REPORT 1381 A STUDY OF THE MOTION AND AERODYNAMIC HEATING OF BALLISTIC MISSILES ENTERING THE EARTH'S ATMOSPHERE AT HIGH SUPERSONIC SPEEDS

• Document Type: CREST
• Collection: General CIA Records
• Document Number (FOIA) /ESDN (CREST): CIA-RDP71B00265R000200130001-4
• Release Decision: RIFPUB
• Original Classification: K
• Document Page Count: 18
• Document Creation Date: December 23, 2016
• Document Release Date: November 21, 2013
• Sequence Number: 1
• Publication Date: January 1, 1961
• Content Type: MISC

, Declassified and Approved For Release 2013/11/21 : CIA-RDP71600265R000200130001-4
REPORT 1381
A STUDY OF THE MOTION AND AERODYNAMIC HEATING
? OF BALLISTIC MISSILES ENTERING THE EARTH'S
ATMOSPHERE AT HIGH SUPERSONIC SPEEDS
By H. JULIAN ALLEN and A. J. EGGERS, Jr.
. Ames Aeronautical Laboratory
Moffett Field, Calif.
Declassified and Approved For Release 2013/11/21 : CIA-RDP71600265R000200130001-4
Declassified and Approved For Release 2013/11/21 : CIA-RDP71600265R000200130001-4
National Advisory Committee for Aeronautics
Headquarters, 1512 H Street NW., Washington, 25, D. C.
Created by Act of Congress approved March 3, 1915, for the supervision and direction of the scientific study
of the problems of flight (U. S. Code, title 50, sec. 151). Its membership was increased from 12 to 15 by act
approved March 2, 1929, and to 17 by act approved May 25, 1948. The members are appointed by the President
and serve as such without compensation.
JAMES H. DOOLITTLE, Sc. D., Shell Oil Company, Chairman
LEONARD CARMICHAEL, Ph. D., Secretary, Smithsonian Institution, Vice Chairman
ALLEN V. ASTIN, Ph. D., Director, National Bureau of Standards.
PRESTON R. BASSETT, D. Sc.
DETLEY W. BRONK, Ph. D., President, Rockefeller Institute for
Medical Research.
FREDERICK C. CRAWFORD, Sc. D., Chairman of the Board,
Thompson Products, Inc.
WILLIAM V. Dtivis, JR., Vice Admiral, United States Navy,.
Deputy Chief of Naval Operations (Air).
PAin, D. FOOTE, Ph. D., Assistant Secretary of Defense, Re-
search and Engineering:
WELLINGTON T. HINES, Rear Admiral, United States Navy,
Assistant Chief for Procurement, Bureau of Aeronautics.,
JEROME C. HIINSAKER, Sc. D., Massachusetts Institute of
Technology.
CHARLES J. MCCARTHY, S. B., Chairman of th6 Board, Chance
Vought Aircraft, Inc.
DONALD L. PUTT, Lieutenant General, United States Air Force.
Deputy Chief of Staff, Development.
JAMES T. PYLE, A. B., Administrator of Civil Aeronautics.
FRANCIS W. REICHELDERFER, Sc. D., Chief, United States
Weather Bureau.
EDWARD V. RICKENBACKER, Sc. D., Chairman of the Board,
Eastern Air Lines, Inc.
Louis S. ROTHSCHILD, Ph. B., Under Secretary of Commerce for
Transportation.
THOMAS D. WHITE, General, United States Air Force, Chief of
Staff.
Hoax L. DRYDEN, PH. D., Director
JOHN F. VICTORY, LL. D., Executive Secretary
JOHN W. CROWLEY, JR., B. S., Associate Director for Research EDWARD H. CHAMBERLIN, Executive Officer
HENRY J. E. REID, D. Eng., Director, Langley Aeronautical Laboratory, Langley Field, Va.
SMITH J. DEFRANCE, D. Eng., Director, Ames Aeronautical Laboratory, Moffett Field, Calif.
EDWARD R. SHARP, Sc. D., Director, Lewis Flight Propulsion Laboratory, Cleveland, Ohio
WALTER C. Wzminms, B. S., Chief, High-Speed Flight Station, Edwards, Calif.
Declassified and Approved For Release 2013/11/21 : CIA-RDP71600265R000200130001-4
Declassified and Approved For Release 2013/11/21 : CIA-RDP71600265R000200130001-4
REPORT 1381
A STUDY OF THE MOTION AND AERODYNAMIC HEATING OF BALLISTIC MISSILES ENTERING
THE EARTH'S ATMOSPHERE AT HIGH SUPERSONIC SPEEDS
By H. JULIAN ALLEN and A. J. EGGERS, Jr.
SUMMARY
A simplified analysts is made of the velocity and deceleration
history of ballistic missiles entering the earth's atmosphere at
high supersonic speeds. It is found that, in general, the gravity
force is negligible compared to the aerodynamic drag force and,
hence, that the trajectory is essentially a straight line. A con-
dant drag coefficient and an exponential variation of density with
altitude are assumed and generalized curves for the variation of
missile speed and deceleration with altitude are 'obtained. A
curious finding is that the maximum deceleration is independent
of physical characteristics of a missile (e. g., mass, size, and
drag coefficient) and is determined only by entry speed and
flight-path angle; provided this deceleration occurs before impact.
The results of the motion analysis are employed to determine
means available to the designer for minimizing aerodynamic
heating. Emphasis is placed upon the convective-heating
problem. including not only the total heat transfer but also the
maximum average and local rates of heat transfer per unit area.
It is found that if a missile is so heavy as to be retarded only
slightly by aerodynamic drag, irrespective of the magnitude of the
drag force, then convective heating is minimized by minimizing
the total shear force acting on the body. This condition is
achieved by employing shapes with a low pressure drag. On the
other hand, if a missile is so light as to be decelerated to rela-
tively low speeds, iven if acted upon by low drag forces, then
convective heating is minimized by employing shapes with a
high pressure drag, thereby maximizing the amount of heat
delivered to the atmosphere and minimizing the amount delivered
to the body in the deceleration process. Blunt shapes appear
superior to slender shapes from the standpoint of having lower
maximum convective heat-transfer rates in the region of the nose.
The maximum average heat-transfer rate per unit area can be
reduced by employing either slender or blunt shapes rather than
shapes of intermediate slenderness. Generally, the blunt shape
with high pressure drag would appear to offer considerable
promise of minimizing the heat transfer to missiles of the sizes,
weights, and speeds of usual interest.
INTRODUCTION
For long-range ballistic trajectories one of the most diffi-
cult phases of flight the designer must cope with is the re-
entry into the earth's atmosphere, wherein the aerodynamic
heating associated with the high flight speeds is intense,
The air temperature in the boundary layer may reach values
in the tens of thousands of degrees Fahrenheit which, corn-
Supersedes NACA Technical Nate 4047 by H. Julian Allen and A.]. Eggers, Jr., 1057.
bined with the high surface shear, promotes very great
convective heat transfer to the surface. Hent-absorbent,
material must, therefore be provided to prevent, destruction
of the essential elements of the missile. It is characteristic
of long-range rockets that for every pound of material which
is carried to "burn-out," many pounds of fuel are required
in the booster to obtain the flight range. It is clear, there-
fore, that the amount of material added for protection from
excessive aerodynamic heating must. be minimized in order
to keep the take-off weight to a practicable value. The
importance of reducing the bent transferred to the missile
to the least. amount is thus evident.
For missiles designed to absorb the heat within the solid
surface of the missile shell, a factor which may be important,
in addition to the total amount of heat transferred, is the
rate at which it is transferred since there is a maximum
rate at which the surface material can safely conduct the
heat within itself. An excessively high time rate of heat
input may promote such large temperature differences as
to cause spelling of the surface, and thus result in loss of
valuable heat-absorbent material, or even structural failure
as a result of stresses induced by the temperature gradients.
For missiles designed to absorb the heat with liquid
coolants (e. g., by transpiration cooling where the surface
heat-transfer rate is high, or by circulating liquid coolants
within the shell where the surface heat-transfer rate is
lower), the time rate of heat transfer is similarly of interest
since it determines the required liquid pumping rate.
