The Aerodynamic Preliminary Analysis System (APAS) utilizes a modified version of the Hypersonic Arbitrary-Body Program (HABP) Mark 3 code in its analysis rationale. Four methods are considered for incorporation into the code as the tangent-cone method. The combination of second-order slender body theory and the approximate solution of Hammitt and Murthy shows the best agreement with the exact numerical solutions and is thus included in the APAS production version of the HABP code

Cruz, Christopher I.

Sova, Gregory J.

English

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NASA Technical Memorandum 4165
Improved Tangent-Cone Method
for the Aerodynamic Preliminary
Analysis System (APAS)
Version of the Hypersonic
Arbitrary-Body Program
Christopher I. Cruz
Langley Research Center
Hampton, Virginia
Gregory J. Sova
Rockwell International Corporation
North American Aircraft Operations
E1 Segundo, California
National Aeronautics and
Space Administration
Office of Management
Scientific and Technical
Information Division
1990

Abstract
The Aerodynamic Preliminary Analysis System
(APAS) utilizes a modified version of the Hypersonic
Arbitrary-Body Program (HABP) Mark III code in
its analysis rationale. Four methods are considered
for incorporation into the code as the tangent-cone
method. The combination of second-order slender
body theory and the approximate solution of Ham-
mitt and Murthy shows the best agreement with the
exact numerical solutions and is thus included in the
APAS production version of the HABP code.
Introduction
The Aerodynamic Preliminary Analysis System
(APAS, refs. 1 and 2) uses a modified version of
the Hypersonic Arbitrary-Body Program (HABP)
Mark III code (ref. 3) in its analysis rationale. An
integral part of such an analysis is the calculation
of inviscid pressure distributions on arbitrary sur-
faces. Vehicle fuselages are often somewhat conical
in shape; thus a method of predicting pressure drag
for such shapes is required.
Four impact pressure methods are evaluated for
their ability to predict the zero-angle-of-attack invis-
cid pressure coefficients of sharp cones with angles of
5°, 7.5 °, 10 °, 12.5 °, 15 °, 17.5 °, 20 °, 30 °, 40 °, and 50 °.
These predictions are then compared with the exact
solution for air. Finally, a method is chosen for use
in the APAS production version of the HABP code.
Symbols
K
Mns
Moc
P
poc
qc¢
Voc
5
0c
0,
7
P_c
pressure coefficient, (p - P_c ) / qoc
constant in Newtonian pressure coeffi-
cient equation
Mach number normal to the shock
free-stream Mach number
local pressure, lbf/ft 2
free-stream pressure, lbf/ft 2
dynamic pressure, 1/2p_cV2
free-stream velocity, ft/sec
impact angle, deg
cone half-angle, deg
shock angle, deg
ratio of specific heats
free-stream density, lbm/ft 3
Description of Prediction Methods
The four impact methods evaluated for pres-
sure coefficient prediction were (1) Newtonian theory
(ref. 3), (2) the original HABP Mark III tangent-cone
empirical method (ref. 3), (3) the Edwards tangent-
cone empirical method (ref. 4), and (4) a combination
of second-order slender-body theory and the approx-
imate cone solution of Hammitt and Murthy (ref. 5).
Modified Newtonian theory yields a pressure co-
efficient which is a function only of impact angle:
Cp = K sin 2 6
where K is equal to the stagnation-point pressure
coefficient (ref. 6).
Both the HABP Mark III and the Edwards ver-
sions of the tangent-cone empirical method calculate
pressure coefficient as a function of Mach number and
impact angle:
48Mn2s sin 2 6
Cp- 23M2s _ 5
The difference between these two methods lies in the
empirical equations for Mach number normal to the
shock. For the HABP Mark III version,
Mns = 1.090909M_c sin 6 + exp(-1.090909Moc sin 6)
For the Edwards version,
Mns = (0.87Moc - 0.544) sin 6 + 0.53
The last method uses a combination of second-
order slender-body theory and the approximate cone
solution of Hammitt and Murthy. The pressure
coefficient found with this method is given by
