nasa report - theory for computing span loads and stability derivatives due to side slip, yawing,...

7/29/2019 NASA Report - Theory for Computing Span Loads and Stability Derivatives Due to Side Slip, Yawing, And Rolling fo

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TECHN I CAL NOTE

' )

:

SPAN LOADSND STABILITY DERIVATIVES

NASA TN 0-4929

LOAN COPY: RETURN TOAFWL (WLlL-2)KIRTLAND AFB, N MEX

E TO SIDESLIP, YAWING, AND ROLLINGOR WINGS IN SUBSONIC COMPRESSIBLE FLOW

M. J. Queijo, .. ,. . .

AERONAUTICS AND SPACE ADMINISTRATION WASHINGTON, D. C. DECEMBER 1968


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TECH LIBRARY KAFB . NM

111111 1111111111 11111111111 111111111111111110131666NASA TN D-4929

THEORY FOR COMPUTING SPAN LOADS AND STABILITY DERIVATIVESDUE TO SIDESLIP, YAWING, AND ROLLING FOR WINGS

IN SUBSONIC COMPRESSIBLE FLOWBy M. J. Queijo

Langley Research CenterLangley Station, Hampton, Va.

NATIONAL AERONAUTICS AND SPACE ADMINISTRATIONFo r sale by th e Clearinghouse for Federal Scientific and Technical Information

Springfield , Virginia 22151 - CFSTI price $3 .00


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THEORY FOR COMPUTING SPAN LOADS AND STABILITY DERIVATIVESDUE TO SIDESLIP, YAWING, AND ROLLING FOR WINGS

IN SUBSONIC COMPRESSIBLE FLOW*By M. J. Queijo

Langley Research Center

SUMMARYA method of computing span loads and some of the resulting aerodynamic derivatives

r wings in sideslip, yawing, and rolling flight is derived. The method is applicable toof arbitrary planform, and accounts for compressibility effects . The span-load

require that angle-of-attack span-load distributions be available for theunder consideration_ The theory developed is a consistent approach, based on the

of a vortex system, for determining the various wing parameters.INTRODUCTION

The aerodynamic span loads and stability derivatives are important in relation tointegrity and inherent stability of airplanes. The lifting surfaces (wing and

surfaces) generally produce the predominant aerodynamic forces; therefore, muchbeen expended in developing methods of predicting the aerodynamics of liftingMost of this effort, however, has been directed toward determination of aero

characteristics associated with angle of attack. Aerodynamics associated withaircraft motions (rolling, yawing, pitching, and sideslipping) have been investigated

a lesser degree, and generally by somewhat cruder methods. This is particularly truer the low-speed regime. For supersonic speeds the nature of the governing equationssuch that it has been possible to obtain equations for loads and aerodynamic derivatives

wings performing various modes of motion. (See refs. 1 to 4, for example.) In cercases i t is possible to use supersonic theory to predict subsonic character

(See ref. 4, for example.)* A preliminary version of the material presented herein was included in a dissertaentitled "A Theory and Method of Predicting the Stability Derivatives CZf3' CZr '

p ' and CyP for Wings of Arbitrary Planform in Subsonic Flow," offered in partialof the requirements for the degree of Doctor of Philosophy in EngineeringVirginia Polytechnic Institute, Blacksburg, Virginia, June 1963.


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There are a number of problems associated with attempting to predict aerodynamiccharacteristics of wings performing the possible modes of motion. One problem is thatof finding an adequate mathematical model for the wing, and the second is that associatedwith solution of the equations which ar ise from use of the mathematical models. Variousmethods for predicting certain aerodynamic characteristics of unswept wings have beendeveloped by a number of investigators, and numerous reports have been published fromwhich certain characteristics can be obtained for specific wings. (See refs. 5 and 6, forexample.) Three general approaches have been used in determining the aerodynamics ofswept wings. These are:

(a) Computations based on mathematical models associated with the use of vortices ,doublets, or other concepts to represent the wing (refs. 7 to 11, fo r example) .

(b) Determination of approximate equations based on treating each wing semispanas one-half of an unswept wing. Th e fictitious "unswept" panels are skewed to simulatea swept wing (refs. 12 to 15, fo r example).

(c) Development of design charts based on tests of a great number of wings withvarious sweep angles, aspect ratios, and taper ratios (ref. 16, fo r example).

The first of these approaches is generally difficult and in some cases involves thesolution of numerous simultaneous equations. For these reasons, only a few aerodynamicparameters (primarilY aerodynamic-center pOSition, CLa, and CZp) have been attacked

--

by fairly rigorous methods. Th e use of high-speed computers to solve many simultaneousequations has permitted the numerical solution of equations better defining the wing boundary, but solutions generally have been obtained for angle-of-attack loading.

The second approach has been quite successful in predicting trends, and with somemodifications has been used to obtain good quantitative results for certain aerodynamiccharacteristics. (See ref. 15, for example.)

The third approach is adequate for engineering data, provided the available dataenvelop the range of geometriC variables of interest. Unfortunately, the amount of dataavailable for some of the wing derivatives is very limited because of the scarcity ofexperimental facilities for determining such derivatives.

The purpose of the present paper is to examine the problem of wing characteristicsin subsonic compressible flow and to develop a consistent method for computing thesecharacteristics.

The method developed herein is a new approach for estimating span loads and thederivatives CZ(3' C Zr' CZp' Cnp ' and GyP for wings of arbitrary planform in sub-sonic compressible flow. I t is based on a vortex representation of the wing which wasfirst developed by the author for sideslipping wings in incompressible flow (ref. 17).2


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are generally applicable to the low angle-of-attack region, where the various wingvary linearly with angle of attack or lift coefficient.

l

aC ll = --p pba-

2YaC ll = -r a rb

2Y

aC n=--p pby

a-2Y

SYMBOLSaspect ratio, b2/Stwo-dimensional lift-curve slope

wing span

wing lift coefficient, L.!py2S2acwing lift-curve slope, ad", per radian

MXrolling-moment coefficient, .!py2Sb2

MZyawing-moment coefficient,.!py2Sb2

side-force coefficient, Fy.!py2S2

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8CyCy P = --

cc

C C l J _ pbc -2Y

pb8 -2Y

cCl- rbcCL 2Vi

DFy

4

wing local chordwing average chord

Section liftsection lift coefficient,.!py2C2

section lift-curve slope,

three-dimensional section lift-curve slope for wing at angle of attack

effective three-dimensional section lift-curve slope for rolling wing

parameter for section lift, per unit lift coefficient

parameter fo r incremental section lift due to Sideslip, for a wing at angle ofattackparameter for incremental section lift due to rolling

parameter for incremental section lift due to yawing, for a wing at angle ofattack

wing root chordwing ti p chordspanwise distance from wing root chord to center of rotation of yawing wingside force

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


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side force associated with a chordwise-bound vortexside force associated with quarter-chord-line vortex

lift

lift per unit length of chordwise-bound vortexlift per unit span of quarter-chord-line vortexlift per unit length of quarter-chord-line vortex

free-stream Mach number

Mach number of free stream normal to wing quarter-chord linerolling moment

yawing moment

rate of roll, radians per secondr yawing angular velocity, radians per seconds wing areau wind velocity in x-direction

free-stream wind velocity relative to wing center of gravitylocal velocity

free-stream wind velocity normal to wing quarter-chord linev wind velocity in y-directionX ,Y longitudinal and spanwise reference axes, with origin at center of gravityx,y distances along reference axes

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Xac chordwise distance between aerodynamic center (a.c.) and moment center(o r center of gravity, c.g., in flight), positive when c.g. is upstream of a.c.

Xc x-distance to wing trailing edgexc/4 x-distance. to quarter-chord line-y spanwise position of centroid of the angle-of-attack span loadingy radius of gyration of angle-af-attack span loadingQI angle of attack (or incidence), radians(3 sideslip angle, radians

r y circulation strength related to spanwise positioncirculation strength related to displacement along quarter-chord line

[) local effective sideslip angle due to yawing, radians infinitesimal displacement in spanwise direction

A sweep angle of quarter-chord line, radianswing taper ratio, Ct/crdistance along the quarter-chord line, measured from wing root chord

p mass density of air

Subscripts:M compressible flowM=O incompressible flowp part due to rolling

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- - - - - - - - - - - - - - - - - - - - - - - - - - -_. -

r part due to yawinga part due to angle of attack(3 part due to sideslip

A star (*) indicates that the quantity has been nondimensionalized by divisionby b/2; fo r example, y* =.-:L-/ .b 2

VORTEX SYSTEM

The vortex system used in this analysis is fo r essentially a modified lifting-linetheory approach, and is illustrated in figure 1. The system consists of a bound vortexalong the wing quarter-chord line and a bound-vortex sheet from the quarter-chord lineto the wing trailing edge. Behind the wing trailing edge, the vortex sheet is in the direction of the airstream. This system was applied to sideslipping wings in reference 17,and proved to be quite accurate in predicting the parameter CZ3 for a wide range ofwings in subsonic, incompressible flow. An appraisal of th:s vortex system indicatedthat it would be applicable to the estimation of other wing derivatives.