These heating problems, of course, have been given con-
siderable study in connection with the design of particular
missiles, but these studies are very detailed in scope. There
has been need for a generalized heating analysis. intended to
show in the broad sense the means available for minimizing
the heating problems. Wagner, reference 1, made a step
toward satisfying this need by developing a laudably simple
motion analysis. This analysis was not generalized, how-
ever, since it was his purpose to study the motion and heat-
ing of a particular missile.
It is the purpose of this report to simplify and generalize
the analysis of the heating problem in order that the salient
features of this problem will be made clear so that successful
solutions of the problem will suggest' themselves.
A motion analysis, having the basic character of Wagner's
approach, precedes the heating analysis. The generalized
results of this analysis are of considerable interest in them-
selves and, accordingly, are treated in detail.
Declassified and Approved For Release 2013/11/21 : CIA-RDP71600265R000200130001-4
Declassified and Approved For Release 2013/11/21 : CIA-RDP71600265R000200130001-4
2 REPORT 1381?NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS
ANALYSIS
MOTION OF THE BODY
Consider a body of mass m enteriNg the atmosphere from
great height. If, at any altitude y, the speed is V and the
angle of approach is 9 to the horizontal (see sketch), the
Impact point (0,0)
parametric equations of motion can be written 2
d21,
C
g+DpV2A sin 0
dt2 27n
072x v2A
?"ern'? cos
dt2 2m
(1)
where
CD drag coefficient, dimensionless
V speed, ft/sec
A reference area for drag evaluation, sq ft
in mass of the body, slugs
mass density of the air, slugs/ft
g acceleration of gravity, ft/sec'
x,y horizontal and vertical distance from the point of im-
pact with the earth, ft
angle between the flight path and the horizontal, deg
(See Appendix A for complete list of symbols.)
In general, the drag coefficient varies with Mach number
and Reynolds number, while the density and, to a very
minor extent, the acceleration of gravity vary with altitude.
Hence it is clear that exact solution of these equations is
formidable. Let us first, then, consider the following
simplified case:
1. The body descends vertically.
2. The drag coefficient is constant.'
3. The acceleration of gravity is constant.'
4. The density as a function of altitude is given by the
relation
P=Por? (2)
where Po and fi are constants. This relation is consistent
with the assumption of an isothermal atmosphere.
2 Properly, the analysis should consider those effects resulting from the fact that the earth
Is a rotating sphere, hut since the altitude range for which drag effects are important is less
than I percent of the radius of the earth, the rectilinear treatment given In this analysis Is
permissible.
3 As is well known, this assumption is generally of good accuracy at the high Mach numbers
under consideration, at least as long as the total drag Is largely pressure drag.
? The acceleration of gravity decreases by only 1 percent for every 100,000-foot Increase In
altitude.
Equations (1) then reduce to the single equation
d'y dV _LCD PoA e?WV2
dt? gm 2m
Noting that
dV
dV ?dy
(It
we let
(3)
V'
and equation (3) becomes the linear differential equation
dZ CD/4A e-a2z+2g=0
dy m
(4)
whieh has the well-known solution
fc?paii coy duopeA
e?PY dy
Z= e m (-2gf e m dy+ermst.)
Performing the integrations, we obtain as the solution of
this relation
(CDPA _auy
_ cop,A !coy 2gv,? om e
Z=V2=eL
" 2gy+conat. (5)
n-1
so that the deceleration becomes, in terms of gravity accel-
eration,
dV
771 CD NA ( C DNA e_a)
_CDpoit 2g \ in
rove Inn v #ntm
g 2 mg
22-1
1 2gy+const. ?1 (6)
As an example, consider the vertical descent of a solid
iron sphere having a diameter of 1 foot. For a sphere the
drag coefficient may be taken as unity, based on the frontal
area for all Mach numbers greater than about 1.4. In
equation (2), which describes the variation of density with
altitude, the constants should clearly be so chosen as to
give accurate values of the density over the range of altitudes
for which the deceleration is large. It is seen in figure 1
that for
and
which yields
p0=0.0034 slugs/ft'
1
0? f
22,000
_
p=0.00&4 e 22,"
the calculated density is in good agreement with the NACA
standard atmosphere values obtained from references 2 and.
3 for the altitude range from 20,000 to 180,000 feet. These
relations have been used ill calculating the velocity and
deceleration of the sphere for various altitudes, assuming
(7)
Declassified and Approved For Release 2013/11/21 : CIA-RDP71600265R000200130001-4
Declassified and Approved For Release 2013/11/21 : CIA-RDP71600265R000200130001-4
MOTION AND HEATING OF BALLESTR; autibi bk.,*
10-2
io-3
o-
10-
0 40 80 120 160 200 240 280
\
\::0034
P 61-11221?M
References
2and 3'
?
?
?
Na
?
\
25
20

5
Altitude, y, feel x 10-3
FIGURE 1.?Variation of density with altitude.
Including gravity
--- Neglecting gravity
V1: 30,000
VE.20,000
V- 0,000
----
0 50 ICO
Altitude, y, feet x 10-3
Fratrun 2.?Variations of velocity with altitude for a 1-foot diameter,
solid iron sphere . entering the earth's atmosphere vertically at
velocities of 10,000, 20,000, and 30,000 ft/sec.
1510
200
3
vertical entrance velocities of 10,000, 20,000, and 30,000 feet,
per second at, 40 miles altitude which, for these cases, may
be consi.ffered the "outer reach" of the atmosphere. The
results of these calculations are presented as the solid curves
in figures 2 and 3.
It is seen in figure 3 that for the high entrance speeds con-
sidered, the decelerations reach large values compared to
the acceleration of gravity. This suggests that the gravity
term in equation (3) may be neglected without seriously
affecting the results.' When this term is neglected the
equation Of motion becomes
250
200
150
0
? -
-e;
0) 100
50
dV dV One A
? ?
?V e-0172
(It dy 2m
V '30,000 Including gravity
---- Neglecting gravity
VE .10,000
(8)
50 100 . 150 200
Altitude, y, feet x 104
FIGURE 3.?Variations of deceleration with altitude for a 1-foot
diameter, solid iron sphere entering the earth's atmosphere vertically
at velocities of 10,000, 20,000, and 30,000 ft/sec.
Integration gives
or
CoPcA
In V? e-PY-I-conest.
2flm
V=eonet.Xe
CDD,,A e_sy
'ern
At the altitude of 40 miles it can readily be shown that the
term
Civ( rov
WM
is very nearly unity so that the velocity may be written
copeA
V=V Re "rn
'It 13 usual to neglect the gravity acceleration a priori (sot e.g., refs. land 4.)
Declassified and Approved For Release 2013/11/21 : CIA-RDP71600265R000200130001-4
(9)
Declassified and Approved For Release 2013/11/21 : CIA-RDP71600265R000200130001-4
REPORT 1381?NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS
and
(IV
C opuA
dt CEp0AV E2 -
e_oe r"
2mg
(10)
where V5 is the entrance speed.
By use of equations (9) and (10) the vertical-descent speeds
and decelerations for the 1-foot-diarsteter. sphere previously
considered have been calculated for the same entrance speeds.
The results are shown as the dashed curves in figures 2 and 3.
It is seen that these approximate calculations agree very well
with those based on the more complete equation of motion
(eq. (3)).
The above finding is important, for it indicates that in the
?general case, wherein the body enters the atmosphere at
high speed at angle OE to the horizontal, the gravity term,
provided OE is not too small, may be neglected in equation
(1) to yield
d2y_CDpinA sin BE
02? 2m
d2x_CDpV2A cos 0,,
0,12 2m
so that the flight path is essentially a straight line (i c.,
0=0), and the resultant deceleration equation becomes
dV CEpAV2
2 m (12)
dt ?
