Construction and Applications of Conformal Maps, Proceedings of a Symposium

(3.26) of a wide degree of generality, are obtained. TiuTimg to other industrial field applications, we now consider fluid flow fields. M o d e m fluid flow owes its rise to the problem of fift of an airfoil. A s is well known, analytic methods are available for determinations of the flow of an ideal fluid about airfoils of a wide variety of shapes, one of the most popular methods consisting in mapping the outside of the airfoil conformally on the outside of a circle. A somewhat related problem of considerable recent industrial and aeronautical interest is the flow of a fluid through a grid of identical blades. T h i s problem is basic to the design of turbines and compressors, and lately nas grown in importance, on account of the gas turbine, for aircraft, locomotives, and stationary power plants. Neglecting compressibility and viscosity effects as well as centrifugal and Coriofis forces i n the rotor, replacing its rotation b y a translation, and disregarding the interference effects of adjacent rotor and stator grids, one arrives at the simplified problem of a flow of an ideal fluid through a stationary grid of identical buckets or blades.

The multiple connectivity of the region outside the blades can be eliminated, say, b y means of the ansformation 2i=e»'"/*, (3.27) here h is the pitch of the grid, assumed parallel to the pure imaginary axis. Figure 3.21, a and b shows le shape of a grid of " b u c k e t s " and the transformed shape. I t will be noted that the values range

b FIGURE 3 . 2 1

iver an interval from 1 :e^""*=l :10*, where d is the transverse dimension of the blades, and the transormed shape is rather unwieldy. One advance i n the 2-plane corresponding to the pitch corresponds to foing once around the origin, while the incoming flow at i » corresponds to a point source at the >rigin. The map of the blade i n the Srplane has a shape so different from streamlined shapes of a single v'ms that the methods that have been developed for mapping the exterior of a single streamlined wing ill tlie outside of a unit circle hardly apply i n this case. It is customarv to plot the flow field on the outside of the unit circle in a ea-plane so that x= — <», 5 = + 00 correspond to a point vortex source at 2 2 = — ^ > 1 > 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)
(3.37)

(see [20]). Hence conformal mapping does not transform solutions of (3.36) into solutions of this e<}uation i n the new plane, and is not as useful for solving stress problems as for potential problems,* S i m p l y connected regions may be mapped on a unit circle i n the 2i-plane. The boundary conditions for F , àF/i>n then lead to an infinite number of equations i n the Taylor series coefficients of/(2i), gizi). Conformal mapping of a sphere on a plane is of interest i n connection with the determination of harmonic fimctions which are homogeneous i n x, y, z. It is evident that if the values of such a fimction are obtained on the unit sphere, they are determined all over space. Â conformal map of the sphere on a plane is obtained by means of the Riemann construction of projecting the points of the sphere from one of its points onto a plane normal to the radius through that point. B y utilizing it, one obtains [21] the following form for harmonic functions v>o, homogeneous of degree zero: ^t>=mi{x+iy)l{R+2)l i?=(x»-hyH2^'^ (3.38) where / is analytic i n its argument. B y inversion and differentiation or by means of Euler's theorem, harmonic functions of positive and negative degree of homogeneity can be derived from (3.38). Homogeneous harmonic functions of degree zero find ready application in the study of conical flow of supersonic linearized compressible fiow. The wave equation ^«+^nr=(-M^—1)^

(3.39)

is transformed into the Laplace equation by putting z=i'jM^—lz'; and the solution (3.38) with z replaced by z' can be used to furnish velocity components and pressure solutions for conical flows. A . Busemann [22], H . J . Stewart [23], and others [24], have utilized (3.38) or its equivalent and conformal mapping to obtain flows around wing edges and 5-wings. The above conformal mapping of a sphere on a plane and the complex-number representation of this plane are also of use i n the kinematics of spherical motion and the dynamics of a rigid body with a fixed point, for instance, a gyroscope supported i n gimbal bearings. A finite displacement i n space corresponds to a linear fractional transformation of the above complex variable, while the determination of the motion corresponding to given rotation components leads to a solution of a R i c a t t i equation [25, 26]. I n conclusion, the above rather hurried and incomplete survey shows that the applications of conformal mapping to industrial problems are many and varied. While it has been successfuUy apphed to many problems, extreme difficulties have thus far hindered its successful application to many others. Conformal mapping must be viewed as but one tool i n the problem of field determination: it competes with and supplements graphical, electrolytic tank, and numerical methods. Industry will foUow with interest the development of rapid electronic digital machines and their harnessing toward the solution of field problems. * A n exception occurs for circular inversion, followed by moltipllcatioD by r*(—z?); thl« does transform solutions of (3.3fi) into new solutions of (3.35) [20].

1 H . Poritsky, The field due to two equally charged parallel couducting c^lindera. J . Math. Phys. 11» No. 3-4 ( 1 9 3 2 ) . 2 M . Abraham Berechnung des Durchgriffs von Verstörkerröhren, Archiv für Elektrotechnik 8» 42 to 45(1919, 1 9 2 0 ; . C. Maxwell, Electricity and magnetism 1, (Oxford at the Clarendon Press, England, 1873), see fig. X I I I ( L i n e s of force near a grating). 4 J . Friedrich, Die maximale Feldstärke und die Kapazität der Drehstromkabel, Z A M M 14» 295 to 310 (1934). 5 H . Poritsky^^ Stresses due to bending and stretching of a strip having parallel edges and internal circular b o u n d a r i e s . Internal General Electric Report (unpublished). Section 3. (61 H . Poritsky, Amplification constant of a grid of circular wires, Internal General Electric Report (unpublislied), Section 9. 7 H . Poritsky, Field concentration near circular conductors, Bui. A m . Math. Soc. 49* No. 6, 417 to 421 (1943). 8 E . 0 . Keller and I. A. Terry, Field pole leakage flux in salient pole dynamo electric machines, Inst, of Elec. E n g . J . , 845 to 854 (1938). [91 H . Poritskv, On the evaluation of Schwartz-Christoffel integrals in flux plotting problems, Internal General E l e c t r i c Report (unpublished). 10 F . W. Carter, The magnetic field of the dynamo electric machine, Inst, of Elec. Eng. J . 1115 to 1138 (1926). 11 H . Poritsky, Graphical field-plotting methods in engineering, Trans. A m . Inst, of Elec. E n g . S7» 727 to 732 (1938). 12 H . Poritsky, Calculation of flux distribution with saturation, A I E K Trans. 70 (1951). 13 H . Poritsky, Stress fields of axially symmetric shafts in torsion and related fields, Proc. of the Third Symposium for Applied Mathematics of the American Mathematical Society, Ann Arbor, Mich. (June 14-17, 1949) 163—186 (McGraw-Hill Book Co., 1950). 114] F . Weinig, Die Strömung um die Schaufeln von Turbomaschinnen (Johann Ambrosius Barth; Verlag, Leipzig, 1935), English translation, BuShips, Navy Dept. (1946). [15] Andrew Vazsonyi, On the aerodynamic design of axial-flow compressors and turbines, J . App. Mech. 7t, 53 t o 64 (1948). [16] C . Concordia and G . K . Carter, D . C . network analyzer determination of fluid flow pattern in a centrifugal impeller, J . App. Mech. 14, A-113 to A-118 (1947). [17] H . Poritsky, H . D . Snively, and C . R. Wylie, A numerical and graphical method of solving two-dimensional stress problems, J . App. Mech. 6» A-63 to A-66 (1939). 18 W. Emmons, The numerical solution of compressible fluid flow problems, N A C A T N - 9 3 2 . 19i H . Poritsky, Stresses and deflections of cylindrical bodies in contact with application to contact of gears and of locomotive wheels, J . App. Mech. 17, No. 2, 191-201 (June 1950). 4 H . Poritsky, Application of analytic functions to two-dimensional biharmonic analysis, Trans. A m . Math. Soc. S f , (20] No. 2, 248 to 279 (1946). 21 H . Poritsky, Homogeneous harmonic functions, Quar. App. Mech. 6, No. 4, 379 to 388 (1949). 22 A. Busemann, Infinitesimal kegelige UeberschallstrÖmung, Schriften der deutschen Akademie der Luftfahrtforschung 7B, 105 to 122 (1943). [23: H . J . Stewart, The lift of a delta wing at supersonic speeds, Quar. App. Math. 4, 246-254 (1946). 24 H . Poritsky, Linearized compressible flow, Quar. Apn. Mech. 6, No. 4, 389 to 403 (1949). 25 F . Klein and A. Sommerfeld, Über die Theorie des Kreisels (Leipzig, B . G . Teubner, 1897-1910). 26 G . Darboux, Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal (Gauthier-Villars, Paris, 1941). h

k

1

4

4*

Conformal Maps of Electric and Magnetic Fields in Transformers and Similar Apparatus G . M . Stein ^

1.

Field Problems in Trannformers

F o r an economic transmission of electric power over long distances the losses i n the line have to be k e p t small. If I, and R represent the voltage, current, and resistance of this line, the equations power transmitted = V / , ^ hnelo88e8=/*i?=(E2^y^

show that low losses are obtained by using a sufficiently high transmission voltage V. T o do this an altern a t i n g current system is used smce its transmission line can be coupled w i t h the power plant, as well as with the consumer, by transformers which boost the voltage F to a high level for transmission and then reduce it again for consumption (fig. 4.1). Such a transformer is a coil w i t h at least two TRANSMISSION TRANSFORMER

FIGURE 4 . 1 .

LINE TRANSFORMER

Electric power transmission

with transformer.

windings, referred to as the high-voltage ( H . V . ) and low-voltage (L. V . ) windings. A s shown i n fig. 4.2 for the example of a shell-tvpe construction, these windings are strung through the openings of an iron core which provides a low reluctance path for the magnetic flux linlung them. The core and coil assembly of liquid insulated transformers is enclosed i n a tank. A s part of the transmission circuit, the transformer appears as a system of reisitors (Hi, i?2), reactors (Li, I^), and capacitors (dy C j , Ct, C4,), as indicated i n the equivalent diagram figure 4.3. E a c h of these circuit elements contains an electric or magnetic field system (fig. 4.2). T h e knowledge of the field distribution is essential for computing the size of these elements. I n particidar, the capacitances Ci, C3, C3, €4, between the windings and between windings and core or ground represent the effect of electric fields. A t the same time these electric fields constitute an insulation problem since their stresses have to be kept under certain limits by proper clearances i n order to prevent an electric breakdown, particularly at the winding comers. A c t u a l l y the voltage changes along the surface of each winding. I n an investigation of electric stresses, however, only a fimited part of the field space i n the neighborhood of the corners is considered at a time, so that all conducting surfaces can be assumed to be at a constant voltage or potential. The no-load inductance Li and resistance Ri correspond to a magnetic field and its losses i n the iron core, and determine the size of the m a ^ e t i z i n g current. The core material has a high flux conductivity or permeability so that the flux remains inside the core, and its boundaries are flux lines. T h e load current / passes through an inductance £3 and a conductor resistance R2- The quantity L2 is also referred to as leakage inductance since i t represents the effect of a magnetic leakage field in and between the windings. These windings lie outside the core; that is, in a space of a very small permeI WestiDKbouse Electric Corporation, Sharon, ^a.

ability as compared with iron. Consequently the magnetic leakage flux tries to reach the core space o n the shortest route, so that the flux lines enter the iron siu'face perpendicularly; this surface is at a constant potential i n a magnetic leakage field. H.V B U S H I N G S

L.V. B U S H I N G S

TANK COVER

OPENING

MAGNETIC LEAKAGE FIELO

MAGNETIC F I E L D IN I

FIGURE 4.2.

Layout of a shell type transformer.

Additional field problems are created where the coil leads pass close to the tank wall and through the tank cover. The high-voltage coil leads usually represent an electric field problem because of their high electric potential, whilst the low-voltage coil leads frequently carry heavy currents the magnetic field of which is reflected i n a lead inductance and i n lead losses. The current distribution i n large c o i l leads oflPers another field problem, especially at the junction point of several conductors. WINDING L.V.

H.V.

^2

"2

'TOO'O'^—WW

1

GROUND

FIGURE 4.3.

EquivalejU impedance diagram

of a transformer,

A device, very closely related to the transformer, is the iron core reactor used for paralleling purposes or for limiting the size of the line cm-rents. T h i s reactor consists of one main winding and of an iron core with a number of air gaps (fie. 4.4a). These gaps contain a m a ^ e t i c field which is quite distorted at their entrance because of the sharp corners. The knowledge of this field distribution is essential for obtaining the magnetic reluctance o f the gaps and the reactance of the apparatus. T h e pole surfaces are at an equal magnetic potential like the iron surface i n a m a ^ e t i c leakage field. T h e excitation of each gap field can be represented by free magnetic charges which are located on those pole surfaces where they form a uniform double stratum [13].* The effect of each double stratum can be replaced b y 'Figures In brackets Indicate the literature references at the end of tbis paper.

a vortex line, that is, b y a current carrying-conductor, located at the ends (fig. 4.4b). These conductors f o r m an equivalent single layer winding which can be used to represent the reactor by a n equivalent two-winding transformer. T h e magnetic field between its equivalent winding and the main winding h a s the same character as the magnetic leakage field i n transformers. IRON

CORE

IRON

CORE

±±±

AIR G A P S MAIN O

WINDING

M A I N WINDING

REACTOR

b

FIGURE 4 . 4 .

EQUIVALENT

EQUIVALENT WINDING TRANSFORMER

Iron core reactor.

These field problems can be simplified b y assuming uniform conditions i n one direction. T h e n the field lines wiU stay i n planes so that each field analysis is reduced to a two-dimensional calculation.

2.

Some General Rules of Field Mapping by the Use of Complex Functions as Solutions of the Field Equations

The proper mathematical method for the mapping of these fields can be selected on the basis of their general ec|uations. Consider first the electric field between the windings (fig. 4.2). This field is fully described b y the vector of the electric field strength and flux density D. They are, i n turn, related according to the material property or dielectric constant k of the field medium, b y (4.1)

D=kE.

This field is excited b y electric charges which are located on the conducting surfaces and which are sources in the field. The per unit source-strength p furnishes inside these sources the divergence relation (4.2)

div D=4xp.

If the reaction of the pulsating magnetic leakage field upon the electric field is neglected, the electric field can be considered as irrotational, requiring curi E=0.

(4.3)

F o r this reason E can be derived from a scalar potential function ^ as E

grad ip.

(4.4)

Since k is mostly invariable, we obtain, from eq (4.1) to (4.4), div (grad
Arp/k.

(4.5)

The magnetic field i n the iron core and the magnetic leakage field are described by the vector of field strength H and flux density B. If n represents the flux conductivity or permeability of the field medium, we write B==^H

(4.6)

The field excitation is represented by the winding currents, which become vortices i n a magnetic field. The current density or vortex strength c furnishes, therefore, inside the windings, the relation curl H=4cirC.

(4.7)

The absence of any true charges m a magnetic field requires div

B=0.

Consequently i t is possible to introduce a vector function A which complies with the identity div(curl A)=Oy (4.9)

B=curl A and which is usually referred to as the vector potential. which may also be written

Equations (4,7) through (4.9) furnish t h e n

curl(curl i4)=4irCM,

(4.10)

curl(curl i4)=grad(div A)—AA,

(4.11)

According to eq (4.9), any sources i n the vector-field of A do not influence the amount of B, so that t h e s e sources can be omitted with div ^ = 0 , whereby eq (4.10) and (4.11) result i n (4.12)

A i 4 = —4TrCM.

The field eq (4,1) through (4.12) apply i n general to three dimensions i n an (x,i/,2)-coordinate system. A two-dimensional field problem i n an (a;,y)-plane reduces the vector-potential to its s - c o m ponent A^, which becomes identical with the scalar flux function ^{x,y) of the field according to (4-13)

while the corresponding scalar potential function ip{x,y) has the same character as i n three dimensions. Under a flux conductivity ¿ = 1 , or m = 1 , eq (4.5) and (4.12) are, therefore, reduced i n two dimensions, that is, for a parallel plane field, to the equations of Poisson: Av9= — 47rp

(no

yf)

inside sources, (4.14)

A^=

—4irc

(no ip) inside vortices,

inside an excitation; while the equations of Ijaplace. A^=0,

A^=0,

(4.15)

represent the conditions outside where the same field may be described b y ^ as well as by ^. Solutions of these Laplace equations are the particular integrals Z=x-\-iy; 'Z=x—iy, w i t h i= ± V ~ l » and any functions/(Z), g(Z)j known as complex functions. They furnish tp and if/ as components of J(Z) =
H=H^+iHy,

and find, according to eq (4.6), (4.9), and (4.13), H^=-

^il^ldy,

Hy = diA/dx.

(4.18)

Since eq (4.7) is reduced outside the vortex excitation to curlir=0,

(4.19)

we can represent, analogous to eq 4.4, fl=—grad

tp,

H^=—à

Hy=

—

à
(4.20)

S y substituting from eq (4.18) and (4.20) into (4.16), we find that dZ

bx

ÖX

ÖX

a n d conclude further that the real and imaginary components v> and ^ of the complex functions /(Z) a n d g(Z) comply w i t h the Cauchy-Riemann equations (4.22) Consequently only those complex f\mctions are considered which are analytic, and only an irrotat i o n a l and sourcdess section of the field is described by them, although an excitation, contracted into points, appears as poles i n the complex function. Because of the synmietrical character of

BOUNDARY

FiauBB 4.5.

General boundary cortditions.

the two media, and let i f i , H2 be two adjacent boundary values of H w i t h normal and tangential components Hi,nt Hi,tt and fla.», I/2.1. T h e n eq (4.19) reqmres that ^1. f =^j. tt

(4.23)

while eq (4.6) and (4.8) are complied w i t h for (4.24) W e let ai and »2 be the angles between Hu H2 and the normal of the boundary, and find the general boundary conditions ^ = i i l . tan » 2 M2

(4.26)

I n the particular case when jut is ^^ry small or very large compared with the angles at or a2 become zero or 90*^, so that the boundary represents a flux line or an equipotentiai fine i n the field on one side. Suppose that a region w i t h such equipotentiai,or flux fine boundaries const, or ^=const, is given i n a IF-plane, and that the corresponding field function FiW)=
M9681—52

35

has been found. O n the other hand any function W{Z) coordinates those boundaries i n the W - p l a n e to another set of boundaries of different shape i n a Z-plane, where the field is now given by J{Z)=F[W(Z)]=
Consequently equipotential and flux line boundaries retain their character i n a transformation W{Z). Such a conformal mapping may, therefore, reduce the field problem i n a region w i t h c e r t a i n boundaries to the analysis of a field w i t h other and usually simpler boundary forms, particidarly i f these boundaries are equipotential or flux lines. This technique has also been applied by L a b u s [1] to boimdaries w i t h a variable potential. The simplest examples of equipotential or flux line boundaries are the straight line or the circle since here the solution is obtained by the principle of excitation images (fig. 4.6). Use signs opposite to the originals for images of sources q i n a medium of a small flux conductivity m and for images o f vortices when m is large. A p p l y the same sign if the two conductivities are interchanged. T h i s l a s t case is confined to problems i n which the resultant excitation of each region is zero, since otherwise the boundaries wiU not coincide w i t h a constant potential or flux. Z

FIGURE 4 . 6 .

SoliUion of field problems by images.

FIGURE 4.7.

PLANE

The excitation of muUiply connected regions.

Depending on the presence of one or several closed boundary lines, a region is simply or multiply connected (flg. 4.7). This discussion pays particular attention to those re^ons whose boundaries are equipotential or flux fines, and whose excitation lies outside i n n openings created by n boundaries. Such regions are distinguished either by a small flux conductivity ^ and equipotentid boundaries, when excited Dy sources, or b y a lai^e ju and flux fine boimdaries under a vortex excitation. The resultant excitation must be finite i n each of at least two openings if the field function /(Z) of the region shall not be a constant. I n general, this function is independent of the location of the excitation i n each opening. The field function/(Z) of an s-times connected region excited by a, /3, 7, . . ., X, . , ., p, ocan be linearly composed of s-1 particular solutions / i , / * , . . ., w i t h excitations a^, . . ., X*, . . ., Pky <^kj by writing /=2<^-tA+const.,

(4.26)

where the constants (7* are derived from the condition a+

. . . +X+

. . . +p + c = 0

(4.27)

/or the resultant excitation in each particular field. Consequently, /(Z) can be divided i n s-1 fields i « . • • i/x, . . .,/p, where/x is excited by + X located i n the corresponding opening, and by — X , located fn place of
ot

P

introduced into eq (4.26), and furnishes

X

Xjt+i

Pt-HI

.

a,_i

ai

X,_i

Xi

Xjt_ 1

P*-i

PI

pi-1

Oft-I

a*

Xjt

.

Xf_x

Xi

P»-1

Pi

.

(4.29)

Xjt_

P*-i

T h e quantities a, . . X, . . ., p,(r and so on, may also represent the difference between the boundary potentials or fluxes if Ct of eq (4.29) is derived by introducing the values of/* on the individual boundaries into eq (4.26). A s a result of eq (4.26), we find that y = const, i n a simply connected region. const, in a doubly connected region, y=C\/i-t-Cy2-|-const, i n a triply connected region, 80 that the field is of zero strength i n a simply connected region, has only one pattern if the space is doubly connected, but may assume an infinite number of patterns i n a triply or more highly connected domain.

3.

Simplications in the Mapping of Rectangular Frames

This group of field problems includes the mapping of doubly and triply connected rectangidar frames (fig. 4.8). The engineer may consider such models as the magnetic field space i n an iron core or as the electric field space between winding and core i n a transformer (as marked i n fig. 4.2). These problems are solved by converting the frames into simply connected convex polygons and by applying the technique developed by H . A . Schwarz i n order to map such polygons i n a Z-plane, Z = x + i y , onto a half-plane (not shown) and from there onto another polygon i n a ¿-plañe t=ip-\-vlf where the field becomes parallel. F o r this purpose, the doubly connected frame with boimdary potentials or fluxes Ay B has to be cut once, and the triply connected frame with boundaries A^ B , C nas to be cut twice along field lines. These field lines are split by cuts at (I, II) and (III, I V ) , which have to be straight if a polygon shall be obtained by these cuts; consequently the frame must be symmetrical to those field lines, and its analysis can be confined to one-half region on either side of the line P-Q. If the frame is not symmetrical, or if the corners are rounded, the mapping may have to be carried out in another way, for example, by the method of S, Bergman, using orthogonal functions and his kernel function [14, 15, 16]. I n particular, the doubly connected frame is mapped on a rectangle, and the triply connected frame is coordinated to a slit rectangular polygon, as shown i n figure 4.8 for boundaiy potentials or fluxes A<^C<^B, A peculiarity of the triply connected space is that the crossing points H a n d Kgo over into the end of the slits. The transformations of both frames have been investigated by S. Bergman [2, 3], who has also made a field plot for a doubly connected frame (fig. 4.17).

Such calculations for the mapping of a whole frame structure are very Interesting from a mathematical standpoint, and are usefu to the engineer for examining the accuracy of simpler solutions. In order to obtain practical technical formulas, the frame must usually be divided into several regions which are analyzed separately. E a c h region should have no more than two vertices with right angles if the mapping shall be described by single periodic functions. Such models are the corners * * X " o r the " T " - s h a p e d spaces between the three branches of a triply connected frame, and are obtained b y neglecting the influence of all vertices outside,—that is, by extending the boundary lines up to infinity. T h e polygon in the parallel field of the t-plane degenerates into a simple strip in the mapping of ' ' Z " and into a slit strip in the mapping of " T " as already mentioned by Labus [1]. A n analysis of the space " contains the analysis of the corner " X " and many other models as particular solutionsj and s h o u l d , therefore, be considered as a fiindamontal problem. Instead of mapping the resultant field, we may also find the flux distribution in a triply connected frame by a transformation of its two symmetrical components with boundary potentials or fluxes A=B

.il// Z-PLANE

2-PLANE

t-PLANE

[_^,

t-

PLANE

j

9

B

C - 4 ( A+B )

0_ _C._4(A+B)-[-

' ' !

J-J-L FIGURE 4 . 8 .

Transfor7nation

of rectangular

FIGURE 4 . 9 .

fraines.

Transformation of component fields in a connected frame.

triply

and a {y2){A-\-B), (fig. 4 . 9 ) . In each case the frame is mapped upon a slit polygon of different shape. For a VÁ)iA B), in particular, this polygon becomes a doubly connected rectanglo with a slit TI-K in its center. If the corresponding frame in the Z-plane is symmetrical to H-K, it is identical with two adjacent doubly connected rectangular frames, so that the doubly connected model is a particular case of the triply connected frame. Similar to the mapping of rectangular frames is the field problem of a periodic series of gaps (fig. 4.10). This model may be a portion of a magnet core, or may represent a pancake form of electric coils.

m

n

Z - PLA

FIGURE 4 . 1 0 .

t-PLANE

Transformation

of a periodic

series of gaps.

F o r reasons of symmetry, only a strip between two adjacent gap centers has to be considered, as shown i n figure 4.10 for the example A<^C<^BThis space can be interpreted i n a Z-plane as a triply connected

p o l y g o n a l r ^ i o n , and is converted into a simply connected space by cuts at infinity ( » ) and at a s y m m e t n c a l fieldline split into I and II. T h e mapping upon the i-plane produces then a sUt polygonal strip region. B y moving one gap end to infinity, we reduce the model i n the Z-plane to a fork-shaped space " 1 ^ ' to b e mapped upon a slit strip. This space **F** is the counterpart of the region " 7 " ^ and is fundamental f o r another line of simpler models. A n analysis of these spaces " T " and " F ^ ' is found i n papers of L . Dreyfus [9], J . Labus [1], E . Weber [12], and others, but has not been carried through yet to a general solution.

4.

The Mapping of the Fundamental Models

I n order to find such a general transformation for the " T " - s h a p e d space, consider this region (fig. 4 . 1 1 ) as a five-angular polygon w i t h two rectangular vertices O i , O t , and with three vertices 1 ) 2 , 0 4 , ft, T b

i

Z - P L A N E

O2

J

=7r

W-PLANE

t-PLANE

FiouRB 4.11.

Transformaiian

of the " T " shaped 9pac$,

which have zero angles and are located at infinitjr. This polygon " J T " i n a Z-plane, Z=x+iyf is first mapped on the upper half of the TF-plane, W—u-{-iVy and then on a slit strip i n a <-plane, t=^-|-tj^ where the region can also be interpreted as a four a n ^ a r polygon w i t h vertices Oi, O ^ , 0$, and K. The two polygon transformations are carried out according to the principle of Schwarz [4] and Christoffel, Consequently, the mapping from the Z-plane upon the H^-plane is represented by (4.30) where the Up locate the branch points i n the IF-plane, and the e, represent the supi>lements of the polygon angles i n the Z-plane, while A , is a dimensional constant. Since three branch points ( O i , O 3 , ft) may be selected freely at « , = + 1, » , 0, we write instead of (4.30), dZ dW

A,

(4.31)

T h e coordinates - U 4 of the remaining branch points Ot and O4, and the constant A , , are derived f r o m the dimensions a, 6, c of the polygon b y developing the transformation (4.31) i n the neighboriiood o f t h e vertices Oj, O4, and Ot, where the polygon can be approximated b y simple strips. T h e transfonnations o f these strips upon the W-plane become the logarithmic functions Z ( T F - * < » ) « ^ l o g W+const., Z(W->0)«^.i^log

at O2

H^+const.,

atOj

(4.32)

«4

« 4 ) as

,

Agi

V(«4+l)(tt3—U4)

log (W-|-ii4)-|-const.,

at O 4

«4

W h e n the strips i n the Z-plane are crossed from b o u n d a i r to boundary, the correspondin^branch p o i n t is circled by a n angle ± x m the IF-plane, — t h a t is, we add a factor e*^' to W or W+U4. B y comparing the corresponding changes on both sides of (4.32), we find that A,

ia/x,

Uz=(u4clay,

«4 =

m +

V ( « / c ) ' + m

(4.33)

m=(l/2)[(a/c)»+(6/

A n integration of (4.31) furnishes then Z

i(2/v)\a

tan--(-^^:pJr)j,

tanh"'T—c t a n - K V ^ r ) + 6

W - l

(4.34)

T h e same method is used for the mapping of the f-plane upon the TT-plane, where « ^ = - | - l , 00, « 4 , —UK locate the correspondmg branch points. Consequently we write (4.35) M a k i n g a development into logarithmic functions i n the neighborhood of O2 and Os, we let a, j3, y be t h e dimensions of the slit strip i n the ^planc, that is, the potential, or flux, differences between the three boundaries, and find, because a+P+y=0, that A,

air.

(7/a)ti4

(4.36)

Under this condition, an integration of eq (4.35) furnishes f=l

logPF+^

l^gTj?)

(4.37)

Z- PLANE

Iv

FiQURB 4.12.

W-PLANE

TramfoTmotion of the fork shaped ap

F o r a corresponding transformation of the fork-shaped space (fig. 4.12) consider its region " F ' ' , as a five-angular polygon, analogous to **T\ but w i t h rectangular vertices at Oi and 0». T h e procedure to be

followed i n its mapping upon a slit strip distinguishes itself, therefore, from that of "T" only b y a different coordination of the boundary points i n the three planes. Consequently, the Schwarz transformation between the Z - and W-planes becomes, according to (4.30),

dW

(4.38)

W(w-u;)

a n d furnishes, b y a development into logarithnuc functions i n the neighborhood of the points Oj, ft, O4, A,

i{a/T\

(4.39)

n=l/2[(a/6)»-(c/6)^+l]

Us=(Ut hi of,

so that the integral of (4.38) can be brought into the for Z

i(2/]r)|a t a n h " * r — b t a n h ' * {'^T)+C

«5—«4

t a n h " ' ^-^^

I W - l

(4.40)

Likewise, we find the transformation from W into t fro: (4.41) and compute, b y developing i n the neighborhood of ft and Os, (4.42)

A^

to obtain, as integral of (4.41), t

/3 log

W+y

(4.43)

T h i s general form does not furnish Z exphcitly as a function of t Consequently, a coordination between points i n the Z - and ^planes has to be established by the help of maps. Such maps may, for example, represent the transformation of the rectangular coordinate system «=const., t J = c o n s t , of the W-plane upon the Z - and ^-planes. However, a direct analytic coordination between these two planes is obtained if the eq (4.37) and (4.43) for t(W) can be solved explicitly for Wit), which may be substituted i n (4.34) and (4.40). Such solutions are found, for example, if two boundaries are at the same potential ( a = 0 ; or j3=0; or 'y=0), or if two potential differences between boundaries become alike (a=)3, or ^=7, or y=a); that is, if the space i n the <-plane is reduced to a simple strip or to a strip slit i n itd center. These special cases appear i n m a n y technical applications. If no detailed field plot is required for the solution of an engineering problem, an analytic answer may be found also for a general set of boundary potentials or fluxes i n the spaces " T " and **F*\

5.

Electric Current Flow and Losses in a Junction of Conductors

A s an example of such an application, the spaces " T " and '*F" shall represent, as shown i n H^ure 4.13, the jimction of three current-carrying electric conductors of the same thickness k and resistivity p for a determination of their losses P. The calculation can be confined to a space defined by the coordinates s/, ^ i n " r * ' and if the field is sufficiently uniform at the space limits so that the losses outside the space may be foimd i n the conventional way. The corresponding region i n the sht strip is designated by the potential drops A , C from point K, If the dimensions a, y of this strip represent the three currents, the voltage drops i n the conductor resistances become (p/h)(pf so that i n a field of direct currents or of single-phase alternating currents, the junction loss P can be written as P={plh){aA+pB+yC).

(4.44)

Three-phase currents should be separated into two sets of components a i ^ a i , and aifiiyt, displaced b y 90 electrical degrees. Their losses P i and P3 are computed individually from the corresponding fielddistribution a n d furnish the residtant three-phase jimction losses as P=Pi+P2.

F o r the solution of these problems, the limits of the jimction are chosen i n one of the three planes, and the corresponding limits i n the other two are derived from the mapping functions. I n the general

I

T I

I

\

T SINGLE-PHASE LOSS

/9

3i

i

P-( ^)(0íA+^B+yC) THREE-PHASE

'y

LOSS

P - P.+ P.

(p^

(P.)

- 1 1 .

FiGUBE 4.13.

Losses in a junction of electric cotiductors,

case of three different currents a, j8, y, this selection is preferably made i n the W^-plane so that the corresponding values of A and j^, and so on, can be derived from (4.34) and (4.37), or (4.40) and (4.43) if the origin of the ^, v?-plane is transferred to iiC b y writing t

for'T",

„ M Í , . o . ( - " S + í . o , [ - | ( 5 + .

(4.45) t

(1/T) j p l o g ( - |

U4

+ 7 .0.

w

(«4

1

for^i?^".

Under the condition that one current is zero or that two currents are alike, we start w i t h a selection of limits A , B, C from which the dimensions s/, ' ^ c a n be computed expUcitly.

6.

Magnetic Fields and Losses in Transformer Cores

A s a second problem consider the magnetic field analysis of transformer cores as mentioned i n connection w i t h figure 4.2. T h i s analysis is a typical example of how the maps of simplified spaces, l i k e " T " and " i " , can be used to get results as accurate and general as by the transformation of a complete frame. I n this particular case, the knowledge of the field distribution is required to find the magnetizing currents and losses supplied to a transformer under no load conditions corresponding to its inductance L i , Za, and resistance i?2, appearing i n figure 4.3. F o r obtaining the magnetizing current, that is, the magnetomotive force i n the core, the field distribution should be computed for a variable permeability /x, using, for example, the method suggested recently by H . Poritsky [5]. Loss calculations, however, may be based on a constant MJ as found by their comparison w i t h test results [6]. If, i n addition to this, the core material is normal silicon steel, the per unit losses ©increase about as the square of the magnetic flux density B, that is, pccB^. Then the total losses P depend on the core fluxes and potential drops
{kXM.

(4.46)

where tpk is merely the conjugate value of i^* for m = 1, and is not identical w i t h the actual magnetomotive force. A s an example, consider the single-phase core-type core shown i n figure 4.14, and divide its space into legs, vokes, and comers. If a is its flux per unit build-up A, and if <pr is its resultant potential drop, its losses become i4.47) P = c o n s t , aiprF o r computing P , let p indicate the per unit loss in the uniformly magnetized leg center, let D be the density of the core material, and introduce an effective core weight G^" and l e n g t h / " , also referred to as the core-loss-weight and core-less-length [6], by writing

(4.48)

G=V'ahD,

where

is given as (4.49)

T h e calculation of P is thereby identified with the determination of the potential drop ipr for any chosen value of a, which will be a = i r in this analysis. B y subtracting a potential drop, obtained in legs

CORNER

i

b>o

OH

^«0

ST

iv

M

-it

W-PLANE

i

7

M I*

N' t-PLANE

FIGURE 4 . 1 4 .

Simplified modtls of a transformer core,

and yokes under the assumption of a uniform field, we reduce the computation of ipr to that of a potential drop ipc which depends only on the dimensions a and c of a single comer, and furnishes (4.50) T h e length e of most core legs is relatively large as compared with the width, so that the field is ractically uniform for y = — a . Consequently, shall be derived from a potential drop ipt on the center ux line ^=ir/2 of an X-shaped space extending into the adjacent leg and yoke up to y = —a and x = — ; that is, we write ¿¡2' (4.51) 1 +

g

The field in the space " Z " may be derived from the results obtained for the space " by substituting the conditions 6=0, ¿ = 0 , or 7 = — a into (4.33), (4.34), and (4.37). We eliminate separate the term for Z into its real and imaginary components, introduce the so-called yoke widening factor X=c/a, and find the a:,y-coordinates of the center flux line ^=TI2 as functions of tp in the form [6]

( i M J a tan

_ i 2r sin g

*

u - i 2(r/X)8in 5)

1

(4.52) y

/I / J

*

-1 2(r/X) cos 5

.

, _, 2r cos 5)

where cosh

a = 1/2 { T - t a n - ' t f * - tan-H^""^*')}

(ip+ifiij)

(4.53)

W e then obtain ia)

d 2a

(4.54)

T h e accuracy of this formula shall be checked with the corresponding results obtained for two whole upper frames (fig. 4.14) one of which has a finite opening width rf, but an infinite yoke height, c= «>, while the opening of the other frame is reduced to a sht, rf=0, whereby c remains finite. T h e frame with c = « represents a fork-shaped space with two equal channels, c=:a, and potentials |3=0, 7 = — a . W e introduce these values into (4.39), (4.40), (4.42), and (4.43), performing the mapping

of the space " F " , transfer the origin i n the Z - and i-planes into the point N; that is, we replace Z Z+d/2 and t by ¿+¥»5/2, and find [6], for « = — « • , Z

.2 i-~|tanh

2 a /cosh i 6 1 tanh" 6 V cosh t— 1 2a

cosh m cosh t~ 1

m=cosh

W2)=2

a

y

-

by

(4.55)

T h e potential drop
L

(4.56)

T h e field in the slit frame with d=0 is transferred from a Z-plane into a parallel field i n a s i m p l e strip of width r in a ^-plane via a IF-plane which contains a slit on its ordinate. These two transformations are performed [6] by W = s i n [ir(Z-c)/2a], (4.57) cosh [
W=~is

and furnish for

s = s i n h (\ir)/2), '

^=0, ia)=2

cosh -1

sin [(l + X)^r/2]' L sin (X7r/2) J

(4.58)

A comparison of ipL in the " i " space (fig. 4,14) with the two half-frame models is made in figiu-e 4.15 b y plotting
SLIT SPACE

r 11 d>o

FIGURE 4 . 1 5 .

Potential

drop in different core models.

width d/a. In the comparison of the spaces " i " and " F " , the error in the Z-space calculation appears to be practically independent of d/a, as shown by the curves (1) for X = 0 0 . T h i s discrepancy is -1-0.42 potential units, except for the smallest values of d/a. W i t h a reduction of X, however, as given by a comparison of the slit space with the space " i " , the error decreases and reaches zero at X=0.75 according to the curves (2) for d=0. T h e curve (2) of the Z-space can also be interpreted to represent only the portion <po of np to the y-axis in a frame of finite d/a. In particular, we find ^o=4.80 at d/a=0 and X=0.75, as weU as for X = 00 and any rf/a, if the correction —0.42 is applied to ipo alone. This result suggests making the correction of
(4.59)

has been derived in figure 4.16 from a potential chart of
tr.

o o o

uf

O

I-

t 3 - 4 YOKE WIDENING FACTOR X

1 1

YOKE (POTENTIAL CHART OF L-SPACE

FIGURE 4 . 1 6 .

Derivation

of the corner potential

drop,

of different center flux lines belonging to X between 0.8 and °D. E a c h curve furnishes a quantity tp' whose plot against X can be approximated by a straight line / = 0 . 2 5 (X-0.9). W e substitute now the approximation 4.8 for
(4.60)

so that
Q U A R T E R L -

S P A C E

( A c c o r d i n g

V>^= 1 5 - 8 4 FIGURE 4 . 1 7 .

Approximate

Bi and 7?2 of the same absohitc amounts the main axes [8]

y> =

and actual fields in a doubly connected

and Z^al-

to

F R A M E S.

B e r g m a n n )

16.24

frame.

A t all other points, this locus is an. ellipse with

5, B

(4.61)

where Bx\, ^jz, By^ represent the components of Bi and ffg. Those two vectors can be interpreted as the gradients of the corresponding potential function ^(x, y) and arc, therefore, derived according 1*1 2

FIGURE 4 . 1 8 .

Components

Ot,-0

of a three phase field.

to (4.21) from the corresponding transformation between the Z-, W-, and i-planes (fig. 4.11) that is, from (4.31), (4.33), (4.35), and (4.36), in the general form

dz

dt

I dZ

dW

dW

(4.62)

where and are given b y (4.33). It has been found by test results (81 that the losses P under such a rotating magnetization of the " T*' shaped space are equal to the sum of the losses P i and Ps i n the two component fields, as i n the electric current flow (fig. 4.13). Consequently, the determination of the three-phase losses is reduced to that of two single-phase fields where the losses are given b y the corresponding potential drops according to (4.46), and can, therefore, be represented as functions of the transformer dimensions [6].

7.

Electric Field Concentration at the Corners of Transformer Coils

F o r discussing the analysis of the electric field of figure 4.2, the winding section of a shell type power transformer is shown i n greater detafl i n figure 4.19. Consequently the region i n the neighborhood of

L.V. WINDING CORE SURFACE

HIGH VOLTAGE

OPENIN

IXXXXXAXX)

xxxxxxxx>

V y X X X y y V5 X X X X X X X X )

FiQUBE 4.19.

Y Y y KXX

L.V. WINDING

Winding section of a shell type power transformer.

the comers Ai and A2 can be considered as an " i " - s h a p e d space, the area between .¿4.3, .¿44, and the core surface becomes a " T^-shaped space, while the region between the lines m and n , including the comers .¿45 and Acf but neglecting the iron surface, appears as a fork-shaped space " P " . These spaces represent, therefore, the clearances between protruding comers of adjacent coils and between winding and core. I t is important to keep the strength of the field i n the neighborhood of such comers under the limit at which undesirable corona dischai^es take place and a break-down becomes likely to occur between conducting surfaces. The voltage stresses at which such limits are reached depend on the coil clearances. A fundamental method for deriving this relationship from the field distribution has been suggested by L . Dreyfus [9], who uses his procedure and the principle of Schwarz [ 4 ] and Christoffel for analyzing a large number of polygonal models with two boundary potentials. Dreyfus takes mto account the fact that the comers of the coils are i n practice slightly rounded, that is, that they never represent the sharp edges of the mathematical model where the field strength becomes infinite. H e defines, therefore, the field concentration at such a comer by the field-strength £ at a point P at a chosen distance s from the theoretical vertex (fig. 4 . 2 0 ) or by the voltage drop between the vertex and P . These two definitions are identical if further, as Dreyfus assumes, the field-configuration i n the neighborhood of each vertex is independent of the main space dimensions. The relationship between £^and s can, therefore, be found for a given model and for given voltage stresses b y developing

the field strength E i n that neighborhood as function of the location of P, Vice versa, this result is then interpreted as a relationship between stresses and clearances for a given field-concentration. Under the assumptions of Dreyfus, i t is possible to represent this concentration b y a single quantity. F o r this purpose, consider the field distribution outside a single comer w i t h the vertex 0, w i t h an angle 5 and supplement € = i r — 5 i n a Z-plane (fig. 4.20). This distribution is obtained b y its m a p -

Z-PLANE

W-PLANE

PLANE

Iv

0

CO

CD

Field ccnceniration at electric windings.

