The aerodynamic design of the oblique flying wing supersonic transport

NASA Contractor Report 177552

The Aerodynamic Design of the Oblique

Flying Wing Supersonic Transport

Alexander J. M. Van Der Velden and Ilan Kroo Stanford University, Palo Alto, California Prepared for Ames Research Center Cooperative Agreement NCA2-343 June 1990 National Aeronautics and Space Administration Ames Research Center Moffett Field, California 94035-1000 TABLE OF CONTENTS Page o iv Nomenclature a local velocity of sound normalized with respect to the freestream velocity a ,b semi-minor, and semi-major axis of elliptic wing AR aspect ratio (dt/dx)/2 half thickness distribution slope of a panel dZ/dx mean line slope of a panel h altitude K lift-dependent drag factor KO Sears Haack factor M freestream Mach number OFW Oblique Flying Wing R range Re Reynolds number S Wing area t/c thickness-to-chord ratio H,V,W perturbation velocity in x,y,z-direction normalized with respect to the freestream velocity W weight U total velocity normalized with respect to the freestream velocity G reek symbot_ I_M 2 Prandtl's compressiblity coefficient: 3/ A panel 1/4c sweep angle 1/4 C quarter chord line of the panel O dihedral, positive with increasing z and y n total propulsive efficiency as used in the Brequet relation slew angle (90°-A) indices: dd drag divergence EAS equivalent air speed f fuel 1 local n number of passengers payload, pressure P powerplant PP Ili|_ Artists Concept of the Oblique Flying Wing OC '_',_ ::,_ C,:x _ '_;_ r',- t_ ",1_ -,_(n 1.1 Introduction Background and Present Work In the late fifties R.T. Jones [25, 1, 13] suggested that aircraft with asymmetrically swept wings (or oblique wings) would offer many advantages over aircraft with conventional wing design at high transonic and low supersonic speeds. The primary advantages of the oblique wing arise from its improved structural arrangement and its reduced subsonic and transonic drag. Recent work by Rockwell and NASA engineers [26, 29, 20] has provided the tools needed for studying the oblique wing for transonic and low-supersonic speeds in greater detail.

In this work we extend these analytic capabilities and apply them to the aerodynamic design of oblique flying wings. Figure 1. shows an artist's impression of the oblique flying wing and Fig. 2 shows a three-view of the baseline Oblique Flying Wing. The baseline wing has a near elliptic planform, which can be swept from 35 ° at takeoff to 70 ° in cruise. The passengers are located at the center of the planform inside the wing structure. Work done by Van der Velden on NASA grant NAG-2-471 J41] indicated that this configuration could provide economical supersonic transportation if the theoretical minimum drag based on potential flow can be obtained.

Prediction Methods Up to now, numerical calculations and windtunnel tests up to Mach 1.4 have been published. Beyond Mach 1.4 only the theoretical minimum potential drag for this configuration is known. At a higher Mach number, the occurrence of shocks and flow separation may limit the applicability of potential flow methods, and we therefore require analysis tools which include these effects. Though the Navier-Stokes equations could be used to analyze the configuration, it is not yet possible to solve for the geometry on the basis of a pressure distribution with this method. Apart from this limitation, thest: :olutions are very time consuming for 3D flow. For 2D flow, fairly rapid solvers exist and they can be used to analyze and design sections. The 3D wing can be analyzed with the Prandtl- Glauert equation for supersonic Mach numbers from 1.5 to 3.0 if the shear-layer is thin and the shocks are weak. The high Reynolds numbers assure that the shear layer is thin for the attached flow, while the s¢.ction data is used to identify the local normal Mach number and recovery distributions that are separation and shock-free.

-n The wing is swept,less hO For lake-oFF E_ -n \ ¢D _E "X t::r .< "x 'X < r_ Mach 2 cruise \ -I \ < (1) Q.

I I Cabin cross section / ,9 15.2 m "////////// = 400 FL / 122 m 1.2 Summary This study describes the aerodynamic design of a thick Mach 2 Oblique Flying Wing (OFW). A preliminary design analysis indicates that the best payload fractions are obtained for a takeoff wing loading of 2 kN/m2 at an altitude of 16 km and an unswept aspect ratio of 10.

An optimization study projects that the highest payload fraction is achieved for a 16% thick root airfoil. This airfoil, and the airfoil family derived from it, are designed with ARC2D, a Navier-Stokes code. The OFW airfoil family allows a good utilization of the passenger cabin and it achieves the required trimmed lift at the 0.32 chord location. This is the rearmost location at which artificial stability and control by a narrow trailing edge flap can be achieved.

Each member of the airfoil family has a different thickness and trimmed drag divergence lift. By selecting the right airfoil at each station along the span it is possible to achieve a Sears-Haack area distribution and an elliptic lift distribution. Such distributions minimize the potential flow drag.

We present a new method for determining a wing's design pressure distribution based on airfoil data. The pressure distribution is calculated from the potential flow velocity perturbations for a given thickness distribution and the prescribed vorticity. The vorticity in supercritical wing region_,_ is based on airfoil transonic normal Mach numbers and includes the influence of local sweep, taper and three-dimensional induced velocities, so that the appearance and the stren_;th of the shock waves can be expected to resemble those of the airfoil. The vorticity in subcritical wing regions is scaled first with simple sweep theory, and then to achieve the desired load distribution. The vorticity di:,tribution is then used to solve for the wing's camb,_r with an inverse panel code. The induced velocity perturbations of this cambered wing are used in the next iteration.

The potential drag of the wing designed in this fashion with a panel code was very ,'lose tn the ones given by R.T. Jc,nes [2] and J.H.B. Smith 13] for a minimum drag oblique wing.

The method was also successful in constraining the local normal Mach numbers to the values that would produc,_" only weak shocks. The moment and force characteristics of the wing at the design conditi,:)n indicate that the wing could be trimmed and controlled without high drag penalties.

2 CONCEPTUAL AERODYNAMIC DESIGN 2.1 Objectives Though there is no one set of missions that can be specified for a new aircraft, it is possible to indicate which range of missions is most likely to result in an economically competitive aircraft. The following section describes the rationale for the selection of mission parameters in this work.

Cruise Mach number We investigated designs optimized for cruise speed between Mach 1.2 and 3.0. The most attention was paid to the Mach 2 cruise speed. At this Mach number the technology risk is moderate and the cost of development can be acceptable. At Mach 3 the aircraft would be twice as expensive [ 12].

Payload Today a proven market exists for long range transports with up to 550 passengers or 100,000 kg payload. Within the geometric constraints posed by the accommodation of passengers there is almost no difference in size between a 1-, and a 100-passenger flying wing, and therefore only aircraft with a high payload will be considered.

Existing payloads are: (Concorde): Too small to be a successful flying wing, but would require the 118 pax least initial investment.

(SST): The target American payload for a supersonic transport. However, in 247 pax view of the growth of the market such a transport may still be too small when it enters the market in 20 years.

(B747): Probably the best payload for a flying wing, but would require the 452 pax highest initial i avestmenc Range In this study a design range of 5000 nautical miles or 9000 km was used, the current range of a B747 with maximum payload. Because all proposed Oblique Flying Wings have a similar level of technology, a good optimization criterion is the payload to takeoff weight IUL_m_INTENTION AI;.LY B_ 5 P_ECED_5.'G PAG_ B;..Ar_K NOT FILMED ratio for a given mission [15]. On closer inspection this criterion encompasses both fuel efficiency and depreciation of the airframe. This economic criterion was used to establish at an early phase which configurations should be developed further, without the need of dubious cost-analyses.

2.2 A Semi-empirical Model for the Conceptual Design The semi-empirical methods developed in by A. Van der Velden in References 4 and 41 assume 1970s technology for all aircraft components. In this way, an unbiased comparison with existing aircraft was possible. A short list of the basic assumptions and methodologies is given below: Geometry To accommodate passengers comfortably, certain minimum geometric requirements have to be satisfied. We selected a minimum passenger cabin aisle height of 75"=191 cm, and a sitting height of 56"=142 cm. We also required that the cargo holds have an internal height of at least 45" with a 6" clearance in order to allow the aircraft to carry the belly containers of today's domestic subsonic transports as well as a range of IATA containers.

The required floor area per passenger is about 0.6 m 2 for an economy layout, and 0.7m 2 for a normal layout according to empirical data from Ref. 5. The cargo floor to passenger floor ratio is approximately 1 to 4 for the configuration presented in Fig. 1.

Aerodynamics The friction drag was calculated for each component based on the Prandtl Schlichting equation. This equation assumes a fully turbulent boundary layer as a function of Mach number and Reynolds number for a Prandtl number of 0.9: ec°mp°nent ]1 0.455 [Lo 10[ R .-2.58 Cfc°mp°nenr- 1+0.18M2/ g (1+0.18M2)2.SJ] We added a form drag according to Ref. 5 pp. 499-501, assuming transition at 5% from the leading edge.

The linear, supersonic, volume-dependent wave drag was based on J.H.B Smith's method

given in Ref. 3. The lift-dependentdrag was taken to be equal to the minimum lift-

dependent dragof anelliptic obliquewing with full leadingedgesuction,accordingto R.T.

Jones[2]. CDwaveis the volume dependent wavedragandCDlift is the lift-dependent

drag asa function of Mach number, slew angle_t and root chord thicknesst from the

expressions developed in Refs.2 and3: 13-(_-4-iAIM+2iA ) CL 2

CDwaVe = _2 Re

4 [13 2{M+iA) 2] CDlift = TRe [_2tM+iA}213/2 M_[b2-a2]sin_cos_

Aab

B :'/'v 2 2 , 2. 2 a cos _+t_ sm B B 2 The wave drag of the other components was calculated from Wards' [28] transfer rule. In particular, we approximated the drag of the nacelles in the presence of the wing by using Swan's [42] estimate. The spillage drag coefficient was based on the engine mass flow ratio, as suggested by Ref 7. The skin roughness drag was based on a material grain size of 17_tm, turbulent flow and the method of Ref. 43. We estimated the drag from fabrication type roughness from Ref. 46. Finally, the flap and systems drag were calculated with the method of Refs 44 and 45.

Weight As a guideline throughout the weight calculations Torenbeek's [5] itemized weight penalty method was used. In Ref. 41 this method was adapted for the OFW. The pressure differential gave the critical load for the passenger cabin, and the bending moment gave the critical load for the outer wing panels. To convert these loads into structural we_j't, only aluminum alloys or materials with the same maximum stress to weight ratio as aluminum alloys were considered.

