NATIONAL ADVISORY COMMITTEE

Transcription

1 NATIONAL ADVISORY COMMITTEE TECHNICAL NOTE NQV SUMMARY OF THE THEORETICAL LIFT, DAMPING-IN-ROLL, AND CENTER-OF-PRESSURE CHARACTERISTICS OF VARIOUS WING PLAN FORMS AT SUPERSONIC SPEEDS By Robert 0. Piland Langley Aeronautical Laboratory Langley Air Force Base, Va. Washington October 194 9

2 ' W! NATIONAL ADVISORY COMMITTEE FOR AWONAUTICS SUMARY OF THE: THMlRETICAL LIFT, DAMPING-IN-ROIL, AND CENTW-OF- RESSURE CHARACTERISTICS OF V'ICXJS WING PLAN FORMS AT SUPERSONIC SPEEDS By Robert 0. Piland SUMMARY l. The informition for the determination of the theoretical lift, damping-in-roll, and center-of-pressure characteristics for a group of plan forms at supersonic speeds has been collected. This information is presented in three forma: first, equations; second, figures presenting generalized curves in which the quantities plotted are combinations of the geometric and aerodynamic parameters; and third, figures presenting curves for representative groups of specific configurations showing variation of the derivatives with Mach number. From this group several observationa have been made as to the effect of taper ratio, angle of sweep, and aspect ratio on the values of the derivatives. IN!I!E?OMJCTION The linearized theory for supersonic flow has been used by various investigators to develop expreasions for the stability derivatives. The information is scattered, however, through a number of reports, not all of which use the same notation. A few of the more important derivatives are therefore collected herein for future comparison with experimental work and to aid designers by presenting the informtion in convenient form. In order to make the paper more useful for purposes of comparison and design work, calculations were made for related groups of specific configurations. The results of these calculations are presented as plots of lift-curve slope, damping in roll, and center of pressure against Mach number. In addition, all available informtion is presented in the form of generalized figures in which the quantities plotted are combinations of the geometric and aerodynamic prameters.

3 I 2 NACA TN 1977 SYMBOLS x, Y, P v M r e c t a coordinates (fig. 1) e a r velocity about x-axis (fig. 1) flight speed stream Mach number speed of sound Mach angle angle of attack (fig. 1) density of air local chord root chord tip chord mean aerodynamic chord x S A h ht D mt taper ratio (ct/cr) wing area aspect ratio (b2/s) angle of weep of leading edge angle of sweep of trailing edge cotangent A cotangent

4 I cl NACA TN U I V complete elliptic integral of second kfnd with modulus k WPm) complete elliptic integral of first kind with modulus k L lift 2 rolling moment (+) M pitching moment about apex lift coefficient c2 rolling -moment coefficient (&b) pitching-moment coefficient / ~ c = 2P %la = - xcp C longitudinal diptance between leading edge of mean aerodynamic chord and center of pressure due to lift, fraction of mean aerodynamic chord

5 I 4 NACA TN 1977 I xcp longitudinal distance between wing apex and center of press-me due to lift, fraction of root chord Subscripts: CP center of pressure 192 regions of wing panel in sketch in appendix PRESENTATION OF RESULTS Expreseiona for various stability derivatives have been obtained by several investigators (references 1 to 13) with the use of the linearized theory for supersonic flow. The equations for determining the theoretical lift, damping-in-roll, and center-of -pressure characteristics of various plan forma have been collected and are given in the appendix with a discussion of the range of applicability for each equation. All wings are considered to be of zero thickness and the equations are given with respect to axes having their origin at the leading edge with the x-axis coinciding with the root-chord line. Table I lists the various specific wings conaidered according to their geometric characteristics. The equations and figure nllmbers given in the table serve as an index to the information presented in the paper. References from which the mterial was compiled are also listed in the table. The types of plan fom for which equations are given are shown in figure 2. Equations for a few plan forma not considered herein may be found in reference 13. Figures 3(a) to 3(f) illustrate the specific configurations considered. Figure 4 shows the variation of the elliptic-fntegral factors E"(l3m) and I(h) with Pm and is to b3 used with the equation8 for the triangular, notched triangular, and swept tapered wings in the subsonic-leadingedge case (Pm < 1). Computational data obtained from the equations are presented in 4x0 forma. Generalized curves (figs. 5 to lo), in which the quantities plotted are combinatlono of the geometric and aerodynamic parameters, permit a rapid estimation of the value of a derivative for any specific wing whose general plan form is considered herein. Curves of the lift-curve slope, damping-in-roll, and center-of -pressure characteristics plotted against Mach number me presented in figures 11 to 16 for the specific configurations listed in table I.