The vortex system adopted for this study allows the possibility of lift generation bythe bound vortices, which are:

(a) The quarter-chord-line vortex, which extends across the entire wing span(b) The chordwise-bound vortices, which are parallel to the wing plane of symmetry

and extend from the wing quarter-chord line to the wing trailing edge.The trailing vortex sheet behind the wing is made up of "free" vortices which are

in the direction of the airstream and hence develop no lift.CIRCULATION DISTRIBUTION

The strength of the chordwise-bound vortices is determined by the gradient of thestrength distribution of the quarter-chord-line vortex; therefore, the lift distribution ofthe wing can be determined i f the vortex strength distribution of the quarter-chord line isknown and i f the wind velocity components are known. The distribution of the wind velocity components relative to the wing can be determined easily for each possible motion ofthe wing. The basic problem in determining wing load distribution, therefore, is that ofdetermining the vortex (or circulation) distribution for all wing motions. The three typesof motion considered in this paper are sideslip, yawing, and rolling.

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Sideslipping WingThe sideslipping wing was studied in reference 17. I t was argued therein that the

circulation distribution in sideslip was essentially the same as that fo r a wing in zerosideslip. Thus zero-sideslip circulation distributions could be used for estimating aerodynamiC loads and the parameter CZf3 for wings in sideslip. The vortex system for thewing in sideslip and the direction of the airflow are shown in figure 2.

Yawing WingThe bound-vortex pattern of the yawing wing is the basic pattern shown in figure 1.

The trailing-free vortex sheet, however, is curved to match the airflow streamlines(fig. 3). An examination of the flow pattern over the wing (fig. 4) shows that there is alateral velocity component resulting from the flow curvature, and that the magnitude ofthe component is a function of position on the wing. The wing therefore can be consideredto be in sideslip, with the effective sideslip angle varying over the wing. The argumentspresented with regard to circulation distribution of the wing in sideslip can be carriedover to the yawing wing .An assumption of the present theory is that the circulation distribution fo r a yawing wing is essentially the same as that for a nonyawing wing.

Rolling WingThe problem of circulation distribution for the rolling wing is somewhat different

from that for the sideslipping and yawing wing. In the case of the rolling wing, the localgeometriC angle of attack is increased by

The primary cause of circulation, and hence the circulation itself, is altered by rolling.The net circulation of a rolling wing, therefore, is made up of:

(a) Circulation due to the symmetric angle of attack(b) Circulation due to the anti symmetric angle-of-attack distribution associated with

rolling velocityThe assumed wing vortex system and the circulation distributions that have been

discussed form the basis for the present theory for the computation of wing aerodynamiccharacteristics.

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PRESENTATION OF RESULTS

General RemarksGeneral equations are derived in appendix A for the span loads and certain aerody

namic characteristics associated with sideslipping, yawing, and rolling for wings of arbitrary planform. These equations are applicable in the low angle-of-attack region, wherethe characteristics vary linearly with angle of attack and lift coefficient. Span loadsassociated with sideslip, yawing, and rolling can be obtained by the methods presentedherein only i f the angle-of-attack span load is known. Such information is available fora wide range of wing geometry (ref. 18, fo r example). The angle-of-attack loading forodd-shaped wings can be obtained by application of the horseshoe-vortex method of references 19 to 22.

The equations of this paper, for incompressible flow, are derived in appendix A.Compressibility effects are derived in appendix B. Some of the pertinent equations aresummarized in appendix C.

Many of the equations derived herein involve the spanwise position of the centroidand radius of gyration of the angle-of-attack loading of the wing semispan. The centroidposition for angle-of-attack loadings has been determined fo r a wide range of wing planforms and is readily available in the literature. (See refs. 8, 18, and 21, for example.)The present theory appears to be the only one in the literature in which the radius of gyration occurs as a factor in determining aerodynamic characteristics. I t was necessary,therefore, to compute the radius of gyration y* for use in this paper. The values weredetermined by plotting the product f - ~ C l )(y*)2 against y* fo r a large number of wings\cCLand performing a mechanical integration to obtain :y* by use of the following equation:

(1)

total of 160 such integrations were performed to obtain the desired coverage of wingplanforms. Values of y* and of :y* are plotted in figures 5 and 6 as functions ofsweep, aspect ratio, and taper ratio for use in this paper and for general information.

the span loads were obtained from reference 18.Span-Load Distributions

The span-load equations derived in this paper are applicable for the determinationthe incremental load due to Sideslip, yawing, or rolling velocity of a wing for which the

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---span load in symmetric flight is known. The span-load equations are summarized inappendix C. Equations (C1) to (C3) are very general and should be used if the sweepangle varies across the wing span. In such instances it may not be possible to find inthe literature the corresponding angle-of-attack loading of the wing. In these instances,as noted previously, the angle-of-attack loading can be determined in a straightforwardmanner (solution of simultaneous algebraic equations) by the method of reference 19, 20,or 21.

I f sweep is constant across the span, equations (C4) to (C6) are applicable and moreconvenient. In these cases the angle-of-attack span-load distribution can be obtainedfrom a number of sources, such as reference 18.

Most of the theoretical angle-of-attack span-load distributions available in theliterature have been determined by use of the Weissinger method. I t is of interest andsignificance that angle-of-attack span loadings determined by the use of horseshoe vortices (as in ref. 19 or 20) will closely approximate loadings from the Weissinger methodi f a minimum of about 20 horseshoe vortices are used to represent the wing. (Seeref. 19.) The horseshoe-vortex method is applicable fo r representing wings having discontinuities in sweep distribution (for example, M- or W-wings), where the applicabilityof the Weissinger method is doubtful. This horseshoe-vortex method was used in reference 22 to determine the span loads of some M- and W-wings, and can be used for wingsof arbitrary planform. The system of horseshoe vortices was also used in reference 17to obtain the loads on Sideslipping wings. In such a case, the loads are obtained fromeither a mechanical integration or from summing terms of a series expansion (ref. 17).

Aerodynamic CoefficientsThe equations derived for the various aerodynamic coefficients are presented in

appendix C. Equations (C7) to (Cll) are very general and are applicable to wings ofarbitrary planform. I f sweep is constant across the semispan, and if the chord is alinear function of spanwise position, equations (C12) to (C16) are applicable andconvenient.

Equations (C12) to (C16) have been used to construct a series of figures from whichnumerical values of the various coefficients fo r a Mach number of zero can be obtainedas a function of wing geometry. The effect of Mach number can be obtained through useof equations (C12) to (C16). The parameters for a Mach number of zero are presentedin the following figures: Cl{3/C L in figure 7; Clr/C L in figure 8; Clp in figure 9;CYp/CL in figure 10; and Cnp/CL in figure 11.

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_ . ,,

DISCUSSION OF RESULTSSideslipping Wing

The span load due to sideslip can be computed by equation (C1) or (C4). No otherappears to be available fo r estimating these loads. Experimental data are alsononexistent , except fo r the sideslip data given in reference 23 for a wing in incom

flow. As shown in reference 17, these data compared well with theequation of the present theory.

The rolling-moment parameter associated with Sideslip fo r a trapezoidal wing inflow is given by equation (C12), repeated here fo r convenience:

Cl

t (3 - 61 -A~ = - - +y* tan A - - - - +0.05CL 2 A(1 + A) A 1 + A

e analysis of reference 17 shows that this equation is quite accurate for wings inflow.