Cp - '/2pocV_ \ \ _ + 1 oc sin2 O, 7./
1 + 7Moc(O_ - Oc) 2cos 20s1 T [-_--- ]i/__- s } -1)
Results and Discussion
Figures 1 to 10 present inviscid pressure coeffi-
cients (zero angle of attack) for sharp cones with half-
angles from 5° to 50 ° and Mach numbers from 1.5
to 25. Each figure contains pressure coefficients cal-
culated with each of the four prediction methods as
well as exact values from the tables of Kopal (ref. 7)
and Jones (ref. 8).
For all cone half-angles investigated, Newtonian
theory underpredicts pressure coefficient throughout
the entire Mach number range. Adjustment of the
Newtonian constant from K = 2 to K = 2("/+ 1)
x ('_ + 7)/('7 + 3) 2 (ref. 1) would give a reasonable re-
sult for Mach numbers greater than 10, when the in-
viscid pressure coefficient is relatively constant with
respect to Mach number.
The HABP Mark III tangent-cone empirical
method does a better job than Newtonian theory,
but at Mach numbers less than 5 it also greatly
underpredicts the exact solutions, as shown in
figures 7 to 10.
The Edwards tangent-cone empirical method is a
vast improvement over the HABP Mark III method.
At smaller cone half-angles, results from the Edwards
method match the exact values closely for Mach
numbers of 1.5 and up. However, as cone half-angle
increases, the discrepancy between the results from
the Edwards method and the exact values grows
larger.
By far, the best of the methods evaluated is that
referred to in figures 1 to 10 as the "2nd Order Slen-
der Body + Hammitt/Murthy" method. With few
exceptions, this method predicts the inviscid pres-
sure coefficient at zero angle of attack with great ac-
curacy. (In most cases, there is less than 1 percent
difference between predictions and the exact values
of Kopal and Jones.) Figures 9 and 10 show the
peculiarities which can occur when this method is
used for large cone half-angles. This degeneration
of calculated pressure coefficient corresponds to the
physical existence of detached shocks for larger cone
half-angles at low supersonic speeds. It is important
to note, however, that even with these discontinu-
ities, this method is still far superior to the other
three methods considered.
Conclusions
The combination of second-order slender-body
theory and the approximate cone solution of Ham-
mitt and Murthy is the superior method of those eval-
uated. It is thus included in the Aerodynamic Pre-
liminary Analysis System (APAS) production version
of the Hypersonic Arbitrary-Body Program (HABP)
code as the tangent-cone method. The Newtonian
theory, original HABP Mark III tangent-cone, and
Edwards tangent-cone methods all have applicability
within given restrictions, but outside of these restric-
tions they may yield misleading results.
NASA Langley Research Center
Hampton, VA 23665-5225
January 4, 1990
References
1. Bonner, E.; Clever, W.; and Dunn, K.: Aerodynamic
Preliminary Analysis System II. Part I--Theory. NASA
CR- 165627, 1981.
2. Divan, P.: Aerodynamic Preliminary Analysis System II.
Part H--User's Manual. NASA CR-165628, 1981.
3. Gentry, Arvel E.: Hypersonic Arbitrary-Body Aero-
dynamic Computer Program (Mark III Version). Vol-
ume I--User's Manual. Rep. DAC 61552, Vol. I (Air
Force Contract Nos. F33615 67 C 1008 and F33615 67 C
1602), McDonnell Douglas Corp., Apr. 1968. (Available
from DTIC as AD 851 811.)
4. Pittman, Jimmy L. (appendix by C. L. W. Edwards):
Application of Supersonic Linear Theory and Hypersonic
Impact Methods to Three Nonslender Hypersonic Airplane
Concepts at Mach Numbers From 1.10 to 2.86. NASA
TP- 1539, 1979.
5. Hammitt, A. G.; and Murthy, K. R.: Approximate
Solutions for Supersonic Flow Over Wedges and Cones.
AFOSR TN 59-304, U.S. Air Force, Apr. 1959. (Avail-
able from DTIC as AD 213 088.)
6. Truitt, Robert Wesley: Hypersonic Aerodynamics.
Ronald Press Co., c.1959.
7. Staff of Computing Section, Center of Analysis (under
the direction of Zden_k Kopal): Tables of Supersonic
Flow Around Cones. Tech. Rep. No. 1 (NOrd Contract
