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7 D-fl143 591 AN EXPERIMENTAHL STUDY OF THE TRANSONIC EQUIVALENCE
RULE i/i
-Ai WITH LIFT PART 2 (..U) NATIONAL AERONAUTICAL
ESTABLISHMENT OTTAWA (ONTARIO) HIGH SPE. V Y CHANUNCLASSIFIED MAR 84 NAE-LR 614-PT-2 NRC-2412 F/G 20/4 N
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1.
11111_L25I -4
MICROCOPY RESOLUTION TESNT CHIARf
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m*National Research Conseil nationaTOOCouncil Canada de recherches Canada
AN EXPERIMENTAL STUDY OFI
THE TRANSONIC EQUIVALENCEPI
RULE WITH LIFT
PART 11
byY.Y. Chan
National Aeronautical Establishment
OTTWADTICOTTAWAELECTE
MARCH 1984JU2098C.,
AppTovrd i
C.2 AERONAUTICAL REPORT
LR-6149Canada NRC NO. 23412
84 07 19 041
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KIA 0R(;
AN EXPERIMENTAL STUDY OF THE TRANSONIC EQUIVALENCE RULE WITH LIFT
PART II
ETUDE EXPERIMENTALE DE L'APPLICATION A LA PORTANCE DE LA REGLE
D'EQIJIVALENCE TRANSSONIQUE
PARTIE II
by/par
Y.Y. Chan
* L.H. Ohman, Head/ChefHigh Speed Aerodynamics Laboratory/ G.M. LindbergLaboratoire d'aerodynamique a hautes vitesses Director/Directeur
.-1
SUMMARY
The experimental study of the transonic equivalence rule with lift,previously reported in Referere 1) has been extended to include a configu-ration of swept-back wing-body and the corresponding equivalent bodies forlift and zero-lift. For zero lift, the wing-body is shown to have higher wavedrag than that of the equivalent body. At lifting condition, the analyses ofthe data, including those of the delta wing-bodies 7epyted in Reference 1,have verified the similitude of drag rise due to lift between the wing-bodies.The additional drag induced by the effective area due to lift on the equiva-lent body was found to agree reasonably with the wave drag generated bylift on the wing-body. The experiment thus verified the area rule with liftwhich must be considered in the optimal design of transonic configurations.
RESUMEr7-
L'6tude expdrimentale de la loi de l'6quivalence transsonique pourla portance, prc~demment signal6e dans la r~fdrence 1, a t 6largie pour in-clure le profil alaire A fleche et les corps equivalents correspondants pour laportance et la portance nulle. Pour la portance nulle, il a t d6montr6 que leprofil alaire pr6sente une train6e d'onde sup6rieure i celle du corps 6quiva-lent. En condition de portance, les analyses de donn6es, y compris celles desprofils alaires delta signa~s dans la r~f6rence 1, ont confirm6 la similitude del'augmentation de train6e caus6e par la portance entre les divers profilsalaires. La trainee additionnelle induite par la surface r~elle en raison de laportance produite sur un corps 6quivalent s'accordait raisonnablement i latrainde d'onde cr6de par la portance sur le profil alaire. L'exp~rience a ainsiconfirm6 la loi des aires pour la portance, laquelle doit 6tre prise en compte ' .dans la conception optimale des profils transsoniques.
* . . -..-- - -~- - - - - - -
. *.--..*-, -
S(iii)
- . I | . V
CONTENTS
Page
SU M M A R Y ........................................................... (iii) .
A PPEN D IX ....... .................................................... (vi)
SYM BO LS ............................................................ (vi)
1.0 INTRODUCTION ...................................................... 1 e2.0 DESIGN OF THE SWEPT WING-BODY MODELS .................................. 1
2.1 Geometrical Design...................................................2.2 Aerodynamic Design ................................................. 1
3.0 EXPERIMENTAL RESULTS AND DISCUSSIONS ............................ 4
3.1 Experimental Program............................................... 43.2 Drag at Zero Lift ................................................... 43.3 Lift ............................................... 53.4 Induced Drag ...................................................... 63.5 Drag Rise of Equivalent Bodies with Lift ................................ 6
4.0 LIFT DEPENDENT WAVE DRAG OF THE SWEPT WING ANDDELTA WING-BODIES ............................................... 7
4.1 Correlations of the Drag Rise of the Wing-Bodies .......................... 7
4.2 Comparison with Equivalent Bodies .................................... 8
5.0 CONCLUDING REMARKS.............................................. . .9
6.0 REFERENCES ........................................................ 10
TABLES
Table Page
1 Dimensions and Parameters of the Swept WIng-Body Model WBS .............. 13
2 Some Geometric Parameters of the Swept Wing and the Delta Wing Models ...... 13
3 Aerodynamic Coefficients of Swept Wing-Body Model WBS .................. 14
ILLUSTRATIONS -
Figure Page
1 The Swept Wing-Body Model and Its Equivalent Bodies ..................... 19
2 Cross-Sectional Area Distributions of the Models With and WithoutEffective Area Due to Lift .......................................... 20
(iv)
S O
,-* * * - - - X. .. . . . . . . . . . . . . . . . . . . . .
.. : -. ;. ', i '= " -. . . - ' ., i - ' :, " •n . . ,- - - . , - . t - ,, . .
. ILLUSTRATIONS (Cont'd) I" Figure Page
3 Functions Due to Lift, E(x), T(x) and F, 2 for Swept-Back andDelta Planform s .................................................. 21
4 Distributions of Effective Area Due to Lift for Swept-Back and .1Delta Planforms at the Design Lift Condition ........................... 22
5 Variations of CL with & and CD with CL for Swept Wing-Body Model WBS 0at Different Mach Numbers ......................................... 23
6 Zero Lift Drag Rise for the Swept Wing-Body WBS and the EquivalentBody BS1 ............................................ .... ..... 24
7 Schlieren Flow Visualization for Wing-Body WBS and Equivalent BodyBS1 at Zero Angle of Attack (Nominal), M. = 0.98 ....................... 25
8 Surface Flow Visualization for Wing-Body WBS and Equivalent BodyBS1 at Zero Angle of Attack, M_ = 0.98 .............................. 26
9 Surface Flow Visualization for Wing-Body Model WBS at Angles ofAttack, M _ 0.98 ............................................... 27."
10 Variations of Lift Curve Slope at Zero Lift with Transonic SimilarityParameter Based on the Wing Thickness to Chord Ratio for SweptWing and Detal Wing Models ........................................ 28
* -- 11 Variations of CD with CL 2 for Swept Wing and Delta Wing Model atMach Number 0.8................................................ . 29.
12 Variations of Drag Factor k with Mach Number for Swept Wing andDelta W ing M odels ................................................ 30
- 13 Drag Rise for the Equivalent Bodies BS1, BS2 for the Swept Wing Case. and B1, B2 for the Delta Wing Case ................................... 31
14 Variations of Wave Drag Factor kw with Mach Number for Swept Wing
and Delta W ing M odels ............................................ 32
15 Correlations of Wave Drag Factor kw with Mach Number for Fixed Lift
Param eter e. z ......................................... ........ . 33r
16 Correlations of Wave Drag Factor kw with Mach Number for Fixed LiftCoefficient CL 2/ A ................................................ 34 O
-p°
17 Comparison of Drag Rise Due to Lift for Swept Wing-Body WBS and Drag-Differential of Equivalent Bodies BS2-BS1 ............................ 35
18 Comparison of Drag Rise Due to Lift for Delta Wing-Bodies WB1, WB2 2,and Drag-Differential of Equivalent Bodies B2-B1 ....................... 36
(v)
v 0i.",,°
'o*
APPENDIX
Appendix Page
Evaluation of Functions T(x), E(x) and F(x) for a Swept Wing .......... 37
SYMBOLS
" . -Symbol Definition
" - a leading edge contour of the wing planform, Equation (Al)
A aspect ratio of the wing, Equation (11) -
b semi-span of the wing, Equation (1)
' creference length = total length of gross wing planform, Equation (1)
CD drag coefficient, Figure 5
CD CD - CDm
CD drag coefficient based on maximum cross-sectional area, Figure 6CC°
CDi induced drag coefficient, Equation (11)
"° CDm minimum drag coefficient
CDw wave drag coefficient, Equation (13)
--. ACD c drag differential of equivalent bodies, Figure 18
- - CL lift coefficient
CL CL -CLm Figure 5
CLM CL at CD -
CL. local lift curve slope
C lift curve slope at zero lift, Figure 10
AD w wave drag rise, Equation (14)
k induced drag factor, Equation (11)
k, vortex drag factor, Equation (12)
(vi)*.G "'
............................................
