamics that have been ignored until now, which follow from the general properties of flow velocity fields and the corresponding equations. The properties are: a. b.
c. d. e.
In steady flows vortices are directed along streamlines. In unsteady flows vortices shed from a body (the so-called free vortices) move along the paths of liquid particles and together with the latter. Velocity circulation over any closed contour is independent of time. A change in the circulation of a bound vortex is accompanied by shedding a free vortex. Because the vortex wake 0"1 does not develop any lift, the pressure across it must stay continuous: (9.1.5) Here the subscripts plus ( +) and minus ( - ) refer to opposite sides of the surface 0"1'
From the Joukowski theorem "in the small" (Belotserkovsky 1967), it follows that the relative velocity of free vortices is equal to zero; in other words, they travel together with the particles of a fluid. The physical nature of the problem under consideration and the required accuracy level determine one more, extremely important stage of the problem formulation, namely, the choice of the flow mode. The major modes are: a.
Noncirculatory flow past a body for which the vortex wake is ignored. This mode is used mostly for analyzing flows past highly elongated bodies as well as stationary oscillating bodies.
248 b.
c.
Method ofDiscrete Vortices Circulatory separated flow satisfying all physically evident conditions, including the requirement of the flow velocity and pressure being finite throughout the flow domain. What was previously said has a decisive impact on the choice of the flow mode. Thus, when studying flow past a thin wing with sharp edges (either leading or trailing, or side edges), one has to admit shedding of free vortices from all of them and impose the Chaplygin-Joukowski condition concerning the velocity finiteness at all the edges. Otherwise, the flow velocity at the edges tends to infinity. Note that in this case even a steady motion of a wing results in pulsating (unsteady) flows. Some simplified modes of circulatory flow past a body for which certain restriction may be lifted. Most popular are the modes that do not require the flow velocity and pressure at the leading and side edges of a thin wing, as well as at a cusp of the surface of a wing to be finite. As a result, the problem may be solved as a steady one, shedding free vortices only from the trailing edge of a wing.
Extensive experience in applying the preceding modes has been accumulated. AJI of them provide satisfactory data with respect to summary quantities; however, they generate locally incorrect data.in the vicinity of sharp edges and cusps.
9.2. FUNDAMENTAL CONCEPTS OF THE METHOD OF DISCRETE VORTICES The formulated aerodynamic problems will be solved with the help of the vortex method, whose major features are described in succeeding text. Both the wing (I and the wake (II are replaced by a continuous vortex surface: the wing (I by a surface formed of bound summary and free vortices stationary with respect to the wing, and the wake (II by a surface of free vortices traveling together with the fluid particles along the streamlines. This continuous vortex layer will be used for calculating the velocity field V whose potential is given by the function
Formulation of Aerodynamic Problems and Discrete Vortex Systems
249
r FIGURE 9.2.
Illustration of the Biot-Savart law.
models are employed as primary elements in problems under consideration. In plane problems, this is a vortex filament of infinite-span. In problems of flow past thin wings at small angles of attack, these are oblique horseshoe vortices. Rectilinear vortex segments-closed vortex polygons (as a rule, triangles or quadrangles) are used in plane problems, general nonlinear three-dimensional problems. The flow velocity fields induced by vortices are calculated with the help of the Biot-Savart formula according to which a segment of a vortex line ds long (whose circulation is equal to n induces at a point M o (whose distance from ds is defined by the radius-vector r) the flow velocity given by
r ds X r dV= - - - - . 41T
,3
(9.2.1)
The direction of ds must be chosen in such a way that the circulation around it, in accordance with the right-hand rule, is positive (see Figure 9.2). If the element ds begins at point M, then r = MM o. Let us denote by n a given discrete vortex entity along which the strength r stays constant due to the known theorems of hydrodynamics concerning vortex filaments (Golubev 1949). Then the velocity field induced by the vortex is given by the formula (9.2.2) The resulting velocity field satisfies the continuity equation throughout the space (with the exception of the discrete vortex itself). AdditionaIly, the flow velocities attenuate as the distance from vortex elements increases, tending to zero at infinity. In steady problems a wake trails into infinity. However, in unsteady problems it changes all the time: oider vortices are carried away by the stream, while newly born ones are shed from the body. Vortex systems
250
Method of Discrete Vortices
must be constructed in accordance with all the theorems about vortex conservation in the framework of a chosen flow mode. At each instant the surface of a body is replaced by summary vortices that are not divided into bound and free ones and are fixed with respect to the body. Curvilinear free vortices are approximated by systems of rectilinear segments. AJthough the number of discrete vortices is finite, it can increase without limit. This is done with the help of a fixed algorithm that ensures fulfillment of the following requirements underlying the efficiency of the method: a. b.
c.
Near a body the dimensions of vortex grids must be approximately equal in all directions. The points at which the no-penetration boundary conditions are met must be located approximately at the centers of vortex polygons (the so-called collocation points). This ensures Cauchy principal values of singular integrals. At the boundaries of surfaces as well as in the vicinity of cusps where the flow velocity may become infinitely large, positions of vortices and boundary points are chosen in accordance with the employed scheme. If the requirement is formulated that the flow velocity is finite at edges and cusps (the Chaplygin-Joukowski (C + Z) condition), then collocation points are placed at and/or near the edges or cusps, i.e., the discrete analog of the fitness condition is employed. Otherwise, vortices are placed here. In what follows, this rule of positioning discrete vortices and reference* points at the edges is called the B-condition of the method of discrete vortices, which was originally formulated in Belotserkovsky (l955c).
The construction of vortex wake pattern is an important stage of solving a problem. In linear steady and unsteady problems this is specified in a most natural manner. However, in the case of nonlinear problems a wake must be constructed gradually: for steady flows, by the method of iterations; for unsteady flows, with the help of time steps. Circulations of the first discrete free vortices shed from a body are calculated; their strengths remain constant as they are carried away by the flow. The preceding method allows all conditions of an aerohydrodynamic problem to be satisfied. Generally, the procedure of solution reduces to implementing (a) the solution of systems of linear algebraic equations satisfying boundary conditions at a body surface and (b) the construction of a vortex wake downstream of a body. -They are called reference points in the present book.
Fonnulation of Aerodynamic Problems and Discrete Vortex Systems
251
9.3. FUNDAMENTAL DISCRETE VORTEX SYSTEMS Consider the velocity field induced by a vortex segment A I A 2 whose strength r is constant and directed from A 1 to A 2 • An arbitrary point A of the segment may be presented in the form
0$
(9.3.1)
t $ 1,
where r A = OA, r l = OA 2, r l2 = A,A 2, and 0 is a certain point of the space. Hence, for the arc element we have ds
(9.3.2)
r l2 dt.
=
Let us next put the clement ds at the point A. Then for the vector r one gets
(9.3.3)
r
=
[( tr l2
r,OOrI2)2 -
r, 2
+
]1/2 0'
(9.3.4)
,
From Formulas (9.2.0 and (9.3.2)-(9.3.4) one gets the following formula for the velocity V at point M o induccd by the vortex segment A I A 2:
(9.3.5)
The primitive of the integrand in the latter equation is readily obtained:
(9.3.6) 1-0
252
Method of Discrete Vortices
or
Let us next introduce a three-dimensional Cartesian system of coordinates OXYZ as shown in Figure 9.1; in other words, let the triplet i,j, k be a right-hand system. Consider a special case when both the point M o and the vortex segment lie in the plane OXZ, and the segment is parallel to one of the coordinate axes, say, A1AzIIOZ. Then
and from (9.3.7) one gets
(9.3.8)
Formulation of Ae,odynamic Problems and Disc,ete Vortex Systems Next we consider a rectilinear vortex of a semi-infinite originates at the point A, = (XI'YI' ZI) and tends to infinity through the point A 2 = (x 2, Y2, Z2)' Then the parameter t appearing in Formula (9.3.0 varies and + oo. Hence, the upper limit in (9.3.6) is equal to + x, substituting the limits one gets
253
span that by passing between 0 and upon
(9.3.9)
However, if the direction at a vortex proceeds from infinity through point A 2 to point A I' then the minus sign is to be used in Formula (9.3.2) for ds, and one gets
r
V=-47T
X
r l2
rIO [
1+
rio· r 12 ]
'120'
.
(9.3.10)
'10'12
Let us consider an example when a vortex originates at point A J = z]>O) and tends to infinity passing through the point A 2 = (X 2 , ZI'O). Point M o = (X O, zo' 0) also lies in the plane OXZ. Then we have r l 2 = (X 2 - xJ)i, rIo = (X o - xj)i + (Zo - ZI)k, r l2 X r J 0 = j(X 2 - XI)(Zo - ZI)' '12 = X2 - X, (it is assumed that X2 > XI), r l2 • rio = (x 2 - xI)(X O - XI)' 0' = (zo - ZI)2, and
(Xl'
V
=
r j-
1
47TZ O -Z J
[1 +
(9.3.11)
Next we consider a rectilinear vortex of an infinite span. It will be assumed that the direction at the vortex is determined by the parameter appearing in (9.3.0, i.e., that Formula (9.3.2) preserves its form. Then the limits of integration in (9.3.5) become - x and + 00. Hence, by substituting t = - x and t = + oc into Formula (9.3.6) one gets
r
r l2 Xr lO
27T
'120'
V=-----
(9.3.12)
In Formulas (9.3.0 and (9.3.3), r l 2 may be assumed to be an arbitrary vector parallel to the direction at the vortex. Let, for example, a vortex pass through point A I = (x" z" 0) and be parallel to axis OZ (and directed in the same direction). Let the directing
Method of Discrete Vortices
254
vector r l 2 coincide with vector k. The flow velocity induced by the vector will be calculated at point M o = (xl,zo,O). The r l2 X rio = -j(x o -XI)' Q' = (x o - X I )2, and r l 2 = 1. Thus, from Formula (9.3.12) one has
r
V= - j - - - -
(9.3.13)
27TX O -X,
When we consider problems involving nonzero-thickness airfoils, we will come across a situation when Al = (x, 0, YI)' r l2 = k, and M o = (xo,O,Yo), Then rlo=(xo-xl)i+(Yo-y,)j, r 12 Xr 10 = -j(xo-x,) + i(yo - y,), and r, 2 • rIo = 0. Thus, we have
(9.3.14)
Let us proceed by considering usual and oblique horseshoe vortices. According to Belotserkovsky (1967), a usual horseshoe vortex (see Figure 9.3a) is a vortex of a constant strength r composed of segment [A,(x l , ZI' 0), AixJ, Z2' 0)] and two semi-infinite rectilinear vortices (A I' + oc) and (A 2' + (0) parallel to axis ox. The direction at the vortex will be specified by vector r l 2' The vortex will be denoted by n(AJ, A 2 )· Let us calculate the velocity V induced by the vortex at point M() = (x o, Zo, 0) lying outside the vortex. Flow velocities induced at point M o by the vortices (AI' A 2 ), (AI' +oc), and (A 2 , +oc) will be denoted by VI 2' Vi' and V2 , respectively. Using Formulas (9.3.8)-(9.3.10), one gets
Formulation of Aerodynamic Problems and Discrete Vortex Systems
255
~r
A;f
(a)
)r
A,
~rr
~~ A,
(b)
rOr
(c)
T FIGURE 9.3. (a) A straight horseshoe vortex; (b) an oblique horseshoe vortex; (c) a closed rectangular vortex frame.
Because (9.3.16) 1 V = jr- [ 47T Zo - Z2
V(X o _X I )2
+ (zo - ZI)2]
(x o - x1)(zo -
=
.~fZ2fPO J 47T
ZI
XI
zd
dxdz [ (x -x) 2 + (zo - z) 2] 3/2 . o
(9.3.17)
An oblique horseshoe vortex (see Figure 9.3b) differs from the usual one in that the segment [A I' A 2] is not parallel to the OZ axis, i.e., Al = (XI' ZI,O), A 2 = (x 2 , Z2'0), and x 2 *- XI' Let the equation of the vortex line (AI' A 2 ) be X(Z) =
i.e., XI = X(ZI) = a + Ztb and point MO<x o, zo, 0) we have
X2
=
a
+ zb,
X(Z2)
= a
(9.3.18)
+ Z2b. Then for the same
256
Method of Discrete Vortices
Hence, by (9.3.7),
r I2
V
1
=j47T A(xo,zo)
[(X o -x 2)b + (zo -Z2) .j(x
o
-X )2 2
+'(ZO
-Z2)2
(9.3.19)
where A(x o, zo) == X o - a - zob == Xu - x(zo). Because for the vortices (A I' + x) and (A 2' + x) the same formulas arc valid, we have
r [ 1 V=j47T Zo -
Z2
_ V(X O -X I )2+(ZO-ZI)2] A(xo,zo)(zo - ZI)
(9.3.20)
Formulation of Aerodynamic Problems and Discrete Vortex Systems
257
Note that if b = 0, Le., II( A I' A 2) is a usual horseshoe vortex, then (9.3.20) coincides with (9.3.17), because in this case x(z) = x I and A = xl) - X(Zo) == xl) - XI' Note also the following circumstance. If point Mo lies at the segment [A" A 2 ], then Formulas (9.3.17) and (9.3.20) must be used in the integral form, and one gets 1 V = jr- [ 47T Zo - Z2
(9.3.21 )
because in this case either X o - X I = 0 or X o - x(z) = 0. Finally, let us consider a vortex of constant strength r having the form of a rectangle in the plane OXZ, whose sides are parallel to the coordinate axes (see Figure 9.3c). Let the corner points of the vortex coincide with the points AI = (XI,ZI'O), A = (X I ,Z2'0) (where Z2 > z,), A J = (X 2,Z2'0) (where x 2 > XI)' and A 4 = (x 2, Zj, 0). The direction at the vortex will be defined by the vector r l 2' By VI 2' V2 J' VJ 4' and V4 I we denote the flow velocities induced at point M o = (xl)' zo, 0) by the vortices (A I' A 2 ), (A 2 , A J ), (A 3 , A 4 ), and (A 4 , AI), respectively. Taking into account the directions of the vortices, one gets from (9.3.7) (see also (9.3.8»,
V23
V
=
r 1 [ -j---47TZ O -Z2
-'
J4 -
r
1
J47TX O - X 2
[
Xo
- x2
V(x o - X 2 )2 + (zl)
-Z2)2
-(z 0 - z2) 2
..../(X o - X 2 )
+ (zo
2
-Z2)
Method of Discrete Vortices
258
(9.3.22)
Hence, for the flow velocity V induced at point M o by the whole of the vortex, we have
v
(x o -
X 2 )2
+ (zo - Z2 )2
(X o -X 2 )(Zo -Z2)
(9.3.23) This formula may also be presented in the form: V
=
.
-J
47T
= .~
J4
r fZ2
7T
z,
1
fZ2fX2 Zr
x,
[
Xo
- x2
2 _/ 2 2 (Zo-Z) V(X O -X 2 ) +(Zo-Z)
[
dxdz 2 (x o -x) + (zo -z) 2] 3/2 .
(9.3.24)
10 Two-Dimensional Problems for Airfoils
10.1. STEADY FLOW PAST A THIN AIRFOIL Let us start by considering two-dimensional steady flow past a thin, slightly curved isolated airfoil (see Figure 10.1). Let the flow velocity be defined by the formula Uo = uxi + uvj. We also assume that the projection of the airfoil onto the plane OXZ occupies the strip - b .:0;; x .:0;; b. Because the airfoil is only slightly curved, the no-penetration boundary condition may be transferred onto the strip - b .:0;; x .:0;; b. This means that the vortex sheet representing the airfoil is located within the strip. The strength of the sheet is independent of z and wilI be denoted by y(x). The no-penetration condition is met at all the points of the strip; in other words, the sum of normal components of the flow velocities induced by the vortex sheet at a point of the strip and of the corresponding component of the oncoming flow velocity is equal to zero. In the case under consideration the method of discrete vortices reduces to the folIowing (Belotserkovsky 1967). The vortex sheet representing an airfoil is modeled by infinitely long vortex filaments of constant strength r k described by the equation x = x k , x k = -b + kh, h = 2b/(n + 0, k = 1, ... , n, and the no-penetration condition (9.1.1) is met at the points X Om = X m + h/2, m = 0,1, ... , n. The normal component Vkm of the velocity induced by the kth vortex at the mth reference point is equal to (see (9.3.13»
(10.1.1) At the mth reference point the normal component Vm of the velocity 259
260
Method of Discrete Vortices
-b
0). 0) -r-T'Ilooooll~""ol-oo-cil'---'6 (c)
FIGURE 10.1. (a) Simulation of a thin, slighlly curved airfoil by a vortex sheet in the interval [-b, bl or the OX axis; (b), (c), and (d) are positions or discrete vortices and rererence points in [ - b, b I ror circulatory, noncirculatory, and finite-velociIy airroil problems. respectively.
induced by a system of discrete vortices is equal to n
Vm
=
E
I~wkm'
( 10.1.2)
k~l
Consider circulatory flow past an airfoil (Figure 1O.1b). According to experimental evidence (Belotserkovsky 1967), the vortex strength must be unlimited at the leading edge (i.e., at the point - b) and limited at t1).e trailing edge (at the point b). Hence, by the B condition of the method of discrete vortices, a vortex must be located near a leading edge and a reference point located ncar a trailing edge; therefore, reference points should be numbered by m = 1, ... , n. If the no-penetration
Two-Dimensional Problems for Airfoils
261
condition (9.1.1) is met at the points, then one gets the following system of equalities: n
L
fkw km
= Vm*,
m = 1, ... , n, (10.1.3)
k=l
or, taking into account Formula 00.1. 1), 1
f
n
__
"
'2 7T k=l
=
k
X Om - X k
V*
m'
m
=
l, ... ,n.
(10.1.4)
From Section 1.3 it follows that if f k = Yn(xk)h (where Yn is an approximate value of Y), then the system of linear algebraic equations 00.1.4) approximates the singular integral equation
- -
fb
1
27T
y(x) dx
-b X o
-
X
= V*(x o),
Xu E
(-b, b).
( 10.1.5)
By Theorem 5.1.1, Relationships (5.1.6)-(5.1.8) hold between a solution to System 00.1.4) and the exact value of the vortex strength that is unlimited at point -b and limited at point b. For noncirculatory flow (see Figure 10. Ie) with the strength y(x) being unlimited at both edges, the B condition requires that discrete vortices be located near both edges; in other words, in this case m = 1, ... , n - 1 for reference points. By employing the no-penetration condition at the points, one gets n - 1 equations for n unknown circulations of discrete vortices. Hence, the number of unknowns is more than that of equations; however, by adding the no-circulation flow condition we have n equations and n unknowns. As a result, one gets the following system of linear algebraic equations: n
L
rkWkm =
V,:,
m=I, ... ,n-l,
k~l
n
(10.1.6)
and, hence, Theorem 5.1.1 is applicable for C = o. For limited flow speeds (see Figure 1O.ld) when the vortex strength is limited at both edges, the B condition requires that reference points lie nearest to the edges, i.e., in this case, m = 0,1, ... , n for the reference points. By using the no-penetration condition at the points, one gets n + 1
Method of Discrete Vortices
262
equations for n unknown circulations of discrete vortices (j.e., the number of equations exceeds that of unknowns). However, it can be made the same by introducing a regularizing term YOn (which is, in fact, a new additional unknown). In other words, we consider the system n YOn
+
L
rkW km
=
v;,:,
m=O,l, ... ,n,
(10.1.7)
k~l
and, hence, Theorem 5.1.1 is applicable in this case too. Thus, Theorem 5.1.1 provides a mathematical foundation for steady unbounded flow past a thin airfoil. The same theorems verify the B condition of the method of discrete vortices. (The B condition is a discrete analog of the Chaplygin-Joukowski hypothesis). Examples of numerical calculations for certain problems were considered previously (see Figures 5.1-5.3). Note that according to Inequality (5.1.30), the method of discrete vortices allows us to calculate summary aerodynamic characteristics to any specified accuracy. The method of discrete vortices has the following import~nt peculiarity: the function y(x) obtained by solving a system of linear algebraic equations is defined by relative positions of discrete vortices and reference points only and is not specified a priori. This is of special importance when one has to solve new problems of aerodynamics that are not yet analyzed from the mathematical point of view. Next we consider a thin airfoil with a flap (see Figure 10.2). In this case the normal component of the oncoming flow velocity suffers a discontinuity of the first kind at the point where the flap is deflected (point q). Hence, at point q the right-hand side of Equation 00.1.5) suffers a discontinuity of the first kind too, and the function y(x) has a logarithmic singularity at that point. By using Note 5.1.4 we see that for the problem under consideration the strength of discrete vortices may be found as follows: 1.
2.
The discrete vortices and reference points must be positioned so that point q lies midway between the nearest discrete vortex and reference point. Then, depending on the type of problem, one has to consider one of the systems (10.1.3), 00.1.6), or 00.1.7). Foundation of the numerical scheme is provided by Note 5.1.4. Calculations show that more accurate results in the neighborhood of q. may be obtained is reference points are chosen in such a way that q is one of them and the normal component of the oncoming flow velocity at the point is assumed to be equal to (V*(q - 0) + V* (q
+ 0»/2.
Two-Dimensional Problems for Airfoils
I •• "
-b
263
1.. ,,,:,,"t ..
••
:t:
FIGURE 10.2. Simulation of a thin airfoil with a flap by a vortex sheet. q is the leading edge of the nap.
3.
If it is required that an equal number of vortices be distributed over the flap and the rest of an airfoil (as in the case of calculating hinge moments due to the flap), then one should make use of Note 5.3.3.
Let us continue by considering steady two-dimensional flow past an airfoil near a solid surface (see Figure 10.3). As earlier, the airfoil is positioned at straight line y = 0, and the nearby solid surface is described by the equation y = - H. In this case the method of discrete vortices is applied as follows (Belotserkovsky 1967): Both discrete vortices and reference points are chosen at the airfoil as was done before; in addition, discrete vortices are positioned at points AkJI = (x b -2H) of the plane OXY, and their strengths are specified as I~H = -I~ in order to ensure no penetration of the line y = -ll. From (9.3.14) it follows that the normal flow velocity component v*f~ induced by the vortex I~/I at the reference point X Om is given by
Thus, for circulatory flow one gets the following system of linear algebraic equations: n
n
L k~
fkw km -
L
fkwl'm
=
In accordance with what was said approximates the equation
__1_
V,;,
m
1, ... ,n.
=
(10.1.9)
k= I
I
[_1_ _
fb y(x) 27T -b Xl) - X
In
Section 1.3, the latter system
Xo ~ X (xl) - x) + 4H
2] d.x
=
V*(Xo). (10.1.10)
Method of Discrete Vortices
264
'1
rt
u
_..t---+~"':o""..q..o-or .......... II z
f/
-'1 -rz -r;y -r" -r!/
FIGURE 10.3. A system of discrete vortices and reference points for an airfoil near a solid surface.
According to Theorem 5.3.1, System (10.1.9) yields a solution converging to the solution y(x) of Equation (10.1.10), which vanishes at point band tends to infinity at point -b. Because Equation 00.1.10) has the form of Equation (5.3.1), for noncirculatory flows with finite velocities one gets systems analogous to (5.3.6) and (5.3.7). Thus, Theorem 5.3.1 provides mathematical foundation of the method of discrete vortices for steady flow past an airfoil near a solid surface. If more accurate calculated results are needed for the immediate neighborhood of airfoil edges, then one has to use unequally spaced discrete vortices and reference points. In this case it is advisable to position the airfoil in the interval [ -1,1] (by putting x = bt) and to make use of Theorems 5.2.1 and 5.2.2 for an isolated airfoil or Theorem 5.3.2 for an airfoil near a solid surface.
10.2. AIRFOIL CASCADES Consider a cascade of thin airfoils (Belotserkovsky 1967) presented at the cross section z = 0 by a system of segments [- b, b] X Yk' where Yk = kl, k = 0, ± 1, ± 2, ... ; I is a fixed positive number and [ -b, b] is an interval of the axis OX. Let us consider steady two-dimensional unbounded flow described in the preceding section. Because the flow past any of the airfoils is subject to the same conditions, the strength of the bound vortex sheet at an airfoil depends on the coordinate x only and is independent of both z and Yk, k = 0, ± 1, ± 2, .... Therefore, the method of discrete vortices is used as follows. Let us replace the continuous vortex sheet simulating airfoil [ - b, b] X Yk by a system of discrete vortex filaments parallel to the axis OZ and
Two-Dimensional Problems for Airfoils
265
crossing the plane z = 0 at points (Xi' Yk' 0), i = 1, ... , n, k = 0, ± 1, ± 2, ... , whose strength is equal to r i = 'Yn(xi)h (points Xi on [ - b, b 1were chosen as in the preceding section). In what foIlows a system of discrete vortex filaments (Xi' y), k = 0, ± 1, ... , is called the ith vortex chain. Flow velocities induced by the chains will be calculated at points (XOj,O), j = 0,1, ... , n. According to (9.3.14), velocity components induced at point (X Oj ' 0) by a discrete vortex filament (Xi' Yk) of strength r i are given by
V.,kij
= -
I~
27T
I~
-2
VV,k'J -
7T
-Yk
(X Oj
2
-
Xi)
7
+ Yi:
= I~ Wx,kij'
-(XOj-X i )
(X Oj
2 = riWy,kij'
2
-
X;) + Yk
(to.2.1)
Hence, the ith chain of discrete vortices induces at point (X Oj ' 0) flow velocity whose components are given by
V.,ij =
ri L k=
~.iJ
Wx,kij'
=
ri L k~
--'XO
-
Wy,kij'
( to.2.2)
'XO
By employing the notion of complex potential (Golubev 1949), it can be readily shown that the normal velocity component ~,ij is equal to ( to.2.3) Thus, the normal velocity component at point (X Oj ' 0), due to all vortex chains forming a cascade of airfoils, is given by ~j =
n
L i=1
1 n I~ w ij = 21 ;~I
7T
L r i coth-(x oj -
Xi)'
(to.2.4)
I
Let us denote the normal flow velocity component at reference points of airfoils by I-j*. The no-penetration condition gives: for circulatory flow, 1 21
n
7T
.L [I cothT(x Oj l~
I
Xi) =
I-j*,
j=I, ... ,n;
(10.2.5)
266
Method of Discrete Vortices
for noncirculatory flow, 1
n
-21.'" r l~
I
7T I
coth-(x u· - x.)