No. 9169), Massachusetts hast. of Technology, 1947.
8. Jones, D. J.: Tables of lnviscid Supersonic Flow About
Circular Cones at Incidence -'/= 1.4. AGARDograph 137,
Pt. I, Nov. 1969.
Cp
.10
.08
.06
.O4
.02
0
2nd Order Slender Body
+ Hammitt/Murthy
........ Edwards Corrected Tangent Cone
Mark III HABP Tangent Cone0
......... Newtonian Theory (K = 2)
o Exact (Kopal, ? = 1.405)
_0
[] Exact (Jones, y = 1.4)
m
i I f l , I , I , I
5 10 15 20 25
Mach no.
Figure 1. Pressure coefficient versus Mach number for 5° sharp cone.
Cp
.12
.10
.08
.06
.04
.02
O
0
2nd Order Slender Body
+ Hammitt/Murthy
........ Edwards Corrected Tangent Cone
Mark III HABP Tangent Cone
......... Newtonian Theory (K = 2)
0 Exact (Kopal, y = 1.405)
, [] Exact (Jones, _, = 1.4)
..._, El"
L I I I I l I I I I
5 10 15 20 25
Mach no.
Figure 2. Pressure coefficient versus Mach number for 7.5 ° sharp cone.
3
Cp
.2OO
.175
.150
.125
.100
.075
.050
0
0
2nd Order Slender Body
+ Hammitt/Murthy
........ Edwards Corrected Tangent Cone
0
[]
Mark III HABP Tangent Cone
Newtonian Theory (K = 2)
Exact (Kopal, 7 = 1.405)
Exact (Jones, T = 1.4)
i Jill Ill Ill II iI .ill • _liJ I ill II Jl liliillllQi_ II "I ll'li lillil I I IIil l" III lillil
, t l I , I J I i I
5 10 15 20 25
Mach no.
Figure 3. Pressure coefficient versus Mach number for 10 ° sharp cone.
Cp
.30
.25 _-
.2O
.15
.10
2nd Order Slender Body
+ Hammitt/Murthy
........ Edwards Corrected Tangent Cone
Mark III HABP Tangent Cone
........ Newtonian Theory (K = 2)
0 Exact (Kopal, T = 1.405)
[] Exact (Jones, _, = 1.4)
.....___ El []
11 li fill .llli111111 .l.lili_ilil. I_I Ill ..... Ill. 111* I i 111 lliill *Jill" IIi
.05 I I , I i I , I ,
0 5 10 15 20
Mach no.
Figure 4. Pressure coefficient versus Mach number for 12.5 ° sharp cone.
I
25
Cp
.5
.4
.3
.2
.1
2nd Order Slender Body
+ Hammitt/Murthy
........ Edwards Corrected Tangent Cone
Mark III HABP Tangent Cone
......... Newtonian Theory (K = 2)
o Exact (Kopal, 7 = 1.405)
[] Exact (Jones, 3' = 1.4)
J I I I I I I I I I
0 5 10 15 20 25
Mach no.
Figure 5. Pressure coefficient versus Mach number for 15 ° sharp cone.
Cp
.6
.5
.4
.3
.2
.1
2nd Order Slender Body
+ Hammitt/Murthy
........ Edwards Corrected Tangent Cone
Mark III HABP Tangent Cone
......... Newtonian Theory (K = 2)
o Exact (Kopal, _,= 1.405)
[] Exact (Jones, 'y = 1.4)
......................................................... ._-,.-.--._.-- .........
0
J I , I i I J I i I
5 10 15 20 25
Mach no.
Figure 6. Pressure coeffÉcient versus Mach number for 17.5 ° sharp cone.
Cp
.7
.6
.5
.4
.3
.2
0
2nd Order Slender Body
+ Hammitt/Murthy
........ Edwards Corrected Tangent Cone
Mark III HABP Tangent Cone
Newtonian Theory (K = 2)
O Exact (Kopal, T = 1.405)
[] Exact (Jones, 7 = 1.4)
, I , I , I i I , I
5 10 15 20 25
Mach no.
Figure 7. Pressure coefficient versus Mach number for 20° sharp cone.
Cp
1.2
1.0
.8-
,6 --
.4
0
IiiiiIII
I I I N
iIIII IIII
0
[]
2nd Order Slender Body
+ Hammitt/Murthy
Edwards Corrected Tangent Cone
Mark III HABP Tangent Cone
Newtonian Theory (K = 2)
Exact (Kopal, _, = 1.405)
Exact (Jones, ? = 1.4)
llllJll .... lllllIllllllNlIilIIlilJllllllliJJllmlJinliJlJllJJlIllllJJiJnl
i I i I i I i I i I
5 10 15 20 25
Mach no.
Figure 8. Pressure coefficient versus Mach number for 30 ° sharp cone.
,
Cp
16
14
1.2 -
10-
8
0
2nd Order Slender Body
+ Hammitt/Murthy
........ Edwards Corrected Tangent Cone
Mark III HABP Tangent Cone
......... Newton°an Theory (K = 2)
o Exact (Kopal, ",f= 1405)
n Exact (Jones, 7 = 1 4)
J-L_
.... _"....... -r....... 3....... -I....... _ ....... [....... ;....... T ....... 7....... ]
5 10 15 20 25
Mach no
Figure 9. Pressure coefficient versus Mach number for 40 ° sharp cone.
2.00
1.75
Cp 150
1.25
2nd Order Slender Body
+ Hammitt/Murthy
........ Edwards Corrected Tangent Cone
Mark III HABP Tangent Cone
......... Newton°an Theory (K = 2)
o Exact (Kopal, 7 = 1.405)
_/\_:_,__j rl Exact (Jones, 7 = 1.4)
1.00 , I i I J I n 1 i I
0 5 10 15 20 25
Mach no.
Figure 10. Pressure coefficient versus Mach number for 50 ° sharp cone.
7_