FiouBE 4.20.

ping upon a parallel field i n a f-plane whereby ZQ and ^0 represent the location of the vertex i n those planes. T h e principle of Schwarz [4] and Christoffel furnishes, for this condition, dZ

A{t-
dt

- i / r

(4.63)

where A is a dimensional constant. If

(dt/dZ) for

t=ip+irp.

(4.64)

W e find E and the distance Z—Z^ between a field point P and the vertex 0] as functions of i~-<po b y (4.63), eliminate i—ipo, substitute Z — Z Q = S , and obtain a constant quantity C=

(4.65)

E\l86]-*^*=(A'ir')

This quantity (7 is a function of \E\, 8, and ¿, but is independent of the location of the field point P , and can, therefore, be considered as a measure for the field concentration at a given vertex. T h e permissible limit for this concentration C should be obtained from test results as function of the vertex angle 5. F o r the example of rectangular comers ( 5 = 3 T / 2 , e = — i r / 2 ) , as found i n the spaces " 7 ^ ' a n d " P ' ' , (4.65) is reduced to (4.66)

C=\E\i3ir8l2y'\

I n a space between several conducting surfaces and comers, the concentration C becomes a t each vertex a function of the voltage stresses and clearances. F o r the example of a " T " - s h a p e d space w i t h stresses a, 0, 7, and clearances o, 6, c, as shown i n figure 4.11 and 4.20, we combine (4.31), (4.33), (4.35), (4.36), and (4.64) into p_

.

dt/dW

1

aW-yU4

(4^^^^

dZ/dW

analogous to (4.62) for the flux density B, We let Ci and Ct be the concentrations at the vertices Oi and O 3 and find Ci b y developing £^and Z of (4.31), (4.33), and (4.67) m the neighborhood of O i , that is, for Z o = 0 and W->1, mto E

1^ a—7U4

1

for z «

IT 3

I+U4

(4.68)

W e eliminate W, and find, for |Z|=s, b y substituting C from (4.66), (7i=(a-TU4) (a)-»/3 [(1 +tfa) (1

(4.69)

+uiJ\-''\

where u% and Ui represent the locations of the branch points i n the W-plane and are given as functions o f the clearances a, 6, c b y (4.33). Since the solution of the integrals for the mapping functions Z{W) and t{W) does not enter this d e r i v a t i o n , the determination of those branch points is the main problem i n finding the field concent r a t i o n . Graphical or numerical methods may be required for this purpose i n more elaborate models. I n order to obtain the field concentration Ci at the vertex 0», we develop E and Z i n its neighborh o o d , and derive (4.70) A corresponding analysis can be made for the fork-shaped space " P " , figure 4.12, i n order to obtain t h e concentrations C\ and at its vertices 0\ and 0^ as functions of the stresses a, /9, 7, and clearances a , ft, c. I n this calculation we derive, from (4.38), (4.39), (4.41), (4.42), and (4.64), the relation .1

E

(4.71)

a n d find

Cfi=(aU5+i3u4) ( a ) - » « [u^u.-u^^

C7,=(a+0ttO (a)-^'3 [(1 - u O ( I - u , ) \ - 1 / 8

(1 - u ^ ] - ' ^

(4.72)

where U4 and are given as functions of the clearances a, 6, c b y (4.39). A number of special conditions for the dimensions a, 6, c, or for the stresses oc, 7, i n the spaces " T " and " i ^ ' , are of particular interest. I n the case of a symmetrical space " T " , that is, for 6 = c i n figures 4.11 and 4.20, we find i n place of (4.69) and (4.70) the relations C

a;

(4.73)

} = » " { { V ^ ' - ^ ] - X f ) ! ' « ' ' [ ' + © • ]

which may be further reduced to 1 1 a ^+(26)»

C|=(?3=0.79 a C,=

Cz=Q{h) ~t/3

= a

0.793 a ( a ) -

1/3

for/3

-1/«

for 0=0,

J

[1

a/2,

+(^)']"'"

y

(4.74)

1

for

^=0.^

We likewise obtain from (4.72) i n the case of a symmetrical space " F " , figure 4.12, that is, for o = c , (4.75)

c, which may be further reduced to C, = C6=0.793 /3

C,=

C,=

1

1

L(2ay

for a = 7

6U

0/2,

-1/6

a(a) -2/3

0.793 p(b)-^^^

1/3

fori3=0. b 2a

-i/fl 1

6 2a

(4.76)

6 1± 2a

for

a=0

w

A number of models with only two bomidaries can be interpreted as special cases of the spaces "T" and 'T', as shown i n figm-es 4.21, I to V I I . If (4.33), (4.69), and (4.70) are applied to the modifications of the space "T", we obtain 1/3

a) ai)

a

for 6 = 0 ,

1=0.793 a[a^+e^±e^a^+e^]~^'\ 0.793 a(a)-«,

3

aii)

,

<7i=a(a)"*",

for b-\-€

for 6 = c = « ;

for c=e=

»

I

T i

c

T

0,

I I

0

•-a

(4.77)

e=0.

1

c

00

1°'

b

0,

I

—

]

-

en

r 1

i

...

k

t 0.

*0

0,

HE

3

Pol

0

T

i

T-O, Q—

— I

, .

J)

0

Coil surfaces related to forms "T"

F I G U R E 4.21.

and

"F."

I n the group of models I V to V I I , figure 4.21, which are modifications of the region F, the spAces V I and V I I are obtained by a refraction of the regions I V and V on the lines m-m and n-n, respectively, whereby the dimensions a, b and c are doubled for the same field configuration, and symmetrical pairs of vertices 0| and O5 are obtained. Consequently, (4.39) and (4.72) are reduced to

av) (VI)

1

1

a

b

C . = 0.793 a

1/3

1

1

a

b

1/3

y

forc=0. (4.78)

(V)

I^'1=0.793

ale^-a^] - 1 / «

for c (VII)

00

00

0.63 a[c2-a^-^/« [ e ^ F V ' e ' - a ' ] " ' ' ' '

Another model, related to the region F , is the series of coils, shown i n figure 4.10, which is obtained from F and a repeated refraction, on the lines m-m and n-n.

8. Magnetic Fields of Air Caps and the Fringing Coefficient This series of boundaries may also represent a magnet core with a number of au* gaps, as shown, for example, i n figures 4.4a and 4.22 for iron core reactors. The problem of finding their reactance is identical with the determination of the linkage between the cod-section and the total flux. F o r this purpose, consider the difference between the actual field and a paraUel field, corresponding outside the gaps to the diagram of an equivalent transformer figure 4b where the effect of the gaps is replaced by that of. an equivalent single layer coil. This difference accounts for the field distortion due to the corners Oi at the gap entrances, and shall be referred to as the fringing field. The flux density in the parallel

field increases, as shown by the dotted line in figure 4.22, linearly in the coil up to a constant value B Q between coil and iron surface, where it rises abruptly to another constant amount, BJ>la, assigned to the inside of the gaps. T h e flux linkage between this parallel field and the coil is found in the usual elementary way [10], like the leakage inductance of a two winding transfonner.

i'lGURE 4.22.

Magnetic Jieid of air gaps.

T h e corresponding linkage of the frinHng flux becomes an intcCTal over the winding section, as far as the fringing field penetrates into the coil. T h i s integration shalll>e left to future developments; that is, the problem shall be confined to those cases where the linkage of the fringing field inside the coil is negligibly small, so that the determination of the linkage with tnc total fringing field is reduced to the computation of the total fringing flux (p^. F o r this purpose, let all gaps be of equal clearance 2a and pitch 26, as shown in figure 4.22. T h e n the field space between the lines m-m and n-n represents a fork-shaped region '*F^\ as shovn in figure 4.12. Since this space is symmetrical to the line p-p, the analysis can be simplified by considering only the space between p-~p and m-m, which is also a modification of the region if one channel width in "P' is reduced to zero (c=0; 7 = 0 ) . T h e transformation of this space in a Z-plane upon a parallel field in a ^-plane becomes thereby a simple strip whose width a represents one-half of the magnetomotive force 2a between two gap surfaces. F o r c=0 and 7 = 0 (4.39), (4.40), and (4.43) are reduced to Z=-il{a

tanh-^^^.^^,-&

Umh-Yf^^^^)}^

f==^log

p^^2

log (6/a). (4.79)

T h e size and distribution of the fringing field shall be investigated by a eomparison of the actual flux density B=Bj+iBy with the density BQ in the parallel field. T h e conjugate value B is given by eq (4.71) for the general space " P " by interchanging A ' w i t h B, and becomes, for c=0 and 7 = 0, (4.80) W e substitute into (4.79), and a=5/?o derive

Z^-i-ia

t a n h - Y 4 5 V 6 tanh"'

(4.81) B

which furnislies on the lines x = +a ^=-<2a ria

and x = — {b — a), that is, for B—B^j the relationships tanh * w — t a n h Bx

^ ( T , » - )?* on m-m, \b Boh

& = ; | 5 " » " - | - ' " - - ( i l ) ! '

(4.82)

° » "

between 2//(2a) and BJBit plotted in figure 4.23 for several values of Consequently', the flux distortion increases with 6/a so that this ratio has to remain under a certain limit for the fringing field to be negligible inside the winding.

I n order to find the fringing fiux ^j?, we select two points R and S on the line m~m at y=—r and 2/=+s, that is, inside and outside the gap in the Z-plane, and consider t=p-\-ia and t= — a-^-ia as t h e corresponding points in the ^plane. W e make r and s large enough so that the field at R and S is practically uniform. T h e corresponding conditions i n the t-plane m a y be written according to (4.79) as «^ = p;^0,

at point 72;

tp=ff'^
log (b¡á),

at point 5^.

(4.83)

Consquently, the fringing flux tpp can be represented as
For this purpose, we substitute Z=a^iy ^-r=^tanh-^

2a

a

In the derivation of ^ = p for y

V

and t=
1+ g

I

tanh ~ i

3(6M

i +

(4.84)

ff—a[r¡a+s¡b].

in (4.79) and find (4.85)

V"gTT"'

r from these functions, we obtain appro:dmately, because of (4.83) 1,

2+1 and combine Q==^

2a

7 r + - tanh-^ - - t a n h - » a

>1,

a

for^=p

(4.86)

We approximate further:

l-4e-^'^^tanh^g=^"^y

^l-(l/g)[l-(a/6)^,

for^-^P

and obtain log g = 2Q + log [l/4a-(a/6)^]

for v^ = p.

We substitute 2 and Q from (4.85) and (4.86), develop (4.85) m a similar way for (JJ p=^|2-tanh

• i+'°>[K-(iy)] 2+log[i(©

1

0,1

z u

SQ4 o

— O N LINE m-m --'ON LINE p-p

i OA 0

DISTANCE

0 5

1.0

FROM

FiaXTBE 4,23.

ff, and derive

a

^¡21

. 0 5

(4.87)

1.5 IRON

2JO

Í5Q3,

0.1 2.5 3 0

S U R T A C L

1

5 RELATIVE

Flux density and fringing

10 15 20 G A P PITCH - t

of gap fields.

(4.88)

T h e mtroduction of these terms into (4.84) furnishes the fringing flux i n the fon

{2a)a Ft

(4.89)

where the quantity Cy shaU be called the fringing coeflScient of the gap magnetomotive force 2a. I n a :eometric interpretation of 6V, the quantity ÔF(4a) represents the necessary increase i n pole-width and epth i n order to include the fringing flux m a uniform gap-fleld. The values of Cy are plotted against b/a i n fl^re 4.23. T h i s fringing coefficient has some similarity w i t h the Carter coefficient [7], which accounts for the influence of slots upon the magnetic flux between magnetic poles.

9. Magnetic Flux Distribution of an Iron Plate and a Double Line of Currents A s an example of conformai maps of simply connected regions where the excitation is found inside the field space, consider the magnetic flux distribution of an iron plate which is placed i n paraUel to a double line of electric currents, as shown i n figure 4.24. Such a double line may, lor example, represent

T

tCAOS

T

°2

t 0

I

1 P Li A T E

Z-PLANE SHEET

IR)

^

X

0+1 SHEET

FiouBB

4.24.

(S)

Magnetic field of plaU and double line.

some heavy loads i n an electrical apparatus. If their currents are alternating, the flux entering the iron w i l l cause an undesirable increase i n the effective load losses and reactance, which can be computed on the basis of the flux distribution [11]. F o r this fleld analysis, let the two currents + / &nd — / be concentrated i n thin filaments with the spacing a which are located on the same side of the plate at its centerline at the distances ai and from the iron center. If the thickness t of the plate is small enough, as compared w i t h the w i d t h 6, and if its permeability is large compared w i t h imity, the plate can be considered as a thin straight equipotential line P-Q i n a Z-plane. The field problem becomes then quite similar to the analysis of the fluid flow around lifting vanes leading to the development of the Kutta-Joukowsky profiles. T h i s flow is excited by a dipole located at a > , and is usually mapped upon the outside of a circle, while two single poles excite the magnetic plate fleld, and their location near the boundaiy suggests a transformation upon a half-plane as the simplest solution. The corresponding mapping function can be found by elementary methods. O n the other hand, the plate may be considered as a bdateral concave polygon with vertices P and Q. Since the transformations of such concave polvgons have found little consideration yet i n the technical literature, i t may be of interest to use this plate as an example. The general transformation of an n-lateral concave polygon i n a Z-plane upon a half W-plane, or upon a circle, has the form

dZ c{w-hr'{w dW

(4.90)

P a r t of this function is the same as for the mapping of convex polygons since the constants (7, and Uw indicate size and angles of the polygon and the location of its branch points i n the W-plane as i n

I n addition to this, the function (4.90) has singularities also at points H and K i n the TT-plane, that is, for W=k and W= k. T h e point H corresponds to <» i n the Z-plane, while K is the image of H with respect to the boundary. F o r mapping the plate field, figure 4.24, upon a half-plane, we choose u,= «> for Q and h=' + i for and obtain A;=^= — i , while P is found at Up=0 for reasons of symmetry. E q u a t i o n (4.90) furnishes then (4.30).

w

1

W

(4.91)

3

F r o m the condition W= « for Z= + b/2 we derive C=+2by

^~2

W'+l'

and conclude 6+2Z 6-2Z

W

(4.92)

The linear transformation W(i)

.t+a IT t—a

would complete the mapping of P-Q upon a circle with the radius a i n a f-plane, as used in flow problems. The function W{Z) of ^.92) transfoi-ms the imaginary axis of the Z-plane into the unit circle of the W-plane, whereby the currents -f-/ and —•/ appear i n both planes from point P at the same angles and $2 with respect to P~Q. These currents represent vortex lines i n the magnetic field sxuTOunding the plate. Consequently, the condition of an equipotential boimdary P-Q i n the H^-plane is complied w i t h b y current images which have the same sign as the originals i n the upper half-plane (R), and are located i n the lower half-plane (S). These additional currents remain images w i t h respect to P-Q i n the Z-plane if this plane is continued from the upper sheet (R) into the lower sheet (S) of a R i e m a n n surface w i t h branch points P and
(4.93)

*2)

represents the distribution in the TT-plane, and w i l l apply also to the Z-plane, when substituting W{Z) of (4.92).

F r o m this field, we derive the flux density Bg at any point Z=x inside the plate as the ratio of the flux ^, enterirg the plate, to the plate thickness The maximmn B „ of B^ is then found at x = 0 . Consequently we substitute (4.94) 0 ) - ^ ( x : y = + 0), B,=4^Ji ^«=^(x;i/ mto (4.92) and (4.93), and obtain [11] Ä=(O.4//0

B,, = {OAIIt)

tanh

V

l"(2x/6)^

tanh"

y

(4.95)

sinh*

for z = 0,

F o r finding the change i n lead reactance attributable to the presence of the iron plate, we substract the field of a double line without plate from the corresponding plate field. T h e flux linkage between the remaining field and the leads is then proportional to that reactance change. I n this way [11], we derive from (4.93) the relation 1^7^=0.2 7 log

1—cos (62+Bi) sin $1 sin

[ 1 + C 0 S (Ö1—dj)]

(4.96)

Of particular interest is the model with an infinite plate width b since it can be considered as two leads near a tank wall, as shown in figure 4.2. This problem could be solved directly in a Z-plane by a p p l y i n g the principle of images in the same way as in the W-plane. Instead of tliis (4.95) and (4.96) furnish, for 6 = qo . 1 - X

B^={OAllt) tan B^^iOAl/t) i^i. =

tanh-^

0.2/ log

2

1-X

(4.97)

1+ X

| ( X + 1/X)+1

If the dimensions i n figure 4,24 becomes negative, that is if the t w o currents + / a n d — / are l o c a t e d o n either side of the i r o n plate, the model c a n be interpreted as the throat of a b u t t w e l d i n g m a c h i n e d u r i n g the welding of two steel sheets [ 1 1 ] , as shown i n figure 4,25. I n this case, the field m a y STEEL SHEETS

A _ ^ _- c TMROAT

i .

f-

TRANSFORMER AC

POWER SUPPLY

SECTION A ' B

\ WELDING SPOT

B

COORDINATE TRANSFORMATION

W-PLANE

(«I

STEEL

FIGURE

4.25.

Location

of steei sheets in weider throat.

be used to find the change in welder impedance attributable to the presence of the plates, since the size of this impedance limits the capacity of the machine. Saturation of the plutes can also be considered b y approximating their magnetization curve as two straight lines. A simple graphical method can be used for plotting such fields, and will be explained for the example of the welder tliroat [ 1 1 ] , in connection with figure 4.26. First, the flux distribution i n the W-plane is constructed by the superposition of the fields of two double lines, that is, of two families of circles, since the points of the resultant field are at their intersections. In order to transfer this field into the Z-plane, wo use the variable 6+ 2Z 6-2Z

(4.98)

as parameter. T h e f, i;-coordinate system appears as a family of hyperbolas in the TV-plane, and as a family of circles in the Z-plane. Those two maps are used for a point-bv-point transformation of the field i n the W-phme upon the Z-plane. T h i s method furnishes the flux distributions, shown in figures 4.27 and 4.28, in tlie two pianos.

F I G U R E 4.26,

F I G U R E 4.27.

System of field plotting for plate and double

Transformed

line.

magnetic field in welder throat with steel sheets.

PLANE,

FIGURE

4.28.

Magnetic field in welder throat with steel sheets.

10. [1]

[2] (3] 4 h

5

6]

X

[91 10 11 [12]

[13]

[141 [15] [16]

Referencei

J . L a b u s , Berechnung des elektrischen Feldes v o n Hochspannungstransformatoren m i t H i l f e der konformen Abbildung, wenn mehrere W i c k l u n g e n verschiedenen Potentials vorhanden sind, A r c h i v fuer Elektrotechnik, B a n d 19, 82 to 103 (1927). S. Bergman^ Über die B e s t i m m u n g der Verzweigunggpunkte eines hyperelliptischen Integrals aus seinen P e r i o d i z i tätsmoduln m i t Anwendungen auf die Theorie des Transformators, Mathematische Zeitschrift, B a n d 19, 8 to 25 (1923). S. Bergman, U b e r die Berechnung des magnetischen Feldes i n einem Transformator, Zeitschrift fuer angewandte M a t h e m a t i k u n d M e c h a n i k , B a n d 5, 319 to 331 (1925). H . A , Schwarz, Über einige Abbildungsaufgaben, J o u r n a l fuer M a t h e m a t i k B a n d L X X , H e f t 2, 105 t o 136 (1868). H . P o r i t s k y , C a l c u l a t i o n of flux distributions w i t h saturation, E l e c t r i c a l Engineering 68, 328 (1949). G , M . Stein, Influence of the core form upon the i r o n losses of transformers, T r a n s . A . I. E . E . 67, P a r t I, 95 to 103 (1948). F . W. C a r t e r A i r gap induction, E l e c t r i c W o r l d and Engineer 38, 884 to 888 (1901). G . Stein a n d E . U h l m a n n , Felderverteilung u n d Drehende Magnetisierung i n Drehstromtransformatoren, i n W . Peterson, Forschung u n d Technik, 70 to 83 (Springer, B e r l i n , 1930). L. Dreyfus, Über die A n w e n d u n g der Theorie der konformen A b b i l d u n g zur Berechnung der Durschlags- und Überschlagserseheinungen zwischen kantigen K o n s t r u k t i o n s t e i l e n unter Oel, A r c h i v fuer Elektrotechnik, 13 123 to 145 (1924). E . G . Reed, T h e essentials of transformer practice, 117 to 123 (D. V a n N o s t r a n d C o m p a n y , N e w Y o r k , 1927). G . M . Stein, T h e voltage drop i n the welder throat loaded w i t h ferromagnetic materials, T r a n s , A . I. E . E . 66, 356 to 362 (1947). E . Weber, D i e konforme A b b i l d u n g i n der elektrischen Festigkeitslehre, A r c h i v fuer E l e k t r o t e c h n i k 17, 174 to 200 (1926). M . A b r a h a m and U. Becker, T h e classical theory of electricity and magnetism, 26 to 32; 127, 128 (Stochert & Co., N e w Y o r k , 1937). S. Bergman, P u n c h c a r d machine methods applied to the solution of the torsion problem, Quar. of A p p l . M a t h . V , N o . 1, 71 to 81 ( A p r i l 1947). See especially the references on page 81. S. Bergman, Complex orthogonal functions and conformal mapping, Note.*? of H a r v a r d U n i v e r s i t y , D i v i s i o n of Engineering Sciences, Contract N O r d 8555 Task F and N R - 0 4 3 - 0 4 6 ( N 5 0 R I 76-16 Office of N a v a l Research). S. B e r g m a n , P a r t i a l differential equations, Advanced Topics, Lecture Notes, B r o w n U n i v e r s i t y , Providence, R . I. Summer Session for A d v a n c e d Instruction and Research in Mechanics, 1941.

5.

Conformai Mapping Applied to Electromagnetic Field Problems Ernst Weber ^

1. Introduction Since the main pm-pose of this symposium is the discussion of the construction and use of conformai maps, particularly i n relation to high-speed computing machines, I should like to select electromagnetic field problems whose solutions, though formally straightforward, can be obtained i n practical form only b y the use of long-winded numerical methods, thus practically crying for the use of reUable computers. Indeed, the solutions of many of these problems, which are of real practical significance, are not even attempted i n analytical form because of the formidable human effort involved; the effort at present is rather spent on repeated collections of masses of empirical data. Three groups of problems will be presented i n order of increasing complexity; the first deals with Laplacian potential fields i n special geometries, the second with potential fields satisfying the Poisson equation even extending to nonlinear forms of it, and the third with electromagnetic wave problems i n certain confined spaces that permit formulation as conformai mapping problems of an unusual type. N o exhaustive presentation can oe given here, but it is hoped that interest can be deflected slightly n'om the most intriguing aerodynamic and h y d r o d ^ a m i c fields of application so frequently associated with applied mathematics, to the not altogether arid topics of electromagnetic fields.

2. Certain Laplacian Potential Problems The simplest and most common problem of electrostatics is the determination of the scalar potential distribution between two arbitrarily shaped conductor surfaces, the "condenser" problem. I n two dimensions the use of analytic functions has become well established either for the direct solution b y the complex potential function P(Z), that is, the conjugate functions and with P(Z)=0(x,v)+i^x,y)

(5.1)

where ^ is the real (physical) potential function and ^ the conjugate stream, or flux function; or for a conformai transformation of a difficult geometry into a simpler one, usually the upper half-plane. Attention will be given here only to the Tatter application, namely the actual conformai transformation or simply " m a p p i n g " of the geometry as an aid i n the solution of the potential problem. Because of its formal simplicity, the Schwarz-Christoffel formula [1,2]* for the mapping of the upper half of the tfvplane into a region i n the Z-plane bounded by straight lines has found wide application in electrostatics and magnetostatics. This mapping function is given i n the form of the integral Z=Cf(w~V)xY'-\w-wii^^-^.

where C 1^2, . polygon; mapping

. . {w-WnY^-Hw^-C,

(5.2)

and C are constants determining the location, scale, and orientation of the polygonal region ; . Wn the locations, along the real axis of the images of the n vertices Z i , . . ., Z„ of the A T , A x , . . .,/3«TtheinternalpolygonanglesintheZ-planeasshowninfiguro5.1. Becausethe of the upper half-plane upon itself is denned i n its most general form by the linear function (5.3)

which contains three arbitrary constants, one can choose i n (5.2) arbitrarily three of the w, values. < Polytechnic Institute of Brooklyn, Brooklyn, N . Y . *FiRure8 fn brackets indicate the Ilterstare references at the end of this paper. 949681—92

— 5

59

There remam, then, ( n + 2 — 3 ) = n — 1 unknown constants which must be evaluated b y integrations a l o n g the real tt-axis between successive vertex images w, and Wr+i, presenting the distance (ww+i—w,) as a n ^^^^v

I

^

^^^^^

implicit function of the corresponding known distance ( Z , + i — Z , ) .

V

^

Because the integrations are w i t h

iiv

Z-PLANE

W-PLANE

w

w

F I G U R E 5.1.

respect to the real variable i/, graphical methods have been used m more complicated cases; convenient computing machines could, of course, readily help m the solution of the usually highly transcendental equations. One can express the conjugate complex field vector dP

dPdw

dZ

dw dZ

(5.4)

i n terms of the complex potential function P , which gives the potential solution in the te?-plane as indicated b y (5.1). T h u s the practically most important physical quantity can be obtained, without actually effecting the total integration (5.2), as long as all the constants have been determined. B u t even assuming the very simple case of a single sharp comer as in figure 5.2,a, it is obvious that the field vector as the negative gradient of the potential function becomes infinite at that corner. Although to be expected, this cannot have practical significance; the slightest fijiite curvature reduces E to finite values, and one would like to solve the problem with reference to figure 5.2,b: W h a t should be the radius of curvature

CrRCte

\

A'

A'

F I G U R E 5.2.

b to assure a maximum gradient E anywhere along the equipotential surface ^ less than (1 +A:)£'o, where EQ is the homogeneous gradient near A'A" and * is an arbitrary real number, usually small?

In the same treatise, H . A . Schwarz [2, p. 109; 3j gives a second-order and second-degree diflPerential equation for the mapping of polygons bounded by circular arcs; however, a rigorous analytical solution o n this basis seems hardly possible. A s an approximation, one has proposed to write down the mapping function (5.2) for the geometry in which any circular arc is replaced by the sharp corner formed by the base tangents; then one replaces the factor (w—w,)^'"^ for this vertex by the factor [(w-pr^-'+Hw-qy'-'l

(5.5)

where py g, and X are real constants (p and q are, in fact, the images of the base points of the circular arc) o f which two can be chosen arbitrarily. It should be possible to choose these parameters such that one can get a very close approximation to the circular contour. T h e actual values would have to depend upon the entire geometry of the region; obviously, an analytical solution even in the simple case of figure 5.2, b, is not inviting, although computing machines might well give good answers. Selecting for simplicity X = l , 5 — p = l , o n e f i n d s [ 4 ] f o r a / 6 = 8 t h a t the actual contoiu* is a wavy line as indicated infigure5.2, c, with the maximum deviation from the circular arc approximately 0.1&. A similar result was found with this roimded corner in a rectangular niche 5]; a better approximation was found in the case of a cylindrical grating [6, 7] with diameters not small compared with the distance between centers of the cylinders. Other applications are the design of test electrodes for liquid dielectrics and gases so that the electric breakdown occurs in the homogeneous section of the field by choosing the above k<0; the more accurate computation of the static amplification factor in vacuum tubes; and so on. F o r electrodes i n gases, it is possible to predict theoretically the break-down voltage, or potential difference, on the basis of the Schumann criterion [8], which in turn was deduced from the original ionization theory of Townsend. One can formulate this criterion as a critical integral.

-0

a d8< In k,

(5.6)

where Jb is a fixed constant for any gas, and a is the ionization coe cient for electrons; a is generally given as a fimction of the field strength E by A Bl exp a (5.7) E E where A and B are constants depending on pressure, temperature, and nature of the gas. T h e integration must be performed along a field line; i n fact, that field line should be selected which gives the largest value for the integral (5.6). If the voltage is left as unknown, (5.6) with the equality sign constitutes an equation from which, for a given A;, the breakdown value of the voltage can be evaluated as function of the geometry. Obviously, if 3ie conformai mapping results in concentric circles for the field lines in the to-plane, these integrations [9] can be performed rather readily numerically; very considerable help can be had from appropriate computing machines. F o r electrodes applied to sheets or disks of solid dielectrics the field distribution becomes particularly important in the precision measurement of the dielectric constant of the solid material. If we take the general two-dimensional arrangement of figure 5.3, a, as illustration, a conformai transformation upon the upper half-plane of the region outside the equipotential electrodes A' and A" leads to figure 5.3, b.

FIGURE

5.3.

T h e potential solution i n the upper half-plane can be obtained either by direct numerical methods i n terms of the original geometric parameters, or by a fiurther transformation upon an ideal paraUel p l a n e condenser as in figure 6.3c, but with a difficult dielectric boundary. It would be most desirable to determine the total capacitance C per xmit depth, normal to the plane of figure 5.3, as a fimction of geometry and ratio of dielectric constants « J M , SO that one could conversely evaluate €2 from a measurement of C. T h e practical importance has led to several empirical relations [10] for different arrang;ements of the electrodes, always assuming them of infinitesimal thickness. It woidd, of course, be of mterest, too, to give the result for electrodes with roimded comers, whereby one could at first assume a s y m m e t r i c a l arrangement so that figure 5.3, a, could be replaced by figure 5.2, b, with a as the half thicwiess of the dielectric sheet. T y p i c a l solutions could be procured by means of high-speed computing machines, which woiüd go far m confirming or replacing the empirical forms.

3. Certain Poisson Potential Problems Problems of electron flow i n vacuum tubes under stationary conditions can be reduced to the potential equation of the Poisson type

(5.8) where p represents the local space charge density and e» the dielectric constant of free space. In (5.8), p must be known as a function of the space coordinates, which does not remain invariant under conformai transformations of the geometry. Indeed, going from the original Z-plane to the uvhalf-plane, the densities are related b y Pn —

dw -2 Pz' dZ

(5.9)

A n interesting example is the arrangement in figure 5.4, a, where represents a vacuum tube grid of high positive potential through which an electron stream passes at comparatively high speed; # a is the

4 iv

4>o

»0

0

\

^

\

. «o

FIGURE

<|)iO

6.4.

collecting anode, generally of varying potential, either higher of lower than * , ; 'i>==0 are the two b e a m forming plates that repel electrons and thus maintain a reasonably uniform flow. Obviously, thejpotential distribution is rather complex but can be obtained without electron flow by direct apphcation of the c o n f o n n d mapping of the polygonal region into the upper half w-plane as i n figure 5.4, b. T h e complete solution i n the presence of the electron stream must be found i n the -wj-plane b y satisfying dZ

Pz

dw

(5.10)

where dZ/dw is known from the geometric mapping as a function of w, and where the density px must be obtained from the continuity relation pV=J,

(5.11)

where in t u m the local velocity V is defined b y the usually stipulated equivalence of e n e i ^ e s 2

V'=\e\,

(5.12)

w i t h {«I the absolute electron charge.

This leads to J

2\e

-11/2

m

-1/Î

(5.13)

w i t h J a fixed reference value to be given along the plane as a function of x; in the first approximation, i t could be assumed constant over the central part and zero over the remainder of the plane This nonlinear problem is solved readily for homogeneous two-dimensional electron flow, when we disregard the plates * = 0 , b y Child's law [11, 12; 13, p. 170], if the initial velocity of the electrons on is assumed as zero; or b y a modification indicating multivalued regions of potential distributions [14, 15; 13, p. 248 to 265], if the initial velocity is assumed finitely large. F o r the nonhomogeneous flow, with the plates * = 0 , the solution i n the «^-plane cannot be given analytically i n any simjSe form, and computing machines appear the only reasonable recourse. T h e most interesting results would be the determination of the actual potential distribution and the dependence of current density «7, or total current, upon the potential values and *oA very similar range of problems can be found in the evaluation of magietic fields within long conductors carrying parallel and uniformly distributed electric current flow. T h e magnetic field B i n this two-dimensional case, as typified b y parallel bus bars of various cross sections, can be derived from a vector potential A: B=curli4, (6.14) where A reduces to a single component i n the direction of the axes of the conductors say Ai, the longitudinal component. Thus one has to solve (5.15)

tiJt

AM.

where /x is the absolute magnetic premeabihty, and Jt is the local longitudinal current density. W i t h i n the conductor cross section, the given Poisson equation holds; outside of the conductors, i t reduces to the Laplacian equation; on the boundary, one has to satisfy continuity of the vector potential as well as of its normal derivative divided by the respective M- I n the case of diincult cross-sectional geometry, one can use conformai mapping to simplify tne boundary conditions, even though the current density does not remain invariant, out rather transforms i n the same manner as the space charge density p i n (5.9). Application to two parallel cylindrical conductors of arbitrary permeabilities and w i t h cross sections i n the form of eccentric annuli [16] permits a much closer evalution of the self and mutual inductances than originally given by M a x w e l l . The mapping is achieved b y

ihx

(5.16)

which converts the annuli into a parallel rectangular strip amenable to Fourier series solution. Again, w i t h the availability of computers assured, many other solutions could be found for combinations of current-carrying conductors, magnetic and nonmagnetic materials characteristic for transformers, relays, and electric machinery [17, 18, 19]. The inherent simplicity of fields w i t h axial symmetry has often led to attempts to construct solutions out of two-dimensional field maps. I t is, of course, possible to write the Laplacian differential equation for cylindrical sj^tems with axial symmetry in the form 1 d0 7 57'

(5.17)

where r is the radial distance from the axis, and z is the axial coordinate. Restricting considerations to regions distant from the axis, one can substitute r=ro4-^ and, for r>ro, interpret the righthand term as an equivalent space-charge density p(r, z). This approximate solution is particularly valuable for computation of end effects [20], taking the two-dimensional Laplacian potential solution ^ ( r , 2 ) for the same geometry for rro b y direct integration of the Poisson equation (5.17) w i t h p(r, z) = H

;—

•

4. Certain Propagation Problems in Wave Guides The propagation of electromagnetic waves through metallic tubes, ov'^wave guides**, of geometric dimensions comparable with the wavelength, exhibits many interesting experimental and a n a l y t i c a l aspects. A comprehensive study of the latter became quite necessary i n connection with the development of short wave radar because of the need for engineering methods of design. The unusual combination of the theoretical physicist J . Schwinger, now of H a r v a r d University, and the scientific electrical engineer N . M a r c u v i t z of Polytechnic Institute of B r o o k l y n , working together at the Radiation L a b oratory of the Massachusetts Institute of Technology, resulted i n a new approach to the solution o f electromagnetic diffraction problems [21, 22, 23, 24], which might well be useful i n other fields of applied mathematics. The particular method of interest here is the substitution of an equivalent static problem solved w i t h the aid of conformal mapping, for the actual dynamic problem presented b y the wave equation. T h i s is made feasible if the electromagnetic field can be expressed i n terms of a scalar two-dimensional wave function ^ or its derivatives and if the boundary conditions can take the form of a homogeneous integral equation. Uy)fUyW)dy'

Mj:^MfUyV{yW,

(5.18)

n-l

where M is a constant,/(y') the unknown field quantity to be determined, and ^0(2/)» ^n(y) are eigenfunctions characteristic for the particular problem and preferably an orthonormal family. This integral equation for the dynamic field quantityy(j/') as solution of the wave equation can be transformed into a n integral equation for the same field quantity/(i/0> but as solution of a static field problem b y being written

Uy)fUy')fiy')dy'

Mj:fiMy)Uy')

i'iKy)ii'iy')\mdy'

= M S ^ < ' (y)ff!:'(.yy(y')dy', n-l

(5.19)

where now jf^j^^ {y) are the static eigenfunctions for the same geometry, w i t h excitation terms on the left-hand side, including, i n addition to the usual first term, terms containing the difference between dynamic and static kernels as ampUtude factors. If this modified static problem (5.19) can be solved easier than the original integral eq (5.18), for example, b y conformal mapping, then considerable advantage is gained. I n order to illustrate this method, a more detailed treatment of aspecific and relatively simple wave-guide problem, first reported i n Schwinger's lecture notes [21] will be given here. A s indicated i n figure 5.5,

FIGURE

5.5.

an electromagnetic wave is presumed to propagate along the positive g-direction within a metallic tube of rectangular cross section (the wave guide); i n the plane z=0 is located a so-called capacitive iris, that is, an infinitively thin metallic shield leaving a uniform slit (window or aperture) parallel to the longer transverse dimension of the wave guide. The presence of the iris, acting as a n obstacle i n the guide, causes a severe local distortion of the incident electromagnetic wave, resmting i n partial reflection a n d in partial transmission beyond the plane 2=0. Because of the usually chosen transverse dimensions of the wave guide, far from the plane 2=0, the reflected and transmitted waves can only have the same character as the incident wave. However, close to the plane 2=0, the field distribution is defined entirely b y the local boundary conditions, requiring vanishing tangential electric field on a l l metallic surfaces (one usually assumes infinite conductivity i n order to escape evaluation of the field within the metal walls), and continuity of electric- and magnetic-field components on the aperture which couples the two wave guide spaces.

Considering only stationary harmonic oscillations i n time of the angular frequency n a t i n g the physical values of the field vectors for this case b y E=3

and desig(5.20)

[E,''^l

where i and H are time-complex vector quantities, we find that Maxwell's [field [equations reduce to (5.21)

curl E

curl Ä = + t w € Ä ,

I n the undisturbed wave guide, the only propagating solutions possible are E^=EiiSm

TI

—a

ex^ {Tiqz)t

q . irx H « = T £ ^ O ' ? x ( ^ ^ 2 ^ ) , k a

where

,

.—7-

2T


TX

H«=—^^o'?jj^cos — e x p ( T i 2 2 ) ,

2T

['•-(DT

if 2T

-}

a

or

(5.22)

(5.23) (5.24)

a<Xo<2a

A s long as Xo>2a, no propagation can take place, q i n (5.23) is imaginary, the electromagnetic waves attenuate; if, on the other hand, Xo<^a, then "higher modes" of field distributions can propagate as well, w i t h sin (2Ta;/a), sin (3Ta;/a), . . . , and individually different propagation constants. F o r conventional use of wave guides, (5.24) must be observed, resulting i n the propagation of the dominant mode (5.22) and of only this one. T h e upper signs i n (5.22) define propagation i n the direction of positive Zj the lower signs corresponding propagation i n the direction of negative 2. The presence of the iris at 0=0 wUl cause higher order modes to appear, but because of the uniformity i n the x-direction, they will show the same dependence upon x as the dominant mode (5.22); they will, however, not have uniform value along the y-direction, but show dependence upon the ycoordinate. Maxwell's field eq (5.21) can still be satisfied b y a single component of the magnetic ij^e H e r t z vector, or perhaps better, b y introduction of a scalar wave function i^(y,2) such that
— xòyp . T X - rr^ Sin 1 „

qoz

a

.104^.

TX

H,

,

.

TX

1^ sm — t

T

TX

T

TX


—

cos

fiùìqa oz

a

(5.25)

a ^

where (v»+g')*(y,2)=o.

(5.26)

The boundary conditions require vanishing tangential electric field o n all metal surfaces, that is, Ep=0 o n 2=constant, and £',=0 on v=constant; this can be summarized into 0^ on

0

on all metal surfaces.

(5.27)

Because of symmetry the electric field i n the plane 2=0 must reduce to E^, so that i n turn the magnetic field i n the aperture must satisfy Hx=0, or also ^=0

on aperture.

(5.28)

Since the x-dependence of the higher modes is the same as that for the dominant mode, one could here select any arbitrary value of x and satisfy the boundary conditions i n y and z for it, being sure that then these boundary conditions are satisfied for a l l values of x. A convenient choice would be x=a/2, since i t eliminates Hp and H^y thus leaving E,

i 01^ 3 dz'

E.

2?,=H.=0

+2 0^'

(5.29)

Solutions of (5.26) would be of the type n=0,l,2,

+i(^q

^«(2/»s)=-4« c o s ^ ^ e x p

(5.30)

w i t h n an integer. A general solution can be constructed as the infinite sum with arbitrary c o m p l e x amplitudes An which must be determined from the boundary conditions (5.27) and (5.28), leading t o a n infinite array of linear relations without good and suflicient reasons for restricting i t to any r e a s o n a b l e finite number of terms. O n the other hand, with the choice (5.24), all modes i n (5.30) except n = 0 do not propagate b u t attenuate i n the 2-direction from z=0 as a source. One can, therefore, construct the total solution f o r z<^0 as superposition of (a) the solution for the dominant mode (5.22) alone w i t h a soUd metal w a l l a t 2 = 0 , where, therefore, ^^=0, and of (b) an infinite sum of mode solutions (30) produced b y the f i n i t e value of Ey i n the aperture of the plane 2 = 0 . Since the only contribution to (b) can come from in the apertiu*e, one will preferably choose to express the electric-field co iponent itself, since i t is r e a d i l y related to ^ b y (5.25) or (5.29). This superposition gives, then, for x=a/2,

Cn cos

(5.31)

exp ± i ^ 2

where the first part is the dominant mode completely reflected at 2 = 0 , and wherejthe Fourier sum h a s the same character as (5.30). One can apply this at 2 = 0 , and if the electric-field component E^ in t h e aperture is designated as proportional to the incident field amplitude EQ, then E,{y,z=0)=Eoe(y)

j:an

=

it-O

COS -

^

(5.32)

1

the coefficients Cn are simply the conventional Fourier series amplitudes and can i n turn be expressed conventionally as integrals of the field eEo over the aperture, so that (5.31) is also, for 2 < 0 , Eosmqz

E,{y,z)

exp

W)dy'+^±,

0 n-l

tt/apert

exp (pnz)(

«(vOco8^cos^dy',

J apert i

(5.33)

propagating

where

Tlx

Pi

3

(5.34)

q

&

and y' is used as integration variable to differentiate from y of the F o m i e r series. T h e i n t ^ r a t i o n s need only be extended over the aperture since only there E^T^O. T h i s has now transformed the boimdary condition (5.28) into a homogeneous integral equation for 6, since from (5.28) a n d (5.29) we have • - D

0=1

fE,iy,z)dzl,o.

or also, upon introducing (5.33),

^ffjapert

b

0 n-l

eiy") c o s ^ d y ' = 0 . 0

apert

(5.35)

I n (5.33), which is of the form (5.18), the first term was suppressed, since its contribution becomes zero w i t h 2 - » 0 . [Obviously, one could arrive at this same result b y defining a Green's function i n (y, 2 ; y\ ^)'\

T h e final solution of (5.35) can now be achieved b y solving the equivalent static problem. Obviously, if ^ = 0 i n (5,26) i t reduces to the Laplacian differential equation i n two variables y and 2, so that analytic functions would be appropriate. O n the other hand, if one takes § = 0 i n (5.34), and rewrites (5.36) w i t h i t m the form (5.19), +¿1.

apart

-
1 g6\

2ni/2

n

cos

nicy b

€{y')

apert

COS

dy

)1 2

1

- S r c o s ^ ^ f .(yOcos^rfy', xn~\ n Ö Japert 0

(5.36)

one has the integral equation for the electrostatic problem which leads to the same electric-field distribution i n the window as the original wave-guide problem.

l'

5

"1" 4

-¥ 1

4

1u

t-PLANE I"

1 t- PLANE

c

W-PLANE F I G U R E 5.6.

