The natural flow wing-design concept

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The Natu aI Flow

Wing-Design Concept

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NASA

Technical

Paper

The Natural Flow

Wing-Design Concept

Richard M. Wood and Steven X. S. Bauer Langley Research Center Hampton, Virginia National Aeronautics and Space Administration Office of Management Scientific and Technical Information Program Nomenclature S wing reference area, ft 2 A cross-sectional area, in 2 t airfoil thickness, in.

a i variables used to define airfoil geom- At change in wing thickness, used in wing etry forward of maximum thickness asymmetry method, in.

(where i = 0, 1, 2, or 3) Ats change in cross-section wing thickness b wing span, in. at centerline, used in wing shearing method, in.

C D drag coefficient, Drag/ qocS x streamwise direction CD,cs cross-sectional inviscid drag coefficient from EMTAC code, f CO, e y spanwise direction CD, e elemental inviscid drag coefficient from z direction normal to wing planform EMTAC code a angle of attack, deg CD, I inviscid drag coefficient, f CD,cs CD, i induced drag coefficient from linear A leading-edge sweep, deg theory Subscripts: CD,o zero-lift inviscid drag coefficient of uncambered wing from EMTAC centerline CD,V viscous drag coefficient lower wing lower surface CL lift coefficient, Lift/q_S max maximum Cp pressure coefficient, (p - P_c ) / qoo root wing root. chord c chord, in.

TE trailing edge di variables used to define airfoil geome- tip wingtip try aft of maximum thickness location (where i = 0, 1, 2, or 3) upper wing upper surface Fc camber function (varies in streamwise Abbreviations: direction) asymmetry (where n = 20, 50, or 90) L/D lift-drag ratio leading-edge bluntness (where n = 1, l wing length, in.

3, or 4) M Mach number EMTAC Euler Marching Technique for Accu- rate Computation p local static pressure, lb/ft 2 LT linear theory Poc free-stream static pressure, lb/ft 2 N-C near-conical wing q_c free-stream dynamic pressure, lb/ft 2 airfoil thickness (where n = 1 or 3)

T(n)

r leading-edge radius, in.

,,o _JtttE,_Tl{.l_ht.t, dkAf_| PRECEDING PAGE BLANK NOT FILMED Abstract A wing-design study has been conducted on a 65 ° swept leading-edge delta wing in which the wing geometry was modified to take advantage of the naturally occurring flow that forms over a slender wing in a supersonic .flow field. Three-dimensional nonlinear analysis methods were used in the study which was divided into three parts--preliminary design, initial design, and final design. In the preliminary design, the wing planform, the design conditions, and the near-conical wing-design concept were derived, and a baseline standard wing (conventional airfoil distribution) and a baseline near-conical wing were chosen. During the initial analysis, a full-potential flow solver was employed to determine the aerodynamic characteristics of the baseline standard delta wing and to investigate modifications of the airfoil thickness, leading-edge radius, airfoil maximum-thickness position, and wing upper to lower surface asymmetry on the baseline near-conical wing. The final design employed an EuIer solver to analyze the best wing configurations found in the initial design and to extend the study of wing asymmetry to develop a more refined wing. Benefits resulting from each modification are discussed, and a final "natural flow" wing geometry has been designed that provides an improvement in aerodynamic performance compared with that of a baseline conventional uncambered wing, linear-theory cambered wing, and near-conical wing.

Introduction transonic and supersonic speeds, only minimal per- formance benefits have been achieved (ref. 5). An- Future supersonic military or commercial aircraft other drawback to variable-camber devices is the in- will be required to have high levels of lifting effi- crease in complexity, wing weight, and the loss in ciency at subsonic and transonic speeds as well as usable wing volume. An alternate approach to the at supersonic speeds; however, the present wing- maneuver-design requirement is to develop a fixed- design philosophies that must be employed to ad- camber wing. In general, these wing-design studies dress these multipoint design conditions vary greatly.

have been fairly successful at their design lift con- A review of the existing wing-design philosophies for dition, but they have suffered severe camber drag subsonic, transonic, and supersonic flight reveals sev- penalties at the lower lift conditions (ref. 6).

eral contradictions as well as several similarities. The To address the need for a multipoint wing-design contradictions exist mainly between the low-speed approach, a wing-design concept has been developed (subsonic and transonic) cruise-design philosophies that contours the upper and lower surfaces of the (refs. 1 and 2) and the supersonic cruise-design meth- three-dimensional wing independent of one another ods (ref. 3). For subsonic and transonic designs, the in order to take maximum advantage of the naturally tendency is to use a lower wing sweep, thick air- occurring flow field and resultant pressure distribu- foils, and blunt leading edges; advanced supereritical- tion. This present approach is similar to the philos- type airfoils are most commonly used. On the other Ophy employed in low-speed, two-dimensional (2-D) hand, supersonic designs typically employ wings hav- airfoil design, but it is counter to studies of tradi- ing higher sweep with thin airfoils and sharp leading tional 3-D wing camber design. The remainder of edges. The supersonic wing-design tendencies are al- this paper will overview the present design approach most solely due to concerns about supersonic wave as applied to delta wings at supersonic speeds. A drag. Wing twist and camber at all speeds are usu- complete review of the iterative computational de- ally provided by linear-theory-type methods.

sign results will be presented and discussed. This paper will summarize the results of references 7 10 At maneuvering conditions, both the low-speed in which the natural flow wing-design concept is de- and supersonic wing-design methods employ variable- camber devices such as leading- and trailing-edge veloped and evaluated. The supporting data for this study were derived both from the application of non- flaps. At subsonic speeds, leading-edge flaps have linear, inviscid, computational aerodynamic methods been shown to be fairly successful (ref. 4); however, at

(refs.11 13)andfrompublished force,pressure, and

and D) of the wing upper surface would have pres- flowvisualization data(ref.8).

sure fields that combine favorably with the local sur- face geometry to produce drag reductions. Figure 1

Wing-Design Philosophy

illustrates how a "near-conical" upper surface wing geometry could reduce the unfavorable drag regions

An extensive survey of theliterature(ref.8) was

of the wing (tic., regions A and C) by moving the

conducted to determine the dominantwinggeomet-

centerline airfoil crest forward and sweeping the out-

ric characteristics (i.c.,leading-edge-sweep andplan-

board airfoil crest line aft to more closely coincide

form)andflowconditions (i.e.,Machnumber)that

with the conical nature of the flow.

should beconsidered in assessing thesupersonic aero-

dynamics of wings.The resultof this effortwasthe

However, the flow on the lower surface of the wing

identification of the delta or triangularwing plan-

behaves quite differently at positive angles of attack,

formasthemostlikelycandidatc forthedevelopment

therefore requiring a different type of geometry. The

of future wing-design methodsbecause of the ex-

flow over the wing lower surface is characterized by a

tensive experimental andtheoretical database avail-

nearly constant compression loading. The magnitude

able.In addition,the empirical correlations derived

of these compression pressures is primarily dependent

for delta wingscouldbc extended to other simple

upon the wing-surface streamwise slope and is not wingplanforms, suchasarrowanddiamond wings, very sensitive to the curvature in the crossflow plane.

throughtile useof simplegeometric andflowcorre-

Based upon these observations, the most beneficial lation parameters.

lower surface geometry would have as large an area as possible with aft-facing slopes to take full advantage

The conventional application of thickness to un-

of the lower surface pressure loading.

cambered delta wings results in a wing that is conical about the wingtip. (See fig. 1.) However, experi- Wing-Design Study mental data (ref. 8) and thcorctical analysis (ref. 11) show that the flow over a swept wing at subsonic, The w{ng-design study has been executed in three transonic, and supersonic speeds tends to be coni- steps: cal about the wing apex and not conical about the 1. Preliminary design wingtip as observed for wings having small values 2. Initial design of leading-edge sweep. The conical nature of the flow field over the delta wing upper surface produces 3. Final design favorable and unfavorable pressure fields, based on The preliminary design of step 1 phase has been drag consideration (fig. 1).

documented in references 7 9 in which the wing For a wing at moderate-to-high lift conditions, the swcep, the design conditions, and the near-conical flow over the wing upper surface may be character- concept were derived based upon considerations of ized by an expansion over the leading edge that is zero-lift wave drag, wing lifting efficiency, and wing followed by a recompression to a more positive pres- loading. The preliminary design phase considered sure as the flow moves inboard and aft.. Through only wing geometries that were symmetric.