6 NACA TN From the variation of the derivatives (figs. 5 to 16) as predicted by linearized theory, several observations have been made as to the effect of sweep angle, taper ratio, and aspect ratio. The following effects were noted between Mach numbers of 1 and 2: (1) For wings with subsonic leading edges, increase of sweep angle decreases the values of At speeds at which the Mach C L, -Czp* line lies behind. the leading edge, however, the lift-curve slope and the -ping-in-roll values are greater with increased sweep. With increased angle of sweep, the center of pressure of a notched triangular wing moves rearward with respect to both the leading edge of the mean aerodynamic chord a i and the apex of the wing 32. a (2) Because any change in taper ratio affectd other wing parameters, the combined effects of taper ratio and sweep angle were noted with aspect ratio remaining constant. For a wing with a supersonic leading edge, an increase in taper ratio (ratio of tip chord to root chord) decreases the value of the lift-curve slope but increases the amount of damping in roll. (3) Increase in aspect ratio increases values of the derivatives and -C and causes the center of pressure of the wing to move 'La 2P rearward. An exception is the triangular wing with a supersonic leading edge where the derivatives CL, and -C are functions of 'IP Mach number alone and the center of pressure is constant at the twothirds -chord point. Cr CONCLUDING RIMARES A group of 52 specific wings representing various plan forms has been conaidered. The variations of the theoretical lift, damping-inroll, and center-of-pressure characteristics of these wings with Mach number are presented. The equationa from which these values were obtained are also presented. From the figures presented, the effect of sweep, taper ratio, and aspect ratio may be noted. The lift-curve slope CL, decreased with increase in sweep angle when the lead- edge was subsonic and increased when it was supersonic. Increased aspect ratio increased

7 the lift-curve slope. In the case of the triangular wing with a supersonic leading edge, however, was independent of changes in cla sweep and aspect ratio. The amount of damping in roll taper ratio, sweep angle, and aspect ratio except in the case of wings with subsonic leading edges in which an increase in sweep decreased the value of -C Again an exception was noted in the case of the 2P - triangular wing with a supersonic leading edge for which -C was not ZP affected by changes in these parameters. The position of the center of pressure moved rearward with increase in sweep angle and aspect ratio, except in the case of the triangular wing in which it was constant at the two -thirds-chord point. -Czp was increased with increase of Langley Aeronautical Laboratory National Advisory Comittea for Aeronautics Langley Air Force Base, Va., August 3, 1949

8 I -' NACA 'I" k n. h r( cu W W n - r n f In W W W + b a, Pi I4

9 8 NACA TN 1977 A group of rectangular wings, illustrated in figure 3(a), ia comi-dered. Variation of EL,, -KIP is ahown Y.- in figure 5 specific curves of % *=, and -C plotted a' cy 2P - against Mach number are shown in figure ll. Equations (3) to (5) are applicable for a range where the Mach line from a tip intersects the trailing edge (PA > 1). Equations (1) and (2) for and xcp 'La are applicable when a kch line from the tip intersects the opposite edge between the midpoint of the aide and the intersection of the side edge with the trailing edge (1 > PA > i). Unswept Tapered Wings "he equations for and C for unswept tapered wings with clu 2P supersonic leading edges which were obtained from reference 5 and an unpublished analysis at the Langley Laboratory are as follows: Limiting condition: PA (3p 4m4-10p2m2-8) 3 (7)

