Copyright ©1996, American Institute of Aeronautics and Astronautics, Inc.

AIAA Meeting Papers on Disc, June 1996A9636617, AIAA Paper 96-2120

Oblique flying wing aerodynamics

Pei LiColorado Univ., Boulder

Richard SeebassColorado Univ., Boulder

Helmut SobieczkyDLR, Goettingen, Germany

AIAA, Theoretical Fluid Mechanics Meeting, 1st, New Orleans, LA, June 17-20, 1996

As a solution to the supersonic transport airplane design problem, the oblique flying wing configuration offers severalaerodynamic advantages. It can minimize the drag over a wide range of speeds, subsonic as well as supersonic. Weanalyze the drag of an oblique wing flying supersonically, based on the linear theory, and provide some newaerodynamic results using theoretical nonlinear design methods. The resulting wing has a high lift-to-drag ratio.(Author)

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AIAA 96-2120

OBLIQUE FLYING WING AERODYNAMICS

Pei Li, Research AssociateRichard Seebass, Professor

Aerospace Engineering SciencesUniversity of Colorado, Boulder, USA

andHelmut Sobieczky

Senior Research ScientistDLR German Aerospace Research Establishment

G6ttingen, Germany

Abstract

As a solution to the supersonic transport airplane de-sign problem, the oblique flying wing (OFW) configu-ration offers several aerodynamic advantages. It canminimize the drag over a wide range of speeds, sub-sonic as well as supersonic. We analyze the drag ofan obljque wing flying supersonically based on thelinear theory, and provide some new aerodynamic re-sults using theoretical nonlinear design methods. Theresulting wing has a high lift-to-drag ratio.

Linear Theory

The supersonic area rule tells us that the wave dragof an aircraft in a steady supersonic flow is identicalto the average wave drag of a series of equivalentbodies of revolution. These bodies of revolution aredefined by the cuts through the aircraft made by thetangents to the fore Mach cone from a distant pointaft of the aircraft at an azimuthal angle 9. This aver-age is over all azimuthal angles. This is depicted inFigure 1. For each azimuthal angle the cross-section-al area of the equivalent bodies of revolution is givenby the sum of two quantities: the cross-sectional areacreated by the oblique section from the tangent to thefore Mach cone's intersection with the aircraft, pro-jected onto a plane normal to the free stream; and aterm proportional to the component of force on thecontour of this oblique cut, lying in the 6 = constantplane, and normal to the free stream.1

The minimum wave drag associated with a given liftrequires that all oblique loadings projected by theMach planes be elliptical. A spanwise elliptical load-ing also minimizes the induced drag. For the opti-mum configuration of a wing, each of its equivalent

bodies of revolution due to lift must then be Karmanogives.2 The wave drag for a wing of given volume,or given caliber, will be a minimum if all the stream-wise distributions of the projected areas are those ofa Sears-Haack body, or a Sears body.3'4

Since we would also like to minimize the friction drag,the aircraft should have as small a wetted area aspossible. This then suggests that the wing also serveas the fuselage. As Jones pointed out long ago,5'6the ideal wing shape for supersonic speeds is an el-liptic wing of high aspect ratio, just as in subsonictheory, but flying obliquely so that its normal Machnumber is subsonic.

At supersonic speeds the wing must be swept to asufficient angle that it functions efficiently as a highMach number subsonic wing in the cross-flow normalto the wing. This then requires a wing derived fromsupercritical airfoils or a full supercritical wing design.Since the wing must house the passengers, it mustbe a relatively thick wing in order not to be so largeas to be impractical.7

The drag of an oblique elliptic wing can be expressedusing our theoretical understanding of drag at super-sonic speeds. The lower bound for this drag is givenby:

D- aSC + ^ y/-2 128qy2 (1)u = qz>fL,f+——£ +——g 4

Tigs Tiql. nlv

Here q is the dynamic pressure of the free stream, Sfthe reference area for skin friction, and Cf the skinfriction coefficient. Thus the first term represents the

Copyright © 1996 by authors. Published by the American Insti-tute of Aeronautics and Astronautics, Inc., with permission

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Figure 1. Fore-Mach cone (above) for the linearizedsolution, intersecting the OFW: detail below.

skin friction drag which we may accurately approxi-mate by the turbulent drag on a flat plate of the samearea and average streamwise chord.

