aerodynamics of tactical weapons to mach number … · aerodynamics at higher angles of attack for...
TRANSCRIPT
AERODYNAMICS OF TACTICAL WEAPONS TO MACH NUMBER 8 AND ANGLE OF -- ETC (U)
UNCLASSIFIED NSWC/TR-O-36-PT-1 NL
uuianuunuuiluamEEahahhEi
-'E l...-,....
111 j$111 11112251.0 : I2.0
OII III IIIICROO.25 RLU 1.4 E CH6
MICROCOPY RESOLUTION' TEST CHART
get
-X-- 77 M
UNCLASSIFIED __
SECURITY CLASSIFICATION OF TIlS PAGE (When. Does Entered)
I'HEORYADNS AN APPCAIO* -
PFRMINGORNZTIOAMAN ADRES PAGE BFORMELEE PROCT. FOSM
2 GVT CCESIN N. 3 MARE S OCATO NER
Naval. Surfac Weapons CenteERED
Dahiren 1VA 22fI CONTRLN OFFICE NAME AND OTADDRESS,-
6ER D ND APPUUTION SATMN(OfhlReot
Il. ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1 DCOTTRUTCT OTTEEN GRAN th UM1e..E, f0520 I lf1itfre Rpef
IS.ZAIO SUPLEENAR NOTESES1. RGAEEET.POECTS
IS. ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ A KEY WORDS UN~uln~ Io Te'ee NUMBERSeew i Ief 7b bokn
Nava Surfac Weap ( o.I ns C *Oener (K21) I42;em idle,~pb b~k.i~
weeaomd. TheTOIN theory NAM & applS iableto boies Cofrol utfion) Ith nERT or SS two 0 piceisrepsiort)
DO ~147 EDTIONOF NO 05 S OOLEE UCLASSIFIEDSCURIT 15a CLSSF C.AON OTN/ DS WNU.be. bgeO
SCH1EDULE
rUNCLASSIFIED
SECURI1T CLASSIFICATION OF THIS PAGE .I .En.,fedJ
20. Abstract (Continued)s effects and much stronger viscous coupling are important. Static aerodynamic prediction capability
at small angles of attack was extended and improved to )a,. = 6, and an empirical code developed forthe Army by G. Aiello of the Martin-Marietta Corp. (Orlando) was adapted for prediction of staticaerodynamics at higher angles of attack for a limited configurational envelope. The Martin-Mariettacode predicts body-tail aerodynamics for arbitrary roll orientation and angles of attack to 451 for0.8 < . < 3. Bodyelone and wing-alone (in the presence of the body) components can be tom-puted/o a = I80and for 0.8 < <3. New individual computational methods are evaluated bycomparison with other computation]I methods an4'experimental data. The resulting extended "com-putr program was segmented in ord to conserve/storage locations. Computational times dependingon .Mach number, configuration, and logram option range from under a second to a minute per Machnuinber 'on the CIC 6700 computer
UNCLASSIFIEDSr.uIgTY CLASSIFICATION OF THIS PAGirUtef Date AM
FOREWORD
This work has been conducted to expand upon previous work and to provide a design tool foruse in estimating the aerodynamics of today's high performance tactical weapons. The resultingcomputer program, into which this work has been incorporated, allows one to predict performanceand to conduct a static and dynamic stability analysis in the preliminary and intermediate designstages without costly and time-consuming wind tunnel tests.
Support for the work was provided by the following sponsors:
The Naval Sea Systems Command under the Surface-Launched Weapons Aerodynamics andStructures Block Program
The Naval Air Systems Command under the Strike Warfare Weaponry/Aerodynamic/StructuresTechnology Block
The U. S. Army Missile Command under Project No. I LI 62303A214The Office of Naval Research under Project No. F414 11
Major procurement contracts from the above funds were let to North Carolina State Univer-sity, Nielsen Engineering and Research, and Lockheed Missiles and Space Co. Minor contracts werelet to General Electric, Armament System Dept., Burlington, Vermont and COMPRO of Fredericks-burg, Va.
This report was reviewed and approved by Dr. F. Moore (K21) and NW. C. A. Fisher (K20).
Released by:Accessionl For
ITIS GRIP&IDTIC TA19
ul, I!-- .--.- ,J F7R.T. RYLNHaJu Strategic Systems Department
Codes
Diii
ACKNOWLEDGMENT
As indicated in the Foreword and the main report body, the majority of the work presentedhere was performed under contract. Much of the initial basic approach and selection of contractorswas motivated by Dr. F. G. Moore.
A majority of the contract requirement statement and program management work was per-formed by Dr. J. Sun, who is currently employed by Hughes Aircraft (Canoga Park). Dr. Sun alsoinitiated the integration of the LMSC pitch damping derivative prediction, Martin-Orlando high-angle-of-attack and high Mach number static load prediction coding into the original aerodynamicprediction code.
Other individuals have contributed significantly to the final form of the computer code.Acknowledgment of their contribution will be made in the user's guide, the second portion of thisreport to be published later.
Contributors to the computations and illustrations in the report were D. Young, V. Morgan,G. O'Connell, K. Buchberger, and G. Haynes of the Aeromechanics Branch.
The contributions of Compo-So-List of Silver Spring, Maryland and the personnel of theProject Publications Branch (E41) in completing this technical report are greatly appreciated.
iv
CONTENTS
Page
INTRODUCTION ........................................................... 1IANALYSIS................................................................1I
CONFIGURATIONAL GEOMETRY.........................................1IREVIEW OF PREVIOUS METHODS......................................... 2
Body-Alone Static Aerodynamics........................................ 2Wing and Interference Static Aerodynamics ................................ 2Dynamic Derivative Computational Methods ................................ 4
HIGH MACH NUMBER STATIC AERODYNAMICS ............................. 4Body-Alone Inviscid Aerodynamics ...................................... 4Wing-Alone and Interference Aerodynamics ................................ 11I
HIGH-ANGLE-OF-ATTACK AERODYNAMICS ................................ 18IMPROVED DYNAMIC DERIVATIVE PREDICTION ........................... 21
LMSC Body Pitch Damping ........................................... 21LMSC Wing-Body Pitch Damping ....................................... 22Supplemental Dynamic Derivative Prediction ............................... 24
IMPROVED TRANSONIC NOSE WAVE DRAG PREDICTION ..................... 26Background........................................................ 26Wave Drag Prediction for a Family of Blunted Tangent Ogives .................. 27Wave Drag Prediction for a Family of Blunted Cones ......................... 29Method of Analysis for a General Nose Shape .............................. 31
IMPROVED TRANSONIC NORMAL FORCE PREDICTION ...................... 32Background ....................................................... 32Analysis Method ........................ ........................... 32Evaluation of Current and Old CN, and CmaO Predictions ...................... 37Modified Functional Form Fit ......................................... 43
SUMMARY OF NEW ANALYSIS METHODS ................................. 43INDIVIDUAL METHODS EVALUATION ........................................ 43
HIGH MACH NUMBER STATIC AERODYNAMICS ............................ 46Body-Alone Inviscid Aerodynamics ..................................... 46
Wing-Alone Inviscid Aerodynamics ...................................... 46
C'omplete Configuration Aerodynamics ................................... 58HIGH ANGLE-OF-ATTACK AERODYNAMICS ................................ 58DYNAMIC DERIVATIVE PREDICTION ..................................... 58TRANSONIC NOSE WAVE DRAG ......................................... 64TRANSONIC NORMAL FORCE PREDICTION ................................ 64
CONCLUDING REMARKS................................................... 66REFERENCES ............................................................ 73LIST OF SYMBOLS......................................................... 78DISTRIBUTION
v
ILLUSTRATIONS
Figure Page
I General Configurational Geometry ....................................... 22 Methods Used to Compute Body-Alone Aerodynamics ........................ 33 Methods Used to Compute Static-Wing-Alone and Interference Aerodynamics ...... 34 Methods Used to Compute Dynamic Derivatives ............................. 45 Tangent Body Geometry ............................................... 56 Wing with Modified Double Wedge Airfoil Section ........................... 117 Wing with Biconvex Airfoil Section ...................................... 128 W ing Pressure Distribution .............................................. 139 Wing-Body Interference ............................................... 14
10 Geometry of Wing Panel ............................................... 1511 Geometry for Biconvex Airfoil .......................................... 1712 Roll Angle and Fin Load Definitions ..................................... 2013 Pointed Tangent Ogive Wave Drag Coefficients (a = 0) ........................ 2914 Transonic Pressure Drag for Pointed Cone-Cylinders .......................... 3015 Separation Pressure Drag on a Cone-Cylinder ............................... 3116 Typical Missile Configuration (Base Configuration) ........................... 3317 Computational Mesh Configuration for Subsonic Mach Numbers ................ 3318 Quadratic Interpolation Formula for Normal-Force-Curve Slope ................. 3419 Load Distribution for Case 7 ........................................... 3820 Comparison of Cone-Cylinder and Cone-Alone Data for a Pointed Conical Nose .... 4221 New Methods for Computing Static Body-Alone Aerodynamics ................. 4422 New Methods for Computing Wing-Alone and Interference Aerodynamics ......... 4423 New Methods for Computing Dynamic Derivatives ........................... 4524 NASA Flared-Body Data Comparison ..................................... 4725 Blunted Cone Aerodynamics ............................................ 4826 Blunted Tangent Ogive-Forebody Axial Force .............................. 5027 Blunted Tangent Ogive-Cylinder, CNa ...................................... 5228 Blunted Tangent Ogive-Cylinders, x, ..................................... 5429 CNO Comparison for a Wing ............................................ 5630 CDW Comparison for a Wing ............................................ 5731 Comparison of Static Aerodynamics for Finned Body at High Mach Numbers
(a = 00 ) ........................................................ 5932 Aerodynamics for TMX-187 (Moo = 4.65, ai = 00) ............................... 5933 CN Comparison for Air Slew Demonstrator ................................ 6034 xcp Comparison for Air Slew Demonstrator Forward of Body Midpoint ........... 6135 Cq Comparison for Air Slew Demonstrator ................................. 6136 Modified Basic Firmer Aerodynamic Comparison ............................ 6237 Army-Navy Spinner Pitch Damping Comparison ............................. 6338 Basic Finner Pitch Damping Comparison ................................. 6539 Cqp Comparison for Army-Navy Basic Finner ............................... 6540 M-1 17 Bomb Nose Wave Drag Comparison ................................. 6641 CAI Comparison for Blunted Tangent Ogive-Cylinders ....................... 6742 CNa and xcp Comparison for Blunted Tangent Ogive-Cylinders .................. 69
vi
ILLUSTRATIONS (Continued)
Figure Page
43 Comparison of Theory and Test Data for Improved 5"/54 Projectile ............... 7044 Comparison of Theory and Test Data for 175 mm XM437 Projectile ............... 7145 Comparison of Theory and Test Data for 155 mm Projectile ..................... 72
TABLES
Table Page
1 CD =f(RN, LN, M.) for Blunted Tangent Ogives ............................ 282 Fit Parameters ...................................................... 303 Cases Calculated for M. = 0.75 and LB =1 ............................... 354 Cases Calculated for M.~ = 0.90 and LB = 1 ................................. 355 Cases Calculated for M.i, = 0.95 and LB = 1 ................... 366 Cases Calculated for M., 1.2 and LB =1I ................................ 367 Nose Geometries ....... ............................................. 398 CN, and Cm,, Values for Nose-Alone Cases 2, 6', and A8 ....................... 399 CN, and Cma, Values for Nose-Alone Cases, 1, 9, 13, A5, and Al10................40
10 CN, and Cmc, Values for Nose-Alone Cases 4 and A3 .......................... 40
vii
iinWID PAW U. I,'.hOf WM)4
INTRODUCTION
This report presents the theoretical basis for the fourth increment in the development of acomputer code for the rapid prediction of aerodynamics of tactical weapons in the region0<M. <8and 0 < < 1800.
The general approach has been to combine existing and newly developed approximate com-putational methods into a single computer program to compute aerodynamics. Computationaltimes, required for the estimate of static and dynamic aerodynamic coefficients for a body-tail-canard configuration for one freestream condition, are in CPU seconds on the CDC 6700 as opposedto minutes or hours required for more detailed physical and numerical models. The accuracyobtained, however, is compatible with that required for preliminary or intermediate designestimates.
The first increment' led to the development of a code for the prediction of static aerodynamiccoefficients to Mach number 3 and angles of attack to 150 for body-alone configurations.
A second increment 2 supplemented the earlier work and resulted in a code for the predictionof static aerodynamic coefficients for body alone, body-tail, and body-tall-canard configurations forthe same range of freestream conditions.
A third increment 3 supplemented the work of the first two increments and extended thecomputational capability to include the prediction of dynamic derivatives, again for the same rangeof freestream conditions.
The prime objective of the current work effort is to extend the capability of the aeropredictioncode to the higher Mach number and higher angle-of-attack region of advanced designs (M. - 8,e - 180'). A secondary objective is to improve the estimate of static derivatives in the transonic
Mach number regime and to improve the dynamic derivative prediction capability at all Machnumbers.
ANALYSIS
CONFIGURATIONAL GEOMETRY
The basic configurational geometry remains the same as for References 2 and 3. For variousMach number and angle-of-attack regions there are geometric restrictions. These will be elaboratedupon in later sections.
Intended applications are for spin-stabilized projectiles, bombs, and rockets (after bum-out) inthe unguided class and guided missiles/projectiles. Inlet and plume effects for various weapons arenot considered. The code, however, has on occasion been used for some nontactical configurations.
The degree of the configurational complexity that can be considered is illustrated in Figure 1.This most complex body-of-revolution consists of a spherical nose cap, two piecewise-continuousnose sections, a straight afterbody, and a boattail, The wing or canard has a trapezoidal planformwith a biconvex or modified double wedge cross section and sharp or spherically blunted leading ortrailing edges. Tip edges are assumed parallel to the freestream. Twist or airfoil distortion isneglected and piecewise similar airfoil shape variation with span is assumed.
REVIEW OF PREVIOUS METHODS
Body-Alone Static Aerodynamics
A summary of the various methods for computing body-alone aerodynamics appears inFigure 2. The majority of the methods used were adapted from those available in the standardliterature. 4 - 9 The empirical and semiempirical schemes used for the transonic lift and wave dragare presented in some detail in Reference 1. The combined Newtonian-perturbation theory, alsopresented in Reference 1, was developed so that reasonable results for static aerodynamics couldbe obtained at low supersonic Mach numbers for blunt-nosed configurations. Mach numbers 0.8and 1.2 are normal division points for the three Mach number regions.
Wing and Interference Static Aerodynamics
The methods used to compute lifting surface alone and interference aerodynamics are shownin Figure 3. Methods adapted from the standard literature were taken from References 10 through16. The remaining methods for wing-body interference, trailing edge separation drag, body basepressure caused by tail fins, and wing drag numerical techniques are detailed in Reference 2.
Figure 1. General Configurational Geometry
2
MACHNUMBER
COMPONENT REGION SUBSONIC TRANSONIC SUPERSONIC
WU AND AOYOMA SECOND-ORDER VANNOSE WAVE DRAG - PLUS DYKE PLUS
EMPIRICAL MODIFIEDNEWTONIAN
BOATTAIL WAVE DRAG - WU AND AOYOMA VANDDEVAN DYKE
SKIN FRICTION DRAG VAN DRIEST II
BASE DRAG EMPIRICAL
INVISCID LIFT AND WU AND AOYOMA TSIEN FIRST-ORDERPITCHING MOMENT PLUS EMPIRICAL CROSSFLOW
VISCOUS LIFT ANDPITCHING MOMENT ALLEN AND PERKINS CROSSFLOW
Figure 2. Methods Used to Compute Body-Alone Aerodynamics
MACHNUMBER
COMPONENT REGION SUBSONIC TRANSONIC SUPERSONIC
INVISCID LIFT AND LIFTING SURFACEPITCHING MOMENT THEORY EMPIRICAL LINEAR THEORY
W D FSLENDER BODY THEORY AND LINEAR THEORY,WING-BODY INTERFERENCE EMPIRICAL SLENDER BODY
THEORY. AND EMPIRICAL
WING-TAIL INTERFERENCE LINE VORTEX THEORYWAVE DRAG EMPIRICAL LINEAR THEORY +
MODIFIED NEWTONIAN
SKIN FRICTION DRAG VAN DRIEST
TRAILING EDGE SEPARATIONDRAG EMPIRICAL
BODY BASE PRESSURE DRAGCAUSED BY TAIL FINS EMPIRICAL
Figure 3. Methods Used to Compute Static-Wing-Alone andInterference Aerodynamics
3
Dynamic Derivative Computational Methods
Methods for computing the dynamnic derivatives are listed in Figure 4. Applicable referencestaken from Reference 3 are 10 through 12 and 15 through 2-6.
