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TRANSCRIPT
* --.... - -- -
NPS67-82-OO2CR
NIIAVAL POSTGRADUATE SCHOOLMonterey, California
" ' =DT-IC
S`~/54 GIM SYSThM
-DD AAUCIAI 4. -rch 1982
Aprvdfor public tIlea~se; distribution unliuit~d
,Prepared for: lrc14~va1 Po~stgraduate School avl- toDI dow
Ioterey, California -93"4
I C ------ T*-RPOR
NAVAL POSTGRADUATE SOWDL14ontaruy, CaVllfor?1i8
p~a~ Admral .JEkeln0., A. Schra'dy
Superintendent Atn rvs
The work reported herein was carried out for the Naval Postgraduate
School by Oded Amichai under Contract Number ?4OOZ28-8lC-H231. The work
presented in this report is in support of solid fuel ramijet research
* ~and the exploration of Navy application's for Advanced Indirect Fire
Support, AIFS. Both projects are funded by the Defense Advanced Research
Projects Agency and are under the cognizance of Professor A. E. Fuhs.
Reproduction of all or part of this report is authorized.
u ~This report was prepared by;
Contracted Research Associate
"Reviewd by:
Ditg I1b Pof!SrActing1 Oa1!rMT1 DQOsa. UWt of Aeronautics
Rsasdby:
M.~- 14 Tll
than ~ ,- ofPea
~ ~ ' .
K-
DISCLAIMER NOTICE
THIS DOCUMENT-IS BEST QUALITY
PRACTICABLE. THE COPY FURNISHED
TO DTIC CONTAINED A SiGar' rFCAN I
NUMBER OF PAGES WHIC, H DO NOT
REPRODUCE LEGI3LY.
- -- I.- .-------*-.- --- --- -- ~ - ---
SECUR TY CLoSSIýICATIOH OV TWS lkQ ý Whe., DX. iP.11vo~d)
REPORT DOCUMENTATION PAGE I BPREA II4STRU!JCGUS____ BEFORE________ N FORM
REPOC" .. 3E" 2N GO0. ZCESO MECIPIENVS CATA.LOG NMBLJMUp
4TITLE ra.,S~brffl 1 1*TfF FM PR SR0 CCvIEIED
PERFORMAN~CE OF SOLID FUEL RANIJET GUIDEDCntatrRprPROJECTILE. FOPR US! 5"/54 SUN SYSTE ________
AIJT44OR~~~gT S. (.ONT~AT~ N
Oded Amichai N002428.-81-C.-H231
QPIE~vOU&INO ORlOANIZPTI' 1AIA4OAN AOXVESS 10. P'!OGF#AM F.LEM,:Nr. PqOJECT. TASK
A MEA It IOR UNI1T NuMOCASNaval Postgraduate School DARPA RDER 4035Monterey, California 93940 'ELEi4ENT: 627022
1 1. C')14TROL1..ING rV14P N AMIE AND A001RESS It. R~fP160T GATE
Defense Adv:ýnced Research Projects Agency March 19B21400 Wilso~n Boulevard NUMSpes;kArlinjtor.,_Virginia 22209 II EUIvCAS.(1te ool
7TMO0iTOPINCZ AG6'F4C' IIAPAE _&ADOMF3SS(I dltImrong fýn CO*Urf? flog Qffic.) 1.SCl1VCAS.(I(i o "
UNCLASSIFIEDf I$*. OVCt.A!SiFICATIOH/ OOWNORAGING
ICWF DIJLEI
Approved for public release, als-cibutiit-ufilimite~d
17. OIS¶'IltUrfOA STATtiMENT (.t In.. abotract .ffaid In~ Black JO. It dlifoetnt fromn R"60v)
6. U~'LEEN1~fNO-ES
1~KtY WORDS5 (C.,rIrd,, r.r 0't ,v..id, It natsar ard ldoIfy Fl. block aousqh.)
guided projectiles, precision guided munitions, guns, gun launched ramjet,soldi fuel ramjets, antiship missile defense.4
0 ABSI NACT (Contin..* on -9mee mid. It .Ioco*.ar Ad fdvntlfy bY block nUMb.,)\fThis rE~ort covers work done on performance analysis of a 5 inch, 54
cali~ber gun-launched guided projectile with solid fuel ramjet (SFRJ).A computer program (TRAIjET) was developed. The program contains ramijet
and trajectory analysis. rhe ramjuet part considers conical shock wave losses,inlet boundary layer losses. normal shock losses, subsonic diffuser recovery,expansion into combustor losses, heat losses at the combustor and nozzle losses
7i 4 A flat earth trajectory with drag and thrust was, nsidered. The variouss
ODDI17 147 EDITION OF ' INOV 55 IS O@iOLETPE UNCLASSIFIEDSICI.IITYe CLASSIFICATION OF Ti-ItS PACE (W71~o Data tntwfd) .
UNCLASSI FIED
20. Abstract continued,drag coefficients which were considered are; cowl drag coeffi:ient, skin 'iragcoefficient, wing (or fin) wave drag coefficient and 4ing (or fir) fr•ctiondrag coefficient. Base drag is assumed to be zec duc tu the jet from ramje'.nozzle.\
tO as founld that the cowl drag coefficient has a major infl4jence on theresul ts Therefore, a separate program (AERO) was developed to calculktte thisparamTter.
,The 50/54 solid fuel ramjet has a capability to produce fuel speciFicimpulse in the order of 400 - 900 sec. dtpending mostly on the flight elt:tude.The thrust coefficient varies in the range of 0.3 t, 0.1 depending on theinternal areas. -- ---
A range in the order of 50 miles can be achieved with the ramjetoperation compared to only 13 miles achieved by thn conventionan projectile.At low-altitude launch, a range of over 18 uwles can be reached in Air-DeTenseScenario.' The ramjet propelled projectile reaches the ranges mentioned aboveat high Mach numbers (Mo >. 1.8).)Ajt is, therefore, clear that the ramjetconcept provides significant imp vement and has an Anti-Ship MissileDefense (ASMD) capability.
II
S_ .U-UNCLASSIFIED
', N OF!. THIS .
""-rS --- ~~ 3r~'-r ,- "- -5 '.
ACKNOWLEDGMENT
The author wishes to express his sincere appreciation to Distinguished
Professor Allen E. Fuhs for his invaluable guidance, help and friendship.
The atthor would also like to express his gratitude for Professor
T. H. Gawain for his assistance and cooperation in calculating the cowl
drag coefficient.
The a.u.hr ^ also gives his thanks to Professor D.W. Netzer for hiF
technical assistance.
The author would like to thank Mr. Uri Katz for his help and for the
fruitful discussions we had during his stay as a research associate ac. the h
Naval Postgraduate School.
The illustrations and graphic arts for this report were accomplished by
Don Jacobs of the N4aval Postgraduate School Educational Media Department.
His help is appreciated.
*0
iii 1)
. 2•1
ABSTRACT
This report covers work done on performance analysis of a 5 inch,
54 caliber gun-launched guided projectile with solid fuel ramjet (SFRJ).
A computer program (TRAJET) was developed. The program contains
ramjet and trajectory analysis. The ramjet part considers conical
shock wave losses, inlet boundary layer losses, normal shock losses,
subsonic diffuser recovery, expansion into combustsr losses, heat losses
at the combustor and nozzle losses.
A flat earth trajectory with drag and thrust was considered. The
various drag coefficients which were considered are: cowl drag coefficient,
skin drag coefficient, wing (or fin) wave drag coefficient and wing (or
fin) friction drag coefficient. Base drag is assumed to be zero due
to the jet from ramjet nozzle.
It was found that the cowl drag coefficient has a major influence
on the results. Therefore, a separate program (AERO) was developed to
"calculate this parameter.The 5"/54 solid fuel ramjet has a capability to produce fuel
specific impulse in the order of 400-900 sec. depending mostly on the
flight altitude. The thrust coefficient varies in the range of 0.3±0.1
depending on the internal areas.
A range in the order of 50 miles can be achieved with the ramjet
operation compared to only 13 miles achieved by the conventional pro-
jectile. At low-altitude launch, a rang. of over 18 miles can be reached
with the ramjet version. Launches at high elevation angles can be
useful in air-defense scenario. The ramjet propelled projectile reaches
the ranges mentioned above at high Mach numbers (Mo0 l.8). It is, therefore,
ivo- -1
clear that the ramjet concept provides significant improvement, and has
an Anti Ship Missile Defense (ASMO) capability.
Vi
II
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I. - - - � ----.-..
TABLE OF CONTENTS
1. INTRODUCTION ....... ... .......................... .. 1
1.1 Background .. .................................. .. 1
1.2 Basic Concept ....... ... ....................... 2
1.3 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3.1 Dimensions ......... ..................... 5
1.3.2 Combustion Process ....... ................. 6
1.3.3 Trajectory ........... .................... 6
1.3.4 Air Defense Scenario ....... ................ 7
1.4 Results. ......... .......................... 7
2. TYPICAL RESULTS ............. ........................ 9
3. DISCUSSION ............. .......................... ..... 19
4. CONCLUSIONS .............. .......................... ... 27
Appendix A. Solid Fuel Ramjet - ..uations -. . . ..... 28
Al. Combustion ............. ........................ 28
Al.1 Computation of fuel - regression rate, weight rate of
burning fuel ar.d fuel - air ratio .... .......... 28
Al .2 Computation of combustor exit conditions .......... 29
Al.3 Computation of nozzle exit conditions . . . . . . . . 3.
'A1.4 C..utati.+n of thrust and thrust coefficient- 31
A2. Chec for Choked Nozzle ........ ................. 35
A3. Heat Losses at the Combustor .................... ... 35
A3.1 Mach number ...... ..................... ... 35
A3.2 Total Pressure. . .................. 38
A4. Computation of Mach Number arid of TotAl Pressure at the Various
Stations of the Inlet ......... 3
ix
S - - ..
A4.I Initial conditions . . . . . . . . . . . . . . . .. . 39
A4.2 Conical shock wave loss ............... 40
A4.3 Boundary layer loss ........ ................. 45
A4.4 Novinal shock lnss ....... ................. .. 46
A4.5 Subsonic diffuser recovery ...... .............. 46
A4.6 Expansion loss ..................... 47
A4.7 Location of anoral shock wave .. .. .. .. .. . .. 49
Appendix S. Trajectory Equations ..... . ................. 50
Bl. Atmspheric Functions ......................... ... 50
B2. Drag. ............. ........................... 50
B2.1 Cowl drag coefficient ..................... ... 50
B2.2 Base drag. ......... ....................... 51
8 B2.3 Skin drg COEfi e ..-....... . . 51
6'1B2.4 Wng ind fin drag coefficients ........ ....... 52
B2.5 Calctiltion of drag; ..... ................. .. 54
B2.6 Drag coeftici•.ut of a conventional projectile without
•3 propultsion ..................................... .. 55
63. Booster ................. .......................... 55
.54B . '-ynaui cs . . . . . . . . . . . . . . . . . . . . . .. . . 56
Appendix C. Flow Chart of the Computer Program ............. ... 57
Cl. Main program ........... ........................ 57
C2. Command subroutines ........... .................... 61
C2.1 I?4LET ........... ........................ ... 61
"C2.2 CORVAL ........ ........................ .. 63
C2.3 TRAJ .......... ........................ .. 64
C3. Individual Subroutines ...... ................... .. 65
C3.1 ATM ................ ...................... 65
C3.2 BOOS ....... ......................... ... 66
X
II _I
C3.3 IrITT . . . . . . . . . . . . . . . . . . . . . . . . .. 57
C3.4 BURN . . . . . . . . . . . . . . . . . . . . . . .. . 68
C3.4.1 INTER.I . . . . . . . .. . . . . . . . . . , 70rIC3.5 NOZZ . . . . . . . . . . . . . . . . . . . . . . 71
C3 5.1 CALCM ........... .................... 72
C3.6 CHOKE ........... ......................... 73
C3.7 HEAT ........... ......................... ... 74
C3.8 INLET ........... ......................... 75
C3.8.1 CONE ......... ..................... 75
C3.8.2 THROAT ...... .................... 76
C3.8.3 NSR ........ ..................... .. 77
"C3.8.4 DIFFUS ....... .................... . 78
C3.9 EXPAN ...................... . ............ 79
C3.lO CHECK ............ ......................... 80
C3.11 RESUL . G.............................. 81
C3C.1 Z TRAJ ... ........... .................... 82
CM 2.11 DPAGG . P .......................... 82
C3.12.2 DVNA . . . . . . . . . . . . . . . . . . . . 84,
'•Appendix 0. Program TRAJET.: Listing .. .. .. .. .. .. .. ... 85
Appendix E. Computer Program List of Symbols ................. 106
Appendix F. Computer Program Users Guide .................. .. 116
AppendiA G. Program AERO: Listing ..... ................. .. 118
Appendix H. Results .................................... 141
References . .Re t . ..... .............. ..... 203
It
XIL
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4. --.-.- , -.-- -- - ..----- I
LIST OF FIGURES
1.1 Schematic View of a Solid Fuel Ramjet ........... 3
1.2 L' S. Navy 5"/54 Semi Active Laser Guided
"projectile (SALGP) ........ ..................... 4
2.1 Solid Fvel Ramjet: Dependence of Fuel Specific
Impulse (I ) on Flight Mach Number at VariousSp
Altitudes ......... ....... ........................ 11
2.2 SFRJ: Dependence of Thrust Coefficient on Internal
Area Ratio (Ao/Ar) at Various Altitudes ....... ... 12
2.3 SFRJ- Dependence of Thrust Coefficient on Inlet and on- Nozzle Area Ratio ..... ... ..................... ..13
2.4 SFRJ: Dependence of Fuel Specific Impulse (I ) onspThrust Coefficient at Various Altitudes .... ........... 14
2.5 SFRJ: Dependence of Fuel Specific Impulse (I ) onspThrust Coefficient at Various Internal Area Ratio . . . . . 15
2.6 Comparison of Trajectory of SFRJ with Conventional
Projectile at Various Conditions .... .............. .. 16
2.7 Solid Fuel Ramnjet Propelled 5"/54 Projectile - Air
Defense Mission .................... 17
3.1 .Solid Fuel Ramjet Propelled 5"/54 Projectile: Design . . . .23
3.2 Solid Fuel Ramjet Propelled 5"/54 Projectile: Design,
Section B-B ....... ....................... 24
xii
1. . . - a -
------ "----."----- *. ...- r.t- - .- • - -- . -- - .. _" . . • " •- -- - - . .
