DEVELOPMENT OF A METHOD FOR PREDICTING THE DRAG DIVERGENCE

MACH NUMBER AND THE DRAG DUE TO COMPRESSIBILITY

FOR CONVENTIONAL AND SUPERCRITICAL WINGS

Richard S. Shevell Fawzi P. Bayan

This Research was Supported by NASA-Ames Research Center

Moffett Field, CA 94035 GRANT NUMBER NAG 2-18

July 1980

i SUQAAR 522

DEVELOPMENT OF A METHOD FOR PREDICTING THE DRAG DIVERGENCE MACH NUMBER AND THE DRAG DUE TO COMPRESSIBILITY

FOR CONVENTIONAL AND SUPERCRITICAL WINGS

Richard S. Shevel 1 Fawzi P. Bayan

Department of Aeronautics and Astronautics Stanford University Stanford, CA 94305

This Research was Supported by NASA-Ames Research Center Moffett Fiel d , CA 94035

~ GRAl'iT NUMBER NAG 2-18

July 1980

Summarv

An analytical method, based upon bo th theory and empirical results, has been developed t o predict the drag divergence Mach number, MDIV and the incremental drag coefficient due t o compressibility, A C D ~ , for an airplane w i t h an optimally designed wing. I t i s assumed that the wing i s the cr i t ical component, i .e . a l l o ther components have higher MDIV The method has been computerised and needs only fou r inputs: l i f t coefficient, CL , sweep angle, , thickness ratio, t /c , and type o f wing, conventional or supercri t ical .

The method i s evahated by comparison t o f l i g h t t e s t drag resul ts for fou r airplanes w i t h different wing geometries. The f o u r airplanes studied were the Douglas DC-9-30, DC-8-50 and DC-8-62 and 2 wide body a i r - c r a f t designated as WB-1 i n this report. Wind tunnel data resul ts for a well developed supercritical wing (SCW) are used-'to establish a simple correction for SCW's. Comparisons between the prediction method and test resul ts are shown over a wide range of CL and Mach number. In general , good agreement i s shown between the method and the test data. Agreement is less good a t low CL's where the lower surface o f the outer wing panels become c r i t i c a l .

Table of Contents

S u ~ a r y ............................... i

I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 1

I I . Development o f the Method for Conventional Wings . . . 2

I I I . Modification of the Method for Supercritical Wings . . e e . 6

IV. Comparison o f the Method t o Test Data . . m . . . . . . e . . 7

A. Drag Divergence Mach Number Results e o e . . 11 B. Incremental Drag Coefficient Due t o Compressibility o o 13

V . Conclusion and Comments . . . . . . . . . . . . 17

References . . . . . . . . . . . . . . . . o . . . e . . . 18

Appendix I - Computer Program . . . . . . . . . . e . . . . o 19

-- L I S T OF SYMBOLS, SUBSCRIPTS AND ABBREVIATIONS P

1. Symbol s

CL F: lift coefficient CD : drag coefficient

R : wing aspect ratio A C D ~ : incremental drag coefficient due to compressibility

e : airplane efficiency factor M : mach number V : local velocity

t/c : wing thickness ratio A : quarter chord wing sweep angle

2. Subscri pts

p : parasite i : induced 00 : freestream N : normal cc : crest critical

D I V : drag divergence

"CM : having the sweep and thickness of the SCW

3. Abbreviations

SCW : supercritical wing FLT : flight

WB : wide body

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Using two-dimensional airfoil theory applied t o modern peaky 'conventional 1 a i r f o i l , pressure distributions were calculated for various thickness ratios, t /c , and l i f t coefficients, CL From these the percentage increase i n local velocity at the crest of an a i r fo i l due t o the l i f t coe f f i c i en t and the thickness ratio were determined. Apply ing simple sweep theory and the Prandtl -Glauert Mach number correction then led t o the expression,

where the subscript N represents the flpw component normal . t o the isobar a t the crest. The isobar has a sweep angle of A . The constant values o f 1.32 and 0.34 were determined from the fairing o f the incompressible pressure distributions on the upper surface of the conventional a i r fo i l s .

