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NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS

TECHNICAL NOTE 2145

LIFT-CANCELLATION TECHNIQUE IN LINEARIZED

SUPERSONIC-WING THEORY

By Harold Mirels

Lewis Flight Propulsion Laboratory- Cleveland, Ohio

I I— <D 28

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D Washington

August 1950

20000803 219 Reproduced From

Best Available Copy

DUO QUALITY DJEEBOEED 4

NATIONAL ADVISORY COMMITTEE FOE AERONAUTICS

TECHNICAL NOTE 2145

LIFT-CANCELLATION TECHNIQUE IN LINEARIZED

SUPERSONIC-WING THEORY

By Harold MireIs

SUMMARY

A lift-cancellation technique is presented for determining load distributions on thin wings at supersonic speeds. The tech- nique retains certain features of the method recently introduced by Theodore R. Goodman, vhile simplifying and generalizing others.

A general expression is derived for the load distribution over a cancellation wing. This expression permits the determination of lift distributions on wings that cannot be solved by cancellation techniques based on the superposition of conical flows. The bound- ary conditions for either a subsonic leading edge or a subsonic trailing edge can be satisfied. Applications of the expression to swept wings having curvilinear plan forms and to wings having reentrant side edges are indicated.

INTRODUCTION

The method of lift cancellation for obtaining the lift dis- tribution on thin wings at supersonic speeds was first suggested in reference 1. The lift distribution on a given wing is deter- mined by canceling excess lift, through the use of a "cancellation wing," on a related plan form having a known loading. This approach has been applied by several authors (for example, references 2 to 4). The expressions provided in reference 1 are applicable for wings that can be generated by the superposition of conical fields.

A procedure is presented in reference 5 for determining lift on a more general class of plan forms than can be handled by coni- cal superposition. The method utilizes a surface distribution of doublets and an inversion by means of Abel's integral equation and is equivalent to a lift cancellation.

NACA TW 2145

This report, prepared at the NACA Lewis laboratory, retains certain features of reference 5 (that is, the use of a surface dis- tribution of doublets and an inversion by means of Abel's integral equation), whereas other features are simplified and generalized. The simplification consists in eliminating steps in the procedure for obtaining lift distributions. The generalization consists in determining a solution that can be made to satisfy the boundary conditions for either a subsonic leading edge or a subsonic trailing edge (Kutta condition). The method of reference 5 yields only the Kutta solution. The lift-cancellation technique developed herein is illustrated by several examples.

In a concurrent investigation (reference 6), source distribu- tions and integral-equation formulations have been applied to obtain the loading on a special series of cancellation wings. Reference 7 employs some of these cancellation wings for the determination of lift and moments on swept wings.

THEORY

The usual assumptions of an inviscid fluid and small perturba- tions are made. The velocity field consists of the free-stream velocity U (taken in the positive x-directionj plus the perturba- tion velocities u, v, and w. The wing boundary conditions are specified in the z = 0 plane.

The local lift coefficient ACp may be expressed in terms

of Au. That is,

AP _ PB"pT = 2(UT"UB) _ 2Au (D

ü P q U ~ U

(All symbols used in this report are defined in appendix A.) Inas- much as the local lift coefficient is directly proportional to Au, Au will be referred to as "lift" in later developments.

Lift-Cancellation Method

The lift distribution on a given wing is to be determined by canceling excess lift on a related wing with a known loading. The method is illustrated in figure 1. The wing for which the lift distribution is desired is shown in figure 1(a) The solution can be expressed as the two-dimensional wing (fig. 1(b)) minus a can- cellation wing (fig. 1(c)). The loading in region I of the

NACA TW 2145

cancellation wing equals the loading in the corresponding region of the two-dimensional wing and the upwash w in region II of the cancellation wing is zero. The loading of the two-dimensional wing minus that of the cancellation wing satisfies the boundary conditions for the flow about the given wing and is the desired solution.

The fundamental problem in the lift-cancellation method is then to determine the lift in region II of a cancellation wing subject to the condition w = 0 in this region and with the assumption of a known loading in region I. Solution of this problem is presented in the following sections.

Derivation of Lift-Cancellation Equations

The lift distribution in region II will be expressed in terms of quantities in region I.

Consider the cancellation wing shown in figure 2. The portion of the leading edge to the left of the origin coincides with a Mach line. The portion of the leading edge to the right (designated r = r-j_(s)) is shown as a supersonic edge, although no restrictions as to a subsonic or supersonic edge are imposed. (A plan-form edge is subsonic or supersonic depending on whether the component of the free stream normal to the edge is subsonic or supersonic.) The line designated r = r2(s) separates region I and region II and is assumed to be subsonically inclined to the free stream at all points. This line corresponds to a plan-form edge of the wing for which the lift distribution is desired.