3.3 Aft Body Fin Design [24]. ........ .................... 25
3.4 Aft Body Fin Design: Section A-A 1241 .... ............. ... 26A4.1 Geometry for Comical Shock Wave Showing Normal Component
of Mach Number ...... ..... ... ........................ 42
A4.2 Geometry for Calculation of Inlet Annular Flow Area
Relative to Inlet Capture Area ....... ................. 43
A4.3 Oblique S~rk Solutions .... ...................... .. 44
B2.1. Scheiattc View of a Wing/Fin ........ .................. 53
G4.1 Geometry for Calculation of Cow•-Drag-Coefficient (Programs
AERO AND COWL) Showing Definitic.n of Symbols ............. ... 139 F
G4.2 Typical Results from AERO ..... .................... ... 140
II
J! I .
I
1 . INTRODUCTION
(1.1 Bacarun
This report covers work done on propulsion and flight me-
chanics of a gun-launched guided projectile. Gun-launched guided pro-
jectiles have been developed by Martia Marietta for the U.S. Army
(Copperhead) [1,Z] and more recently, also, for the U.S. Navy [3,4]
(5"/54 Mark 46). The USN round has solid rocket propulsion.
The addition of a -,iouid fuel ramjet (LFRJ) to the Navy'sversion, was examined in th.! past by Brown [5]. This report concentrates
on the addition of a solid fuel ramjet (SFRJ), instead.
It is believed that solid fuel ramjet has some potential
advantages compared with the liquid fuel. Some of these advantages
are:
-Simple design
-High reliability in operation
-Low cost
-Fuel control system not neededOn the other hand, there are alsu a few disadvantages to SFRJ compared
to LFRJ. Some of these are:
-Difficult to control magnitude of -.hrist
-Dificuliesin achieving high combustion efficiencies
In both cases, the addition of propulsion improves dramatically,
the performance of the projectile by multiplication of range and enhanced
maneuverability. Even more; to operate and produce thrust, the ramijet
eng~ine depends only on its forward motion at supersonic speeds and does
no t employ any moving parts. This fact, which is especially emphasized
in SFRJ, leads to some advantaaes of the ramjet concept over the other
1f
propulsion alternatives, at supersonic soeed. On the other hand, the
ramjet engine requires an auxiliary booster to accelerate it to its
supersonic operating reoime. The boost required causes some systemdifficulties. But, while solving these problems, the ramjet system I -
becomes even more attractive for use with gun-leunched guided pro-
jectile, like the U.S. Navy 5"/54.
A computer program was developed to analyze the performance
of the SFRJ, The computer program was written for the IBM-370
computer at the Naval Postgraduate School, Monterey, California. HTPB
was selected as a fuel, but performance with any other fuel can be tested,
using the same model. A flat earth trajectory with drag and thrust was
considered. Using solid fuel, a thrust-equal-dray trajectory is more
difficult to achieve withi SFRJ. Therefore, most of the results given
in this report eliminate this case, and the exact value for drao, at
each point, was calculated. However, if desired, it is possible to
change the air mass flow, in order to obtain thrust-equal-drag flight.
The computer program can calculate this case diso.
1.2 Basic ConceptJ i
The ramjet engine consists of an air inlet, which serves as
a diffuser, a combustion chamber, and an exhaust nozzle [6,7]. The
diffuser admits air to the engine, which is mixed with fuel (solid or
liquid) at the combstor. After the burning process, which adds heat
to the flowing air within the system, the gases are transfered to the
"nozzle. The nozzle ccnverts part of the thermal energy into kinetic
energy to produce thrust.
six The areas inside the ramjet engine are usually divided into
six stations, as illustrated in Figure 1.1. Station 0 defines the cross-
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section area of the stream tube captured by the inlet of the projectile.
Stations 1 and 2 identify the diffuser. Station 1 itself is at the
throat of the diffuser, but sub-station IC is after the conical wave;
sub-station 11 and S1 are ahead of the normal shock wave, located at
the throat, or at the actual place, respectively; sub-stations 12 and
S2 are as above, but behind the normal shock wave. Stations 3 and 4
refer tQ the entrance and to the exit of the fuel grain within the
combustor, respectively. Note that both A3 and A4 increase with time
as fuel burns. Station 5 and 6 belong to the nozzle's throat and to
the exit of the nozzle, respectively.
1.3 Data
1.3.1 Dimensions
In order to be compatible with the Navy's 5 inch, 54 caliber,
Mark 46 gun mount, as modified for gun launched guided projectiles
(Figure 1.2), a set of requirements were adapted initiall-y-; Thise were:
a. External shape of existing 5" guided SAL projectile
b. Length - 60.97"
c. Length of combustion chamber - 23"
•,d. Total weight - 104.7 lb.
e. Muzzle velocity - 2500 ft/sec.
Typical values for the internal areas in the ramjet within the
• projectile are (units - sq. in.):
A A A A2 A3 A5 A6
19.3 5.2 2.6 4.3 8.2 7.5 13.1
Refe1," to Figure 1.1 for definition of the areas. Ar is a reference area.
For typical flight Mach number of: M0 = 3, the appropriate Mach numbers
at the main stations are typically:
45
; MC M 11 M12 M2 M3 M5 M6
2.2 2.1 0.56 0.3 0.1 1 2
1.3.2 Combustion Process
Some of the losses in the total pressure were taken to be
constant. These are:
a. Inlet boundary layer losses (Tr'D; typically = 0.93).
b. Subsonic diffuser recovery (r" typically = 0.93).
c. Nozzle losses (n; typically = 0.96).
These values are typical and are not expected to vary too much.
All the other losses in total pressure, which are dominant to
, the projectile per'ormancevere calculated. These are:
a. Conical shock wave losses (TrC).
b. Normal shock losses (rl,S).
c. Luýses due to expansion into the combustion chamber (e.
d. Heat losses in the combustion chamber (vh).
See section A 1.4.2 for definition of various r . Combustion efficiency
was taken constant: 0.90 1.9Air heat capacity ratio was also taken constant: ya = 1.4; however, the
value for the gas heat capacities ratio of the combustion products (",f)
was calculated from thermodynamic data for Hydroxy Terminated Polybutadiene,
HTPB, burned in air 18]. The stoichiometric chemical reaction of HTPB
burning with oxygen is as follows:
4 C-73H 0301 + 393 02 - 292 CO2 + 206 H20 (1.3.1)
1.3.3 Trajectory
In the trajectory part of the program, the various drag coef-
ficients were calculated. Those are:
a. Cowl drag coefficient (CDN), [9-.12]. .
6
... . ...-..-...... .._,
b. Skin drag coefficient (CDs), [13-15].
c. Wing (or fin) wave drag coefficient (CDww), 16 - 20].
d. Wing (or fin) friction drag coefficient (COWF), [13, 19].
Base drag [13, 21-23] is assumed to be negligible due to the jet fr M ramjet
nozzle.
The model, which was chosen to calculate the cowl (nose) drag
coefficient, was based on a theore~ical development done by T. H. GawainL91. The modified program (AERO) is listed in Appendix G. However,
program AERO as it is, appears to be too long to be used directly in the
main program (TRAJET). Therefore, best fit curves for calculated results
from AERO were used in TRAJET.
Skin drag coefficients were calculated for either laminar or
turbulent flows. The same routine was also used to calculate wing or fin
friction drag coefficients. To calculate the wing wave drag coefficients,
a psuedo 3-dimensional model was developed.
The program alsc has an option to calculate a trajectory of a
projectile without propulsion. In this case, the drag coefficients which
are calculated are:
a. Nose drag coefficient (a different model than the above).
b. Base drag coefficient
c. Skin drag coefficient (as above).
1.3.4 Air Defense Scenario
- -K In the air defense scenario, the program takes into account only r
cases in which the projectile exceeds a Mach number of at least 1.8. This
value of minimum Mach number (XM0 ) can easily be changed.
1.4 Results!ach iection of the program was developed, tested and run sep-
arately. The ramjet part was first run without thi trajectory part using
7
14\
constant altitude (Typically - 10,000 ft). The same was done with the
trajectory part usinq vacuum case, thrust-equal-drag flight, or constant
thrust case. The final version was programmed to give the following
optional printings:
a. Loop on all possible values of AG/Ar and As/Ar and print
summary tables only.
b. Print detailed, time dependent, tables for any specific area
ratio chosen:
- Results from combustion process (file name: CMB D)
- Results from trajectory process (file name: TRJ 0)
- Various drag coefficients (file name: ORG 0)
c. Detailed print of every step during the calculation, for
I checkup.
d. Variation of the above:
Detailed print of cases that were found not to be
suitable:- Reasons only
- Full detailed parameters
- Loop on Mach numbers, also (output of subroutine
CALCM)
i 'Iii
- -
2. TYPICAL RESULTS
Figure 2.1 presents the dependence of the fuel specific impulse (Isp,in sec.) of the ramjet on the projectile Mach number at various altitudes.
It appears that the 5"/54 ramjet has a capability to produce fuel specific
impulse in the order of 400 - 900 sec., depending mostly on the flight al-
titude. The dependence of Isp on the flight Mach number is weak.
Figures 2.2 and 2.3 present the dependence of the thrust coefficient
(Cf) on the internal area ratios Ao/Ar and As/Ar. In figure 2.2, the change
I i of Cf with altitude and with Mach number is also presented. The thrust coef-
ficient (Cf) varies in the range of 0.3±0.1 while Ao/Ar changes from 0.25 to
0.40 and A5/Ar changes from 0.42 to 0.26,IrThe correlation between the fuel specific impulse (I )I and the thrust
sp.
coefficley, (Cf) is presented in figures 2.4 and 2.5. in both figures, a Mach
number of M0 = 3.0 was selected. In figure 2.4, the correlation was checked
at various altitudes and at various Ao/A 5 area ratios. In figure 2.5, various
"internal area ratios (Ao/Ar, A5 /Ar) are presented.
More detailed results are presented in Appendix H. The dependence of the
projectile performance on the other internal area ratios (AI/A 0 , A2 /Ao, A3 /AV)
was also checked. Some typical results are presented in that Appendix.
An altitude vs range dependence for various elevation angles is presented
"in figure 2.3, A range of over 80 km can be achieved with the ramjet operation,
compared to only s~ightly more than 20 km achieved by the conventional pro-
"jectile. A low-altitude launch (in this figure, an elevation angle of 15°
was selected) is also presented reaching a range of over 30 km with the ramjet
operation. The high elevation angles are mostly used in air-defense scenario.
The drag of the projectile when the ramjet is not operating, for example,
9
;i•
iiII'
I-I I-I I I -I- I I I I I- I I -
after burnout, was not determined. The computer program does not account
for the drag increase due to the ramjet not operating. Consequently some
of the trajectory curves in figure 2.6 are in error. However, for tra-
jectories at low gun elevation, the ramjet burns all the way to splash.
These trajectories are accurate, For all trajectories, the curves are
accurate to the point of ramjet burnout. The trajectories of interest to
air defense are accurately calculated. Trajectory of thrust-equal-drag
(vacuum) case is shown for comparison.
Results for air defense scenario are presented in figure 2.7. Only
ranges where the projectile Mach number exceeds at least 1.8 where con-
sidered. The area ratios where chosen as specified in the figure. The
Stwo cowl angles, shown in figure 3.1 were 20° and 9.5° respectively.
