APPENDIX
APPENDIX AERODYNAMICS EQUATIONS LIFT sin a + € sin a where
= - 0.355
Cl M < 1.0 Co = 0 irAR - 0.153 BAR M > 1.0 C = linear interpolation with respect to 3 from C = 0 at 3=0 to [0.955-O.35/M)] _ _ Co = e at AR - 0.13 M > 1.0 6 > ± - AR , _ Ft).955-0*.35/M)] , - e INDUCED DRAG C = %C tan a Di L where M < 3.0 j K = 0.25 (1 + M) M M * 3.0 j K = 1.0 M ZERO-LIFT BODY DRAG C = C + CD + CD
DOB D B FB BB
P Body Pressure Drag M < 0.8 { C = 0 DpB 0.8 < M < 1.2 CD = linear interpolation with respect to M from J R pB Cn „ = 0 at M = 0.8 to B P CD M = l pB ' M ^ 1.2 | CD = CD from numerical integration of pressure R R I distribution on body Body Base Drag Cp ep = Cp from Prandtl-Meyer expansion of forebody pressure distribution RA Unless Cp < Cp when, C C PBASE " ?2-D 0.91 M - 0.20 M + 1.51 (From fig. 5, p. 34, ref. 8) Body Friction Drag S / WET\ 1 4- ?l 1 S C/ * BODY. \ REF/ fn - 0 455 M < 0 ' . 8 i FB i \ 0 . ^4 6 7 Y 2 5 8 - J- M 2 1 (logio Re) - (l 4
2 * *o y
where (MAC) BODY 2(.ZTt/Z) Re = P M a and 0 0 0 i A
(I)
BODY CD_ = linear interpolation with respect to M from 0.8 < M < 1.2 R
CD = c at M = o.s to
FB DpB
CD = CD at M = 1.2
FB FB = M * 1.2 { CD™ CD™ from numerical integration of local skin- re rt> friction coefficients on body Body Bluntness Drag
M < 0.8 CD = o
BB CD = linear interpolation with respect to M from 0.8 < M < 1.0 I RR RR CD =0 at M = 0.8 to BB CD = CD at M = 1.0 RR RR L>D DP irr.
NOSE M > 1.0
CD
BB S REF where 1/2
±-\ -4-.3.15
8 1 0 ,0.5 ' " M A X " ' ' " NOSE ( .
'SKIN \1000 ZERO-LIFT FIN DRAG CD F BF where C CD = D Xc X F VERTICAL FINS CANARD HORIZONTAL FINS computed separately by the following equations.
Fin Pressure Drag M < 0.8 { CD = 0 F 0.8 < M < 1.0 CD p = linear interpolation with respect to M from p CD p = 0 at M = 0.8 to CD = C at M = 1 pF °pF '° 5/3 FIN z M = 1.0 C = 3.4 -, cos A u D c pF \c/ , . S FIN T TV FIN REF 1.0 < M < M • CD p = linear interpolation with respect to M from SA P C = C at M = 1.0 to Dpp DpF C at M = M Dpp SA a !_ FIN
> MSA C
D e S FIN REF Fin Friction Drag 1 + 2 - REF .
C = 0.455 DpF 0.467 2 58 - ( l (lo R e ) - l glo where (MAC) FIN Re = P M a, n o ° ° y, Fin Bluntness Drag
M < 0.8 j C = o
DBF 0.8 < M < 1.0 CD = linear interpolation with respect to M from J RC BF.
Cn , = 0 at M = 0.8 to Dr Df
CD = C at M = i.o
BF DBF FIN where 1 2 c o s r 725 cos - r -'^ NOSE REFERENCES 1. Gregory, Thomas J.; Petersen, Richard H.; and Wyss, John A.: Performance Tradeoffs and Research Problems for Hypersonic Transports. J. Aircraft, vol. 2,1965, pp. 266—271.
2. DeYoung, John; and Harper, Charles W.: Theoretical Symmetric Span Loading at Subsonic Speeds for Wings Having Arbitrary Plan Form. NACA Rep. 921, 1948.
3. Lawrence, H. R.: The Lift Distribution on Low Aspect Ratio Wings at Subsonic Speeds.
J. Aero. Sci., vol. 18, no. 10, Oct. 1951, pp. 683-695.
4. Puckett, A. E.; and Steward, H. J.: Aerodynamic Performance of Delta Wings at Supersonic Speeds. J. Aero. Sci., vol. 14, no. 10, Oct. 1947, pp. 567-578.
5. McDevitt John B.; Rakich, John V.: irFlieSA'ero'd.ynamic Characteristics .of, .Several Thick Delta Wings at Mach Numbers to 6 and Angles of Attack to 50. NASA TM XM 62,1960.
6. Jorgensen, Leland H.: Elliptic Cones Alone and With Wings at Supersonic Speeds. NACA Rep. 1376, 1958.
7. Van Dyke, Milton D.: The Slender Elliptic Cone as a Model for Nonlinear Supersonic Flow Theory. J. Fluid Mech., vol. 1, May 1956, pp. 1—15.