General solution for load distribution on cancellation wing. - The upwash field in the z = 0 plane (due to an arbitrary distribu- tion of vorticity Au and Av) may be written, from reference 8,

w = 2«

[(y-yQ)Av + (x-x0)Au] dx dyQ ^

[(x-xo)2-ß2(y_yo)2]3/2

The symbol | designates the finite part of an infinite integral, as defined in reference 9. Application of the finite-part concept to linearized supersonic-wing theory and the evaluation of the finite part of an infinite integral are discussed in references 8 and 10. For the present, it will suffice to state the fundamental definition of the finite part of an integral with a 3/2-power singularity, namely,

NACA TN 2145

r f(x0)dxc (x"xo)

372 [f(x0)-f(x)]dx0 2f(x)

^2 (x-x0)

372" (x-a)' a ^> a

By a transformation to the Mach coordinates of reference 11,

(3) CO to ■«J

x = I (s+r) M

7 =M (s_r)

r = § U-ßy)

s - I (x+ßy)

2B

-\

elemental area = -5 dr ds J

(4)

equation (2) becomes

w = M_ 8ir

P dr0 dsQ

|]r-r0)(s-s0)]3/2

(5)

Upon substitution of the limits of integration, as indicated in figure 2,

r

M J 8*

V

ns r1

us. (s-s0)

3/2

I^Pdr

Jrl<so> (r-rQ)V2

OS Or

ds.

(^ö175 w

0^ ^o

r]>0)

372

y

(6)

NACA TN 2145

Integrating by parts, noting that A<p = 0 at rQ = ^(SQ), and

recalling the definition of the finite part (equation (3)) yield

r- CV)

or

rx(s0)

^ °

r or.

= lim \ rQ->r

ACp Ö72

^(r"ro) Jr^..) rl(so)

Ay dr0

(r-ro)

1 2

or

u

Acp dr0

ri(s0)^°

(r-r0)3/2

(7)

Thus

^Bdr0 = 1

7 3/2 2

os pr

ds^

0 (s-sn)

372 TAB ) 1 o

ACp dr

(r~ro)

(8)

Similarly, reversing the order of integration (with appropriate changes in limits of integration), integrating by parts, and then returning to the original order of integration establish the identity

dso

nT ÖACp .

, >-r0)3/2 rl(so)

os r2

as. Afp drr

J (s-s0)

3/2 (r-r0)3/2

0 ° Jr^Bo)

(9)

WACA TN 2145

The right side of equations (8) and (9) are identical. Equation (6) can now be written as

w = - 2L 8«

r>s pr ds,

0 (-o) Tß

ACP dr0

(r-r0) ' <^u ~ ^rl(so^

For points in region II, w - 0 and equation (10) becomes

(10)

M CO CD -«1

0 = ds. ACp dr.

(s-s )3/2 (r_r )3/2

VSo>

(11a)

or

where

0 = G(r,sQ) dsQ

(s-s0)3/2

0(r,so) = &<P dr,.

, ,(r-r0)3/2 rl(s0)

V °'

(lib)

(12)

Equation (lib) is an integral equation for the unknown function G(r,s0). The solution (appendix B) is

G(r,sQ) = 0 (13)

Thus,

0 =

rl<so)

A<P dr0 372 ~

(r-r0)

rr2(so) r\r

•J rx(s0)

ACPldro ~( ^3/2 +

r2(so>

^ 372

NACA TN 2145

or

nT

*2<»o>

*PlI ^o

(r-r ) '

rr2(BQ)

r,(s ) lv o'

(r-rJ (14)

The right side of equation (14) vill be considered known. Equa- tion (14) is then an integral equation for Aqp-Q. The solution (appendix B) is

Acpn

^r-r2(s) *2(s)

ACpT dr

Ms) (r-ro)A/r2(s)-r0

(15)

Equation (15) indicates that the doublet strength in region II, namely ACpj-p can be obtained by a line integration along sQ = s

in region I. The geometric interpretation of the various terms in equation (15) is shown in figure 3.

It can be shown, by expanding #Pj in a Taylor's series about

r = r2(s), that equation (15) yields a continuous solution

(Atpjj = A9j) at r = r2(s). (A discontinuity in A*? implies a

lifting line (reference 12) and is unrealistic.)

The lift distribution in region II can be expressed as

dtfPi Au II

rII ~5x~

^11 Or + ^11 ös 5r~~ cix ös c)x

-äGl^-n or, from equation (15),

—*- Au M II

= (j? + SsJ AJr-r2(s)

Nr?(s)

AqPj dr0

rx(s) (r-r0)/^r2(s)-rc

(16)

NAGA TN 2145

Differentiation yields (See appendix C.)