I
}
I
=
V* }
j=I, ... ,n-l,
'
n
j
(10.2.6)
n;
=
for flow with finite velocities,
j
=
0,1, ... , n. (10.2.7)
According to Section 1.3, for I~ = Yn(x)h, Systems 00.2.5)-00.2.7) approximate the singular integral equation 1 b -j 21
7T
y(x)coth- (xu - x) dx
l
b
=
V*(x u),
Xu
E
(-.b,b), (10.2.8)
which may be reduced to (5.3.9). In fact, Equation 00.2.8) may be written in the form I
-I jb
2-b
(X U -X)Coth(7T(X U -X)/I) Xu - X
_ * y(x) dx - V (xu), (l0.2.9)
Hence, it suffices to denote K(x u, x) = (xu - x)coth(7T(x u - x)/l) and = 1/7T, where the kernel is an analytic function. Because it is physically obvious that, subject to corresponding additional conditions, Equation 00.2.8) has a unique solution for each index K = 0,1, -1, solutions to Systems 00.2.5)-(10.2.7) converge to the corresponding exact solutions of the equation, and Relationships (5.1.6)-(5.1.8) are valid for them. Thus, the method of discrete vortices is also fully verified for cascades of airfoils. K(x u, xu)
10.3. THIN AIRFOIL WITH EJECTION The. effects of both air ejection and injection (suction) are used to increase lift of aircraft wings during take-off and landing. In order to calculate an increase in lift due to ejection, we simulate an airfoil by a vortex sheet whose strength is to be found.
Two-Dimensional Problems for Airfoils
o
267
,
"
f
FIGURE 10.4. An airfoil with a sink at point q. The discrete vortices and reference points are distributed in such a way that point q is one of the reference points.
Thus, consider a thin airfoil (a plate) located at the segment [ -b, b] of axis OX with air ejected at point q E ( -b, b) (see Figure 10.4). If fluid is ejected on the upper surface of an airfoil, then, due to the no-penetration condition, there are no singularities in the flow velocity distribution at the lower surface of the profile. The flow velocity pattern at the upper surface of an airfoil in the neighborhood of point q is determined by the expression
v" =
Q/(27Tr).
(10.3.1)
where v" are the radial velocity components induced by a sink, r is the distance between a point under consideration and the sink, and Q is the flow rate of the sink. Because the sink lies at the axis OX, the flow velocity induced by it at the axis is paralIel to the latter and, taking into account the flow direction, is determined by the formula Vxs = Q/[27T(q - x)].
( 10.3.2)
Let y(x) be the strength of the vortex sheet at point x E [ - b, b]. Then, at the upper and lower surface of a plate the vortex sheet induces tangential velocities (Nekrasov 1947) v.~.-= ±y(x)/2.
( 10.3.3)
where the plus (+) and minus (-) signs refer to the upper and lower surface, respectively. Tangential flow velocities due to a sink are given by Formula 00.3.2), i.e., (10.3.4 ) For the lower surface of an airfoil, one has
1/,,1' + v.~ == cp(x),
( 10.3.5)
where, according to the problem formulation, cp(x) is a smooth function in the neighborhood of the sink.
Method of Discrete Vortices
268
From (10.3.3)-(10.3.5) one gets for the value of y(x) in the neighborhood of a sink, y(x)
Q =
7T(q-X)
¢(x) -
2cp(x)
= --.
q-x
(10.3.6)
where ¢(q) = Q/ 7T. At the upper airfoil surface, the summary tangential flow velocity due to the vortices and the sink has a singularity of the form of (10.3.6). The strength y(x) of the layer under consideration must satisfy both Equation (10.1.5) of the theory of airfoils and the conditions at the edges of an airfoil. The function y(x) will be found with the help of a numerical method developed on the basis of the method of discrete vortices. Let us choose the points where discrete vortices, {Xi' i = 1, ... , n}, and the reference points, {X/O, j = 0, 1, ... , n}, are located, as was done when proving Theorem 5.1.4, i.e., so that point q is one of the reference points and the relative positions of discrete vortices and reference points repeat the ones used in the preceding sections. While constructing the system of linear algebraic equations for determining the strength I~ of the discrete vortices, the no-penetration boundary condition is met (or all reference points x O/ except point x O/ = q, which coincides with the sink. In fact, it is nonsensical to speak abo~t a normal velocity component at the point where a sink is located. Thus, we deduce that in the case of a circulatory problem when y(x) is unlimited at the leading edge and limited at the trailing edge, there are n discrete vortices and n - 1 linear algebraic equations (defined by reference points X Oj ' j = 0,1, ... , n, j -=1- jq) for calculating the latter. The value of circulation over the whole of the airfoil, f~b y(x) dx, is unknown. Therefore, to make the problem well-posed we note that the strength of a discrete vortex located near a sink may be assumed to be known and equal to f j = ¢(q)h / (q - x j ) or f j ; I = ¢(q)h/(q - x j ,,), because function ¢(x) entering 00.3.6) is smobth in the neighborho"od of point q (by all means, it belongs to the class H). Thus, for a circulatory problem one has to consider the system of linear algebraic equations n
L
fiw ij =
Iij* -
Ijqw/ qj ,
j=I, ... ,n,j-=1-jq,
(10.3.6')
i= I i */q
where I~ = Yn(x;)h. In .this case mathematical foundation of the chosen numerical scheme is provided by Theorem 5.1.4. Calculations carried out with the help of the proposed mode demonstrate good conv(;rgence of an approximate solution to the exact one (see Figure 10.5).
269
Two-Dimensional Problems for Airfoils
~.-o-
r 11.0 11,0
,
.
iOJ.a i s
--o-
•
-.---v-.~~ .....!o-.....--o-o_ .--<10_.
I
i
~
2,0
o
J
~~~
h..
1lf-....-.
I
~2
0,3
y\
-2,0 6uxPb au
I~ ~\=
C1If(JKOH
0,5
l
~iS
~
cmt/Kt/N
7~~
l
~
",,-
~
f
i,
-6,0
FIGURE 10.5. The vortex sheet strength distribution along an airfoil with a sink al point x I for a circulatory problem. The dashed and solid lines correspond 10 the exact and numerical solutions, respectively.
Consider next the problem of flow with finite velocities past an airfoil. For a given oncoming flow one has to find such a sink flow rate Q at point q for which the strength y(x) of a vortex sheet is limited at both the leading and the trailing edge. By the B condition of the method of discrete vortices, in this case one has to choose j = 0, 1, ... , n, omitting, however, the reference point number jq corresponding to the sink. Thus, we arrive at a system of n equations with n unknowns: n
L i~
fiw ij =
Jtj*,
j
=
0,1, ... ,n,j i=jq.
By Theorem 5.1.5 the approximate solution Yn(x) the solution of Equation (to.1.5):
y(x)
(to.3.7)
I
2 2 2 Vb - x 7T
q-x
Jh
=
Ii/h converges to
(q - xo)V*(x o ) d.x o . -h Jb2-x~(x-xo)
(to.3.8)
Hence, the sink flow rate ensuring flow with finite velocities past an airfoil
270
Method of Discrete Vortices
is given by the formula (see 00.3.8» o ) d.xo . Q = 7Tt/J(q) = 7Tlim y(x)(q -x) = -2-/b V 2 _ q2 fb V*(X 2 -b
x--+q
Jb
- Xl;
(10.3.9)
Finally, of interest is a vortex sheet corresponding to noncirculatory flow past an airfoil at a fixed sink flow rate Q at point q. In this case the B condition requires that the no-penetration condition be met at points x o/, j = 1, ... , n - 1, j -=f- jq. A'isuming that the discrete vortex strength f/ q is known, one gets n - 2 equations for n - 1 unknowns. The system of equations may be augmented by digitizing the equation fb y(x) d.x
=
0,
(10.3.10)
-- b
which is, in fact, the condition of no-circulation flow past an airfoil. Thus, we have to consider the system n
I~wi/
L i~
I
=
JIj* -
f/'Iw/q/'
j
=
1, ... ,n - 1,j -=f-jq,
i,;'iq n
L i
~
fi
=
(10.3.11 )
1
i*/q
Similarly to Theorems 5.1.4 and 5.1.5, it can be shown that a solution to the latter system of equations converges, with the same estimates as in the case of the theorems, to the solution of Equation 00.1.5):
( 10.3.12) If an airfoil contains several sinks, then one has to employ Theorem 5.1.6 a'nd the corresponding notes. In order to usc unequally spaced points when considering problems of flow past an airfoil with a sink, one has to usc the results of Section 5.2.
Two-Dimensional Problems for Airfoils
271
10.4. FINITE-THICKNESS AIRFOIL WITH A SMOOTH CONTOUR In the three preceding sections of this chapter we dealt only with very thin airfoils. However, of great practical importance are airfoils having a finite thickness (Kotovskii et at. 1980, Matveev and Molyakov 1988). Let an airfoil contour L be described by a plane simple curve x = xU) and y = y(t), t E [0, l], on the plane OXY, which may be either unclosed or closed (Muskhelishvili 1952). In the former case, it is supposed that at point M(t) lying on the curve (whose radius-vector r M = xi + yj), r M = JX'2(t) + y'2(t) i= 0 on [0,/], both x'(t) and y'(t) meet the Holder condition of degree a, i.e., x'(t),y'(t) E H(O') (Muskhelishvili 1952). In the latter case, if point M(x(O), y(O» = M(x(l), y(l» is not a cusp (i.e., if the airfoil has no sharp edges), the x'(O + 0) = x'(l - 0), y'(O + 0) = y'(l - 0). However, if the point is a cusp, then there exist one-sided unit vectors tangential at the point; otherwise, there exist T(O + 0) = [x'(O + O)i + y'(O + O)jl!rMro+o) and T(l - 0) = [x'(l - O)i + y'(l - O)j]jrM(I 0). When L is a closed curve, then we put I = 27T. Now let the airfoil be immersed into steady ideal incompressible flow characterized by the free-stream velocity vec)9l~l]o = iUx + jUy • Let the surface of the airfoil be modeled by a vortex ;JilCtwhose densiiy at point M(t) = M(x(t), y(t) of the curve L is equal to y(t) (Belotserkovsky 1967). Then the condition that the normal flow velocity component be equal to zero at point M o of the airfoil L (the condition of airfoil no-penetration at point M o), i.e., the condition according to which the sum of the normal velocity components Uo and the velocity V induced by the airfoil vortex layer is equal to zero, may be represented in the form (10.4.1 ) where
Method of Discrete Vortices
272
Hence, Equation (l 0.4. 1) may be written in the form
where to E (0, l) and f(to) = - Uo • 0M o' The latter equation is valid for both unclosed and closed curves L. If L is an unclosed curve, then Equation 00.4.2) may be written in the form I
-2 7T
1I K(to,t) y(t)dt =f(to), to t
0
to
E
(0, I),
( 10.4.3)
where
If functions x'(t) and y'(t) belong to the class H(a), then
(10.4.4 )
toE(O,I).
From this equation it follows that Equation (l0.4.3) may be presented in the form
-1
21T
where
11 0
Y ( t ) dt to - t
+
1/K,(to, t)y(t) dt = f(to), 0
to
E
(0, I), (10.4.5)
Two-Dimensional Problems for Airfoils
273
Let x"{to),y"{to) E H(a) on [0,1]. Then according to Muskhelishvili (952), K(to,t) and K1(to,t) also belong to H(a) on [0,1], uniformly with respect to each coordinate to the other. Let now L be a closed curve (l = 27T). Then Equation 00.4.2) cannot be presented in the form 00.4.3) with a continuous kernel. K{to, t) for x", y" E II on [O,/l, because the kernel in Equation 00.4.2) tends to infinity for t = to and t = to + I. Therefore, first Equation 00.4.2) is written in the form 1 27T
1
27T
0
K(to, t) sin«(to - t)/2) y(t) dt
=
f(to),
to
E
[0,27T], (10.4.6)
where
Functions x2{to, t) and Yit o, t) suffer discontinuities of the first kind, because x2{to, t) ---) 2x'(to) for t ---) to and x2{to, t) ---) - 2x'{to) for t ---) to + 27T; however, function , MM 0 is continuous at point to and does not vanish on [0, 27T l. Consider the functions
r;
Y3(tO,t)
=
t - t Y2(tO,t) - 2cosTY'(to)'
(10.4.7)
Because 2 cos(~)(to - t)x'(to) ---) 2x'(to) for t ---) to and 2 cos(~) (to - t)x'{to) ---) - 2x'(t o) for t ---) to + 27T, the functions x 3(t o, t) and Y3(t o' t) are continuous at point to' Thus, for a continuous curve, Equation 00.4.2) may be written in the form
274
Method of Discrete Vortices
-
1
21T
1 cot--A(t(p to - t t)y(t) dt 27T
0
2
( 1004.8)
Now we note that AUo, to) = L to E [0, 21T 1 Hence, Equation 0004.2) for a closed curve L finally may be written in the form
1
27T 1 127Tto - t -4 --y(t)dt+ K,(to,t)y(t)dt=f(to), 1T 0 2 0
where to
E
(1004.9)
[0, 21T ],
If x"(t), y"(t) E H(O') on [0, 21T], then by using the results of Muskhclishvili (952) again one can show that the kernel of Equation 0004.9), K1(to,t) E 11(0') on [0,21Tl Now we can proceed to using the method of discrete vortices to solve numerically problems of steady flow past finite-thickness airfoils. If L is an unclosed curve, then Equation 0004.5) is a singular integral equation of the first kind with a Cauchy kernel on [0, l], and, hence, discrete vortices and reference points must be distributed along [0, l] as was done in Section 10.1 (see Systems 00.1.3), 00.1.6), and 00.1.7). If the first and second derivatives of the functions y(t) and xU) are small in absolute values, then Equation 0004.5) may be replaced by the equation
1/
1 y( t) dt 21T () to - t
-
=
f(to)·
(1004.10)
Two-Dimensional Problems for Airfoils
275
However, if an airfoil contour is described by y = y(x), x E [-b, b), and y'(x) and y"(x) are sufficiently small in absolute values, then Equation OO.4.tO) may be written in the form (to an accuracy of second-order infinitesimal quantities) (10.4.11 ) This equation describes a thin, slightly curved airfoil. Some examples of choosing vortex modes for such airfoils are presented in Figures to.6 and to.7. Next we consider the situation when L is a closed contour. In this case discrete vortices are placed at points (x(t), y(t» and reference points are placed at points (x(to), y(to», i = 1,2, ... , n, where I; E [0, 27T] (interpreted as points of a unit-radius circle) divide the latter into n equal parts and t Oi are the middles of the parts. The flow conditions are formulated as follows: 1.
If flow is known to be noncirculatory, then one has to consider the system of linear algebraic equations n
YOn
+
L
fiw ii =
-
Jtj* ,
j
1, ... , n,
=
;= I
n
(10.4.12)
2.
where YO Il is a regularizing variable. The results of using this scheme to calculate flow past a circular cylinder are presented in Figures 10.8 and 10.9. If a point of the contour is known where the flow strength y(/) vanishes, then the positions of discrete vortices are chosen in such a way that the point is one of them. Let the number of the point be equal to n. Then the system n-I
YOn
+
L
I~W;i
= -
Jtj*,
j
=
1, .. _,n,
(10.4.13)
;-1
is to be considered, and the distributed strength of the vortex layer is found from the formula i
=
1, ... ,n. (to.4.14)
276
Method of Discrete Vortices , -f
a
L
0
,.
0
,
II D
0
I
(a)
,
-f
L.-.o-..-.o
-f
f II
0
P
0
0
• (c)
0
T1
It
0
P
0
11 0
P
0
(b) 0
I
0
, f
, f
FIGURE 10.6. Distributions of discrcte vortices (0 0 0) and reference points (x X x) for a thin, slightly curved airfoil considered for equally spaced grid points. (a), (b), and (c) correspond to the circulatory, noncirculatory, and finite-velocity problems, respectively.
FIGURE 10.7. Distributions of discrete vortices (0 0 0) and reference points (x X x) for a thin, slightly curved airfoil considered for unequally spaced grid points. (a), (1::-), and (c) correspond to the circulatory, noncirculatory, and finite-velocity problems, respectively.
10,5, METHOD OF A "SLANTING NORMAL" In the preceding section Equation 00.4.2) was derived appropriate for the case when the airfoil contour L is a smooth closed curve and the nondimensional thickness is sufficiently large (no less than 6-10%). However, for smaIler nondimensional thicknesses, one has to retain a comparatively large number of discrete vortices. Even more complicated is the situation when a contour contains angular points or sharp edges. In the latter case the solution contains singularities due to the points, and the regular kernels of Equations 00.4.5) and 00.4.9) have non integrable singularities at the points. Although the method of discrete vortices aIlows us to solve the problem numericaIly, it is necessary to deal with a rather large number of discrete vortices. Therefore, the foIl owing approach is often used. It is especially useful for the cases when an airfoil has sharp edges or when its nondimensional thickness is small (Shipilov 1986).
Two-Dimensional Problems for Airfoils
277
)<'IGURE 10.8. Distribution of lip along a circle. The dashed and solid lines correspond to the exact and numerical solutions, respectively.
~
~n0rox. solution
f.f!
e
if!
FIGURE 10.9. The dependence of coerticicnt C,. on the angle of attack a for a lOok. symmetric airfoil.
An unclosed cUIVe L, is chosen within an airfoil, with respect to which the closed contour L is divided into two unclosed smooth contours L + and L - placed above and below the cUIVe L], respectively (see Figure 10.10). If contour L contains a sharp edge, then it is a common end of the CliIVes L t , L -, and L I' CUIVe L I is such that the normal to it at each point M o crosses the CliIVes L + and L - at only one point, Md and Mo·, respectively, thus establishing a one-to-one correspondence between the points of the CliIVes L], L t and L I ' L -. Let the parametric equations of the CliIVes L" L t have the form x = x,(t), y = y](t) and x = x "(t), y = Y I(t), respectively, where t E [-1,1]. Let us denote the unit vector of the outer normal to the cUIVe L on L I by i . Then the no-penetration II condition 00.4.1) for the contour L may be written in the form
ott
V(M(J)' o~,=
"
-u()·
O~II·'
Md
E
L
+,
M o E L-,
(10.5.1)
278
Method of Discrete Vortices
----~~
r
r"
v
~ .....
I~ -
~
FIGURE 10.10. The principle of transference of boundary conditions from Ihe surface of an airfoil onto the middle line. At point M(I the boundary condition is determined by vectors n,t,: and nMIl , respectively.
where points Mol ELL are taken at one and the same value of the parameter to' Let point Mo E L, correspond to points M(: and Mo , and let 0 Mil and T Mil be unit vectors of the normal and the tangent to curve L I at point M o (0 M'II and eM' II. are directed to one side from L I)' Then one can write
( 10.5.2) Instead of the velocity field V disturbed by the airfoil L, we shall consider the velocity field V, induced by a vortex sheet of strength y(t) and a sheet of sources of strength /L(t) distributed along the curve L]. We shall also require that Equalities 00.5.0 hold for the velocity field VI at the points of the curve L,. Thus, at the points outside L, one has
Xr/.t dt,
(10.5.3)
where M o et L i , w(t o, t) is a factor before y(t) in 00.4.1), and V is the gradient sign; also,
(10.5.4)
Two-Dimensional Problems for Airfoils
279
where VI +( M o) is the flow field velocity VI at point Moon the side of 0 Mo-' As far as the vector field VI is continuous, it approximates sufficiently well Equation 00.5.1) on curves L' and L' and, hence, on curve L; in other words, the no-penetration condition may be approximately satisfied for the contour L. Calculated data confirm this conclusion (Shipilov 1986). Let us replace the vectors ott - entering Equalities 00.5.4), according " of velocity fields induced by both the to 00.5.2). By using the properties vortex and the source layer, one gets for points of L I (Belotserkovsky 1967, Tikhonov and Samarsky 1966)
1 xln-- ·rJ..r dt rMMo to
+
E ( -
b'(to)jl 27T 1, 1).
a
1
p,(t)--In--I a'TM " rMM "
'r~,
dt
( 10.5.5)
Unlike the tangential component, the normal velocity component induced by the vortex layer has a singularity at the points of curve L I' and an opposite statement is true for the components of a simple layer potential. Hence, Equations 00.5.5) form a system of two singular integral equations in the functions y(t) and p,(t). However, it is more convenient to pass from System 00.5.5) to the system
2S0
Method 01 Discrete Vortices
x
t
1 JL(t)_d_ (In__) dn Mo
-I
r MMo
(10.5.6) Next consider the case when the contour L is symmetric with respect to L], and M; and Mr; are the points where the normal to L, at point M o crosses the CUNes L + and L - respectively (see Figure 10.10). Then, it can be assumed that in Equalities 00.5.2),
at (t)
=
-a (t),
(10.5.7)
and hence, System 00.5.6) becomes
1)
d ( + bl(tO)jl JL(t)-In-- r/.t dt 7T
-I
dTM
o
r MMIi
1)
+ a'(tO)jl JL(t) -d- ( In-- r/.t dt 7T
-I
dn Mo
r MMIi
=
f7(to)
+ 11 (to),
=
It (to)
-
II (to)· (10.5.8)
This system is a diagonal singular integral system because the first equation is singular with respect to one function, and the second equation is singular with respect to the other function. If under the circumstances L] is a segment of a straight line, then w(Mo, t) . T M = a/ dn II • • M(J (In(t/rMMo ) ;;;; 0 for M o, MEL], and, hence, System 00.5.8) splIts IOto two independent equations
Two-Dimensional Problems for Airfoils
281
( 10.5.9) each of which is a singular integral equation of the second kind on a segment with variable coefficients. The solution to System 00.5.9) must be chosen subject to the condition that the disturbed flow velocity vanishes at a sharp edge and is limited at all the other points. An example of both analytic and numerical solutions to system 00.5.9) for the case when L is an ellipse is presented in Shipilov (986).
10.6. A PERMEABLE AIRFOIL Recently, much attention has been paid to calculating aerodynamic characteristics of parachutes and delta-wing planes (Belotserkovsky et al. 1987) whose surfaces are partially permeable. Therefore, of practical interest is the problem of flow past an unclosed contour L when the corresponding airfoil is permeable. In this case the boundary condition 00.4.1) for the velocity field is presented in the form (10.6.1)
where W(M o) is the cross-flow velocity at point Mo of the curve L. It is supposed that W(Mo) = W(M o) = "M". Usually, the function W(M o) is assumed to depend on the pressure drop, i.e., W(M o) = F(tip(Mo and the pressure drop is expressible through the vortex sheet strength y(t o) at point M o of an airfoil, which corresponds to the value of the parameter to (Belotserkovsky et al. 1987). Thus, by modeling the contour L by a vortex layer of strength y(t) we arrive at a solution to the singular integral equation (see Equation (10.4.5»
»,
to
»
E
(-1,1).
(10.6.2)
As a rule, the function F(tip(Mo is obtained experimentally and depends on the material used to manufacture an airfoil. The concrete method for !>olving Equation 00.6.2) numerically depends to a large extent on the form of the function F 1( yU o
».
282
Method of Discrete Vortices
»
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
Method of Discrete Vortices
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
Method of Discrete Vortices
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).
Three-Dimensional Problems
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
Method of Discrete Vortices
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,
Method of Discrete Vortices
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
Three-Dimensional Problems
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
Method of Discrete Vortices
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~
<
-aj::: laj;;: I· =
(11.2.16)
I
;'i-j kTm
This inequality expresses the Hadamard criterion. Because for any numbers a; and b; the equality (11.2.17)
Three-Dimensional Problems
293
holds where bn + I = 0), System (11.2.1) is equivalent to a system of the form 01.2.10) and, hence, is also well-conditioned. Note 11.2.1. By employing Formula (9.3.17) it also can be shown that the system of linear algebraic equations (11.1.2) for the problem of circulatory flow past a rectangular wing is well-conditioned.
