lifting-line theory for a swept wing at ......the transonic flow about a lifting, high aspect ratio,...

QUARTERLY OF APPLIED MATHEMATICS 177JULY 1979

LIFTING-LINE THEORY FOR A SWEPT WING AT TRANSONICSPEEDS*

By

L. PAMELA COOK

University of California, Los Angeles

Abstract. The boundary-value problems describing the first-order corrections

°(jR-jRl"AR)

to two-dimensional flow about a lifting swept wing at transonic speeds (M„ < 1) arederived. The corrections are found by the use of the method of matched asymptoticexpansions on the transonic small disturbance equations. The wing is at a sweep angle of0((1 — M*)l/2) in the physical plane, hence of 0(1) in the transonic small disturbanceplane. As has been noted for subsonic flow, the finite sweep angle necessitates theintroduction of terms 0((AR)~l In AR). These terms arise naturally in the matchingprocess. Of particular interest is the derivation of the near field of a skewed lifting-linewhich is found by Mellin transform techniques. Also of interest is the fact that theinfluence of the nonzero sweep angle can be completely separated from the unsweptsolution.

Introduction. Prandtl's classical lifting-line theory gave aspect ratio corrections tothe two-dimensional flow in the cross-sections of an unswept finite span wing. The induceddownwash at each spanwise station was obtained. The method involved solving anintegral equation. Van Dyke [13] systematized Prandtl's approach by recognizing that itcould be considered as a singular perturbation. He considered the inviscid incompressibleflow about an unswept wing with (aspect ratio)"1 as the small parameter. Inner and outerexpansions were constructed and matched. The inner problem consists of two-dimensionalflow at each spanwise station and corrections; the outer flow represents the flow past abound vortex which sheds a vortex sheet and corrections. The advantage of this method isthat there are no integral equations to be solved. Cook and Cole [6] extended Van Dyke'sapproach to describe compressible (transonic) flow about an unswept lifting wing. In thiscase the inner region represents the two-dimensional (nonlinear) transonic flow on a cross-section of the wing and corrections. The outer region represents the (linear) flow past abound line vortex trailing a vortex sheet and corrections. It was shown that for similarairfoil sections the computations need be carried out at one spanwise station only since thecorrections can be scaled to be independent of z*.

In this paper we extend the techniques of Van Dyke, and Cook and Cole, to describethe transonic flow about a lifting, high aspect ratio, yawed wing. Investigations have beencarried out on the swept wing for incompressible (linear) flow, for example Burgers [7],

* Received June 15, 1978.

178 L. PAMELA COOK

Thurber [12]. Thurber points out that a new term which is order of (AN)'1 In (/*/?) arisesin the induced downwash. A similar term arises naturally by the matching process in thetransonic (compressible) flow. Cheng and Hafez [3] have treated swept wings in transonicflow. Their primary interest was in wings for which the (tangent of the sweep angle)3 is »<5, where S « 1 is the ratio of the thickness of the airfoil to its chord. In this paper we areconcerned with the case where the (sweep angle)3 = O(S).

In Sec. 2 the three-dimensional transonic small disturbance theory boundary valueproblem for a lifting yawed wing is formulated [4], The sweep angle is 0((1 - AfJ)1/2) inthe physical plane, hence 0(1) in the small disturbance plane. In Sec. 3 the asymptoticexpansions as B = (AR)81/3 -» where 8 is the typical flow deflection, are constructed.Shock waves can appear in the inner flow which describes the flow in cross-section planesand corrections. The outer expansion starts with the potential of a skewed bound vortex.The inner limit of the outer expansion is found in Appendix A by Mellin transformtechniques. Matchings of the inner and outer expansions are shown. The boundary-valueproblems describing the inner potentials <p0, <Pi, are discussed in Sec. 4. is describedby the usual two-dimensional transonic small disturbance equation (in terms of coordi-nates measured from the wing). ^ , which is the 0(AR'1 In AR) term, arises naturallyfrom the matching process, and is described by a linear equation whose coefficients dependon tp0 (variational equation). <p2 is described by the same equation as ipt except that theequation is forced by spanwise variations in . <?2 can be broken up into two pieces, <p2il)which is independent of sweep angle, and <p2w which is the sweep correction. Airfoils ofsimilar section are discussed in Sec. 5. In particular, for similar sections the <p0, <Pi, <?2(1)problems can be scaled to be independent of spanwise variation. Thus they need only beevaluated at one spanwise station.

There are nonuniformities in the expansion as —> 1, as well as near the wing tips.

2. Formulation of boundary value problem. In this section we formulate a boundaryvalue problem for the first-order transonic disturbance potential:

<p(x, y, £■ K, A, B).

In this restricted problem the tangent flow boundary condition is applied in the planeof the wing and the trailing vortex sheet is assumed to lie wholly in the plane of the wing.The Kutta-Joukowsky condition, that the flow leave the trailing edge smoothly, can thusbe applied unambiguously at the trailing edge of the projection of the planform onto theplane of the wing. The self-induced downward motion and the rolling up of the trailingvortex sheet is not considered, so that presumably there is a non-uniformity of theapproximation in the flow near the vortex wake toward downstream infinity. It is notexpected that this will cause any appreciable error in the life distribution over the wing. Infact in the transonic region corresponding to high subsonic speeds, which is consideredhere, propagation of disturbances in the upstream direction is relatively weak. Theboundary value problem formulated here is thus the transonic equivalent of linearizedlifting surface theory. The lifting-line theory which is constructed is the approximation forlarge aspect ratio.

Let the units of length be chosen so that the wing chord at the midspan is one and theextent of the wing in the z direction is 2b cos /3 (see Fig. 2.1). The wing surface is then givenby

lV(x, y, z) = 0 = y - 8FUil(x - z tan A, z/b) + ax, (2.1)

LIFTING-LINE THEORY 179

0(v'l - Ml)

Fig. 2.1. Wing in physical coordinates.

where A is such that the line x = z tan A joins the z extremities of the wing, Fuj describesthe geometry of the upper and lower surfaces respectively, a is the angle of attack, and 8the thickness ratio. The projection of the leading and trailing edges onto the plane y = 0 isgiven by the functions xLE,TE(z/b).

The transonic expansion procedure is based on the limit process 5 10, —> 1 (cf. [4])with the transonic similarity parameter K, the angle of attack parameter A, the aspect ratioparameter B all fixed. In addition, the coordinates (x, y, z) are held fixed, to account forthe relatively large lateral extent of the disturbance field. Thus, the potential is represented

$(x, y, z; M, a, 8, b) = U{x + 52/3 cp(x, y, f; K, A, B)

+ S"3 ct>lIXx, y, K, A, B) + •••, (2.2)

where K = transonic similarity parameter1 = (1 - M£)/82/3, A = angle of attackparameter = a/8, B = aspect ratio parameter = b81'3, {y = 81/3y, z = 8l/3z) = transversecoordinates, and tan = sweep angle = 5"1/3 tan A. (See Fig. 2.2).