Au /\|r-r2(s)

nr2^

II

Aux dr JE SL

r^s) (r-rQ) AJr2(s)-r0

1 - dr2(s)

pr2(s)

ds 2ß«A/r-r2(s)

(ßAux-AvT)dr v I I' o

Ms) y\/r2(s)-3 (17a)

Equation (17a) is the desired expression for the lift distribution in region II in terms of quantities in region I.

Consider Au-j-j to consist of two components, AUJJ* anc*

Au *', where AuT ' and AuTT" are the first and second terms

on the right side of equation (17a), respectively. Investigation of the integrals indicates that at r = r2(s), Aujj1 = Auj;

whereas Aujj", in general, has a half-order singularity.

When region II is to the right of region I (fig. 3(b)), the integration for AU-Q is conducted along the line rQ = r and

may be written as

Au II - /Js-s2(r)

p s„(r)

AuT ds I o

s-i(r) (s-s0) /\/s2(r)-sc

1 - ds2(r)

ps2(r)

dr 2ßn/Js-s2(r)

(ßAuI+AvI)ds0

, , /\ls2(r)-so s1(r) V

(17b)

NACA TN 2145

Discussion of equations (17a) and (17b). - In the paragraph preceding equation (2), the line r = r2(s) was described as sub-

sonically inclined at all points to the free stream. This condition is necessary so that the inner integral in equation (lla) (that is, G(r,sQ)) can be equated to zero for all points in region II. If

this restriction on r = r2(s) is not satisfied, the development beyond equation (lla) becomes invalid. The derivation of cancella- tion equations vhen r = r2(s) is supersonic was not undertaken

because such problems can be solved more simply by other methods.

In regard to the boundary conditions, it has been assumed that Au-j. is specified. Equations (17a) and (17b), however, indicate

that a knowledge of Av-j- is also required in order to obtain a

solution for AUJJ. Two possibilities exist, as illustrated in

figure 4. In the first case (fig. 4(a)), region I is upstream of region II (along the line r = r2(s)) and Avj- is uniquely defined

by the specified AuT according to the relation

Auj dxQ (18a)

The integration is conducted along lines of constant y. In the second case (fig. 4(b)), region II is upstream of region I (along the line r = r2(s)) and the expression for Avj in region I

(for y<0) is

^ - £ AUJJ dxQ + Auj dx0

My) (18b)

Equation (18b) indicates that a knowledge of AuI3- is required in

order to find Avj. But Avj must be known (equation (17a)) before

AUJJ can be found. Thus, the solution for AUJJ from a specified

Auj is not unique for the configuration of figure 4(b) and an

additional boundary condition must be imposed. The line r = r2(s),

however, corresponds to a plan-form edge of the airfoil whose load

10 NACA TW 2145

distribution is desired. The situation indicated in figure 4(b) occurs vhen r = r2(s) corresponds to a subsonic trailing edge.

The additional condition to be imposed is therefore the Kutta con- dition. In terms of the cancellation wing, this condition requires that the perturbation velocities be continuous in crossing r = r2(s),

Solution for AUJJ satisfying Kutta condition at r = r2(s). -

It will nov be shown that when the Kutta condition is imposed at r = r2(s), the appropriate Avj distribution is such as to make

the second integral in equation (17a) identically zero; that is,

Or2(s)

r-^s)

ßAUj-Av.

[r2(s)-ro" T/2 ^o' » 0

ÖACpj or, inasmuch as ßAuT-AvT = M —-—} x x or„

pr2(s) ÖACpj

orT"dro

ri(s) t2(-)-o] TJz = 0

This concept and its proof follow from a suggestion of H. S. Eibner of the NACA Lewis laboratory.

Thuf, from equation (7), (12), and (13),

J rx(s) (r"ro) I72 = 0 (19)

for all points (r,s) in region II. Therefore,

f r~(s) oACpj

dr.

U MB) (r-rj I72

P1

Jr2(s)

oA<P- II

(r-r0)

dr_

172 (20)

NACA TN 2145 11

Upon taking the limit as r approaches r2(s), equation (20) becomes

r2(s)

ÖA«Pj drr

ri(s) [M^-O]

I72 lim r->r?(s)

r

Jr2(s) (r-r0)

I72 (21)

However, dA<Pj/dr0 must be continuous in the vicinity of r2(s). (The perturbation velocities on the basic ving can be discontinuous only along Mach lines or along plan-form edges. Inasmuch as r = r2(s) is neither of these cases, all derivatives of A<Pj must be continuous in the vicinity of r = r2(s).) When the Kutta con-

dition is imposed, hif?11/brQ is therefore also continuous (and bounded) in the neighborhood of r = r2(s). Then, using a mean value for dA<Pjj/dr0,

vr

lim r-^r2(s)

ÖACp- 'II dr.