1l.3. A WING OF AN ARBITRARY PLAN FORM In this section we consider wings of complicated plan forms as well as a schematic flying vehicle as a whole. Let us start by considering a plane wing in the form of a canonical trapezoid (a trapezoid lying in the plane OXZ whose two sides are parallel to the axis OX; see Figure 11.1). We will show that the results obtained previously for a rectangular wing are also valid for this case. Then, we will continue by considering a complicated lifting surface divided into canonical trapezoids (see Figure 11.2). Thus, we will consider the linear problem of steady flow past a canonical trapezoid (J whose sides lie on the straight lines Z = 0 and z = I, and the leading and the trailing edges are specified by the equations (11.3.1) The oncoming flow velocity will be specified as Uo = Uri + U"j, Ux > O. In Belotserkovsky (1967) it was proposed to solve the problem of finding the strength of a continuous vortex sheet modeling trapezoid (T by using the method of slanting horseshoe vortices constructed in the following way. Each side edge of trapezoid (J is divided into n + 1 equal parts. The grid points are connected by straight lines on which the bound vortices (the vortex segments of slanting horseshoe vortices) are located. Segment [0, l] of the OZ axis is divided into N equal parts, and straight lines parallel to the OX axis are drawn through the grid points (including the end points z = 0 and z = l). The points of intersection of the straight lines parallel to the OX axis and passing through the middles of the segments formed by the division of segment [0, l] of the OZ axis and of the straight lines connecting the middles of the corresponding segments of the side edges of trapezoid (J are assumed to be the reference points. The following approach will be used for describing discrete vortices and reference points. Let D = [0, 1] X [0, l] be a rectangle lying in the plane OX I Z and F be the mapping of the rectangle onto the trapezoid described by Formula 0.2.16). Let us next choose points x} = ih], X6i = xl + h 1 /2, hi = I/(n -+- 1), i = 0,1, ... , n, Zk = (k - l)h 2 , ZOk = Zk + h 2 /2, h 2 = liN, k = 1, ... , N + 1. In what follows it is assumed that hlh k :'5: R < +oc,
294
Method of Discrete Vortices
OI'""'T------------;
FIGURE 11.2. Thc vortcx system for a schematic aircraft in the framework of the circulatory problem.
h = max(h" h z ), k = 1,2, for h ~ O. By Ai k = (Xi k' Zk), Ai Ok = and Aoi,ok = (XOi,Ok' ZOk) we denote 'points ~f the can~nical trapezoid (J that are, respectively, images of point A), k = (xl, Zk)' A), Ok = (xl, ZOk)' and A~i,ok = (Xl;i' ZOk) belonging to rectangle D, obtained by using the mapping F. By n ik we denote the slanting horseshoe vortex incorporating the bound vortex (Au, Au. I)' whose strength is f ik = Y(Xi,Ok' ZOk)(X i + I,Ok + Xi,Ok)' Note that by the definition of the mapping F, (Xi,Ok, ZOk)'
( 11.3.2) where J(ZOk) is the Jacobian of the mapping F. Hence, the strength of vortex I1 ik is given by (11.3.3) where 'Pik = Y(Xi,Ok,ZOk)J(ZOk), i = 1, ... ,n, k = 1, ... ,N. Flow velocities induced by the slanting horseshoe vortices will be computed at reference points Ijm = (XOjO m, ZOrn), which are images of the points Ij~ = (xA j , ZOrn)' Note that the line of the bound vortices
Three-Dimensional Problems
295
(Ai k' Ai k e,), k = 1, ... , N, is the image of the straight line Xl = xf for the'mapping F, and, hence, the equation of the line is given by x;(z) =x(xf,z) =a(xf) +zb(xf), a( xf)
=
+ xf (a
a
t -
a _),
b(x:)
=
b +xf(b+- b ). (11.3.4)
Let v;{m be the normal velocity component induced by the slanting horseshoe vortex Il;k at point Ijm, and let /l}m be the normal velocity component induced at the point by the entire system of slanting horseshoe vortices. Then, by using Formula (9.3.20) one gets n
/l}m
=
;~I
wikjm =
n
N
L L
v;{m
=
k~1
N
L L ;~l
f ik wi!:' ,
k~l
1 [ 1 47T zOm - Zk.
r(Ai,k,Ij,m) AOj,Om,;(Zom -
1
zd '
From Formula (t 1.3.5) and the results of Section 3.4 it follows that if function cp(x l , z) = y(x(x', z)z)J(z) is such that (ICP/ dZ E H* on D, then
(11.3.6) where the quantity Ia(x Oj ' zom)1 IS the same as O(X Oj ZOm) entering Equation (3.4.3), Because '-ve consider circulatory flow past a canonical trapezoid IT, or by the B condition discrete vortices must be located near the leading edge,
296
Method of Discrete Vortices
and reference points must be located near the trailing edge. Therefore the work of Belotserkovsky (I967) proposed to calculate circulations f ik of the discrete vortices from the following system of linear algebraic equations n
N
jm -'" - " ' - fikWik ;= 1 k~ 1
V* jm'
j=I, ... ,n,m=I, ... ,N, (11.3.7)
where J.j:' is the normal component of the oncoming flow velocity at the reference point ~m' Now we see that taking into account 01.3.6), the latter system of equations approximates the integral equation
(11.3.8)
Next, following the procedure of Sections 11.1. and 11.2, we will show that in the framework of the method of slanting horseshoe vortices for circulatory flow, System 01.3.7) 0) is well-conditioned, (2) approximates an integral equation any solution of which vanishes at the side edges, and (3) approximates an integral equation any solution of which vanishes at the trailing edge and tends to infinity at the leading edge of the trapezoid (T. Thus, the method of slanting horseshoe vortices will be shown to single out the solution to Equation (I1.3.8) that corresponds to the physical flow pattern of the problem under consideration. To prove the first statement, we have to apply again the transform (I 1.2.17) to the summands r.;'~ 1 f ik wi!:' for a fixed k, having denoted the quantities Y(X i . Ok ' ZOk)J(ZOk) and wit by Uk and hi' respectively. Then we have to employ Formula (9.3.20) and note that
jm _ Wi'l,k
jm_ W ik -
f
Zhlf
Zk
(I)
XX1,j,Z
x(x!,z)
dxdz
(1139)
[ 2 "]2'" (XO-X) +(ZO-Zr
when~ X(X;',
z) and x(xl-t l' z) are equations describing the lines on which the bound vortices (Ai, k' Ai. k + I) and (Ai ~ l,k' Ai l,k' I) lie. Finally, it suffices to show that the Hadamard criterion holds for the transformed system (as was done before for System (I 1.2.10». T
297
Three-Dimensional Problems
In order to prove the second and the third statements, we note that
A~j,om.i + 2AOj.Omib(xl)(zom -
+ (ZOm
-
Zd
2
2
Zd
I
0 (Xi)
(11.3.10) Therefore, System 01.3.7) may be written similarly to System 01.1.6):
_ r( Avp.' Ijn,) AOj,Om,i( ZOm
I~Oj'Om'il
),
Zp.) j = l, ... ,n,m = 1, ... ,N.
(11.3.11)
By solving System 01.3.11) with respect to er.f ~ I 'Pik hI) and passing to the limit, we deduce that System (11.3.1]) approximates the integral equation
where z
fjJ( z, zo)
=
Jzo(l - zo)
dv
jo ...; v(l-v)(v-z
o)
,
and Oz<x o, zo) is the limit of the right-hand side of System 01.3.11) for n, N -4 00. From Equation 01.3.12) it follows that
(11.3.13) As far as for the trapezoid under consideration, J( z) > 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
Method of Discrete Vortices
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
Three-Dimensional Problems
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
Method of Discrete Vortices
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.<
The Jacobian of the mapping is equal to z) = x~ (z) - x~ (z). Similarly to the way it was done before for a canonical trapezoid, we choose points x,' and X~i' i = 1, ... , n.. on segment [0,1] of the axis OX' separated' by dista~ces hf and points Zk and ZOk' k = 1, ... , N., on segment [I~, I;] separated by distances If trapezoids 0;' and 0;, are located one after the other in the oncoming flow direction (i.e., if = k = 1,2), then h?, = h~, and, hence, N. = Nv ' Thus, coincidence of the lines of free semi-infinite vortices on (I, and (I is ensured. Next, the mapping F, is used to specify the slanting horseshoe vortices ni,k, on the trapezoid (I. as well as the reference points ~,m .. Let us denote by
hf
I; I;,
the normal velocity component induced by the slanting horseshoe vortex n ivk " of trapezoid (Iv with intensity [i,.k, at point lj,m, of trapezoid 0;" by Vj:m, denote the normal velocity component induced by the entire system of slanting horseshoe vortices on (Iv, and by J1j,m, denote the normal velocity component induced by the entire system of vortices on (I at the
Three-Dimensional Problems
same point
301
IJ,m: Then, p
/Ij,m, =
p
/Ij~m, =
L v~
I
n.
N"
L L L /Ii):'. v- I
i,,~
(11.3.24)
I k,,= I
Because we consider a circulatory problem (i.e., the solution being sought must be unlimited at the leading edges of trapezoids and limited at their trailing edges), by the B condition the following system of linear algebraic equations must be considered for obtaining unknown circulations of discrete vortices (Belotserkovsky and Skripach 1975):
i,
= 1, ... ,n" m, = 1, ... , N.,
€
= 1, ... ,p,
(11.3.25)
where /Ij,*m, is the oncoming flow normal velocity component at the reference point IJ,m: However, if a noncirculatory problem is considered for which the solution must be unlimited at both the leading and trailing edges, then by the B condition one has to consider the system
j, = 1, ... , n, - 1, m, = 1, ... ,
L i~ -=
I:,m,
=
0,
j, = n" ME = 1, ... , N.,
€
N.,
€
= 1, ... ,p,
= 1, ... ,p. (11.3.26)
I
By isolating the summands corresponding to trapezoid 0"" on the left-hand sides of Systems (11.3.25) and (11.3.26), we deduce that the sought functions y(x, z) behave as needed at both the leading and the trailing edges.
12 Unsteady Linear and Nonlinear Problems
The ability of the method of discrete vortices to provide simultaneously a discrete model of the physical phenomenon under consideration and a method for solving numerically the corresponding mathematical problem is fully displayed in the analysis of unsteady linear and nonlinear, ideal incompressible flows past lifting surfaces. However, the systems of integral and differential equations describing the problems remain virtually unstudied. Here we shall try to fill the gap.
12.1. LINEAR UNSTEADY PROBLEM FOR A THIN AIRFOIL Consider a thin airfoil traveling through an ideal incompressible fluid with average translational velocity Va at an angle of attack 0'. Let us introduce a frame of reference OXZY fixed at the airfoil, i.e., in fact we shall consider unsteady flow past the airfoil. In the case of unsteady flow, circulation around an airfoil varies with time. By the theorem about the constancy of circulation along a material loop (which does not come across singularities), free vortices that are shed from the airfoil continue moving with the flow and their circulation is time-independent (Golubev 1949). Free vortices remain parallel to the bound vortices at the airfoil and, similarly to the latter, are parallel to the OZ axis. As far as the problem is considered in the linear formulation and for small angles of attack only, free vortices may be assumed to move in the plane of the airfoil (i.e., in the plane OXZ) with speed Va (Belotserkovsky, Skripach, and Tabachnikov 1971). Hence, a free vortex shed from the airfoil at instant T travels, by instant t, through distance Va(r - T). 303
304
Method of Discrete Vortices
It is required to calculate the strengths of the bound and free vortex sheets. This will be done by employing the method of discrete vortices (Belotserkovsky and Nisht 1978). Let the airfoil occupy the segment r- 1, 1] of the axis, ox. Because a free vortex sheet sheds from the trailing edge of the airfoil, the flow velocity at the latter must be finite, and hence, a reference point and a discrete vortex must be located near the trailing and the leading edges, respectively. Thus, discrete vortices and reference points may be conveniently distributed in the following way. The discrete vortices are at points Xi = -1 + (i - 3/4)h, h = 2/n, i = 1, ... , n, and the reference points are at points X oJ = x j + h/2 = -1 + (j - t)h, j = 1, ... , n. The discrete time spacing, I1 t, will be chosen in accordance with the formula Uo 11/ = h.
(12.1.1)
For simplicity, we put ~) = 1. Therefore, the coordinate of a free vortex shed from an airfoil at instant ts' at instant t r is equal to gsr =x n
+ (tr - t, + I1t) =x n + (r - s + 1) 111.
(12.1.2)
Let the circulation of the discrete vortex located at instant r at point Xi be equal to rir , and let the circulation of free vortices shed prior to the instant from the airfoil be A" s = 1, ... , r. By employing the no-penetration condition at reference points X o ' j = 1, ... , n, we arrive at the system of equations ) n
L i-I
r
I~r wij
+
L s~
A, w sjr = - J.j* ,
j
=
1, ... ,n, r
=
1,2, ... , (12.1.3)
I
where wij is the normal velocity component at reference point x j induced by a unit-strength vortex placed at point Xi' and w sjr is the normal velocity component induced at the same point by a unit-strength vortex located at point {". In the latter system of equations Tin i = 1, ... , n, and A r are unknown at the reference instant r, while A I " ' " A r -I are determined at the preceding reference instants and remain unchanged as time passes (these are circulations of the free discrete vortices). Hence, the system contains n + 1 unknowns and n equations. Let us supplement the system under consideration by an equation for the constancy of circulation around a material loop encompassing both the airfoil and the wake. If an airfoil starts,moving from the state of rest, then the equation has the form n
L i= I
fir
+
E As s= I
=
0,
r
=
1,2, ....
(12.1.4)
Unsteady Linear and Nonlinear Problems
305
Thus, to solve the problem, one has to consider the system of linear algebraic equations n
L
r
I~rw;j +
;=]
L s~
n
j= 1, ... ,n,r= 1,2, ... ,
Aswsjr = -Uyn I
r
L F;r + LAs =
( 12.1.5)
r = 1,2, ....
0,
s=]
;~]
By putting fir = Y(X;, tr), A, = 8(ts) tJ.t, and taking into account Formula (9.3.13), the latter system of equations may be rewritten in the form
j
=
1, ... , n, r
=
1,2, ... ,
n
L y(x;, tr)h + L 8(t.) tJ.t =
r = 1,2, ... , f(tr) = Uyr . (12.1.6)
0,
s=]
;~]
This system of equations must be solved step by step for r = 1, r = 2, etc.; hence, it is convenient to rewrite it in the form
j n
L
=
1, ... , n, r
=
1,2, ... ,
r-]
y(x;, tr)h + 8(tr) tJ.t
=
-
L
8(1') tJ.t,
s=l
r
=
1,2, ....
( 12.1.7)
The matrix of the latter system has the same sign as that of System (5.1.4); hence, it is both well-conditioned and solvable, and by Formula
306
Method of Discrete Vortices
(5.1.13) one has 'V(x t) - __1 [In' I i', h I,
X
(
I)
[n
j-I
27Tf(/,) -
h_
1
"_[(n'.I) '- h I,O)
Xi - X Oj
8(t,.) dt
r-l
L - - - - '- - - -
,\_ I X Oj -
X" -
(t r
-
Is
)
+ d/)
i= 1, ... ,n,r= 1,2, ... ,
8(/,)
=
1 dl
-
_[1"- I)
[n, , _1[ ) ' , '1)
I,n - I
'-
j~'
h
I,O}
h
8(t,) dl
r-l
X
(
27Tf(t,) -
L
s~ 1 X Oj - X" -
+:~8(t.)Ml
_
X",,-X Oj
i
=
n
'
t,
(I, -
+ 1, r
)
=
+ dt)
1,2, ... , (12,1.8)
where
On the other hand, by the results obtained in Section 1.3 and supposing that the functions y(x, t) and 8(t) belong to class H* at the corresponding sets, we deduce that System 02.1.6) approximates the following system of integral equations:
Unsteady Linear and Nonlinear Problems
f
307
y(x,t)dx +i' 8(T)dT -----Xo - X 0 X o - 1 - (I - T)
l
- I
27Tf(t),
=
X
t
oE
( -
1,1), t
0,
( 12.1.9)
y(x,t),d.x+ {8(T)dT=0.
-I
~
0
Let us show that the latter system of equations has a unique solution meeting the required boundary conditions. It is physicaIly obvious that the first equation of the system must be considered with respect to the function y(x, t} unlimited at the point x = - 1, as a singular integral equation of index K = 0 on [- 1, 1]. Therefore, by solving the equation with respect to y(x, t), one gets 1
y(x,t)=fl(x,t)+~
[!2;-x 1/ -l+x
7T
xt
1
7T
+ Xo
dx o
1 -xo (x -xo)(x o -1 - (t - T»'
I
1 -2
8(T)dT
0
[!2;-x fl + -1 X
-I
1
+ X o 27Tf(t) dx o
1-
- - - == 2
X
o
x -
X
o
[!2;-
x --f(t). 1+x
(12.1.10) Let us recalI some formulas presented in Gakhov (1977), Muskhelishvili (1952), and Prudnikov, Brychkov, and Marichev 1983):
Ixol < 1,
f
l
-I
1
~
V~
dx X
o - x ==
II
dx
f-I(x-b)~
7T,
=
1 -1 + xb + arcsin x - b _ 2 -";b -1
7T
+---r=== 2
";b
-
1
(12.1.11)
Method of Discrele Vortices
308
where the upper and the lower signs correspond to b > 1 and b < - I, respectively. From the latter integral we get
f
I
- I
V+ I
X
1-
X
o
~
o b -
X
=
_ 7T
+ 1TV b + 1 b - I '
o
b> I,
b > 1.
(12.1.12)
Formulas 02.1.11) and 02.1.12) yield
(T+X;
l
f V~
dx o
(x - xo)(x o - I - (I - T»
I
=_/2+(t-7) 7T V t-T 1+(/-T)-X·.
(12.1.13)
Hence, Formula 02.1.10) becomes
y(x, t)
=
1 fl(X, I) + -
1T
{8--1 V + X / l+x ()
2
(I - T) 8 ( T) dT I-T l+(t-T)-X (12.1.14)
By substituting this expression for y(x, t) into the second equation of System (12.1.9) and using the second equation of System (12.1.12), we get
I/2 1
+ (I - T)
o
1-
8(T)dT= -f2(t),
T
f2(/)
=
t
f,(x, t) dx
=
27Tf(/). (12.1.15)
-I
This !s the Abel equation (Goursat 1934). Let us rewrite it in the form
/8 ( T) dT 1o ~ t T
=
-
f2 ( I) ..[i -
I M
/ { 0
v2 1
V(t -
..;t=1..[i8(T)dT. (12.1.16) T) + 2 + 2
Unsteady Linear and Nonlinear Problems
309
Then, by applying the inversion formula for the Abel integral (Goursat 1934), we get d7
8(t) + -1- /
/2 7T fc)~
=
_1_ [ -fz(O) _ /27T
Ii
iT -a ( 0 a7
~ ) 8(s) ds h + 7 - S + /2
rf~( 7) d7] 0
~
( 12.1.17)
.
From Formulas (12.1.16) and (12.1.17) we deduce that Equation (12.1.17) has the form 1 / U (0) / U'(7) d7 8(t)+-l K (t,s)8(s)ds= -/2~-lii ~, viol - 7
27T 0
K(/, s)
=
/ 1 [/2 + s
d7 7 -
S
Ii
+ 2]
~ 1-
.
(12.1.18)
7
~h+7-S
The kernel K(t, s) is continuous because it always may be written in the form 1
K(/,s)
=
m
M] . 1o yz(l-z)y2+z(/-s) [ y2+Z(/-S) +v2 (12.1.19)
Hence, Equation (12.1.18) is the Volterra equation of the second kind having a unique solution for any integrable right-hand side (Goursat 1934). Thus we deduce that the system of equations (12.1.9), which describes the problem under consideration, has a unique solution obtainable from Equation (12.1.18) and Formula (12.1.14). Figure 12.1 presents the types of vortex systems.
12.2. NUMERICAL SOLUTION OF THE ABEL EQUATION It is obvious that by solving the preceding linear unsteady problem by the method of discrete vortices, we develop at the same time a numerical method for solving the Abel equation (12.1.15). An analogous numerical scheme for solving the equation was proposed by L. N. Poltavsky, and
310
Method of Discrele Vortices LarDAI:
010
a
'0
-f
o
D
f
(a)
(b)
FIGURE 12.1. Distributions of discrete vortices (0 0 0) and calculation points (X X X ) over an airfoil (the segment [ -I, I]) and in the wake. (a) and (b) correspond to equally and unequally spaced points over the airfoil, respectively.
calculations carried out by E. B. Rodin demonstrated rapid convergence of the process. Rodin was also the first to extend the method onto the two-dimensional Abel equation. Consider the equation
/cp(s) ds 1o
<X
(
-
1- S )
t
0<0'<1,
f( ),
(12.2.1 )
which has the solution _ sin 0'7T [f(O) cp(/) - -7T- II-ex +
i
/,(s) ds ] )1 ex - S
l
0
(I
(12.2.2)
for any f(d having a continuous derivative function !(d (Privalov 1935). The numerical method for solving this equation is constructed as follows. We start by considering conjugated sequences of numbers. Let us consider the sequence of numbers {A ik }, k = 1,2,3, ... , i = 1, ... , k, subject to the conditions
2.
Akk
*- O.
(12.2.3)
For the sequence {A ik } and the number K *- 0 we define the sequence {Ark}, where k = 1,2, ... , i = 1, ... , k, in the following way: a.
Quantity Ark satisfies the system of equations k
L
AijAjk
= K,
k
=
1,2, ... ,
j~l
which always has a solution, because Akk *- O.
(12.2.4)
Unsteady Linear and Nonlinear Problems
b.
311
By definition,
Definition 12.2.1. Sequences {A7k} and {A ik } are called conjugated sequences with the conjugation index K.
Consider the sum k
A;,-A sk
L
+ A;,J+IA,-,!,k +,,, +A;kAkk ==
A;/,A pk '
(12,2.6)
p~s
By making use of Properties (12.2.3)-(12.25) we get from (12.2,6), k
k-s+ 1
L
A;p\'k
=
p=s
L
ArjAj,k-s-r I
=
K,
(12.2,7)
j= 1
This identity will be caIled the fundamental identity for conjugated sequences of index K, Next consider the system of linear algebraic equations k
L
A7k X ,=A k ,
k
=
1,.",n.
( 12.2.8)
'~l
Let us multiply both sides of the latter system by Akn and sum the result with respect to k from 1 to n: n
k
n
k=l
'~l
k~l
L Akn L A7k X, = L
AknA k ·
Next we invert the order of summation on the left-hand side of the equality: n
n
LX, L A7k Akn '~l
k~'
n =
L
AknA k ·
( 12.2.9)
k=l
By the fundamental identity 02.2.7) we get the equality
(12.2.10)
312
Method of Discrete Vortices
valid for any n. Therefore, substituting n - 1 for n and subtracting, we obtain
(12.2.11)
Now we will consider Equation (12.2.1). Let us divide the semi-axis = [(i - l)h, ih), i = 1,2, ... , and choose in E j an arbitrary point ~i. Consider the sequence {A ik = [{(k - i + Oh}" {(k - i)h}"l;a} that satisfies Equalities (12.2.3). Let us construct the conjugated sequence {Ark} (with index K = 7T/sin a7T). Consider the system of linear algebraic equations [0,00) into semi-intervals E j
k
L
Ark 'Ph ( g,.)h
=
k
f(kh),
=
1,2, ....
(12.2.12)
;=1
By Formula (12.2.11) we have
Using Relationship (12.2.3), we get
Let us start by supposing that f(O) = 0 and !"(t) is continuous. Then, Formula (12.2.13) becomes _ sin7Ta
'Pn( ~n) - - -
7T
n
L j 1
ii
r[u -
h
+ Ojh] ds
(nh - s)
{j - l)h
=
1)h
I- "
o < OJ
< 1,
(12.2.14) because
1
jh
Ak = I
(i _ l)h
ds (kh - s) I "
Unsteady Linear and NonLinear ProbLems
313
Now it may be readily shown that sin CX7T nh Icp(nh) - CPh({;n)1 ~ h--f~axl 7T
sin
0
ds (nh-s)
I-a
CX7T
~ --f~ax(nh)ah = O(h).
(12.2.15)
7T
where f~ax
Next let f
"*
=
max f" ( t ) , IE[O,nhl
nh
~
T < +00.
O. Then,
(12.2.16) and, hence, Icp(nh) - CPh({;n)1 ~ O(h) +
lAin sin CX7T --If(O)1 - -
1 I (nh) -a
h
7T
I .
(12.2.17)
In accordance with the previous definition of Aln , we have
where 0 < On < 1, n = 1,2, .... With analogous estimates, these results may be transferred onto the equation 1
cp(s) ds
----;::-a + 1o(t-s)
1K(t, s)cp(s) ds 1
=
(12.2.19)
f(s),
0
for which one has to consider the system k
L i=1
k
Ai,k CPh( gJh
+
L i=l
K(kh, {;;)cp( {;;)h
=
f(kh),
k
=
1,2, ....
Method of Discrete Vortices
314
Next consider the two-dimensional Abel equation
cp( ~ , TJ) d ~ dTJ
x y
11 o (x -
0
0
13
a
(12.2.20)
f(x,y),
=
(y - TJ)
where 0 < 0', f3 < 1. Representing the integral entering 02.2.20) as an iterated one and solving the resulting Abel equations sequentially, we get sin 0'7T sin f37T [ f(O,O)
cp(x,y)
= ----
7T
x
7T
1-13]
Y
ex
1 (fi ( ~, 0) d~ x
1
~
+
Y
0
1:)1- 13
x-~
(12.2.21) Let us divide the semi-axis OY into semi-intervals E./ = [(I" - J)h .. , ih), i = 1, 2, ... , and divide the axis OX into the sets "E;= [(j - J)hx,jh),j = 1,2, .... Considerthesequenceofnu~bers{Aik>{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
Method of Discrete Vortices
.-
..... ~ ./
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
Unsteady Linear and Nonlinear Problems
"
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
Method of Discrete Vortices
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)
Unsteady Linear and Nonlinear Problems
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)
Method of Discrete Vortices
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
Unsteady Linear and Nonlinear Problems
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
Method of Discrete Vortices
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
Method of Discrete Vortices
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
Method of Discrete Vortices
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.
Method of Discrete Vortices
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
boundary conditions 03.3.1) may be written in the form
iJ
+
Hy(z cos~ - x cos~)
+ Oz( x coslry -
y cos~).