Since shock waves introduce entropy changes only to the third order, it is sufficient toconsider a potential $>. The first-approximation potential <t> satisfies the well-knowntransonic equation

(AT - (7 + \)<px)(pxx + <j)yy + </>££ = 0. (2.3)

1 In a more complete theory of the second transonic approximation, representations such as = 1 - K52'3+ Kj(K)S"3 - ■ • • are used to describe the similarity parameter K. Similar representations are used for (j7, z).

180

/ \ -*te — CTE(z/fl) + z tan 0

Xle = Cle(z/B) + z tan 0

Fig. 2.2. Wing in transonic coordinates.

The boundary conditions for our problem will now be discussed. First of all there is nodisturbance at upstream infinity, so that

0* » <t>y , <t>£ -* 0 as x —> — (2.4)

Also, at downstream infinity pressure disturbances must die out. Since the transonicpressure coefficient is given by

€p = = (pHlP/2)5V3 = _2(/>x ' (2'5)

this means that

4>x — 0 as x -> co. (2.6)

The boundary condition of tangent flow becomes

<t>y(x, 0±, z) = df"'' (^ - f tan I3,z/B) - /I, (2.7)

over Cle(z/B) < x - z tan 13 < CTE(z/B), where /3 = tan-1(<51/3 tan A). At the trailing edgethe Kutta-Joukowsky condition can be expressed

[(Px]te ~ 4>x(xte{z/B), 0+, z) — <px{xTE{z, B), 0—, z) = 0. (2.8)

Note that the Kutta-Joukowsky condition implies zero pressure perturbation at thetrailing edge.


The vortex sheet lies in the plane / = 0, with x — z tan /? > Cte(z/B), |z| < B cos 0,and stretches to downstream infinity. There is no jump in pressure across the vortex sheet,so

[<t>x]v. = 0+, z) - 4>x(x, 0-, z) = 0, (2.9)

[0]#« = [4>]te = r(£) for |z| < B cos /?. (2.10)Here r(z) is the spanwise distribution of the (perturbation) circulation around a cross-section of the wing

r(£) = i<t>xdl= r:/B> [<t>x]w dx = [0]„ , (2.11)J JXLEU/B)

where [(t>]w = 0x(x, 0+, z) - 4>x(x, 0-, z) for |z | < B cos /?, CLE < x - z tan 0 < CTE .Note that the total lift L of the wing is obtained by spanwise integration of the circulationdistribution,

1*1* fBcosf}L = PMl,i // [0X] dx dz = paLPbl,i / r(£) dz. (2.12)

JJW J -Bcos/3

The problem for the potential 0 is made complete by appending to the differentialequation (2.3) the shock jump conditions which must hold across any shock waves whichoccur. Since (2.3) is a perturbation version of the continuity equation (the mass flux vectorcan be shown to have the expansion pq = p„t/{i(l + 5*/3(K<px - (7 + l)0|/2) + • • •) +5qr} + • • •) it can be written as a divergence,

^ = 0, (2-13>

where V = (8/By, d/dz) = gradient in a transverse plane, qr = (<t>y , fe) = velocityperturbation in a transverse plane. The integrated form of this divergence expression musthold across shock waves. If the shock surface is given by

S(x, y, z) = x - g(y, z) = 0, (2.14)

the local normal to the shock surface n is given by

n-|V5| = i - 81/3Vg in {x, y, z) variables. (2.15)

The integrated form of (2.13) is

+ 1K<px - 2: -[qT]-Vg = 0, (2.16)

where [ ]„ = ((downstream value) - (upstream value)) across the shock. Since there is nojump in tangential component of velocity across the shock it follows that [qr]s X Vg = 0,or [qr] is in the direction of Vg. Also

[0*]8Vg + [qT], = 0, (2.17a)or alternatively

[<t>], = 0, (2.17b)

182 L. PAMELA COOK

guarantees no jump in tangential velocity component. (2.17) provides a differential ex-pression for the shock geometry and a shock polar can be found from (2.16),

K I 7 + 1 / 2— V* [0i]s + [V</>]s2 = 0 (shock polar). (2.18)

The differential equations, boundary conditions, and shock relations define a problemwith a (presumably) unique solution. In order to obtain lifting-line theory we study thedependence of the solution of the problem on the aspect ratio parameter B, which becomeslarge.

The formulation of the restricted lifting surface problem and the treatment of thelifting line as an approximation for B -> c° follow closely the ideas of Van Dyke for theanalogous problem in incompressible flow.

3. Lifting-line expansions. In order to study the dependence of our first-order tran-sonic disturbance potential <j)(x, y, z; K, A, B) on the aspect ratio B » 1, we consider thedistinguished limit of the equation (2.3) under the boundary conditions (2.4), (2.6), (2.7),(2.8), as B -> oo. There are two basic limiting processes. In the outer limit x* = x/B, y* =y/B, z* = z/B are held fixed as B -> oo, and the planform shrinks to a line (of singularities).In the inner limit a = x — z tan /3, y, z* = z/B are held fixed as B —> <*>, and so theboundary value problem for 0 becomes essentially two-dimensional.

In the following section we summarize the form of the inner and outer expansions.Correctness of the expansions is shown by matching them, and the far fields of the innerexpansion problems are found by the matching. Fig. 3.1 shows the wing geometry in the a,y, z coordinates.

B cos/3

a = CteXz/B) (K*-(y+ l)0(j)0(T(T + 0jj + 0«+2tan/30ff; = O

Fig. 3.1. Boundary value problem in a, y, : coordinates


Inner expansion (a = x - z tan /J, y, z* = z/B fixed). The inner expansion has the form

4>(x, y, z; B) = y\ z*) + y; z*) + ^4>2{a, y, z*) + ••• (3.1)

where the equations governing the 0, are

{K* - (7 + 1 )<t>0a)<t>Q(T(, + <t>oyy = 0, (3.2a)

(iK* - (t + l)4>0a)<l>l<ra - (7 + l)<t>0aa<t>u + yy = 0, (3.2b)(K* - (y + ^)<P0a)<t>2aa ~ (T + + fa™ = 2 tan (3 </>0(TZ* , (3.2c)

where K* = K + tan2/?. The tangent flow boundary conditions from (2.7) are:

0O;(*, 0±; z*) = ^ (a, z*)- A, (3.3a)

0±; z*) = 0, 02j(ff, 0±; z*) = 0, (3.3b,c)

on -cos 13 < z* < cos /J, CLE(z*) < <j < CTe(z*)- The Kutta-Joukowsky condition must besatisfied for each term of the inner expansion:

[4>Qa]TE = 0, [0j CT] tb = o, [</>2 ^te — 0, (3.4a,b,c)

where the trailing edge is given by -cos /3 < z* < cos /3, a = CT£(z*). The conditions atinfinity are obtained by matching to the near field of the outer expansion.