The gun elevation angle was varied from 70 to 80°. The change of atmosphericconditiovis with the altitude was taken into account. In table 2.1, some
"typical results for "Surface-to-Surface Mission" are presented.
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w 3
= !-~* 4J
I. ~~C,* ,
cm0
Lie.i
II
(W j)3flLL1
¶ 17
TABLE 2.1
Solid Fuel Ramjet Propelled 5"/54 ProjectileSurface-to-Surface Mission
Ranges (kin.) vs Gun Elevation Angles
Elevation
Angle '1*° 250 45" 650 800
"a. Ramjet(l) 15.6 49.9 80.3 15.9 5.6
b. Projectile 9.2 17.5 20.2 16.3 7.6I W IIU I..
Propulsion
Note: 1. Area ratios as in Figure 2.7
I.
,I
I
3. Discussions
Looking back at figure 2.1, the dependence of Isp on altitude and on
Mach number should be explained. We shall do that by using the equations
described in Appendix A.
From equations: (l.4.9a), (1.4.12), (1.4.16)
together with equations: (1.1.1), (1.1.2), (1.1.3)
one obtains:
Csp f X 3 (3.1)
where: X3 k P 0.4M 1. 4 (3.2)
and: C A k2 (3.3)
where: X2 = klMo[X -l] (3.4)-0.4
1 4 Mo+k 5 (PO ) (3.5) "
The parameters k, to k5 are functions of the various area ratios, the
temperature of air (TO), the heat capacity ratio of air (Ya) anti the
perfect gas constant (Ra). These parameters are assumed to be constants
in discussing the influence of the change in altitude and in Mach number
on the value of iSp. The altitud dependence is mainly due to change in
atmospheric pressure (Po). In the conditions :hosen for figure 2.1, the
dependence of Isp on pressure is approximately PO0O4 at M -3. That means
that, in the region mentioned, I pressure dependence Is mostly due to
change of X3 (equations 3.1 & 3o2). Frm-2 the same equations, the Mach
number dependence of Isp can also be explained. At high altitude, the
1.41change of X3 (Mo') is very close to the Cf dependence on Mach number
(M- 1 .3) and therefore I is almost constant while changing MO. On the•, ( O1 .35•
other hand, at sea level, the change in X. , is snmtaller than that of
-214Cf(Mo ) and therefore Is changes with M0 as shown in figure 2.1.19
1I 14
-i #,--tet-%=_& ~.
Figures 2.2 - 2.5 present similar dependences of the ramjet performance,
and can well be understood using the same equations.
Testing these results, together with those presented in Appendix H,
the design of the ramjet internal area ratios can be completed. The
results are listed in Table 3.1.
20II IK
jti
20
- - ,~----'. -- -
Table 3.1: Ramjet Design
Dimensions:
External diameter = 5"
Total length = 60.97"
Total weight = 104.7 lb (47.5 kg)
Area Ratios:
Ao/Ar A1/A0 A2 /A0 A3/Ar A5 /Ar A6 /A1
0.25 0.47 0.887 0.426 0.26 1
ReferLeice Area:
Ar 19.3 sq, in. (124.5 cm
Combustor
Solid fuel: Hydroxy Terminated Polybutadiene (HTPB).
Fuel weight;o..3 kg jFuel density: 971.56 kg/m 3
Fuel specific impulse (Isp): 400 - 900 sec. I
Booster
i Booster weight: 2 kg
Booster density: 1650 kg/mr3
Booster specific impulse (Isp): 240 Sec
Performance
Muzzle velocity: 762 m/sec
Velocity after booster: 863 m/sec
Thrust Coefficient (Cf): 0.2 - 0.4
21
LL'
In figures 3.1 - 3.4, the designed ramjet concept is presented. The
guidance and the control sections as well as the warhead were not redesigned.
The location of the tailfin is described in figures 3.3 - 3.4 [White,24].
The configuration of the solid fuel ramjet presented here, is in accordance
with the design of the liquid fuel ramjet done previously by Brown (5].
The new design is also in agreement with the Navy's requirement to be
compatible with its 5"/54 Mark 46 gun mount, as modified for gun launched
guided pritjecti les.
Figur'e 2.6 presents a full-range comparison of the SFRJ 5"/54 pro-
.jectile performance with that of the conventional projectile. The
improvement in oerformance of the ramjet-.propelled, luided projectile is
very significant. The extended range of the ramjet concept can be used in
a "Surface-to-Surface Mission" (Table 2.1). The main improvement of the
ramjet concept might be in an "Air-Defense Mission" (figure 2.7). A
minimum Mach number of M0= 1.8 was assumed for both the ran'et concept
and the conventional projectile. At lower altitudes, the projectile is
limited by burn-out of the fuel. At higher altitudes, the performance
boundary is due to ramjet unstart. The corventional projectile is limited
by the Mach number decay to valIU '. Ies than 1.8. IT+ is slf-evient
that the ramjet concept provides significant improvement, and, therefore,has significant ASMD Capability. I.
2Z
ii-. II
i 'a
4%I Jr CJ
9=
U-J
ii I
C.0.
~Jl - 23
MAXIMUM OUTSIDE DIAMETER
V.. .4.96 .
ALL DIMENSIONS IN INCHES
1.50
tV
0.076SECTION B-6
I I.
a Figure 3.2 Solid Fuel Ramjet Propelled V'/54
Projectile Uesigri, Section B-B
2I4
-H-'cV
-l
II =_j.I(D
o. ,
lI I
MAXIMUM OUTSIDE DIAMETER
MAX.MU,-4.96 1IrALL DIMENSIONS IN INCHES
ALLOWABLE AREAFOR EXHAUST
S....-=-NOZZLE
* /iSi•~.50 ,
S0.-219
SECTIOK A-A
"t*..
Figure 3.4 Aft Body Fin Design: Section A-A t2 4 1
S26i lk
Yi
4. Conclusion
Based on the computer model discussed above, the ramjet-propelled,
guided projectile provides significant improvement over the conventional
projectile. The fuel specific impulse of the SFRJ is in the order of 600
±200 sec, depending mainly on the altitude of the projectile. The
appropriate value of the rocket is only 300 sec. The thrust coefficient
varies from 0.,2 to 0.4, depending on the atmospheric conditions (altitude)
and on the geometry of the projectile (internal areas). Therefore, the
ramjet-propelled, guided projectile reaches a range of about 80 km compared
to range of slightly more than 20 km in the conventional projectile. The
improvement of the ramjet concept might be in both "Surface-to-Surface
MissiGn' and in "Air-Defense Mission". It provides an ASMO weapon which
is cmipiefwntary tO gulded misies=-
More work is required to design an optical system incorporated into
the inlet. For satisfactory ramjet performance at the flight Mach numbers,
the lens must be a conical shape.
2 I.
K
'I
.27 - " - .- _ ,
I,
Apvendix A: SOLID FUEL RAMJET: EQUATIONS
Al. Combustion
A1 .1 Computation of fuel - rearession rate, wei ht rate of
burning fuel, and fuel -. air ratio
Define G as the weioht flow rate of air per unit through the
entrance port to the combustor; see figure 1.1. Hence G is given by:
WaG = (1.1 .1)
3and has dimensions of lb/sec.in2 . The simple form of the regression
rate of a solid fuel, 9', is given by:
9' aGn (1.1.2)
Where a and n are empirically determined constants. The dimensions for
t are in/sec. Knowledge of the temperature dependence of the regression
rate will allow the use of a more accurate model instead of equation(1 .1.2). I.:
The weight of fuel burned per- unit of time is Ls follows:
W f = pf tW03L 3 (1.1.3)Where Pf is the density of the fuel in 1b0in3, D is the inside dia-
lb 3
meter of the fuel grain. Note that 03 increases as the fuel burns.
The length of fuel grain is L3 inches. For the case of HTPB, the value
of 0.0351 Wb/in 3 was taken. In an actual solid fuel rainjet, P varies
along the grain; t is largest in the region immediately downstream of
the entrance port of the combustor. However, for the ramjet model
develooed here, the value of t is assumed to be constant along the grain. 'i' ~~Consequently: ,
D3 D30 + 2 tdt (1.1.4)
28
The integral form for the change in grain internal diamc - is used
since t may vary with time. The initial grain inside diameter is 0
inches, and the inside area for fuel grain as a function of time can be
written as:
A [ A3 0 + 2 ý d (r.d.5)
By definition, the fuel - air ratio is:'fi f = -- (1 .1.6)
Wa
The value for Wa is obtained from weight flow through the inlet, and the}a
value for R is calculated using equation (1.1.3). The total mass flow
through the nozzle is given by:
W •T : Wf + W~a a �a(I + f) (1.1.7)
For HTPB burning in air, the Ttoichiometric value for f is 0.0728; high
combustion efficiency is difficult when f is less than 0.025.
A1.2 Computation of Combustion Exit Condition
Combustor exit conditions are specified by four quantities as
"follows: stagnation temperature, TT4, OR; stagnation pressure, PT4'
psi; ratio of heat capacities, yf; and gas constant, Rf, '"/OR. In
the computer program, the appropriate mks units are used, i.e. 0 k-:,V K t respectively. To determine the exit conditioris, two input quan-
tities are needed. These are fuel - air ratio and stagnation temperature
at the combustor inlet, TTO. Note that TT3 =.TTo has been assumed.
From the thermodynamic data for HTPB burning in air, one determines
TT4 (th), Yf, Rf. The symbol TT4 (th) is a theoretical temperature which
results from 100% combustion efficiency. Introducing the definition of
combustion efficiency yields:
29
--- -tI I I lI i I I l
TT4 [T TT4(t TT 01 TTO(12)
As discussed previously, a constant value of nT equal to 0.9 has been
assumed.IThe value for T4 is calculated based on one-dimensional, choked
nozzle flow. A certain value of PT4 is required to force a certain weight
flow, WT' through the nozzle. Assume that yf remains fixed through the
nozzle. Define a function of yf as:
[ 2 ] J�Yf + 1)/[2(yf - 1)]i 9 : ~ f ( 1 . 2 . 2 )Tf+
Define a characteristic nozzle velocity,
C*', m/sec:• :c * : ( 1 .2 . 3 )
I:
e
where g is the acceleration of gravity and has value of 9.807 m/sec
The required value for PT4 is given by:WTc* •T (1,2.4)
The decrease of flight stagnation pressure, PTO' by inlet and combustor
losses must not be too large. If the inlet and combustor do not provide
the required PT4' the inlet will unstart and Wa will decrease.
I,.,.1
301 I s
II
A1.3 Computation of Nozzle Exit Conditions
The relation between the area ratios and the Mach number is well
known by the formula:
A5 M, f + I)/2: (Yf + l)/[2(yf - ()]A .M6 y f " + ---/2 (I1.3.1)' l + M62
A5, A6 are the areas at the throat and at the exit of the nozzle, re-
spectively. Knowing yf, the exit Mach number (M6 ) can be calculated
for any nozzle area ratio (As/A 6 ). This indirect calculation is done
in subroutine CALCM, using Newton - Raphson's iteration routine.
The tuLal presure at the exit of the nozzle (PT6) is defined by:
, $" ~PT6 =' PT4 Tn ( 32I "., where the total pressure at the exit of the combustor (p4) was calculated
previously (e.g. 1.2.4).
IiKnowing the total pressure at the exit of the nozzle (P6 and the
Mach number at this point (M6 ), the exit pressure (p6 ) can be calculated;
) - \-f/(yf - ) (P6 PT6 1 + 2 M6 (1.3.3) VAl .4 Computation of Thrust aod Thrust Coefficient
A1.4.1 Thrust Coefficient (Cf)
The thrust of the engine is the net rate of change in
I ,' momentum at a steady state condition, and is given by:
F P6 A6 +'6U - poAo - 0 - po(A 4 - A0 ) + po(A4 -A6 )
p6 A6 + it 6 U6 - PoA6 - o0Uo (l .. 1)
.j3 (3
where U., U6 , Ito$ are the velocities and mass flow at the inlet
entrance and at the nozzle exit, respectively.
From the continuity equation, the fol'iowing relation arrives:
thU M 0I 2 A (1.4,2)
Substituting for the density (p) from the perfect gas equation of state:
P '1 (1.4.3)
gives: rtU =T U2 A (1.4.4)
From the definition of Mach number and speed of sound:
U2 = M2a2 ; a2 = -yRT (1.4.5)
Therefore: rhU D . 2yRTA
MU oM2yA (1.4.7)
Substituting (1.4.7) into (1.4.1) gives:
F = P6A6(1 + YfM6 + YaM) (1.4.8)
The thrust coefficient is defined
C Ff qo-Tr (1.4.9)
21 2
where: qo = ½PoUo = "YaPoMo2 (1.4.10)0i
Combining equation (1.4.9) and 1.4.10) gives:CF F (1.4.11)
Cf "-
r r
Substituting for the thrust from equation (1.4.8) turns equation (1.4.11)
into:"t-'f12A6/Ar rPT61Po 2) 2Ao ( ,
Cf Y M 2 0T61 p 6 +a 0
41 32
"iK
A1.4.2 Pressure Losses
The pressure ratios in the above formula:
PT6/PTO* PT6/P6
can be substituted by a function of pressure losses across the ramjet:
Dr6 PT6 PT4 PT3 PT2 PTO (1.4.13)
PO PT4 -T3 PT2 PTO PO
We define the pressure losses as follows:
PT6Tn Nozzle lossesPT4
PT4*.•4 Rayleigh flow lossesPT3.. .
