1、Improved Method for Prediction of Attainable Wing Leading-Edge Thrust Harry W. Carlson, Marcus 0. McElroy, Wendy B. Lessard, and L. Arnold McCullers April 1996 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NASA Technical Paper 3557 Improved Method
2、for Prediction of Attainable Wing Leading-Edge Thrust Hary W. Carlson Lockheed Engineering 6 Sciences Company . Hampton, Virginia Marcus 0. McElroy and Wendy B. Lessard Langley Research Center 0 Hampton, Virginia L. Arnold McCullers ViGYAN, Inc. 0 Hampton, Virginia National Aeronautics and Space Adm
3、inistration Langley Research Center Hampton, Virginia 23681 -0001 - April 1996 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Available electronically at the following URL address: http:/techreps.larc.nasa.govAtrsAtrs.html Printed copies available f
4、rom the following: NASA Center for Aerospace Information National Technical Information Service (NTIS) 800 Elkridge Landing Road 5285 Port Royal Road Linthicum Heights, MD 21090-2934 Springfield, VA 22161-2171 (301) 621-0390 (703) 487-4650 Provided by IHSNot for ResaleNo reproduction or networking p
5、ermitted without license from IHS-,-,-Contents Abstract . 1 Introduction . 1 Symbols 2 Present Method Development . 3 Normal Airfoil and Flow Parameter Derivation 3 Theoretical Two-Dimensional Airfoil Analysis (Cp. lim = Cp. however, this process depends on the skill and experience of the computer c
6、ode user. The present method described in this paper provides a better solution in which the theoretical two-dimensional airfoil matrix is expanded to include a leading-edge radius of zero. With this change the method is applicable to a continuous range of leading-edge radii from zero through the st
7、andard values to very large val- ues approaching half of the wing maximum thickness. Expansion of the two-dimensional airfoil matrix to include variations in location of maximum thickness was accomplished by a revised relationship between stream- wise airfoil sections of the wing and the derived two
8、- dimensional sections, a relationship that results in much closer representation of the real flow over a lifting sur- face. Revision of the attainable thrust prediction method also provided an opportunity to take advantage of infor- mation relating to the effect of Reynolds number on attainable thr
9、ust that was not available before publication of reference 1. In reference 1, the two-dimensional experimental data used to define limiting pressures were restricted to R I 8 x lo6 (based on the chord). The present method discussed herein makes use of data obtained up to R = 30 x lo6. Because revisi
10、ons to the previous method are quite extensive, the development of the present method is cov- ered in detail, even at the expense of some repetition. Some examples of the application of the present method to data for wings and wing-body configurations are given. Correlations are included for data pr
11、eviously used in references 6 and 8 and for new data as well. In addi- tion, instructions are given for the evolution of the sys- tem to accommodate new two-dimensional airfoil data, as it becomes available, so as to provide a more exact and more complete formulation of attainable thrust depen- denc
12、e on Mach and Reynolds numbers. Symbols b wing span, in. CA axial- or chord-force coefficient CD drag coefficient *CD drag coefficient due to lift, CD - CD,o c, o drag coefficient at a = 0“ for configuration with no wing camber or twist CL lift coefficient Pat pitching-moment coefficient normal-forc
13、e coefficient pressure coefficient limiting pressure coefficient used in defini- tion of attainable thrust 2 vacuum pressure coefficient, - Y M2 local wing chord, in. S average wing chord, - , in. b section axial- or chord-force coefficient change in section axial- or chord-force coef- ficient relat
14、ive to a = O0 section theoretical thrust coefficient (from linearized theory for zero-thickness airfoils) section attainable thrust coefficient mean aerodynamic chord, in. exponents used in curve-fit equation for attainable thrust factor exponent used in curve-fit equation for lim- iting pressure co
15、efficient parameter used in curve-fit equation for lim- iting pressure coefficient attainable thrust factor, fraction of theoreti- Lt cal thrust actually attainable, - = Ln t t,n parameter used in curve-fit equation for attainable thrust factor constants used in airfoil section definition free-strea
16、m Mach number equivalent Mach number replacing Mn to account for Cp,lm f Cp,vac normal Mach number (fig. 2) attainable thrust parameter, Kt 1 + (17 theoretical thrust parameter, dynamic pressure Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-R Reyno
17、lds number based on mean aero- dynamic chord r leading-edge radius, in. r - 11 C ri leading-edge radius index, - (2 l2 S wing area, in. 2 s spanwise distance, in. t section theoretical leading-edge thrust t* section attainable leading-edge thrust X, Y,Z Cartesian coordinates, positive aft, left, and