(13.3.2)
Aerodynamic Problems for Blunt Bodies
329
Because the Laplace equation is linear, the potential may be presented in the form:
6 =
U,
L i~
U;$i· I
Then the expressions for forces and moments acting upon a body become (Belotserkovsky 1967)
dU
6
Mx
=
-
L -jA i4 i~
6
I
6
dU
6
L
I
U3U;A iZ ,
6
6
L
-
i~
I
6
U6U;A i4 +
L
U4U;A i6
i= I
I
6
- L
UJU;A il
+
L
U1U;A iJ ,
i~'
I
6
6
- L i~
I~
i= I
L -jA iS
i~
+L 4,U;A i5
6
UZU;A;3 +
i= I
i~
U,U;A i6 I
i~
- L My = -
6
L
-
U1U;A iZ + I
L i~
UZU;Ail' I
( 13.3.3)
Method of Discrete Vortices
330 where A·= -p 'I
f
a
(T
an M
(13.3.4)
dUM
are the so-called virtual inertia coefficients (integration is implemented ovcr the surface of a body, Le., the outcr side of u). It should be noted that in thc case of a body moving through water, virtual inertia is oftcn of the same order of magnitude as the mass and thc inertia momcnts of the body itself. Using the theory of functions of a complex variable, many authors have calculatcd virtual inertia coefficients of various bodies (Riemann and Kreps 1947). However, until now all attempts have failed to construct a unified technique for calculating them for arbitrary bodies participating in arbitrary motions. Nevertheless, this can be done with the help of thc mcthod of discrete vortcx pairs (for two-dimensional flows) or by cmploying the method of closed quadrangular-triangular vortex frames (for three-dimensional flows), as discussed in the following text. Thus, coefficients Ai} are calculated by means of finding thc flow potential
a
cosnx,
-- =
a
y cosnz - z cos ny,
a
x cosny - y cosnx.
-- =
-- =
a
-- =
an
cos ny,
=
a
-- =
cos nz,
z cosnx - x cosnz,
(13.3.5)
Let us denote j
=
(13.3.6)
1, ... ,6.
Then the problem of finding virtual inertia rcduces to solving the six integral equations j
=
1, ... ,6, M o E
U,
(13.3.7)
where V,(Mo) is specified by Formula 03.2.6), i.e., wc arrive at the solution of integral equations with rcspect to the double layer potential jump.
Aerodynamic Problems for Blunt Bodies
331
In the two-dimensional case the lattcr equation has the form 1 -f 2 7T
,2
(0
MM n
M'
n
Mn
) - 2(r
MM n ' 4 MMn
0 M
)(r
MM n '
0) Mn
g
r
u
(M) d
(TM
Mo E
=f(M) 0
(T.
,
(13.3.8)
If (T is thc intctval [ -1,1] of the OX axis lying in the plane OXY, then the equation has the form 1 JI "2 g(x) ( 7T -I X
and if
dx _)2 o
=
X
X
f(x o),
o E (-1,1),
(13.3.9)
is a unit-radius circle centered at thc origin of coordinatcs, thcn
(T
OoE[0,27T]. (13.3.10)
In thc three-dimensional case Equation 03.3.7) has thc form 1 r -f 4
2
(0
MM n
7T
0 M'
Mn
) -
3(r
r
(T
MM n ' 5 MM "
0 M
)(r
MM n
,0
Mn
)
g
(M)d
Mo E
(TM
if.
=f(M)
0'
(13.3.11)
If (T is the quadrant [ - 1, 1] X [ - 1, 1] of thc plane OXY bclonging to the spacc OXYZ, then 1 -4 7T
fl fl -I
- I
g(x, y) [
dxdy (x o - x)
2
+ (Yo - y)
2]3/2 =
f(x o, yo),
xo'Yo E (-1,1).
(13.3.12)
13.4. NUMERICAL CALCULATION OF VIRTUAL INERTIA COEFFICIENTS: SOME CALCULATED DATA Let us present the main idea of numerical calculation of virtual inertia in thc two-dimensional casco Let a c10scd contour (T be spccified in thc parametric form x = x( 0), y = y( 0), 0 E [0, 27T), where x( 0) and y( 0) arc periodic functions with the period 27T. Let us next choose points 0" ... , On interpreted as points of a circle dividing the latter into equal arcs, and
332
Method 01 Discrete Vortices
points 00l'o .. ,OOn' which are the middles of arcs 0,02, ... ,OnOI. Points M Ok = (X(OOk),y(OOk»' k = 1, ... ,n, are assumed to be the reference points, and by the following system of linear algebraic equations is substituted for Equation (13.3.8): n
L
m = 1, ... ,n,
Wkmg k = 1m'
(13.4.1)
k~1
where 1 A
_
Wkm -
-
27T
J
,2
MM Om
(0
M'
n MOm ) - 2(rMM llm '
0
M
)(rMMOm ' n ) M ilm d
4
lIM'
'MM Om
Uk
It is known (Sedov 1971-72) that a double potential jump at a surface of a body is determined to an accuracy of a constant. Hence, System (13.4.1) is ill-conditioned, and in order to single out a unique solution it suffices to specify a value of the potential jump at a point of the surface of a body:
In fact, to solve aerodynamic problems, not the function g(M) itself, but its derivative is needed. Hence, the function may be ascribed an arbitrary value at a point of the surface. System (13.4.1) subject to Condition (13.4.1) has the form n
L
wkmgk
=
1m'
m
=
1,2, ... , n.
(13.4.3)
k=1 k-Fq
This system has more equations than unknowns and has no solution. In order to obtain a well-posed problem, one has to introduce a regularizing variable 'YOn (Lifanov 1980a). This results in the system n
'YOn
+
L
wkmgk = 1m'
m
=
1, ... ,n.
(13.4.4)
k=1 k *q
The matrix of the system may be shown to be well-conditioned for any n (Dvorak 1986).
Aerodynamic Problems for Blunt Bodies
333
Because the value of the potential at the outer side of contour calculated by employing the formula (Tikhonov and Samarskii 1966)
(J
is
the following formula must be used for calculating the value of the potential at point MOm of the contour:
( 13.4.5)
Note 13.4.1. According to Sedov (1971-72), the velocity field due to the potential of a constant-strength double layer positioned at a given surface (J coincides with the velocity field induced by a vortex filament of the same strength placed at the boundary of surface (J (if the latter is unclosed). However, if (J is an unclosed curve, then the corresponding velocity field coincides with the velocity field induced by a pair of discrete vortices of the same strength, which is equal to the double layer potential strength. Therefore, for a two-dimensional problem one has (13.4.6) where W km is the normal velocity component at point MOm induced by a unit-strength vortex placed at point Mk(x(Ok)' y(Ok»' k, m = 1, ... , n. Therefore, this method for calculating aerodynamic characteristics with the help of system (13.4.4) is called the method of discrete vortex pairs (see Figure 13.1). Note 13.4.2. If the contour (J contains angular points, then points Mk are chosen in such a way that the angular points are among them; in other words, discrete vortices are placed at the angular points. Let us consider the three-dimensional problem. In accordance with Note 13.4.1, we start by considering closed quadrangular-triangular vortex frames. Therefore, surface (J is modeled by a system of such frames, and reference points MOm are placed at the centers of the frames (see Figure 13.2). Then, by fulfilling the no-penetration condition at the points, we arrive at the system of linear algebraic equations of the form (13.4.0, where w km is the normal velocity component at point Mom induced by the kth vortex frame of unit strength. This system is also iII-conditioned, and the condition of the form (13.4.2) singles out its unique solution again.
334
Method of Discrete Vortices !I
o'----....j--;'-=
fo'IGURE 13.1. Simulation of a closed contour by the method of vortex pairs.
!J
6
FIGURE 13.2. Simulation of a surface by closed vortex frames with the reference points positioned at the centers of the frames.
Using the latter condition, we arrive at a system of the form 03.4.3), which has no solution because it has more equations than unknowns. Then, introduction of the regularizing variable 'YOn allows us to obtain a system of the form (13.4.4), which is well-conditioned and in which _
_1 J
W km -
-
47T
Uk
2 'M
MUm
(0
M'
n MOm ) - 3(rMM om ' 5
0
M
)(rMM nm ' n ) MOm d
(TM'
'MM om
(13.4.7) Note that W km is equal to the normal velocity component at point MOm' induced by the vortex filament of unit strength located at the boundary of surface (Tk' It should be stressed that all the filaments must be similarly oriented.
Aerodynamic Problems for Blunt Bodies
335
A"
KfJtJ"1
1.51---4---+---
O'5L..-_--...,.L.--_---JI---_.......l_ _--1_ _--l
o
50
100
N
FIGURE 13.3. Convergence of calculated virtual inertia coefficients of an ellipse as a function of the number of discrete vortices. The solid and the dashed lines correspond to calculated results and the exact solution (Riemann and Kreps 1947), respectively.
Ir
~
lY
2b
/
z
V
~
o
~ l.,../ 0.5
V
1/
V
V *;:/
1/
Jr11
~ 1.0
1.5
q/tJ
.FlGURE 13.4. Virtual inertia coefficients for ellipses.
In order to demonstrate efficiency of the present method, it was used to calculate virtual inertia coefficients of ellipsoids, rectangles, and parallelepipeds (see Figures 13.3-13.7).
13.5. NUMERICAL SCHEME FOR CALCULATING SEPARATED FLOWS In separated flows vortex sheets develop not only at a body surface, but also in the wake (see Sections 9.1 and 13.1). The latter sheet may also be modeled by free discrete vortices. Let us start by considering two-dimensional flows. In this case, angular points of contour (J coincide with the flow separation points: it is quite
Method of Discrete Vortices
336
B
m
Jr
2lJ
k~
..4
!r;,
-
~ 0.5
tJ
~
~
1.5
1.0
FIGURE 13.5. Virtual inertia coefficients for rectangles.
--
a
1.50
1.12
o
U;~
\\.
' ..
'-.l.
.
1
2
1
2
II
FIGURE 13.6. The effect of a screen on the virtual inertia of a square. Calculated results and the exact solution are shown by crosses and the solid line, respectively (Riemann and Kreps 1947).
natural for an ideal fluid, because otherwise the flow velocity disturbances at the points tend to infinity. The vortex sheet at a body surface and in the medium may be modeled by pairs of discrete vortices, as shown in the work of Belotserkovsky, Lifanov, and Mikhailov (I 985); however, most often the simulation is carried with the help of discrete vortices. The approach is described here in detail. Figure 13.8 presents the spatial distribution of discrete vortices. At the initial instant they are positioned only at the contour itself; the angular points bf the latter are among the points occupied by discrete vortices (these are the discrete vortices assumed to be free, in accordance with Section 13.2). At the next instant, the free vortices are displaced into the
Aerodynamic Problems for Blunt Bodies
O,OS
-100
337
z
~
-
b
-
ISO
200
2S0
H,
FIGURE 13.7. Calculated values of the virtual inertia for a cube, '\66' depending on the number of vortex frames at the surface of the cube.
~~!----
x
FIGURE 13.8. Distributions of discrete vortices (eeel and reference points (x X X l in the case of the unsteady problem for a contour containing angular points.
flow along the local velocity vector v (the displacement being equal to v d'T, where 'T is the time step) without changing their circulation, because the medium is ideal. At the second calculation instant, one has to deal with discrete vortices of unknown circulation located at the contour and free vortices of known circulation located near angular points. Subsequently, the free vortices travel with the fluid particles, newly born free vortices shedding from each of the comers at each reference instant. The Lord Kelvin (Thomson) condition (according to which circulation around a closed contour embracing both a body and the wake stays constant) is met at each reference instant. Thus, at each reference instant one has to consider a system of linear algebraic equations with respect to circulations of discrete vortices located at the contour, for which the number of equations exceeds the number of unknowns by 1, i.e., a system of equations generally has no solutions. By introducing a regularizing variable, one may obtain a system of linear algebraic equations possessing a unique solution. Thus, for the scheme shown in Figure 13.8 we get the system of
338
Method of Discrete Vortices y
FIGURE 13.9. Distributions of closed vortex frames at the surface of a body and in the wake in the case of three-dimensional unsteady problems.
equations 4 'YOn
+
L
r-
I
L L
Wk,m
8R
sW/m
+ 27Tfm,
R= 1 s= 1
k=l
m n
L k~l
4
f[
=
= 1, ... ,p, r = 1,2, ... ,
r- 1
L L
8R ,s'
(13.5.1)
R~ls~1
which corresponds to the rth reference instant. The positions the free discrete vortices occupy at the next instant are determined by the formulas
(13.5.2) For, separated three-dimensional flows the discrete vortex sheet at the body and in the wake is represented by closed vortex frames (see Figure 13.9). At the initial instant, 'T = 0, the vortex wake is absent, and one has to solve the problem of noncirculatory flow past a body. Hence, the system
Aerodynamic Problems for Blunt Bodies
339
~~·.·1-..
. ..
.
.-I--..-....-~........
........, ... .....,...-
....-...-~ •
...............,-....-
"
e
Co II ~
10 .. ,.
.....................................-~
...-
•
~
-..
•• -.....-
II'
••
.A • • ..
....-
_..__.....,...-..-.....-
~
-........ .... ........................- ....----.... .
---_
- -__"'-
•
_..--
__
--...,.......
..--".--..-..............--... ~ ", ..... -~~,~~",', «
-
...........
_._
.. _ .....--......""'"'-"'
,
, .... ," ,' .....
. ·0'····----· · · · · · · · . .. _- .. _.,- .. -..
,~~~ ~~~~--'.~-_._._---,~~~...-~~
'~///~
_--"'" " ..
~
, / / ,• • • • • • • • t t " "
•• ••
,
~~,-~"",",
- -"""
_
"""
•• ,
. ._,
•• ~..
'--"--~""''''''''-. -,..,..------~
."''''' ,,- __
n ...... : • ,
........ _,...........
::.".
~..,
' .... , ,.... "............ " ~
.
'\"" ~ -. ~ '-'
.- ---
~
........................'""'---....-.e---.'"'""li:
Pm
"
.. .
...... "
--, •••• , '-"Ill"" "",- . , ...... " "IIl._ , ' . - . . . ~.-
"'---.--._-~,.,,-.
" ' 6 1 1 ' . - _ . __ " ,.~..
---
~
....
"
"IIl
__ .. ',."
.. , - - _ . t
-,.
__ . _
...
'. '~~:~~::::" .. ,.-.. .. "","""",--'-_--......
'- ... ,
--._, "-"~"""''''''''''''''''''''''''---... ":-:::.......--::..::.-.--:'-" ....•• "
••:.
::-::::::::::--::~=:..:
•
/ ~
• ..... _
."
....-
....~---""'~,.-~
~------
"",
~ ...._.-.-
.- -.-_.. .-
FIGURE 13.10. Symmetric unsteady separated flow past a square (the initial period of the flow).
___
.... _..
~_
•
FIGURE 13.11.
....
...... _........ ......i:""-. ,
....
~
.... _ _ .... - -
Asymmetric unsteady separated flow past a square.
of linear algebraiC equations for calculating the unknown circulations of the vortex frames has the form (13.4.4). If a vortex sheet sheds from the angular line L p of surface (J, then the side of a vortex frame lying at the body surface and bordering on L l> must lie at L p • The vortex frames bordering on the line L p on its opposite sides must have a common side
340
Vortice~
Method of Discrete
..
"",,~~~_~_~_~~~~,~~~,,~~~~~,,\\.9.
, ,,, """", ...... - - ..
,,""'-. "" ",
.. ....
4.....
'\ '\
... """"""",
", ,
"
.
~."'\\\\\\\.\.'
,.
••• " \ \ \ \ \ \ \ " , , •••
"'·f
\\\\\\,"' .... .. '".",\\", •...•. ,. ·"'11"""""----
...... ..... "
~.
'
~~
~
" ,
............ ,
..... , , , "
.. ,.... ...........
1It
•• -
,~
,\\ ••• , ' ~
_
.
P.""t • • 1.1.,"" _•••. , __- •" •••• ,. ""'-0,, .... '01
- _
................................... ~
~
,
• • #'
.-.;.
~
..
~~
. ... ~""""'\\\\'" .,,'" . " ' \ \ \ \ \ \ , . \ ..... . .. "" '\~\\\\ \"" .. " . ''''' ""\\\,,,.-
".'
....
.. - ....... -,,,,, '" "' .... , , ,
-.
"
~
I
'-' - - - -
11
_
.....
"
"
til
"
.... ~~-~~-----~,~
..
~,~,~~~--~~----."
"
"
~,
M'"
"
,
,~~.-~~
,.,~,-~
--~~-~'
FIGURJo: 13.12. Flow field in the case of unsteady flow past a triangle.
.. ....
. ·. . ·. ··· ... . ·.
-.~., DDDDD ~~~~
... , 00000 I \~", , .. , ... ,,""" , " , 00000 .\"", . ,, ,.. ","" ....... ,. . . " DDDDD DDDDD
(a)
... .. .. . . . .. ..
... "" "_ ,-,
. .
..
::::;~:~::::""'~r;;::: : :::.:= .~::':""~~ ~3::: " :::
\ , ..... ,
",,, !
.. . .. .. ..
.·..... " .. ............ ..... · ......... ··. ...... - :: : .....1----1: . : : . : 00000
., •••• DDDDD ., ••••
~) 0
Z
..
.... 1----1,.,::::: DDDDD) : : : : : : : : DDDDD
\----1: : : : : : : : DDDDD (c)
,
4
'I"
a
(d)
FIGURE 13.13. correspond to T
=
Flow field in the case of unsteady flow past a building. (a), (b), and (c) 0.5,2.5, and 5.0, respectively; (d) shows the flow velocity variation in a gust.
(see Figure 13.9). For 'T > 0 a free vortex frame of circulation 8;,. equal to the difference of circulations of the corresponding vortex frames at the surface (J sheds into the flow from this common segment: r k- 1 _
m
rk- 1
}
(13.5.3)
Circulations of the free vortex frames do not vary as time passes. and each
Aerodynamic Problems for Blunt Bodies
341
p
o
A
1
-2
- "0' A
U.
"
D
,
D
. ----0 . ....... ~
~ .
o
•
FIGURE 13.14. Pressure distribution at the surface of a square. The solid line shows calculated results, ••• shows experimental data of Sluchanovskaya (1973) and 0 0 0 shows experimental data of Gorlin (1970).
p
- A()C fj
~ K.. o
'\"
A
1
u.
-
~o
,
-0
~
-2
FIGURE 13.15. Pressure distribution at the surface of a rhombus. The solid line shows calculated results and 0 0 0 shows experimental data of Gorlin (1970).
FIGURE 13.16.
Unsteady flow past a cube. The form of the vortex sheet for
T
= 6.
342
Method of Discrete Vortices
~
w
y ~---~
-..~
.....
~
~-.~
~~-..-..
.....
.............
~~.-.-..~
~~~..-.
FIGURE 13.17. The vector flow vclocity field in the plane OXY for
o L-_-!-__'::-_--!I~-..l--~'_-~' Z
3
lJ
FIGURE 13.18. The drag coefficient
S
T
=
6.
.L
6
ex for a cube versus T.
of the vortices displaces along the local flow velocity vector (see (13.5.2». The preceding numerical technique for calculating separated unsteady flows was used to compile programs and to calculate two-dimensional flows past a square, a triangle, and building-like figures (Figures 13.10-13.15) and three-dimensional flows past a cube (Figures 13.16-13.18).
14 Some Questions of Regularization in the Method of Discrete Vortices and Numerical Solution of Singular Integral Equations
14.1. ILL·POSEDNESS OF EQUATIONS INCORPORATING SINGULAR INTEGRALS While considering singular integrals, one often comes across situations when, due to measurement or calculation errors, the density of an integral is only known approximately. The operation of calculation of a regular integral is known to be stable in all commonly used metrics; in other words, small variations in a regular integral correspond to small variation in its density (Tikhonov and Arsenin 1979). In Luzin (1951) and Khvedelidze (1957) it was shown that singular integrals, and hence equations incorporating the laUer, are stable in the metrics L z with a corresponding weight at a given curve. On the other hand, Luzin (1951) constructed an example of such a function f( 0), f7T 1T f( 0) dO = 0, which is continuous on [ - 7T, 7T] and has a limited variation, whereas the function -1 y(Oo)
=
-2 7T
0 0 1 cot--f(O) dO 2 27T
0 -
(14.1.1)
0
suffers infinite discontinuities at a set that is dense all over [ -
7T, 7T ].
343
Method ofDiscrete Vortices
344
This example shows that singular integrals are unstable in uniform metrics, Le., if f( (J) is a continuous function close to zero in a uniform metric, then y«(J) may differ from zero by a however large quantity. Singular integral (14.1.1) proves to be unstable also in the case when the function f l ( (J) - fz( (J) is smooth and has small values in a uniform metric. The following theorem is true. Theorem 14.1.1. For any positive numbers € and M there exist such periodic functions !J{ (J) and f 2 ( (J), infinitely differentiable on [0,2 7T], that the corresponding functions Yl«(J) and yz<(J) (see Formula (14.1.1) are also infinitely differentiable, and the inequalities
(14.1.2)
hold. Proof. In the theory of trigonometric series it is shown (Luzin 1951) that if function f( (J) belongs to the space L z on the segment [- 7T, 7T], then it may be represented by the Fourier series
f«(J) -
L
(ancosn(J+bnsinn(J),
rr
f«(J) d(J
=
0, (14.1.3)
-11"
n=l
and the conjugated function y«(J) given by the formula
y«(J) -
L
(-bncosn(J+ansinn(J),
(14.1.4)
n=l
also belongs to L z on [ - 7T, 7T], and the equalities 1
f«(J) = 27T
1 cot--y«(Jo) d(Jo 2 11"
0
8 - 80
2
(14.1.5)
and (14.1.0 hold. The function constructed by Luzin (1951) is such that the corresponding Fourier series converges to it uniformly. Let us denote by fn( (J) [Yn( (J)] the nth truncated sum of the Fourier series for function f( (J) [Y( (J )]. Function Yn( (J) is conjugated with respect to function fn( (J), and hence the two functions satisfy Relationship (14.1.5). Because function fn«(J) converges uniformly to function f( (J), a number n l( €) may be found for a given
Some Questions of Regularization
345
number € > 0, such that for all m, n > n 1(€) the inequality ( 14.1.6) will be valid. On the other hand, function Y( 0) is a limit of the sequence of functions Yn ( 0). Let point 0 1 be the point where function Y( 0) suffers an infinite discontinuity. Hence, at the point 1Yn( 0,)1 ---) 00 for n ---) 00. Thus, if we consider numbers m and n satisfying the latter inequality, and fix n while m tends to infinity, then IYn(OI) - Ym(O,)1 ---) 00 for m ---) 00, Le., for a given number M > 0 there exists m(M) such that
Thus the theorem is proved, because functions fn( 0), f m( M)( 0), Yn( 0), and Ym( M / 0) are infinitely differentiable. • As far as a singular integral over a circle is expressible via an integral with Hilbert kernel, it is unstable in a uniform metric. In Gakhov (1977), Ivanov (1968), and Muskhelishvili (1952) it is shown that a singular integral over a segment or a system of nonintersecting segments may be reduced to a singular integral over a circle. Hence, the integrals are unstable in a uniform metric. Because solutions to singular integral equations are expressed through singular integrals, the former are also unstable in a uniform metric. In practice, as a rule, one has to solve numerically singular integral equations whose right-hand sides and regular kernels are known approximately in a uniform metric. Therefore, the methods of numerical solution of these equations must be stable in a uniform metric because in practice one has to compare calculated results mainly in this metric). Such solutions may be constructed by the method of regularization developed by Tikhonov (Tikhonov and Arsenin 1979). Maslov (1967) has shown that a certain modification of the regularization proposed by Tikhonov for limited operators is applicable to singular integral operators that are unlimited operators in metric C. However, a reasonable use of singularities present in kernels of singular integral equations allows us to employ the simplest method of regularizing a numerical solution without constructing the regularizing operator explicitly. Originally, the idea of such regularization was used for integral equations of the first kind with a logarithmic singularity in works by Tikhonov, Dmitriev, and Zakharov (Dmitriev and Zakharov 1967, Tikhonov and Dmitriev 1968), who called it natural regularization or self-regularization. Later it was proved (Arsenin et at. 1985,
346
Method of Discrete Vortices
Lifanov 1980a, Matveev 1982b) that the method of discrete vortices, treated as a method of solving singular integral equations numericalIy, and its generalizations described in the preceding chapters, are also a method of natural regularization. Because the preceding regularizing methods of numerical solution are based on using various quadrature formulas for estimating singular integrals, in the folIowing text the quadrature formulas will be shown to possess the regularizing property in both uniform and integral metrics.
14.2. REGULARIZATION OF SINGULAR INTEGRAL CALCULATION In this section the folIowing problem is considered (Matveev 1982b). Let a singular integral (one of those considered before) be calculated with the help of one of the previously mentioned quadrature formulas with (1) its density at the grid points known exactly and (2) the density known with a certain error whose value is known in a uniform metric. What will be the difference between the calculated values? By what means may the difference be made smaller by increasing the number of points? . Consider a singular integral on a segment in the case of using equalIy spaced grid points. The folIowing theorem is true. Theorem 14.2.1 (Matveev 1982b). Let the density cp{t) of singular integral (5.1.1) ocer segment [a, b] be known at the equally spaced grid points E = {t i = a + ih, h = (b - a)/(n + 0, i = 1, ... ,n} with an error Ei such that IE,I ::s:; E, i = 1, ... , n. Let Sn(t o), t Oj = t j + h/2, j = 0,1, ... , n, be the value of the quadrature sum for the integral calculated by using Formula (1.3.6) at the grid points Eo = {t Oj ' j = 0,1, ... , n} when the values of cp(t) are substituted by cp(t) + Ei . Then the inequality (14.2.1) holds where j = 0,1, ... , nand O(E In n) = 2d3 + In n).