PT3Se Combustor expansion losses,PT2
PT2ITr= D Inlet losses =(conical wave loss )*(boundary layer loss)* .-
PTO
normal shock loss )*(subsonic diffuser recovery) = CnD'TNShD.
Therefore:
PT6 PTO i- PTO
SO n •h ie rC 'D' 'NS '0" PO losses IpO
Finally, substituting for total pressure ratios PPTO'PO; PT6/P6) gives:
I •
r Y33>1 6 I I I I I I I I I ¶
This relation can be used in equation (1.4.12) which calculates the
thrust coefficient of the system.
Al.4.3 Computation of Thrust
The thrust can easily be calculated from the thrust coefficient,
Susing eq. (1.4.9):
F = Cf qo A r (1.4.9a) •
The dimensions of F are Newtons (after multiplying eq. (1.4.9a. by
the acceleration of gravity, g). The fuel specific impulse, 1W in
N/kg/sec, is defined by:
= (1.4.16)
The specific fuel consumption, SFC, in kg/hour/N iv given by:
"V SFC - 3600/1 (1.4.17)
Ramjet perform-ance is specified In tt ms of the performance pareineters
Cf, F, Isp and SFC.
ii3
i.
K f
,1,"NJ
A2. Check for Choked Nozzle
The total pressure at the throat of the nozzle is given by:
PT5 =PT4 Tn(.1
Again, PT4' is the total pressure at the exit of the combustor, and is
calculated by eq. (1.2.4). 1n is the nozzle loss. It follows that:
P5 = PT5 f +
The pressure at the throat of the nozzle (p5 ) should be equal or greater
than the atmosperic pressure (po).
P5 p0 (2.3)When inequality (2.3) is satisfied, the nozzle is choked.
A3. Heat Losses at the Combustor
A3.1 Mach Number
a. ContinuitZ
At the combustion chamber, fuel is added and the continuity
equation is:
03 U3 A3 (1 + f/a) = P4 U4 A4 (3.1.1)where A., and A. refer to the entrance and to the exit of the combustor,
respectively. By assuming that A3 a A4 , the continuity equation
(eq. 3.1.1) can be written as follows:
Su4"03 (1 4 f/a) = (3.1.2)P4 3
Replacing the velocities (U3 and U4 ) by the appropriate Mach numbers
(eq. 1,4.5), turns eq. (3.1.3) into:
P30( + f/a) 4/T4 Yf Rf (31.1.3)j
35
Where Ra, Rf are the gas constants of air and of the combustion products,
respectively. Hence:
3M A/- / fTf 'I + f /a
b. Momentum
Applying the conservation law of momentum to the discussed
proble: P3 + 3 3 P4 + 0 2 U (3.1.5)
From the definition of Mach number (eq. 1 .4.5) and the perfect gas
equation of state (eq. 1.4.3):Pi Ui2 Yi P. Mi2 (3.1.6)
"Substituting equation (3.1.6) in equation (3.1.5):
P4 1 + Ya M3 (3.1.7)
P3 1 + Yf M4
Again, from the perfect gas equation of state:
- I1.
i3 P3 T3
Substit-tion of (3.1.8) into (3.1.7):
T 2_ _P T3 \ ,1 + Y aM 3 .
(3 .1 .9 )"3"V + Yf M42
- Substituting (eq. 3.1.9) turns equation (3.1.4) into:
fM 4 •+ a-a (3.1.10)
K $6 ----.-.--.-----. *
Replacing the temperatures T3 and T4 by the appropriate total temperatures:3 4
Ti TiT , + --- M2
and assuming that the change in total temperature up to the entrance of
the combustor, is negligible (TTo = TT3 ), one obtains:
2). Yfout2I(3\ TT4I1 a M3 1+ .- M ý4 (YfR f ' 3 .M6 'TO/ \ + fYfM4 y- Va (7~+ f/a)
The Mach number at the exit of the combustor (M4) can be
"solved knowing the conditions at the throat of the nozzle:
A5 S (Yf +l)/ 2 (yf+l)/[2(yf-l)]A_ (yf +1
""M4 2 (3.1.12)
The computation is again indirect, using subroutine CALCM. Knowing
M4 , Equation (3.1.11) is used to compute M3 . The solution is received
by iteration. As first approximation, M can be solved fmm the
following equation:
S( +YM 32 ) B (3.1 .lla)
',.where: (i+ 'yM2) 4TT4 ½"w e e L"f 4 ) ( Ya R (I + f/a) (3 .1.13)
B 1 /a
(i ' consequentTy:
+BB -4)a(
M3N
'The subscript N shows that the computation was made from the nozzle
i *I Li37
direction. Now, B is changed to be:ij + "--- 1-M 2
y -4
+ - (3115)
This expression is consistent with equation (3.1.11). Substituting
back into (eq. 3.1.14) gives an improved value for M3 . This procedure
can be repeated several times, but it was found that even after two
interations, the value received for MMN is accurate enough, due to the
small ciiange in B resulting from (eq. 3.1.15).
A3.2. Total Pressurei By definition:
P T i = 1 i2S(3.2.1)
Consequently:
PT- P3 1 (3.2.2); PT4 T4 F Yf- .-- yf/ (TYfl !
Substituting for (p3/P 4 ) from equation (3.1.7) results: .II ~y /(Y )(r y,-l 2 a a
T3N 2 (3.2.3) f.
Where the subscript N was defined previously.
38 ! /I;
III I,A4. Computation of Ma&.h 4urriber and of Total Pressure at the Various
Stations of the Inlet '
f ,A4.1 Initial Conditions
In the previous sections (Al - A3) the pressure conditions at
the combustor and at the nozzle region were calculated. Here, the
pressure conditions at the inlet will be calculated independently.
Knowing the total pressure conditions at the various stations of the
inlet will allow the c&eck of whether the inlet can supply the amount
of air needed by the combustor. As will be seen afterwards, this check
will also allow to specify the location of the normal shock wave at the
h.4; inlet.
A Assuming that the statIc pressure (p.), the static temperature
(rO) and the flight Mach number (140 ) are known from the trajectory pat t
"1 of the program, the total pressure and the total temperature can be
calculated:Ya 2 ']Ya(Ya" a* a
PTO PO 1[ + (7.I U
TTO=TO[i a M 21 (4.1.2)
The weight flow through the inlet is given by:
a 0 0Ac (4.1.3)pL LUsually, when flight Mach number is equal or greater than the inlet
design Mach number, the value for Ac/A 0 is unity. But for flight Mach
Snumber less than design Mach number, Ac/A 0 becomes less than 1.0. A
value of 0.9 was selected as a constant value for AC/A 0 . The additive
drag due to A /A0 < 1 was ignored.
1I~ 0i
I3
----- " •-' -- ' ,' ----- .. 4-• •',-? %•'- -- ". -'- - •'-- " '•-'• " ' " " ~ -
A4.2 Conical Shock Wave Loss
In this section, the conical shock wave loss will be computed;
the calculation results include the total pressure, Mach number anda-' area behind the conical wave (P 1c MC, AIC, respectively).
6- A4.2.1 Pressure
The pressure coefficient can be defined as follows:
Cp Po (4.2.1)i•-Ya po M'7
For a cone, the pressure coefficient (C ) can approximately be for-p
mulated as: 0.09 [ l 6Cp 0.083 + I (4.2.2)
M 0
where a is the cone half angle. The difference in pressure on surface
and behind shock wave is ignored in this model.
Knowina C the pressure ratio (Pic/Po) can be calculated (A..2.1)
' yi :P I C -Y 2
C P -1 + C - M02 (4.2.1a)p0 p2 7 0
The same pressure ratio, is also related to the Mach number, normal tothe conical shock wave (Mn). Using (eq. 2.48a) in ref. [161, one can
get:
+IC1 2a (Mn2 -1) (4.2.3)
0 a: PO +YaI'
Knowing (Plc/pa) from equation (4.2.1a), Mn can be calculated from
equation (4.2.3)"ip- Cl 4-1~ (4.2.3a)
I"M 1 +Pc 1 •,-M n PO 2'Ya
40
After computing the pressure ratio due to conical shock wave (equation
4.2.1a) and the Mach number normal to the cone (equation 4.2.3a), one
catt use equation (2.54) in veference [16] to compute the total pressure
ratio at the conical shock wave:
PTic r 2-Y 2 1l/(-Ya1 +1 M 2 a/N-PI 1 ) ," -1 (4.2.4)
P~ ~ '''a M +2n
From equation (4.2.4) one obtains:
PTIC =PTO "C (4.2.4a)
A4.2.2 Mach Number Downstream of Conical Shock Wave
Using figures 4.1 ard 4.2, the wave angle (6) can be defined
as:12 = axcsin t(4.2.5)
On the other hand, the deflection angle (e) is defined from equation t.'
(4.10) in i.ference [16]:! ~ ~(Mn21
6 = arctan (cot W j (4.2.6)
M0 ( Ya+cos(2B)}+2
The Mach number behind the conical shock wave (M.c) may, therefore, be
"-obtained using equation (4.7) in reference 116]:
MIC 2 Y i- (4.2.7)
Th rli [ (B-e) M .a
The relation between the deflection angiE (9) and the wave angle (6) for
various Mach numbers are shown in figure 4.3 (reproduced from figure 4.27Z
in veference [161).
i I
wo
dcr
C-4,
*let
43)
LA..
- O(83,!N NOU03U'3
44I
A4.2.3 Area RatioUsing figure 4.2, the area ratio behind the conical shock
wave can be obtained:
AIC -tan cos(e) 4.2.8)
0W tans )
Equation (4.2.8) gives the area normal to the flow at the inlet lip.
A4.3 Boundary Layer Loss
A4.3.1 Mach Number
The area ratio is related to the appropridte Mach number by
by the formula:
A1 M IC (+)2(4.3.1)
Where A is the area at the throat of the inlet. Similarly,I
-. l(-y+1)/[2(y -1)] KI A1* (ya+l)/2 a--Y (4.3.2)
Where A1 nd ., relates to the area and to the Mach number ahead
. ~~I + M
of the normal shock wave. Dlviding.these two equations gives: I'Y I 1 (Y +])/[2(-y I)]SA 11 1C I +- -"'- M 2 "
ll I 1 + aca(4.3.3)
Ii,
45 1 '
Knowing MIC, Ya' AIC, Al,, equation (4.3.3) can be used to calculate Ml1,
indirectly, by subroutine CALM, which was mentioned previously. M11should be supersonic (MI > 1) to prevent unstart conditions.
A4.3.2 Pressure
The total pressure in front of the normal shock wave (PT11) is
received from the connection:
PT11 = D' (4.3.4)
Where TV, the boundary layer loss is assumed to be 0.93.
A4.4 Normal Shock Loss
The Mach number, behind the normal shock wave is defined as:
1 - -- 1 (4.4.1)i i
1 :M 2 _• 1' 4.4.-
2
a
•]The total pressure behind the normal shock wave is defined as:
YY+ / Ya(Y-1 ( I(a-l
-- 1 2 f M 2 Y_-_PT12 PTIJ(4.4.2)
1 a +111 Y+11+ -+- MA4.5 Subsonic Diifuser Re¢ovsrv
Similav to equation (4.3.3) ere can obtain:
SYa-1 22 (Ya l )/r 2 (Ya-l)]A2 M12 1 2 _ (4..1
Knowing MI,, ,a A12 , and A2 . the Mach num.ber At the exit of the inlet
ca. be coT, uted using subroutine CALCO.. Th; tU-tal pressure at this station
is defined as:
PT2 PT12 'D't (4.5.2)
,46
Where the subsonic diffuser recovery (irD") is assumed to be 0.93.
A4.6 Expansion Loss
A4.6.1 Mach Number
On their way to the combustor, the gases coming frmo the inlet
expand at station 3; a sudden change in area from A2 to A3 occurs. The
sudden change in area acts as a flameholder by creating a hot recirclation
region. In this section, the loss in total pressure due to this expansorn
is calculated.
