. S. D e p a r t m e n t o f
.«aliiiuul Applied
Bureau
Miitliemalie»
of
C o i n n i e r c e
Stundardn
Series
•
18
UNITED STATES DEPARTMENT OF C O M M E R C E NATIONAL BUREAU O F STANDARDS
C o n s t r u c t i o n o f
•
a n d
• Charles Sawyer, Secretary
A. V . A * U n , Director
A p p l i c a t i o n s
C o n f o r m a i
M a p s
Proceedings of a Symposium
Held on June 22-25, 1949, at the Institute for Numerical Analysis of the National Bureau of Standards at the University of California, Los Angeles
Edited by E . F. BBCKBNBACH
Vational
Bureau
of
Standards
A p p l i e d M a t h e m a t i c s Series • 1 8 Issued December 26, 1952
FOR S A L E B Y T H K S U P E R I N T E N D E N T O F D O C U M E N T S , U. S. G O V E R N M E N T P R I N T I N G O F F I C E , W A S H I N G T O N P R I C E $2.25 (Backram)
2S,D.C
I
H
Preface The development of calculating machines, which showed sturdy growth during the first part of the present century, has recently received a startling stimulus with the completion and successful operation of several fully automatic, electronically sequenced, general piu*pose digital machines capable of performing at fantastically high speeds. I n recognition of the need for basic research and training i n the types of mathematics which are pertinent to the efficient exploitation and further development of such machines, the National Bureau of Standards established the Institute for Numerical Analysis, as a section of the National Applied Mathematics Laboratories, i n 1948 on the campus of the University of California, L o s Angeles. I n this endeavor, the Bureau operates with the cooperation of the University, and with the support of the Office of N a v a l Research of the N a v y Department. A t the end of its first year of existence, the Institute was host to two symposia, one on conformai mapping and the other on the M o n t e Carlo method. The papers in the present volume were prepared for presentation at the conformai mapping symposium. Conformai mapping was chosen as a symposium topic for several reasons. T h i s is a very old and classic branch of mathematics, going back even to Ptolemy. There are many existence theorems, starting w i t h Riemann's, yet little attention has been given to constructive details. The differential equations involved are far from trivial, but they are not so complicated as to preclude the possibility of their being handled b y machines now coming into use. M a n y excellent mathematicians are interested i n the field, and it was felt that the symposium would serve well to introduce them to, and interest them i n , the use of high-speed machines for the approximate solutions of mathematical problems. Finally, there are many important physical applications of confonnal mapping. I deeply appreciate the encouragement given b y John H . Curtiss, chief of the National Applied Mathematics Laboratories, and by H u g h Odishaw, assistant to the Dii'ector of the National Bureau of Standards, during the organization of the symposium and the preparation of this volume; and the help given b y Cornelius Lanczos, Alexander Ostrowski, and Wladimir Seidel, who served with me as a committee of the Institute in arranging the symposium. The contributors to this volume also have been most helpful i n promptly submitting manuscripts. E. F. Los Angeles, California. September 1949.
BECKENBACH,
Editor.
Contents Page
Preface __ 1. O n network methods i n conformai mapping and i n related problems R. V. M i s E s 2. F l o w patterns and conformai mapping of domains of higher topological structure R.
7
COUKANT
3. Some industrial applications of conformai mapping H.
iii 1
16
PORITSKY
4. Conformai maps of electric and magnetic fields i n transformers and similar apparatus... _ G. M .
STEIN
5. Conformai mapping applied to electromagnetic field problems ERNST
31 59
WEBER
6. O n the use of conformai mapping i n two-dimensional problems of the theory of elasticity _. I. S. SOKOLNIKOPP 7. Conformai mapping related to torsional rigidity,-principal frequency, and electrostatic capacity G . SZEGÒ 8. F l u i d dynamics, conformai mapping, and numerical methods
71
79 85
A N D R E W VAZSONYI
9. The use of conformai mapping i n the study of flow phenomena at the free surface of an infinite sea
87
E U G E N E P. C O O P E R
10. A n application of conformai mapping to problems i n conical supersonic R. C . F . B A R T E L S and O.
flows.
