nasa technical note e./ - corethe swedish saab 35 draken and saab 37 viggen (ref. 4), and the...

N A S A T E C H N I C A L NOTE N A S A e./

DESIGN CHARTS OF STATIC AND ROTARY STABILITY DERIVATIVES FOR CROPPED DOUBLE-DELTA WINGS IN SUBSONIC COMPRESSIBLE FLOW

by John E. Lumur

Lungley Reseurch Center Lungley Stution, Humpton, Vu.

TN - D-5661

NATIONAL AERONAUTICS AND SPACE A D M I N I S T R A T I O N WASHINGTON, D. C. 9 FEBRUARY 1970 /iiI i

https://ntrs.nasa.gov/search.jsp?R=19700009361 2020-03-23T20:48:41+00:00Z

1. Report No. 2. Government Access ion No. NASA TN D-5661 I

4. T i t l e and Subt i t le

DESIGN CHARTS OF STATIC AND ROTARY STABILITY DERIVATIVES FOR

CROPPED DOUBLE-DELTA WINGS I N SUBSONIC COMPRESSIBLE FLOW

7. Author(s)

J o h n E. Lamar

.

9. Performing Organizat ion Name and Address

NASA Langley Research Center

Hampton, Va. 23365

2 . Sponsoring Agency Name and Address

National Aeronautics and Space Admin is t ra t ion

Washington, D.C. 20546

5. Supplementary Notes

6. Abstract

3. Rec ip ien t ' s Catalog No.

5. Report Date

February 1970 6 . Performing Organizat ion Cot

-8. Performing Organ i ro t ion Ret

L-6716 0. Work U n i t No.

126- 13- 10-01-23 1. Controct or Gront No.

-3. Type o f Report ond P e r i o d C

Technical Note

4. Sponsoring Agency Code

A n evaluation of a modified vers ion of t h e Mul thopp subsonic l i f t ing-surface t h e o r y was made by

comparing t h e theoret ical values w i t h experimental data. Near zero l ift, t h e theory was found to predict

reasonably adequately t h e l i f t -curve slope, aerodynamic center, damping in rol l , damping in pi tch, and

lift coefficient due to p i tch rate for delta and cropped delta p lan forms and also t h e l i f t -curve slope for

double-delta planforms. Based on t h i s theory, a series of design char ts has been prepared fo r cropped

double-delta p lan forms in subsonic compressible flow.

17. Key Words Suggested by Au lhor (s ) 18. D is t r ibu t ion Statement

Design c h a r t s Unclassified - Unlimited Cropped double-delta wings

Static and ro ta ry derivatives

I

9. Security C lass i f . (of t h i s report) M. Secur i ty C lass i f . (of t h i s page) 22. Pr ic

Unclassified Unclassified $:

'For sale by t h e Clearinghouse fo r Federal Scient i f ic and Technical In fo rmat ion Springfield, V i r g i n i a 22 151

DESIGN CHARTS OF STATIC AND ROTARY STABILITY DERIVATIVES

FOR CROPPED DOUBLE-DELTA WINGS IN

SUBSONIC COMPRESSIBLE FLOW

By John E. Lamar Langley Research Center

SUMMARY

An evaluation of a modified version of the Multhopp subsonic lifting-surface theory was made by comparing the theoretical values with experimental data. Near zero lift, the theory was found to predict reasonably adequately the lift-curve slope, aerodynamic ten

ter, damping in roll , damping in pitch, and lift coefficient due to pitch rate for delta and cropped delta planforms and also the lift-curve slope for double-delta planforms. Based on this theory, a series of design charts has been prepared for cropped double-delta plan-forms in subsonic compressible flow.

INTRODUCTION

The National Aeronautics and Space Administration has programs underway to provide aerodynamic design information on aircraft configurations and components for speeds ranging from low subsonic to hypersonic. Information has already been published that indicates the consideration which has been given to both fixed and variable geometry wings for use in the design of supersonic transport and military aircraft. (See, for example, refs. 1 and 2.)

A class of fixed wings, the cropped double-delta wings, has found recent application in the design of these aircraft. Examples of proposed and actual aircraft using the concept of cropped double-delta wings are the latest Boeing supersonic transport (ref. 3), the Swedish SAAB 35 Draken and SAAB 37 Viggen (ref. 4), and the Lockheed A-11, also designated YF-12A and SR-71 (ref. 4). A search of the literature has indicated the existence of some design data, but only a few systematic investigations have been performed on cropped double-delta planforms in the subsonic and supersonic speed regimes. (See, for example, refs. 5 and 6.)