The proof of the theorem folIows from Formula (1.3.2). Note that the latter estimate depends only on the relative position of grids E and Eo for n ---) 00, and hence, the grids may be translated along segment [a, b] (see notes concerning the choice of grids in Section 1.3). Note 14.2.1. Estimate 04.2.0 cannot be perfected as far as the order of magnitude is concerned.
347
Some Questions of ReguLarization
In fact, let n = 2m and €; = - €, i = 1, ... , m; €; = €, i = m 1, ... , 2m. Then
L
€.h
;~l
t i -tom
n
h
m =
2€i~1
~
2€ In(n
I
m-l
-tl-1m---t-
i
~
+
2
2€ k~O 1 + 2k
+ 1).
(14.2.2)
If n = 2m + 1, then by putting €i = €, i = m + 1, ... ,2m + 1, onc gcts an estimate of the form 04.2.2) again. Note 14.2.2. By denoting the value of a singular integral at points t Oj by l(to), one gets
(14.2.3) In the latter estimate the first summand, O(tOj)' depends on the form of the function cp(t), and hence, this circumstance must be taken into consideration when choosing grids E and Eo. Thus, if cp(t) has an integrable singularity at point q E (a, b) (as in the problem of an airfoil with a flap), then E and Eo must be chosen in the folIowing way: 1. 2.
Point q lies midway bctween the nearest points belonging to E and Eo· Point q coincides with one of the points E if cp(t) suffers a discontinuity of the first kind at the point; however, in the latter case one must take [cp(t; q - 0) + cp(t; If + 0)];2 at point t i q = q.
The behavior of quantity O(tOj) was analyzed in the preceding chapters. The second summand O( € In n) depends only on the form and relative position of the grids E and Eo. Next we consider a singular integral over a circle. In this case the folIowing theorem is true. Theorem 14.2.2. Let us denote by Sn(toj) the quadrature sum for an integraL over a circle at point t Oj ' caLcuLated with the heLp of FormuLa (1.2.9), where cp(t) + €; is substituted for cp(t). Then estimate 04.2.0, where O( € In n)
=
27T€(3 + In n)
(14.2.4)
348
Method of Discrete Vortices
is valid for ISn(to) - Sn(toj)l, where Sn(tOj) is specified by the same formula.
The proof of the theorem follows from refinement of Formula (1.3.2). It suffices to note that in Formula 0.2.8) we have C = 1T, because sin x ;:::.: 2X/1T for 0 $ x $ 1T/2. Note 14.2.3. An analogous statement is true for the quadrature sum for an integral with Hilbert kernel, obtained from Formula (1.5.4) or (2.1.19). In fact, in this case the inequality
Ln
I
k= I
21T cot Ok - °OJ 12 n
21T(3
$
+ Inn)
( 14.2.5)
holds. Note 14.2.4. Theorem 14.2.2 is valid also for the interpolation quadrature formula for a singular integral over a circle. In fact, we have
I
~ 1 1121Titki '- - - - - - - < t k - tOm 2n + 1 -
k=O
2n L k=O
1T/(2n + 1) Isin( Ok - 00m)/21
$
21T(3
+
In n).
(14.2.6)
Next consider interpolation quadrature formulas for a singular integral over segment [ -1,1]. Thus, let
Let
Q'
cp(t)
=
w(t)l/J(t),
w(t)
=
(l -t)"'(1 + t)13,
l/J(t)
E
(14.2.7)
H.
= f3 = -1j2. Then, for m = 1, ... , n - 1, n
L
k= I
Icos 2k -
1T/n
1 1T - COS-1T 2m 2n 2n
1
=
1T $
1T/n
n
"2 kL:1 I. 2( k Sin
- m) - 1 4n
I . 2( k + Sin
m) - 1 1T 4n
I
1T/n
n
"4 kL:1 12( k
I
- m) - 111T 4n
::::; D -1/ 2, -1/2(m, n),
I . 2( k +4m) SIO
n
1
1T
I (14.2.8)
Some Questions of Regularization
349
where the quantity D -1/2, I/im, n) satisfies the relationships: 1.
For
tOm E
[-1 + 8,1 - 8],
(14.2.9)
D 1/2, -1/2(m, n) = 0l,o(ln n).
2.
For
tOm E (
-1, -1
+ 8) or
8, 0,
10m E (I -
1
D_ I / 2,_1/2(m,n)= .f
V 1 - t~m
(14.2.10)
02o(lnn). '
Let now a = f3 = 1/2. Then, for m = 0, 1, ... ,n + 1, 7T
n
L k=l
k
--sin 2 - - 7 T n+1 n+l 2k 2m - 1 cos 7T - cos 2( n + 1) 2( n + 1)
I
1
=
7T
k
7T • 2 --SIO --7T
n
L "2 k ~ I
I.
Sm
n+l 2( k - m) + 1 7T 4( n + 1)
n+l 2( k + m) - 1 7T Sm 4( n + 1)
II'
7T
-
4
I
k7T
7T
< -
I
--sin 2 - n+I n+l
Ln -,-------'-'---'--,--,.---.,--------,k =1
/2(k-m)+11 4( n + 1)
I'
7T Sm
2(k+m)-1 4( n + 1)
7T
I (14.2.11)
where the quantity D I / 2 , I/im, n) satisfies Relationship (14.2.9) for any tOm; in other words, the quantity of the order of In n entering this relationship is independent of 8. Finally, consider the case when a = 1/2 and f3 = -1/2. Then, 47T
n
L k= I
k
---sin 2 7T .---=2..:.:n-=-+-;-::I'------'2=-=n-:........:+~I-___;:_;_ 2m-- 1 cos 2k 7T - cos 2n + 1 2n + 1
I
I
Method of Discrete Vortices
350
47T
7T
~ "4 k
---sin z 2n + 1 2n
+1
7T
r:] --,-1-2(-::k---m=)..:....+--'---:-'11;-----,/-.-=..:..:2(-::k'--+~m-)---1 --;""/ n
47T
$
k
+
D ljZ , -I/z(m, n),
2
7T SIO
4n + 2
7T
( 14.2.12)
where the quantity D ljz , . 'jim, n), m = 1, ... , n, satisfies Relationship 04.2.9) for all lorn E (-1, + 8,1) and Relationship 04.2.10) for all tOm E [-1, -1 + 8]. Thus, the validity of the following theorem ensues from Formula 04.2.3) and the results obtained in Chapter 1 and this section. Theorem 14.2.3. All the quadrature formulas constructed previously by using two grids possess the property of regularization in a uniform metric. The number of points n forming the grid E is a regularizing parameter. Let us explain the idea of this theorem. Let functions cp(t) and fjJ(t) be specified at points of grid E with an error € in a uniform ~etric; in other words, let the function (j;(t) [~(t)] be specified. Let O(tOm, n, €) be the absolute value of the difference of a singular integral at point tOm and the quadrature sum calculated by using function (j;(t) [~(t)]. Then, for any number Ll > 0 there exists such a number dLl) > 0 and such a function n( €) ~ ~ for € ~ 0 that for any positive number € < d Ll), the quantity O(t om ' n(€), €) does not exceed the number Ll at the corresponding set. Thus, for a singular integral on segment [-1,1] when cp(t) E H* on [ - 1, 1], the preceding set is the segment [ - 1 + 8, 1 - 8]. For a singular integral at a circle when cp(t) E H, the set is the whole of the circle, etc. (see the results obtained in Sections 1.2 and 1.3 and in this section). Note 14.2.5. Quadrature formulas of the form (1.3.17), constructed by using either the grid E or the grid Eo, also possess regularizing properties.
14.3. METHOD OF DISCRETE VORTICES AND REGULARIZATION OF NUMERICAL SOLUTIONS TO SINGULAR INTEGRAL EQUATIONS As. far as singular integral equations are unstable in a uniform metric, the question arises of whether the preceding methods of approximate solution of the equations are stable in the metric. If the approximate methods were developed on the basis of some quadrature formulas for singular integrals, it is clear that the quadrature formulas must be stable
351
Some Questions of Regularization
with respect to small variations in the integral density in a uniform metric. Upon passing from singular integral equations to Fredholm integral equations of the second kind, one may use any quadrature formulas possessing regularizing properties, because a method of approximate solution of Fredholm equations of the second kind developed on the basis of the formulas is stable. It is of interest to analyze stability of direct methods of numerical solution of singular integral equations, developed on the basis of quadrature formulas possessing regularizing properties. It turns out that the use of such arbitrary quadratures may result in developing unstable methods of approximate solution of singular integral equations. For example, by using the quadrature formula of the form (1.3.17) on the set E = {t i = a + ih, h = (b - a)/(n + 1), i = 1, ... , n}, we deduce that the system of linear algebraic equations j
=
1, ... , n,
(14.3.1)
contains n unknowns and n equations. The system approximates the characteristic singular integral equation (5.1.1) on [a, b], but its determinant vanishes for any odd n. In Ivanov (1968) it is shown that a similar situation may take place for a singular integral equation of the second kind on a circle. By using two grids for constructing a numerical method of solving singular integral equations, one may develop welI-conditioned systems of linear algebraic equations providing a method that is stable in a uniform metric. Originally, the idea of using a pair of grids was formulated in Belotserkovsky (1955) while constructing the method of discrete vortices for solving aerodynamic problems. In this section we will show that the methods of solving singular integral equations, described previously and based on the ideas of the method of discrete vortices, are in fact methods of regularization for the equations. 1. Let us consider the problem of stability of the preceding methods with respect to small variations in right-hand sides. We will start by analyzing Equation (5.1.1) and presenting Systems (5.1.3)-(5.1.5) in matrix form: A'Pn
=
fn'
(14.3.2)
where A is a matrix of coefficients appearing in the preceding systems of equations, and 'Pn and fn are the columns consisting of unknowns and right-hand sides, respectively. Next consider the system (14.3.3)
352
Method of Discrete Vortices
where En is a column of random errors, such that each of the coordinates is smaller in the absolute value than the number E > O. The following theorem is true. Theorem 14.3.1. For a difference of solutions to Systems 04.3.2) and 04.3.3), the inequality (14.3.4) holds, where {3~1)(tk) has the following properties: 1.
For all points t k positive number,
E [ -
1 + 8, 1 - 8], where 8 is an arbitrary small
(14.3.5) 2.
For all points t k
E [
-1,1], n
L
~3~1)(tk)h
$
BI •
(14.3.6)
k=1
(here constants D I , (; and B I are independent of n).
Proof. System 04.3.3) has the solution ( 14.3.7) where elements of the inverse matrix A I are specified by Formulas (5.1.13). By this formula and Equation (14.3.7) the validity of Inequality (14.3.4) ensues from Equations (5.1.14), (5.1.15), and (5.1.23)-(5.1.26), because 1 nih _len) _pn)
h K,k.
L
J= I
h K,Ojlt - t k
$ .1
ml)(tdln n.
(14.3.8)
OJ
Theorem 14.3.1 is also true for the characteristic equation at a circle (6.1.1) if System (6.1.2) is substituted for Equation 04.3.2). However, in this case, 3~1)(tk) .:::;; D z for all t k at the circle where the constants D z is independent of n. This conclusion follows from Formulas (6.1.6), (6.1.10), (6.1.1 i), and 04.2.4). Theorem 14.3.1 is valid also for characteristic equation (6.2.0 with the Hilbert kernel. In the latter case either System (6.2.6) or System (6.2.19) must be substituted for Equation 04.3.2), and the matrix A - I is given by
Some Questions of Regularization
353
Formula (6.2.13) or (6.2.23), respectively. Then one has to employ Formula 04.2.5) according to which ,~~l)( Ok) $ D 3 for Ok E [0, 27T], where D 3 is independent of n. • Next we consider Equation (5.1.1) for the case when [a, b] = [ -1, 1] and unequally spaced grid points are used. Let us write the systems of linear algebraic equations (5.2.2), (5.2.15), and (5.2.16) in the form (14.3.9) and the system incorporating errors on the right-hand sides in the form ( 14.3.10) Then the following theorem is true. Theorem 14.3.2. For the difference of solutions to Systems (14.3.9) and (14.3.10), the inequality (14.3.11) holds where the quantity ~1~2)(tn) possesses the following properties: 1.
If
K =
1, then
( 14.3.12)
2.
for all tk E [ - 1, 1]; If K = - 1, then
t k E[-1+8,-8], tk
3.
4.
E
(14.3.13)
[-1, -1 + 8] U [1 - 8,1]. (14.3.14)
If K = 0, then the quantity i{~2)(tk) satisfies Inequality (14.3.13) for points t k E [-1,1 - 8] and Inequality (14.3.14) for points t k E [1 8,1]; For all K, n
L k~l
m)(td a 2
k
.:s; B 2 "
(14.3.15)
Method of Discrete Vortices
354
Here D4 , D s, D 6 , and B 2 are constants independent of both nand k. The validity of Theorem 14.3.2 follows from the results obtained in Section 5.2 and Formulas (14.2.8), (14.2.11), and (14.2.12). Note 14.3.1. Estimates (14.3.4) and (14.3.10 cannot be refined. In the simplest way this may be demonstrated for an equation with the Hilbert kernel. Let us choose points Ok in such a way that the point 7T is one of them for k = k 1T • The errors will be selected as follows. If 00j < 7T, then En,j = - E; if 00j > 7T, then En,j = E. Then the following inequality holds at point 7T:
L cot 7T -
°OJ
n
Ij=l
2
27T En,j -
n
I~ B
38
In n,
(14.3.16)
where the constant B 3 > 0 is independent of n. Analogous examples may be considered for other cases also. 2. Let us continue by considering the problem of st,ability of the numerical methods under consideration with respect to small variations in the elements of matrices of systems of linear algebraic equations. Let us consider the following system of linear algebraic equations: (14.3.17) where matrix Al may be obtained from matrix A (see System (14.3.2) as follows. If uniform grids E and Eo are used in a singular integral equation on a segment, then t l and t Oj are substituted by
(14.3.18) respectively, where the number E must be sufficiently small (say, E ~ 1/8), so that points () and tOj are not coincident, and h is substituted by (14.3.19) If ail equation with the Hilbert kernel i~ considered, then, in an_alogy to Formulas 04.3.18) and (14.3.19), we put Ok = Ok + EI,n,kh and 00j = 00j +EI,n,Ojh, ~ = 27T/n, and for an equation over a circle, t k = exp(iO k ) and tOj = exp(i OOj)'
Some Questions of Regularization
355
The following theorem is true.
Theorem 14.3.3. For a difference of solutions to Systems 04.3.2) and 04.3.17), the inequality l'Pn(td - ~n(tdl ~ ~~~')(td(€ Tln,.=€
+ TIn,. +
TI:,.)ln n,
1 - (€ In n )n 1 - € In n Inn
(14.3.20)
holds, where the quantity ~3~)(tk) is analogous to the quantity ~~l)(tk)'
Proof. System 04.3.17) is solvable, and (14.3.21) where matrix A 1 1 may be obtained from matrix A - I by replacing the factors~ /(n) and I(n). K = 0 1 -1 by the factors j(n) and j(n). obtained K,k K.OJ' '" ~ K,k K,OJ from the original ones by replacing t k and t Oj by tk and (OJ, respectively. By the choice of points (k and (OJ' one gets n
j(n) K,k
=
Tl
l(n)(1 + f3K,k ) K,k
(1 +
0'
k,m ) ,
m=l
m*k n
j;~Jj
=
/;~Jj(1 + 13 ,oj) K
n (1 + O'Oj,m),
m~l
m *j
(14.3.22) where the constants B 4 , B 5 , and B 6 are independent of n. Let us prove Formulas 04.3.22) for jJ,nl and jJ~(~j in the case of equally spaced grid points. We have [see (5.1.12)]:
( 14.3.23) and hence, Formula 04.3.22) is valid.
356
Method of Discrete Vortices
Thus,
where B 7 is independent of n. The validity of Theorem 14.3.3 follows from Formulas 04.3.21)-04.3.24) and Theorem 14.2.1. • Note 14.3.1. If € In n < 1, then the factor preceding d~)(tk) in Inequality 04.3.20) is of the same order of magnitude as € In 2 n. If [a, b] = [-1,1] in Equation (5.1.1) and unequally spaced grid points are under study, then one has to consider the system of linear algebraic equations
( 14.3.25) where matrix A, is obtained from matrix A entering System 04.3.9) as follows. If we denote t k = cos O"k' t Oj = cos O"Oj' and a k = a(Ok)' then (k = cos(Ou + €"n,k h ), (OJ = cos(OI,Oj + €l,n,OLh ,), hI = 7Tln and Ok = a(Ok + €I,n,khl) must be substituted for tk and t Oj ' Next, matrix A tk , t Oj ' and ii k are to be substituted for t k , t Oj ' and a k entering matrix A, respectively. Then, an inequality of theJorm 04.3.20) is also valid for the absolute value of the difference r/Jn - ~n' where the quantity ~~~')(tk) has the same properties as quantities D'j2, 1/2(m, n), D -lj2, 'jim, n), and D- 1j2 , -Ijim,n) for K = 1, -1, and 0, respectively, and the solution is limited at point t = 1. Let us consider the differences between solutions cp(tk) to Equations (5.1.1) and (6.1.1) and CPk(Ok) to Equation (6.2.1) and the corresponding solutions to systems of linear algebraic equations of the form (14.3.3) and (14.3.17); in other words, we will analyze the quantities O(l)(€,n,t k ) and 0(2)(E, n, t k ), where 0(1) = Icp(t k ) - q)n(tk)1 and 0(2) = ICP(rk) - ePn(tk)l. The following theorem follows from Theorems 14.3.1 and 14.3.3. Theorem 14.3.4. Let equally spaced grid points be chosen for Equation (5.1.1), and let function f belong to the class H on [a, b]. Then for any Ll > 0 there exist such a number d Ll) > 0 and such a function n( E) ~ ao for € ~ 0 that for any positive number € < d Ll), the following inequalities are valid: a.
For all the points t k number,
E
[a
+ 8, b - 8], where 8 > 0 is a fixed small i
=
1,2.
(14.3.26)
Some Questions of Regularization
b.
For all the points t k
E
357
[a, b],
i
=
1,2.
(14.3.27)
To prove this theorem it suffices to use the inequalities
and Theorems 5.1.1,14.3.1, and 14.3.3. For all t k , Inequality 04.3.26) holds for equations on a circle as well as for equations with the Hilbert kernel. If [a, b] = [ -1,1] in Equation (5.1.1) and the grid points arc unequally spaced, then by O(i)(€, n, t k ), i = 1,2, one has tQ denote absolute values of the differences fjJ(tk) - (frn(t k ) and fjJ(t k ) - (frn(t k ) for which Theorem 14.3.4 also holds. In accordance with the latter theorem, the preceding algorithm for obtaining approximate solutions to Eqs. (5.1.1), (6.1.0, and (6.2.0 is a regularizing one. The regularizing parameter n is equal to the number of equations of a system of linear algebraic equations substituting for the original singular integral equation. Numerical methods for full singular integral equations of the first kind on a segment-circle and with the Hilbert kernel possess an analogous regularizing property.
14.4. REGULARIZATION IN THE CASE OF UNSTEADY AERODYNAMIC PROBLEMS In the preceding section we have shown that the method of discrete vortices is in fact a method of regularization of numerical solutions to iII-posed (in a uniform metric) problems of steady two-dimensional flow of an ideal incompressible fluid past an airfoil. Presently, there are no analogous mathematical proofs for unsteady problems of flow past an airfoil. Moreover, in order to ensure convergence of a numerical solution in the case of a nonlinear unsteady problem, one has to take into consideration the following specific feature of the problem (Belotserkovsky and Nisht 1978). From Formula (9.3.14) it follows that flow velocities induced at a point approaching a vortex filament tend to infinity. One can come across a similar situation in the case of a nonlinear unsteady problem, when the solution may start oscillating at a segment. In
358
Method of Discrete Vortices
order to suppress the oscillations one has to artificially limit flow velocities in the neighborhood of the vortex axis. However, the following simple and most efficient method of regularization may be proposed. Let € be the shortest distance between reference points and discrete vortices at an airfoil (both the points and vortices may be unequally spaced). For a chosen vortex system one cannot be sure that the flow velocity field will be determined correctly within the neighborhood of the vortex axis. Therefore, if in the process of solution the distance between free vortices becomes less than €, the induced flow velocities must be limited. According to (9.2.1), disturbed flow velocities vanish at the axis of a vortex filament. Hence, the flow velocity field within a circle centered at the vortex axis, whose radius is equal to €, should be either determined by means of linear interpolation between the values of the velocity at the circle and at the vortex axis or just put equal to zero. Calculations show that due to discreteness of the scheme with respect to both coordinates and time, free vortices sometimes "jump across" the airfoil surface. However, this deficiency of the method was removed by introducing a condition according to which such a vortex returns to its original position at the next step.
Pan IV: Some Problems of the Theory of Elasticity, Electrodynamics, and Mathematical Physics
15 Singular Integral Equations of the Theory of Elasticity
15.1. TWO-DIMENSIONAL PROBLEMS OF THE THEORY OF ELASTICITY Following M. M. Soldatov (Belotserkovsky, Lifanov, and Soldatov 1983), we will demonstrate how two-dimensional problems may be reduced to singular integral equations. The basic relationships of the theory of elasticity are given by the Kolosov- Muskhelishvili formulas (Muskhelishvili 1966, Parton and Perlin 1984), which determine stresses through two functions of a complex variable, cp and fjJ, analytic in the domain D of the plane OXY: (T
+
+
2i'TXY
x
O"y - O"x
O"y = 4Re[cp'(z)], =
2[.Zcp"(z)
+
fjJ'(z)].
(15.1.1)
In the absence of bulk (mass) forces, the functions satisfy the homogeneous biharmonic equation of the problem; thus, it suffices to meet the boundary condition at contour L bounding the domain D: tEL,k=1,2.
(15.1.2)
If the latter condition is written in displacements, then k = 1, and
K)
=
I, =
K
3 - 4v
for plane strain,
= { ;::
for plane stress,
2t-t(u
+ iv),
(15.1.3) 361
362
Method of Discrete Vortices
where v is the Poisson coefficient, f.J- is the shear modulus, and u and v are displacements along the axes OX and OY, respectively. However, if Condition 05.1.2) is given in stresses, then k = 2, and
KZ =
-1,
(15.1.4)
where (Jx' (Jy' and T xy are the normal and tangential stresses in the coordinate system OXY; I = cos(n,x) = dy/dsand m = cos(n,y) = dx/ds are the direction cosines, and C z is a complex constant determined by the condition to be presented in the following text. For a singly connected domain, contour L in 05.1.2) is passed around counterclockwise; in the case of a multiply connected domain, contour L is passed around in such a way that domain D stays to the left. For functions cp(t), fjJ(t), and cp/(t) and complex conjugated functions fjJ(t) and cp/(t), limiting values are taken at points located inside and outside D in the cases of solving the internal and the external problem, respectively. Analytic functions cp( z) and fjJ( z) are sought in the form
cp(z)
=
1 -.! -tw(t) dt, - z 27ft
L
fjJ(z)
1
=
- 2. 7ft
! C~ - iw'(t) dt, t -
L
Z
zED,
(15.1.5)
where the auxiliary function w(t) is defined by the integral equation
Kk -
2
C w( t) +
~! w( T ) d (In T - i ) + 27ft
L
T -
t
Kk
+C
27fi
1 (T-t) X!w(T)d(ln(T-t»+-.!W(T)d ~ =fk(t), L 27ft L T - t
k
=
1,2,
(15.1.6)
obtained from 05.1.2). Here C is a complex parameter determining the type of the resulting equation. For C = - Kk' k = 1,2, one arrives at the Fredholm equation of the
Singular Integral Equations of the Theory of Elasticity
363
second kind (Muskhelishvili 1966): KkW(t)
K + _k.jW(7)d
27Tt L
(T-f) 1 (7-t) I n - - + -.jw(7)d -=----=- =fk(t). 7-t
27Tt L
7-(
(15.1.7) Due to the presence of an eigenfunction, the latter equation is hardly solvable by the method of formal quadratures, because the determinant of the corresponding system of linear algebraic equations is equal to zero, and calculated values of the function we seek are unstable (Parton and Perlin 1982, 1984). One comes across similar complications when reducing the problem to other regular equations of the Muskhelishvili and Sherman-Lauricella types [see Parton and Perlin (1984) where some methods of removing the drawback are considered, such as fixation of W at some points and exclusion of the corresponding equations, and the use of errors of quadrature formulas for refining the structure of algebraic equations]. If C = 0, then Equation (15.1.6) is a degenerate singular integral equation of the second kind. A nondegenerate equation of the second kind may be obtained, for instance, for C = i. However, here we analyze the reduction of Equation (15.1.6) to a singular integral equation of the first kind, which is achievable for C = K k • Then, Equation 05.1.6) reduces to
j
(
-Kk. w(7)Re -d7) 7Tt L 7 - t
+ -1. j _ w(7)d 27Tt L
(7-=----=-t ) 7 -
=
fk(t),
k
=
1,2.
(
(15.1.8) As we see, this equation contains Hilbert's kernel. Let D be a singly connected domain whose contour L is a Lyapunov curve, i.e., is given by the same parametric equation, x = x(O), y = y(O), as in 00.4.1). Then we have
°
d7) x'(x -x) +y'(y -y) 0Re ( - - = O ~ dO=cot---OK1(0,Oo)dO, 2 7-( (x-x o) +(Y-Yo) 2 7=X+iy,
t = X o + iyo' 0,0 0
E
[0, 27T].
° °
(15.1.9)
Note that the function K\(O, 00) is periodic in both and 0 with the period 27T, K\(Oo, 0 ) = 0.5, and belongs to the class H on [0, 27T) (see Section 10.4).
°
Method of Discrete Vortices
364
Let us analyze the properties of Equation (15.1.8) for the case when L is a circle and the equation has the form
k
=
1,2.