Jr2(s) (r-r0)V2

lim r-»r2(s)

= 0

Therefore

'ÖACp II ST

>r2(s)

dr.

°/(r2(s)<r0<r) | (r-r0)V2 r2(s)

MB)

dACP, dr.

l^)-r0f/Z = 0

(22)

(23)

which was to be proved.

12 NACA TN 2145

The solution for Aujj that satisfies the Kutta condition at

r = r2(s) is then, from equations (17a) and (23),

Au II /^r-r2(s)

r~(s) Px2 AUj drQ

J M»> (r-ro)A/r2(s)_rc

(24a)

for the wing of figure 3(a). Similarly,

^s2(r)

AuII = AJs-s2(r) AuT ds I o

<h(r) (s-sQ)/^s2(r)-sc

(24b)

for the wing of figure 3(b),

An alternate derivation of equations (24a) and (24b) (appen- dix D) indicates that only solutions satisfying the Kutta condition will result from the integral equation formulations of reference 5.

Sidewash in region II. - An expression for AVJJ can be obtained by differentiating equation (15) with respect to y. The result is

pr2(s)

Av Vr-r2(s)

II AvI dro

J

1 + dr2(s)

rx(s)

M-P(S)

(r-rQ) /\Jr2(s)-rc

ds ßAUj-Avj dr.

2«/y/r-r (s) I /\/r2(s)-rQ °

to

Similarly, for region II to the right of region I,

NACA TN 2145 13

AvII =

ds2(r) 1 + —i— ds

S2(r) /^s-s2(r) I Avj dsQ

(s-s0)A/s2(r)-sc sx(r)

2rt^s-s2(r)

ps2(r)

ßAux+AvT ds

ßAUj+ÄVj

A/s2(r)-s0 S]_(r)

When the Kutta condition applies, these equations become, respectively,

A/r-r?(s)

r2(s)

AvT dr JE 2_

(r-r0)^r2(s)-rc

'^(s)

and

Avn =

^s2(r)

/^s-s2(r) Avj ds0

8 (p) (s-s0)A/s2(r)-sc

It should be noted that when r = *"2(s) corresponds to a subsonic

trailing edge, Av-j-, as well as AV-J-J, is not generally known.

The preceding expressions are therefore primarily useful for those problems where r = r2(s) corresponds to a subsonic leading edge.

APPLICATIONS

The loading in region II of a cancellation wing is given by the line integrals of equation (17a) or (17b). When the Kutta condition is imposed at a subsonic trailing edge, the expressions reduce to equations (24a) and (24b). These equations can be used to find the load distribution on a large variety of wings that cannot be solved by cancellation techniques based on conical

14 WACA TN 2145

superposition. Wings vita curvilinear plan forms or arbitrary camber are examples. In each case, however, the solution for the J related wing must be known. S

The equations are applied in several illustrative examples. Only the solution associated with the cancellation wing is con- sidered. The complete solution consists in the loading of the related wing minus the loading of the cancellation wing.

Leading-Edge and Side-Edge Cancellations

In these cases, the lift to be canceled is upstream or to the side of the plan form for which the loading is desired (figs. 5 and 6).

Tip region of swept wing. - The loading in the region influ- enced by the side edge (IIa and II^ of fig. 5) of a swept wing having a subsonic leading and a supersonic trailing edge can be obtained by canceling excess lift on a triangular wing. The Kutta condition is applied across the portion of r = r2(s) influencing region 11^,. The lift to be canceled in region I is (reference 13, equation (23))

, H62x m HeVr) (25)

AjeV-ßV Ve2(s+r)2-(s-r)2

where H and 0 are constants defined in appendix A. The doublet distribution in region Ia, again from reference 13, is

W = H/\/e2x2-ß2y2 « £1 /\fe2(s+r)2-(s-r)2

Ia M

from which

&VT =g^ag -Hß2y_= -Hß(s-r) (26)

Ia * ^92x2-ß2y2 A/02(s+r)

2-(S-r)2

The sidewash distribution in region It (that is, Avj) could be

found by an integration of the type indicated in equation (18b). A knowledge of AVj, however, is unnecessary in the present problem

because the Kutta condition is applied for region 11^.