• 4
SYMBOLS (Cont'd)
Symbol Definition
wave drag factor, Equation (13)
K transonic similarity parameter for equivalent body, Figure 6
transonic similarity parameter based on wing thickness, Figure 10
Kt0body length, Equation (10)
M_ free stream Mach number
q free stream dynamic pressure, Equation (14)
Sc geometric cross-sectional area, Equation (10)
S effective cross-sectional area, Equation (10)
Sw wing area, Table 2
t maximum thickness of wing section
- x,y,z Cartesian co-ordinates
. x/c 0 , Equation (10)
x/Q, Equation (10)
a angle of attack
atm a at CD M
angle of attack parameter, Equation (10)
- ratio of specific heat, Equation (10)
ei -ratio of transverse length scales, Equation (10)
wing sweep parameter with respect to the reference length CO, Equation (10)
wing sweep parameter with respect to the body length
0; body thickness parameter, Equation (10)
slender ratio of equivalent body, maximum diameter/!, Figure 6H ¢velocity potential, Equation (2)
(vii)
................... ..-. ".. ".-....". - . ."
A. -1 -
AN EXPERIMENTAL STUDY OF THE TRANSONIC EQUIVALENCE RULE WITH LIFT
PART II
1.0 INTRODUCTION
In a previous report (Ref. 1) an experimental investigation of the transonicz area rule withlift was presented. The purpose of the experiment was to examine the validity of the theoreticalresults on the extension of the classical area rule with lift( 2 ). The extended rule is given in a form ofI
flow similitude. It states that the flows are equivalent if the distributions of sources Sc (x) and thedoublets D0 (x) along the body axis are identical for the same transonic similarity parameter based onthe maximum cross-sectional area of the configurations. The sources are related to the geometry of thebody and its effective enlargement due to lift, and the doublets to the lift disttribution along the con-figuration. The effective cross-sectional area ,'ue to lift has been shown to be appreciable and inducesadditional wave drag in the transonic flight range( 3 ).
This early investigation was concerned with two delta wing-body combinations whose dragcharacteristics were determined at transonic speeds. The models had the same cross-sectional areadistribution but different wing span and thickness. The equivalent bodies for zero lift and for a speci-fied lift were also included in the investigation. The delta wing was chosen because the planform shapeallowed the design of identical effective cross-sectional areas due to lift for models with different swept-back parameters. However, most of the transonic aircraft have swept-back wing with moderate or largeaspect ratios. From a practical point of view, it is essential to evaluate the applicability of the extendedarea rule to such a configuration. A swept-back wing-body model was thus designed to be investigatedin the transonic Mach number range as in the previous experiment. The geometric and aerodynamicparameters of the swept wing model were designed to be closely to those of the delta wing, so that thedata from both experiments could be compared directly.
In the delta wing experiment, the wave drag and induced drag due to lift were correlated tothe square of lift and the inverse of lift curve slope. For a thin wing without leading edge suction, theinduced drag factor, henceforth called drag factor, defined in this manner would approach unity 4 ).For the delta wings with round leading edges, the drag factors ranging from 0.6 to 0.8 were obtained.Based on the drag factor defined in such manner, the wave drag of the delta wing due to lift wasdeduced and the results for two wings were found to be closely similar throughout the Mach numberrange studied. The inclusion of the lift curve slope in the drag-lift correlation, however, is later foundto be less desirable as the former varies appreciable in the transonic range. The drag factor defined insuch a manner will no longer be a simple correlation function of the drag-lift polar at a given Machnumber. An alternative approach is to replace the experimental lift curve slope by its theoreticalslender wing value of 7rA/2, where A is the aspect ratio. The drag factor is then in its conventionalform and depends only on the drag-lift correlation at a given Mach number. The data of the deltawings in Reference 1 and the present data of the swept wing are analyzed in this report using this formof the drag factor. The wave drag due to lift is also deduced in this form. The results of all three wing-bodies are then presented together for comparison.
In what follows, the model design and the aerodynamic consideration of the swept wing- •body will be first presented. The experimental results and their analyses, including the data of thedelta wing-bodies, are then given in detail and related to the extended equivalence rule with lift. Thebasic concepts of the theory can be found in Reference 1 and will not be repeated herein.
2.0 DESIGN OF THE SWEPT WING-BODY MODELS
2.1 Geometrical Design
The swept wing-body has a cylindrical central body similar to that of the delta wing-body
Model WB2. The span, the aspect ratio and the angle of sweep of the leading edge of the swept wing0'"S
. . . .
-2-
are taken to be the same as those of the delta wing Model WB2 before the wing tip round off (Fig. 1). AThe mean aerodynamic chord is chosen to be three inches so that the wing area will be close to thatof WB2. The wing has a taper ratio of 0.4 and the same symmetric Boerstoel profile as that of thedelta wings. The thickness to chord ratio of the section is constant along the span and there is no wingtwist. The thickness of the wing is determined by matching the cross-sectional area distribution fromthe wing tip forward as close as possible to that of the delta wing models. The resulting area distribu-tion for tic = 0.057 is shown in Figure 2. The additional area at the rear of the hump comes from theouter portion of the swept wing downstream from the leading edge of the wing tip. Reduction of theadditional area to fit the area contour of the delta wing cannot be achieved without altering thegeometrical layout of the planform. It is preferable, however, to keep the planform characteristics ofthe swept wing close to those of the WB2 wing for aerodynamic consideration, thus no further modifi-cation is done to the cross-sectional area distribution. The resulting equivalent body is thus not exactly .-
identical to that for the delta wing. The complete body shape is shown in Figure 1 as Model BS1. Thewing tip is rounded off to ease the local spanwise loading and the adverse effects due to sweep (5 ) . Thewing tip contour is designed to be
x/c - 0.8] x/c0 > 0.8_ 0.2 sin (1) (1b 0 .2 y/b > 0.8
where b is the semi-span and the reference length c o is that of the delta wing model, taken as a scale*. for the present wing design.
* The basic dimensions and the nondimensional parameter of the final design for the wing-body model are given in Table 1. The parameters of the wing sweep X and the thickness r" have valuesonly slightly different from those of the delta wing Model WB2. the product of these parameters X Tis nearly identical for both models. Some of these geometrical parameters, which are important for thecorrelations of the experimental data, are listed in Table 2 for both the swept wing and the delta wingmodels.
2.2 Aerodynamic Design
When the wing-body is at lifting condition, the flow about the configuration is governed bythe effective sources and doublets distributed along the body axis. The strengths of these singularitiesare given explicitly in terms of three functions F(x), E(x) and T(x) depending on the load distributionof the wing(I ,2). For the present planform, the load distribution is unknown and has to be determinedexperimentally or theoretically. However, it is noted that these functions are not too sensitive to the
* . details of load distribution for a given planform. An approximate expression of the load distributionreflecting all essential characteristics of the wing loading has been shown to be adequate(3 1. Theexpression for load distributions given in Reference 3 is adopted herein. The jump of the velocitypotential across the wing plane can be written as follows,
(x- x )3/2
-= _x-x 3/ 2 [a O + a, (xt-x) + a 2 (x t - x) 2 + ... ] (2- .(xt - Y / (2) O
for x, < x < x t
where the subscripts Q and t denote the leading and the trailing edges respectively. The wing loading isgiven by the x-derivative of the potential jump and thus gives a square root decay of loading as the
*Q. leading edge is approached. The coefficients in the polynomial are determined by the conditions at the ,trailing edge at which it is assumed that the spanwise loading is elliptical and the chord-wise loadingdecays linearly. In the wake the load is zero. These conditions yield respectively
2 _ 2)11a o = - 1
i-r
S-3-
3 aoa, =
• 2 t XQ
(3)a2 0
With the conditions at the trailing edge specified, the potential jump can be written as
1 (x- xQ) 3 / 2
•'" ¢ II = -(1- y2)1/ 2 [ 5(xt- x) + 2(x- x,)] (4)e. .:..7r 5s/ 2
where c = xt - xk, is the local chord. The x and y derivatives of the potential jump are given as
15 2)/2 /2 -x)Cx II = - (1- y2 1 2 (x- x )1 (xt x (5)2 , 5 /2 Q t:
2irc 51
dx~ dxQdx 5-- 2-".-y 5 1 de 3 1 dx, dy dy
"- [[ y = ¢ I +(6) --Y 2 2 c dy 2 x- x¢ dy 5(x --x ) + 2(x x
At the wake trailing behind the wing, the conditions in Equation (3) give
-.. = 2(1 2)1/2
7T
I =0 (8)
= - - (9)
* matching continuously with the expressions in Equations (4), (5) and (6) at the trailing edge. It shouldbe noted that the spatial variables x and y in this section (and in the Appendix) have been scaled bythe total length co and the semi-span b of the wing planform respectively. With the load distributionspecified, the functions F(x), E(x) and T(x) are evaluated numerically. Some details of the numericalprocedure are given in the Appendix. The results are shown in Figure 3 with the abscissa scaled by theplan form length c,,, so that a direct comparison with the delta wing case can be made. For the swept
*wing, the x-derivative of the function F(x), F, (x), relating to the axial load distribution, peaks at 9about the mid point of the planform length, while on the delta wing, it reaches a maximum near thetrailing edge. The function T(x), relating to the load distribution over the wing surface, behavessimilarly for both planforms. The function E(x) signifies the cross-flow kinetic energy and has thesame trend for both cases.