experimental observations, the location of the recom- Step 2 in the design process was the initial design pression region has been observed to lie along a ray (refs. 9 and 10) in which an iterative computational emanating from the wing apex. If the upper sur- design was conducted using a full-potential-based face is divided into four quadrants, defined by the in- computational method (ref. 12). The objectives of the tersection of the airfoil maximum half-thickness line initial design were to identify the aerodynamic per- (crest line) and the crossflow recomprcssion line, two formance trends associated with variations in airfoil favorable and two unfavorable performance regions shape (thickness, bhmtness, and maximum-thickness are identified. The two unfavorable regions, which position), the spanwise variation in airfoils, the verti- contribute to the drag, are the inboard forward re- cal shearing of airfoils, and the redistribution of wing gion (A) and the outboard aft region (C) of the wing.

volume. In this phase of the design, independent The inboard forward region (A) of the wing experi- variations in all geometric parameters for the wing ences a recompression of the flow prior to the airfoil upper and lower surfaces were investigated.

crest line; this results in larger pressure coefficients acting on a forward-facing wing surface. On the other Step 3 was the final design (ref. 10) in which the hand, the outboard aft region (C) of the wing is char- wing surface was further refined through small per- acterized by a rearward sloping surface that com- turbations of all geometric variables. Within this bines with the high negative pressure coefficients to section of the design process, the primary computa- produce high drag levels. The other two regions (B tional tool was an Euler code (ref. 13). The figures of merit foreach of these design phases weretotal aero- Wing upper and lower surface asymmetry was also investigated as a means to better align the

dynamicforcesand momentsand wing sectional

forces.Wing surface pressure distributionsandde- wing streamwise surface slopes with the naturally tailed flowfield informationwerealsoexamined to occurring pressure distribution (fig. 2). Camber and assess the adequacy of the computational fluid dy- twist were not used to create wing upper to lower surface asymmetry because they would not allow for namics (CFD)methodology.

accurate and independent control of the wing upper

Geometry

and lower surface geometry. The method for wing upper and lower surface asymmetry developed for

In orderto implement the proposed wing-design

this study allows for complete control of the wing

concept in a logicalfashion, a method wasdeveloped

upper and lower surfaces independent of one another.

for generating winggeometry that wouldallowfora

Wing upper to lower surface asymmetry was created

broad range of analytic wingsurfaces tobedeveloped

in a two-part process.

froma fewinput parameters. Selected asthe foun-

dationof themethodwasthe modifiedNACAfour-

Part 1 in the process was directed at increasing digit airfoilseries whichcanbc usedto define a wide tile slopes on the upper surface leading edge (forward rangeof analyticairfoil shapes.(Seesketch A.) As of tile airfoil maximum thickness line). Part 1 was notedin sketch A, theairfoilforwardandaft sections used to modify the cross-sectional surface slopes by areeach defined by a polynomial equation (rcf. 14). redistributing a percentage of the local thickness Theairfoilsexamined in this studyhadvariations in from the lower surface to the upper surface as defined maximum thickness from0.02c to 0.08c, a maximum- by a camber function F,. that varies in the streamwise thickness positionfrom0.2cto 0.6c, andleading-edge direction (constant at a given cross section). The radii from0 to 0.012c. Perturbations in typicalwing cross-sectional contour method is depicted in the geometries wereobtainedby varyingall airfoil pa- upper half of figure 2 and shows that the wing ramctcrsboth independently and in combinations. leading and trailing edges remain at the same vertical Airfoil thickness, maximum-thickness position,and position. However, a result of part 1 is that the

leading-edge radii parametrics werestudied in addi-

magnitudes of all lower surface streamwise slopes are tionto spanwise variations ofalltheparameters. The reduced and all upper surface streamwise slopes are

airfoil thickness-to-chord ratio wasincreased in the

increased. Note that this would be less than optimum spanwise directiontocreate a wingofnearly constant for tim wing upper and lower surfaces in region D as

thickness andthus increased the forward-projected

shown in figure 1.

areaof the wingat the leadingedge.The sweep of

To correct, this deficiency, a cross-section shearing

the airfoil crestline wasincreased to betteralignit

method was employed, as part 2, to modify the with the conicalflow overthe wing uppersurface.

streamwise surface slopes on the wing in regions that

Theairfoil bluntness wasreduced inboardto reduce

are dominated by streamwise flow. (See the lower

thebowshock andwasincreased spanwise to control

half of fig. 2.) The cross-section shearing method the leading-edge expansion characteristics.

was defined such that the cross-sectional thickness at the wing root was centered about the y-axis. The value of the vertical displacement required to center x the root thickness of the cross section about the axis is then applied to each point in the cross section.

The result of this two-part process for wing surface Forward of maximum-thickness location: asymmetry is as follows: symmetric airfoil at the

og ff.13 wing root, increased slopes on the wing upper surface

z-a +alT+a2 +a 3 .

leading edge, increased forward-projected area on the wing upper surface, reduced slopes on the wing upper Aft of maximum-thickness location: surface aft of the airfoil crest line, and a larger region x _ _ x)3

z= 0+dl(l- )+d2(l )2+d3( 7'

with aft-facing slopes on the wing lower surface.

Step 1: Preliminary Design Study Leading-edge radius: The preliminary wing design effort was focused at dcvcloping an understanding of the basic acro- " - a--Q-0 = 1.1019 ( / - 2 L]2 c ] dynamic characteristics of wings at supersonic speeds.

Of particular interest was the influence of wing Sketch A leading-cdge sweep angle and wing airfoil profile.

Based uponthe predicted dragresultspresented in

Study parameters • Airfoil contour

reference 7 andthewing-design space-concept results

• Airfoil distribution

of reference 8, a baseline standarddelta wing (no

• Sweep of airfoil

twistandcamber) wasestablished that consisted ofa

maximum-thickness 65° sweptleading edge with a 4-percent-thick blunt, line

modifiedNACA four-digitairfoil with a maximum Airfoil maximum-

thickness located at 20percentchord.

thickness lines: _- Conceptual projected plan form Standard A _ _._ing

In addition,if the high-lift,low-lift,andzero-lift

Proposed --_\ _ - ___ Section A-A

dragdataofreferences 7 and8 arereviewed, a vahm

of/3 cot A of 0.6 (composed of a wing leading-edge

sweep of 65° and a Machnumberof 1.62)would

providean excellent opportunityfor high levelsof

aerodynamic performance.This selection is based

upontherationalethat a 65° sweptdeltawingwith

anaspect ratioof 1.86 provides a design that balances

zero-liftdrag and drag due to lift compared with

a moreslendergeometry.(Seeref. 8.) At a Mach

numberof 1.62,the effectof vacuum pressure limit

will beminimal,thusproviding a 70-percent increase

Resulting

in uppersurfacelifting potentialcompared with a

A _-- wing trailing- edge geometry

75 ° sweptwingandan8-percent decrease compared

with a 55 ° sweptwing.

Sketch B

Thebaseline near-conical wingwasselected based

radii from 0 to 0.012c, and airfoil asymmetry from

upontheanalysis presented in reference 9. Tile base-

0 to 90 percent.

line near-conical geometry (neithertwist nor cam-

ber) consisted of a 65 ° swept leading edge with a

In order to point out the benefits due to this 4-percent-thick, modified NACA four-digit airfoil unique wing-design philosophy throughout a large lift with a maximum-thickness position varying linearly range, lift coefficient values of 0.1 and 0.3 were chosen in the spanwise direction from 0.2c at the wing root for design points. Presented in figure 3 are predicted to 0.6c at 66-percent semispan location. The near- aerodynamic characteristics of the baseline standard conical wing concept is derived from considerations and the baseline near-conical geometries. The figure of matching the wing upper surface geometry to the shows that the lift characteristics are the same for naturally occurring flow characteristics. A schematic the near-conical and standard wings; however, the of the near-conical geometry method, which is pre- drag was reduced considerably, thereby increasing sented as sketch B, depicts a 65 ° swept delta wing the L/D of the near-conical wing for both cruise with a standard airfoil distribution and the near- and maneuver lifting conditions (CL = 0.1 and 0.3, conical concept of redistributing the airfoils in the respectively). Note that the drag characteristics of spanwise direction to create a near-conical geometry.

the near-conical wing do not include a base drag As shown in sketch B, the resulting wing geometry increment associated with the wingtip base area. A has two base areas located at the wingtips. These conservative estimate of 0.002 for this coefficient is base areas result from truncating the airfoils (which derived by assuming a base pressure coefficient of wn-ap around the airfoil maximum-thickness line) at -0.2.

the wing trailing edge.