Note that the problem for <fi0 has the form of the usual two-dimensional transonicproblem for the flow past a lifting airfoil. At this point the influence of sweep angle is notfelt. The correction term , which is <9(log B/B), enters solely because of a nonzerosweep angle and is needed for matching with the outer solution. Corrections for the finiteaspect ratio appear in both 4>! and $2 although more strongly in 02 since some of the 02correction remains even in the case that the sweep angle goes to zero.

Outer expansion (o* = <r/B = (x — z tan /3)/B, y* = y/B, z* = z/B = z/b fixed asB -♦ oo). in this limit all lengths are measured relative to the span. The expansion has theform

<t>(x, y, £) = Mx*, y*, z*) + ^-^(jc*, y*, z*) + ^v>2(x*, y*, z*) + • • • (3.5)

where the equations satisfied by the <pt are:

^*^0,1 a* a* + ^0,1 y*y* + ^0,lz»z* ~ 2 tan = (3.5a)

+ <p2y*y* + v2z*z* ~ 2 tan ^ ^2a*z* = (T + 1 Pq' (3.5b)

As in the inner expansion the 0(log B/B) term is introduced for matching. However, inthe outer expansion the 0(log B/B) terms remain even in the case that the sweep anglegoes to zero. Note that, if written in x*, y*, z* variables, Eqs. (3.5, 3.5a, 3.5b) are preciselythe same as for the unswept wing. The nonlinearity appears in the 0(\/B) term and thedependence of the outer expansion on sweep angle appears from boundary conditions andmatching. For a fixed a, a* -» 0 as B -> <», hence the image of the wing collapses to a line.Its effect on the outer flow can be represented in the x*, y*, z* coordinates as a skewed linevortex shedding its vorticity downstream and higher order singularities. The boundaryconditions on the <p, are

184 L. PAMELA COOK

*P\(J* r*->co 0, (3.6)

WioU = 0. (3.7)The remaining conditions are obtained by matching with the far field of the innerexpansion. This matching determines the type of singularities needed in the <pt . We findthat:

_ J_ r y* T(j) (. , ** - S sin 0 \^ 4tt y*2 + (z* - scos0)2 V j(x* - s Sin ay + Kv*2 + K(z* - ~s cos fi)2' *J*2 + (z* - j cos /?)2 \ ^x* - j Sin ̂ + Ky*2 + AT(z* - 7 cos /?)2

7* 7(g)>>*2 + (z* — j cos /3)2= - r47r

(l + »• + (?•-.. cos fl) tan fl V (J g)^ jK*(z* - s cos 0)2 + 2c*(z* - ^ cos 0) tan 0 + Ky*2 + <x*2)'

which is the superposition of elementary horseshoe vortices distributed along x* = z*tan 0, —1 < z*/cos 0 < 1, and trailing off parallel to the a* axis,

= °1JL P ®i(g) d. n9.4tr K* [AT*(z* - J cos 0)2 + 2<r*(z* - j cos 0)tan 0 + + a*2]3'2*"

which is the potential in a*, y*, z* coordinates of a distribution of divortices along thelifting line, and

_ ~ . S2(j) ^ [A"*(z* - j cos /?)2 + 2o-*(z* - 5 cos 0)tan 0 + A>*2 + <x*2]3/2

= £!. r4tt A-* i.,

r4tt A"*

, ^ -- i Sz(^) i' ' [K*(z* - 5 cos 0)2 + 2<r*(z* - j cos 0)tan 0 + Ky*2 + <x*2]3/2

+ zL f1 iM.47r J-t _y* + (z* — i- cos /3)2

•fl+ -»costf)ung =U + rf (3.10)^ ^jK*(z* - s cos 0)2 + 2<r*(z* - s cos 0)tan0 + Ay2 + <j*2'

where <pp2 is a particular solution of (3.5b) whose behavior is specified as (a*, y*) -> 0,cT*/y* fixed.

Expansions of the integrals in (3.8, 3.9, 3.10) are worked out in general in Appendix A.In particular it is found that

= " (72xcosS/} ) + (2^8 V ^ ~ J°{2*}y* + W ln (3"''>a* 2Di(z*/cos 0) 7i (cos 20+1)

<Pi= , ~ ' +^-0 +tanff . —r— J-+ 0(y*\nr*), (3.12)2TjK*r*2 cos 0 2t 4ir(K*)3/2 cos 0 v ' v '

= g* £)2(z*/cos fl) ,y* S2(z*/cos ft)*'2 2iryjK* r*2 cos 0 2irjK* r*2 cos 0

+ - + j^} + I",'). (3-13)


where

J, - + J-L-J r yM™Mds + Jt!.x4r cos /3 J-coe/) z ~s

Jc0(z*) is given in Appendix A (A.9), ( )' means d/dz*( ), 6 = arctan^A^Vo-*),r* = (A:V2 + <r*2)1/2 and

A = X±J-()'{!2SZ! cos » - 5213"4A"* \2ttcos /? / I r* 4r*

n y + 1 (( To VV/ + cos (26 + 1), „ , 02 , cos 20 1+ tan " / J 1 2 108 ' + 2 + ~8 8 cos 4#

Matching. The boundary value problem described by the nonlinear equation (3.2a)and its associated boundary conditions (3.3a, 3.4a) has the form of the boundary valueproblem for two-dimensional transonic flow past a lifting airfoil. In particular the far fieldis given by [5]

T06 , (T+ 1) (T0\log/- „ , l/ D0 „ , E0 . „00 - - -X— + \^r- J cos 9 + -) cos 6 + r —r sin 62ir 4K* \2tt / r r \2ir^K* 2xv'A*

"T6iXFii(fe)*COS3,,}+0(1T)aS'--"' <3I4>where r = (cr2 + K*y*2)1/2, 6 = tan-1 (^K*y/a), and r^z*) is the circulation at the spanwisestation z*; DJ^z*) and E0(z*) are the doublet strengths at the station z*.

The behavior of 0!, 02 as r -> co is one of the main results of matching. In particular weexpect that

0, = Alz*)a + Blz*)y + sm26 - ^ 6 + ■■■ ; (3.15)

that is, a uniform flow, then a circulation term. The term with sin 26 arises naturally in Eq.(3.2b). We also find that

7^^>1|og' + 8»''+ *•*' 17 +

+ tan ((£ )')' )('°^(cos2»+l) + P + co|2» _ co|4»)

D'o cos 26 + 1 , £; sin 2<9 \ , 1 , \2tt(K*)3/2 2 271- (AT*)3/2 2 / \ /■ g r) ' ( }

•as r -> as.The term 0(y log /■) arises from the forcing part of the equation, 0O(7.* . In the

matching this term combines with the B^y term in 0t. The A2, B2 terms represent apossible uniform flow, r2 a circulation, and the 0( 1) terms multiplied by tan /3 arise fromhigher-order forcing terms in 0O(7Z* as well as from 0Off0^ and 4>2a4>Qa„ terms.