(2)

Very little theoretical work has been done to determine compressibility effects one derivatives investigated in this paper. The Prandtl-Glauert transformation is appli

for symmetric flow, bu t there are a number of fundamental questions regarding itsi f the flow about the wing is not symmetric. The Prandtl-Glauert transforma

can be applied either as a change in section lift-curve slope or as a change in wingin symmetric flow. In unsymmetric flow, a geometric change in accordance

the Prandtl-Glauert transformation results in a distorted wing shape - that is, thewing semispans are different from each other. This basic problem of appli

can be avoided, as was done in reference 14, by assuming that each wing semispans one-half of a wing in symmetric flow. The compressibility corrections are then

to each wing semispan and the loads of each semispan are used to obtain the wing

A somewhat simpler approach was used in reference 24, where the author simplyto modify the section lift-curve slope as i f the actual wing were in symmetric flow.I t does not appear possible to obtain a direct evaluation of the estimated compres

effects on the values of Cl{3/CL fo r an isolated wing, since high-speed data are

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generally obtained for wing-fuselage combinations. However, wing-fuselage data wouldbe expected to have about the same trends as wing-alone data. Figure 12 shows experimental compressibility effects for several wing-body combinations and compares theresults with those predicted by the wing-alone theory of reference 14 and by equation (2)of the present paper. The data and theories all show an increase in the magnitude ofCl f3 /C L with increase in Mach number. The effect on wing-fuselage data is somewhatsmaller than that predicted by the wing-alone theories. Reference 14 suggests an empirical method fo r accounting for the fuselage. However, as pointed out in that reference,additional effort is required to define more precisely the effects of the fuselage onCf3/CL'

Yawing WingThe span loads due to yawing can be determined from equations (C2) and (C5). The

author could find no other theory or experimental results to compare with these equations.The derivative obtained fo r the yawing wing is Cl r ' For a trapezoidal wing in

compressible flow it is given by

C lr =G(l + tan2A) _ .J!.. tan A +.J!!J.1 - \ ) ~ ( y * ) 2 +(!. tan A _ tan2A)(y*)2CL 2A 1 + \ 4A2\1 + \ J A 1 + \ 2

t tan A 9(1 - \ ) ~ -* ~ t a n A 3 1 - * -* 3x:c 9+ - - Y + -- - - -- xacY + + ---=-----=-2A(1 + \) A2(1 + \)2 2 A 1 + 2A(1 + \) 4A2(1 + \)2(3)

Although this derivative is important in determining the dynamic lateral stability of aircraft, there is little published information on theoretical or experimental values of Cl .rSome incompressible-flow information is available from references 5 and 12. However,reference 5 is applicable only to unswept wings. Figure 13 shows comparisons of experimental data (incompressible flow) with the theory of reference 12 and with equation (3) ofthe present paper. Both theories fi t the data equally well.

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Reference 24 and equation (3) both predict an increase in ClrlCL with increasein Mach number, as shown in figure 14, but no experimental data are available forcomparison.

Rolling WingThe equation for additional load due to rolling velocity is , fo r a trapezoidal wing in

compressible flow,

rrA (4)

2 +

This type of loading is amenable to solution by the Weissinger method (ref. 25). Com-parisons of span load due to rolling as obtained by equation (4) and from reference 25 areshown in figure 15. The agreement between the two theories is quite good for incompres-sible flow.

The span load distribution can be used to obtain the parameters CZp ' CYP' andCnp' and these are given by the equations

Cy--p = y*tan ACL

(5)

(6)

(7)

Comparisons between equation (5) and experimental results for CZp are shown infigure 16. Generally there is good agreement between experiment and equation (5), par-ticularly at low aspect ratios.

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The derivative Cy generally has no effect on the dynamic lateral stability of anpairplane and therefore is not important in itself. However, its value is of importance intransforming certain of the other derivatives to various systems of axes. The basic reference fo r theoretical values of this parameter in incompressible flow is reference 12.Theoretical and experimental results from reference 12 and values from equation (6) areshown in figure 17. Both theories generally underestimate this parameter. The presenttheory indicates no effect of compressibility on Cy p/C L , whereas reference 24 indicatesthat CYp/C L decreases with Mach number. There appear to be no available data tosubstantiate either theory.

The wing contribution to Cnp is important with regard to dynamic stability ofairplanes. Several methods have been developed for predicting this derivative in incompressible flow. These methods are compared with each other and with experimental datain figure 18. Equation (7) indicates no compressibility effects on Cnp/CL, whereas thetheory of reference 24 indicates a decrease with increasing Mach number. However,there appear to be no experimental data available for determining compressibility effectson Cnp/C L .

CONCLUDING REMARKSA theory and method are developed fo r computing span loads due to sideslip , yawing,

and rolling for wings in subsonic compressible flow. The method is applicable to wingsof any planform, provided the angle-of-attack load distribution is known. Such information is available in the literature for a wide variety of wing planforms, or can be obtainedby straightforward methods. The span load (o r circulation) distribution is used to determine the stability derivatives due to Sideslip, yawing, and rolling. Derivatives estimatedby the method are compared with other theories and experimental results where availableand applicable. I t is not possible at this time to evaluate the present theory relative toothers because of the limited amount of data available - particularly with respect to thederivatives associated with yawing and rolling.

Langley Research Center,

14

National Aeronautics and Space Administration,Langley Station, Hampton, Va., July 8, 1968,

126-13-01-50-23.


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..

APPENDIX ADERIVATION OF EQUATIONS FOR INCOMPRESSIBLE FLOW

The vortex system used in this ana lysis includes a bound vortex along the quarterchord line , a bound vortex sheet from the wing quarter-chord line to the wing trailingedge, and free vortices (along streamlines) behind the wing. Interaction of the relativewind with the bound vortices produces forces and moments. Equations for these forcesand moments are derived in the following sections. In the derivations, reference is madeto the right wing semispan unless otherwise noted.

Sideslipping WingThe equations for span load and rolling moment for wings in sideslip were derived

in reference 17. A few of the equations are included here for completeness and becausethey are required for the compressible-flow development.The incremental loading due to sideslip is given by

(A1)

and is anti symmetric over the wing. The resulting rolling-moment parameter isC C l C d --l 1 cc cC= -!S ( ~ ) tan A - c* L Ciy*dy* + 0.05CL 2 0 \cCL Ci 4 dy* (A2)

where the term 0.05 is a correction obtained in reference 17.For wings with straight leading and trailing edges the sweep angle is constant and

c* can be expressed as a function of wing span:

c* = 4 ' _ (1 _ A)yJA(l + A) r J (A3)Equation (A1) therefore can be written as

(A4)

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APPENDIX Aand equation (A2) can be written as

C S . l ~ C C ) s . 1 ~ j d ( : ~ l ) = _ an A y*dy* + 3 1 _ (1 _ \)y* L (1 y*dy* + 0.05 (A5)CL 2 0 cC L (1 2A(1 + \) 0 dy*Equation (A5) can be integrated, with the result

Cl (3 = _.!I 3 + y* (tan A _ ' " l . . . : : . 2 : ) ~ + 0.05CL A 1 + \ ~ (A6)Yawing Wing

The analysis of the yawing wing follows closely that of the wing in sideslip (given inref. 17) but is more complex because the airflow velocity and direction relative to thevortices are functions of both the spanwise and the chordwise position on the wing.

The total wind velocity at any point on the wing (see fig. 4) is given by1V l = r(D - y)-cos 0

The components parallel and normal to the wing plane of symmetry are

u = -Vl cos 0 = -r(D - y)and

v = -Vl sin 0 =-r(D - y)tan 0 =-rx

(A7)

(AS)

(A9)

For the right span of the wing, the component of velocity normal to the wing quarter-chordline is given byVN = -u cos A - v sin A

or, with substitution of equations (AS) and (A9),VN = r(D - y)cos A + rX c/4 sin A (A10)

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APPENDIX AI f the velocity at the wing center of gravity is denoted by V, then

V = rDand equation (AlO) can be expressed as

VN = (V - ry)cos A + rX c/4 sin A (All)The lift per unit length of the quarter-chord-line vortex is given by

(A12)

As discussed previously, a basic assumption of this analysis is that the circulationdistribution is that due to angle-of-attack loading, and is not altered by yawing. The liftper unit length of the quarter-chord-line vortex therefore is given by

(A13)

Similarly, the lift per unit span is given by

A unit length of quarter-chord-line vortex covers a span equal to cos A, so that= ly cos A. Using this equality, and noting that VN = V cos A for the angle-of-attackcase, it can be shown that

This simply indicates that in computing span loads the same circulation is used whetherone deals with the free stream and the spanwise direction or with the quarter-chord lineand the velocity normal to the quarter-chord line. Equation (A13) therefore can bewritten, per unit length of quarter-chord-line vortex:

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APPENDIX Aor, per unit span of the vortex:

VN ( )=p- - ry cos A. ya (A15)