10 NACA TN The variation of and -PCzp with PA for various taper and -C for a group ratios is shown in figure 6. The curves of C L ~ of unswept tapered wing8 (fig. 3(b)), plotted against Mach number are shown in figures l2(a) to 12(d). Equations (6) and (7) are applicable when the Mach line from the apex intersects the trailing edge (PIA 2 2). Triangular W ings The equations for CLuy 9, and Czp for triangular wings with subsonic or supersonic leading edges which were obtained from references 6, 7, 8, and 9 are as follows: Limiting condition: BA $ 4 (a) Subsonic leading edge Limiting condition: PA 2 4 PCz, = -&I( en) 32 (b) Supersonic leading PCLa = The variation of ~CL,~ k, and -PCzp with PA is shown in Cr figure 7. The variation of CL~, %y and -Czp with Mach number is C r

11 10 NACA TN 1977 shown in figure 13 for the configurations shown in figure 3(c). Equations (8) to (10) are applicable when the Mach line from the apex lies ahead of the leading edge (b < 1). Equations (11) to (13) apply when the Mach lines from the apex lie behind the leading edge (b > 1).,, Notched Triangular Wings The equations for %, and Cl for notched triangular wings P with subsonic or supersonic leading edges follow. They were obtained from references 10 and 11. No equation was given for for wings with supersonic leading edges. clp (a) Subsonic leading edge Limiting conditions: Pm = < 1 PCL, Pm 2 IN1 - m ( COS-^( (1 + N) 32 -N) + N(l - N2) 1/21 (14) xcp = N(4 - N2) + (2 + N2)(1 - I@)-1/2co~-1(-N) 3(1 - N2)[N + (1-8)-1/2cos-1(-N~ > Limiting conditions: = 1 (b) Supersonic leading edge l = > N = > - l

12 NACA TN I. The variation of - xcp 5 and -El, with l3m for various EL, cr c 225 values of PA is given in figure 8. The parameters Ch, c,, c and -C for the group of wings illustrated in figure 3(d); plotted 2P against Mach number, are shown in figures 14(a), 14(b), and 14(c). Equations (14) to (16) are applicable for a range of Mach numbers for which the Mach line from the apex lies ahead of the leading edge (l3m < 1) and the Mach line from the trailing edge of the root chord lies behind the trailing edge (Pm 2 N). When this latter condition is violated a value of the derivative is obtainedwhich has the significance of an upper limit (reference 10). This region (Pm < N) is indicated by a -- bashed curve. Equations (17) and (18) for CL, and -, respectively, c, A are applicable when the Mach line from the apex intersects the trailing edge of the wing. The relation between XCp anii Bcp E expression C r is given by the Swept Untapered Wings The equations for C L ~ and CiP for the case in which the Mach line from the apex lies on the leading edge (Pm = 1) were obtained by modifying equatiom in reference 12 in the following mnner:

13 i 12 NACA TN 1977 For the lift case, where subscripts 1 and 2 refer to the regions of the wing panel shown in the following sketch: If the taper ratio is 1 and the Mach line coincides with the leading edge (m = l), equation (9) of reference 12 becomes d b +mc and equation (12) of reference 12 becomes Performing the operations indicated in equations (a), (b), and (c) yields an expression for CL, in terms of P and A. Similarly, modifying equation (19) of reference 12 in the preceding mnner gives - v 2dV (d) czl - 3SbV

14 NACA!I" and equation (23) of reference 12 becomes The amount of damping contributed by the tip section, however, is seen to be negligible and consequently the expression derived for this region has been omitted from the expression for. Performing the operations c2p indicated in equation (d) and taking the derivative of C2 with respect to give the equation for C 2v The equations for CLa and C2 derived are as follows: P Condition of applicability: Pm = 1 r - The equations for C L ~ and C2 P case in which the Mach line from the apex crosses the trailing for a swept untapered wing for the edge ( P > $ + i) were obtained from an unpublished analysis by Mr. Sidney M. Harmon of the Langley Stability Research Division and are as follows: 2 1 Limiting condition: P > + - m

15 14 NACA TN 1977 P4m4( - 4p4m4 + 8P%n2)c0s-~ The variation of PCLU and -pcz, with b for various values of PA is shown in figure 9. No equation is available for the case in which the Mach line from the apex cuts the side edge of the wing; consequently the points obtained from equations (19) and (20) have been faired to the curves obtained from equations (21) and (22). These regions are indicated by dashed lines. No equations were available for Xcp for wept untapered wings. curves of CL and -Czp, varying Cr with Mach number, for the swept untapered wings shown in figure 3(e) are presented in figures 15(a), l5(b), and 15(c). U. Swept Tapered Wings Equatiorm for and C2 for ewept tapered wings obtained P from reference 12 and from the previously mentioned unpublished analysis follow (no equation is given for, for the case of a wing with a c2p supersonic leading edge):