Matching is carried out in each cross-section plane z* fixed, \z*\ < cos 0, with the helpof an intermediate limit. In a class of limits intermediate to the inner and outer limits,

r (<r2 + K*?Y» . n , „" m" —m—,s fixed as B -* •

186 L. PAMELA COOK

where 1 « X « B. Thus in the intermediate limit

r = \{B)rx -.oo, r* = L = ^^-rx - 0.

The calculations are simplified if the far-field of the inner expansion is merely written interms of outer ( )* coordinates and a direct comparison of the inner (r* —> oo) and outerexpansion (r* -> 0) is made.

Now, writing r = Br* in (3.14), (3.15) and (3.16), the expansions take the form:

Inner: 0 = —^ 0 + '°S r* + + A2<j*

+!Yfe1 (£ W-- ̂ ■♦»»4- cos 0 , £° sin e _ iv 7 + i , r; .

2irjK* r* 2*JK* r* 2x 4 2 2tt Sm

+ ^>/7 7 + 1 W T0 VY/'log r* (cos 20 + 1) , 62 , cos 20 cos 40 \lan nUiHUsJ) \ 2 + y + -s HD'o cos 20 + 1 Eg sin 20

27t(A-*)3/2 2 2ir(K*)3/2 2 J) '

where we chose /?, = (tan 0/^K*)(T'o/2tt) in order for the expansion to remain valid. Alsowe chose = 0; the justification for that will be that the expansions match.

Outer <p = -(- 7° )fl + tan// 7o ) -j^y* In r* - J0y*\2ircos/3/ \27rcos/3/ JK*

log BS 33, cos 0 _/ _JD[_tanJ3_( .B I 2irjK* cos 0 r* \2ircos0' 4tt(K*)s/1 cos 0

+ 1ll y + 1 Y 7° Vigil! cn.n _ !_±j/ To V cos 30B l\ 4K* A 27r cos 0 J r* 16A"*V 2x cos/3/ r*

3l>2 cos 0 S2 sin 0 _ y2 .2x cos 0 r* 2ir JK* cos /? r* 2ir cos 0

I ♦ -,„</7+ 1f( 7o VVf (cos 20 + 1) log02 cos 20 cos40\+ tan ̂ ^ j \ j 2~ ~~8 T~/

■)}■£>; (cos 26> + 1) 6^ sin 20 ..4,r /-JffWZ 4^ e^*\3/2/f (J.18J4tt (AT*)'" 4x (K*):

Matching is accomplished to all orders shown if

7o(z*/cos 0) = JD^ = 7 + 1 (r0(z*))2 S2cos 0 cos 0 4 JK* 2tt 'cos/? °'

"•" sSy" ^A'' "•


The essential matching for the completion of the boundary value problems for0i , 02defines B^z*) and B2(z*) in terms of the first inner circulation r0(z*). For /3 = 0, Bl = 0,and B2 agrees with the induced downwash of the trailing vortex sheet as in the classicallifting-line theory.

4. Boundary value problems for 0O , 0i , 02 • As a result of the asymptotic match-ing of the previous section, complete boundary value problems can be formulated for thefirst-, second- and third-order inner potentials <t>0 ,<t>i , 02 • The problem for 0O is the two-dimensional flow past an airfoil at the same shape and angle of attack as the actual wing ata given z* station. The 0(log B/B) correction, 0, , corresponds to a perturbed two-dimensional flow past a flat plate with induced downwash at infinity, due to the trailingvortex sheet. The 0(1/B) correction, 02 , is a new type of term. It corresponds to the flowpast a flat plate with an induced 0(y In y)-type behavior at infinity. There are alsospecial shock conditions to be considered for each of 0i , 02 •

For 0O we have (see (3.2a), (3.3a), (3.4a), (3.14))

{{K* - (7 + 1 ))<p0(T)4>0oo + 4>oyy = 0, (4.1 )with the following boundary conditions:

cp0a 0 at infinity, (4.2)" r)F

<Mff,0±,z*)= —*±(a,z*)-A, (4.3)CO

the Kutta-Joukowsky condition

[^cÎte = 0, (4.4)and the shock conditions which (cf. (B.8, B.9)) are integral forms of the conservation formof (4.1):

K(t>cr _ 7 t 1 <t>la + [<t>oyf = 0, [0O] = 0, (4.5)

on the first-order shock locus

The shock geometry is such thatSo:* = s„(f). (4.6)

goiy) = - [0oj]/[0.off]- (4.7)

These shock wave jump conditions must apply locally across any shock waves that appearin supersonic zones of the solution. The shock locus g0(y) is not known in advance andmust be found as part of the solution. Fig. 4.1 shows the boundary value problem for </>„ •

For a given airfoil shape the boundary value problem for 0O cannot, in general, besolved analytically. Instead, one of the standard computational schemes for resolvingshock waves is used ([2, 9]). Alternatively one could consider shock-free transonic flowsand the corresponding airfoils generated from the hodograph solutions, for exampleGarabedian [8],

The correction flow 0! satisfies a boundary value problem which is linear, assuming0o , go are known. The equation is (3.2b)

(K* ~ (y + l)0Off)0iffff ~ (7 + lîffÔffff + 4>lyy ~ 0 (4.8)

L. PAMELA COOK

00(7 * 0

<t>oy= $pL -A[<t>oa\ = 0

a

Cte(z*)Cle(z*)

<t>oy = (dFu/da) - A (K* - (7 + 1 )0o<r)0o<7<r + <t>0yy = 0

Fig. 4.1. Boundary value problem for 0O •

or in conservation form~ (y + • + 0iyy = 0.

The boundary condition on the slit a* = 0, CLE(z*) < a < CTE(z*) is that of a flat plate

<M<r, 0±,z») = 0. (4.9)However, there is a downwash at infinity toward the flat plate, as determined by matching(3.17), (3.18). We have

7<4'10)

as r -> 00, where r^z*), the induced circulation, is to be found. The boundary conditionson the trailing edge and wake are identical to those of </>0:

[<Ai<t]te = 0 (Kutta-Joukowsky condition), (4.11)

[01 Iwake = COnSt. = r,(z»). (4.12)

The shock conditions for the linear equation require special treatment. Details aregiven in Appendix B. The results are

K4>oo <Ao(~ ' ^ J [01 a] + [0oa][^*0iff ~ (J + 1)0o<t0i<t] + 2[0Oy][0ly]

= [0<b][0<W _ ^~2 0?ff] + [0oJ^*0oTO + (7 + 1 )0q<T0Offff] + 2[<t>0ay]j . ^

[0i] = — ̂i[0off] , (4.14)

where all jumps are applied on the zeroth-order locus a = £o(v)- The shock locus has beenrepresented as

* = go(y) + 2*) + ^ g2(y; z*)+ . (4.15)


An overall conservation theorem has also been derived to act as a supplementary shockcondition:

Is { [<m} dy = 0 (4.16)where w* = -K* + (y + 1 )0O(T . This condition guarantees that there are no source termsat infinity in the solution to the boundary value problem for^ . The equation to be solved(4.8) is a mixed-type equation, but it is linear and the regions of ellipticity or hyperbolicityare known in advance from the 0O solution. Fig. 4.2 illustrates the boundary valueproblem for <pi .