HIGH MACH NUMBER STATIC AERODYNAMICS
This work was performed under contract by F. De Jarnette of North Carolina State Universityduring FY 77-FY 80.
No real gas effects are considered, although some real gas effects become important aboveM = 5. Strong viscid-inviscid interactions are neglected. Lifting surface-body interference is
neglected. The body base pressure, wing base pressure, body and wing friction routines are thesame as those reported in References I and 2.
One consideration is the Mach number range of validity of thc high Mach number predictionmethod. This will be discussed in more detail in the methods evaluation section. A Mach number,M = MV. will be the dividing Mach number at the lower range limit and is a code input.
Body-Alone Inviscid Aerodynamics
The method used for pointed bodies with attached shock waves is based on a modification ofthe method given in Reference 27. For blunt bodies, a modified Newtonian pressure distribution is
matched to the second-order shock-expansion pressure distribution. This method is a modificationof that used in Reference 28.
COMPONENT \REGION SUBSONIC TRANSONIC JSUPERSONICBODY-LONEROLLEMPIRICAL
ROLL DAMPING *SURFACE EMPIRICA
BODY-ALONE MAGNUS THOYEMPIRICALMOMENT
WING AND INTERFERENCE ASMDZRMAGNUS MOMENT ASMDZR
BODY-ALONE PITCH EMPIRICALDAMPING MOMENT
WING AND INTERFERENCE LIFTING LINEARPITCH DAMPING MOMENT SURFACE EMPIRICAL THEORYTHEORY
Figure 4. Methods Used to Compute Dynamic Derivatives
4
The perturbation theory of Reference 4 breaks down when the local slope is greater than the
Mach angle. For more slender bodies, other nonlinear terms, which are neglected by perturbationtheory, become significant at higher Mach numbers even though the local slopes are smaller thanthe Mach angle. Modified linear theory of the type presented in Reference 29 uses the local 1 in aniterative solution and avoids the slope-Mach angle limitation. The approach of Reference 4 isprobably not applicable at Mach numbers above 4. The current approach is given in Reference 30.
For a body at small angles of attack, the deviation of free streamlines from meridian lines is
small and hence is neglected.
From the method of characteristics, for a body of revolution on a free streamline
ap 2yp a5 I ap (I)as sin2m as CosM. aC,
ap 2yp 6 sinsin6 (2)ac - sin 2 p W, r
C, refers to the left running or outgoing characteristic. Incoming characteristics are neglected(i.e., reflection from a bow shock). The angle, b, is the angle of the local streamline tangent; s isthe distance along a streamline. The original body is replaced by a tangent body as shown inFigure 5.
TANGENT BODY
rrr '-- C 3 4
___________________________ __. _.I_
A-A
-- A
Figure 5. Tangent Body Geometry
5
P
The solution starts as a cone solution or as a solution that matches the Newtonian pressuredistribution. The match-point procedure will be discussed in detail in a later section.
At a comer such as 1-2, the terms on the right-hand side of Equation I are neglected com-pared with the left-hand side. Thus, at the comer, the well known Prandtl-Meyer relationshipholds.
On the straight section 2-3, ab/as = 0. If one considers a free streamline close to thesurface 2-3, and assumes that the pressure is constant on that streamline, that the C, characteristicsare straight, that the surface Mach number is a constant, and that AC, is a constant, then one canderive approximately (at a = 0 meridian and streamlines coincide)
p = k, + k2ek3s. (3)
Here, s' = x'/cos 8 is the distance from the corner 1-2, x'is the corresponding horizontal distancefrom the comer, and k, = Pa is the asymptote for the pressure distribution. For 8 > 0 on Section 2-3and s' very long, Pa equals the cone pressure at the freestream Mach number. For 8 < 0, the assump-tion is made that Pa = p., or the freestream pressure. This last assumption can be relatively poor.For 8 < 0, a pressure distribution approximating the recompression on an conical boattail isis needed. In general, Equation 3 must be considered heuristic. The remaining constants can bedetermined from a knowledge of the pressure, P2, and the pressure gradient, (ap as)2 .
The pressure distribution is then given by
P = Pa + (P2 -Pa)e - 7 (4)
(P2-Pa) (5)
where P2 is obtained from the Prandtl-Meyer relationship for an isentropic expansion. Also,
P 2 -- V 1 = --( 2 - -6 1 ) ( 6 )
and
. .= - tan" 1 (M2 -) - tan- -P i (7)
6
........--.-. ~.
where
2'yp (8)sin 2j(
sin- M (9)
(i + M 2) -1/-1 (10)
and po is the stagnation pressure behind a bow shock.
For a compression comer, the isentropic relationship is used. If an isentropic solution forP2 is not obtainable, P = Pa is assumed for the straight section. The term (ap/as) 2 is obtained fromthe approximate isentropic relationship from Reference 27. Thus
as L 2 sinS 1 - sin 2 ) + B2 E_ as X 1() + X2 (11)
where
7pM2
2(M2 - 1)
and7+1
I+ ] (13)2
The same relationship is used for compression corners.
The pressure relationship breaks down when 17 < 0 and an asymptote cannot be reached.When i < 0, p = Pa for an expansion corner and p = P2 for a compression comer are chosen.
The pressure, which is used for load integration, is evaluated at the tangency points (C) inFigure 5.
Equation 4 provides a solution along a meridian plane surface streamline. For small angles ofattack, a pressure coefficient expansion is assumed of the form
7
Cp = (Cp)a= 0 - A sin 2a cos ( + (r cos 2 (p + A sin 2 )sin 2 a/2
= Cpa + (Cp2 -Cpa)c - r (14)
where € is the azimuth angle measured from the leeward plane
A(x) = - I \ (15)
F(x) = a2CP) (16)a 0.0'= 0
and
A(x) = (a 2 o,) (17)
The streamlines make an angle, e, with the meridian lines of the order e - sin sin a. s' isalong a surface streamline. Furthermore, P2 and its oa derivatives are independent of since e issmall; Pa, however is dependent on 0.
It may be readily shown that the loading functions (Cp),=O, A, r, and A have the samefunctional form as Equation 4; i.e.
(Cp)o 0 (Cpa)a= 0 + (Cp 2 -Cpa)a= 0 e'n (18)
A = Aa + (A2 -A,) e- (19)
r = + (W2 - ra) e-r (20)
A = Aa + (A2 - A ,) e-'? (21)
Cp2 is given in terms of Cp1 from the Prandtl-Meyer relationships. Differentiation of thePrandtl-Meyer relations with respect to c once and then twice leads to the following relationships:
A2 = X2 A, (22)
A2 = X2 A l (23)
r12 = 2 {ri + , [G(M 2) - G(MI)J (24)
8
--al
where
G(M) = - (25)
(M2 - 1)3/2
and
X - _ P.\o/(26)LM VM 2 - 1
The expressions for (Cpa)a=O, Aa., ra, and Aa are obtained from an approximation for theasymptotic pressure coefficient; i.e.,
Cpa = (Cpa)a=o + ACp (27)
where (Cpa)a = 0 is zero for 6 < 0 and for 6 > 0 is obtained from an approximate relation for the conesolution taken from Reference 31 where
(Cpa),=n = sin2 6 + Qn [ + I (28)(,y- I )K 2 + 2 K
where
K 2 = (M -) sin26 (29)
Equation 28 is accurate except upon approaching the detachment Mach number.
Next, ACp, the angle-of-attack dependence of Cpa, is given by a blend of slender body andNewtonian approximations; i.e.
ACP -sin 2otsin 26 cos 0 + sin2 jcos 2 6 [2 0 ( - tan2 6) - 2 + sin 2]
(30)
From Equations 15 through 17 and 30 one can obtain
Aa = sin 26 (31)
r. = 2 cos 2 6 [(2 - tan 26) (32)
9
and
A; 2Cos 2 6 (2 1~ l- tan 2 6) -2 + 2#)] (33)
In order to start the solution, Cp I A1.I , and 1 must be specified as initial values. For apointed or truncated body, these relations are given by Equations 28, 29, and 3 1 through 33. For ablunt body (not necessarily spherical), the modified Newtonian pressure distribution may bematched to Equation 14 at some match point. The match point is taken as M = 1.1, as determinedfrom the isentropic pressure relationship combined with the modified Newtonian pressure relation-ship evaluated at a = 0.
The modified Newtonian pressure distribution is given as
Cp = C,,, (cos a sin 6 - sin a cos 6 cos q) 2 (34)
Comparing Equations 14 and 34 yields
C11 = Cpo sin 26 (35)
A1 = CPO sin 5 cosb (36)
F = 2C cos 2 6 (37)
and
A, = sin2 6A = -2C sin (38)
Evaluation of the loading function Equations 18 through 21 at the end of an interval (point3 in Figure 5) continues the solution.
Integration of the pressure distribution yields the loading aerodynamic coefficients as
CA = 8 RR' (C + +_+4 sin2 a dx (39)
CN = 4sin2ce f RAdx (40)0
and
Cm(about nose) = -4sin 2c J RA(x + RR') dx (41)0
All dimensions above are in calibers. The reference area coincides with the maximum noseradius. The second term in Equation 41 is neglected.
10
iWIIII
Wing-Alone and Interference Aerodynamics
The geometry used for the wing is the same as that used by Moore. 2 Two basic airfoil shapesare used-the double wedge and the biconvex. Each of these airfoil shapes may have a sharp orblunt leading or trailing edge. Figure 6 shows the planform and airfoil sections for the doublewedge, and Figure 7 shows them for the biconvex case. No twist or airfoil distortion is allowed.Piecewise similar airfoil shape variation with span is assumed.
The theory used to determine the pressure distribution is based on two-dimensional supersonicflow properties. At a given point on the surface, the slope of the surface relative to the undisturbedfreestream is determined first. If this angle makes a compression surface, the pressure is calculated
from tangent wedge theory. In this theory, the angle between the surface and the freestream isthe wedge angle used to calculate the pressure. If the angle indicates an expansion surface, thepressure is calculated from Prandtl-Meyer theory for an expansion from freestream propertiesthrough this angle. These angles are illustrated in Figure 8, and the two theories are referred to asthe tangent-wedge and Prandtl-Meyer theories. For blunted leading edges, modified Newtoniantheory is used.
Y
K L M N
FF
LE b/2
/ -A... . -
G H I J-
SECTION F-F (SHARP LEADING AND TRAILING EDGES)z
9 (Y D y
XA D
SECTION F-F (BLUNT LEADING AND TRAILING EDGES
fly)
E-. X
Figure 6. Wing with Modified Double Wedge Airfoil Section
II
y
C, N
F FV / --- b12- b/
-A-4
G jC,
SECTION F F (SHARP LEADING AND TRAILING EDGES)
z
Cy) V
x
SECTION F-F (BLUNT LEADING AND TRAILING EDGES)2
Figure 7. Wing with Biconvex Airfoil Section
12
V
-6
EFF
PRANDTL-MEYEREXPANSION (bEFF < 0)
EFF
WEDGESOLUTION(OEFF
> 0
Figure 8. Wing Pressure Distribution
Here
n unit vector normal to the surface
V. = unit freestream vector and
6eff
= effective deflection angle = sin- (-fi
For a wing connected to a body, the two-dimensional theories described above are in errornear the body and the tip of the wing. However, Hilton 32 showed that for a rectangular wing theseerrors cancel each other (see Figure 9). For wings other than rectangular ones, the errors do notexactly cancel, but the predicted forces and moments have been found to be reasonably accurate. 3 3
If the wing pressure is calculated by two-dimensional theory (shock-expansion):
1. The pressure near the body is too low because of wing-body interference
2. The pressure in the tip region is too high because of tip losses
These effects cancel each other and the two-dimensional theory gives the correct force on arectangular wing.
The body ahead of the most forward wing-body junction point affects the wing loading asdoes the body close to the root chord junction line. The wing affects the body loading downstreamfrom the forward wing-body junction point. Slender wing-body interference approximations usedin Reference 2 are valid only for the low Mach number flow regime.
13
M.
BODY
\\ /\ /', /
RECTANGULAR WING /
\ /\ /\ /
Figure 9. Wing-Body Interference
The double-wedge wing shown in Figure 6 has three flat panels on the top and three flat panelson the bottom. The pressure on each panel is determined by the tangent wedge or Prandtl-Meyertheory, and the pressure is constant over each panel. If the leading edge is blunted, modifiedNewtonian theory is used to calculate the pressure over the blunted portion in the same mannerused by Moore. 2 If the trailing edge is blunted, the empirical two-dimensional base pressures givenby Moore2 are used and then extrapolated for the higher Mach numbers.
Each flat panel is represented by the x, y, and z coordinates of the four comer points of thetrapezoidal surface as shown in Figure 10.
Referring to Figure 10, V 1- 4 is the vector connecting points I to 4 and V 2 -3 connects 2 to 3.ThenIA
VI-4 = AX4 " + Ay 4j + AZ4k (42)
wher,,j, and k are the unit vectors in the x. y, and z directions, respectively, and
AX 4 = X4 - Xl
Ay4 = Y4 - Y= b/2 (43)
AZ 4 = Z4 - Z I
and
V- 3 = Ax 3i + AY3J + Az3 1k (44)
14
4
3
Y
2 x
Figure 10. Geometry of Wing Panel
where
AX 3 = X3 - X2
AY 3 = Y3 - Y2 = b/2 (45)
Az 3 = Z3 - Z2
The unit vector normal (outer) to the panel is then
VI V-4 X V2-3IVI- 4 X V2- 3 (46)
Relative to the wing coordinate system, the unit vector parallel to the freestream velocity vector is
c = Cos (o + )1 + sin ( + )j (47)
where a is the angle of attack and 5 is the deflection angle of the wing. The effective angle betweenV. and fi is called 6eff
where
sin 8 ,ff = - oo (48)
Performing the indicated vector operations with the equations above, one obtains
[cos (a + )(Az 4 - AZ3 ) + sin (a + 6)(Ax 3 - Ax 4)Jb/2 49sin ef =D(4
where
D = [b/2(Az 3 -Z4)12 + [-Ax 3 Az 4 Ax 4 Az 3
] 2 + [b/2(Ax 4 -Ax3)12I1 / 2 (50)
If 5tff > 0, the tangent wedge theory is used with a wedge angle of 6eff- The explicit wedge solutiongiven by Reference 34 is used here. If 6 eff < 0, the Prandtl-Meyer expansion is used with 16 effI as
the expansion angle from freestream conditions.
Once the pressure on each panel is calculated, the axial force, normal force, and pitching
moment can be calculated. The effective area for the axial force on a panel is
SA = z 2 -z I +z 4 -z 3 1b/4 (51)
and therefore the axial force is
A = PSA (52)
where p is the pressure on the panel. Note that SA may be positive or negative. The effective area
for the normal force on a panel is
SN 'x2 - x 1 + x4 - x3 ]b/4 (53)
and the normal force is
N = +PSN (54)
where the + sign is used for panels on the lower surface and the - sign for panels on the upper
surface of the wing.
For biconvex airfoils, it is convenient to divide the wing into a front half for x < C/2 and arear half for x > C/2. The chordwise pressure distribution is calculated for the chord at midspan.The geometry of the wing is shown in Figure 1I.
The airfoil section for the front half of the wing is considered first as a cross section perpendic-
ular to the leading edge. In this cross section the airfoil is represented by a circular arc from the
midchord position forward to the nose. If the leading edge is sharp, this circular arc extends to the
leading edge itself. However, if the leading edge is blunted, the circular arc blends into the cylin-
drical leading edge. From the geometry of similar triangles, the slope of the circular arc at its mostforward position is given by
-2(cos AL: Cvg/ 2- rLl: ) (rL -
sin 61 (55)
(COS^X I ag/2 - r 1 )2 + (rL. -
16
YY
CHORD AT MIDSPAN
X,
PLAN VIEW
z
CIRCULAR ARC
rLE/
AIRFOIL OF FRONT HALF AT MIDSPAN(X', Y', Z COORDINATES)
Figure 11. Geometry for a Biconvex Airfoil
17
and the coordinates of this point are
xI = rLi (l -sin 61)/cos ALE(56)
zi = rL1 COS6 1
Now consider the airfoil section in a plane perpendicular to the y-axis. In this plane the circulararc described above is approximated by the parabola
tan 6, cos ALE Cg tavg
z = (x 2+----(5
This expression can be used to determine the effective angle for the tangent wedge or Prandtl-Meyer theories as
tan 61 cOsALE (x - C os" (ae+6)
-± sin (a + 5)
sin 5eff (12 (58)
[I + (tR_-_tt)11
where the + sign corresponds to the lower surface and the - sign to the upper surface. For the
front half of the wing
Cavgx1 < x 2 (59)
The pressure is calculated at 10 positions along this arc and integrated by Simpson's rule to obtainthe axial force, normal force, and pitching moment.
The rear half of the wing is handled in the same fashion as the front half.