From equation (A14) it can be shown that

(A16)

Substituting equations (All) and (A16) into equation (A15) results in

1 ~ ~ C C l ) _ y =- P (V - ry) + rX c/4 tan A V - _ - cC L2 cCL a

(A17)

per unit span.The lift of each chordwise-bound vortex must be found through an integration, since

the lateral velocity varies along the length of the vortex. The lift is given by

or, with substitution of equations (A9) and (A16) ,

d ~ C C l ) 1 - SXC/4 \CC L al2 = -- prVcCL x dx2 Xc dy (A18)

The local section load coefficient is given by

or using equations (A17) and (A18) and normalizing with respect to the semispan givesd!.CCl )

( ccl ) (1 rb * rb * ta A) ~ C C l ) rX6/4 rb * a dx*cC L a,r = \ - 2V Y + 2V xc/4 n V::C L a - J x ~ 2V x dy* (A19)18


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APPENDIX A

Since d ( _ ~ C Z ) !dY* is not a function of x, the integral of equation (A19) can be evalu\CC L Ciated easily, and equation (A19) becomes

The incremental load parameter due to yawing is given by

Similarly, for the left wing semispan,

In general, the rolling-moment parameter for a wing is given by

(A23)

he rolling-moment parameter due to yawing is given by

Cz 1Sl ceZ--.!: = -- y*dy*CL 4 -1 esc rbL 2V:

(A24)

of symmetry and the use of equations (A21) and (A22) with equation (A24)in

(A25)

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

APPENDIX AEquation (A25) is perfectly general an d somewhat complex for an arbitrary wing. I f thewing leading and trailing edges are straight, however, some simplification is possible.In this case (see fig. 3),

and3x* = x*/ - - c*c c 4 4

Substituting these relations into equation (A25) results in

(CC l )j* 3 - 3 9 dVCL a+_@(y*-y*\tanA--x*--CJY*dY*2 ) 2 ac 16 J dy* (A26)Integration of equation (A26) is very lengthy and tedious. The solution of equa

tion (A26) is given by

lr 1 9 tan A 1 - .\ 27 1 - .\ - 2 3 1 - .\ 1 - 2C ()J t= -(1 + tan2A) - -- +-- - - - (y*) + - -- tan A - - tan2A (y*)C L 2 2A 1 + .\ 4A 2 1 + .\ A 1 + .\ 2

3 tan A 9(1 - .\) -* ta n A 3 1 - .\ x* -* ac 93x*+ 2A(1 +.\) - A2(1 + .\)2 Y + -2- - A 1 +.\ acY + 2A(1 +.\) + 4A2(1 + .\)2

(A27)Rolling Wing

In th e analysis of the rolling wing the additional local angle of attack due to rollingmust be considered. This additional local angle gives rise to an incremental circulationand therefore an incremental lift. Thus the net lift at any spanwise position is given by

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APPENDIX A

(A28)

This can also be expressed as

(A29)

where (cla)p and (cla)a are the local three-dimensional section lift-curve slopes inrolling flight and at angle of attack, respectively. As a first approximation, it is assumedthat the local lift-curve slope is equal to the lift-curve slope of the wing semispan. Withthis assumption,

(A30)

The additional lift due to rolling is antisymmetric over the wing; therefore, the effectivelift-curve slope of the wing semispan in roll is apprOXimately the lift-curve slope of awing having the geometry (A and A) of the wing semispan. A convenient and accurateexpression for wing lift-curve slope in incompressible flow is (from ref. 14)

(A31)

Therefore, for the wing in roll the lift-curve slope is given approximately byA

) "2 ao(CLa p = 1/22 A )2 +Jcos A J

21


25/79

APPENDIX Aor, i f ao assumes its theoretical value of 27T,

(CLa)p =- - - - - - - ' -7T ' -A- - -1- /-2

~ 2 C ~ S S (A32)

2 +

Using relations (A30), (A31), and (A32) with equation (A29) results in

1/2

1 + _2_+ = - [ ( 2 _ C _ ~ S - = - A _ ) 2 _ + - - - , ~ ~ pb y*CL 2V

(A33)

Since the lift coefficient given by equation (A33) is all associated with the angle-of-attacktype of loading, it ca n be directly related to circulation through equation (A16), so that

(A34)

I t is now necessary to consider forces and moments due to interaction of the velocity components with the vortex system having the circulation distribution given by equation (A34). The velocity components are the rectilinear velocity V and the velocity(normal to the wing plane) py.

The interaction of the velocity V '.'lith the quarter-chord-line vortex results in awing rolling moment given by

S/2MX = - pVryy dy-b/2 (A35)or using equation (A34), nondimensionalizing, and taking the derivative with respect topb/2V gives

22

----- - - - ---


26/79

APPENDIX A

From the definition of y* in equation (1), it is seen thatC - 1 rrA ('1*)2lp - -"2 2 1/22+( A )+42 cos A

(A36)

Next the interaction of the rolling velocity py with the vortex system is considered. This velocity is normal to the wing vortex plane. Since flow through the vortexsheet is not permitted, only the interaction of the additional velocity py with thequarter-chord-line vortex and the extreme wing-tip chordwise-bound vortices need beconsidered.

The force caused by interaction of py with a chordwise-bound vortex is given by

or, with substitution from equation (A16),C l ) - -3 eC L a(fy) 1 ="8 PPyVccC L dy dy

This force is in the wing (or vortex) plane and is in the y-direction (perpendicular to thechordwise-bound vortices). One approach to evaluating this force fo r the wing-tip vortices is to perform the integration

C C l )-b/2 eC L a(fy) 1 = 2S pCPyVcCL d dy(b/2)-E Y

23

-----


27/79

APPENDIX Aor

This equation can be evaluated by parts, with the result

Y C ( ~ C l ) [ { ~ ~ ) Jb/2cCL Q' ~ C C )l~ e c ~ ~ ( b / 2 ) - E S

/2 (CCl) ( dC)- - c +y- dy(b/2)- eCL Q' dySince all the functions involved in this equation are regular and well-behaved in the indicated limits, it is apparent that A(;y)t will approach zero as approaches zero.Therefore, from the analysis made in this paper, the wing-tip vortices will contribute noresultant force or moment due to rolling velocity - that is, A(fy) 1 = O.

The next possibility to consider is the interaction of the rolling-velocity componentswith the circulation on the quarter-chord line. The associated force is in the wing planeand is perpendicular to the quarter-chord line. The magnitude of the force is

per unit of vortex length. This force results in a side force which can be written as

Fy =Vp pb y*ry tan A2Vper unit of span. A resulting side-force coefficient is given by

S/2 FyCy = dy-b/2 1.PV2S. 2Using equations (A34) and (A38) with (A39) and retaining up to first-order terms inpb/2V results in24 .

(A37)

(A38)

(A39)


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APPENDIX A

Cy = CL -_- L y*tan A dy*S ~ C C l ) bo cCL (l I 2VThe coefficient of side force due to rolling is found to be

which integrates to

- - p = an A -_- y*dy*y Sl(CC l )CL 0 cC L (l I

Cy--p =y*tan ACL (A40)

The force given by equation (A37) also results in a yawing moment which is given by

S/2 ( . )( fyhMZ = x /4 sm A - Y cos A-- dy-b/2 c cos Aor, proceeding as before to form the yawing-moment coefficient,

Integration of this equation results in

(A41)

25


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APPENDIX BCOMPRESSIBILITY EFFECTS

The effects of compressibility on the span loads and derivatives can be estimated byproper treatment of the local span load. As a first approximation , the load coefficient ata particular spanwise position is assumed to increase in direct proportion to the totalwing lift coefficient. At a constant angle of attack, therefore,

(CL)M _(c L) M=O - (B1)Note that this assumption indicates no effect of compressibility on the center of pressureof the span load.

A convenient equation for wing lift-curve slope, given in reference 14, is

(B2)

Combining equations (B1) and (B2) and letting ao assume its theoretical value of 27Tresults in

Equation (B3) can be used to modify the local span load and resulting derivatives toaccount for Mach number effects.