Assume now the cross sectional geometry of figiu'e 5.5 as shown i n figure 5.6, a, and call the complex variable (5.37) I t is then needed to find an electrostatic field distribution which results in the yet unknown field strength Ep=Eoe i n the aperture and generally satisfies the integral eq (5.36). Introducing a complex potential function P as i n (5.1), and observing that the electric field is defined b y either conjugate function Ep

ay

ÒZ

ay

(5.38)

we see, b y comparison w i t h (5.29) and reference to (5.27), that the flux function is here of primary importance. Simplification of the boundary results, provided that a conformai transformati
C ^/{t+m)it-D

and

^ 6 / 2t f = - c o s h M j-j T

\t+wi

l—m

j-¡— l+m

(5.39)

o r also i

l+m

2

, TÍ" , l—m

cosh

6

2

(5.40)

A further transformation into the rectangular strip of the tü-plane shown i n figure 5.6, c, is given b y dw

D

dt

(¿a-1)*

and b

t ü = - cosh"^ t+ibt

(5.41)

or also t 949581—92

0

cosh

TtP

T

(5.42) 67

T h e choice of points 3 and 4 at ± 1 i n the i-plane completely determines the location of points 2 a n d 5; introducing ¡=-\-ihi, and í * = + ' i ( ^ i + ^ ) , respectively, into (5.40), gives two transcendental e q u a t i o n s for I and m. B y equating (5.40) and (5.42), one has directly 10=— c o s h "

eosh - J -

-cosh-^i

2 -

(5.43)

X

for the mapping of the given geometry into the simpler one of the w-plane, where the boundary c o n d i t i o n s now read »^=0,

on u—0:

^=0,

on ??=0 and o n «? = 6.

(5.44)

Analytic functions satisfying (5.44) can readily be given, namely, F=a
I

smh

— \ - a 2 % smh —^—|-*

•

(5.45)

from which ^ = 3 P can also be computed. T h e amplitude coefficients are unknown and must, of course, eventually be determined so as to satisfy (5.36). B y means of (5.43), the analytic solution P is transformed into the original f-plane, thus P ( f ) = a o - c o s h - ' d+axi{e''-

l ) » + a 2 i 2 0 ( ^ 2 - 1)*H

X

Rationalizing will then give

.

(5.46)

(y, s ) = 3 P ( i ' ) and also, i n accordance with (5.38), ^.(y,2)=-^3P(r),

(5.47)

which can be resolved into a Fourier series of the type

E,{y, z)= A + S A , cos ( ^ )

exp ( ^ ) + S

5 » cos ( ^ )

exp ( - ^ )

(5.48)

where the coefficients are hnear combinations of the amphtude values ao,ai, . . . of (5.45). F o r 2 = 0, (5.48) m u ^ l represent the electric field i n the aperture, since it is different from zero only there, so t h a t with (5.32) one obtains A,=

C,^^{

eiyOdy',

A,+B„

= C n ^ ^

(

^(y/) cos ^

^y'.

(5.49)

Using (5.38), one can express ^ from (5.48) (one could, of course, have reversed the procedure equally well), and obtain ^=^2—5o—S

^ c o s ^ ^

An e x p ^ ^ ^ ^ — B „ e x p ^ — j

(5.50)

where one can use (5.49), except that BQ and Bn are still unknown. T a k i n g the value at 2 = 0 , where ^ = 0 o n the aperture, and separating known from unknown coefficients, one has - 5 o + 2 S

—

nx

Bn cos

o

x

i (An+B„) iTTi n

cos

o

(5.51)

and with (5.49) this is seen to have the right-hand side identical with 2B^lb times the right-hand side of the integral equation (5.36). It has, therefore, been demonstrated, that the static solution (5.45) or (5.48) defines identically the desired djmamic electric field. I n order to actually determine the Fourier coefficients, one needs now to compare the left-hand sides of (5.51) and (5.36) as well, obtaining, with (5.49) , the relations

^n = ~^OT

0

LPn

AJ

J apert

^(,')cos^f-rf,'=-^(^, + B , ) ( l - i )

from which it follows that (5.52)

Since these Fourier coefficients are actually found in (5.49) as linear combinations of the static mode amplitudes a©* <^it • • •» the relations (5.52) represent the final linear system of equations from which the a a can be evaluated. Obviously, the advantage of this method rests upon the fact that the approximation even by one or two terms in (5.45) is a very good one. W i t h the knowledge of the values, one can then determine BQ in (5.51) by comparing with the first term in (5.36) and b y using (5.49), namely,

W)dy'=-i^f

BO=—^eJ Og_

Japert

(5.53) 2

since AQ is known as a linear combination of the a„ values. Actually this would be generally the most desirable result, since it determines the amplitude of the propagating mode contribution b y the iris in accordance with (5.33); for applications to microwave theory it permits the computation of the reflection coefficient of the dominant and propagating mode and the evaluation of the equivalent circuit aspect for engineering designs.

5- References [1

2 3 4 5 6 7 8] 9 [10] 11 12 13 14 [16 16 17 18 19 20 21 22 [23] [24]

E . B. Christoflfel, Annali di Mat. [2] 1, 95 (1867). H . A. Schwarz, J . f. Math. 70, 105 (1869). H . Bateman, Partial diflFerential equations of mathematical physics, p. 504 (Dover Publications, New York, 1944). L. Dreyfus, Arch. Electrot. 13, 131 (1923). M , Walker, Conjugate functions for engineers (Oxford University Press, London, 1933). H . W. Richmond, Proc. London Math. Soc. [2] 23, 389 (1923). W. R. Smythe, Static and dynamic electricity, p, 98 (McGraw-Hill, New York, 1939). W. 0. Schumann, Elektrische Festigkeitslehre (Springer, Berlin, 1923). J . G . Hutton, Trans. A I E E 66, 1674 (1947). A. H . Scott and H . L. Curtis, Edge correction in the determination of the dielectric constant, J . Research N B S 22, 747 to 775 (1939) RP1217. D . C. Child, Phys. Rev. 32, 492 to 511 (1911). I. Langmuir, Phys. Rev. [2] 2, 450 to 486 (1913). K . R. Spangenberg, Vacuum tubes (McGraw-Hill, New York, 1948). B. Salzberg and A, V. Haeff, R C A Review 2, 336 to 374 (1938). C . E . Fay, A. L. Samuel, and W. Shockley, Bell. Sys. Tech. J . 17, 49 to 79 (1938). T . J . Higgins, J . Math, Phys. 21, 159 to 177 (1942). B. Hague, Electromagnetic problems in electrical engineering (Oxford University Press, London, 1929). L. V. Bewley, Two dimensional fields in electrical engineering (MacMillan, New York, 1948). E . Weber, Mapping of fields (Wiley, New York, 1950). J . C, Maxwell, A treatise on electricity and magnetism. Vol, I, p. 305 (Clarendon Press, Oxford, 1892, 3d ed.). D. S. Saxon, Notes on lectures by J , Schwin^er, M I T Radiation Laboratory Report (1944). N, Marcuvitz, Waveguide handbook (Radiation Laboratory Series, McGraw-Hill, New York, 1951). N. Marcuvitz, Wave guide circuit theory; coupling of wave guides by small apertures. Dissertation 1947, Polytechnic Institute of Brooklyn. N. Marcuvitz, The representation, measurement, and calculation of equivalent circuits for waveguide discontinuities with application to rectangular slots, Polytechnic Institute of Brooklyn, Report No. R-193-49, PIB-137 (1949).

On the Use of Conformai Mapping in Two-Dimensional Problems of the Theory of Elasticity I. S. Sokolnikofif 1

1 • Two-Dimensional Problems The object of this synoptic paper is to mdicate some of the uses of conformai maps i n the solution o f a wide class of two-dimensional boundary value problems of the theory of elasticity. The phrase " two-dimensional boimdary value problems" is used here i n the sense that the mathematical formulation o f such problems requires only two independent variables. Accordingly, the theory sketched i n this essay is concerned not only w i t h the analysis of the states of plane stress and plane strain, but it also includes a variety of problems which, from a physical point of view, are three-dimensional. A m o n g the latter are problems concerned with the deflection of thin elastic plates and Saint Venant's problems o n torsion and flexure of elastic cylinders. Since a systematic treatment utilizing the conformai mapping techniques of those two-dimensional elastic problems that are reducible to the solution of the problems of Neumann and Dirichlet already appears i n text book literature [12],* the discussion of this paper w i l l be confined to the boundary value problems involving the biharmonic equation and its generalized form that figures prominently i n anisotropic elasticity. T h e basic theory sketched i n this paper belongs to N . I. Muschelisvili, who gave a rigorous analysis of the fundamental boundary value problems of two-dimensional isotropic elasticity i n a series of papers that culminated i n a prize-winning monograph entitled " S o m e Basic Problems of the Theory of Elast i c i t y " [11]. The importance of this work may be surmised from the fact that considerably more than one hundred research articles concerned w i t h the generalizations and applications of the theory to specific problems have appeared since the time of its publication [13]. T h i s paper is concerned w i t h a general survey of those phases of the theory that are related to the application of conformai mapping techniques to two-dimensional problems of isotropic and anisotropic elasticity.

2.

Formulation of Problems

We supp>ose that a closed two-dimensional region R is occupied by an elastic medium having at least one plane of elastic symmetry, which we take as the (x^a:^)-plane of the rectangular coordinate sjTstem. F o r simplicity of exposition we suppose that the region R is simply connected, although the extension of the theory to multiply connected regions presents no insurmountable difficulties. T h i s region R, finite or infinite, is assumed to be bounded by a rectifiable Jordan curve C, which has a uniquely defined normal, except possibly at a finite number of points of C. W e consider two problems. W h a t is the state of stress and deformation i n the region R when: Problem A. The boundary C is subjected to the action of prescribed forces i n the (x',a:*)-plane? Problem B, T h e points of the boundary C a r e subjected to prescribed displacements in the (x',x*)-plane? The components of the displacement vector u at a typical point {x^yix^ of R are denoted b y u ^ , a = 1, 2, and those of the stress tensor b y Ta^(x^a:^). Throughout this paper we suppose that the indices a, ^, 7, . . ., have ranges 1, 2, and we use the customary summation convention i n regard to repeated indices. Also we use a comma preceding an index to denote the partial derivative w i t h respect to the variable symbolized by that index. Thus Z7.1

and

> University tiH Califomis, Los Angsles, Calif. 'Figures in brackets indicate the literature referenoM at the end of this paper.

Ua.»t

0^,

If the medium contained i n i2 is i n equihbrium under the action of prescribed external forces Xa(s) distributed along the boundary C, s being the arc parameter, then the T ^ ^ satisfy the system of p a r t i a l differential equations, Ta(?.3=0,

ini?,

(6.1)

while on the boundary C we have the condition (6.2) where the are the components of the unit normal exterior to R, Moreover, if the functions Ta$(x^tX^) are of class in Ry then the eqs (6.1) imply the existence of A i r y ' s stress function ?7(a;',a;^), such t h a t Til

r 22

(6.3)

u .I2t

T\2

and i t is well known [5] that i n the case of an isotropic medium the function biharmonic equation vVC/(xSx2) = 0

where ^^=5^^^^

U{x\x'^)

satisfies t h e (6.4)

m/?,

two-dimensional Laplacian operator.

F o r the anisotropic elastic medium, w i t h one plane of elastic symmetry coincident w i t h the (x^,x^)plane, the corresponding stress function U satisfies a differential equation of the form [4] C ^ ' U

à*U

à*U

, C

àix'Yàx

à{x')

d(x')'d(x=0

àx'à(xy

à{xy

(6.0)

where the constants Ci are real functions of known elastic moduli of the medium. We consider first the formulation of the boundary value problems A and B i n terms of the stress function U for the isotropic case. The general solution of the biharmonic eq (6.4), i n terms of two analytic functions tp(z) a n d xiz) of the complex variable z=x^+'u^, has the form [2, 9, 10] 2ü = z
x{2)+W),

where bars denote the conjugate complex values. N o w if we insert the relations (6.3) i n the boimdary conditions (6.2), form the expression and make use of the obvious relations between the tangential and normal derivatives, we get , d if^{U,,+iU,^=Xx+iX2,

(6.6)

Xi+iXj,

(6.7)

where d/ds represents differentiation along the arc-length of the contour (7. The right-hand member i n the expression (6.7) is a known function of the points of C , a n d integration from an arbitrary point a=9o yields the boundary conditions i n the form U,i+iU,2=i

\

{Xi+iX^d8+const.=fi{8)+ij2{^)+con8t.,

where yi(«) andya(«) are known functions along C making use of the formula (6.6), yields

(6.8)

O n the other hand, a n elementary calculation,

SO that the boundary condition (6,8) for the determination of unknown functions tp{z) and ^(2) = x'(3) assumes the form
If the functions/«(«) satisfy the Lipschitz condition. \m-f(82)\<

M\8,-82

M>0,

a>0.

(6.9)

for all pairs of points s=Si, s=S2 along C, then it is not difficult to prove that the analytic fimctions ^ and ^, continuous in a closed region 7?+C', can be determined so that the solution of the boundary v a l u e Problem B is essentially unique [10]. F r o m the fact that the components of the stress tensor Ta0 are expressible i n terms of the second derivatives of ^ ( 2 ) and xi^) [see (6.3) and (6.6)], it follows that the constant of integration in (6.9) c a n be fixed arbitrarily without altering the state of stress in the medium. Accordingly, we take the v a l u e of this constant to be zero, and consider the functional equation connected with Problem A in t h e form: on C .

(6.10)

T h e formulation of Problem B in terms of analytic functions
Kip(z)~

(6.12)

z ip'(z) — ypiz)

T h e nonessential linear terms representing rigid body motion are omitted in (6.12), and K represents a certain rational fimction of Lame's constants X and Consequently, if the displacements along C are specified as functions of the arc-parameter s, i n the form. u:

then for the determination of
=

2n{gi+ig¿,

(6.13)

which has a structure similar to (6.10). Functional equations of this form have received considerable study during the past several years. W e shall be concerned here only with the practically important case of solutions that are continuous i n the closed region R+Cy and shall indicate in the following section a mode of attack on the problem that utilizes the conformal maps. T h e corresponding problem in anisotropic elasticity is sketched i n section 4.

3.

Solution of Functional Equations

T h e method of deducing a formal solution of (6,10) and (6.13) immediately suggests itself when the region R is bounded b y a unit circle |2|<1, and when the functions/« and gat appearuig i n the righthand members of these equations, are of bounded variation. Since the structures of these equations are similar, we confine our attention to the boundary value problem A. T h e unknown functions

(6.14)

and since/i and/j, regarded as functions of the polar angle 0 defined b y 2=e**, are of bounded variation, one can write (6.15) in which the Fourier coefficients Cn m a y be regarded as known constants. Inserting expansions (6.14) and (6.15) in (6.10) and comparing like powers of on both sides, one obtains expressions for the de> We omit the details of IntegratioD, since they present no Interest. we have «-(6X+flM)/(3X+^).

In the c u e of plane strain we have <*CX+3ii)/(X4-^), while for the state of plane

termination of unknown coefficients a„ and 6« in terms of known constants c,. T h e justification o f vahdity of the procedure used above i n deducing the formal solution depends, of course, o n the nature of the apphed stresses. It presents no difficulty when the functions Xa{s) are sufficiently smooth. If the contour C bounding R is not a circle, one can introduce an analytic function 2 = w ( f ) t h a t maps the region R conformally on the imit circle 1^1 < 1 - L e t the transforms of ip{2) a n d ^ ( 2 ) be denoted respectively b y

(6.16) where o'=c*' represents a point on the boimdary 7 of the circle |rl = l> and/i+i/2 is the transform of t h e right-hand member of (6.10). T h u s , it can be regarded as a known function of the angular variable d. T h e method of solution in series, indicated above, can now be applied to determme the analytic functions

i^i(f)

n-0

and

It is not difficult to show that the coefficients

Ut)

ay

r
n-O

and 6„ can be determined explicitly in terms of the

Fourier coefficients of the fimctions/i+i/2 and w((f) for a suitably restricted simply connected region R can be approximated b y a polynomial 2=co»(f) of sufficiently high degree, in the sense that the polynomial 2 = a ) « ( f ) maps the region |f K l on some region Rn which can be made to approximate the region R as closely as desired, then the practical value of the method of solution outlined above becomes immediately obvious. Another mode of attack on the problem is to reduce the determination of functions ^(2) and ^ ( 2 ) to the solution of certain integral equations. There are several such reductions, and we sketch here only one of these. T h i s particular reduction appears to be most useful in obtaining numerical solutions [10]. T h u s consider the eq (6.16) and the equation

obtained from (6.16) b y forming its conjugate. If both members of these equations are multiplied by (l/2Ti)[
(6.17) 1 2

< r - f "^2

r U
2

where

If we now make use of Cauchy's integral formula a n d the numerical result

2xt valid for every function

analytic in | f | < l and continuous in |f | ^ 1, we get

The complete equivalence of the simultaneous integro-differential eq (6,18) with the functional eq (6.16) follows from the theorem of H a m a c k [11, 12] whenever the functions/i (tf) and/2(d) satisfy the Lipschitz condition i n the interval (0, 2x). The first of eq (6.18) determines V i ( f ) , to within a constant that has no effect on the calculation of stresses, and the second yields if'i(r). We can reduce the first of these equations to an integral equation of the Fredholm type b y differentiating i t with respect to f. W e rewrite i t first i n the equivalent form ^ » ( f > + 9 i v (^^^^^ ^ ; ( < r ) r f a + x o , ( r ) + ^ = ^ ( r ) ,

(6.19)

where Xs^I(6)/w'(0)» differentiate (6.19) with respect to f, and allow f to approach an arbitrary point o- = T of f 1 = 1. This leads to

or ^'M+J^f^

K(r,


+

\o>'lT)=A'{r),

(6.20)

where TTf

X

-

I)L{TJ
^

f.

^ a>(o-) —fa)(T)

. 3"

w (

—T

The term involving X i n (6.20) can be removed b y defining a new function * ( f ) s ^ ( j ' ) - [ - X w ( f ) , so t h a t the integral equation for the determination of *(J*) has the simple standard form

*'(r)+2^J^i$:(T,
(6.21)

If the mapping function co(f) is rational, this equation can be solved i n closed form b y elementary methods which utilize the properties of Cauchy*s integrals and the theorems of residues [11]. I n other cases i t can be solved b y numerical methods.

4.

The Case of an Anisotropic Medium

A generalization of Muschelisvili's approach to two-dimensional problems of isotropic theory to the anisotropic case was first made b y S. G . Lechnitsky [4] but because of the difficulty of exchanging scientific information with the workers i n the Soviet U n i o n , quite independent and often parallel courses of investigation of anisotropic problems have been pursued i n this country and i n England [1, 3, 6, 7. 13, 14, 15, 16, 17]. I n this section we give only the briefest of indications of the role that conformal mapping theory can be made to play i n obtaining effective solutions of two-dimensional problems of anisotropic elasticity. A s was indicated i n section 2, the stress function U of the anisotropic theory satisfies the differential eq (6.5), which can be written i n the form Z>4D3D2D,Z7=0,

(6.22)

where d

d

a n d the constants /i^ are the roots of the characteristic equation C8M*+C4M'+C3M'+C2M+CI=0.

(6.23)

4

The general solution of eq (6.22), when the roots of (6.23) are distinct, has the form C7=|:F,(X'+M<X»),

where the functions F< are of Class (7*.

(6.24)

The case of multiple roots is of relatively trivial interest because i n that circumstance the p r o b l e m can be reduced to the isotropic case by a simple linear transformation. F r o m the fact that the energy of (leformation is non-negative, it follows that the roots are n e v e r real; and since the coefficients i n (6.23) are real numbei-s, we conclude that the roots are conjugate complex numbers. Thus we can write [4]

T h i s permits us to introduce two complex variables Zi and 22 defined b y the formulas Zl^X^

+

fXlX

x' + {ai + i^{)x^'

y with T i :

XÌ X

xJ+^xî, and Z2=X^

+

ti2X^

a;^+(a2+x02)x2

XÌ

with T.:

x^ +

a2X^,

X

xl+ixl,

Since

U{x^, x^

is a real function, the solution (6.24) can be written i n the form (6.25)

U=F,{z{)+F2(zè+F,{zx)+F2{22h

where bars, as usual, denote the conjugate complex values. It follows from the definition of the variables Zi and 22 that their domains Ri and R2 are obtainable from R b y the homogeneous deformations Ti and T2. Also, from the differentiability of U{x^, x^) i t follows that Fi(zi) and ^ 2 ( ^ 2 ) are analytic functions defined i n the regions Ri and ^ 2 » respectively. We readily deduce from relations (6.3) that the stresses T^P are expressible i n terms of the derivatives of functions Fi{zi) and ^ 3 ( 2 3 ) ; and the substitution of such expressions i n the boundary condition (6.2) yields the formulas f^iF,(zd+ix2K{z2)+7^F^+Mz^^

(6.26) X2(8)dS^j2{s)y

where fi (s) and Ti (s) are known functions on C, The formulation of the boundary value problem B leads to a quite similar functional equation, which there is no need to record here. The mode of solution of equations (6.26), following the pattern of section 3, inunediately suggests itself. Thus if is a simply connected region, one can map the regions and ^ 2 conformally o n a imit circle |f| < 1 i n the new complex J'-plane w i t h the aid of the mapping functions 2i=w,(f),

"2(r)

i n such a way that the correspondence of the boimdary points of the regions R, i?,, and R3 is preserved. If this is done, the resulting functional equations can be solved either b y the methods of infinite series, or the problem reduced to the solution o f Fredholm's integral equation m the manner described i n section 3. The difficulty i n solving problems A and B b y this method obviously lies i n the construction of mapping functions a ) i ( f ) and W 3 ( J ' ) . Some aspects of such mapping have been dealt w i t h b y V . M o r k o v i n i n a doctoral dissertation that was concerned w i t h the deflection of thin anisotropic plates under loads distributed over their faces [7].

5.

Concluding Remarks

The foregoing account contains a brief and quite incomi)lete survey of the use of conformai maps i n the solution of two-dimensional boimdary value problems i n the theory of elasticity. However, the

c e n t r a l idea of replacing the differential equations and the boundary conditions b y an equivalent funct i o n a l equation may find uses i n the fields of apphed mathematics other than elasticity. T h e difficulties encountered i n obtaining explicit solutions b y the general methods sketched i n this paper are generally traceable to the determmation of conformai mappmg functions i n a sufficiently simple form. I t was p o i n t e d out above that if the mapping function z=
6. 1 2 3 4 [5] 6] 7] 8] 91 (10 11 12] 13] (14) 15 16] 17]

References

A. E . Green, Proc. Roy. Soc. [A] 173, 173 to 192 (1939); 174 (1940); 184, 181, 231, 289, 301 (1945). E . Goursat, Bui. de la Soc. Math, de France 26, 236 (1898). S. Holgate, Proc. Roy. Soc. [A] 185, 35, 50 (1945). S. G . Lechnitsky, Eine statisch flache Aufgabe aus der Elastizitätstheorie eines anisotropen Körpers. Applied Mathematics and Mechanics (Leningrad) 1, No. 1, 90 (1937). A . E . H . Love, A treatise on the mathematical theory of elasticity, p. 204 (Cambridge University Press, England, 4th ed. 1934). R. Morns and G . H . Livens, Phil. Mag. Ser. [7] 38, 153 to 179 (1947). V. Morkovin, Quart, of Appi. Math. 1, 116 to 129 (1943). N. I. Muschelisvili, Bui. Acad. Sci., U S S R (1919), pp. 663 to 686. See also [9). N. I. MuscheUsvili, Math. Ann. 107, 282 to 312 (1932). N . I. Muschelisvili, Zeitschrift für angewandte Mathematik und Mechanik 13, 264 to 282 (1933). N . I. Muschelisvili, Monograph published by Academy of Sciences, USSR, 635 (1949). I. S. Sokolnikoff, Mathematical theory of elasticity, 97 to 263. (McGraw-HUl, New York, 1946). I. S. Sokolnikoff, Bui. Am. Math. Soc. 48, 539 to 555 (1942). A partial but representative bibliographical list is contained in this paper. I. S. Sokolnikoff, Bui. A m . Math. Soc. 47, 391 (1941). A. C . Stevenson, Proc. Roy. Soc. (A] 184, 129 to 218 (1945). A. C. Stevenson, Phil. Mag. Ser. [7] 34, 105 (1943). G . I. Taylor, Proc. Roy. Soc. [A] 184. 181 (1945).

7«

Conformai Mapping Related to Torsional Rigidity, Principal Frequency, and Electrostatic Capacity G . Szeg51

1.

Introduction

I n a project sponsored b y the Office of N a v a l Research and conducted at Stanford University since 1946 oy G . Pólya and G . Sze^ö, the task was set to i n v e s t i a t e relations between certain geometrical a n d physical quantities associatea w i t h a given solid (surface) m space or a given domain (curve) i n the plane. Various mathematicians from other universities, i n particular M . Schiffer from H a r v a r d Univ e r s i t y and Ä. Weinstein from the N a v a l Ordnance Laboratory, and graduate students from Stanford U n i v e r s i t y participated i n this studv. A n accoimt of the principal results of the past years is given i n a book [2]* b y G . Pólya and G . Szegö. I n the present lecture only a part of these relations is discussed, namely those which involve conformal mapping. Three quantities are considered: (1) the torsional rigidity of a cylinder, (2) the princ i p a l frequency of a vibratmg membrane, (3) the electrostatic capacity of a solid (surface) w i t h rotational symmetry. A s to (3), the relation of the electrostatic capacity to conformai mapping is well known i n the case of an infinite cyhndrical solid. T h e solids (surfaces) of revolution studied here constitute a n essentially different case. The three quantities mentioned can be characterized as extrema of certain problems i n the calcidus of variations. We give these characterizations i n a slightly modified form which might be useful i n apphcations. Two types of conformai mappinjg; are considered. L e t F be an arbitrary closed curve i n the complex z-plane, a n d ^ o ^ n arbitrary pomt m the interior of T represented b y z=ao* T h e " i n t e r i o r mapping function", 2 = 2<(f)=ao+aif+a2f*+

+anr

+

(7.1)

maps the interior of the unit circle l i ' K l onto the interior of F i n such a manner that {=0 is transformed into z—€to. T h e condition a i > 0 normalizes this mapping. T h e "exterior mapping function". (7.2)

2 = 2.(f)=6.f + 6o+6_,f-'+6_8r-*+

maps the exterior of the unit circle J f | > l onto the exterior of F i n such a manner that the point at infinity is transformed into itself. T h e mapping is nonnalized b y the condition 6|>0. The constant GI^^TÌ is called the interior radius of F w i t h respect to the point po; we denote the maxim u m of r^, for a l l positions of po i n the interior of F, b y f. T h e constant bi=r is called the exterior radius or transfinite diameter of F.

2.

Torsional Rigidity

(a) L e t Z> be a plane domain bounded b y the smooth curve F. with D is defined b y p

IUI

r

A

>

*

J

>

l

v

*

T h e torsional rigidity P associated

'

where dc is the area-element of D and both integrations are extended over D. T h e admissible functions y satisfy the boundary condition / = 0 on F. F o r any chosen admissible fimction /, the right-hand expression i n (7.3) yields an upper bound for 1/P. 1 Stonlcird Uxuvontty, Stanford, Calit *Flfar«8 In brackets bidicate the literature referenoes at the

of this paper.

Instead of choosing/ we prefer to choose its level curves 'ii(x,y)=X, where the parameter X runs f r o m 0 to 1. T h e curve « = X contains the curve u = X ' if X > X ' . The limiting case X = 0 corresponds to s o m e point po i n the inteiior of D, while the case X = l corresponds to the boundary curve r . We 8etf=k{u), where hiu) is a smooth function satisfying the boundary condition A.(1)=0. Then we find

M1)=0,

(7.4)

B{\)h{\)d\y

where ^ ( X ) = / ! g r a d u\ds,

-B(X)=/|grad u\-^ds.

(7.5)

B o t h integrals have to be extended over u=\; ds is the arc-element of this curve. (b) N o w we determine h{u) i n the most advantageous way, that is, so that the quantity on the right-hand side of (7.4) should be a minimum. Introducing the function

C(\)=£

(7.6)

B{()dt,

we obtain on integrating by parts : £

Bi\)hi\)d\

=

<7(X)Ä'(X)rfX,

so that by Schwarzas inequality we have

Here the equality sign holds if A(X)= I

(7.7)

T h i s is the function h{\) for which the right-hand side of (7.4) is a minimum. inequahty

This results i n the

The functions A{\) and (7(X) occurring i n this inequality have a simple meaning: ^(X) is the " f l u x " of u across the level curve u=\; C{\) is the area of the domain bounded by the level curve u=\. (c) We apply this result by choosing for the level curves the curves corresponding to the concentric circles |f| = X i n the conformai mapping z=zt(f). Then dn

.1

dn

Because of the conformality the ratio ds/dn is invariant, that is, ds/dn=\dd/d\t of the imit circle. Hence -4(X)=2TX,

where dB is the element

C(X)=irSn|aJ*X»», n-l

(7.9)

and we obtain the interesting inequality ^ S S

zrf^k«l'l«»l^

(7.10)

n - i 111-1 fn-\-n

Fron^ (7.10) we conclude, for instance, that

(l/2)Tr*^ (l/2)Tr?.

O r : Of all domains D with given

interior radius rt relative to a fixed interior poird p^y the torsiorial rigidity P is a minimum for ahout the point po*

a circle

(d) A less informative inequality arises b y choosing for h{\) that function which is the " b e s t " i n the special^case when D is the unit circle; this is the function A(X) = 1—X*, so that b y (7.4) we have

W h e n we choose the level curves as above, this leads to the inequality (7.12) f r o m which the inequality t h a n (7.12); indeed,

(1/2) xf* follows again. mn m-\-n

^2

It is easy to see directly that (7.10) is sharper n

m+1

n+1

(e) T h e characterization (7.3) of the torsional rigidity can be extended to multiply connected domains whose boundaries consist of a smooth curve F and of a finite number of mutually exclusive curves which are situated i n the interior of F . The following modification [3] of (7.3) is then necessary. T h e admissible f i m c t i o n / must be 0 on F and constant i n the closed interior of the other curves, and the integrals i n (7.3) must be extended over the complete interior of F . (The contribution of the constant parts to the numerator in (7.3) is zero.) L e t po be again a point i n D. W e define Green's function of D relative to po i n the usual way as a function G harmonic i n J9, having a source-point at po, and vanishing on the complete boundary of Z>. I n the vicinity of po we have <7=log^-log^-|-j/, where r< is a suitable positive constant and ^ a regular harmonic function vanishing at poTi is ageneralization of the interior radius introduced above. We prove now the validity of the inequality

(7.13) The quantity

P^(l/2)irr*^(l/2)irr; i n this more general case. F o r this purpose we choose for tt(a:,v)=X the levelTcurves of Green's function, u=e~^t so that Igrad u|=c~*|grad (?|=Xd(?/dn; hence again . A ( X ) = 2 x X . p A s to C(X), this is the area of the domain u{Xjy)^ \ log (1/X). T h e Green's function of this domain relative to p^ w i l l be tf-log so that its interior radius is Xrf. C(\) ^ T(Xr()', so that indeed

i=log — l o g

^+9^

According to a generalization [4] of the area-theorem we have 0 -2Tr

3.

'^^-2'^-

Principal Frequency

The principal frequency A of a membrane D can be defined b y A*

(7.14)

where the admissible function / satisfies the same condition as i n the problem of torsional rigidity. Choosing again some level curves u{x,y) = \, a,ndj=h(u), we obtain the bound

A*^'^, £

. B{\) (A(X) )«dX

A(1)=0,

(7.15)

where ACk) and B{\) have the same meanmg as in section 2. W e apply this general result by takin; again the ciu^es corresponding to the concentric circles i n the interior mapping; hence (7.9) holds an^ we have 1

r'(A(X))»X»"-»dX

"

¡¿0

a

(7.16)

It seems difficult to determine the " b e s t " function h{u) in a convenient form. T h u s we proceed as in section 2(d) and choose for A(X) the fimction which fimiishes the solution of our problem in t h e special case when D is the unit circle with center at po. T h i s is the function (7.17) where j is the least positive root of the Bessel function Jo1 A

2

a

n-1

^2n-i.

T h e resulting inequality

M,=j\j,{j\)n'd\.

(7.18)

furnishes, among other things, the following: X
4.

Electrostatic Capacity

(a) L e t S be a surface of revolution in space, C its electrostatic capacity with respect to the infinitely large sphere. L e t r be the exterior radius (transfinite diameter) of the meridian curve of S . We establish then the following relation between the quantities 0 and 7: (7.19)

C^^r;

the equality sign holds if the given sm*face is a circular disc arising b y the rotation of a finite segment (needle) about its axis of symmetry. (b) Seeking in general upper bounds for the capacity of a surface iS, we may use Dirichlet's principle:

in^y/rigrad/PrfF,

C

(7.20)

where the admissible function / is defined i n the exterior of 5, satisfying the c o n d i t i o n / = 1 on S a n d / = 0 at infinity; dV is the volume-element. We choose a set of level surfaces u(x,y,z) =\0^\^ly where X = l corresponds to S and X = 0 to the infinitely large sphere. Also we set/=A(w) and determine h(u) in the most advantageous way. T h i s results in the mequality (4T)

-1

(7.21)

£{A{\))-'d\

The last integral is extended over the level surface •u=X with the surface element dc. T h i s quantity AiS) corresponds to that mentioned in section 2 and is denoted b y the same letter. (c) We apply this result to a surface of revolution b y choosing the level surfaces as follows. We consider the exterior level curves of the meridian curve mentioned i n section 1; these cmves correspond to |f| = X"* in the conformal mapping z~Zt(t)' N o w we rotate the level curves about their axis of symmetry, which we assume to be the real axis of the complex ;?-plane. T h i s yields the following general ineauality for the capacity of a solid of revolution i n terms of the exterior mapping 2 = 2 j ( n of the meridian curve: d\

XM(X)'

20

(7.22)

where A f (X) is the mean value : M(X)=-

{'

Jo

^z.{\'h'')de,

(7.23)

Inserting the power series expansion (7.2) for z,(\) and performing the integration, we obtain 6.=r

«=1,-1.-3,-5,

M ( X ) = ^ S ^

(d) N o w we apply a theorem of O . Golusin [1] stating the inequaUty e.(fi)—2.(f^ provided |f,| = | f , | = X ~ ' > l . S

Choosing f i M

-»+1

(7.25)

è6.(l-X')=r(l-X»), f j=X

èrd-X»),

we obtain n=l,-1,-3,-5,

t

«

f

(7.26)

O m i t t i n g the terms 6 i = r , dividing b y X ' , and integrating from 0 to X, we find h

x~"

n

n

or ilf(X)^(2/T)r(X-iH-X).

1,-3-5,

#

4

(7.27)

T h u s from (7.22) we have

20

(2/T)r(l+X»)'

(7.28)

which estabhshes (7.19).

5.

References

[11 G . Golusin, Recueil Mathématique (Mat. Sbornik), N. S. 1» [61], 183 to 202 (1946). 2) G . Pâlya and G . Szegfi, Isoperimetric inequalities m mathematical physics, Ann. Math. Studies 27 (1951). 3] G . Pôlya and A. Weinstein, On the torsional rigidity of multiple connected cross-sections, Ann. Math. S2, 154 to 163 (1950). [4] G . Szegö, Sitzungsberichte of the Berlin Academy, p. 477 to 481 (1928).

8.

Fluid Dynamics, Conformai Mapping, and Numerical Methods Andrew Vazsonyi *

1.

Introduction

T h e theory of perfect fluids is governed hy the Laplace equation of potential theory. Potential theory is one aspect of conformal mapping, and conformal mapping can be interpreted as the geometry of perfect fluid theory. M o s t applications of conformal mapping to perfect fluid theory use analytical concepts, and the geometry of conformal mapping appears as a visual aid. However, certain numerical methods depend more heavily on the geometry of conformal mapping. One often has to solve a series of flow problems w i t h identical or similar boundary conditions. Numerical methods usually do not perm i t simplifications just because the boundary conditions are similar, and so the problem arises whether special procedures can be devised which w i l l permit saving i n the amount of the numerical work to be performed. T a k i n g as an example the use of the method of finite differences, one reahzes that a considerable amount of work could be saved if the geometrical boimdaries could be made simple. T h i s simplification of boundaries can often be accomplished by conformal mapping, as, for instance, i t is possible to change a comphcated boundary into a straight line. T h e method of finite differences yields answers much more readily w i t h such boimdary conditions. A few examples w i l l make the apphcation of this method clear.

2.

Perfect Fluid Flow About Arbitrary Airfoils

Suppose one has to compute the flow w i t h the aid of the method of finite differences. I n the vicinity of the airfoil, a very fine lattice w i l l have to be used i n order to get accurate solutions. However, it can be shown that w i t h the aid of Joukowski's transformation, most airfoils can be transformed into approximate circles. W i t h a logarithmic transformation, an approximate circle can be transformed into an approximate straight line. T h e transformed flow along the straight line can be determined w i t h the aid of a fairly uniform and coarse net, and, as a consequence, the amount of numerical work c a n be reduced.

3.

Compressible Flow Around Airfoils or in Channels

I n this problem one might have to solve the compressible flow equations w i t h identical geometrical boundary conditions but for different M a c h numbers. If one attacked the problem directly by the method of finite differences for each M a c h number, a large amount of numerical work would have to be performed. However, w i t h the aid of conformal mapping one can transform a channel or an airfoil into an exact straight line. The mapping function can be computed by using the method of finite differences. Once the transforming function is established, the transformed compressible flow equations have to be solved i n the transformed plane w i t h straight line boundary conditions. T h i s method saves a considerable amount of numerical work [1, 2].*

4.

Perfect Fluid Flow Centrifugal Impellers

Often the shape of centrifugal impellers is close to logarithmic spirals. B y the use of a logarithmic transformation, the impeUers can be transformed into approximate straight lines. Because of the simplification i n the boundary conditions, the flow problem can be solved more readily i n the transformed plane. A g a i n , the conformal mapping saves a considerable amount of numerical work. 1 U . 8. Naval Ordnance Test Station. Pasadena. Calif. 'PIgurea In brackets indicate the literature references at the end of this paper.

5.

Perfect Fluid Flow Through Cascades of Arbitrary Airfoils

T h i s problem requires the solution of Laplace's equation under very intricate geometrical b o u n d a r y conditions. However, it can be shown [3] that, with the aid of the tgh z function, the Joukowski transformation, and the logarithmic transformation, the cascade can be transformed into an approximate straight hne. The flow problem can be solved i n a new plane w i t h the method of finite differences. W h e n one is required to make a complete study of a series of cascades w i t h fairly similar airfoil shapes, a tremendous amount of numerical work can be saved. The conformal mapping can be accomphshed i n a purely geometrical fashion with the aid of charts prepared i n advance.

6.

Summary

It can be seen that, w i t h the aid of judiciously selected conformal maps, the numerical work to be performed i n the solution of flow problems can be considerably reduced.

7.

References

[1] H . W. Emmons, The theoretical flow of a frictionless, adiabatic, perfect gas inside of a two-dimensional hyperbolic nozzle, N A C A Technical Note 1003, May 1946. |2] H . W. Emmons, Flow of a compressible fluid past a symmetric airfoil in a wind tunnel and in free air, N A C A Technical Note 1746, Nov. 1948. [3] A. Vazsonyi, On the aerodynamic design of axial-flow compressors and turbines, J . Appi. Mech. 15» 53 to 64 ( M a r c h 1948).