Now, again, if the density relation given by equation (2) is
used and it is noted that
dy
di dV . dV
. m --V sin ?
sin BE dt " 4
equation (12) becomes
dV CEp0A dy
V 2m sin OD
which can be integrated to yield
Copoit coy
V= VE e zams'"E
and the deceleration is then
(13)
dV ?
di CEpolIVE2_py e-Pv fini sin ok
e 8 (14)
2mg
The altitude yi at which the maximum deceleration occurs
is found from this relation to be
1 ) CDp0A
`n OM sin 05
(15)
If y, is positive the velocity V, (from eqs. (13) and (15)) at
which the maximum deceleration occurs becomes
V1=VEela20.61 V,,
and the value of the maximum deceleration is
dV dV
(dt) (dt)
_ 'eV E2 sin 85 (17)
II max 11 2ge
If equations (13) and (14) are rewritten to make the
altitude reference point yi rather than zero, then
Cop GA e_mvi+ ao
V= V,, e 2sm sin Op
and
dV
CnoInA ?13(v11-40
dt CEEGAVE2 -0 tv 1+ AY I in Sin Pg e
e E
2mg
respectively, where Ay is the change in altitude from
Substitution of equation (15) ? into these expressions can
readily be shown to give
V0,,, -ie-Pou=p? (flay) (1.8)
W=e
and
(dVIdt)
\ Jav_e-SAYea-e-06/4.= Pt (Say)
(d VAR)
g )1
(19)
Equations (18) and (1.9) are generalized expressions for veloc-
ity and deceleration for bodies of constant drag coefficient
and, together with equations (15) and (17), can be used to
determine the variation of these quantities with altitude for
specific cases. The dependence of F' and F" (3Ay) on
gay is shown in figure 4.
Fl (JCL y)
"8:.??
F" () Sy)
-20 -10 0 10 20 30
Bay
FIGURE 4.?Variations of r (flay), F" (flay), and F" (flay) with flay.
4.0
5.0
The maximum deceleration and the velocity for maximum
deceleration as given by equations (17) and (16) apply only
if the altitude y, given by equation (1.5), is positive. Other-
wise the maximum deceleration in flight occurs at sea level
with the velocity (sec eq. (13)) .
CopnA
V=Vo=VEe? 213rnsin 0
E
Declassified and Approved For Release 2013/11/21 : CIA-RDP71600265R000200130001-4
(20)
Declassified and Approved For Release 2013/11/21 : CIA-RDP71600265R000200130001-4
MOTION AND HEATING OF BALLISTIC MISSILES 5
and has the value Total heat input.?The time rate of convective heat trans-
fer front the air to any element of surface of the body may be
Copst expressed by the well-known relation
(V
?r _ dt Opp.AV .2 -0,,E (21)
MAZ g 0 ',mg e?,m,,
HEATING OF THE BODY
It was noted previously that for practicable rocket missiles,
it is vital that the weight of the missile be kept to a minimum.
The total heat transferred to a missile front the air must be
absorbed by some "coolant? material. Since this material
has a maximum allowable temperature, it follows .that it
can accept only a given amount of heat per unit weight.
Hence, the total heat input to the missile must be kept at a
minimum for minimum missile weight.
Often the coolant material is simply the shell of the missile
and as such must provide the structural strength and rigidity
for the missile as well. The strength of the structure is dic-
tated, in part, by the stresses induced by temperature gradi-
ents within the shell. Since these temperature gradients are
proportional to the time rate of heat input, the maximum
time rate of heat input is important in missile design. The
heating, of course, varies along the surface but, since the
shell transmits heat along as well as through itself, the
strength of the structure as a whole may be determined by
the maximum value of the average heat-transfer rate over
the surface. This is simply the maximum value of the time
rate of heat input per unit area. On the other hand, the
structural strength at local points on the surface may be de-
termined primarily by the local rate of heat input. Hence,
the maximum time rate of heat input per unit area at the sur-
face element where the heat transfer is greatest may also be
of importance in design.
If liquid cooling is employed, the maximum surface heat-
transfer rates retain their significance but, now, in the sense
that they dictate such requirements as maximum coolant
pumping rate, or perhaps shell porosity as well in the case of
transpiration cooling. Whichever the case, in the analysis
to follow, these elements of the heating problem will be
treated:
1. The total heat input.
2. The maximum time rate of average heat input per unit
area.
3. The maximum time rate of local heat input per unit area.
Since it is the primary function of this report to study
means available to the missile designer to minimize the heat-
ing problem, the analysis is simplified to facilitate compari-
son of the relative beating of one missile with respect to
another?accurate determination of the absolute heating of
individual missiles is not attempted. With this point in
mind, the following assumptions, discussed in Aupendix B,
are made:
1. Convective heat transfer predominates (i. e., radiation
effects are negligible):
2. Effects of gaseous imperfections may be neglected.
3. Shock-wave boundary-layer interaction may be ne-
glected.
4. Reynolds' analogy is applicable.
5. The Brenda number is unity.
where
1/ heat transferred per unit area, ft-lb/ft2
A
It convective heat-transfer coefficient,
f t2ft-lb sec?R
(22)
fr, recovery temperature, ?R
temperature of the wall, ?R
time, sec
and the subscript / denotes local conditions at any element of
the surface dS.
It is convenient in part of this analysis to determine the
heating as a function of altitude. To this end, noting that
?dy
dt? ?
V sin OE
we see that equation (22) may be written
dll 14(T,?T.),
(23)
dy V sin BE
With the assumption that the Prandtl number is unity, the
recovery temperature is
71,. 7',
2 ? 2
where
Al Mach number at the altitude y, dimensionless
'Y the ratio of specific heat at constant pressure to that
at thnstant volume, 0,10?, dimensionless
static temperature at the altitude y, ?R
so that
(T,-71.),= T?T?,-1-7+1 M2T
It is seen that for large values of the Mach number, which is
the case of principal interest, the third term is large com-
pared to reasonably allowable values of T? T.. It will
therefore be assumed that.; T? T. is negligible ? so that
(71,?T.) Al2T (24)
2
Moreover, since.
Af2T=-- Vt
?1)0?
equation (24) may be written
,
(Ty?T) V2 (25)
2C?
t should be noted that without this assumption, the heabinput determination would he
greatly complicated since the changing wall temperature with altitude would have to be con-
sidered to obtain the heat Input (sec ref. 1). For high-speed missiles which maintain
high speed during descent, the assumption is obviously permissible. Even for high-speed
missiles which finally decelerate to low speeds, the assumption is generally still adequate
since the total heat Input is largely determined by the heat transfer during the high-speed
Portion of flight.
Declassified and Approved For Release 2013/11/21 : CIA-RDP71600265R000200130001-4
Declassified and Approved For Release 2013/11/21 : CIA-RDP71600265R000200130001-4
6
REPORT 1381? NATIONAWADVISORY COMMITTEE FOR AERONAUTICS
Now the local heat-transfer coefficient h1 is, by Reynolds'
analogy, for the assumed Prandtl ntunber of unity
1
Cii17
2 fi -dr Pi I
(26)
where O, is the local skin-friction coefficient based on con-
ditions pp VI, etc., just outside the boundary layer. Thus,
since (T,? Tip) is essentially constant over the entire surface
S, the rate of total heat transfer with altitude becomes from
equations (23) through (26)
V '
dy a dy 40,, sin BEI C flC p IVdS
where Q is the heat transferred to the whole surface S. This
equation may be written
(IQ_ pV'S
dy ? 4 sin 0 E
wherein C,, is set equal to C? and
Sat() V
(27)
(28)
The parameter C/ is termed "the equivalent friction coeffi-
cient," and will be assumed constant,' independent of alti-
tude, again on the premise that relative rather than absolute
heating is of interest. With equations (2) and (13), then,
equation (27) is written
dQ CI' S p 17R2
o e_aue flmsin OE (29)
dy ?
4 sin BE
C Ep0A rov
Comparison of equation (29) with equation (14) shows that
the altitude rate of heat transfer is 'directly proportional to
the deceleration, so that
? dQIdy mg (0e'S
-RIVIdt\ 2 sin O,kCA
(30)
and therefore the maximum altitude rate of heat transfer
occurs at the altitude yi (see eq. (15)) and is given by
(d Q\ (d Q\ (3 te7R2 (Cd.' S
\AY Lax
(31)
it follows, of course, that the altitude rate of heat transfer
varies with incremental change in altitude from yi in the same
manner as deceleration, and thus (see eq. (19))
(dQ1dOtiv_F? )
134
(dQ1dy),
(32)
This assumption would appear poor at first glance since the Mach number and Reynolds
number variations are so large. Analysis has indicated, however, that the effects of Mach
number and Reynolds number variation are nearly compensating. The variation In C"/ for
typical conical missiles was found to be, at most, about 50 percent from the maximum CY In
the altitude range In which 80 percent of the heat is tranXihrred.
