1、LOAN COPY: RETURN TO KIRTLAND AFB, N MEX AFWL (WLIL-2) THEORY FOR COMPUTING SPAN LOADS AND STABILITY DERIVATIVES DUE TO SIDESLIP, YAWING, AND ROLLING FOR WINGS IN SUBSONIC COMPRESSIBLE FLOW * * . -3,; . by M. J. therefore, much effort has been expended in developing methods of predicting the aerodyn
2、amics of lifting surfaces. Most of this effort, however, has been directed toward determination of aero- dynamic characteristics associated with angle of attack. Aerodynamics associated with other aircraft motions (rolling, yawing, pitching, and sideslipping) have been investigated to a lesser degre
3、e, and generally by somewhat cruder methods. This is particularly true for the low-speed regime. For supersonic speeds the nature of the governing equations is such that it has been possible to obtain equations for loads and aerodynamic derivatives for wings performing various modes of motion. (See
4、refs. 1 to 4, for example.) In cer- tain limiting cases it is possible to use supersonic theory to predict subsonic character- istics. (See ref. 4, for example.) * A preliminary version of the material presented herein was included in a disserta- tion entitled “A Theory and Method of Predicting the
5、Stability Derivatives CzP, C1, Cnp, and fulfillment of the requirements for the degree of Doctor of Philosophy in Engineering Mechanics, Virginia Polytechnic Institute, Blacksburg, Virginia, June 1963. Cyp for Wings of Arbitrary Planform in Subsonic Flow,“ offered in partial Provided by IHSNot for R
6、esaleNo reproduction or networking permitted without license from IHS-,-,-There are a number of problems associated with attempting to predict aerodynamic characteristics of wings performing the possible modes of motion. One problem is that of finding an adequate mathematical model for the wing, and
7、 the second is that associated with solution of the equations which arise from use of the mathematical models. Various methods for predicting certain aerodynamic characteristics of unswept wings have been developed by a number of investigators, and numerous reports have been published from which cer
8、tain characteristics can be obtained for specific wings. (See refs. 5 and 6, for example.) swept wings. These are: Three general approaches have been used in determining the aerodynamics of (a) Computations based on mathematical models associated with the use of vortices, doublets, or other concepts
9、 to represent the wing (refs. 7 to 11, for example). (b) Determination of approximate equations based on treating each wing semispan The fictitious “unswept“ panels are skewed to simulate as one-half of an unswept wing. a swept wing (refs. 12 to 15, for example). (c) Development of design charts bas
10、ed on tests of a great number of wings with various sweep angles, aspect ratios, and taper ratios (ref. 16, for example). The first of these approaches is generally difficult and in some cases involves the solution of numerous simultaneous equations. parameters primarily aerodynamic-center position,
11、 CL, and C ) have been attacked by fairly rigorous methods. The use of high-speed computers to solve many simultaneous equations has permitted the numerical solution of equations better defining the wing bound- ary, but solutions generally have been obtained for angle-of-attack loading. For these re
12、asons, only a few aerodynamic ( IP The second approach has been quite successful in predicting trends, and with some modifications has been used to obtain good quantitative results for certain aerodynamic characteristics. (See ref. 15, for example.) The third approach is adequate for engineering dat
13、a, provided the available data envelop the range of geometric variables of interest. Unfortunately, the amount of data available for some of the wing derivatives is very limited because of the scarcity of experimental facilities for determining such derivatives. The purpose of the present paper is t
14、o examine the problem of wing characteristics in subsonic compressible flow and to develop a consistent method for computing these characteristics. The method developed herein is a new approach for estimating span loads and the Cnp, and CY for wings of arbitrary planform in sub- derivatives CIp, Cz,
15、 Czp, sonic compressible flow. It is based on a vortex representation of the wing which was first developed by the author for sideslipping wings in incompressible flow (ref. 17). P 2 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Results are general
16、ly applicable to the low angle-of-attack region, where the various wing characteristics vary linearly with angle of attack or lift coefficient. SYMBOLS A aspect ratio, b2/S a0 two-dimensional lift-curve slope b wing span L wing lift coefficient, - 12 CL 2PV s wing lift-curve slope, - “L, per radian
17、a, MX rolling-moment coefficient, -pv 12 sb cz 2 c =- P pb a- 2v aCz c =- lr rb a- 2v MZ 12 -pV Sb 2 yawing-moment coefficient, Cn 2v FY side-force coefficient, - CY 12 -pv s 2 3 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-wing local chord wing a
18、verage chord Section lift section lift coefficient, TPV 12 c acl a 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
19、for incremental section lift due to sideslip, for a wing at angle of attack parameter for incremental section lift due to rolling parameter for incremental section lift due to yawing, for a wing at angle of attack wing root chord wing tip chord spanwise distance from wing root chord to center of rot
20、ation of yawing wing side force Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-S U V VN V *7y X7Y side force associated with a chordwise-bound vortex side force associated with quarter-chord-line vortex lift lift per unit length of chordwise-bound v
21、ortex lift per unit span of quarter-chord-line vortex lift per unit length of quarter-chord-line vortex free-stream Mach number Mach number of free stream normal to wing quarter-chord line rolling moment yawing moment rate of roll, radians per second yawing angular velocity, radians per second wing
22、area wind velocity in x-direction free-stream wind velocity relative to wing center of gravity local velocity free-stream wind velocity normal to wing quarter-chord line wind velocity in y-direction longitudinal and spanwise reference axes, with origin at center of gravity distances along reference
23、axes 5 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-1ll11 I I I l1111l1ll1l ,1l111l Il11111111ll11ll11ll I , , , . . - . Xac XC xc/4 - Y - Y CY P rY rt 6 E A x 5 P chordwise distance between aerodynamic center (a.c.) and moment center (or center o
24、f gravity, c.g., in flight), positive when c.g. is upstream of a.c. x-distance to wing trailing edge x-distance. to quarter-chord line spanwise position of centroid of the angle-of -attack span loading radius of gyration of angle-of -attack span loading angle of attack (or incidence), radians sidesl