LAPORTE
11. O n the Helmholtz problem of conformai representation ALEXANDER
91 105
WEINSTEIN
12. Some generalizations of conformai mapping suggested by gas dynamics
117
LIPMAN BERS
13. The use of conformai mapping to determine flows with free streamlines DAVID M .
125
YOUNG
14. Conformai mapping i n aerodynamics, with emphasis on the method of successive conjugates
137
I. E . G A R R I C K
15. O n the convergence of Theodorsen's and Garrick's method of conformai mappings. . . . . . _____ __ _ A . M . OSTROWSKI 16. O n a discontinuous analogue of Theodorsen's and Garrick's method
149 165
A . M . OSTRO WS K i
17. O n conformai mapping of variable regions
175
S. E . W A R S C H A W S K I
18. A variational method for simply connected domains A . C . ScHAEFFER and D . C . S P E N C E R
_
_
189
•
Page
19. Some remarks on variational methods applicable to multiply connected domains M . ScHiFFER and D . C . S P E N C E R 20. The theory of Kernel functions i n conformal mapping S. B E R G M A N and
M.
199
SCHIFFER
21. A new proof of the Riemann mapping theorem P. R .
193
207
GARABEDIAN
22. The Kernel function and the construction of conformal maps
215
ZEEV NEHARI
23. A n approximate method for conformal mapping
225
L E E H . SWINFORD
24. O n the effective determination of conformal mapping MAX
SHIFFMAN
25. The difference equation method for solving the Dirichlet problem P. C .
227 231
ROSENBLOOM
26. Relaxation methods as ancillary techniques
239
RICHARD SOUTHWELL
27. Conformal invariants L A R S V . A H L F O R S and
243 A.
BEURLING
28. O n subordination i n complex variable theory E . F . B E C K E N B A C H and
E . W.
247
GRAHAM
29. Asymptotic developments at the confluence of boundary conditions
255
HANS LEWY
30. Monodiffric functions : R u F U s ISAACS 31. Recent contributions of the Hungarian school to conformal mapping.
257 267
G . SzEGi)
32. Bibliography of numerical methods i n conformal mapping W.
SEIDEL
269
1.
On Network Methods in Conformal Mapping and in Related Problems ^ R, V , Mises *
A fairly general type of problem of mathematical physics can be written i n the form i(u)=0,
B(ti)=0,
where L is some differential operator in n dimensions, and B{u)~0 stands for a set of so-called boundary conditions, that is, of equations for u and its derivatives valid i n subspaces of less than n dimensions. A s the simplest case, one may consider the ordinary problem of conformal mapping: £ ( i i ) = A ( u ) , n = 2 [B{u)=Q denoting given li-values on a given contour]. There are, however, mu0, with v=0 corresponding to a boundary line; otherwise, nothing need be changed i n our preceding description. F o r example, a Moebius strip is represented by figure 2.10, and the limiting case where one of the infinite slits falls into v=0 is represented in figure 2.11, with the boundary indicated by shading. F o r such half-plane domains we may again stipulate a normalization similar to that used for genus zero: We may place the map of a fixed boundary point of G at the point at infinity on v=0 and fix the left end of one interior slit at w=0, v=l. The mapping theorem. We now consider the class of all slit domains B with r such boimdary slits and w i t h characteristic number Z , and state the following result. Theorem: Every Riemann domain G, orientable or not, with characteristic number L and with r boundarieSf can be whipped conformally on a slit domain B. The mapping on a normalized half-plane slit domain is uniquely determined if the point at inûnity is to correspond to a fixed point 0 on the boundary of G.
It m&y be added that in this theorem the slit domains could be replaced by half-plane slit domains provided G possesses at least one boundary element not equivalent to a single point.