The total heat input to the body at impact follows from equa-
tion (29) (integrating over the limits 0 y< co) and is
cDp,A )
(d
C invE2 (1 e Saisin OE
The impact velocity, V, (the velocity of body at y=0), is
c?,,,A.
Vo= VE
(33)
so that equation (33) may be written in the alternative form
tn S\ /2-7 2_Ty
167 T op A) vE V? )
Maximum time rate of average heat input per unit area.?
To determine the time rate of average heat transfer per unit
area, equations (25), (26), and (28) with equation (22) may
be shown to give
dEl0v=1 (2, ' (35)
dt 4 f
(34)
which, together with equations (2) and (13), becomes at
altitude y
ac,,,?.4 .0
Cf p0VE3 e_e 249. sItsoe "
dt 4
(36)
The maximum time rate of average heat transfer per unit area
is found from this expression to be
(dH?) _CH.) 13 ( Cit )
dt mV1 sin OE
? ?dt ODA (37)
and it occurs at the altitude
1 30Dp0A
Ihr-fi n(2fim. sin OR)
(38)
where the velocity is
V2= V E (39)
As with altitude rate of heat transfer, it can be shown that
(dHavIdt)a,
?F" (Say) (40)
(dguldt) 2
Equations (37), (38), and (39) apply if the altitude for maxi-
mum time rate of average heat transfer per unit area occurs
above sea level. If y2, by equation (38), is negative, then
this rate occurs at sea level and is, from equation (36),
3CEpuA
Crioc\ d Hav\ cyf poi Ea 2fin I sin DE
RI)?a?x dt )0 4 e
(41)
Maximum time rate of local heat input per unit area.?
The elemental surface which is subject to the greatest heat
transfer per unit area is, except in unusual cases, the tip of
the missile nose which first meets the air. It seems unlikely
that a pointed nose will be of practical interest for high-speed
missiles since not only is the local heat-transfer rate ex-
ceedingly large in this case, but the capacity for heat reten-
tion is small. Thus a truly pointed nose would burn away.
Declassified and Approved For Release 2013/11/21 : CIA-RDP71600265R000200130001-4
Declassified and Approved For Release 2013/11/21 : CIA-RDP71600265R000200130001-4
MOTION AND HEATING OF BALLISTIC MISSILES
Body shapes of interest for high-speed missiles would more
probably, then, be those with nose shapes having nearly
hemispherical tips. The following analysis applies at such
tips.
It is well known that for any truly blunt body,, the bow
shock wave is detached and there exists a stagnation point
at the nose. Consider conditions at this point and assume
that the local radius of curvature of the body is a (see sketch).
Sow shock wove-.
Stoonotion
streo m line
ite nose
The bow shock wave is normal to the stagnation streamline
and converts the supersonic flow ahead of the shock to a low
subsonic speed flow at high static temperature downstream
of the shock. Thus, it is suggested that conditions near the
stagnation point may be investigated by treating the nose
section as if it were a. segment of a sphere in a. subsonic flow
field.
The heat-transfer rate per unit area. at the stagnation
point is given by the relation
dig Nu,k,(T w?T,)
dt a
where k, is the thermal conductivity of the gas. at the re-
covery temperature (i. e., total temperature) T? encl./Vitt, is
the Nusselt number of the flow. If the flow is assumed to
be laminar and incompressible,' Nu, is given, according to
reference 5, by the relationship
Nur=0.934 lle,?Prn
We retain the assumption that the Franckl number is unity,
note that Re.= PValp,, and substitute equation (25) into
equation (42) to obtain
did p
V'
dt 1 a
Now it is well known that at the high temperatures of
interest here, the coefficient of viscosity ?, varies nearly as
the square root of the absolute temperature and is given by
the relation
At =2.31 X 10-871,1/2
If this expression is combined with equation (25) (neglect-
ing Te), equation (43) may then be written
(43)
(1.11Vs
di' =6.8X10-6-111
a
(44)
The assumption of constant density certainly may Invalidate thisanalysis for any quantl.
tative study of the relatively neold.wall" flows of Interest here. For the purpose of studying
relative heat transfer it should, however, prove adequate.
? The constant In ciliation Oki Is obtained with the assumption of incompressible Pow In
the stagnation region. The effects of compressibility and dissociation orthe molecules of
air In the region tend to increase the value of the constant by as much as a factor 1 In the
speed range of Interest In this report. For the comparative purpose of this report It is un-
neceniery to take these effects Into amount.
7
which, when combined with equations (2) and (13), becomes
dH=6.8X10-6 ire? 1
dt a
3C0P.A e?fiu
2 24n2 sin BE
6 e
(45)
The maximum value of dig :WI can readily be shown to be
(d11,\
d t Lark dt)3=6.8 X10-8 Mm sin OR
(46)
3eCpaA "1"
which occurs at the altitude
y3:4 in 30DpA
Vol sin BE)
corresponding to the velocity
1/3-= a-20.8517n
(47)
(48)
The manner in which the heat-transfer rate per unit area at
the stagnation point varies with incremental change in alti-
tude from y, can be shown to be
(daidt),,, - 7
pay 4 (1e?
? Oav)
=
= " 094)
011,Id e 6
(49)
The dependence of F" (flay) on PAy is shown in figure 4.
Equation (46) applies only if y3 is above sea level. If y3,
from equation (47), is negative, then the maximum heat-
transfer rate per unit area at the stagnation point occurs at
sea level and is
3CopoA
OIL\
k. dt )?,w8?\ dt a
0-6.8 X 10-0 v?Ra e wisin oE
DISCUSSION
(50)
MOTION
The motion study shows some important features about
the high-speed descent of missiles through the atmosphere.
The major assumptions of this analysis were that the drag
coefficient was constant and the density varied exponentially
with altitude. It was found that the deceleration due to
drag was generally large compared to the acceleration of
gravity and, consequently, that the acceleration of gravity
could be neglected in the differential equations of motion.
The flight path was then seen to be a straight line, the missile
maintaining the flight-path angle it had at entry to the
atmosphere.
For most missiles, the maximum deceleration will occur
at altitude. One Of the most interesting features of the
flight of such a missile is that the maximum deceleration is
independent of physical characteristics (such as mass, size,
and drag coefficient of the missile), being dependent only on
the entry speed and flight-path angle (see eq. (17)). The
missile speed at maximum deceleration (eq: (16)) bears a
fixed relation to the entrance speed (61 percent of entrance
speed), while the corresponding altitude (eq. (15)) depends
on the physical characteristics and the flight-path angle but
not on the entrance speed. It, is also notable that for a
given incremental change in altitude from the altitude for
Declassified and Approved For Release 2013/11/21 : CIA-RDP71600265R000200130001-4
Declassified and Approved For Release 2013/11/21 : CIA-RDP71600265R000200130001-4
0 REPORT 1381?NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS
maximum deceleration, the deceleration and speed bear
fixed ratios to the maximum deceleration and the entry
speed, respectively (see fig. 4 and eqs. (19) and (18)), hence,
the deceleration and speed variation with altitude can readily
be determined.
If the missile is very heavy, the calculated altitude for
maximum deceleration (eq. (15)) may be fictitious (i. e.,
this altitude is negative) so the maximum deceleration in
flight; which occurs just before impact at sea level, is less
than that calculated by equation (17) and is dependent on
the body characteristics as well as the entry speed and flight-
path angle (see eq. (21)). However, the variation of speed
and deceleration with altitude from the fictitious altitude
given by equation (15) can still be obtained from figure 4.
HEATING
Total heat input.--In the heating analysis, a number of
simplifying assumptions were made which should limit its
applicability to the determination of relative values of heat-
ing at hypersonic speeds. It is in this relative sense that the
following discussion pertains.
In considering the total heat transferred by convection
to a missile, it is evident from equation (33) that the course
the designer should take to obtain the least heating is
affected by the value of the factor
CDp0A
?B 01)
/97n sin OE
To illustrate, first consider the case of a "relatively
heavy" missile for which this factor is small compared to
unity (the term "relatively heavy" is used to denote that
the denominator involving the mass is very large as compared
to the numerator involving the drag per unit dynamic
pressure, CA). Then
cap?,4
1?e pm sln OR
is small compared to 1. If this function is expanded in
series and only the leading term retained, equation (33)
becomes
CISPoVE2
Q=
40 sin OE
(52)
For the relatively heavy missile, then, the least heat will be
transferred when c,15 is a minimum?that is to say, when
the total shear force acting on the body is a minimum.