The second correction to the two-dimensional flow, 02 , satisfies a linear equationwhich is forced by the zeroth-order flow. The equation for 02 is

(K* - (y + l>£oa)<t>2oo ~ (7 + 1)00^2(7 + 02yy = 2 tan 0 0O(TZ* , (4.17)

where 0O is assumed known. The boundary condition on the slit y - 0, CLE(z*) < a <CTe(z*) is that of a flat plate,

<M<r, 0 ±; 2*) = 0. (4.18)The condition at infinity, which is determined by matching, is:

r'02 ~ tan 0 ' y log r - J0{z*)y

cos 26 cos 4(A+ -x- +4K /\\ 2x/ / \ 2 2 8 8 /

+ Mf^(^2.+ l)+s^sin2«}-ap» + o(l^), (4.19)where r2(z*), the induced circulation, is to be found, and

* +i C ^^ ~ SJF ( + r;(l + W«'(COS-<i -

./ri(j) - ri(z*),

a—1-« _ n—ds-COSP I L J I

2tt \\ ' W 'r (4.20)

The conditions on the trailing edge and wake are identical to those for 00,01 , namely:

[02(7]te = o , (4.21)

[02 ]vs = r2(z»). (4.22)

The shock conditions for the forced equation are worked out in Appendix B. Theresults are:

K*<t>oa ~ <t>la [02ff] + [0O„][**02„ ~ (7 + l)0Oa(A2ff] + 2[4>0y][<t>2y]

= _^2Cf){[A_*0O<t(J _ (y + 1 )0Oo0O(T(t][0O(t] + [0Offff] - 1JYL 0O_

+ 2[0oy][0O(ry] + 2 tan/3 [0O(T2.][0OJ} , (4.23)

190 L. PAMELA COOK

*'~-,y2^)an'i'-Yu + ••• !(t = ^o(j) (Jump conditions

given)

01 y — 0 —/ ' ) [0i<r] 0

[0.] = r,

c"(z*»Cte(z* )

(A:* - (7 + 1 )0o<t)0i<tct - (7 + ' )4>ia<t>oao + <t>iyy ~ 0

Fig. 4.2. Boundary value problem for <p, .

[0z] = -g2[0off] , (4.24)

where again all jumps are applied on the zeroth-order locus a = g0(y)- As for 0,, an overallConservation theorem has been derived for <p2 to act as a supplementary shock condition,

f { K02ff + 2 tan 0 0O,*] - ^4 [**<M dy = °- <4-25)s0 l0offJ

This guarantees that there are no source terms at infinity in the solution 02 . Fig. 4.3illustrates the boundary value problem for 02 •

Note that as in the unswept case, if the flow about the airfoil represented by (j>0 isshock-free, then so is that represented by 0, , 02. The uniqueness of the solution for the

To fll , - IV2^K* y^^ogr-J0y- + 0(\)

02y = 0

a = g0(y) (Jump conditionsgiven)

[02 a] = 0

— cr

Cle(z*Y[02] — r2

cte(z*)

(K* - (7 + 1 )0ocr)02cT(7 - (7 + l)0offff02a + <t>iyy = 2 tan /3 <t>oa:*


shock-free case, when $0 and hence , cf>2 are shock-free, has been proved [5], That is, if<t>°.1,2, 0i,2. are two solutions of the problem respectively, then

— = 0*1,2 ~~ 01,2

satisfies

(A"* - (7 + l)<j>oa) H a a ~ (T + 1)0o<t<x S a + Hyy = 0,

Ay|j7_o = 0, Cle(x*) < cr < Crf;(z*),

[HJt* = 0, H = - (I\/2ir)0 + 0(1//-) as r CO,as in [4],

For computational ease, since the 02 problem is linear, it can be broken down into twoproblems. We have 02 = 02(1) + <f>22\ where 0i" satisfies:

(.K* - (7 + l)0O(r)0(2™ ~ (7 + 1 )<t>ooo<t>2o + 0$y = 0, (4.26)

0$ 1^=0 = 0, CUz*) <CT< Cr^z*), (4.27)

{4>2o]te = 0, (4.28)

= ~ 2^ f 1*^1^ ~ 6 + °{X/r) as^°°' (4"29)and across the zeroth-order shock

[**<£„, - (^- - (7 + 1)00,02',] + 2[0oy][02p]

~ |[^*0O(T<r — (7 1 )0o<T0ocrff][0O(r]

**0o„ — 2 0Off t0Off(r] "I" 2[0oj?][0o<tj7]J* > (4.30)

[04"] = SWM, (4.31)where we also have g2(y) = g£l) 4- g^2\ and 4>\2) satisfies

{K* - (7 + l)0off)0(iToo - (7 + l)0oa02a + 0$y- = tan /? 0O(rZ*, (4-32)

02^-0 = 0, CL£(z*) < a < CTe{z*), (4.33)

[02cr]r£ = 0, (4.34)

_2g+i) , e20i2> = tan 0 £/log r + 7S(z*)/ + tan /?{(I^1)(( g:)2) ( log ^ (co« - ' - + ^

+ co|20 _ co^40 (cos 2d+l)+ 4^°)3/2 sin 2d

-Vqf>8+°(^) asr^oo, (4.35)

192 L. PAMELA COOK

where

JKZ.) = - (l +"■<«»» -=")«"!)

~\{f + /" )r:(.s) ln|r* - J di)" - cos8 J cos/? '- cos/3 J cos/3 y

" "S" (0+ ^)n # ■in(^* ~tan 0)) ■The shock conditions for fa2' are (4.23) and (4.24), with fa replaced by fa2\ g2 by gi2).

That is, the fa problem separates into two pieces. The first piece, 0i", g2l), is preciselythe correction that arises if there is no sweep. If sweep occurs, a second term fa2) mustbe superimposed on fa21] to correct for the sweep angle. The induced circulation then is(r.log B/B) + (IT + YP)(\/B).

Numerical results have been found for the unswept case by Small [10, 11]. Thusessentially $0, fa" are known. Computation of 0! will present no new difficulties as is alsoprobably true of fa2).

There are singularities in the small-disturbance approximations both at the tips of theairfoil and at the foot of the shock. We have not dealt with those specifically.

5. Similar sections. The planform with similar cross-sections is especially amenableto lifting line analysis since, by suitable scaling, the problems for fa , <j>i, 02" becomeindependent of z*.