%-..•*L"
-4 -0
The design conditions for the lifting case are the same as those of the delta wing,
M =0.975
= 0.0711
- 0.2024
The effective area due to lift can now be calculated from the expression (Eq. (25) of Ref. 1) for thewing body.
)- S (x) 1 (-.- (C(x)) + T(x) + 2 E(x)- co 87( + 1)X 2
(10)
where the tilded x is scaled by the body length Q and the barred x by the planform length c,). Theresulting effective area is shown in Figure 4. The area dis' ribution for the delta wing is also shown forcomparison. Because of the nearly symmetrical form of the loading functions F (x) and T(x) for theswept wing, the effective area is more evenly distributed at both sides of the peak, while for the deltawing the distribution is highly weighted towards the trailing edge. With the effective area due to liftadded to the basic geometric cross-sectional area S,(x), the final area distribution of the equivalentbody for the lifting case is shown in Figure 2. The corresponding model, designated BS2, is shown inFigure 1. Note that the nose piece is common for all models.
3.0 EXPERIMENTAL RESULTS AND DISCUSSIONS
3.1 Experimental Program
The experiments were performed in the tran;onic test section of the NAE 5-ft (1.5 m) X5-ft (1.5 m) blowdown wind tunnel. The experimental set-up was similar to that of the delta winginvestigation. The forces and moments of the model were measured by a one inch Able MK226-component internal balance. The base pressure was also measured to complement the balance dragmeasurements. The static pressure distribution along the top and bottom walls were monitored byone inch static pressure pipes. Boundary layer transitions on the wings and the nose of the body wereleft free as in the delta wing experiments. Surface flow visualizations were obtained for certain Machnumbers and angles of attack. Schlieren visualizations of the flow field were also taken for cases withMach number close to unity. The test Mach number ranged from 0.8 to 1.1. The Reynolds numberwas fixed at 8 X 106 per foot or 2 X 106 based on the mean aerodynamic chord of the swept wingmodel. The raw data were reduced by the same computer program as used for the delta wing investi-gation. The results are presented in terms of CI1 , & and CI) in Figure 5 for the wing-body Model WBS. SThe same data are also listed in Table 3. These data are further analyzed in terms of parameters of theequivalence rule and compared directly with those of the delta wing models.
3.2 Drag at Zero Lift
The transonic drag rises for the wing-body Model WBS and the equivalent body BS1 are .shown in Figure 6. The drag rise of the swept wing Model WBS follows closely to that of the deltawing Model WB2 up to a value of K, the similarity parameter of about - 1.0, corresponding to aesMach number of 0.98. At higher Mach numbers, the wave drag of the swept wing model increases-.
steadily and is higher than those of the delta wing models.
- - - - .~. . . . . . . . . . .
-5-
For a swept wing with moderate aspect ratio and sweep-back angle, tile flow about tile wingretains some two-dimensional sectional characteristics and at high Mach numbers shocks with nearlyconstant strength extend along the full span. This shock formation can be seen from the Schlierenflow visualizations, a typical one shown in Figure 7, taken at a Mach number of 0.98, shows an ob-lique shock at the trailing edge turning rapidly normal to tile flow. For a delta wing, the flow abottthc wing is more three-dimensional and nearly conical, thus the shock decays along the span. Thesurtace flow visualization in Figure 8 shows, at the outer portion of tile swept wing, that the shockinteracts strongly with the houndary layer causing local flow separation. The strong shock and theflo w separation on the swept wing may contribute to the higher drag in comparison with the deltawing.
The drag rise Of the equivalent body for the swept wing-body 13S1 is also shown in Figure 5.For Mach number less than unity, the drag rise of BS1 is about half of that of WI3S, although thegeneral shape of these two Curves are closely similar over the range of the Mach number considered.The same discrepancy has been observed in the original area rule study ' and in more recent ex per-niunts, . The Schlieren t)icture of BS1 in Figure 7 shows a near normal shock formed not far fromthe end of the hunip. The shock decays faster than that of tile WBS and thus is weaker. The surfaceflO%% visualization indicates no boundary layer separation at the aft portion of the hump (Fig. 8).
Figure G also shows that the drag rise of BS1 is appreciably lower than that of B1, theequivalent body for the delta wing models. Comparison of the Schlieren pictures of BS1 in Figure 7and that for B1 in Figure 9 of Reference 1 clearly indicates that the gradual reduction of the hump r•at the downstream side helps to weaken the shock. It may be recalled that the body contour aheadOif the maximum cross-section is practically identical for both models.
For tle far field, the data of the static pressure distributions along the upper and the lowerwalls are found to he too scattered for analysis to establish flow similarity. Thus no study of the farfield similarity was made for the swept wing model and its equivalent bodies.
3.3 Lift
The lift versus angle of attack curves for all test Mach numbers are plotted in Figure 5.Slight nonlinearity of the lift curve above two degrees angle of attack can be observed for all casesespecially for those between Mach number 0.9 to 1.0. The nonlinear increase of lift has been shown tobe due to tile tip vortex sheet, which induces a vortex lift, usually associated with slender deltawings'" 1. Another significant aerodynamic chracteristic of the swept wing observed in the experimentis the strong upwash over the outer portion of the wing, induced by the spanwise component of thetrailing vorticity shdded(l from the wing I ". This effective incrase of angle of attack along the spancauses the formation of a strong shock and flow separation at the outer portion of the wing, whichAlan lhe clearly seen from the surface flow visualization (Fig. 9). At 2.9 degree angle of attack forinstance, trailing edge separation starts from the wing tip and spreads inward to about one-third oft he wing span. Strong shock-boundary layer interaction is indicated by the rapid turning of the surfacestreamline at the separation region. Stronger interaction can be seen as the angle of increase to 5.5dhegree. The S shape of surface streamlines at the mid portion of the wing indicates the formation of asecond shock ahead of the main shock at the inner portion of the wing, the existence of which hasI een shown both ex perimentally and theoretically( I I j 2 "
The lift curve slopes at zero lift are correlated against tile transonic similarity parameterbased on the wing thickness in Figure 10. The delta wing cases are also shown for comparison. Thepresent data show a small rise between Mach numbers of 0.92 to 0.96 with a peak at 0.94. The restof the data vary only slightly through the Mach number range. The values at both the subsonic and ..supersonic ends approach those of the delta wing W11. At the low Mach number end, the values of', 'A agree well with that predictedI by subsonic linear theory( 1 3 At Mach number unity, the tran-
sonic correlation of C( -(t/(')I versus A(t/() / is close to that of the delta wing WB21.
* 07
.6 . -- - L- -: - - :
~4 .6.
3.4 Induced Drag
Before the wave drag becomes apprecible. the drag force acting on the wing-body is com-posed of the induced drag due to lift, the skin friLtion and the pressure drag caused by boundarylayer displacement or separation. The latter two components can be identified as viscous drag as theyexist only in viscous flows. At zero lift, the total drag consists of the viscous drag only. With the viscousdrag at zero incidence subtracted, the drag coefficient is plotted against the square of lift coefficientfor Mach number 0.8 in Pigure 11. The upper and the lower limits of the induced drag, correspondingto zero and full leading edge suction respectively, are also shown in the graph. The delta wing resultsfrom the previous experiments are also plotted for comparison. The drag polar of the swept wing isslightly lower than that of the delta wing WB2, indicating that the planform shape has only a smalleffect on the induced drag of these two wings having nearly the same aspect ratio. The parabolicdrag polar is linear up to C1 I 2 about 0.1, above which the viscous drag is anticipated and the correla-tion becomes nonlinear. The classical expression for induced drag as a function of lift and aspect ratiocan be written for the linear portion of the drag polar as
CI) = k (11)..-'i rA
where k is the induced drag factor. The coefficients of the parabolic relation has been obtained by apolynomial fit of the data and the coefficient of the second degree term yields k/7rA. This form is
used in the present analysis in preference to k/CL , employed in Reference 1. The latter involves the
lift curve slope CL - which varies appreciably in the transonic range, thus the drag factor k is no longer .':-
a simple coefficient of drag-lift correlation.