9. Use of Conformai Mapping in the Study of Flow Phenomena at the Free Surface of an Infinite Sea Eugene P. Cooper ^ The problem of the entry, from ah:, of large, high-speed solid bodies into water is essentially a potential problem, since the enects of gravity, viscosity, and compressibOity are minor and the motion of the water starts from rest. The determmation of the shape of the free water surface d m i n g the entry is a major part of the problem. One might think that such determination, i n the case of twodimensional bodies, could be accomplished b y the use of the Helmholtz-Kirchoff theory. T h a t this m a y not, i n fact, be done is due to the circmnstance that the free surfaces are not streamhnes and that, at any particular time, the fluid velocity v varies widely over the stu^aces. The potential time-derivat i v e term i n Bernoulli's principle is essentially involved. I n the case of a body h a v i n g straight sides, the Helmholtz method, using Schwarz-Christoffel transformations, would be applicable i f the flow were linearly bounded both i n the potential, or «vplane, and i n the f-plane, where i ' = l n (~~dz/dw)y but, as a matter of fact, i n the entry problem, the flow is not hnearly boimded i n either of these planes. There does exist, i n the entry problem, a fairly simple conformal mapping which yields a linearly bounded flow region. I f the complex velocity is v=^Vj—i v,, and the complex position is z=z+iyf then, i n the plane of

h=J*')Jdv/dzdZf

(9.1)

the flow is linearly bounded. Thus the A-plane might be chosen to play a rôle similar to the {'-plane of the steady-state, free-streamline problem. Since, however, there seems to be no way to choose another linear conformal map to correspond to the linear potential map of the steady-state problem, the Helmholtz method docs not lend itself to sunple modification to treat this sort of transient flow problem. A limiting case of the entry problem is that of glancing entry. T h e question naturally arises : Is the steady state implied b y 0® entry angle an appropriate description of the limit of glancing entry i n the sense that the flow i n this limit is equivalent to the steady-state flow? The a a w e r to this nuestion is generally i n the negative, and may be illustrated b y the character of the solutions to the steaay-state problem. W e now investigate such solutions. A s an idealized body let us consider a half-plane whose edge is parallel to the undisturbed water surface. L e t the body move w i t h constant velocity, V, p a r a l l d to tnis surface, or consider the body fixed w i t h the fluid (water) flowing against it w i t h velocity, F , i n the x-directions at infinity. This problem was first solved b y H , Wagner w i t h reference to the theory of the landing of seaplanes. T a k i n g V — 1 for convenience and introducing Vx+i a^id the complex potential function w=t^+iXj we obtain the mappings as shown i n figure 9.1. r o i n t 3 is a stagnation point whose position is to be found. T h e only reasonable physical assumption w i t h regard to the oehayior of the free streamline, 1-2 is that i t is parallel to the plate at an infinite distance up the plate, forming a jet of w i d t h 5. B y the Schwarz-Christoffel transformation we have

dw

di

t—a

Ò

ir(l — a )

t—\

(9.2)

and 1

dt V

1

(9.3)

i n which r=a-(l-aM(-a),

(9.4)

a n d the constant a is determined by the indicated " j u m p " i n the f-plane, namely, a = c o s fii > U . 8. NftTft] Ordnuoe Test Station, Pasadena. OBIU.

(9.5)

PHYSICAL

PLANE

^ - PLANE

(^^( Velocity riol)

(Streom Function)

W-PLANE

INTERMEDIATE t - P L A N E t = t^tily

FlQURE 9.1.

Integration of (9.3) and use of dzjdt—{dzldw){dwldt)

dx dt

gives

i (1 — cos i3) - sin^ i3 (f

—

V^' — 1)

¿-1

7 r ( l — C O S /3)

(9.6)

and 5

dt

7r(l

—

cos

sin 0 — s i n j3 cos j3 (t— ¿-1

V*'—l)

(9.7)

Thus, b y elementary integration, y is given as a function of x through an intermediate i)arameter, t. T h i s is the equation of the free water surfaces. W c only need to take care in eq (9.6) and (9.7) to choose the appropriate values of the many-valued functions resulting from the integration. N o w consider the limit i—* oo (point 5). W e have

dt

->-^

(9.8)

and

dx

~

5

sin i3

W~*7Kl—cos Thus y

CO as In X, for

» .

1

(9.9)

t—1

T h i s is also true for x-^—

<»

W e see, therefore, that it is impossible to satisfy the geometric boundary conditions on the free surface at infinity and that, i n particular, the solution is characterized b y a parameter 5 which is not determinable. T h i s result is not surprising since there is no characteristic length i n the problem. We m a y , therefore, not expect definite surface dimensions to be determinable. N o w one may modify the physical situation by introducing a characteristic length as i n figiu:e 9.2. W e have taken the case ^ = 9 0 ° for simphcity. Using the mapping techniques as above, with 1 —sin y

a

(9.10)

1+sm y

we find {l +

d2__./à\ dt^

H J

a)-i^/{t-ay-{l-\-à)

'

(9.11)

¿-1

T h u s the introduction of the characteristic length I does not suffice to yield a physically meaningful solution. T h i s is not brought out in Milne-Thompson's book Theoretical Hydrodynamu.fi*\ in which the problem is discussed. ||]

FIGURE 9.2.

It has not been found to be possible to obtain a physically meaningful solution to any problem of this type when the sea is infinite. G . Kreisel, in a wartime report, discussed the rôle that gravity might play m problems of this type, and showed that a gravitational term, however small, would guarantee the existence of a solution, meeting all the boundary conditions for the problem of the infinite halfplane. However, the introduction of such a term spoils the mapping technique. If the sea is considered not to be infinite, then the mapping technique will yield a solution satisfying all boundary conditions. A . E . Green carried through the analysis for the case i n which the sea had a finite depth with a flat, sohd bottom. T h e mappings introduce elliptic functions in this case and make the analysis very involved. T h e author has carried through a similar analysis replacing the sea by a two-dimensional jot of finite width H. In this case the mappings and integrations introduce elementary functions only. F o r a plate of inclination /3 the surface shape is given parametrically b y dt {t+b){t-\)

— cos Q

tdt {t+h){t~\)

i sin â

1

di+Ce'^

(9.12)

in which b and C are determinable constants. T h e parameter H essentially characterizes the solution and, as >CDJ the logarithmic divergence characteristic of the infinite sea case appears. In this limit the jet is defiected through an angle of approximately 1°. T h e author has also carried through the case of the finite plate, with ^3 = 90°, dipped in the two-dimensional jet. Again, elementary solutions may be obtained in terms of the jet width H with the solution diverging as H - * <».

^

10.

An Application of Conformal Mapping to Problems in Conical Supersonic Flows R. C . F. Bartels and O. Laoorte ^

1 • Introduction A general method employmg the facilities of conformal mapping is here presented for treating the linearized equations of the supersonic flow past conical bodies i n three dimensional space. T o illustrate the method, the flow past a delta wing whose leading edges he w i t h i n the M a c h cone is determined. T h e work as presented here appears i n part i n two earlier reports [1]* and [2] ' having limited distribution. The transformations of the linearized equation are equivalent i n the end result to those employed b y Busemann [3] and Stewart [4] i n their important works on the subject. However, the presentation given here brings i n sharp focus the analogy between the problem for the conical flow and the classical problem for a minimal surface. T h e components of velocity are represented i n complex form b y integrals similar to those of Weierstrass and, as a consquence, are completely described i n terms of a single fimction of a complex variable. Formtdated i n this way the corresponding boundary value problem for the flow past a body is amenable to treatment by methods i n the theory of analytic functions of a complex variable utilizing conformal mapping. I n the treatment of the problem for the delta wing, i t is shown that the boundary conditions prescribed on the wing surface and the M a c h cone are not sufficient* to determine the flow uniquely. O n the other hand, it is shown that when these conditions are supplemented b^ an additional condition requiring the normal force coefficient of the wing to be finite, a flow without discontinuous vortex sheets is completely determined for all angles of attack and yaw.

2.

Mathematical Formulation

Consider the irrotational flow past a slender pointed body placed i n a uniform stream moving w i t h supersonic speed, the vertex of the body pointing i n the direction of the oncoming stream. Liet X, F , and Z be the coordinates of a right-handed, rectangular Cartesian coordinate system w i t h the o r i ^ n at the vertex of the body, and the positive direction of the Z-axis coincident w i t h the direction of motion of the undisturbed stream ahead of the body. Further let the constants w„ and M denote the velocity and M a c h number, respectively, of the undisturbed stream, and let « , t?, and w denote the components of the disturbance velocity i n the directions of the X, F , and Z axes, respectivelv, so that the components of the total velocitv of the flow i n these directions at any point are u , Ü, and w+ti?,, respectivdy. T h e n , i n accordance w i t h the linear theory,^ the components of velocity t*, and w are given by the equations U

^•^

ax

„=

àY

«,=_^t àZ

(10.1)

where the potential {X, Y, Z) of the disturbance velocity satisfies the linear differential equation àX^^àY'

/3

0,

(10.2)

w i t h 0=(M^—1)*'*, and the boundary condition that on the surface of the body the normal of the total velocity be equal to zero. Thus if 5(.Y,F,Z) = 0

(10.3)

< University of Mlcbisan, Ann Arbor. Miob. * Ficures in brackets indicate tbe llteraturv references at the end of this paper. s The reports (1 and 3] will henceforth be referred to as B B Report and C M Report, respectlTely * See references [fi, 6], and [7, p. 24].

»49581—52

91

represents the equation of the body surface, the boundary condition becomes ''5X+''5F+^'^+'^')5Z=^ T h e characteristics of the hyperbolic differential eq (10.2) are the cones of revolution w i t h axes parallel to the Z-axis and w i t h semivertex angle m defined by the equation ^=cot"'/3.

(10.5)

T h e downstream halves of these cones are significant physically since each envelops the domain w h i c h is influenced by a small disturbance originating at its vertex. These semicones are identified w i t h t h e well-known M a c h cones of sonic disturbance i n a stream of uniform supersonic velocity. If i t is assumed that the body is entirely enclosed w i t h i n the M a c h cone attached to the vertex o f the body having the equation Z^-fi^(X^+Y^=0, Z^O, (10.6) then, as is well known from the properties of the solutions of eq (10.2), this surface constitutes a boundary separating the region of the constant state of the flow ahead o i the body from the region of the disturbed flow adjacent to it. According to the theoiy of oblique shock waves, such a transition can occur i f and only if the velocity is continuous. Therefore, at points of the M a c h cone the potential ^(A^, Y, Z ) must also satisfy the condition: ^ u=v=w=0. (10.7) T h u s i n the linear theory the problem of determining the flow past a slender pointed body l y i n g entirely w i t h i n the M a c h cone of its vertex consists of determining the function ^{X, Y, Z) which satisfies the differential eq (10.2) i n the region between the body and the M a c h cone together w i t h the boundary conditions (10.4) and (10.7) at points of these two surfaces, respectively. I t is easily verified by differentiation that the fimctions Uj r, and iv corresponding to a solution it>(X, Y, Z) o f the differential eq (10.2) are also solutions of this equation. Therefore, i n the region between the M a c h cone and the surface of the body, r, and w satisfy the three equations 5Ht 5Z2

(10.8)

0,

and so on, where i n the last two equations,- u is replaced by v and w, respectively. However, i n view of eq (10.1), these three functions are not entirely independent of one another, but, i n addition, satisfy the foUowing three equations i n the same region: bw

5r_5tt

5 w _ bv

bu

bY

bZ~bZ

bX~bX

bY

0.

(10.9)

These are simply the conditions that the velocity field defined by the functions u, r, and w be irrotational. If the surface of the body is conical, that is, if JS(Z^,F,Z) i n eq (10.3) is a homogeneous function of Xj y , and Z , it is easily verified that the functions ultX,tY,tZ)t v(tX,tYftZ)t and w(tXjtYftZ) w i l l , for arbitrary positive values of the constant t, satisfy the differential eq (10.8) and (10.9) and the boundary conditions (10.4) and (10.7), provided that u{XyYjZ), v(XyY,Z) and w(-X',F,Z) do likewise. T h i s observation led B u s e m a n n ' to conjecture that i n this case the velocity field is conical w i t h respect to the origin, or vertex, of the body; i n other words, that the velocity is constant along rays emanating from tms point. This is equivalent to assuming that the components of velocity u , r, and w are homogeneous functions of X, F , and Z of degree zero, or that the velocity potential ^, apart from an additive constant, is a homogeneous function of degree one. Under this assumption, eq (10.1) and (10.9) are readily solvable for the fxmction 0, giving the expression
Moreover, it is also easily verified that i n system of differential eq (10.8) and (10.9) the functions « , and w determined from will satisfy the system (10.8) and (10.9).

Yv+Zw+consi.

(10.10)

this case the function 4> defined in terms of a solution of the b y the relation (10.10) will satisfy eq (10.2), and conversely, a solution of the differential eq (10.2) b y means of eq (10.1) I n other words, the conical flow fields defined by a solution

* It should be remarked that in the linear two-dimensional flow over a wedge the corres; lines cannot be satisfied. >See(Sl.

\g condition of continuity of the velocity across the M a d i

o f either eq (10.2) o r the system of eq (10.8) and (10.9) are identical. T h i s conclusion is not altogether o b v i o u s at first, since the two systems of differential equations are not of the same order i n the function ^. I n view of these observations i t follows that if the flow over a slender conical body is assumed to b e conical, the problem of determining this flow is equivalent to either: (1) determining a single homoeneous function {XtYyZ) of de|;ree one satisfying the differential eq (10.2) and the boundary conditions 10.4) and (10.7) ; or (2) determining the three homogeneous fimctions u , v, and w of d ^ e e zero satisfying t h e system of differential eq (10.8) and (10.9) and the same boundary conditions. T o these conditions w i U be added the requirement that the functions r, and w and their derivatives of the second order b e continuous throughout the region between the M a c h cone and the conical body * and at all regular p o i n t s ' of the body surface. I t w i l l be seen i n the subsequent sections that the second form of the p r o b l e m leads to an elegant formulation i n terms of functions of a complex variable.

f

3.

Transformation of the Equations for a Conical Flow

Since the functions v, and w i n a conical flow are constant along a ray through the origin, these quantities are i n general fimctions of any two independent parameters which define uniquely the direction of the ray. L e t these be taken as the parameters { and i; defined b y the relations ^ X R+Z

Y R+Z

,

R=^JZ^-P^{X^+Y^

(10.11)

Z=R-

(10.12)

Then Z = - B J - ,

0X

Y=R^y

i3X

X

^

where 1 X - | (i-f'-i?*). I t is evident that eq (10.11) and (10.12) set up a one-to-one correspondence between points of the region enveloped b y the M a c h cone (10.6) and the points of the (f,i?,i?)-space, R^Oy l y i n g w i t h i n the circular cylinder of unit radius about the i?-axis. T h i s correspondence is such that cones i n u i e (X^Y^Z)space w i t h vertices at the o r i ^ and contained entirely within the M a c h cone are mapped into cylinders i n the (f,i;,iî)-space l y i n g entirely w i t h i n the i m i t circular cylinder; the M a c h cone itself corresponds to this unit cylinder as a limiting case. Since rays through the origin i n the space of (X, Y^Z) along which the functions u, v, and w are constant are mapped into lines parallel to the R axis, the "flow'* defined i n the image space b y these functions is twondimensional i n character, the same values being defined at corresponding points i n all planes parallel to the ({,i;)-plane. Therefore « , r, and w are functions of f and ij and are defined over the domain of the (t,ir)-plane into which the region between the M a c h cone and the conical body are projected b y the transformation (10.11). T h i s domain is i n general m u l t i p l y connected (see fig. 10.1). T h e external boundary of the domain is the i m i t circle w i t h center at the origin corresponding to the M a c h cone; the internal boimdary is a curve whose shape is determined b y the shape and orientation of the conical body. I n order to translate the analytical properties of the fimctions u , u, and w i n terms of the parameters { a n d 1?, consider first the differential equation of the type i n (10.2) and (10.8). I t follows, on substituting X, F , and Z from (10.12), that a solution * of eq (10.2) which is a homogeneous function of degree n i n the variables F , Z satisfies the equation n(n4-l) X=*

(10.13)

of the homogeneous functions u , v, and w is zero, these fimctions are solutions of the Laplace equation obtained from (10.13) b y setting n = 0 . Hence, tt, p, and w are • It is well known that the velodty of the flow In (be raglon between the Mach cone and the conical body is not necwesarOy continooos If the body has a shani traiUnc edge. In this case a sarfaoe along which the velocity Is disoontlnuons la admissible, the soKsUed trailing vortex ibeet. This case will not be eoDsidend here. ' A reffolar pobit (or ray) of the sarCace Is one at whldi the sarCaoe has a oonttntums nonnal vector. > T h e Dvameters C.« are analogoos to the coordinates of the stereographic projection of the point (XfR^.YfR'.Z/Fri, where ff's-JTH-VH-Z>, on the unit 8pber« with center at the origin onto the V)-plane. Equations 00.11) and 00^2) are obtained In a similar nuumer if the sphere Is replaced by the hyperboloid Z>~0>{X*-\-Y^-I and the center of projection la taken as (0.0,-1). See for example. Laporte and Rainlcb |9) for an application of the ttenocrapbio paruneters In the theory of minimal sorfaoea.

harmonic fimctions of the variables { and 1/ in the multiply connected domain of the -plane. Such II II a triplet of harmonic functions will define an irrotational conical flow in the region between the M a c h cone and conical body, provided that in this region they also satisfy the system of eq (10.9). V

FIGURE

10.1.

f-pfane.

The'jvelocity potential 0 of .a conical flow is a solution of eq(10.13) for n~l. T h e analytical character of this function will be discussed briefly at the end of this section. T h e triplet of harmonic functions ti, tJ, and w can be expressed i n an elegant form b y introducing the complex variable f={+ii?. T h u s if the harmonic functions -u, P , and w are considered as the r e ^ parts respectively of the three analytic functions ?7(f), y ( f ) , and W{^) of f, it can be s h o w n " that u, v, and w will satiny eq(10.9) i n the region between the M a c h cone and the conical body if, and only if, the functions C7, V, and W are given b y the integral formulas

(10.14) where F{^) is an arbitrary function of f which is analytic in the corresponding multiply connected domain in the f-plane. T h e conical flow problem is therefore equivalent to that of determining a single function F{^) which is analytic in the domain of the i'-plane and such that the real parts u, and w of the complex fimctions U, V, and W defined by (10.14) are single-valued and satisfy the boundary conditions (10.4) and (10.7) on the surface of the conical body and the M a c h cone, respectively. For convenience these conditions may also be expressed in terms of the parameters { and rj by making the substitutions (10.12). Corresponding to eq(10.4) the fimctions u, P , and w satisfy a condition of the form Au+Bv+Ciw^+w)==0,

(10.15)

where A, By and C a r e in general fimctions of ? and 17, at points of the curve forming the interior boimdarj^ of the domam m the ^-plane. O n the other hand, it follows from (10.7) that u, v, and w vanish on the unit circle forming the outer boundary of this domain; that is,

u—9=w=0

for

If = 1 .

(10.16)

The function F(f) is therefore determined by the values of u, and w on the boundaries of the domain in the f-plane. This form of the conical flow problem has the obvious advantage of being susceptible to treatment by the methods employed in the theory of functions of a complex variable. Of particular importance in this regard is the method of conformai mapping. • 6 B . Report, sectioiis 1 and 3.

It should be remarked that the functions u, Vj and w corresponding to an arbitrary analytic function F(^) determine a flow w i t h conical properties. I n this respect the role of the fimction F(^) i n the theory of irrotational (linearized) conical flows is analogous to a n d as important as the rôle of the complex potential i n the theory of the irrotational motion of a n incompressible fluid. Of course, the functions u, V, and w obtained b y an arbitrary choice of F ( f ) will not necessarily vanish on the M a c h cone i n accordance with (10.15).'** Incidentally, for the simple case F ( r t = a o P * , where do is real, the formxdas (10,14) yield the familiar Kârmân-Moore solution for the flow over a circidar cone [10]. It shoidd also be remarked that i n general the function F{0 is single-valued i n the domain of the ^"-plane, and continuous on the unit circle. T h e fimction F ( f ) is also continuous at every point of the interior boundary of this region which corresponds to a regular point on the surface of the body. These properties readily follow w i t h the aid of the theory of analytic functions and the corresponding properties of the fimctions u , t?, and w within and on the boimdaries of the region of flow. The analogy between the fonmdas (10.14) and those representing the solution of the m i n i m a l surface problem as formidated b y Weierstrass " is self-evident. I n the Tatter case i t is shown that the coordinates of a point o n the minimal surface can be expressed as the real parts of a triplet of analytic functions, s a y / , ( f ) , ( 7 1 = 1,2,3), given b y formxdae very similar to (10.14). I n this cx)nnection the function defined b y (10.10) is analogous to the Sttitzfunktion of P a m v i n and M i n k o w s k i which p l a j ^ a n important part i n this theory.** T h e value of the Stiitzfunktion at a given point of the minimal surface m a y be defined as the scalar product of the imit normal vector at that point and the position vector of the point, giving the distance from the tangent plane at the point to the origin of the coordinate system. I n the case of a conical flow, i t can be shown that the position vector {X, F , Z) of a point i n space is parallel to the normal vector at the corresponding point on the hodograph (surface) of the flow i n the space of (tt,y,w). Thus the coordinates of the points of this hodograph surface are defined b y the real parts of the integrals i n (10.14), a n d the potential 4> is the scalar product of the position vector of this point and the normal vector ( X , F , Z ) . There is however an important difference between the two theories which makes the conical flow problem more difficult. I n order to determine a minimal surface i t is sufficient for the coordinate functions to satisfy** the relation (r)]*=0. Since this relation is invariant with respect to conformal transformations i n the f-plane, the Weierstrass formulae also enjoy this property of invariance. Equations (10.14) lead to an analogous relation between the derivatives of U, V, and namely,

[f/'(r)i^+iF'(f)]'-^npF'a)p=o. However, this single condition is not sufficient to insure that iz, r, and w satisfy aU three eq(10.9)." T h e remaining relations among the derivatives corresponding to (10.9) are not invariant with respect to arbitrary conformai transformations i n the f-plane. A s consequence the formulas (10.14) are also not invariant under such transformations. Returning to the velocity potential 0, we can obtain particular forms of this function from eq (10.13) b y the method of separation of variables. Thus, i n the notation of (10.12), w i t h {=p cos w, n = p sin « , particular solutions of (10.13) for which the corresponding functions u, w, and w i n (10.1) vanish on the M a c h cone are obtained i n the form: ** /2'*(a„ cos mw+6m sin mw) Qr(0,

m= 0 , 1 , 2 , . . . ,

(10.17) F

where t=(l — X)/\=Z/R, and Q7(t) is the associated Legendre function of the second k i n d ; am and 6« are arbitrary constants. Particular solutions are also obtained b y replacing the functions Q^{t) b y the associated Legendre function of the first k i n d Pr(0» but the corresponding functions v, and w do not v a n i s h on the M a c h cone.*" If n—l, the potential functions i n (10.17) are homogenous of degree one i n Xj F , a n d Z , and define conical flows satisfying the condition (10.7) o n the M a c h cone. Hence, a general representation of the velocity potential of a conical flow is of the form


cos mù)+bn sin mw)

Qr(Q,

1» It can be shown that all three components of velocity will vanish on the unit circle If! - 1 If any one of the three vanishes everywhere on the circle. II B l a s c h k e n i l . p . 238. u Courant'Hilbert [12], p. 44; also Laporto and Rainich [9], p. 157. >• Courant-Hllbert 112], p. 135. 1« B B . Report, p. 10. » Compare with Hayes [13) and Poritsky [14]. 1* T h i s fact was not observed by Poritsky [141-

4.

Application of Non-Euclidean Rotations of Space and Their Equivalence With Conformai Transformations of the r-Plane

I t is advantageous always i n the treatment of boundary value problems for differential equations to consider those coordinate transformations which preserve the form of the differential equation. T o this end, consider i n particular those linear, homogeneous transfoiTnations of space which leave t h e differential eq (10.2) unchanged. This set of transformations contains the following two d i s t i n c t subgroups: Pitch ^ i 8 r i = / 3 7 c o s h a — Z s i n h S ,

(10.18)

PY fiinh a+Z cosh a. and cosh rf; - Z smh 4',

fiXi=pX Yaw

i

(10.19)

pY.^pYy Z,

fiX sinh i + Z cosh 4'

where a and are real. These two families of transformations constitute the so-called Lorentz groups of non-Euclidean rotations of space about the X~ and F-axes, respectively; a and 4( arc referred to as the non-Euclidean "angles of r o t a t i o n " i n pitch and yaw, respectively. A n y solution of the differential equation (10.2) is transformed b y these equations into a solution of the same equation i n terms of the variables Xi^ Yu Z i . Moreover, the M a c h cone (10.6) is also invariant with respect to these transformations. A s a consequence the velocitj' potential 0 of a linear, supersonic flow field bounded b y a given M a c h cone is transformed by eq (10.18) and (10.19) into the velocity potential of another such field bounded b y the same M a c h cone. The foregoing transformations of space can, of course, be accomplished b y a transformation from the f-plane to the frplane with f i = { i + 2 i 7 i , where {i and rii are related to XijYx.Zy i n precisely the same manner as { and 17 are related to X^Y^Z b y eq (10,11) and (10.12). These transformations are especially'' simple. They are simply conformal transformations of the interior of the unit circle i n the f-plane onto the interior of the unit circle i n the fi-plane. This is made evident b y substituting i n (10.18) a n d (10.19) the expressions for XyY^Z i n (10.12) and the corresponding expressions for XxyYiyZy. After some rearrangement of terms equations (10.18) and (10.19) can be shown to be equivalent to the following two relations, respectively: (10.20)

where Ò = tanh5 ^

(10.21)

and (10.22)

where a=tanh|-

(10.23)

Unfortunately, the Lorenz transformations (10.18) and (10.19) are not conformal transformations of space. Consequently, they do not preserve the form of the boundary conditions (10.4) pertaining to the body surface. Moreover, eq (10.9) are also not invariant under these transformations. A s a consequence, formulas (10.14), as has already been stated, are not invariant w i t h respect to the corresponding conformal transformations (10.20) and (10.21) of the f-plane. F o r purposes which w i l l appear later it will be advantageous to determine the change i n the boundary conditions and the formulas (10.14) corresponding to two successive Lorenz rotations, one i n pitch followed by one i n yaw. Consider first the transformation of the boundary condition (10.4) corresponding to the successive applications of eq (10.18) and (10.19) i n that order. After rearrangement of the terms, the boundary condition on the surface Sx{X,,YuZx)^S(X,Y,Z,)

= 0

(10.24)

in the space of (A't^Fi^Zi) becomes

where = fiu cosh ^+pv

Pu 1

=

sinh a sinh ^—w cosh a sinh 7,

(10.26)

/3v cosh a — w sinh a,

= —i3u sinh ^—fiv sinh S cosh J + K ; cosh S cosh J , and cosh a sinh J/

0 « u

(10.27)

w« sinh a, w « cosh a cosh ?

T h e corresponding transformation of the integral formulas (10.14) can be obtained b y the successive application of the conformal transformations defined by (10.20) and (10.22) taken i n that order. A s a result one is led to the following equations i n which ?7i(fi), F i ( f i ) , Wt(fi) are the analytic functions of [i whose real parts are respectively tne harmonic functions UuVuWi defined i n (10.26): /[rn(l +

Vxitù

ir,,(l-fD+

2T„MF,(fO
i r « ( l - f D +

2T«r,]F,(fOrfrx,

f î ) + i r a , ( l - f D+

2r„MF,(fOrff„

fS +

/[r«(l+fD

T^i(fi)

^/[r»(l +

+

(10.28)

where Fi (fi) is an arbitrary analytic fimction i n the domain of the f i-plane, and the constants T,^ are defined as follows: Tu=aT

sinh' sinh 7,

TM=1,

TM=
ra8= T cosh

Tij=
(10.29)

w i t h a, S, 0y and r given by -I

tanh a=p

tan a,

i = c o s ' a—jS* sin' a,

r

sin 2 a • • H

sin 2M

(10.30)

T h e conical flow problem can therefore be rephrased as a boundary value problem for the analytic function Fxii^ i n the domain of the fi-plane corresponding to the region between the surface (10.24) a n d the M a c h cone i n the (.X'i,yi,Zi)-space. T h e function Fi(fi) is determined by the condition that the single-valued real parts«i,t»i, and Wiof the analytic functions ^/i(fi), V i ( f i ) , W\{S\\ respectively, vanish o n the M a c h cone, and consequently also on the unit circle in the h-pI&i^C) satisfy the condition (10.25) on the surface (10.24). Thè latter condition can, of course, be expressed as a relation of the form (10.15) between the functions on the curve forming the interior boundary of the domain in the fi-plane.

5*

Formulation of the Problem for the Deha Wing as a Boundary Value Problem for an Annular Region

T o illustrate the general theory, let the conical obstacle be a delta wing formed b y a sector of a plane with vertex at the origin and mclined at an an^le a with respect to the Z-axis, the direction of the uniform flow. T h e angle of flare of the wing, that is, the angle of the sector, will be taken as 2Y, and the angle of yaw as ^ (see fig. 10.2). T h e angles a, 7, and 4f are so restricted that the wing is always contained within the M a c h cone (10.6). Traces of the wing sector and M a c h cone in the plane Z=$ are shown in figure 10.3a. If the equation of the plane containing the wing sector is taken as F = Z tan a,

tiie wing corresponds to an arc of the circle/^ {•+,»-2

FiGUBE

10.2.

+

(10.31)

1=0,

The delta wing at an angle of attack and

yaw,

i n the"i[-plane; the wing edges correspond to the end points ({J, 17J) and ( f i , i?i) of| this arc, and the whole lies w i t h i n the unit circle |r{=l (see fig. 10.3b).

T R A C e OF M A C H C O N E C

t^- PLANE

4K

0

T R A C E OF WING P L A N E

Z-i d

FIGURE

"

10.3,

Z-PLANE

Correspondence between the iXYZ)-8pace

•

2.-PLANE

and the f, f i , z, planes.

B y an appropriate choice of a and J i n equations (10.18) and (10.19) defining the non-Euclidean rotations of space, the wing sector can be made to coincide w i t h the .^iZi-plane and to be symmetrically located w i t h respect to the F i Z i - p l a n e . T h i s corresponds to a mapping of the arc of the circle (10.31) w i t h the given end-points onto a symmetrically placed segment of the {i-axes i n the fi-plane (see fig. 10.3c) b y means of the conformal transformation (10.20) and (10.22). T h e coordmates of the end1' Thlfl circle cats the unit circle orthoBonally. distinct lines Is a circle orthogonal to the unit circle.

Indeed, the image of every plane passing through the origin and intersecting the Mach cone In two

points of this segment are denoted by ( ± i i * , 0 ) . I t is readily shown that the appropriate constants a a n d 6 i n eq (10.20) and (10,22) to bring about the desired mapping, as well as the coordinate of the end-points of the segment are expressible i n the form a=tanh

b—tanh

a

-^7

Jts

tanh ^>

(10.32)

where the three quantities a, y, and J are related to geometric parameters of the wing a, 7, and ^ b y the equations tanh a=p tan a (10.33) and tanh ( J ± y) csch a=tan

(^ ± y) esc a.

(10.34)

I n this connection, an arbitrary choice of the so-called non-Euclidean angles of attack, 5, semiflare, Y, a n d yaw, J , w i l l determine a set of angles a, 7, and ^, which are consistent w i t h the requirement t h a t the wing be contained entirely within the M a c h cone. Thus, i n view of the work i n the preceding section, the problem of determining the conical flow past the delta wing can be formulated i n terms of the function Fi(fi) which is analytic i n the doubly connected region of the f r p l a n e boimded externally b y the imit circle |J'i| = l , and internally b y the segment ^{i^ of the real axis. T h e function is determined b y the requirement that the real parts t/i, Vi, and Wi of the functions Ui(^i), F i ( f i ) , and TFi(fi), respectively, defined b y (10.28), satisfy the following boundary conditions: for

f, = 1 ,

(10.35)

a n d , according to (10.25) and (10.27), sinh S,

for

Ifigfi,

(10.36)

It is to be noted that the parameter a i n eq (10.29) and (10.30) is the angle of attack of the delta wing. T h i s is evident o n comparing eq (10.33) w i t h the first of eq (10.30). I n the procedure of solving this boundary value problem, let the doubly connected region i n the fi-plane be mapped conformally onto the annular region i n the plane of the complex variable z^z+iy (see fig. 10,3d) b y means of the relation _cn{zi;k)+ik'sn(zi;k) dn{zuk)

'

(10.37)

with 2 , = - ^ - l o g 2,

(10.38)

where the modulus k of the Jacobi elliptic function is defined b y /:=tanh 7 ;

(10.39)

k' is the complementary modulus, and K is the value of the complete elliptic integral of the first k i n d w i t h modulus k. T h e n the points of the unit circle |ri| = l correspond to the points of the unit circle = 1, and the points of the segment ^{IB i n the f r p l a n e correspond to the points of the circle 2K

where K' is the value of the complete elliptic integral of the first kind with modulus k'. I n particular, the end-points (±{IB,0) of the segment corresponding to the wing edges are mapped into the points z=±ro. " Cf.

C M Report, section 3 .

949581—52

8

A s a consequence of (10.37) a n d (10.38), the mtegrals (10.28) can be written i n the f o n Ui(z)=f{Tn

c n Zi+k'Ti2

sn Zi+t^

y\i.z)=f{rn

c n Zx+k^r^-i

sn Z I + T J S d n Zi)J(z)dz,

Wx(z)

/3/(T8I c n Zi+k'T2z

d n z^j{z)dzt (10.40)

sn Zi + r^i d n z^J{z)dz,

where the function/(g) is analytic i n the annulus ro<|2|
V i = 4 w , sinh a, on|2 P

It is important to observe that the function j(z) is necessarily continuous at points of the u n i t circle |3| = 1. T h i s follows from the continuity of Uj, Wx and their derivatives of the second order o n this circle. T h e function is also continuous at all points of the circle |T|=ro, save possibly at the p o i n t s 2 = ±ro corresponding to the wing edges. F o r , w i t h the exception of these points, Wi, t?i, and their derivatives are continuous o n the wing surface. It should be remarked that the circles concentric w i t h the circular boimdaries of the annulus are mapped b y the transformation (10.37) into a set of algebraic curves of the fourth degree i n the f-plane, which i n turn correspond to elliptic cones i n the {Xx, F i , Zi)-space. Use of this fact was made i n the treatment of the problem of the conical flow past elliptic cones,*'

6.

Characterization of the Function

I t is readily shown that the function/(s) determined b y the conditions (10.41) and (10.42) is expressible i n terms of elliptic functions. However, these conditions are not sufficient to specify the function i{z) completely, b u t must be supplemented b y additional conditions specifying the order of the singmarities at the wing edges.'*' T h e representation of/(s) is unique if, for example, i t is further required that the normal force coefficient of the wing be finite.'* T h e essential arguments leading to these conclusions w i l l be outlined briefly i n this section." T o this end, let the function h{z) be defined b y the equation M2)=2^2^F,(2), and let its values i n terms of the complex variable ^{zx)^h{z)

Zx=^'X-\-iy\

(10.43) of eq (10.38) be denoted b y

(10.44)

B y virtue of the properties of the function Vx{z) w i t h m and on the boundaries of the annular region ''o
reflection ^ that the corresponding analytic extension of the function p{zi) is doubly periodic w i t h t h e periods 42? and 2iK' and, w i t h the possible exception of the points g|= ±2nK+{2m+i)iK\ m,n= 0,1,2, . . ., analytic i n the whole of the 2|-plane. Moreover, p(si)=p{2i), or, what is the same thing, is real for real values of Zy Hence, p(2i) is an elliptic fimction of Zx and is determined at all points o f the 2i-plane b y its values i n the upper half of the period rectangle: 0^a;<4.K, -~K'
2i+T|,

cn 2 , +

T2,

dn s,)/(s).

(10.45)

the zeros of p(zi) include those of the function A:'sn 2 i + c n 2 i + T J J d n 2i, provided/(s) is not infinite at t h e same points. I t is easily seen that these zeros are simple and lie o n the real axis of the Zi-plane a l o n g which y(2) is continuous, and that they correspond to the following two points on the unit circle i n the i'-plane: te (10.46) where x = c o s " ' ((tan a)/fi]; the two corresponding" points i n the period rectangle of the 2i-plane w i l l be denoted b y Zi=ci a n d 2 i = c l . However, neither the order of these zeros of p(2i), nor the orders of its poles, are prescribed b y the boimdary conditions. N o r , for that matter, are the disposition and order of other possible zeros of p(2i) prescribed b y these conditions. I t is possible to show that, as a consequence of the boundary condition (10.42), the residue of p(2i) at each of its poles is necessarily equal to zero, but, other than this, no further limitations are placed o n the orders of the poles b y the boundary conditions formulated for the general problem. Conversely, let the eUiptic fimction p(2i) have the following properties: (a) periods AK and 2iK'; (b) a t most two poles, each w i t h residue zero, i n the period rectangle at 2 | = î K ' a n d Zi=2K+iK' ; a n d finally (c) a t least two simple zeros a t the points Zi=cj and z[=c' on the real axis i n the 2rplane and corresponding to those defined b y (10.46). T h e n there is determined a function/(2) b y (10.46) for which the boimdary conditions (10.41) a n d (10.42) are satisfied and which i n turn determines a conical flow past the delta wing. Since the complete specification of a n elliptic function requires not only the location of the zeros and the poles, but also information o n their respective orders, i t is evident that the function p(2i) possessing the foregoing properties is not imique. Hence the conical flow past the delta w i n g is not completely determined b y the boundary conditions i n the general formulation of the problem. O n the other hand, i t can be shown that the normal force exerted o n a finite portion of the wing bounded b y its edges is finite only if the order of the poles of p(2j) is at most two. Since the residue of p(2i) at each of its poles is zero, the order of its poles i n this case is exactly two. T h i s proves to be sufficient to determine completely the elliptic function p(2j) as weU as the function y(2). Hence, among all conical flows past the delta wing satisfjring the boundry conditions (10.4) and (10.7), there is a unique one for which the normal force coefficient of the wing is finite.

7.

Construction of the Elliptic Function p(2i) for the Case of Finite Lift

T h e construction of the elliptic function p(2i) for the case of finite lift can be made to depend on the fundamental pair of elliptic fimctions w i t h the primitive periods 4K and 2iK': p,(2,)=/r s n ^ 2 i — c n Zi dn Zi

a n d p2(2,)=^ 8n*2i+cn Zi dn Zi

(10.47)

E a c h of these functions has a single pole of second order w i t h zero residue i n a period parallelogram, p,(2i) having its pole at Zi=iK\ and pj(2i) at Zi=2K+iK\ I t follows from the general theory of elliptic functions that any fimction of this class w i t h the periods 4K and 2iK^ having second order poles at the points Zi=iK' and Zi=2K+iK\ at each of which the residue is zero, and having no other poles i n the period parallelogram, can be expressed as a linear Cf. Bieberbach [15], p. 210. Wbittaker and Watson (16). p. 430, et aeq.

combination of pi(ei) a n d pt{zi) plus a constant. be of the form

Therefore the required eUiptic function p(2:i) m u s t

p(2,)=5"{X,Pl(20+X2P3(2,)-l}, where B"y X i , and X3 are constants. I t is not difficult to show that coefficients Xi and X3 are u n i q u e l y determined b y the two zeros Zi=c[ and Zi=Ci corresponding to (10.46).** Indeed, i t can be shown t h a t p{zi) vanishes at these two points if and only if sn^iH- TucnsiH- rasdn^i) {k'snzi—Tijcnsj—

rjsdnsi),

.B' is a n arbitrary constant. 'inally, the constant B' is determuied b y the boimdary conditions (10.41) and (10.42). he aid of the second of ea no.40). ea (10.45) and (10.48). i t can be shown that 1

sinh 5

(10.48) Thus,

(10.49)

where E' is the complete elliptic integral of the second kind corresponding to the modulus k\

8.

Summary of the Resuhs for the Deka Wing

T h e fimctions u , t», and w corresponding to the elKptic function p(2i) i n (10.48) are readily calcidated w i t h the a i d of eq (10.26), (10.40), a n d (10.45). Since i n the linear theory of supersonic flow t h e pressure at points on the wing surface and, consequently, the force exerted on any portion of this surface, is essentially determined b y the v d u e s of the component w of the disturbance velocity, the attention may be restricted to this component alone. T h e following is a summary of the significant results obtained for the delta win^, the intermediate steps i n the derivatives being omitted.** I t is advantageous m the formulation of these results to introduce the real variable 2{.

Pi

where, as {j ranges over the s e g m e n t — ^ ^ { i » i n the fi-plane corresponding to the wing surface, the variable pi ranges over the interval — 1 ^ Pi ^ 4-1. Then the values of the component w at points on the upper and lower surface ^ of the wing are expressible i n the form k cosh

sin'a

w

B

2

\

-

k'^

E ' W c o 8 h 2 ^ - ^ (2

2

(10.50)

VI

here l + k

sinh 4^

ct)E%

2

E ' j r ' cosh 2 ^ - ^ ( 2

the positive sign of the first term i n the brace giving the values of w on the positive (or upper) face of the wing, and the negative sign the values of w on the negative (or lower) face. Since the wing edges correspond to p i = ± 1 , i t follows from (10.50) that the component w is infinite at both edges, except at an edge for which ^±tanh J = 0 . I t follows from (10.34) and (10.39) that the exceptional cases correspond to 4^= ± 7 ; that is, the component w is finite at an edge Iving i n the F Z - p l a n e , B y means of the result m (10.50), the nondimensional normal force coefficient Cn of the wing, defined as the force per unit area of the wing divided b y the stagnation pressure, can be written i n the form Cn =

where C

TÜ

&m

2a,

kk'^ (T —
••CM Report[21,pp.»-37. » For further detsUs see C M Report [3], pp. 28-^. " According to the convention adopted in this report, the "upper" surfooe of the wing Is the high pressure side,

9.