Sideslipping Wing

(B3)

Reference 17 shows that fo r a Sideslipping wing at an angle of attack in incompressible flow the local span-load parameter is given by

26


30/79

l \ : )-. J

(CCl) (CC ) (ce l)T ~ , " = or ,,(I + tan A) _ !l. c * ~ d 7 "4 dy*The Mach number perpendicular to the wing quarter-chord line is given by

MN =M cos(A - (3)Combining equations (B3), (B4), and (B5) results in

(Cc

l\ 2 + ~ ~ ) 2 J1/ 2c) = cos A + 4M,{3,CL 2 + ~ c o ~ A t ~ _M2cos2(A _,J }1/2 (C;rt(1 + tan A) -t+4

The span-load increment due to sideslip is found by differentiating equation (B6) with respehigher order terms in {3 (letting (3 approach zero). The result is

[2 ~ 1 / 2 2 + _A _ + 4 cCl~ C l ) = (cos A) ( c ~ I) tan A_l. c* d(T )"

c{3 M fI 2 J /2 \ C CL 4 dy*2 + ~ c o ~ ,J (I - M2COS2A) +J

[ 2 Il1/2}+ C ~ r ) " A2M2(tan A ) ~ + ( c o ~ A) +

2 1/2 2[ ( c o ~ A) (I - M2COs2A) + + [ ( c o ~ A) (I - M2cos2A)+


31/79

I t should be noted that for zero sideslip, equation (B3) reduces to

2 t2Cl)M,a= 2 + ~ ) +4(C l) a 2 ~ 1 7 2

2 + L ~ c o ~ A) (1 - M2cos2A) +Using relation (BS) with equation (B7) and dividing through by total lift coefficient results i

dLCCL\ A2M2tan A\cCr)M a [cc!..\ /C C l ~ = ( ~ C U tan A - c* d y * ' + ~ C J M , Q )2 2 2 2 + ~ (cC L i3 M \cCrjM,a _A _ _ A2M + 4 \C

cos A

The rolling-moment parameter due to sideslip is (from ref. 17)Cl i3 = _.! r1[ cClJCL 2 Jo ~ C L f 3 } y*dy* + 0.05

or, using equation (B9),

li3 1 1 cC l d -=--C(c ~ ~ C C l ) _ = __ 3 C L MCL M 2 SO (ECJM,Q tan A -"4 c* dy* ,a1y*dy* + 0.05

_.!S1(CCl) A2M2tanA2 0 \CCL M a l1 /2r 2, [ ( c o ~ f/ -A2M2 + 4J l + [ ( c o ~ ti) -A2M2 +

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


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Nco

The correction factor 0.05, as discussed in reference 17, is independent of aspect ratio orpressibility effects can be related to geometric changes by the Prandtl-Glauert relations, aeffect on the factor 0.05, it would appear that the factor is not a function of Mach number.

Performing the integration as was done for the incompressible-flow case yields (for a

(c {3 1 3 - 6 1 - ACL M = -"2 A(l + A) + y* tan A - A 1 + A 1 -* A2M2 tan A+O.05- 2 y 1 / 2 [~ c o ~ S-A2M2 + j L ~ c

Comparison of this equation with equation (A6) shows that

A2M2tan A1 / 2 )~ c o ~ J -2M2 +J 2 + [ ( c o ~ AY -A2M2

Yawing WingThe span load for a yawing wing in compressible flow can be determined in a manner

sideslip case. In yawing flight, however, the effective Mach number varies across the wingmal to the wing quarter-chord line is given by equation (All), and the corresponding Mach

MN = Mtos A - (y*cos A - x ~ / 4 sin


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t. )o Using equations (B3) and (B12) with equation (A20) results in

cClJ =\ C C ~ ) M , r , a t 1/)L2 + ~ c o h ) + j 0 * ~ * -X ~ / 4 tan ( ~ ) a -~ ~ ~ X ~ /{2 + A2 _ A2 M 2 ~ o s A_ b (y*cos A X ~ / 4 sin Aycos2A cos2A [ 2V , 'Differentiating equation (B13) with respect to rb/2V, letting rb/2V approach zero, and pslip case, results in the following span-load parameter:

C C c c ~ ) = (-y* + x ~ / 4 tan A) ( ~ C l ) _. ! ~ X * )2 _ ( x * ) j d ~ ~ ~ ) M , aL 2V cCL M 2 c/4 cM ,a+ ~ ~ C l \ A2M2 (-y* + x ~ / 4 tan A)

\CCJM a f( 1/2r, G c o ~ At -A2M2 + L2 + ~ c o ~ At -A2MThe rolling-moment parameter due to yawing is found by substituting equation (B14) into eqforming the integrations in the same manner as was done in the incompressible case . The


34/79

c,.,......

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

C Z r ~ U1(1 t 2A) 9 t a nA1 - \ 2 7 ( 1 - \ ) J ( ~ ) 2 (31-\t A=- + an - --+--- Y +---an -CL M 2 2A 1 + \ 4A2 1 + \ A 1 + \+ - y + -- - --- x y + +t tan A 9(1 - \) -* (tan A 3 1 - :+< -* 3x:c _2A(1 + \) A2(1 + \)2 2 A 1 + \ ac 2A(1 + \) 4A+ A2M2 _(y*)2tan2A + (Y*t (1 + tan2A) + x:cy*tan A

2 It ]1/2{ 1/2}~ C O ~ S-A2M2 +J 2 + ~ " c o ~ At -A2M2 + .Comparison of this equation with equation (A27) shows that

(CZr\ = (CZr\ + A2M2 - (y*)2tan2A + (y*)2 (1 + tan2A) + x=cy*\C L-; M \C L) M=O 2 1/2r It~ C O ~ S A2M2 + j t + ~ C O ~ Af -A2MRolling Wing

In compressible flow, equation (A28) becomes

( c z ) M , ~ l h)M,a,p = (cl) M,aC (cz)M,aJ


35/79

APPENDIX BThe compressibility correction to (CZ)M,P is based on the aspect ratio of a wing semispan, as discussed previously; therefore, from equation (B2),

and

(B17)

I t can be shown that equation (B16) becomes

(B18)

In rolling flow the velocity perpendicular to the wing quarter-chord line is given by

and the corresponding Mach number is1/2

MN = M ~ o s 2 A + ( ~ ~ t ( Y ' ) ~ (B19)Substituting equation (B19) into (B18), differentiating with respect to pb/2V, and lettingpb/2V approach zero results in the span-load parameter

32


36/79

APPENDIX BThe circulation for the rolling wing in compressible flow is found by use of equa

tions (A16), (B18), and (B19), and is

(B20)Substitution of equation (B20) into (A35), nondimensionalizing, taking the derivative withrespect to pb/2V, and letting pb/2V approach zero results in the rolling-momentparameter

(B21)

Performing the integration results in

(B22)

Comparison of equations (A36) and (B22) shows that

(B23)

The side force due to rolling is found through use of equations (A38), (A39), and(B20), and is (for pb/2V approaching zero)

Cy- - p =y*tan ACL (B24)

33


37/79

APPENDIX BComparison of equations (B24) and (A40) indicates that there is no compressibility effecton the parameter Cy p CLo Similarly, it can be shown that there is no compressibilityeffect on the parameter Cnp/CLo

34


38/79

c:..:I01

APPENDIX CSUMMARY OF EQUATIONS

General Equations for Additional Local Span LoadingSideslipping wing:

d ~ C C l ) A2M2tancC ccL M,a _ l_ 1/2( ~ \ = ( ~ C l ) tan A - 4 c ' dy' + ~ C J M , a ~ ) 2 j f~ C L i J ) M \!,CL M,a _ A2M2 + 4 2 +'cos A

Yawing wing:

(ec ) ~ c e l )CC lrb = (-y* + X ~ / 4 tan A) ~ ~ C l ) _. ! ~ x * )2 _ ~ x * ) ~ d \ C C L M,aL 2V eCL M 2 \ c/4 CM ,a

+ ( ~ C l \ A2M2 (-y* + x ~ / 4 tan A)\ C C ~ ) M a It 1/2r' ~ O O : S _A2M2+j t + ~ C O ~ S -A2M2

Rolling wing:)

1TAy*C lCl __?c pb - (EC L M,a \1 f-L)L< A 2 ~ 2 + 4\ 2V)M,a 2 + 4\008 A


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c..:>0) Equations for Additional Local Span Loading fo r Swept Wings

Sideslipping wing: 8eeeel ~ ~ e e l ) 3 G 3 cC l M Q!- - = -- ta n A - 1 - (1 - .>t)y* 'cC L {3 M eC L M,a A(l + .>t) dy*

f.eCl )+ ,eCL M a It ]1/2r" 2' ~ c o ~ S -A 2M2+ J L 2 + ~ C O ~ A ) -A 2M

A2M 2tan A

Yawing wing:

( ~ C C C ~ \ = _ {Y* - y* -dtan A x ~ J t a n ' : : \ ( ~ e l ) L 2 ~ ) M j eCL M,a

2 G J{3 (-*) 3 * 9 ~ 1 - (1 - .>t)y* - Y - y* tan A - - x - 1 - (1 -A(l + .>t) 2 2 ae 4A(1 + .>t)

Rolling wing:

C C l ) _A2M2&* - [(y* -d tan A - x ~ ~ t a n + -C { L M,a 2 1/2 2 1/~ c o ~ A) - A2M2 + 2 + ~ c o ~ A) - A2M2 + j

)el _ _l y* 1/2(c Pb) - eC L M,a 2 + G . ( ~ ) 2 _ A2M2 + J2V M L4 \COS A 4 J


40/79

C.:I-l

General Equations fo r Aerodynamic DerivativesSideslipping wing:

1 d-V 1 eC le L \ l 11 A2M 2ta' 1( ; ~ 8 M ' . tan. -t " ' ~ Y ' M'Y'dy' + 05 -., 0 ;;CLM,. ~ , : , A)' -A'M' +rYawing wing:( ~ ~ t ,%f ~ Y ' -x;/4 t a n ' ) ( ; ~ 8 M ' . + ~ ~ X ; / 4 ) ' - ( X ; ) r ~ ) M + ' d Y '

Rolling wing: 11 ce l 2 y * - x ~ / 4 t a n A + - - A M2, 0~ ' C J M ' . ~ , o ~ S-A'M' +r , +~ , o ~ .) ' - A'M'+(C'pk -% ( F" r r'( , ~ ~ L (y


41/79

00 Equations for Aerodynamic Derivatives of Swept WingsSideslipping wing:

A2M2tanCZ(3 - - . ! ~ 3 + y* (tan A - ! i : + 0.05 - y* U )2 ~ 1 / 2 { ~C - 2 A(l + A) \ LA- _ 2M2 + 4 2 +L \cos A

Yawing wing:

C - 1CL - '2(1 + tan2A) 9 1 - :\. t A 27 - :\.)J( -*) 2--- an +--- y2A 1 + :\. 4A2 1 + :\.(3 1 - :\. t A t a n 2 A ~ (-*) 2 tan A 9(1 - :\.) ~ - * (tan A 3+ --- an - -- y + - - y + -- - -A1+:\. 2 2A(1+:\.) A2(1+:\.)2 2 A3 x ~ c 9 A2M2 -(y*)2tan2A + (Y'*) 2 (1 +tan2A

+ 2A(1 +:\.) + 4A2(1 + :\.)2 +-2 2 j1/2{- A 2M2 + 4 2 + ( ~ )cos A \-COS A

-- - - . - . - - .


42/79

C...:lCD

Rolling wing:

- ~ - - - . .- - - - - .-.-

Cl - 1(-*)2--"2 Y rrA2 + A )2 _ A 2M2 J /2L\2COSA -4-+J

Cyp =y*tan ACL

~ Y * ) 2 + ~ Y * ) 2 _ (Y*) ~ t a n 2 A + x ~ c y * t a n j


43/79

REFERENCES

1. Jones, Arthur L.; and Alksne, Alberta: A Summary of Lateral-Stability DerivativesCalculated for Wing Plan Forms in Supersonic Flow. NACA Rep. 1052, 1951.

2. Margolis, Kenneth: Theoretical Lift and Damping in Roll of Thin Sweptback TaperedWings With Raked-In and Cross-Stream Wing Tips at Supersonic Speeds. SubsonicLeading Edges. NACA TN 2048, 1950.

3. Harmon, Sidney M.: Stability Derivatives at Supersonic Speeds of Thin RectangularWings With Diagonals Ahead of Tip Mach Lines. NACA Rep. 925, 1949. (Supersedes NACA TN 1706.)

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

5. Pearson, Henry A.; and Jones, Robert T. : Theoretical Stability and Control Characteristics of Wings With Various Amounts of Taper and Twist. NACA Rep. 635,1938.

6. Weissinger, J.: Der schiebende TragfHigel bei gesunder Stromung. Bericht S 2 derLilienthal-Gesellschaft fUr Luftfahrtforschung, 1938-1939, pp. 13-51.

7. Multhopp, H.: Methods for Calculating the Lift Distribution of Wings (SubsonicLifting-Surface Theory). R. & M. No. 2884, Brit. A.R.C., Jan. 1950.

8. DeYoung, John: Theoretical Antisymmetric Span Loading for Wings of ArbitraryPlan Form at Subsonic Speeds. NACA Rep. 1056, 1951. (Supersedes NACATN 2140.)

9. Falkner, V. M.: The Calculation of Aerodynamic Loading on Surfaces of Any Shape.R. & M. No. 1910, British A.R.C., Aug. 1943.

10. Simpson, Robert W.: An Extension of Multhopp's Lifting Surface Theory to Includethe Effect of Flaps, Ailerons, etc., Rep. No. 132, ColI. of Aeronaut., Cranfield(Engl.), May 1960.

11. Weissinger, J.: The Lift Distribution of Swept-Back Wings. NACA TM 1120,1947.12. Toll, Thomas A.; and Queijo, M. J.: ApprOXimate Relations and Charts for Low

Speed Stability Derivatives of Swept Wings. NACA TN 1581, 1948.13. Queijo, M. J.; and Jaquet, Byron M.: Calculated Effects of Geometric Dihedral on

the Low-Speed Rolling Derivatives of Swept Wings. NACA TN 1732, 1948.14. Polhamus, Edward C.; and Sleeman, William C., Jr.: The Rolling Moment Due to

Sideslip of Swept Wings at Subsonic and Transonic Speeds. NASA TN D-209, 1960.

40

l


44/79

15. Campbell, John P.; and Goodman, Alex: A Semiempirical Method for Estimating theRolling Moment Due to Yawing of Airplanes. NACA TN 1984, 1949.

16. Anon.: Data Sheets - Aerodynamics. Vols. I, IT, ITI, Roy. Aero. Soc.17. Queijo, M. J.: Theoretical Span Load Distributions and Rolling Moments for Side

slipping Wings of Arbitrary Plan Form in Incompressible Flow. NACA Rep. 1269,1956. (Supersedes NACA TN 3605.)18. DeYoung, John; and Harper, Charles W.: Theoretical Symmetric Span Loading at

Subsonic Speeds fo r Wings Having Arbitrary Plan Form. NACA Rep. 921, 1948.19. Campbell, George S.: A Finite-Step Method for the Calculation of Span Loadings of

Unusual Plan Forms. NACA RM L50L13, 1951.20. Gray, W. L.; and Schenk, K. M.: A Method fo r Calculating the Subsonic Steady-State

Loading on an Airplane With a Wing of Arbitrary Plan Form and Stiffness. NACATN 3030, 1953.

21. Diederich, Franklin W.; and Zlotnick, Martin: Calculated Spanwise Lift Distributions,Influence Functions, and Influence Coefficients fo r Unswept Wings in Subsonic Flow.NACA Rep. 1228, 1955. (Supersedes NACA TN 3014.)

22. Diederich, Franklin W.; and Latham, W. Owen: Calculated Aerodynamic Loadingsof M, W, and A Wings in Incompressible Flow. NACA RM L51E29, 1951.

23. Jacobs, W.: Systematische Druckverteilungsmessungen an Pfeilfliigeln konstanterTiefe bei symmetrischer und unsymmetrischer Anstromung. Bericht 44/28,Aerodynamisches Institut der T. H. Braunschweig, Nov. 8, 1944.

24. Fisher, Lewis R. : Approximate Corrections fo r the Effects of Compressibility onthe Subsonic Stability Derivatives of Swept Wings. NACA TN 1854, 1949.

25. Bird, J. D.: Some Theoretical Low-Speed Span Loading Characteristics of SweptWings in Roll and Sideslip. NACA Rep. 969, 1950. (Supersedes NACA TN 1839.)

41


45/79

x--0>c::.-,

Q) 0L :l U"'0

,'-Ol Q)c: --:.0 0g :l0

Olc:

42

(/'lU-0>\ Jc:::l0Ll,(/' l

"'00L :U

' \"\

c:nc:


46/79

""")

/

v

rWing leading edgeQuarter-chord-l

Chordwise-bou

I"" Trav

Figure 2.- Comp lete vortex system for a wing in sideslip.


47/79

"'"'"

Quarter - chord-I ine vorteChordwise-bound

D


48/79

C]1

Center ofgravityAerodynamiccenter

x\

I .