The fuel weight was calculated from the Breguet equation. We added to the cruise fuel weight a range increment of 1200 km for diversions and a 7% allowance for climb: _/-0.23(R+1.2) with R in Mm Instead of using the usual Class II itemized weight penalty method and iterating to obtain the correct weight, we can model the takeoff weight more conveniently by expressing the takeoff weight by using a Class I method: /)wpp w to+/)woe /)woe /)woe Wto= Wfixed + 0(wto) - _ Wto + 0---if-S+ _ + Wp + wf

F/D

The variables and their values are given below: Wfixed fixed weight items 500 kg /)WOe variation of empty weight with wing area. This includes midsection, /)S outboard panels, flaps and vertical tails. 35.2 kg/m 2 /)wpp variation of nacelle and engine weight with lift-to-drag and takeoff _wto) weight at h=16500m cruise. This includes nacelles, pivots and engines: 1.02 An expression for the power plant weight fraction with constant total weight and variable altitude is given in section 3.3.

variation of the empty weight with takeoff weight. This includes the gear, apu, instruments, and hydraulic, pneumatic, electrical and anti-icing systems. 0.071 variation of the empty weight with number of passengers. This includes the operational items, furnishings and equipment, and airconditioning.

Propulsion We assumed a conventional turbofan layout with 3D inlet and variable geometry exhaust.

Engine performance was estimated from isentropic work relations which were corrected with realistic efficiency factors. FAR 36 stage 3 Noise requirements determined the bypass ratio of the engines. The noise levels were determined by.an empirical relation from Ref.

4. between jet exhaust velocity and velocity profile.

Stability and Control Ref. 41 indicated that sufficient control authority over an unswept OFW existed at a center of gravity location of 32% mac with a 10% simple slotted trailing edge flap actuated by a stability augmentation system at VEASmin = 130 m/s during climb. Vertical tailplanes with a planform area of 5% of the wing planform area located near the wing tips provide adequate control authority in case of engine failure.

2.3 Payload Size Affecting Wing Geometry The wing can be tailored in such a way that we obtain a Sears-Haack area distribution, elliptic lift and maximum utilization of the volume for payload and fuel. This was accomplished with the following approach: The total empty weight of the OFW for a given payload relates directly to the wing planform area. So the maximum payload fraction is obtained at the maximum payload floor fraction Sp/S. For a given payload the required wing size can be determined by the assumption of the Sears Haack Area distribution and given minimum internal dimensions to accommodate the payload.

Fig. 2.1 shows the available fraction of the planform area used by passengers and cargo plotted against the maximum root section thickness and the root taper ratio of 0.7 for an assumed NACA airfoil. The taper ratio of 0.7 was selected because it gave the least variation in thickness-to-chord ratio over the span of the passenger cabin. Section 2.5 will show that the design is constrained by the maximum local thickness-to-chord ratio. It is therefore advantageous to use this maximum over as large an inboard span as possible to maximize the available payload area and volume. Starting with a root chord of given thickness, the wing is tapered up to the point where the internal height is less than that required to store cargo, from this point on, the wing is tapered so it joins with the tip geometry. The tip geomeu'y is similar to that of the NASA oblique wing demonstrator [20].

To provide some idea of the aisle layout, a 15.2 m root chord wa introduced. This value was used for the baseline aircraft of Fig. 1. Fig 2.1 shows that, for a given payload, the required area of the flying wing increases significantly below a root thickness of 2.2 m, but it does not decrease significantly beyond a root thickness of 2.4 m. If we consider the necessity of an additional aisle, then a thickness of 2.3 m becomes even more app,. "ling. It is also clear that the volume, and hence the wave drag, does increase quickly for a vehicle that becomes smaller but increases in volume. This would be the case if we where to use a higher thickness.

The preceding data is not adequate to allow us to decide on a final geometry for the aircraft.

However, it has been shown that for an initial taper around 0.7 and a maximum thickness around 2.3 m a good utiliTation of the available volume and area is achieved. In Section 3.3 we will present further proof of the validity of this choice.

Fig. 2.1: OFW planform utilization Sears Haack area distribution, NACA 00XX-30 basic thickness distribution .35 Note: Indicated are the minimum number .30 " of aisles required for a_ .25 Sp/S / I _ _ o5c .2o o_ PAYLOAD i f 0.2C ,.

.15 span 0.85 1.

I _ 2 _ 3 aisle required .lO 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 tmax (m) Fig 2.2: Effect of technology on payload fraction R=9000 km, Mach 2 8Woe 60.

8S 50.

kg/m2 0.14 40.

30.

0.16 Wp 20.

Wto 10.

O° 3.5 4.0 4.5 5.0 5.5 q L/D 2.4 The Impact of Technology on Economy For conventional aircraft, wing loading is a powerful means to change the aircraft's performance. For an oblique flying wing designed to accommodate passengers, this is not the case, and the range of available wing loadings will be primarily a function of desired operating costs.

One way of looking at the economy of a configuration is through parameters such as payload fraction, fuel used per passenger and structural weight per passenger. As discussed by Torenbeek [151 and Van der Velden [121 the payload fraction has the most influence on the aircraft's economic performance since it contains terms expressing the fuel and depreciation cost in about the right ratio. For a supersonic aircraft with a range of 9000 km a payload fraction of at least 12% should be achieved if the aircraft is going to be economically competitive. A payload fraction of 16% represents a fraction not yet obtained by subsonic technology.

In section 2.3 a maximum thickness of 2.3 m was selected and this corresponds to a payload floor fraction of about 0.235. In section 2.2 the area required for one passenger and cargo was found to be 0.8 m 2 for a half normal, half economy layout. The total weight of a passenger and cargo is about 97 kg, which results in a payload loading (pl) of 119.38 kg/m 2.

The required wing area can now be expressed as: S= Wp/(pl Sp/s) Inverting and dividing by the takeoff weight gives: WtO/s= 28.0 [mt°/mp] The expression found for the wing loading results in values of the wing loading between 233 and 175 kg/m 2 for target payload fractions between 12% and 16% respectively. To investigate whether such wingloadings can actually be achieved from a technological perspective, we simplified the class I weight model further by expressing the tu:" three terms as a fraction of maximum takeoff weight: _Woe Wto= 0.179 Wto+ _ S+ 1.69 Wp + wf Fig. 2.2 shows that for a target payload fraction of between 12% and 16% and its appropriate wing loading, only current technology is required. The variation of operating empty weight with wing area can also be interpreted as an average structural weight per unit

areamultiplied by the ratio of total wettedareato the referencearea.Sincespanloading

wings do not exhibit a relative increasein structuralweight with an increasein size,this

number representsa level of structuraltechnologyindependentof size. The structural

weight per unit areavariesfrom componentto component, but for the OFW describedin

Ref. 41 an averagevalue of 35 kg/m2 was found. An earlier parametric study of 7

supersonicdesigns[18] found that their averagestructuralspecificwing weight was 38

kg/m2within a 10%range.

This meansthat the OFWs of interestwill havemaximumwing loadingsof around200

kg/m2 for a rangeof 9000 km and a payloadfraction of 14% using only conventional

technology.This is the samepayloadfraction thatthe B747achievesandmorethan50%

higherthanis projectedby thecurrentHSCT [16] studies.It is alsointerestingto notethat

this numberis notvery sensitive toexpected changes in technology level.

2.5 Factors Determining Wing Size The range of possible wing sizes for a given payload is limited by a number of constraints: Passenger floor For the near optimum root thickness of 2.3 m, a nearly fixed ratio between passenger floor area and total wing platform exists, which determines the minimum size of any vehicle.

Thickness-to-chord ratio For a given lift coefficient, freestream mach number, and a limit on local Mach number, there will be a limit to the thickness-to-chord ratio for which we can design the wing. For a freestream Mach number of 0.6, a normal local Mach number of 1.2, and limitations on the pitching moment this thickness is about 16%. This result is discussed in more detail in chapter 3.

Lift-to-drag ratio There will always be a lift-to-drag ratio poor enough to render a given range prohibitive.

For the oblique flying wing and a range of 9000km this lift-to-drag ratio is near 6.5.

'1' I I r I I I I i l I I I i I l 1 I I I I 'I I I I @.5 1 1.5 2 2.5 S ( 1E_8@m2 ) Fig. 2.4 Parametric study II L/O 12- M2 :588 R:gMm

.se

T:2.3m 10- .75 8 - _- 2.5 P_.P5 6 / __-_oaR B - H_ b/a 8.5 I 1.5 2 8.5 S (18188_2) Fig. 2.3 shows the influence of size on the performance of Mach 2 oblique flying wings with the following parameters assumed constant : altitude h=16.5 km maximum thickness tmax=2.3 wingloading W/S= 2 kN/m 2 propulsive efficiency 1"1--0.45 The lift-to-drag ratio was calculated as a function of the ellipse ratio and the wing planform area. The linearized expression for wing weight as presented in section 2.2, was used to find the required fuel weight and therefore the required lift-to-drag ratio to obtain the specified wing loading with a given payload.

Starting at point A in Fig. 2.3, an acceptable design, increasing the ellipse ratio with given payload is possible up to point B, where the maximum thickness constraint limits further reduction of the wing size. It is obvious that the smallest configuration with the highest payload fraction occurs at point C. This configuration accommodates 500 passengers. In this figure, lines of constant payload fraction are nearly horizontal. Increasing the lift-to- drag ratio by increasing the aspect ratio for a given payload will directly increase the payload fraction for a given altitude.

To be competitive, a new supersonic aircraft should have a payload fraction for this range which is similar to that of the B747. This means at least a payload fraction at least greater than 10%. A payload of 500 passengers constitutes an optimum since the payload fraction does not increase beyond this point, while the number of aircraft required by the market will go down.

For a given aspect ratio and aerodynamic refinement, the wing loading can be changed only by increasing range or by making the structure heavier (Fig. 2.4). Again we are bound by the minimum size aircraft constraint. If we start at A we can reduce the size of the vehicle until we are limited by the size of the passenger floor or the maximum section thickness. It is clear that the best configurations must employ very thick sections.

To obtain an economically competitive aircraft, the configuration should have a planform area between 800-1500m 2, aspect ratios between 8 and 12, and root thickness-to-chord ratios of at least 14%.

Fig. 2.5: Wave drag and induced drag of an oblique wing as a function of sweep angle and Mach number ellipse ratio 8, t/c*=0.153 Lift dependent drag factor K I I Drag due. to volume, factor KO. I r_ .

.......... i.......... !............ i,f

.......... ........... z.0 ....

0.6 .

.......... i............ i....... ii...........

0.4 .

I

0.2 ° .2 2.0 2.8 .2 2.0 2.8 Mach Mach The wave drag of the wing is the average wave drag of the individual area distributions of its equivalent bodies for a full roll of the Mach-plane.