18、 up, respectively (fig. 2) x distance behind wing leading edge a angle of attack, deg Y ratio of specific heats, 1.4 how- ever, as discussed previously, the connection between two-dimensional airfoil sections and the three- dimensional wing sections is made through theoretical leading-edge thrust co
19、efficients and not angle of attack. To find an appropriate three-dimensional wing section angle of attack to match a two-dimensional section angle of attack would be difficult, if not impossible, because of the extreme variation of upwash just ahead of the wing leading edge. The theoretical thrust c
20、oefficients provide a better connection because of the dependence of these coefficients on linearized theory singularity strength, which is a measure of pressure levels in the vicinity of the section leading edges. When pressure limiting has only a small effect, as it does for low Mach numbers and l
21、ow angles of attack, the subsonic airfoil computer code gave values of theoretical thrust greater than that for a zero-thickness airfoil. Thus, Kt can be greater than 1.0 with a maximum value that tends to increase with increasing airfoil thickness. Because experimental data show little or no eviden
22、ce of the theoretical benefit of air- foil thickness on attainable thrust given by the two- dimensional airfoil computer code, the attainable thrust factor Kt, as shown in figure 4, is restricted to values of 1.0 or less.2 In retrospect, an alternative procedure could have been applied. An attainabl
23、e thrust factor defied as the ratio between thrust coef- ficients with and without pressure limiting (2n sin2 a), replaced by c: for M, = 0) would automatically limit Kt to values less than 1.0. Although this alternative procedure has some attractive features, the resultant method would not be expec
24、ted to give signif- icantly different results. As shown later, experimental data are used to calibrate the method. A different calibration would compensate for changes in the Kt factor. After the vacuum pressure-limited thrust coefficient data are determined for the .wide range of airfoil sections d
25、epicted in figure 3, the next step is to represent the data by empirical equations for use in automated calculations. The representation process is quite involved and was developed after considerable trial and error. For the inter- ested reader, a discussion of the strategy employed is given in appe
26、ndix A. As discussed in appendix A, the use of a theoretical thrust parameter Ptt and an attainable thrust parameter Pat provided the means of incorporating a range of airfoil geometric properties in a simplified rep- resentation of the attainable thrust factor K,. Results of the data representation
27、 are shown in figure 5 in the form of Pat given as a function of P,. Each of the five plots in figure 5 shows results for a given value of the radius index. The curves shown in figure 5 represent a fairing of the data provided by a single equation derived in appendix A to cover Mach numbers ranging
28、from 0 to nearly 1.0, maximum thickness ratios from (TIC), = 0 to 0.15, locations of maximum thickness from q = 0.1 to 0.5, and leading-edge radius indices from 0 to 1.2. The equation is with Kt limited to values less than 1.0 where The limitation of Kt to values no greater than 1.0 permits attainab
29、le thrust to equal, but not exceed, theoretical thrust values defined by lifting surface theory. In the curve-fitting exercise, primary attention was given to representation of factors near the middle of the Pat range. In addition, the greatest emphasis was placed on data Provided by IHSNot for Resa
30、leNo reproduction or networking permitted without license from IHS-,-,-representing nominal airfoil parameters of (TIC), = 0.09 and q = 0.5. Thus, the system is less accurate for extreme airfoil shapes, particularly for thin airfoils with forward locations of maximum thickness and sharp or nearly sh
31、arp leading edges. In figure 5, the decrease in Pat with increasing P, is clearly shown, as is the strong dependence on Mach number. A comparison of the plots in figure 5 shows the effect of increasing leading-edge radius. A sharp leading edge (fig. 5(a) produces a substantial level of attainable th
32、rust. Through an oversight, the previous attainable thrust method of reference 1 did not account for any of this thrust. For a leading-edge radius of zero, the pre- vious method gave a thrust of zero. Sketch F is an exam- ple of the variation of attainable thrust with increasing leading-edge radius
33、for a 9-percent thick airfoil with maximum thickness at the 50-percent chord station at an angle of attack of 12“ at M, = 0.5. Sketch F In figure 5 the considerable dependence of the attainable thrust parameter on maximum thickness and its location is not clearly evident. Sketch G shows the variatio