-v

fX ac

y 1c"-4

Figure 4.- Details of wing geometry and wind velocity components for yawing win g.


49/79

:I 1 1 Em 1 1 1 1 1 130 1 1_5

t--- 04

y*_3

:Ir-----r-----,I ~ [ 1'---'----'11'---'----'11r----r----11I ~ I 1 ~ 3 0 1] 1 1 1 1 1 1 1 1 1 1 1 1451 1

:I 1 1 1 1 1 1 1 1 1 1 1 \601 10 I 2 3 4 5 6 7Aspect ratio, A

(a) II '" O.Figure 5.- Spanwise location 01 centroid 01 angle-ol-attack loading.

46


50/79

- - -- - - - _ - ___

A,deg

:I I I I I I I I I I I I 1451 I.5

--30

.3

y:I...---------r-- I...------r---II ~ I I ~ I 1"'----'----1 r----.------II r---.-------. I I-:I I 1 I I I I I I I I I 130 I I:I I I f I I I II 145/ I I I I:I I I . [ I I I 161 I I I I I Io I 2 3 4 5 6 7

Aspect ratio, A(b) A = 0.25.

Figure 5.- Continued.

47


51/79

.5A, deg

4-45

.3

r . 5 1 ~ I ~ I ~ I O 4] ~ I I ~ ~ I...----r------"II ~ I I ~ I I ~ I 1r----r----f30I::I I I t I I I I I I I I 45 1 I:I II BE j601 II IIJo I 2 3 4 5 6 7

Aspect ratio, A(e) A = 0.5.

Figure 5.- Continued .

48

--- - -


52/79

y.:I 1 I I I I I I I I I I I 0 I 1:I 1 1 1 1 1 1 1 1 1 I ] 30 1 1:I 1 1 [tEfft] 1 1 1 4 5 1 1:I 1 1 FFffl 60 1 1 I I I I I0 I 2 3 4 5 6 7Aspect ratio, A

(d) A= l.0.Figure 5.- Continued.

49


53/79

:I 1 1 1 1 1 1 1 1 1 1 1 1-301 1:I 1 [ 1 I 1 1 1 1 1 1 10 1 1 J

y*

:I I tHfl 1 I I l ~ 1 1.5 -! - - - -! - - - - ....- 45--r -4

:I-------o-- E t E ~ 1 6 0 Ir----r----il r----r-----I r----r----ll1o I 2 3 4 5 6 7Aspect ratio, A(e) A = 1.5.

Figure 5.- Concluded.

50

-- . .


54/79

- ---

.51 l i t b 1 1 1 1 1 I ~ ' : ~ I4 ~ ' - - - - - - ' - - - - ' - - ~ ' - - - - - - - - I . . . . -:I 1 1 Fi 1 1 1 1 1 1 1 l30 1 1:I 1 [ 1 1 1 1 1 1 1 1 1 1011

y*

:1 1'---'---[1"---'---11"---'---11"---'---1 r-----r---I '------'-----"301

:I 1 1 [ 1 1 1 1 1 1 1 1 145 1 I:I 1 1 [ 1 1 1 1 1 1 1 1 1

601 1

'0 I 2 3 4 5 6 7

I

Aspect ratio, A(al A = O.

Figure 6.- Spanwise location at radius at gyration tor angle-at-attack span load.

51


55/79

y*-

52

:1 1 1 1 1 1 1 1 1 I I ! 1-30 1 I:I 111111111111 0 8:Ir--r-- F==F==I r==r=-Ir---r--I~ 1 3 0 1:I I I I 1 1 1 1 1 J 1 1 1451 1:I 1 1 [ I .I I J I I I I 160 I Io I 2 3 4 5 6 7

Aspect ratio, A(b) A =0.25.


.


56/79

A,deg

:I 1 1 [ 1 1 1 1 1 1 1 1 1- 451 1:I 1 1 [ 1 I 1 1 1 1 1 1 1-30 I 1:I I 1 1 1 I 1 1 1 1 I 1 10 1 1"'*Jr----r----II r----r----II1'----'----'11'----'----'11r-----r------11 .------r-----!I1 ~ 3 0 1

:] 1 I 1 1 I I 1 1 I I 1145

1 1

:I I I [JJJd I I I I Iwl 1'0 1 2 3 4 5 6 7Aspect ratio, A(c) A =0.5.

Fgure 6.- Continued.

53

- -


57/79

A,deg

.5 1 I I 1- -+ - - -+ -1I---+---+-I--+--+--1-4 - - - -+ - -11-45 1 I4:I---r-Ir-r--I r====t===1 F===+===I I ~ I j r--r- i_30 I I] I I I I I I I I I I I 1

0

1'1*

:1- - - - - - - - r - - -..----r---t I..----------r--I'---------'----I I..----------r--I'---------'----I ~ . . - - . - - - - - - - - . 3 0 I

0] I I 0 I I t I I 1 , 45 1 I:I I I Elff1 I I I I 160 1 I' 0 I 2 3 4 5 6 7

54

Aspect ratiO, A(d) ]x = l.0.


. ~ _ _ _


58/79

11., deg

61 I I I I I I I I I I I I I I.5y*:"--------'--1..------r---Il r----r----I r----r----II --r- jr----r-----130I] I I [ i l l t I I I I 1451 I.61 I I kFEfj I I I I 160 I I50 I 2 3 4 5 6 7Aspect ratio, A

(el A =1.5.Figure 6. - Concluded.

55


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A,deg2

-45I--- ' 30

0 V -V ---I-- 0/ ' 30- I - - - r-2 V I- 45l--v 604 !--V-.6 (0) x=o.

A, deg.2 - -45--- -30/ ----'. / --- 0o ,/ .....-- 30. / -V , / ......---I--".,/ - 45---:2 / ' ,/ ......---V 60L-- ....

-4

V V-.6(b) X=0.25.

2 3 4 5 6 7Aspect ratio, AFigure 7.- Variation of C [ ~ / C L with aspect ratio, sweep, and taper ratio. M = 0; x ~ c = o.

56


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CZ{3CL

.2

o

-4

-;8

.4

0

-4

-;8

-1.2o

III

/V

I--I ----V - I - - -V V J,..---/ . / -/ 0' I--f- -V,-V I-- I--/v V f--"./ L.--V -V /v/ ......-

. /f...-- 60/

(c) x= 0.50.

-L-;: .--. / " .--- - >----V ...--I-- f--"t:=- I - - f- - 60V V I--. /(d) X=1.0.

2 3 4Aspect ratio, AFigu re 7. Continued.

Adeg-- 45- -30

030-45

A, deg-45-300

3045

5 6 7

57


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58

.4

o

-.8

-1.2o

/II

I---"'"""

V V - f - - !----I - - -r;/V I - -V r-f.--

V f.-- 60....--V(e) A= I. 5.

2 3 4Aspect ratio, A

Figu re 7.- Concl uded.

A,deg-45 ; - --30 r---

030 r---45 r--

5 6 7


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"" ...........r-- A,deg---3 '" 60- . 0 ............-30 -- t---- 45-45 r----r-- r - - 30C l rC .2L .1o (0) A=O.

.6

.5 \"-."- A,deg-..-4 \ 60\ "'-3 ' \ "- ..........r--r--- 45"- --.r-- t--- 30.2

r-- 0-1 T 30 -L -45(b) A= 0.25.2 3 4 5 6 7Aspect ratio, A

Figu re 8.- Variation of C1r/CL wi th aspect ra tio, sweep, and taper ratio M= 0; Xac = O.

59


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.8

\'" A, deg.6 0 '" "'-....I'-..-\ 60'-.....Cz rC .4L "-..... ----- - 45-30 -r.:::::: -- 30 '---45 t------.2o (c) A =0.50.

1.0

i - - - 0 ' \\ '"\ \ ""'" A,deg........'\ 60""-'" I"--......r--I - - -.8

.6

.4

30 r-- - 45t---45 r- 30'" t-- i - - ----2 (d) =1.0.2 3 4 5 6 7

Aspect ratio, AFigu re 8.- Conti nued.

60


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1.2

.8

4

01\'"30-45

(e) X =1.5.

r---- A, deg-...:::::: 60

r--.. r--r-- - 1 0 - 45f'-.- I- .t - - 30- - I -

2 3 4 5 6 7Aspect ratio, AFigure 8.- Concluded.