For configurations with wings and canards, the analysis described above is used to treat each
configuration like a separate wing. The contributions for each component are then combined todetermine the overall forces and moment.
HIGH ANGLE-OF-ATTACK AERODYNAMICS
Two basic approaches have been used to predict high angle-of-attack aerodynamics.
The first of these is an empirical approach. 35 Body-alone, fin-alone, fin-in-the-presence-of-a-body, and interference aerodynamic coefficient components are obtained from empirical functional
18
form fits for a family of body-tails for a given range of angles of attack, Mach numbers, and rollangles. No control deflections are considered.
The second approach is much more elaborate. 36 It uses the semiempirical or heuristic cross-flow analogy with wing-alone loading and other data for body vortex parameters. Vortices fromthe nose, body, and lifting surfaces are tracked in a stepwise manner with distance along the bodycenterline. Control deflections are considered. Roll moment and side moment coefficients arepredicted as well as longitudinal aerodynamic coefficients. Body-canard-tail configurations as wellas body-tail configurations are considered.
Currently, the aeroprediction code contains the coding from References 35 and 37. Themethod is restricted to body-tail configurations. However, it requires about one-tenth of thestorage required for the code in Reference 36, and computational times are much faster. For thecurrently used method, the range of input parameters for a cruciform configuration are
1. Mach number: 0.8 to 3.0
2. Angle of attack: 0' to 180' for isolated components (roll angle = 00) and 00 to 450 for
body-tail combinations at arbitrary roll angles from 0' to 180'
3. Tail: Trapezoidal planform, edges parallel to body centerline
a. Leading edge sweep angle: 0' to 70'
b. Taper ratio: 0 to 1
c. Aspect ratio (twn fins): 0.5 to 2.0
4. Nose length (pointed tangent ogive): 1.5 to 3.- calibers
5. Cylindrical afterbody: 6 to 18 calibers length and with the end of the body parallel tothe tail trailing edge
6. Total span-to-diameter ratio (two fins): I to 3-1/3.
Roll angle definition and positive fin load orientation is shown in Figure 12. Note that 0 ismeasured 1800 from the definition in the aeroprediction code (i.e., the windward side instead ofthe leeward side).
The axial force coefficient is assumed to be contributed entirely by the body.
The total normal force coefficient is given by
(CN)TOT = CNB + I1CNT 1 +CNT 3 ISin
+ [CNT + CNT 4 I CoS 1ST/Sef + 1BT (60)
19
CNT CNT2
1 2
/0
xCIYT4 CYT3
Ad C-
4 3
V. sin.,
Figure 12. Roll Angle and Fin Load Definitions(Looking Forward)
where ST is the fin planform area, Sref is the body reference area, CNB is the body-alone normalforce, Coefficients, CNTi, are the individual fin loads in the presence of the body and other fins,and IBT is the tail-to-body carryover normal force coefficient.
The longitudinal center of pressure (from the nose) for the entire configuration is given by
(Xcp)TOT = XcpBCNB+[Sini [(xCP)T1CNT 1 + (Xcp)T 3 CNT3
"Cos [(xCP)T2CNT 2 + (xCP)T4 CNT41] ST/Sref
+ IB-rxcpI}/(CN)TOT (61)
All xp values are given in calibers: XcpB is the body-alone Xc, (Xcp)Ti are the individual tail Xcpvalues in the presence of the body, and Xcp, is the center of pressure for the carryover load.
If one neglects the small body frictional roll coefficient and the tail carryover onto the body,one can estimate the roll coefficient as
C - ST/Sr f CNTi sin (i2-i+ ) 1 +t Ycp (62)
Tail carryover data were unavailable for side force and side moment coefficient computa-tions. Details of the individual component functional forms are available in Reference 35 and willnot be presented here.
20
IMPROVED DYNAMIC DERIVATIVE PREDICTION
A portion of this work was performed under contract by L. Ericsson of Lockheed Missileand Space Co. (LMSC). 38 - 40 Only Cmq + Cmor prediction is considered. In fact, the CmG or con-tribution term of constant vertical acceleration is not considered. Restrictions on the LMSC codeare
1. Body or body-tail configurations
2. Aspect ratio, Aw, of the wing-alone limited to less than 2.3
3. Initial cylindrical radius for a spherically blunted or truncated body limited to less than 0.25calibers
4. Contribution of afterbody and boattail or flare are neglected at M. > M* (see Equation 64)
5. Dynamic derivative evaluated in the neighborhood of a = 0.
At low Mach numbers, one has the option of using the older method. At higher Mach numbers,,4the contribution of the lifting surfaces needs to be supplemented in regions where the LMSCroutine is invalid or not used.
LMSC Body Pitch Damping
In subsonic flow Cmq + Cm& is given by a relation based on slender body theory
(Cmq + Cm&)B = -4(0.77 + 0.23 M.)(Qv0.77 + 0.23 M2 - xcg) 2 (63)
Here V is the body length in calibers and xcg is the moment and rotation reference point fromthe nose in calibers. The range of validity is M.. < 1.
Hypersonic flow approximations are applied above a certain Mach number, M. = M* associatedwith an effective hypersonic similarity parameter of 0.4 such that
V* = 0.4 csc 0* (64)
0* = tan- (65)
where LN is the length of the nose in calibers. If MV*< 1.5, then M * = 1.5 is chosen. For M0*> M*,an embedded Newtonian approximation is used. For I < M. < W* a linear interpolation is usedwhere
(Cmq +Cmd,)B = M* (Cmq +Cm&)B(M_=I)
+ ("' ) (Cmq + Cm)B(M_9*) (66)
21
For M. 1> * the computation proceeds as follows:
fLN (x - Xcg + RR')R'
(Cmq + Cm&) B = -1 CpOC7 dx (67)0f (I + R1 2 )
where R and R' are the body radius in calibers and x derivative, respectively, x is in calibers, 6 isthe local surface angle associated with R' (see Figure 5), and CPo is the stagnation pressure behind abow shock
C = -- + 3 15 M2 (68)
Equation 68 is approximate for large Mach numbers. Also
[1.01 + 1.31 [log (10 M. sin OF"' M. sin 6 > 0.4C 1 1.625 M.0 sin 6 < 0.4 (69)
Note that flared bodies are not considered.
LMSC Wing-Body Pitch Damping
For M. < 1, slender wing-body theory is used where
(Cmq +Cm&)w = (Cmq +Cm)wB - (Cmq +Cm&) B (70)
where the subscript B refers to the body, w to the wing, and wB to the wing-body combination.Finally
(Cmq + Cm)wB = -4 (I -0.2332) {[xw Vl - 0.232 - Xcg]2
Cr- -- C ID (1. - ~ ] 21
-4(15) 2 Kma [ x - C1D (I - "-,) (71)--4 -- cg
Here D is the reference diameter of the body, Dw is the mean diameter of the body near thewing, Crw is the root chord, and xw is the x distance in calibers from the nose to the beginning ofthe root chord. Other variables are
=/ I/l -Nei (72)
2bwAw= aspect ratio = (73)
Crw(l + X)
22
= Ctw./Crw taper ratio (74)
CID = Cr,/D { - (I-X) - 2Cr tanp 77 (75)
-1 (76)
t/D=xwD + Crw CrwDteID D + C/D (77)
K2a = 21 + (AO/4) + / + )2 (78)
and
A = (A/Aw)AW
The parameter, A, is the aspect ratio for the planform obtained by extending the leading
and trailing edges to the body centerline, i.e.
(A/Au) + Dw I Db (80)
bw I + X
For supersonic flow, I < Mo < V/l + (A w / 4 )- 2 Equation 70 again applies
(Cmq +Cmd)wH = -4(bw/D)2Kma [te/D - xcg - D-
xwD + Crw+ _ D c- Xg (Cmq + (Cm )B (81)
For hypersonic flow, M. > VI + (Aw/4) 2
mrw bw f* I + X(Cmq Cm w =8/r Dj D -- 6 + 4t*(Q*-a*)
Mq MOWD D
A a* 2 - (0 X)(2t* 2 - 8/3t* a*+a*2 ) (82)
xwD + Crw I CrwD c g (83)
23
a CrW + 2- wb tan 2j (84)2D \2Cr w I
f* is the dynamic pressure ratio across the bow shock and is defined as
1I.0 KN < 1.25
KN- 1.25
f = o 12.669724 1.25 < KN l< 1] (85)
.17 KN > 11
KN = M. sin 6'n (86)
ON = sin-1 x/CDN/(C-yCpo) (87)
0 N is an effective cone angle for the bow shock in which the wing is immersed and CDN is the
nose wave drag where
CDN = 8 f CyCp sin 2 5RR' dx (88)0
Supplemental Dynamic Derivative Prediction
The LMSC work was assumed to be able to replace the empirical prediction of Reference 17for the body-alone pitch damping coefficient. Since the LMSC prediction neglects the effects ofan afterbody and a boattail at supersonic Mach numbers, the method of Reference 17 often pre-dicts better values for the pitch damping (as will be shown later). For this reason and because ofthe other limitations indicated in the last two sections, the pitch damping LMSC prediction forbody-alone and body-tail is currently an option in the aeroprediction code. Even when the LMSCpackage is chosen, the various elements of the package will be bypassed when any of the coderestrictions are not met.
At higher Mach numbers, where M. > MQ, the potential models for determining the liftingsurface roll damping and pitch damping derivatives are assumed to be invalid and are supplementedby the strip theory methods.
For a wing, the strip loading for Cqp and Cmq is assumed to be proportional to the localchord such that
_Q - N(CNa)w 0bw/ 2 (y + w2 C d
D 2 ( _W) .22w (I + X) 0
N(CN)w (bw + Dw) 3 X - (Dw) 3 + 4± X) [(bw + D w) 4 - D 4 1 (89)
6D 2bw(l + X) 4 bw W
24
For the wing-alone case, D is replaced by bw for determining CP. Equation 89 predicts highvalues of Cep. This expression was subsequently replaced by an expression that assumes the loadinggoes to zero at the tip as /I - (y/bw/2) 2 - 0 such that
Cp= 411 \ (w) 2 W) + 13 (90)
N(CI~ (b + W (90)
where C,,f is the reference length and N is the number of fins.
(- X)4 3
1 (1 -X)i12 = 6 - " 6
7r 2 (1 -X)13 =
16 15
Similarly, for zero roll angle and zero angle-of-attack position
mq 4(CN)w f bw/ 2 C((y)=m - -D(I)Cw+) f (Xcg - w - y tan A1 - x) 2 dxdyD2 (bw/2-)Crw (I + X) f ,o
(CNadw A (Xcg - Xw)4
3 (bw/2)Crw(I + X)D 2 an AxL
-xCg - Xw 2 tan A2 1/)I
_ an A - Crw ( I -N) L(xcg - Xw - Crw)4
[tan AI w/2
(xc Xw- XCrw- bw/2 tan Ai)4]} (91)
Here (CNa)w is the wing-alone CN, for two fins and A, = ALE is the leading edge sweep angle.For the wing alone case, D is replaced by Crcr = 2/3 (C2w + C'w + Crw Ctw)/(Crw + Cw). Equa-tions 90 and 91 are applied to both the canard and the tail and added together. Interference isneglected as was the case with the high Mach prediction of F. De Jarnette.
For the wing-alone potential problem for constant roll rate, the loading is zero at the root andthe tip. However, for strip theory, the lift problem loading does not vanish at the wing tip.
25
The CQp and C,1,q models given in Equations 90 and 91 will be evaluated in more detail later.
IMPROVED TRANSONIC NOSE WAVE DRAG PREDICTION
Background
In Reference I ti"' inviscid axial force was assumed to be contributed by a linear superposition.. of component parts of the body in the transonic flow regime. The nose wave drag was based on a
fit to a small disturbance potential model solution for a family of pointed tangent ogive noses.The nose wave drag was fitted as a function of Mach number and nose length. The effect of nosebluntness was ignored.
A separate nose wave drag uncoupled from the boattail and wake effects is valid for supersonicMach numbers. It is also valid for subsonic Mach numbers where a long cylindrical afterbodyfollows the nose section.
When the cylindrical afterbody is not long, the boattail wake and nose flow fields are stronglycoupled.
Currently, the boattail wave drag is based upon a small disturbance potential solution thatassumes a long cylindrical afterbody ahead of the boattail, and a supersonic Mach number close to
1.0. The boattail drag is assumed to decay linearly to zero at M. = 0.95 from the predicted valueat M. = 1.05.
The base drag prediction is empirical and is determined by the expression CAB = -CpBA(M.)(RB/Rref) 3 where CpBA(M,,) is the base pressure coefficient for a long afterbody with no boattail.This empirical relation is approximate and is valid for very small boattail angles and a long afterbody.
For a body with strongly coupled nose, boattail, and wake effects, it would probably be bestto vary the sum of the wave drag and base drag for Mach numbers below M. = 1.0 in a mannersimilar to the variation of CpBA(M-). However, this approach was not implemented due to a lackof data.
The remainder of the discussion deals with the case of a separable nose wave drag.
In Reference 4 I, it is shown that the variation of the nose wave drag is shape dependent. Fornoses with a finite angle at the shoulder, the pressure drag does not go to zero as the Mach numberis decreased due to the effect of the flow separation at the shoulder on the pressure distributionon the nose.
In Reference 41, it is indicated that the nose shapes of interest are bounded approximatelyby families of cones and ogives. This is the approach taken here.
26
/k.- _..
Wave Drag Prediction for a Family of Blunted Tangent Ogives
This work was performed under contract by Nielsen Engineering and Research (NEAR).Initially the wave drag was to be based on interpolation between tabulated wave drag values for afamily of blunted tangent ogives. The matrix of independent variables for this situation is
M.o = 0.7, 0.95, 1.05. 1.2
(nose length)LN = 0.75, 1.0, 1.25. 1.5, 2.0,3.0.4.0, 5.0
(nose radius) RN = 0. 0.05, 0.1. 0.15. 0.2, 0.25, 0.30, 35, 0.4, 0.45, 0.5
The wave drag is based on pressures generated by a time asymptotic solution of the Euler equations.Details are reported in Reference 42. Actual computations were made at RN = 0.025. 0.05, 0.15.0.25, 0.35. and 0.5. and L\ = ., .3.0. 8.0. The computations at RN = 0.5 are of course independ-ent of nose length. The lull geometric matrix was determined by cross plotting and graphicalextrapolation and interpolation. ('omputations for short noses with small bluntness did not agreewith the full potential model of Reference 43 or data. Therefore, portions of the wave drag matrixwere recomputed using tile South-Jameson code.43 The new matrix generated is previously
unpublished data and is given in Table I for 2 RN and 2 LN.
First. for variable geometric parameters, the wave drags at a given Mach number are deter-mined by a four-point Lagrange interpolation. Next. for variable nose bluntness, the wave drags ata given nose length are determined b, a five-point Lagrange interpolation. Finally, for a given nosebluntness, the final wave drag is computed by a five-point Lagrange interpolation.
Two problems arise in implementing this wave drag algorithm. For blunter noses, the wave
drag is still significant for Mach numbers below M_ = 0.8. In this case, a quadratic decay to zerowave drag at M. = 0.5 is assum1ed. For pointed ogives, the apex angle may lead to a detached shockat M. = 1.2 for LN - 3. Small disturbance potential solutions, however, are possible betweenM_ = i.2 and the detachment Maclh number provided L., is limited to alues greater than about1.75. For LN = 1.75, tle initial angle is approximately equal to the Mach wave angle at detachment.This coincides with a Mach number of the order 1.5. !igure 13 shows the approximate variation
with Mach number for a few pointed tangent ogives. I-or M_ < 1.2. Ct) is given by the NEARtabulated data. For M. > 1.5 computations are established using a second-order shock-expansioncomputation program described in an earlier section. Note that the second-order shock-expansionmethod may not be applied below conic shock attachment Mach numbers with any accuracy. Apotential solution of the type described in Reference I will fill the gap between M. = 1.2 and theattachment Mach number. For LN 1.75. however. the initial slope will be greater than the Mach
wave angle, somewhere between M_ = 1.2 and the shock attachment Mach number.