The cross-sectional area distributions and the drag buildup for CL = 0.1 for the baseline standard Step 2: Initial Design Study and baseline near-conical configurations are shown The initial design phase of the study was under- in fignlre 4. The plot of cross-sectional area distribu- taken using a full-potential code (ref. 12) for the anal- tion shows that the near-conical geometry has less ysis tool. Perturbations in the baseline near-conical volume in the front half of the wing and greater wing geometry were made holding the volume to a volume in the rear half of the wing compared with nearly constant value. The modifications made to the standard wing. As a result, the total volume the baseline near-conical wings were variations in of the near-conical wing is slightly increased over airfoil thicknesses from 0.02c to 0.08c, maximum- that of the standard wing. However, the plot of sec- thickness locations from 0.2c to 0.6c, leading-edge tional drag shows that the improvements due to the rejected because of its large reduction in volume com-

near-conical geometry compared with the baseline

pared with the baseline geometry. A comparison of wingareevidentoverthe full length of the wing.

the drag data of figure 6 shows that the drag re- The surface pressure distributions at x/l = 0.4 duction resulting from the modified thickness (T(3)) and 1.0 for CL = 0.1 are shown in figure 5. These is primarily due to reduced drag at the wing apex data illustrate how the spanwise pressure distri- (0 _< x/t <_ 0.2).

butions are altered slightly because of geometry modifications, with the primary difference being an Spanwisc surface pressure distributions at x/l = increased leading-edge expansion. This increased 0.4 and 1.0 are shown in figure 7 for the base- leading-edge-expansion pressure acting on the mod- line near-conical configurations and the modified- ified surface contour for the near-conical geometry thickness near-conical wing with the 3- to 6-percent- results in reduced drag.

thick airfoil distribution. The predicted pressure distributions for the two wings arc very similar.

The rest of this section of the paper will present the predicted effect of symmetric and asymmetric Leading-edge bluntness variation. Changes wing surface contouring between the upper and lower in leading-edge radius (bluntness) were examined on wing surfaces. All modifications have been performed the baseline near-conical wing. Figure 8 presents on the near-conical wing, and comparisons between the effect on drag due to changes in the leading- the near-conical wing and the modified near-conical edge bluntness at lift coefficients of 0.1 and 0.3.

wings will be made.

In addition to the leading-edge bluntness variations Wing thickness variation. The natural flow shown in figure 8, several methods were investigated that reduced the bluntness in the spanwise direction.

wing-design philosophy suggests that improved aero- dynamic performance would result from an increase These tapered bluntness methods had an increase in airfoil thickness in the spanwise direction by allow- in drag and thus were not considered for further ing for an increased forward-facing area to be located analysis. The data of figure 8 show that increasing on the wing upper surface for the low pressures to act bluntness in the spanwise direction reduces the drag for low-lift conditions but has little or no effect upon. To study this effect, a wide range of thickness modifications were computationally evaluated, and at high-lift conditions. At low-lift conditions, the selected results from these analyses are presented in leading-edge expansion is concentrated at the leading figures 6 and 7. edge and combines with the various leading-edge shapes to provide the different drag characteristics.

The cross-sectional area distributions and drag However, at high-lift conditions, the leading-edge distributions in the streamwise direction are pre- expansion extends farther inboard. As a result, the sented in figure 6 for the baseline near-conical wing percentage of the leading-edge expansion that acts and two modified-thickness near-conical wings hav- on the leading edge is significantly reduced, and thus ing a thickness that varied from the root to the tip the infuence of leading-edge geometry on drag is of 0.02c to 0.08c and 0.03c to 0.06c, respectively. In reduced. The drag data for all modifications show the present study for slender swept wings, the gen- results similar to those observed for the thickness eral characteristics of the wing upper surface pressure modifications in that the total drag reduction is due at all streamwise stations (i.e., the expansion region primarily to lower drag over the forward portion of near the leading edge) are assumed to be nearly inde- the wing. The leading-edge radius modification that pendent of the airfoil geometry; therefore, an increase has the lowest drag was found to be a variation from in thickness in the spanwise direction would create an 0 at the root to a maximum value of 0.012c at the improved surface for the wing upper surface flow to tip.

act upon compared with a traditional design. An additional constraint of the thickness study was to The surface pressure coefficients are shown in fig- maintain a constant wing volume; thus an increase in ure 9 for the baseline near-conical wing and the thickness outboard must bc accompanied by a corre- near-conical wing with the leading-edge radius dis- sponding reduction in the inboard thickness.

tribution varying from 0 to 0.012c. A review of the pressure data for these two geometries shows The 2- to 8-percent-thick configuration was found similar trends and levels and supports the observa- to have a reduction in volume of 19.3 percent and tion that the reduced drag for the modified-bluntness at CL = 0.1 a reduction in CD of 17 counts, and wing is primarily due to the reduced apex drag. At the 3- to 6-percent-thick configuration had a reduc- tion in volume of 4.8 percent with a reduction in x/l = 0.4, the full-potential code predicts an ex- pansion followed by an abrupt recompression on the CD of 4.1 counts. Figure 6 illustrates these results; upper surface and an expansion spike on the lower however, the 2- to 8-percent-thick configuration was

surface at the leadingedge of tile modifiedleading-

tor with wing asyminetry would give a more negative

edgewing (N-CB(1)). This erraticpressure distri-

zero-lift pitching moment for all wings and possibly

bution is mostlikely caused t)y the inability of the

result in an increase in trim drag. The corresponding full-potential codeto accurately resolve theexpand- drag is seen to decrease slightly for the 20-percent ingflowovera wingwith a sharpleading edge.Tile asymmetric configuration and increase for configu-

spanwise pressure distributionfor tile baseline near-

rations with greater asymmetry compared with tile

conical wingat x/l = 0.4 is well-behaved and shows

baseline near-conical wing.

a very gradual expansion at the leading edge on both At the high-lift condition (CL = 0.3 in fig. 11), the lower and upper wing surfaces. At the aft stream- the 90-percent asymmetric configuration was found wise location (x/1 = 1.0), where both wings have to obtain significantly greater lift and drag in the bhmt leading edges, nearly identical pressure load- forward region of the wing than the other three ings are predicted.

configurations. The drag buildup data of figure 11 As mentioned previously, the method for asym- show that the 20-percent asymmetric wing has lower metric wing surface contour and shearing was (level- drag compared with the other wings. The large oped to allow for detailed control of the wing upper increase in drag with large amounts of asymmetry and lower surface geometries during the design pro- predicted by the full-potential method is a result of cess. This is in contrast to typical design methods the inability of the codes to resolve the flow about that warp tile wing through twist and camber applied a sharp leading edge. The increase in drag with to the mean chord plane. The thickness distribution increasing wing asymmetry and the increased lift is then wrapped about the mean chord phme.

loading on the forward portion of the 90-percent asymmetric wing at the high-lift condition compared Asymmetric wing contouring and shearing.

with the near-conical baseline wing raises doubts Two approaches were _sesse(t in the application of about the ability of the full-potential method to the asymmetric wing surface contour method to the accurately resolve the flow field about these sharp natural flow wing-design concept. The first approach leading-edge wings. (Scc sketch C.) Note that as was to increase the amount of asymmetry in the wing asymmetry is increased, the wing leading edge streamwise direction, referenced to the wing apex, begins to develop a reduced leading-edge radius on in order to incremue the wing leading-edge surface the lower half of the leading edge and an increased slopes on the outboard portion of the wing to match leading-edge radius on the upper half of the leading the increased upwash angle and resultant expansion edge. (See sketch C.)

pressures. The second approach was to impose a constant asymmetry over the complete wing in an The surface pressure coefficient (Cp) distributions effort to control the magnitude of the upwash as it are shown in figure 12 for the CL = 0.1 condition at x/l = 0.4 and 1.0 for the near-conical and the increases in the spanwise direction and, thus, control 90-percent asymmetric near-conical configurations.

the flow expansion about the wing leading edge.