From the continuity equation:
: 2U2A2 -- 13 3 A3 (1.4.2a)
K From perfect nas equation of state:
-P (1.4.3)
and from tie defifrtion of Mach number and speed ol sound:M = U/a; a (i.4.5)
Equtior, (1.4.2a) turns, therefore, -into the form:
Pn, P3~T 2 2~ A 2 .- M3 aTr3 A3 (4.6.1) K
For sudden expansion of ani incompressible fluid, the change In static
pressure going from small avtea A2 to large area A3 is given by (16, 25]:
-P2 - 2A2.(1.A 23 ) (4.6.2)Sq2 23
Where A2 3 ' A2 /A 3. Also, for the incompressible case, the stagnation
"L•., pressure ratio ib given by [16, 25]:
Pf3 q2/p2SPT3 (1 -1P -A) 2 (4.6.3)
!Accord.rg to equation (4.6.3) as M2 decreases PT3 approaches PT2" The
model for su~den expansion with compressible flow is much more complicated.
47
II
It was assumed that static pressure is constant in the expansion. This
assumption, which is reasonable for low values of M2 , is a conservative
one, i.e. pT3gPT2 is lower.-
Substituting: T2 /T3 = I + • , + e 3 2
and P2 =P3 turns equation (4.6.1) to:
/ ~ a 2A2 3 1 +-2 -M 3 (4.6.4)
Solving equation (4.6.4) for M3:
W hr -e:13 - (4.6.5)
Where: A2M •
-M 2 (1 + aM22 ) (4.6.6)
The subscript I shows that the computation was made from the inlet
direction.
A4.6.2 Pressure
By assuming again that p2 P3 " SI II•PT_3 :PT3/P3 (4.6.7)
" .•iPT2 PT2/P2
and therefore:
"/ 1 + Mg a-P3i pT2 I(4.6.8)
Where subscript I is as defined previously.
" I ~4
A4.7 Location of Normal Shock Nave
The main problem in computing the Mach numbers and the total pres-
sures at the inlet arises from the fact that the exact location of the
normal shock wave is not known, and must be found. The way of solving
this problem is as follows:
First, solve for two extreme conditions by assuming that the normal
shock wave is located at the throat and at the exit of the inlet, respec-
tively. After kiowing the lower and the upper values for M3 1, PT31'
iteration can be made to find the exact location Gf the normal shock
wave. The criteria for this iteration is matching of values for M3,
PT3 from both the inlet and the nozzle directions, i.e.;+ i MI " 311' T31 PT3
I, M31 3N' PT31 PT3N(4.1
I 1L
SI J
!'
.111I
SF I H
* 1
II
APPENDIX B: TRAJECTORY EQUATIONS
BI. Atmospheric Functions
Best fit curves were calculated for basic atmospheric functions,
pressure, density, temperature and viscosity of air as a functlon of
the flight altitude. The basic formula which was used for this process
is as follows:
F = A exp(-B X 10-6h) (1.O )
Where h is the altitude in meters, and A,B,C are numerical parameters.
The appropriate atmospheric functions are as follows:
PO - 1.03322 X 10 exp (-59.148 X 10" hIO0) (1.2)
PO ' 1.224845 exp(-29.0i44 X 03 h)
*-iT - 288.16 exp(-13.232 X 10-• h 1.0709)
When: h 0 - 11,000m (1.4)
T - 217.240, whei h - 11,000..32,00Wr (l.4a)
LJ ,1793 X 10~ exp(-4 374 X 1G 0 .S-
W~hen; h 0- 11 ,OOn (00M
VO 1.41.724 X '10-, 1 vh.* i n >_ i0 , ,
"In these fo•,•Ie, the pres'iure (p,ý has dimensi,5rs of kg/m 2 , the / 1<density (Ip) is .qiven in kg/m4, the temperature (")' is given in OK,
i , .2and the viscosity (v,) is giver in kg/(m.sec) (o-: ,-sec/m ).
B2.
1B~ Cowl Drao, Coeffiient
2.1.1 It was fcound Ohat the cowi drag coefficient has a strong
inf!uence on the resuitsi Therefore, a new model for this cowl drag
"coefficietit was developed. The mode) was based on a theoreticalde~elnpm,ýPt done p;'eviouýIy by Prof. T. H. Gawain [9]. The main
i i 50
difference between this development and the classical theory, is that
the boundary conditions are applied at the body surface rather than
along the axis.
2.1.2 The model, which originally was developed for simple cases
(cones, etc.) was nmdified to fit the shape of the projectile, illustrated
in Figure 1.1.
2.1.3 The modified program is listed in Appendix G. In the combined
program (TRAJET) an interpulation procedure was used as a subroutine in
order to simplify the calculation process.
B2.2 Base Drag
After checking the influence of the nozzle exit area (A6 ) on the
perform.nance, it was aecided to allow A6 ts reach the lxiuimi value
possible (A), in order to reduce base drag. In this case, the base drag
is negligible.
B2.3 Skin Drag Coefficient
The Ruynolds number is well known to be:PoU L.Re (2.3.1)
SThe transition Reyi-uvods nwTmer of
Re* - 2*106
is usually taken as criterion for transition between laminar flow
(lower values) and turbulent flow (I .-: values). The incompressible
laminar skin friction coefficient is related to the Reynolds number
as follows:
- DS,L " l.328/-7L (2.3.2)
Where the subscripts DS,L stand for skin drag coefficient, and laminar
flow, respectively. On the other hand, the turbulent skin frictioncoefficient (CDST) is calculated indirectly from the formula:
i51
LI) 3
1, og10 O(CDST ReL) 0.242 (2.3.3)
The computation is done in subroutine CALDC, which works in a similar
way to subroutine CALCM which has been described earlier concerning the
calculation of various Mach numbers.
B2.4 Wing and Fin Drag Coefficients
The wings and the fins of the projectile, also contribute to
drag. Basically, each of these drag coefficients contains two parts:
-Wing/fin wave drag.
-Wirtg/fim frt-ction drag.
2.4.1 A psuedo 3-dimensional model was chosen to simulate the wave
coefficient, The basic formulae used in this calculation are:
M /an -lv(Mo)= ta-1 7 (M 2-l) - tan- M02- (2.4.1) .,
a -l an aya+1 -/(ya+1)
pO/PTO = (1 + a +12-" MO2)Ya/ (2.4.2)
The effective span could be taken as (see fig. B2.1):
W - (2.4.3)
Substituting: tanu = 9/c , sin1j = 1/M gives:
b' b c (2.4.4)
The drag coefficient would, therefore, be:
S2 'oI/T0 - Po2/PTO tb' (2.4.5)
The w•y th 4s simulation uses the above equations could easily be
understood when looking at the formulae together with the flow chart of
the appropriate subroutine (Appendix C). Interference drag is ignored.
52
II
JC
I.J~
I?
Ii
I'
Ii
IFigure B2.1 Schematic View of an Wing/Fin:
b=span, c'-chord, tuthickness
Li
2.4.2 The friction drag coefficient was calculated using the
existing model for skin drag coefficient. (See Section 82.3)
B2.5 Calculation of Drag
Define the dynamic pressure (9) as follows:
Po2o 2/ q •½p1 0 (2.5.1)
When q is in units of kg/(m.sec 2), (or: N/m2 ). Also, define the following
geometrical units: jAp = R2 ; Sp = 27RL (2.5.2)
Where R,L are the radius and the length of the projectile, respectively.
Similiarly, for the wings or the fins:
SIi S,, = nbc (2.513)
Where n is the total number of wings/fins (a value of 8 was taken fur nw,
A and b,c are the span and the chord uf the wing/fin (see fig. B2.1)
Consequently, the drag (D) is given by:
p APW + SF (2.5.4)"I D = q Ap ~CMN + 'p CBS + WW (CDWW + COWF) (2.4
""2.6 Drag Coefficient of a Conventional Projectile Without
* .?!2Propul sion .
,! I~. The program has an option to calculate also the trajectory of a
projectile without a propulsion. The projectile is a conventional round.The formulae used to claculate the drag coefficients in this case are:
*1g. 2.6.1 Nose Drag
CON (0,083 + O.096/Mo2 )(a/10) 6 9
I WWhere a is the cone half angle.
2.6.2 Base Orag
COB (0.6837 - 0.3165 M + 0.0525 M2)(2/f,)
A 54I
-- 4I
2.6.3 Skin Drag
Skin drag is calculated as discussed in section B2.3.
0 = Drag = q Ap (C + C.B) + SpD.
p DN DO + PD
; b3. BoosLer
The projectile has an initial muzzle velocity of 2500 ft/sec. Part
of the combustor volume can be used as a booster to accelerate the
projectile even more so that starting the ramjet will be easier-
Define exhaust velocity (U e) as:
e 2 sp,B 9 (3')
Where Isp, is the specific impulse of the booster's fuel (in sec) and [is the acceleration of gravity (in m/sec).
From Newton's law [6, p. 323]!F = it 8ue = (M di t) d•lu (3.2)
l8
SWhere m p and it are the mass of the projectile and the mass flow of
the booster respectively.• : dt"••' TdU• = ffB Ue mp - M t (3.3)
Consequ',rtly:
AU = U(T) - U(O) -- UeZn m (3.4)
Where T is the booster burn time. Hence,m
AI e mp -m, (3.5)
AU is the change in initial velocity due to tihe booster, wher-e m5 is the
mass of the booster.
I.
K! -55
II
B4. DynamicsThe flat earth trajectory with drag and thrust is well known.
The differential equations of motion are:
= -q + (F-D) sine/m.dt 2 p (4.1)dt
mp- (F- cose (4.2) t
where y is the altitude, t is the cime. g is the acceleration of
gravity, F is the thrust, 0 is the drag, 0 is the clevdtion angle,
"m is the projectile mass.
For numerical solution of equations (4.1) and (4.2) a finite difterence
form can be used as follows:
xj+2 - (F-D) cose t . 2xj+r - (4.3)
The Yj+ 2 = [-g + (F-D) sinfmpJ] at 2 + 2yj+1 -yj (4.4)
The values for j -1 are from initial conditicns, i.e.
-o, y 0 (4.5)
x2 * cse At, Y2 " sn * .. t (4.6)
arct Xm+J9 (4.7)
Having calculated xJ+2 and -j+ from (eq. 4.3, and 4.4), one obtained new
values for trajectory parameters using (eq. 4.7). Also thrust, drag
and projectile mass are updated for j+2 .
56
.1 56
t
Appenc.x C: iwCato
C 1. -4ain P ram
FDATTA/
(100))
BOOS
IN-IT
(15)
ci I
-BUBURNH
7olRi'ght I, -4rong >2 120 ~I
CR
'Thrt1~t YES
NO
i'Rght; w 3.
rorig t
LI-I -
< V73:0119
IiA
IT,
C2
Right WWrng12
XAN
I~gh CHC w
j IIr
T1L gh W 120
CORVAL I
I ~Right w 2"
Wrong 120
RR33I
"/5
RRSU L
L.,.. ag. W C
wrong.
Rig~~ -Eý
2
PR)ILOO
o VEV
7•hrust'YYE
07i 5 choose.
NOnew 21O I
NO1,o55A
'10
C2. cojmand Subroutines
CONE
MIN
Riht W
WRing 120R...
THROAT (1)
Rihtight 12
NSR( 1(M 2I pT2I
3' :::: T12)
!J 'DIFFUs
02(M2, P T2)
. @Rturn
IN
ii MIN
A _3IPTI
THROAT ( 2)"I
(MR "
i ~NSR (2)M22, T22):
ExPAN(M 31'PT31 i
< wron 2
R
P_3 12)-
<P,
YES
I I
"I 1I
' 62
I'
•CORV• L
choose:
Ts
THROAT (3)
"L p
NSR (3)
V £!~s2YPTSQ KDIFFUS (3)
: N ,PT2) i KEXPii
-I3 ?M~ 'T31) -t W
rW] o h 120 IR
T31 NO
~T3NSES change: ,
Return 5
52
63
I st time
, : linitial
conditions
ATM%I
I IS~DRAGG
I ! ~DYNA o
II
'4 .
pii
. 64
I t .. --....-- *
C 3. Individual Subroutines
ATM
altitude
|y
computeiP0 (kg/mr)
i0 ,
compute*300 %(kg9/rn
TO ( OKI
0
compute:PO (N-sec/mr
0
i i4-j
"65
-- ,19,
BOOS
I f~PBISP, B
$
compute:WB (kg) I
compute:,,Ueq (m/sec) I-
I *I
compute:&U (m/sec)
n I Return
compute:
-c-ontinue
I 6I'!
III
'~II
I,
IbNIT
P T0 , To., "
compute :I - I
I compute:
~TO
.'i
I Return
I I6II
ii I
" I. -- -i i i iti i
BURN
- ~~w (-1
compulte:
A3 W
L
IYE
I'a 9NL
THERMO INTER
r4P
"Returnfl
I9I P
! i 69
-- C..