27
Table 1. CL) = '(RN, LN, M.) for Blunted Tangent Ogives
2RN 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.02 LN
M. = 0.8
1.5 0.040 0.038 0.037 0.035 0.032 0.028 0.023 0.018 0.015 0.030 0.0922.0 0.025 0.022 0.016 0.010 0.005 0.002 0.000 0.000 0.000 0.012 0.0922.5 0.010 0.006 0.004 0.002 0.001 0.000 0.000 0.000 0.002 0.015 0.0923.0 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.005 0.019 0.0924.0 0.000 0.000 0.000 0.000 0.000 0.000 0.002 0.007 O.O2 0.028 0.0926.0 0.000 0.000 0.000 0.000 0.000 0.000 0.002 0.008 0.023 0.045 0.0928.0 0.000 0.000 0.000 0.002 0.005 0.010 0.015 0.022 0.037 0.055 0.092
10.0 0.000 0.000 0.000 0.002 0.005 0.010 0.010 0.024 0.042 0.062 0.092
M. = 0.95
1.5 0.160 0.197 0.203 0.203 0.198 0.189 0.180 0.167 0.154 0.150 0.2792.0 0.080 0.137 0.145 0.138 0.130 0.118 0.103 0.085 0.070 0.1.'2 0.2792.5 0.040 0.080 0.090 0.089 0.075 0.070 0.063 0.047 0.038 0.110 0.2793.0 0.015 0.052 0.062 0.062 0.050 0.042 0.034 0.028 0.033 0.116 0.2794.0 0.000 0.15 0.027 0.027 0.020 0.015 0.012 0.008 0.050 0.132 0.2796.0 0.000 0.000 0.005 0.002 0.002 0.003 0.010 0.027 0.075 0.152 0.2798.0 0.000 0.000 0.002 0.002 0.004 0.015 0.018 0.055 0.088 0.160 0.279
10.0 0.000 0.000 0.000 0.002 0.009 0.015 0.025 0.055 0.090 0.164 0.279
M - 1.05
1.5 0.280 0.333 0.335 0.333 0.329 0.322 0.313 0.300 0.285 0.270 0.4052.0 0.200 0.268 0.272 0.208 0.202 0.252 0.238 0.221 0.207 0.249 0.4052.5 0.155 0.220 0.225 0.222 0.213 0.201 0.185 0.170 0.162 0.248 0.4053.0 0.135 0.165 0.175 0.175 0.167 0.152 0.135 0.130 0.157 0.249 0.4054.0 0.110 0.130 0.133 0.130 0.122 0.111 0.100 0.102 0.161 0.252 0.4056.0 0.078 0.080 0.076 0.070 0.064 0.062 0.070 0.105 0.170 0.257 0.4058.0 0.055 0.050 0.045 0.043 0.047 0.055 0.070 0.105 0.177 0.262 0.405
10.0 0.036 0.032 0.025 0.027 0.037 0.050 0.070 0.105 0.182 0.267 0.405
M = 1.2
1.5 0.411) 0.457 0.408 0.470 0.4o8 0.460 0.448 0.458 0.418 0.414 0.5502.0 0.331 0.383 0.307 0.393 0370 0.364 0.348 0.335 0.328 0.388 0.5502.5 0.283 (.338 0.332 0318 0.300 0.2x 0.275 0.275 0.290 0.387 0.5503.0 0.247 0.280 0.273 0.260 0.245 0231 0.223 0.232 0.285 0.387 0.5504.0 0.194 0.205 0.100 0.175 0.100 0.155 0.1165 0.204 0.280 0.387 0.5506.0 0.108 0.102 0.()92 0.087 0.086 0.102 0.135 0.10 0.278 0.387 0.5508.0 0.065 0.060 0.050 0.048 0.063 0.095 0,130 0.185 0.275 0.38 7 0.550
10.0 0.0318 0.038 0.038 0(1045 0.051) 0.090 0 130 0.185 0 -7 3 0.387 0.550
" .. . . . .. id I i IIII .. . . . .. ..2" ,.II I1 i I 3 7 . . . . a : - _ _. ---8
0.8- o AFATL DATA (REF 4410 HIGH MACH AND NEAR
0.7 COMPUTATION POINTSNEAR (0.8 - M " 1.21
HIGH MACH (M ' MDCTACHI
0.50-5 -- V...-W. =
NEAR APPROX SHOCK/ s-- - DETACHMENT FROM
0.4 / ./ " 'd -, ==CONE TABLESo 0.4
L10.3 LN = 1-5
HIGH MACH
0.2- L =/N
01 LN 3
0.0-06 1012 2.0 3.0 4.0 5.0 5.6
M
Figure 13. Pointed Tangent Ogive Wave Drag Coefficients (a = 0)
Wave Drag Prediction for a Family of Blunted Cones
As indicated in the introduction and at the beginning of this section, the nose wave drag isshape dependent. The approach taken here is that the nose is either a blunted tangent ogive or ablunted cone. The only alternative would be to use a code such as that presented in Reference 43to generate families of solutions for various nose shapes.
Figure 14 shows the pressure drag versus Mach number variation for various pointed cones.The pressure drags are established by integration of the pressure data from Reference 45. Note
that the transonic flow regime or shock attachment does not start until M. > 1.2 for angles above 1 5".
Data were available up to 6 = 40'. However. 6 = 250 is about the maximum practical anglefor many applications. Note that the pressure drags are higher than those for tangent ogives of thesame length. Noses such as secant ogives may be fitted by a cone with the effective cone angleapproximately equal to the mean value.
The total forebody drag (pressure drag plus Viscous drag) is relatively independent of bluntingfor Mach numbers below shock attachment. 46 Data in Reference 46 are for cones instead of cone-cylinders. The drags are different for the two cases. however, it is a;;umed that the effect ofblunting is the same.
I-or Mach numbers below M_ = 0.9, separation effects at the cone-cylinder junction affect thepressure on the nose. As the Mach number is lowered, the pressure drag does not go to zero. The
2'9
0.7
0.6
0.5-- 25'~SONIC CONDITIONS
ON SURFACE OF
0.4 INFINITE CONE
0.3
0.2 15
0.1 /0 0-------------------
6= 5'
0.3 0.5 0.7 1.0 1.5 2.0M
Figure 14. Transonic Pressure Drag for Pointed Cone-Cylinders
residual drag was termed viscous drag in Reference I. However, the magnitude indicated inReference I. is incorrect, since the data used are based on cones rather than cone-cylinders.
The drag behavior below M,, = 0.7 was established by the assumed asymptote
CAI, = (A po, + C'Aple - /M
All of the undetermined coefficients may be determined from CAP data at M. = 0.7, 0.8, and 0.9.Table 2 gives the resulting fit parameters.
Figure 15 is a plot of CAp,, against 6 compared with the asymptote from Reference 1, whichis assumed to apply below M. = 0.8 only. The asymptotic behavior determined here is indicated inFigure 14 by dashed curves. For all practical purposes. the asymptote for CAp is reached by M_ = 0.5.
Table 2. Fit Parameters
5 CAp, (ApI A
5 0.00041 10.812 10.09210 0.00780 13.50o 5.38115 0.024o4 75.810 6.46820 0.04871 41.728 5.66525 0.08037 22.11) 4.760
30
0.202 CONE DATA
C.)
0.1 CONE-CYLINDERDATA
0-0 10 20 30 40
CONE HALF ANGLE
Figure 15. Separation Pressure Drag on a Cone-Cylinder
The pressure drag for cones may then be given by table lookup between the limits0.5 < M., < 1.5. By limiting the angle to 20', the Mach number range becomes 0.5 < M.o < 1.2.
Method of Analysis for a General Nose Shape
At the beginning of this section, it was indicated that the wave drag is dependent on shape.It remains to estimate the wave drags for nose shapes that are neither blunted tangent ogives orblunted cones. Unfortunately. iot enough numerical or test data are available to test any empir-icism. It would be advantageous to consider a power law base as well if these data were available.
One possibility is a linear interpolation as given in the relation
= I(Caw)T.o.(R I - R ') + 2Rs(Caw)c()] (92)R . + R'I
where (Caw)T.O. is the tangent ogive value of wave drag, 5s is the shoulder angle, and 61 is theinitial nose angle for a pointed or truncated nose. Here 5, is the initial angle at the end of thespherical cap for a spherically blunted nose and (C,w)c(6) is the cone wave drag evaluated at 6.R 1 and R' are slopes corresponding to 6 and 6, .
Thus, when the shoulder angle is zero, a tangent ogive value is computed. When the shoulderand initial angle are equal, the cone value is computed. For a truncated body, the radius RN isassumed to be that of a pseudospherc that could be added.
31
Unfortunately, not enough experimental or numerical data are available to test the heuristicexpression.
IMPROVED TRANSONIC NORMAL FORCE PREDICTION
Background
In Reference I, (,, and C,, were assumed to be contributed by a linea, m perposition ofcomponent parts of the body in the transonic flow regime. Nose properties were assumed to beindependent of other parts of the body.
Boattail properties were assumed to be independent of the nose and afterbody. The nose-alone CN, was based on the cone-alone normal force data and xcp is given by slender body theory.
The afterbody normal force. CN&, is given as a function of afterbody length and Mach numberindependent of other parts of the body. Both CN, and x, are given by the transonic small dis-turbance theory of Reference 6. For low Mach numbers, CNI is predicted from data correlation.
For the boattail, CNI is based on data correlation: xcp is given by slender body theory. A longafterbody is assumed.
This combination of methods used in Reference I was necessitated by the lack of a theoreticaland experimental data base at the time Reference I was written.
The method of Reference 47 was intended to replace that of Reference 1. The work wasperformed under contract by NEAR.
Analysis Method
The method of Reference 47 solves the Euler equations by an implicit time asymptoticmethod at a I' angle of attack. The resultant CN, and C,,,, were to be fitted as a function of bodygeometric parameters and Mach number. Geometric lengths were given in caliber dimensions. Theparameter ranges to be considered are taken from Reference 47 and are shown in Figure 16 forbodies with blunted tangent ogive nose shapes.
A number of problems have arisen in implementing this method. For a fairly coarse grid of48 (points along the body) by 12 (points normal to the body) by 19 (equally spaced circumferentialplanes), approximately 3900 ('DC 7600 CPU seconds are required per configuration. A typical gridtaken from Reference 47 is shown in Figure 17. Needless to say, the computational time isprohibitive. For each Mach number, 19 or 20 configurations were considered. In some cases, a com-pletely converged solutior was not obtained. In other cases, the grid was too coarse (for longerbodies).
The functional fit form is a truncated Taylor series with free coefficients fitted by least squares.The functional form is shown in Figure 18. The configurations actually computed are given inTables 3 through 6. 0 in Figure 18 refers to reference conditions, ARN = (RN - .25)/.5; ALN= (LN -
2 .5)/5 AL A (LA - 2.0)/5; and AO 0 = (0 B- 5 )/ 1 5 are the quantities given in Figure 18.
32
RUB
D
NOSE AFTERBODY BOATTAIL
PARAMETER RANGE: 0.75 M <- 1.2
0.025 - RN - 0.5
1.5 < LN _ 5.0
0 LA 5.00 <:LB <_2.0
'
0 0 -B 10 °
Figure 16. Typical Missile Configuration (Base Configuration)
-14 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 14
x
Figure 17. Computational Mesh Configuration for Subsonic Mach Numbers
33
z
z ca 0
o z .1
Z E
zzzz I
I-
4 -
0 z z ~
1-j cc .
m Z NO
zz
z E
2~ ZN 3
zct, z z-.c
z 0
+U +
N N <1
i~-0O
zz I- 34
Table 3. Cases Calculated for M. = 0.75 and LB = j
Geometric Parameters Euler Code Values Least-Square Values
Case OB dCN dCN
RN LN LA (Degrees) LB da Xcp da Xcp
1 0.25 2.5 2.0 5 1 1.542 -0.4714 1.5776 -0.32622 0.05 2.5 2.0 5 1 1.750 0.04893 1.7747 0.21463 0.50 2.5 2.0 5 1 1.195 -1.925 1.1821 -1.96984 0.25 1.5 0.25 5 1 1.403 0.07262 1.4097 -0.13115 0.25 5.0 5.0 5 1 1.478 -2.280 1.4771 -2.2896
6' 0.05 2.5 0.25 0 1 2.393 1.328 2.3513 1.21067 0.50 1.5 5.0 0 1 1.935 0.8494 1.9309 0.87738 0.50 5.0 0.25 0 1 2.085 1.154 2.0847 1.19919 0.25 2.5 2.0 0 1 2.202 1.147 2,2267 1.0143
10 0.05 5.0 5.0 0 1 2.295 2.491 2.2955 2.5022
11 0.50 1.5 5.0 10 1 0.6884 -9.249 0.7051 -8.999212 0.50 5.0 0.25 10 1 0.2472 -29.78 0.2479 -30.134813 0.25 2.5 2.0 10 1 0.9802 -3.503 0,9470 -3.586415 0.05 1.5 0.25 10 1 0.9875 -0.5368 0.9872 -0.6565A3 0.25 1.5 2.0 5 1 1.548 -0.4303 1.5073 -0.5041A5 0.25 2.5 5.0 5 1 1.622 -1.326 1.6094 -1.3903A8 0.05 2.5 2.0 10 1 1.246 -1.464 1.2478 -1.5871AIO 0.25 2.5 0.25 i0 1 0.8119 -2.269 0.8249 -1.8410All 0.10 1.5 0.25 0 1 2.124 0.0821 2,1458 0.2236
Table 4. Cases Calculated for M.. = 0.90 and LB = 1
Geometric Parameters Euler Code Values Least-Square Values
Case 0 B dCN dCNRN LN LA (Degrees) LB da Xcp da Xcp
1 0.25 2.5 2.0 5 1 1.498 -0.8088 1.4542 --1.09392 0.05 2.5 2.0 5 1 1.689 0.0848 1.6834 -0.06173 0.50 2.5 2.0 5 1 1.305 -1.973 1.3115 -1.84704 0.25 1.5 0.25 5 1 1.479 -0.03936 1.4685 -0.31155 0.25 5.0 5.0 5 1 1.437 -2.764 1.4424 -2.7339
6 0.05 2.5 0.25 5 1 1.494 0.3263 1.6030 0.64617 0.50 1.5 5.0 0 1 2.124 1.178 2.1156 1.16138 0.50 5.0 0.25 0 1 2.008 0.7239 2.0057 0.73589 0.25 2.5 2.0 0 1 2.240 1.113 2.2653 1.0701
10 0.05 5.0 5.0 0 1 2.302 2.433 2.2981 2.4400
11 0.50 1.5 5.0 10 1 0.6003 -12.23 0.6023 -12.309712 0.50 5.0 0.25 10 1 0.4092 -18.29 0.4096 -18.409013 0.25 2.5 2.0 10 1 0.9875 -3.926 0.9687 -4.020815 0.05 1.5 0.25 10 1 1.032 -0.8020 1.0293 -0.8941
A3 0.25 1.5 2.0 5 1 1.388 -1.751 1.4182 -1.3427A5 0.25 2.5 5.0 5 1 1.562 -1.808 1.5659 -1.7509A8 0.05 2.5 2.0 10 1 1.369 -1.125 1.3755 -1.0947AIO 0.25 2.5 0.25 10 1 0.8812 -2.247 0.8938 9900AI l 0.10 1.5 0.25 0 1 2.366 0.8041 2.3546 , 441
35
Table 5. Cases Calculated for M. = 0.95 and L =1
Geometric Parameters Euler Code Values Least-Square Values
Case 013 dCN dCNRN LN LA (Degrees) LB d xcp da cp
1 0.25 2.5 2.0 5 1 1.641 -0.5432 1.5336 -0.92622 0.05 2.5 2.0 5 1 1.833 0.2568 1.8396 0.14363 0.50 2.5 2.0 5 1 1.361 -2.192 1.3652 -2.11124 0.25 1.5 0.25 5 1 1.352 -0.7351 1.3685 -0.85395 0.25 5.0 5.0 5 1 1.543 -1.964 1.5733 -1.8363
6 0.05 2.5 0.25 5 1 1.652 0.3024 1.6443 0.38217 0.50 1.5 5.0 0 1 2.225 1.048 2.1860 0.96788 0.50 5.0 0.25 0 1 2.118 0.6828 2.0962 0.64219 0.25 2.5 2.0 0 1 2.193 1.021 2.3506 1.2406
10 0.05 5.0 5.0 0 1 2.330 2.506 2.3057 2.4914
11 0.50 1.5 5.0 10 1 0.5024 -20.17 0.5382 -18.572912 0.50 5.0 0.25 10 1 0.7661 -8.846 0.7820 -8.595213 0.25 2.5 2.0 10 1 1.252 -2.260 1.0861 -3.423015 0.05 1.5 0.25 10 1 1.323 -0.4737 1.3319 -0.4645
A3 0.25 1.5 2.0 5 1 1.442 -1.717 1.4924 -1.3205A5 0.25 2.5 5.0 5 1 1.632 -1.639 1.6299 -1.6229A8 0.05 2.5 2.0 10 1 1.456 -0.8952 1.5107 -0.6385AlO 0.25 2.5 0.25 10 1 0.8763 -2.764 0.9269 -2.2428All 0.10 1.5 0.25 0 1 2.471 0.8912 2.3989 0.8074
Table 6. Cases Calculated for M., = 1.2 and LB = 1
Geometric Parameters Euler Code Values Least-Square Values
Case 0B dCN dCNRN3 LN LAd daNRN LN LA (Degrees) LB do Xcp d- cp
1 0.25 2.5 2.0 5 1 2.136 0.1233 2.1773 0.60582 0.05 2.5 2.0 5 1 1.989 0.7118 1.9209 0.33013 0.50 2.5 2.0 5 1 2.167 0.0200 2.1823 0.03134 0.25 1.5 0.25 5 1 2.740 0.6582 2.7247 0.70385 0.25 5.0 0.50 5 1 2.334 1.403 2.3160 1.2874
6' 0.05 2.5 0.25 0 1 2.500 1.494 2.5538 1.59697 0.50 1.5 5.0 0 1 2.584 0.8771 2.6007 0.93268 0.50 5.0 0.25 0 1 2.592 0.7101 2.6184 0.75639 0.25 2.5 2.0 0 1 2.562 0.8747 2.4646 0.5613
10 0.10 5.0 5.0 0 1 2.180 -1.888 2.1965 -1 7688
11 0.50 1.5 5.0 10 1 1.983 -1.035 1.9505 -1.166112 0.50 5.0 0.25 10 1 1.894 -0.9524 1.8917 -0.921613 0.25 2.5 2.0 10 1 1.833 -0.6771 1.9130 -0.296715 0.05 1.5 0.25 10 1 2.176 0.7843 2.1743 0.8177
A3 0.25 1.5 2.0 5 1 2.247 -0.0886 2.2748 -0.1955A5 0.25 2.5 5.0 5 1 2.265 -0.0367 2.2831 0.0177A8 0.05 2.5 2.0 10 1 1.751 0.0354 1.7516 0.1103AIO 0.25 2.5 0.25 10 1 2.323 0.7546 2.2464 0.4127All 0.10 1.5 0.25 0 1 2.846 1.153 2.8508 1.1936
36
The Taylor series is used for the nose-alone; afterbody-alone, and 0.5-, 1.0-, and 1.5-caliberboattails. In the actual computations, only a 2-caliber boattail is considered. The loading on0.5-, 1.0-, and 1.5-caliber boattails is obtained by integrating over the truncated portion of the full2-caliber boattail.