(15.1.10)
If an internal problem formulated in stresses is self-balanced, then both the force and the moment vector due to external loadings must vanish. Therefore, Equation 05.1.10) is to be supplemented by conditions imposed on the right-hand side. The condition that the vector of external forces equal to zero is fulfilIed automaticalIy if f2( 0) is a unique function. However, to be unique, the function 12 must be periodic on L; then 12(0)
=
f2(27T).
(15.1.11)
From Equation 05.1.10) it folIows that the latter condition does not impose any restrictions on function w. According to Parton and Perlin (1982), the condition that the vector of the moment due to external forces equal to zero,
Re !12(t) dt
=
0,
(15.1.12)
I.
results, in the case of a circle, in the relationship
(for Kk = K 2 = - 1). Here the left-hand side is a fulIy imaginary expression for any w, and that is why Equation 05.1.12) does not impose any restrictions on w. Continuing the analysis of the problem on a circle, we conclude that the homogeneous equation 05.1.10) has (for Ik == 0) eigenf.'nctions whose form is determined by presenting w( 0) in the form of, for instance, the Fourier series. For k = 1,2 the eigenfunctions of Equations 05.1.8) and 05.1.10) are given by the complex constant w=a+ib
and for k
=
(15.1.13)
2 are given by the function (15.1.14)
365
Singular Integral Equations of the Theory of Elasticity
[for the homogeneous equation 05.1.10), whereas for Equation 05.1.8) they are given by the function w( T) = iazT, where a, and a z are arbitrary constants]. Thus, Equations 05.1.8) and 05.1.10) must be solved simultaneously, subject to certain auxiliary conditions "cutting off' the eigenfunctions 05.1.13) and (I5.1.14). The conditions may be presented in the form
f
L
W(T) dT
=
0,
k
=
1,2,
T
W(T) W(T)] { [ ~dT+~dT =0,
k = 2.
( 15.1.15)
D. I. Sherman was the first to introduce the left-hand sides of the latter equations into integral equation 05.1.7) to ensure uniqueness of its solution (Parton and Perlin 1982). For a circle Conditions 05.1.15) acquire the form 27T
1o w(O) dO 1m
t
o
7T
0,
k
=
1,2,
w( 0) e iO dO = 0,
k
=
2,
=
(15.1.16)
and hence W in the form 05.1.13) does not satisfy the first of the latter two equations, whereas in the form 05.1.14) it does not satisfy the second one. From Equation 05.1.10) it follows that
(15.1.17) and hence the integrals of both the real and the imaginary part of function fk are equal to zero. If the problem is solved in stresses, then Condition 05.1.17) determines the complex constant C z in 05.1.4). However, if the problem is solved in strains, then the condition is fulfilled automatically for a function f1 that is analytic within the domain D and continuous on L. For a circle, Condition 05.1.12) results in the relationship {27T
10 (fR sin o
(J -
ft cos 0) dO
=
0,
fz =fR
+ ift·
(15.1.18)
Method ofDiscrete Vortices
366
For an arbitrary smooth closed contour L, integration of 05.1.10) with respect to 0 results in the relationship (15.1.19)
analogous to Condition 05.1.17) on a circle. Thus, the second major problem for a region bounded by a closed Lyapunov contour L is reducible to integral equation (15.1.8) and Conditions (15.1.11), 05.1.12), (15.1.15), and (15.1.19), allowing us to determine the function w. In the case when L is a circle, the conditions must be replaced by (15.1.10), (15.1.11), and 05.1.16)-(15.1.18). These equations are singular and have the Hilbert kernel. Let us demonstrate the application of the numerical method developed in Section 6.2 to solving the two-dimensional problem of the theory of elasticity by considering Equation 05.1.10) subject to Conditions 05.1.11) and (15.1.16)-(15.1.18) for k = 2, i.e., K k = -1. We start by singling out the real and the imaginary parts of Equation 05.1.10) and Condition (15.1.16). Then we arrive at the system of equations
27T [0 - 00 ] 1 27T 1o w R(0)cot-2--sin(0+Oo) dO+ 0 wj(O)cos(O+Oo)dO
27T
1o wR(O)cos(O+ Oo)dO+
127T [0 - 00 0 wj(O) cot-2
+ sin(O+ 0 0 ) ] dO
(27T
Jo wd O) dO = 0, o
(15.1.20) where the subscripts R and I denote the real and the imaginary parts of the corresponding function, respectively. Suppose functions fiO) and fj(O) belong to the class H on [0, 27T). Choose points 0; and 00 ;, i = 1, ... , n, as was done in Theorem 6.2.1. Replace System 05.1.10) composed of integral equations by the following
Singular Integral Equations of the Theory of Elasticity
367
system of linear algebraic equations:
j
+ f32 +
f3 3 sin
1, . .. ,n,
=
00}
j = 1, ... , n,
2w
n
L
[Wn,R(Oi)sin 0i - wn,/(O})cos 0d -;;
=
O.
(15.1.21 )
i= 1
-Here f31' f3 2, and f33 are regularizing factors that make the system well-conditioned. Note that for n ---) 00 all three factors tend to zero, when and only when all Conditions 05.1.17) and 05.1.18) are met. In fact, by summing up first the first n equations of System 05.1.21) and then the second group of n equations, one gets (15.1.22)
where (Xi ---) 0 for n ---) 00, i = 1,2,3,4. Then, by multiplying the first n equations by cos 00}' j = 1, , n, respectively, and the second group of n equations by sin 00}' j = 1, , n, and summing all the 2n equations, one gets (taking into account Condition 05.1.18) (15.1.23 )
where
(Xi ---)
0 for n ---)
00,
i
=
5,6,7,8.
368
Method ofDiscrete Vortices
The statement made previously for f3], f32' and f33 follows from Systems 05.1.22) and (15.1.23). Note that the regularizing factors f3], f32' and f33 may be introduced in an alternative manner; however, in any case, they must tend to zero for n -+ oc, and the system must remain well-conditioned. For instance, one may take for the first n equations f3], 00j f32' and OJj f33' and for the second group of n equations f3], (27T + 00j)f32' and (27T + 00j)2f33. If a circle is loaded by uniformly distributed concentrated forces, then the functions fR( 0) and f/( 0) suffer discontinuities of the first kind at the points where the forces are applied. In this case 0; and 00j are chosen in such a way that the discontinuity points coincide with the reference points 00j' and fR( 0) and f/( 0) are represented by the arithmetic mean of one-sided limits. If a domain under consideration has symmetry axes, then the system of integral equations 05.1.20) may be transformed into a system of singular integral equations of the first kind on a segment, and either all or some of the conditions imposed on w will be satisfied. If the resulting system of equations is solved numerically, then one has to use the results obtained in Chapter 5, with the points of the sets E and Eo chosen for each of the equations with respect to the function in which an equation is singular. Thus, generally, the sets of points 0; and 00j chosen for w R and WI differ. Figures 15.1 and 15.2 present calculated results for System 05.1.21) for a circle subjected to different loadings. In all cases the calculated data are stable and thoroughly convergent for the order of the system varying from 30 to 110. This conclusion is also substantiated by comparing with exact solutions. The problem is similarly solved for any singly-multiply connected domain whose contour is composed of a finite number of closed Lyapunov curves. To solve the problem one has only to specify the contour in parametric form.
15.2. CONTACT PROBLEM OF INDENTATION OF A UNIFORMLY MOVING PUNCH INTO AN ELASTIC HALF-PLANE WITH HEAT GENERATION Following Saakyan (Lifanov and Saakyan 1982, Saakyan 1978) we will show how the problems formulated in the headings of Sections 15.2 and 15.3 may be reduced to singular integral equations. Let a rigid punch whose base has an arbitrary smooth configuration move over the boundary of an elastic half-plane with velocity Vo less than or equal 'to the Rayleigh wave speed. At the same time, the punch is being indented into the half-plane by force P (see Figure 15.3). It is supposed that there is dry friction between the punch and the half-plane; in other words, the tangential stresses within the contact zone
Singular Integral Equations of the Theory of Elasticity
FIGURE 15.1. Approximate values of functions WR and with two concentrated balanced compressing forces.
FIGURE 15.2. Approximate values of functions trated balanced extending forces.
WR
and
wI
wI
369
for the case of a circle loaded
for the case of three concen-
are proportional to the normal pressure. Hence, heat is generated within the contact zone whose quantity is proportional to the speed of relative motion, the friction coefficient, and the normal contact pressure (Aleksandrov et al. 1969). The portions of the punch and the half-plane that are not in contact are supposed to be thermally insulated. Let us choose a stationary system of coordinates 0 1 XI YI and a moving system of coordinates OXY attached to the punch. Obviously, the two systems are related by the formulas X = XI - Vot and y = y" and in the latter system of coordinates the quantities being sought arc independent of
370
FIGURE 15.3. force P.
Method of Discrete Vortices
Motion of a rigid punch over an elastic strip in the presence of indenting
time; in other words, a steady-state situation will be analyzed. Heat generation within the contact zone results in the appearance of heat fluxes Q,(x) and Qix) directed into the half-plane and the punch, respectively, and related to the contact pressure p(x) by the relationship (15.2.1)
where f3 is the friction coefficient. According to the Fourier heat conduction law, one has
k
=
1,2,
where A1 and A2 are the heat conductivity coefficients, and T1(x, y) and T 2 (x, y) are the temperatures of points belonging to the half-plane and the punch, respectively. We will suppose that deformations of the elastic half-plane do not affect the temperature field. Then the problem splits into the following two problems: the problem of calculating the temperature field and the twodimensional problem of elastodynamics in the presence of a temperature field. Let us start by considering the former problem. First we construct the influence function for the problem; in other words, we construct a solution to the heat-conduction problem for an elastic half-plane, with a point source of unit heat flux moving along a thermally insulated boundary at a constant speed Vo. Then we arrive at the heat equation
(15.2.2)
Singular Integral Equations of the Theory of Elasticity
371
subject to the boundary conditions
(15.2.3 )
nx
for YI = 0, and l , YI' t) < oc for YI -4 -oc. Here T(x l , YI' t) is the temperature of points belonging to the half-plane, and p, c" and Al are the density, specific heat capacity, and heat conductivity of the material of the half-plane, respectively; 8(x) is the Dirac function, and ~ is the coordinate of a heat source at the instant t = 0. In the movable system of coordinates Equations 05.2.2) and 05.2.3) become
(15.2.4) rJT(x, y)
I y~
rJy
1
--8(x-O 0
( 15.2.5)
AI
nx,
and Y) < oc for Y -4 -oc. The following complex Fourier transform with respect to the variable x may be applied to the latter boundary problem: d 2 T(y, a) - - : ; - 2-
dy
• -
(0'2
_
+ iO'K)T(y, a)
dT(y, a)
for y
dy
T(y, a) < oc
for y-4
0,
(15.2.6)
0,
=
-00
=
,
(15.2.7)
where a is a complex parameter of the transform, K = pC, vol AI' and T(y, a) is the Fourier temperature transformant given by the formula
By solving Equation 05.2.6) subject to the boundary conditions 05.2.7) one gets eia~
T(y, a)
=
ey~(a)
-
AI 71( a)
,
(15.2.8)
372
Method of Discrete Vortices
where 1]( a) = Ja 2 + i a K. The function 1]( a) branches at points a = 0 and a = - i K in the plane of the complex variable a = (J + iT. In order to isolate a single-valued branch of the function, the plane a must be cut along the lines connecting the branching points with the infinitely distant point and lying in the upper and the lower half-plane, respectively (see Figure 15.4). This cut allows us to choose a single-valued branch of the root, meeting the condition 11( a) ~ Ia I for a ~ ± 00 along the real axis. Let us next construct the influence function for the two-dimensional problem of elastodynamics in the presence of a temperature field generated by a moving concentrated source of strength S. The motion is accompanied by those of concentrated normal P and tangential Q forces. Then we arrive at the Lame equations written for the stationary coordinate system 0 1 XI Y I in the presence of a temperature field:
( 15.2.9)
where u(xl' YI' t) and
V(X I , YI' t) are the displacement components, O(xJ, Yr. t) = nul aX I + 8vI8YI is the volumetric expansion, p is density, ,\ and /L are the Lame coefficients, Y = (3,\ + 2/L)a" at is the linear expansion coefficient, and T(x l , y\, t) is the temperature determined
beforehand. By using the Duhamel-Neumann relationships rJV
u. y
=
,\0
+ 2/L- - yT
8YI'
the boundary conditions may be presented in the form
for Y
=
0; (15.2.10)
Singular Integral Equations of the Theory of Elasticity
373
FIGURE 15.4. The complex variable plane with cuts (see the dashed lines) singling out the unique branch of the function l1(a) = Ja 2 + iak.
Upon passing again to the moving system of coordinates, we get for the latter two equations,
(15.2.11)
8v
AO+21-t--yT=-P8(x-O
ay
au + -8V) ay ax
I-t ( -
=
Q8(x -
fory=O,
O·
(15.2.12)
By differentiating the first of Equations (15.2.11) with respect to x, the second with respect to y, and summing the results, one gets the following equation in O(x, y):
(15.2.13) Let us subject Equations (15.2.11)-(15.2.13) to the generalized Fourier transform given by the formula
Method of Discrete Vortices
374
Then we get a system of ordinary differential equations in the variable y with respect to the functions u(y, u), v(y, u), and O(y, u), subject to boundary conditions obtained from 05.2.12). Let us find some special solutions to the system, meeting the latter boundary conditions (taking into account 05.2.10». To do this we first observe that if at the real axis the function 71( u) assumes the values chosen for the single-valued branch of 71( Q') beforehand, then the generalized Fourier temperature transform coincides with the complex Fourier transform. However, in order to solve a contact problem one has to consider displacements of the half-plane's boundary points only, i.e., to consider u(O, u) and v(O, u) only. By subjecting u(O, u) and v(O, u) to the inverse Fourier transform, one obtains the influence functions u(x,O) and v(x, 0) for the two-dimensional problem of elastodynamics formulated for a half-plane in the case of uniformly moving concentrated forces and a heat source. Next we use the values of the integrals 1
-f 27T
x
ieiCT~
-e -x u
1 -f 27T
x
-x
iCTX
eiCT~ _e-
iCTX
lui
1 dcr = "2sign(x -
1 du
=
-In
7T
0,
1
Ix - {;=I
+ C,
known from the theory of generalized functions (Brychkov and Prudnikov 1977) where C is a certain constant. In the theory of generalized functions, the latter is equal to the Euler constant; in the two-dimensional problem of elastodynamics for a half-plane it is equal to infinity, because the system of forces applied to the half-plane is unbalanced. Then the influence function finally becomes
u(x,O)
=
P Q -7TvO v]-sign(x - 0 + vI-In 2/L
/L
1
Ix -
/:1
~
( 15.2.14)
Singular Integral Equations of the Theory of Elasticity
375
where
(1 + ki) - 2k,k 2 k l (1 - ki) 1 +q
v2
=
-2-k,-(-I-----=k---=i-) ,
i = 1,2,
C2
=VIL/P.
Next we consider the contact problem. Evidently, the effect of a punch on a half-plane is equivalent to applying unknown normal p( ~) and tangential q( ~) contact stresses and a heat source Q,( ~) to the boundary of the half-plane, all of them distributed over the segment [-a, a]. By the principle of superposition, the displacements of boundary points of the half-space may be obtained by integrating Equations 05.2.14) with respect to ~ and replacing P, Q, and S by p( 0, q( 0, and QI( 0, respectively. Then one gets
U(X,O)
=
1 21L
-7TV O V , -
X
fb-a R, ( ~ -
fa
p(Osign(x -
Od~
-a
x) Q I(
0
dt
+ c 3'
376
Method of Discrete Vortices
7TVO v]
v(x,O)=--2/L
fa
1
fa
q(Osign(x-Od~-v]In -a /L -a
1
lX
-
~
IP~d~
( 15.2.15) where c 3 and C 4 are certain infinite constants. The contact conditions are given by equalities
v(x,O)
=
Ixl < a,
f(x) - d,
(15.2.16)
where f(x) describes the base of a punch, d is the punch immersion, and T] and T 2 are the temperatures of the half-plane and the punch, respectively. Suppose the punch is much longer than the contact zone, and hence it may be replaced by a half-plane as far as its temperature is concerned. Then the temperature of the punch's boundary points is given by (15.2.17) where C s is also an infinite constant. For the temperature of the boundary points one gets from (15.2.8),
x
t
+ c6 ,
R(s - x)Q](s) ds
-a
R(s - X) = _1 27T
t' [_ Va -co
__ 1 ]eia(S-Xl da:. (15.2.18) lal
1
+ iaK
2
If one considers a harmonic (in time) temperature distribution within the punch and the half-plane and singles out the wave pro~agating into infinity, then by equating the temperatures of the punch and the half-plane within the contact zone (see Equations 15.2.17) and (15.2.18), respectively) in the limit of the wave's frequency tending to zero, one arrives at the heat contact equation 1 '7TA 2
fa -a
Inlx - slQis) ds
= -
1
?TAl
fa
Inlx - sIQ](s) ds
-a
1 - -f A]
a
-a
R(s - X)Ql(S) ds (15.2.19)
Singular Integral Equations of the Theory of Elasticity
subject to the condition that the infinite constants eliminating: 1
fa
Q,(s) ds = Al - a
1
Cs
aQz(s) ds.
f
"z
377
and c 6 are self-
( 15.2.20)
-a
EXcluding the quantity Qz(s) appearing in Equation (15.2.19), using relationship Equation (15.2.1), and differentating the equation with respect to x, one gets
r
A] + Az 7TA] Az
Q](s) ds _ f3 Vo
-a
1
+-
S -
fa
A1
X
7TA z
aR(s - x)
ax
-0
r -a
p(s) ds S -
X
Q](s)ds=O.
(15.2.21)
Then by substituting the expression for the displacement v(x,O) for q(x) = f3p(x) into the first contact condition (15.2.16) and differentiating
with respect to x, we obtain
y
a
iJRz(s -x)
A] /L
-a
ax
- v] v z - f
Q](s) ds = -f'(x). (15.2.22)
The latter equation should be supplemented with the equation of the punch equilibrium:
r
p(x) dx
= P.
(15.2.23)
-a
Thus, we have derived a system of singular integral equations of the first and second kind, Equations 05.2.21) and 05.2.22), which, subject to Conditions 05.2.20) and 05.2.23), has a unique solution. Let us introduce the nondimensional quantities entering the latter system of equations in the following manner: ~=
x
-, a
l/J(
n
a =
pp(x), i = 1,2,
Method of Discrete Vortices
378 /-to
cp( 0 = -pf'(x),
Then by introducing for the sake of convenience a new function x( ~) = (1 + A)Qj( 0 - ~ P< 0, we arrive finally at the following system of singular integral equations:
f
x(s) ds +
l - I
_7T_
~
S -
1
+
A -
x [X(s) +
l l/J(s) ds f -I-S-'---~- +
7Tf3 v ol/J(S)
f
dR(s -
l
d~
I
~l/J(s)]
v
0
lb' = 0,
2 fl
- -1-+-A
-I
dR 2(s d~ 1
X
[X(s) + N(s)] ds = - -cp(
0
0, (15.2.24)
VI
subject to the conditions
t
t
X(s)ds=O,
l/J(s)ds= 1.
( 15.2.25)
-I
-I
The first equation in (15.2.24) is a singular integral equation of the first kind with respect to the function X( s), and the second equation is a singular integral equation of the second kind with respect to the function l/J(s). Because in both functions the index 1 solution is sought, we deduce (by using the results of Sections 5.2 and 7.1) that Equations (15.2.24) and Conditions 05.2.25) must be replaced by the following system of linear algebraic equations: 7T ; ,
- 'n
[1
k= I
+
7T
~i
'T k -
L
7T~
1
+
+ -1+ A m
A
p~
I
°
p
aR( Tk
-
0 ]
d~i
aR(tp
-
at:
~,
~i)
t/J.*p --
xt
0,
i=I, ... ,n-l,
Singular Integral Equations of the Theory of Elasticity
379
j=I, ... ,m-l, n
m
7T
L xtn
k=1
(15.2.26)
=0,
where 'T k and ~i are the same as in System (5.2.3) and t p ' Sj' and a p are the same as in System (7.1.9). The number a is given by the formula 1
a
=
-
-arctan( {3vo )' 7T
Thus, the contact problem of motion of a punch has been reduced to that of solving a system of linear algebraic equations. System 05.2.26) was solved numerically for a variety of speeds of motion of a punch and the following values of the parameters: A = 35, {3
=
cilc~
=
0.275,
0.27,
The contour of the base of the punch was described by the function f(x) = 0.lx 2 • Figure 15.5 presents distributions of heat fluxes directed into the half-plane and into the punch (see the solid and dashed curves, respectively). Contact pressures calculated at the points corresponding to the roots of the polynomial Pio' -]- a)(x) were compared with the values of the function a -sin 7T0' -P(x) = w(x) [50] - 20'(1 P 57T0]
+ a) - (1 + 20')x
-x 2 ],
which gives an accurate analytic solution for the contact pressure for a similar problem without heat generation, obtained by the method of orthogonal polynomials. The comparison presented in Figure 15.5 shows that the heat generation does not practically affect the contact stresses distribution (for VO/C2 = 0.2 the difference is less than 0.03%). Figures 15.6 and 15.7 show the contact pressure distributions and their regular portions for different speeds of motion of the punch. We see that an increase in the speed of motion results in a decrease in pressure in the
380
Method of Discrete Vortices
,, \ \
!\ \
...
""
'-
1-----1'+------1
FIGURE 15.5. Distributions of heat fluxes directed into the half-plane (solid line) and into the punch (dashed line) for V[1/C 2 = 0.25.
0.75
~
,-
1 2 3
J
0.5
/)
~
FIGURE 15.6. Distributions of contact pressures and their regular portions for various speeds of the punch. 1 indicates VO/c 2 = 0.44 X 10- 5 , 2 indicates VO/c 2 = 0.5, and 3 indicates VO/c 2 = 0.8.
central part of the contact zone. At the same time, the stress <X'ocentration coefficients at the ends of the contact zone decrease.
15.3. ON THE INDENTATION OF A PAIR OF UNIFORMLY MOVING PUNCHES INTO AN ELASTIC STRIP Let us consider a pair of punches moving along the edges of an elastic strip whose thickness is equal to 2d (see Figure 15.8). The forces P and Q are applied in such a way that the summary moment acting on a punch is equal to zero.
Singular Integral Equations of the Theory of Elasticity
o
-f
381
0,5
FIGURE 15.7. Distributions of contact pressures and their regular portions for various speeds of the punch. 1 indicates II;I/C2 = 0.44 X 10-5, 2 indicates 1I;)/c 2 = 0.5, and 3 indicates VU/c 2 = 0.8.
O:T/,
:&
-0
I
I
0 'b
. _ _ . .l...-. _ _ ~ I I
I '77li'-~-""'.J
FIGURE 15.8. Motion of rigid punches over the boundary of an clastic strip in the presence of indenting forces P.
It is supposed that the tangential stress q(x) acting at the contact surface between a punch and the strip is subject to the Coulomb law q(x) = {3p(x), where {3 is the friction coefficient. Let us derive the governing equations. We will start by choosing a stationary system of coordinates 0 1 XI YI and a moving system of coordinates OXY attached to a punch and determined by the formulas x = x I Vc)t and Y = YI' As far as the strip is symmetric with respect to the line Y = -d, we will consider its upper half only. Let us use the results presented by Saakyan (1978), who considered the problem of concentrated forces and heat sources moving along the edges of an elastic strip at speed Vo' By using the principle of superposition in the absence of heat sources, one may immediately derive expressions for the vertical component of the boundary point displacements (in the
Method of Discrete Vortices
382
presence of distributed normal and tangential stresses): 1
V(X,O)
1 K](x - s)q(s) ds - -a /L
f /L
= -
a
f
a
Kz(x - s)p(s) ds, (15.3.1)
-a
where
e-i(TU
- (1
k; =
+ k~)sinh( (Tkzd)cosh( (Tk]d)] - - d(T, (TLl
n 1-
z
-%-' c;
i = 1,2,
cf
A+2/L
= --p
C
z = /L z p
A and /L are the Lame coefficients and p is the density of the strip material. Within the contact zone we have the usual contact condition (Shtaerman 1949)
v(x,O)
=
f(x) - 0,
-a < x < a,
(15.3.2)
where f(x) is a function describing the punch base and 0 is the measure of identation. AJthough f(x) == () for the punches shown in Figure 15.8, we preserve the function f(x) in what follows. The functions K](u) and Kiu) may be presented in the form
Singular Integral Equations of the Theory of Elasticity
383
where
are regular functions in the neighborhood of zero and Co is an infinite constant. It can be easily shown that all derivatives of the functions arc also regular in the neighborhood of zero. By substituting 05.3.3) into (15.3.1) and (15.3.1) into the contact condition 05.3.2), and taking into account that q(x) = f3p(x), one gets
f
a [
-a
1
In-I-_-I S
X
=
+
1]
-°
7Tf3 0 sign(x - s) - -R(x - s) p(s) ds 2 0,
/L
-(8 - f(x) - coP), OJ
where
P
=
r
(15.3.4)
p(x) dx.
-a
After differentiating the latter equation with respect to
f
a -a
[1
-- - S -
X
1aR(x - s) ]
0,
ax
p(s) ds
+
7Tf300p(X)
X,
one obtains
=
-
/L
-f'(x). OJ
(15.3.5)
384
Method of Discrete Vortices
Upon passing to nondimensional quantities x =
a~,
p( 0 =
d* = d/a,
ap(x)/P, and g = P/(/La), Equation (15.3.5) and Condition (15.3.4)
become
f
l
f N( ~ 0] -]
p(s) - - ds
1
+ 7Tf300p( 0 + -
_]s-~
f'(
I
s)p(s) ds
=
0
---
gO]
(15.3.6)
t
p(s) ds
=
1.