References

[1] O. Laporte and R. C . F . Bartels, Ân investigation of the exact solutions of the linearized equations for the flow past conical bodies, Engineering Research Institute, University of Michigan, Bumblebee Report No. 75 (1948). This paper is based on work done for the Bureau of Ordnance under contract N O r d 7924. (2J R. C . F . Bartels and O. Laporte, Supersonic flow past a delta wing at angles of attack and yaw. Engineering Research Institute, University of Michigan, C M Report No. 471 (1948). This paper is based on work done for the Bureau of Ordnance under contract N O r d 7924. [3] A. Busemann, Infinitesimal conical supersonic flow, Schriften der Deutschen Akademie für Luftfahrtforschung 7 B , No. 3, 105 to 122 (1943). 4 H , J . Stewart, The lift of a delta wing at supersonic speeds, Quart. Appl. Math. 4 , No. 3, 246 to 254 (1946). 5 H , Glauert, The effect of compressibility on the lift of airfoils, Proc. Roy. Soc, (A) 1 1 8 , 113 (1928). 6 L. Prandtl, Über Strömungen, deren Geschwindigkeiten der Schallgeschwindigkeit vergleichbar sind. Jour, of the Aero. Research Inst., Tokyo Imp. Univ., No. 65, 14 (1930). 7] R. Sauer, Theoretische Einführung in die Gas Dynamik (Springer, Berlin, 1943). 8] A. Busemann, Drücke auf kegelförmige Spitzen bie Bewegung mit Ubershallgeschwindigkeit Z. A . M . M . 9, 496 to 498 (1929). [9] O. Laporte and G . Y . Rainich, Stereographic parameters and pseudo-minimal hypersurfaces, Trans. Amer. Math. Soc. 39, 154 to 182 (1936). (10] T . V . Kàrmàn and N . B. Moore, Resistance of slender bodies moving with supersonic velocities with special reference to projectiles, Trans. Soc. Mech. Engr., 303 to 310 (June, 1932). (11] W. L. Blaschke, Vorlesungen über Differentialgeometrie, / (New York, 3d ed., 1929, Dover Publications reprint 1945). [12] R. Courant and D . Hilbert, Methoden der Mathematischen Physik, //, (Berlin, 1937). [13] W. D . Hayes, Linearized supersonic flows with axial symmetry, Quart, of Appl. Math. 4 , 255 to 261 (1946). 14 H . Poritsky, Linearized compressible flow, Quart, of Appl. Math. 6, 389 to 403 (1949). 15 L. Bieberbach, Lehrbuch der Funktionen théorie. 1 (Leipzig, 4th ed., 1934). 16 E . T . Whittaker and G . N . Watson, A Course of Modern Analysis (Cambridge, 4th ed., 1935). 17 O. Laporte, R. C. F . Bartels and R. C. O'Rourke, An investigation of the exact solutions of the linearized ecjuations for the flow past conical bodies. Part III: Supersonic flow past an elliptic cone at an angle of attack, Engineering Research Institute, University of Michigan, C M Report No. 675 (1949), This paper is based on work done for the Bureau of Ordnance under Contract N O r d 7924. [18] C . J . Bouwkamp, A note on singularities occurring at sharp edges in electromagnetic diffraction theory, Physica 1 2 , 467 to 474 (1946). [19] J . Meixner, Die Kantenbedingung in der Theorie der Beugung elektromagnetischer Wellen an Vollkommen leitenden ebenen Schirmen, Annalen der Physik 4 4 1 , 2 to 9 (1950).

11.

On the Helmholtz Problem of Conformal Representation Alexander Weinstein '

1.

Introduction

The theory of jets and wakes was fu^t mvestigated b y Helmholtz. The mathematical nature of this theory leads us to a general mapping problem which is essentially different from Riemann's probl e m . I n the case of the latter a given s i m ^ y connected domain is to be mapped onto a standard domain such as a circle, a half-plane, or a strip. B u t i n the Helmholtz theory only a part of the boundaries of the domain to be mapped is given, this part consisting of the walls of a nozzle or the profile of an obstacle. The remaining part of the boundary, called the "free'' boundary, is not given geometrically, but is only defined imphcitly by an isometric (more properly "quasi-isometric") condition, which may be formulated as follows: the magnification coefficient of the arc element da of the free boundary is to remain constant i n the mapping. (This property of the free stream fines follows immediately from Bernoulli's theorem.) It w i l l be seen that the question of the existence and uniqueness of a solution i n the Helmholtz problem cannot be settled i n the same way as i n the Riemann case. F o r instance Schwarz's lemma is not applicable to the former since we are no longer dealing w i t h the mapping of the same domain by two supposedly different functions. Indeed, a priori there m a ^ exist at least two distinct domains w i t h the same fixed walls but w i t h different free boundaries which can be mapped onto the standard domain. The purely mathematical aspect of the general Helmholtz problem has been intensively investigated i n the last 25 years. T h i s research has led to the development of methods essentially different from those employed i n Riemann's problem. A general review of this work has appeared recently [16].* A review of some other questions i n the theory of jets and wakes can be found i n [19] and [20]. I n the present paper we shall confine ourselves to the discussion of the case of polygonal boundaries. T h i s is of particular importance for purposes of mmierical computation, since it is clear that a numerical solution i n the case of curvilinear walls should start from polygonal walls as a natural approximation. T h e solution for the polygon also has great theoretic interest, since it is the parallel i n tne Helmholtz problem of the Schwarz-Christoffel transformation i n the classical theory of conformal representation.

2.

The Jet Problem

L e t us consider for the sake of definiteness a symmetric jet i n the physical 3-plane. z—x+iy. The upper half of the jet is shown i n figure 11.1. We assume that the given boundaries (or walls) consist of the real axis WQ and of a waU W that has a horizontal tangent at x = — <».

z-PLANE FIGURE

11.1.

1 Institute for Fluid OTiumloi and Applied Mftthanutla, Unlrenlty of Maryland. College Park, M d . OfSoe of Naval ReMaroh. > FIgura In bimckets indicate the literatore ntoranoM at the end of this paper.

T h U work wai done under the iponMrsblp of

The free boundary A is not given geometrically, b u t we assume that i t has a horizontal tangent at x=+ 0 0 . Helmholtz's problem can then be formulated as follows: T o determine the free lines A i n such a way that the domam J bounded b y Wo, W, and A can be mapped b y a schhcht function/(s) = +i}^ on the strip S bounded b y the lines i^=0 a n d ^ = T / 2 . I n this mapping the segment i^=x/2, O < 0 < a > shall correspond to A and the magnification \df/dz\ along A shall be a positive constant which is not given but has to be determined. I n what follows corresponding lines are denoted b y the same letters and w i l l be called walls a n d free lines, even if located i n the/-plane (fig. 11.2). T h e quasi-isometric condition -1

(11.1)

dz

is characteristic for the problem and gives an implicit definition of the free line.

w

A

df dz

-CONST.

TT

1 z-PLANE

f-PLANE FIGURE

11.2.

FIGURE

11.3,

We consider immediately the case of a polygonal channel (fig. 11,3), T h e oldest method of approach i n this case is the Helmholtz-Kirchhoff hodograph method. L e t us introduce the w-plane, where dj

(11.2)

dz

is the complex velocity, and u and v are the components of the physical velocity vector. I n the case of polygonal channels the Kirclmoff method of solution is based o n the assumption that the hypothetical solution has a schhcht image i n the w-plane. This image w i l l obviously be a sector w i t h slits radiating from the apex of the sector. T h i s apex is the origin and is the image of a l l vertices of the channel. T h e circidar arc corresponds to the free boundary (fig. 11.4). T h e radius of the sector corresponds to the magnification \djldz\ of the free boundarv and the end point of a sUt corresponds to the maximum of the velocity on a side of the polygonal wall between two stagnation points. WhOe we do not know the domain i n the 2-plane, we now know, according to Kirchhoff, the d o m a i n i n the/-plane and i n the w-plane, and i t is easy to obtain an explicit formula for w(/). This djjdz is a known fimction w(/). Hence dz/df=ljw(f)f and z is finally obtained b y a quadrature 1

d/.

(11.3)

I n order to obtain w{f) explicitly, we replace w by Z = l o g w. T h e domain i n the Z-plane corres} )nding to our sector is bounded b y one vertical segment and a certain number of horizontal segments, T h e mapping from Z onto / is clearly a repeated application of the Schwarz-ChristofFel formiDa. F r o m the preceding remarks it would seem that the Schwarz-Christoffel formula is sufi&cient to solve the Helhmoltz problem for polygonal channels. However, this is not the case. I n the Kirchhoff method for jets the domain i n the w-plane is known exactly only i n the trivial case of n=l{n is the number of sides of the polygon). F o r n > l we do not know the lengths of the radial segments c u t out of the sector, or, what amoimts to the same thing, we do not know tne coordinates of the end-points of the segments i n the Z-plane. These coordinates are the parameters of the problem and should serve for the determination of a polygon of prescribed shape. However, this method of approach has n o t been followed through b y the author of this paper. Instead of the inverse method of Kirchhoff outlined above, the inverse method of L e v i - C i v i t a , Cisotti, and V i l l a t has been used. T h e latter method applies also to the case of curvilinear boundaries, which cannot be treated analytically b y the Kirchhoff method.

0

W

(-00) w-

w0

PLANE

FlGUBB 11.4.

3.

The Inverse Problem

L e v i - C i v i t a and V i l l a t gave a general solution for the inverse problem; namely, a general formula for the inverse function z(f) which is defined i n the strip. A l l functions defined i n the physical 2-plane can be considered also as functions of / i n S, T h e complex velocity w=df/dz=u—iv considered as a function of / has a constant absolute value o n the line A which corresponds to the free boundary. Jjcvi-Civita puts ^=^=Me"'=Me-'+";

(a.= « + i T ) .

(11.4)

I n this formula, 6 denotes the angle which the velocity makes w i t h the ar-axis. I n the /-plane, ^ is a harmonic fimction of and V*, which vanishes on the straight wall W©. Its conjugate T vanishes on A ; this implies the condition dd/rf^=0. The values of 6 considered as a function of 0 and ^ are not given on the w a l l W i n the/-plane as long a s / ( 2 ) has not actually been determined. T o any arbitrary choice of the fimction B{, f) vanishing o n WQ and satisfying the condition d$/d}l/=0 o n A . A n explicit representative of 6 b y a Poisson integral can be obtained by using i n place of / a new independent variable f={+'ii?=pe*' introduced b y L e v i - C i v i t a and defined b y the formulas (11.5)

r'-i

d^

(11.6)

T h e one-to-one correspondence between the/- and f-plane is indicated i n figure 11.5. Again the same notations wiU be used i n both planes. B y Schwarzas reflection principle the harmonic functions d and T c a n be defined i n the entire circle |f| ^ 1. F o r |f| = l , ö and r satisfy the conditions r(x-
T(2T-
dir -

a) = - d{a),

e(2ir-
= e(
(11.7) (11.8)

T a k i n g arbitrary values for ^(
(11.9)

I n this way we see that the general expression for w (considered as a function either of f or of/) contains an arbitrary function ${
r - 1 rfr

il

r ' - i

where

(11.10)

f

fodfl

(11.11)

w

is the ordinate of the separation point of the free line. remark that w i l l be used later.

Before discussing these formulas let us make a

W

A 1

t - PLANE FIGURE

11.5.

F r o m (11.9) it follows that the harmonic function $ is completely determined by its values on the wall W. I n particular, its values on the free boimdary lie between the m a x i m u m and m u i i m u m of its values on W. I n view of (11.8) a level line d=const. intersects the real axis 17=0 (i. e., the free boundary) i n one point only. I t follows (Theorem of Boggio) that the free boimdary of a jet corresponding to a concave channel (fig. 11.1) is necessarily convex w i t h respect to the flow. T h i s proof was communicated to the author by H . Lewy. The famous general formula (11,10) of L e v i - C i v i t a leads to numerous results obtained b y an appropriate choice of the arbitrary function 6 along the wall Win t h e / o r f-plane. However (11.10) does not yield directly a domain with a prescribed wa 1 i n the physical 3-plane. T h i s problem w i l l be discussed i n the next paragraphs. The method used w i l l be a method of continuity, which is based on a continuous change of the arbitrary function until the prescribed walls are obtained. T h e solution of this problem has required several significant changes i n the standard procedure of the classical method of continuity. Theorem of Boggio,

4.

Polygonal Channels

L e t us consider i n some detail the case of a polygonal wall, which was the first i n chronological order of the problems considered i n this theory. L e t (fig. 11.3) Zo — Uof Zl=Xx+iyij

. .

Zn=X„+iynt

Zn+l=~

<^ +iyn

(11.12)

be the vertices of the given polygonal waU of a channel that is concave w i t h respect to Wo. L e t U be the lengthof thesegment3*0fc_i, ( ¿ = 1 , 2 , . . . , n) and let ^it be its direction, where
A polygonal channel can be represented

the total curvature of the wall. the coordinates

¡üt h, ij, . . . » Int

i n a 2n-|-l dimensional space.

Pit Pit

b y a point P w i t h

• . • » ^1

(11.15)

T h e region M occupied b y the points P is described b y the inequalities ^>0,

Z»>0,

Oá/5*
(11.16)

I t follows from (11.9) that a n y assumed flow w i t h a polygonal channel is given b y taking a n arbitrary positive constant ju and by putting 2n

(11.17)

where the points f i, fa, . . . , f « , (0
r'-irfr

(11.18)

T h e formula of Cisotti reminds us formally of the classical Schwarz-Christoffel formida

( f - f i ) - ^ ^ . . . (r-f»)-^- d i ,

(11.19)

w h i c h gives the mapping of the circle ({"I < 1 o n a polvgon i n the 2-plane. I t should be noted that i n the Schwarz-Christoffel formula the points f* are allowed to lie o n the lower half of the circle |f | = 1 as well as o n the upper half, and that M ÌB merely a positive constant. L e t us recall here that Schwarz^s own work was based o n Riemann's mapping theorem, the reflection principle, and Schwans's lemma. I n fact the fundamental discoveries of Schwarz were inspired b y his desire to determine thè parameters i n the Schwarz-Christoffel formula. T h e determination of parameters was investigated independently b y Schiaffi b y a method called today the method of continuity, b u t he d i d n o t complete his proof (for an historical account and a n alternative proof see [6]). O u r approach also uses a method of continuity but one which is greatly modified as compared w i t h the work of Schlafii and his successors. I n order to prove a certain functional determinant different from zero, Schiaffi computes i t explicitly ; but we replace this procedure b y an indirect proof, dependmg o n a certain boundary value problem (see sec. 7). I t would be of both practical a n d theoretical interest i f the determinant arising i n the Helmholtz problem for polygonal jets could be computed explicitly b y a method similar i n principle to that of Schiaffi. Such a treatment woidd presiunably be based o n the theory of hypergeometric functions of several variables or perhaps higher transcendental functions. Schlàfli's computation was anjJyzed b y J . Kampé de Fériet [17). T h e formula of C i s o t t i contains 2 n - h l parameters Ci, J s , . . . f
(11.20)

A channel given b y a Cisotti's formula can therefore be represented also b y a point Pc i n a space of 2 » + 1 dimensions. T h e region Mc occupied b y Cisotti's points Pc is limited b y the ineqiialities 0
M>0,

0 < < r i < < r 2 < . . . <
(11.21)

T o each Pc corresponds a single point P given by the formulas I

sin ^ ^ " s i n
) + M g s sin i öjt

1

sin C A + s i n
5.

tan
(11.22)

Existence Theore:

The existence theorem for a flow w i t h a given channel states conversely that to each P corresponds at least one Pct or i n other words that the 2 n + l nonlinear eq (11.22) possess at least one solution for a n y prescribed values of the left-hand sides. The introduction of the quantities ft, /32, • • • , as additional parameters, at first glance apparently superfiuous, will prove very useful, since it will allow us to consider walls with variable sides and variable angles [6]. The existence theorem has been proved for all channels of total curvature less than x. The proof may be outlined as follows: T a k i n g ^1—^2= . , . fin=0 and giving arbitrary values U) fx, (TI, , . .,
The corresponding variation of parameters is then obtained b y solving the 2 n + l linear equations 5M + ± | ^ 5 a , - h ± ; | ^ 5 ß „ i-i

bffj

(¿ = 0 , 1 , 2 ,

...,n);

^§i = dßu

(t = l , 2 , . . . , n ) .

(11.24)

I n order to prove that these equations have a solution we have to prove the following fundamental theorem. Theorem A .

The functional

determinant

D

Öjlojljj

. , . jln,

ßij

' . .

jßn)

(11.25)

(This theorem is the only one proved by Schlafli i n his famous paper.) Theorem A is a consequence of the theorem of local uniqueness and will be discussed later. I t holds for aU polygonal channels of total curvature less than x. Moreover, we have to show that during the deformation process the corresponding formula does not degenerate before the polygonal channel Pi is reached. B y nondegeneration we mean that the parameters do not approach indefinitelv close to the boundary points of the region Mc. T o establish this last result we have to prove the following fundamental theorem. Tiever vanishes in the interior of the region Mc,

Theorem B .

For all points P located in a closed subregion of the region M, 1

,n-l)

the qttantities (11.26)

admit positive upper and lower bounds which are continuous functions of IQ, /J.

In. A , ß. Theorem B expresses the a priori limitations of all possible solutions of the eq (11.22) and fortu-

nately follows here b y an elementary estimation of the right-hand side of these equations. T h i s is due to the fact that W is convex. I n the case of the Schwarz-Christoffel formula the corresponding theorem B is more diflicult to prove for nonconvex polygons (see [6]). The proof of theorem B for convex polygons applies without any essential modification to the case of jets w i t h convex walls. T o prove the existence of a formula Ph corresponding to the polygonal channel Pi we proceed as follows: L e t the deformation process be represented b y a line L in the region M and suppose that all points on L except the end-point P i have corresponding formulas. Theorem B shows that during the ) See [9,12].

process of deformation of the channel the corresponding Cisotti point Pc remains i n a closed domain i n the interior of Mc. Hence as P approaches P , along L we can choose among the corresponding points Pc a subsequence converging to a certain limit point Pc. We recall that to Pc there corresponds one point P that aepends continuously on P c . Therefore, the image of Pc is P i . This establishes the existence of at least one set of parameters corresponding to any polygonal channel P i for which theorems A and B are established. The classical continuity method as outlined above is based o n the validity of theorems A and B . However, the theory of topological degree of the mapping of spaces of finite or infinite dimension as developed originally b y Brouwer and brilliantly extended b y Leray and Schauder yields the following remarkable result. T h e existence theorem follows from theorem B alone and is independent of theorem A . Theorem A , however, is needed for uniqueness and will be investigated now.

6.

Local Uniqueness ^

We therefore turn our attention to the much-discussed question of uniqueness: Is a jet (or wake) uniquely determined b y the given rigid wall? U n l i k e that i n the Riemann problem, the question of uniqueness is here more difiicult than the question of existence, and there are many cases i n which the existence but not the uniqueness of the flow has been proved. L e t us first consider the following theorem of uniqueness i n the small for sjonmetric jets: There is no second jet infinitely close to the given jet with the same walls b u t w i t h an infinitesimally different shape of the free boundary. I n order to prove this proposition let us denote hy fi{z) the complex potential of a jet which is infinitely close to the given one. L e t (11.27)

5/=/i(2)-y(2)

be the corresponding variation of/(z). T h e theorem of uniqueness i n the small states that if both jets have the same wall, this function vanishes identically: (n.28)

5/=0.

T h e variation df has been defined here as a function of z i n the domain of the given jet. Since we have a one-to-one correspondence between z a.ndf=f{z), w h e r e / is the complex potential of the given flow, the variation Sf can be considered as a function of / (or of f) and can be computed b y differentiation of (11.10) for fixed value of z. I n this way we obtain the equation 0 = - ^ -

ri^d/-hi6io

or

sJ=wr^^df-iwao,

(11.29)

where Sw and SIQ are the variation of w and lo vnth respect to the parameters at, , , . ,
where

'f-'l'-;

(11.3.)

T h e variations 5Zo, 6/i . . . , 6Z„ of the orifice, and the lengths of the sides of the given wall, are connected with the variations of the parameters Spi, dtri, . . . , 5
(¿=0,1,2, . . . ,n),

(11.32)

w h i c h are obtained by putting Spi= . . . =5/3„=0 i n 11.24. L e t us assume now that the equation SjmO is a consequence of the equations Uo=0, 5li=0, . . . , ¿¿»=0, which express the fact that the second assumed jet has the same walls as the given one. F r o m (11.30), (11.17), and (11.31) i t follows immediately that 6M = 0, «86e[0. 7, 8,0,10, It, and 12].

5
. . . ,

6a» = 0.

(11.33)

T h i s result means that the system of homogeneous equations obtained b y putting i n (11.32) the lefthand sides equal to zero admits only the trivial solution (11,33). T h e eq (11.28) yields therefore the foUowmg result. T h e determinant of the system (11.32), which is obviously equal to the functional determinant (11.25) i n theorem A , is always diflFerent from zero.

7.

Boundary Value Problem ^

The proof of the fundamental eq (11.28), which plays a central role i n this theory, has been established b y studying the boundary conditions satisfied b y the variation hj. Since, b y assumption, both jets have the same wall, the p o i n t s / = / ( 2 ) a n d / i = / i ( 3 ) , corresponding to the same point, a , o n the ^

W

f

A

w f-PLANE FIGURE 11.6.

wall W {or i n the physical ^-plane, both lie on the line W (or i n the/-plane so that the varia tion 3/is real (fig. 11,6). W e have, therefore, the boundary condition (11.34)

on W and i n the /-plane. L e t us imagine for a moment the mapping of a half-plane / (instead of tlie circle of the SchwarzChristoffel formula) o n a polygon. T h e n we would have b y similar argument 5^=0 everywhere o n the boundary and we would conclude immediately that 6^ is identically zero. T h i s would give us the required proof. B u t this reasoning would fail i n the case of the Helmholtz problem, for a point z o n the first free boundary A does not necessarily lie on the second free boundary (fig, 11.7). W

A

Z - PLANE FIGURE 11.7.

Fortunately we have there a natural boundary condition which holds whether or not we have the same rigid wall for both jets. I n fact, from (11.29) we immediately obtain, by differentiation, the fundamental identity, ^ ^ ^ - ^ ^ ^ 5 / = « l o g « . ,

(11.35)

the real part of which, on the free boundary i n the /-plane, yields the following natural boundary condition: ^Jf—^H=-ilog^, See [A, 9].

{dn^d^ld,

(11.36)

where 1/M {d6/d4>) is the curvature K of A. Moreover, for an unvaried separation point, the imaginary p a r t of the same identity (11,36) yields the supplementary condition d(64d d4>

0,

for/=/o.

(11.37)

W e note that the unknown constant 5^ appears explicitly i n (11.36). The uniqueness i n the wall is thus reduced to proving the following: Theorem 1. {Local uniqueness.) The harmonic function 6^ satisfying the boundary conditions (11.34), (11.36), (11.37) is identicaUy zero. The proof of theorem 1 has been obtained b y considering first the special case i n which 3M is equal to zero [5]. T h i s case is that of neighboring jets having not only the same walls but also the same w i d t h at infinity downstream. T h i s assimiption is an artificial restriction but, as w i l l be indicated, the complete proof of the fimdamental theorem 1 may be shown to follow from this simpler and apparently special case [9]. P u t t i n g 3/i=0 and denoting i n this case the corresponding harmonic function 5^ b y j3, we formulate the following special case of theorem 1: Theorem 2. {Local uniqueness in the restricted sense.) The harmonic function 0(^,^) satisfying the boundary

conditions

P=0

on W a n d Wo

dß^de_ dn

do

ß

on A

(11.38) (11.39)

6 identicaUy in the strip S.

classical anDhcation of Green's formula, due to Fourier

n.{{

aß Ò4>

aß

+

de d^

(11.40)

T h e proof of theorem 2 would be trivial if the function dö/d^ were shown to be always negative, but unfortunately i n our problem this usually is not the case. I n fact, i n the important case o f a channel w i t h walls concave w i t h respect to the fluid, dejdi^ is always positive (see theorem of Boggio); i n this case the left-hand side i n (11.40) is the difference and not the sum of two positive numbers. T h e proof of theorem 2 presents therefore a major difiiculty. Proofs valid for more and more extensive classes of given walls have been established successively b y Weinstein [5], H a m e l [7] and [8], W e y l [10], and Friedrichs [11].

8.

Jacobi's Transformation of Ecpiation (11.40) *

T h e latest proof of theorem 2 is that of Friedrichs, who showed that under certain conditions the lefth a n d side of (11.40) can be written as the sum of two positive quantities. T h i s transformation holds provided that there exists a harmonic function U m. S which is always different from zero and which satisfies the natural boundary condition dU^d± dn

di^

U

on A.

(11.41)

Assuming the existence of U, Friedrichs puts i n (11.40) ß=Ufi (Jacobi's multiplication variation), simple computation shows then that (11.40) is transformed into the equation a

t/(i?;+nj)rf^*=0

A

(11.42)

from which follows inunediately the required equations 1 7 ^ 0 and fi^O. L e t us consider now the question of the existence of the harmonic function U. It has been pointed out by C . Jacob [14] that the components of the velocity cos e and i?=/i**«'sind satisfy the natural boundary condition (11.41), a fact easily verified by differentiation. L e t us assume the existence of a component au+bv (where a a n d 6 are two constants satisfying the equation a*-|-6^=l) which never vanishes on the given wall W. T h e n by (11.8) this component is everywhere different from zero and we may take for U the function U=au+bv

(11.43)

I t follows that theorem 2 is valid for a large class of channels, in particular for all concave channels w i t h walls of total curvature less than T . •Se«(ia

9.

Reduction of Theorem 1 to Theorem 2 ^

L e t us consider a channel for which the local miiqueness i n the restricted sense (theorem 2) has been established. W e shall prove that theorem 2 implies theorem 1. L e t us introduce i n theorem 1 as a new imknown the harmonic function 0*=6^+(^—(1/2) T)6 l o g W e have, b y (11.34), (11.36), a n d (11.37), the following boundary conditions for P*: log

P* /3*=0,

dp*^de dn d

d
drj^

/I,

on Woi

(11.44)

on W;

(11.45)

on A ;

(11.46)

for / = / » .

(11.47)

The conditions (11.45) and (11.47) show the existence of a level line P*=0 different from W a n d passing through the separation point/o. T h i s line must have another point i n common either w i t h W o r w i t h Wo or w i t h the free boundary A. L i the first case, we inunediately have the result /3*=0. I n the second case we would have 6ti=0 so that p* wmdd satisfy the conditions of theorem 2 a n d therefore would vanish identically. F i n a l l y , i n the t h i r d case p* satisfies the conditions of theorem 2 i n a certain subdomain S* of S, which again yields the result p* = 0. Since p*^0 impUes Sn=0 a n d 6^=0, theorem 1 is proved.

10.

Uniqueness in the Large

8

L e t us consider again for the sake of definiteness the class of aU polygonal channels for which t h e existence of a jet and the local uniqueness have been established. T h e uniqueness i n the large o r the uniqueness i n the ordinary sense is obvious i n the trivial case of a straight wall running parallel to the a!-axis. T h e corresponding points i n the domains M and Mc w i l l be donated b y 0 and Oc, respectively. L e t us assume for a moment the existence of two points P¿" and Pc* i n Mc corresponding to a single point P i n i l f . W e w i l l establish a contradiction b y linking P w i t h 0 b y a line i n M which lies i n t h e interior of a closed subdomain of AÍ. T o this line correspond two lines P¿" Oc and P¿* Oc i n Mc w h i c h lie, b y theorem 2, i n the interior of a closed subdomain of Mc* L e t us consider the fiirst point Qc w h i c h these lines have i n common. L e t Q be the corresponding point o n the line PO. Clearly theorem I does not hold for Q. W e have thus a contradiction w i t h the assumptions, a n d the uniqueness i n t h e large is therefore proved. T h e uniqueness i n the large for curvilinear walls has been established along the same lines b y L e r a y [13] i n the theory of wakes. I t is remarkable that the theory of L e r a y for wakes, which is based o n the methods of L e r a y and Schauder, establishes the existence theorem for a larger class of problems than that for which uniqueness has been proved. H i s results have been summarized i n [16].

11.

Concluding Remarks

A s mentioned at the beginning of this paper, we have confined ourselves to the case of polygonal channels which chronologically was the first case to be investigated. A s was pointed out before, a topological method of continuity was developed later b y Leray and Schauder which pennits a direct treatment of curvilinear walls. W e believe, however, that the theory of polygonal channels is n o t superfluous, first because i t is the natiu'al approach for numerical computations and for a l i m i t i n g process leading to curvilinear channels (A. Weinstein [5]); and secondly because the topological method itself uses a limiting process starting w i t h the case of a finite number of parameters so that the passage through the finite dimensional case seems unavoidable. Besides the methods of continuity there is another approach to the Helmholtz problem which was proposed somewhat later b y LavrentieflF [15]. LavrcntieflF's niethod remains more i n the realm of c o n formal representation than the methods of continuity which use diversified mathematical tools. I t should be observed that Lavrentieff's method applies only to wakes while the methods of continuity apply equally well to both wakes and jets and cover essentially the whole domain of Helmholtz theory. Moreover, i t should be admitted frankly that the author of the present review d i d not check i n detail T See (9,131. • [See 12,13].

the stringency of a l l Lavrentieff's arguments. A short account of Lavrentieff's theory is to be found i n the report [16]. B r i l l i a n t l y ingenious as is the method of Lavrentieff, on the whole it seems that the methods of continuity furnish a greater contribution to the general development of the mathematics.

The author expresses his gratitude to R . J . Diamond, National Bureau of Standards and California Institute of Technology, for his invaluable aid i n preparing the present report for publication, and also to Professor E . F . Beckenbach, National Bureau of Standards, for his kind interest in providing facdities for this work.

12 • References 1 H , Helmholtz, Ueber Discontinuierliche Flûssigkeitsbewegungen, Monatsberichte der Berliner Akademie (1868). 2 G . KìrchhofT, Zur Theorie freier Flüssigkeitsstrahlen, Journal fur die reine und angewandte Mathematik 70» 289 (1869). 3 T . Levi-Civita, Scie e leggi di resistenza, Rendiconti del Cìrcolo Mathematico di Palermo 3 2 » 1 (1907). 4 H . Villat, Sur la resistance des fluides, Annales Scientifiques de l'Ecole Normale Supérieure 2 8 , 203 (1911). 5 A. Weinstein, Sur Tunicite des mouvements glissants. Comptes Rendus de l'Acaclemie des Sciences de Paris 1 7 6 , 493 (1923); F i n hydrodynamischer Unitätssatz, Mathematische Zeitschrift 1 9 , 265 (1924). 6 A. Weinstein, Der Kontinuitatsbeweis des Abildungsatzes für Polygone, Math. Zeit. 2 1 , 72 (1924). 7 G . Hamel, Ueber einen hydrodynamischen Unitätssatz des Herrn Weinstein, Proceedings of the Second International Congress of Applied Mechanics, 76 (Zurich, 1926). [8] G . Hamel, E i n hydrodynamischer Unit&tssatz, Proceedings of the Second International Congress of Applied Mechanics, 489 (Zurich, 1926). [9] A. Weinstein, Sur les jets liquides à parois données. Rendiconti d. R. Accademia dei Lincei, 119 (1926); Sur le théorème d'existence des jets fluides, ibid, 157 (1927). [10] H . Weyl. Strahlbildung, nach der Kontinuitatsmethode behandelt, Nachrichten der Gesellshaft der Wissenschften, 227 (Gottingen, 1927). 11 K . Friedrichs, Ueber ein Minimumproblem mit freiem Rande, Mathematische Annalen 1 0 9 , 60 (1933). 12 J . Leray et A. Weinstein. Sur un problème de représentation conforme posé par la théorie de Helmholtz, Comptes Rendus de l'Académie des Sciences de Paris 1 9 8 , 430 (1934). [13] J . Leray, Les problèmes de représentation conforme de Helmholtz, Commentarii Mathematici Helvetici 8 , 149 and 250 (1935-6). [14] C . Jacob, Sur la détermination des fonctions harmoniques par certaines conditions aux limites, Mathematica 1 1 , 160 (1935). [16] M . Lavrentieff, Sur certaines propriétés des fonctions univalentes et leurs applications a la théorie des sillages, Recueil Mathématique 4 6 , 391-458 (Moscou, 1938). [16] A. Weinstein, Nonlinear problems in the theory of fluid motion with free boundaries, Proceedings of the First Symposium on Applied Mathematics, Ann Arbor, Michigan (1949). American Mathematical Society, Vol. 1, p. 1, 1949. [17] J . Kampe de Fériet, I^s fonctions hypergéometriques de plusieurs variables et la représentation conforme d'un demi-plan sur un polygone, Annales de la Société Scientifique de Bruxelles 4 9 , 55 (1929). [18] A. Weinstein, Sur la formule de Schwarz-Christoffel, Comptes Rendus de la Société de Physique de Genève 5 2 , 62 (1936). [19] T h . von Karman, The engineer grapples with non-linear problems, Bulletin of the American Mathematical Society 4 6 , 745 (1940). [20] G . Birkhofif, Hydrodynamics, chap. 2, Princeton University Press, (1950).

i

12.

Some Generalizations of Conformal Mapping Suggested by Gas Dynamics' LiDman

1.

Bers*

Introduction

The theory of two-dimensional subsonic gas flows is of comparatively little interest to the aerodynamicist, since a l l real flows are three-dimensional and since spectacular deviations from the laws of classical hydrodynamics occur only at supersonic speeds. T o a mathematician, however, this theory m a y seem rather attractive, for its very similarity to classical hydrodynamics suggests various generalizations of conformal mapping as well as of other aspects of complex function theory. T o review some of these generalizations is the a i m of this paper. W e shall emphasize particularly the problem of a flow around an airfoil which involves difliculties of a computational nature.

2.

Tlie Differential Equations of Gas Dynamics^

We consider i n the " p h y s i c a l " 2-plane (z=x+iy) a steady potential flow with velocity components u=u{x,y)f v=v{Xyy), and densityp= p{x,y). T h e condition of vanishing vorticity, a n d the continuity equation i m p l y the existence of a "complex potential" such that the "complex velocity" (12.1)

w=u—iv=Qe~*'*

satisfies the relations

If we assume that the density depends on the pressure only, Bernoulli's equation implies that the density is a given function of speed: P = P(2),

(12.2)

a n d from the preceding relations we obtain for ^ and ^ the basic (nonfinear) equations pia)
(2'=*»J+¥>D.

p(2)^.= - l f .

(12.3)

If we now consider F i n the U'-plane (hodograph plane), we obtain for tp and ^ a system of linear equations, as was first observed by Molenbroek [37] and Chaplygin [21]. I t will be convenient to set log w

y=a+ip=i

and consider a and p as independent variables. <Pa=Tmh.

(/3=log g),

Tlie desired equations read ^ f i = - U m a ,

where Ti =

1

-t

p

T2=—

(12.4)

1 d{pq) p

—7-

dq

• Syrftcuse University, Syracuse, N . Y . 1 Some new results announced In this paper (sections 2 and 6) have been obtained during work on a research contract with the Office of Naval Researcli CN«-onr-248). > A rather complete biblihy on the mathematical theory of gas flows will be found In |24].

I n the case of an incompressible fluid we m a y set p s l . T h e n T I S T S S I and eq (12.3) and ( 1 2 . 4 ) are C a u c h y - R i e m a n n equations showing that is an analytic funcition of both z and 7 . F o r a n i s e n tropic flow of a perfect ^ s the pressure is proportional to p*, where is a constant (the ratio of specific heats), A:>1. If the units are chosen properly, the resulting density-speed relation is

1

Ar

A*->

1

(12.5)

2

and the equation obtained from (12.3) b y eliminating ^ becomes 1

fc+1

2

,

1 A

k

^

,/,

^ - 1 ,

2

2

ifc-M 2

F r o m the mathematical point of view there is no need to restrict ourselves to the (p,g) relation (12.5). W e m a y assume instead that (12.2) is an arbitrary sufficiently smooth function satisfying the inequalities dp <0, dq

P>0,

d(pq) dq

>o

in an interval 0
1

i^(pg)y^^v P

dq

/

T h i s is equivalent to the eUipticity of systems (12.3) and (12.4). A m o n g all functions p(q) satisfying our conditions the one given b y 1

(i+ar^ (with Q,= +co) is of special interest. tion o f minimal surfaces.

(12.6)

I n this case the equation for ^ obtained from (12.3) is the equa-

(12*)*^y(Pw+(l+¥>D*>w=0,

(12.7)

while syste: 1 (12.4) is trivially reducible to C a u c h y - R i e m a n n equations. T h e observation due to Chaplygin that the theory of minimal surfaces may be described as a theory of a certain (fictitious) gas flow is of considerable importance. I n the study of minimal surfaces it s u ^ e s t s new problems and methods. I n applied gas dynamics it leads to an important approximate method for computing slow gas flows [21, 29]. I n the general theory of nonlinear eUiptic equations, eq (12.7) m a y well assume the part played b y Laplace's equation in the development of the theory of linear partial differential equations. A s i n the case of Laplace's equation i n two variables, all problems relating to the equation of minimal surfaces m a y be formulated in terms of analvtic functions. A t the same time it is h i g U y probable that general properties of eq (12.7) are shared b y a wide class of nonlinear equations. W e might note at once that i n several problems to be discussed below definitive results have been obtained thus far only for the case of minimal surfaces.

3.

Solutions of the Gas Dynamical Equations as Generalized Analytic Functions

T h e class of functions F=ip-\-i^ whose real and imaginary parts are connected b y system (12.3) or b y system (12.4) may be considered as a generalization of the class of analytic functions. I t is natural to ask which properties these "generalized analytic functions" share with ordinary analytic functions, and i n which ways they behave differently. T h u s we may note at once that mappings b y pairs of solutions of either (12.3) or (12.4) are interior transformations in the sense of Stoiloff [40], i . e., topologically equivalent to those b y holomorphic functions. I n the case of system (12.4) elimination of one of the unknown functions leads to a linear partial homogeneous differential equation of elliptic t ^ e , and it is well known that the theory of these equations is very similar to that of the Laplace equation. I n the case of analytic coefficients there also exist integral operators (developed b y Bergman [2,3] and others) which transform analytic fimctions of a

c o m p l e x variable into solutions of linear partial differential equations. Bei]gman and his collaborators Ixave developed this method into a powerful tool for investigatmg such equations, i n particular the equat i o n s of gas dynamics i n the hodograph plane. Equations (12.4) do not, of course, form the most general linear elliptic system, a n d the special f o r m Of these equations leads to a n integration theory (due to Gelbart a n d the audior [18, 19, 20]) w h i c h paraUels dosely that of analytic functions. L e t F(y)=
J yo

+

is path-independent, a n d considered as a function of 7 is again a solution of (12.4). e v e r y complex constant o, a "formal power"

a-Z^''K7o,7)=w!r...

f

I n particular, for

aidxy)

(12.8)

is a solution of (12.4), and so is the sum of a imiformly convergent "formal power series" i:a..Z<»>(7o,7).

0

(12.9)

Theorems analogous to the binomial theorem, to the fundamental theorem of algebra, and to various interpolation theorems, hold for "formal polynomials" (finite formal power series). If i n (12.9) we choose for a . the coefficients of the T a y l o r series defining the trigonometric, exponential, and hyperbolic functions, we obtain solutions of (12.4) which are of the separated form, say 4f=A{a)B(ß), T h i s includes, i n particular, the solutions of the hodograph equation for ^: ^ = 4 ' , ( i ) ( a cos na+b sin no), w h i c h have been extensively investigated (for the case (12.5) b y Chaplygin a n d others). T h e central result of tfals theory, however, is the expansion theorem which states that every- solution of (12.4) c a n be represented, i n the neighborhood of any regular point, b y a formal power series. T h i s was proved originally under the assumption that Ti(ß) a n d Tt(ß) are ancdytic functions. Recently the author was able to show that the expansion theorem remains vaUd even if these coefficients are assumed to be merely twice continuously differentiable [17]. I n the case of system (12.3) the nonlinearity of the equations precludes the possibility of a similar I t J parallelism to the theory of analytic fimctions, and i n general we m a y expect a considerably more heated theory. Results obtained for minimal suHaces, however, show that i n some problems M theorems (though not proofs) o n solutions of the nonlinear system (12.3) are simpler than those o n analytic functions. T h u s the C a u c h y - R i e m a n n equations possess a n infinitude of solutions regular i n the whole plane, and the study of entire functions is m itself a major mathematical discipline. F o r the eq (12.7) of minimal surfaces, however, a theorem of S. Bernstein [9] states that every solution regular jor all finite values 0 / X
(x«+y')+0(l),

where Ay B , C are constants. If Zo is an isolated singularity of a {multiple-valued) solution of {12.7) y and if tpx and
4.

Mapping Problems Related to Flows in Jets and Channels

If we study not the whole class of solutions of the gas-dynamical equations, but specific flow problems we are necessarily led to mapping problems. Consider, for instance, a jet flow out of a symmetrical vessel bounded b y two straight walls. T h e domain of the flow i n the 2-plane is not known, since we do not know the free boundaries along which the speed must be constant. It is seen at once, however, that the fimction F maps this doniain onto a rectilinear strip, whereas w maps the same domain onto a sector. T h e flow will be known once we find the mapping of the domain i n tlie w-plane onto the domain in the F-plane. If the fluid is incompressible, the mapping is to be conformai and m a y be found b y elementary means. If the fluid is compressible, the mapping functions *»(a,/3), ^(a,j8) are to satisfy eq (12.4). T h i s particular mapping problem has been solved completely b y Chaplygin [21] with the aid of the particular solutions mentioned above. Chaplygin's result suggests a more general question. Is it possible to map a given domain in the {a,ff)-plane {which satisfies the hypothesis of Riemann^s mapping theorem) onto a rectilinear strip, by means of fmictions satisfying system (12.4)? A n affirmative answer was given b y Schapiro [38] not o n l y for system (12.4) but also for the general quasi-linear homogeneous elliptic system. T o the author's knowledge only a sketch of the proof (which is based o n some earlier results of Lavrentieff [30]) has been pubhshed to date. A n even more difficult mapping problem arises i n the computation of a flow i n a channel. H e r e a curvilinear infinite strip i n the 2-plane is to be mapped onto a rectilinear strip i n the F-plane. I n the case of a n incompressible fluid the mapping is to be conformai; i n the case of a gas the mapping functions must be solutions of system (12.3). Recently Lavrentieff [31,32] showed that Riemann^s mapping theorem holds for a wide class of nonlinear systems of partial differential equations. T h e equations of gas dynamics for the subsonic range belong to this class. Gas-dynamical motivation, incidentally, is clearly visible in Lavrentieff's ai^^ument. A detailed discussion of the remarkable results achieved b y Lavrentieff and his collaborators is beyond the scope of this paper.

5.

The Mapping Problem of Airfoil Theory m

Another problem which i n the case of an incompressible ffuid leads to conformal mapping is that of a flow around a n airfoil. W e assume that a profile P possessing a sharp traUing edge at ZT is given in the 2-plane; we wish to determine a ffow past P which is horizontal at infinity a n d satisfies at Zr the Kutta-Joukowski condition. T o solve this problem the domain E{P) exterior to P is mapped conformally onto the domain | r | > l i n the auxiliary f-plane ( f = f + i i y ) i n such a way that f = dj;/dz^O for 2 = 00. It follows at once that in the i"-plane F will be of the form

F=A(f-|-i-|-2ism5logf)

A>0,

(12.10)

where {=e~** is the imBge of ZT- T h u s the flow will be known once the conformal mapping of | f | > l onto E{P) is found. W e shall show that the corresponding problem for a gas also leads to a m a p p i n g problem, though this time the mapping is determined not b y differential equations as i n the cases considered before, but rather b y a certain parametric representation.* W e first assume that the flow exists and introduce i n E{P) the Riemannian metric [10] dS

dx^+dy^+-

p dq

(sin adx

cos

adyy.