The interpolation procedure proceeds as follows: At Mo = 0.6, the nose-alone, afterbody-alone, and boattail-alone CN, and Cma values are obtained by the method of Reference 1. Next, atM. = 0.75, 0.9, 0.95, and 1.2, the nose-alone; afterbody-alone; and 0.5-, 1.0-, and 1.5-boattail-length contributions to CNa and Cma are determined from the Taylor series. At each Mach number,the boattail-alone contribution is determined by a four-point Lagrange interpolation for the boattaillength of interest. For a zero length boattail, CN, and Cm contributed by the boattail are zero.Finally, the nose-alone, afterbody-alone, and boattail-alone contributions are determined at theMach number of interest by a five-point Lagrange interpolation in the values at M. = 0.6, 0.75,0.9, 0.95, and 1.2. Since two different methods are used, a smooth merge would not be expected atM. = 0.6. At M., > 1.20 the code as currently written would transfer to a potential code for thelow supersonic range. Again, a smooth merge would not be expected for the current new coding orthe older coding at M. = 1.20.
At subsonic Mach numbers, the hemispherical end configuration leads to poor pressure dis-tribution prediction on the aft end of the body, especially for 0 B = 00. Some guidance from flowvisualization is needed in choosing a free streamline wake boundary which may be approximated bya solid surface. The next best thing is an extended smoothly varying stinglike wake free streamline.The alternative is to couple a boundary layer and transonic computation such as used in Reference48. The model of Reference 48 would have to be extended to include a recirculation model near thebase or a knowledge of the free streamline shape.
Examination of dCN/dx in Reference 47 (see Figure 19) indicates that this model predicts aboattail loading at 0
B = 0. This is questionable, particularly for M. = 1.2, for the example shown inFigure 19. Unfortunately, the 0 B = 0 numerical data are included in the Taylor series fit. Thetruncated portions of a 2-caliber boattail are not the same as those of 0.5-. 1.0-, and 1.5-caliberboattails. However, the prohibitive computational cost led to the current approach.
A proper evaluation of the current method would ideally require extensive dCN/dx distribu-tion data or Co data that could be used to determine dCN/dx. The majority of the data availableare total load data for complete configurations. However, the basic input data to the fits needto be examined and evaluated as well.
Evaluation of Current and Old CNO and Cma Predictions
One limit that one could examine is the case of a long constant radius afterbody. In principle,the nose and boattail should be isolated in this case. Indeed, this was the assumption of Reference I.
Nose-Alone Contribution. Nose geometry considered could be listed as (see Tables 3-6) inTable 7.
37
0.07 - - 0.07 -
0.06 M RN LN LA LB B 0.06 !M, RN LN LA LB B
0.05 1075 05 15 50 20 0 005 00 05 15 5.0 2.000.04
0.04 0.04
0.03( 0.03
0.02 7- 0.02 - R
dCN 0.01- 0. 0 0.01
00 0.03
0.03 0.02
dN 0.02 0.01-
0.0201 0.02
0.00 0.01-
-0.04 -0.04
0 12 3 4 5 6 7 8 90 1 2 3 4 5 6 7 8 9
(a) M.o = 0.75 (b) M. = 0.90
0.07
0.70.06- R N L, L LBn0.6 lyo .05 12 .5 1.5 .0 2.0 00.05 -o.5 0.5 1.5 5.0 2.0 o 0.04
0.04 0.03
0.03 0.02
dC N 0.02 d -0.0
dx 0.01- 2 0.00
0.00- 0.01
10.010.02-
0.02- 00.03-1
0 03 /0.04--
0 1 2 3 4 5 6 7 890 1 2 345 6 7 89
xx
(c) M.o 0.95 (d) M.o 1.20
Figure 19. Load Distribution for Case 7
38
Table 7. Nose Geometries
RN LN Case(s)
0.05 1.5 150.05 2.5 2, 6', A80.05 5.0 10 (not for M. 1.2)0.1 1.5 All0.1 5.0 10 (only for M. = 1.2)0.25 1.5 4, A30.25 2.5 1,9, 13, A5, A100.25 5.0 50.5 1.5,2.5,5 (7, 11), (3), (8, 12)
The hemispherical cases shown in Table 7 are used in the fits. In actual usage, a hemispherical
cap is given by RN = 0.5, LN = 0.5.
Values of Cn a and Cma for the nose alone were taken from unpublished data obtained from
G. Klopfer. Computational data for cases 2, 6', and A8 are tabulated in Table 8. Cases 2, 6', and A8have different values of LA and 0B. However, the data seem to indicate that the assumption of
nose contribution independent of afterbody and boattail is not too bad.
Cases 1, 9, 13, A5, and AIO also have a common nose. The nose-alone contribution is tabu-
lated in Table 9.
Again, the trend toward nose-alone values independent of the aft portion of the body is
apparent. An exception is for cases 4 and A3. For a nose as blunt as in cases 4 and A3, the after-body and boattail apparently influence the nose. Case 4 has a very short LA = 0.25, as opposed toLA = 2 for case A3.
However, the numerical data base available to take advantage of this result is far too small.The values at M.o = 1.2 are suspect since for case 10 a value Of CNa = 3.03, which is much too high,
is obtained for the nose. Including dependence of CNa, and Cma on LA and OB in the curve fit forthe nose, distorts the contribution of these elements.
Table 8. CNa and Cma Values for Nose-Alone Cases 2.6'. and A8
CNa C maM ./Nose-Alone
Case 2 Case 6' Case A8 Case 2 Case 6' Case A8
0.75 2.31 2.28 2.14 -2.86 -2.79 -2.76
0.9 2.24 2.17 2.33 -2.87 -2.75 -3.150.95 2.45 2.46 2.44 -3.37 -3.39 -3.351.2 2.28 2.28 2.25 -2.97 -3.14 -2.95
39
Table 9. CNc and C.. Values for Nose-Alone Cases 1, 9, 13, A5, and A I0
CNO, CmaI
M. /Nose-AloneCases 1,9, 13, A5, AI0 Cases 1,9, 13, A5, A10
0.75 2.07, 2.07, 2.08, 2.06, 2.07 -2.04, -2.04, -2.05, -2.01, -2.010.90 2.12, 2.15, 2.16, 2.13, 2.23 -2.01. -2.03, -2.11, -2.00, -2.250.95 2.38, 2.30, 2.38, 2.41, 2.40 -2.58, -2.29, -2.58, -2.57, -2.521.2 2.73, 2.78, 2.72, 2.64, 2.80 -3.21, -3.46, -3.24, -3.30, -3.54
Table 10. CNQ and Ca Values for Nose-AloneCases 4 and A3
CNa Cma
MCase 4 Case A3 Case 4 Case A3
0.75 2.01 2.05 -1.43 -1.480.9 2.45 2.72 -1.95 -2.110.95 2.47 2.74 -2.01 -2.151.2 2.94 2.74 -2.52 -2.28
A better approach would have been to use a potential model such as developed in Reference 49for noses plus very long afterbodies. A possible fit would be
(CNa)N = F(RLN) + G(LA, OB)
N, LN) LA +KI (93)
The last term would provide correction terms for short LA bodies. For supersonic Mach numbersthe second term in Equation 93 would not be needed.
Another question that arises is to what extent the nose loading is dependent on the shape ofthe nose. Reference I bases the nose loading on the cone-alone data from Reference 46 where
(CNG)N = Cl (M.,) tan 6* + C2 (M.o) (94)
where 6" is the angle at the shoulder. Thus, for the family of blunted tangent ogives, the predictionis (CNa)N = C2 (M.) or independent of the nose parameter For tan 6* -- 0, (CNa)N - 2.0 is thetrend of the data for cones. However, C2(M.) is above 2.0 for higher Mach numbers; this is just aproperty of the force fit.
Computations from Reference 47 and data from Reference 49 indicate that (CNa)N forblunted tangent ogives is above 2.0. Data from References 45 and 46 indicate that (CNa)N is lessthan 2. Reference 46 indicates that bluntness does not greatly affect (CNa)N in the transonic flowregime for cones for Mach numbers below the shock attachment Mach number for a pointed cone.
40
t-igure 20 is a compar- n,.olv.c the in tc.rAtcd data kio Reference 45 for cone-cylinders andthe valuCS obtained in Re ferctnk 1 101 coneS alonC, lic fit Iron Reference I is low for (",a forthe three Cc 111glc, plottCd l l,, cr. \,p t5 preidicted 1y slender-body theory is too high com-pared with data, cxccpt bcho, , I1 01C one.m2le. Percentagcewise the deviation is not too greatfor x, 7 rih in tetrated ,It I rt 11 RcfcreI-c 45 1a1,, solm e scattcr but follows a general trend withthe slender-hodv lntit clear\h ,' AIp I ()K I Aed I) - 0.
Afterbody-Alone Contribution. If, TC tI, data trends and computational trends are even lessclear. Reference I assutmes tfhat the car', t' cr from the nose or hoattail is negligible and (('N A.€ and(Xp)A/LA are a ft'Lction of ac rbh0L IC kngthf and Macth number alone. The data of Reference Ialso indicate that (-Na is tototonic, itcreaing to an aswnptote as LA grows. Tihe (('NQ )A fromReference I appears to he representative of tte ',+ buLdlip on the afterbody for a long afterbody
with a slender nose and a very small angle at the Shoulder.
For cone-cylinders. (\, buildup is available from Reference 45. For ogive-cylinders, only
('N, as a fIt n ction of aftcrbod lentI (Reference 50) i,, available. Unfortunately. nose-alone dataare not available in References 50 or 5 1, which also considers power-law nose shapes. In Refer-ences 50 and 51, the ttiinintn afterbody length is 4. )ata frotn References 45 and 50 indicatethat CNo continues to increase witi alterbody length in an almost linear manner rather than goingto an asymptote. The magri itude of the linear slope cannot be accounted for by a viscous crossflowcorrection, which is also linear with L sitte it would reqItire angles of attack of the order of10' or more. The tagnitude for a 10 cone-cylinder is also larger in Reference 45 than in
Reference 1. 'hte (('Na).\ contribution for a 10- cone at M = 1.1 increases with LA to a maxi-tILutn at about 1.5 caliber, tlhen dowIt to a nt mininturn at about 3 calibers, and then increases in asteady linear manner with increasing L,\ . The data for Reference 50 are only for LA of 4 calibersor greater.
Whereas. the available data attd Reference I predict an incremental (CNI )A. which is positive,
tite computtations of Reference 47 predict bothf positive and negative values. The computations ofReference 47 predict positi e increments ott the afterbody only for a 00 boattail angle.
No comment w ill be made (o th. xI prediction since the ('N, prediction accuracy is uncertain.
Boattail-Alone Contribution. Ilere tHIe data available ate almost nonexistent. Reference I usesa extremely limited data base froM C,,Ceti tafly otte source3 2 to establish an empirical prediction.Reference I indicates that tile correlatiott is,ippfemetted with data for the 5"/54 improved and175 mm projectiles. This would bias t fie correlat ion for these projectiles since it is assumed that thenose and aftcrbody prediction, are correct. Acain. x I,s given by a slender-body estimate. Titeexpression given in Reference I is not correct and should be
II bS, I (Vol IN fj - S(95)
41
Ilk .... .... .... soft 1
2.0 0 REF 46 INTEGRATED DATA
0 0106 15
0 .O %0 20SRE 1
- A(DATA F~IT)
1.5 A 10
o1 0C. .-. N " O 1"
0 0 01
1.0," 20
0.70 1.0 1.2 1.4 16
0.70-
o 0-0-O------------
SLENDER BODY THEORYA
00.65 A 0 A 0
0 6 A 00 0
0.70 1.0 1.2 1.4 1.6
Figure 20. Comparison of Cone-Cylinder and Cone-Alone Data for a Pointed Conical Nose
42
here I xp ) is the center of pressure in calibers. k is the body length in calibers, LB is the boattail
length in calibers, (Vol),, is the boattail volume. Srd. i(r,4, SB = irR , and RB is the cylindrical
radiuS at the end ot the hody in calibers.
I-or a finite angle hoattail, the computations of Reference I predict a downloading. The
computations of Reference 47 predict positive and negative loading for a 0' boattail, which is
really a continued afterbody.
Modified Functional Form Fit
A prclininar analysis ot the prc%[()lI paragraphs indicates that both the original and the
more recent prediction methods hasC some shortcomings. The NEAR functional form is invalid
outside the fit range. namncl 5-caliber afterbody lengths. The older computation is at least bounded
br longer afterbodics. (Currentls, the user has a choice as to which method to use for prediction
purposes. f-or afterhodics longer than 5 calibers, the older method is used.
T he N1 AR code Aas dcened to b1 inaccurate for supersonic Mach numbers because of theM_ = 12 estimatc for the individual hod% components. The M. = 1.2 values that are used in the
final interpolation are currentl, determined fron functional form fits to computations generated
b, an improCd Van )yke hybrid potential model (Reference 53). The new functional form fitswere generated from a much larger data base than the original form fits. The total computational
cost was about one hour of (l)( C 700 execution time. The ('Na and xcp predictions at M. = 1.2compare well with those ot the original potential comput at ion.
SUMMARY OF NlF% ANALYSES METHODS
1liC body-alone analysis methods ar suminaried in Figure 21. The Mach region divisionpoints are nornallN at W_, = 0.8. 1.2. dnd somew'here in the region 2 < MQ < 3 as determined by
the user.
Nes nethod, for tonlputing the wing-alone and interference aerodynamics are listed in
I igure 22
I inally. the ne methods Used to compute the dynamic derivatives are summarized in
I igure 23. [he inpuLt option refcrs to use of [ MS( or other prediction methods of Cmq + Cmd for
bod'y-alone or hod -tail . ontfigurations.
At high angles ol attack. th1w hody-alone or body-tail computations are based on the empiri-
lsrn of Reference 35 ()nls static derivatives arc determined.
INIDIVIIDUAL METHODS EVALUATION
In this s.ton the individual, new analsis methods will be evaluated by comparison withother analytical methods. numerical nethods, and experimented data where available. No com-
parison tort lite Oldert ithods, ill be considered except where applicable.