In order that a part of this void might be filled, a systematic investigation using the modified Multhopp subsonic compressible lifting-surface approach of reference 7 (after it was shown to be applicable) with the appropriate boundary conditions was undertaken

to determine the lift-curve slope, aerodynamic center, damping in roll, damping in pitch, and lift coefficient due to pitch rate for nine families of cropped double-delta planforms. The purpose of the present paper is to present the resul ts of this investigation in design-chart form. The design charts presented are for the attached-flow condition only and do not include the effects of leading-edge separation, which are discussed in reference 8.

SYMBOLS

b2aspect ratio, -S

wing span, feet (meters)

Liftl i f t coefficient, qoos

lift-curve slope, - per degreea @

lift coefficient due to pitch rate, - per radian

rolling-moment coefficient, Rolling moment

goosb

damping-in-roll parameter, - per radian Pba -2v

pitching-moment coefficient about F / 4 point, Pitching moment q . p

damping-in-pitch parameter, - per radian - CYF’

root chord

-c = Mean geometric chord of the total wing b/2

subsonic free-stream Mach number

2

m number of span stations where pressure modes are defined

N number of chordal control points at each of m span stations

P roll rate, radians/second

q pitch rate about E/4, radians/second

qcc f ree-stream dynamic press u re, pounds/f oot (newtons/m eter2)

S total wing area, feet2 ( m e t e d )

V free-stream velocity, feet/second (meters/second)

X,Y rectangular Cartesian coordinates nondimensionalized with respect to b/2, where origin is in plane of symmetry at half root chord (positive x, aft; positive y, along right wing panel)

Xac aerodynamic center, in fractions of F', measured from leading edge of C'

(positive aft), ~

-aCm + 1. aCL 4

yb spanwise location of leading-edge break, feet (meters)

a angle of attack, degrees

A outboard leading- edge sweep angle, degrees

A' = tan-l(tan A/p ) , degrees

x overall taper ratio, Tip chord Root chord

X inboard leading-edge sweep angle, degrees

x' = tan-l(tan x / p ) , degrees

3

METHOD OF ANALYSIS

The method used in this paper for predicting lifting pressures, hereafter called the present method, employs the modified Multhopp approach. (See ref. 7.) Briefly, this method employs an acceleration potential (developed from a sheet of pressure doublets) in conjunction with the linearized Euler equations to relate the pressure difference across the wing to the downward velocity over the wing surface. The effects of compressibility are accounted for by using the Prandtl-Glauert rule.

Formulation of the problem of determining the pressure difference across the wing leads to an integral equation to which the solution in closed form is difficult, except for certain classes of wings, because the answers sought (surface loadings) a re a part of the integrand. Also adding to the difficulty of solution is the presence of a second-order singularity which is in the integrand. For these reasons it has been necessary to resort to an approximate solution which makes use of a finite, rather than unlimited, number of boundary points over the wing at which the flow is constrained to be tangent. Coupled with this finite number of boundary points is a series representation of the lifting pressures which have the same number of unknowns as there a r e boundary points and whose values are determined by a process of matrix algebra.

In reference 7 the accuracy of the numerical results of the modified Multhopp method was shown to be dependent on the combination of the number of spanwise stations m where each of N chordwise control (boundary) points was located. Two methods of estimating the appropriate combinations of N and m were studied in reference 7; one was concerned with computing the correct leading-edge thrust from the chord loadings and the other with seeking the aerodynamic center for which convergence had .occurred.

The converged-aerodynamic-center method was selected to determine the appropriate pair of N and m to be used in this report because it involved less computational labor. Upon examining the results of reference 7 and making use of the method described therein, a pattern of N = 8 and m = 23 was postulated to be sufficient for the kinds of planforms to be studied herein. In order to verify that this pattern would give results which lay in a converged-aerodynamic-center region, the aerodynamic center was calculated for several different sets of N and m and is plotted as a function of m in figure 1 for one of the families of cropped double-delta planforms for which design charts were prepared. A +2-percent E' er ror band is plotted about the N = 8 and m = 23 results, and for m > 11 the aerodynamic-center values appear to lie within this band. However, the predicted results obtained with the N = 8 and m = 23 pattern a r e not necessarily the same as would be found for an infinite number of points, but the two answers should compare closely. (The N = 8 and m = 23 control-point pair was also used in predicting the rotary aerodynamic characteristics.)