____..... .
, -= ,
* 'I
hoose DRAGjz.to D RAG
compute cpute
•t TT (h
C DO•
ompotmp
te
il
, ~ co mpute :
•,.,
. 4A?-
y .Pf T4
ERcampute L
K 71
i
/
CALal
AN, AM
'y MN
I i" Iassum~ :MM
C 1
7)
J /
< s change-R, tui
noteA: I. M - area at known and at unknown mach number, respective.Ly.
y - y or ",f N bmN -known mach nme
2. CALCD work-i in a similiar way.
72
CCHOKE
Scompute:PTsP5
K
NO-
I,,.
_.
I..
-I
It
7344.
HEAT
A3 A5 Y
C-ALCI*M 4
Right!
IO g
C3. 8 INU.T
r~. (.YE
Crcompute:
copute:
r ompute:
P,¶c
M " 1 % A,1**
Tl I
CR-I-,1
It -6
..... .. .
NSR
"Orro
pp
I
~Rarturn
FDI.FFUS
-or
AM2' a2
CALCM-ýM 2
Right/!4rong 2
I : Returni
K. 78
EXPAN'
M2 ,-y a
compute:
S31
A, cmpute""T31
i It.
.4
*i
N 7I i9
- . ---- - . - .
°II
~T3II CHECK
ppTt 3N
YES
,1
I • ' *
%.T
1'80
I POfbo AA0'ref )6
M- 'Y 'y y
compute:Cf,q0
r compute:Thrust, ,
SFC
81,
i !
.1
'II
81
C3. 12 TRAJl
Mo.To
C ~ombustio NOmue
INTERcCompute:
S copue I c
DRAGWW
Y loiM 0
Lb,c ,t ,A ref
PRNW -Y1 oe
[rims suedo 3
y - +
.1<* 83
*PAN T
Y2- 2 ~ - I- 1
-. --- • .•
DYNA
'I'
SA3
I1compute:
y3
Scompute:
If
compute:
"I TI8
Return
"-' compute,
84,
I APPENDIX D:
PROGRAM TRAJET: LISTING
I
II
,o ,.
I85
-- - . .. Z - --
FILGI TRAJEr FORTRAN A NAV'AL POSIORAOU&TE SCHOOLCcc~cceccccr~cc~cccc cc cc~ccc~ccciicr..cccc c"cCCCCcccccccct~cCCCCcCrcCC YR&000101
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a-:
APPENDIX E: COMPUTER PROGRAM LIST OF SYMBOLS
PROGRAM 0UlOS UNITS MEANING.
SYMBOL SYMBOL
GEOMETRICAL SYMBOLS
APEF Ar m 2 Reference area
AJ A. m 2 Area at station i3{
A 2 Area behind a conical shock waveAIC A Ic
m 2 Area ahead of a norm al shock wave
AJ2* Aj2 m 2 Area behied a normal shock wave
AJj2
A30 A 0 m2 Initial area at station 3A30 A30m
AIAJ AilA. - Area ratios
SL3m Length of combustion chamber
L3 3 ,.
pm Length of projectile
LPR Lp ,- 1~iUS f nyniptile
1RPRRALFA
deg. Inlet cone half angle
*U;hen: J-l ,2.S, the shock wave is at station 1 2 real case, respectively.
if
'Nii~
• r . -,
K
PROGRAM EQUATIONS UNITS MEANING
SYMBOL SYMBOL
ATMOSPHERIC SYMBOLS
TO T OK Static temperature at altitude Y3
PO PO kg/m 2 Static pressure at altitude Y33
RHOA 00 kg/m3 Air density at altitude y3
MUA I0 N.sec/m2 Air viscosity at altitude Y3U0U 0 m/sec Projectile muzzle velocity
MO M0 - Projectile initial mach number0 2
GRAV g m/sec Gravity (9.807)
GA ya - Air heat capacities ratio (cp/Cv)
"GAI (yal)/2
GA2 - (Y -1)/2
GAI2 (ya+l )/[2(ya-1)]
GA2 -a/(Ya- )
107
Ii
II
'V-.t .1 - --- -
PROGRAM EQUATIONS UNITS MEANING
SYMBOL SYMBOL
COMBUSTIOtM CHAMBER'S SYMBOLS
RHOF kg/m3 Fuel density
ETAT T'- Burning efficiency
A,N A,N Burning rate parameters
WA •a kg/sec Air mass flow
WF kg/soc Fuel mass flow
WT kg/sec Total mass flow
F F _
RDOT rm/sec Burning rate
CSTAR rn /Sec 'g*9*T~4 / f(when:r r(f)RF Rf m/1K Hot gas KR )ole
SK H g const ant[ -MW(kg/mole)*g(m/sec ).GF Hot gas heat crparities ratio (CP/CV)GF'i - - (yf+l)/2
GFZ - - (Yf-l)/2
GF12 - (yf+l)/[2(yf-])]
GF3 Yf/(Yf-I )
i 108
I, i
PROGRAM EQUATIONS UNITS MEANING
SYMBOL SYMBOL
THERMODYNAMIC SYMBOLS
TJ T K Static temperatare at station j
TTJ TTj Total temperature at station j
PJ P. kg/m Static pressure at station j
P TJ P kg/m 2 Total pressure at station j
MJ M. Mach number at station jPTJI, PTJ2 PTjlI'PTj2PT•l, ,MJ2 MJ1 'MTj f1 As above with AJ1, AJ2
MJl ,MJ2 MJ1 'Mj1
T3M T3M kg/m 2 Maximum T3 available
M3N, M31 M3 N,M3 1 M3 calculated from nozzle and inlet
direction, respectively.
109I
' 1'
it j
11
i .9
PROGRAM EQUATIONS UNITS MEANING
SYMBOL SYMBOL
LOSSES SYMBOLS
PIDI IT BoLndary layer losses
PID2 7D" -Subsonic diffuser recovery
PIN -ITn Nozzle losses
-rC - Conical wave losses
-NS - Normal shock losses
•e - Expansion losses
rh -Heat losses
I1
' f
t..
a -
L 1 1
PROGRAM EQUTIONS UNITS MEANINGI SYMBOL SYMBOL R MY
RAMJET PERFORMANCE SYMBOLS.
cr Cf - Thrust coefficient
Thrust F N Thrust(or kg)
ISP I N/kg.sec Fuel specific impulseISP sp (or sec)
SFC SFC kg/hour/N Specific fuel consumption
,I
I •
V 1
4.'.
II~ 11
PROGRAM EQUATIONS UNITS
SYMBOL SYMBOL
TRAJECTORY SYMBOLS
Drag D N Drag
WPR W kg Mass of projectile
wB WB kg Mass of booster
OELU AU m/sec Change in initial velocity due tobooster
U U m/sec U + AUT0
DELT AT see Change in time
,!TI t sic Time
TOB t sec Time of burning" BOB
TITA 6 deg. Gun elevation angle
I; TETP ap deg. Projectile second cowl angle
DRAG COEFFICIENT
CDN C Nose drag coefficient
_DWW C Wing wave drag coefficientCDWW DWW
CDWF COWF Wing friction drag coefficient
C. Skin drag coefficient (laminar/turbulent)
CDB C0B _ Base drag coefficient
'I
I-
liz ~ !
JL
PROGRAM ý9 I ONs UNITS MEANING
SYMBOL SYMBOL
MATHEMATICAL SYMBOLS
PI - 3.14159
IP_ Printing parameter: > 0 combustionresults together with trajectory(on different files):
= 0 working results only;
= I also reasons for not running
= 2 also full reasons for not running
= 3 also loop on mach number (CALCM)
-l trajectory prints only
ITRA Loop paramerer:
I fl - +1 single value for AO/Ar, A5/Ar
= -l loop on Ao/Ar, A5!/A,, and
print summary, only.
ILOO Check parameter:
< 1 regular run
> 1 doesn't have a solution
IL, ILO Trajectory printing parameter
S ' ~ (prints every 11-0 point).iK
IRAM Ramjet parameter:
0 ramjet in operation
1= projectile without propulsion
"xM .M Stopping roach number _
N 113
-,• uMO MO
PROGRAM EQUATIONS MEANING
SYMBOL SYMBOL
SUBROUTINES
INIT - Computes initial conditions
BURN -Computes combustion chamber's performanceTERMO Thermodynamic comand subroutine
INTER Computes, by interpulation, thermodynaic
conditions, or cowl drag coefficients.
NOZZ Computes nozzle performance
CALCM Computes oach number indirectly
RESUL Computes ramjet performance
CHOKE Checks if nozzle is choked
INLET Co,,-- nd subrouti ne for inlet
CONE Computes conical wave loss
THROAT Computes boundary layer loss
NSR Computes normal shock loss
DIFFUS Computes subsonic diffuser performance
MIN Command subroutine to compute situation
when normal shuck is at point 2
CORVAL Command subroutine to compute situation when
normal shock is at the correct place
EXPAN Computes losses due to expansion into the
combustion chamber
HEAT - Computes heat losses at combustion chamber
CHECK - Check for pressure capability
TRAJ - Command subroutine to compute trajectory
ATM - Computes atmospheric conditions as a function
of altitude (y)
114
PROGRAM EQUATIONS MEANING
SYMBOL SYMBOL
BOOS - Computes booster performance
DRAGG - Computes drag
CALCD - Computes skin drag coefficient, indirectly
DRAGH - Cowl drag command subroutine
DRAGWW -Computes wing/fin wave drag coefficient
(command subroutine)
PRANT Computes Prandtl-Meyer angle (,v)
from a given mach number
APRANT Computes mach number from a given Prandtl-
"Meyer angle (v)
PRES - Pressure ratios formula
DRAGPR - Computes drag coefficients of a projectile jwithout combustion
DYNA - Computes the dynamics of the projectile
DATTA - Initial data
PRILOO - Prints ramjet perfornance
PRIN - Prints detailed values when does not fine
"I ! solutior!
I1
, i
APPENDIX F: COMPUTER PROGRAM USERS GUIDE
Fl. Input Data
AOAR, AMAR, AIAO, A2AO, A3AR
ALFA, TETA, TETP
IPR, ITRA, XMO, IRAM
P101, PID2, PIN
Options
ALFA = Inlet cone half angle
TETA 2 Gun elevation angle
TETP - Second cowl angle
IRP = 0 clean print of RAM + TRAJ
S=l, 2• 3 more details on RAM
- rml only
ITRA - I works on one set of ddta
= -- loop on AOZA5 K
IRA" - 0 Ramnjet
SI1 Projectile without Propulsion
"F2. Execution Commands (For use with IBM 370)
F2.1 0 2enin JLJ G? Cotý d s' • LLXXXXP
! Password
GLOBAL TXTLIB FORTMOD? MOD2EEH
J Q F.2 ýýo lation
FORTGI TRAJET
"F2.3 Run on T-minal
FILEDEF 02 DISK TRJ D(RECFM F BLOCK 132 PERM
F:LEDEF 03 DISK DRG D(RECFM F BLOCK 132 PERM
116
i
FILEDEF 05 DISK INP 0
FILEDEF 06 DISK CMB D
EXEC RUN TRAJET
XEDIT CMB 0
(or: XEDIT TRJ 0)
(or: XEDIT DRG 0)
Note: CMB D will contain the combustion process results.
TRJ D will contain the trajectory part.
ORG D will contain drag coeffic'ents.
F2.4 Hard Copy
FILE
PRINT CMB D
PRINT TRJ D
PRINT ORG 0
117
.ii.