43
MACHNUMBER LOW HIGH
COMPONENT REGION SUBSONIC TRANSONIC SUPERSONIC SUPERSONIC
SECOND-ORDEREULER SECOND-ORDER VAN SHOCK-
NOSE WAVE DRAG PLUS DYKE PLUS EXPANSION PLUSEMPIRICAL MODIFIED
NEWTONIAN NEWTONIAN
SECOND-ORDER SECOND-ORDER
BOATTAIL WAVE DRAG WU AND AOYOMA VAN DYKE SHOCK-
EXPANSION
SKIN FRICTION DRAG VAN DRIEST II
BASE DRAG EMPIRICALEULER OR WU SECOND-ORDER
INVISCID LIFT AND EMPIRICAL AND AOYOMA TSIEN FIRST- SHOCK-
PITCHING MOMENT PLUS EMPIRICAL ORDER CROSSFLOW EXPANSION
VISCOUS LIFT AND ALLEN AND PERKINS CROSSFLOWPITCHING MOMENT
Figure 21. New Methods for ComputingStatic Body-Alone Aerodynamics
MACH
NUMBER LOW HIGHCOMPONENT REGION SUBSONIC TRANSONIC SUPERSONIC SUPERSONIC
INVISCID LIFT AND LIFTING SURFACE EMPIRICAL LINEAR THEORY SHOCK EXPANSIONPITCHING MOMENT THEORY STRIP THEORY
LINEAR THEORY,SLENDER BODY THEORY AND LNA HOY
WING BODY INTERFERENCE SLENDER BODYEMPIRICAL THEORY AND EMPIRICAL
WING TAIL INTERFERENCE LINE VORTEX THEORY
LINEAR THEORY + SHOCK EXPANSION - MODIFIEDWAVE DRAG EMPIRICAL MODIFIED NEWTONIAN NEWTONIAN STRIP THEORY
SKIN FRICTION DRAG VAN DRIEST
TRAILING EDGE SEPARATION EMPIRICALDRAG
BODY BASE PRESSURE DRAG EMPIRICALCAUSED BY TAIL FINS
Figure 22. New Methods for Computing Wing-Alone
and Interference Aerodynamics
44
z 8X- 00. 6
LU LU ci-I0 I~z ~Lz CC-u
C.)F-0 x). F-_LU zcr 0. i
oJ o uj Z
c> 0 0 0
0 w -J z 4 2oi 0 zLU LU i -j < 2 A(.4 04A N 4 CLLU
F- 0<. F- L cc cc z
0 0 ccLU
LU cc LU LU CLujLLuLU -i :) -i XI L
o U ) 0-1 L
0 u4 0. C. z E
LU LU LU C
2 8
u> 2U
zL LU LU _r
0 x-C 0~22 8
LL .JU, .LZ 0
c LU z LU jU) z
L0 0O 0L~ 0 J 0 0.U22 7
oi Z~~~ifZUJI ~
0 o 0U 0 u o. 0. a -Z -- co L 0
0___ __ _ __ _ 0 w__0_________
z Z m4 -1 z Z <4a
HIGHlIMACH NUMBELR STATIC AERODYNAMI(S
O ne consitlerat ionl is ik de'ttinhilat ion of' the Macli n umber MI k4, below which potentialthcor\ \\ ill be LuSCL and ab)oV ewliich tile sCcondl-0Irder shock-expansion and strip theories will
bc Used.
-Body-Alone lnyiscid Aerodlx mimics
Flhe 1lioh Macht body-alone routinck waiS OJ alnatedI adl compared with thle potential codes fromRel kIciiuces I and 53 and with c xpcriintal datia. Space limitations permit only a small portion ofhis Cx aluationl to be presetcd here. OriginAk It st as hoped that thc High Mach body-alone code
couLld be CXtCndcd downkl to M_ I .?. i.., to be USed throughou1.t the supersonic flow regime. Thisdid not p rove ttl be feasi I c.
Aft1l1.111h purely inviscid data f integrated prcsLUrC data) are not rcadily available. data for twocon fienrations. a blunt-no0sed. flared b)ody and a 1 1-.5 bluntcd conical :lose, for which integrated
*prcssure' data arc ax ailable wetre takcn fromn Ref-eren-ce 28. Comparisons made between these data.the rcSulltS From thc IHighi Mach codc. and thc two potential codes of' References I and 53 are given
* in Fienres 24 and 25. As canl be secn. the Hligh Mach code docs a creditable job for all functionsf -or thecse particular Conf iura tion s and Mach number range shown. Inl general, the drag. CA . pre-diction was found to showv bctter agremn thnta civd o h P n predictions.
[rom Reference 44, the forebody drag, C At ' was available for blunited tangent ogive-cylinderbodics with aftcrbodv. lengths of' 0 cylindcrs. C'omparisons of ('At. for four noses are given inFienUrc 26. Note that t hc Ilit'h Mach prctfdictionl accuracy for C'At: is quite poor, below M_ 1 .5and improves with Mach number. [or khe low supey-sonic Mach range, the methoq of Reference Iis best for blunted tangent-ogive noses. TFile nornmal force derivative, CNI. comparisons for the
sankfou boiesareprecutd i l'enr 2? whle downstream fromn the nose-afterbody junc-tionl ts Livein in 1-i"cre 28. These comparisons inidicate that the C., prediction using the High Machmetlhodology degrades sharply wijth bluntness: nlone of thel methods appear to compare well with thedata in predicting x,;)- Also, the treild toln .X as prcltfe! by thle High Maclh code is incorrect forlie b lii ter con fil Urlat i ois. Ilue poorest pre dict ion is for It igh l Curved hod ies and bodies with
lt' atixe: slopecs.
)neC clear concIlusion is tihat thflit NI1'l MaC It otl -alone predliction Shoultd not be used belowM_ 2. I lowever. 10 ot her rap)id coin p)ut atonal choice Currently is available for predicting CN, and
XI at higlh Macli 11i nitibers. Inl th liec delit' d ividling poinit between high Macli n1umber and low
supersonic: NiaCh n11.1tb11 Is anl inputL~ choice where a range of'2 _ MV _< 3 is reconmmlended.
Wing-Alone Inv iscid Aerodynaimic,,
I or the range o a) dphlabilit\ . tie ,trip met hod predlicts almost conistaint O3(\ anI OCNI for a
c"ILvii lm~ (',r I lmick nes-i )-cliOrt ratio. 1o prtopt'rlN cx aluate thle method. more exact solutionsorCc enittenta Idtat a x o nld be requiired. UiifortunlyO,. neither was available. Instead, comparisons
xx ith 111tlit potent ial I o lut in o I Re fere nec 2 were made for M_ >_ 2 as shown in Figure 29) for f3CNG*I 1le conipa rvils art shown~l to inpllrox, e ith increasng as peel ratio and Macli number, as Would be
C.\ peted. I I.In nrc 301 compares, ( ,, I r, a stinicitritcal diamiond (dtouble wedge) airfoil where
(rk(x% aks ) ()- Im rAl a"ses shlci. \!a inl as onec would expect, better agreement occurs for thefargeLr aispect ratios.. \lihoweli flit- ( \,, prif~iti ion is high. iinterference effects arc not considereda id tenet' inl mane11 east's f lit pre fiction of (*,,, at liieli Machi niumbers is tquite atltiate.
4o
A
_ _ _ _ 0
01
c 00Z C)
0
CC0 C
E0II 0
(0 0N d o
(0 -~ N3 (3SQN 1A10U1) X
100
50
> >
0 00
0 CD,
(EMI iNC)) V
x f47
A . .
11.50 BLUNTED CONE NOSE, RN = 0.175
1.2 .
0.8
0.4-
0.0 0.4 0.8 1.2 1.6 2.0
x
0.25
3 0.20 - -- - - - -0 .
0.15 0 0
0.10 -0 NASA/LANGLEY DATA (REF 28)
~ - AEROPREDICTION0.05 - HIGH MACH CODE
-. - IMPROVED Van DYKE (REF 53)
0.0- I I i0.0 1.0 2.0 3.0 4.0 5.0
Mo
Figure 25. Blunted Cone Aerodynamics
48
11.50 BLUNTED CONE NOSE
2.0
01.6 0
II1.2
Z 0.8 -0 NASA/LANGLEYZ, DATA (REF 28)
- HIGH MACH
0AEROPREDICTION0.4- IMPROVED Van DYKE
(REF 53)0.0- I ! I I
0.0 1.0 2.0 3.0 4.0 5.0
Mo00
1.0
0 0 0
0.8 0 0~00z2 0.60
! 0.4Ux
0.2
0.0 I I I I0.0 1.0 2.0 3.0 4.0 5.0
Mo0o
Figure 25. Blunted Cone Aerodynamics (Continued)
49
0.5-
0 DATA (REF 44)0.4- - AEROPREDICTION
-- IMPROVED Van DYKE*-HIGH MACH CODE
0.3 ---U.
0.2--- -
0.1
0.0-
(a) LN -1.843, RN 0. 125, a 0
0.5-
0.4-
0.3
0.2
0.1-
0.0-0 12 3 4 5
Mc 00
(b) LN = 2 , RN =O0, 0
Figure 26. Blunted Tangent Ogive-Forebody Axial Force
50
0.6 -
0.5--
0.4
0.3
0.2 0 DATA (REF 44)- AEROPREDICTION
- -IMPROVED Van DYKE0.1 -.- HIGH MACH CODE
0.0 i
0 1 2 3 4 5Mo
(c) LN - J85Q RN =0.375, t 0=
0.5-
0.4-
U-040.3--
0.2-
0.1-
0.0 - ! - I I0 1 2 3 4 5
(dI LN =2.357. RN 0-.250, o0
Figure 26. Blunted Tangent Ogive-Forebody Axial Force (Continued)
o DATA (REF 44)-- AEROPREDICTION
4 .... IMPFOVED Van DYKE---- HIGH MACH CODE
z0
3-
2 3
1 -
0- I I II0 1 2 3 4 5
(a) LN = 1.8 43 , RN = 0.125, at= 0
5
40
3 0-- ...
2
1--
0- I I It0 1 2 3 4 5
(b) LN 2, RN =0, C= 0
Figure 27. Blunted Tangent Ogive-Cylinder, CNO
52
5-0 DATA (REF 44)- AEROPREDICTION
4--- IMPROVED Van DYKE
HIGH MACH CODE
3 3
z 00
(c) 0 -a 1.85 ,R .7 ,o 0
2 -- *° " -° ..
1-
0 - I I I Io 1 2 3 4 5
(C) LN = 1.859. RN = 0.375, a 0
5-
4-
2-
1-
0I I I Io 1 2 3 45
d) LN = 2.3 57 , RN = .250, a =
Figure 27. Blunted Tangent Ogive-Cylinder, CNG (Continued)
0 DATA (REF 44)-1.2 - AEROPREDICTION
- - IMPROVED Van DYKE--- HIGH MACH CODE
-0.8 0xU 0
-0.4 0. 0
0.00.0 0.5 1.0 1.5 2.0 2.5 3.0
M 00
(a) LN 2 , RN O,c=O
-1.2
-0.8 0
x
0.4 -- ""
0.0- I I I I0.0 0.5 1.0 1.5 2.0 2.5 3.0
M
(b) LN =1.843, RN =0.12 5 ,a=0
Figure 28. Blunted Tangent Ogive-Cylinder, xcp (From Body-Nose Junction)
54
... .. ..... .. .. .. .= ....... -..... ,- I
-1.4-
-1.2
-0.8
-0.4 0 DATA(REF 4 4 )- AEROPREDICTION
-- IMPROVED Van DYKE-.- HIGH MACH CODE
0.0-,III0.0 0.5 1.0 1.5 2.0 2.5 3.0
M 00
(c) LN =2.357, RN =0.250, a 0
-1.2
CL -0.8x0
- 0.4
0.0-0.0 0.5 1.0 1.5 2.0 2.5 3.0
(d) LN = 1.859, RN =0.375, =0
Figure 28. Blunted Tangent Ogive-Cylinder. x, (From Body-Nose Junction) (Continued)
55
4.5-
4.0 --
3.5-
3.0-
0.5 0 --- STRIP THEORY0--0POTENTIAL THEORY
0.0-0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
(a) X, 0.99, A, 1, A 1=0
4.5 -
4.0-
3.5-
3.0-
0.51 0 --- STRIP MODEL0- POTENTIAL MODEL
0.0 - i i i
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
(b) X,, 113, Aw = 2, A, 4 5'
Figure 29. CNO, Comparison for a Wing
56
40 4
c 30
0
05o J --( StRiP MOtif
00 - - + + -- - - 4 -00 0 10 1 20 25 30 35 40
M
(a) X =0.99, A, = ,A, =
45
o4
ll 35
C 3.
05 t CI-. STRlP MODEL
0-0 POTENTIAL MODEL
00 05 10 15 20 25 30 35 40M,
(b) X = I/3, A, =2, A =45°
50 T -__ _ - - - -------------
451IC
35
"C ::0 } ll I'TIPMOD0,
00 05 10 15 20 25 30 35 40M.
(c) X =0.40, A, = .A, =60°
Figure 30, Cl>w Comparison for a Wing
57
Complete Configuration Aerodynamics
I-igure 31 compares data5 4 and the High Mach computation for a typical body-tail-canard
con figuration. lhe configuration shown in Reference 54 has body strakes not accounted for bytile present combined theory, which partially accounts for the higher experimental values of ('N*
Data %%ere also available at N1, = 4.65 f)r tile configuration shown in Figure 32. The body has
a spherical Cap ,ith a O.l-caliber radius. followed by a conical frustum 3.11 calibers long and5.74: inclination, folloved by a conical frustum 2.13 calibers long of inclination 2.360. followed by
a con,,tant radius afterbodv of 2.07 calhiers, and ending with a 2-caliber, 100 inclined flare. Tilewing cross section is a rectalgular slab 0.002 calibers thick with a leading edge sweep of 70' and a
trailing edge sweep ol 0'. The root chord and tile tip chord are 2.46 calibers and 0.44 calibers,
respectisely. The noinal reference diameter is 0.25 ft. Reference 54 indicates that the end of tilebody is a skirt rather than a flare. lhis could account for the difference in drag values.
HIGH ANGLL-O--.).rTA(K ..\EROI)YNAMICS
lhc approach is purel\ empirical as earlier indicated. One configuration that lies in the geo-metric and Mach niimber angle-of-attack envelope is tile Air Slew demonstrator configuration.( omuparisons .ith cxperiinental data are given in Figures 33 through 35. The normal force coeffi-
cient, (,,. t predicted adequately and x),, is predicted best at higher angles of attack. However, the
\ prcdi tion is (I the rilgit order of Huagniitude on.1.
-n o.t cr conpjarison is made for a modified basic finner configuration (Figure 36). The_omp.irion ,is slightl better for X in this case even thougt this vehicle differs from tie first only
JlIltlN in losc length. Note that the aerodynamics for complete configurations are predicted to
o: 4 i ,nlI. [le method of Reference .50 gives predictions to a = 1 0' but no roll orientationdependent acrd. namics arC coMIp uted . The current method gives only the individual component
aerodv naniics for angles of attack greater than 45' . Tie one exception is for the axial force coeffi-
cient prediction. 1he current method assumes that the body contributes the majority of tile axialforce or drag.
l)YN.,\\IIC DERIVATIVE PREDICTION
[lie I NIS( pitclh damping predict ion algorithin was compared very extensively with data anda purely empirical ( henoral I lectric prediction program (under contract to General Electric for tile
evaluation. (oniparisons given in Reference 57 are too numerous to be included here. Tile coi-Clusion, of R. WhVte of (.1 .5 7Wils thlat the LSM(' algorithin was adequate for most applications.
I or body-alone cases, ('m prediction was modified as earlier indicated. When tile value of
('M predicted is less negative than that predicted h t lie (G.F SPINNFR program and not within
75 p'rcent of lhe SiPINNI R value, tle SPINNER value is chosen.
(omparisons for the Army-Navy Spinner test vehicle are made in Figure 37. The originalL.MS( prJiction is shown to depart considerably from the data and the SPINNE.R prediction.
58
O.B
: 0.8 -0
z X,
A - 0 .4 -
CCN
z 0.2 -
U
o THEORYS----DATA (REF 54)
0.0.1
2.0 3.0 4.0 5.0M :, NASA TMX-1751
Figure 3 I. Comparison of Static Aerodynamics forFinned Body at High Mach Numbers (a = 00)
COO CN, PER tad) Xcpli (FROM NOSE)
THEORY 0.48 9.4 0.60
DATA (REF 54) 0.36 10.0 0.59
Figure 32. Aerodynamics for TMX- 187(Mo 4.o5,a 00)
5()
DIMENSIONS IN INCHES
FIN HINGE LINE
145.6 M
168.0
15
12-
9
6 --
0 AEDC DATA (REF 55)3- 0 COMPUTED
- DATA FOR 0.1875 SCALE OFMODEL SHOWN
-' M3=13, o=0- i I
0 10 20 30 40 50 60
S(deg)
Figure 33. CN Comparison for Air Slew D~emonstrator
60
0 COMPUTED
0jAEDC DATA IREF 66)
-05
-1.0
20
S.5M~ 1.3, o 0
LENGTH OF BODY, 10 CAL
3.0 i
0 10 20 30 40 50
(deg)
Figure 34. x, Comparison for Air Slew DemonstratorForward of Body Midpoint
0.0-Q COMPUTED
QAEDC DATA (REF 55)
-0.2--
-0.3- 0
-0.4- 10
M_~ 1.3o 22.5
-0.6 - I I0 10 20 30 40 so 60
(deg)
Figure 35. CQ Comparison for Air Slew Demonstrator
61
D- 0.15 ft
20
16
12z
8 /
4 0 -- C EXPERIMENTAL (REF 56)
- THEORY
0- I 1 I I I I0 10 20 30 40 50 60 70 80 90 100
1 .0 -
Ci -o-- 00.8 /
0
0.6
cc 0.4--U-
0. WM0- 2.5, 0)x
0
0 10 20 30 40 50 60 70 80 90 100
ANGLE OF ATTACK (deg)
Figure 36. Modified Basic Finner Aerodynamic Comparison
62
I Lq
I 0
2
a
I 0.