NACA W 2145 15

The loading in region II& is obtained "by substituting equa-

tions (25) and (26), with r replaced by r0, into equation (17a), vhich yields

Au II. H02/\/r-r2(s)

^2(s)

(s+r ) dr v o' o

t-^]

ri(s)

pr2(s)

(r-rQ) Ajr2(s)-ro^02(s+ro)

2-(s-ro)2

H 1

2ä ̂ r-r2(s)

[e2(s+r0)+(s-r0)]dr0

/Jr2(s)-r0Aje2(s+r0)

2-(s-r0)2

(27)

For region II-^, the Kutta condition applies and

_ ^2(s)

H02/^r-r2(s)

(r-r0)Ajr2(s)-ro^e2(s+ro)2-(s-ro)2

Au (s+r0) drQ

II>

Ms) (28)

Equations (27) and (28) reduce to elliptic integrals of the first, second, and third kind upon transforming the variable of integra- tion from rQ to (0o according to the relation

ro = iTe [(1-0)8 +agdfe2] (29)

where

a2 = (l+0)r2(s)-(l-0)s

Equations (27) and (28) may then be written

16 NACA TN 2145

Au "a " ^S^l^^Hi^)-^^]

11 *,k) - F(0,k) H/^sIT

I" _ dr2(s) L I ds .

jr#Jr-r2(s) 2|~E^|,k) - E(0,k)] -

J (|,k) - *(*,*) (30)

and

Au Hi

JP24|r-r2(B) J(s+r)(l+e)riI(^n>k\ . n(0,n,kjl - -.Afro" a L \2 / J A/S6

where

[r(|,k) - F(0,k)]

■ $

-e 4se

a2

0 = sin-1A

a = (1+e) r-(l-0)s

ax = (1+e) ri(8)-(l-e)8

(31)

n = a„ « (1+0) r„(s)-(l-e)s

Eeentrant side edge. - A plan form has a reentrant edge if a line of constant y intersects the plan form at more than two points.

The load distribution in the region influenced by the reentrant side edge is to be determined for the wing of figure 6. The side

NACA TN 2145 17

edge is first, for simplicity, the straight line r = Kgs, which is a subsonic trailing edge across which the Kutta condition is applied. The side edge then alternately becomes a subsonic leading and a subsonic trailing edge. The load distribution in region I is

simply the Ackeret value Auj = ^—, and Avj = 0. Regions IIa, r* 8.

lib, and- He are considered separately.

Eegion IIa.

From equation (24a) with Auj = 2<xU

pKgs

Au II.

t^T-K2B 2aU dr0

Jt _s ß(r-ro)^K2s-ro

» i°^ tan" ß«

I J(K2+i)s 1 r-K2s

(32a)

or, in x,y-coordinates

A„ 4dü -1 » I x+ßy Auiia = istan \temper (32b)

Eegion IIb.

A knowledge of Av-r is required. From equation (18b),

Av xb 3y 4aU ß*

py/m2

tan'

■ßy •i y/mg

2-U =MJVi V Ko-l ^2J

(33)

18 NACA TN 2145

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NACA TN 2145 19

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20 NACA TN 2145

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22 NACA TN 2145

induces lift in region I*. The cancellation of lift in region I' induces lift in region I", and so forth. lach of these cancella- tions is handled as previously described. These computations are very tedious vhen lift is induced upstream of a subsonic leading w edge (for example, region I' of fig. 8), inasmuch as a knowledge 3 of the sidewash (AvT,), as well as of the lift distribution

(Aujt), is needed in order to continue the cancellation process.

Numerical methods are generally required.

Successive cancellations are discussed more extensively in references 3 and 4.

SUMMARY OF ANALYSIS AND APPLICATIONS

A general expression was determined for the lift distribution over a cancellation wing. The expression is valid when the plan- form boundary (on cancellation wing) separating the region of zero upwash from the region for which the lift is specified is everywhere subsonically inclined to the free stream. This expression permits the determination of lift distributions on wings that cannot be solved by cancellation techniques based on conical superposition. The boundary conditions for either the flow about a subsonic leading edge or a subsonic trailing edge can be satisfied.

The lift cancellation technique was illustrated for swept wings having curvilinear plan forms. Leading-edge and trailing-edge cancellations were considered. In addition, the loading in a region influenced by a reentrant side edge was found.

Lewis Flight Propulsion Laboratory, National Advisory Committee for Aeronautics,

Cleveland, Ohio, January 16, 1950.