4

I:

Although the present method w a s developed for steady, longitudinal, lifting problems, it can be used to compute rotary stability derivatives because of the relationship developed in the lifting-surface theory between tangential-f low boundary conditions and the downwash produced by a sheet of pressure doublets. The manner in which computation for rotary stability derivatives is made is based on recognizing, as pointed out in reference 9, that the steady-state rolling and pitching maneuvers are equivalent to linear twist and parabolic camber, respectively. This equivalence can be seen by realizing that the upwash on the upgoing wing in steady, rolling flight is just py and the tangent of the angle that the section makes with the free stream is just py/V. (The tangent of an angle is approximately equal to the angle in radians for small angles.) Integrating this expression along the chord gives a straight mean camber line, but one that is twisted along the span. A similar procedure can be used to obtain the mean camber line for the pitching motion about the reference point. The upwash is just qx for the upgoing part of the wing. The tangent of the angle that it makes (small angles of attack) with the free stream is just qx/V. Integrating this along the chord results in a parabolic camber.

VERIFICATION

In order to verify the applicability of the present method for predicting the static and rotary aerodynamic characteristics of the cropped double-delta planforms, a comparison of the results from this method with experimental data (refs. 10 to 18) and with results from other theories (refs. 19 and 20) is presented in table I for delta, cropped delta, and double-delta wings. It should be noted that in this table no comparisons a re presented for cropped double-delta wings because experimental data were lacking. However, comparisons of data for the other wings with results from the present method indicate that this method was applicable to wings with breaks in the leading edge and cropped tips and, hence, to cropped double-delta wings. Results from the present method a r e also compared with experimental data in figure 2. The theoretical methods selected for comparison are those which predict only the rotary stability derivatives since the theoretical methods which predict the static stability derivatives were compared with results from the present method in reference 7. In examining table I, it should be remembered that the theoretical method of reference 19 has an inherent restriction associated with it because its derivation was based on wings with vanishing aspect ratios.

Static Stability Derivatives

Generally good agreement is found between the theoretical and experimental values for CL,, where the theoretical values have variations of about rtl0 percent from the experimental values. (See fig. 2.) Also, the predicted Xac values generally agree well with the experimental values and have variations of about 4 0 percent from the

5

experimental values or *5 percent of E. No comparison between the theoretical and experimental location of Xac could be made for the double-delta wings of A = 1.60 because reference 18 reports that the pitching-moment data were questionable, and hence not presented, for small angles of attack.

Rotary Stability Derivatives

Comparison of results from the present method with experimental data and results from the other theories for the rotary stability derivatives is restricted to the delta and cropped delta planforms since (1)no experimental rotary derivatives were found for the double-delta wings and (2) the theoretical methods of references 19 and 20 were not readily (if at all) applicable to the double-delta wings.

Damping in roll C .- The present method generally predicts values of CZP(2)

about 20 percent more negative than those measured experimentally for a wing of A I 1.0. A large part of this loss in theoretical damping may be due to the wing tips of the experimental model experiencing a stall condition for which the present method does not account. The actual individual differences between the experimental data and values from the present method a r e smaller for all configurations than those resulting from use of the method of reference 19, which was developed for slender wings only. However, when experimental data and values from the present method a r e compared with values from the method of reference 20, it is found that the predicted values of reference 20 agree almost as well with experimental data as values from the present method for the delta wings and agree slightly better with experimental data for the cropped delta wings. Hence, either the method of reference 20 o r the present method could be applied with almost equal accuracy for delta or cropped delta wings but the present method is more easily applied to cropped double-delta wings.

Damping in pitch (Cmq) .- The present method generally predicts values of Cm

q which are about 30 percent less stable than the experimental values; however, the results obtained by this method agree better with experimental data than those obtained by the method of reference 19, which predicts results that a r e consistently more stable than experimental data and results from the present method. The method of reference 19 is based on slender-wing theory and is obviously outside of its range of applicability.

Lift coefficient due to pitch ra te CL9).- The present method generally predicts ~~

( __ values of CL about 25 percent larger than the experimental values; however, the

q resuIts from this method agree better with experimental data than the results predicted from the method of reference 19. The method of reference 19 is again obviously outside its range of applicability.

6

General observation.- If a consistent set of experimental data and the corresponding predicted values from the present method, as given in table I, were plotted against aspect ratio, it would be found that the correct trends are predicted by the present method for all three rotary stability derivatives.

DATA PRESENTATION

Because of the multiplicity of planforms that can be encountered in connection with composite wings, such as the cropped double-delta planform studied here, design charts cannot be expected to eliminate the need for computer calculations to determine accurately the aerodynamic characteristics of any particular planform. However, design charts which broadly cover the range of planform parameters of general interest for aircraft which must fly at both subsonic and supersonic speeds a re felt to be useful in selecting the general type of planform which might best satisfy the various aerodynamic requirements for the preliminary design thereby reducing the computer study requirements. Design charts, which illustrate the degree to which some of the more important aerodynamic parameters a re affected by the various geometric variables, a r e presented herein.