-/ L
U).
IrI If cc
U) W -
W3 + btu
63
The revised LMSC prediction improves the prediction. Comparisons with data when the center ofrotation is shifted downstream are better.
In Figure 38, the LMSC model is shown to predict the increased damping at M.= 1.0 for the
basic finner and follow the data fairly well at the higher Mach numbers. The strip theory predictionfor lifting surface contribution to the pitch damping is seen to be adequate. As was the case for
t wing ('N,, prediction, tae strip theory prediction obtained is improved for higher aspect ratios
and Math numbers. Unfortunately, data fo, bodies with two sets of lifting surfaces were notavailable.
Figure 39 shows that the strip theory estimate is adequate for C p predictions at high Mach
numbers. The comparison with data and the merging with the potential model at lower supersonicMach numbers is shown to be adequate. Again no data are available for vehicles with two sets of
lifting surfaces. As was the case for the wing-alone static coefficient comparisons, the accuracy
obtained with potential theory is better for higher aspect ratio and Mach number.
TRANSONIC NOSE WAVE DRAG
Inviscid pressure data are very scarce, particularly for the transonic case. Figure 40 shows the
comparison between limited data, the old theory, and the current theory for a M- 117 bomb nose.The numerical integration of pressure data was somewhat sensitive since there were not quite
enough pressure tap locations. However, the theoretical prediction, which is based on a full poten-tial solUtion, is much better than the original prediction based on the solution of the full Eulerequations. The original Euler computation did not have enough grid resolution to accuratelydescribe the pressure variation on the noses of blunt pointed bodies. The current prediction is seento be an improvement over the prediction of Reference I.
Figure 41 compares the older prediction method, the current method, and data from Refer-ence 44. Again, (AI is the forebody axial force coefficient. The values at Moo = 1.2 labeled potential
theory are determined from supersonic small disturbance theory for the pressure drag plusVan )riest's estimate of friction drag. )eviations of the supersonic small disturbance values from
the curves at M_ = 1.2 provide a measure of tile discontinuity in the two methods at M_ = 1.2. For
the blunter configurations, a definite improvement is obtained by using the new table lookupprocedure. The older method neglected the effect of bluntness.
'I RANSONIC NORMAL FORCE PREDICTION
As indicated earlier, the Ni'AR envelope of hod parameters is limited. The afterbody length,.,N, has a maximum of 5 calibers.
Figure 42 shows a comparison between the older colflptitation. the NEAR computation. andan improved N[AR computation with data. ile set of bodies from Reference 44 with 6-caliber
atterbodies is just beyond the NEAR envelope maximun of 5. In general, the CN prediction forthis set of data is poor for the N [AR computation and improved by the revised fits at the M, = 1.2
interpolation point. lhie NEAR x,,, prediction. on the other hand. is better than the older methodand considerably improved by the revised curve fit at supersonic Mach numbers. As earlierindicated, the modified NFAR functionali/ation is an input option for LA less than 5 calibers.
64
-600
0 STRIP THEORY PREDICTION
0l DATA (NAVORDI
A DATA (BARLI
-500-- OLD THEORY (REF 3)
-NEW THEORY (LMSCI
SEE FIG 39 FOR CONFIGURATION
-400-
L) E
E
300- A"..
-300
-200-
-100 I0 1.0 2.0 3.0
Figure 38. Basic Firmer Pitch Damping Comparison
30 0
0
20-
0
10- 0 BRIL BALLISTIC RANGE6 NSWC WINO TUNNEL0 AEOC WIND TUNNEL0 STRIP COMPUTATION
- THEORYREF 3) 0
0 -
0 12 3
Figure 39. C~p Comparison for Army-Navy Basic Finner
65
I -/
010
10 13
M
Figure 40. M-1 17 Bomb Nose Wave Drag Comparison
Additional comparisons for projectile shapes are given in Figures 43 through 45. All threecases have data with considerable scatter. There seems to be no sufficient justification for recom-mending any of the methods over the other. Currently the choice of algorithm is a program inputoption. For afterbodies longer than 5 calibers tile older method is automatically used.
CONCLUDING REMARKS
The capabilities of an earlier aerodynamic prediction methodology have been extended tohigher Mach numbers and angles of attack. At small angles of attack, the inviscid static aero-dynamic prediction has been extended to higher Mach numbers (approximately M.. = 6) and thepitch damping coefficient is computed by a new routine for Mach numbers from 0 to approxi-mately 6 for body-alone or body-tail configurations. Use of this routine is a program option.Also for small angles of attack, the roll damping and pitch damping contributions of lifting surfacesare estimated by a strip theory at higher Mach numbers (to approximately M. = 6). This extendsthe capability for the roll damping computation. The capability for prediction of the pitch dampingis extended to higher Mach numbers for configurations with body-canard-tail configurations andfor body-tail configurations when tile iww pitch damping option is not chosen.
The transonic nose wave drag routine has been replaced. Significant improvement is obtainedfor blunted tangent ogive and blunted cone noses. Also the transonic normal force predictionroutine has been changed. Use of this routine is a program option.
An empirical high angle of attack routine has been adapted for body-tail configurations.This routine is also an input option. The prediction technique is applicable for a limited envelopeand 0.8 < M_ < 3. Body-tail static aerodynamics are predicted to 450 angle of attack with
66
00
04
0 0 110
-~I
-z
00
2 0 >I T
0, 00
-- C
CiCci-
672
2 >0
0.0 Dz °
j i
9 00
csc
Cu
0L--4-68
\\ o 3-\\ 0
TRANSONIC CN AND X.
0 0 %*
o DATA IAFATL-TR-77-811- - AEROPREDICTION (OLD)
-- NEAR0 --- IMPROVED AEROPREDICTION
0.0 0.6 0.7 0.8 0.9 1.0 1.1 1.2
-2.5
0 LN =3 ,RN =0'=0/S-2.0--
LA 6/
-U1.5 I
U 0 ~
0 10' /00
m -0.5-0
~ 0.5
0.0 0.6 0.7 0.8 0.9 1.0 1.1 1.2
Figure 42. CNa, and x, Comparison for Blunted Tangent Ogive-Cylinders
69
K5> 4FRN LN L-A LB 0'
U0.0828 2.75 1.45 1.0 7.5'
00.0
c 0 - 0 00 1z0
0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
0-J
C(n0 BL SPAK0RNG
WY -2 0 AECWNDTNE
U. - MROE NA
0 ILSPR AG0. 0.6 0 1.0 WI.2 1.41N 6 1.8 2.
07
> 3.0--00
00Lz 0
002
-q -J RN '-N LA LB EI8
1. 0 0.04 2.9 1.6 1.0 80
0.0 0.5 1.0 1.5 2.0 2.5 3.0
3.0 -
2.0-0 0
1.0
0.0
LI--
CL0
-2.R~
--REF 47 (INTERPOLATED EULER. NEAR)IMPROVED NEAR
3.0I0.0 0.5 1.0 1.5 2.0 2.5 3.0
Figure 44. Comparison of Theory and Test Data for 175 mm XM437 Projectile
71
Kil
4 -
S 3
U-l
-- ,-;rnnrCI
0 15ImM00. .8 10 1. .4 16 1.A.
4-EI CLD OE
0. 0. 0.0 2.4 1.45 82518 .
4O
2
0
0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
Figure 45. Comparison of Theory and Test Data for 155 mm Projectile
72
arbitrary roll orientation. Individual components (including the body-alone case) aerodynamics
are predicted to 180' for the plus position roll orientation.
All of the now routines have been i-corporated into the existing code. Input logic and optionshave been improved. A detailed description of the computer program and a listing will be given in a
later publication. Computation times depend on configuration, code option, and Mach number.
A computation time can range from under a second to between 30 seconds and a minute per
Mach number on the CDC 6700.
REFERENCES
S1. F.G. Moore, Body-Aloe Aerodynamics of Guided and Unguided Projectiles at Subsonic,
Transonic. and Supersonic Mach Numbers, NWL TR-2796 (Dahlgren, Va., November 1972).
2. F. G. Moore, Aerodynamics of' Guided and tnguided Weapons. Part I - Theory and Applica-
tion, NWL TR-3018 (Dahlgren. Va., December 1973).
3. F. G. Moore and R. C. Swanson, Jr., Aerody"namics of Tactical Weapons to Mach Number 3
and Angle of Attack 150: Part I- Theory and Application, NSWC/DL TR-3584 (Dahlgren,Va., February 1977).
4. M. D. Van Dyke, "First- and Second-Order Theory of Supersonic Flow Past Bodies of Revolu-
tion," JAS, Vol. 18, No. 3 (March 1951), pp. 161-179.
5. H. S. Tsien, "Supersonic Flow Over an Inclined Body of Revolution," JAS, Vol. 5. No. 12
(October 1938), pp. 480-483.
6. J. M. Wu and K. Aoyoma, Transonic Flow-Field Calculation Around Ogive Cylinders hy
.Vonlinear-Linear Stretching Method, U.S. Army Missile Command. Technical Report
RD-TR-70-12 (Redstone Arsenal, Ala., April 1970). Also AIAA 8th Aerospace Sciences Meet-ing, AIAA Paper 70-18 ) (January 1970).
7. F. R. Van Driest, "Turbulent Boundary Layer in Compressible Fluids," JAS, Vol. 18. No. 3.pp. 145-160, 216.
8. J. H. Allen and F. W. Perkins, (haracteristics of" Flow Over Inclined Bodies of Revolution, Na-
tional Advisory Committee for Aeronautics (NACA) Rm A50L07 (Moffett Field, Calif., 1965).
9. E. S. Love, Base Pressure at Supersottic Speeds on Two-Dimensional Airfoils and on Bodies of
Revohtion With an(d Without Turhulenrt Boundary Layers, NACA TN-3819 (Hampton. Va..
1957).
10. 11. Ashley and M. Landahl, .lerodinamnics of Wings and Bodies, Addison-Wesley Publishing
Co. (Reading, Mass., 1965).
11. W. R. Chadwick, "The Application of Non-Planar Surface Theory to the Calculation of
External Store Loads." AL4A J urnal, Vol. I I (March 1974), pp. 181-188.
12. R.T. Jones and ('. Cohen, High Speed Wing Theorv, Princeton Aeronautical Paperbacks,
Number 6 ( 1960).
73
1 3. D. R.jChapmanW.R. Wnibrow, and R. H. Kester, ExperimentalInvestigation of Base Pressureon ButTaln-dcWnsaSuesncVlcteNC Re.119(Moffett Field, Calif.,1952).
14. W. C. Pitts, J. N. Nielsen. and G. F. Kaattari. Lijt and Center of Pressure of Wing-Body,-TailCombhinations at Subsonic, Transonic. and Supersonic Speeds, NACA TR-l 307 (Moffett Field,Calif., 1957).
1 5. J. C. Martin and 1. Jeffreys, Span, Load Diribution Resulting From Angle of Attack, Rolling.atid Pitching for Tapered SwepthacA Wings Wit!h Streamnoise Tips. NACA TN-2643 (Hampton.Va., 1952).
16. F. S. MalvCStuto, K. Margolis, and H. Ribner. Theoretical Lift and Damping in Roll at Super-some Speeds of Thin Sieeptback Tapered Wings Withi Streatniwise Tips, Subsonic LeadingEdges, and Supersonic Leading Edges. NACA TR-970 (Hampton. Va.. 1952).
17. R. H. Whyte. SPIN -73. An 11pdated Version of' the Spinner Computer Program. PicatinnyArsena! TR-4588 (Dover. N.J.. November 1973).
18. V. F. Lockwood, Effects of Swceep on the Damping-in-Roll Characteristics of Three SweptbackWings 1laig an Aspect Ratio of 4 at Transonic Speeds, NACA Rm L50J 19 (Hampton, Va.,December 1950).
19, G. J. Adams and 1). W. D)ugan. Theoretical Damping in Roll and Rolling Moment Due to[iffmretiial Winig hicidencwe fi)r Slender Crucilbrm? Wings and Wing-BodY Combinations,
NACA TR-1088 (Moffett Field, Calif., 1952).
20. J. N. Nielsen, isl eoyais McGraw-Hill Book Co.. Inc. (New York. 1960).
21. 1. J. Cole and K. Margolis, Lift and Pitching Moment at Supersonic Speeds Due to ConstantVertical Aceceleration Jo~r Thin So'eptbacA Tapered Wings Wit/i Streamvise Tips. SupersonicLeading, and Trailing Edges, NACA TN-3 196 (Hampton. Va.. July 1954)
22. J. C. Martin. K. Margolis, and 1. Jeffreys, (Calculation of' Lift anid Pitching Moments Due toI ngle of' A~ttack and .SteadY Pitching Velocity' at Supersonic Speeds for the Sweptbackl'apered Wings Wit/i Streamwiise Tips and Supersonic Leading and Trailing Edges. NACATN-2699 (11ampton. Va.. June 1952).
23. J. C'. I vvard. Usc oJ Source I)istrihttioi for Evaluating Theoretical A erod 'viiainics of' ThinF'inite Wip . at Supersonic Speeds, NACA Report 951I (C'leveland, Ohio, 1950).
24. 1. J. C'ole and K. Mai golis. L.ift and Pitching Momnent at Supersonic Speeds Due to ConstantVcrt( a! .1 celerationi fr Thiin .S'ep thacA Tapered Wings Wit/h Streaniwise Tips. SupersonicLeading. and Trailing Edges. NACA TN-3 196 0I lanipton. Va.. July 1954).
25. F. S. MalvCStutO and I1- M. Hoover. Lift and Pitching, Deri,'atircs of Thin SwepthacA TaperedWings Wit/i StrcumOinis Tips .nd Sub soniic Leading Edges at Supersoniic Speeds, NACAI'N-2 294 0 1am pton. Va., Februa ry 1951I)
260. Douglas Aircraft Co.. Inc.. USAF Stability and] Control I)ATCOM. Revisions by Wright-Pat tcrsoi Air Forcc Base (Dlay ton. Oh io. July 1963). 2 Vols.
74
27. C.A. Syvertson and 1). I1. lennis..i Second-Order Shock-Lxpansion Method Applicable to
Bodies ojfRe'ohition .'ar Zero Lift, NACA TN-3527 (Moffett Field, Calif., January 1956).
28. C. M. Jackson. W. C. Sawyer. and R. S. Smith. A4 .lethod for Determining Surface Pressure on
Blunt Bodies of Rcrohution at Small Angles of A ttack in Supersonic lo'1ow, NASA TND-4865
(Hampton, Va., 1968).
29. R. T. Stancih, "Improved Wave Drag Predictions Using Modified Linear Theory," J. Aircraft,
Vol. 16, No. 1 (January 1979). pp. 41-46.
30. F.R. De Jarnette, C.P. Ford, and D.L. Young, "Calculation of Pressures on Bodies at Low
Angles of Attack in Supersonic F-!ow,'" Jmrnal of Spacecraft and Rockets, Vol. 17, No. 6, pp.
529-53 6.
31. M. L. Rasmusson, "On Hypersonic Flow Past an Unyawed Cone," AIAA Journal. Vol. 5,
No. 8 (August 196'), pp. 1495-1497.
32. W. F. Hilton, Hih Speed.4rod.ynamics, Clh. 12, Longmans. Green and Co. (New York, 1951).
33. W. P. Helms, R. L. Carmichael, and C. R. Catellano ,4n Experimental and Theoretical Investiga-
tion of a SYmmetrical and a ('ambered Delta W ig ('Cotnifuraion at Mach Numbers from 2. 0
to 10. 7, National Aeronautics and Spae Administratio n (NASA) TND-5272 (Moffett Field,
Calif., March 1969).
34. A. G. Hanmit and K. R. A. Murthy, "Approximate Solutions for Supersonic Flow Over Wedges
and Cones," JAS, Vol. 27. No. 1 (January 1960), pp. 71-73.
35. G. F. Aicllo. A 'rod.v'namnic Methodology (Bodies with Tails at Arbitrary Roll Angles. Transonic
and Supersonic), Martin-Marietta Corp. OR 14. 145 (Orlando, Fla., April 1976).