34、n of attainable thrust with increasing thickness for an airfoil with a leading-edge radius index of 0.3 and maximum thickness at the 50-percent chord station at the same flow conditions. The nearly linear dependence of thrust on thickness clearly illustrates the importance of thickness and its front
35、al projected area in the develop- ment of thrust. The effect of the location of maximum thickness on developed thrust is illustrated in sketch H for the same nominal conditions of maximum (TIC), = 0.09, ri, = 0.3, a = 12“, and M, = 0.5. Benefits of a more forward loca- tion of projected frontal area
36、s on which thrust is devel- oped are clearly shown. However, these thrust benefits are achieved at the expense of a tendency toward increased profile drag for such sections. Sketch G Sketch H Equivalent Mach Number Concept (Cp,b f Cp,vac) Equation (1) was developed to account for the reduc- tion in
37、attainable leading-edge thrust resulting from the application of realistic constraints on local pressure coef- ficients. A limiting pressure defined by the vacuum pres- sure coefficient has been shown to have a powerful effect on the amount of theoretical leading-edge thrust that can actually be rea
38、lized. However, even more severe limits on achievable thrust are experienced in the real flow over airfoil sections when the local flow lacks sufficient energy to negotiate turns about the airfoil surface without becoming detached from that surface. Establishment of values for these more severe limi
39、tations is addressed in the following section of this paper. Before that, a means of application of equation (1) to the estimation of attain- able thrust for values of limiting pressure other than the vacuum limit is developed. Equation (1) can be used for a full range of limiting pressures between
40、0 and Cp,vac by substitution of a properly defined equivalent Mach number Me for the normal Mach number M,. The substi- tute Mach number is defined by the following logic. As Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Sketch I illustrated by the
41、 pressure distributions shown in sketch I for a given airfoil section at the same positive angle of attack, the pressure coefficient at any point on the airfoil will vary with Mach number according to the Prandtl- Glauert rule. Thus, if the limiting pressure Cp,lim also changes in accordance with th
42、e Prandtl-Glauert rule the attainable thrust factor Kt will be the same at all Mach numbers because both cy and ct !, will have the same Mach number dependence. Then with Me selected SO that Cp,lim(Me) = Cp,vac(Me), the appropriate value of Kt for the normal Mach number under consideration is calcul
43、ated by substitution of Me for Mn in equa- tion (1). The required Me is determined by setting Cp,vac(Me) = Cp,lim(Me), the intersection point of the curves shown in sketch I, and by solving for Me. Thus, and after the solution for the equivalent normal Mach number, Experimental Two-Dimensional Airfo
44、il Analysis (Cp,lim Calibration) To define practical values of the limiting pressure coefficient, an incomplete version of the present method was applied to experimental two-dimensional airfoil data for symmetrical sections. (See refs. 10-18.) Correlations of axial-force coefficients predicted by th
45、is incomplete present method with experimentally determined axial- force coefficients, as shown in the examples of figure 6, were used to determine, by trial, values of Cp, that would match the experimental trends. For these symet- rical sections, AcA is simply the negative of ct ,. The example corr
46、elations in figure 6 were chosen to represent the procedures that were applied to the large amount of data available in the references. These data had a range of airfoil maximum thicknesses from 4 to 15 percent of the chord and locations of maximum thickness from 10 to 42 percent of the chord. Howev
47、er, leading-edge radius indices had only a small range of 0.24 to 0.33. Mach numbers ranged from 0.03 to 0.90, and Reynolds num- bers varied from less than 1 x lo6 up to 30 x lo6. To evaluate limiting pressure coefficients, equa- tions (1) and (2) were combined in a computational pro- cess in which
48、Kt and cZ;. were calculated for a series of trial Cp, values to find the value that most closely matched the experimental data. In matching the trial curve fit to the experimental data, particular attention is given to breakaway of the experimental axial force from the theoretical leading-edge full-
49、thrust curve. For most of the plots, this breakaway point can be established with reasonable certainty. For other plots a breakaway point of the experimental data is not readily obvious. The prob- lem occurs because axial-force data were not presented directly for some of the experimental investigations; the axial force had to be derived from lift- and drag- coefficient data. For some of these data, as in
