Afterbody pressure distributions and wave drag were calculated using a second-order theory for a variety of conical boattails at zero angle of attack. Results are presented for Mach numbers from 1.5 to 4.5, area ratios from 0.200 to 0.800, and boattail angle from 3 degrees to 11 degrees. The results indicate that for a given boattail angle, the wave drag decreases with increasing Mach number and area ratio. The wave drag, for a constant area ratio, increases with increasing boattail angle. For a specific Mach number, area ratio, and fineness ratio, a comparison of the wave-drag coefficients for conical, tangent-parabolic, and secant-parabolic boattails showed the conical boattail to have the smallest wave drag

Jack, John R

English

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TECHNICAL NOTE 2972
THEORETICAL PRESSURE DISTRIBUTIONS AND WAVE DMGS
FOR CONICAL BOATTAILS
By John R. Jack
Lewis Flight Propulsion Laboratory
Cleveland, Ohio
Washington
July 1953
.. . . . .. . .. . . . . . ,- . . ..- .- ..-. . ... .... . . . .. . .. .- --
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TKXKKML NOTE 2972
TEIEOREIIICALPRESSURE DISZKIBUITONS AND MNll IWWS
FoR CONICALBomTAm3
By John R. Jack
Afterbody pressure distributions and wave drags were calculated
using a second-order theory for a variety of conical boattails at zero
sugle of attack. Results are presented for Mch numbers from 1.5 to
4.5, area ratios from 0.200 to 0.800, and boattail angles from 3° to II”.
The results indicate that for a given boattail angle, the wave drag
decreases with increasing Mach nuriber~d area ratio. The wave drag,
for a constant srea ratio, increases with increasing boattail angle.
For a specific l&ch number, area ratio, and fineness ratio, a comparison
of the wave-drag coefficients for conical, tangent-parabolic, and secant-
parabolic boatta~ showed the conical boattail to have the smallest
wave drag.
INIROIXXTION
One of the major components of a missile configurateion is the
afterbody section known as the boattail. There are, however, little
data available to serve either as a guide in the design of boattails
for supersonicbodies or as a basis for esthating the aerodynamic loads
and wave drags associated with boattails. It has generally been assumed
that boundary @er effects render the potential flow computations
toward the rear of the missile meaningless. However, avafible qyzri-
mental data indicate that potential theory does predict the boattail
chsxacteristics adequately for most design purposes. An investigation
was therefore undertaken at the IIMR Lewis Laboratory to study systemat-
ically the variations of pressure distributions and wave drags of ~
conical.boattaila with Mach number, area ratio, and boattail angle .
.
1)
‘After completion of the work presented h this report, a report of sim-
ibr content came to the attention of the author (ref. 1). However,
since reference 1 does not present say pressure dlstributions, which
are quite valuable for structural desiw and for esttit ing base pres-
sues, ptilication of the present report was considered warranted.
— .— ---
.2
Rressure distributions and wave-drag
numbers fran 1.5 to 4.5, area ratios
angles from 3° to U_”.
NACA TN 2972
b
coefficients are-presented for &h
from 0.200 to 0.800, and boattail
MEI!HODOF COMWIMTION
Pressure distributions for each boattafi were calculated using the
second-ofier theory developed in reference 2. It was assumed that the
boattails are preceded by a cylindrical.section of sufficient length to
give uniform flow at the free-stresm lhch number at the beginning of
each boattail. The calculating procedure followed was that presented
ti reference 3, in which the approximate boundary condition at the sur-
face of the body is used to obtain the perturbation velocities, and the
exact isentropic pressure relation is used for evaluating the pressure
coefficient at each point on the bdy. ti each case the solution was
carried downstream to the petit at whSch the radius of the I&ch cone from
the beginnhg of the boattail has grown to ten times the local radius
of the bcattail. As indicated in reference 2, the second-order solution
could be carried beyond this petit by extending the tables used in the
computations;however, the area ratios of practical interest correspond
to boattai.1.lengths which, in general, are within this Mmitat ion. !l?he
procedure presented in reference 3 proves to be an expedient means of
obtatiing both a first- and a second-order solution. The average time
for calculating the combined first- and second-order solutions was
apprccdmately 11 cmput er hours.