16 NACA TN rl VI1 VI1 n x x + W.r cu V x + rl W m 4 g 8; II 3 rl x It d 3 a m I d 3 d.rl, m l + n rl + d W n t i + IVI n rl I d + 3 * F: -I- CE! +T -+ 3 Id W rl- - 3 rl I 1: 3 m (rl' rl I a W 3 + n rl + d W n rl + a.vl n rl + d I 3 + k IW I +

17 16 NACA TN 1977 rl + N14 I+\ a I

18 NACA TN The variation of Pcl_l and -Ezp with Pm for vazious values U PA for swept tapered wings of taper ratio 0.5 is shown in figure 10. of Curves of C L ~ and -Czp plotted against Mach number are shown in figure 16 for the configurationa of figure 3(f). Epuatiom (23) (for CL,) and (24) (for -C2 ) are applicable for the case in which the P Mach line from ths apex lies ahead of the lead- edge and the Mach line from the trailing edge of the root chord lies behind the trailing edge. Violation of the latter condition will give values of the derivative which have the significance of an upper limit (reference 12). These regions are indicated by the dashed parts of the curves. Equation (25) (for CL,) is applicable when the Mach line from the apex intersects the 2 trailing edge (P > + l+n ). The two prts of the curves obtained In from the equations for C L ~ have been faired together over the region where the Mach line from the tip crosses the side edge of the wing.

19 18 NACA TN 1977 REFWENCIS 1. Lagerstrom, P. A., and Graham, Martha E. : Low Aspect Ratio Rectangulm WFngs in Supersonic Flow. Rep. No. SM-l3llO, Aircraft Co., Inc., Dec Douglas 2. Bonney, E. Arthur: Aerodynamic Characteristics of Rectazlgular WFn@;B at Supersonic Speede. Jour. Aero. Sci., vol. 14, no. 2, Feb. 1947, pp Lageratrom, P. A., and Graham, Martha E.: Some Aerodynamic Formulaa in Linearized Supersonic Theory for Damping in Roll and Effect of Twist for Trapezoidal Wings. Rep. No. SM-13200, Douglas Aircraft Co., Inc., March 12, Harmon, Sidney M. : Stability Derivatives of Thin Rectangular Wings at Supersonic Speeds. Wing Diagonal8 ahead of Tip Mach Lines. NACA 'I" 1706, Tucker, Warren A., and Nelson, Robert L.: The Effect of Torsional Flexibility on the Rolling Characteristics at Supersonic Speeb of Tapered UnrJwept Wings. NACA TN 189, Brown, Clinton E. : Theoretical Lift and Drag of Thin Triangular Winga at Supersonic Speeds. NACA Rep. 839, Ribner, Herbert S., and Malvestuto, Frank S., Jr.: Stability Derivatives of Triangular Wings at Superaonic Speeds. NACA TN 1572, Puckett, Allen E.: Supersonic Wave Drag of Thin Airfoils. Jour. Aero. Sci., vol. 13, no. 9, Sept. 1946, pp Brown, Clinton E., and Adama, Mac C. : Damping in Pitch and Roll of NACA Rep. 892, Triangular Wings at Supersonic Speeds. 10. Malvestuto, Frank S., Jr., and Mazgolia, Kenneth: Theoretical Stability Derivativea of Thin Sweptback WFngs Tapered to a Polnt with Sweptback or Sweptforward Trailing Edges for a Limited Range of Supersonic Speeds. NACA TN 1761, Puckett, A. E., and Stewart, H. J.: Aerodynamic Performance of Delta Wings at Supersonic Speeds. Jour. Aero. Sci., Vol. 14, no. 10, Oct. 1947, pp

20 12. Malvestuto, Frank S., Jr., Margolis, Kenneth, and Ribner, Herbert S.: Theoretical Lift and Damping in Roll of Thin Sweptback Wings of Arbitrary Taper and Sweep at Supersonic Speeds. Subsonic Leading Edges and Supersonic Wailing Edges. NACA TN 1860, Jones, Arthur L., and Alksne, Alberta: The Damping Due to Roll of Triangular, Trapezoidal and Related Plan Forms in Supersonic Flow. NACA TN 1548, l9h.