(12.11)

B y virtue of the general imiformization theorem we can map E{P) onto 1^1 > 1 b y a mapping which is conformal with respect to this metric (and normalized so that f = » ,òé/^>a:>0, ò?/òy=0 for 3 = < » ) . Next we set 1 d{p q) hdq (12.12) dq J q w^—q*^* s

L rfg J

(12.13)

(12.14)

* For a gas every flov problem leads to a new mapping problem, while in the theory of Inoompreasible fluids the solution of one mapping problem Klr«a all Ible flows past a given airfoU.

I t turns out that m the f-plane w* is a regular analytic function, whereas ^ a n d ^ satisfy the linear partial differential equations n=n.. V',= - T ^ 6 (12.15) a n d the conditions ^ = 0 o n If 1=1; ^f>0, ^,=0 at f = " (12.16)

as w e l l as the K u t t a - J o u k o w s k i condition a t the point f = 6 " ' * into which our mapping takes ZT- (This means e"** is a branch point of the level line ^ = 0). If /i is the value of g* corresponding to the m a x i m u m attained b y the speed g, we also have that ax lw*(f) = M

(12.17)

F i n a l l y , the inverse maping of |f|>l onto E(P) is given b y Chaplygin's integral (12.18) Note that T is a known function of g, hence also of q*=\w*\. B y virtue of (12.1), (12.12), and (12.13), w is known once w* is, and so is p. In order to find the flow aroimd P we must find an analytic function w*{^)j |f|>l, satisfying condition (12.17) and such that after and ^ are determined b y (12.15) and (12.16), the transformation maps the domain |f|>l onto the domain E{P). T h i s mapping problem simplifies considerably in the case of the density-speed relation (12.6), i . e., in the case of a minimal surface. In this case we have (12.19) 9

2

(12.20)

l + d + gV'

and

T-1.

(12.21)

Hence F is an analytic function of f, and must be of the form (12.10). written in the form 1 1 f ^ r ) d!;-^fF'H;)w*d!;, 2J w*

T h e mapping (12.18) m a y be (12.22)

and is a mapping b y means of two (nonconjugate) harmonic functions. I t is easy to see that (12.22), together w i t h the relation ^=9JF(f), constitutes a special form of Weierstrass' parametric representation of m i n i m a l surfaces.

6. The "Indirect" Problem T h e "indirect problem" of airfoil theory consists i n finding a method for generating flows around closed profiles whose shapes are determined a posteriori. T h e solution of this problem i n the case of a n incompressible fluid presents no diflSculties, since i t is easy to obtain analytic functions mappinglf |> 1 onto domains bounded b y simple curves which look hke airfoil profiles. F o r the case o i a m i n i m a l surface the indirect problem was solved b y Tsien [41] for circulation-free flows. T h e general problem was solved b y the author [10] and i n a more elegant way b y Gelbart [26], whose method was also arrived at independently b y L i n [36] and Germain [28], Gelbart i n effect sets 2

w

*^r=/(f),

and writes (12.22) in the form

/(f) where Fif) is given b y (12.10). /(f)=af+61og f +

4 j / '(f)

H e then observes that whenever/(f) satisfies the conditions S

n-0

a

^ + ^ 1 si sin 6

1 - ) - = « . 4 a/ a

2im)\<\F'{i;)

(12.22) maps |f| = l onto a closed curve. If this curve is simple, v>=5R FQ;) represents the potential of a flow around the curve. In the case of a general density-speed relation the indirect problem presents considerable difficulties. F o r the case of eq (12.6), methods for constructing airfoil flows were proposed by Bergman [4, 5, 6, 7, 8], Cherry [22], and lighthill [34^ 35]. These authors work in the hodograph plane which necessitates the use of multiple-valued functions. Their methods are rather laborious and make.more or less expUcit use of special properties of eq (12.6). The considerations of the last section sugeest another method for constructing symmetrical flows. (The asymmetrical case involves difficulties w & c h have not yet been overcome). This method is equivalent to that proposed b y Christianovitch [23], and independently by the present author [10], (The treatment of circulatory flows in Christianovitch's paper contains a mistake.) We begin b y choosing a function w*(f) which is continuous on |f| = l , regular analytic for I f l ^ l , different from 0 except at f = l , r = — 1 , and admits a Laurent expansion w*(t)=ao+^

%

n-2 $

an real.

The function should also satisfy condition (12,17), ju having the meaning explained above. There are no difficulties in constructing such functions. F o r instance, let Z = Z ( f ) be a function mapping f |> 1 onto the domain exterior to a profile P' in the Z-plane, symmetrical with respect to the A-axis (with Z ( a > ) = CO, Z ' ( o o ) > 0 ) . The function

where £ is an appropriately chosen positive constant, satisfies our conditions. Next we set q*~ fw*|, determine q from (12.12), and obtain T a s a known function of { and 17, (i^-\-n^ ^ 1)Then we solve equations (12.15) under the conditions (12.16). (The arbitrary multiplicative and additive constants entering in

l onto the "exterior" of a closed curve P. If this curve is simple, we have obtained a gas flow past P . We may be sure that this curve will be simple if arg w* (e*") is monotone for 0 < w < T . This can be achieved by choosing the auxiliary profile P' as convex. In order to integrate eq (12.15) it is convenient to introduce new coordinates {i, iji by the relation

(f+^'/)(6i+^nl) = l , and to set where T„ is the value of T for f = « » , and w and x are required to be regular functions of For x (iit Vi) we obtain a perfectly well-behaved linear partial differential equation which is to be integrated within the unit circle with continuous boundary conditions of the first kind. Thus the construction of a symmetrical gas flow (with a prescribed maximum local speed) past an airfoil has been reduced to the solution of the Dirichlet problem for a linear equation of elliptic type. The theory of this problem is well known, and so are numerical methods which in principle permit us to compute the solution. The effective computation of the solution, however, would involve an immense amount of labor. The practical appUcation of the method described here must await the further development of computation techmques. It is hardly necessary to point out that what is desired is a machine capable of solving a large number of simultaneous linear algebraic equations,

7.

The "Direct" Proble

The direct problem of airfoil theory (determination of the flow around a given profile) has been solved thus far only for the case of a minimal surface. We want to show, however, how in the general case this problem may be reduced to a functional equation for a function of a single real variaWe [12]. We shall restrict ourselves to symmetrical flows. Assume that we know the flow and the mapping described in section 5. Without loss of generality we assume that the length of P is 2ir and write the equation of this curve in the form z^Z(a)

= ZT+£

e*^^^^da\

0 <
Our mapping takes a point e*" of the circle into the point Z[/(a>)] of the profile. metry we have /(0) = 0, /(2T) = 2 T .

(12.23)

B y reasons of s>Tn(12.24)

W e shaU see that the knowledge of the function
dip

dtp

dff

T 0)

1

(12.25)

I f we know q and tp along If | = 1 we can obtain/(w) b y a simple integration, the multiplicative constant entering i n tp being determined b y the second condition (12.24). T n u s we have described a n analytic process which leads from the function/(6)) back to the same function. This m a y be written symbolically i n d i e form /=T(y), (12.26) where T is a certain rather complicated operator. If we Imew that for a given profile a gas flow w i t h arbitrarily high subsonic speeds exists, then we would know that the functional eq (12.26) is solvable. T h e corresponding existence theorem, however, has been proved thus far (by F r a n k l and K e l d y s h [25]) only for flows w i t h "sufficiently s m a l l " free stream speed." I t m i ^ h t well be that the proof of the existence of the flow w i l l depend upon a direct proof of the solvabihty of (12.26), as was the case for m i n i m a l surfaces. I n the case of m i n i m a l surfaces the preceding considerations simplify considerably and can be carried o u t for asymmetrical (in particular circulatory) flows as well. I t turns o u t that the functional equation (12.26) m a y be written out expUcitly as a nonlinear integral equation. If the profile has a sharp trailing edge of opening ax ( 0 < a < l ) and no intruding comers, the equation reads: ''co8»(^^-r^e*^>|d(r /(o,)=T[y(«)]

1 2irJo

(12.27) sm^

sm^

where

M
X = 2 -^-/maxj s m ^

eL/(«)l rf,

cos

T h e proof that this equation has a solution has been given under fairly general conditions o n the profile f l 4 , 15]: the function 9 (0-) is absolutely continuous and satisfies a uniform Hdlder condition, and the VH^ofUt is ''essentially convex." B y this is meant that the negative variation of 8((«)» we compute successively the functions /,=T(/.-,), »=1,2,..., in the hope that the sequence {f,} converges to the desired solution. T h e necessary calculations can be handleid o n a n ordinary desk machine. Computations have been performed for a sUghtly modified form of equation (12.26) i n which the speed at infinity is prescribed instead of the maximum local speed, and the position of the front stagnation point instead of the angle of attack. (This last modification seems to be essential for the convergence.) T h e results of our computations [11, 13] present strong numerical evidence of the convergence of the method. T h e theoretical verification of the convergence, however, seems to be an exceedingly difficult problem. * Bee also [30] where % more spedal result Is proved for minimal surfaces wlUi the aid of an InteKro-dlfferentlal equation dlflbreot from the one used by the present author. ' A different method Usr mAviag approximately the direct problem for mftiimal soilaoie b HJ9a In |t, 37J.

I n principle, the same method could be applied to the functional equation (12.26), except that a t each step one would have to solve a Dirichlet problem for a linear partial differential equation. N u merical computations w i l l have to be postponed until adequate automatic calculators are i n operation. Since this lecture was dehvered, considerable progress has been achieved. T h e analogy between solutions of equations of the form (12.4) and analytic functions (§ 3) appears today as a very special case of the general theory of "pseudo-analytic" functions developed b y the author [42, 43] (cf. also the work of P o l o i i i [48, 49, 50]). T h i s theory appUes to the most general linear eUiptic system of two partial differential equations for two unknown functions ^ (x,v), 4^ (x^y). It includes a complex differentiation and integration, a classification of singularities and branch-points, a C a u c h y i n t ^ a l formida, T a y l o r and Laurent series, etc. The conjecture (stated i n § 3) concerning removable singularities of solutions of the non-linear system (12.3) for which qp{q) is bounded has been verified b y F i n n [45]. I n fact, F i n n obtained a more general result [46]. Independently of the work of Shapiro ( § 4 ) , Dressel and Gergen [47] obtained an extension of Riemann's mapping theorem for certain linear systems of partial differential equations. T h e existence theorems for minimal surface stated i n § 7 have been shown to hold without the "essential convexity" condition [44 . V e r y recently the "direct prob e m " (see sec. 7) has been solved b y M . Schiffman for smooth profiles, and a little later b y the author also for profiles w i t h a sharp trailing edge. Schiffman used variational methods, whereas the present author treated equation (12.26) b y the Schauder-Leray method.

8. References 1 S. Bartnoflf and A. Gelbart, N A C A Tech. Note 1171 (1947). 2 8. Bergman, Math. Sbomik 2, 1169 to 1198 (1937). S. Bergman, Trana. Am. Math. See. 53, 130 to 155 (1943). Ì S. Bergman, N A C A Tech. Note 972 (1945). 5 8. Bergman, N A C A Tech. Note 973 (1945). 6 S. Bergman, N A C A Tech. Note 1018 (1946). 71 S. Bergman, N A C A Tech. Note 1096 (1946). 8 S. Bergman, Trans. A m . Math. Soc. 61, 452 to 498 (1947). S. N. Bernstein,^omm. Soc. Math. Kharkov, 16, 38 to 45 (1915).

h

21] 22 23 24 25 26 27 281 29 30 31 32 33 34 35 36 37) 38 39 40 41 421

L. Bere, N A C A Tech. Note 1006 (1946). L. Bere, N A C A Tech. Note 1012 (1946). k. Bers, N A C A Tech. Note 2056 (1950). L. Bers, Proc. First Symp. Appi. Math., American Math. 8oc. 1, 1 (1949). L. Bere, Trans. A m . Math. Soc. 70, 465 to 491 (1951). L. Bere, Annais of Math. 63, 364 to 386 (1951). L. Bere, Amer. J . Math. 7Ï, 705 to 712 (1950). L. Bere and A. Gelbart, Quart. Appi. Math. 1, 168 to 188 (1943). L . Bere and A. Gelbart, Trans. A m . Math. Soc. S6, 67 to 93 (1944). L. Bere and A. Gelbart, Annais of Math. 48, 342 to 357 (1947). S. A. Chaplygin, Trans. Moscow Univ. Math. Phys. See. 21, 1 to 121 (1902). (Also N A C A Tech. Note 1063 (1944)). T . M . Cherry, Proc. Roy. Soc. I M , 45 to 79 (1947). S. A. Christianovitch, T r u d i C A H I , 481 (1942). R. Courant and K . O. Friedrichs, Supereonic flow and shock waves. (Interecience Publisbere, Inc., New York, 1948). F . Frankl and M . Keldysh, Izvestya Akad. Nauk SSSR 13, 1 to 121 (1934). A. Gelbart, N A C A Tech. Note 1170 (1947). A. Gelbart and D . Resch, N A C A Tech. Note 2057 (1950). P. Germain, Comptes Rendues (Paris) 532 to 534 (1946). T . V. Kirmàn, Journal Aero. Sci. 8, 337 to 356 (1941). M . A. LavrentiefT, Mat. Sbomik 42, 407 to 424 (1935). M . A. Lavrentieff. Mat. Sbornik 21 (63), 285 to 320 (1947). M . A. Lavrentieff, Izvestya Akad. Nauk SSSR, Ser. Mat., 12, 513 to 554 (1948). J . Leray and J . Schauder, Ann. Ecoie Norm. Sup., 51, 46 to 78 (1934). M . J . LighthUl, Proe. Roy. Soc. 191, 341 to 369 (1947). M . J . Lighthill, Proc. Roy. Soc. IW, 135 to 142 (1947). C. C . L i n , Quart. Appi. Math. 4, 291 to 297 (1946). P. Molenbroek, Arch. Math. Phvs. S, 157 to 186 U 890). Z. Y . Schapiro, Doklady Akad. Nauk. SSSR 30, 685 to 687 (1941). N. A. Slioskm, Trans. Moscow Univ. Math. Phys. Sec. 7, 43 to 69 (1937). Stolloff, Leçons sur les principes topologtques de la théorie des fonctions analytiques. (Ganthiere-Vil lare, Paris, 1938.) H . 8. Tsien, Jour. Aero. Sci., 6 , 399 to 407 (1939), I^. Bere, Proc. Nat. Acad. Soi. 30, 130 to 136 (1950). L. Bere, Proc, Nat. Acad. Sci. 87, 42 to 47 (1951). L. Bere, BuU. A m . Math. Soc. 57, 463 (1951). R. S. Finn, Bull. A m . Math. Soc. 57, 178 (1951). R. S. Finn. Bull. A m . Math. Soc. 57, 466 (1951). F. G. Dressel and J . J . Gergen, Duke Math. J . 18, 185 to 210 (1951). G. N . Poloiii, Doklady Akad. Nauk, 58, 1275 to 1278 (1947). G. N. PoloSii, Dokladv Akad. Nauk, 60, 769 to 772 (1948).

13.

The Use o l Conformal Mapping to Determine Flows With Free Streamlines David M . Y o u n g '

1.

Introduction

1. Notation and Terminology. Conformal mapping is the classical method used to determine plane flows w i t h free streamlines i n an ideal incompressible fluid. T h e physical assumptions that are made i n order to permit a mathematical treatment of such flows are well known and are given i n [7].* W e define: i n the physical plane; i=i+i>i} vector; W=U+iV; W is the complex potential of the velocity field; that is, (7=velocity potential, F = s t r e a m fimction, dWldz=i\ « = l o g f. T h e conformal mapping of the coordinate lines of the H^-plane into the Z-plane gives streandines a n d equipotentials. Similarly, isobars (lines of constant pressure) and isoclines (lines of constant velocity direction) are obtained b y the conformal mapping of the coordinate lines of the f-plane into the Z-plane. I n free streandine theory, both conformal mappings are achieved indirectly b y methods t h a t we now describe. 2. The Analytic Method. L e t us consider flows whose f-domain, caUed the "hodograph", is a circular sector of angle ir/n. A circular sector may be mapped onto the upper half-plane of the variable tby Z=x-\-iy;

Xf y are coordinates {""^i? is the velocity

t

^(r+f-),

(13.1)

or b y t

(13.2)

r

W e map the W-domain onto the upper half-plane of the variable T by a Schwarz-Christolfel transform a t i o n . I n a complete flow which is bounded b y streamlines, the Tr-domain is bounded b y parallels to the U-9^is; hence, dW/dT w i l l be a rational function of T. Since T and t are variables, each of whose domain is the upper half-plane, we have T

At{i:)+B

(13.3)

Ct(S) + D

N o w (13.3) represents the most general conformal mapping of the upper half-plane onto itself. A, B, C, a n d D are real parameters (with AD—BCT^O) so far undetermined, whose evaluation will be treated i n sections 3 and 4. Once we have the parameters of (13.3), we can obtain l F ( f ) and Z ( f ) as follows: Wit)

dW dT

dT d(, di

(13.4) (13.5)

V The work described herein was done under the direction of Professor Garrett BLrkhoff, at Harvard University, for Naval Research Contract N5orl*76» Pro)ect 32. The work of part I V was done mainly by the author. ^FUrures In bradcets indicate the literature references at the end of this paper

Relations of the form (13.1) through (13.5) connecting W , i", and Z are usually regarded, i n the classical literature, as the solution of the mapping problem. W h e n mmaerical results are desired, however, there are several practical difficulties to be overcome. These are described i n subsequent sections.

2. Manual Computation and the Use of Tables Unfortunately, the parameters A, By C, and D of (13.3) may not be expressible directly i n terms of the given physical dimensions. T h u s i n the problem [17] of uie divided jet (fig. 13.1) one is usually given the length of the plate, and 5, the specification of the 3.

The Determiriation

of Parameters.

FlOURB 13.1.

position of the plate relative to the axis of the incoming jet. T h e narameters, however, are k n o w n functions of and as, the angles of the branches of the divided jet. W e have 1 TSJ-\-

(13.6)

2 Cj log tan ^ a¡

and 2

JZ^[hs sin (ai+a^)log s m ^ —

+sin

(ai—aj)

2 hi cos ( a , — o i ) + 2

log sm

2

cos (oj—a,),

(13.7)

where

C/=C08 Uj, hr

Ci-C,

0%—Ca

Si=áa 3

a,, h,=

hi

1,

O2—C\

O2—6j

Given a i , a nomogram (fig. 13.2) is constructed by calculating by (13.6) and (13.7) values of p and q corresponding to given values of as and «3- The values of ota(p,2) and «3(2),?) are then obtained b y inverse interpolation. The parameters Ay B, Cy and D may then be determined. 4. Another Example—A Generalized Riabouchinsky Flow. For a generalized Riabouchinsky flow [1, 15] the difficulty of determining the parameters is much more acute. In this problem the flow is desired around a pair of 45** wedges placed symmetrically in a channel of infinite length (fig. 13.3). By symmetry we need treat only one-quarter of the fiow, so that the hodograph is again a circular sector (fig. 13.4). The IT-diagram is bounded by a rectangular polygon (fig. 13.5). Sudi a domain may be

a_« ConttoAt

a^«Conttont

FiouRB 13.2.

Lin« of Symmetry FiauBB 13.3.

upper half-plane of the variable T by a Schwarz-Christoffel transformation so that except for at most square roots.'

Thus

(13.8)

and

1

(13.9)

«^ere m

1 - r 1 + f*'

I

2

ri-pjri

1 '

r i - » n c

—L i + « » * J

W PLANC F I G U R E 13.4.

F I O U B E 13.5.

The parameters depend upon Vj the limit of the velocity at large distances from the wedges, and upon vc, uie velocity at point C (fig. 13.3). Physicidly, one is given L/W and D/W, where L=proiected length of each wedge; l^=half width oi channel; 2>=half total length of cavity. As in the divided jet problem, a nomogram can be constructed by computing the values of L/W and D/W for given values of vc and and using inverse interpolation. The construction here is much * Mora fcoenUy. let OS consider those cases where we can by syminetiT treat 0 ^ H the part constdered Is bounded by free itnunltaus and Its hodoeraph is a olrenlar sector of angle wjn with n rational, then Z may be expressed In the form ImfR(h y p ( 0 Mt where RH) is a raUonal taKtion u i d P(0 is a polynomial. X is by definition a byperelllptic integral. When P(0 is of degree 4 or less, / is an eQiptlo integral.

ore difficult, since the values of LjW and DjW are expressible only as untabulated definite integrals; 2-J

L

1+ m 1— m

2 C

A+C + 1

W

m a»

(13.10)

drtiy

m;2 - è

(13.11)

cos 0
D

(13.12)

where tan 2 9 = r

2 A-\-C

a

The above integrals must be evaluated b y a numerical integration procedure (sec. 9), which requires 2 days of hand computation for each pair of values vc and . 5. EvcUiiaiion of the Iviegrals {IS4)
HODOGRAPH F I G U R E 13.6.

If n is irrational, the integrals (13.4) and (13.5) represent untabulated functions. T o obtain numerical results one must either use complex numerical integration (sec. 9) or else construct tables of special functions of a complex variable. Whether or not tables are worthwhile depends o n the generality of the functions, the number of real parameters involved, and hence the size of the tables, and upon problems of interpolation. W e consider special examples. (1) Circular and Hyperbolic Functions. I n cases where n = l (fig. 13.6) the numerical work could be greatly reduced if two-parameter * tables of inverse circular and hyperbolic functions were available w i t h a sufficiently fine mesh to permit convenient interpolation. F o r example, for the K i r k h o f f Rayleigh fiow [7] (fig. 13.7) we have B.

Untabulated Functions.

z(r)

+ cos a r + 1 ) ' ' 2 s i n * 1

r^-2

f — cos a a ( r ^ - 2 c o s a * f + 1)

4 sin^ a

r log

4

r\W)=^[h±^h

h = nlT+Cy

J+const.,

(13.13)

(13.14)

M=const

Formulas (13.13) and (13.14) are rather compficated, and to compute Z{W) requires 67 operations. I n terms of complex circular functions the formulas become much simpler: f = tan 0 sin a + c o s a,

(13.15) (13.16)

a

0=-+-

Z

1

cos

-™sm' a 2T

cos

(13.17)

M

(

1

in

a {

4

i—7— <^ sin a + 2 cos {2—a)+-r cos (4<^ 2 sin

+const.

(13.18)

* B y elementary functions is meant complex alRcbraic, trigonométrie, exponential, and loRaritbmio functions. « Since the complex variable Z varies, two real variables 9t(Z) and 3(Z) are involved. For purposes of discussion in this section, the term "parameters" also will Include both 9t(Z) and 3 ( Z ) .

T o use formulas (13.15) through (13.18) one must evaluate complex direct and inverse fimctions. T w o to three significant figures can he obtained b y the use of Kennelley's tables an [11]. The direct complex fimctions can be evaluated to foiu* places b y the use of the W P A tal

FlQURB 13.7.

for the corresponding real functions, although i t woidd be desirable to extend the range of the ai^imient f r o m 2.0 to at least 10. Tables for the complex inverse functions woidd be desirable. W i t h a mesh of .01, about five volumes of 200,000 entries each would be required. B y the use of these tables the l a b o r needed to evaluate Z{W) could be reduced to 20 simple operations. (2) Incomplete Beta Function. F o r a simply connected now with circidar sector h o d o ^ a p h of angle x/n, where n may be irrational, i t is shown i n [9] that the integrals may be expressed m closed f o r m i n terms of elementary functions and the incomplete beta function, defined b y

j: s -

(13.19)

Also when n is rational, n—r/s, and s is large, the elementary solution i n closed form is possible b u t involves many complex partifd fractions. I n these cases the use of the incomplete beta fimction w o i d d be much more convenient. F o r example, the problem of the infinite wedge (fig. 13.8) can be solved w i t h the aid of the incomplete beta function, with

z(r)

8=r

2[Bu„(8)-e"''B,U-s)],

FIGURE

9

l/m+l/n=l.

(13.20)

13.8.

Three real parameters are involved. About five volumes of 200,000 entries each would be necessary t o tabulate this function. Higher order interpolation i n three variables would be required, (3) Jacobian Elliptic Integrals (5]. F o r the generalized Riabouchinsky flow, Z(f) and Z{W) can be expressed i n terms of elliptic integrals of the third kind, defined b y n{z,n,K)

dt {\-Vnt^[{\~t^{\-KH^\^

(13.21)

The Jacobian eUiptic integrals of first, second, and third kinds are of wide generality and occur i n m a n y other problems/ Tabulation of Jacobian elliptic functions (an elUptic function is the inverse of a n elhptic integral of the first kind) for real and purely imaginary a r ^ m e n t was started b y the M a t h e m a t i c a l Tables Project of the National Bureau of Standards but is now mactive. F o r the elliptic functions and elUptic integrals of the first kind three parameters are involved, hence about five volumes per function would be necessary for complete tabulation. F o r the elUptic integral of the third kind four parameters are involved. Since i t does not appear practical to construct four parameter tables, tabulation of (13.21) for real and purely imaginary arguments would seem to be called for. E v e n if this were done, three real parameters would be involved. Additional formulas would be used for complex values o f 2 . About eight volumes of tables would be required. 6. Interior Streamlines and Equipotentiais. Isobars and isoclines may be computed readily when the parameters of sections 3 and 4 have been found, provided the integral (13.5) can be evaluated directly or b y use of tables. O n the other hand, the problem of getting interior streamlines and equipotentials is i n many cases much harder; yet the distinction is generally overlooked. Although W{i) may always be obtained directly, and usually i n closed form b y (13.4), i t is not usually easy to compute f(ÍÍO. Smce i(T) may always be obtained i n closed form, i t is sufficient to get T(W)f which m a y be the solution of a complex algebraic or transcendental equation.

W PLANE

•

FIGURE

13.9.

FIGURE

13.10.

Thus let us consider the problem of the planning forms [21] (fig. 13.9). The IF-diagram (fig. 13.10) may be mapped onto the upper half-plane Thy dW dT

and

W=\ogT+T,

where get

(13.22)

W=U+iV,

(13.23)

T=R+iS

T o solve for T(W), we can separate this equation into real and imaginary parts and, eliminating S, F = ( É 2 ( t ' - « ) - i ? 2 ) * + a r c tan

R^^ R

This may be solved for R by an iteration process [4, p. 171]. S =

(e}^^-^-R^K

(13.24)

S may then be obtained from (13.25)

• We estimate that i t would take about three times as long to obtain streamlines and equipotentials in this way as to obtain isobars and isoclines. I n other cases the ratio of times would be even greater, since we might be led to solve two simultaneous real transcendental equations. • The following has been proved b y E . Zarantonello, Harvard University. Theorem: Let the flow be simply connected and bounded b y two isobars and two Isoclines. Further, let the angle between the isoclines be a rational multiple of w. If there are a finite number of point sources, sinks, and vortices i n the flow, then there exists a parameter t such that t V - K i ( * ) , f - E i ( » ) , and 2 - E 1 C * ) , where are elliptic Integrals of the first, seoond, or third kinds.

Another method that may be used to solve (13.23) is to integrate the differential equation dT

T

dW

T+l

b y step-by-etep numerical integration i n the complex plane. I n some cases the determination of ^{W) may be straightforward but laborious, as i n the HelmholtzR a v l e i g h flow where the evaluation of ^(W) consumed about 30 percent of the time required to get Z(W).

One can get the free streamline b y noting that i t is also the isobar (fl = l . Then, having determ i n e d the boundary of the flow, one might get the interior streamlines by the finite difference methods, section 11, or b y use of an electrolytic tank [19]. A n alternative procedure for obtaining interior streamlines and equipotentials is to compute values of z and W for equally spaced values of $8(log T) and 3 G o g f), and plot these i n the s-plane. These w i l l be nodes of a grid of isobars and isoclines. H a v i n g the values of W for points of an irregularly spaced grid i n the 2-plane, we m-ay get streamlines and equipotentials by inverse complex interpolation. T h i s method would seem to be laborious when more than three significant figures of accuracy are required.

3.

The Use of Large-scale Calculating Machines

If the interiors of flows w i t h free streamlines are to be computed, large-scale computing machines seem called for even i n the simplest cases. The M a r k I Automatic Sequence Calculator of the H a r v a r d Computation Laboratory was used 8] to compute the Kirkhoff-Rayleigh flow for a = 9 0 ° , 45'', 15° (fig. 13.7). T o compute 1,600 values required ten 24-hour machine d&ya; the machine computing, checking, and printing each value. We estimate that i t would take at least one computer a year to do the same work w i t h a desk machine and tables. I n those cases where Z(f) and/or Z{W) are expressible i n terms of elementary functions, the adaptation to large-scale computing machines is straightforward i n principle. The choice of scale factor and mesh size can be foreseen b y computing selected values b y hand. 8. The Use of Large-Scale Computing Machines to Determine Parameters. F o r problems i n which the determination of the parameters presents a serious difficulty, such as i n sections 3 and 4, a large-scale computing machine would be particularly useful i n evaluating definite integrals needed to construct the nomogram. The definite integrals may be evaluated b y real numerical integration, modified near the s i n ^ l a r i t i e s of the integrand (see Appendix I). The procedure may be adapted to some of the more flexible machines so that the integrals may be evaluated for many values of the parameters and an accurate nomogram constructed. 9. Complex Numerical Integration. Although complex numerical integration can be arranged very conveniently for machine comi)utation, i t has not previously been used for fiow problems w i t h free streamlines. T o obtain isobars and isoclines, we may use formula (13.5), 7.

The Need of Large-Scale Computing Machines in the Simplest Cases.

The integrand is often algebraic or can be expressed i n terms of exponentials, logaritlims, and trigonometric functions which the machine can compute. Numerical integration may also be used to enable the machines to construct tables of special functions of a complex variable, such as those given i n section 2, A . T o perform the complex numerical integration one may use the ordinary ® formulas for real numerical integration; or, if sufficient storage capacity is available i n the machine, a five-point formula, wluch utihzes the fact that the integrand is an analytic function, may be used. We have

The error ^ is not greater than (AV1890)|/"(Q • ScarborouRh \4], pages 117-14A. ' T h e error estimates are derived from results riven in (13] and (141: /"(t) is a point in the domain of convexity corresponding to the domain oovered by /**(Z). where Z covers a convex region oontainlng the five points;/•({) has an analogous interpretation In the error estimate lor f
949581—52

10

We have also

+y(Z,.,,,)]4-tl5L/(Z,_,.,)-/(Z,^,.,)l}.

(13.28)

Here the error is not greater than (AV360)|/"({) T h e above formulas appear preferable to those formulas for real integration with corresponding error terms since points closer to the region of integration are used. In some problems even when the substitution method could be used, numerical integration m a y be preferable. T h u s i n the Kirkhoff-Rayleigh flow, with a = 4 5 ° (fig. 13.7) numerical integration could be used for W with four-place accuracy, provided the mesh size is less than one-fourth the distance f r o m the singularity (the stagnation point). W e estimate that 50 percent of the multiplications could be saved. T h e values near the stagnation point could be obtained b y series development. 10. Unified Coding. A s a substitute for four parameter tables, we have prepared a unified c o d i n g scheme for a high-speed, large-scale computing machine, to obtain isobars and -isoclines for a n entire class of problems. T h u s , for the class of flows bounded b y free streandines, whose hodograph is a circular sector * of angle x/n, Z may be obtained as a n integral of the form n jfTT—ne""'"

sinh

r = c o s h w,

u>=n log f.

(13.29)

Here there are 3 or less values of the real parameters nj. W e propose to compute the integrand and to perform numerical integration syBtematically for l
4.

Approximate Numerical Methods

A s a n alternative to the analytic method, finite difference methods have been used [2, 3] to solve flow problems w i t h free streamlines. One great advantag;e of these methods is that one can start w i t h the physical dimensions, avoiding the difi&culties discussed m sections 3 and 4. A x i a l l y synmietric flows [12] and flows w i t h curved boundaries, whose analytic solutions are not y e t known, have also been solved b y finite difference methods. F o r the generalized Riabouchinsky flow (sec. 4), streamlines and equipotentials were obtained i n a week b y D . .Mien and G . Vaisey b y using finite difference methods. W h e n the values of the parameters thus obtained were used as a first approxunation, 2 months were required to obtain the correct parameters and to compute the free streamline and the length of the plate b y the analytic methods. B y the use of an electrolytic tank,* L . M d a v a r d obtained a third solution of the flow problem. W e give a comparison of the values of V^/Vf and F c / F , : 11. Finite Difference Methods.

A n a l y t i c method Relaxation methods Electrolytic tank

Vr

Vr

0.707

0.921

.714 699

. 907 . 906

W h e n the values of Vc/Vf and V„/Vf found b y finite difference methods for the analytic solution were used, the length of the plate and the length of the cavity were found to be about 7 percent too low. Since those values of the parameters are oidy about 1% percent different from the correct values, we conclude that the physical dimensions are very sensitive to changes i n the velocity ratios. A practical diflSculty with finite difference methods for flow problems w i t h free streamlines is that the free streamlines must either be obtained b y other methods, or found by successive guesses." These require great skill and experience [3]. 12. Analysis of Error. I n addition, the relaxation methods have two apparent sources of error: the well-known error due to the finite difference approximation to the second partial derivatives discussed further i n section 13 and i n [22]; and an error, which has received little attention, caused b y the < This class Is somewhat less general than that treated In this paper, and does not include the Riabouchinsky flow. • For a description of the electrolytic tank, s e e [19]. 1« V f - i f r e e streamline velocity; Va>=Iimiting velocity at large distance from the wedges; Vr«»the velocity at C, (see fig. 13.3). T h e values of F w / V / a n d Vcl Vt given for the finite difference methods and for the electrolytic tank were obtained by numerical difTerentiation, which process is estimated to be accurate to within about 1 percent. T h e length of the channel was taken as finite and equal to the distance t>etween the walls. Although the values obtahied analyticaUy apply to a channel of infinite length, the velocity ratios would change less than 0.5 percent if the length had been taken as finite. i> Since free streamlines have in general infinite curvature near the ends of a plate, they cannot be obtained by any nonzero mesh.

necessity of replacing an unboimded region b y a boimded region, and correspondingly assuming artificial boimdary conditions. F o r the finite difference solution, the velocity was assmned constant on an artificial boimdary DE, figxire 13.3, eight half-diameters upstream from the plate. A variation i n velocity of 3 percent on BE was found b y the analytic method. I n the finite difference solution, i t was also assumed that the velocity potential is constant on the artificial boundary DE. I t can be proved analytically that a solution where both the velocity and the velocity potential are constant on the line DE does not exist. W h e n only the potential was assumed to be constant on DEy analytic expressions for the mapping functions were obtained. I t was found that the effect of the artificial boundary on the physical dimensions was less than 1 percent. Next, for the case without the artificial boundary, and with the correct parameters, the values of Z(W) corresponding to a rectangular region R i n the W^-domain were calculated b y numerical integration. Interior values of Z , for a square grid i n i?, were obtained b y finite difference methods w i t h the aid of the Electric Computing Board of the Watertown Arsenal [101. I n this combined finite difference and analytic solution, the errors due to artificial boundary conditions and incorrect free streamlines were eliminated. 13. Error of Finite Difference Approximation. Since we use conformal mapping frequently, i t seemed of interest to determine the error due to the finite difference approximation of the finite difference methods i n the simple case of the mapping of the unit circle onto the unit square. M e s h sizes of A.= l/2,1/3,1/4, and 1/6 were u s e d , " and the values of R and 6 were compared with the exact values foimd b y elliptic functions. F o r the finest mesh (A=l/6) the maximum error i n R and $ was 3 percent (most of the values were much more nearly exact). The numerical results and computational details are given i n Appendix I I .

Appendix 1. Numerical Integration Near Singularities The procedure for evaluating
0(z)

a)

(13.30)

i

N o w G{z) is integrable i n closed form, and F(z)=f{z)—G{z) may be computed for a suitable grid i n z near z—a. T h e n S F{z)dz m a v be found b y numerical integration (see sec. 9). Finally, we have ff{z)dz=f

F(z)dz+f G{z)dz. 2. Case of One Singularity of Fractional f(z)=F(z)-\-Qiz), where "

Order.

líf(z)

has a pole of order k at 2 = a , we may write

G{z)={z — a)-*{ao+ai{z — a)+ . . .

+ar(z—aY},

(13.31)

a n d where r is chosen so that F{z) is small. T h e n G{z) may be integrated i n closed form, and F{z) = f(z)~G(z) m a y be integrated numerically as i n (13.30). If r is chosen large enough so that Flz) is very small, i t is very hkely that the error i n the numerical integration of F{z) w i l l be small. However, because of the factor of fractional order, the higher derivatives of F(z) w i l l have a singularity at 3 = a , and hence the formal error estimates for numerical integ r a t i o n will be inapplicable. Other methods may be employed which are more laborious but which avoid the above difficulty. O n e method is to change variables. Thus if k=rls (r and s integers), we may let (13.32)

a = {z—a)',

a n d then the integrand has a pole at
h(z).

J> For the mesh A * 1/6, the finite dlflerenoe solution was obtained by using the electric computer of the Watertown Arsenal [10]. u If there exist real constants m and Af so that 0<m< Um ''f'>-''«*'> Z—0


we may replace z by g(z) In (13.30) and (13.31). This may sometimes be more convenient In practice.

(13.33) (13.34)

t This may be transformed to the differential equation dh(z) dz

g(z)~h(z) a

which may be solved b y step-by-step methods, starting from the initial conditions h{a)=g(a),

h'(a)=l

g'(a).

3, Integration Near Two SingtUarities of Half Order Very Near Together. L e t the singularities be at z~ai and 2 = 0 3 , and let b be any convenient point near ai and O j , and near which J*f(z)dz is desired. We may write f{z) — G(z)+F(z)j where Giz)=(z-a,)-Hz-a2)-H<^o-tCy{z-b)+

in which r is chosen so that G{z) is small. found as before.

Appendix 2*

. . . +0,(2-6)'},

Since G(z) m&y be integrated i n closed form, Sf{z)dz

m a y be

Approximate Conformai Mapping of the Unit Square Onto the Unit Circle

The radius R is determined as follows [2]. W e find b y finite difference methods a function such that ^ = l o g r = l o g V ? + P on the boimdary of the square (see fig. 13.11), and ^ is harmonic i n the square. Then 7?=r c"*.

FIGURE

13.11.

B y using symmetry, we can find the angle B, which is the harmonic conjugate function of log i?, by solving a problem of mixed type (see fig. 13.12). Since the Watertown Arsenal machine is best adapted to solve the first boundary value problem, we solved the equivalent problem (fig. 13.13).

d=4 5'

FIGURE

13.12.

FIGURE

13.13.

The numerical results are given below. The conformal mapping of the unit square onto the unit circle A comparison of Ui6 approximate values of *'R" and 9 obtained by finite difference methods with the exact values obtained by elliptic funotions

v-0

R Exact

0 0 0 0 0

t-H

9-H

9

9

R

R

9-\

9

jlndeter) minate 0 0

Exact

.IMM

45.00 0.21845

*-H

.1M66

46.

Exact

.23180

45.

0.32740

.23390

45.

.32880

30D20

46.

31180

4A.

k-H

30062

46.00

.34556

17.71

Exact k-H

46967 47290

46.00 45.00

48937

26.25

.46760 .46683

45.00 45.00

48951

25.83

.65337

Exact k-H

.62715

45.00

.64280

30.18

k-H k-H k-H

.63 I.I I

46.

.62725

45.

Exact

-71180

45.00

.73860

45.

.74 IVI

*-«

9-H

9-H

9-H

9-H

21S65

k-H Exact

»-H

k-H k-H k-H

h-H k-H k-H k-H

71370

Exact k-H

k-H k-H k-H Exact k~H

k-H k-H k-H

.80079

I.I o I Mill I I II I H.I I I Mill

.34635

18.36

0.43643 .43860

. 64209

.43550 .51730

18.00

. 51910

17.37

.80960

32.00

46.00

.80965

31.93

45.00 1. MM I 46.00 45.00 46.00 45.00 1.00000

62.68

32.63

0 0

10.70

.64540 .64360

0 0

48

17.19

.74981

.706

.68060

16.44

44

16.96

.83730

.80750

9.33

0.89800

34.48

.80660

9.