.
Wave-tig coefficients for each boattail were obtained by graph-
ically integrating the pressure distributio~ over the boattail surfaces.
The wave-drag coefficient was based on the maximum area of the boattail
and is deftied by
where Cp is the exact isentropic pressure coefficient, r is the local
boattail radius, and ~ and R are the maxhum and minimum boattail
radii, respectively.
I?XSQECSAND DEXUSSION
Pressure distributions and wave drags. - The variations of pressure
coefficient in the axial direction for the conical boattaiM me pre-
sented in figure 1 for selected values of boattail angle and free-stresm
—
..
.
c
NACA TN 2972
&ch nmiber. In each case, the pressuxe
the Prandtl-Meyer value at the beginning
coefficient has
of the boattail
3
a~roxhmt el.y
because of the
~ans ive turning through the boattail mgl.e, which is fold.owedby a
co~tinuous compression along the boattail surf~.ce. The pressure
coefficient level increases with increasing Mach nuniberfor a given
boattail angle and, for a given I&h nuniber,decreases as the boattail
angle increases.
The dependence of the pressure distributions upon fineness ratio
and area ratio msy be found by correlating boattafl angle with fineness
ratio and area ratio. The variation of fineness ratio and area ratio
with boattail angle is given in figure 2.
Graphical integration of the pressure distributions presented in
figure 1 over the boattail surfaces yields the variation of wave-drag
coefficient with area ratio and with boattail.angle (figs. 3 and 4,
respectively). The dashed lines presented in figme 3 represent the
limit@g area ratio to which the theory of reference 3 can be applied
without extending the tables used in the computations. The wave dxag
coefficientspresented in figure 4 were obtained by extrapolating some
of the data of figures 3(a) and 3(b). For a given lkch number, the
coefficient of wave drag increases with ticreasingboattail angle and
decreases with increasing area ratio.
To extend the I&ch number ~nge investigated, the wave-drag coef-
ficient has been plotted as a function of the reciprocal of the I&ch
nuiber (fig. 5). The results of the present calculations were extra-
pcilded to 1~~ = O by using the concepts of Newtonian flow theory
which predict a zero drag coefficient for all boattail angles as
% +- (ref. 4). For a specific boattail angle, the coefficient of
wave drag decreases as the Mach nuniberis ticreased.
Body contour effect. - To obtain the effect of body contour on
pressure distributions and wave drags, a tangent- and a secant-parabolic
boattail contour each having the same len@h and area ratio have been
investi~ted and the results compared with those obtatied from the cor-
respon~g inscribed conical boattail. Design parameters for these
boattails were arbitrarily chosen and are:
l%chnmber, ~........ . . . . . . . . . . . . . . ..O .2.5
Ftieness ratio, L/~...... . . . . . . . . . . . . . ..”2”255
Arearatio, A/~ . . . ...08. ....~~””””””o”o-2oo
Secant-parabolicboattail leading-edge angle, deg . . . . . . . . . 3.5
Conical boattail angle, e,deg . . . . . . . . . . . . . . . . . ..7
The contours considered for the comparison are slike inasmuch as they
me members of the psrabolic fsmily, with each having a clifferent expan-
sion sz@.e at the leading edge of the boattail. Defining equations for
these boattail contours axe as follows:
_.— .— . .-
—.
4Tangent-parabolic:
NACA TN 2972
2
ii=l ()- 0.02719 +%
Secant-parabolic:
2
~=1
()
- 0.06116 = - 0.01364 ~
% m
conical:
t=l - 0.1228 ~%
Pressure distributions for the tangent- and secant-parabolic
tails are ccmpared with the pressure distribution for the conical
tail h figure 6. Graphical.integration of these pressure distributions
to obtain the respective wave drags shows the conical.boatt-ail.to have
the least wave drag (~ = 0.0465). The tangent-parabolicboattail hm
a drag 1.27 times the conical boattail drag; while the secant-parabolic
boattail &rag is 1.02 times as large.