The average wave drag at C is much higher than at A because the wave drag is quadratically related to the amplitude of the areadistribution.

_b" SF-B: shortest obliq uivalent body Mach angle / LEB: longest equivalent body Mach 1.2 I-_ Mach 2.0 [_ Mach 2.8 ]5 2.6 Effect of Cruise Mach number, Sweep and Altitude In Fig. 2.5 the potential drag for the OFW is depicted as a function of wing sweep and Mach number. The factor KO represents the ratio of the volume-dependent wave drag to the Sears Haak area distribution wave drag. The factor K represents the lift-dependent drag. The total potential drag can now be expressed as CD= KO CDsear s Haack+K CL 2 It is clear that in- or decreasing sweep by more than a few degrees from the optimum sweep angle results in very high drag penalties. The supersonic area rule, as presented by Robert T. Jones [19] and Harvard Lomax [221, provides us with an explanation. It states that the wave drag of a supersonic configuration is related to the average wave drag of all the equivalent supersonic bodies. The beginning and end of an equivalent supersonic body is determined by the intersection of a line along which pressure differences can travel [M+ to M-] and the body center line.

For a Mach number of 1 all equivalent bodies have the same length and the average wave drag of the wing is just the minimum, KO=I. Increasing the Mach number will increase the difference in length between the shortest equivalent body and the longest equivalent body.

Since the wave drag is quadratically related to the amplitude of the area distribution, the average wave drag of the area distributions is much higher at Mach 2.8 than at Mach 1.2.

Further increase of the Mach number will make the leading edge supersonic. For a supersonic leading edge, the equivalent body length associated with the M- lines will go to zero for an oblique wing of infinite aspect ratio. For an unswept aspect ratio of 10.2 the volume-dependent wave drag will be 25 times higher than the minimum value obtained at 90 ° sweep.

Using the aerodynamic model of section 2.2, we can calculate the lift-to-drag ratio of the configuration as a function of Mach number and the Mach number normal to the average sweep angle. For each Mach number, a different wing normal Mach number gives the maximum value of lift-to-drag.

Mach<l.6 Mn=0.7 1.6<Mach<2.2 Mn=0.6 Mach>2.2 Mn=0.5 The sweep of the long axis of the ellipse can be found by taking the inverse cosine of the Fig. 2.6: Effect of Mn Number on Lift-to-Drag ratio h=16.500m, S=1500m2, ell=8, W/S=2kN/m2

i ...... 1-;- Mn=0 I

16.

........ \ .....!................... :...................:...... i._._ B_ Mn=0.6 I

14.

:: :: :" IT Mn 07 I.

E3 12.

10.

, 1.0 1.5 2.0 2.5 3.0 Math fraction on 2.7: Effect of Altitude and Payload Fig.

Lift-to Dra.ci ratio _p/Hto L/O m 1_'- L ,/D R - g ['lr_ =L+62 i !

12- 1(3- 1L+ 8 - 1 [ 1 1 I I 1 ] r 1 1 1 ] I I | l l 1 I I 1 1.5 _ 2.5 3.@ -] M_CH ratio of normal to freestream Mach number. It is clear from Fig. 2.6 that designing an airfoil for Mn=0.6 will result in good lift-to-drag ratios over the entire Mach range.

In Fig. 2.7 the lift-to-drag ratio and the payload fraction of the configuration with varying altitude are shown based on the full model described in 2.2. Even though the maximum lift-to-drag ratio is obtained for the highest altitude considered for every Mach number, an altitude of 16 km will result in near maximal payload fractions over the entire range of Mach numbers considered. Beyond 16.5 km the engine weight will increase almost as quickly with altitude as the fuel weight saved by the lift-to-drag ratio increase. [Note: 1] From these considerations it follows that a CLM 2 of 0.3 will be usable for any OFW cruising at Mach numbers between 1.5 and 2.8. Since we established in section 2.3 that for a given altitude, maximum lift-to-drag ratio and payload fraction are interchangeable we could use lift-to-drag ratio as the new objective function.

If we were to change the design range, the initial cruise wing loading and altitude would change, but, to first order, not the design lift coefficient. Kuchemann & Weber [10] as well as Torenbeek [ 15] have pointed out that the optimum ratio of the cruise lift coefficient and the lift coefficient at which the maximum lift-to-drag ratio is achieved remains constant for long range aircraft. Since increasing the range will mean taking on additional fuel with constant wing area the maximum lift-to-drag ratio will remain constant except for the influence of altitude on Reynolds number. The following chapters may therefore be applicable to any range between 6 Mm and 12 Mm.

Note: Ill In a private communication Robert T. Jones disagreed with this statement.He agreed that one needs a greater engine diameter for flight at higher altitudes, but he argued that the loads on the engine didn't change so the weight of the engine shouldn't change either. The bigger engine should than be downrated for operation at lower altitudes.

3 DEVELOPMENT OF THE AIRFOIL 3.1 Objectives and Requirements In the previous chapter we showed that an airfoil for which CL M 2= 0.3 and which has a thickness of at least 14% will give the configuration an economically competitive payload fraction.

The airfoil must be designed with a sufficiently small transonic wave drag. For current subsonic transports, a drag rise of 20 counts (0.002) is considered acceptable for the high- speed cruise condition. According to simple sweep theory, a 70 ° swept wing with airfoils operating at this drag rise condition will have a total drag increase of only 1%.

The airfoil should be designed for Mach numbers between 0.5 and 0.7, to be used for the full range of OFW sweep angles and Mach numbers. We will attempt to reach this goal with just one airfoil-family design. In section 2.2 we mentioned that if control authority is provided by a narrow trailing edge flap, the location of the liftvector is limited to 32% of the chord.

Other considerations in the design were related to the utilization of the geometry by passengers. The relative thickness distribution over the chord was chosen to enable maximum use of the cabin Since passengers only accept very small inclinations of the cabin floor, the airfoil bottom was designed in such a way that for most of the operation the floor will be level. The airfoil section should also provide adequate space for the landing gear and systems.

3.2 Baseline Airfoil Design We designed the airfoil for a Reynolds number of 200 million, and set the transition ,:t 2% from the leading edge, to simulate the cruising conditions. For a two aisle layout the best utilization of the cabin is obtained with the maximum thickness at 35% of the chord.

At Mach 0.7, the upper surtace normal Mach number distribution needs to be close to the maximum shockfree value over half the chord if we want a resultant lift vector at 32% of the chord. We found that an airfoil with a flat pressure distribution at Mach 0.7 had the same drag-rise CL M 2 between Mach 0.5 and Mach 0.7.

The recoverywasdesigned with Head's[ 17] turbulentboundary layermethod.Wedefined

the separationpoint at H=2.0. Low-speedairfoil designswill havethe highestpressure

gradientsat the beginningof the recovery. But such steepgradientswill changethe

locationof the shockat the higherMachnumbers,andthereforetherecoveryhasto start

more gradually.The lower surfaceand the aft uppersurfacearethen tailored to get the

appropriate pitchingmomentandthickness.

TheOW701014 [20] airfoil hadtherequiredflat supercritical pressure distributionatMach

0.7. We thereforemodified the pressure recoveryof this airfoil to satisfy the pitching

moment requirements,while leaving the first half of the upper pressuredistribution

undisturbed.Thesemodificationsweredone with Panda,an interactive subsonicairfoil

designprogram. After eachmodification,we analyzed the airfoil pressure distributionfor

Mach0.5, 0.6 and0.7 at the requiredlift with ARC2D. For mostof this designphasea

coarsegrid (Fig. 3.1) wasusedsincethis enabledus to run jobs of lessthan200seconds

on the CrayXMP. As canbe seen,mostof the grid pointsarein the boundarylayer.

Fig. 3.2 shows the iso-Mach lines at the design condition of Mach 0.7. The rapid

accelerationof the flow at the noseandthe constantlocal Mach numberof 1.2 on the

rooftop arevery notable.The wavinessof the lines is causedby grid coarseness. It is also

possible to look at the boundarylayer velocity in more detail. Figs. 3.3a,b show the

nondimensionalizedvelocity and Mach number acrossthe boundarylayer during the

recovery.Theflow is fully attached andtheboundarylayeris very thin.

Figs.3.4a,b,c,d showthe 16%OFWairfoil pressure distributionat thedragdivergence lift

coefficientfor Mach 0.50,0.6, 0.65and0.7.For all of theseMach numbers, the required

trimmedlift wasjust obtainedwith the present baseline design. Onecanclearly observe the

shocktravellingfromthe leadingto thetrailingedgeastheMachnumberincreases.

Figs.3.5 a,b,cshowthecharacteristics of theairfoil designaccordingto ARC2D.We have

also included results from an analysisdone with the BAUER-code.The BAUER code

predictssignificantlylessdragthanthe ARC2D code.However,oncethe ARC2D grid is

refinedfive-fold, thedragof thesectiondecreases by 30countsandthepredicteddragsare

very close. The pressure distribution, and therefore the pitching moment and lift

characteristics, remained unchanged by thegrid refinement.

2O

GR]O Fig. 3. i F I .5O -0.50 -0.25 0100 0.25 0.50 0.7'5 x Fig. 3.2: ;_ines of Constant .'lach Number O. 700 M_ @ 1.50 °

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3.30" o o 2.00MIO' Re I .47-t0:' T_,me i93x40 GRID 7" CD _ \, \ o \,, "3'. 5 @'. 6 0'. 7 0'. 8 0'. 9 0.0 ]'. I 0'.2 0'.3 0.4 × Fig. 3.4f d_ 0.600 M.

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\ O.O £'.1 0'.2 0'.3 0'.4 0'.5 0'.6 0',7 Or.8 0'.9 Characteristics of 16% OFW airfoil Fig. 3.5" Re 2e8, transition at 2% chord, 193x40 grid a: Lift 1.1 1.0 --...B--- ARC2D M0.6 ARC2D M0.7

__._ A°c2o Mos

i ............... • ................ - .............

.9 LBAUER MO.7 CL .8 .7 .6 .5 : grid refined : 500x100 .4 • o .3 .2 0. 1. 2. 3. 4. 5. 6.

oc (degrees) b: Drag .025 .020 CD .015 .010 .005 .000 .2 .3 .4 .5 .6 .7 .8 .9 1.0 1.1 CL c: Pitching moment -.02 -.03 -.04 £M -.05

ii i iiiiiiiiiiiiiiii il • • iii

-.06 -.07 ................ _,.: .............................. _._ ..........................