(15.3.7)
-]
where
x {' [ 2k]k 2 tanh( uk]d*) - (1 + k~~tanh( uk 2 d*) -oc;
_
4k]k 2 tanh( uk]d*) - (1
+ k~) tanh( uk 2 d*)
+ k 22 ) ] -i(]"(~-s)d 2 e u. (1 + k~)
2k 12 k - (1 4k]k 2
-
From Gakhov (1977) and Muskhelishvili (1952) it is known that a solution to singular integral equation (15.3.6) may incorporate a singularity at the ends of the interval ( -1,1) (see (7.1.2»: p(O
=
w(Ocp(O
(1 -
=
O"p + ~r]-"cp(O,
(15.3.8)
where rp( ~) is a limited function and the number a is given by the relationships f300 + cot 7T0'
=
0,
0>0'>-1.
(15.3.9)
385
Singular Integral Equations of the Theory of Elasticity
FIGURE 15.9. Contact pressure distribution for J!; l /c 2 = O.S: I indicates dla = 1.5, 2 indicates dla = 3, and 3 indicates dla 24.
Singular integral equation (15.3.6) may be solved numerically as shown in Section 7.1; in other words, we pass to a system of linear algebraic equations with respect to the values of the function 'P( ~) at the corresponding grid points:
f' (Sj)
--gOI
'
j=],oo.,n-1,
n
L ai'Pj =
j
1,
=
n.
(15.3.10)
i= I
Here ti' i = 1, ... , n, are the roots of the Jacobi polynomial - I a\( 0 and Sj' j = 1, n - 1, are the roots of the polynomial p~ f,l+a)(o, 'Pi = 'P(t). In this case, a i may be written as
p~a,
00"
p~_f,l+a)(ti)
1 ai
= nsin7TCx p(l+a,-a\(t)' n I
i=l,oo.,n,
I
because according to Bateman and Erdelyi (1953) and Erdogan, Gupta, and Cook (1973), d dx [p~a,(3)(x)]
=
1 "2(n
+
a
+ f3 +
1)P~~;
1,f3t
I)(X). (15.3.11)
Numerical calculations were performed for the following values of the constants: the number of reference points n = 10, the Poisson coefficient for the strip material JJ = 0.31, the friction coefficient f3 = 0.27; the values of Volc2 (where c 2 is the Rayleigh wave speed) and the ratio of the strip width to the length of the contact zone, diu, were varied. Figures 15.9 and
Method of Discrete Vortices
386
"
~
---
7
f
~ ~
43
,,
o
-f
FIGURE 15.10. Values of the function "'( ~), for V; j lc2 indicates dla = 3, and 3 indicates dla 24.
=
0.5: I indicates dla
=
1.5, 2
p Hft--=---t----+47lJ---4--~J-l
-f
-45
o
FIGURE IS.H. Contact pressure distribution for dla indicates Volc2 = 0.2, and 3 indicates Volc z = 0.9.
f ,
=
100: I indicates V;jlc z = 0.01, 2
15.10 show the contact pressure and the function cp( ~) versus ~. Here cp( + 1) and cp( - 1) are the stress concentration coefficients whose values increase with an increase in the width of the strip. Additionally, we note that for diu = 24 the plot of function cp( 0 virtually coincides with the straight line corresponding to the solution of an analogous problem for an elastic half-plane. Hence, for an elastic strip whose width exceeds the length of the contact zone by a factor of 25 or more, the interaction of the punches is negligibly small. It should be noted that exceeding the limiting value of diu causes the influence of the second punch to become negligibly small. Also note that diu depends, though only slightly, on the speed of motion of a punch: the higher the speed, the larger the critical value of diu. Figure 15.11 shows the contact pressure versus ~.
16 Numerical Method of Discrete Singularities* in Boundary Value Problems of Mathematical Physics
In this chapter, folIowing Gandel 0982, 1983), we show how a wide class of boundary problems of mathematical physics may be reduced to singular integral equations and solved by employing the numerical method of discrete singularities developed previously.
16.1. DUAL EQUATIONS FOR SOLVING MIXED BOUNDARY VALUE PROBLEMS Let us consider dual equations of the form
Ao+
L
(Ancosny+Bnsinny) =0,
Y
E
CE,
n=l
(16.1.1)
bAo
+
L
(1 - €n)(nA n cos ny + nBn sin ny)
=
f(y),
Y
E
E,
n=l
* The method of discrete singularities in the theory of elasticity and electrodynamics is a numerical method similar in its physical and mathematical structure to the method of discrete vortices in aerodynamics.
387
Method of Discrete Vortices
388 m
E
U (O'k' 13d,
=
CE
= [ - 7T, 7T] \
E,
k=1 -
7T
<
<
0'1
131
< ... <
«
am
13m <
( 16.1.2)
7T.
Here the smooth function f(Y), Y E if = u 7:.1 [ai' 13k l, the constant b, and sequence En, n E N == {l, 2, ... }, are given, and En tends to zero for n ---) ao no slower than O(n -2). The coefficients An, An' and En' n EN, must be found. The possibility of using the method of discrete singularities for numerical solution of the preceding equations was considered in Gandel (1982, 1983), where some mixed boundary problems for the Laplace and Helmholtz equations were reduced to dual equations under analysis. Let us start by considering some of the problems and writing the corresponding dual equations. 1.
The simplest example is the following mixed boundary-value problem for the Laplace equation inside a circle: r < R,
(16.1.3)
= 0,
cp
E
CE,
(16.1.4)
= f( cp),
cp
E
E,
(16.1.5)
Llu = 0, Ulr~R
aaur I
r" R
where f( cp), cp E E, is a given smooth function. The solution u = u(r, cp) will be sought in the class of functions that are twice continuously differentiable inside the circle and remain continuous up to its boundary. The harmonic function is presented in the form u(r, cp) = A o
+
t
n
~
I
(~ r(A n cos ncp + E" sin ncp),
and the unknown coefficients A o, An' and En' n E N, will be found from the dual equation of the type (16.1.1) and (16.1.2): Au
+
L
(An cos ncp + En sin ncp) = 0,
cp
E
CE,
cp
E
E, b
n=1
L
n(A n cos ncp
+ En sin ncp)
=
Rf(cp),
n~l
derived from boundary conditions (16.1.4) and 06.1.5).
=
0,
En
== 0
umerical Method of Discrete Singularities
389
Let us consider the problem of a stationary continuum limited by two infinitely long coaxial cylinders at whose external surface and a portion of internal surface composed of a finite number of longitudinal strips temperature is maintained constant, and at the remaining portion of the internal surface heat flux is specified. Let the axis of the cylinder coincide with the OZ axis of the Cartesian system of coordinates, and let the cross section by the plane OXY be a ring whose internal and external radii are R] and R 2 , respectively. In the cross section we introduce polar coordinates r and cpo Next the internal cylindrical surface is divided into two sets of longitudinal strips {r = R cp E CE, Z E R} and {r = R], cp E E, Z E R}, where R is the set "of real numbers. For the sake of simplicity we will consider the case when the temperature is independent of z. Let the temperature field between the cylinders be described by u = u(r, cp), R] < r < R 2 • Thus, we arrive at the boundary-value problem flu
0,
(16.1.6)
Ulr~R2 = 0,
(16.1.7)
=
cp
E
CE,
( 16.1.8)
cp
E
E.
(16.1.9)
Here f is a smooth function, and u = u(r, cp) is sought in the class of functions that are twice continuously differentiable inside the ring and remain continuous up to the boundary. A function that is harmonic inside the ring and meets Condition (16.1.7) has the form
x (An cos n cp + B n sin n cp), and the unknown coefficients A o, An' and B n, n EN, may be found from the dual equation of the type 06.1.1) and (16.1.2): A
o -_::...-+ In R 21R I
oc
1+(RIR)2n
=]
1 - (R] 1 R 2)
L n
I
2
2n (nAn
COS
ncp + nBn sin ncp)
=
Rf( cp)
390
Method of Discrete Vortices
cp
L
Ao+
(Ancosncp + B i ] sin ncp)
E
E.
0,
=
n~l
cp
3.
E
CE,
derived from boundary conditions 06.1.8) and 06.1.9). Next we consider a three-dimensional mixed boundary-value problem for the Laplace equation, namely, the problem of stationary temperature distribution within a finite cylinder at whose side surface composed of a finite number of longitudinal strips and the two bases, temperature is maintained constant, whereas at the remaining side surface heat flux is specified. We will seek a function u = u(r, cpz) that is twice continuously differentiable inside the cylinder, remains continuous up to its boundary, and meets the conditions flu = 0, ul z () = ulz=1f = 0, Ulr=R = 0,
aaur I
ro
1
R
= Rf(cp,z),
r
< R, 0 < z <
r
< R,
°< °<
/I,
(16.1.10) (16.1.11 )
z < H, cp
E
CE,
(16.1.12)
z < H, cp
E
E.
(16.1.13)
A solution to the Laplace equation 06.1.10) meeting Condition (16.1.11) will be sought in the form
u(r, cp, z)
=
L k=l
7Tkz
uk(r, cp)sin--, /I
r
< R, 0 < z < H,
wht;re uk(r, cp) satisfies both the Helmholtz equation
r < R,
(l6.1.14)
'umerical Method of Discrete Singularities
391
and the boundary conditions corresponding to (16.1.12) and (16.1.13): cp
E
CE,
(16.1.15)
cp
E
E,
(16.1.16)
where the Fourier series coefficients
kEN.
The function Uk
uk(r, cp)
=
=
uk(r, cp) is sought in the form
u~lJ(r)
+
L
(u~I~(r)cos ncp
+
u~2~(r)sin ncp).
/l~l
By (16.1.14) the unknown functions u~Z(r), k, n E N, i = 1,2, and n = 0, i = 1, are solutions to the modified Bessel equation
w"
+
1
1Tk
(( Ii
-Wi -
r
2
)2 + 72 n
)
w = 0,
I w( +0)1 < oc.
Hence, u~Z(r) coincide, to an accuracy of a constant factor, with modified Bessel functions x
In(x)
=
(x/2)2s-n
L '( s=o s. n
+ s )".
where x
=
1Tk -r
H'
Thus, the sought after functions uk(r, cp), kEN, may be presented in the form
u (r m) k
,..,..
=
I O(1Tkr/lI) A I o( 1TkR/H) kO
I (1Tkr/H) n~l In ( 1TkR/H) x
+ " -n - - -
X(A kn cos ncp + B kn sin ncp),
392
Method of Discrete Vortices
and by boundary conditions 06.1.15) and (16.1.16), the unknown coefficients A kO , A kn , and B kn , n EN, must be found from the dual equation A kO
+
L
(A kn cos ncp
+ Bkn sin ncp) = 0,
cp
E
CE,
n~l
(7TkR/H)I~( 7TkR/H) - - - - - - - - A kO + I o(7TkR/H)
(nA kn cos ncp
'X
(7TkR/H)I~(7TkRIH)
n=l
n1n(7TkRIH)
L --------
+ nBkn sin ncp)
=
fk( cp),
cp
E
E.
Let us show that this is a dual equation of the type (16.1.0 and (16.1.2). In fact, from the recurrent relationship for modified Bessel functions (Bateman and Erdelyi 1953),
one gets
(7TkR/ H) I~( 7TkR/ H) -------- =
nIA 7TkR/H)
where
E
n
1-
En'
(7TkR/H)I n+ l( 7TkR/H) = ---------n~(7TkR/H)
and from the asymptotic equality In(x) - (x/2Y In! for a fixed 00, it follows that En = O(n 2). A number of mixed boundary-value problems for a plane layer may be reduced to the dual equation 06.1.0 and (16.1.2) also, if the sought after solution is a periodic function of one of the Cartesian coordinates. The simplest example of the problem is the two-dimensional problem of a stationary temperature distribution within a uniform layer between a pair of parallel planes when a given temperature is maintained at one of the planes as well as at a periodically repeated system of strips of the other boundary plane, and heat flux is specified at the remaining portion of the boundary. For determining the temperature field u = u(x, z), - 00 < x < + 00, o :;:'; i! :;:'; H, one has the following boundary-value problem:
x*-O and n ---)
4.
au
=
0,
u(x
+ 27T, z)
=
u(x, z),
ulz~1f=O,
.1 E
R, 0 < z < H, (16.1.17)
-7T:;:';X:;:';7T,
(16.1.18)
Numerical Method of Discrete Singularities
-
-rJu
rJz
I
z~ 0
=
[(x),
393 X E
CE,
(16.1.19)
X E
E.
(16.1.20)
The harmonic function (16.1.17), whose period is 27T in x and that meets Condition (16.1.18), has the form z ) u(x,z)= ( 1 - - A o + H
x
sinh n( H - z)
L.
SInh
n= 1
nH
(Ancosnx+Bnsinnx).
From boundary conditions (16.1.19) and (16.1.20) for z = 0, one gets the following dual equation of the type (16.1.0 and (16.1.2), which may be used for determining the unknown coefficients A o' A,,, and B n , n E N:
Ao + L
(Ancosnx
+ Bn sin nx) =0,
X
E
CE,
X
E
E,
n~l
1 HAo +
L
coth n/l(nA n cosnx + nBII sin nx) = f(x),
n~l
5.
where b = I/H, En = 1 - coth(nH) = O[exp( -2Hn)]. Finally, we will consider a more complicated boundary-value problem reducible to a system of two dual equations of the type under consideration. Specifically, we will consider the mixed boundary-value problem for a stationary equation describing a layer limited by a "double lattice." Let the layer be located between the planes z = ± H /2 of the Cartesian system of coordinates and let the "lattices" be composed of periodically rcpeated systems of strips whose edges are parallel to the axis OY and that are positioned at the upper and the lower boundary planes, respectively. For the sake of simplicity we will consider the plane problem for the Laplace equation. The mathematical formulation of the problem is as follows. We seek 27T-periodic function in x, u
=
u(x, z),
u(x
+ 27T, z) = u(x, z),
X E
R,
Izl
~ H/2,
that is continuous up to the boundary and satisfies the conditions Llu
=
0,
au (x'"2 H) a;
=
fl(i),
Izi < H/2,
(16.1.21 ) (16.1.22)
394
Method of Discrete Vortices
H)
u(X , 2
au (x, - 2 H) a; u(X, -H/2)
=
(16.1.23)
0'
= fz(x), =
0,
X E
Ez,
(16.1.24)
X E
CEz,
(16.1.25)
where E i = U ;:':,JO'ik' f3ik), -7T < ail < f3il < ... < aim < 7T, i = 1,2; /;(x), X E Ei , are given smooth functions. A solution to the problem is sought in the form
u( X, z)
=
a o + b oz +
L {( an cosh nz + bn sinh nz )cos nx n=1
+(cn cosh nz + d n sinh nz)sin nx},
Izl.:::;; H/2.
The first dual equation is derived subject to Conditions 06.1.22) and 06.1.23) for z = H/2: .
(16.1.26) bo +
L
[n(A n
+ Bn)cos nx + n(Cn + Dn)sin nx]
n=l
- L {( €~An
- n< Bn)cos nx
+
(€~nCn -
n=1
(16.1.27) the second one is obtained from Conditions 06.1.24) and 06.1.25) for z = -H/2:
X E
CEz,
(16.1.28)
Numerical Method of Discrete Singularities
bo -
L
[n(A n - Bn)cosnx
395
+ n(Cn - Dn)sin nx]
n=1
+
L
{(€~nAn + €;nBn)cosnx + (€~nCn + €;nDn)sinnx}
n~1
=f2(x),
X
( 16.1.29)
E £2'
where An = all cosh nH/2,
B n = bn sinh nH/2,
C n = C n cosh nH/2,
D n = d n sinh nH /2,
€~
€;
= 1 - tanh nH /2,
= coth nH /2 - 1.
From these examples it follows that in order to obtain solutions to the preceding boundary-value problems of mathematical physics in the form of Fourier series, one has to find their coefficients from the dual equations of the type 06.1.1) and 06.1.2). However, often there is no need to calculate all the coefficients, because only one or some of the first coefficients are of physical importance as, for instance, in the case of studying wave diffraction (see paragraphs 3 and 4). On the other hand, in the case of the heat conduction problems considered before, it suffices to know the function being sought (temperature) only at the portions of the boundary where it is not specified (but the heat flux is given). It is clearly desirable to find the values independently, without calculating the coefficients of the series. Thus, for applications a method of solving dual equations is needed that would be sufficiently versatile and would allow us to answer questions of physical importance without carrying out extremely complicated calculations. A method for solving dual equations of the type under consideration, based on reduction to the Riemann-Hilbert problem, was developed in connection with solving problems of electromagnetic wave diffraction on a lattice (Agranovich, Marchenko, and Shestopalov 1962; Shestopalov 1971). Another numerical method of solving dual equations of the form 06.1.1) and 06.1.2) as well as of their continual analog-the corresponding dual integral equations-based on the reduction to a singular integral equation on a system of segments (Gandel 1982, 1983) and its subsequent solution by the method of discrete singularities (Lifanov and Matveev 1983b) is discussed in the next section. The method is quite simple and, independently of the number of segments composing the major set £ (see (16.1.2), results in solving similar systems of linear algebraic equations. It should also be noted that all the quantities discussed in the preceding text may be readily expressed through the solution to a singular integral equation.
Method of Discrete Vortices
396
16.2. METHOD FOR SOLVING DUAL EQUATIONS Let us continue by reducing the dual equation 06.1.1) and 06.1.2) to a singular integral equation on a system of intervals. Denote U(y)
L
== Au +
+ B n sin ny),
(An cos ny
yE[-'lT,7T], (16.2.1)
n~l
dU(y)
F(y)
== - -
x
L
=
n~l
dy
(-nAn sin fly + nBn cos ny),
Y
E [ - 'IT, 7T ].
(16.2.2)
For the problems of mathematical physics under consideration U(y), Y E [ - 7T, 7T], is a continuous function, and
F(y)lyc
F. =
O(q
1/2),
(16.2.3)
where q is the distance from point y to the boundary dE of set E. By Equation 06,.1.1), F(y)
=
0,
F(y) dy
=
0,
eE,
(16.2.4)
k=1,2, ... ,m.
(16.2.5 )
Y
E
and
!
f3k
"'k
Let us add the relationship Au
+
L
(_l)n An
=
0,
(16.2.6)
n=l
obtained from Equation 06.1.1) for y = 7T. Then the latter equation will be replaced by Equations 06.2.4)-(16.2.6). Taking into account Equation (16.2.4), one gets from Equation 06.2.2) for n EN, An
=
-
1 -f F(y)sin nydy, 7Tn
(16.2.7)
F.
Bn =
1 -f F(y)cos nydy, 7Tn f.'
( 16.2.8)
Numerical Method of Discrete Singularities
397
Also taking into account Equation 06.2.7) and using the known expansion of function g(x) = x/2, x E ( - 7T, 7T), into the Fourier series, we get from 06.2.6), 1 Au
=
-
(16.2.9)
-2 fF(y)ydy.
7T E
Thus, all the unknown coefficients are expressed by function F(y), E, which is to be found. Note that function U(x) is also directly expressible by F(y):
y
E
U( x)
f7T
=
F( y) dy,
X E [-7T,7T],
-7T
and by 06.2.4) and (16.2.5), U(x) is continuous. Next we apply the Hilbert transform for periodic functions to function F(y), y E [-7T,7T]: 1
(I'F)(x) == 27T
f7r F(y)cot-Y- x dy, 2
-7T
which transfers cos ny into - sin nx and sin ny into cos nx. Taking into account (16.2.4), one gets from (16.2.2), ex
L n= 1
(nAn cos nx + nBn sin nx)
=
-
1 Y- x - f F(y)cot-- dy. (16.2.10) 27T I, 2
Let us substitute the sum r:~~ 1 (nAn COS nx + nBn sin nx) appearing in 06.1.2) by (16.2.10), and let us substitute all the remaining required coefficients entering the other summands on the left-hand side of (16.2.2) by their presentations through F(y), Y E E (see (16.2.7)-(16.2.9». After some evident manipulation, we conclude that function F(y), Y E E, satisfies the singular integral equation
1
-f
F(y)dy
7TEY-X
1
{
bY} F(y) dy
+ - f K(y - x) + -
2
TTE
=
-f(x),
XE
E,
(16.2.11) where
K(x)
=
1 x 1 "2cot"2 - ~ -
L n~l
En
sin nx.
Method of Discrete Vortices
398
Let us describe the class of functions in which a solution to Equation (16.2.11) must be sought. Let us denote the construction of function F(y) in interval (ak' 13k ) by Fk(y); in other words, Fk(y) = F(y)lyE(t>k.fh)' By 06.2.3),
Y
E
(ak' 13k)' k
=
1,2, ... ,m,
(16.2.12) where Uk(y), Y E [ak' 13k], are Holder-continuous functions, nonzero at the ends of the interval. Also, m additional conditions (16.2.5) must be met:
k
=
],2, ... , m.
The characteristic equation for (16.2.1l) in the said class of functions, subject to the additional conditions 06.2.5), has a unique solution. Approximate values of the functions uk(y), k = 1,2, ... , m, may be readily obtained by using the numerical method proposed in Lifanov and Matveev (983) and previously described in Section 5.4. As applied to the case under consideration, the major result may be formulated as follows. Let
i = 1, ... ,n k , (16.2.13)
j=1, ... ,nk- 1,
k = 1, ... , m,
(16.2.14)
and K(x),f(x) E Hr(a). If the singular integral equation 06.2.1]) subject to additional conditions 06.2.5) is uniquely solvable in the class of functions determined by Conditions (16.2.12), then at a sufficiently large N] =
min n k ,
l,,;k,,;m
the system of linear algebraic equations [with respect to unknowns
Numerical Method of Discrete Singularities
399
j = 1, ... ,n p
-
1,
j=n p ,p=1,2, ... ,m, (16.2.15)
has a unique solution, and
Iu
k. nk.
(y(n,» l
- Uk (y(n,» I
I
5,
0
-In na ) ' ( n r+ k
k
1,2, ... ,m, (16.2.16)
=
for N] ---) oc. Having obtained the values of Uk,n,(Y?,», i = 1, ... , n k , k = 1, ... , m, one may find approximate values of the sought after coefficients A o' An' and En' n EN, by using Formulas (16.2.7)-(16.2.9), as well as approximate values of the functions Fk(y), Y E (O'k> 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<
(y(n,» i
1
_ Uk,nk. (y(n,» i
-
i = 1, ... ,nk' k = 1, ... ,m.
,
Then approximate values of the functions Fk(y), Y the formulas O'k
E
(O'k' 13k)' are given by
< Y < 13k , k
=
1, ... , m.
( 16.2.17)
Following Longhanns and Selbermann (1981), one arrives at the error estimate
Wk
=
V(
13k -
y)(y - O'd· (16.2.18)
400
Method of Discrete Vortices
Then by using Formulas (16.2.7)-(16.2.9), one obtains the following approximate values of the coefficients A q , Bq , q E N, and A o:
_ 1 mIn, Bq = - '"- -n '" qk=l
_ Ao =
1 -
-
U
k,n,
(y(n,»cos qy(n;) I I'
ki=l
1
n "
2 '-
-
n,
"u '- k,n, (y(nk»y~nk). i i
k=lnki=l
Taking into account (16.2.16), the latter formulas permit us to conclude that (16.2.19) where it is supposed that ndNl :<:;: D* < +00, k = 1,2, ... ,m, N 1 ---) 00. Next we write a system of singular integral equations whose solution is obtained by solving the system of dual equations (16.1.26), (16.1.27), (16.1.28), and (16.1.29). Acting similarly to what was done when reducing the dual equation (16.1.1) and (16.1.2) to a singular integral equation, we introduce two functions F 1( x) ==
L -
n[( An + Bn)sin nx - (C n + Dn)cos nx],
Ixl:<:;:
7T,
n=l
(16.2.20)
F2 (x) ==
L -
n[(A n - Bn)sin nx - (C n - Dn)cos nx].
(16.2.21)
n=l
From (16.1.26) and (16.1.28), one gets f:(x) =0,
(16.2.22)
xECEi ,i=I,2,
together with m, + m 2 relationships
j
(3.,
cr"
, F;(y) dy
=
0,
k
=
1, ... ,m i , i
=
1,2.
(16.2.23)
401
Numerical Method of Discrete Singularities
All the sought after coefficients a o, b o' An' En' en' and D n, n E N, may be expressed through the functions F;(y), Y E E;, i = 1,2, which are found from a system of singular integral equations. After some manipulation, we arrive at the final result. The function F/x), i = 1,2, is sought in the class of functions pr~sentable in the form
where $;(x) E H and does not vanish at the boundary aE; of the set E i • The sought after functions Fi(x), i = 1,2, meet Conditions 06.2.23) and the system of singular integral equations
--1 f '7T
Kz(x, y)Fz(ydy
--1 f '7T
=
-!,(x),
X E
E"
fz(x),
X E
E z , (16.2.25)
(16.2.24)
£2
Kz(x,y)F,(y) dy
=
E,
where K (x y)
"
=
1 y-x 1 1 -cot-- - - - + 2 2 Y- x .2
1 Kz(x,y)
=
-
2
n= I
Y
ox;
n'-::l
In
sin n(y - x) + 2H '
Y
x
L
"€
€Zn
sinn(y -x) + - , 2H
4e
nlf
From the general theory of singular integral equations it follows that in the class of functions under consideration, characteristic equations have unique solutions if the corresponding additional conditions 06.2.23) are met.