In considering composite planforms, a variety of geometry controlling parameters are available for selection as the independent variables. However, since all of the configurations studied have straight unswept trailing edges, the following variables are the ones selected for use herein:

(1) Inboard leading-edge sweep angle (x) (2) Outboard leading-edge sweep angle (A)

(3) Spanwise location of the leading-edge break yb

(4) Overall taper ratio (A)

These variables, and in particular x and A, were used to develop design charts at M, = 0. However, the charts can be used for any subsonic Mach number if the wing geometric properties a re corrected in accordance with the Prandtl-Glauert rule. Hence, the design charts are presented in t e rms of these corrected variables x' and A' and also in terms of yb and A .

Wing geometric characteristics.- Unfortunately, the design charts based on the four variables just discussed do not allow convenient identification of other dependent parameters of interest, such as aspect ratio and mean geometric chord. Therefore, charts of wing geometric characteristics have been prepared which show the variation of inboard sweep angles and mean geometric chord with aspect ratio; these a re presented as figures 3 and 4, respectively. Since the wing trailing edge is straight, a knowledge of the

7

mean geometric chord allows a ready determination of the moment reference point F'/4 with respect to the trailing or leading edge of the root chord.

Wing aerodynamic characteristics .- The static aerodynamic characteristics for-

the cropped double-delta planforms are presented in figures 5 and 6. Figure 5 shows the lift-curve slope and figure 6 the aerodynamic center. The rotary aerodynamic characteristics for the cropped double-delta planforms are shown in figures 7, 8, and 9. Damping in roll and damping in pitch are shown in figures 7 and 8, respectively, and lift coefficient due to pitch rate is presented in figure 9.

Figure 10 is a summary of data which presents the aerodynamic-center locations for cropped delta wings as a function of X. This figure is included herein because it shows an interesting occurrence; that is, regardless of the leading-edge sweep angle, a cropped delta wing has its aerodynamic center at approximately 25 percent of the plan-form mean geometric chord when the taper ratio is 0.32. Reference 21 reports a similar finding except that the taper ratio required was slightly different.

The reasoning used in reference 21 was that if planforms with limiting aspect ratios of zero and infinity both had their aerodynamic centers at the 25-percent mean-geometricchord location, then perhaps, by suitable planform variation, the aerodynamic center could be kept at the 25-percent mean-geometric-chord point within these two aspect ratio limits. By using this basic reasoning, which does not hold for a delta wing (see ref. 22), the taper ratio required for a cropped delta wing was determined to be -0.303.

CONCLUDING REMARKS

An evaluation of a modified version of the Multhopp subsonic lifting-surface theory was made by comparing the theoretical values with experimental data. Near zero lift, the theory was found to predict reasonably adequately the lift-curve slope, aerodynamic center, damping in roll, damping in pitch, and lift coefficient due to pitch rate for delta and cropped delta planforms and also the lift-curve slope for double-delta planforms. Based on this theory, a se r i e s of design charts has been prepared for cropped double-delta planforms in subsonic compressible flow.

Langley Research Center, National Aeronautics and Space Administration,

Langley Station, Hampton, November 21, 1969.

8

REFERENCES

1. Ray, Edward J.: NASA Supersonic Commercial Air Transport (SCAT) Configurations: A Summary and Index of Experimental Characteristics. NASA TM X-1329, 1967.

2. Spearman, M. Leroy; and Foster, Gerald V.: A Summary of Research on Variable-Sweep Fighter Airplanes. NASA TM X-1185, 1965.

3. O'Lone, Richard G.: SST Redesign Permits Titanium Gains. Aviat. Week Space Technol., vol. 89, no. 25, Dec. 16, 1968, pp. 41-43.

4. Taylor, John W. R., ed.: Jane's All the World's Aircraft, 1968-1969. McGraw-Hill Book Co., Inc.

5. Tumilowicz, Robert A.: Characteristics of a Family of Double-Delta Wings Designed to Reduce Aerodynamic Center Shift With Mach Number. M.S. Thesis, Polytech. Inst. Brooklyn, 1965.

6. Benepe, David B.; Kouri, Bobby G.; Webb, J. Bert; et al.: Aerodynamic Characteristics of Non-Straight-Taper Wings. AFFDL-TR-66-73, U.S. Air Force, Oct. 1966. (Available from DDC as AD 809409.)

7. Lamar, John E.: A Modified Multhopp Approach for Predicting Lifting Pressures and Camber Shape for Composite Planforms in Subsonic Flow. NASA TN D-4427, 1968.

8. Polhamus, Edward C.: A Concept of the Vortex Lift of Sharp-Edge Delta Wings Based on a Leading-Edge-Suction Analogy. NASA TN D-3767, 1966.