NACA TN 2145 23

a-L

a2

AC,

APPENDIX A

SYMBOLS

The following symbols are used in this report:

(l+0)r-(l-0)s

(l+0)r1(s)-(l-0)s

(l+0)r2(s)-(l-0)s

local lift coefficient,

constant

root chord of swept wing

E(0,k)

F(|ZU)

PB-PT

elliptic integral of second kind,

r>sin 0

E(0,k) =

\j

>\/l-k2ü)02

A/I-WO2

d(^

elliptic integral of first kind,

psin 0

F(0,k) = dtt»

J o Aj(l-0)o2)(l-k20)o

2)

G(r,s0) function of r and s0 defined by equation (12)

H

K

k

2aU

ßE (%,'{l-Sc

slope of plan-form edge in r,s-coordinates, dr/ds

modulus of elliptic integrals

24 NACA TN 2145

M Mach number

m slope of plan-form edge in x,y-coordinates, dy/dx

n parameter of elliptic integral of third kind

p local static pressure

1 TT2 q = 2 pU

r>ro s>so

x>xo

Z>Z0

Mach coordinate system (equation (4))

U free-stream velocity

u,v,w perturbation velocities in x-, y-, and z-directions, respectively

Au = ^"Uß (proportional to local lift)

&v = vT-vB

Cartesian coordinate system

a angle of attack

ß = A/M2-I

5 semivertex angle of triangular wing

0 ß tan 5

IT(0,n,k) elliptic integral of third kind,

psin 0 dü)o

(l+n(ü02) A|(l-k2Ö"(l-(0o

2)

NACA TN 2145 25

P density-

V

T

Eegions:

I

area of integration

amplitude of elliptic integrals

perturbation velocity potential

doublet strength, ^T-Cpg

integration variable

region on cancellation wing for which loading is specified

■^a^b» * • * subdivisions of region I

II region on cancellation wing for which w = 0

t

III

• • subdivisions of region II

additional region on cancellation wing for which w = 0

Special designations:

r = r^s) r as function of s along plan-form boundary 1

s = sx(r) s as function of r along plan-form boundary 1

y = yi(x) y as function of x along plan-form boundary 1

x = x-L(y) X as function of y along plan-form boundary 1

r = r2(s) r as function of s along plan-form boundary 2

s = s2(r) s as function of r along plan-form boundary 2

*

and so forth.

26 NACA TN 2145

Subscripts:

1,2,3 refers to plan-form boundaries 1, 2, and 3, respectively

I,II refers to regions I and II, respectively

B bottom surface of z = 0 plane

T top surface of z = 0 plane

o variable of integration .

NACA TN 2145 27

APPENDIX B

SOLUTION OF INTEGRAL EQUATIONS

Consider the following integral equation (in the notation of the appendix in reference 5), where the function f(x) is assumed known and the function u(|) is to be determined:

f(x) -

r>x

n(i) ae

ox

[u(|)-u(x)] d| 2u(x)

(x-e) 3/2 (x-a) 1/2 (Bl)

After an integration "by parts, equation (Bl) may he written

f(*) u(a) = I u'(e) dt Tx^j1^" |_7xT|7l72

(B2)

Equation (B2) is now an integral equation of the Abel type. The continuous solution for u(|) is (reference 14)

nz

u(z) = 2« f(x) dx

(B3)

evaluated at z = £. This result is presented in reference 5.

Equation (lib) corresponds to equation (Bl) with u(|) = G(r,s )

and f(x) =0. The solution for G(r,s0), according to equa- tion (B3), is then

G(r,sQ) = 0 (13)

28 NACA TN 2145

Equation (14) corresponds to equation (Bl) vith

i = rQ a = r2(s0)

^r2(so)

x = r f(x) = -

. vJr1(s0)

A<P dr

(r-r0) 3/2

u(l) = A<PI3.

The solution for A<PJJ according to equation (B3) is then

nz

t& II = 25

^2(so) dr &CpI dro

, ,/\F?| , , (r-r0)3/2

r2(sQ) Jr^s«,)

After reversing the order of integration and integrating,

ACPn = /\/z-r2(s0)

Pr2(so) ACpj drQ

rl<so)

(z-rQ) ^(r2(s0)-rc

Equation (B5), evaluated at z = r and s0 = s, yields

ACpii = l\[r^r2(s)

pr2(s) ACPj drQ

*l(s) (r-r0)A/r2(s)-r0

(B4)

(B5)

(15)

The derivation of equation (15) is similar to that for equa- tion (16) of reference 5.

H to to -j

NACA TN 2145 29

APPENDIX C

H

DIFFERENTIATION TO OBTAIN AUJJ

The differentiation indicated in equation (16)

2ß . (b d

is to be conducted.