Method of Discrete Vortices
402
As long as a singular integral over only one of the sets E i , i = 1,2, is present in each of the equations composing a system, an approximate solution to the boundary problem discussed in the preceding section (paragraph 5) may be obtained by the method of discrete singularities. Accordingly, the system of linear algebraic equations is composed of systems of such equations for each of Equations (16.2.24) and (16.2.25) (see (16.2.15». By solving the system of n linear algebraic equations with n unknowns thus obtained, we first find approximate values of the functions usk(Y), then those of Fsk(Y) = U,k(y)[(/3sk - y)(y - ask)] 1/2 and Fs' S = ],2, where Fsk(Y) is the restriction o[function Fs on the interval (ask> /3sk). Now we can find approximate values of both coefficients of the series for function u(x, z) and the values of the function u(x, z) itself at the portions of the boundary where it was not specified beforehand:
u(x, -
H) 2
=
rF
2 (y)
dy,
-7T
XE[-7T,7T].
Note concerning a continuum analog of the dual equations (16.1.]) and (16.1.2): Let us consider the dual integral equations (Gandel 1983) 1 --f ..n;; 1 r;;;V27T
f
-
x
Q( A)e iAX dA
x
IAIQ( A)(1
CE,
=
0,
X E
=
f(x),
x Ec,
(A)
-x
+
E(
A»e
iAX
dA
(B)
x
where E = Uk~l (ak,h k ), -00 < a l < hi < ... < am < hm < +x, CE = R \ E; [(x), X E E, and E(A), A E R, are given functions, and E(A) = 0(A- 2 ) for A -+ 00, while Q(A), A E R, is an unknown function. One comes across these equations when considering boundary problems of electrostatics-electrodynamics in the case when the boundary is an isolated lattice composed of a finite number of plane strips. Usually, to solve such a problem it suffices to calculate the values of the integral
..n;;f ]
x
.
_
Q(A)e'AxdA=U(x)
-x
on the left-hand side of the first equation for x
E
E.
403
Numerical Method of Discrete Singularities
Acting in a formal way, we denote dU(x) F(x) == - dx
i
=
f AQ( A)e iAX dA, {2;-x x
and derive a singular integral equation for function F(x), x be sought in the class of functions presentable in the form F(x)
=
m
,
Vnk~ ,I(bk
X E
adl
R,
X E
-
E
E, that will
E,
where
0,
=
X E
k
By applying to the function F(x), x (flF)(x)
=
E
CE,
(C)
1, ... ,m.
(D)
=
R, the Hilbert transform
~ fX F(y) dy 7T -y + x -x
and taking into account (C) and the fact that
H: exp(iAy)
>4
i[(IAI;A)exp(iAx)],
we get 1 1 F(y) dy f IAIQ(A)eiAXdA=-f ' {2; -x Yx
7T
E
Then, using the presentation for F(x), x (C), one obtains AQ(A)
1
= -.-fF(y)e- iAY
i{2;
X E
R.
X
E
R, and taking into account
dy,
A ER.
E
By using the latter two relationships and Equation (B), one may derive the singular integral equation 1 F(y)dy -f + f K(y 7TEY-X
E
x)F(y) dy
=
f(x),
X E
E,
404
Method of Discrete Vortices
which function F(y), Y
E
K(z)
E, satisfies. Here
=
l o c i AI . -.f -€(A)e-JAZdA 2m _ oc A
is a kernel coinciding with the Fourier transform of function €(A)sign A, A E R, to an accuracy of the factor l/U..,l2;). It should be remembered that the required function is sought in the class indicated previously and meets Conditions (D), thus ensuring unique solvability of the corresponding characteristic equation. After obtaining the function F(y), Y E E, the sought after function U(x) may be calculated by using the formula U(x) = f:1 F(y) dy, x E R.
16.3. DIFFRACTION OF A SCALAR WAVE ON A PLANE LATTICE: DIRICHLET AND NEUMANN PROBLEMS FOR THE HELMHOLTZ EQUATION Let us consider two important examples of boundary 'problems of mathematical physics resulting in dual equations of the type 06.1.1) and 06.1.2). SpecificaIly, we wiII consider three-dimensional Dirichlet and Neumann problems for the Helmholtz equation in the case when the boundary is formed by a plane lattice. Both the problems of diffraction of a plane monochromatic wave on a plane ideaIly conducting lattice (numeriral solution of these problems is discussed in the foIlowing section) and the problems of diffraction of acoustic waves on "soft" and "rigid" lattices reduce to the stated two boundary problems. Both the Dirichlet and Neumann problems considered in this section reduce to a mixed boundary problem for the Helmholtz equation in a half-space, whose dual equations have the form (16.1.1). The method described in the preceding section was used to solve model problems numericalIy; the solutions agreed favorably with accurate solutions. 16.3.1. Formulation of the Dirichlet Problem and Derivation of Dual Equations Let a lattice formed of periodicaIly repeated (the period being equal to 21) systems of strips whose edges are paraIlel to the OX' axis be considered in the plane OX'Y' of the Cartesian system of coordinates {(x', y', z'): z'
=
0,
-00
< x' <
+00, y'
E
CE'},
Numerical Method of Discrete Singularities
405
where E = Uj~IEj, Ej = (O'j,f3/), eE' = [-I,Il\E', -I < O'{ < f3; < ... < 0';" < f3;" < I. Let a plane monochromatic wave = i exp( -ikz'), where k = w/c is the wave number and c is the speed of light, fall onto a lattice from the half-space z' > 0 perpendicularly to the former. Let us denote by U o the sum of the incident and reflected waves in the upper half-space for the case when the first boundary condition U oIz' ~ 0 = 0 is met throughout the plane:
u
U
ikz o = ie-- '
-
ie ikz ' = 2sin kz',
z'
~
(16.3.1)
O.
If the total field vanishes at the lattice, then it may be sought in the form UO
ulolal =
{
+
z > 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' <
x,
z'
~
0,
and meet both the Helmholtz equation
z'
~
(16.3.2)
0,
and the matching conditions
rJu-\ -
rJz' z'~o
rJu'l =2k+--
az' z'~o'
y' EE',
( 16.3.3)
y' EE',
(16.3.4)
which together with (16.3.2) ensure validity of the Helmholtz equation for the total field Llulolal + k 2 u\Olal = () throughout the space outside the lattice. At the lattice itself the total field vanishes: y'
E
eE'. (16.3.5)
AdditionaIly, in order to ensure that a solution to the boundary problem under consideration is unique, both radiation conditions at infinity and Meixner conditions at the ribs of the lattice must be met (Hanl, Maue, and Westphal 1964; Shestopalov et al. 1973).
Method of Discrete Vortices
406
Solutions to the Helmholtz equations in the upper (z' > 0) and the lower (z' < 0) half-space meeting the periodicity conditions have the form u =(y' , z') = A 0±e ± ikz'
x
+
7Tny' cos-+B I
(
" e ~ y"z' A £...
1:
n
1
7Tny' ) sin-. I '
n= I
z'
~
0,
V
(7Tn/I)2 - k 2 and the radiation conditions will be fulfilled where 'Yn = if the branch of the radical is chosen in such a way that
<0
for n < klj7T,
'Yr,>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, < ... <
am
< 13m < 7T.
Then, Equations 06.3.8) and 06.3.9) become An
+
L
(An cos ny
+ En sin
ny) = 0,
Y
E
CE,
n=1
(16.3.11 )
Y EE.
( 16.3.12) These equations correspond to Equations 06.1.1) and 06.1.2) for b
=
-iK,
f(y) ==
K,
».
where En = O( K 2 1(2n 2 As was shown in the preceding section, all the sought after coefficients An' En' n EN, and A o are expressed by Formulas 06.2.7)-06.2.9) through solution F(y), y E E, to the singular integral equation 06.2.11), where one should put b
=
-iK,
f(x)
=
K,
n EN,
Method of Discrete Vortices
408
and the choice of the branch of the radical is determined by Conditions 06.2.5). The Meixner conditions determine behavior of both the wave field and its first derivatives in the neighborhood of a strip edge. These conditions generate presentations 06.2.12) for construction of function F(y) in intervals E j • 16.3.2. Model Problems Consider the function
g(x) =
(
-1],
0,
X E [-7T,
-~, 0,
xE(-I,l), xE[l,7T],
which after being expanded into the Fourier series becomes 1
g(x)
=
-
-
oc
L
-
4
n= I
XE[-7T,7T], (16.3.13)
because according to Bateman and Erdelyi (953),
fl
1
7T
~
V1
n = 0,1, ... ,
n
-I
where
/1
1 t t - x
L OC
and because for x
E ( -
1, I),
1
t
dt --1· 7T-I~t-x-' I
J
hence we have
L 00
=
1f
1+-
n=]
I
7T-1
t
[1
t-x
1]
~2 -cot-- - - - dt I-t
2
2
t-x
(16.3.14)
(Bateman and Erdelyi 1953).
NumericaL Method of Discrete SinguLarities
409
Thus, from 06.3.13) it follows that
1
00
$,(n)
-4 + "~ --cos nx
xE[-7T,7T]\(-I,I), (16.3.15)
0,
=
n
n= I
and employing (16.3.14), one gets
X E
(-1,1), (16.3.16)
where
f(x)
4+ b + -J 1 [1-cot-t - XI] - --
t dt
I
:=; - -
4
7T
- L
-]
2
t - x
2
€n
~
xE[-I,I].
(16.3.17)
n~]
Finally, we arrive at the following result. The sequence Au
= 1/4,
n EN,
gives a solution to the dual equation 06.1.0 and (16.1.2) with the right-hand side f(x). Note also that in the case under consideration a solution to the corresponding singular integral equation (16.2.10 is given by the function
x
F(x)
=g'(X)IXE(-],]) =
~.
(16.3.18)
In what follows we will briefly discuss the model problem on a system of intervals £ = E] U £2 (for m = 2). It is obvious that the function F(y) = F](y) + FiY), where
Fk(y)
Ck(Y - (a k + 13d 2» 13k - y)(y - ad
=
{
J(
0,
'
(16.3.19)
Method of Discrete Vortices
410
k = 1,2, and C I' C 2 are given constants, is a solution to the integral equation 06.2.11) with right-hand side
f(x)
Y- x
1
=
-
b
- f F(y)cot- dy - -fyf(y) dy 27T E 2 27T E 1
x
+ 7TfF(y) L
L
En
sin n(y -x) dy,
X E
E, (16.3.20)
n=l
and meets Relationships 06.2.5). By substituting F(y)l y < Ek = r,,(y) (see 06.3.20) by the corresponding expressions 06.3.19), we calculate fk(X) ==!(X)IXEL k , k = 1,2:
1
{I
I}
Y -x - -fF(Y) -cot-- - - - dy 7T F 2 2 y-x
+ -1 f F(y) [ 7T E
L x
n=1
En
bY] dy,
sin n(y - x) - -
2
where k, [ take on the values of 1, 2 and k *- [. The first two integrals may be expressed explicitly: ~f F,,(y) dy ---7T Ek Y - X
=
Ck ,
All the remaining integrals were calculated approximately by using Gaussian quadratures. For one of the intervals, E = ( - 1, 1), the model problem was solved numerically by employing a high-speed supercomputer*. Table 16.1 presents approximate values of un(yfnl), i = 1, ... , n, along with the accurate values u(yfn l ) == yJn l calculated for the number of grid points n = 20. The two sets of data differ (in absolute values) by less than 2 x 10 -5 . • This is probably the first mention in press about the Russian supercomputer on which General Belotserkovsky has worked (G. Ch,).
Numerical Method of Discrete Singularities
411
TABLE 16.1
1 2 3 4 5 6 7 8 9 10
, un(yfn )
un(y,!n l
u(yln)
0.99693 0.97235 0.92389 0.85266 0.76042 0.62945 0.52249 0.38269 0.23344 0.07846
0.99692 0.97237 0.92388 0.85264 0.76041 0.64945 0.52250 0.38268 0.23345 0.07846
-0.07846 -0.23345 - 0.38269 -0.52251 -0.64945 -0.76041 -0.85264 -0.92389 -0.97236 -0.99692
II
12 13 14 15 16 I7 18 19 20
u(y,
-0.07846 -0.23344 -0.38268 - 0.52250 -0.64945 -0.76041 -0.85264 -0.92388 -0.97237 -0.99692
Then we have calculated approximate values of the sought after coefficients A o, A q , and B q , q = 1,2,3, by using the formulas
_ Bq
1"X =
"i...J U n (y(n»cos qy(n) iL I ,
-
nq
q EN,
i~1
Ao= - -
1
n
"
2n.'-I~
I
U (y(n»y(n) n
I
I'
As before, approximate values of the integrals at the segment ( - 1, 1) were calculated by employing the weight function (l - y2)-1/2 and Gaussian quadrature formulas. Table 16.2 shows approximate values of coefficients Ao' A" A2 , and .4 3 and numerical values obtained by using the fonnulas of the accurate solution to the model problem A o = 1/4 and A q = cfJI(q)/q, q = 1,2,3, and employing the Bessel functions tables (Abramowitz and Stegun 1964). The difference between the approximate and accurate values of the coefficients does not exceed 12 X 10- 6 • To the same accuracy, coefficients HI' H2 , and H3 were equal to zero. The model problem for two intervals was solved numerically for C 1 = 2, C 2 = 0.5, a l = -3, 131 = -2, a 2 = -1, 132 = 0.5, n l = 4, and n2 = 6. Table 16.3 permits comparison of approximate values of uk,n,(Yi(n k ) , i = 1, ... , nk> k = 1,2 with the accurate ones Uk(y;n k » = Ck(y;n d (13k + a k ) /2). The absolute error does not exceed 2 X 10 6 throughout the domain.
Method of Discrete Vortices
412 TABLE 16.2 q
.4,
A,
q
.4,
A,
0 I
0.250008 0.440062
0.25 0.4400506
2 3
0.288364 0.113010
0.2883624 0.1130197
TABLE 16.3 k
I 2 3 4
1
=
k=2
uk. ",(y,!",)
Uk(y,l"d
0.923879 0.382683 -0.382683 -0.923879
0.923879 0.382684 -0.382683 -0.923879
I 2 3 4 5 6
Uk' nk(y,l"d)
Uk(y,l"d)
0.362221 0.265164 0.097056 -0.097058 -0.265164 -0.362220
0.362222 0.265165 0.097057 -0.097057 -0.265165 -0.362222
16.3.3. Neumann Problem Let a plane monochromatic wave = i exp( - ikz ') fall onto the lattice {(x', y', z'): -oc < x' < 00, y' E E' + 2nl, nEZ, z' = 0} from the halfspace z' > 0 perpendicularly to the lattice. By Vo we denote the sum of the incident and reflected waves in the upper half-plane in the case where the second boundary condition
v
-avo
az'
I
z··~o
=0
'
Vo = 2i cos kz', z'
~
0,
(16.3.21 )
is met at the whole of the plane OX' Y' . If the normal derivative of the total field vanishes at the lattice, then the field is sought in the form
v.
_
total -
{Vo + V+, V-,
z' > 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)
Numerical Method of Discrete Singularities
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
Method of Discrete Vortices
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
Numerical Method of Discrete Singularities
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:
Method of Discrete Vortices
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
Numerical Method of Discrete Singularities
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
Method of Discrete Vortices
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
Method of Discrete Vortices
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
fl( M) d(I =
(17.2.8)
O.
u
If (T is an open curve and the second derivatives of the functions presenting it in the parametric form belong to the class H( 0') on [ -1,1], then, similarly to what was done in Section 10.4 for an airfoil with an unclosed contour, one can show that Equation 07.2.7) transforms to the equation 1 JIg ( t ) dt -
27T
-I
(to - t)
2
+
J
I
-1
K(t IP t)g(t) dt
=
1(/0)'
(17.2.9)
whose solution has the form g(t) = ~ t/J(t), where t/J '(t) is a function belonging to the class H on [ - 1, 1]. If (I is a closed contour, then it transforms to the equation
1
1 27r g( t) dt • 2 87T 0 SIO «(to - t)/2)
-
+
[27TK(to' t)g(t) dt = 1(/0), ()
(17.2.10)
whose solution is periodic and has a derivative that belongs to the class H on [0, 27T].
17.3. MIXED BOUNDARY VALUE PROBLEMS In electrostatics one comes across problems for the Laplace equation subject to the boundary conditions
(17.3.1 ) where (II and and (II U (I2 =
(I2 (I.
are curves crossing each other at the end points only,
424
Method of Discrete Vortices
A solution to the problem is sought in the form
(17.3.2) Then boundary conditions (17.3.1) result in a system of two integral equations with respect to two functions /L(M), ME 0"1 and g(M), ME 0"2:
(17.3.3 ) The first equation of the latter system has a logarithmic singularity on and hence: must be differentiated with respect to a parameter. As a result, one gets a diagonal system of two singular integral equations. A unique solution to the system is singled out with the help of an integral relationship for functions /L(M) and g(M), obtained by fixing an arbitrary value of the parameter in the first equation (17.3.3). Thus, one is looking for a solution (/L(M), g(M» to System (17.3.3), where function /L(M) tends to infinity at the end points of the curve 0"1 as a reciprocal of the square root of the distance from the end points, and function g(M) tends to zero at the end points of curve 0"2 in the same manner. 0",
Conclusion
The advent and wide use of high-speed supercomputers in practically all areas of human activities* is one of the most remarkable manifestations of the scientific and technological revolution. The importance of quantitative description and mathematical modeling is constantly growing. A new method of research based on numerical experiment has emerged and is developing aggressively; it has promoted establishing close relations between the physical, mathematical, and computerized approaches to studying natural phenomena and on developing a universal language of communication for all three. The most practical proved to be discrete methods of description that are most natural, result in simpler mathematical logic, and are adequate to computer languages. Fundamental problems of mathematics and applied problems of mechanics, electrodynamics, mathematical physics, and the like tend to be brought closer together, to interweave, and to affect each other favorably. Numerical experiments become a virtual laboratory of a mathematician. First, they permit an a priori check of the facts forming the foundation of a theory. Second, realistic computer calculations help develop, verify, and perfect numerical methods. Presently, the methods are required to be much more strict and versatile, because they form the production basis of modern science. The methods must not only be correct, but also highly efficient, stable, universal, and logically simple. This book is dedicated to detailed discussion and verification of a method for solving numerically (with the help of a computer) both oneand multidimensional singular integral equations characteristic of various -This remark by General Belotserkovsky about supercomputers was not in the original book published in 1985. At that time there were no supercomputers in Russia. It is amazing how quickly they have penetrated all areas of human activity in Russia (G. Ch,).
425
Method of Discrete Vortices
426
areas of mathematical physics and applications. The methods of using the equations were illustrated by considering problems of aerodynamics, elastodynamics, and electrodynamics. Under present conditions, the transformations to which the original ideas of the method described in this book were subjected should be considered natural. 1.
2.
3.
4.
Heuristic analysis and numerical experiments used for perfecting the method by applying it to special test problems of the theory of airfoils possessing exact solutions. Generalization of the method, its extension onto wings of an arbitrary plan form, and development of the method at a new level with the help of logical considerations and numerical experiments. Rigorous mathematical verification by means of constructing special quadrature formulas for both singular integrals and numerical solution of the equations. Generalization of the method onto a wider class of singular integral equations and its applications in a variety of new areas.
Viability of the method is mostly due to its logic, versatility of the employed approach, and stability of calculated results. In order to further promote development of the body of mathematics under consideration as well as to meet everyday needs of a number of applied sciences, we would like to draw the reader's attention to the following problems: 1.
2.
Many applied problems reduce to solving integral equations and their systems. Because an effective method for solving singular integral equations has been developed and widely tested, it is worthwhile to seek ways to derive equations belonging to the type, some of which were presented in the foregoing text. Construct formulas of the type of quadrature formulas of the method of discrete vortices, for the singular integral f(to)
=
f
I.
y( t) dt to - t
'
(C.l)
which would ensure uniform convergence of a fixed accuracy for the cases when: a. L is a segment/circle, and y(t) suffers discontinuities of the first kind at a number of internal points. b. L has points of bifurcation (say, L is T- or + -like in shape), At some of the points y(t) suffers power-law integrable disc. continuities.
Conclusion
3.
427
Construct quadrature formulas of the type used in the method of discrete vortices for the integral of the theory of a finite-span wing,
A
y(x,z)
=
if'T(ZO -Z)
(
1
2
Xo
+
,
-x
)
V(X O -xt + (ZO -Z)
2
dxdz, (C.2)
which would ensure uniform convergence at reference points, depending on differential properties of the regular part of function y(x, z). For example, if (T is the rectangle [-b, b] X [-1,1], let us suppose that y(x, z) has the form
.;' 'f2-x I" -
4.
Z"
--
b +x
u(x, z),
where function u(x, z) [the regular part of function y(x, z)] has alI the derivatives of the rth order that belong to the class H( a) on (J. Consider Problem 3 for the two-dimensional singular integral =
B
if [(x o, Yo, (J)u(x, y) dxdy 'T (x o - x)
2
+
(y - Yo)
2
(C.3)
'
where 0 is the polar angle of point (x, y) with respect to point Yo)' and [(x o, Yo, (J) is a characteristic of the integral B. Consider the same problem for the integral
(Xl!'
_if
Q(x, y) dxdy
G -
,
'T[(X O 5.
,
xt + (Yo - yr]
(CA)
3/2.
Construct numerical methods of the type of the method of discrete vortices for the singular integral equation ay(to)
+
b y(t) dt -1. f.K(to,t)y(t) dt t 7T
L
10
-
I.
=
[(to)
(C.5)
(I- is one of the curves mentioned previously), ensuring uniform
convergence in cases when: a. [(to) suffers one or a number of discontinuities of the first kind or power integrable discontinuities. b. Both kernel K(co, t) and function [(to) suffer discontinuities at the same points 10 , c. a and b are complex numbers.
Method of Discrete Vortices
428
6.
Construct for the Abel equation l_'P_(_7_)_d-:::-: + fl K (t , 7) 'P( 7) d7 = f( t ) fa(t-7) a
7.
(e.6)
a numerical method analogous to the one constructed in Section 12.2, whose convergence rate would depend on differential properties of f(O and K(t,7). For equations
if
Y(X,Z)2 u(zo - z)
a. b. c.
(1 + V
x(~
-x 2 )dxdZ (X o - x) + (zo - z)
Prove the existence and uniqueness theorems (under corresponding additional conditions), Construct and substantiate numerical methods ensuring uniform convergence over the entire region (J, Do the same in the presence of a summand with a regular kernel in Equations (e.7) and (e.8).
Perhaps you will succeed in finding such a system of orthogonal (on the rectangle [-b,b] X [-i,1] of the plane OXY) polynomials Pn,m(x,z) of degree n in x and m in z, and with the weight w(x, z) = (Vb 2 - x 2 2 - Z2 )- '. Using such a system and the presentation
Vi
G(X, z)
=
(e.9)
w(x, z)u(x, z),
where u(x, z) is a function of the class H on the rectangle, it will be possible to construct an interpolation polynomial un(x, z), i
by using roots
(Xi'
=
1,oo.,n,j
=
l,oo.,m,
(e.lO)
z) of polynomial Pn,m(x, z). Then, with the help of the
Conclusion
429
polynomial, it will be possible to construct the system of algebraic equations n
m
L LU n(X;'Zj)A7:F
k
=!(XOk,ZOm)'
=
1, ... ,n, m
=
l, ... ,m.
;= I j= I
(C.ll) Here
Atr are coefficients of the quadrature formula for the integral f
bfl
-b
-I
w(x,z)u(x,z) ------------,30;-"/"'"0
[(X O-x) 2+ (Zo -Z) 2]
~
dx dz,
(C.12)
constructed with the help of polynomial u/x, z) at point (X Ok , X Om ), It should be noted that points (X Ok , zOrn) must correspond to the roots of the polynomial-function
Qn,m(XO' zo) =
f
bfl
-b -I
w(x, z)Pn,m(x, z)
[
0
(X o -xt +
(Zo
-z)
2]3/2 dxdz.
(C.l3)
On the other hand, the use of two-dimensional splines also seems to be attractive. Both theoretical analysis and numerical experiments in this area would be of interest. 8.
Mathematical verification of the method of discrete vortices in the framework of the nonlinear unsteady problem (see Section 12.4).
In addition to the development of the preceding approaches, numerical experiments have inflicted more radical changes upon numerical methods, because derivation of integral equations for solving nonlinear problems of the theory of wings (and even more so for the study of separated flows past various lifting surfaces) proved to be not only quite laborious but also unnecessary (Belotserkovsky 1968, Belotserkovsky and Nisht 1978). The experience of using discrete models considered in this book permitted us to describe various phenomena by using one and the same language at all stages of research -starting from the construction of a physical scheme and finishing with computer simulation. A vortex proved to be not only a clear-cut physical concept, but also a convenient mathematical abstraction ensuring, in particular, stability of calculations. Figure C.l shows an example of calculated separated flow past a plane that is perpendicular to the oncoming flow and travels with speed U o' A<; usual, the plate is replaced by a system of discrete vortices, and the
Method of Discrete Vortices
430
..
• • •r•.; :.:
.
..
- :~.I. ~ :... ~:..':':,-A,-. ~ _. ,- .'. • ".-.I...~...• • :..(,.... ..\ .~ • •• :: .... ~,.... .. .". : .:• •!~~:• •-:- e.,::. .\,...f,· . ... .,: ~.....' .....'
. . . e)er:.:.-. -:.:l..
~" .. \.:'>.......:;~
.!.·::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
Method of Discrete Vortices
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.
References
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433
434
Method of Discrete Vortices
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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
Method of Discrete Vortices
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
Method of Discrete Vortices
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
Method of Discrete Vortices
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
Method of Discrete Vortices
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
Method of Discrete Vortices
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