9. Etkin, Bernard: Dynamics of Flight - Stability and Control. John Wiley & Sons, Inc., c.1959.

10. Tosti, Louis P.: Low-Speed Static Stability and Damping-in-Roll Characteristics of Some Swept and Unswept Low-Aspect-Ratio Wings. NACA TN 1468, 1947.

11. Lange and Wacke: Test Report on Three- and Six-Component Measurements on a Series of Tapered Wings of Small Aspect Ratio (Partial Report: Triangular Wing). NACA TM 1176, 1948.

12. Jaquet, Byron M.; and Brewer, Jack D.: Low-Speed Static-Stability and Rolling Characteristics of Low-Aspect-Ratio Wings of Triangular and Modified Triangular Plan Forms. NACA FtM L8L29, 1949.

13. Lange and Wacke: Test Report on Three- and Six-Component Measurements on a Series of Tapered Wings of Small Aspect Ratio (Partial Report: Trapezoidal Wings). NACA TM 1225, 1949.

9

14. Goodman, Alex; and Jaquet, Byron M.: Low-Speed Pitching Derivatives of Low-Aspect-Ratio Wings of Triangular and Modified Triangular Plan Forms. NACA RM L50C02, 1950.

15. Berndt, Sune B.: Three Component Measurement and Flow Investigation of Plane Delta Wings at Low Speed and Zero Yaw. KTH-Aero TN 4,Div. Aeronaut., Roy. Inst. Technol. (Stockholm), 1949.

16. Goodman, Alex; and Adair, Glenn H.: Estimation of the Damping in Roll of Wings Through the Normal Flight Range of Lift Coefficient. NACA TN 1924, 1949.

17.Hopkins, Edward J.; Hicks, Raymond M.; and Carmichael, Ralph L.: Aerodynamic Characteristics of Several Cranked Leading- Edge Wing- Body Combination at Mach Numbers From 0.4 to 2.94. NASA TN D-4211,1967.

18.Wentz, William H., Jr.; and Kohlman, David L.: Wind Tunnel Investigations of Vortex Breakdown on Slender Sharp-Edged Wings. Rep. FRL 68-013 (Grant NGR-17-002-043),Univ. of Kansas Center for Res., Inc., Nov. 27, 1968.

19.Ribner, Herbert S.: The Stability Derivatives of Low- Aspect-Ratio Triangular Wings at Subsonic and Supersonic Speeds. NACA TN 1423, 1947.

20.Queijo, M. J.: Theory for Computing Span Loads and Stability Derivatives Due to Sideslip, Yawing, and Rolling for Wings in Subsonic Compressible Flow. NASA TN D-4929, 1968.

21.Berndt, Sune B.: Note on the Position of the Aerodynamic Center of Pointed Wings at Subsonic Speeds. KTH-Aero TN 5, Div. Aeronaut., Roy. Inst. Technol. (Stockholm), 1949.

22.Jones, Robert T.: Properties of Low-Aspect-Ratio Pointed Wings at Speeds Below and Above the Speed of Sound. NACA Rep. 835, 1946. (Supersedes NACA TN 1032.)

10

----- ----- - ---- -----

-----

TABLE I.- WINGS AND REFERENCES USED IN THE VERIFIING PROCEDURE AT LOW SUBSONIC MACH NUMBERS

Geometric parameters I StaAa;abt;ty I Rotary stability Iderivatives

4 1 d 4 d +

Delta wing

0.25 0 86.4 0 0 0.006 0.466 _.___

,0065 .459 _ _ _ _ _ .50 0 82.9 0 0 ,0125 ,464 -0.01

.0125 .443 -.046 ..-__.___-.049

1.00 0 76 0 0 .024 ,400 -.035 .022 ,402 ...._

,023 .421 ...__

,023 .414 -.os7 -.098_ _ _ _

1.07 0 75 0 0 ,024 .450 -.075 ,024 ,410 -.093

.._- -.lo5

1.67 0 67.4 0 0 .034 ,391 _ _ _ _ _ ,034 ,386 _ _ _ _ _

2.00 0 63.4 0 0 .0425 ,346 -.117 ,037 ,375 _..__

.0385 ,315 -.153 ..._ -.196

..... _ _ _ _ -.161

2.31 0 60 0 0 .042 ,400 -.17 .042 ,368 -.171 _ _ _ _ -227 ..... ...- -.160

2.5 0 58 0 0 ,045 ,374 __.__

,045 .363 .....

3.00 0 53 0 0 ,050 ,320 -.166 ,048 ,355 _ _ _ _ _ ,050 .352 -204 ...___ _ _ _ -.295

..._ -.221

4.00 0 45 0 0 ,057 ,362 -.22 ,058 .336 -.241

.... -,393

..._ -.278

Cr red delta wings

0.54 0 15.9 0 0.303 0.016 0.289 _ _ _ _ _ ,014 2 5 6 _ _ _ _ _

.89 0 61.4 0 ,303 .025 ,262 .....