The second term is the induced drag for an elliptical-ly loaded wing, where L is the lift. We recognize thisexpression as the induced drag of the wing in theflow normal to it, that is as L2/(7tc7n£>2), where qn is thedynamic pressure of the normal flow and b is the un-swept wing's span. We then interpret the productqntr as qcos^K to2 or gs2, where K is the sweep an-gle and zero for an unswept wing, and s is the spannormal to the free stream.

The third and fourth terms are the wave drag due tolift and the wave drag due to volume, where V is thewing's volume and P2 = M? - 1. The two lengths, //and lv are the averages over all azimuthal angles ofthe individual lengths of the equivalent bodies of rev-olution, appropriately adjusted for the variation of the

component of the force lying in the 0 = constantplane.

As noted earlier, for the minimum wave drag due tolift, the loading in each oblique plane must be ellipti-cal, that is that the equivalent body of revolution ineach azimuthal plane is a Karman ogive. This isreadily realized in an oblique wing with an ellipticplanform. With the chord varying elliptically, everyloading projected by the oblique Mach planes is alsoelliptical.

For the minimum wave drag due to volume eachequivalent body must be a Sears-Haack body. Thisminimizes the wave drag for a given volume. It is,however, the wing's thickness that is set by passen-ger height, not the volume. Design studies have indi-cated that OFWs have excessive volume. If weminimize the wing's thickness, that is the caliber ofthe equivalent body, then for the same caliber bodyas the Sears-Haack body, the volume in Eq. (1) isreduced by V(8/9).

The lengths in the last two terms are the averageover all azimuthal angles of the effective length forlift, and volume, for each azimuthal angle, as deter-mined by the supersonic area rule. Thus // is the ac-tual length for that azimuthal plane divided by cosQ.To calculate these lengths we must determine theangle at which the tangent to the Mach cone cuts theplane of the wing.

For simplicity we assume that the wing lies in a hori-zontal plane. We recognize, but ignore, the fact thatthe wing must incline its lift vector slightly to offsetthe leading edge suction which occurs on only oneside of the wing. In practical cases this results inwing plane inclination of less than two degrees.

If we write down the expression for the fore Machcone depicted in Figure 1, and consider its apex tobe at a large radial (and thereby axial) location, thisequation becomes that for its tangent plane. Thisplane intersects the horizontal plane and thereby thewing, in a line that makes an angle, <p, given by

tancp = ±psin9,

with the y-axis. Now that we know the angle cut bythe tangent to the Mach cone, we also know thelength of the equivalent body of revolution for thatplane is

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The loading on the cut made through the wing iscomposed of two parts: that due to the lift; and thatfrom the leading edge suction; we ignore this latterforce. With these approximations we may determinethe two lengths // and V

(2)

-1 ,1 _ 1 f* dQ/:" 2*Jo /4(0) (3)

2 + 3/774 4 2 7/2'

2b (sin*,) (1-m )

where m = p cott,. For a wing with subsonic leadingedge, m<\.

The drag arising from the lift of an elliptic wing, i.e.,the sum of the induced drag and wave drag due tolift, can also be determined by applying Kogan's the-ory.8 And this theory can be used to show that an ob-lique, elliptically loaded wing has the minimuminviscid drag for a given lift.9

The approximations made here are equivalent totreating the oblique wing as a lifting line. Thus wemay use Eqs. (1) and (2) to determine the invisciddrag for an oblique lifting line. This gives, for the in-viscid flow past a lifting line,

D = (4)nqs

This is the result first derived by Jones, using theprinciple of combined flows.10'11

The linear result for an arbitrary elliptic wing is verymuch more complex.9 We simply give the inviscid

here for an oblique wing of large aspect ratio:

(cosA)2(1 + m2)

-m2-G

where the m of Eq. (4) is modified to be

(5)

m = pcotX, 1-32 (cosX)

and

G -_!_.[:1-m2 L

4(cosX)2(1 +m2)"2

Here A is the swept wing's aspect ratio, s2 divided bythe wing area.

An oblique elliptic wing simultaneously provides largespan and large lifting length. The reduction in thewave drag of an oblique wing of finite span comesfrom being able to provide the optimum distribution oflift and volume in all oblique planes. The aerodynam-ic advantage of the oblique wing over the swept wingat moderate supersonic Mach numbers stems fromthis fact.