This result is as would be expected, if one notes that requir-
ing B1, or, in other
words, when this missile is "relatively light." In this event,
CDpOA
_e del sin OE,1
and equation (31) can be approximated
mV 2
4 E CDA
For the relatively light missile, then, the least convective
heating is obtained when C/S/CDA is a minimum. This is
at first glance a rather surprising result, for it indicates
that the heating is reduced by increasing the total drag,
provided the equivalent frictional drag is not increased
proportionately as fast. Physically, this anomaly is
resolved if the problem is viewed in the following way:
The missile entering the atmosphere has the kinetic energy
imVE2 but, if
copo4
' )
Omidneg_c# o
e 2
VE
is small, then nearly all its entrance kinetic energy is lost,
due to the action of aerodynamic forces, and must appear
as heating of both the atmosphere and the missile. The
fraction of the total heat which is given to the missile is,'?
from equation (33),
I (C/S\
\CDA)
Thus, by keeping this ratio a minimum, as much as possible
of the energy is given to the atmosphere and the missile
heating is therefore least.
In order to illustrate these considerations in greater detail,
calculations have been made using the previously developed.;
equations to determine the heat transferred by convection
to a series of conical missiles. Two classes of missiles have
been considered. Missiles in the first class were required
to have a base area of 10 square feet. Missiles in the
second class were required to have a volume of 16 cubic
feet. Gross weights of 0, 1,000, 5,000, 10,000, and 03
pounds have been assumed, and the entrance angle, Off,
has been taken as 30? of arc in all cases. Missile heating,
up to the tirna of impact, has been calculated as a function
of cone angle for entrance speeds of 10,000, 20,000, and 30,000
feet per second. In these calculations the pressure drag
coefficient was taken as constant for a particular cone at the
value corresponding to the entrance Mach' number (a value
of TE=500? R was assumed throughout). These coeffi-
cients were determined from reference 6 for cone angles of
10? and greater. For cone angles less than 100, reference 7
was employed to determine these coefficients (base drag
was neglected in all cases). The total drag coefficient was
taken as the sum of the pressure drag coefficient plus the
skin-friction coefficient, the latter coefficient being taken at
its value for maximum total heat-input rate with altitude.
The boundary layer was assumed to be wholly turbulent
since the Reynolds number, based on length of nin along
the surface of a cone and local conditions just outside the
boundary layer, was always greater than about 6 X10? and,
in fact, was of the order of billions for the more slender cones.
Turbulent-boundary-layer data were obtained from refer-
ences 8 and' 9, and Sutherland's law for the variation of
Viscosity with temperature was used in obtaining "equiv-
alent flat-plate" heat-transfer coefficients.
I' Note that even that If all drag Is frictional drag, only half the heat is transferred to
the body. The other half is contained In the boundary layer and is left In the air In the
body wake.
Declassified and Approved For Release 2013/11/21 : CIA-RDP71600265R000200130001-4
Declassified and Approved For Release 2013/11/21 : CIA-RDP71600265R000200130001-4
MOTION AND HEATING OF BALLISTIC MISSILES
----Missile denser than steel
40
30
20
Missile eight 0 lbs
Missile weigh?:co lbs
------
10 ?
?.,
40,000
Missit weight 0 lbs
20 40 60
Cone angle. degrees
(a) V= 10,000 ft/sec
(b) Fe 20,000 ft/see
(e) VE=30,000 ft/sec
Flamm 5.?Convective heat transferred at impact to conical missiles
of sante area entering the earth's Atmosphere at an angle of 300 to
the horizontal and velocities of 10,000, 20,000, and '30,000 ft/sec
(base area--10 sq ft).
(c)
80
100
17
6?
Missile weight r co lbs
---------
5
4
3
2
1
0
I 28
E. 24
a 20
16
C);
112
ti; 8
4
0
12
60
50
40
30
20
10
----Missile denser than steel
10,000
0 ------------
(a)
(b)
0
CC)
20 40 60
Cone angle, degrees
(a) 1',= 10,000 ft/sec
04 VE=20,000 ft/see
(c) VE=30,000 ft/sec
80
9
100
. FIGURE U.?Con vee live heat transferred at impact to conical missiles of
same volume entering the earth's atmosphere at an angle of 30? to
the horizontal and velocities of 10,000, 20,000, and 30,000 ft/sec
(voltnne=16.34 Cu ft).
Missile heating calculated in this manner for the fixed-base-
area and fixed-volume cones is presented in figures 5 and 6,
respectively. Curves for missiles having densities greater
than steel are considered improbable and are shown as
clashed lines. It is clear that for both classes of bodies, when
the missile is relatively heavy, the optimum solution is
obtained by making Of'S as small as possible (small Colic
angle case) and this optimum is accentuated with increase in
speed. On the other hand, when the miSsile is relatively
light, reduced heating is obtained by making Cr/5MA as
small as possible (the large cone angle case). It is noted
Declassified and Approved For Release 2013/11/21 : CIA-RDP71600265R000200130001-4
Declassified and Approved For Release 2013/11/21 : CIA-RDP71600265R000200130001-4
10
REPORT 138I--NATIONAL 'ADVISORY COMMITTEE FOR AERONAUTICS
also that, in general, the advantage of reduced heating of the
relatively light, blunt cones is more pronounced in the fixed-
base-area case than in the fixed-volume case.
Maximum time rate of average heat input per unit area.?
It was previously noted that the maximum time rate of
average heat input per unit area may be of serious importance
in determining the structural integrity of missiles entering
the atmosphere at high speeds." In order to illustrate this
fact, consider the case of a missile having a shell made of
solid material and assume that the rate of heat transfer per
unit area does not vary rapidly from one surface element to
the next. Then the rate of transfer of heat along the shell
will be small compared with the rate of transfer through the
shell. The shell stress due to heat transfer is that resulting
from the tendency toward differential expansion through the
shell and it is proportional to dT?idn where T0 is the tempera-
ture at any point n Within the shell and is measured per-
pendicular from the shell surface. We define lt,, as the
thermal conductivity of the shell material; then the rate at
which heat transfers through the shell per unit area is lc,,(d11,1
dn) and this must, at nr-0, equal the rate of heat input per
unit surface area. For the missile considered as a whole,
the maximum value of the average thermal stress in the shell
is a measure of the over-all structural integrity and the
maximum value of this stress will occur at the surface when
dH??_.1 (dQ\
di
is a Maximum.
The course the designer should take to minimize the ther-
mal stress for the missile as a whole is dependent, as for the
case of total heat input, upon whether the missile is relatively
heavy or light. For the relatively heavy missile the value
of B, given by equation (51), is small compared to unity.
The maximum value of the average thermal stress in this
case is proportional to (see eq. (41))
idll?A poV
dt 4
and, hence, the least average thermal stress is obtained by
making cy a minimum. 011 the other hand, for the rela-
tively light missile the maximum value of the average.
thermal stress is proportional to (see eq. (37))
(dHaA _( flmVna sin OR
dt VIA) 6e
and, hence, the least average thermal stress occurs when
Of'/CDA is a minimum.
In order to illustrate these considerations in greater detail,
the maximum values of the time rate of average heat input per
unit area have been calculated for the constant-base-area and
the constant-volume cones previously discussed in the section
on total heat input. These values were. determined in much'
the same manner as those of total heat input, with the excep-
tion that Of' Was evaluated at y2 (rather than yi), given by
equation (38) when it applies, and otherwise at yo=0. The
results are shown in figures 7 and 8. It is seen that the
(54)
(55)
II This Is the common case when the shell inaterial itcts as structural support and must
also transport or absca b the heat.
8
6
4
----Missile denser than steel
/Missile eight co lbs
5,000
0
0
X
U
.0 0)
7 NV) 80
40
e, 20
e
w 0
cr.