Results from the full-potential solver show that at The baseline near-conical wing was used as the x/1 = 0.4, the configuration has more negative Cp basis for this initial asymmetric wing contour study.

values on the lower surface at the leading edge than An extensive number of asymmetric wing contours on the upper surface. The large spikes in the C_ data were evaluated in the initial design phase. These at the leading edge of the wing are often seen in the analyses identified the constant asymmetric wing data from this code because of difficulties in solving contour method as providing improved performance the fllll-potcntial equations near regions where tile compared with the streamwise variation methods.

geometry changes rapidly, i.e., sharp leading edges.

Presented in figures 10 and 11 are the predicted In order to resolve these discrepancies, additional lift and drag characteristics for several constant analysis of the asymmetric wings will be performed asymmetric wing contour surfaces at lift coefficients with the Euler solver in the final phase of the study of 0.1 and 0.3, respectively. Figure 10 illustrates how in order to minimize errors obtained at the leading changes in wing asymmetry affect the total lift and edge of the wing. (See the appendix.)

drag buildup of the near-conical wings. Wing asym- metry is seen to reduce the lift contribution on the Step 3: Final Design Study forward region of the wing and results in most of the The wing geometry variables in the final design lift coming from the aft region of the wing for low-lift study were selected based upon the full-potential re- conditions (CL ----0.1).

sults obtained in the initial design phase of the study.

Although pitching moment is not directly ad- A constant wing thickness of 4 percent and a wing dressed in this study, the aft movement of the lift vec- thickness that varied linearly from 3 percent at the Presented in figures 13 and 14 are Euler-predicted lift and drag buildup plots for wings with 20-, 50-, and 90-percent asymmetry at lift coefficients of 0.1 and 0.3, respectively. A comparison of the Euler- Ze predicted pressure distributions and flow field char- aeteristics with the full-potential results shows that the Euler results are smooth and continuous about the leading-edge region whereas the flfll-potential re- sults are erratic. (Sec figs. 12 and 22.) As a result of _.y this analysis it was concluded that the Euler method provides an improved model of the flow about the wing. At low lift, the 50- and 90-percent asymmet- ric wings showed large reductions in drag. At high lift, the 20- and 50-percent asymmetric wings showed small reductions in drag, and the 90-percent asym- metric wing again showed large reductions in drag.

The drag reduction at the high-lift condition was ex- pected; however, the increased drag reduction with increased wing asymmetry at low lift was not ex- pected. Based upon the Euler analysis results pre- sented in figures 13 and 14, the 90-percent constant asymmetry was selected for further analysis.

Thickness and leading-edge bluntness modifica- tions were also studied in this phase of the design and resulted in the selection of a leading-edge bhmt- ness variation from 0 to 0.012c (B(1)) and a thick- \ \ ness variation of 3 to 6 percent (T(3)). Shown in I figure 15 are cross-section cuts and streamwise cuts through the basctinc standard wing, baseline near- /-- 90-percent asymmetry conical wing, and the baseline near-conical wing with and shearing bhmtness, thickness, and asymmetry modifications.

Cross-section cuts are presented for x/l = 0.25, 0.50, Sketch C 0.75, and 1.00, and strcamwise cuts are presented for 2y/b = 0, 0.2, 0.4, 0.6, and 0.8. The sketches in figure 15 show that the near-conical wing has an root to 6 percent at tile tip were selected for fllrther evaluation. The near-conical method initially em- increased thickness in the spanwisc direction, com- pared with the standard wing, that creates an in- ployed was maintained, and two leading-edge blunt- ness distributions were selected. The first bluntness crease in leading-edge bluntness on the outboard por- was a constant 0.009c and the second varied between tion of the wing and thus an improved surface for the 0 and 0.012c along the span. Wing asymmetry and flow to expand about.

shearing was the only geometry variable that was ex- The selected leading-edge bluntness modification tensively studied in the final design phase.

(B(1)) has a reduced bluntness at the wing apex and an increased bluntness on the outboard portion of The full-potential results obtained in the initial the wing. This modification results in reduced design phase failed to provide a clear understand- drag because of a combination of lower pressures at ing of the influences of wing asymmetry on the aero- the apex and reduced forward-facing slopes on the dynamic performance of the wings. As a result, an leading-edge lower surface which the positive pres- in-depth evaluation was performed with the Euler sures act on.

method in which both constant and varying asym- metric contouring and shearing methods were evalu- The selected wing thickness modification (T(3)) has reduced thickness inboard and increased ated. The Euler analysis presented in figures 13 23 thickness-to-chord ratio outboard compared with the confirmed the full-potential results which concluded baseline near-conical wing. This results in a geom- that only the constant asymmetry methods provide etry that has increasing leading-edge bluntness in improved aerodynamic performance at both the low- the spanwise direction. The modified-thickness_wing lift and high-lift conditions.

Cross-section spanwise cuts and streamwise cuts

hasa nearlyconstantdimensional thickness andan

through the baseline standard, baseline near-conical,

increased bluntness in the spanwise direction;thus

and natural flow wings are shown in figure 19(a). The

it is well-tailoredto the naturallyoccurringnear-

sketches show that the combination of a near-conical conicaluppersurface flow.

airfoil distribution with the selected bluntness, thick-

Thenear-conical methodandthe selected blunt-

ness, and asymmetry modifications creates a three-

nessand thickness modifications weredirectedat

dimensional wing geometry that is well-tailored to

contouring the wingleading edgeanduppersurface

match the naturally occurring flow field, as discussed

geometry, andthis resulted in symmetric wings.In

previously. To provide additional insight into the

an effort to modify the wing lower surfacegeom-

three-dimensionality of the geometry, elevation cuts

etry whilemaintainingthe preferred uppersurface

through the three wings are presented in figure 19(b).

characteristics, the wingasymmetric contouring and

Note that both the baseline standard and baseline

shearing methodwasused.Themathematical mod-

near-conical wings are symmetric about the horizon-

elingmethod selected for thestudydid not allowfor

tal plane. The elevation cuts show that the near con-

an independent designof the wingupperandlower

ical wing has a significantly improved upper surface

surfaces; however, the methoddid allowthe desired

geometry compared with the standard wing; how-

character for eachsurface to bedeveloped in the de-

ever, the lower surface of the near-conical wing is not

signprocess.The selected asymmetric wing geom-

properly contoured to match the expected constant

etry has90-percent volumetricasymmetry at each

pressures.

cross section.The asymmetric winghasa largere-

gionofaft-facing slopes onthelower surface, reduced As discussed previously, the lower surface of the

aft-facing slopes ontheuppersurface rearward of the

wing should have aft-facing slopes to take advan-

airfoil crestline,andincreased forward-facing slopes

tage of the positive pressures in reducing the drag on the uppersurface in frontof theairfoil crestline.

and creating lift. A review of the natural flow wing geometry shows an upper surface that has a

The predicted aerodynamic characteristics of the

nearly constant leading-edge shape along the en-

selected thickness andbluntness modifications along

tire span. Figure 19(b) shows that the upper sur-

with thosefor 90-percent asymmetry are shownin

face forward-sloping area is increased and the upper

figures16 18. At low-liftconditions (fig. 16),it was

surface rearward-sloping area is decreased compared

foundthat modifyingtile thickness resultedin the

with the near-conical wing. The elevation cuts for

largestreductionin drag;whereas wing asymmetry

the natural-flow wing also show that the lower sur-

andmodifyingtile leading-edge bluntness provided

face geometry is dominated by a large region with a

only smallreductions compared with the baseline

rearward slope, thus providing a much improved sur-

near-conical wing. At high lift (fig. 17), asymme-

face compared with the standard and near-conical

try provides the largestreduction in dragcompared

wings.

with thebaseline near-conical wing.Thedatain fig-

ure 18showthat the thickness modification yields

Figures 20 and 21 illustrate the lift and drag the highest (L/D)max,whichoccurs near e L = 0.15.

buildup for the baseline standard, baseline near- At higher values of CL, the asymmetric wing has the conical, and natural flow wings at lift coefficients of best performance.