As the Mach number increases, the wave drag sets in. The wave drag due for lift, as well asthe induced drag, is related to the square of the lift force' 1 5 ). The expression in Equation (11) cantherefore be used for both the induced drag, also called vortex drag, and the wave drag, with thecoefficient k containing both components. The variation of k against Mach number for the swept wingand the delta wings are shown in Figure 12. At the low Mach number end, the contribution comessolely from the vortex drag. For the delta wings WB1 has a lower induced drag factor than WB2,which has a larger aspect ratio. This may be caused by the higher suction generated by the thickerwing section of the WB1 model. As the Mach number increases, the drag factor for the WB1 modelincreases to a maximum and then drops sharply to a level as low as that before the drag rise. For WB2,the drag factor peaks at about the same Mach number of 0.96, drops slightly and then increases gra-dually as the flow becomes supersonic. The swept wing WBS has the lowest value of k before drag rise.However, it also has the highest value after drag rise. Before the drag rise, both WB2 and WBS show adip of drag factor implying a reduction of vortex drag. This slight reduction of drag is caused byfavourable interference between the wing and the body of certain configurations and has been 1-7observed in other tests of wing-body configurations( 7 .1 6).
3.5 Drag Rise of Equivalent Bodies with Lift
The transonic drag rise of the equivalent body of revolution for zero lift of the swept wingconfiguration BS1 has been discussed in Section 3.2. It was shown in Figure 6 that the drag rise of BS1 6is much lower than that for the wing-body Model WBS. The drag rise characteristics of the equivalentbody BS2, with the effective area due to lift forming a bigger bulge, would be expected to be similar.This indeed is the case when both data are plotted in terms of the similarity parameters for axisym-metrical bodies as shown in Figure 13. The data are well coalesced even though these two bodies arenot exactly geometrically similar. The data of the equivalent bodies for the delta wing-bodies, B2 andB1, with and without lift respectively are also shown in the same figure. Again, the data are wellcorrelated. The lower drag rise of the BS1, BS2 bodies in comparison with the B1, B2 bodies is due tobroadening of the hump on the formers (see Fig. 2) that allows a more gentle recompression at therear of the hump. The resulting rear shock is comparatively weaker than that of the B1, B2 bodiesand no boundary layer separation has been observed on the BS1, BS bodies.
•*. ,
-7- :Al
4.0 LIFT DEPENDENT WAVE DRAG OF THE SWEPT WING AND DELTA WING-BODIES
The lift depending wave drag which is the centre of the present study will be analyzed inthis section. The data for the delta wing-bodies from Reference 1 will also be considered, together -with the present swept wing-body data.
4.1 Correlations of the Drag Rise of the Wing-bodies
The wave drag at lifting condition can be deduced by subtracting the vortex drag portionfrom the total drag. Since the vortex drag does not change with Mach number, its value at Mach '0number 0.8, the lowest Mach number of the experiment, is taken to be the vortex drag. No wave dragis present at this Mach number. The wave drag factor is therefore
k = k - kv (12) - -
where k, is the vortex drag factor and kw the wave drag factor. The wave drag data is now expressedin the same form as the vortex drag in terms of lift and aspect ratio as
CL2CD =kw - (13)w 7rA
The resulting wave drag factor for the swept wing and the delta wing-bodies are shown in Figure 14.It should be noted that the drag coefficient in Equation (13) is based on the gross wing area.
Since the lift is mainly carried by the wing, the scaling length for the drag rise due to liftshould therefore be the overall chord c. of the wing planform(1 7). The wave drag rise written in termsof the drag coefficient CD based on wing area is thus
W
AD w 4CDW
qc0 A- ) (
C
where S, is the maximum cross-sectional area of the model. Substituting the wave drag coefficient inth orm of Equation (13) from the experimental data into Equation (14), the drag rise takes thefollowing form
AD, 4 L
qc() (p2)2 irA (15)qco (o -.
The expression can be further written in terms of the lift parameter e z , which control both the7.
source and the doublet strength due to lift (see Eq. (10) and Eqs. (12), (13) of Ref. 1),
AD~, k,_ __ )__ _ & 2
A (X2(Y) ( _c()2=)42 2--"(16)qc1 2 47r"/+ 1)M' (')-
C
"-.:-:
,- %°~ - . - -~-
K I Ka.7
Since all wing-body models have the same maximum cross-sectional area, thus TX is the same for allcases. If the drag rise is directly related to the lift parameter, then the correlation of the wave dragfactor as shown in the bracket of Equation (16) must be a function of Mach number only. The corre-lations in such form for the data of three models as a function of Mach number are shown in Figure 15.
* .- The data for the swept wing Model WBS and the delta wing Model WB2 collapse well through thewhole Mach number range. These two models have nearly the same span but different root chords andplanform shapes which seem to have little effect on the correlation.
The wave drag factor for the delta wing Model WB1 with a short span does not follow thecorrelations of the other models below Mach number 0.98. The correlation has much higher valuesthan the other data. For Mach number greater than 0.98, the correlation drops to the same level as
- the others and then tends to increase again at the end of the experimental reach number range. Thisshows that the drag rise for wings with different spans cannot be compared for the same lift parameterin which the lift is scaled by the total planform length. It would be more al)pro)riate in the drag riseregion if the lift is caled by the span. This form is readily given by Equation (15) and the wave dragfactor correlated in such a manner is shown in Figure 16. The data of the delta wing WB1 now co-alesce much better in the drag rise region with the data of the other two wings with larger spans. Thiscorrelation implies that the drag rise due to lift is directly related to Ct - A in a form similar to thatof the vortex drag.
From the Schlieren flow visualization taken during the experiment, the distinct character-istics of the flow with respect to these two regions of correlation can be observed. At the drag riseregion, the shock waves formed about the wing are situated ahead of the trailing edge. The strength ofthe shock for this type of transonic flow, with a supersonic pocket terminated by a shock, dependsstrongly on the sectional characteristics of the wing. The dependency on the sectional geometry can bereadily seen from the fact that for the present models having the same cross-sectional area, the corre-lation factor A? 2 in Equation (15) can be shown to be proportional to the mean thickness of thewing. As the Mach number increase to 0.98 and above, the shock moves to the trailing edge and the
.* flow over the wing is fully supersonic except at a small region around the leading edge. The charac-teristic of the wave drag is then changed and follows the classical results of wave drag for supersonicflow.
It may be recalled that the wing-body WB1 is designed to have the same effective cross-sectional area due to lift as the Model WB2 by manipulating the special relations of the lifting function
i F, T and E for the delta planform so that the flow similitude may be established (see Eq. (10) andalso Section 3.1 of Ref. 1 for details). The similitude of flow field does seem to exist for the far fieldas shown in Figure 21 of Reference 1. However, the similitude of drag rise in the form of Equation (16)does not seem to be applicable and the alternative form Equation (15), should be used.
We may now summarize the conditions of application of the area rule with lift. For configu-rations with nearly the same sweep-back parameter X, the drag rise is directly related to the square of
the lift parameter c . For configurations with large difference in X, the drag rise is correlated to the
square of the lift coefficient and the aspect ratio as CL 2 /A. When the flow over the wing is supersonic,
the lift parameter e - should be used regardless of the value of X. The planform shape has only minor
effect on the drag rise.
4.2 Comparison with Equivalent Bodies
The drag rise of the equivalent body at zero lift has been shown to be much lower than that. of the wing-body for the swept wing experiment (Section :3.2). A direct comparison of the drag rise
of the equivalent body including lift with the wing-body at lifting condition will therefore not bemeaningful. However, the differential of the drag rise between these two (e(quialent bo(dies is Causedby the additional effective area due to lift. Thus a comparison of the drag rise differential due to liftas given in the precee(ding section is justified and will provide an assessment of the range of application
e . ."..- .i
* . . . - ,. D
-~i' -- 9-
of the theory. For comparison, the drag rise is now given in terms of the drag coefficient based on themaximum cross-sectional area, which is the same for the wing-bodies and the equivalent body withoutlift. For the wing-body the drag rise is calculated from the experimental correlations given in Figures
&15 and 16 at the design lift condition with iz equal to 0.2024.