2
cc
300
200
1,000
100
/Missile weight co lbs
0,000
5 000
1,000
Missile weight .0 lbs
(b)
/Missile weighi co lbs
Mi site weight lbs
1,000
(c)
20 40 60 100
Cone angle, degrees
(10 1'= 10,000 fth;ce
(b) VE= 20,000 ft/see
(c) V= 30,000 ft/sec
FIGURE 7.?Maximain average rate Of convective .heat transfer to
conical missiles of the same base area entering the earth's atmosphere.
at an angle of 30? to the horizontal and velocities of 10,000, 20,000,
and 30,000 ft/sec (base area= 10 sq ft).
80
maximum values of average thermal stress are reduced for
both the slender cones and blunt cones as compared to the
relatively large values of this stress experienced by cones
of intermediate slenderness.
Maximum time rate of local heat input per unit area.?
Perhaps even more important than the maximum value of
the average shell stress is the maximum stress that occurs
in the shell at the surface element of ? the missile nose,l2
where the local heatAransfer rate is probably the greatest,
for, in general, this latter stress is many times larger.
fact, this rate of loeal heat input can be so large as to promote,
i'In this report we are concerned only with bodies [(wings or stabilizers are used, their
leading edges are similarly surface elements which experience Intense beat transfer.
Declassified and Approved For Release 2013/11/21 : CIA-RDP71600265R000200130001-4
Declassified and Approved For Release 2013/11/21 : CIA-RDP71600265R000200130001-4
MOTION AND HEATING OF BALLISTIC MISSILES
----Missile denser than steel
0
n
T?
Missile weight= rd lbs
80
60 /// ?
?
S.
?
?\12P00
40
?
S.
?
r?-?
5,000
a,
fl
cx
300
i Missile weight :0 lbS
/
/,....??
/i \
200 // ?
??
/ ?
N
? 10' 000
S.
Missile weightrOlbs
3/4 0 20 40 60 80 100
Cone ongle, degrees
(a) V g=10,000 ft/sec
(b) VE= 20,000 ft/sec
(a) 17E=302000 ft/sec
FIGURE 8.?Maximum average rate of convective heat transfer to
conical missiles of the same volume entering the earth's atmosphere
at an angle of 300 to the horizontal and velocities of 10,000, 20,000,
and 30,000 ft/see (volume=16.34 Cu ft). ?
(c)
temperature gradients through the shell that are intolerable
even with the most highly conductive materials (copper,
silver, etc.).'3 Thus some additional means of cooling, such
as transpiration cooling, may, in any case, be vequired in
'this region.
It was stated previously that pointed-nose bodies are un-
desirable due, in part, to the fact that the local heat-transfer
rate per unit area at the tip is excessive. The validity of this
statement is demonstrated by the results of the analysis.
It is clear (see eq. (44)) that since the local transfer rate varies
inversely with the square root of the tip radius, not only
should pointed bodies be avoided, but the rounded nose
LI See reference 1 for further discussion.
11
should have as large a radius as possible. The question
then arises; if the nose radius is arbitrarily fixed, what course
is available to the missile designer to minimize the problem of
local heating at the stagnation point? From both equations
(46) and (50), it is seen that for an arbitrary nose radius, if
the mass, entry speed, and flight-path angle are fixed, then
the only way to reduce the stagnation rate of heat input
per unit area is to increase the product CDA. In fact, a
relative stagnation-point heat-transfer rate per unit area, 4'
may be expressed in terms of B (see eq. (51)), if it is defined
as the ratio of the maximum stagnation-point heat-transfer
rate per unit area for a given missile to the maximtun rate
the same missile would experience if it were infinitely heavy.
For the infinitely heavy missile, the maximum rate occurs
at sea level and is (see eq. (50))
so that from equation (50)
acop?A
4,=e 20m5the,??e_
(56)
if the given missile also attains its maximum rate at sea level
(i. e., y3=0; eq. (47)); whereas
p3m sin E
3e0DPA 113e13
(57)
if the given missile attains its maximum rate above sea level
(eq. (46), y3 positive). The variation of 4, with 1/B is shown
in figure 9. Clearly, the high pressure drag shape has the
advantage over the slender shape in this respect.
In order to illustrate these considerations in greater detail,
again consider the constant-base-area and constant-volume
cones discussed earlier. Assume the pointed tips of all the
1.0
.8
0
6?6
0
Cc .2
3
6 9 2 15
'
A
Flount; 9.?Variation of relative heat-transfer factor with
Declassified and Approved For Release 2013/11/21 : CIA-RDP71600265R000200130001-4
Declassified and Approved For Release 2013/11/21 : CIA-RDP71600265R000200130001-4
12 REPORT 1381?NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS
cones are replaced by spherical tips of the same radius a.
The relative effect of varying the cone angle on the stagna-
tion-point heating can than be assessed by determining the
variation of the product
\. di )tha
This product has been calculated for the various cones, as-
suming CD to be unaffected by the addition of the hemi-
spherical tip (the tip radius may be arbitrarily small), and
the results are shown in figures 10 and 11. It is seen again
that the missiles having large cone angle (high drag co-
efficient) are considerably superior.
Missile denser than steel
Missile weight, lbs,
4
10,000
,-5,000
C.3
.0
T
?
40
Stagnation point heot-transfer rote,Iii
0
20
0
(o)
Missile weight, lbs
cco
r- 5,000
' 000,
0
(b)
Missile weight, lbs
riot)
-------------
20 40 60
Cone angle, degrees
(a) VR=10,000 ft/sec
(6) VE= 20,000 ft/sec
(c) Ve=30,000 ft/sec
FIGURE 10.?Maximum rate of convective heat transfer to the stagna-
tion point of spherically tinned cones of the same base acea entering
the earth's atmosphere at an angle of 300 to the horizontal and
velocities of 10,000, 20,000, and 30,000 ft/sec (base area=10 sq ft).
0
n co
4--
----Missile denser than steel
Missile weight, lbsi
" 60
i?1Cfi 40
120
2 0
0
CI? 120
80
40
Missile weight, lbs
(b)
Missile weight, lbs
/10,000
/5,000
0 20 40 60 80 100
(e) Cone angle, degrees
(a) T= 10,000 ft/sec
(b) Ve=20,000 ft/sec
(c) V=30,000 ft/sec
FIGURE IL?Maximum rate of convective heat transfer to the stagna-
tion point of spherically tipped cones of the same volume entering
the earth's atmosphere at an angle of 30? to the horizontal and
velocities of 10,000, 20,000, and 30,000 ft/sec (volume= 16.34 cu ft).
DESIGN CONSIDERATIONS AND CONCLUDING REMARKS
In the foregoing analysis and discussion, two aspects of the
beating problem for missiles entering the atmosphere were
treated. The first, concerned the total heat absorbed by the
missile and was related to the coolant required to prevent
its disintegration. It was found that if a missile were
relatively light; the least required weight of coolant (and
hence of missile) is obtained with a shape having a high
pressure drag coefficient, that is to say, a blunt shape. On
the other hand, it was found that if the missile were relatively
heavy the least required weight of coolant, and hence of
missile, is obtained with a shape having a low skin-friction
drag coefficient, that is to say, a long slender shape.
The second aspect of the heating problem treated was
concerned with the rate .of heat input, particularly with
regard to thermal shell stresses resulting therefrom. It ?was
seen that the maximum average heat-input rate and, hence,
maximum average thermal stress could be decreased by
Declassified and Approved For Release 2013/11/21 : CIA-RDP71600265R000200130001-4
Declassified and Approved For Release 2013/11/21 : CIA-RDP71600265R000200130001-4
MOTION AND HEATING OF BALLISTIC MISSILES 13
using either a blunt or a slender missile, while missiles of
intermediate slenderness were definitely to be avoided in
this connection. The region of highest local heat-transfer
rate and, hence, probably greatest thermal stress was
reasoned to be located at the forward tip of the missile in
most cases. This was assumed to be the case and it was
found that the magnitude of this stress Was reduced by
employing a shape having the largest permissible tip radius
and over-all drag coefficient; that is to say, the blunt, high
drag shape always appears to have the advantage in this
respect.