"111
K1- ~I II
S.. . .. . . .. ... .. .. .. . . .~~~.. .. , . ........ _ -.- . ... ,- • - - _ -
APPENDIX G:
GI. PROGRAM AERO(N LISTING
-V
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Aa36
II tic~ nLAJ .1O0C TO 3026 A! C 1050
wRl JC 2. 102'. 1iA,Q3ALXtJI.~IJl *R1 3j) A93
3024 1C'4~ *.3a¼XlX5~~t.ZX , ,QQldGO yrl 3777
Q 9148AM 345110
II-PJ2P ~L:3 .OA6QO 3140
CC Tr) 3241 AE310
CALCULTE G(43 AND RP(J) P'I5 CYLINJORM[AL. Mo~SECTION Aq~ 317
308I1 r.I[Nfn.,30i~ [Tr, 31213.qU)f30 3101 1J-JAP.JiJL.
r30
3101 PINIP2'h)d.L.JG T-1 3121 %RJ4.. 1 h.11 ) L ~ f L 0 . A E RQ)3 2 5c 3
F 14. 3161 & fU 11 fl
3121 [0 9(11 Lr)SflTI GO I0CALCKtATr R(43 AND o~iJ) :OY4ONICAL TAIL A54050J
A ,6 J3 fL40
A pqRJ33L O 0
'X AL -- I L CL 34Dv 3141 t-J i(PJ)C E9L
A~3!Al o 33s0
A 370)0
'tztp110
LAY231380
004 U 3'413"
313lS¶EjO; ~ O31 AICK,'ATC P(43 AND RPEJi FO SECOND M!D0SE710% g34
A , 0345.0
G OO TU zi4 A E ' 0 1 35*a~AFt1 01550
CALC[L AT!cA 4JI i&NO RtP(J PjOR SPLONO CONICAL T4 IL AF'93560
* O ji)4:.!PPVT2UeCJ)-7LkCL2i
AE'318)50
p IA". 03600
123
.'--------'-.-,--.-00 b-
TAIL
,ULU.&!r kkil ANC ROMJ N OGIVALT'L 00664
G00 TO 320531 t64001
. P TOP T L .G T . O . .A N .O L .L E .l .') 0U I 3 h 9O
PC ?AvI' ,O~ ..3
00 1 t Jll,'L;&11 -XL K(1ACL )942 1)133
32?1 COPNT?~'J 14 k 0T 6
C'1MPLE~tICI 'ESSAGE A~)S1
~O~TI* ~AjQ'I T 790
I " Q If! T1 4 *32 ot I TRC AlVQQMPEt Ae5 0 110'0
3 &1 %O WT I .'C-SA $ ýA01 W .
t o 1 1001 2 @1 AFA 01160
r-GO UL T TOO A'903tJ0
TO 3@6 TAF403'*S0
31 OA44ZII 3kl~ aO..,2I
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3 3346-5 c A-3 45 504 a
Go 'T.3 012 3 1 1R 4 4
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lp.,r A gfO')2lo
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- - - - - - - -
FILE: AMRO FOR TRAN 4 PINVAL POSTO'40OU471 SCHOOL
IF II 6 -xf li.?i.4t.O.0IGO TO 3307 hfRO41,30
3307 A.4gIJ-IrII-l4T~v~A(JI-AITl*T 55Q043410
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IFIAIJ- 11Jl.NE0.l G TO 33*5 A~04
3327 U41F:'1I1 AFRO-Po
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FILE: AFRO' FOKYfAN A NAVAL, 0SGIAOV&Tt SrHOL
tFC 01.0 oM TO 3355 IO00
3353 F3 @A{ 3Q353-,E10,31 24(rs" 311?;. Apooc
335 xA C.C.A/44 AFO SOI
336t V'15 - A411CULATIOH CONPLETED'1 A.-isGC TO loci, A(0Ql 110iAfuLA&TE ilUtPUY A,:RUS05O
A 1)135L R(FC 2.I 0t A I..*. J 0tXt&RI4.A~
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3691i ;1fT1 f3J."P179111t' .,e.0
w~j~ nU PUT OP LOT TO ILE 019i~lAFP15610
GO 1261 1: kz4 '
4 Af! 85E9
- --------- - -
G2. PROGRAM COWL
ADJUSTMENT OF AERO FOR DO-LOOP ROUTINE
12
I~if
I'
I :
,I
4l
Kj
FILE$ COWL FIIIIhAM A% NAVAL PoSTORAAUATh $CNOIL cwol
*llI45ý
CALL nt1AAC m.A~piWIvm
C. Ls MAWVn..ACDW rn 1nlnIR
- SHI'IYN rjwr) infl
33 .rrl cI ',IMPICI r
tt4A
4 rrl11 a
OIENOINAiZ.T?.~7Zt1?hP~Pul~htPf Crhd~iLnO
Sc'IWti )1 1C0
C 07m FO A'32.1 70.PIQIA(OIF405Gt101KPM I C'34 IZkOI
C -'0.12 71.6 a-0.11`70-4.5 OFS.,-O.2ZSO012
flFCg-0.2S40. C r' II
c G A40 iYo
C ATA ttL7.YSO/ M~9~,&,OjMoC &t.NS2 ý/61 guiO
Cnag
-' UP :RWI ANCLUAFI
*LuOPLFIXO OAT(Nfll C-4 )1A
k~o~ftL*wT~t
JAR- liLJ .12ma )I
Ig4LL *ONiHCL Cnw'iit'34o0
274 621T4U Csfc-a- (IwO '15,-'".C W.*f r)44ftW) St
ZI-JoL Jntnn49-3 01 01i~AL
t3 * '¶ibs (0
us ~ F X '03 *L oj 032 r3ov
3021 C4TPI rw0026
10241 0p)n0 00nI0 U 01CO~~I ou EIN'> Ia!. rl~d-)WS
ind I&G 3 *0 IIF .
a,~#AFPj N 0j n " I' IA 4,1rI4rj ~ i
30I -I-.-1rnTn32
FILSA COKe.: fORTRAN A M&VA%, POSTGRAOJITE SCICO3.
RFIJJ.o -COWOO7IO
310t .NI~ AVTmfO63101 tTINUE T.M.ICC TO 3121
311CahI'lw *flU Noi6Q 0
CALCULATE Rlfl hANO RAW)j FOR CONICAL, ?AIL. Rao4 o
7AT-Ok/CL
A0 ELi JL C L 2w~o 6l0DO J+A* I IJ I-A- +gLogggo
APtT C 0 R87
31,41 ¶O.ILi- aJOCbF4l(NA4t*NC.F.LT.P&IGO TO 3123' COwoe90
RILP)t.OLCwo oqio
* CALCULATE 0(J) ANC RAW~) FOR SECUW NI1OSSCTION ~WOU94313C GC TO3 3241 ~fI )6
1123118
AIJISISlNL-L awoe 10
OwomCbpo125 F9,i'A..o 0~wJqG T6 '21,LCfwo 101).
E C&LC&,AtE FII N RPIJ QtSCNCOIAr4 Clwo 1 0665C312? "vT2~-Q-'L2/Ct., Cowl) Ia o
00 3179 4.48% RN ,) O11
COwu 1100
324 6C ASkfX4l . WRo0117%j~~mi 8fTASUA1XAbXTi.
'.9 3254 Fflfl&T(' .XA 280(101 )
OACI1IU. 0oapCAO'0. 28 ZsII8
¶- 00 b3 4
*. eNLp: nwt
OW11
R< 1 5
I J1.16.010GO O 3 n129 ~
1343~~~ -WA,1
TO L 3
234 CO.S 0R T I) ~oaUa t550
314 ~ j.Z Go TO 3321 l.15iCO'1540
00 33ý1 J-IJ
T. 4 ' .*.t3I¶1.A5283.
*FT T- G TC 32ew rv iO 630
330 L t ( T I 0,I0(11t In 3307 M X6iQ
IM C0ii0 11
STO ttf I r OwO I ý4
hst112 (A) 1 67
WitkpEJ A 'R"- G 91, '8,
I "I. eulJI TO*f~l 333? Ia(IitI~ mE~ Xo
3U a 994 1O%33 crdowto
p37 ~If w 4
.,~~~c~ . ,) C0 ,3
33294 -F0 CIC AP X2100F*104
K~fr. i T l3 O 1 1lEd
33 SýJ$1 t 5awa go 01
co o 0A194
X. .4
CA L W PRhI -SL 0142 14St0*
J-1I1 * W.30t
FILE' COWL- !(opYRAK A OAVAL pI3I~chROUATUI SCiCt.'"oLl
AL AAA4LK I.A.AllILt) voii~s
ALL APhT('T pjt£OO
CALL D2SI ~. I §j2PO0
QTi ' rA- GwfQ
ENO sRwOC. 30ý
S~gRC-JTINEf PRANTI.NII.GA.ANIJ ~OwO~l~ T&S~~$~I~fbi A .. 'i~CflOfl3O
TJ- nsi4
A S 'I.JC N sAA EAAN-'i I ) GI.F. Wt-AT& I -IN PINz5 o
tNO Cot ol V1170055I C. 01.O RE, 1.,PI00it 1k G ', ý ,1 1 J
ss .- TkZ .q1 INýE~AIJ 0 '.0 .,30
IFIF-.F' .11 T 56 Iw
/1:0 o 556nw131
G3. PLOT ROUTINES FOK USE WITH AERO [9]
-PREPLOT P (PLOT ON PRINTER)
-PREPLOT G (PLOT ON PLOTTER)
-CHARTS (CONTROL)
13
I 'I
I1 I
° • I -I;
I ' .
I hI13: I
Mitt: PqEPtaTP FOR~TRAN A NAVAL POSTIRAOUATe SCHOOL
JI/ANICZ SO 41" 3250.0253).'ANICNAI XflSA'.CLASS-A PRPo00loIt milC FRX P *a' 0I2
ii P9PT.$Vfl CPRO
1PEC1PICATION STktEFNI4TS pqcC @6900060
IMPLICIT poLt(A4.0 2NTGR021tC 0090
C PRCOOIOOREA0 5.1 COO) XN.IL1OL.NL.NA0ý.tNI~i,K'40IS1.KrAIL, '001
r.000 4 P AT(1 3.3t...S.412!31.ZI2K,A2)/IX.SA4#ttX.4014.S) - PvcOO0LNP2.NL-2 *R:'j a I
1020 ,6.06 P~O M.Q
106, FflA . i. R ' J-'IN'11AQY IF PESULTS'/J/ *RoilOO1w'T 1.IIXMXL,)t .NL .N A .N8.N4T&A 8, KNOSIC KT A I L,64PA&IK A P :16J*11)
tI PRI0O*
-. f?%I ' CM-.A2.Y16,0-1. 2A41'O'. NAS ,P.##TMS. PA, 'i 00*3Is04.Fo4 TI.C2'9?4T4'A.F.I *R!0O260
L100 rOPPA a~, I & &C-A vST EN.' ,Ffl.4,' I0.%flA4it/) P.3C2AI~- CALL PLO PIK.¾,NLP2.41 P, RFl('2q0
lmP -I )^&A 1I6~I *6910" T'1140 P'fl( b, AXA F035COFIIETGO SE'IP130610 If X' UC¼[email protected],t j 11 Onto4/jICAL LAtPI (XQ AO. NL P2.4) PRIPO 4
120 Pf o In I',60ýA,6x1ALI FOACFCOEFFICIENT7 0GA VS VI/1 PRC003I063q , C . ,') .APT 17:o jg097
1ia20' OAa c'k K I2~ C4 D Z/ p 1CALL
0LniPIK.QOaZ NLP2,4I pPtrou*&
wal F1.12001P4 9 004
//C0.VSLN 0Pl.3
11 AT1[I 11 i
"Z a"' . 133 A4
.1133
nut PCT PrA'TRAM A t4j-AtA K~STGRA0UATE SCH03L w.
SPE-G1FICATION SYATE'4r4TS p~
C OImfVNSIOK 5, ,314
.KAP.C4A.MCs.CA2. PP"0JL30
104ao CfINT [%[J F OkO1
LOAD f06
'lNOtI I *t...¶JMMAl,,K (TE, PLTSqlI?4,.Q~ XL, LNANrT pft*3QZLO
A C &2t XAC NLj/-i3/ PoRt~o 230
q :0,&IL bF~ P 354.*Na
*T Fa) 516.1 * VRCOIJ70
CALL "LI'l" &7I* ... P3. ~23O....U..''A LL 00 0 3,OLt)TGIXII1Z
PB 00 350
~13
itSI P11933 0*
1f4 .SYSK *Ct~ IVPftF0OIRýEY4 HA IlC 9 o
Rot 000CUI'v'
3 gO.0::lC G b;':C IPS:i!Fle:LsA
1? ;! 4 '~tF 2 jA AtG~hObLd01ý5 X j~;Ml~Aj88 S ~ U T0 itjjt/)isX1.51-LIN 0O 0ISG~t Ej ,IS , (CL'304. .40 kh -SS%; Q
0x -N'lY ' .tCC.
IF:.S82 0f[:1;j:~ t: jSI % 423 6XTQtO i L14.L FCTS0LNU
1 xxrep ua l :,PAS$), uciYlujuO*4 X1%I?41 00 USNL Cti l .LF I S 2.l4,1PAC0111, .ISThOjN T.f UA L ~oUIUj9
Zy ~ ~ ~ ~ ~ ~ S .~fCR 10k CSIsLvf 1s0¶1 S.-otOP 1S 1,1?MlaIS fl, :1. onuhtSuA I¶I
.9 ISM flSP 88 fm lyl: AOIIAPE.LCTFf&R 00.1 U 0009JU30 i(.'S 00 w.CNft-35 IISI ZESCRvt5.P~d QZvl
32 )!III.16 DI LIT1!50.0I FP)IR .'LSENN103C#~lt.TCPO 013. 54Q100$" Ie X VI!