61


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62

Cz P

o

-.1

-.3

-.4

o

-:1

-.2

-.3

-.4

-.5o

A,deg0 -45 !------0 -:30

r---.. 0 01':::. :30 ----- b. 45r--. D 60--I--h -I----r----b......""" ---- - PI--- I-- fa- I-- t::..--P(0) ),. =0.

I:'--.."" -- roi'-.. -------- I.nk\. ......... -.......... -(b) ),.= 0.25.

2 3 4 5 6 7Aspect ratio, A

Figure 9. - Variation of Clp with aspect ratio. taper ratio, and sweep.

-- -- '--- - ----- --


66/79

Cz P

o

-.1

-4

-.5o

-.1

Cz p

-.4

'"

;::-.....

2

r---.....I---r---.....

(c) A =0.50.

,

'" I----

(d) A= 1.0.3 4

Aspect ratio, AFigure 9.- Continued.

10 A, deg-450 -30 0l::,. 30t::.. 45D 60

t--- r-- -f.oI-- -at--- L..........1-

A, dego -450 -30 0D. 30t::.. 45D 60

I--. t---t--- \0--- i'-.......kJP:::......."

k->5 6 7

63


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64

o

-.1

-.2

Cz -.3p

-4

-.5

-6'0

1&

2

t'--...r-....

(el A = 1.5.3 4

Aspect ratio, A

Figure 9.- Concluded.

A,deg0 -450 -30 06 30

45D 60

-.............r--I'--- t::::::::::-............

iCt

t.....'-......1

5 6 7


68/79

- - - ~ ... ._ _. _

.8 A,deg' - - --6045 c---30

o 0-30 r---45-.4

-.8 (a) X = O.

1.2 I IA,deg

8 60

4 4530

o 0-30 -

..:4 -45

(b) X=0.25.2 3 4 5 6 7

Aspect ratio, AFigure 10 .- Variation of CYp/CL with aspect ratio. swee p. and taper ratio. M= ; x:c = O.

65


69/79

66

1.2

.8

.4

o

-.8

1.2

.8

4

o

-4

-8'0

A,deg

60

I(c) A=0.50.

A,deg- 60

(d) A = 1.0.2 3 4 5Aspect ratio, AFigure 10.- Continued.

..

45300

-30-45r---

45300

3045 -

6 7


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1.2 A,deg60

.845

430

o 0-30

-4 - 4!: -

(e)A=1.5.2 3 4 5 6 7

Aspect ratio, AFigure 10.- Concluded.

67


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68

o-:1

-:3

o-.1

-:2

-.3o

A,deg \ -300 t....-

45

60

2

Ir-45j

30 J(a) X=O.

A,deg

0 f---P--301-45 L 304560

(b) x : 0.253 4 5 6 7Aspect ratio, A

Figure 11 .- Variat ion of Cnp/CL with aspect ratio, sweep, and taper rat io. M= 0; xac = o.


72/79

-

o .A, deg--1

\ -30 ~ - 4 5 Or-- \ 30

t-45:2 60(c) ~ : : 0 . 5 0 .

-.3

o A,deg-30 -

0 :1 45 ,.30} 45 I - - - -:21- r--f--.

{ d ) ~ = I . O -- 60-:30 I

A, deg II -OlI30 / r- 30

I45 f - - -

-45--:2

(e) :: 1.5. 602 3 4 5 6 7

Aspect ra t io, AFgure 11.- Concluded.

69


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- - - -Expe r imen t , ref. 14--- Theory, ref. 14-- ----- Theory, eq. (2)

1.8 IA=4 II A=6r----- A= 3.6 A=45/ /A=0.6 X =0.6 //1.4 V..- / V-- 1..// ..----_/ --.0

r----- A=4A = 32.6 / / A=4X = 0.6 / j A= 45X= 10 /1.4

10..-/1- ---- ..- VI - ~ ---V

4 I

(CZf3/CL)M 1.4( CZf3/CL)M=O 1.0

r--- A=4 A=4A= 45 A=45 // X=0.3X= 0.6 / /r----- ..- ---.- V ..-- : ; / ..- ve--- -- -1.4 A=2.31

A=4 A =52.4r--- A= 60 X= 0)..=0.6 ~ . . - ---::;::::: = ' ~ I - - V---.0.6

A=4A=2 A= 36.9 ,X=O /A= 45 ./I!r----- X= 0.6 -- ..-- - - -..-I" f-

l.4

4 .8 1.2 o .4 .8 1.2M MFigure 12 .- Experimental and calculated efects of compressibility on C L '

70


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Cz r

Cz r

.6

(a) A = 2 .6/ (c) A= 2.61r - - - A= -45 .A. =45A= 0 A= 1.0--" ----- 1:'\ - ---

.2

-- 0 - I ----.-_ (:'1- - i.==-- ! - - --oEq . (3)------- Theory of ref. /20 Data, ref. 12.6

l7/(b)A=5./6 (d) A = 1.34 / Vr - - A= 0 A=60 /A : 1.0 A:: 1.0 /-- / --6-- 1(:'1 (D __ -.2 --- .- I--I--- L G) --- --...: r---1-0- - - ~ .2 .6 o .2 4 .6

Figure 13.- Theoretica l and experimenta l va riat ions of C1r with CL for several representative wings in incompressible flow.

71


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1.8

1.4

1.0

1.4

1.0

1.0

1.4

1.0

1.4

1.0o .2

A=4A= 0A = 1.0 /

-- q. (3)- ----- Ref. 24I

A=41\.=45A= 1.0b=-=""-=

A=4A= 60A= 1.0

A=6A=45X= 1.0

A=21\.= 45A= 1.0

-::-:::-:-:=- ::::-::-=4 .6

Mach nu mber, M

LL VV

..-:::::V " -""-

---

----- -

V...---

.8

Figure 14.- Estimated effect of Mach number on the parameter Ctr {CL'

1.0


76/79

1.6

1.2

.8

4

o1.6

12

- pbc 2V 4

o2.0

1.6

1.2

.8

//

/V

J/II

...-..../ \II

/ ...-...." II/- \ // A= 1.5 \ / A= 3.5A =45 A= 600.5 V = 0.5Lc----.

r\ Eq . (4) / '1 - - - - - - -Ref. 25 / - "- \j I I ,II LII jA = 3.5 I A = 3.5A= 45 A= 45= 0.5 I ~ = O

' / \j / \\1//" A--,i V/ - , ~ jt 1/

I f IA=6 I A = 3.5A =45 A= 0= 0.5 / 0.5II4 .8 1.2 o 4 .8 1.2

y* y*

Fi gure 15.- Computed span- load distri bution due to ro lling ve loc ity. M = O.73


77/79

o

-.1 I!,....:2

Experiment Eq.(5) A A,deg 00 g 2.61 -45 , - . /0 5.16 0 ....4 0 ' 2.61 45fl. l: ! 1.34 60-.5

2 3 4 5 6Aspect ratio, A

Fgure 16.- Ca lculated and experimenta l va lues of damping-in-roli parameter Cl p' M = 0; A = 1.0.

4


78/79

-.Jc:.n

__- _

--------- ---- ~ - - - - - - -

......~ ~(c) A- - 0 I:) 0

CYp 0A-

- [ ~ [ ; = t - - - t - - t - - + - ~ - - ~ - - r _ _ t -r- _ ... -... I IA - 261 I I(0) A : -450 . Eq. (6)I--- = 1.0 I--- ______ Theory of ref. 12r-- (d) Ao Data, ref. 12 Ar I " I(b) A =5.16

A-a- 1.0 ~ V '

-!4

.8

!4

L "-..... ....I IyP 0

- ---

-4o .2 4 .6 o .2CL CLFigure 17. - Theoretical and experimental variations of CyP with CL for several representative wings in inco


79/79

.2

:J .

-.2

.2

Cnp 0 I..:)===:.

o

( ) A =2.61A= -45 0). = 1.0["- t- .-r - - r - -

(c) A=2.6 1A= 45). = 1.0

r - - f0- - - -t ---JP--.2 4

(b) A= 5.16A=O).= 1.0

1:'1 1':.'l.' (D -- t - -r--- -

Eq. (7)---- - - - Theory of re f. 12

0 Data, ref. 12I I(d) A = 1.34A= 60),=1.0 0

- - r---o- .- - - - - . --t - - t-- ---- I - -.6 o .2 4 .6

Fi gure 18.- Theoretical and experimental variat ions of Cnp wit h CL fo r seve ral represen tative wings in incompressible flow .

nasa report - theory for computing span loads and stability derivatives due to side slip, yawing,...

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