Combining equation (1) w i t h the one-dimensional compressible flow equations and reca1 1 ing that

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conventional wing, where M represents the Mcc cal cul ated w i t h ccAscw

the sweep and thickness r a t i o of the supercritical wing, and M CCactual

represents the calculated Mcc o f the conventional wing w i t h i t s own sweep and thickness. Then mul t iplying the f l ight tes t Mach number a t various values o f AC by th i s r a t i o , we can o b t a i n new curves for each conventional wing a i r c ra f t , for a given CL , which i n effect represent the drag rise curve o f the conventional wings w i t h the sweep an.d thickness of the supercritical wing. This i s done for a complete range of 1 i f t coefficients.

DC

The graphical results are shown i n Figure 2 for a typical CL of 0.4. Results for additional CL's are shown i n Figures 3 t o 4. The dotted lines represent the f l i g h t tes t data o f the conventional wings adjusted t o the sweep and thickness r a t i o of the SCW. The solid l ines are the f l i g h t tes t data for the conventional wings and the wind tunnel data resul ts for the supercritical wing designated by SCW. As can be seen i n Figure 2, the steep por t ion of the three dotted lines come very close together, which shows how well the method developed i n Section I I can correct for thickness and sweep as f a r as MDIV i s concerned. The agree- ment i s no t as aood a t low AC ' s for reasons discussed l a t e r .

Figure 2 alsoshows, and this i s true for a l l l i f t coefficients studied, t h a t there is a difference o f approximately 0.06 i n Mach number between the band containing the three dotted lines and the supercritical wing wind tunnel resul ts . Thus us ing the method described i n Section I I , we can calculate AC versus Mcc for a supercritical wing by add ing 0.06 t o the freestream Mach number. DC

IV. CornDarison o f the Method t o Test Data

TO compare the prediction method w i t h test data Mcc i s calculated from equation (2) for various CL's and for the sweep anales and thicknesses o f the three conventional wing a i r c ra f t . In addition, a f o u r t h conventional wing a i rc raf t , the DC-8-62 ,-63 was examined. The DC-8-62,-63 a i r c ra f t have

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the same a i r f o i l s , sweep and thickness except for a s l igh t average thickness reduction due t o wing t i p extensions. I t was predicted t o have the same Mc, as the DC-8-50. M,, i s then entered i n the ratio M,/Mcc and, using Figure 1, i s obtained versus M,. aCDC for the SCW was determined by the same method w i t h 0.06 added t o Mcc . The graphical results o f

i n dotted lines. The s o l i d lines represent the test results, f l i g h t t e s t s fo r the DC-8, DC-9 and W B 4 and wind tunnel te3ts for the SCN. In Figure 8, the DC-8-50 i s not shown a t CL = 0.5 because no f l i g h t da t a ex i s t a t this high CL .

versus M, obtained by the method are shown i n Figure 6, 7 and 8,

- A. Drag Divergence Mach Number Results (Figure 5 )

As noted ea r l i e r , we define MDIV as the p o i n t on a curve o f A C D ~

versus M, a t constant CL a t which the slope becomes O . 05. Figure 5 shows MDIV versus CL for three conventional wing transports. The DC-8-62,-63 has a b o u t the same MDIV and i s no t shown separately. The resul ts show good agreement between the prediction method and the f l i g h t t e s t d a t a a t CL's a t and above the design l i f t coe f f i c i en t s for these a i rcraf t o f approximately 0.3 t o 0.35 for the DC-8-50, DC-8-62 and the DC-9-30 and 0.4 for the WB-1. A t CL = 0 . 4 5 , the error is of 0.5% for the Dougl as DC-9-30, 1.2% for the Douglas DC-8-50 and 0.8% for th-e W B 4 wide body a i r c ra f t . A t lower CL the agreement i s n o t as good. A t CL = 0.25 , the error between the actual and the predicted is about 2.7% for the Douglas DC-9-30, 3.% for the Dougl as DC-8-50 and 5.1% for the WB-l wide body a i rcraf t .