Figures 2 and 3, taken directly from Jones,12 makeclear aerodynamic advantages of the oblique wing.The first compares the minimum inviscid drag due tolift of an oblique elliptic wing with that of a delta wingof the same span and streamwise length. The sec-ond figure compares the wave drag due to volume ofan oblique elliptic wing and a swept wing having thesame aspect ratio and the same thickness. Overmost of the Mach number range, the drag of the ob-lique wing is less than half the drag of the sweptwing. As Eq. (1) shows, the wave drag of a wing hav-ing a given volume varies inversely with the fourthpower of the length. It is evident that the reduction ofwave drag due to volume is in part due to the greaterlength of the oblique wing in the flight direction.

As described earlier, we see that the oblique ellipticwing with an elliptic load has the minimum drag (in-duced and wave drag) due to lift. To achieve the ellip-tic load distribution, twist variation along the wingspan, or bending the wing up at the tips, is needed.

Smith13 noted that the Sears-Haack area distributionis the product of an elliptic distribution and a parabol-ic distribution. The lengths of the chords cut by paral-lel planes on an elliptic planform are distributedelliptically. Thus, if all sections of wing were parabolic(biconvex), each area distribution of the equivalentbody of revolution due to volume would be the prod-uct of an elliptic and a parabolic distribution, and thewave drag of the wing would be a minimum for given

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volume.

To fix our ideas on the relative size of the variouscontribuitons to the total drag, let us consider a con-crete example.

McDonnell Douglas Aerospace West studies14'15

provide guidance on how many passengers a realis-tic OFW might carry, how much it might weigh, andat what speeds and altitudes it might fly for the Machnumber range 1.3-1.6. We choose a freestreamMach number of ^2 for simplicity and a sweep angleof 60 degrees for ease of control and aeroelastic sta-bility. Higher speeds are possible with more sweep,but the wing's control becomes increasingly difficult,with 60 degrees being judged acceptable in previousstudies.

We use these studies as a guide to conclude that a800 passenger aircraft with a 5200 nautical milerange will have a wing with a maximum chord ofabout 55 feet and a span of about 475 feet. This pro-vides an aspect ratio of 11 and a wing area of 20,518square feet.

This OFW transport will enter cruise at about 1.5 mil-lion pounds and leave cruise at a weight of 0.9 mil-lion pounds. Thus its lift will vary considerably. Fornominal conditions we take the weight to be 1.2 mil-lion pounds, the volume to be 85,800 cubic feet andthe altitude to be 42,000 feet.

Using Eq. (1) and the nominal conditions, we calcu-

late the turbulent skin friction drag on a flat plate,and more directly, the other terms to conclude thatthe drag in pounds is:

D = 4.28 x 104 (skin friciton) + 1.62 x 104 (induced) +3.65 x 103 (wave-lift) + 1.08 x 104 (wave-volume),

giving an overall maximum L/D of 16.3. Here we useSmith's formula13 for the skin friction drag coeffi-cient, which gives the almost same result as thewell-verified method of Sommerand Short.16'17

If we correct this lifting line result using Eq. (5), thedrag due to lift increases from 1.99 x 104 to 2.07 x104 pounds; i.e., by 4%, reducing the maximum L/Dto 16.2. This is nearly double the maximum L/D forswept wing aircraft of the same swept span andlength, as reported by Kuchemann.18

Nonlinear Theory

Following simple sweep theory, we examine the air-foil thickness possible for a given section lift coeffi-cient and Mach number based on the normalcomponent of velocity. At cruise conditions, the flowover an OFW is that behind the nearly conical shockwave emanating from the leading tip. The wing isswept so that the component of this flow normal tothe wing's leading edge will be sufficiently subsonicthat a thick, shock-free airfoil may be found. Wehave assumed this sweep to be 60 degrees, and afreestream Mach number of V2, giving a normalMach number of 0.707, and a tangential component

.008

.006

CD

.004

.002

1.0 1.2 1.4 1.6 1.8 2.0M

Figure 2. Drag due to lift; oblique elliptic wing anddelta wing, M= V2 (Ref. 12).

Figure 3. Wave drag due to volume of oblique ellip-tic and swept wings as a function of Mach number(Ref. 12).

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Figure 4. Embedded shock wave in a conical cross-flow.

of 1.23.

Boerstoel19 provides guidance on how thick a non-lifting airfoil might be if designed to be shock free.His results, and those of others, suggest it should bepossible to design an 18% thick shock-free airfoil fora normal Mach number, Mn, of 0.76. This suggeststo us that a 17% thick airfoil with a C/ of 0.6 might beshock free for a Mach number of 0.7. We presumehere that, at their design point, optimum OFWs willhave neighboring lift coefficients for which they areshock free, but not optimum, as is often the case forsupercritical airfoils and wings. Conversely, we pre-sume a shock-free design will have a neighboringL/D that is even higher than that when it is shockfree.