For the swept wing-body the comparison is shown in Figure 17. The drag rise differentialagrees very well with the drag rise to lift for Mach number below 0.96. For Mach number greaterthan 0.96, the drag rise due to lift of the wing-body is lower than the drag rise differential of theequivalent bodies. A similar comparison for the delta wing models is given in Figure 18. For the ModelWB1, the drag rise up to Mach number of 0.96 is evaluated with CL 2/A equal to 0.0078 corresponding
to F of 0.0829. For the Model WB2, the value of CL 2 /A yields e of 0.2024 the design value. If
the drag rise is compared with the same e -Z m this region, the drag curves will look like the wave dragT
factor curves in Figure 15. The comparison in Figure 18 shows that for delta wings the equivalentbody differential gives a much higher drag rise than that generated by lift. However, it should be notedthat flow separation occurs at the rear of the hump of the erquivalent body with lift, B2 at the higherMach numbers. This undoubtly will give higher drag differential .:n the ideal case without separation.From these comparisons we may observe that the addition'. drag induced by the effective area due tolift on the equivalent body is close to the wave drag generated by lift in the drag rise region. At theupper end of the transonic range the differential drag of the equivalent bodies is higher than thatgenerated by lift.
:': -5.0 CONCLUDING REMARKS
The experimental study of the transonic equivalence rule with lift presented in Reference 1has been extended to include a swept-back wing-body and the corresponding equivalent bodies ofrevoluation for lifting and non-lifting cases. The results of the analysis of the wave drag and drag risein the transonic range are summarized as follows:
1. For zero lift drag the swept wing-body has the same drag rise characteristics as the deltawing-body with the same sweep-back parameter X up to Mach number about 0.97. At higherMach number, the drag of the swept wing-body is higher than that of the delta wing-body.
2. The equivalent body for the swept wing-body at zero lift has much lower wave drag thanthe wing-body. The high wave drag of the wing-body is attributed to the shock formationalong the span and the flow separation at the outer portion of the wing.
3. The drag rise due to lift is directly related to the square of the lift parameter e 7 for con-
figurations with the same sweep-back parameter A. For configurations with large differencein!, the lift parameter is in the form of CL 2 /A. When the flow over the wing is supersonic,
the lift parameter E -z should be used regardless of the value of X. The planform shape has
. -only small effect on the drag rise.
4. The additional drag induced by the effective area due to lift on the equivalent body agreesreasonably with the wave drag generated by lift on the wing-body.
From these results, we may conclude that the area rule for zero lift does not work well forswept-back wing-bodies with moderate aspect ratio as observed by other authors in area rule experi-
ment( 6 .8). The area rule due to lift as stated in (3) and (4) above, works reasonably well. It is thereforeV. essental to include the additional effective area due to lift in the consideration of area rule for the
t' "design of transonic configurations.
*. .. - 4. . - - % . . : .. • • . . . . -. ,o .- , .- . , . , . .-. - - : .' . -o r - , .:b . . . " " • . . . .
-10-
6.0 REFERENCES
1. Chan, Y.Y. An Experimental Study of the Transonic Equivalence Rule withLift.NAE, Aeronautical Report LR-609, National Research CouncilCanada, March 1982.
2. Cheng, H.K. Transonic Equivalance Rule: A Nonlinear Problem Involving Lift.Hafez, M.M. J. Fluid Mech., Vol. 72, Part 1, 1975, pp. 161-187.
3. Cheng, H.K. Lift Corrections to Transonic Equivalence Rule, Examples.AIAA Journal, Vol. 15, No. 3, March 1977, pp. 366-373.
4. Relf, E.F. Note on the Drag of Thin Airfoils.Perf. 1048, (Rept. 15582), British Aeronautical Research Council,January 1953.
5. Lock, R.C. Theory of Aerodynamic Design for Swept-Winged Aircraft atBridgewater, J. Transonic and Supersonic Speeds.
IN "Progress in Aeronautical Sciences, Vol. 8." Ed. D. Kichemann,Pergamon Press, Toronto, 1967.
6. Whitcomb, R.T. A Study of the Zero-Lift Drag-Rise Characteristics of Wing-BodyCombinations Near the Speed of Sound.NACA Report 1273, 1956.
7. Gustavsson, A.L. Transonic Wing Tunnel Tests of the PT3 Model, A SupercriticalMattsson, R. Wing on a Simple Body.
Tech. Note AU-1134, Part 1, The Aeronautical Research Instituteof Sweden, 1974.
8. Agrell, N. Investigation of the Transonic Drag Characteristics for Non-slenderMattsson, R. Wing-Body Combinations and their Equivalent AxisymmetricNyberg, S.-E. Bodies at Zero Lift.
Proceedings of 11th Congress of the International Council of theAeronautical Sciences, Vol. II, September 1978, pp. 292-304.
9. Kiichemann, D. The Aerodynamic Design of Aircraft.
Pergamon Press, Toronto, Chapter 4, 1978.
10. Cheng, H.K. On Lifting-Line Theory in Unsteady Aerodynamics.USCAE 133, Department of Aerospace Engineering, University of '"Southern California, January 1976.
11. Schmitt, V. Pressure Distributions on the ONERA-M6-WING at TransonicCharpin, F. Mach Number.
AGARD-AR-138, Experimental Data Base for Computer ProgramAssessment, May 1979.
12. Treadgold, D. Some Aerodynamic Interference Effects that Influence the Tran-Wilson, R.H. sonic Performance of Combat Aircraft.
AGARD-CP-285, Subsonic/Transonic Configuration Aero- -
dynamics, September 1980.
13. Schlichting, H. Aerodynamics of the Airplane.Truckenbrodt, E. McGraw Hill, New York, 1979.
14. Frick, C.W. The Experimental Aerodynamics of Wings at Transonic and Super.sonic Speeds.IN "Aerodynamic Components of Aircraft at High Speed." Ed.A.F. Donovan and H.R. Lawrence, Princeton University Press,Princeton, New Jersey, 1957.
15. Jones, R.T. Aerodynamics of Wings at High Speeds.Cohen, D. IN "Aerodynamic Components of Aircraft at High Speed." Ed.
A.F. Donovan and H.R. Lawrence, Princeton University Press, -
Princeton, New Jersey, 1957.
16. Allis, A.E. et al Experimental Investigation of Parametric Models in Support of aNASA Advanced Transport Technology Program.Vol. 1, Data Summary, NASA CR-112130, October 1972.
17. Spreiter, J.R. On the Range of Application of the Transonic Area Rule.NACA TN 3673, 1956.
.'
................................. .. .
-13-
TABLE 1
DIMENSIONS AND PARAMETERS OF THE SWEPT WING-BODY MODEL WBS
Body length 2 = 17.5 in. (444.5 mm)
Body radius r = 0.875 in. (22.23 mm)
Semi-span b = 7.56 in. (192.0 mm)
Mean aerodynamic chord cm= 2.98 in. (75.7 mm)
Root chord Cr = 4.2857 in. (108.8 mm)
Taper chord X = 0.4
Leading edge swept angle A = 38.4350
Aspect ratio A = 5.04
Wing thickness t/c = 0.057
Wing swept parameter X = 0.4320
Thickness parameter r = 0.0275
TABLE 2
SOME GEOMETRIC PARAMETERS OF THE SWEPT WING AND
THE DELTA WING MODELS
WB1 WB2 WBS -
co (in.) 6.000 6.000 7.715
b (in.) 4.044 7.286 7.560
Sw, (in. 2 ) 25.155 45.278 45.106
S, (in. 2 ) 3.647 3.647 3.641
co / V 0.343 0.343 0.441
A 2.600 4.690 5.068
0.231 0.416 0.432
0.0497 0.0276 0.0275
rX 0.0115 0.0115 0.0119,. o oo1-.
.