-.08 .2 .3 .4 .5 .6 .7 .8 .9 1.0 1.1 CL Fig. 3.6: OFW family of airfoils .08 N .06 .04 .02 .00 -.02 12% -.04 16% -.06 I I I I I I I I I -.08 1.0 .0 .1 .2 .3 .4 .5 .6 .7 .8 .9 X/C 3.3 Thickness Trade-off After the baseline t/c=16% airfoil was designed, it was possible to design a family of airfoils with the same pitching moment characteristics but different thicknesses. Since the upper surface of the airfoil is already designed close to the optimum, only the lower surface remains as a variable. In addition, in order not to disturb the flow on the upper surface we had to maintain the lower surface leading and trailing edge geometry.

Airfoils of this kind can be obtained by adding or subtracting a thickness distribution represented by a transformed sine function over a 0 to n range. The added thickness distribution is zero on the trailing and leading edge and reaches a maximum at 40% of the chord. Adding this thickness distribution to the lower surface did not change the location of the center of pressure. These airfoils are shown in Fig.3.6. Instead of referring to them by their actual thickness t/c we use t/c*,.which is defined as 1.5 times the section volume parameter (=area/t c). This variable has more physical significance than t/c if we want to relate the achievable lift to the section thickness.

Figs. 3.4 e, f show the 12% and the 14% thick airfoils with their pressure distributions at ot=3.3 ° and M 0.6. In Fig. 3.7 and 3.8 one can see the lift, drag and pitching moment of these airfoils for the drag-rise condition, with and without trim. We trimmed the airfoil with a 10% narrow trailing edge flap. As expected, lift is lost when thickness is increased. Since we know the drag-rise characteristics of the airfoil and the CL at which drag-rise occurs for each airfoil, we can now model drag-rise in the aircraft polar.

To select the best maximum thickness for our configuration, we express the change in takeoff weight relative to the baseline configuration (point C in Fig. 2.3) as a function of the maximum thickness-to-chord ratio. The size of the passenger cabin and the takeoff weight are fixed.The maximum thickness ratio influences the takeoff weight because it 1) changes the planform area required to accommodate the passengers, and this area determines the structural weight fraction, and 2) changes the lift-to-drag ratio, and this ratio determines the fuel weight fraction.

The lift and drag coefficient at the drag-divergence condition can be expressed as: CL ar=CL ad,2D cos 2A CD dd=(CD parasite,rel-CD wave.ref)+CD wave,ref +K CLal where:

CLdo,2o

Drag diverergence CL for varying t/c* from Fig. 3.7 CD par asite,re t-CD wave,ref 0.00413 from Ref. 41 CD wave,ref 0.00133 from Ref. 41 or Section 2.2 0.370 from Ref. 41 or Section 2.2 K The required ratio of planform area to reference planform area for a constant cabin floor area follows from the relation in Fig. 2.1. For a given ellipse ratio this corresponds to a certain t/c* which, in turn, can be converted to a lift-to-drag ratio by the previous relations.

The fuel weight fraction can be calculated with the I3requet equation. The change in planform area will also influence a fraction of the structural weight. If we assume a constant total weight and a varying payload fraction, the structural weight fraction that depends on the wing area will vary linearly with the planform area. For a constant total weight the powerplant weight fraction will increase linearly with drag and inversely with ambient density:

IWpPref0 fCL.fCO}:IWP ref COS)

w ,j lW- o, f a cD--6-7 fCL lWtoro----TCD , The combined effect on the weight fractions due to variation of the thickness ratio is shown in the last column of table 13.3.1. As observed before qualitatively, the best root thickness- to-chord ratio is around t/c*=O.153. Serious penalties can be expected for deviations of more than 10% from this wdue.

table 3.3.1: Structural and Fuel Weight Fraction as a Function of Thickness Ratio [S l=[Spref I Wf Ws [tmax] I_-?JL--_-oJ [t/c*] [L//DIDD [W-go] [W-_o] tWtoj[Wpp] [W_Wf+Wpp] 2.1 1.81 0.104 11.6 0.38 0.33 0.19 0.91 2.2 1.24 0.132 10.7 0.41 0.22 0.12 0.76 2.3,ref 1.00 0.153 9.89 0.43 0.18 0.10 0.72 2.4 0.92 0.166 9.29 0.46 0.17 0.09 0.72 2.5 0.85 0.195 7.82 0.51 0.15 0.09 0.76 2.7 0.73 0.229 6.01 0.62 0.13 0.08 0.83 Fig. 3.7: Thickness tradeoff ACD=0.002 and trim @ 0.32c 1.1 1.0 .9 E

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.6 .5 .4 .08 .10 .12 .14 .16 .18 .20 t/c* (=1.5 AREA) Fig. 3.8: Airfoils analyzed with ARC2D M0.6, alpha=3.3, grid:193x40 1.1 1.0 ........... i..... _____i ............ " ........... _._.._.._.o..o..

.9 ......

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.5 .08 .09 .10 .11 .12 .13 .14 .15 .16 t/c* (=1.5 Area) 3O 4 THREE DIMENSIONAL WING DESIGN 4.1 General considerations The following considerations play an important role in the determination of an acceptable pressure distribution on the Oblique Flying Wing: 1. The wing has to provide sufficient resultant lift at 32% chord location. This can be achieved by scaling the airfoil pressure distributions with simple sweep theory.

2. The loading of the wing should be nearly elliptic in both the y- and x-projection to minimize the induced drag of a supersonic oblique wing. It can be shown that for a high (unswept) aspect ratio oblique wing, the loading is near-elliptic in the x-projection when it is elliptic in the y-projection I2].

3. A Sears-Haack area distribution minimizes the volume-dependent wave drag of the wing [3, 48].

4. In the design of the airfoil we used a Navier-Stokes code. This code models shocks and shock-induced flow separation. We designed the airfoils in such a way that shocks gradually move to the tailing edge and gradually increase in strength with increases in angle of attack. The linear theory that is used to design the three-dimensional wing does not model these non-linear effects, but if the wing has the same transonic local normal Mach numbers distribution as the airfoils, we can expect it to have similar non-linear behavior.

5. To prove that the three-dimensional wing is separation free we wJuld have to do a three dimensional boundary layer analysis. In the absence of this analysis we should at least show that the flow has a positive velocity in the direction normal to the isobars.

6. The prescribed pressure distribution should result in a geometrically realizable w, qg. It is for this reason that we propose to specify the thickness distribution and warp the mean camber surface of the wing to satisfy all other requirements.

4.2 Wing Planform and Basic Thickness Distribution We find the wing planform and the airfoil selection along the wing span from the following considerations:

a) A minimim sizewing withoutexcessive dragis obtained if thedragdivergence lift

coefficientCL,dd(t/c*) is achieved over thewholewingspan(Fig. 3.7).

b) A wing with minimuminduceddragis obtainedwith elliptic loading.Fora given

CL(t/c*)c attheroot wecanfind theCLCata givenspanwise locationy.

c) A wing with minimum volume-dependent drag is obtained with a Sears Haack area distribution. For a given A(y)= 5- [ c2 t/c*] at the root we can find the A(y) at a given spanwise location y If we select a root t/c*--O.153, as suggested by the analysis in section 3.3, we know the distribution of CL,ddC and A over the span. We now have two known quantities (i.e.

CL,dd and A) and we have to solve for the two unknows t/c*(y) and c(y).

In Fig. 4.1 the spanwise distribution of chord, thickness-to-chord ratio and local lift coefficient are depicted for a wing with a Sears Haack area distribution and elliptic loading with OFW-airfoils. The wing planform resembles an ellipse, but is slightly more tapered.

The higher taper ratio, and therefore decreased wetted area, is possible because the airfoils become thinner near the tips. Thinner airfoils have higher lift coefficients, so less chord is required to carry a given load.

To simplify the manufacturing the wing consists of a limited number of linearly tapered sections instead of a continually curved leading and trailing edge. It is clear that deviation from the ideal area-distribution will result in extra volume-dependent wave drag. However, such small deviations from the elliptic loading will not cause significant additional induced drag.

To evaluate the trade-off between the number of linearly tapered sections and the drag we wrote a program to calculate of volume-dependent wave-drag of area distributions based on the Eminiton-Lord method [21]. This analysis showed that the wave drag of a linearly tapered oblique wing is almost twice that of a Sears Haack body. This would mean a decrease of 18% in lift-to-drag ratio of the OFW. Even the absolute wave drag of a linearly tapered wing was a third higher than that of a Sears-Haak area distribution although the volume was less. With the OFW wing design presented in Fig. 4.2 we obtained a wave drag that was no more than 13% higher than the theoretical minimum (AL/D=2%).

Fig. 4.1: Wing design on the basis of the airfoils 1.4 passenger 1.2 cabin 1.0 .8 .6 ,_-°°,°°, .4 Chord [c(y)]/[c(root)] .2 Thickness [t/c*]/[t/c*(root)] actual tip Lift [Cl(y)]/[Cl(root)] geometry .0 1.0 .0 .1 .2 .3 .4 .5 .6 .7 .8 .9 y/b Fig. 4.2: OFW planform and thickness distribution .4 .2

0.0

I I OFWA14 OFWA12 "mOFWA16 J o -.2 N -.4 -.6 -.8 1.0 .0 .1 .2 .3 .4 .5 .6 .7 .8 .9 2y/b 4.3 From a Two-Dimensional Pressure Distribution to a Three-Dimensional Wing Introduction In the 1960's Lock [23] developed techniques to transfer the pressure distribution from an airfoil to a wing based on local normal Mach number. He pointed oot that the major disadvantage of the transformation lies in the fact that the magnitude of the velocity determines the local Cp, but its direction determines the local normal Mach number.

Therefore, this transformation can only be made if one assumes that the local velocity has the same direction as the freestream velocity. This assumption is incorrect for low-aspect ratio wings as well as near the tips of high-aspect ratio wings. Another disadvantage is that if the upper surface pressures are determined by the airfoil normal Mach number distribution, the lower surface pressures, and therefore the loading (Cp,lower-Cp,upper), are determined by the thickness distribution. To control the loading, the thickness distribution has to be changed which may result in unrealizable geometries. This is probably the reason that simple sweep theory is often used to transfer pressure distribution from airfoil sections to the wing. Simple sweep theory relates the wing pressures to those of the airfoil with the following expression: Cp,3D = Cp,2D c°s2Aref Instead of using the angle between the quarter chord line and the Y-axis as the reference sweep angle, Boppe [36] proposes using the angle between the line of two-dimensional shock locations projected on the wing and the Y-axis (see Fig. 2). Although this will assure a correct transformation based on normal local Mach number near this line, the transformation is incorrect for other locations and, in some cases, the shock may well appear elsewhere.