0.1 and 0.3, respectively. The results presented in figures 20 and 21 show that the natural flow wing All analysis results presented previously were for when compared with the baseline near-conical wing single geometry modifications to the baseline near- has significantly lower lift over most of the wing, but conical wing. The final step in the design process there is a significant increase in the sectional lift (as was to combine the various geometry modifications indicated by the increased slope in the plot of CL to further refine the wing. A comparison between against x/l) over the final 10 to 15 percent of the the baseline wing, the baseline near-conical wing, wing. This increase in the sectional lift results in a and the final design natural flow wing is presented similar increase in the sectional drag at the trailing in figures 19-23. The final design natural flow wing edge of the natural flow wing. A review of the lift is defined by a near-conical airfoil distribution that varies from a mm×imum thickness position of 0.2c at and drag buildup results for the isolated geometry modifications (see figs. 16 and 17) shows that the in- the root to 0.6c at 0.66 semispan positions. The thickness distribution varies from 0.03c at the root creases in drag and lift at the trailing edge are due primarily to asymmetry. A review of the wing ge- to 0.06c at the tip, and the leading-edge radius ometry (figs. 15 and 19) shows that the combination varies from 0 at the root to 0.012c at the tip. The of asymmetry with the near-conical method not only selected wing asymmetry is a constant 90 percent and produces the desired large forward-facing streamwise sheared.

slopes onthe uppersurface but alsocreates a region that accounts for nonlinear flow effects, leading-edge of aft-facing areanearthe trailingedge onthe upper thrust, and vortex flow was selected for the design.

surface located inboardof y/(b/2) = 0.78. The pre- (See ref. 15.) The linear-theory cambered-design pro- ccss was conducted on a baseline standard wing at a dicted surface pressure data of figure 22 show that series of lift coefficients between 0 and 0.3. Neither this aft-facing region produces an additional expan- pitching moment nor geometry constraints were im- sion over the upper surface inboard region of the wing posed in the design process so as not to restrict the at the trailing edge which combines with the aft- facing upper surface to create a drag increase. This drag reduction potential of the design. The linear- increase in drag is shown in figure 20 by the abrupt theory-predicted performance of all designs was then compared and evaluated over the lift coefficient range increase in slope in the plot of CD,I against x/l for from 0 to 0.3. Based upon this analysis, the camber the natural flow wing.

surface at CL ----0.16 was selected as best and would The Euler-predicted longitudinal aerodynamic be evaluated with the Euler method of reference 15.

characteristics of the baseline, near-conical, and final design natural flow wings are presented in figure 23.

Sketches of the geometries of the natural flow The plot of lift against angle of attack shows that wing and linear-theory cambered wing are presented all three wings vary linearly for lift coefficients up in figure 24. A review of the cross-section cuts for to 0.3 and have nearly equivalent lift-curve slopes.

the two _dngs shows that the linear-theory wing The data in figure 23 show that the final design nat- is severely warped compared with the natural flow ural flow wing has higher L/D values and lower drag wing. This large amount of wing warp is typical of compared with the baseline near-conical wing for lift linear-theory designs and results from the tendency coefficients greater than 0.05. Compared with the of these methods both to distribute the predicted standard wing, the natural flow wing produced a loading equally between the wing upper and lower drag reduction of 0.0012 at the low-lift condition and surface and to align the wing leading edge into the of 0.0060 at the high-lift condition. Note that the upwash field. Despite the extreme warpage of the base drag increment mentioned earlier would tend to linear-theory design, a close examination of the ge- reduce the magnitude of these benefits. Compared ometry of the two wings shows that the natural flow with the baseline near-conical wing, the natural flow wing has an increased upper surface forward-facing wing provides a drag reduction of 0.009 at a lift co- area for values of x/l between 0 and 0.5 and a re- efficient of 0.1 and of 0.0050 at a lift coefficient of duced upper surface forward-facing area for values 0.3.

of x/1 greater than 0.6. The combination of the larger upper surface forward-facing area (increased The large drag reductions achieved at lifting con- leading-edge thrust) and the fiat lower surface ge- ditions are a result of improvements in thc drag-due- ometry for the natural flow wing compared with the to-lift characteristics as well as a reduction in zero-lift linear-theory wing should provide improved perfor- drag for the natural flow wing design compared with mance at all lift conditions.

the baseline near-conical and standard wings. The predicted L/D characteristics show that the natu- The EuIer-predicted cross-sectional lift and drag ral flow design provides a 15-percent improvement in coefficient distributions for the natural flow and maximum L/D compared with the baseline standard linear-theory cambered wings are shown in figures 25 design and a 10-percent improvement compared with and 26 for lift coefficients of 0.1 and 0.3, respectively.

the baseline near-conical design. The natural flow A comparison of the lift distributions of 0.1 and 0.3 wing was found to reduce the drag coefficient from for each wing shows that they follow the same trends.

the baseline and near-conical wings by 13 percent and The lib data show that the natural flow wing is more 10 percent, respectively, at a lift coefficient of 0.1 and aft-loaded than the linear-theory design which has a by 14 percent and 12 percent, respectively, at a lift near-linear distribution. A further review of the wing coefficient of 0.3.

geometry sketches of figure 24 shows that the reduced loading at the wing apex (x/1 = 0.1) of the natu- Comparison of Natural Flow Design ral flow wing is due to the increased wing leading- With Linear-Theory Cambered Design edge upper surface slopes compared with the linear- To further evaluate the relative merits of the nat- theory wing. The increased slopes would reduce the ural flow design, a cambered wing derived by lin- leading-edge expansion and resultant upper surface ear theory has been developed and analyzed with loading at this x/l location. Additionally, the up- the Euler method of reference 13. In order to fully wash field would be significantly reduced. A further assess the merits of linear-theory design methods, review of the geometry presented in figure 24 shows a state-of-the-art, linear-theory wing-design method that the leading-edge droop of the linear-theory wing increases alongthespan(increasing x/l) from 0 to a wing-design philosophy were presented and applied very large negative angle. The result of this geom- to fiat wings. The present study is an extension of etry is that at low-lift conditions, the linear-theory the previous effort to include variations in maximum- wing would be more highly loaded at the apex than thickness location, thickness, leading-edge bhmtness, the natural flow wing; and as lift would increase, the and wing asymmetry.

streamwise loading on the linear-theory wing would An initial design phase employed a nonlinear full- become a nearly constant level similar to that ob- potential analysis method to assess the merits of the served for the natural flow wing.

natural flow design approach as well as the effect Linear-theory and Euter-predicted drag charac- of thickness, bluntness, and wing asymmetry mod- teristics for the baseline standard wing, linear-theory ifications. However, if the leading edge does become cambered wing, and natural flow wing are pre- sharp (as is the case for the highly asymmetric ge- sented in figure 27. Also shown for comparison are ometries and for low values of leading-edge radius), linear-theory predictions for cambered designs with the full-potential analysis is questionable. Therefore, C L = O.1 and 0.3. Note that the linear-theory re- in order to more accurately predict leading-edge ef- sults of figure 27 include both leading-edge thrust fects, an Euler analysis method was employed; and and vortex flow increments. The results presented the resulting benefits due to modifications in geome- in figure 27 show" that the natural flow design has try were assessed and rated for overall performance.

lower drag at CL = 0.1 and 0.3 than all other wings A "natural flow" wing, which was a combination shown. Euler-analysis results for the baseline stan- of the optimum thickness, bluntness, and asymmetry dard wing compare well with the linear-theory results modifications, was analyzed using the Euler method because of tile mild surface curvatures. However, and compared with the baseline standard wing, the the Euler analysis of the linear-theory design clearly baseline near-conical wing, and a_'ing developed us- shows a significant loss in performance at high-lift ing linear-theory design methods. The natural flow conditions from that expected from linear-theory es- wing was found to reduce the drag coefficient from timates. These analyses show that the performance the baseline, near-conical, and linear-theory-designed of the highly warped (twist and camber) wings pro- wings by 13 percent, 10 percent, and 2 percent, re- dueed by linear-theory design methods is severely de- spectively, at a lift coefficient of 0.1 and by 14 per- graded as the wing pressure loading is increased and cent, 12 percent, and 10 percent, respectively, at a begins to violate the linear-theory assumptions. The lift coefficient of 0.3. These values do not take into natural flow wing was found to reduce the drag co- account a base drag penalty that would be present for efficient from the linear-theory wing by 2 percent at the near-conical and natural flow wings. An accurate a lift coefficient of 0.1 and by 10 percent at a lift determination of this penalty will require experimen- coefficient of 0.3. Again, it should be noted that no tal measurements.

base drag penalty for the natural flow wing has been included in the comparisons.