* @4
-14- 0
TABLE 3
AERODYNAMIC COEFFICIENTS OF SWEPT WING-BODY MODEL WBS
MACH Nlo= 0 .79 9 M=31 MACH NO0e.847 M=31I /FT ( M 7.79 /' * *( P) =76 76
AL PH AS C. 9 c r) ALHAB CLB CD
*.. -3.0844 -0o2'78 0.0171 -2 o9679 -0.2309 0.0170-?07762 -0.2192 0.0157 o-2.714 -0o2079 0.0158"-2.65 -0.19o 0.0147 -2o !4q9 -001R?3 00147
-2.2431 -0.1739 0.019 -2.1 = "6001692 0.0140-1o9400 -001502 0.01-1 -1.8309 -0.1461 0.0132-1 .6465 -0o1295 0.01?4 -1.525Q -0.1248 0.0126-1 .3659 -0.1034 0.0011q -1 .?446 -0.0993 0.0,0!-0661 -0.0767 0.0116 -C.469 -0.0715 0.0116-0.7s80 -0 0614 0o0113 -0.6628 -0.0508 0.011 ?-0.5167 -0.0441 0.0111 -0. 61 -0.031 0.0112-0.2085 -0.0130 0.0!11 -00 Of 43 0.0050 0.01!I0.0O50 0.0058 0011! 0.2016 0.0153 0.01110o35- 0.0185 0.0113 Oo.4934 0.03A9 0 .01120o6461 0.0481 0.01 ! 0*77R,3 0.062F 0.01130.9163 0.0681 0.0116 1.0520 o.3791 0.0117S01o2'9q 0.1014 0oO117 1 o3390 0.0991 0o01211.5098 0o1192 0.0123 1 .6354 0ol268 0901171 .7883 0.1428 0.0130 1.940Q 0o1591 0.01342.0789 001647 0.0139 2*2307 0.1780 0.01432.3840 0.1056 0.0149 2o5754 O 2153 0. 01552.0 e6631 0.21 R6 0.0162 F.t22 2 0o. 270 0o01682 . 9i99 0.2449 0 *0176 3. 1099 0.26,36 0 e01 1843o2312 0.2627 0.0191 3.4094 0.2Q12 0.02023.'5179 0.2838 0.0210 3.7043 0 .3158 0 o022"3.793q 0.3006 0.0228 4o0002 0.349.3 0.0246
4.eo08P 0 . 32P7 0 .04t 4.o979 0o3745 0 o 0 2734.o676 0.3413 090275 4.o702 0.3P3 0o02984.6769 0.3R30 0o0306 4.o8757 0o425 0.03744.o613 0o4056 0.036 5.1695 0049533 0e0 " . -
5.2C; 85 0o 427E 0.036< 5o4409 0o4715 0.04075 .406A 0.4477 0.0400 5.701!8 0.5029 0.0456
MACH Nreo0.o94 M-31 MACH NC*=0.912 M=31E/T( M) = 7 74. E/FT( M) =7. 70
ALDHAR CL q Co ALPHAR CLR Co
-3o0891 -0.2719 0.0186 -301551 -002959 0o0208-2o7755 -0.2362 0o0167 -2.e119 -O 2'554 001 P2-2.4570 -0.2013 0.0152 -'P.4990 -0.2243 0.0163-?.2?71 -0.1-12 000143 -2O2508 -0 el16 0 O01 ;?-19039 -0.1517 0.0134 -1.9402 -0.1699 000141-1.6057 -001282 0.0128 -1.610 -0.1378 0.0131-1 3318 -001147 0.0122 -1 93301 -001116 0 0012_6-1.0573 -0.0(37 00118 -1. 0574 -0.0899 0.0120-0.7598 -0.06'S 0.0114 -3 07506 -0o0637 0.0117-0.4460 -0o0361 0.0112 -0o4627 -090411 0.0115
. -0.1449 -0.0075 0.0111 -0 .1567 -0.0148 0.01140o.184 0.0115 000113 0.1318 0,0094 0.01140.4064 090277 0.0113 O. 43v7 0.0396 0.01140.7210 0.0590 0.0116 0.7206 0.061A 0o01161m0060 0.009 0o0119 1.0279 0o0018 0.01 191.3039 0. 1104 0.0122 1 .3065 C0.11?1 0.01?5 @l1o6029 0.191 0.0128 1.5Q85 0o1381 0.01331.8920 o.l54 0.0!17 1 * 8_953 0.126 0o0142
/*2.2026 0.1921 0.0149 2.2104 0.2022 0.0194 "?e5132 0.2 11 090163 2.5 20 0.2339 0901712•8197 0.2623 0.0191 2e8464 0.2'91 0.01033.1044 0.2894 0.0199 ?oI'56 0o3069 090217".4207 0.32?67 0.02-?2 _.493 0*3353 0o0244107193 0.3685 0.0260 7o7"456 0. 3687 0o027-4.0430 0.3990 0.02Q3 4. 03'1 094034 0o02144.0327 0.4?13 0.0326 4.3468 0.4403 0 059.4.6165 0. 44s86 0.0366 4 .6C 31 0.475 p 0.04114.e9165 0o4769 0o0411 4.955? 0.5061 0.04615.265 0.5160 0.046Q 5025195 065161 0o0511'5.51050 0. 5768p 0.0517 5.5468 0.561' 0o05645.7660 0.569_ 0 .0568 9.r00q 0.5'745 0.0604 ""
* Ha
TABLE 3AERODYNAMIC COEFFICIENTS OF SWEPT WING-BODY MODEL WBS (Cont'd)
MACH N0*=0,.03j M=3 1 MACH Nn.=0.9)4R M=31Q F/FT VA )=774 0
ALDHA CD ALPHAS CL9 co
-3.04055 -0.2895 0.0226 -3.I'60 -0.3018 006-2721 -0243 00019n -2.85RO -0.2718 00O36-'.3914 -0.2170 0.0176 -2.5274 -0.2744 0.0210-201697 -0.%1Q52 0.0165 -2.2AS2 -0.209 2 0.0194-1I.8557 -0.1703 0.0151 - 1.9651 -0.1719 0.0178-195415 -0.1403 0.0140 -1.6644 -0.1524 0.0165-1.2720 -0.1175 0.0133 - 1 *3 46 -091243 0.0194-0.9P27 -0.0975 0.901'26 -1.*0q99 -001365 0.0146-0.6490 -000971 0001,20 -0.7821 -0.0735 0.0139- 0 9 26 -0.0289 0.0119 -0.5012 -0.0524 10.017S-0.0923 -0.0182 000118 -0.0197? -0.0219 0*0l,30.'2254 000203 000118 0.0966 O.007F 0.0133
'05235 00480 00120 003()21 0.0073 47 001350lo8216 0.0777 0.0124 0.7013 0@0666 0001381.1157 00105F0 0.0128 1*0024 090054 001i441.3975 0.1266 0.0136 102P6 .16 005* .23 0.1647 060148 1 O5862 0*1;1 1 0 e01642.0267 0 1'976 000160 108948 0.1826 0.0177?93314 0.2294 0.0178 201,783 0.20is8 00 1932.628q5 0.*2597 0.0199 ?'.44930 002446 0002162 *9 34P 0.2944 0.0223 2.%7995S 0.2750 0.0241 S39240? 0.3287 0.0?52 3.0918o 30303! 0.02O'67305?50 0*3611 0.0285 :3.4 044 093397 0.03013.9440 00.3973 0.0323 3.7002 0.3716 0.03394.13?4 0.4223 0o0 'A6 1 '3.9802 0.39q 0.073694 443f6 0a4'5'3 2 0.0405 4* 293c; 094290 0.04164e7507 0.4903 0.0459 4*5q03 0.4540 0904585e0416 0.50833 0.0500 4.8651 0 94795 0005005o.2'q16 00.51 P6 0.05;32 5.1662 0.5011 0.05485.662 0.606 0.0602 45.4475 105211 0005914*5o8384 0.*575 1 090616 5.701 0 0.5479 0 *0639)
'44H JO..95 M1 ACH NO.=0.979 M=3 19_/v'TeM)=7*73 I=/ FT ( M) 766ALDHAV3 CLRr !) hL PH AS CL8 9Co
-70.14-28 -0.2848 0.90295 -2.9850 -0.2585 0.0324-2.8377 -0.2919 0.02619 -2.6660 -0.2264 090299-2. 502' -0.2165 0.02t?4 4 -29?475 -0.1946 0*0278-2o21940 -0. 1972 0.0210 - 2.1295 - 0.917'98 0,0268*-1 o9594 -0.1641 00021? -1 .8015 -0.1492 0.0250-1*67654 -0.1415 00019Q -1.04862 -0.1192 0*023M-103AR7 -001171 00190q -1.2250 -0.1061 0.0231-10P60 -0.0878 0.0180 -099394 -0.0779 0.0223-0077450 -0.0607 0.0172 -0.6176 -0.0435 0.0216- 0.@4'98 -0.0386 0001458 -0.3461 -0.0274 0.0214-0.2091 -0.0194 090167 -1.0480 -0.0087 0.0214*e OR- 11 0.0054 0.0166 0.2375 000115 09021600.3596 0.0236 0.0169 09510? 0.0'316 00021B*0.6739 0.63599 090173 0,8293 0.06F54 0.0221I *Qi08 0.0709y 000178 1.1127 0.0896 0902291.2489 0.1058 0.0185 194417 001301 0.02391 e50;16 0 @1~ 7R$ .10197 1.'1pl59 001502 0.0248I 1.04R6 0 a1 95 0.0210 2.0151 0.1737 0.02630 2.26 .18I14 0.*02?23 ?930P,7 0.20,43 0.0280R*2o44529 00.22c~ 0.0245 '5 9q7 00.2i() 000297.78 0.51 0.0?70 298835 ?.513 0 0 -10 e2978 000298 3.21?9 0.2980 000347'3. 73q59 0 318q4 0. 0329 -,09212 0. 3340 00.03'rf? .6416 0.3479 0.*0,2q-6 3.8129 00'9 0.0407 -3.*943 0 0 ol7 ';9 0 00'a4 401 041 003A40 0904414.2649 0 a4119 090 4"49 4 olCT5 004141 0.04774.587 0.4464 100F7 4 of, PR- 094446 0.0051 q4o8463 004710 0.0524 4oqc4R 0.4717 0.0 06015 .1 4A7 0 9496 P 0.90570 7; 2037 0. 4069 0906n45.4217 0.5120 0 00608 6 '.57c5 0 .520S 0 0 0 900 5.7* 05329; 0.0#;5 1 Fe 04 ti .4 1 0 0.0q7*.-
TABLE 3
AERODYNAMIC COEFFICIENTS OF SWEPT WING-BODY MODEL WBS (Cont'd)
MqArH Wl%=0.907 M=31 MACH M=l.1 =3 1
ALP PAR c FC ) ALPH AR CL~ RC 0
-2,ap3A - 00.2 623 00393 - 9~2 -00.26'8 n 0174.-'676 -0.2"" -1.0377 -23 -o?? .0 7?4q
0 .7 7pc -0.02379 3.0 31-2 -?.97 -3.1992 0.03,202 1 i2.'37 -0.01 ?q1 00099 ? 00618 -0.o173? 0 0_1
-1.*7q981 -3.1462 30 0?9_ -1.97924 -1 i -2 325-1.aq7gk -0.*1?3 2 0 032'73 1-.45 51 -0. 1192 0. 0299- 1*'362 -0.1012 0?0"66 -1.01611 -0.00pt 00?