,023 ,257 .....

.99 0 45 0 .56 ,0115 2 2 0 -0.065 ,0255 ,174 - ,098

.._. -.095

1.33 0 66.6 0 ,125 .030 ,367 _ _ _ _ _ .031 ,336 .....

1.33 0 61 0 ,250 .032 ,322 .....

.032 ,219 _.___

1.33 0 45 0 ,500 .033 ,267 .....

,032 .203 _ _ _ _ _ 1.34 0 58 0 .303 ,034 264 _ _ _ _ _

.032 ,258 ...__

1.99 0 45 0 .36 .036 2 5 0 -.16 ,043 246 -.I85 _ _ _ _ _ _ _ _ _ -.179

3.00 0 45 0 .15 .050 .340 - 2 2 .053 .305 -.241

.... -.239

D ,le-delta wing

1.30 82 60 0.500 0.024 0.647 .025 ,583

1.60 75 65 ,474 ,0365 _ _ _ _ ,032 .455

1.60 80 65 .320 ,036 ... -_ _ _ _ .032 ,488

Unpublished experlm Present method

10 Present method Method of 19

10 11 15 Present method Method of 19

12,14 Present method Method of 19

15 Present method

10 11 Present method Method of 19 Method of 20

12,14,16 Present method Method of 19 Method of 20

15 Present method

10 11 Present method Method of 19 Method of 20

12,14,16 Present method Method of 19 Method of 20

15 Present method

15 Present method

12, 14 Present method Method of 20

13 Present method

13 Present method

13 Present method

15 Present method

12, 14 Present method Method of 20

12,14 Present method Method of 20

17 Present method

16 Present method

18 Present method

11

xuc ' : 2%E'bund percent 0 u h u f the N=8

m =23point

50 I

xuc I 40 percent E

30

20

70./ c;50 i.I

~

I

40 i xuc I

percent Z 30

I I

20

1~

h

Ti, YJ i

I

IXac i I percent E I

I1I

~II I i I

f6 a 24 32 36 40 44 m

(a) Different transformed leading-edge inboard sweep angles a re used. A ' = 5 0 O ; A = 0.10. 3= 0.50. s b/2

Figure 1.- Effect of number of chordwise and spanwise stations on t h e aerodynamic center for typical cropped double-delta planforms.

12

m

IlI i/ E' bond about the N=8 m = 23point

N ,.. 0 2 1 0 4 0 6

8 , j

u1 II I I

i ~

i

I

I I

t I

'J c?, h 0 I I

Ir;.lT

I6 h

/. I

4 8 12 16 20 m 22 28 32 36 40 44

(b) Different leading-edge break ratios are used. x' = 85'; A ' = so;A = 0.10.

Figure 1.- Concluded.

13

I + Ii Y I I I

i I ! 1

iP II

~

, -

0 .o/ .L .V"

easured

I

I I

I

I

I

pY

1 1

30 40 50 70 ~xac'!!easured ,percent E

(a) Static derivatives.

F igure 2.- Predict ion accuracy of modified Multhopp method at N = 8 and m = 23 for some aerodynamic character ist ics of delta and delta-like wings.

14

0%error rror ‘1,T 4I

I

I

-24

-.04 -.08 -.I2 -I6 (Cm, I

mea ?d)~

I

-

(b) Rotary derivatives.


15

90

80

70

60

50

40

30

20

X,'deg

0

- /O

-20

-30

-40

-50

-60 0 / 4 6 7 8 9 10

(a) A ' = go;h = 0.1.

Figure 3.- Geometric character ist ics of cropped double-delta planforms - re la t ionsh ip of parameters.

16

I

2 4 5 7 8 9 10 P A

(b) A ' = 50'; A 0.25.

Figure 3.- Continued.

17

(c) A ' = 50'; A = 0.50.


18

I -4

X '.

3 4 5 6 7 8 9 10 / A

(d) A ' = 60'; h = 0.1.


19

I ,

,, I

i Ii ~

! ?

0 / 3 4 5 6 7 8 9 10 P A

(e) A ' = 6oo; A = 0.25.


20

-- -b 2

a050 I 2 3 4 5 6 8 9 IO

P A

(f) A ' = 60°; A = 0.50.


21

0

,

5 6 7 8 9 IO P A

A ' = 72'; h = 0.10.


22

I

90

80

70

60

50 i

40 I I-

30

20

X,'deg /O

0

- /O

-2C

-3G

-4c

-5c

- 6 C 0 4 5

PA

(h) A ' = 720; A = 0.25.