These results provide us with rather crude, but useful,
bases for determining shapes of missiles entering the atmos-
phere which have minimized heat-transfer problems. If
the over-all design considerations of payload, booster, et al,
dictate that the re-entry missile be relatively heavy in the
sense of this report, then it may be most desirable to make
this missile long and slender, especially if the entry speed is
very high (say 20,000 ft/sec or greater). Perhaps the slender
conical shape is appropriate for such a missile. It seems
clear, too, that the tip of this missile should be given the
largest practicable nose radius in order to minimize the
maximum local heat-transfer rate and hence maximum local
alien stress problem. Even then it may be necessary to
employ additional means to minimize the heat-transfer rate
and, hence, thermal stress encountered in this region (e. g.,
by transpiration cooling).
Let us now consider the case where the over-all design
conditions dictate that the re-entry missile be relatively
light, in the sense of this report. This case will be the more
usual one and, therefore, will be treated at greater length.
A shape which should warrant attention for such missile
application is the sphere, for it has the following advantages:
1. It is a high drag shape and the frictional drag is only
a few percent of the total drag.
2. It has the maximum volume for a given surface area.
3. The continuously curved surface is inherently stiff
and strong.
4. The large stagnation-point radius significantly assists
in reducing the maximum thermal stress in the
shell.
5. Aerodynamic forces are not sensitive to attitude and,
hence, a sphere may need no stabilizing surfaces.
6. Because of this insensitivity to attitude, a sphere
may purposely be rotated slowly, and perhaps even
randomly" during flight, in order to subject all
surface elements to about the same amount of
heating and thereby approach uniform shell
heating.
Note that If rotation is permitted, slow, random motion may be required in order to
prevent Magnus fonts from causing deviation of the flight path from the target. It should
also be noted that at subsonic and low supersonic speeds gun-fired spheres, presumably not
rotating, have shown rather large lateral motions In flight (see ref. 10). It Is not known
whether such behavior occurs at high supersonic speeds.
On the other hand, the sphere, in common with other very
high drag shapes may be unacceptable if:
1. The low terminal speed cannot be permitted (e. g.,
because of excessive wind drift).
2. The magnitude of the maximum deceleration is
greater than can be allowed.
The first of these disadvantages of the sphere might be
minimized by protruding a flow-separation-inducing spike
from the front of the sphere to reduce the drag coefficient
to roughly half (see ref. 11). Stabilization would now be
required but only to the extent required to counterbalance
the moment produced by the spike. Special provision would
have to be made for cooling the spike.
Both of the disadvantages of very high drag shapes may
however be alleviated by using variable geometry arrange- ?
ments. For example, an arrangement, which suggests itself
is a round-nosed shape with conical afierbody of low apex
angle employing an extensible skirt at the base. With the
skirt flared, the advantages of high drag are obtained during
the entry phase of flight where the aerodynamic heating is
intense. Late, the skirt flare may be decreased to vary the
drag so as to produce the desired deceleration and speed
history. If the deceleration is specified in the equation of
motion (see motion analysis), the required variation of drag
coefficient with altitude can be calculated.
The examples considered, of course, are included only to
demonstrate some of the means the designer has at hand to
control and diminish the aerodynamic heating problem.
For simplicity, this problem has been treated, for the most
part, in a relative rather than absolute fashion. In any
final design, there is, clearly, no substitute for step-by-step
or other more accurate calculation of both the motion and
aerodynamic heating of a. missile.
Even from a qualitative point of view, a. further word of
caution must be given concerning the analysis of this paper.
In particular, throughout, we have neglected effects of
gaseous imperfections (such as dissociation) and shock-wave
boundary-layer interaction on convective heat transfer to
a missile, and of radiative heat transfer to or from the
missile. One would not anticipate that these phenomena
would significantly alter the conclusions reached on the
relative merits of slender and blunt shapes from the stand-
point of heat transfer at entrance speeds at least up to about
10,000 feet per second. 11 cannot tacitly be assumed,
however, that this will be the case at higher entrance speeds
(see Appendix B). Accurate conclusions regarding the
dependence of heat transfer on shape for missiles entering
the atmosphere at extremely high supersonic speeds must
await the availability of more reliable data on the static and
dynamic properties of air at the high temperatures and
pressures that will be encountered.
AMES AERONAUTICAL LABORATORY
NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS
MOFFETT FIELD, CALIF., Apr. 28, 1953
Declassified and Approved For Release 2013/11/21 : CIA-RDP71600265R000200130001-4
Declassified and Approved For Release 2013/11/21 : CIA-RDP71600265R000200130001-4
A
13
APPENDIX A
SYMBOLS
reference area for drag evaluation, ft'
body factor, dimensionless
(See eq. (51).)
drag coefficient, dimensionless
skin-friction coefficient based on conditions just
outside the boundary layer, dimensionless
equivalent skin-friction coefficient, dimensionless
(See eq. (28).)
ft-lb
specific heat at constant pressure' slug ?R
ft-lb
Co specific heat at constant volume,
slug ?R
F' ,F' functions of 13Ay, dimensionless
(See eqs. (18), (19), and (49).)
acceleration due to force of gravity
(taken as 32.2
sec'
11
in
Nit
Pr
Re
14
ft-lb
convective heat-transfer coefficient,
ft' sec ?R
ft-lb
heat transferred per unit area,
ft'
ft-lb
thermal conductivity,
sec ft' (?R/ft)
mass, slugs
Mach number, dimensionless
Nusselt number, dimensionless
Prandtl number, dimensionless
total heat transferred, ft-lb
Reynolds number, dimensionless
surface area, ft'
temperature (ambient temperature of air at
altitude y unless otherwise specified), ?R
time, sec
ft
velocity, ?
sec
X,y
A
0
a
horizontal and vertical distance from impact
point, ft
ft'
variable of integration,
sec'
constant in density?altitude relation, ft-'
(See eq. (2).)
ratio of specific heat at constant pressure to
specific heat at constant volume, C,10,
dimensionless
increment
distance within the shell measured normal to
shell surface, ft
angle of flight path with respect to horizontal,
deg
slugs
coefficient of absolute viscosity,
ft sec
slug
air density,
ft'
radius, ft
relative heat-transfer factor, dimensionless
(See eqs. (56) and (57).)
SUBSCRIPTS
0 conditions at sea level (y=0)
1 conditions at altitude yi (eq. (15))
2 conditions at altitude y2 (eq. (38))
3 conditions at altitude y2 (eq. (47))
conditions at entrance to earth's atmosphere
local conditions
recovery conditions
stagnation conditions
wall conditions
conditions within the shell of the missile
Declassified and Approved For Release 2013/11/21 : CIA-RDP71600265R000200130001-4
Declassified and Approved For Release 2013/11/21 : CIA-RDP71600265R000200130001-4
APPENDIX B
SIMPLIFYING ASSUMPTIONS IN THE CALCULATION. OF AERODYNAMIC HEATING
As noted in the main body of the report, the heating
analysis is simplified by making the following assumptions:
I. Convective heat transfer is of foremost importance;
that is, radiative effects may be neglected.
2. Effects of gaseous imperfections, in particular dis-
sociation, may be neglected.
3. Effects of shock-wave boundary-layer interaction
may be neglected.
4. Reynolds' analogy is applicable.
5. Prandt1 number is unity.
The restrictions imposed by these assumptions will DOW be
considered in some detail.
In assumption 1, two simplifications are involved; namely,
(1) radiation from the surface of the body is neglected, and
(2) radiation to the body from the high-temperature dis-
turbed air between the shock wave and the surface is neg-
lected. The first simplification may be justified on the
premise that the maximum allowable surface temperature
will be about the same for one body as compared with
another, irrespective of shape, and, consequently, radiation
away from the surface will be approximately the same.
Hence, neglecting this form of heat transfer should not
appreciably change the relative beating which is of principal
interest in this paper.
The second simplification of ignoring radiative heat trans-
fer from the disturbed air to the body is not so easily treated.
At ordinary flight speeds this form of heat transfer is neg-
ligible since it is well established that at temperatures not
too different from 'ambient temperature, air is both a poor
radiator and a poor absorber. At the flight speeds of
interest, temperatures in the tens of thousands of degrees
Fahrenheit may be easily obtained in the disburbed air flow,
especially about the heavier blunt. bodies. At these temper-
atures it does not, follow, a priori, that air is a poor radiator.