8. Love, Eugene S.: Base Pressure at Supersonic Speeds on Two-Dimensional Airfoils and on Bodies of Revolution With and Without Fins Having Turbulent Boundary Layers. NACA TN3819, 1957.
9. Shapiro, Asher: The Dynamics and Thermodynamics of Compressible Fluid Flow. The Ronald Press Co., N. Y., 1954.
10. Koelle, Heinz Hermann: Handbook of Astronautical Engineering. McGraw-Hill Book Co., Inc., 1961.
11. Eckert, Ersnt R. G.: Survey of Heat Transfer at High Speeds. ARL Rep. 189, Aeronaut. Res.
Lab., Office of Aerosp. Res., Wright—Patterson Air Force Base, Ohio, Dec. 1961.
12. Truitt, Robert Wesley: Hypersonic Aerodynamics. The Ronald Press Co., N. Y., 1959.
13. Eminton, E.; and Lord, W. T.: Note on the Numerical Evaluation of the Wave Drag of Smooth Slender Bodies Using Optimum Area Distributions for Minimum Wave Drag. J. Roy. Aero.
Sco., Jan. 1956, pp. 61-63.
Figure 1.- Aircraft configurations, = A 75° 5^/5 = 0.0935 2^/2=0.667 W =500,0001 b GTO Figure 2.- Nominal configuration; all body cross sections are elliptical.
Figure 3.- Body shape parameters
A=75
°
ESTIMATE MACH 0.6 1.3 5.0 0-6 (SHARP LEADING EDGE) lift.
DATA (REF. 6) ESTIMATE SHAPE o D
o
.4 A
o
.3 .2 j I I i 8 12 16 20 a, deg Figure 5.- Effect of shape on lift; M = 1.97.
DATA (REF. 6) ESTIMATE SHAPE O D
o
o
Figure 6.- Effect of shape on induced drag; M = 1.97.
ESTIMATE MACH A =75° 0.6
<] 0
d/b = 2.0 1.3 5.0 .4 .3 .2 .05 .10 .15 C C D~ D(BASE) Figure 7.- Effect of Mach number on induced drag.
O DATA (REF.6) THEORY (VAN DYKE) CROSS SECTION CROSS SECTION .16 d/b = 6, A = 7I.6 = 3, A=76.7° .12 Cp . 0 8 _GL-00.
. 0 4 .25 .50 .75 1.0 0 .25 .50 .75 1.0 y/a, SEMISPAN STATION e Figure 8.- Forebody pressure distribution; M = 1.97, a = O
SECTION A-A
^
^8, DEFLECTION ANGLE ^ (EXPANSION) %/ SECTION A-A Figure 9.- Afterbody-pressure-calculation geometry.
O DATA (REF.6) M=l,97 REYNOLDS NUMBER = 8 x I0 ESTIMATE ALTITUDE = 13,000 ft .12 1 / A = TAN' [7.34(a/b)~' 2] .10 .08 .06 FOREBODY PRESSURE DRAG .04 .02 TURBULENT SKIN FRICTION DRAG i i i i i i I 2 3 4 5 ELLIPSE AXIS RATIO (a/b)
Figure 10.- Comparison of forebody zero-lift drag with experiment; no base drag
is included; CD is based on cross-section area.
TYPICAL FLIGHT PATH NOMINAL CONFIGURATION (1000 q PATH ABOVE M = 2) (WITH Z = 100 ft) .10 PRESENT ESTIMATE O AREA RULE WAVE DRAG (WING APPROX~20 STATIONS) (+BODY SKIN FRICTION) .08 n AREA RULE WAVE DRAG(WING APPROX-10 STATIONS) (+BODY SKIN FRICTION) .06 FIN AND BLUNTNESS AFTERBODY PRESSURE .04 FOREBODY PRESSURE BODY SKIN FRICTION .02 0 4 6 8 10 MACH NUMBER
Figure 11.- Zero-lift drag versus Mach number; CD . is based on theoretical
body plan area.
\ NOMINAL CONFIGURATION A = 75° STT-/S = 0.0935 ITT/I- 0.667 500,000 Ib 8 WGTO = . PQ = 7.0'lb/ft (L/D) MAX
•[
4 — i
y '
i i 1 1 1 1 0 2 4 6 8 10 12 MACH NUMBER
Figure 12.- Effect of Mach number on CL/D)MAX-
(L/D) MAX 1.2 = 0.0935 ITT/I = 0.667 60 65 70 75 80 85 SWEEP (A), deg i i i 9 8 7 6 5 4 3 2 1 FOREBODY ELLIPSE RATIO, Q/b
Figure 13.- Effect of body sweep on (L/D)MAX-
8r (L/D) MAX A = 75°
f'=°
.02 .04 .06 .08 .10 .12 .14 FATNESS RATIO, S-^/S I | i i i | 12 8 6 5 4 3 FOREBODY ELLIPSE RATIO, d/b
Figure 14.- Effect of fatness ratio on (L/D)MAX-
(L/D) MAX A =75° if =0.0935 O .4 .5 .6 .7 .8 .9 BREAKPOINT LENGTH RATIO, l^/l 2 3 4 5 6 7 FOREBODY ELLIPSE RATIO, d/b Figure 15.- Effect of breakpoint length ratio on (L/D)MAX- (L/D) MAX 0 400 800 1200 1600 GROSS TAKEOFF WEIGHT (W ), 1000 Ib G T O i i i 100 150 175 200 225 250 275 BODY LENGTH, ft Figure 16.- Effect of gross takeoff weight on (L/D)MAX; gross body density 7.0 lb/ft .