* ~~~-3 09'8 -3.r8 0 00 389 3071 0 ?0_28?'7 -Jo0714 0.*0253 -0.567 -O.04P1
-0 934QO5 -080?76 0.0 0.O9 -13,-) ~ 1 o3?74-0 .3f)6I? -0.3059 0.32Y4 8 0.007? -303.347 'N ,n?72 -09242? 00.01 C6 0.0?g00 .:301il 010140 00 273005116 0.0374 0 .0 291 1, 575R )01 3.4 ;?: 0.02760.03 0 0.*06905 0 .0299 Co 17!)6 0031,F 0.02A0100961 00 o85c 0.0262 1 *1 755 0.3 Q4 0p1.398<)A0 0.1118 0072 1.e46R6 3')76 1 o 029QI1.7028e 0.1435 0.002?8 1 1, 774-; 00.1 1 0 6 ' c<I1o9766 09 1632 0o029? 2 *0611 A 0.1 A ' 00'?? *2.0P4 0.1998 000310 ? .3q4 1 3?9 7 0 00,402. 5S12 092247 0o03 2 6 2*64c;0 0, 2-210 0. n0X94?2.8715 0.2513 0.0345 2 0'982p 00.P1,0 0077473e1I954 0*2A93 00 37? '302533 0 02?9et 0.104003.4c00 0032OLO 0.0401 7 t54 so 0.3254 0.0429377 757 0.3454 3.3 428 3903912 10.'25 47 0.a14c574 0 8p;.9 0 *379)7 0.0464 4. 191~4 0 33875 0.e041)?4o3F833 .0.4111 0.0501 4.462R 0.o4';> 0.00934 06677 3.42q9 0.05S33 4o7986 0044R7 000971a 094?39 1 o448 4 0.05055 9. 026 1 0 o4 69 F 00ee5.2671 094918 09062? F.3346 0. 4q94 0o.061505.92 0.s~ 5126 0.0663 9.6338 0.5?'75 0 00730F.8045; 0.51S9 0.0704 c.9740 0.15990 F 0.0740
MACH Nlo.lo036 M=31 MACH No 1 .054 M=3 1q9FFr( M) 7.o67 OcfFT ( M) =7 .64
ALPHA9 CLP SC!) AL PH At cLp C!)
-390404 -0.2596 000389 -2.00906 -00.2974 00.0378-?97351 -0 271,3 0.o076 7 -206077 -1.2322 0.00391-? -241 6% -0.2021 0*0349 -2.3774 -0*20'2 0.033P-291671 -0.1804 00 0?76 -2.1413 -0o1771 0.0325-108651 -0.1552 0.03?4 -1.8R297 -0.1507 000310-1 .c732 -1113 0 oC3 11 -1.9F;0 44 - 0.012 ?1 0. 0299-1.o2P07 -0.1051 0. 0301 -1I.24P2 -0.1074 090292-099A49 -000810 0.0296 -0.9537 -0.0792 0.0284-0.6938 _0409A5 09 0?91 -0.6660 - 0.o0 585 0.027R-lo.4043 -0.0306 000287 -0.3717 -00015 3.0?74-0.1215 -000119 3902R6 -0 36,07 -0.0205 0.0027 20. 1544 000049 00.029 082-36n 0.0247 0.02720.4662 0.0?58 060?qa 0.95003 00 0?99 3.02750.7797 000688 000292 0.7959 0.0629 0402q01 v0475 000898 0.0297 1.0866 000912 0602871.3508 0.11,55 000336 1.3664 00.15! 0.02041*62SOO0 .1337 000315 1.6912 no.1478 0.0307I oqiq0 001650 0.0330 1 m9757 no 1716 0 On: I?2e22?29 1 73 0 034? 2 c272?6 0.1984 007342o5179 0.2172 000399 295834 0.2,312 00 395cp2OR1 42 0.2427 0.0379 2.8q701 0.2973 0.03'747391440 0.02%;31 0.0435 3.o1951 0 e2732 000,304?.4257 0.3070 0.0427 "2o4445 0.3007 0.041R-1.0'161 0.37772 0904505 i.75s1 003-215 0.0450300910 003499 0 004&A1 40 097 0.3614 000486493112 0.3852 000519 4.o3f-9PS 0 Q3 C)77 00 05 140,5900 004060 0 00992 496629 0 4?297 0 0305984.8924 0.o47 AF 00 0993 4. 936A 0.4426 009s93502109 Is.4774 0o,0646 5 02' P 0 94 7 f) 0906755 *4 0817 0.4q16 000681 0 579347 0 o4q68 0006%315e7?37 0.0 50 0*0713 c9.,77 20 0 E 1 1 090713
r..!Tr I1 U T ,T. lT__,U . 1 ... . n . . . -, ..
iO. - 17-
TABLE 3* AERODYNAMIC COEFFICIENTS OF SWEPT WING-BODY MODEL WBS (Cont'd)
MACH Nn-= 108 M=31l / F ( M ) = 7 ,T7 5
ALOH A5 CL8 C -
-3.0082 -0o2647 00 i-. '. " - ,6 ? P '
O o 7 _74 0 0761-2.528 -0.1994 0.0'41~P I~ 5 -Ool 014 0 e0130-1 .8-79 -001,65 0o0316-1.5?70 -0. 37 0.030 5::::-1.*219 -0.,045 0.*0?95-e.9 60 -0 .0766 0.0286
6-.31? -0604q5 090280--0. 534 -0.080 0.0277
-C. 0669 -0.0068 Oo02750.2?, 0.0218 0o0276.5?78 0.050 0•0?78 S0.80Q0 0.0608 0028311074 0.0919 0.02901.4029 0o1178 000291.70p1 0 1,06 0.0311loPO34 0916q2 0.0-752- PI R 9 S 0007,392.5541 0.2082 060154
? 0 8 0.o?4 F)7 0.e0:!7P39 1702 0o2707 o 0"49'9-. 4680 o.oQgo 0.o425 r-. 757 0 .26 1 0e04534e 035f- 0o 34 0 0.0478
.3 4 3-1"I 0.3760 0.0514a-- or& 0 .4 349 0.05;5149311 0o4281 0.05885 s 3? 26 0 o .,56 4 0.06705 *. Ooa 0*4A46 0.0e676
*5..'.
0 -_19-0
17. 50
* . K 700" 444.5 mm
1778mm 1 .4m
WING -BODY WBS 192.02 mm
1. 7 50"44.45 mm
DIA
6.00"15 2.4m m
BODY BSI 3. .99-06mm1.750"
44. 45 mmDIA
_ t 2.5 2 MX DIA5 4.6 6m m
BODY S2 9797 mm8248"46.35 mm
DIA
2.3 2 6 MAX D IA59. 08 mm
FIG. 1: THE SWEPT WING-BODY MODEL AND ITS EQUIVALENT BODIES
0 -20-
0
I-
loo,00.D
0
14,,
w
LL
U.
- 0- / z
0
M C-)
N 0
z w
0 u
U) U.