Data on the properties of air at these temperatures are
indeed meager. Hence, it is clear that calculations of
radiative heat transfer from air under these conditions must,
at best, be qualitative. Nevertheless, several such calcula-
tions have been made, assuming for lack of better informa-
tion that air behaves as a grey body radiator and that
\Vein's law may be used to relate the wave length at which
the maximum amount of radiation is emitted to the temper-
ature of the air (this assumption, in effect, enables low-
temperature data on the emissivity of air to be used in
calculating radiation at high temperatures). In these
calculations effects of dissociation in reducing the temper-
ature of the disturbed air have also been neglected and
hence from this standpoint, at. least, conservative (i. e., too
high) estimates of radiative heat, transfer should evolve.
The results of these calculations indicate the following: (I)
Radiative heat transfer from the disturbed air to the body
is of negligible importance compared to convective heat,
transfer at entrance speeds in the neighborhood of, or less
than, 10,000 feet per second; (2) Radiative heat transfer, in
the case of relatively massive blunt bodies, may have to
be considered in beat-transfer calculations at entrance speeds
in the neighborhood of 20,000 feet per second; (3) Radiative
heat transfer, in the case of relatively massive blunt bodies,
may be of comparable importance to convective heat transfer
at entrance speeds in the neighborhood of 30,000 feet per
second. From these results, we conclude, then, that the
neglect of radiative heat transfer from the disturbed air
to the body is probably permissible for all except, perhaps,
very blunt and heavy shapes at entrance speeds up to 20,000
feet per second. However, this simplification may not be
permissible, especially in the case of heavy blunt bodies
entering the atmosphere at speeds in the neighborhood of, or
greater than, 30,000 feet per second.
In assumption 2, the neglect of effects of gaseous imper-
fections, particularly dissociation, on convective heat transfer
would appear to be permissible at entrance speeds up to
and in the neighborhood of 10,000 feet per second, since at
such speeds the temperatures of the disturbed air are not
high enough for these imperfections to become significantly
manifest. On the other band, as the entrance speeds ap-
proach 20,000 feet per second, temperatures of the disturbed
air may easily exceed 10,0000 Rankine, in which case appre-
ciable dissociation may be anticipated, inside the boundary
layer for all bodies, and inside and outside the boundary
layer in the case of blunt bodies. The magnitude of these
effects is at present in some doubt (see, e. g., the results
of refs. 12 and 13.) lienee, for the present, the neglect of
effects of gaseous imperfections on convective heat transfer
is not demonstrably permissible at entrance speeds in the
neighborhood of 20,000 feet per second or greater.
In assumption 3, it has been shown by Lees and Probstein
(ref. 14), and more recently by IA and Nagamatsu (ref. 15),
that shock-wave boundary-layer interaction may signifi-
cantly increase laminar skin-friction coefficients on a flat
plate at zero incidence and Mach numbers in excess of about
10. Lees and Probstein found somewhat the opposite effect
on heat-transfer rate in the case of weak interaction. It is
not now known how this phenomenon depends upon body
shape or type of boundary layer. However, it is reasonable
to anticipate that there will be some effect, and certainly if
the skin-friction coefficient is increased in order of magnitude
at Mach numbers approaching 20, as indicated by the
results of Li and Nagamatsu for strong interaction, then
the phenomenon cannot be presumed negligible. Hence,
we conclude that, from this standpoint, also, the convective
15
Declassified and Approved For Release 2013/11/21 : CIA-RDP71600265R000200130001-4
Declassified and Approved For Release 2013/11/21 : CIA-RDP71600265R000200130001-4
16 REPORT 1381?NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS
heat-transfer &imitations of this report may be in error at
entrance speeds of the order of 20,000 feet per second or
greater.
The assumption that Reynolds' analogy may be used to
relate skin-friction and heat-transfer coefficient does not,
especially in the light of recent work by Rubesin (ref. 16),
seem out of line with the purposes of this paper, at least at
entrance speeds up to and in the neighborhood of 10,000
feet per second. However, it does not follow, a priori, that
this assumption remains valid at substantially higher
entrance speeds, especially in view of the imperfect gas and
shock-wave boundary-layer-interaction effects already dis-
cussed.
The assumption of Prandtl number equal to unity would
also appear permissible for the analysis of relative heating
of missiles at the lower entrance speeds considered here.
However, in view' of the questionable effect (see again refs.
12 and 13) of dissociation on Prandtl number, it is not clear
that this assumption is strictly valid at the intermediate
and higher entrance speeds treated in this report.
From. these considerations it is concluded that the sim.pli-
lying assumptions made in the main heat-transfer analysis
of this paper will not significantly influence the results at
entre:nee speeds in the neighborhood of or less than 10,000
feet per second. However, at entrance speeds in the
neighborhood of and greater than 20,000 feet per second,
these results must be viewed with skepticism. More accu-
rate calculations of heat transfer at these speeds must,
among Other things, await more accurate determinations of
both the static and dynamic properties of air under these
circumstances.
REFERENCES
1. Wagner, Carl: Skin Temperature of Missiles Entering The Atmos-
phere at Hypersonic Speed. Tech. Rep. No. 60, Ord Res. & Dev.
Division; Dept. of Army, Oct. 1949.
2. Diehl, Walter S.: Standard
NACA Rep. No. 218, 1925.
3. Warfield, Calvin N.: Tentative Tables for the Properties of the
Upper Atmosphere. NACA TN 1200, 1947.
4. Grimminger, G.: Probability that a Meteorite Will Hit or Penetrate
a Body Situated in the Vieinity of the Earth. Jour. Appl, Phys.,
vol. 19, no. 10, Oct. 1948, pp. 947-956.
5. Sibulkin, M.: Heat Transfer Near the Forward Stagnation Point of
a Body of Revolution. Jour. Aero. Sci., vol. 19, no. 8, Aug. 1952,
pp. 570-571.
6. Mass. Inst. of Tech. Dept. Df Elec. Engr., Center of Analysis:
Tables of Supersonic Flow Around Cones, by the Staff of the
Computing Section, Center of Analysis, under direction of
Zdenek Kopal: Tech. Rep. No. I, Cambridge, 1947.
7. Eggers, A. J. Jr., and Savin, Raymond C.: Approximate Methods
for Calculating the Flow About Nonlifting Bodies of Revolution
at High Supersonic Airspeeds. NACA TN 2579, 1951.
8. Van Driest, E. R.: Turbulent Boundary Layer in Compressible
Fluids. Jour. Aero. Sci., vol. 18, no. 3, Mar. 1951, pp. 145-146.
9. Van Driest, E. R.: Turbulent Boundary Layer on a Cone in a Super-
sonic Flow at Zero Angle of Attack. Jour. Aero. Sci., vol. 19,
no. 1, Jan. 1952, pp. 55-57.
10. Richards, Elizabeth: Comparative Dispersion and Drag of Spheres
and Right Cylinders. Aberdeen Proving Ground, Md., Ballistic
Research Laboratories, Rep. 717, 1950.
11. Moeckel, W. E.: Flow Separation Ahead of a Blunt Axially Sym-
metric Body at Mach Numbers 1.76 ,to 2.10. NACA RM
E51125, 1951.
12. Moore, L. L.: A solution of the Laminar Boundary-Layer Equa-
tions for a Compressible Fluid with Variable Properties, Includ-
ing Dissociation. Jour. Aero. Sci., vol. 19, no. 8, Aug. 1952, pp.
505-518.
13. Crown, J. Conrad: The Laminar Boundary Layer at Hypersonic
Speeds. NAVORD Rep. 2299, Apr. 15, 1952.
[4. Lees, Lester, and Probstein, Ronald F.: Hypersonic Viscous Flow
Over a Flat Plate. Princeton University Aeronautical Engineer-
ing Laboratory Rep. 195, Apr. 20, 1952.
15. Li, Ting-Yi, and Nagamatsu, Henry T.: Shock Wave Effects on
the Laminar Skin Friction of an Insulated Flat Plate at Hyper-
sonic Speeds. GALCIT Memorandum No. 9, July 1, 1952.
16. Rubesin, Morris W.: A Modified Reynolds Analogy for the Com-
pressible Turbulent Boundary Layer on a Flat Plate. NACA
TN 2917, 1953.
Atmosphere?Tables and Data.
? U.S GOVERNMENT PRINTING OFFICE 1061 0-614060
Declassified and Approved For Release 2013/11/21 : CIA-RDP71600265R000200130001-4