QUARTERLY OF APPLIED MATHEMATICS 177 JULY 1979
LIFTING-LINE THEORY FOR A SWEPT WING AT TRANSONIC SPEEDS* 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) are derived. The corrections are found by the use of the method of matched asymptotic expansions on the transonic small disturbance equations. The wing is at a sweep angle of 0((1 — M*)l/2) in the physical plane, hence of 0(1) in the transonic small disturbance plane. As has been noted for subsonic flow, the finite sweep angle necessitates the introduction of terms 0((AR)~l In AR). These terms arise naturally in the matching process. Of particular interest is the derivation of the near field of a skewed lifting-line which is found by Mellin transform techniques. Also of interest is the fact that the influence of the nonzero sweep angle can be completely separated from the unswept solution.
Introduction. Prandtl's classical lifting-line theory gave aspect ratio corrections to the two-dimensional flow in the cross-sections of an unswept finite span wing. The induced downwash at each spanwise station was obtained. The method involved solving an integral equation. Van Dyke [13] systematized Prandtl's approach by recognizing that it could be considered as a singular perturbation. He considered the inviscid incompressible flow about an unswept wing with (aspect ratio)"1 as the small parameter. Inner and outer expansions were constructed and matched. The inner problem consists of two-dimensional flow at each spanwise station and corrections; the outer flow represents the flow past a bound vortex which sheds a vortex sheet and corrections. The advantage of this method is that there are no integral equations to be solved. Cook and Cole [6] extended Van Dyke's approach to describe compressible (transonic) flow about an unswept lifting wing. In this case 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 a bound line vortex trailing a vortex sheet and corrections. It was shown that for similar airfoil sections the computations need be carried out at one spanwise station only since the corrections can be scaled to be independent of z*. In this paper we extend the techniques of Van Dyke, and Cook and Cole, to describe the transonic flow about a lifting, high aspect ratio, yawed wing. Investigations have been carried 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 (/*/?) arises in the induced downwash. A similar term arises naturally by the matching process in the transonic (compressible) flow. Cheng and Hafez [3] have treated swept wings in transonic flow. 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 are concerned with the case where the (sweep angle)3 = O(S). In Sec. 2 the three-dimensional transonic small disturbance theory boundary value problem for a lifting yawed wing is formulated [4], The sweep angle is 0((1 - AfJ)1/2) in the physical plane, hence 0(1) in the small disturbance plane. In Sec. 3 the asymptotic expansions 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 planes and 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 transform techniques. Matchings of the inner and outer expansions are shown. The boundary-value problems describing the inner potentials
2. Formulation of boundary value problem. In this section we formulate a boundary value problem for the first-order transonic disturbance potential: 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 describes the geometry of the upper and lower surfaces respectively, a is the angle of attack, and 8 the thickness ratio. The projection of the leading and trailing edges onto the plane y = 0 is given 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 ratio parameter B all fixed. In addition, the coordinates (x, y, z) are held fixed, to account for the relatively large lateral extent of the disturbance field. Thus, the potential is represented $(x, y, z; M, a, 8, b) = U{x + 52/3cp(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 attack parameter = a/8, B = aspect ratio parameter = b81'3, {y = 81/3y, z = 8l/3z) = transverse coordinates, 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 to consider a potential $>. The first-approximation potential (AT- (7 + \) 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 no disturbance at upstream infinity, so that 0* » €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 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 [ [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 ^ = 0, (2-13> where V = (8/By, d/dz) = gradient in a transverse plane, qr = ( [0*]8Vg + [qT], = 0, (2.17a) or alternatively [ 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 +2— 1 V*/ [0i]s + [V]s2= 0 (shock polar). (2.18) The differential equations, boundary conditions, and shock relations define a problem with a (presumably) unique solution. In order to obtain lifting-line theory we study the dependence of the solution of the problem on the aspect ratio parameter B, which becomes large. The formulation of the restricted lifting surface problem and the treatment of the lifting line as an approximation for B -> c° follow closely the ideas of Van Dyke for the analogous problem in incompressible flow. 3. Lifting-line expansions. In order to study the dependence of our first-order tran- sonic disturbance potential 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 LIFTING-LINE THEORY 183 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) 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*) < [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 at infinity are obtained by matching to the near field of the outer expansion. Note that the problem for ^*^0,1 a* a* + ^0,1y*y* + ^0,lz»z* ~ 2 tan = (3.5a) + As in the inner expansion the 0(log B/B) term is introduced for matching. However, in the outer expansion the 0(log B/B) terms remain even in the case that the sweep angle goes to zero. Note that, if written in x*, y*, z* variables, Eqs. (3.5, 3.5a, 3.5b) are precisely the same as for the unswept wing. The nonlinearity appears in the 0(\/B) term and the dependence of the outer expansion on sweep angle appears from boundary conditions and matching. 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 line vortex shedding its vorticity downstream and higher order singularities. The boundary conditions on the *P\(J* r*->co 0, (3.6) WioU = 0. (3.7) The remaining conditions are obtained by matching with the far field of the inner expansion. This matching determines the type of singularities needed in the which is the potential in a*, y*, z* coordinates of a distribution of divortices along the lifting line, and _= £!.~ r. S2(j) ^ 4tt A-* i., [A"*(z* - j cos /?)2 + 2o-*(z* - 5 cos 0)tan 0 + A>*2 + + 47rzL J-tf1 _y* + (z*iM. — i- cos /3)2 •fl+ -»costf)ung =U + rf (3.10) ^ ^jK*(z* - s cos 0)2 + 2 where = " (72xcosS/} ) + (2^8 V ^ ~ J°{2*}y*+ W ln (3"''> a* 2Di(z*/cos 0) 7i (cos 20+1) + - + j^} + I",'). (3-13) LIFTING-LINE THEORY 185 where J, - + J-L-J r yM™Mds+ Jt!.x 4r 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 + + tan "n y + 1 (( To /VV/ J 1 + cos 2(26 + 1),108 ' „ +, 022 +, ~8cos 20 81 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 value problem for two-dimensional transonic flow past a lifting airfoil. In particular the far field is given by [5] T06 , (T+ 1) (T0\log/- „ , l/ D0 „ , E0 . „ 00 - - -X—+ \^r- J cos 9 + -) cos 6 + r —r sin 6 2ir 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 spanwise station 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 we expect 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 a possible uniform flow, r2 a circulation, and the 0( 1) terms multiplied by tan /3 arise from higher-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 help of an intermediate limit. In a class of limits intermediate to the inner and outer limits, " m"r —m—,s( r = \{B)rx -.oo, r* = L = ^^-rx - 0. The calculations are simplified if the far-field of the inner expansion is merely written in terms of outer ( )* coordinates and a direct comparison of the inner (r* —>oo) and outer expansion (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 4- cos 0 , £° sin e _ iv 7 + i , r; . 2irjK* r* 2*JK* r* 2x 4 2 2tt Sm ^>/77 + 1 W T0VY/'log r* (cos 20 + 1) , 62 , cos 20 cos 40\ + lannUiHUsJ) \ 2 + y +-s H D'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. Also we chose = 0; the justification for that will be that the expansions match. Outer £>;(cos 26>+ 1) 6^ sin 20 ■)}■.. 