(L/D) MAX GROSS TAKEOFF WEIGHT - GROSS BODY DENSITY = THEORETICAL BODY VOLUME I 2 4 6 8 1 0 1 2 1 4 16 GROSS BODY DENSITY (p ), Ib/ft G i i i 400 250 200 175 150 BODY LENGTH, ft
Figure 17.- Effect of gross body density on (L/D)MAX> gross takeoff weight
500,000 Ib.
(L/D) MAX A I I I | | | | V 0 3 0 0 0 3500 4 0 0 0 4500 5 0 0 0 5500 6 0 0 0 RADIATION EQUILIBRIUM TEMPERATURE, °R 15.0 5.0 2.0 1.0 .50 .25 .10 BODY NOSE RADIUS, ft Figure 18.- Effect of maximum allowable leading-edge temperature.
A - 5 5 « L E PLANFORM AR = 0.933 X = 0.200 t/C = 0.040 5 r (L/D) MAX
t
0 .05 .10 .15 .20 HORIZONTAL FIN AREA/BODY PLAN AREA
Figure 19.- Effect of horizontal fin size on (L/D)MAXJ maximum leading-edge
temperature = 4350° R.
A = 60° LE PROFILE AR = 1.200 X = 0.400 t/C = 0.040 5 r (L/D) MAX
i
0 0.5 1.0 1.5 2.0 VERTICAL FIN AREA/BODY PROFILE AREA
Figure 20.- Effect of vertical fin size on
maximum leading-edge
temperature = 4350° R.
= 50° A L E
pLAr
AR = 1.675 X = 0.200 t/C = 0.040 5 r MACH ^-6.0
4 _ EZTT i—
12.0 (L/D) MAX 3 - 0 .025 .050 .075 .100 CANARD AREA/BODY PLAN AREA
Figure 21.- Effect of canard size on maximum leading-edge
temperature = 4350° R.
1600 M=I2.0 h= 115,000 ft 0=0° NOMINAL CONFIGURATION A = 75° = 0.0935 X = 0.667 N IOOO X £=180 ft I500 X Figure 22.- Planform temperature contours, °F.
2 8 0 0 LOWER SURFACE UPPER SURFACE 2 4 0 0 ' . 2 0 0 0 LJ OL QL LJ Q .
q = 1000 Ib/ft e = 0.8 8 0 0 TURBULENT BOUNDARY LAYER
T
WEDGE, 8 = 7.65°
a B
10 12 14 16 18 20 22 24 MACH NUMBER
Figure 23.- Surface radiation equilibrium temperature; 25 feet aft of leading
edge.
ALTITUDE, 1 0 0 0 ft 2 4 0 0 2 0 0 0 M = 12.0 1200 X = 25 ft € = 0.8 TURBULENT BOUNDARY LAYER WEDGE, 0 = 7.65° B
f
10 20 30 LIFT LOADING (L/S), Ib/fr Figure 24.- Effect of altitude and lift loading on lower surface temperature.
M = I2 .01 T r ALTITUDE = I40.0OO ft NOMINAL CONFIGURATION BASIC BODY -.02 - BASIC BODY BASIC BODY + HORIZONTAL FINS + HORIZONTAL FINS -.03 - SHT=O° + FLOATING CANARD +FIXED CANARD = -a CA -.04 - L i i i i .05 -5 0 10 15 20 25 a, deg Figure 25.- Longitudinal stability.
M = I2 ALTITUDE = 140,000 ft 'MAX NOMINAL CONFIGURATION (FLOATING CANARD) Figure 26.- Horizontal fin control power.
-.01 - "CG -.02- M = I2 -.03- ALTITUDE = 140,000 ft NOMINAL CONFIGURATION (FLOATING HORIZONTAL FINS) -.04 - L I I I I .05 -5 10 15 20 25
a.deg
Figure 27.- Floating canard control power.
.002 M=I2 ALTITUDE = 119,200 ft .001 a = 10° ~^T-\ > "CG -.001 A = 75° STT/S = 0.0935 -.002 W = 500,000 Ib GTO -.003 .4 .7 .8 .9 BREAKPOINT LENGTH RATIO, Figure 28.- tffect of body shape on stability.
CONFIGURATION ANALYTICAL MODEL I I Figure 29.- Directional stability model.
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