21 20 NACA TN m v -- A rl v RE.Po ng 4 3-Irlrlrl i I f m N

22 NACA!L" I. A I\ I' Y I. I.

23 22 NACA TN 1977 i 4 =e, 1 Figure 2.-Various types of plan form for which equations are given-

24 NACA TN t I ' 4=3 2& I A=# I I A=& 1 I A=0 (a) Rectangular wings. Figure 3.- Specific configurations for which curves are presented. Numbers within plan forms correspond to wing numbere in table I.

25 (b) Urnwept tapered wings. Figure 3.- Continued.

26 NACA TN I I // I f - A=.W A=4 (b) Concluded. Figure 3.- Continued.

27 26 NACA TN 1977 (c) Triangular wings. Figure 3.- Continued.

28 NACA TN ~- I * M I 6944 O A= 4 A-4 (d) Notched triangular wings. Figure 3.- Continued.

29 28 (e) hept untapered Figure 3.- Continued.

30 (f) Swept tapered wings; X = 0.5. Figure 3.- Concluded.

31 Figure 4.- Elliptic-integral factors of the stability derivatives that determine their variation with Mach number.

32 NACA TN I. Figure 5.- Variation of PCL,, xcp/crj and +CzP with PA for rectangular wings.

33 32 NACA TN QY Figure 6.-Variation of j3c and -pc, with j3a for various taper La P ratios.

34 I NACA "N cs I./ Figure 7.- Variation of /~CL,, xcp/cr, and +3C, winge. P with PA for triangular

35 NACA TN P.3 2./ b LW 2 Q? / C - Figure 8.-Variation of BCL,, +.p/~r, xc,/f, and 4, P notched triangular wings. with prn for

36 Figure 9.- Variation of BC and 4 C z wlth Bm for wept untapered La P wings with supersonic leading edges.

37 36 NACA TN 1977 Figure lo.-variation of PCh and +C, with pm for swept tapered P wings. Taper ratio, 0.5.

38 . NACA 'I" (a) Variation of CL, and +p/cr with Mach number. b, sp/cr, and 4 Figure 11.- Variation of C with Mach number for lp rectangular winge of aspect ratio 2, 3, 4, 6, and 8.

39

40 NACA TN '..8 M. (a) Taper ratio, 0.8. Figure 12.- Variation of C+, and -C with Mach limber for unswept U zp tapered wings.

41 40 NACA TN 1977 M (b) Taper ratio, 0.6. Figure 12.- Continued.

42 NACA TN M M (c) Taper ratio, 0.4. Figure 12.- Continued.

43 42 NACA TN 1977 M (d) Taper ratio, 0.2. Figure 12.- Concluded.

44 NACA TIV I. 7 I / M Figure 13.- Variation of CL, qp/cr, and 4 2 with Mach number for P triangular wings.

45 44 NAcA vb 4 M Fisre lk.-vmiation of CL,. xcp/cr, - /E, and -C with Mach xcp 2P number for notched trianslar wings.

46 (b) Angle of aweep, 45O. Figure 14.- Continued.

47 46 NACA TN 1977 (c) Awe of Sweep, 60'. Figure 14.- Concluded.

48 47 (a) Angle of sweep, 30'. Figure 15.- Variation of C and -C with Mach nuniber for swept La zp untapered wings.

49 Q (3 I.4.3.2,I */ M (b) Angle of sweep, 45'., Figure 15.- Continued.

50 UCA Q. u I.2./ M c (c) Angle of sweep, 60. Figure 15.- Concluded.

51 50 NACA TN 1977 M I I I I I I I I I /i M '2 (a) Angle of sweep, 30'. Figure 16.- Variation of C b and 4 with Mach number for swept 2P tapered wings. Taper ratio, 0.5.

52 I I t. I (b) Angle of sweep, 45O. Figure 16.- Continued.

53 ~~ ~ 52 NACA!I" 1977 (c) Angle of sweep, 60'. Figure 16.- Concluded. NACA-Langley