This report covers the tests of a 2" diameter model of the

4. 5" H. E. Rocket, T38E3, to determine the force and moment coefficients and the location of the center of pressure. The tests were made in the 14" diameter working section of the High Speed Water Tunnel at the California Institute of Technology. The work was authorized by a letter of January 31, 1944, from Dr. E. H.

Colpitts, Chief of Section 6.1, National Defense Research Committee, New York City

Robison, Gerald B.

Caltech Authors - Main

E:This copy not to be released 
security regulations write 
· on copy issued:"Reg rade 
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Securit 72 
R.ESEAQCH ~DEVELOPMENT 
THE HIGH ~PEED WATER. TUNNEL 
CALJFOQ,NfA JN0TlTUTE OF TECHNOLOGY. 
PASADENA, CALJ POQNlA 
.S E.CTTON N<2 G>.l -Sr 2.07-1919 
LABO~ATOR..Y N~ ND-41 
.OFFICE OF SCIENTIFIC RESEARCH AND DEVELOPMENT 
NATIONAL DEFENSE RESEARCH COMMITTEE 
DIVISION SIX- SECTION 6.1 
FORCE TESTS 
OF THE 
4.5" ROCKET, T38E3 
ROBERT T. KN APP 
OFFICIAL INVESTIGATOR 
THE HIGH SPEED WATER TUNNEL 
AT THE 
CALIFORNIA INSTITUTE OF TECHNOLOGY 
HYDRODYNAMICS LABORATORY 
PASADENA, CALIFORNIA 
Section No 6 i-sr207- i9i9 Report Prepared by 
Laboratory No . ND-4:1. 
May i J :1.945 
~his docuaent contains information affe c ting 
the national defense of the United States within 
the aeaning of the Espionage Act , ~0 U.S C 31 
and 32 , as aaended . The transaission or the 
P~val••ihn n¥ itc ~nn~on~~ in ~nv MAnnA~ ~n ~n 
Gerald B. Robison 
Hydraulic Engineer 
''-' i F I f EY'f I t t 
~ONF I DENT I AI: 
F I G. l- 2 11 D I AMETER MODEL OF 4 . 5 11 H.E. ROCKET, T~E3 
F IG . 2 - 211 DIAMETER MODEL OF 4.5 11 H . E. ROCKET, T38E3 
..- G9NF19ENTIAl:~ 
-CONFIDENTIAL 
FORCE TESTS 
OF THE 
4.5" ROCKET, T38E3 
. PURPOSE OF TEST 
·This report covers the tests of a 2" diameter model of the 
4. 5" H. E. Rocket, T38E3, to determine the force and moment coef-
ficie~ts and the location of the center of pressure. The tests 
were made in the 14" diameter working section of the High Speed 
Water Tunnel at the California Institute of Technology. The work 
was authorized by a letter of January 31, 1944, from Dr. E. H. 
Colpitts, Chief of Section 6.1, National Defense Research Com-
mittee, New York City. 
DESCRIPTION OF MODEL AND PROTOTYPE 
Figures 1. 2., and 3 are views of the 2" diameter model. 
Indents and screw openings are visible in the fuse cap, and the 
construction of nozzles in the base is also shown. An outline 
F IG. 3 - 2" DIM..1ETER MOD EL OF 4.5" H.E. ROCKET 
S HOWING NOZZLE S 
CONFIDENTIAL 
CONFIDENTIAL 
-2-
drawing of the model is presented in Figure 4. Comparative out-
lines of the 5" SSR RocketJand 3 . 5" Ro cket with Nose No. 54) 
referred to herein) are shown in Figure 5. 