Consider a wing surface given by (2.1) such that the planform has a chord c(z*). If weassume that

^(a,z*)-A = GU°/c{z*)),OG

and if we scale 0O, a, y by the chord so that

fa = c{z*)U2, Y), 2 = g ~ *Cl* ~ Cte) , Y = y/c(z*),

then the problem (4.1), (4.2), (4.3), (4.4) for fa becomes

(K* ~ (7 + 1 WosWoiz + ôvv = 0,

<Aoy|y = o = Gii,((2), 1/2 < 2 < 1/2,

IM] 2 -1/2 = 0.y-o

This problem has no explicit dependence on z*. The circulation r0(z*) for fa is T0(z*) =C(Z*)[ô]z-l/2 ■

y-oA similar scaling works for fa , fal). For fa let fa = c'(z*)c(z*)\p0, 2, Y as before; the

scaled fa problem from (4-8), (4-9), (4-10), (4-11), can be written

((A"* — (7 + 1 )*Ao2 )Âi2: )j; + V'lyy = 0,

&y|y-o = 0, —1/2 < 2 < 1/2,


[V'is] 2-1/2 — 0,y-o

lAi Y tan /3 2^* + ''' as/*->°°.

Thus, the 4>i problem is z*-independent. The circulation Ti(z*) for 0! is I\(z*) =)c'(^*)[Âi]2_x/2- F°r <t>2. let <t>zl) = d(z*)c(z*)\p2n\ \p0, 2, Y as before, where

d(z*) =

Then the i/4" problem can be written, from (4.26), (4.27), (4.28), (4.29), as

(k* - (y + wmmh + mr = o,#V|r-o = 0, -1/2 < 2 < 1/2,

û) Z[ôj2_i/2 asrôo, [^2Xi] s-i/2 = 0.L y=o v-0

Thus the t/41' problem is z*-independent.Unfortunately the same process does not seem applicable to <^21 which is described by

(4.32), (4.33), (4.34), (4.35).

Appendix A: The near field of a skewed lifting line and singular behavior of vortex-line integrals. The solution of

ox*x* tPoy*y* tPoz*z* 0*

on IR3 - |(.x*, z*)|x* - z* tan /3 > 0, | z* \ < cos/3}, the equation satisfied by the first termin the outer expansion, <po , for which

Vox« —» 0,X* —»± CD

[fox*] = 0 across {(x*, z*)| x* — z* tan /? > 0, \z* \ < cos /?}, consists of a superposition ofelementary horseshoe vortices distributed along x* = z* tan (3, - cos /? < z* < cos (3, andtrailing off parallel to the x axis. So,

<P 0 = ±r4t J _!

y* y(s)y*2 + ^z* _ ^ cos

ds, (A. 1)-liinl ) ds((x* — s cos (3)2 + Ky*2 + K{z* — s cos /3)2)172/

where 7(5) is the distributed vorticity. At the wing tips y is zero, t(±1) = 0.Since we are interested in the behavior of <p0 as we approach the lifting line, let a*

measure the distance from the lifting line,

a* = x* - z* tan /3,

and change variables in (A.l) so that

t = z* - s cos /3.

194 L. PAMELA COOK

Then, we are interested in the behavior, as a*, y* -> 0, a*/y* fixed, of

1 j ro'P+z* (z*-t\ y* (. a* + t tan/? \IJ0 7\ cos B J v*2+ t2'\ (K*t2 + 2ct*t tan ,8 + Ky*2 + a*2)1" JVo ~ 4ttcos (3 (J0 ' \ cos /3 / y*2 + t2 \ (K*t2 + 2c* t tan 0 + Ky*2 + <r*2)1/2/

j*coB0-2* (z*-r\ y* (t , <r* -7 tan 0 ,A ^

J„ 7\ cos/? / >>* + r2 ' V(K*t2 - 2c*t tan/? + Ky*2 + <r*2)1/2/ T) ' 1 '

where K* = K + tan2/?, and the change of variables r -» — r was made in the last integral.Changing variables once more in order to isolate the small parts of <p0, let p = r/y*, fory* > 0; then

1<Po -

dr

47rcos /?[ h1{y*p)fl(p)dp + f h2(y*p)f2(p)dp\ (A.3)

J o J 0 '

where hi(y*p) = 7((z* T ,F*p)/cos /?)//(cos 0 ± z* — y*p), H is the Heaviside unitfunction2,

f( . 1 / >4 ±Cp \^ 1 + p2 \ (,K*p2 ± Dp + E)1/2J '

and A = a*/y*, C = tan /?,£> = (2a*/y*) tan /?, E = K + (a*2/y*2) are all fixed in the limitthat a*, y* -> 0.

Now (A.3) has the form for which the asymptotic expansion as/* -» 0 can be foundusing Mellin transforms [1], That is, if M denotes the Mellin transform, then since

Wp)T y''y{-£j)+o^''n

I)+ 0(1/»*)'„ - „ p2 \ sjEJ

where ( )' means d( )/dz*, we have that

r(co»0±z*)/y ( z* T v*o\wi-/. •>'is analytic for Re s > 0, and its analytic continuation to Re ^ > —2 is analytic with theexception of poles at the nonpositive integers, and

^ L 1 + p2 (' + (K*p2 ± Dp + EY'2) dp(K*p2 ± Dp + Ef

is analytic for 0 < Re j < 2, and its analytic continuation for Re s < 3 is analytic with theexception of a pole at s = 2. Also,

M[hj\ 1-5] M[jf j] -» 0 as |lm j| -►

Thus,

4x cos.2

4x cos

Vo = / hj(y*p)fj(p)dpOS p j= i J o

J—r i 2iri r-Mfa 1 -s] M[fj; s] ds'Ua fJ j = \ J


for 0 < v < 1,

= ~ d rnc ft E Residue(M[/iy; 1 - + 0(y*2 In/") (A.4)HIT COS p j.i s-1,2

as y* —> 0. The proof can be found in Bleistein and Handelsman [1],To find the explicit terms in the expansion note that

M[fh s] = jg {j + p2 (' + (K*p2 ± Dp + Ey/i) ~ Ps 3( 1 ± dp

+ j" p-3( 1 ± ^Hip- 1 )dp, s

so that

M[f/, s] = bj + 0(s - \) as s-> 1,(A.5)

M[f/,s] = ~(\ ±-j^)y^ +dj+0(s-2) as 5-2,

where

i = r 1 (. , A ±CP \' J, I + p2 V (K*p2 ± Dp + E)l/2J p '

dj = /„ 1 + p2 0 + (K*p2 ± Dp + £)1/2) ~ pO JK*) H^ ~ X">dp~

Similarly,

dpcos 0 / \ cos /3 / \ cos /?/

z* \/cos± z*V~s 1 _ ^ ,/ z* V cos /3 ± z*\ 2~s 1cos/? A >>* / 1 — j ^ \cos 0/ \ _y* / 2 — ̂ '

so that

mA • i — d = —^,1.cos fi/s-l (A6)

M[hj\ 1 - s]= ±y*y' (^j) jzrj + ei + _ 2) as 5 - 2<

«|A/i 1 - s] - ~y{^n)jzr; + 0(1) as i.

where

196 L. PAMELA COOK

So, from (A.4), (A.5), (A.6),

<Po ~ W cos ft § { bjy (cos ft) ± (cos ft ) (' ±47

+ 0(y*2 In j*) as y* -» 0. (A.7)

Integrating by parts once in , using the fact that7(± 1) = 0, and combining terms, we get

+ >-ru* + coB0)/y* r \ COS|(J // 7 dp

" (2* —C.Ofift)/V* P

(•cos/3 ^

/*/ -cos/3

(cosft) 7 (cosft) ^ z* \\ , U^Tl *+ 'CiT w //+ 00,

where

c0 - bt + b2 , ci — di d%

These last two constants, c0 , cx , can be calculated explicitly to give

Co = -2 tan~l(yjK* y*/v*).