4tt4,r (AT*)'"/-JffWZ 4x4^ (K*):e^*\3/2/f (J.18J Matching is accomplished to all orders shown if 7o(z*/cos 0) = JD^ = 7 + 1 (r0(z*))2 S2 cos 0 cos 0 4 JK* 2tt 'cos/? °' "•" sSy" ^A'' "• LIFTING-LINE THEORY 187 The essential matching for the completion of the boundary value problems for0i , 02 defines 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 classical lifting-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 the first-, second- and third-order inner potentials {{K* - (7 + 1)) K(t>cr_ 7 t 1 goiy) = - [0oj]/[0.off]- (4.7) These shock wave jump conditions must apply locally across any shock waves that appear in supersonic zones of the solution. The shock locus g0(y) is not known in advance and must 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, be solved analytically. Instead, one of the standard computational schemes for resolving shock waves is used ([2, 9]). Alternatively one could consider shock-free transonic flows and the corresponding airfoils generated from the hodograph solutions, for example Garabedian [8], The correction flow 0! satisfies a boundary value problem which is linear, assuming 0o , go are known. The equation is (3.2b) (K* ~ (y + l)0Off)0iffff ~ (7 + l^iff^Offff + 4>lyy ~ 0 (4.8) L. PAMELA COOK 00(7 * 0 a Cte(z*) Cle(z*) 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 7<4'10) as r -> 00, where r^z*), the induced circulation, is to be found. The boundary conditions on the trailing edge and wake are identical to those of 0: [ K4>oo ~ ' = [0 [0i] = —^i[0off] , (4.14) where all jumps are applied on the zeroth-order locus a = £o(v)- The shock locus has been represented as * = go(y) + 2*) + ^ g2(y; z*)+ . (4.15) LIFTING-LINE THEORY 189 An overall conservation theorem has also been derived to act as a supplementary shock condition: Is { [ (K* - (y + l>£oa) cos 26 cos 4(A 4K /\\ 2x/ / \ 2 + -x-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- 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. The results are: K* + 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) 01y — 0 —/ ' ) [0i [0.] = r, c"(z*» Cte(z* ) (A:* - (7 + 1)0o Fig. 4.2. Boundary value problem for [0z] = -g2[0off] , (4.24) where again all jumps are applied on the zeroth-order locus a = g0(y)- As for 0,, an overall Conservation theorem has been derived for f { K02ff + 2 tan 0 0O,*]- ^4 [** 2^K*To y^^ogr-J0y-fll , - IV + 0(\) a = g0(y) (Jump conditions given) 02y = 0 [02 a] = 0 — cr [02] — r2 Cle(z*Y cte(z*) (K* - (7 + 1 )0ocr)02cT(7- (7 + l)0offff02a + — = 0*1,2 ~~ 01,2 satisfies (A"* - (7 + l) = ~ 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 **0o„ — 2 0Off t0Off(r]"I" 2[0oj?][0o {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) , e2 0i2>= 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) " -- cos/3cos8 J cos/3cos/? 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 precisely the correction that arises if there is no sweep. If sweep occurs, a second term fa2) must be 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]. Thus essentially $0, fa" are known. Computation of 0! will present no new difficulties as is also probably true of fa2). There are singularities in the small-disturbance approximations both at the tips of the airfoil 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 amenable to lifting line analysis since, by suitable scaling, the problems for fa , ^(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 + 1WosWoiz + ^ovv = 0, [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'(^*)[^Ai]2_x/2- F°r 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, ^u) Z[^oj2_i/2 asr^oo, [^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 <^21which 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 term in the outer expansion, Vox« —» 0, X* —»± CD [fox*] = 0 across {(x*, z*)| x* — z* tan /? > 0, \z* \ < cos /?}, consists of a superposition of elementary horseshoe vortices distributed along x* = z* tan (3, - cos /? < z* < cos (3, and trailing off parallel to the x axis. So, y* y(s) -liinl ) ds,ds (A. 1) ((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 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/? \ dr Vo ~ 4ttcos (3 IJ0(J0 7\' \ cos /3B J/ y*2v*2+ + t2'\t2 \ (K*t2 + 2ct*t2c* t tan ,80 + Ky*2 + a*2)1" j*coB0-2* (z*-r\ y* (t , „ - „ p2 \ I)+sjEJ 0(1/»*)' where ( )' means d( )/dz*, we have that wi-/. r(co»0±z*)/y ( z* T v*o\ •>' is analytic for Re s > 0, and its analytic continuation to Re ^ > —2 is analytic with the exception 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 the exception of a pole at s = 2. Also, M[hj\ 1-5] M[jf j] -» 0 as |lm j| -► Thus, Vo= / hj(y*p)fj(p)dp 4x cos.OS p j= i J o 2 J—r i 2irir-Mfa 1-s] M[fj;s] ds 4x cos'Ua fJ j = \ J LIFTING-LINE THEORY 195 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 + p20 + (K*p2± Dp + £)1/2)~ pO JK*) H^ ~ X">dp~ Similarly, dp cos 0 / \ cos /3/ \ cos /?