25、ip angle, radians circulation strength related to spanwise position circulation strength related to displacement along quarter-chord line local effective sideslip angle due to yawing, radians infinitesimal displacement in spanwise direction sweep angle of quarter-chord line, radians wing taper ratio
26、, Ct/cr distance along the quarter-chord line, measured from wing root chord mass density of air Subscripts: M compressible flow M=O incompressible flow P part due to rolling 6 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-r part due to yawing a! p
27、art due to angle of attack P part due to sideslip A star (*) indicates that the quantity has been nondimensionalized by division by b/2; for example, y* =L. b/2 VORTEX SYSTEM The vortex system used in this analysis is for essentially a modified lifting-line- theory approach, and is illustrated in fi
28、gure 1. The system consists of a bound vortex along the wing quarter-chord line and a bound-vortex sheet from the quarter-chord line to the wing trailing edge. tion of the airstream. and proved to be quite accurate in predicting the parameter wings in subsonic, incompressible flow. that it would be
29、applicable to the estimation of other wing derivatives. Behind the wing trailing edge, the vortex sheet is in the direc- This system was applied to sideslipping wings in reference 17, for a wide range of An appraisal of this vortex system indicated C The vortex system adopted for this study allows t
30、he possibility of lift generation by the 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. T
31、he 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 the strength distribution of the quarter-chord-line vor
32、tex; therefore, the lift distribution of the wing can be determined if the vortex strength distribution of the quarter-chord line is known and if the wind velocity components are known. The distribution of the wind veloc- ity components relative to the wing can be determined easily for each possible
33、 motion of the wing. The basic problem in determining wing load distribution, therefore, is that of determining the vortex (or circulation) distribution for all wing motions. The three types of motion considered in this paper are sideslip, yawing, and rolling. 7 Provided by IHSNot for ResaleNo repro
34、duction or networking permitted without license from IHS-,-,-Side slipping Wing The sideslipping wing was studied in reference 17. It was argued therein that the circulation distribution in sideslip was essentially the same as that for a wing in zero sideslip. Thus zero-sideslip circulation distribu
35、tions could be used for estimating aero- dynamic loads and the parameter C for wings in sideslip. The vortex system for the wing in sideslip and the direction of the airflow are shown in figure 2. ZP Yawing Wing The bound-vortex pattern of the yawing wing is the basic pattern shown in figure 1. The
36、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 a lateral velocity component resulting from the flow curvature, and that the magnitude of the component is a function of position on
37、 the wing. to be in sideslip, with the effective sideslip angle varying over the wing. The arguments presented with regard to circulation distribution of the wing in sideslip can be carried over to the yawing wing. , An assumption of the present theory is that the circulation dis- tribution for a ya
38、wing wing is essentially the same as that for a nonyawing wing. The wing therefore can be considered Rolling Wing The 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 local geometric an
39、gle of attack is increased by PY A=- V 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 antisymmetric angle-of -a
40、ttack distribution associated with rolling velocity The assumed wing vortex system and the circulation distributions that have been discussed form the basis for the present theory for the computation of wing aerodynamic characteristics. a Provided by IHSNot for ResaleNo reproduction or networking pe
41、rmitted without license from IHS-,-,-PRESENTATION OF RESULTS General Remarks General equations are derived in appendix A for the span loads and certain aerody- namic characteristics associated with sideslipping, yawing, and rolling for wings of arbi- trary planform. These equations are applicable in
42、 the low angle-of -attack region, where the characteristics vary linearly with angle of attack and lift coefficient. associated with sideslip, yawing, and rolling can be obtained by the methods presented herein only if the angle-of-attack span load is known. Such information is available for a wide
43、range of wing geometry (ref. 18, for example). The angle-of-attack loading for odd-shaped wings can be obtained by application of the horseshoe-vortex method of ref- erences 19 to 22. Span loads The equations of this paper, for incompressible flow, are derived in appendix A. Compressibility effects
44、are derived in appendix B. summarized in appendix C. Some of the pertinent equations are Many of the equations derived herein involve the spanwise position of the centroid and radius of gyration of the angle-of-attack loading of the wing semispan. The centroid position for angle-of-attack loadings h
45、as been determined for a wide range of wing plan- forms 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 gyra- tion occurs as a factor in determining aerodynamic characteristics.
46、 It was necessary, therefore, to compute the radius of gyration ?* for use in this paper. The values were determined by piotting the product (%)(Y*) against y* for a large number of wings and performing a mechanical integration to obtain ?* by use of the following equation: CCL (?*I2 =lo 1 (CCl)(Y*)
47、2dY* q A total of 160 such integrations were performed to obtain the desired coverage of wing planforms. Values of y* and of ?* are plotted in figures 5 and 6 as functions of sweep, aspect ratio, and taper ratio for use in this paper and for general information. Data for the span loads were obtained
48、 from reference 18. Span- Load Distributions The span-load equations derived in this paper are applicable for the determination of the incremental load due to sideslip, yawing, or rolling velocity of a wing for which the 9 Provided by IHSNot for ResaleNo reproduction or networking permitted without
49、license from IHS-,-,-span load in symmetric flight is known. The span-load equations are summarized in appendix C. Equations (Cl) to (C3) are very general and should be used if the sweep angle varies across the wing span. In such instances it may not be possible to find in the literature the corresponding angle-of-attack loading