0

7 8 9 10

23

0 I 2 3 4 5 6 7 8 9 10

/ A

(i) A ' = 720; A = 0.50.


24

tt I 7

I l l 050

/ / I 075

1 i

C0 I 2 4 5

(a) A ' = No; A = 0.10.

Figure 4.- Geometric character ist ics of cropped double-delta planforms - mean geometric chords.

25

bP - i

I

' iII I

I I I' 2 3 4

(b) A ' = 50'; A = 0.25.


26

5

......1..

/ 2 3 4 5

(c) A ' = 50'; A = 0.9.


27

ii

0.25

I 1I 1

0.50

I /I I

0 . I IC

I "-" : : I

ITV I

.illI I '

2 3 4

(d) A ' = 60°; h = 0.10.


28

_.. .....

II

/4 -yb b/2

I ' .0.25

/: - 0.50

l l i1.2 I l l5 / / I 1

/C

5

6

E' 7

6

5

4

3

".--I

J-j/' , ..i

I- -2b -I 13

bA

I

2 i

GI / !

f-I

00 2 3 4 5

(e) A ' '= 60°; A = 0.25.


29

050

075

I O 0

0 / 2 3 4 5

(f) A ' = 60'; h = 0.50.


30

L

- 0.50

I l l I l l 0.75

I l lI l l

I 00 I I !

I

I I

II I I

1 J

.Ti

r /

J I

It I x

i, I ! ;z c

I

I

2 3

(g) A ' = 72"; h = 0.10.


31

I

E'

LL

t j I

-

I

I

i I I II I I

2 3 4 5

(h) A ' = 72"; h = 0.25.


32

...

1.5 I

/7 i l l I ’

16

/5

/4

13

/2

I /

IO

E.‘ 9

8

7

6

5

4

3

2

/

0 0 2

T

3 4 5

(i) A ’ = 720; A = 0.50.


33

~

I

I 1

/___"..

IIIT'

x.. 1'

1. I " 1 . i P

I I

I j1R

C> 'I! I

r; I

I.

)&;I

2

-0 .I .2 .3 4

-yb b/2

(a) A ' = 5Oo; A = 0.10.

Figure 5.- Li f t -curve slope fo r cropped double-delta planforms.

34

. (6) A' = 50'; A = 0.25.


35

X,'deg

LO u,b/2

(c) A ' = 50'; h = 0.50.


36

I I - I I I1 I I I I I II

0 85 0 80 0 75 A 70 A 65 a 55 ' n 50 ' 0 0

40 20

n O I

I

!

! I

E f

I 1,

! iiII

i 'I

iI ~

I II ~

I

I

I ~

.I .2 .3 4 .5 .6 3 b/2

(d) A ' = 60'; A = 0.10.


37

"0 .I .2 .3 4 .5 .6 .7 .8 .9 LO

(e) A ' = 60°; h = 0.25.


38

c; I X:deg o 85

80_.

0 75 A 70 A 65 a 55 n1 : c

.2 .3 4 .6 .7 E .9 LO 2 b/2

(f) A ' = 60'; h = 0.50.


39

.I

I

I ~

Ii

I i +I !

I iLI 1

I

1 i I i-I

j 1 ..

.9

(g) A ' = 720; h = 0.10.


40

(h) A ' = 720; A = 0.25.


41

w

0 75 A 70 4 6 0 a 40 n 20

.03 0 0

0 5

02

.01

n -0 .I .2 .3 .6 .7 .8

(i) A ' = 720; A = 0.50.


42

.9

x

X,'deg

o 85 58 80

-+A A 7.5

XUC /

percent

(a) A ' = 50°; A = 0.10.

Figure 6.- Aerodynamic center for cropped double-delta planforms.

43

I

I

b X; deg o 85

80 0 75 A 70

xac I

percent

(b) A ' = 50°; A = 0.25.


44

xideg o 85

80 0 75 A 70 A 60 Ll 55 D 45 o 40

Xac , percent Z

-0 ./ .2 .3 4 .5 .6 .I .8 .9 LO u, b/2

(c) A ' = 50'; A = 0.50.


45

X i deg o 85

80

XOC

'Ic.5 '1

(d) A ' = 60'; A = 0.10.


46

X: deg o 85

80 0 75 A 70 A 65 d 55 0 50

Xac , percent E'

(e) A ' = @lo;h = 0.25.


47

X: deg 0 85

80 0 75

60" A 70 A 65 Ll 55 D 50 o 40 0 20 0 0

Xac I

percent E

./ .2 .3 .4 .5 .6 :7 .8

(f) A ' = 60'; A = 0.50.