First

/\Jr-r2(s) Ws)

**1 ^o

^(s) (r-r0)Ajr2(s)-r0

(16)

2ß A "M &UII

dr2(s) 1 1 ds

2itA/r-r2(s)

pr2(s)

^ dro

(s) (r"ro)/\/r2(s)-ro

^9(s)

4r-r2(s> fö A d

är ös A<PT dr I o

ri(s) (r-r0)^/r2(s)-rc

Inasmuch as ACp,. is a function of rQ and s,

(Cl)

pr2(s)

Ö 3F

M-o(s)

ACPX dr0 A^ drc

r (s) (r-r0)^r2(s)-r0 Jr (s) (r-r0)2A/r2(s)-r0

(C2a)

30 NACA TN 2145

and

^2(s) Nr2(s)

ACPI ^o d

*■ | , , (r-r0)AJr2(s)-r0 , x (r-r0)Jr2(s)-r0 rr(s) V

CD -*1

lim ro-»r2(s)

1 dr2(s) 2 ~~ds~

ACp dr I o

^(s) (r-r0)[r2(s)-r0]3/2 +

<ACpI>r0 = r2(s) dr2(s)

(r-ro)^r2(s)-i ds

y

^ACPl)r0 = ri(s) dri(s)

[r-r^ s )J /^r2(s)-r1(s) ds

(C2b)

However, (A<Pj)r _ r (s) = 0 and, by integration by parts,

fvr2(s) .(a)

A<PT dr 1 Q = _2

13/2

ÖACpj ACpj

Lo^ + (r-r0)_, drv

r,(s) (r-r0)[r2(s)-r0]3/2 Jr (g) (r-rQ)A/r2(s)-rc

-i r.

lim ro->r2(s)

2ACpT

(r-r0)A^r2(s)-r0

*•]».

so that equation (C2b) can be written as

NACA TN 2145 31

^r2(s) ^2(s)

Si ACpj drc _^— drn ds °

P1(8)(r"roWr2(8)-ro Jri(.) (r"ro)Vr2(s)-rc

pr2(s)

dr2(s) ds

ÖACPj A<Pj- "Sri + (r-r_) dr.

Ms) (r-r0)^r2(s)-r0

(C3)

Upon substituting equations (C2a) and (C3) into equation (Cl) and integrating by parts those integrals containing tfpj, equation (Cl) finally reduces to

Au II hjr-r „(s)

\T2(B)

AuT dr I o 'ri(s)(r-r0)Vr2(s)-rc

^o(s)

dr2(s)

ds (ßAux-Av_) dr I I7 o

2ß*Vr-r2(s)Jr (s) /\/r2(s)-ro (17a)

32 NACA TN 2145

APPENDIX D

ALTERNATE DERIVATION OF SOLUTION SATISFYING

KUTTA CONDITION AT r = r2(s)

The integral equation formulation in terms of ACp (equa- tion (14)) resulted in a solution that was continuous in ACp (equation (15)) but discontinuous, in general, in the derivative

-r^ = Au (equation (17a)) at r = r2(s). In order to obtain a

solution continuous in Au, an integral equation may be formulated that is similar to equation (14), but in terms of Au rather than A*?. The inversion shown in appendix B should result in a solution that is continuous in Au but discontinuous in the derivatives of Au at r = r2(s).

Consider equation (10) for the v distribution in the z = 0 plane. This equation will be differentiated with respect to x using a technique introduced in reference 15 (equations (l) to (3)). The expression for w at any point (r,s) is, from equation (10),

w = - M_ 8*

AT> dr ds ^o o

(s-sj 3/2(r-rJ3/2

(Dl)

where T is the area abc in figure 9(a). The wing is moved upstream a distance dx (fig. 9(b)), keeping the coordinate system fixed in space. The expression for the upwash at (r,s) now becomes

w + |2 dx = - M. ox 8jt

Acp + |£? dxl dr ds ox0 loo

(s-So)3/2(r-r„)3/2

A«P dr ds o o (s-s„)3/2(r.I.n)3/2

(1X3)

NACA TN 2145 33

The second term on the right side of equation (D2) is zero because A<P = 0 along the leading edge. Subtraction of equation (Dl) from equation (D2) then yields

dw Sx"

2L 8it

Au dr ds

(8-8 )3/2(r-r )3/2 (D3)

ÖW For points in region II of a cancellation wing, ^ = 0. Thus, for the wing of figure 3(a), dx

0 =

rl(so)

Au dr (D4)

This equation is the same as equation (lla) except that Au replaces A<P. The inversion by Abel's integral equation, for AUJJ in terms

of Auj then gives (from equation (15))

M-o(s)

Au hjr-T (s)

II Au, dr

I o

Ms) ^r-ro)^2(s)-ro

(24a)

Inasmuch as the integral equations of reference 5 are formulated in terms of Au and are inverted by means of Abel's integral equa- tion, only solutions satisfying the Kutta condition will result therein.