.88860

A

36.84 1. I m I

36.46

.83506

0

24.83

A

.83335

1. M.U.I

0.83538

6.96

A

1. M i l

10.92

.74965

20.03

45.00

0.64360 .65040

56332

2a 46

.86802

10.98

.90606

4.12

30.35

.86797

10.83

.90668

4.06

31.47

1.00000 1.00000

12.17 11.25

1. M i l l

20.76

1. M_MJ

31.28

1.00000 1.00000

11.90 IZW

1. n o t

5.46

1.00000

6.19

1. M i l l

6.30

0.96337

I. M I.M

3.06 1 I.I.I It

1. Ml I.I

197 1. M.M.I

L37

1,35

I I Ml IMM Mill I.M I \ M.rM

References 1 D . Riabouchinsky, On steady fluid motion with free surfaces, Froc. London Math. Soc. 19» 206 to 215 (1920). 2 R. V . Southwell, Relaxation methods in theoretical physics (The Clarendon Press, Oxford, 1946). 3 R. V . Southwell and G . Vaisey, Fluid motions characterized by free streamlines, Phil. Trans. Roy. Soc. [A] 24#, 117 to 161 (1946). 4 J . B. Scarborough, Numerical mathematical analysis (Johns Hopkins Press, Baltimore, M d . , 1930). 5 E . T . Whittaker and G . N . Watson, Modern analysis (Macmillan Co., New York, N . Y., 1943). V . L. Streeter, Fluid mechanics (McGraw-HiU, New York, N . Y., 1948). 7 L. M . Milne-Thomson, Theoretical hydrodynamics (Macmillan and Co., Limited, London, 1938). 8 Harvard Computation Lab. Report No. 1, Helmholtz-Rayleigh flow streamlines and equipotentials (April 1949). This report was prepared for the Atomic Energy Commission under Contract AT(30-1)—497 Code H U X . [9: Birkhoff, Plesset, and Simmons, Wall efl'ects in cavity flow I. [10 O. L. Bowie, Electrical computing board for the numerical solution of partial difl'erential equations, Watertown Arsenal Laboratory Report W A L 790/22. 11 A. E . Kennelley, Tables of complex hyperbolic and circular functions (Harvard Univ. Press, 1914). 12 G . Shortley, R. Weller, P. Darby, and E . H . Gamble, Numerical solution of axisymmetrical problems with applications to electrostatics and torsion, J . App. Phys. 18« 116 to 129 (1947). [13] A. Lowan and H . Salzer, Coeflicients for interpolation within a square grid in the complex plane, J . Math. Pbvs. X X I I I , 156 to 166 (1944). [141 P. Montel, Sur quelques propriétés des différences divisées. Journal de Mathématique Pures et Appliquées, 9 Série, 16, 219 to 231 (1937). 15: N . Simmons, The geometry of liquid cavities with special reference to the effects of finite extent of the stream. 16 R. von Mises, Z. der Deutsche Ing. 61, 469 to 473 (1917). 17 A. E . H . Love, On the theory of discontinuous fluid motions in two dimensions, Proc. Camb. Phil. S o c , 7, 175 to 201 (1891). [18] National Bureau of Standards, Projects and Publications of the National Applied Mathematics Laboratories (Jan.March 1949). [19] L. Malavard, Le technique des analogies rhéoélectrique. Travaux du Laboratoire de Calcul Expérimental Analogique (Paris, 1949). [20] National Bureau of Standards, Tables of circular and hyperbolic sines and cosines for radian arguments, M T 3 (1939) Supt. of D o c , Wash., D . C . [21] G. GreenhiU, Theory of a streamline past a plane barrier. Advisory Committee for Aeronautics. Report No. 19, p. 35 (London 1910). [22] L . Fox, Some improvements in the use of relaxation methods for the solution of ordinary and partial differential equations, Proc. Roy. Soc. [A] 196, 31 to 59 (1947). r

'

F

h

^

•

4

4

4

14,

Conformai Mapping in Aerodynamics, with Emphasis on the Method of Successive Conjugates I. E . G a r r i c k ^

1. Introduction Some remarks of Dr. Hugh Dryden, Director of Research of the National Advisory Committee for Aeronautics, made in a portion of his recent Wilbur Wright Memorial Lecture [l] may well serve to introduce this paper. "One of the surprising developments in aeronautics is the wide utility of potential theory of a nonviscous fluid. A few decades ago classical hydrodynamics was considered as a piwely mathematical discipline, having little or no connection with the behavior of a real fluid. The practical field of hydraulics was based almost completely on empirical experiments with little underlying theory. With the advent of the airplane, and experimentation on its component parts, it was foimd that predictions made on the basis of the mathematical potential theories of classical hydrodynamics were strikingly accurate in many instances, especially as bodies of better aerodynamic form were discovered. As a result, aeronautical engineers have developed a considerable degree of confidence in the predictions of potential theory." Of great significance in potential flow theory is the theory of an incompressible fluid, even for wide subsonic flow ranges of a compressible fluid. Moreover, in the theory of the incompressible fluid that of the two-dimensional or plane flow field plays a leading rôle. It is in relation to this phase of the theory that the significance of complex variable theory and conformai mapping directly enters the aerodynamical or fluid dynamical scene. It is now over a century since Riemann's fundamental theorem of conformai mapping was first established. This theorem permits us to find the flow pattern and pressure distribution about a compUcated profile from that of a simple or standard flow pattern. For the actual achievement of this residt for an arbitrarily given single wing profile, the older methods, associated with the various mathematical proofs of the theorem, are entirely too slowly convergent in relation to a life span. Such is also the case with methods and criteria based on area distortions (associated with the names of Koebe, Bieberbach, and Hohendorf). Especially popidar in the literature has been the indirect procedure of defining classes of airfoil shapes by chosen transformations, and also the approximate methods based on thin ainoils. Practically all of the procedures prior to 1931 dealing with the direct problem have been either very laborious or approximate and not readily adapted to successive improvement. In 1931 and 1932 at the Langley Aeronautics Laboratory of the National Advisory Committee for Aeronautics, Theodorsen and the present author introduced a method for the direct treatment of arbitrary wing sections, that has remamed outstandmg in its convenience, simplicity, and generality of application. This work represents in a real sense a contribution to mathematics arising out of aerodynamics. It has been often referred to by now in the scattered literature. For the sake of completeness and to provide a proper background for some of the material to be discussed, some of the main features of this work will be reviewed. These features should find applications in many other boundaryviJue problems of current applied mathematics.

2.

Review of Features of the Theodorsen-Garrick Method

The method as applied to wing sections (exterior potential field problem) [2]* consists of two main parts (fig. 14.1): (a) Transformation of a given wmg-section profile into a near circle; (b) Transformation of the near circle into a circle. Remarks on (a). A few appropriate remarks may first be made on the transformation to a near circle. The achievement of a near circle from the given profile is highly desirable in the further developments, for it will effectively replace the given profile. It is actuaDy desired to map the profile into a t Lancley Aeronautical Laboratory, National Advisory Conimfttee for Aeronautics, Langley Field. Va. 'Fignivs in brackets Indicate the literature references at the end of tbls paper.

"star-shape'' contoiu'i that is, one in which a radius vector from the or^in intersects the boundary cm:ve only once. Although we speak of a transformation of the given profit into a near circle, actually equivalent to this clearly is any transformation of the region of the given body to a near standard region, ¿-PLANE

Z -

PLANE

f(2)

Z -

PLANE

FtauKE 14.1.

as for example, a near half-plane or near line-region, which standard boimdary can be transformed directly to a circle. For airfoil shapes the well-known transformation 2 ' = f + a V f , leading to elliptic cylinders and Joukowski airfoils, has proved to be convenient and simple enough for use as an initial transformation to invert the profile into a near circle. In conjunction with this inversion transformation it is important to choose the origin of coordinates so that the portions of sharp turning of the profile, near the leading and trailing edges, have a parabolic or elliptical appearance relative to their focii. These focii then serve to determine the proper choice of origin. For other types of bodies, other initial transmrmations may be considered appropriate. For example, even for wing profiles, the generalization of the Joukowski transformation that leads to finite angles at the trailing edge, the Karman-Trefitz transformation (based on a skeleton profile of circular arcs instead of a line segment) may be considered, although its inversion is much more difficult. In this coimection, the usefulness of lists or dictionaries of prepared conformai transformations is emphasized. Finally, it is remarked that even a residting near circle may be improved by the choice of its associated origin of coordinates (conformai centroid). Remarks on (b). As noted, the near circle in the 3'-plane replaces our ordinal profile. The next step is a decisive one, the conformai transformation of the near circle into a circle in the 2-plane. For this purpose the transformation from z' U> zis expressed in the form of the integral transcendental function

a4.1a) or (14.1b)

log|=/(2),

where f(z) is a power series in l/g,

Applying the transformation ( 1 4 . 1 ) to the correspondence of the boundaries of the circle of radius i?, z—Re^'^y and the near circle 2'=a€*''"'*, where ae* is the radius vector and B is the polar angle, and separating real and imaginary parts, we come directly to the following Fourier series relations: 2ü(«fi cos » ^ + 6 » sin n^); 1

bn cos n^-ha* sin n^).

(14.2)

where £2-

and

i{=a«^.

The mterchange of coefficients exhibited in the two infinite series is characteristic of conjugate Foiuier series. (Observe Uiat the constant ^ 0 is associated with the logarithm of the radius of the circle, and is given by

^ is associated with the logarithm of the radius vector in the s'-plane: and ^—6 is the angular displacement or ai^idar distortion of corresponding points on the boxmdaries. The correspondence of fields at infinity has been selected in this work as 2 = 2 ' and (dz'/dz) = l, thus fixing the choice of ^ 0 and also leading to the fact that I

(
The association of two functions y(^) and g{ip) defined by two conjugate Fourier series may also be expressed in an integral form: '31' 1 c —

(14.3b)

2

It mav be observed that the relations essentially express the imaginary part of the boundary values, along a circle, of a complex fxmction, in terms of the real part, and vice-versa. The study of properties of a Fourier series in terms of conjugate series has led to some very profoimd mathematical developments in the theory of Fourier series. The integral relation may broadly be considered as an operation which generates functions conjugate to given ones.

3.

Method of Successive Conjugates

We may observe that the function yjf is originaUy given as a function of B, denoted by ^(fl), where d is the polar angle in the plane of the near circle. To denote ^ as a fimction of the polar angle tp of the circle, we will employ the notation ^(y). Thus (14.4) The basic integral relation corresponding to the series in eq (14.2) may be written, with tp—B designated by the function c(^), as cot

(14.5)

2

This relation is not a definite integral, but a functional equation, a nonlinear integral equation, since actually if{B) is known initially and not '^{ip). The process of solution which we have described as the "method of successive conjugates'' is as follows: Writing, in eq (14.5), * ( ^ ) = ^ ( ^ ) = ^ [ ^ — « ( ^ ) ] , we have 3»

(14.6)

«(v»0

An initial selection of the function c(^), denoted by eo, is inserted on the right-hand side. This function may actually conveniently be chosen to be zero if nothing better is known. Evaluation of the integral (see sec. 4) then leads to a new function ci(^) on the left-hand side. Repetition of the process l ^ d s ultimately to: e,(v>)] cot

2

dip.

Investigation of conditions on the given function 4f{B) and on the initial choice CQ necessary and stifficient

139

for convergence of the process is an exceedingly interesting theoretical matter. Although such conditions if sufficiently broad would be both gratifying and helpful, for practical evaluations they are not required, since at every stage the process represents a solution of a proper mapping problem and is self-checking, the deviations of tn+i and u measuring the convergence. (The appropriateness of the choice of the initial transformation to a near circle and of the origin of coordinates axe also reflected i n the convergence.) Conditions sufficient for convergence of the method of successive conjugates have been examined theoretically by S. E . Warschawski [3] i n an interesting contribution to this subject. H e defines a nearly circular curve i n a definite quantitative manner and establishes several sufficient conditions for convergence and also gives various estimates of the n t h approximations for t(
4«

Numerical Evaluation of Conjugate Functions

I n the following there is developed briefly a convenient and direct procedure for evaluation of conjugate functions. I n essence the procedure is simply a residt of direct harmonic analysis of a given function, and subsequent synthesis of its conjugate function. It is somewhat surprising that although classical i n principle i t appears to be of such recent vintage. The method has been given, at the suggestion of the writer, i n another development b y N a i m a n i n an N A C A wartime report [61.* (For related work done independently i n England and France, see [20].) L e t there be given a single-valued continuous bounded fimction ^(^), of period 2ir, whose conjugate fimction is sought. L e t the interval x < ^ < x be divided into 2N equally spaced intervals of extent 2w/(2N) = wlN=Qj and let the 2N values of the given function be written as —

^(-2)=*-i. * ( 0 ) = 4 ' Of

^(2)=*if

^{kq)

= ^^

^(Nq)='if{ir)^'^

rf}

where the 2N values correspond to integral values of the index k fro Observe that, because of periodicity, we nave

(14.7)

(AT—1) to AT, including 0.

L e t the original function ^ (^) be represented b y (or approximated by) the unique finite trigonometric series with graph passing through the 2N chosen points: (14.8a) where Z here, as elsewhere i n this section, means summing over the 2N values of s from — (N— 1) to including 0. T h i s form corresponds to the real series *(v»)=S

cos 8ip+b, sin Sip),

(14.8b)

where We first solve for the coefficients .¿1, i n terms of the 2N values of

Consider the following sum of 2N terms each term consisting of a product of

-2

*

and a unit vector e'""*',

^ a

«

where 5=

S

B y direct sununing of the geometric progression or otherwise, we obtain 0

for s 7^ m,

2N

for s = m.

B=2NS:=

T h e n the desired evaluation of the coefficients is (14.9) E q u a t i o n (14.9) corresponds directly to harmonic analysis of the function ^(^); and, as mentioned later, (14.9) determines the various coefficients of the transformation eq (14.1). T h e finite trigonometric series conjugate to the function ^(v>) given i n eq (14.8) m a y b y definition be expressed as « ( ^ ) = S ' ( « » 8 " ! s 0 , sgn 8 = — 1 fors<0. W i t h the aid of eq (14.9) let us find the value of e(v>) from eq (14.10) at the position ^ = 0 ; that is, let us find the value of 1 i ikMt «0=2

The coefficient of

sgn « = 2 ] ^ 2 ' sgn « ^

^te

i n this expression is 2?^ S ' (sgn «)«


I

and examination shows its value to be

N

^"^

^'N

=

T cot ~

= 0

2N

for k odd, for ifc even.

Finally, grouping terms with corresponding positive and negative indices, we have « 0 = - ^

S(*.-*-Ocot|j,

fcodd.

(14.11)

F r o m the integral relation we have, i n general, ¿

=

r

X'' ^ í * ' )

^

(14.12a)

M^p' + vd-^i^'-ip)]

cotfrf^,

(14.12b)

and ,(0)=-l

( ' [^(^)->^(-^)] c o t f

(14.12c)

Comparison of eqs (14.11) and (14.12c) shows that the value of €o is obtained b y the simple geometrical rectangular rule of summing ordinates at odd stations times the width of the double interval between even stations. I n the harmonic analysis of periodic functions various writers have pointed out (for example, [7]) the pecuUar distinction of the trapezoidal rule, which yields a result corresponding to o n e given i n (14.9). Here, however, we have something even more simple, the rectangular rule. I t is observed however that, as long as we wish our evaluation of e(
(14.13)

where the sum is taken for the odd values of k, and the 2N values of * are designated

I t is known that the value of «(*») at ^ = 0 may depend markedly, i n fact exclusively, o n the local behavior of the function ^(v) near ^ = 0 F o r example, if ^(^) or its derivative d'^ldip should have a finite discontinuity at ^9=0, e(0) or, respectively, the derivative at ^ = 0 , w i l l be infinite (such cases m a y arise at sharp comers). T h u s «(0) m a y depend critically on the nature of the curve near *(0). However, the procedure for numerical evaluation just given shows that the ordinates of the curve at ^(0) do not even enter the problem. F o r the class of functions composed of a finite number of harmonics (actually equal to 2 i V - l ) , the result is exact for any such series passing through the chosen odd points and through an a r b i t r a n l y selected additional set of points intermediate to the odd set. T h e result must be considered as a property of the very useful class of curves defined b y finite trigonometric series. A c t u a l l y , for arbitrary curves, the local slope of the ^(^) curve at ^ = 0 may be significant; and the result assumes that any fairing of the curve between odd points is such t h a t no harmonics higher than 2A^-1 are introduced b y the process of fairing. It m a y also be remarked that the values of the derivatives dylz/dtp and dtjdtp, which are required i n a study of the velocity field of a flow pattern, can also be similarly developed i n terms of the chosen ordinates. Further remarks:

5.

Application of Conjugate Functions in Various Integral Equations With Singular Kernels

The process of generating a conjugate function from a given one and the associated numerical procedures can be of great utility i n other problems i n applied mathematics. I t perhaps m a y not b e considered out of place merely to indicate i n this paper the scope of such applications b y the example of how certain important integrals i n aerodynamics may almost automatically be handled with the conception of conjugate functions. * Consider the equation ,(g)^ir ^

'

IT Jo

., COS 5 — C O S a

(14.14) '

where Cauchy principle values are imphed. B y defining/(tr) to be an even function of a, and adding an i n t ^ a l expression easily identified by symmetry to be zero, we obtain sin Ö—sin
g(e) sin e

cot

2 a(r.

(14.15)

The functions g(B) sin B a n d / ( 0 ) are therefore related by conjugate Fourier series, so that conjugate Fourier series of J{B) S I

(14.16a)

C o n v e r s e l y the reciprocal relation (see eq (I4.3b)) may be employed to 3deld the solution of the integral equation as /(ö)=conjugate Fourier series of [g{B) sin

tf].

(14.16b)

F

T h i s procedm*e of relating conjugate functions may thus serve broadly as a fountain-head for rel a t i n g m a n y types of functions inventing symmetrical singular kernels; for there are numerous such kernels i n applied mathematical problems reducible to the cotangent b y various operations; for example. 1

x—y and so on.

1 cos ^—cos

V—Ö log sin 2

1—cos 1—cos

{iff—9) {fp+9)

The equation /({) 2

useful in aerodynamics and given a detaUed treatment by Sfihngen in [8], is another example for which the above procedure is directly appHcable.

6.

Remarks on the Evaluation of Flow Quantities in the Flow Field

The flow pattern for a given profile is determined directly in terms of the conformai mapping functions and the standard flow pattern past the circular cylinder. As has been described in [2], the boundary surface velocities and pressures are conveniently determined in terms of ^(^), t{*f>), and their first derivatives. In order to determine velocities and pressures in the flow field, if desired, one procedure is to utilize directly the conformai mapping fimction, the various coefficients of which may be determined from the ^(^) function as mentioned following eq (14.9). Another procedwe, described in [9], avoids the explicit determination of the mapping function, and utilizes an integral relation (generalized Cauchy formula), dip

log 1

(14.17)

R

This relation leads directly to the association of a given value in the 2-field with a value in the s'-field and by an iterative process described in [9] to the reverse.

7.

Application to the Flow Past Bodies of Revolution

It is intended to present in this section only the high spots of an application of conformal mapping to three-dimensional axial flow past bodies of revolution. The method has been developed by C. Kaplan of the Langley Aeronautical Laboratoiy and details are given in [10]. The equation of continuity apphed for this problem leads to the following equation for the velocity potential, replacing Laplace's equation in 2-dimensions: Ò /

Òip

ò x V'-^-^ò-pVòp òx

Ò /

Vo,

Òip

(14.18)

where x is the coordinate distance along the main axis and p is a perpendicular distance in a meridian plane. The basis of the method consists in the observation that the form of eq (14.18) is invariant under a conformal transformation: 2 = a: + t p = y ( { + t „ ) = / ( f ) ;

that is,

where /)=p(f,ij). Further, it is noted that the particular conformai transformation of the given meridian profile into a circle essentially yields the simplification that the parametric equations of the given profile for the two variables x and p are given i n terms of one variable |, where the radius vector i n the plane of the circle may be denoted as The circle itself corresponding to the meridian profile is given b y the coordinate i j = 0 . The actual evaluation of the coefficients of the conformai transformation is readily performed b y the procedures already discussed, i n particular from the harmonic analysis of the '*'(*») function; see eq (14.9). T h e boundary conditions to be satisfied on i j = 0 take simple forms i n terms of the velocity potential; and a process of iteration is then used to obtain the velocity potential, i n w h i c h successively one additional coefficient of the conformai transformation is introduced, the differential equation and the boundary conditions being satisfied successively to this order. T h e process is repeated, bringing i n as many terms of the transformation as are deemed desirable. I n each step certain imiversal functions occur which need be evaluated only once for all shapes and b y elementary intégrations (see [10] for details). It is to be noted that the problem of transverse flow may also be considered i n the same way, but details for this problem have not been published.

8.

Treatment of Poisson'g Equation in Some Plane Flow Problems of a Compressible Fluid

Partial differential equations more general than Laplace's equation often arise i n flow problems. I n the theoretical treatment of the difficult flow problems of a compressible fluid, iterative methods are often employed i n which expansion is made of the velocity potential, or stream function, i n terms of the M a c h number of the main stream, or i n a parameter representing the thickness or camber of the profile (see [11]). I n much of this work, the Poisson equation V2^=/(x,y) plays a central rôle. The function on the right-hand side may be associated with density changes i n the fluid and regarded as the effect of a continuous distribution of sources i n the field of flow. A procedure for treating this equation w i t h the aid of complex variables and conformai mapping [12] is as foUows: (a) I t is convenient to introduce the complex variables 7 i n place of x, in the plane of the given profile. (b) The profile is mapped onto a circle i n the Z-plane by a conformai transformation, thereby transforming/(ar,y) into a function of Z and Z . (c) The effect of a single singularity i n the external flow field (source) is found, satisfying also the boundary flow conditions. (d) Double integration of this effect over the field of flow for total effects of the distribution function is next: E v a l u a t i o n of this surface integration is accomphshed first b y the use of Stokes* theorem to yield a single integration over the boundary circle and a control circular boundary at infinity. I n this process i t is convenient to make use of various elegant results of vector analysis i n the plane employing functions of a complex variable [13]. (e) E v a l u a t i o n of the resulting line integrals is then systematicafly performed b y means of the device of creating analytic functions b y replacing Z by WjZ. Consequently the line integrals are evaluated b y Cauchy's residue theorem. I n brief, as stated i n [11], " T h e device of introducing z and 0 as independent variables, then utilizing the conformai mapping of the plane of the obstacle into the plane of a circle, and finally replacing the double integrals b y line integrals thus enables one to evaluate the first effects of compressibihty on the flow past an arbitrary shape. The point of interest to an applied mathematician is that here is a method whereby a Poissons' equation involving rather compUcated boundary conditions can be solved with the aid of analytic functions of a single complex variable. The subject is certainly worthy of further investigation."

9.

Conformai Mapping of Biplane Wing Sections

A generahzation of the methods described for single airfoils (simply connected regions) to biplanes or slotted airfoils (doubly connected regions) has been developed i n [14]. The essential features of this work are parallel to the monoplane case and consist of three parts:

(a) an initial transformation of a biplane arrangement of two arbitrary line segments onto two circles or onto an annular region; proper inversion of this transformation may yield for a given biplane ttrrai^ement a "near" annulus; (b) the transformation of the nearly annular region to an exact annular region; and (c) the utihzation of the flow pattern of two circular cylinders, in particular as developed by Lagally. The problem draws on the theory of eUiptic functions in a natural way and represents, we beueve, one of the most informative and appropriate applications of these ubiqmtous functions. In particidar we may mention only that the integral relationship central to the discussed method of successive conjugates is here replaced by two simultaneous relations representing the interrelation of four boundary functions for the two boundaries. A great deal of numerical effort is yet required to bring out properly the scope of systematic investigations in this work.

10.

Lattices or Cascades of Airfoils

The problem of determining the flow past infinite rows of airfoils symmetrically arranged is of considerable interest and utihty. It is noted that many approximate methods exist, in particular a convenient semi-graphical one being given in [15]. Treatments of lattices of arbitrary airfoils have been given by HoweU [16] and hy Garrick [9]. In the latter reference, the essential features of the method are parallel to those of the smgle airfoil case. In fact, the flow field of the lattice of airfoils can be transformed to the plane of a single circle, in which two (branch) points are distinguished, the neighborhoods of which represent the flow regions far ahead and far behind the airfoil lattice. The initial transformation of a lattice of lines is inverted to yield a ''near" circle, and the near circle transformed to a circle by the methods described. It is to be noticed that improvements in the initial transformation are desirable and should be sought, particularly to take care of highlv cambered airfoils more conveniently, in order to reduce the amount of subsequent calculations. In this connection and in order to gain useful insight into the method and results good use can be made of the inverse method of defining and creating airfoil lattice arrangements by choosing various appropriate functions S^(^) instead of working with 4f(0). This procedure would in a sense yield a glossary or dictionary of prepared mapping arrangements which would facihtate further systematic developments.

11 • Creation of Wing-Section Profiles of Desired Properties A growing hterature is developing on a so-called inverse problem, the finding of profiles to go with prescribed pressure distributions. In the opinion of the writer several diflSculties arise or exist in defining this problem, to satisfy both the mathematician and the aerodynamicist. For one thing, attempts have not been successful in making precise statements of the problem in regard to uniqueness, closure, proper trailing edge, leading edge contours, avoidance of grotesque nonstreamline figures most likely to be subject to separated flow, or of no physical significance as figures eight (or worse). For another, the prescription of pressure distributions with respect to a reference chord leads to nonuniqueness ; and prescription with regard to normals to the boundary surface leads to indefiniteness, since the physical boundaries are being sought. Another difficulty is the fact that our insight and knowledge of flow behavior are not developed to the point that an exactly defined desirable pressure distribution can be specified. We are perfectly content to obtain desired types of pressure distributions. Thus the problem retains elements of compromise and quaUtative features. Perhaps the safest and surest procedure would be to proceed from a given profile and a given pressure distribution to a quaUtatively modified neighboring pressure distribution and a correspondingly modified profile (see for example [17]). In this regard ^(v?) and t(
12,

Velocity Correction Formulas

The conception of "velocity correction formulas" has arisen i n attempts to associate the velocities and pressures i n the flow field about a body i n a compressible fluid with those about the same body in the incompressible fluid. The search for such simple concepts has of course been greatly prompted by the difi&cult mathematical nature of the flow equations of a compressible fluid and b y the fact that no exact relevant solutions are known for closed bodies. The procedures leading to velocity correction factors are best developed b y use of the hodograph variables. I n [21] a discussion is given with spcMcial reference to correction formulas of Prandtl-Glauert, von Kérmân-Tsien, Temple-Yarwood, and GarrickEaplan. Since these correction factors attempt to relate the solutions for the compressible and incompressible fluids, the conformai mapping methods, associated with the latter, are pointed up. Comparison of results obtained by these simple and cheap methods, with those obtained by long and tedious iteration methods, leads to the conclusion that the factors can be remarkably good for subcriticaJ subsonic flows. Such a comparison is given i n [11, p. 41] where it is stated: " T h e agreement between the two methods over such a wide range of thickness coeflicients and stream M a c h numbers is remarkable. Indeed the development of velocity correction formulas and their use i n the prediction of compressibility effects should be considered as an outstanding achievement of theoretical aerodynamics. For, consider that the problem of compressible flow involves a nonlinear differential ecjuation for which very httle mathematical treatment is available, nevertheless with the aid of a few simple ideas and very little labor the essential residts can be obtained b y means of velocity correction formulas." Though no completely logical justification can be offered for such overall corrections, they would seem to have at least as much merit as "sampling processes" that have been mentioned for some partial differential equations. The topic is an intriguing one and deserves to be made more precise. It has been brought up here mainly because it calls attention to additional usefulness of conformai mapping.

13.

Concluding Remarks

It has been the aim of this paper to sketch general conformal mapping methods of applied mathematics which have arisen and have been developed because of incentives of aerodynamical investigations. It need hardly be stated that the flow problems and methods bear a close relationship to the classical boundaiy-value problems of Dirichlet and Neumann and to the determination of Green's functions. Avoiding burdensome details as far as feasible we have indicated how the methods may be adapted to routine numerical procedures and have pointed to some fields of endeavor to be filled out b y numerical applications. Numerous broad aerodynamical applications have been completely omitted. A few among these concern such problems as various flow interference effects including body-wing interactions, and boundary jet corrections for wind tunnels, phases of nonstationary flows, boundary layers, downwash and loading distributions, " c o n i c a l " flows at supersonic speeds, and bodies i n ciu^ed flows. The importance of having available exact results for potential flow is not to be underemphasizcd. These results serve as a standard to which to refer results of all approximate methods of a nonviscous flmd, and further, as the ideal to which to refer results, both experimental and calculated, that take into accoujit viscous effects of a real fluid. Finally, another objective of this paper has been to emphasize that conformal mapping is not circumscribed i n its utility to classical two-dimensional incompressible flow problems, for its applications may deal also with flows i n space and with the flow of a compressible fluid.

14.

References

(1) H . L. Dryden, The aeronautical research scene—goals, methods and accomplishments, 1949 (37th) Wilbur Wright Memorial Lecture, London, April 28, 1949, Royal Aero. Soc. 2 T . Theodorsen and I. E . Garrick, General potential theory of arbitrary wing sections, N A C A Rep. No. 452, 1933. 3 S. E . Warschawski, On Theodorsen's method of conformal mapping of nearly circular regions. Quart, of Appi. M a t h . 3, No. 1, 12 to 28 (April, 1945). [4j H . Wittich, Bemerkungen zur Druckverteilungsrechnung nach Theodorsen-Garrick Jb. dtsch. Luftfahrt-Forsch. I, 52 to 57 (1941). See also same author, Z. angew. Math. Mech. 25/27, 131 to 132 (1947), and Fiat Review of German Science 1939-1946. Applied Math. Part I, Practice of conformal mapping, by K. Ullrich, 93 to 118 (1948). [61 E . Study, Vorlesungen Über Ausgewählte Gegenstände Der Geometrie. Zweites Heft, Herausgegeben unter Mitwirkung von W. Blaschke. Konforme Abbildung Einfach-Zusammenhängender Bereiche. (B. G . Teubner, Leipsig u. Berlin, 1913). [6] I. Naiman, Numerical evaluation by harmonic analysis of the c-Function of the Theodorsen arbitrary-airfoQ potential theonr, N A C A Wartime Report L-153 (originaUy A R R No. L5H18, Sept. 1945). m W. E . Milne, Numerical calculus, p. 302 (Princeton University Press, Princeton, N. J . , 1949). [8 H . SAhngen, Die Lösungen der Integralgleichung

9(x) TragflQgelthi

m di 2rJ-a X - l

[9] I. E . Gftrrick, On the plane potential flow past a lattice of arbitrary airfoils, N A C A Rep. No. 788» 1944. 10 C . Kaplan, On a new method for calculating the potential flow past a body of revolution, N A C A Rep. No. 752, 1943. 11 C . Kaplan, A review of approximate methods in subsonic compressible flow, N A C A - U n i v e r s i t y Conference on Aerodynamics: A compilation of the papers presented, Durand Reprinting Committee, Cal. Inst, of Tech., 29 to 47 (June, 1948), [12] C. Kaplan, On the use of residue theory for treating the subsonic flow of a compressible fluid, N A C A Rep. No. 728, 1942. 13 L. M . Milne-Thomson, Theoretical hydrodynamics (MacMillan and C o ^ London, 1938). 14 I. E . Garrick, Potential flow about arbitrary biplane wing sections, N A C A Rep. No. 542, 1936. 15 S. Katzoff, R. S. Finn, and J . C. Laurence, Interference method for obtaining the potential flow past an arbitrary cascade of airfoils, N A C A Tech. Note 1252, 1947. 16 A. R. Howell, Note on the theory of arbitrary aerofoils in cascade. Royal Aircraft Estab., Note No. E3859, Mar., 1941. 17 T . Theodorsen, Airfoil contour modifications based on c-curve method of calculating pressure distribution. N A C A Wartime Report L-135 (originally A R R L4G05, July, 1944). [18] T . Theodorsen, and I. Naiman, Pressure distributions for representative airfoils and related profiles. N A C A Tech. Note 1016 (1946). 19 I. H . Abbott, A. E . VonDoenhoff, and L. S, Stivers, Jr., Summary of airfoil data, N A C A Rep. No. 824. 1945. 20 S. Goldstein, Low-drag and suction airfoils, the Eleventh Wright Brothers Lecture, Jour, of Aero. Sciences, No. 4, 189 to 214 (April, 1948). [21] I. E . Garrick and C. Kaplan, On the flow of a compressible fluid by the hodograph method: I—Unification and extension of present-day results, N A C A Rep. No. 789, 1944; II—Fundamental set of particular flow solutions of the Chaplygin differential equation, N A C A Rep. No. 790, 1944,

are:

Other references and bibhographies are contained i n the foregoing references.

T w o notable omissions

T h . von K i r m ^ n and E . Trefftz, Potentialstrommung um gegebene Tragfl&chenquerschnitte, Z. F . M . t . 111 (1918). L . Lichtenstein, Neuere Entwicklung der Potentialtheorie; Konforme Abbildung. Encyklop&die der Math. Wissenschaft Bd. II, Analysis, Teil III, No. 3 (C3), 1919.

15.

O n the Convergence of Theodorsen's and Garrick's Method of Conformai Mapping' A. M . Ostrowski*

Introduction We consider a simple, closed, rectifiable curve C in the w-plane, tr= pc**, containing the origin in its interior. The equation of C in polar coordinates may be (15.1) 9 being the vectorial angle, where P(tf)=lg p is uniform, continuous and periodic with period 2 T . Let «'=y(2),

/(0)=0,

/'(0)>0,

2=rei9

(15.2)

give the conformal mapping of the interior of C upon the interior of i?., the unit circle. and is boimded, we obtain for C the parametric representation ^=<^(*>)=arg/(e"'),

p = |/(OI

If P'i?) exists (15.3)

In order to obtain a functional relation implying the unknown function B{ip) we consider the branch of the function Ig (/(2)/2), which is real for 2=0. This branch is in regular and uniform and remains continuous on the boundary.' Since for 2=c** we have 9ílg:ííf)=F(^(^)),

3lgM=^(^)

for an unknown integer m, we see that the function 0(^)—^ is the conjugate function to P(9(^)), that is to say that the Fourier series of (^(v>)—^ is an allied series to that of F(0(^)). The function B{tp)—tp is a periodic fimction with a period 2 T and represents the boundary values of the harmonic function ^ Ig {J{z)lz); since according to our assumptions this hannonic fimction vanishes at the origin, we have, by Gauss' mean value theorem.

X

2v

(15.4)

tp)dtp=(i.

can

•known e{ip)—ip

t

P(^^-O)]cot^á<.

(15.5)

Equation (15.5) is a nonlinear integral equation for 6{ip) and the starting point of Theodorsen's and Garrick's method for the determination of B{ip) [8, p. 11]^ and [9, p. 8 and 9]. As soon as 9{ip) is determined the functions/(3)/s and/(2) are obtained for instance from Poisson's integral and the problem of conformal mapping can be considered as solved. In the last 40 years very much subtle and ingenious work has been spent on the right-hand integral in (15.5). It was thus all the more a siuprise to the pure mathematicians that the frontal attack launched on the integral equation (15.5) by Theodorsen and Garrick in order to obtain numerical results, met > The iM'eparatlon o( this paper was sponsored (tn part) by the Office of Naval Research. 'National Bureau of Standards, Los Angeles, Calif., and University of Basle, Basic, Switterland. * Compare raferenoe C3|. tFifforea In b r a c t s Indicate the literature references at the end of this paper.

with full success. In many contours used in aerodynamics the first two steps alone of the method of successive approximations elaborated by Theodorsen and Garrick give an approximation suflficient for all practical purposes. Theodorsen and Garrick start from an initial value ^o(^) and derive a sequence of functions 9,(^) by the recurrence formula -Lj;im(^+o)

P{e,{9-m

cot I

A,

0,1,2,

(15.6)

In all the examples treated by them this sequence $,(^) converges rapidly to the function 6i
(0<Ö<2x)

(15.7)

0(ei).

(15.8)

and proves then

Under the further assiunption that for the same value of c (15.9) Warschawski proves even (15.10) further, under the additional assimaption (15.11) he proves that 6^,M converge imiformly to ö'(^) and that even (15.12) In his discussion Warschawski obtains not onlv the general estimates (15.8), (15.10), and (15.12) but also the corresponding inequahties with explicitly worked out niunerical coefficients.^ The leading idea of Warscnawski's method was the use of Parseval's formida and the inequalities between the mean values of the squares of conjugate functions derived from it, and the same type of argument is used in the present paper. Warschawski estimates in the first line the mean values of (B,~ fff, {$',~ 0^ and i^—B^^i from these estimates appropriate limits tor \9w—6\,\K—B'\ are easily derived. The choice 0O(^)B^ adopted throughout by Warschawski is even from the theoretical point of view an imnecessanly restricted one. ^though Theodorsen and Garrick usually start with this initial value the^ point out themselves in a certain example that this choice is a "relatively poor one". They say explicitly, "Indeed, the closer the choice of the function 'ZiO) is to the final solution 1(9), the more rapid is the convergence".* We show in this paper that the results of Warschawski can be proved and improved also imder the hypothesis of a more or less arbitrary initial fimction 6oOp). We prove that (15.10) remains valid if the inequality (15.9) is replaced by the assumption that |P''(0)| is uniformly bounded and that already under this assumption tlie sequence converges io$'; we obtain (16.13) T h e estimate (15.12) can be derived under the hypothesis that both \P"{B) and P'"(B) are uniformly b o u n d e d — i t is not necessary that they are " s m a l l " . A s a matter of fact we prove the ore general theorem i n which under the hypothesis (15.7) and the assumption that the first m derivatives of P(B) are imiformly bounded, the uniiorm convergence of the first m-1 derivatives of B, to the corresponding derivatives of B is shown. However all these results, although important from the theoretical point of View, cannot be immediately applied to the problems of practical computation. I n practice we do not work w i t h B, as they are derived successively from formula (15.6) but w i t h modified functions which are obtained from the > Further he iiiTestlntes what range of valaes of • wonld ensiua that the oontoors corres; * |0» p. 13]. T h e last remark must not hoveTer be taken too literally.

idlng to the approxlxnatlng functions #>(i») would be star^ehaped,

integral i n (15.6) by approximate integration and undergo under circumstances certain smoothing processes. F o r that reason i t is practically more important to obtain a good estimate of error of the initial function do and to derive an estimate for the corresponding function ^ i , estimates which may then be used at each step of the computation. Such estimates are given i n theorem 1, formula (15.54), and i n theorem 2, formula (15.60). In our discussion we concern ourselves with a generalization of equation (15.5) i n which

cot

^dt.

(15.14)

I n this case we camiot any longer assume the existence of a solution of (15.14) which followed i n the case of equation (15.5) from the general theory of conformal mapping. However, it is possible to prove the existence (and the uniqueness) of this solution under hypothesis (15.7) and the existence of its derivatives up to the (m — l ) s t order, if P{d) has bounded first m derivatives. We deal from the beginning w i t h (15.14), although i n the case of (15.8) where the existence and the properties of a solution ${ip) can be taken as granted, the discussion can be a little simplified. T h e results are contained i n the theorems 1-3, which are proved i n sections 4 to 6 of the paper after some lemmata are developed in sections 1 to 3. I n section 1 we give a table of notations which are employed throughout the paper without being each time introduced anew. I n this paper we deal only w i t h the theoretical aspect of the method. I n the following paper* we develop a practical procedure for this method.

1.

Notations*

Lemmata 1-3

I n what follows we use throughout the following notations: 37=

/ ( * . - « ) ) cot I d<,

(15.15) (15.16)

2 P«=P»>(^)=P<*>((?.(^)) D!f'=^!^ieifU-'en'

F,=P?> D.=^Dr

(Jfc=0,l,...;.p=0,l,...),

(15.17)

(it=o,i,...; 1^=0,1,...),

(15.18) (15.19) (15.20) (15.21) (15.22)

2 Lemma 1.

We have if P'{B)y . . . , P^'^^B) and 6\ , , ,

(15.23)

2

are continuous

dm P((?W)=P'(e)fl<->+B„(«). m dip

(15.24)

where

(15.25) B„(0)=TOP''(«)e'e'"-"(m>3), • Bee paper 16 In this volume.

(15.26)

Atn(e)=T,P'^(e)C„.Áe% Í.-2

and Cm,» ore polynomials in 6', ...

. . . ,

Í?*-»»),

(m>3),

with integral numerical coefficienis.

, d^"*

In particular

we have (15.28)

T h e proof b y mduction is immediate. Lemma


T >

2. (Minkowski's we have

inequ/iliiy).

IJ }\{.*p)y /a(^), - • • , /*(*») or^ integrable in the square (15.29)

T h i s follows at once from the integral form of the so-called general M i n k o w s k i ' s inequality [4, p. 3 1 , 146]. Lemma 3. (Warschawski*s inequality) [10, p. 18]. IJ J(
j(
j(€fdt

dt

Jo

(15.30)

r2T i J(
ProoJ: Under the hypothesis J

from (15.30). the identity

J(ip)dip=0 there exists avalué i¡f with /(^)=0, and (15.31) follow:

T o prove (15.30), we can assume without loss of generality that ^ < ^ < ^ + 2 T .

From

V+2r

J(id'~J(
£j(t)rit)dt

2 I

4'

J(t)f(t)dt

it follows that JM'-My\<2£\ff'\dt.

fi
JW-MYl


J

V / ' l dt+ 2J^"\JJ'\di

j =

dt

X

mm,

and (15.30) follows from the Cauchy-Schwarz inequality.

2.

Properties of Conjugate Functions.

Lemmata 4 - 5

If U(
''^+

sin vfp)

(15.32)

can be written down although this formula is to be understood in the well-known symbolic sense. In this case there exists also a function V(
cos v+a, sin v
(15.33)

i

V(^) is defined uniquely except on a zero set. A n y of the functions V'(^)+const. is then called the conjugate or allietl function to U(tp). A m o n g a l l these conjugate fimctions the functions corresponding to the development (15.33) are characterized by the equation (15.34) A periodic function with period 2x satisfying (15.34) w i l l be called normed. formula that if V() is conjugate to U(
It follows from Parseval's

(15.35) If y is a normed conjugate to

we have i n particular from (15.34) and (15.35): (15.36)

^(V^<^(Í7^.

If among the fimctions conjugate to Z7(^) there exists one Vi(v) which is piecewise continuous, then only the functions Vi(«>)+const. are to be called conjugate to L^(^) and m a y only be subject to modification i n the points of discontinuity. In this sense the following theorem is true: Lemma 4. Ij U(
I n applying this result repeatedly it is generalized i n the following way: Lemma 5. If U( are absolutely continuous, F****(v>) is integrable and F***(v>) is a normed conjugate function to ¿7***(^).

If U(*p) is periodic w i t h period 2 T and U'{^) function, V'(^) to Ui
is integrable, then we obtain a normed conjugate

F(^)=rC7(^),

(15.37)

where the integral on the right (15.15) converges for almost a l l values of v>. - I n particular this converges c e r U i n l y for a l l values of for which UM has a finite derivative or more genersJly a Lipschitz condition of positive order.

3.

Le

M i l l

ata on Certain Recurrent Formulae.

Lemma 6. Let «, a and b be three constants with 0 < < < 1 , a>0, an infinite sequence of nonnegative numbers satisfying the inequalities +

X,+i
b^pr,

b>0,

Le I I I 1 1 ata 6-8 ami X,(p=0,

1, 2,

. .) be

(15.38)

Then if

+a(l-€) X

(15.39)

1

is the one positive root of the equation

(15.40) we have for all Xw-

X < m a x (Xo,\). Proof:

If we have for one v: Xw<\,

it follows from (15.38)

X+i<«X-|-a+6Vx=X. * For proof, see |7, p. 2231.