NASA Langley Research Center Concluding Remarks Hampton, VA 23665-5225 A novel wing-design concept is presented in which March 11, 1992 a natural flow wing-design approach is employed that uses a near-conical thickness distribution to Acknowledgment match the wing upper surface contour to the conical nature of the flow at supersonic speeds. In previous Brian E. McGrath of the Lockheed Engineering & studies conducted by the authors, the description of Sciences Company has made significant contributions the delta-wing planform selection criteria, the design to the theoretical analysis presented in this report.

Mach number selection criteria, and the near-conical His work is gratefully acknowledged.

Appendix

into several patches using a cubic spline routine to define the surface within each patch. The desired

Theoretical Analysis Methods

grid clustering is then set up on the body surface. An The initial theoreticalanalysiswas conducted elliptic grid generator is implemented to generate the with the full-potentialmethodof reference 12,and grid for the flow field calculations between tile body the finaltheoretical analysis usedthe Eulermethod surface and an appropriately dlosen outer boundary.

of reference 13; both weredeveloped by Rockwell

The number of patches, points per patch, and

International undera grantwith the NASALangley

points in the nornml direction could be varied to

Research Center. The input geometry and grid

cluster the grid in regions where more refinement to generation are common to both codes and allow for the grid was necessary. The fldl-potential code was a straightforward comparison of the results from the executed on the Control Data Corporation VPS-32 two analysis tools. Skin-friction drag was calculated computer on a 4-patch grid with 12 points in the by using the method described in reference 16 and first and fourth patches and 20 points in the sec- added to the inviscid drag predictions from the full- ond and third patches with 22 points in the nor- potential and Euler routines.

real direction and 20 input geometry planes. (See The full-potential code employs the conservative, fig. Al(a).) The average execution tiine was approx- steady form of the full-potential equations devel- imately 140 Central Processing Unit (CPU) sec. The oped to solve predominantly supersonic flow with full-potential code was also run on a Cray Y-MP embedded subsonic regions. Tile theory of char- computer. Initially, the same gridding method was acteristics is used to monitor the type-dependent employed and average CPU times were from 20 to flow field, and a conservative switching method han- 70 sec. A refined grid with 3 patches (20 points in tile dles the transition between the supersonic march- first and third patches and 30 points in the second ing algorithm and the subsonic relaxation procedure. patch) and 100 input geometry planes (fig. Al(b)) varied in CPU time from 200 to 700 sec depending The finite-differenced equations are solved by using an implicit approximate factorization method. A on angle of attack. The significant increase in CPU finite-volume, multizone implementation of a total time was mainly due to increased input/output (I/O) variation-diminishing (TVD) formulation (based on time. The Euler solver, with tile three-patch re- Roe's method in ref. 17) is used to solve the Euler fined grid mentioned previously, had execution times equations across the entire Mach number range. An that averaged between 300 and 1300 CPU see. Euler infinitely large time step (causing the transient terms results were obtained for the linear-theory-designed in the discretized equations to vanish) and a space- wing with a four-patch grid. The 4-patch grid had marching method are used in supersonic regions of 28 points in the first and fourth patches and 9 points in the second and third patches. The second and the flow; a finite time step and a relaxation method third patches were applied locally at the wing lead- are used in subsonic flow regions.

ing edge in order to provide a high definition of the The wing geometries are defined using a routine thin geometry.

written by the authors which takes advantage of the Both codes were modified to print out sectional analytic description of the modified NACA four-digit airfoils. The wings can be easily generated with values of lift, drag, and pitching-moment coefficients the capability for varying airfoil maximum-thickness as well as the longitudinal summation of these values.

This was done so that the effect on the forces due to location, airfoil thickness, leading-edge bhmtness, geometry modification could be better understood.

wing asymmetry, and leading-edge sweep. The wing This then allowed the authors to determine which geometry definition is input to the code as a set of discrete points in a crossplane at various streamwise modifications provided the most drag reduction in locations and is identical for both the flfll-potential the wing design. Both codes were modified to output incremental force and moment buildups as well as and Euler codes. The griding routine inside each code then divides tile cross-sectional input points cross-sectional area, wetted area, and volume.

X X o '_ cj | o X × e_ i N N References 9. Wood, Richard M.; and Bauer, Steven X. S.: Evaluation of a Three-Dimens{onal Empirically Derived Wing at Su- 1. Jameson, Antony: Iterative Solution of Transonic Flows personic Speeds. AIAA-88-0481, Jan. 1988.

Over Airfoils and Wings, Including Flows at Mach 1.

Commun. Pure _ Appl. Math., vol. XXVII, no. 3, May 10. Bauer, Steven X. S.; Wood, Richard M.; and Brown, S. Melissa: A Natural Flow Wing Design Employing 3-D 1974, pp. 283--309.

Nonlinear Analysis Applied at Supersonic Speeds. AIAA- 2. W'oodward, F. A.: An Improved Method for the Aero- 89-2167, July/Aug. 1989.

dynamic Analysis of Wing-Body-Tail Configurations in Subsonic and Supersonic Flow. 11. Siclari, Michael J.: The NCOREL Computer Program for Part I---Theory and Application. NASA CR-2228, Pt. I, 3D Nonlinear Supersonic Potential Flow Computations.

1973.

NASA CR-3694, 1983.

Part II Computer Program Description. NASA CR-2228, 12. Shankar, Vijaya; Szema, Kuo-Yen; and Bonner, Ellwood: Pt. II, 1973.

Full Potential Methods for Analysis/Design of Complez 3. Carlson, Harry W.; and Miller, David S.: Numerical Aerospace Configurations. NASA CR-3982, 1986.

Method for the Design and Analysis of Wings at Super- 13. Szema, Kuo-Yen; Chakravarthy, Sukumar; and Shankar, sonic Speeds. NASA TN D-7713, 1974.

Vijaya: Supersonic Flow Computations Over Aerospace 4. DeCamp, Ronald W.; and Hardy, Richard: Mission Adap- Configurations Using an Euler Marching Solver. NASA tive Wing Advanced Research Concepts. A[AA Atmo- CR-4085, 1987.

spheric Flight Mechanics Conference A Collection of 14. Abbott, Ira H.; and Von Doenhoff, Albert E.: Theory of Technical Papers, Aug. 1984, pp. 465 470. (Available as AIAA-84-2088.) Wing Sections. Dover Publ., Inc., c.1959.

5. Covell, Peter F.; Wood, Richard M.; and Miller, David S.: 15. Carlson, Harry W.; and Walkley, Kenneth B.: Numeri- Investigation of Leading-Edge Flap Performance on Delta cal Methods and a Computer Program. for Subsonic and and Double-Delta Wings at Supersonic Speeds. NASA Supersonic Aerodynamic Design and Analysis of Wings With Attainable Thrust Considerations. NASA CR-3808.

TP-2656, 1987.

1984.

6. Mason, W. H.; Siclari, M. J.; Miller, D. S.; and Pittman, J. L.: A Supersonic Manuever Wing Designed for Non- 16. Sommer, Simon C.; and Short, Barbara J.: Free-Flight linear Attached Flow. AIAA-83-0425, Jan. 1983.

Measurements of Turbulent-Boundary-Layer Skin Friction in the Presence of Severe Aerodynamic Heating at Mach 7. Wood, Richard M.; and Miller, David S.: Impact of Airfoil Numbers From 2.8 to 7.0. NACA TN 3391, 1955.

Profile on the Supersonic Aerodynamics of Delta Wings.

AIAA-85-4073, Oct. 1985. 17. Roe, P. L.: Approximate Riemann Solvers, Parameter 8. Wood, Richard M.: Supersonic Aerodynamics of Delta . Vectors_ and Difference Schemes. J. Comput. Phys., Wings. NASA TP-2771, 1988. vol. 43, no. 2, Oct. 1981, pp. 357 372.

Region Surface slope Pressure Drag Expansion region A Positive Hi_a High Compression region B Positive Low Low Airfoil maximum-thickness line C Negative Low High D Negative High Low Cross-flow re,compression line Lower surface Upper surface Standard win_

Nat_ia.'_

,,D Figure 1. lllustrat_ion of natural flow wing-design concept• Chordwise variation Spanwise v_ No asymmetry Asymmetry _X F c = (1- -_) (_/')apex + (_) (-_'Z)TE

Y._r--'(Pc+_)_p_

+ "2)lower Ylower=t(Fc 1 Shearing

%='_Fc

y=y- At s Figure 2. Wing asymmetry and shearing method• Baseline Near-conical a, deg

I

.4 Baseline Near-conical \ \ \ \ L/D

I 1 I I

0 .1 .2 .3 .4

%

Figure 3. Full-potential predicted aerodynamic characteristics of baseline standard and baseline near-conical wings for M = 1.62.