Ci = In

So finally,

tan 0-JK* tan/s/ Z «r« \ 4K«2tan ft + JK* JK* llnl* + 7») ~ ~Tl

7 (cos ft ) tan ft 7 (cos ft) + ^ _ >'* fco8^ 7 (cos ft )27rcosft 2tJK* cos ft ^ r 4x cos ft i _C08g z* - j

9 Cta" ^ R {v (-^V + 'n(cos2ft - z*2)1/2)2xJ/^*cosft I \cosft/v '

, (_J_) _ Y z* \/•cos/3 T \cos £ / 7 \ COS ft / , , , / 2* \ .

rfs + r w?)1

V cos ft / , / JA:* - tan ft V/2 , *9, * ̂2,cosff '"Ijlf + tanJ +O0"ln^X


or more simply,

— y( Z* \ y'( Z* )Vcos 0 / tan (S Vcos 0 J * .Vo = a— + -772 a y* In r2ir cos 0 2ttJK* cos /?

I S \r co»fl y \ o )_cosp_ds_

4tt COS/3J-C08/3 z s

,!*

where

•/S(Z*) ° 2,J*"1+ l"(«ncos-(i - *•■))»)

1 / rantl r-co.H\ /"' (C0S ) *! ( cosrf) V!. ,+ 2 U. + I JI -F7T, 7> <*'

,( Z* \

+ ST^ff" {-''W - ,an "> + (' + (A.9)

The other integrals in (3.9), (3.10) have the form <r*/47r, or >>*/47r times the form

* - f. -»W ds.-i (a*2 + Ky*2 + K*(z* - j cos/?)2 + 2(t*(z* - j cos/3)tan/3)3/2

This integral can be treated in precisely the same way as tp0 • With z* — s cos /3 = r,

i = 1( iIzjl)

/ COS/?/ \

cos/3 Wz._co9(s (ff*2 + A>*2 + A>2 + 2<t*7 tan /3)3/2 V '

With p = r/>>*; £>, £ as before,

( z* -y*p\ J z* +>"*-2 f r<i,+co^>/>" COS/3 / ncosp-z*),y* «\ cos 13 J \

~ cos /3 \ J o (K*p2 + Dp + E)3/2 p J0 (K*P2 - Dp + E)"2 pj

_ ^"2 (if0 3Cj(y*p)(fj(p)dp) ,x y ̂ i ^ n 'cos 0 \ /rt J o

where

W*> = ̂ ^)«co.0 **•-,>), W = ± ^ + Er

198 L. PAMELA COOK

Then / ♦ \y.~0. ̂ + ' w + °<''V)'

so that

/<cos/3 ±z»)/y» / z* T dp

is analytic for Re .y > 0 and its analytic continuation to the negative half plane is analyticexcept for poles at the negative integers,

M[5j\s] = /"J n

"Sdp„ (K V ± Z)P + E)

is analytic for -0 < Res < 3 and its analytic continuation to the whole plane has poles atthe integers, and

M[Kj\ I - s] M[$j\ s] >■ 0.

t f Xj(y*p)Zj(p)dp7 = 1 0

2 fv+i<x> ^

n v T 2tt/7 1 - s]M[$j\ s] dsCOS p \ fr1 J '

Im s

So,y* 2 / 2

= cos ft

.,*-2 / 2

for 0 < f < 1,,,♦-2 2

as >>* --> 0.

Since

then

and

-j—- T Residue M[K,\ 1 - jJA/pF,; j] + 0{y * In y*) (A.10)cos ft s = li2

/*(cos/i+z* >/.v* /•»*_!_ l'*n\

M[3C,, 1 - s] = Jo cos'/J J ^

M[3Cy; I - s] = + 0(1) as j-»l, (A.I

A/[3C,; 1-5] = ±-vV(;*C2°S^) + 0(1) as ,-^2. (A.12)

So, from (A.10), (A. 11), (A.12),

* = K :os /J T*5 S(z*/cosft) + —— tan ft ^ g'(z*/cos ft) + 0(ln r*).

So, (3.9) is

b = *! Pi(zVcosfl) <r*2 tan/3 Pj1 2*r*^K* cos/3 27rr*2 K*3/2 cos 0 ^ } ' (' J)


as y* -» 0, and (3.10) is

= <r* 3D2(z*/cos 0) + y* S2(z*/cos /?)2 2 ̂ K*r*2 cos 0 2^K*r*2 cos 0

_ i . g* tan 0 Sa ^*2 tan %2tt cos ,3 01 2nr*2(K*)3/2 cos0 27rr»2(A:*)3/2 cos/J

+ 0(y* In/*) (A.14)

as >>* -♦ 0.

Appendix B: Shock relations. The shock conditions for <p(x, y, z; B) are the con-servation form of the full equation (2.3),

[*0X - ^^±1 ti~\a[4>x}e + [$$ + [fa]* = 0 , (B. 1)

and the condition of no tangential jump across the shock,

[<!>], = 0 , (B.2)where the shock locus is given by

x = G(y, z; B) .

In the new coordinates a = x — z tan /?, y, z, these conditions become

\k*4>„ ~y + 1 Ms + [4>y]2. + Ml - 2 tan /?[</>£]8[<M« = 0 , (B.3)

[<t>]s = 0 , (B.4)where the shock locus is given by

o" = g(y. B) = -z tan /3 + G(y, z; B) . (B.5)

In inner variables a, y, z* = z/B, the expansions of 0, and of the shock location 5, are

4> = + y, z*) + (B.6)

* = go(y, Z*) + Z*) + 1 g2(y, Z*)+ ... . (B.7)

For any function f(x, y, z; B) with an expansion of the form (B.6),

[/]» = f((go + gi + jft + ■■■) ,9,*,b)

- /((ft + ft + ^ ft + • ■ •) . y, z; b)

So, expanding / in terms of the /, for large B, and expanding the /, in their right- and left-hand Taylor series about g0 , we obtain

[/]« = [/ok + iglUoaU + LAU + J" ift[/off]s„ + [/2]s„} + * • ' ■

200 L. PAMELA COOK

Substituting the appropriate expansions of the form (B.6), (B.7) into the shock relations(B.3), (B.4) gives

K*<t>0o -) 0o<7 J + ^ ("Y 1 )0o<70i<7]so

"t" ^ [/w*02<7 (T "t" ' )<t>oa(l>2a]s0 ■(" ̂ Si g ft)[^*0o<7<7 (*V 1 )0o<70o<7<7]s

' {[<Ao<7]»0 + '°gg ^ [02<r]«„ + ( Si + ~§Si)[<t>Offffk}

+ [0Oj?]so "I ^ [0O.pk [0lj»k 2 ~ [02>~]so [<Ms0

+ ft + j S*) 2[0„,k [<fioay)So - [0o,.]So [<t>oa]s0 + o(( 2) = 0 ,

and

[0«k + ^ [0,k + j [^]»o + ft + j ft) [<MS„ + o(( 2) = 0.