Data pertaining to the prototype were given as follows : 
Length overall 
Maximum diameter 
Distance) nose to center of gravity 
Weight witho ut motor powder 
Velocity at 70° Fahrenheit 
NOS[ NO l ~l DWG. ND•I)41;J 
8.HZ ~ 
STA~IIitO IOOV SICTIOH CWO NO •Iil<1411 
, ... ,.~ 
3i.4<0 inches 
4. SiS inches 
16 . 19 inches 
42.5 pounds 
ex:>.o f t I sec 
A'TERIODY NO llil OWQ. ND • I54I • U 
z.uz· 
- -- ---·~ ------
r------- ,.,.. t .. 
f-- ------------ 31 440 
FIG. 4- OUTLI'NE DRAWING OF 4.5 11 H.E. ROCKET, T38E3 
~±=-- ~--~~·-" 18J-
:!! " ROCKET 
1-- ------
2 4 84 " 
F IG. 5 - COMPARATIVE MODEL OUTLINE S 
4.5" H.E. ROCKET , T 38E3; 5 "S SR ROC KET, MODELS 2::. AND 25; 
AND 3 .5" ROTATING R OC KET 
CONFIDENTIAL 
CONFIDENTIAL 
-3-
TEST RESULTS 
The results of the tests are shown in Fiqures 6 to 8. inclu-
sive. These include flow diagrams obtained from model performance 
in the Polarized Light Flume. 
0.5 
0 .4 
u" 
.., o.z 
i 
.. ~ 0 .1 
0 v 
01 
~ 0 
D 
r-
• 
~ 
; 0 . I 
0 
0 
01 i\ 
_\ 
\ 0 
.Q 2 
0 .10 
/ 
:,...' 
/ 
v 
v 
v 
v 
f 
u 0011 
.; 
" ~ o.oe 
/ v v 
/"" li / 
v / 
/ v 
I D.DZ 
""-r-... 
/ 
/ 
• • • 10 
0 v z • • • YAW ANOLl IN DtiiiiUS, 'If" YAW ANGLE: IN D£c;tt[tS1 y 
FIG. 6 -4.5" H.E. ROCKET, T38E3 
RELATION OF CROSS FORCE AND MOMENT COEFFICIENTS 
TO YAW ANGLE 
-
~ ,__ -~ 
• • YAW ANGLE, DlGMI!I 
YAW NOLL, DE:GR lS 
" 
-
. 
u 
= "' 0 
O.l. t 10 ,_, t t.a s • 
IIIEVNOLDS NUMilR, ._ 
F I G, 7 - 4. 5 11 H. E • ROCKET , T 38E 3 
RELATION OF DRAG COEFFICIENT TO YAW ANGLE AND REYNOLDS NUMBER 
AND OF ECCENTRICITY TO YAW ANGLE 
./ 
I) 
CONFIDENTIAL 
COHFIDEhTIAL 
-4-
't ' o• 
FIG. 8- FLOW DIAGRAMS OF 4.5" t-I,E. ROCKET, T38E3 
DISCUSSION OF RESULTS 
Definition of terms, coefficient and center-of-pressure ec-
centricity formulae, and a general discussion of static stability 
are given in the Appendix to this report. 
The coefficient curves have resemblances to those of the 5" 
Rocket, tested previously, as may have been expected from the 
general similarity of their shape. See Figure s. A minor differ-
ence appears in the moment coefficient curve, Figure 6, which 
shows a small amount of static stability up to about 3/4 of a 
degree of yaw for the 4. 5" Rocket. This minor stabilit·{ was noted 
only because of readings taken at i/2 degree of yaw, a procedure 
not normally followed. However, it may be noted that a similar 
sm.all static stability was obtained for the 3.5" Rocket with Nose 
No. 54* at yaw angles of less than i-i/2°. The positive slope and 
values of the remainder of the moment coefficient curve indicate 
the static instability of the projectile and its increasing amount 
with increase of yaw angle. 
The small static stability for yaw angles below 3 / 4° is indi-
cated again by the positive values of the center-of-pressure ec-
centricity, Figure 7. 
The cross force coefficient, Figure 6, is an average of 5% 
greater than Mode:!. 25, and iO% greater than Model 2i of the 5" 
Rocket. 