.020

Baseline Near-conical

.015

All 2 .010 .005

I I I I I

.2 .4 .6 .8 1.0 x/l .0100 Baseline .... Near-conical .0075 ,,_'¢ CD ,I .0050 .0025

I I I I I

0 .2 .4 .6 .8 1.0 x/1 Figure 4. Cross-sectional area and full-potential predicted drag buildup for baseline standard and baseline near-conical wings for M = 1.62 and C L = 0.1.

Upper surface -.50 - Lower surface -.50 -- -.25 -.25 - _ |

cpo

-.q c. 0

.25 .25 - I 1 I .50 I 1 I 1 I .2 .4 ,6 .8 i!0 '500 .2 .4 .6 .8 1.0 yl(b/2) y/(b/2) -.5( - _ -.50 -- I 'l I I -.2_ I I i -.25 -- I i"* I I

''/ \

%

l_ Cp 0 .25 - .25 I I I [" J I ! I l I •500 .2 .4 .6 .8 1.0 .50 .2 .4 .6 .8 1.0 yl(b/2) y/(b/2) (a) Baseline wing.

(b) Near-conical wing.

Figure 5. Full-potential predicted surface pressure distributions for baseline standard and baseline near-conical wings at x/l = 0.4 and 1.0 for M = 1.62 and CL = 0.1.

.020 Geometry (t/C)roo t (tic)tip Near-conical 0.04 0.04 N-C T(1) .02 .08 N-C T(3) .03 .06 .015 A/l 2 .010 .005

I I ] I I

.2 .4 .6 .8 1.0 x/l = : z== = .0100 m Geomelxy (t/C)roo t (t/c)ti p Near-conical 0.04 0.04 N-C T(1) .02 .08 N-C T(3) .03 .06 .0075 C D, I .0050 .0025

_r I i i i

.2 .4 .6 .8 1.0 x/l Figure 6. Cross-sectional area and full-potential predicted drag buildup for variations in airfoil thickness for M = 1.62 and C L = 0.1.

.... Uvper surface -.50 - __-.----. l_x_wer surface -.50 - -.25 -- , '_ j I -.25

.............. -I cp o

Cp 0

_ ' _ /X, ._ .__

50 it -.50- I' / \ -.2s- ,, _. ',

-.25 . ... :, / ,. ,, .... --- -

......... - --_,,,.-.,_ ,f_ -- _p ,., -- !

c_o _ .25-

.25 - .50 I I I I 1.0 I • .2 .4 .6 .8 1 I I .50 _ .8 1.0 yl(b/2) .2 .4 .6 y/(b/2) (b) N-C T(3) wing.

(a) Near-conical wing.

Figure 7. Full-potential predicted surface pressure distributions for near-conical wing and near-conical wing with airfoil thickness variation from 0.03c to 0.06c at x/1 = 0.4 and 1.0 for M = 1.62 and CL = 0.1.

.008 Geometry (r/C)root (r/c)tip Near-conical 0.009 0.009 N-C B(1) 0 .012 N-C B(3) .003 .012 / .006 N-C B(4) .005 .012 2

J

.004 CD,I .002

I I I ! I

.2 .4 .6 .8 1.0 (a) CL = o.1.

.O4 Geometry (r/C)roo t (r/c) tip Near-conicaI 0.009 0.009 / N-C B(1) 0 .012 / N-C B(3) .003 .012 / .03 CD ,I .02 .01

"q I I I I

.2 .4 .6 .8 1.0 x/l (b) CL = 0.3.

Figure 8. l_fll-potential predicted drag buildup for variations in leading-edge bluntness for M = 1.62 and CL = 0.1 and 0.3.

.... Upper surface

-.50 - _ Lower surface -.50 -

-.25 -- -.25 -- ¢,

.... ,. ....... ,j

c o ............. ' c.o-_ ....... _-'I, '50 1 I I I I _/ \_ "50 I I I I I ' 0 .2 .4 .6 .8 1.0 / \ " 0 .2 .4 .6 .8 1.0 y/(b/2) '/ _ y/(b/2) -.50 -- :, / \ -.50 - ,t %

,, / \ "

-.25- ,' . / \ __ -.25- '' ...... ". , I / _ ..." " '',. ' I .25 - .25

• 5°0 ._ .' .'6 .' ?.0 5°0 ._ :.' .' .'8 ?.0

y/(b/2) y/(b/2) (b) N-C B(1) wing.

(a) Near-conical wing.

Figure 9. Full-potential predicted surface pressure distributions for near-conical wing and near-conical wing with leading-edge bluntness variation from 0 to 0.012c at x/l -- 0.4 and 1.0 for M = 1.62 and CL = 0.1.

Geometry Asymmetry, .100 Near-conical 0 N-C A(20) 20 percent / -_ N-CA(50) 50 .075 .... N-C A(90) 90 /_,

///

.050

cL

,///'

.025 , , .2 .4 .6 .8 1.0 x/l .008 Geometry Asymmetry, percent Near-conical 0 .006 CD, / .004 .002

I I I I I

.2 .4 .6 .8 1.0 x/l Figure 10. Full-potential predicted lift and drag buildup for variations in wing asymmetry for hi = 1.62 and C L -- 0.1.

Geometry Asymmetry, .300 n percent /,.,, Near-conical 0 /_ N-C A(20) 20 /// N-C A(50) 50 Z// .225 N-C A(90) 90 / .150

cz

.075

I ! I I

.2 .4 .6 .8 1.0 x/l .08 Geometry Asymmetry, percent Near-conical 0 N-C A(20) 20 N-C A(50) 50

/

.06 m N-C A(90) 90

/

.04 CDj S / • /S" • _jS** .02 0 .2 .4 .6 .8 1.0 x/l Figure 11. Full-potential predicted lift and drag buildup for variations in wing asymmetry for M = 1.62 and CL = 0.3.

.... Upper surface -.50 - -- Lower surface -.50 -- -.25

CpO

I I I I 1 .2 .4 .6 .8 , .0 y/(b/2) % Cp0 .25 - .25 - I I I I I

5°o '2 '4 ', '8 ,!o 5°0

.2 .4 .6 .8 1.0 y/(b/2) y/(b/2) (a) Near-conical wing.

(b) N-C A(90) wing and shearing.

Figure 12. Full-potential predicted surface pressure distributions for near-conical wing and near-conical wing with 90-percent asymmetry and shearing at x/l = 0.4 and 1.0 for M = 1.62 and C L = 0.1.

.10o percent z z Geometry Asymmetry, //fH" N-C A(20) 20 N-C A(50) 50 N-C A(90) 90 Near-conical 0 / .075 .050 m

%

.025

I I I

.2 .4 .6 .8 1.0 x/l .o10o m Geometry Asymmetry, percent Near-conical 0 /1 N-C A(20) 20 /t,(, • N-CA(50) 50 /i .0075 -- ---N-CA(90)90 // /,,,, CD d .0050 .0025

I I I I I

.2 .4 .6 .8 1.0 x/l Figure 13. Euler-predicted lift and drag buildup for wing asymmetry variations for M = 1.62 and C L -- 0.1.

.300 Geomelry Asymmetry, percent ,/_ Near-conical 0 ,jr N-C A(20) 20 //

- _ N-CA(50) 50 //

.... N-C A(90) 90 /

%

.075

"7 I ! ! 1

0 .2 .4 .6 .8 1.0 x/l .0500 Geometry Asymmetry, percent Near-conical 0 N-C A(20) 20 .0375 C D j .0250 .0125

I I I i I

.2 .4 .6 .8 1.0 x/l Figure 14. Euler-predicted lift and drag buildup for wing asymmetry variations for M = 1.62 and C L : 0.3.

i

CP_

_v

v Z Z Z r_ .100 m Geometry ,,_t Near-conical /_' N-C BO) - /_ N-C T(3) .z/_ N-C A(90) J_" .075

//

.050

CL

.025 m

I I I

.2 .4 .6 .8 1.0 x/l .0100 Geometry Near-conical N-C B(1) = _ N-C T(3) z .... N-C A(90) ,/_l,)'," .0075 CD, I .0050

"y'''"

.0025

I I I I I

0 .2 .4 .6 .8 1.0 x/1 Figure 16. Euler-predicted lift and drag buildup for near-conical wing and near-conical wing with optimum bluntness, thickness, and asymmetry variations for M = 1.62 and CL = 0.1.