Boppe used an inverse panel code to find the slopes of the wing camber surface after he determined the desired pressure distribution based on simple sweep theory. It could be argued that with a non-linear inverse design code such as Boeing's A555 [38], a wing with any thickness and pressure distribution can be designed. There are however two major disadvantages in using non-linear inverse codes. First, they require more storage and speed than a linear potential code. Another disadvantage is that the perturbations due to thickness and camber can no longer be added, depriving us of a powerful tool in wing design. We will show that it is possible to use this linearity to determine the pressure distribution for a low-drag, geometrically-realizable wing.

Overview Unlike previous method, ff_e present method accounts for the influence of local taper, local sweep and three-dimensional induced velocities to specify the wing pressure distribution based on airfoil data. In an iterative way, the pressure distribution is calculated from the potential flow velocity perturbations for a given thickness distribution and the prescribed vorticity. The present method combines the philosophy of Lock's and Boppe's method to prescribe the vorticity on the wing. The vorticity in supercritical wing regions is based on airfoil transonic normal Mach numbers and includes the influence of local sweep and three- dimensional induced velocities, so that the appearance and the strength of the shock waves can be expected to resemble those of the airfoil. The vorticity in subcritical wing regions is scaled first with simple sweep theory, and then to achieve the desired load distribution.

Fig. 4.3 presents the computational algorithm. The airfoils, which were designed with a Navier-Stokes code [24], were analyzed with the potential flow code that was used for the three-dimensional wing geometry, at the same Mach number and angle of attack. The vorticity and the local nomlal Mach number predicted by the potential flow code were used to specify the vorticity (Uvort) distribution on the wing. In turn, the vorticity distribution is used to solve for the wing's camber with an inverse panel code. The induced velocity perturbations of this cambered wing are used in the next iteration.

Like those mentioned in the introduction, this method does not include a three-dimensional boundary layer analysis. Such an analysis is required if we want to check for flow separation. In the absence of such an analysis we would at least have to show that there is a positive velocity in the direction normal to the isobars according to potential flow. If a three-dimensional boundary layer analysis shows separation, the effective boundary layer may be added to the geometry and the present method repeated. 1¢ the separation occurs because the induced velocities increase the adverse pressure gradients relative to the airfoil's, the airfoil pressure recovery must be redesigned.

Detailed Description From the arguments mentioned in the introduction it follows that specifying the vorticity distribution based on local normal Mach number will generally not result in a wing that has the desired load distribution. On the other hand, if one specifies the vorticity based on simple sweep theory one cannot control the location of the shocks, and the non-linear pressure distribution may look very different from the linear pressure distribution.

Fig. 4.3: Determination of the wing camber distribution ARC2D 193x40 grid pressure distribution WlNG3D 20 panel pressure distribution op m Irrl I -2.0 3.

/ -1.5 analyze section for -1.0 -.5 same angle .0 of attack by panel code .5 M < 0.95 1.0 .0 .1 .2 .3 .4 .5 .6 .7 .8 .9 X/C Input"

I

Mn, 4 Uvort for each airfoil Location of airfoil_ Determine Uvort for each panel:

on the 3D wing, --tw

wing elanform Transformation method 2D Flow supercritlcal A: Mn2D=Mn3D geometry Program settings I _""_ subcrltlcal 4Uvort B: simple sweep theory and scaling for elliptic load UvortB < Uvort(Mnl.0) Iteration file: critical C: weighted average A,B perturbations UvortB < Uvort(Mnl.0) due to lift and thickness Inverse Panel Code

I

i: :::: _:.._.._ _ :.::. :_:.::, ...........

We therefore propose to use different criteria for different parts of the wing: To satisfy the loading constraints we scale the vorticity of wing regions with subcritical pressures based on simple sweep theory. To obtain elliptic loading on the wing, it may be necessary to scale this distribution again with a correction factor MY). To avoid the occurrence of shocks and separation the local normal Mach number should be limited to positive subsonic values: Uvort,3 D (Mn=0.0) < Uvort,3 D < Uvort,3 D (Mn=l.0) Uvort,3 D = _. Uvort,2 D coS2Aref but; If the transonic normal Mach numbers of the wing are the same as the corresponding airfoil Mach numbers, the non-linear effects on the three-dimensional wing will correspond approximately to those of the airfoil. Therefore it is sufficient to transfer only the supercritical pressures with the local normal Mach number criterion: Un3D MI,n,3D "-- a3 D - M2D" Using second-order small perturbation theory, we can write expressions for the local normal velocity and the local speed of sound to obtain the local normal Mach number. All the velocities are normalizexl with the freestream velocity: Un,3 D = 5/{(l+u)cosA+vsinA} 2+w2 Local normal velocity UI = 3/(l+u)2+v2+w 2 Local velocity a= "_/I'/M 2+_(7-1 )( 1- U '2) Local velocity of sound [301 M is the freestream Mach number, In these expressions: Ais the panel quarter chord sweep angle, anu the normalized x, y, z- perturbation velocities.

ti,W,V are Fig. 4.4 shows the panel geometry representing the wing. For the upper surface, the perturbations on each panel of the aircraft can be expressed as: = =U'+u Utop Uthic k + Ulift + u vort vort, .... V' R 1 and Vtop Vthic k + Vlift (dt/dx)/2 sin® u vort R1 - u vort ' Wtop = Wthic k + Wlift + (dt/dx)/2 cosO - u vort R2 = W' - u vort R2.

For the lower surface they can be expressed as: Ubo t = Uthic k + Ulift - u vort = U'-u vort ' Vbot = Vthick + Vlift + (dt/dx)/2 sinO + u vort RI = V' + u vort R1, and Wbo t = Wthic k + Wlift - (dt/dx)/2 cosO + u vort R2 = W' + u vort R2.

In these expressions the subscripts thick and lift refer to the perturbations due to thickness and lift on the panels due to the other panels. R 1 and R 2 are the tangents of the vorticity in the YZ and the XY plane respectively: (Y2-Y1) (X3-X 1-X4+X2+X2-X 1) (Z2-Z 1) (X3-X 1-X4+X2+X2-X 1) R2= R 1 = ((y2_Y1)2+(Z2_Z1)2) ((Y2-Y 1) 2+(Z2-Z1) z) For a planar wing these expressions are simplified: Wthick=Ulift=Vlift=0. We can now write the equation for u vort,3D based on local normal Mach number: M2D = MI,n,3D M2D a3D-Un,3D = 0 M2D 2 [1/M2+l/2(Y--1) (1-(l+u)2-v2-w2)l - [{ (l+u)cosA+vsinA}2+w2}] = 0 a u vort2+b u vort+C = 0 where: a = - Q- Q R12- Q R22- C 2- R12 $2+ 2 R 1 S C- R22 b = -2 Q+ 2 v' R 1 Q+ 2 W' R 2 Q- 2 C2+ 2 v' R 1 $2+2 R 1 S C+ 2 U' R 1 SC+ 2 W' R 2 C _ M2D2/M 2 - 2 Q U'-Q u '2- Q v '2- Q w '2- C 2- 2 u' C2- u '2 C 2- v '2 s 2- 2 v' S C - 2 u' v' SC- w '2 Q = I/2(Y--I) Mto p , C = cosA, S = sinA.

-b + _/[b2-4ac] The solution is: Uvort,1,2 = 2a The largest u root violates the small perturbation assumption, so we select the one with the smallest absolute value. Problems may occur when the actual normal Mach number of the panel is far from its design normal Mach number.

Fig. 4.4: Oblique wing sign conventions

Y

Cy pitch: Cm I Cz

X

Cx roll: Cr yaw: Cn

/

Z

Y

normal to mean plane, proj. ZY Panel with positive perturbations that satisfy the boundary conditions.

For example; when the normal component of the freestream Mach number on a panel exceeds 1.0 and a subsonic normal Mach number is prescribed on the top and the bottom of the panel, we are unable to resolve the discrepancy by changing the vorticity. Such problems can be solved by increasing the panel sweep.

We can now compute the target pressure distribution over the wing, using a first or a second order approximation: First order: Cp = -2u Second order: Cp = M2u2+( 1-UI 2) = M2u2+ 1- [( 1 +u)2+w2+u 2] .

The final perturbation U 1 can be corrected with Riegel's correction [37] to improve the pressure coefficients estimates near the leading edge: U21,top,Riegels = Ui2/(1 +( (dt/dx)/2 + dz/dx) 2) U 2 = Ul2/( _ l,bot,Riegeis 1+( (dt]dx)/2 dz/dx)2).

Which expression is superior depends on the code that is used to calculate the wing camber. If the code uses only the first order relation to determine the panel mean line slopes than we should use ACp (=4Uvort) as input. The final result will still be second order accurate if the final perturbations are combined to a second order accurate pressure distribution. If the panel slopes are determined by higher order relations, the second order expression for ACp should be used. The inverse code will return the required induced velocities and incidence (dZ/dx) for each panel. These values are used for the next iteration.

The perturbations to start the iteration can be obtained from an analysis of the wing with the original airfoil camber.

The panel code used in this paper was written by Ralph Carmichael and later modified by the authors for oblique wing research. The modifications include the method described in this paper and an improved drag calculation. The original code WING3D is based on work by Woodward [39] and Carmichael. WING3D solves the Prandtl-Glauert equation in subsonic and supersonic flow: ( 1-M2)_xx+t_yy+_zz=0.

The perturbations are defined as: u=_/Sx, v=5O/Sy, w=5_/Sz.

Sources, sinks and vortices that are distributed continuously over each panel are solutions

to the Prandtl Glauert equations.Sincewe know their inducedvelocity distribution in

space,we can usethem to constructthe flow field aroundthe wing. For this purposean

aerodynamicinfluence matrix [AI] containing the induced u-perturbations for each

singularityis setup, invertedandmultiplied throughby a columnvector representing the

wing meansurfaceboundaryconditions.In this way we obtainthe u-perturbations at each

panel: [4 Uvortl = [A CP]= [AI] -1 [dZ/dx-O_l.

It is also possible to solve ffpr the wing camber if the u-perturbations are known: [dZ/dx-0_]= [AI] [4 Uvort 1.