The normal component of the flow accelerates overthe wing to become locally "supersonic." The returnof this component to "subsonic" cross flow is normal-ly through a shock wave, just as it is on supercriticalbut not shock-free airfoils. This cross-flow shockwave adversely affects the boundary layer and,thereby, the wing's lift and drag, just as it does onsubsonic, supercritical airfoils and wings (see, e.g.,Ref. 20).

We can fix our ideas for supersonic flow by consider-ing supersonic conical flow past a wing with sub-sonic leading edges and conical camber. Such awing will, unless designed using special tools, have across-flow shock wave like that depicted in Figure 4.

While this flow is supersonic, the cross-flow plane

equations are mixed, being hyperbolic outside theconical shock wave and inside the local "supersonic"cross-flow region, but elliptic elsewhere.

Thus, the fictitious gas method proposed bySobieczky21 for the design of supercritical airfoils,which has been extended to more general gas lawsand applied to three-dimensional wings,22"" appliesto conical supersonic flows as well. This extension tosupersonic flows was first demonstrated by Sritha-ran.26 Recently, we have suggested that this methodmay apply to fully three-dimensional supersonicflows.2' The studies reported here derive from thework reported in Refs. 27 and 28.

The axial component of the flow over the obliquewing causes a disturbance to the Mach number dis-tribution of the normal component along the span.We may approximate this axial flow by that over aslender body of revolution whose cross-sectionalarea equals that of the wing, by using the linear the-ory. The resulting variation of the normal Mach num-ber in the span direction is shown in Figure 5. Wesee that there is essentially a linear variation aboutthe mid-chord value (/= 0) in the normal componentof the Mach number, requiring different airfoil de-signs, or at least thickness, along the wing. We rec-ognize, then, that the upper surface curvature andthickness of wing sections should be decreasing to-ward the wing trailing tip, in order to avoid creating across-flow shock or increasing a shock's strength ifone has already formed. Our objective is to find outhow well we might do in designing an OFW using su-percritical airfoils that we develop and then appropri-ately blend to form the wing.

M,

1.2

1.0

0.8

0.6

0.4

0.2-5.0 -2.5 0.0

y2.5 5.0

Figure 5. Estimated variation in the normal Machnumber component along the wing span; Mn =0.707 at y = 0. Axial flow at Mt = 1.23 direction fromright to left.

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Figure 6. Pressure contours and grid in the subson-ic flow, and the characteristics pattern in the super-sonic flow for a shock-free airfoil designed for OFWapplications: Mm = 0.707, C, = 0.6.

We need to remark here that two goals come imme-diately into conflict. The OFW design that has theminimum induced and wave drag due to lift, has anelliptic load. The optimum area distribution to mini-mize the wave drag due to volume, or due to thick-ness, is the Sears-Haack body for volume, and theSears body for thickness. In our studies we have en-deavored to minimize the drag due to lift, acceptingthe resulting wave drag due to volume.

Results

At first we perform a preliminary airfoil design usingour fictitious gas method. A 17.4% thick baseline air-foil is generated by Sobieczky's geometry tool28 fora flow of Mm = 0.707 with C/ = 0.6, which corre-sponds to the normal flow component of the super-sonic flow over the swept OFW. For choosing thefictitious equations used in the Euler solver, we pre-scribe a new energy equation27 to change the equa-tions inside a local supersonic region so that theyremain elliptic there. This results in a shock-free flowwith a smooth sonic line, but the wrong gas law, in-side the supersonic region. The correct mixed typestructure of the transonic flow is recovered in thenext step: supersonic flow recalculation by means ofthe method of characteristics, using the just calculat-ed data on the sonic line for the initial values. Thisrecomputation of the flow with the correct equationsof state has a lower density in the supersonic flowand provides a modified, and thinner, airfoil design.The result is a slightly flattened section (see Fig. 6)with a thickness of 17%.

Figure 6 shows the pressure contours in the subson-

ic flow and the characteristics in the local supersonicregion for this new airfoil. In some cases we will findlimit lines in the real supersonic flow calculation, indi-cating a smooth flow does not exist. Then a new de-sign must be attempted.

For the airfoil considered above we have selected adesign lift coefficient appropriate for an OFW trans-port. The OFW's lift coefficient for its supersonicMach number will be lower by the square of the ratioof the normal to freestream Mach numbers, or in thiscase, by a factor of 4. Thus the wing design will havea lift coefficient, CL, of about 0.15.