.300 N-C B(1) j' - -- N-C T(3) 2p .225 - = -- N-C A(90)/- /,, .150 m

CL

.075 ,2 .4 .6 .8 1.0 x/l .O500 Geometl_ Near-conical N-C B(1) j .0375 i, ss_ CD ,I .0250 .0125

I I I ! I

.2 .4 .6 .8 1.0 x/l Figure 17. Euler-predicted lift and drag buildup for near-conical wing and near-conical wing with optimum bluntness, thickness, and asymmetry variations for M = 1.62 and CL = 0.3.

10 _ Geometry Near-conical /_ N-C B(1) /// - N-C T(3) /._ o_, deg

I 1 I I

0 .1 .2 .3 .4

%

G¢_mell'y Near-conical N-C B(1) N-C T(3) N-C A(90) L/D

I I I I

0 .1 .2 .3 .4

%

Figure 18. Euler-predicted lift and drag characteristics for near-conical wing and near-conical wing with optimum bluntness, thickness, and asymmetry variations for M -- 1.62.

r_ Z f_jl _jjl Z I I

I'

I 3_ .100 -- Baseline Near-conical Naturalflow .075 -- Geometry /// .050 --

CL

.025 .2 .4 .6 .8 1.0 xll .0100 -- Geometl'y Natural flow f/ .0075 --

//

ct),i .0050--

.0025

/

I I I I I

.2 .4 .6 .8 1.0 xll Figure 20. Euler-predicted lift and drag buildup for baseline standard, baseline near-conical, and natural flow wings for M = 1.62 and CL = 0.1.

.300 -- GoDmell-y Baseline J Near-conical // .225 -- .150 --

cL

.075 --

Jl I I I

.2 .4 .6 .8 1.0 x/l .0500 -- Geometry Baseline Near-conical - -- Natural flow ,_'j" J CDj .0125

I I I I I

0 .2 .4 .6 .8 1.0 x/1 Figure 21. Euler-predicted lift and drag buildup for baseline standard, baseline near-conical, and natural flow wings for M = 1.62 and CL = 0.3.

.... Upper surface Lower surface -.50 - -.50 -- -.25 -.25 - Cp0 CpO__ .25 - _ .25 - I I I I I .500 I I I I I •50 0 .2 .4y/(b/2j6 .8 1.0 .2 .4/(b/2_6 .8 1.0 -.50 -- -.50 -- j- '1 -.25 /i -.25 -- _- _, f

\

Cp0 CpO .25 .25 - "500 .[2 .]4 .16 .18 11.0 "500 .]2 .]4 .16 ./ 11.0 y/(b/2) y/(b/2) (a) Near-conical wing. (b) Natural flow wing.

Figure 22. Euler-predicted surface pressure distribution for near-conical wing and natural flow wing (i.e., near-conical wing with optimum bluntness, thickness, and asymmetry variations) at x/1 = 0.4 and 1.0 for M=1.62 and CL = 0.1.

G_omel:ry Baseline Near-conical Natural flow a_, deg Geometry Baseline Near-conical - _ Natural flow \ LID Figure 23. Eulcr-predicted lift and drag characteristics for baseline standard, baseline near-conical, and natural flow wings for 3I = 1.62.

Natural flow .... Linear theory 1 m J z 0 z 0 ,.P (c) x/l = 0.40.

Co) x/l = 0.20.

(a) x/l = O.10.

i I -1_ I I

-1; 1 2

-1 o 1 2 1 2 Y Y 1- z 0 ,p,c" (cO x_l= o.6o.

I I I I I -1 o 1 2 3 4 5 Y ] -- z 0 .i 2 - I _ _ _ _ _ _

5'

;5--"

(e) x/l = 0.80.

I I I I I -10 1 2 3 4 5 Y 1 -- _S s __jm__-_._ z 0 , f.,¢.,¢ f Sf_'_ (t) x/l = 0.99.

I I I I I -10 1 2 3 4 5 Y Spanwise cuts of natural flow wing and linear-theory cambered wing. Design CL = 0.16.

Figure 24.

.10 CL .05

_T- I I I 1

.2 .4 .6 .8 1.0 x/l .010 i j CD, 1 .O05

I I I I J

.2 .4 .6 .8 1.0 xtl Figure 25. Euler-predicted lift and drag buildup for natural flow wing and linear-theory cambered wing for M = 1.62 and CL = 0.1.

.30

Linear theory z

Natural flow l/

, • p/ jJ_'/ sf_

i I i I

0 .2 .4 .6 .8 1.0 x/l .050 ,,e ,t CD, 1 .025 #,.._' _'P J' /

I i I ! 1

.2 .4 .6 .8 1.0 x/l Figure 26. Euler-predicted lift and drag buildup for natural flow wing and linear-theory cambered wing for M = 1.62 and CL = 0.3.

(CD)LT = (CD,i)LT + (CD.o)EMTA C + CD, V (CD)EMTA C = (CD,I)EMTA C + CD, V .07 m

%

Baseline .06 [] . EMTAC Design C L = 0.16 Natural flow

A

Baseline Design C L = 0.1(3 .05 Design CL = 0.16 Design C L = 0.30 .0_

%

.03 .02 .01 I I I I I ".I 0 .1 .2 .3 .4

%

Figure 27. Linear-theory and Euler-predicted drag characteristics for uncambered standard wing, linear-theory cambered wing, and natural flow wing at M ---- 1.62.

Form Approved REPORT DOCUMENTATION PAGE OMB No. 0704-0]88 Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources.

gathering and malnta_nlng the data needed, and completing and reviewing the collection of information Send comments i'egardlng thls burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202 4302, and to the Office of Management and Budget, Paperwork Reductlon Project (0704-0188), Washington, DC 20503.

1. AGENCY USE ONLY(Leave Mank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED May 1992 Technical Paper 4. TITLE AND SUBTITLE 5. FUNDING NUMBERS The Natural Flow Wing-Design Concept ¼rU 505-61-71-01 6. AUTHOR(S) Richard M. Wood and Steven X. S. Bauer 8. PERFORMING ORGANIZATION 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) REPORT NUMBER NASA Langley Research Center Hampton, VA 23665-5225 L-16837 10. SPONSORING/MONITORING 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) AGENCY REPORT NUMBER National Aeronautics and Space Administration NASA TP-3193 Washington, DC 20546-0001 11. SUPPLEMENTARY NOTES 12b. DISTRIBUTION CODE 12a. DISTRIBUTION/AVAILABILITY STATEMENT Unclassified Unlimited Subject Category 02 13. ABSTRACT (Maximum 200 words) A wing-design study has been conducted on a65 ° swept leading-edge delta wing in which the wing geometry was modified to take advantage of the naturally occurring flow that forms over a slender wing in a supersonic flow field. Three-dimensional nonlinear analysis methods were used in the study which was divided into three parts preliminary design, initial design, and final design. In the preliminary design, the wing planform, the design conditions, and the near-conical wing-design concept were derived, and a baseline standard wing (conventional airfoil distribution) and a basctine near-conical wing were chosen. During the initial analysis, a full-potential flow solver was employed to determine the aerodynamic characteristics of the baseline standard delta wing and to investigate modifications of the airfoil thickness, leading-edge radius, airfoil maximum- thickness position, and wing upper to lower surface asymmetry on the baseline near-conical wing. The final design employed an Euler solver to analyze the best wing configurations found in the initial design and to extend the study of wing asymmetry to develop a more refined wing. Benefits resulting from each modification are discussed, and a final "natural flow" wing geometry has been designed that provides an improvement in aerodynamic performance compared with that of a baseline conventional uncambered wing, linear-theory cambered wing, and near-conical wing.

14. SUBJECT TERMS 15. NUMBER OF PAGES Supersonic; Wing design; Computational fluid dynamics; Drag reduction 43 16. PRICE CODE Aq_ 20. LIMITATION 17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION OF REPORT OF THIS PAGE OF ABSTRACT OF ABSTRACT Unclassified Unclassified _ISN 7540-01-280-5500 'Standard Form 298(Rev, 2-89) Prescribed by ANSI Std Z39-18 298-102 NASA-Langley. 1992