Application of the Method First we investigate the pressure distribution over a 16% thick airfoil with ARC2D, a Navier-Stokes code and WlNG3D, a potential flow code. Fig. 4.5a shows the pressure distribution at _x=3.3 ° and Mach 0.6 calculated with ARC2D. We used a coarse 193 x 40 grid. There is a weak shock just aft of the leading edge. The center of pressure of this section is at 32% of the chord. Fig. 4.5b shows the pressure distributions calculated using WING3D with and without Riegel's correction. The distributions look similar to the one generated by ARC2D, though the pressures are a bit too high on the first part of the upper surface, resulting in an undcrprediction of the lift coefficient by about 7%, and the center of pressure is at 36% of the chord. In this case, probably due to the coarse panelling, Riegel's correction does not increase the accuracy. We will use the vorticity and the local normal Mach number distribution of the sections analyzed with the potential flow code for the inverse design. The reason the potential flow solution is preferred over the ARC2D solution can be easily understood in the two-dimensional case. If the pressure distribution from 4.5b were used to inversely design the camber distribution, we would get back the same airfoil. This airfoil _ould give the non-linear pressure distribution of Fig. 4.5a if it were analyzed with the ARC2D. To first order, this is also true for wings.

Next, we apply the method to the 72 ° swept oblique flying wing SST. Fig. 4.6a shows an artist's impression of this transport. Because the normal component of the freestream Mach number changes as much as 0.4 from the leading to the trailing edge, the effective local sweep and taper must be included in the transformation. In addition, the low aspect ratio may induce significant three dimensional lateral velocities that change the local normal component of the flow.

Oneinversedesignof the oblique flying wing wasbased on the methodpresented in the

previoussectionandanotherusedthe simple sweepmethod.Both designshada straight

32%chordline. Because the referencesweeps on the forward andaft wing aredifferent,

Boppe'smethodresultedin a wing with unequal lifts on the wing halfs.Forcingthe normal

Mach numberto be the sameeverywhere on the wing, asLock proposed,would be very

difficult because of thelargevariationof thefreestream normalMachnumbercomponent.

For an elliptically loadedwing this would result in intolerablechangesin the thickness

distribution as well as largewavedrag. We limited the numberof panelswith thickness

representing the designto 400to improvetheconditionnumberof theaerodynamic matrix

andtoreducethecomputationtime (20 panelsof the samewidth in spanwise directionand

20 panelsof the samefractionalwidth in chordwise direction).

Fig. 4.6b showsthe cruisepressuredistribution. The suctionpeakon the uppersurface

decreases from the forward tip to the aft tip. This is because the sweepangleof the panel

increasesin this direction, and more suctionis requiredto producethe sametransonic

normal Machnumbers.In Fig. 4.6d we seethat the normalMach numbers on the leading

edgeof the presentmethoddesignarethe sameasfor the airfoil, while the simple sweep

designovershoots this targetby asmuchas0.1. The simplesweep designalsohasa weak

shocknearthe 60% chordlocation of the aft tip. Fig. 4.6c showsthat the control of the

normal Mach numberby the presentmethodis achievedby varying the load distribution

from tip to tip with respectto the simplesweeploaddistribution. The loadingis decreased

significantly on the leadingedgeof the forwardtip andthe60%chordlocationof the aft

tip. Because the centerof thewing is untapered andthe lateralinducedvelocitiesaresmall, its loaddistributionsarevery closeto thedistributionsaccordingto simplesweeptheory.

In Fig. 4.6f the effective angle of attack of the meansurfaceis shown.The camber

distributionresembles thelinear antisymmetric distributionsuggested by previousoblique

wing research[20,29]. The forward-facingwing tip is substantiallymorecamberedthan

the rearward-facingtip. Using the scaling factor, X, we were able to producea nearly

elliptic load for a planarwing asshownin Fig.4.6e.

4.4 The Calculation of Drag with Coarse Paneling We designed the wing for target pressure distributions, but after this is done we still have to analyze the wings off design characteristics. If we use pressure integration to calculate the drag of wing, very dense paneling is required with any panel code. More than ten panels on the first 1% of the chord are needed to get the leading edge suction right, and Fig. 4.5: The Pressure distribution of the OFWA 16 a. Analyzed with ARC2D. (193x40) grid CL=0.81 Cm=-O.05 /alpha=3.3 de(]., Mach=O.6 un 0.600 M.

3,30" o 2,00.10' RI 1,47"101 T_ml Ig3_ 40 GRID

i __

.2"7 \\ \ &- 0.0 0'.1 0'.2 O'.] 0'.4 0.5 0'.6 0'.7 0'.8 0'.9 × b. Analyzed with Wingbody. (20 panels + thickness) CL=0.76 Cm=-0.08 /alpha=3.3 dog., Mach=0.6 cp -2.0 -1.5 -1.0 -.5 .0 .5 1.0 .0 .2 .3 .4 .5 .6 .7 .8 .g 1.0 x/c Fig. 4.6a: The oblique flying wing b: Upper surface pressure distribution forward tip aft tip Cp,up -'30 present method: simple sweep -.25 2y/b=-0.85 theory: 2y/b=-0.35 .'---0_ 2y/b=0.35 -.20 2y/b=0.85 -.15 -.10 -.05 0.00 .05 .10 .

X/Croot c: Chordwise load distribution forward tip .25 2 /b=-0.85 ACp 2 '/b=-0.35 2 '/b=0.35 .20 2 '/b=0.85 .15 reduced i.

.10 .05 .00 -4. -3. -2. -1. O. 1. 2. 3. 4.

d: Upper surface normal Mach number distribution 1.3 M n 1.2 1.1 1.0 .9 .8 .7 .6 .5 .4 X/Croot e: Spanwise load distribution forward tip aft tip .22 .20 CLC .18 .16 .14 .12 _ target elliptic l ................... _ present method design ............

.10 .08 .06 - .0 -.5 .0 .5 1.0 f: Camber distribution t0.

isochord= .0250 isochord= .1750 , isochord= .3250 isochord= .4750 6.

di/dx isochord= .6250 (deg.) isochord= .7750 4.

isochord= .9250 2.

O.

-2.

-4.

"6.

-8.

-1.0 -.5 .0 .5 1.0 2 y/b even with such fine paneling the induced drag cannot be predicted with an accuracy of more than about 10%. The Woodward code assumes that the wing is thin (i.e. dt/dx<l) and therefore additional errors would occur if we were to use fine paneling near a rounded nose. In this section we will investigate whether it is possible to get reasonably accurate results for the drag with coarse paneling.

Lift-dependent drag The induced drag of each wing strip can be expressed as: n CDi/q=_(ACPi dZ/dxi Area i + ATx,ile/q) i=l where: Area. is the projection of the panel area on the XYplane ATx,i le/q is the leading edge suction in X-direction R.T. Jones [25] gives the following expression for the X and Y component of suction force on leading edge panels of wings with subsonic leading edges: ATx, i/q= -2_x ,_-l-m 2) limx__xlU2(x-xle), and ATy,i/q=ATx, i/q tan A Multhopp [ 11 ] and Garner [91 evaluated this limit for incompressible flow assuming that u- perturbation on the chord is expressed in Fourier coefficients: u=b/_:c [7 cot (0/2) +4l.t {cot (¢_/2)-2sin_}+_c{cot (_/2)-2sinO--2sin2_}..]

so; limx_>xleU2(x-xle)=(b/Tzc)2 [_,+4_t+_ ..... ]2 The first two terms of this expansion can be related to the lift and pitching moment coefficients of a strip formed by the corner points of the panels over the le_qth of the chord, as follows: ¥ =CLc/2b and la = CM b4 "c/2b In this expression the peturbations other than 7 represent the u-distribution at the ideal angle of attack. Therefore, we could also write the limit as: 2c 2 __ 2 limx_>xle u (x-xle)= _ ICL+ _ CM1/41 _=-ICLi/CMI/n,rcf]

Thevariable_ is theratio of pitchingmomentto lift attheidealangleof attack,e.g.,the

conditionwhereu--0on the leadingedge.It shouldbeevaluated with the same codeand

panelingasis usedfor thedragcalculation.Fora wing with a narrowchordit is reasonable

to usethe 2D valueof _ sincethereis not muchinducedcamber.Forwingswith significant

inducedcamberthevalueshouldbefoundby: (CDi,ell-CDi,ref)

CLi= CL [1-_/14

CXsuc,ref The ref(erence) condition denotes the Mach number and angle of attack for which the wing loading is near elliptic, and full suction is assumed (_=0) Since this expression does not include wave drag due to volume, it can only be used for subsonic Mach numbers.

Generally, a discontinuity in _ is observed between subsonic and supersonic speeds. The supersonic value can be found by setting the induced drag equal on both sides of the speed of sound. The reference pitching moment is the weighted average pitching moment due to the u-perturbations around the quarter chord of the strips.

In Fig. 4.7a the lift-dependent drag for a flat straight wing, a flat delta wing and the OFW are compared with their theoretical values as presented in Ref. 14 and Ref. 2. The values predicted by the panel method correspond quite well up to Mach 3 .The induced drags of the OFW and the delta wing are about the same above Mach 2. If we unsweep the wing the induced drag can be significantly lower for speeds below Mach 2 (see also Fig. 2.5).

Volume-dependent drag For thin wings the volume-dependent drag can be expressed as: n CDvol/q=i_ 1 (CPt dt/dx i Areai) and: Cp t =-2 Uthic k Fig. 4.7b gives the volume-dependent drag for a flat straight wing and a flat delta wing.

The drag values are very close to those predicted by potential flow theory [14, 3]. For subsonic speeds the volume-dependent wave drag is correctly integrated to about zero. The oblique wing has a lower volume-dependent drag below Mach 2, but above this Mach number the drag rises sharply, while the volume-dependent drag of the delta wing keeps dropping. One could reduce the volume-dependent drag somewhat by sweeping the wing

further,but it would not bepossibleto keepthelocal normalMachnumberstransonicdue

to theeffectof wing taper on the effective sweep of each panel.

4.5 Evaluation of lhe Forces and Moments We can now analyze the OFW with the improved WlNG3D code. The sign convention is given in Fig. 4.4. All forces and moments were calculated on the basis of second-order pressure coefficients. The design condition for the wing analyzed with the linear potential flow code was CL=0.07, Cm0.32 = -0.0035, and an elliptic lift distribution was required.

Fig. 4.8a shows that the lift increases linearly with angle of attack. The required cruise lift is achieved at the reference angle of attack (0_=0). The potential drag (Fig. 4.8b) is within a few percent of the theoretical minimum given in Refs. 2 & 3. The wave drag due to volume 0.0013, and the lift-dependent drag value of 0.37 are very close to the values calculated earlier in the conceptual OFW study [ 411. The side force (Fig. 4.18c ) is the component in the Y-direction of the leading edge suction. It increases quadratically with the lift coefficient. The value of the sideforce is zero at cruise conditions, and is at most one percent of the value of the total lift at the off-design conditions.

At the cruise Mach number of 2, the pitching moment (Fig. 4.8d) is close to zero and the aircraft has a static margin of 0.13. At subsonic speeds, the static stability is almost neutral.