P
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2 P
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Q
iiMiiiin^ '
2'
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^
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—;
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een eiven [9, 101. F o r the general misaligned position shown i n fig\ire 3.11, C , no solution is known to the author, though the form (3.16) is quite appUcable. The field o f figure 3.11, A , incidentally, is of interest i n connection w i t h the capacity of a microphone. JFrom the above i t is clear that analytical methods, at least as developed thus far^ have only limited power i n solving the compUcated field problems arising i i i electric machines. E l e c t n c a l engineers have resorted extensively to the use of "flux p l o t t i n g " ox free-hand dmwing of the flux lines and e q u i p o t ^ t i a l s . A s is weU known, these curves, when drawn for constant equal increments A^==A^, form a curvilinear set of 87naU squares, A certain aptitude, somewhat between mechanical drawing ability and artistic drawing, is required for successful flux plotting, and with practice people possessing such aptitude can learn to draw flux plots for a great variety o i cases w i t h relative ease. Examples of free-hand plots are shown i n figures 3.12, 3.13, 3.14, 3.15.
FlQUBB 3.12.
FIGURE 3 . 1 3 .
Flux dittribiUion
in a slot.
Flux p u l 8 a U o n - B C M ^ - . 4 9 3 .
F l u x plotting has also been appUed with relative success for magnetic fields i n current-carrying regions where ^ satisfies not the Laplace equation (3.18) but the Poisson equation 5^4-^=4x7 01^ ' ày^
(3.19)
FiouBE 3.14.
Fltix di$tribution at no load in air gap of synchronous salient-pole machine*
where / is the current density. One method consists i n superposing rf^i, any analytically determined solution of (3.19), with 1^2, a solution of (3.18), which is chosen so that ^=1^1+^2 satisfies the proper boundary conditions; ^2 is obtained b y means of a flux plot. Figure 3.16, taken from [11], shows a n example of this procedure for a rectangtdar coU w i t h uniform current density i n a slot. O n l y the plots of i^i and of ^1+4^2 are shown i n figure 3.16. A n alternative method of procedure consists i n starting w i t h a " k e r n e l " , or point, of vanishinjg field strength, and drawing lines of " n o w o r k " ; that is, orthogonal trajectories of the flux lines.^ A p p l i cation of Green's theorem for constant /, j
jv^dxdy
4
IT/j
jdxdy
0^ on
de,
(3.20)
relates the gradient of ^ (and hence the spacing of the lines of constant ^) to the area subtended by the lines of no work. A t the boundary of the coil the lines of no work and the flux lines merge into a small square pattern of lines of constant ^, V'. A n example of a construction of this type is shown i n flgure 3.17; here the kernel is at the upper left-hand boundary of the flux plot. The graphical superposition method may be apphed to solutions of the equation 2,
(3-21)
> They would be lines of constant magnetic potential 9, were It not (or the fact that in current-carrying regions the field la rotational and 9 does not exist
I FIGURE 3 . 1 5 .
Flux disiribtUion , (3.25) one mav obtain for v> the eq (3.22) with (x, y) replaced b y (r, z) and A=r*'""'''. T h e solution of
a-nd a point vortex sink at z%=R, and the l o w is described b y w=^-f
(«1—it;,)ln(s2+i?) + ( t f i + i t i ) l n ( z 2 H - I W + (
Ma+ÌP2) In (
—^)
+ ( — t f 2 — ù ' 2 ) In ( 2a — 1/^), (3.28)
tvhere {UxyV\) is the velocity of the incoming flow, and (wa,???) the velocity of the deflected flow. However, the details of the mapping function z{z^ and its determination for a given blade shape, as well as the determination of 7?, have been carried out i n relatively few cases and many shapes still involve exceedn g l y laborious calculations. Also of interest is the converse problem, where the blade shape is not given but where the pressure is prescribed as a function of the distance along the blade, usually from conditions requiring that there should not be too adverse a pressure gradient, resulting i n imstable boundary layers causing the flow to depart from the boundary. The art of calculating the blade shape for the converse problem is but little advanced. F o r cases where the pitch h is relatively small compared to the dimension of the blade, and where the angle of deviation is rather large, it turns out that in the gj-plane the points Rj—R Ue very close to the u n i t circle. A s a result even purely numerical methods run into a considerable amount of diflSculty since the calculations have to be carried out to a large number of places. A m o n g analytical and numerical methods of carrying out this mapping and determining the flow, m a y be cited [14, 15]. A n appUcation of the d-c board to the determination of the flow through a grid has been described hy C . Concordia [16]. When one takes into account compressibility effects, boimdary layer effects, and phenomena introduced by the rotation of the rotor, the problem becomes considerably more complicated. However, even i n the simple ideal case described above it must be kept i n mind that as the moving blades pass the stationary blades, the region over which the flow has to be determined is quite complicated and is 949581—52