•water Tunnel Tests ot the 3.6• Rotatin& Roeket•, Seetion 
llo. 6.t-sr207-1270, April 21, 1944 
COdFIDENTIAL 
CO.,F I DEtn i A~ 
-5-
The drag coefficient~ c0~ is shown for angles of yaw from 0° 
to 9° in Figure 7 and also its manner of variation with Reynolds 
number (same Figure) The drag coefficient of the 5" Rocket at 
C0 yaw~ is approximately iS% 1 ess than that of the 4 S" Rocket 
However~ the difference becomes negligible for yaw angles of S0 
and greater in the case of Model 25~ and is reduced to an average 
of 7% for yaw angles of 2° and above fbr Model 2i of the 5" Rocket 
The maximum water Tunnel velocity used in the tests was 
60 ft/sec This is equivalent~ for the same Reynolds Number to 
a velocity of 350 ft/sec in air with a temperature of 60° Fahren-
heit The ratio between prototype velocity in air and water 
Tunnel velocity for this model for the same Reynolds number is 
5 83 
Flow line drawings , Figure 8 show slight disturbances at 
th~ junction of the nose and fuse structure~ and still smaller 
disturbance at the 1ndents and screw openings Turbulence at the 
rear is typical of a br~nt -end afterbody and very similar to that 
of the S" Rocket 
CONFIDENTIAL 
CONFIDENT IAL 
APPENDIX 
DEFINITIONS 
YAW ANGLEJ 1/J 
The angleJ in a horiz~ntal planeJ which the axis of the pro-
jectile makes with the dir~ction of motion. Loo king down on the 
projectileJ yaw angles in a clockwise direct1on are positive (+) 
and in a counterclockwise directionJ negative (-). 
PITCH ANGLE, a 
• The angleJ in a vertical planeJ w~ich the axis of the pro-
jectile makes with the direction of motion . Pitch angles are 
positive (+) when the nose is up and negative (-) when the nose 
is down : 
LIFT , L 
The forceJ in poundsJ exerted on the projectile normal to 
the direc t ion of motion and in a vertical plane. The lift is 
positive (+) whe n ac t i n g upward a n d negative (-) when acting 
downward. 
CROSS FORCE , C 
The forceJ in poun dsJ exerted on the projectile normal to the 
direction of motion a nd in a horizontal plane The cross force is 
positive when acting in the same direction as the displacemen~ of 
the projectile nose for a positive yaw angleJ i . e.; to an observer 
facing in the direction of travelJ a positive cross force acts to 
the right 
DRAG, 0 
The forqe J i~ poundsJ 
the direction of motion . 
direction opposite to the 
MOMENT , M 
exerted on t he projectile parallel with 
The drag is positive when acting 1n a 
direction of motion 
The torqueJ in foot poundsJ tend1ng to rotate the projectile 
about a transverse axis Yawing moments tending to rotate the 
projectile in a clockwise direction (when looking down on the pro-
JeCtile) are positive (+)J and those ~ending to cause counter-
clockwise rotation are negative (-). Pitching moments tending to 
rotate the projectile 1n a clockwise direction (when looking at 
the projectile from the port s1de) are positive (+)J and those 
tending to cause counterclockwis e rotation are negative (-). 
CONFIDENTIA L 
CONFI DENT IAL 
-b-
In accordance with this sign convention a moment has a de-
stabilizing effect when it has the same sign as the yaw angle or 
the opposite si g n of the pitch ang le 
In all model tests the moment is measured about the point of 
support Moments about the center of gravity of th e p r ojec t ile 
have the symbolJ Meg · 
NORMAL COMPONENT, N 
The sum of the components of the drag and cross force act ing 
normal to the axis of the projectile . The value of the n o r mal 
component is given by the following: 
N = D sin ~ + C cos ~ ( i) 
in which 
N Normal component in lbs 
D Drag in lbs 
c Cross force in lbs 
~ Yaw angle in degrees 
CENTER OF PRESSURE, CP 
The point in the axis of the projectile at which the resul -
tant of all forces acting on the 'p.rojectile is ·a?plied . 