The yawing moment (fig. 4.8e) reaches its minimum at the condition of elliptic loading.

The yawing moment increases when the angle of attack is increased from the condition of elliptic loading. In this case, the increased upwash on the aft wing half tilts its lift-vector forward, which creates a positive yawing moment. The yawing moment also increases when the angle of attack is decreased from the condition of elliptic loading. When the angle of attack is decreased, the lift on the aft wing half is re&_ced much _aster than the lift on the forward wing half. The resultant of the induced drag shifts in the positive Y-direction and creates a big positive yaw,ng moment. The rolling moment (Fig. 4.8t3 is ahnost zero at the required lift and increases with angle of attack because the rear wing loading i_lcreases due to the upwash from the fo_"ward wing half.

The lift-to-drag ratios presented in Fig 4.8g predict a maximum value around 11 for Mach 2.0, close to the values predicted by the method in Ref. 3 (point C in Fig. 2.2) for the same parasite drag.In Fig. 4.8h the relation between K and CL is shown. At the design angle of attack, the lift distribution is more elliptical and the K factor approaches the theoretical minimum [2]. The force and moment curves are qualitatively similar to those published in Reference 6 of a highly swept oblique wing.

Fig. 4.7: Aerodynamic comparison for selected wings a.

.8 K ............... ................ ................ i/P.

.7 OBLIQUE Panel i i _/ / _ • ................ theory : : A ./';. / _7 .6

_.M<, 3.0 ............... :: ............... "i ....._i_'y"_'Z -_'

.5 .4 .3 - .......... ::;i_:_::'''_'_:_ .... !......... .i,:,:_ ........... i................ i..............

.2 ::i ::::::: ::!:::::::: iii il :::::::::::::::::::::: :i: :ii :: ::::: .1 .0 .0 .5 1.0 1.5 2.0 2.5 3.0 b° .012 m

I---,_ .EoT,.Go_ Ii i

o > r-_ O .010 I_ _ I.i..._. ........... - ................ :................ _ ............

................................ i............... i............... i............... i...... .7.. .....

.008 .006 ii:i :iiii iii iiii:;; ;i;iii ;: .O04

-. -. ...___. _.-. _

.002 .000 •0 .5 1.0 1.5 2.0 2.5 3.0 MACH Rectangular AR=8 t/c*=0.03 Delta AR=I t/c*=0.04 / Same volume Oblique AR=1.05 t/c*=0.048 sweep 72.5deg/ Fig. 4.8: Warped OFW in potential flow Mach 2 AR=1.05 72.5 deg sweep Cref= 2.4 Bref=2.5 Sref=6.1 / Xref=0.32c Yref=Cenler span a: Lift .20 , I----_ Mach 0.6 Wingbody 20x2Opanels + thickness I J .15 .10 ._1 O .O5 0.00 -.05 -.10 ° -5. -4. -3. -2. -1. O. 1. 2. 3. 4.

alpha b Potential drag .018 .016 .014 .012 .010 .008 CL E3 .006

iiiiiiiiiiii i iiiiiii

.004 .002

.............. vou e

.000 -.10 -.05 0.00 .05 .10 .15 .20 CL c" Side force .005 .000 -.005 -.010 -.015 -.020 -.025 .20 -.10 -.05 0.00 .05 .10 .15 CL d: Pitching Moment .03 .02 .01 .00 -.01 -.02 .20 -.05 0.00 .05 .10 .15 -.10 CL

e" Yawing moment

.020

............. ° ............................... • ...................

.015 t- ................ • ................ - .... °°° ..... °° ........ ° ......... _ ................ ! ..............

.010 O .005 .000 -.005 -.05 0.00 .05 .10 .15 .20 -.10 CL f: Rolling moment .008 .006 .004 .002 Or .000 -.002 -.004 -.006 i J -.008 -.10 - 05 0.00 .05 .10 .15 .20 CL g: Lift-to-drag ratio with a parasite drag of 0.00413 15.

..... ° ............

10.

a ° "l ° -5.

| | | | I i-.-_-.-- Mach 2 Wingbody 20x2Opanels + thickness i Mach 0.6 Wingbody 20x2Opanels + thickness

I

| -10.

-. 1 0 -.O5 .2O 0.00 .05 .10 .15 CL h" Lift dependent drag factor K 2.5 _ Mach 2 Wingb;ody 20x'_Opanels ;+ thickn;ess In 2.0 _."-'" _ Mach 0.6 Wingbody 20x2Opane!s + thickness JR K

I

1.5 ._[_\ i i iR.T. Jones[2 ] ' ' I ii ii I

lo ...... __,Theoret,ca, min,ma I _ ..... ........ { ......... I

.......... i......... _ ........ !......... i......... i......... ;........ i......... .

o.ao i i :: i i i _ _

i ; ; ; ; ; ; ; ; • .0 .20 .00 .02 .04 .06 .08 .10 .12 .14 .16 .18 CL 4.6 Control Authority Control authority is the capacity to maneuver and trim the aircraft by generating moments and forces. One can obtain the desired pitching and rolling moments by deflecting the 10% trailing edge flap suitably along the span. In this way, the flap deflections can be directly used to control the aircraft by an artificial stability and control system. The required flap deflections are found by setting up a matrix with the partial derivatives of the rolling and pitching moments with respect to a suitable deflection along the span, inverting this matrix and then solving it for the required pitching and rolling moment. As an example we will calculate the symmetric deflection (s) and an asymmetric (negative on leading tip) deflection (a) required to balance the aircraft due to a 1 degree angle of attack change at CL=0.0065 and Mach 2: [:] 8Cm/Ss 8Cm/Sa ]-1[ CM required [-0.00435-0-00139]-1 0"00325 ]j0"095] = 8Croll/SsSCroll/Sa Cronrequircd =[ 0.0053 -0.0057 [ [-0.00100][-2.61] To trim the aircraft we need to deflect the flap at the leading tip 2.7 ° downward, and we need to deflect the flap at the trailing tip 2.5 ° upward. Between the trailing and leading tips the flap setting varies linearly. For a trimmed OFW, the lift gradient is 10% higher than for an untrimmed OFW and the induced drag remains approximately the same.

Recent research [27] has revealed that high side accelerations in pitch maneuvers are unacceptable to pilots. In the case of the OFW, for a 1.3 g pull-up maneuver at cruise a sidefore of 0.03g is created. This sideforce can be easily compensated by deflecting the two most inboard vertical tailplanes (shown in Fig. 1) in the same dire'_tion. Since each fin has an area of 1.7% of the wing planform and a lift gradient of 0.04/o, we would only have to deflect them by 0.2 °. However, higher side forces can be expected during subsonic maneuvering.

Yawing moments are generated by deflecting the fins on each wing half in opposite directions. The yawing moments due to changes in the angle of attack are of the same magnitude as the side force. But, the one engine out condition is the most important, and this condition dictated the size of the vertical fins as described in Ref. 41 .The trailing fin gives the aircraft a directional stability of Cnl3=0.04/rad, slightly higher than the minimum value recommended in Ref.5.

Conclusions An earlier study [41] suggested that the Oblique Flying Wing (OFW) would be economically attractive if it could achieve cruise lift-to-drag ratios that are comparable to those of other supersonic configurations. Because the OFW is able to adjust its sweep angle for each Mach number, it achieves a higher lift-to-drag ratio than any existing configuration up to Mach 2.0. Fig. 5.1 compares the lift-to-drag ratio of the OFW to that of the B747 and the Boeing High Speed Civil Transport [16].

The structural weight of the OFW will also be less than that of conventional configurations because the cabin is used as part of the wing. The wider span does not lead to a higher bending weight because the load is distributed over the span. (Fig 5.2).

Improvements in lift-to-drag ratios and empty weight will lead to better payload fractions.

Since the payload fraction is directly related to Direct Operating Costs [15, 12] the improved payload fractions will result in lower DOC and therefore better economic performance. Fig. 5.3 compares the OFW and the 1989 HSCT designs on the basis of payload weight fractions. The OFW achieves payload fractions that are very close to that of the B747, while the HSCT configurations [34, 35] have significantly lower payload fractions.

Fig. 5.4 gives a general comparison between the Oblique Flying Wing, the current Boeing High Speed Civil Transport and the Boeing 747. We see that the OFW is lighter, carries the same payload and requires less runway than the B747, while being almost as fast as the HSCT.

Current and Future Work Current research on the Oblique Flying Wing includes an environmental and an operational impact study. The research discussed for this paper and the current work was funded by NASA AMES.

Main Advantages of the Oblique Flying Wing m L/D C |A. takeoff __ ................................. _ CI CMliamxbRange 25.

OFW'" ""_ ....... i-- _" i i | D. Boomless 20.

...... .......... ..... , 15. . " ........ ..........

E °

I Con ,;r:,i:i: ........ [

.4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0 Mach Fig. 5.1" The OFW has better performance and requires less fuel than other aircraft because it has higher lift-to-drag ratios.

Bending k load of Bending load of spanloader nal configuration Lift

,_'_i _t t, tC

_-_ Ciiiiiii_iab iiiiiiiiii;i;;i;iiiiiiiiili[_i Weight Weight Fig. 5.2: The OFW has less structural weight than conventional configurations because the load is distributed over the span and the cabin is part of the wing.

Payload MTOW

- .... • I,anoe='OOOn l

14% /1_ B7_47 : : 12% .,,°,==,=° ..... ,,'_ ........ ° ........ °..,° .............. •.°.°.o,o_..., .... • .................

: B 1080-8:_7 10% ................. !"_ .... ;'"_ ...... i!" ' '!_ e, ng ......

8% .... ° ............ "_ .......................... * ..... , .............................

Concorde ................. ::R_,3,5QO.om.. ! ............... i. " .......

6% :.................. :, .:. _. .: ...... , .... ,°.,= ,°,, ...... • ...... _ ,* ...................

4% MDD " 2% ,° ........ , ...... ,_ ................. -,,°° .............. _ .................. • ..... _.., .....

0 1 2 3. 4. Mach Fig. 5.3: The payload fraction of the OFW will be comparable to that of existing subsonic aircraft and 50% higher than that of the current High Speed Civil Transport designs.

lOOm cru_ I Range=5000nn_ Type OFW B 1080-827 B747 I Mach2.0 Mach2.4 Mach 0.85 Speed Payload 462pax 247pax 452pax 300tons 340tons 360tons Weight Runway 2000m 3600m 3300m Fig. 5.4" The OFW accommodates twice as many passengers, has less takeoff weight and requires only half the runway length of the current HSCT Boeing design.

References 'The supersonic flying wing', Aerospace America, october 1 Jones, R.T.