(15.41)

If on the other hand X r > X , we have by (15.38)

therefore X,+i
and we have in any case

X + i < n i a x (X,X). (15.41) follows now by induction. Lemma 7.

Let X,{v=Ofl,

. . .), be a sequence of nonnegatwe numbers satisfying the recurrence formula

X„+i<Xn + (7on'V^+<^in''+<72V^(S/«'X;y where Co, Ci, C^t i j , r, are positive constants with IJT = 1, O - ^ O , p a positive constant>2
( n = l , 2 , . . ,).

Xn
(15.42) and c a

(15.43)

Put

(15.44) and take an integer n^ with 1

«o>(^')'"*

(15.45)

T h e n we will prove (15.43) with a = m a x (x^

. . . , ^ - . i ^ '

(15.46)

We prove this by induction, assuming that (15.43) is true for I, . . . , n and showing that then (15.43) is also true for n + 1 . B y (15.46) it is sufficient to assume n>no. T h e n we have to show that

-X',+eon'V^+6'xn'+<7,Vn(i: v**'X;)'<«(n +1) and by our assumptions it is sufficient to show that

this will follow certainly from

that is by (15.44), from a

(71+

lY'-'-n'-^'-n'^-Ciifii

But we have

and we have therefore only to show that a(pn''-C,-yJn)>C,n\

°^^^'>n^-%

.^->ni-'+5^-

(15.47)

But from (15.45) and (15.46) we have

and by adding we obtain (15.47). Lemma 7 is proved. Lemma 8. Let X (v=0, 1, . . . ) a ««fi^nctf of nonnegative numbers satisfying the recurrence formula

P^€iX?y+C,

Xl^^<€Xl+C,n^Xn+Ct(^^

( n = 0 , l , , . ( 1 6 . 4 8 )

where t, Cu Cj, Cz, i ; , r are positive constants and nr = \y c < l . Then the sequence X is bounded. Proof. On applying lemma 7 to the sequence X\ with
then ^Xn is bounded as n-> <» and the same is true for

v^^^f^ • XI satisfies therefore an inequahty

of the form XU,<^l+a

and our assertion follows from lemma 6.

4. Theorem 1.

Theorem 1.

Convergence of

B,{tp)

Let P($) be periodic with period 2 T and have a uniformly bounded derivative P'(B),

P'(6)l <e,

€<1,

(15.49)

where the constant e is positive and <1.^ Let 0o(f) 7(^) ^ « two functions of ^ absolutely continuous^ vhiU and y'(le in the square in any finite interval. Let further Bo{ip)~^ andy(ip) — ip be periodic with period 2 T . Define, starting with Bo(ip)j the sequence of functions ö « ( ^ ) ( n = 0 , 1, . . .) by the recurrence formula

( n = 0 , l , . . .).

Bn+iM--yM+TP(Bn) a.

(15.60)

Then the integral equation

(15.50a)

BW)-TP{e)=y{ip)

has one and only one continuous solution B{
(15.51)

D,<Ee 2T IK<2^é'+^^<2n+~

. 2T

B,+I-B,\<2^TE(^€'+j^^^^^

B^-B

B,-B\

<

1 - V Í

V

This condition is a very sharp one; In 1M7 H . Wittlcb |13] gave wltbout koowln« Warsehawskl't paper as a sufficient oondlUon t
M9581—52

11

(15.52) (15.53)

(15.54)

(15.65)

^/^/K'<ß^'+Y^^
(15.56)

T h e part j8 of the theorem follows at once from the lemma 5. W e have b y (15.50) for n = l , 2, . . . Proof:

9n+l(^)-
and therefore b y induction i n virtue of (15.19) that is (15.51). A g a i n i t follows from (15.50) b y lemma 5 eUi-y'==TP'{en)eL

and therefore b y (15.36) Thence b y (15.29) and b y (15.20) and (15.21) i n iterating this repeatedly

and (15.56) is proved. I t follows now b y (15.29)

and (15.52) is proven.

B u t then, b y lemma 3 Ö„+i-Ö„|„I?:<2-^xE(^M+Y^)

that is (15.53).

I t follows that the series

^o+S

(Bw+i—0,) is convergent

uniformly i n a n y finite

interval, that is, that the sequence B,(ip) tends uniformly to a l i m i t 6(ip). W e have then

r ^ J , ? /

=

—

T

w

T

—

*

'

and (15.54), (15.55) are proved. $ is measurable, P{B) measurable and bounded and T{P{B)) exists almost everywhere. by (15.50), (15.49), and (15.36) f o r n ^ « : ^|J^mP{B)) + y-Bf<

^|UaT{P{B))-T{P(ßM+

V ^ ( ö « + i - ö ) ^ < ,^^{ß-B,f+

Then

V^(Ö«+i-^^-*0,

and (15.50a) follows. T o prove finaUy the uniqueness of 9(v»), assume that there are two solutions B{tp) and B*{tp) of (15.50a). T h e n we have B- B*=T{P{e) - P(0*)),

and therefore b y (15.49) ^{B-By
J^XB" By < ^/{P{B) - P{0^)f

I t follows, since B-B* is continuous. 0-^=0.

5. Theorem 2.

A,

Theorem 2.

Convergence of

e^ifp) and e'X<^

Suppose that beyond the hypotheses of theorem 1 P"{$) exists and is bounded:

(15.57) Then we hare

^«+1-01. <^2jrEE'nt'' ne

en~e\<'^'2irEE'

(1-0

(n>0),

(15.58)

(n>0),

(15.59)

(n>0).

(15.60)

B. Suppose that beyond the hypotheses of A, 0o(ip) and y'{) are absolutely continuous, while flj(^) and y^i^) exist and are integrable in the square in any finite interval. Then d'() and we have

-Oil)

(15.61)

=0(nt'')

(15.62) (15.63)

Bu (-1—^n--Oi^fiit^)

-0(V^€")

(15.64) (15.65) (15.66)

Proof.

Under the hypotheses of A we have B:^,-B:=TiP'iBn)

B: - P X ^ - O B : . ^ ,

and therefore a l l B^ (^)^re'absolutely continuous, while Bl (tp) are integrable w i t h their squares. iyn'<^/iP'(Bn)B:

Further

- T O - i ) < ? :-i]^

The bracketed expression is equal to p'iBn) (B: -

- i)+(P'(
and its modulus is b y (15.49) and (15.57) is less than or equal to

therefore b y (15.29)

í>:<*2):-i+i7aV-^(
now - / D Í = í í ' « * : t-hen b y lemma 3 and b y (15.51) Bn-^Bn-l\<^|21rE

g,^,i--'

and (15.67) becomes b y (15.56) gl
(n>0)

(»>o).

(15.67)

W e have therefore

+ that is (15.58).

F r o m lemma 3 we have now »+1

that is (15.59).

Bn I < V2irZ>«P: <E'^j2iE

71«"

(n>0),

F i n a l l y we have Bn-B\<J^\e,^l-Br\<E'^}2'KE

2

(»i>0),

I — »

and (15.60) follows at once.

T h e part A of the theorem 2 is proved.

U n d e r the hypotheses of part B we have

and b y (15.36)

T h e bracketed expression is equal to p:'^^{e:,^-B:y+2BLP:^^{Bu,~B:)+B:'^^^^

and its modulus is i n virtue of (15.57)
B y (15.29) i t now follows that (15.68) where 5=v^(
(16.69)

r=V^[(F:'+t-p:'Wi.

(16.70)

Here we have p:\,-p:\<2r„. P u t now for i ' = 0 , 1 . . . (15.71) T h e n we have from (15.70) (15.72)

T<2vtUn.

F o r the different terms of S we obtain b y lemma 3 a n d (15.56) and (15.58)

«:+.-e:i
^'zm

e:y=^0(.nU'D:),

V ^ ^ l ( < ' . + i - W 1 < ^2T D . D:

+1

=0(F«)

and therefore b y (15.68), (15.69), and (15.72) D:'^,'-éD:=O{n'€-D':)

+

(An%^^/D:'J+0(Un)+0{V,}

a 5.73)

To express Um and V« we start fro

From this and by lemma 3 we have

l«:i;r>' and by (15.71) and (15.58)

C7.=0(l)+0(s«*V5jy-

(16.74)

Put now U\ = Un-x+*Un-t+

• • • +«-'l7o=S«-'-*C^-, y»0

(n=l,

2 . . . ( 1 6 . 7 5 )

Then we have by (15.74) f o r n > l : (15.76)

W^, = 0 ( l ) + 0 [ ( S / e ^ V S j ) From C i - Y ' ^ n P X + P I O w e have by (16.29), and (15.36)

+

by (15.71).

Thence, since by (15.71) and (15.21)

we obtain

and we have finally

V.=0(1)+0[(zj Introduce this and (15.74) into (15.73).

Put now J^=z,,

v'Sj)'"

We obtain

Then we have

for suitable constants cu c^, C|.

From lenmia 8 we have now x,=o(i),

D:=O(I),

that is (15.61). By lenmia 3 and (15.58) it follows further fl:+,-d; = 0(n€i). Therefore S

(dJ+i—^0 converges uniformly and ^i(^) is uniformly convergent to a continuous function.

which must be 9'(tp). We see further that $1 must be uniformly bounded.

We return now to (15.67) and use that €^ is uniformly bounded.

Then it follows from (15.51)

In iterating this we obtain

2?:<(2?í+íw;)«"=0(n€»), that is (15.62), and therefore by lemma 3 1+1

BL-B'=0

and the theorem 2 is proved.

6.

Theorem 3»

Convergence of B^^^)

Theorem 3. Let for an integer m>l the function P(B) be periodic with period 2v and suppose that has bounded derivatives up to the order m: P^W
(0
..

m),

(15.77)

where i ; i = € < l . Let Bo(»

(;i=0,l, . . . , m - l ) ,

DS^=0(n^€*) D

continuous

(15.78) (15.79)

Oil),

e:ru-(>lr'=0(w+it')

(M=O, .

m-2),

(15.80) (15.81)

ffW_flW

= 0(n'+»€")

( 0 n

m-l

(AI = 0 ,

,m-2),

(15.82)

n\

2 e V .

(15.83)

We prove first the properties of the B,{(p) under a. They are true by hypothesis for i ' = 0 ; suppose that they are true for a certain value of then for this value B,+i{). On the other hand PX 3 and proceed by induction. We assume that our theorem is already proved for all smaller values of m and that therefore we have Proof.

D»> = 0(n'^€-)

( X = 0 , 1, . . . , m - 2 )

(15.84)

e + i - « . * ' = 0(n^+*«")

(

in-a

(X=0, . . . , m - 3 )

j i \

^a>_^w = 0 ( ^ x + i ^ » )

(x=0, . . . , m - 3 )

It follows then that each of the m—1 sequences ( X = 0 , 1, . . . . m - 2 ) converges uniformly to a l i m i t which is respectively B^M, that therefore B(^) has m—5 derivatives, and further that a l l expressions

1^1=0(1)

(X=0, 1, . . . , m — 2 ;

n = 0 , 1, . . . ),

t h a t is are boimded from above b y a constant independent of n and X. Since Bn+i—Bn+i is conjugate to P „ + t — P „ we have b y lemma 5 for k=m—l

or m

Therefore b y (15.36)

I P(^.+0 l^dtp

d

i Pi»n) dtp

a n d b y lemma 1 and (15.29)

where T,"', T ? ' designate respectively the first and second square root to the right. W e discuss first r,<»>. B y lemma 1 we have T^" = 0,

a n d for Ji;>3 b y (15.26) i n the notation of lemma 1 TIT < k - v C ^ ^ I ^ ^ X i T e r ^ ^ P r f i : ^ ^ +

I ^ ( 9 . + 0 - AMY

(fc > 3).

W e have obviously for k=2, m = 3 b y (15.77) and (15.90) p:+.«;v.-p:c=p:+.(c+.-o+(p:-i-p:)C=o(i9:+,-o;i)+r)(ie.H.,-#,i), v . ^ ( / ^ + . e . - p : o ' = O ( D : )+0(0.),

r^»=0(n«»). F u r t h e r for fc > 3 i n virtue of (15.77)

'.+1

0 . + ' » 2 ^ - '

0( 0.^-,» - 0." - " I)+0(10» a n d this is b y (15.86) w i t h X = 0 and b y lemma 3 equal to 0(

- e,

e; 11+1

e:i+iai9;.Mi 1«.»+-."

I (19,+. - 0.1+19U. - «: D),

It follows now by (15.29)

(15.95)

^j:MPu^e:^^^

Further it follows from (15.77) and (15.59) that P*^«?»+i)=P^(
(M=0,1, . .

m-1)

and therefore in virtue of (15.27), by lemma 1 and by (15.90) and (15.28):

^«-,(^«+I) = SP^H
P^(en)C„M^U

• .

(15.96)

e+-.'0+O(e-),

(15.97)

e+-,^0 + O(l),

M

M-2

where Am-i{B) is the expression from lemma 1. Again we have by (15.86) and (15.87), since the expressions

(7„_,.,(^,+ „ . . ., Birn-Om-iMr C^M^u

. • *.B^^'?)-C^M.

• .

M

Ck,M are polynomials,

e-")=o(n"*"%")

• . .,e-^0 = o ( n ~ / )

(M = 2, . . . . m - 1 ) (M = 2, . . .,m).

Therefore we have A«-i(e«+i)=X-i(O+0(n""%-),

^«(e»,+i)=^«((?0+O(l),

V^l^-.-i«'-+0-^-.-i«'-)]'=O(n"-^")

(15.98)

V^[^«(^«+i)-^«(M'=0(l).

(15.99)

If now i t = m , it follows from (15.93), (15.95), (15.85), and (15.99) rj"» = 0 ( l ) + 0 ( V ^ € ^ V ^ ^ n ^ ' " ' ) »

(i5:=m>3).

(15.100)

If on the other hand we have k=m— 1, T ? * is given by (15.94), and we can therefore assume A:>3, m > 4 ; but then (15.84) is applicable for X = 2 and for x = ^ — l = m — 2 and (15.90) is applicable for X = 2 . We have then

v^(p:+i^:+ie+-i"-p: (»:<'."-")»=o{n^ « - ) + o ( n i « » ) = 0 ( 7 1 -

«-).

In combining this with (15.93) and (15.98) we obtain finally r i ~ - » = 0(n"-=^€'»)(^ = m = 3, 4, . . .),

(15.101)

since this is true also for m = 3 in virtue of (15.94). On the other hand we have for T\

and therefore by (15.29) and (15.59)

Introducing this and (15.101), (15.100) into (15.91), we obtain I>«Vi"<«Z>^-" + ^ ^ n " - ^ « ) + 0 ( V n e " ) V ^ ^ * " - " %

(15.102)

Düt^

tD£^ + 0(1)^-Oi^ßtt^ZÏÏ^^

+ OCVi^/V-^ö/-«»),

(15.103)

O n the other hand i t follows fro F-1

that -1

(15.104)

F-0

(15.102) assumes now the form

or if we put generally Z?*"~" = e ' X

(»»==0, 1, . . .), -1

and b y lemma 7 w i t h ^ = 0 we have

and this is (15.78) w i t h M = » » - 1 . I t follows now from (15.104) for k^m(15.103) becomes i n introducing (15.104) w i t h k=m • B—1 i?.^i«I>.^ + 6 V ^ « » S Z > ^ +

B-l

»

\ that ^ f l ^ " " ' is 0 ( 1 ) .

0(l)<eZ?^>+<7S€*^.'^+0(l),

»-0

F-O

(15.105)

and i f we apply the lenuna 8 i n replacing there « b y « * we see that Dir* = 0(1). and that is (15.79). W e see now that the sequence ^ J " ' " is uniformly convergent to a continuous function which must be and (15.82), (15.83) follow now immediately from (15.81). T h e proof of theorem 3 is completed.

7,

References

[1] S. N . Bernstein, Sur l'interpolation trigonométriqué p&r la méthode des moindres carrés, C . R. Acad. Se. de r u . R. S. S. 4 , (1934). [2 C . Carathéodory, Conformai representation, Cambridge tracts (Cambridge University Press, Cambridge, 1932). [3 I. E . Garrick, Conformai mapping in aerodynamics with emphasis on the method of successive conjugates, see this volume. 4 G . H . Hardy, J . E . Littlewood, and G . Pölya, Inequalities (Cambridge University Press, Cambridge, 1934). 5 I. Naiman, Numerical evaluation of the «-integral occurring in the Theodorsen arbitrary-airfoil potential theory, originally issued April 1944 as Advance Restricted Report L4D27a. [6] I. Naiman. Numerical evaluation by harmonic analysis of the «-function of the Theodorsen arbitrary-airfoil potential theory, N A C A Wartime Report L~1.53 (originally A R R No. L6H18, Sept. 1945). 7 W. Seidel, Über die Ränderzuordnung bei konformen Abbildungen, Math. Ann. I M (1931). 8 T , Theodorsen, Theory of wing sections of arbitrarv shape, N A C A Report No. 411, 1931. [9 T . Theodorsen and I. E . Garrick, General potential theory of arbitrary wing sections, N A C A Report No. 452, 1933. [10 S. E . Warschawski, O n Theodorsen's method of conformai mapping of nearly circular regions. Quart. Appl. Math., in. No. 1, 12 to 28 (April 1945). [11] E . J . Watson, Formulae for the computation of the functions employed for calculating the velocity distribution about a given aerofoil. Aeronautical Research Council, R. and M . No. 2176, M a y 1945. [12] H . Wittich, Bemerkungen zur Druckverteilungsrechnung nach Thoodorsen-Gamck, Jb. dtsch. Luftfahrt-Forsch. I, 52 to 57 (1941). [13] H . Wittich, Konforme Abbildung einfach zusammenhängender Gebiete, Z. angew. M a t h . Mech. vol. 25/27, 131 to 132 (1947)

•49081—92

12

163

I

16.

O n a Discontinuous Analogue ot Theodor sen's and Garrick's Method ' A. M . Ostrowski '

The integral equation e{^)-y{^)=^±£

cot | dt

[P(e(^+t))-P{e{
discussed i n the preceding paper ' is solved theoreticaUy b y a method of successive approximations w h i c h involves the repeated computation of the singular integral occurring i n the equaUon. I f this i n t e ^ a l is replaced b y a n approximating stun, i t is very diflBcult to estimate 3 i e influence of the approxi m a t i o n errors on the convergence of the process. W e w i l l therefore consider i n what follows a n approximating problem obtained b y replacing the singular integral i n our integral equation b y a sum and give a procedure for the complete solution of this problem. The difference between this solution a n d the solution of the original mtegral equation can then be discussed b y using the results of our preceding paper a n d involves only a single discussion of the approximation errors. O u r procedure depends upon the two foUowing theorems. Theorem 1. Consider for a positive, odd integer p=2n+l

the two sets of p values

2FT 9'f

P

— ( " = 1 1 2 , .

.

(16.1)

Then 3m p

X — Xp

(16.2»)

is a trigonometric polynomial of degree n satisfying the conditions gp=g(x,)

( v = i , 2 , . . ,,p),

(16.3)

The normed * conjugate trigonometric polynomial to g(x) is x—x,

COS—s y(x)=i

i:gp

X — X »

COSP—r—

^

^

(16.4»)

¿1

The values off(x) fp=f{x.)

(>'=1,2,. . .,p)

(16.5)

are connected with the g,by p linear relations f.^i^ci^^g,

(M =

1,2,. . .,!>)

> The preparation of this paper was sponsored (in part) b / the Office of Naval Research. > Nationa] Bureau of Standards, Los Aneelcs, Calif, and UnfTersity of Basle, Basle, Switurland. • A. Ostrowski. On the converKence of Tneodorsen's and Qarrick's method of conformai mapptntt. < The normed conjugate trigonometric poljmomial Is one, whoso Fourier expansion has the constant term«0.

(16.6)

with an antisymmetric matrix A,

the elements a^, of which depend only on the difference p. —v.

We have

(16.7) tan(M — y)

{fji — p even)

2p

pa^

(16.8*)

(n — v odd)

(t^-y)^^

cot

Proof. I t is verified immediately that the expression g(x) given i n (16.2°) satisfies the conditions (16.3), since if we take z=x^, the nmnerators sin p{x^—Xp)l2 vanish for IIT^V, while the l i m i t of

the coefficient of is obtained b y the BemoulU-L'Hôpital rule as 1. Consider now the expression .

i>-i

U=2e

(16.9*»)

+1.

In using here Euler's formula we obtain U

I

. cos —

sin pu+i(cos i i — c o s pu) sm u

U—€

sin u

so that sm pu sin u

m

cos u — c o s pu sin u

SU

(16.10°)

O n the other hand we obtain from (16.9**) p~i 2

/><(P-1)«_ 1 6

i

(16.11°)

irai

and see that both expressions (16.10*») are trigonometrical polynomials of degree 7i=(p—1)/2 in 2u and further that 3 U is the (normed) conjugate tr^onometric polynomial o f 9?U. It follows that the expression (16.2*') is a trigonometric polynomial o f degree n and that the expression (16.4**) is the (normed) conjugate trigonometric polynomial to g{x). If we put now x^ into (16.4**), we obtain the formula (16.6) where are given b y cos

X — Xp

V

sm

cos p

x—x, 2

(16.12°)

x—x, 2

If y>=v, we determine a,r i n making x^Xp and see at once b y B e m o u l l i - L ' H d p i t a l rule that If li^Vf we have ^cos

cos (M

(M-")-

sin (M

t« a

2

{n-yi—i-iy-'

J COS

sin (M—v)

v) —

and the relations (16.8**) follow immediately. Theorem 2. For a positive even iiUeger p=2nthe . yW ^ -

vjT

a„=^.

-

V

expression

X~Xp ^ X — Xr f?r sm 1) — 2 ~ cot — 2 "

(16.2")

onometric polynomial of degree n satisfying in the points (16.1) the conditions (16.3) nxfrmed conjugate trigonometric polynomial to g{x) is given by

fix)

.

X—Xp\

1 —COS p — —

2

. Z — Xp

) cot

2

(16.4-)

arid its values (16.5) are connected with the Op by p linear relations (16.6) where the matrix Ap is again an antisymmetric matrix in which the depend only on p.—p and are given by ' {jjL—p even)

0 * V = i ( l - ( - i r ' ) c o t ( M - F ) ^

2

X

-cot(^—v)-

(16.8*) (M — v o d d )

The relations (16.3) are in this case again immediately verified. Consider the expression ,g«p-n).. + g * ( y - i ) « _ 2 e - ^ " 2 sin u whose real and imaginary parts are Proof.

^ r r sin (p+l)ttH-sin ( p — l ) u 9Jf7= — i , ^ ^=sm^ 2 sin u

cot u (16.10«)

COS (p+ l)tt+cos ( p — l ) u — 2 cos u

3 ^

(16.9*)

(1—cos pv) cot u

2sm.u

O n the other hand, using Euler's formula for sin u, we obtain «-1

t7=l+2Xl«"'"+«""'' I—1

»

$RC/=1 + 2 S cos 2int+cos 2nu,

3 C / = 2 S sin 2vu+sin 2 n u . (16.11*)

r-l

vol

normed trigonometric given by 1—COS p

2

cot

2

and the expressions (16.8*) follow inamediately. The theorem II is proved. The formulae (16.8**) show that in the case of an odd 2>=2n+l we have exactly n=(p—1)/2 different |a„r|5^0, while for even p=2n we have exactly [(p/4] values of 1«^»! different from zero. This makes the case of an even particularly adaptable for use on high-speed computing machines with restricted high-speed memory. This case presents further the advantage that m usuig the formula (16.6) the computing work is reduced by M- O n the other hand for an odd p the interpolating polynomial (16.2°) is the most general trigonometric polynomial of degree Cp—1)/2, uniquely determined by (16.3). We obtain an interesting transformation of formulae (16.6) if we define /„, a^p for aU integer values of M and v as periodic functions with period p. Then the formulae (16.8*) and (16.8*) give wie right values of a„» for all integers M and P. It follows further from the same formulae that (16.13)

0^+

W e can now transform the formulae (16.6) into the following formulae p-i 2

f^—^ot^+pigt^+p—gi^f)

( M = I , 2 , . . . ,p)

(16.14)

Indeed, in the case of an odd p==2n-\-l, (16.6) can be written in the fom

I——B

and (16.14) follows at once from (16.13) and a^=0. *ThflM formula© (ie.8«) hare been Efven by E . J . Watwn, Formulae for the oomputatton of functions employed fbr calculatlnc the Telocity distribution about a glTen aerofoil, Aeronautical Research Council, R. and M . No. 2178, M a y 1046; I. Naiman, Numerical evaluation of the integral ooenrlnK in the Tfaeod£«sen arbitrary-airfoil potential theory, originally issued April 1944 as Advance Restricted Report I.4D27a; I. £. Oarrick, Conformal mapping in aerodynamicR v i t h emphasis on the method of successive conjugates, see this volume.

If p is even and of the form 4m, we have from (16.6)

S

-(2m-l)

(M=1,2,. ,

ccm+rQ^+p

,,p)

and (16.14) follows again i f we use that b y (16.8*) a ^ ^ + 2 « = 0 and ( p — l ) / 2 = 2 m — 1 + J i . F i n a l l y , i f we have p=^7n+2, we obtain from (16.6) 2m+2

/„=

S

<X^+'9M+' ( M = 1 , 2 , . , . , y ) .

-(2in-l)

B u t here cc^^2m+2=o^mt+im^O because the difference of the indices is even, and ci^,t+2m+i=ocut^ipf2i vanishes b y (16.8') too. Thus the limits of the s u m are ± ( 2 m — 1 ) = ± (p—4)/2, a n d (16.14) follows again if we use that 2 m — 1 is the greatest o d d number • < (p—1)/2. I n the case of a n even p the formidae (16.14) show i n connection w i t h (16.8*) that the expressions of the/n w i t h even M only contain the w i t h odd v and vice versa. W e are now going to derive some orthogonality relations for transformation (16.6). W e introduce from now o n the notation

t

Su=SM^)=-

^ p

«(a:r)

(16.15)

F - i

where the index x is explicitly indicated i f u contains more than one variable. from the theory of roots of u n i t y

F o r instance, we have

( ± ¿ = 1,2,. . . , p - l ) .

^^*»'=0

We notice first that if we take a l l gp=l,

(16.16)

we must have

B u t then we obtain from (16.6) ¿ « ^ = 0 f-1

( M = 1 , 2 , . . .,i?),

(16.17)

( i ' = l , 2 , , . .,3?).

(16.18)

and since A , is antis3rmmetric ^

0^=0

W e now write formulae (16.2°), (16.2°) i n the form 9i^)=^Ts9pp(x)

(16.19)

S
(16.20)

and (16.4°), (16.4') as /(3;)=^

P

P

F o r a n odd i > = 2 n + l we have from (16.10**) a n d (16.11°) sm

X — Xp

V—K-

iPp{x)=

= 22

cos «(a: - zO + 1 ,

(16.21)

s m - ^

x — x,

cos — l^.(x)=

¿á

sin?

cos p 2

x — x^ ¿é

—

n

= 2S8Ín K(X-X,).

(16.22)

^'

We have then b y (16.16) 5'(^,+i^,)=iS (2 ¿ • See, In the case of an even p. the references In footnote 5.

e'^f'-'.) + A = 2

¿¡

1 = 1,

and therefore Further, we have

and therefore o n takmg the real part on both sides i n the cases of + and — sign, respectively, (16.24)

1

Sh*.^

5^,^^+5^,^„=2^/z,)-1

(16.25) (16.26)

S f , f ^ = ip^ixp) - 1

= *»,(XM)

(16.27)

- 1

I t follows now from (16.19), (16.20), and (16.27) „—..^Sf(,x)=^S'i:

9.9,M,=\

i:

9.9'S4>,4>,=^

9.9'(v,{x,)-l)

}

(16.28)

Sf=Sg'-(S9)',

(16.29)

±fi='±9i-l(±9X

Consider now the case of an even p=2n; ^ , = s i n n(x — Xp) cot

i n this case we have i n (16.19) and (16.20): -1

x~x.

-

=l+2Sco9

^

K-1

K(X —Xr)+cos

n(x—Xr)

rt-1 —X l ^ r = ( l — c o s n(x—Xr)) cot — ^ = 2 y ] s i n i : ( x — x ^ + s i n n(x — x » ) . X

(16.30) (16.31)

It follows fro: «-1

S{ip.+ih)=

1 + 2 S ^ c * ' * ' " ' ' +Sc'"<'-'-' = 1

that the formulae (16.23) remain valid for an even p.

Further, we have

l+2S«"''*'5'e'*'+2S«=*=*''''5:€±*'^+4S«"'**''*^^^ K-l

X-1

C.X-1

n-1

n-1

«-1

X-1

I n virtue of (16.16) the second, third, fifth, sixth, seventh and eighth terms of the right-hand side are zero and there remains fi-i

K.X-l

In taking here on both sides the real part we obtain in the cases of -)- and — sign, respectively

l-h(-lr+^ x,^+COS n ( z . -

=2vJ^(x,) — 1 — (— 1)' ^^

(16.32) 5^,l^.=

^.(x^)-l-(-l)'-^''

(16.33)

We denote now by S'tp and S'V* respectively 'SV=iS^(a:j.-i)

V

V «-1

From (16.31) we see that

«-1

(16.34)

(c—M even);

^r(x,0=O it follows from (16.34) and (16.33) that {v or M even);

(V, M odd).

(16.35)

-S'V,^«=0 (v or M odd); - S V r ^ , = ^ , ( x J — 2 (v, M even), and therefore in using that
(16.36)

'S*^,lfv=0

1

1

5V.IAM=^.'(XJ-2

1 "

p—1

1 ^ P «.X-I

«,X-1

px-1

P

/

The argu lent is completely symmetric if we interchange the x, with odd and even indices. obtain p 2

2

1

«-1

p

I

0 / 2

p

Thus we

2

«-i

/

(16.37)

.-1

(16.38) In what follows we shall consider the/, and as the components of two vectors {, ij in the p-dimensional space. It follows then from (16.29) and (16.38) that both for odd and even p (16.39)

i\<\v

If P is a given function, we shall understand by

P(fi)

the vector with the components

P(gi),

P(9t)j

' ' • t P(9P)-

We are now going to prove the following theorem. Theorem 3. Suppose that for a positive integer p a square matrix A that for any vector { in p-dimensional space Rp we have

of order p has the property

(16.40) Let P(x)

be a continuous function for which P'(x)

P\X)

exists almost everywhere and satisfies

<€,

0<€<1

(16.41)

for a positive € < 1 . Let finally ri be an arbitrary vector in Rp. Then the vector equation i-ri

=

AP(0

(16.42)

has a solution ( which is uniquely determined and can be obtairied as the limit of a sequence of vectors deduced by the recurrence formula

{ , + , - „ = AP(€,) jrom an arbitrary

initial"

vector {©;

(16.43)

have

(16.44)

1 Proof.

It follows from (16.41) by the mean value theorem that for arbitrarj- x and x* P{x*)\
x~x

We have therefore for two arbitrary vectors { and {* the inequaUty (16.45)

\PiQ-p{e)\<^\i-i*

Suppose now that { and f* are two solutions of (16.42).

Then we have

i-i*=AiP(i)-p{e)),

and it follows from (16.40) and (16.45) that

{ - r i < | P ( Q - W <*U-«*I; we see that f—f* must vanish. The uniqueness of f is proved. We write now (16.43) for n and n — 1 and subtract. If follows then from (16.45) i

f.=^(P({.)-P({..,)),

f.l)l<e|{.-{.-

+1

Iterating this, we obtain P + l l+

l

ip\

Çp + H

(p>0, n = l , 2 ,

. ),

(16.46)

and we see that the series ^ ( { , + 1

in) is absolutely convergent

Therefore with n—> » {. = { o + i : ( ^ + i - W

converges to a vector { as asserted in our theorem. Finally, we obtain from

(16.42) follows then from (16.43) with n

OD

and from (16.46) the estimate 1

*"

Theorem 3 is proved. We return now to the discussion of the matrix Ap in order to determine its eigenvalues. It We consider first an antisymmetric matrix C y^, defined by 6^

C, that is (16.47)

(M,»'=1,

and assume that from è

yp^Vw ( M = 1 . . . .,!>)

always follows p

..-I

(16.48)

T h i s can be stated differently if we consider the vectors { = ( x i , . . . , x,), v=(yij • - • tVp) and define generally for a vector 17= (yu . . . , y^) as its trace trj the vector, all components of which are equal to (16.49)

P A vector ij with tij=0 will be called a diagonal vector. B y our assumption it follows from

(16.50) alwavs that (16.51) If we take now 17'= (1,1, . . .,1) it follows from (16.51) \Crt' C belonging to the eigenvalue 0. Further it follows now

0, i ^ ' = 0 ^ a n d 17' is an eigenvector of

±

(16.52)

We see from (16.52) that f in (16.50) is always a diagonal vector ti=tCv

(16.53)

= 0,

O n substituting in (16.48) the expression of r^, we obtain 2

1

1^

^ •

P

(M?ÍX)

(16.54)

O n multiplying (16.50) on both sides b y C' we obtain in virtue of (16.54) Ü'f=i?-íi7

and since C

C

(16.55)

17+Í17.

Suppose now that 17 is a diagonal vector.

T h e n from (16.50) and (16.55)

(16.66) and we see that — 1 is an eigenvalue of to which belongs the (2> — 1)-dimensional set of diagonal vectors as eigenvectors. We have by (16.56) for any vector 17 with tr¡=0: C(Cfj±iv)=±i(Cv±iv)

and we see that Crt+irj, if it does not vanish, is an eigenvector of C to the eigenvalue i and CTJ—ÍTI, if it does not vanish, an eigenvector to the eigenvalue — i . From

1

'

l[(Crj + 2i

iv)-(Cri-iv)]

follows that these eigenvectors to the eigenvalues ± i , together with the vector 17', form a complete system of J7 independent vectors. If in particular p is odd and C" is a real matrix, i and —i are then eigenvalues of multiplicities (p—1)/2. Consider now for an even p—2n

and assume that fro

an antisymmetric matrix £ ) = ( 5 ^ ) defined b y D'^ I'M

. . . , p)

\w

(M=1,

• . . I

P)

— D, that is

always follows Xp

n

1 n

x-i

(16.57)

T o state this i n vectorial [language consider again the vectors ( = ( x i , a n d define now as the trace tiijof 17 the vector tri = {ziyZ2, . . . , 2 , ) , where

T h e n it follows from

1 "

1

w x-i

W x-i

i=Bii

"

always

2

(f=l»

•

,

1? * = ( ! / ! ,

•

•

•

I

VP)

(16.58)

(16.59)

tr,\\

A vector 17 with
i7'=(l,0,l,0, . . . ), a n d from (16.59)

Z?i7'=Z?i7''=0;

(16.60)

and ^i" are then two independent eigenvectors of Z> to the eigenvalue 0 F r o m (16.60) i t follows further S W = S * M 2 « - i = o «-1 <-l

(M=I,.

.

.,p)

(16.61)

O n putting the expressions of x^ into (16.57) we obtain at once 1 n

1

±

(16.62)

—

u — v ^ O ,

11 —

v

even

odd

(16.63)

and therefore for any vector T?

t h a t is, since (16.64) It follows now at once that all diagonal vectors are eigenvectors of therefore, i n virtue of

to the eigenvalue — 1, and

V(Dyi±iyi)=±i{D-n±iit),

for any diagonal vector 1;, Z?jy+iiy, if i t does not vanish, is an eigenvector of D to the eigenvalue % and Dvj—iijf if i t does not vanish, an eigenvector of D to the eigenvalue — i . A n d exactly as for the matrix C we see that +i and —i are eigenvalues of D of the multiplicities n ^ , n _ with n + + 7 i _ = 2 n — 2 and possess correspondingly n+, n _ independent eigenvectors. If i n particular Z? is a real matrix, we have n+ = n_=n—1. I n applying our results to the matrix Ap, we see that it has only linear elementary divisors and its fundamental poljTiomial is either X(X^+1)" or X*(X^+1)" according as we have ^ = 2 n + l or 2n. W i t h choice of appropriate coordinates the matrix Ap becomes a diagonal matrix with the diagonal ( t , — t , . . . 1,0) or ( i , — i , . . . ,i,—i,0,0) for odd and even Consider now the recurrence formula (16.43) i n which P(x) = ax,

a>l.

O n choosing appropriate coordinates, we separate this formula into p recmrence formulae 3

I

or \~ia

\

\—%a/

and we see that w i t h n—>«» the convergence of Xj'"* is only possible for special values of yi and xf^. follows that i n the convergence condition (16.41) of theorem 3, M a x |F'(x)|
It

(16.65)

17.

11 On Conformal Mapping of Variable Regions'

S. E . W a r s c h a w s k i *

1 • Introduction Simultaneously w i t h the development of the principal existence theorems of conformal mapping m u c h attention was given to the related problem of the study of the dependence of the mapping function on the region. If £ is a simply connected region and w=f{z) is a function which maps the xmit circle s K l conformally onto B, the problem is to study the change which/(g) imdergoes by a continuous deformation of the region R. Generally speaking one can s a y — a s might well be expected—that the mapping fxmction varies continuously with the region, and there are many results i n the hterature on conformal mapping that substantiate this statement and give it its precise meaning. It is the object of this paper to review these results w i t h a particular emphasis on such parts as may possibly be of help i n problems connected w i t h the effective computation of the mapping function. The results i n the literatxire may be classed into two types: (i)tnose of a " q u a l i t a t i v e " nature i n form of convergence theorems which establish the convergence of the mapping functions of a sequence of regions tending to a limit region; (ii) theorems of a more " q u a n t i t a t i v e " type which give actual estimates for the modulus of the difference of the mapping functions of two regions that are "close" to each other. The results of the first type are for the most part well known and shall be reviewed here only very briefly. L e t R„(n=\y 2, . . .) denote a sequence of simply connected regions all of which contain the origin w = 0 , and let the function w=Jn{z) map the circle |2|<1 conformally onto i?„ so t h a t / „ ( 0 ) = 0 a n d /1(0)>0. C . Carath^odory [5]* established a necessary and sufficient condition i n order that the fimctions/„(2) converge uniformly i n any circle | 2 | ^ r < l to the mapping function/(3) of the " k e r n e l " R of the regions i?„ (Carath^odory's " K e r n e l Theorem"), If tlie regions R^ and R are bounded b y closed Jordan curves C» and (7, respectively, and if the convergence C„—>C is suitably defined, then/^(g)-3/(2), uniformly in l ^ l ^ l ; see Courant [9] and [10] a n d K a d 6 [22]. Kad6's theorem is formulated i n a particularly simple way: In order that Jn{2)'-*f{z) uniformly

in 1^1 ^ 1 it is necessary and sufficient that the Frechet distance'

of Cn and

C approach

0 as

n—> 0 0 . F o r further results on the case treated b y C a r a t h ^ d o r y and a discussion of the equivalence of the theorems in Courant's and Rad6's papers, see Markouchevitch [16]. (See also Gattegno and Ostrowski 12], sec. 14.) Warsc lawski extended these considerations to the derivatives of the mapping function. In [29], (sufficient) conditions are established under which the d e r i v a t i v e s ( 2 ) of any fixed order k converge uniformly in \z\ ^ I to y**^ (2). I n [27 it is shown that a necessary and sufficient condition i n order that fi (z) converge t o / ' (s) i n the mean, that is. /-(r«")-/'(re<')|dd->0,

as n - > a ) ,

uniformly for 0 _^ r <^l is that (i) _ C „ >C (in the sense that the Frechet distance approaches 0) and (ii) L where Z„ and L are the respective lengths of C„ and C. Furthermore, convergence m the mean o f order p for every j)>0 is established there for curves Cn and C w i t h continuously tumiiag tangent line. I n the present paper we are interested principally i n " q u a n t i t a t i v e " results: G i v e n two regions Rx and R2, which have the origin i n common and whose boundaries Bx and B2 are " c l o s e " to each o t h e r . (One may define a distance Z?(5i, B^ between Bx and B2 and require that it be "small".) If fi(z) and/a(2) map the circle \z\ 0 , /'2(0)>0, it is desired to find an upper bound for the difference 1/1(2)—/a(2)IT h i s boimd s h o u l d be valid either i n any circle | 2 | ^ r < l (in the general case) or i n the closed circle \z\ ^ 1 (in case > The preparation of tbls paper was sponsored (in part) by the Offioe of Naval Research. ' 'National Bureau of Standards* Ix>8 Angeles, Calif., and University of Minnesota, Minneapolis. Minn. tFiRuree In brackets indicate the literature referenoes at the end of this paper. * T h e Frechet distance d of two Jordan curves C\ and Ct is defined as follows: For any contlnuom one-to-one transformation of Ci onto Cs the distance of dlnK points has a maximum. The greatest lower bound of these maxima for all possible transformations Is d.

of Jordan domains); i t m a y be expressed i n terms of the distance D(Bij B^) and other suitable parameters, and should approach 0 w i t h D{Biy B^. A n analogous problem m a y be posed for the difference of the derivatives and for the integral means

of the f\mctions/i(2) and/zC^) and of their derivatives. I n all instances the problem is simplified i n the case of nearly circular regions; that is Ri or i?2 is a circle. There are several sharp estimates for this special configm-ation. F o r this reason the present report is divided into two parts: the first contains the results on the nearly circular case and the second deals w i t h bounds obtained for the general situation of two arbitrary regions. •

2.

Nearly Circular Regions

1. Bounds for the mapping function. L e t R denote a simply connected region i n the plane of the complex variable w which contains the origin and whose boundary B is contained i n the r i n g i - 6 ^ k l ^ i

for some e, 0 < e < l . Suppose that the function z=tp{w) maps R conformally onto the circle I ^ K l such that 0 . I t is to be expected that the difference l^jCtt?)—w| will be "small*when c is " s m a l l " , and i t is desired, therefore, to find an estimate for this difference. T h i s problem has been considered i n the literature principally i n two forms: (a) T o determine an upper bound for | ^ ( t ( ? ) — i n terms of e and p which is valid i n any circle tt)|