CENTER-OF-PRESSURE ECCENTRICITY, e 
The distance between the center of pressure (CP) and th e 
center of gravity (CG) expressed as a decimal fraction of th e 
length (1) of the project1le The center - of-?ressure eccen t ric1ty 
is derived as follows: 
( 2) 
in which 
e = Center-of-pressure eccentricity 
1 Length of projectile in feet 
leg Distance from nose of projectile to CG 1n feet 
lcp Distance from nose of projectile to CP ,in feet 
CONFI DENTIAL 
CONFIDENTIAL 
-c-
COEFFICIENTS 
The three force and moment coefficients used are derived as 
follows: 
in 
D 
p v2 AD 
-
( 3) Drag coefficient) 
2 
Cross force coefficient) Cc c 
p v2 AD 
2 
( 4) 
M 
v2 
ADl p -
2 
Moment coef~icientJ· ( 5) 
which 
D Measured drag force in lbs 
c Measured cross force in lbs 
p Density of the fluid in slugs/cu ft = w/g 
w Specific weight of the fluid in lbs/cu ft 
g Acceleration of gravity in ft/sec 2 
AD= Area in sq ft at the m~ximum cross section of the pro-
jectile taken normal to the geometric axis of the pro-
jectile 
V Me an relative velocity between the wat er and the pro-
jectile in ft/sec 
H MomentJ in foot-poundsJ measured o.bout any particular 
poin t on the geome tric axis of the projectile 
1 = Overal l leng th of the projectile in feet 
CONTROL ANGLE 
In considering the effect of rudders on static stability) 
either in yaw or pitchJ the term "control angle" is used to denote 
the yaw below which a given rudder setting with opposite sign to 
the yaw will t end to return the project il e t o zero yaw) and above 
which the yaw wil l further increase . The control angle 1s useful 
for indicating th e effect1veness of rudders and for comparing th e 
static stability of d1fferent projectiles w1 th equal rudder 
se ttin gs. 
t'MJ I= I OI=NT I A I 
,. 
CONFIDENTIAL 
-d-
RUDDER EFFECT 
The total increase or decrease in moment coefficient, at a 
given yaw or pit~h ongleJ resulting from a given rudder setting 
This increase or decrease In moment coefficient is measured from 
the moment coefficient curve for neutral rudder setting 
REYNOLDS NUMBER 
In 
inertia 
utility 
comparing hydraulic systems involving only friction and 
forces, a factor called Reynolds number IS of great 
This IS defined as follows: 
R lV 
v 
(6) 
In which 
R Reynolds number 
1 Overall length of projectile feet 
V Velocity of proJectile, feet per sec 
v K1nematic v1scos1ty of the fluid, sq ft per sec IJ./ p 
p Mass density of the flu1d 1n slugs per cu ft 
1J. = Absolute V1scos1ty In pound-seconds per sq ft 
Two geometrically similar systems are also dynamically simi-
lar when they have the same value of Reynolds number For the 
same fluid In both casas a model with small linear dimensions 
must be used with correspondingly large velocities It is also 
po3sible to compare two cases with widely differ1ng fluids pro-
vided 1 and V are properly chosen to give the same value of R 
CA VITATION PARAMETER 
In th@ analysis of cav1tation phenomena, the cavitation 
parameter has been found very useful . This 1s defined as follows: 
in 
K·= PL PB 
v2 (7) 
p 
-
2 
which 
K Cavi tat1 on parameter 
PL= Absolute pressure in the undisturbed liquidJ lbs/sq ft 
P8 = Vapor pressure corresponding to the water temperature) 
lbs/sq ft 
V Velocity of the projectile) ft/sec 
CONFIDEtHIAL 
CONFIDENTIAL 
-e-
p mnss density of the fluid in s.lu.;s per cu ft w/g 
w weight of the fluid 1 n 1 bs per c u f t 
g acceleration of gra vity 
Note that any homogeneous 
tat ion of this parameter. 
this parameter in terms of 
set of units can be used in the compu-
ThusJ it is ofte n convenient to express 
the headJ i.e.J 
where 
K ( 8) 
~I Submergence plus the barometric headJ ft of water 
h 8 Pressure in the bubbleJ ft of water 
It will be seen that the numerator of both ~xpressions is simply 
the net pressure acting to collapse the cavity or bubble The 
denomina tor is the velocity pressure S1nce the entire variation 
in pressure around the moving body is a result of! the velocityJ it 
may be con~idered t hat the velocity head is a measure of the pres-
sure available to open up a cav1tation void From this point of 
viewJ the cavitation parameter is simply the ratio of the pr·essure 
available to collapse the bubble to the pressure available to open 
it. If the K for incipient cavitat1on is consideredJ it can be 
interpreted to mean the maximum reduction in pressure on the sur-
ieee of the body measured in terms of the velocity head. ThusJ 
if a body starts to cavitate at the cavitation parameter of oneJ 
it rr.eans that the lowest pressure at any point on the body is one 
velocity head below that of the undisturbed fluid. 