Report Documentation Page
Nallona' AeronauTIcs an(_
S_a(e Ad nn,p,s1,_lon
1. Report No. / 2. Government Accession No. 3. Recipient's Catalog No.
NASA TM-4165 1
4. Title and Subtitle
Improved Tangent-Cone Method for the Aerodynamic Preliminary
Analysis System (APAS) Version of the Hypersonic Arbitrary-
Body Program
7. Author(s)
Christopher I. Cruz and Gregory J. Sova
'9. Performing Organization Name and Address
NASA Langley Research Center
Hampton, VA 23665-5225 11.
12. Sponsoring Agency Name and Address 13.
National Aeronautics and Space Administration
Washington, DC 20546-0001 14.
5. Report Date
February 1990
6. Performing Organization Code
8. Performing Organization Report No.
L-16404
10. Work Unit No.
506-49-11-01
Contract or Grant No.
Type of Report and Period Covered
Technical Memorandum
Sponsoring Agency Code
15. Supplementary Notes
Christopher I. Cruz: Langley Research Center, Hampton, Virginia.
Gregory J. Sova: Rockwell International Corporation, North American Operations, E1 Segundo,
California.
16. Abstract
The Aerodynamic Preliminary Analysis System (APAS) utilizes a modified version of the
Hypersonic Arbitrary-Body Program (HABP) Mark III code in its analysis rationale. Four
methods are considered for incorporation into the code as the tangent-cone method. The
combination of second-order slender body theory and the approximate solution of Hammitt and
Murthy shows the best agreement with the exact numerical solutions and is thus included in the
APAS production version of the HABP code.
17. Key Words (Suggested by Authors(s))
Hypersonics
Tangent cone
18. Distribution Statement
Unclassified--Unlimited
Subject Category 02
19. Security Classif. (of this report)unclassified 120"SecurityClassif(°fthispage)Unclassified 121N°'°fPages22"Price8 A02
NASA FORM 1626 OCT 86
For sale by the National Technical hfformation Service, Springfield, Virginia 22161-2171
NASA.Langley, 1990





Improved tangent-cone method for the Aerodynamic Preliminary Analysis System (APAS) version of the hypersonic arbitrary-body program

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