We use this shock-free, redesigned airfoil as thecenter section to create an OFW of elliptic planformwith a 10:1 axis ratio, providing an unswept aspectratio of 12.7. We calculate the three-dimensionalflow using the code CFL3D,29'30 which was devel-oped at NASA Langley. To determine the blending ofwing sections, we used the variation of the normalMach number along the body caused by the tangen-

0 *""""" a b c

CFigure 7. Wing geometry parameters: planform,twist distribution and thickness factor along span;support airfoils and their blending weight.

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tial flow component of free stream by using the lineartheory for bodies of revolution, as shown in Figure 5.

To achieve the elliptic load distribution twist mustvary along the wing span. This variation is linearnear the center, strongly decreases at the trailing tip,and slightly increases at the leading tip. But differentMach numbers and sweep angles require differingtwists. An elliptic loading is best realized by bendingthe wing up at the tips. For simplicity we have usedwing twist in our studies.

In Figure 7 we depict the blending of supercriticalsections and the variation of twist used to achieve anearly elliptic loading at the shock-free design point.The twist was varied from -10 to +5.5 degrees; thewing section thickness parameter varied from 0.3 to1.5 from the trailing to the leading tip. Between thecenter section and trailing tip, three support airfoilsare blended.

The resulting inviscid lift-to-drag ratio is a most un-satisfactory 14.1. This should not be surprising. Wehave designed a wing that would perform well in asubsonic normal flow of Mn = 0.707 and then variedits twist and thickness to achieve an elliptic load atdesign conditions corresponding to a wing swept at60 degrees in a Mach >/2 flow. There is a relatively

Figure 8. Lift-to-drag ratio as a function of angle ofattack, * denotes the shock-free airfoil design point.

strong shock wave on the upper surface for most ofthe span of the wing. That is our sectional shock-freedesign method failed.

We are accustomed to the observed fact that highlyefficient supercritical wings (and airfoils) frequentlyhave nearby shock-free flows, usually at a lowerMach number and/or CL. Consequently, we explorethe variation in UD of this wing design with angle ofattack.

).0 0.2 0.4 0.6 0.8 1.0x/c

0.100.080.060.040.020.00

Load

-5.0 -2.5 0,0 2Tb

Figure 9. Pressure distributions at five span stations and isotachs on three grid surfaces for OFW with ellip-tic load distribution obtained with varying wing sections and nonlinear twist distribution, UD = 22.5.

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Figure 10. OFW upper surface inviscid flow, MM = 1.414, L/D = 22.5: Surface streamline; surface characteris-tics (Mach conoid traces on wing surface) delineating regions of dependence; cross-flow shock formation vi-sualization.

Figure 8 shows this variation of the lift-to-drag ratiowith angle of attack. Surprisingly, we find that thenonlinear optimum inviscid L/D of about 20 occurs ata Ci much lower than the design C^. A lower CLmeans a lower flight altitude and thereby lower struc-tural requirements for pressurization. It also means afuller realization of leading edge suction,31 and aweaker shock wave down the wing.

Then, at this new design condition, we changed thetwist distribution from that in Figure 7 to one that var-ies from -6 to +3.5 degrees between the trailing andleading tip to provide a near elliptic load. We alsocarefully varied the sensitive ordinate of the supportairfoil in the trailing portion of the wing to obtain theresults shown in Figure 9. By this process we haverecovered a nearly elliptic loading and the shockwaves on the trailing portion of the wing are relativelyweak. The inviscid L/D of this wing is 22.5. Estimat-ing the skin friction as before, we find L/D = 10.8.

Van der Velden,32 in his recent studies of such air-craft, finds an L/D of 11 for a 250 passenger, Mach1.6 OFW. The aircraft has four engines and a verticaltail. This aircraft uses a 19% thick wing swept to 68degrees and has a span to chord ratio of 7.4. On theother hand, Saunders, et al.33 and Kennelly, et al.34

find numerically, and verify experimentally, a value of8.5 (± 0.5), for the wing alone, on a realistic, 450 pas-senger, Mach 1.6 OFW. This 17% thick wing, againswept to 68 degrees and with a span to chord ratio of7.3, has been numerically optimized with realisticconstraints. Our studies, at a lower Mach number,and without constraints on the wing shape, give re-sults that seem intermediate between these two oth-er studies.