/ z* \/cos± z*V~s 1 _ ^ ,/ z* V cos /3 ± z*\ 2~s 1 cos/? A >>* / 1 —j ^ \cos 0/ \ _y* / 2 —^' so that mA • i — d = —^,1 «|A/i 1 - s] - ~y{^n)jzr;.cos fi/s-l + 0(1) as i. (A6) M[hj\ 1 - s]= ±y*y' (^j) jzrj + ei + _ 2) as 5 - 2< where 196 L. PAMELA COOK So, from (A.4), (A.5), (A.6), + 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 ^ (cosft) 7 (cosft) ^ z* \\ , /* / -cos/3 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*). tan 0-JK* tan/s/ Z «r« \ 4K«2 Ci = In tanft + JK* JK* llnl* + 7») ~ ~Tl So finally, 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 \ COSft / , , , / 2* \ . rfs + r w?)1 Vcos ft / , / JA:*- tan ft V/2 , *9, * ^ 2,cosff '"Ijlf + tanJ +O0"ln^X LIFTING-LINE THEORY 197 or more simply, — y( VcosZ* 0 /\ tan (S y'( VcosZ* 0 J) * . Vo = 2ir cos 0a— + 2ttJK*-772 cos /?a y* In r I S \ ,!* r co»fl y_cosp_ds_ \ o ) 4tt COS/3J-C08/3 z s where •/S(Z*)° 2,J*"1+ l"(«ncos-(i - *•■))») + 21 U./ rantl + Ir-co.H\ JI /"' (C0S -F7T,) *! ( cosrf) 7>V!. <*' , ,( Z* \ + ST^ff" {-''W - ,an"> + (' + (A.9) The other integrals in (3.9), (3.10) have the form -»W ds. *- f.-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, ( iIzjl) / COS/?/ \ i = 1 cos/3 Wz._co9(s (ff*2 + A>*2 + A>2 + 2 _ ^"2 (if0 3Cj(y*p)(fj(p)dp), cos 0 \x /rty^ i ^J no ' where W*> = ^^)«co.0 **•-,>), W = ± ^ + Er 198 L. PAMELA COOK Then / ♦ \ y.~0. ^ + ' w + °<''V)' so that / is analytic for Re .y > 0 and its analytic continuation to the negative half plane is analytic except for poles at the negative integers, M[5j\s] = /" "Sdp J „n (KV ± Z)P+ E) is analytic for -0 < Res < 3 and its analytic continuation to the whole plane has poles at the integers, and M[Kj\ I - s] M[$j\ s] >■0. Im s So, y* 2 / 2 = t f Xj(y*p)Zj(p)dp cos ft 7 = 1 0 .,*-2 / 2 fv+i ,,♦-2 2 -j—- T Residue M[K,\ 1 - jJA/pF,;j] + 0{y* In y*) (A.10) cos ft s = li2 as >>*--> 0. Since /*(cos/i+z* >/.v* /•»*_!_ l'*n\ M[3C,, 1 - s] = Jo cos'/J J ^ then M[3Cy; I - s] = + 0(1) as j-»l, (A.I and 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*5S(z*/cosft) + —— tan ft ^ g'(z*/cos ft) + 0(ln r*). So, (3.9) is b = *! Pi(zVcosfl) as y* -» 0, and (3.10) is = as >>*-♦ 0. Appendix B: Shock relations. The shock conditions for [*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 y + 1 \k*4>„~ Ms + [4>y]2.+ Ml - 2 tan /?[£]8[ [ 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" ^ [/w*02<7 (T "t" ' ) ' {[ + [0Oj?]so"I ^ [0O.pk[0lj»k 2 ~ [02>~]so[ + ft + j S*) 2[0„,k [ [0«k + ^ [0,k + j [^]»o+ ft + j ft) [ Collecting terms of the same order gives: 0(1): K*4>o„~ 1~- [ok=0, (B.9) K*(j)0a 2 0<>a [0i 1.2 + ft [K*0 0(7(7 (7 + 1 ) + 2[0Oy]so[01y]so 2ft[0oy]So[0o<7y]so 0, (B.10) [0l]»o = -ft[0Off]»„ , (Bll) °(i): /w*0o<7 2 0°ff [02CT]s„+ [^*02,7 - (7 + 1 )02 g2\ [K* ^*00(7 2 0O(tJ + 2[0oay]«o[^O^l^o 2 tan (} [ [02k = -£2[0o The zeroth-order conditions (B.8), (B.9) are the usual two-dimensional shock conditions where K* is the adjusted similarity parameter. The 0(log B/B) conditions (B.10), (B. 11) LIFTING-LINE THEORY 201 involve three unknowns, , [$iy]% , gi, and are essentially the same as the first-order correction for the unswept case (which is 0(1/B)). The 0(1/B) correction is the same as the 0(log B/B) correction except that the extra term —2tan/3[02.]So[oa]s„ turns up to correct for the sweep angle. As in the unswept case, there is another shock condition to be checked which arises by applying Green's Theorem to the equations for 0 = ff $ ■ , K*c{>iy)dady = 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. The integrals over B, W and S0 vanish from the boundary conditions. Since dy/da = -[0O(T]So/ [ £ ^ M / dy = o. (B.14) For 0 = ff $-((w*(p2a- 2(tan /3)0oz»).K* = \ K* + / ((-w* 02 ~ Ay In r + Jy + Bd + • • ■ — w*(t>2aa+ K* Rx + JK*y + ^-K*y - 2(tan /3)a r„' + • • • ~ K*Aj>+ K*Ayin />+ + - * + f8?" + JK*y + Bx-t2K*- 2(tan 0)aTo'd + ■■■, ■ [ ((—w*02(T- 2 tan 0 So we have f {~w* + 2 tan 0 <£0,.]—[K*4>,y] dy = 0. (B15) J o L0OO-J References [1] N. Bleistein and R. Handelsman, Asymptotic expansions of integrals, Holt, Rinehart and Winston, New York, 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 transonicflow, 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 Reprinting Committee, 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 Plasma Dynamic Conference, June 1974 [10] R. D. Small, Transonic lifting line theory: numerical procedure for shock-free flows, to appear, AIAA J., June 1978 [11] R. D. Small, Calculation of a transonic lifting line theory, Studies in Transonic Flow VI, UCLA School of Eng. and Applied Science Report, April 1978 [12] James K. Thurber, An asymptotic method for determining the lift distribution of a swept-back wing offinite span. Comm. Pure Applied Math. XVIII, 733-756 (1965) [13] M. D. Van Dyke, Perturbation methods in fluid mechanics, Parabolic Press, 1975