48

.9

J

xoc , percent

0 ./ .2 .3 4 .5 .6 .7 .8 .9

(g) A ' = 72"; A = 0.10.


49

I11111 I I I I I I I I 11111111 -I

x;deg o 85

80 0 75 A 70

xoc , percent

.0 ./ .2 .3 .4 .5 .6 .7 .8 .9 LO

(h) A ' = 72'; A = 0.25.


50

11111111111.1111111111.1111111111111111111 I 111 I I I I I111111I I II II 111 I I II I II I II 11111111111111111

x: deg o 85 0 80 0 75 A 70

Xac , percent 2'

=0 .2 .3 4 .5 .6 .7 .8 .9 LO

b/2

(i) A ' = 720; A = 0.50.


51

0 ./ .2 .3 .5 .6 .7 .8 .9 LO

(a) A ' = No; A = 0.10.

F igure 7.- Damping-in-rol l rotary derivative for cropped double-delta planforms.

52

e

0 ./ .2 .3 4 .5 .6 .7 .8 .9 LO u, b/2

(b) A ' = 50'; A = 0.25.


53

0 ./ .2 .3 .5 .6 .7 .8

-yb 6/2

(c) A ' = 50°; A = 0.50.


54

.9

(d) A ' = 6oo; A = 0.10.


55

I

(e) A ' = 60'; A = 0.25.


56

-----.-. 111.111111. "111111 I 1 1 1 1 1 1 1 1 1 1 I 1.11111. I 1 111 I IIIIII I I I 111 1111 I I I I I I ' I I II I II

(f) A ' = 60'; A = 0.50.


57

(g) A ' = 720; A = 0.10.


58

./ .2 .3 .4 .5 .6 .7 .8 .9

u, b/2

(h) A ' = 720; A = 0.25.


59

0

(i) A ' = 720; A = 0.50.


60

-.--.. I... -1111...1.-..11111...1.11111111.. 11111111.1111.. IIl111.11.1111.111111111111111 I11111 1111111 I I I I 111 I I II 111 I 1 1 1 11111111 I I 111111II I 111 I '

-1 .1 0 .I .2 .3 4 .5 .6 .7 .8 .9 l.0

(a) A ' = 50°; A = 0.10.

Figure 8.- Damping-in-pitch rotary derivative for cropped double-delta planforms.

(b) A ’ = 50°; A = 0.25.


-.3

N m q

-4

-.5

-.6

- .7

b/2

(c) A ' = 50'; A = 0.50.


(d) A ’ = 60’; A = 0.10.


1- -7\ X'

-.3

-4 f lmo

-.6

-.I

0 .I .2 .3 4 .5 .6 .7 .8 .9 ub-6/2

le) A ' = @lo;h = 0.25.


10

(f) A ' = 600; A = 0.50.


x:deg o 85

(g) A' = 720; A = 0.10.


0 .I .2 .3 .5 .6 .7 .8 .9 LO yb-6/2

(h) A ' = 72O; A = 0.25.


X:deg

o 85 80

o 60

b/2

(i) A ' = 720; h = 0.50.

Figure 8.- Concluded. Q)W

0 ./ .z . 3 4 .5 .6 .7 .8 .9 yb

b/Z

(a) A ' = 50°; A = 0.10.

Figure 9.- Rotary derivatives of l i f t coeff icient due to p i tch rate for cropped double-delta planforms.

70

U0 ./ .2 .3 I .5 .6 .7 .8 .9 LO

u,b/2

(b) A ' = SOo; A = 0.25.


71

0 ./ .z .3 .5 .6 .7 .8 .9 LO

(c) A ' = 50'; A = 0.50.


72

I I I

0 ./ .2 .3 .4 .5 .6 .7 .8 .9 LO

(d) A ' = 60°; h = 0.10.


73

I

(e) A ' = 60°; A = 0.25,


74

./ .2 .3 .5 .6 .7 .8 .9 LO

(f) A ' = 6oo; A = 0.50.


75

0

0 ./ .2 . 3 A .5 .6 .7 .8 .9 LO

(g) A ' = 72"; h = 0.10.

F igure 9.- Continued.

76

./ .2 .3 4 .5 .6 .7 .8 .9 LO

A 6/2

(h) A ' = 72'; A = 0.25.


77

0

"0 ./ .2 .3 .5 .6 .7

-' b 6/2

(i)A ' = 720; h = 0.50.


78

.9

= 1.0

=O

XUC 8

percent C I

I 2 .3 .5 /I

Figure 10.- Aerodynamic center for cropped delta planforms.

NASA-Langley, 1970 -1 L-6716 79

AERONAUTICSNATIONAL AND SPACE ADMINISTRATION WASHINGTON,D. C. 20546

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