Collecting terms of the same order gives:

0(1): K*4>o„ ~ 1~- <tia [0oo-]so ' [0oj~]so — o, (B. 8)

[</>ok=0, (B.9)

K*(j)0a 2 0<>a [0i<rk + [K*<t> 1(7 (7 + l>/>oo4>l<rk [0 0(t]s

+ ft [K*0 0(7(7 (7 + 1 )<Po,<t>Ooao][<P«a<t>oo ]s0 + [0 0(7(7]. K*(p 0(7 2 0°ff1.2

°(i): /w*0o<7 2 0°ff

+ 2[0Oy]so[01y]so 2ft[0oy]So [0o<7y]so 0, (B.10)

[0l]»o = -ft[0Off]»„ , (Bll)

[02CT]s„ + [^*02,7 - (7 + 1 )02<t]so [0o<r]8„ + 2[0Oy]«o[02y]«o

g2\ [K*<t>oa„ - (7 + 1)00<70 0(7(7 k [0 00"]5"o [0O(7a]jo

^*00(7 2 0O(tJ + 2[0oay]«o [^O^l^o 2 tan (} [<t>op*]s0 [(frocAso , (B. 12)i"o

[02k = -£2[0o<rU • (B. 13)

The zeroth-order conditions (B.8), (B.9) are the usual two-dimensional shock conditionswhere K* is the adjusted similarity parameter. The 0(log B/B) conditions (B.10), (B. 11)


involve three unknowns, , [$iy]% , gi, and are essentially the same as the first-ordercorrection for the unswept case (which is 0(1/B)). The 0(1/B) correction is the same asthe 0(log B/B) correction except that the extra term — 2tan/3[</>02.]So[</>oa]s„ turns up tocorrect for the sweep angle.

As in the unswept case, there is another shock condition to be checked which arises byapplying Green's Theorem to the equations for <pi, <£2, which are in divergence forms. Theequation, shock conditions, and boundary conditions for <t>i are the same as those for thefirst-order correction in the nonswept case, hence [5]:

0 = ff $ ■ , K*c{>iy) da dy = f (- w*c/)ItT ,K*4>iy) • h dl

= f \a dy + K*4>ly-da, y = ̂ Ky,<7={j^ ,

where D is the region bounded by the body B, the zeroth-order shock 50, the wake W. Theintegrals over B, W and S0 vanish from the boundary conditions. Since dy/da = -[0O(T]So/[<Aoy]So, we have

For <t>2,

0 =

£ ^ M / dy = o. (B. 14)

ff $-((w*(p2a - 2(tan /3)0oz»). K*<t>2y) da dy

= \ K*<t>2y da + [ K*<f)2yda + [ (-w*<t>2a - 2tan0(pOz, , K*4>2y)'ri dljb Jw J s0

+ / ((-w*<t>2a - 2(tan/J)0Oz«> + K*<t>2w) ddSR

where SR = a2 + y2 = R. The integral over the body is zero since <f>2$\ B = 0; the integralover the wake is zero since [</>2y]w = 0. Since

02 ~ Ay In r + Jy + Bd + • • ■— w*(t>2aa + K*<t>!<;y - 2 tan 0

~ + (k* + (y + l)r £ ) + K*Ay In r +

Rx+ JK*y + ^-K*y - 2(tan /3)a r„' + • • •

~ K*Aj> + K*Ay in /> + + - * + f8?"

+ JK*y + Bx-t2K* - 2(tan 0)aTo'd + ■■■,

■ [ ((—w*02(T - 2 tan 0 <j>oz')<r + K*<f>2yy) ddJsR

= O - 2 tan 0 T0'(z*) f R6 cos 6 ddR^„0.

202 L. PAMELA COOK

So we have

f {~w*<t>la - 2 tan 0 <t>02., K*^)-ndl = 0,J «

or

J o+ 2 tan 0 <£0,.] —[K*4>,y] dy = 0. (B15)

L0OO-JReferences

[1] N. Bleistein and R. Handelsman, Asymptotic expansions of integrals, Holt, Rinehart and Winston, NewYork, 1975

[2] J. W. Boerstoel and G. H. Huizing, Transonic shock-free airfoil design by an analytic hodograph method,AIAA paper No. 74-539, presented AIAA 7th Fluid and Plasma Dynamic Conf. 1974

[3] H. K. Cheng and M. M. Hafez, Transonic equivalence rule: a nonlinear problem involving lift, J. Fluid Mech.72, 161-187 (1975)

[4] Julian D. Cole, Modern developments in transonic flow, S.I.A.M. J. Appl. Math. 29, 763-787 (1975)[5] L. Pamela Cook, A uniqueness prooffor a transonic flow problem, Indiana Univ. Math. J. 27, 51-71 (1978)[6] L. Pamela Cook and Julian D. Cole, Lifting line theory for transonic flow, S.I. A.M. J. Appl. Math. 27, 1978[7] William Fredrick Durand, ed., Aerodynamic theory: a general review of progress; Vol. II, Division E,

General aerodynamic theory, perfect fluids, Th. von Karman and J. M. Burgers, 1943, Durand ReprintingCommittee, Calif. Inst, of Technology, pp. 197-201

[8] P. Garabedian and D. G. Korn, Analysis of transonic airfoils. Comm. Pure Appl. Math. 24, 848-851 (1971)[9] E. M. Murman and J. D. Cole, Inviscid drag at transonic speeds: studies in transonic flow HI, UCLA School

of Eng. Rpt. 7603, Dec. 1974; also AIAA paper No. 75-540, presented at AIAA 7th Fluid and PlasmaDynamic Conference, June 1974

[10] R. D. Small, Transonic lifting line theory: numerical procedure for shock-free flows, to appear, AIAA J., June1978

[11] R. D. Small, Calculation of a transonic lifting line theory, Studies in Transonic Flow VI, UCLA School ofEng. and Applied Science Report, April 1978

[12] James K. Thurber, An asymptotic method for determining the lift distribution of a swept-back wing offinitespan. Comm. Pure Applied Math. XVIII, 733-756 (1965)

[13] M. D. Van Dyke, Perturbation methods in fluid mechanics, Parabolic Press, 1975

lifting-line theory for a swept wing at ......the transonic flow about a lifting, high aspect ratio,...

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