00 C~0cnCd5
N 2
* -21-
0.2
E SETWN 0
0.1
D C)ELA WIN
00-3
T
0.1
DELTA WING
4.0-
SWEPT WING
30-
DELTA WING
20-
-1 00
*0 -.0 0,2 04 0 6 X/c 0 8 10
FIG. 3: FUNCTIONS DUE TO LIFT, E(x), T(x) AND Fx 2 FOR SWEPT-BACK ANDDELTA PLAN FORMS
0 -22-
0.3-
S DELTA WINGlift
0.2-
0.1I,(CLD
0r 0.2 0.4 0.6 0.8 1.0 1.2
FIG. 4: DISTRIBUTIONS OF EFFECTIVE AREA DUE TO LIFT FOR SWEPT-BACK ANDDELTA PLAN FORMS AT THE DESIGN LIFT CONDITION
J. -23-
'A
cc2
2, c e
C 9
0 00 7
a 0
00 Z
--- --- --
FIG 5:VAIATONSOF6L I0 0AN CD WIHC O WP IGBD OE B
ATIFRN MC UBR
. 24
CI
0o M
-J-
0U.
o cc
4 4-
U- '
N, -U
'0 -25-
EQUIVALENT BODY BSI, a=-O.05*
* 01
WING-BODY WBS, a 0O.O01
FIG. 7: SCHLIEREN FLOW VISUALIZATION FOR WING-BODY WBS AND EQUIVALENT
BODY BS1 AT ZERO ANGLE OF ATTACK (NOMINAL), M_. 0.98
'0 -26-
WING-BODY WlBS
EQUIVALENT BODY BS1
FIG. 8: SURFACE FLOW VISUALIZATION FOR WING-BODY WBS AND EQUIVALENTBODY BS1 AT ZERO ANGLE OF ATTACK, M_. 0.98
0
0.=2.940
n=5.51 0
FIG. 9: SURFACE FLOW VISUALIZATION FOR WING-BODY MODEL WBSAT ANGLES OF ATTACK, M_, 0.98
'0 -28-
0
C,,>
- 0
Z Z
cc~
w
L= I
0
cv0
~ 0 U.
0
w zj
0;~C ODC w
K 0 00I
-29-
\~ w
au IL 'l
- 0
C)W
- II~ -0
'S 0LL
U),
z 0Jo 0 0 U
II 4
7 7 00 0 010 0 d U
K -30-0
0
* II
CO w
C'CC
a.a- i
wU) (no
co LL
a) -
0 cc
0
0 0
4
C)) 0C~U)
-~ LL
-' v w7 - -.- -:* - ~1 ~ - 7 . z
Jo.o
40 w 00 U
4 00
c~ Lju
z-i X 0
(L >W*
0 w 3wO 0
0 Q0 0
4 UU
co o
-5 I- IZI-22 22
owi
- 32.
0
-o 0
7 ZU
LA W
0
U) C.4
0 0WL
0CU-
U)q
0If LOZ)0
* 0Je4
* -33-
00
L-J
0
c.
0
*LU
)0) 0
-00
(n(.~ U- -
LLJ 0
Loi
.9.cj
-~ .4l
6 .34- :
0
C.)
' IC)LU
00
\C.)
0 0
-0 0
U.
U_
0)0
- 0 00).
m OD
LL
O~L w. _
CD.
30
6 0
*... .,. .-- ~ - . -- . -
4 -35-0
wU-UL.0
U)
0
CD 0
cco .o . .
/ Ui 000 >-
M 0
I -U.- -
a D~00LI
w 0
Z00W Co NI,ucQ I.C.
C- 0.O 0
4w 4e
0 0
C 0?
i-L-
I 0.
0 -36-
I-
m co U- 0
0 C0
40D
0 0
0 .C.
0 0~ '
00 z
0
- U SLL.
O 0 0 wO
0 -a 0 U
00in m
cw woL - - oc
0
0a, 0
-0 O4- (D 0~Uc
*Qcn.
LLS
-37.
APPENDIX
EVALUATION OF FUNCTIONS T(x), E(x) AND F(x) FOR A SWEPT WING
The functions F(x), T(x) and E(x) constitute the expression for the effective area due tolift (Eq. (10) in the text). All involve integrations in the y-direction and both T(x) and E(x) involvedouble integrals. These integrals are evaluated numerically as follows.
The function T(x) has the form (Eq. (6) of Ref. 1)
' a 2 a 2 1T(x) = 4 f f 0 n (x,s) (x,y) n i dsdy (Al)
a a
where a, and a, are the y-co-ordinates of the leading edges for each side of the wing respectively. Theintegrals can be evaluated in sequency as
,,,.x_1 2 ( 2 Yl
T(x) = (X,Y) f x (x,s) Qn i ds dy (A2)
As x appears only as a parameter, the integrands are now written without x and the square bracket forshort. For the integral with respect to s the integrand 0, (s) is written for the integral As by the trape-zoidal rule as
.n+ 1 nox (s) + (s- sn):, :n s s -ns
(A3)
ox - ( sn ¢+ s
n S
where*X
n n+ nX
n+1 n
and* a2 - a,
As s s~J - S = n+J - n =
.+]-N- 1
Substituting Equation (A3) into Equation (A2) and evaluating the integrals, we have* .
N-I
1 -- s Sn) I n (y) + kn I2 n(Y) (A4)
*<.-.-.
-38-
where
I 1 (y) =[(y*- sn. In Iy -s,.. I - (y - s,, n Iy-sn 1 + (s +Isn
=ny 1/ (Y - n+l) 5n . ly-s n+ (y n n+
The integral with respect to y can then be written with the trapezoidal rule as
Tx 41r M= [0 [(Y)I(Y)] m +I1 + [0' (Y)I(y)] 2 A5
It is explicit that n y m for n =m.
The function E(x) has the form (Eq. (7) of Ref. 1)
1 a 2 a 2 1E(x) =- f f R0 (x,y) I ,, RD(x's) IJY Qn dsdy (M6)
41r' (y -S)
and can be evaluated in the same way as T(x) with OI (x,y) IIreplacing II~(x,y) D
The function F(x) involves a single integral (Eq. (5) of Ref. 1)
F(x)= f 10 0 dy (A7)
al
In calculating the effective area due to lift, the x-derivative of F(x) is used. Again applying the trape-zoidal rule, the F, (x) can be written as
M Ym Ym da2 daFX (x) O M~ + + OX] 2 + i00x9a 2 1 dx -sa I(x d (A8)
For a fixed x station, the numerical procedure is carried out along the y-direction froma, (x) to a2 (x over the wing and the wake. The velocity potential and its derivatives are given inEquations (4), (5) and (6) over the wing, Equations (7), (8) and (9) for the wake. The resulting func-
* tions are shown in Figure 3.
REPORT DOCUMENTATION PAGE / PAGE DE DOCUMENTATION DE RAPPORT
REPORT/RAPPORT REPORT/RAPPORT
LR-614 NRC No. 23412la lb
REPORT SECURITY CLASSIFICATION DISTRIBUTION (LIMITATIONS)
CLASSIFICATION DE SECURITE DE RAPPORT
Unclassified 3 Unlimited2
TITLE/SUBTITLE/TITRE/SOUS-TITRE
An Experimental Study of the Transonic Equivalence Rule with LiftPart II
4
AUTHOR(S)/AUTEUR(S)
Y.Y. Chan5
SERIES/SERIE
Aeronautical Report6
CORPORATE AUTHOR/PERFORMING AGENCY/AUTEUR D'ENTREPRISE/AGENCE D'EXECUTION
National Research Council CanadaNational Aeronautical Establishment High Speed Aerodynamics Laboratory7
SPONSORING AGENCY/AGENCE DE SUBVENTION
8
DATE FILE/DOSSIER LAB. ORDER PAGES FIGS/DIAGRAMMES
COMMANDE DU LAB.
84-03 45 189 10 11 12a 12b
NOTES
13
DESCRIPTORS (KEY WORDS)/MOTS-CLES
1. Wings (swept-back) 3. Transonic equivalence rule2. Wings (delta)
14
SUMMARY/SOMMAIR E
The experimental study of the transonic equivalence rule with lift,previously reported in Reference 1, has been extended to include a configu-ration of swept-back wing-body and the corresponding equivalent bodies forlift and zero-lift. For zero lift, the wing. body is shown to have higher wavedrag than that of the equivalent body. At lifting condition, the analyses ofthe data, including those of the delta wing-bodies reported in Reference 1,have verified the similitude of drag rise due to lift between the wing-bodies.The additional drag induced by the effective area due to lift on the equiva-lent body was found to agree reasonably with the wave drag generated bylift on the wing-body. The experiment thus verified the area rule with liftwhich must be considered in the optimal design of transonic configurations.
15
r........ * *.* *-.~r - , , - -,IV41A rrr -
-4 ,
t -01
*&r~R h~ittwt
*LAI
.1> '~' '