Figure 10 shows the intersection of the Mach conoids

with the wing's surface, at the new design condition.As evident in the pressure distributions at down windspan stations (see Fig. 9), a cross-flow shock formson the wing's upper surface. Points in the decelerat-ing flow, in this case those behind this shock waveare influenced by the wing's trailing edge.

Conclusions

The linear theory demonstrates that an oblique ellip-tic wing may have quite high lift-to-drag ratios atmoderate supersonic Mach numbers. Our results in-dicate that an elliptically loaded OFW can be ob-tained by twist and thickness variations along thewing span. The cross-sectional airfoil is designed tobe shock free using our fictitious gas method, and dif-ferent support airfoils are blended along the wing tominimize the cross-flow shock. An initial candidatedesign gives a lift-to-drag ratio of 10.8. Higher valuesshould be possible. We conjecture that the fictitiousgas method may be extended directly to three-di-mensional supersonic flows.27 Such an extension orsome other robust optimization tools should be ableto improve considerably on our modest results.

We suggest that a Mach 1.4 OFW makes a goodcandidate for a large, and yet supersonic, transportaircraft. It is unlikely, however, that such a radical de-sign would first appear as a large aircraft. Thus, if anOFW were to be introduced, it is more likely to ap-pear first as a smaller supersonic transport or as acargo aircraft. Further studies are needed to deter-mine whether or not an OFW is attractive enough tojustify an experimental aircraft program.

Acknowledgments:

This work was partially funded by the German Alex-

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ander-Von-Humboldt Foundation through a 1991Max Planck Research Award, and by a grant fromthe late W. Edwards Deming. The authors thankmany colleagues, including Tom Galloway, Tom Gre-gory, Robert Kennelly, Mark Page, Steve Morris, BobLiebeck, Blaine Rawdon, Alex Van der Velden, andMark Waters for their valuable comments on this pa-per.

References

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2. Von Kdrman, Th., and Burgers, J. M., Aerodynam-ic Theory, W. F. Durand ed., Vol. 2, Springer, pp.172-175, 1934.

3. Sears, W. R., "On Projectiles of Minimum WaveDrag," Quart. Appl. Math., Vol. 4, No. 4, pp. 361-366,1947.

4. Haack, W., " Geschossformen kleinsten Wellen-widerstandes," Lilienthal-Gesellschaft fur Luftfahrt,Bericht139, pp. 14-28,1941.5. Jones, R. T., "Wing Plan Forms for High-SpeedFlight," NACA TN863, 1947.

6. Jones, R. T., "Effects of Sweepback on BoundaryLayer Separation," NACA Rep. 884, 1947.7. Li, P., Seebass, R., and Sobieczky, H., The Ob-lique Flying Wing as the New Large Aircraft," ICAS-96-4.3.2, 20th International Council of the Aeronauti-cal Sciences Congress, Sorrento, Italy, September 8-13, 1996.

8. Kogan, M. N., "On Bodies of Minimum Drag in aSupersonic Gas Flow," Prikl. Mat. Mekh., Vol. 21, No.2, pp. 207-212, 1957.

9, Jones, R. T., 'The Minimum Drag of Thin Wings atSupersonic Speeds According to Kogan's Theory,"Theoretical and Computational Fluid Dymamics, Vol.1, pp. 97-103, 1989.

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28. Sobieczky, H., Choudhry, S. I., and Eggers, Th.,"Parameterized Supersonic Transport Configura-tions," 7th European Aerospace Conference, Tou-louse, October 1994.29. Thomas, J. L, Taylor, S. L, and Anderson, W. K.,"Navier-Stokes Computations of Vortical Flows overLow Aspect Ratio Wings," AIAA J., Vol. 28, No. 2, pp.205-212, 1990.

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32. Van der Velden, A., 'The Oblique Flying WingTransport," New Concepts in High Speed TransportDesign, H. Sobieczky, ed., (to appear).33. Saunders, D. A., Kennelly, R. A., Cheung, S. H.,and Lee, C. A., "Oblique Wing Design Experience II"(in preparation).34. Kennelly, R. A., Jr., Bell, J. H., Buning, P. G.,Carmichael, R. L., Lee, C. A., McLachan, B. G.,Saunders, D. A., Schreiner, J. A., Smith, S. C., andStrong, J. M., "Integrated Test and Analysis of a 'Re-alistic' Oblique All-Wing Supersonic Transport Con-figuration" (in preparation).

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Copyright ©1996, American Institute of Aeronautics and Astronautics, Inc.

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