REFERENCES

1. Lagerstrom, Paco A.: Linearized Supersonic Theory of Conical Wings. NACA TN 1685, 1948.

2. Coleman, T. F.: Supersonic Lift Solutions Obtained by Extending the Simple Linearized Conical Flow Theory. Rep. No. CM-440, North American Aviation, Inc., Feb. 20, 1948. (Bumblebee Proj. Contract NOrd 9784.)

3. Cohen, Doris: The Theoretical Lift of Flat Swept-Back Wings at Supersonic Speeds. NACA TN 1555, 1948.

34 NACA TN 2145

4. Cohen, Doris: Theoretical Loading at Supersonic Speeds of Flat Swept-Back Wings with Interacting Trailing and Leading Edges. NACA TN 1991, 1949.

5. Goodman, Theodore E.: The Lift Distribution on Conical and Nonconical Flow Regions of Thin Finite Wings in a Supersonic Stream. Jour. Aero. Sei., vol. 16, no. 6, June 1949, pp. 365-374.

6. Eibner, Herbert S.: Some Conical and Quasi-Conical Flows in Linearized Supersonic Wing Theory. NACA TN 2147, 1950.

7. Eibner, Herbert S.: On the Effect of Subsonic Trailing Edges on Damping in Eoll and Pitch of Thin Sweptback Wings in a Supersonic Stream. NACA TN 2146, 1950.

8. Eobinson, A.: On Source and Vortex Distributions in the Linearized Theory of Steady Supersonic Flow. Rep. No. 9, College Aero. (Cranfield), Oct. 1947.

9. Hadamard, Jacques: Lectures on Cauchy's Problem in Linear Partial Differential Equations. Oxford Univ. Press (London), 1923, pp. 133-135.

10. Heaslet, Max. A., and Lomax, Harvard: The Use of Source-Sink and Doublet Distributions Extended to the Solution of Boundary Value Problems in Supersonic Flow. NACA Eep. 900, 1948. (Formerly NACA TN 1515.)

11. Eward, John C: Use of Source Distributions for Evaluating Theoretical Aerodynamics of Thin Finite Wings at Supersonic Speeds. NACA Rep. 951, 195^

12. Mirels, Harold, and Haefeli, Rudolph C: Line-Vortex Theory for Calculation of Supersonic Downwash. NACA TN 1925, 1949.

13. Heaslet, Max A., and Lomax, Harvard: The Calculation of Down- wash behind Supersonic Wings with an Application to Triangular Plan Forms. NACA TN 1620, 1948.

14. Whittaker, E. T., and Watson, G. N.: Modern Analysis. The MacMillan Co. (New York), 1943, p. 229.

15. Eward, 'John C: Theoretical Distribution of Lift on Thin Wings at Supersonic Speeds (An Extension). NACA TN 1585, 1948.

NACA TN 2 145 35

CT

y

w=-aU

w H

y s

*-y

x \ "^ N«^—Wach w=-aU \ Au=0 \

line

(a) Given wing.

w=-aU Au=Auj

^ T

" \ * \

w=-aU I Au=AuT

/ /

/

w=-aU Au=Auj i

(b) Two-dimensional wing.

w=? Au=Au.

I

(c) Cancellation wing.

Figure 1. - Superposition to obtain lift on given wing by canceling lift on two-dimensional wing. (Given wing equals two-dimensional wing minus cancellation wing.)

36 NACA TN 2 145

r,r0

U

*-y,y0

r=r1(s)

H

-si

Figure 2. - Typical cancellation wing.

NACA TN 2 145 37

(a) Region I intersected by right forward Mach line from (r,s),

s=s1(r)

(b) Region I intersected by left forward Mach line from (r,s).

Figure 3. - Geometric interpretation of terms in equations (17a) and (17b).

38 NACA TN 2145.

*-J,Jo

x^^y)

x=-ßy

r,r0

H

(a) Region I upstream of region II (along r=rg(s)).

x=-ßy

*- y,y0

x=x1(y)

(b) Region II upstream of region I (along r=rg(sj),

Figure 4. - Possible relations between regions I and II in regard to determination of Av-j-o

NACA TN 2 145 39

w

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40 NACA TN 2145

NACA TN 2 145. 41

■y.y

(a) Regions II and III continuously interacting.

(b) Regions II and III not continuously interacting.

Figure 7. - Typical cancellation wings for canceling lift downstream of subsonic trailing edge of swept wings.

42 NACA TN 2145

Figure 8. - Successive cancellations.

NACA TN 2 145. 43

Leading edge

/r

r'ro X,X0

y.yQ

S.S.

(a) Original position of wing.

Leading edge

/f *-y.y0

r,r0 X(x0 s,s

(b) Wing moved upstream distance dx.

Figure 9° - Areas of integration relating to equations (Dl) and (D2),

NACA-Langley - 8-1-50 -1050