2
constantly changing from one instant to the next. Usually the rotor blade pitch is not equal to that o f the stationary blades, but they may have a ratio of two rather large integers. The situation is s o m e w h a t analogous to the one obtained for the field of a rotor of an electric motor rotating past the stator, t h o u g h , due to the multiple connectivity of the region of flow, it is more comphcated even for the c o n d i t i o n s of ideal flow. Turning to miscellaneous further applications of conformai mapping, we observe that conformai mapping can be used in combination with other methods for simphfying comphcated boundaries, provided that the differential equations are not rendered a great deal more complex as a result of t h e mapping. A s an example, i n [17] a nmnerical procedure is used to solve A i r y ' s differential equation f o r a region shown i n figure 3.22. First, by means of a free-hand flux plot, the region i n question is mapped u p o n
FXOTTBB 3.22.
a strip i n the 2i-plane, thus straightening out the partly semicircular, partly straight, upper boundary. The ratio r=\dzi/dz\ of the small square side i n the 2i-plane to that of the g-plane is obtained from t h e )lot and smoothed out by curve construction, and the differential equation of A i r y , or rather its e q u i v a ent equations (3.29) are replaced by V F (3.30)
i
Equations (3.30) are then solved by partly numerical, partly analytical methods. Similarly in [18] conformai mapping is used i n the problem of determining flow of a compressible fluid past an airfoil in a uniform channel. F i r s t the flow of an incompressible fluid is used to transform the boundary of the airfoil into a rectilinear one (since i n the to-jA&ne this boundary becomes const.) ; then the equations of motion are expressed in terms of the new coordinates and solved by methods o f relaxation. Including the effect of compressibility one is led to an equation of the form (3.22), where h is a proper function of the gradient m a ^ i t u d e of 0. A conformai transformation puts (3.22) into a similar form except that A is now a function of the product of the magnitude of the gradient of ^ and of the square side ratio. F o r numerical solutions this extra comphcation offers no great inconvenience. A n interesting application of conformai mapping occurs i n extending the Hertz theory of contact stresses to take account of surface irregularity. I n the two-dimensional case the relation between the normal displacement v and the normal force r over the contact interval | x K a is given by (see [19]) const
la
s ds.
(3.31)
When V is given over the contact interval and F is sought, one has to solve the integral eq (3.31). introduce the analytic function J{^)
J _ " ^ F{s) h i {z
s)d8f
We
(3.32)
and note that the real part of / is proportional to v over the contact interval, while its imaginary part is proportional to the force integral F(8)dslá.
(3.33)
One can solve (3.31) very conveniently by means of a conformai mapping i n which the contact interval and ellipses confocal with its end points are transformed into lines parallel to the real and pure imaginary
B y resolving both v and F over the contact interval in Fourier series in the transformed variable, w e readily obtain the solution of (3.31). T h e integral eq (3.31) is closely related to the equation (3.34) w h i c h occurs i n the determination of the induced drag of an airfoil due to trailing vortices. I n general stress problems under no body forces one encounters the repeated Laplace equation (3.35) a n d , i n particular, as stated above, the two-dimensional case of (3.35) is satisfied by A i r y ' s function. T h e general solution of (3.35) can be expressed thus: F=9î[/(3)+ii;(2)],
(3.36)
w h e r e g are arbitrary analytic functions of 2, and z is the complex conjugate of z, Conformal mapping, w h i l e it transforms the Laplace equation into another Laplace equation, does not preserve the repeated L a p l a c e equation (3.35). However, it transforms (3.36) into SR[/(3l) + 2(2i)