36. C. A. Smith and J. N. Nielsen, Prediction of.1,'rodlmnamic Characteristics of Crucifortn Missiles
to Itigh Angles of Attack Utilizing a Distributed Vortex Wake, Nielsen Engineering and
Research (NEAR) TR-208 (Mountain View, ('alif., January 1980).
37. G. C. Winn, Lser's Manual fi.r the Martin-Marietta Iligh Angle-of-Attack Aerodynamic
Methodology f)r Bod.Y-Tail Missiles, Redstone Arsenal Technical Report T-78-63 (Redstone.
Ala., June 1978).
38. L. E. Ericsson, Modification of Acrodnliamic Prediction of the Longitudinal Dynamics of'
Tactical Weapons. Lockheed Missile and Space ('ompany LMSC-0646354 (Sunnyvale, Calif.,
June 1979).
39. Addendum to LMSC-0646354, LMSC-0646354A (Sunnyvale, Calif., August 1979).
40. L. F. Ericcson and S. R. I)eLu. U(scr's Mannalifr LAiSC (hneralized Wing-Body Subroutine,
LMSC-0646354A (Sunnyvalc, Calif., August 1979).
41. 'tngim'ering Design lamlbook lc'si, of, .1 crodinamnicalli Stabilized 1-ree Rockets, U.S.
Army Materiel (ommand AMCP 706-280 (Washington, I).C., July 1968).
42. 1). (haussCC. ImIrd 7"raswic .NOs' )drav l:stimates fin. the .\'S1C Missile ('omputerProgram. NSWC/I)l. TR-3030 (l)ahlgren, Va.. April 1978).
75
43. J. 1). Keller and J. C. South, RA.'BO) .1 FOR TRAN Program frbr Invlscid Transonic FlowO1r .lxi\vtnmturic lodie.'s, NASA TMY-72331 (Ilampton, Va., February 1976).
44. Acrodynamt( ('haractristi.s o 1 2-. 3-, and 4-alihcr Taiigent-Ogive ('linders with Nose
Bubess Ratis o 0.90, 0.25, 9. 0. and 0. 75- at M1ach vin bers froin 0.6 to 4.0, AFATL-TR-77-8 ([glin AFB, 1:13., January 1977).
45. R.L. taniner and AD. Leff. Li'war.lcrdi nalnit Loads on Conc - inders at Mach Numbers
-from Q 7 to 2,0(. NASA (CR-4 13 (Washimnt, ).C'., March 1966).
46. R. V. Owens. A rodl'namic ('/aracteristics of Shcricall.i Blunted ('ones at Alach Vumbers
Irom 0.5 to 5.0, NASA TN-I)-3088 (Huntsville. Ala., December 1965).
47. (. Klopfer and 1). Chaussec..'ttnc'rical Solution of Three-Dimensional Tratsonic Flows
Around Axisyminctric Bodies (it Aine ol1 AttacA, NEAR TR-176 (Mountain View, Calif.,
February 1979).
48. G. D. Kuhn and J. N. Nielsen, "Prediction of' Turbulent Separated Boundary Layers," AIAAPaper 73-663, Presented at the AIAA 6th Fluid and Plasma Dynamics Conference, PalmSprings, Calif. (July 1973).
49. T. Hsich, "Perturbation Solutions of Unsteady Transonic Flow Over Bodies of Revolution,"AIAA Journal, Vol. 16, No. 12 tl)ecember 1978), pp. 1271-1278.
50. H. Gwin and D. J. Spring. Stability ('haracteristics of a Family of Tangent Ogive-CvlinderBodies at Mach Vionhers Iroln 0.2 to 1.5. U.S. Army Missile Command RG-TR-61-1 (Redstone,
Ala., 1961 ).
51, 1). J. Spring, The 1 f/c ct oj .Vose Shape and 4Aterhodj" Length on the Normal Force andNeutral Point Location of Ax.isyinmetric Bodies at Mach Numhers from 0.8 to 4.5, U.S.
Army Missile Command RF-TR-64-1 3 (Redstone Arsenal, Ala., July 1964).
52. W. D. Washington and W. Pettis, Jr., Boattail tl'f'cts on Static Stability at Small Angles ofAttack, U.S. Army Missile ('omnmand RD-TM-68-5 (Redstone Arsenal, Ala., July 1968).
53. L. Devan, "An Improved Second-Order Theory of Inviscid Supersonic Flow Past Bodies ofRevolution," Paper 80-0030, Presented at the AIAA 18th Aerospace Sciences Meeting,
Pasadena. Calif. (January 1980).
54. 1. 0. Nichols, .Atnalsi.s and ('ompilation of Ilissile Aerodynamic Data, NASA CR-2835
(Washington, 1)'.., November 1Q75).
55. J. M. Whoric and F1. S. Washington. Acrodiamuic ('haracteristics of the Air Slew Demon-
strator llodels at Maclh .\',nhr rom .6 to 1 1.3., AF.I)C-TR-76-92 (Tullahoma, Tenn.,
August 1 976).
7 6
- -- . ~ . 'At
56. W. B. Baker, Jr., An Aerodynamic Coefficient Prediction Technique for Slender Bodies withLott Aspect Ratio Fins at Mach Numbers ftrm 0.6 to 3.0 and Angles of Attack from 0 to180 Degrees, AEDC-TR-97 (Tullahoma, Tenn., March 1978).
57. R. Whyte, 1. Burnett, and W. Hathaway, Evaluation of' the Computation of Pitch DampingbY Subroutine LMSC, General Electric Co., Armament Systems Dept. (Burlington, Vt.,November 1979).
77
LIST OF SYMBOLS
A Lifting surface aspect ratio
b Span of two fins (excluding body)
C1 Distance along characteristic of the first family
CA Axial force coefficient
CAl: Forebody axial force coefficient
CAp Pressure axial force coefficient
CDo Zero angle-of-attack drag coefficient
CDW Wave drag
Cq Roll moment coefficient
C~p Roll damping coefficient
Cm Pitching moment coefficient
Cmq + Cmi Pitch damping coefficient
CN Normal force coefficient
CNB Body-alone normal force coefficient
CNTi Isolated ith fin (in presence of the body) normal force coefficient (cruciform
configuration)
(CN)TO T Body-tail total normal force coefficient
Co P Pressure coefficient
Cr Fin root chord
C t Fin tip chord
D Body reference diameter
Dc , D, Mean body diameter near a canard and tail, respectively
IBT Body-tail normal force coefficient interference
K Hypersonic similarity parameter
Q Body length (cals.)
LA, LN, LB Lengths of afterbody, nose, and boattail respectively (cals.)
M Mach number (local)
M. Freestream Mach number
p Static pressure
r Cylindrical cooruinate (cals.)
R Body cylindrical radius (cals.)
78
RN Body, nose spherical radius
s Arc distance ilong body
x Body coordinate (parallel to body axis)
xc I)istance from nose to moment reference
xC, x' Distances to fin apex for canard and tail, respectively, fin nose
xCP Distance fromn moment reference to center of pressure
XCPB Buody-alone x, from nose
(x p)Ti Isolated im fin (crucilorm configuration) xp from nose
(x, ) I)T Total body-tail x, from nose
Xcp I Interference xp from nose
y Distance out from a wing root
a Angle of attack
-y Calorically perfect gas heat capacity ratio
6 Local body angle
0 B Boattail angle
X = /C r
A AV, Leading edge sweep of lifting surface
sin- 1 _
Roll angle
Superscripts
Differentiation with respect to x
Subscripts
a Asymptotic value
w Wing or tail value
o Stagnation value
1, 2 Values upstream and downstream from a body comer respectively
a Partial differential with respect to a(a - 0)
Reference area based on I). Moment reference length based on D. Dynamic derivative momentcoefficients based on (radian rate) 1)
2 V_
79
DISTRIBUTION
Commander Office of Naval ResearchNaval Sea Systems Command 800 N. Quincy StreetWashington, DC 20360 Arlington, VA 22217
Attn: SEA-62R41, Mr. L. Pasiuk Attn: Mr.D. SiegelTechnical Library Dr. R. Whitehead
Technical LibraryCommander
Naval Materiel CommandWashington, DC 20360 Commanding Officer
Attn: Mr. S. Jacobson (MAT-032) Naval Air Development CenterDr. John Huth Warminster, PA 18974Technical Library Attn: Mr. S. Greenhalgh
Mr. C. ReitzCommanderNaval Air Systems Command Technical Library
Washington, DC 20360Attn: AIR-320, Mr. W. Volz Superintendent
AIR-3 20, Dr. H. Mueller U.S. Naval AcademyAIR-330D, Dr. W. H. Clark Annapolis, MD 21402AIR-530, S. Loezos Attn: Head, Weapons Dept.Technical Library Head, Science Dept.
Dr. A. MaddoxCommander Technical LibraryNaval Weapons CenterChina Lake, CA 93555 Superintendent
U.S. Naval Postgraduate SchoolMr. R. Meeker Monterey, CA 95076
Mr. Lloyd Smith Attn: Technical Library
Mr. R. E. SmithMr. H. Schafer Officer in Charge
Technical Library Naval Intelligence Support Center4301 Suitland Road
Commander Washington, DC 20390
Pacific Missile Test Center Attn: J. B. Chalk
Point Mugu, CA 93041 Technical Library
Attn: Mr. J. RomMr. G. Cooper Commanding OfficerTechnical Library Naval Ordnance Station
Indian Head, MD 20640
Commander Attn: Technical Library
Naval Ship Research and Development CenterWashington, DC 2007 Director, Development Center
Attn: Dr. T. C. Tai Marine Corps Development and EducationMr. M. J. Malia CommandTechnical Library Quantico, VA 22134
NASA Langley Research Center Mr. E. LuceroLangley Station Mr. L. TisserandHampton, VA 23365 Mr. G. J. Pietrangeli
Attn: Mr. J. South Technical LibraryMr. L. SpearmanMr. C. M. Jackson, Jr. Raytheon CompanyMr. W. C. Sawyer Missile Systems DivisionDr. R. C. Swanson, Jr. Hartwell RoadMiss E. J. Landrum Bedford, MS 01730Technical Library Attn: Mr. D. P. Forsmo
Technical LibraryVirginia Polytechnic Institute and State
University McDonnell-Douglas Astronautics Co. (West)Department of Aerospace Engineering 5301 Bolsa AvenueBlacksburg. VA 24060 Huntington Beach, CA 92647
Attn: Dr. J. A. Schetz ttn Dr. CA Xerko
Dr. C. tt. Lewis Attn: Dr. J. XerikosTechnical Library Technical Library
North Carolina State University McDonnell-Douglas Astronautics Co. (East)
Department of Mechanical and Aerospace Box 5 16Engineering St. Louis, MO 63166
Box 5246 Attn: Mr. J. WilliamsRayleigh, NC 27607 Techn Mr. S. Vukelich
Attn: Dr. F. R. De Jarnette Technical LibraryTechnical Library Lockheed Missiles and Space Co., Inc.
The University of Tennessee Space Institute P.O. Box 1103Tullahoma, TN 37388 Huntsville, AL 35807
Attn: Dr. J. M. Wu Attn: Mr. D. AndrewsMr. C. Balasubramayan Technical LibraryTechnical Library Lockheed Missiles and Space Co., Inc.
University of Notre Dame P.O. Box 504
Department of Aerospace and Mechanical Sunnyvale, CA 94086
Engineering Attn: Dr. Lars E. Ericsson
Box 537 Mr. P. Reding
Notre Dame, IN 46556 Mr. H. S. Shen
Attn: I)r. R. Nelson Technical Library
Technical LibraryNielsen Engineering and Research, Inc.
Applied Physics Laboratory 5 10 Clyde AvenueThe Johns Hopkins University Mountain View. ('A 95043Johns Hopkins RoadLaurel, MD 20810 General Electric Co.
Attn: l)r. L. L. Cronvich Armament Systems DepartmentMr. F. T. Marley Burlington, VT 05401Mr. J. C. Hagan Attn: Mr. R. Whyte
CAL SPAN Advanced Technology Center Attn: Mr. D. G. MillerP.O. Box 400 Technical LibraryBuffalo, NY 14225
Attn: Mr. B. Omilian Honeywell Inc.600 Second Street
Northrop Services, Inc. Minneapolis, MN 55343Huntsville, AL 35810 Attn: Mr. S. Sopczak
Attn: Mr. W. Boyle Technical Library
Science Applications Inc. Pacifica Technology680 E. Swedesford Rd. P.O. Box 148Wayne, PA 19087 Del Mar, CA 92014
Attn: Mr. P. Murad Attn: Dr. H. T. Ponsford
Vought Corporation Rockwell InternationalP.O. Box 5907 Missile Systems DivisionDallas, Texas 75222 4300 E. Fifth Avenue
Attn: Mr. F. Prillman P.O. Box 1259Dr. W. B. Brooks Columbus, OH 43216Mr. R. Stancil Attn: Mr. J. E. Rachner
Technical LibraryHughes Aircraft Corp.Canoga Park, CA 91304 Boeing Computer Services, Inc.
Attn: Dr. J. Sun P.O. Box 24346Technical Library Seattle, WA 98124
Attn: Mr. R. WyrickSandia Laboratories
Albuquerque, NM 87115 Motorola Inc.Attn: Mr. R. La Farge Missile Systems Operations
Mr. R. Eisler 8201 East McDowell Rd.Technical Library P.O. Box 1417
Martin Marietta Aerospace Scottsdale, AZ 85252
P.O. Box 5837 Attn: Mr. G. H. Rapp
Orlando, FL 32805Attn: Mr. G. F. Aiello Weapons Systems Research Laboratory
Technical Library G.P.O. Box 2151, Adelaide, S.A., 5001Salisbury, S.A., Australia
Business and Technology Systems, Inc. Attn: Mr. L. M. SheppardSuite 400 Aerospace Building Mr. K. D. Thomson10210 Greenbelt Rd. Technical LibrarySeabrook, MD 20801
Attn: Dr. J. B. Eades, Jr. Messerschmidt-B61kow-Blohm GMBHUnternehmensbereich Apparate
Lawrence Livermore Laboratory MUnchen 80-Postfach 801149Earth Sciences Division Bayern. Federal Republic of Germany (DFR)University of California Attn: Dr. H. G. KnocheLivermore, CA 94550 Dr. Gregoriou
Chief of S and R Division Arnold Engineering Development CenterDevelopment Center USAFMarine Corps Development and Education Tullahoma, TN 37389
Command Attn: Mr. J. UsseltonQuantico, VA 22134 Mr. W. B. Baker, Jr.
Technical LibraryCommanding GeneralBallistic Research Laboratory Commanding OfficerAberdeen Proving Ground, MD 21005 Air Force Armament Laboratory
Attn: Dr. C. H. Murphy Eglin Air Force Base, FL 32542Mr. L. McAllister Attn: Dr. D. DanielMr. A. Platou Mr. C. ButlerMr. R. McCoy Mr. K. CobbDr. R. Sedney Mr. C. MathewsI)r. W. Sturek Mr. E. SearsMr. C. Nietubicz Mr. F. StevensTechnical Library Dr. L. E. Lijewski
Commanding General USAF AcademyARRADCOM Colorado Springs, CO 80912
Picatinny Arsenal Attn: Technical LibraryDover, NJ 07801
Attn: Mr. A. Loeb Commanding OfficerMr. H. Hudgins Air Force Wright Aeronautical LaboratoriesMr. G. Friedman (AFS()Technical Library Wright-Patterson Air Force Base, OH 45433
Attn: Dr. G. Kurylowich
Commanding General Mr. D. SheredaU.S. Army Missile R and D Command Mr. J. JenkinsDROMIl-TDK Mr. D. Hoak
Redstone Arsenal Mr. G. Fleeman
Huntsville, AL 35809 Mr. M. PinneyAttn: Mr. R. Deep Advance Research Projects Agency
Dr. D.L.Spingary Department of DefenseWashington, DC 20305
('ommanding General Attn: Technical Library
Frankford Arsenal NASAPhiladelphia, PA 19104 Washington, DC 20546
Attn: Mr. W. Gadom,,ki Attn: Technical LibraryTechnical Library
NASA Ames Research CenterCommanding Officer Moffett Field, CA 94035Harry Diamond Laboratories Attn: Dr. G. ChapmanWashington. DC 20013 Mr. V. L. Peterson
Attn: Technical Library Technical Library
Raytheon Company G20
Spencer Laboratory G30Box SL7162 G40Burlington, MS 01803 K
Attn: Mr. S. Pearlsing KI0 (2)Mr. P. Giragosian K20
K21K21 (Devan) (40)
Defense Technical Information Center K22 (2)Cameron Station K23 (2)Alexandria, VA 22314 (2) K24
RLibrary of Congress R44 (5)Washington, DC 20390 U20
Attn: Gift and Exchange Division (4) U23
GIDEP Operations Office X210 (6)
Corona, CA 91720
Defense Printing ServiceWashington Navy YardWashington, DC 20374
Local:E41G