'High Speed Wing Theory' Princeton Aeronautical Paperbacks 2 Jones, R.T.

'Lift/drag ratios of optimized slewed elliptic wings at 3 Smith, J.H.B.

supersonic speeds' The Aeronautical Quarterly. Vol. 12 pp.

201-218 August 1961 4 Van Der Veiden, A. 'Design of a Small Supersonic Oblique-Wing Transport Aircraft' Journal of Aircraft Vol 26,#3 March 1989 Also Engineer's thesis U. of Technology Delft, 1986, 318pp 5 Torenbeek E., 'Synthesis of subsonic aircraft design', Student edition, Delft University of Technology/Martinus Nijhoff, 1982 'An experimental investigation of three oblique wing and body 6 Jones, R.T et alli combinations', NASA TM-X-62256, 1973 Notes:'Propulsion of Aircraft' University of Technology 7 Wittenberg H.

Delft. (In Dutch) Dictaat D-32A 1983 9 Garner H.C. 'Some remarks on vortex drag and its spanwise distribution in incompressible flow' JRAeS vol 72 pg624 1 0 Kuchemann, D., 'Aerodynamic design of aircraft', Pergamon Press 'Methods for calculating the lift distribution of wings' 11 Multhopp H.

R&M 2884 Jan 1950 12 Van Der Velden, A. 'An economic model to evaluate high-speed transport aircraft designs', NASA CR177530.

'Slewed Wing Supersonics',The Aeroplane, March 1961.

13Lee G.H., 14 Schlichting/Truckenbrodt 'Aerodynamik des Flugzeuges' zweiter band Springe_ "'erlag Berlin 1969 'Fundamentals of Conceptual design optimization of subsonic 15 Torenbeek E.

transport Aircraft', Delft University of Technology dept. of Aerospace Eng. 1980.

'High-Speed Civil Transport Study'.NASA CR 4233, 9-'89 16Boeing C. A.

1 7 Head M.R 'Entrainment in Turbulent Boundary Layers', British Aeron. Research Committee, reports and memo 3152, 1960 18 Van der Velden,A. 'Design of Oblique, Swept and Delta Wing Supersonic Aicraft', Unpublished Delft University of Technology, 1985 parametric investigation including:'Boeing l-2a, 3-2a, 5-3a, SXJET,Concorde,MDD AST,SCAT 15F' 'Theory of wing-body drag at supersonic speeds' NACA 19 Jones, R.T Report 1284 'The Aerodynamic Design of Oblique Wing Aircraft' AIAA- 20 Kroo, I 86-2624 AIAA/AHS/ASEE Aircraft Systems Design and Technology Meeting October 20-22 1986 Dayton Ohio 'On the numerical evaluation of the drag Integral' RAE RM 21 Eminton, E 3341, 1961 'The wave drag of arbitrary configurations in linearized flow 22 Harvard, L.

as determined by areas and forces in oblique planes. NACA RM A55A18 TIB 4620, March 1955 'An equivalence Law Relating Three- and Two Dimensional 23 Lock, R.C Pressure Distributions'. RAE Reports and Memoranda 3346 May 1962 'Efficient Solution Methods for the Navier-Stokes Equations'.

24 Pulliam,T Lecture notes for the Von Karman Institue for fluid dynamics lecture series: Brussels Jan 1986 25 Jones,R.T. 'Aerodynamic design for supersonic speed'. Advances in aeronautical sciences, vol 1. Pergamon Press 1959 26 Mehta, U. 'The computation of flow past an oblique wing using the thin- layer Navier-Stokes equations. Nasa TM 88317 1986 W., Gilyard G.B, Maine T.A 27 Kempei R., McNeill 'A piloted Evaluation of an Oblique-Wing Research Aircraft Motion Simulation With Decoupling Control Laws' NASA TP 2874 Nov. 1988 28 Ward. G.N.

'Linearized theory of steady high-speed flow.'Cambridge University press 1948 29 Rockwell 'Oblique Wing Research Aircraft Phase B Preliminary Design'. Rockwell Intl. 15 April 1987 30 Liepmann, H.W, Roshko, A.

'Elements of Gasdynamics', Galcit aeronautical series, 1957 J.W° 34 Graf, D.A.,Stroup, 'Economic feasibility of high-speed civil transports:,AIAA Aircraft design and operations meeting, Atlanta Georgia.

Sept 7-9. 1988 Winston, M., Morris, S.

35 Morris, C., 'Some key considerations for high-speed civil transports'.

AIAA-88-4466 'X-29 Aerodynamic Design and Performance' AIAA 36 Boppe, C.W Professional Study Series - Aerodynamic Analysis and Design 6,7 October 1988 Palo Alto 'Airfoil sections'London, Butterworths 1961 37 Riegels,F.W.

'Recent applications of CFD at Boeing' Professional Study 38 Tinoco, E.N Series - Aerodynamic Analyis and Design 6,7 October 1988 Palo Alto California 'Analysis and Design of Wing-Body Combinations at 39 Woodward, F.A.

Subsonic and Supersonic Speeds'. Journal of Aircraft vol 5.

no 6 1968 'The Conceptual Design of a Mach 2 Oblique Flying Wing 41 Van der Velden, A.

Supersonic Transport', NASA CR177529 May 1989 'A discussion of selected aerodynamic problems on integration 42 Swan, W.C of systems with airframe on transport aircraft' Boeing Co Sevier,J.R.

43 Czarnecki,K.R, 'Investigation of effects of distributed surface roughness on a turbulent boundary layer over a body of revolution at a Mach number of 2.01' NACA 4183, 1958 'Die Bestimmung der flugzeugpolaren fuer entwurfszweke' 44 Fiecke, D.

DVL 1956 'Aircraft drag prediction for project appraisal and performance 4 5 Butler,S.F.

estimation' AGARD CP124 1973 Sevier,J.R. , CarmeI,M.M.

4 6 Czarnecki,K.R.

'Effects of fabrication type roughess on turbulent skin friction at suprsonic speeds' NACA TN 4299 'The aerodynamic design of winged aircraft at transsohl • and 48 Lock,R.C.

supersonic speeds'. Journal of the Royal Aeronautical society June 1963 Software References The following software tools were used for the parametric studies: AVSAD (by A.J.M. Van der Velden, TUDelft/UC Berkeley/Stanford U Pascal, IBM XT) General parametric configuration design program Detailled description in Chapter 2 -code made available upon request The airfoil was designed with the following software: PANDA (by. I.Kroo, Stanford AA, Fortran, Macll): Interactive inviscid analysis of a 2D airfoil by Riegels Method. Weber R&M 3026, a modification for thin airfoil theory. 180 points to define the airfoil. -code available from author ARC2D (by T.Pulliam, NASAS AMES, Fortran, CRAY XMP) Solution of the Euler and thin layer Navier Stokes equation. The airfoil was defined with a 193x40 C-grid generated by HYGRID and later with a 500x100 C-grid, first mesh point 5e-7c from the wall for the NS solution.

LBAUER (by L.Bauer, Fortran, VAX) Solves the 2D linear potential equation for transsonic flow over the airfoil geometry + boundary layer. We used an 160x80 O-grid.

To design and analyse the 3D Oblique Flying Wing the following software tools were developed: WAVE (by A.J.M. Van der Velden, Stanford AA, Fortran, MaclI) Eminton Lord code voor volume-dependent wave drag -code made available upon request PANEL (by A.J.M. Van der Velden, Stanford AA, Fortran MaclI) Panneling of configurations based on arbitrary input comer points and defined transformation and transformation of 2D-->3D pressure distributions with second order CP's. Output for Wing3D, SHADE and QUICKPLOT-code made available upon request WING3D (by R. Carmichael, modified by Van der Velden, Fortran, MaclI VAX) Woodward linear potential flow code for subsonic and supersonic flow, with second order pressure distributions and leading edge suction correction. -code made available upon request Aknowledgements The authors wish to thank prof. Dr. Harvard Lomax and Dr. Unmeel Metha of the Fluid Dynamics Division of NASA AMES, Ralph Carmichael of the Aerodynamics Division of NASA AMES and prof. Dr. Robert T. Jones of Los Altos for their assistance in the completion of this work. Special thanks to Dr Evelin Sullivan for improving this paper.

Report Documentation Page

Nalot_l A_onau_¢$ S.pa ce Adm #mNallon 2. Government Accession No.

3. Recipient's Catalog No.

1. Report No.

NASA CR- 177552 5. Report Date 4. Title and Subtitle June 1990 The Aerodynamic Design of the Oblique Flying Wing Supersonic Transport 6. Performing Organization Code 8. Performing Organization Report No.

7. Author(s) A-90168 Alexander J. M. Van Der Velden and ilan Kroo 10. Work Unit No.

505-60 9. Performing Organization Name and Address 11. Contract or Grant No.

Stanford University NCA2-343 Pa]o Alto, California 94053 13. Type of Report and Period Covered Contractor Report 12. Sponsoring Agency Name and Address National Aeronautics and Space Administration 14. Sponsoring Agency Code Washington, DC 20546-0001 15. Supplementary Notes Point of Contact: Unmeel Mehta, Ames Research Center, MS 202A-1, Moffett Field, CA 94035-1000 (415)604-5548 or FTS 464-5548 16. Abstract The aerodynamic design of a supersonic oblique flying wing is strongly influenced by the requirement that passengers must be accommodated inside the wing. Recent research has revealed that thick oblique wings of very high sweep angle can be efficient at supersonic speeds when transonic normal Mach numbers are allowed on the upper surface of the wing. The present work was motivated by the ability to design a maximum thickness, minimum size oblique flying wing.

To achieve these goals, we used a 2D Navier-Stokes solver to design airfoils up to 16% thickness with specified lift, drag and pitching moment. A new method was developed to calcu- late the required pressure distribution on the wing based on the airfoil loading, normal Mach number distribution and theoretical knowledge of the minimum drag of oblique configurations at supersonic speeds. Using an inverse potential flow solver we calculated the wing mean surface for this pressure distribution.

The lift-to-drag ratio of this wing was significantly higher than that of a comparable delta wing for cruise speeds up to Mach 2.

18. Distribution Statement 17. Key Words (Suggested by Author(s)) Unclassified--Unlimited Oblique, Flying wing, Supersonic transport aircraft, Computational fluid dynamics, Transonic Subject category---05 airfoil design, Wing design 19. Secudty Classif. (of this report) 20. Security Classif. (of this page) 69 A04 Unclassified Unclassified 21. No. of Pages ] 22. Price NASA FORM 1626 OCT B6 For sale by the National Technical Information Se*'vice, Springfield, Virginia 22161