The shape and size of the cavitation bubbles for a specific 
projectile are functions of the cavitation parometer. If p 8 is 
taken to represent the gas pressure within the bubble instead of 
the Japor pressure of the waterJ as in normal investiga t ions) the 
value ot r· obtained by the above formula will be applicable to an 
air bubble . In other wordsJ the oehavior of the bubble will be 
the sa~e whether the bubble is due to cavitationJ the inject1 on of 
exhaust gasJ or the entrainment of air at the time of launching. 
The following chart gives values of the cavitation parameter 
as a function of velocity and submergence in sea water. 
GE~ERAL DI~CU~~IGN OF ~TATiC STABILITY 
Water tunnel tests are ade under steady flow condltionsJ 
conseiuently the results only indicate the tendency of the steady 
state hydrodynamic couples and forces to cause the projectile to 
return to or move away from its equilibrium position after a 
,..,.. ... ~ I r'\ r"".ITI A I 
CONF IDEN TI AL 
-f -
disturbance Dynamic couples and forces 1ncluding either positive 
or negative damping are not obtained If the hydrodynamic moments 
are restoring the projectile J then it is said to be statically 
stable , if nonrestoring statically unstable In the discussion 
of static stability the actual motion following a perturbation is 
not considered at all In fact the pro ectile may oscillate con-
tinuously about an equilibrium position without remaining in it 
In this case 1t would be statically stable but would have zero 
damping and hence , be dynamically unstable With negative damping 
a proJectile would osc1llate With continually increasing applitude 
following an initial perturbation even though it were statically 
stable E~ui~ibrium IS obtained if the sum of the hydrodynamic. 
buoyant , and propulsive moments e~ual zero In general. propul-
sive thrusts act through tke center of gravity of the projecti}e 
so only the first two Items are important 
If a projectile IS rotating from Its equ1librium posi t ion so 
as to increase its yaw angle posit1vely the momen t coefficient 
must 1ncrease negatively (according to the sign convention adop-
ted) 1n order that 1t be stat1cally stable Therefore, for pro-
jectiles without contrcls or with' f1xed control surfaces, a nega-
tive slope of the curve of moment coeff1cient vs yaw gives static 
stab1lity and a pos1t1ve slope g1ves 1nstabil1ty For a pro-
jectile w1thout controls static stability IS necessary for a 
successful flight unless stability is obtained by spinning as in 
the case of rifle shells For a proJeCtile with controls. s t abi l-
izing moments can be obtained by adJusting the control surfaces, 
and the slope of the moment coefficient as obtained with fixed 
rudder position need not give static stability Where buoyancy 
either acts at the center of gravilY oi can be neglected. equil-
ibrium is obtained when the hydrodynamic moment coefficient equals 
zero For symmetrical projectiles this occurs at zero yaw angle 
i e when the proJeCtile axis IS parallel to the traJectory For 
nonsymmetrical projectiles _ such as a torpedo when the rudders are 
not neutral the moment is not :tero at zero yaw but vanishes at 
some definite angle of attack Where buoyancy cannot be neglect ed 
equilibrium is obtained when CM = -Csuoyancy and the axis of the 
proJectile IS at some angle with the trajectory 
For symmetrical projectiles the degree of stability or In 
stability can be obtained from the center-of pressure curves If 
the center of pressure falls behind the center of grav1ty a re-
storing moment ex•sts giving static stability If the center of 
pressure falls ahead of the center.of gravity the moment is non-
restoring , and the projectile will be st~t ~ally unstable The 
degree of stability or Instability IS indicated approximately by 
the distance between the center of gravity and the center of 
pressure In general for nonsymmetrical projectiles the cross 
force or lift IS not zero when the moment vanishes so that the 
center of pressure curve is not symmetrical and the simple rules 
just stated cannot be used to determine whether or not the pro-
Jectile will be stable In ·such cases careful Interpretation of 
the moment curves IS a more satisfactory method of determin1ng 
stability relat1onshi~ 
CONF IDEN TIAL 
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GQMFI9ENTIAl 


Force Tests of the 4.5" Rocket, T38E3

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Force Tests of the 4.5" Rocket, T38E3

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