2P.Ij? Exc 0 G-*L~; ISYSCL L L tAT Out.i~ 4L* ~ 5 K~Pj 31 ,5LTQfl1 00.7fl000
,fie &C aR 0 4S0IPCL:,UELLTICSjct r6 IC40I0I xxv£ 1a; 0a aUP uo Y'.UOg
TR" In 0 , R" 00 1,P" 1 4ACC 11 5, f 1 "' Nk -'s v UA% 3 Y" , ý.' I4
L 04.U
IIMM ONF-'OPAPOWI
11 1
FILEp CHAIMS FIEC A NAVAL PVSTZRAOUATIE SCHOOL
CCCtqTOct ERRAO~n~q CW, ECL Q PPfIaTRC~~ -PR~T4TA
E FP4
E t f. P c I v A IN T O *t P iO rfl R ... 4~TEC14jLE P RFPLO IP
CGCI21 -0O-'LOI;NTERt("TI 4U5ropy p L0 LE CAM P F mRTRA NA A FILE PT OIFOOL A PLOT FORTRAN A (REPLACE -..XAE c tiJpKIT PL.C1 tPORTR'A
136
G4. Results from AERQ/COWL
G4.1 The symbols used in programs AERO and COWL are defined in AERO
and are presented in Figure G4.1. The values of a, b1 , c1 , b2 , c2 are
normalized with respect to r,. NA, NB, NC, NB2 , NrC2 are the appropria.f
numbers of points used in the program. The cowl angles aI l were
selected as 20°, 9.5 , repectively.
To create flow at thi first cowl in Figure G4.A which .is the same as
for a rainjet inlet, an extension to the body was used. The extension con-
sists of the cone of length a and cylinder of length bI in FlTire G4.1.
The cone angle, 81, is d deg;aec. The value of bI was varied until the
pressure at the first cowl was equal to antient; a value of bI equal to
7r, gives this condltion.
G4.2 The normaliz.d values of the. various variables which were
selected are as follc.:
"a - 7.'81 , b 7. 0.2 ,b 2.13 , C2 1.06 units
- DL 0.0736, - DL2 0.177 units. JThe appropriate numbers of noints are:
NA 203, NB 200, NC =6, NB 61, N% 30t2
The dimensional values are:
C 0.40, b2 -4.22, c. - 2.1 inch
* r 1 - 1.98, r, - 2.13, r 3 - 2.48 inch
A(..3 It wai found that the cowl drag coefficient is sensitive to the
magnitude of projected cowl area. For example, another combination of
these variables:
137
I
a 7.111 b~ . 7.~c 0. 38, b2 - 2. 16, C2 1.46I,; • =7.13 b1 7., • • .8,BZ = .60, NC =.40
SNA =196, NB = 193, HC 2 , 2 * 60, NC2 = 40
-DL. 0.138(20°). -DLz 0.243 (9.50)
Gives highe.r values of the cowl drag coefficient:
)Mo 3.0 2.3
G4.2 0.073Z 0.0953
G4.3 0.1214 0.1562
Therefore, attention was mde to 3elect an inlet shape as smooth as
G4.4 AERO can produce also graphical results. A typical example is
illustrated in Figure 4.2.
G4.5 Cnmter Guide
Use opening conuwnds and compilation similar -W that defined in
F2.1, FZ.2.
V The routine to run AERO on the terminal is defined in prog .
itself.
The routine to run COWL on the terminal is as follows:
FILEDEF 02 DISK OCU D (RECFM F BLOCK 132 PER4
EXEC RUN COWL
•PRIT OUT 0 optional• XEDIT OUT 0
138!1
I
CM
-1-
-- '' a c
6.M K
xkk, J r_
p aw 2,p
139
Ul
_--
Lo~ Cyl
-4 It
- ~ B 0
J -p
CIO
-3 ./ S
cu.VI- 4
G4-O
It' IQ
U. ,j jg M 'I., ~, C 4
11 . 1 I
1=1 MISi
140.
I~I,
APPENDIX H: RESULTS
Hi. SUMMARY
Ao/Ar A5/Ar A1/A1 A2iAo AiAr /A
0.25-0.40 0.25-0.40 0.47 0.887 0.426 9.5 15
25
35
4C
I 45
K0.25-0.40 0.25-0.40 0.50 0.827 0.426 9.5 7
25
45• ~65 -
"" •0.25-0.40 0.25-0.40 0.42 0.827 0.426 9,5 45
0.58
0.25-0.40 0.25-0.40 0.47 0.75 0.426 9.5 45
0.827
fl 0.91
0.25-0.40 0.25-0.40 0.47 0.827 0.27 9.5 45
"0.32
0.47
0.25-0.40 0.25-0.40 0.47 0.827 0.426 6.5 45
11.5
114
Flit': CMr3.' 0 A NAVAL. P(1ItGOADUATF SCHOODL PAGf 001
AA/20, A%/AR 1f(% Axlf Vnk To X114491 VIOAI TFP TFTA
()o 24- CO .g 0: -A (.46 ., .ý1;Q dF#00 0 ?q4 (1 00 k 9OF~* ,I'
o.2Pr-O .ZSn,'0O 0.31 0195*2 O.25 ' *'*w5 0.188 p404 0.l'71`60Z 0 O F ~r06 -0. .10r*13 0.sliew -0 t). 1oC.~ 0U C .?50U04 0: 363r +0 1) ?4;IF#C 'o'K k) 9F40 : QF 4-0 8: 14j , : '~ :''
C. 19cr'). 50. +0 0). I I7r~n 0.! -+0s5 +I5'04 0, 19I.Ov N1. 'CS 0.1l 15.41 0 . 5r
C.~~ ~~ a42U U,%VC) 5 1,' r02 0. I1 14)5 "@W*8504 0. 97"*? aI *01 j 01 ýqr 0:53*)~I 5~0V.~~~~iJ0~3c'*o a .%t sol*5 ..V ""o I 0 25. * I. INV -'~ 0?')* 1 1Ii1
C. 31QE '4) ,. 4 t)~ ' a .1, ~ f,2 '0 '.1Fo% Ci.1 Q~' '15 0 0.13'r 40 -0.a 10'5#04 ON or 1 0. 5)vl # to
Ptr. o s ahll? u.E't0 3. 7,'01 0. '1 r*05 0. 0r 4(1 '. a '3Jn 0.4 45- 0: A51-01. a15n~ 0 51r '.i.:
0. jc '2'.O (.*1.. 0 .11 +0 0.16 1 0 ft0 '04I 0.1 7.0 to 0. C6 5 0 :l Q I8q0 .)50,401 0. %' +3fC .W.0 tu 0: 7.75*) t .I a '~ C .j3'0 Ol0*2 0.14).0 0.1 !'Iri.? 9: ~ I~l .3*
C. +(3 '3+J)044 .71*?0 . 6I1 0 .1 lr. 1 Ia~0 0 Sr :0 .I*5 0. )*4 C..50
0I 01.):1c Vfusa .:C 0. I'Ll F :110 0. 1p3 f #G5 0. -11 04 .qio'. fa 0. 1 S5V#.07
C. )6(1-+,),) G.P 4 t.0'lI u: It fir* t * 0 Q *0'. '. sr ~*n4 0.10 7 0. R10 0.) 'I 0. soc')
U. '3t'E1 4.1,0 '4) 1 (5'042 t).1 F 94* 0, K3 53 '04 01.1 PIC *,I U 4) 0 0. OCM '*4f 0.9-,Q 3 0 0~U
l.4 U-14r 0 -2ktv u.0 :43 3) 1) 0.850%0)r0 , 02S."1 :n1
Us 0 61 ,m0H1 ; .'t-o -,( rz 090 " .r:1
st(ir Cu?. 0 A) £NiVAt P0S10R4004VF SCMI~flI. neOct
0.ZaWai fcll 0 5fo 4r29. bvijF'0 069e,0 n.14F#0 0.50 11 l:oo4 0.910.' 0. !%~OF62
C *0tto aJ!C 0. 114vtoof 0. ~31r6 IT It ie -Q~tiJ 0. ,qr-0; .q5k. 0.5%O'O.It . . V~ Ic. 0.03 ;or+ 0.4OU it8 8:H. 0jl.3 §* 5
0.1.1 0* 4 0.r* * 3:44 (* 3''t 4 aS1t, 5fag30 c0 0.%ti' i. o$ 01 tAT #3 3IJQI'D 0.71.5r00,02 0.jI-i ,1 404 o -SFn 0.%):)0.20Cd Cjig~~on 6. 4..t20! Y,&. *0 .IEt' 0.10t7o es,#ns#05q0. ;f C.qOr,0:2ZO.itEZ0
C.?q0( 0143 C.? 110 ) CIS l~ * n[! "1 1:141F: ty :, tl O. 051f 0:u1531 t O' S1ý oaoRr, 4. ~tt oTr0l '.5~20:.*00'03a O:'0, 9-4 . l 0 .o 03.oYS04 0.t Vr*02 0., Ž$01 .c [or*t 0 jq5Qr~j+f 0.?sr'60
0..""l .75..t 410 IC f#4) 0. 1FOO735 U.J67
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References
1. P. H. Morrison and 0. S. Amberntson - "Guidance and Control of a
Canon-Launched Guided Projectile," Journal of Spacecraft and Rockets,
Vol. 14 (6), pp. 328-334 (1977).
2. P. H. Morrison - "A Lesson Learned about Cann n-Laa•ched Guided Pro-
jectiles," Journal of Guidance and Control, Vol. 3(2), pp. 154-157 (1980).
3. Surface Warfare, pp. 18-19 (October, 1980).
4. "Martin Marietta 5" Guided Projectile," Maritime Defense, Vol. 4(4),
p. 128 (1979).
5. G. L. Brown - "Propulsion for Ramjet Propelled Guided Projectile for
5"/54," Master's Thesis, NPS, Monterey, CA (December 1980).
6. Philip P. Hill and Carl R. Peterson - "Mechanircs and The..modynamirs
of Propulsion," Addison-Wesley Publishing Company, Inc. (1965).
7. E. A. Bonney, M. J. Zucrow, C. W. Besserer - "Aerodynamics, Propulsion,
Structures and Design Practice," D. Van Nostrand Company, Inc. (1956).
8. 0. R. Cruise - "Theoretical Computation of Equillibrium Compositions,
Thermodynamic Properties and Performance Characteristics of Propellant iSystems," NWC-TP-6037 (Naval Weapons Center, China Lake, CA 93555,
April 1979).
9. T. H. Gawain - "On the Inviscid Supersonic Flow About a Slender Body
of Revolution," NPS-67-81-007 (Naval Postgraduate School, MWnterey, CA
93940, April 1981).
10. James E. Calogeras and Edward T. Meleason - "'Wind-Tunnel Investigation
of Techniques for Reducing Cowl Drag of an AxIsymetric External-
Compression Inlet at Mach 2.49," NASA TM X-1516, March 1968.
203 g
11. Frank G. Moore and Roy C. Swanson, Jr. - "Aerodynamics of Tactical
Weapons to Mach Number 3 and Angle of Attack 150, Part I-Thecry and
Application," NSWC/DL-TR-3584, February 1977.
12. Fred R. DeJarnette and Kenneth M. Jones - "Development of a Computer
Program to Calculate Aerodynamic Characteristics of Bodies and Wing-
Body Combinations," NSWC/DL-TR-3829, April 1978.
13. Sighard F. Hoerner - "Fluid-Dynamic Drag," Hoerner Fluid Dynamics,
P.O. Box 342, Brick Town, NJ 08723 (1965).
14. A. E. Fuhs - "Ramjet Propulsion Lecture Notes," Naval Postgraduate
School, Monterey, CA 93940 (January - March 1981).
15. William W. Hager, Fred R. DeJarnette and Frank G. Moore - "Optimal
Projectile Shapes for Miniu., Total Drag," NSWC/nL-TP-3597S May 1977.
16. H. W. Liepmann and A. Roshko - "Elements of Gasdynamics," John Wiley
& Sons, Inc. (1957).
17. John J. Bertin and Michael L. Smith - ':Aerodynamics for Engineers,"
Prentice-Hall, Inc., Englewood Cliffs, NJ 07632 (1979).
18. Ralph M. Rotty - "Introductici to Gas-Dynamics," John Wiley & Sons, Inc.
(1962).
19. Frank G. Mooi'e - "Aerodynamics of Guided and Unguided Weapons, Part I-
Theory and Application," NWL Technical Report TR-3018, December 1973.
20. Frank G. Moore - "Aerodynamics of Guided and Unguided Weapons, Part II-
Computer Program and Usage," NWL Technical Report TR-3036, January 1974.
21. W. G. Rogers and Spyridon Livanis - "Ballistic Performance Evaluation
of Swedish Base Bleed Adapted to 5"/54 Projectile," IHRT 628, Jul' ?.
22. Joseph Huerta - "The Effect of Rocket Nozzle Geometry and Secondary
Flow Field on Base Drag at Mach Numbers 2.50 and 3.00, an Experimental
Investigation," BRL-MR-1829 (AD-815444), March 1967.
Z04
I I I II I
°)z
23. George 0. Shrewsbury - "Effect of Boat-ail Juncture Shape on Pressure
Drag Coefficients of Isolated Afterbodies," NUA-TM-X-1517, March 1968.
24. J. S. White - "Aerodynamics and Control of a 5"/54 Gun Launched Missile,"
Master's Thesis, NPS, Monterey, CA (December 1980).
.25. W. Kaufmann - "Fluid Mechanics," McGraw-Hill Book Co., 1963.
205
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205
Nt
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