The reason for this large difference at low CL i s believed t o be due t o the lower surface o f the outer panel of the wing becoming cr i t ica l a t lower l i f t coe f f i c i en t s . Because a.11 these a i rcraf t have sweep, they have considerable wing twist and a t low CL the local l i f t coe f f i c i en t s on the outer panels becomes very low. W i t h the significant camber of the a i r fo i l s , the lower surfaces have lower Mee's than the upper sur- faces a t these conditions. This was n o t considered i n the method which considers the pressure d i s t r i b u t i o n on the wing upper surface only. Note that the WB-1 h.as a la rger e r ror a t low CL probably because I t has a larger sweep and hence more twis,t

-11-

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To account for the effects of the lower surface o f the outer panel becoming c r i t i c a l a t low CL , the method is corrected by decreasing the slope of MDIV o r Mcc versus CL by two thirds below CL = 0.35. By doing this we reduce considerably. the discrepancy between the predicted values and the tests. A t CL = 0.2, the error i s now of about 0.1% for the Douglas DC-9-30, 0.6% for the Douglas DC-8-50 and 1.5% for the WB-1 wide body a i rc raf t . The CL o f 0.35, .below which the MDiV improves more slowly than the upper surface pressures would predict, is a rough average of the design CL o f these aircraft . Aircraft designed for c ru i se a t lower CL would cal l for an appropriate change i n this correction.

B. Incremental Drag Coefficient Due t o Compressibility (Figures 6 , 7 and 8)

A t higher cruise l i f t coefficients the agreement is very good, especially a t higher Mach numbers on the steep por t ion o f the drag curves. For lower l i f t coef f ic ien ts the agreement i s no t as good due t o the fact that the lower surface of the outer panel of the wings becomes c r i t i c a l , as discussed earl ier.

A t higher l i f t coefficients, > O 3 5 , the errors i n M, fo r a given A C D ~ are w i t h i n 1% t o 2% while even a t lower l i f t coe f f i c i en t s ( 2 5 t o -30) they are usually well w i t h i n 3% i n sp i te of the cri t ical lower surface. A t lower Mach numbers the . Douglas DC-8-50 exhi bits an unusual early drag r i s e due t o wing-pylon interference which was cured i n t h e cutbakk pylon design o f the DC-8-62,-63.

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Computer Program

The l is t ing and a sample r u n o f an interactive computer program which i teratively solves equation 2 for Mc, and then finds i s given i n Appendix I . The program is written i n Basic.

V . Conclusion,ad Comments

The resul ts of Figures 5, 6 , 7 and 8 , show t h a t the method developed for the prediction of drag divergence Mach number and the incremental drag coefficient due t o compressibility for modern conven- tional wings and well developed supercritical wings, yields good resul ts , cer ta inly. sa t i s factory for advanced design studi es . I t s sim- p l ic i ty - on'ly fou r inputs - makes i t very easy t o use, and allows a r a p i d approximation t o these two important design parameters.

I t shoul d be noted that i t i s easy t o produce a wing o r an airplane t h a t has a higher A C D ~ and/or a lower FlDIV than predicted herein. I t i s d i f f i cu l t t o be much better. Fowever, the DC-8-62 and WB-1 demonstrate a more gradual early drag r i s e than predicted by this method.

The SCW considered here has maximum allowable a f t loading. A

p a r t i a l use o f this feature wil 1 produce an MDIV greater than the conventional, b u t less than the SCW cases described i n th i s method.

-17-

References

1. "Evaluation of Methods for the Prediction sf Airplane Performance ( W i t h Special Reference t o the NASA-Ames GASP Program) , ' I Stanford University, Department of Aeronautics and Astronautics, August, 1978. Unpublished report, prepared for MASA-Ames Research Center under Grant Number NASA-219Q.

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