SUMMAlwaE’
Afterbody pressure distributions
l.atedusing a second-order theory for
and wave drags have been calcu-
various conical boattails at zero
angle of a%ck. The following conclusions have been rbached after
mm the res~ts for a ~ch ntier range from 1.5 to 4.5 and for
boattail angles from 3° to KLO:
1. For a given boattail angle, the pressure coefficient level
ticreased with increasing Mach number and for a given &h number
decreased as the boattail angle increased.
2. The wave-drag coefficient for a given boattail angle decreased
with ficreasing I&h number and area ratio. For a given area ratio
and increasing boattail.angle, the boattail wave-drag coefficient
ticreased.
3. For a I&ch nunber of 2.5, an area ratio of 0.200, and a fineness
ratio of 2.25, a comparison of the wave drags for a conical, a tangent-
parabolic, and a secant-parabolicboattail.shows the conical boattail
to have the smallest wave drag.
Lewis Flight Propulsion Laboratory
National Advisory Committee for Aeronautics
Cleveland, Ohio, April.28, 1953
boat-
boat-
——
—
—
wWA TN 2972 5
REFERENCES
1. Huth, J. H., and Dye, H. M.: Axial and Normal Force Coefficients for
Pointed Bodies of Revolution at Super- and l&personic Speeds.
Psx’tu - Boattails. RM-905, U. S. Air Force Proj. RAND, The RAND
Corp., Aug. 5, 1952.
2. Van Dyke, Milton D.: First- and Second-Order Theory of Supersonic
Flow Past Bodies of Revolution. Joux. Aero. Sci., VO1. 18, no. 3,
Mar. 1951, PP. 161-178.
3. Van Dyke, Milton D.: Practicsl Calculation of Second-Order Super-
sonic Flow Past Nonlifting Bcxliesof Revolution. N21CATN 2744, 1952.
4. Grinmd.nger,G., Williams, E. P., and Young, G. B. W.: Lift on
hclined Bdies of Revolution in Hypersonic Flow. Jour. Aero. Sci.,
vol. 17, no. M_, Nov. 1950, pp. 675-690.
—— –-—
—--—— --
—— —
—-
NACA TN 2972
-. 3
-. 2
-.1
0 1
(a) lb - 1.5.
0
(c) l% - 3.5.
-.06
-.04
-.02
“’=”’”*
v J. z 5 4 5 6
.7
\
L I 1 I I 1 1 1 1- 1
Axial station, *
(d) l% - 4.5.
Fl@.re 1. - Variation of pressure coefficient in axial direction
for conical boatt.ails.
— —.
NACA TN 2972 7
.
.
1.0
.8
$ .6
.2
0
I
e
/
-L!iIDJ:A,D
1+----L .+
\
.$=(l.2&my
\ G,
deg
\
\
3 .
\
\ \
\
11
2 4 6 8 10
Fineness ratio, L/~
Figure 2. - Relations between boattail angle, area ratio, and
ness ratio.
fine-
.— ——... .
8“
.16
.12
.08
.04,
0
(a) Mach nuniber,1.5.
.12
.08 1--~
I
\ A
.
\
\— — — .— 3
0 .2 .4 .6 .8 1.0
Area ratio, A/k
(b) Mach number, 2.5.
Figure 3. - Variation of wave-drag coefficient with area ratio. (Dashed
lines represent limiting area ratio for theory of ref. 3.)
.———
. ..-. —.
.
,,
.06
.04
.02
0
.06
.04
.02
0
(c) Mach number, 3.5.
I \
<
1
\
3
\
—_
.2 .4 .6 .8 1.0
Area ratio,A/%
(d) Mach nuuiber,4.5.
Figure3. - Concluded. Variation of wave-drag coefficient with
ratio.
area
——
——--— —___ ._ ._
10 NACA TN 2972
- .8
0 2 4 6 8 10 12
E@attall angle, f3,deg
(b) Mach nuuiber,2.5.
Figure 4. - Variation of wave-drag coefficient with boattall angle.
.
.
.
—
NACA TN 2972 11
.
.06
.04
#
.02
0
(c) Mach number, 3.5.
0
0,
.06
q
g
s
.04
.02 T
o 2 4 6 8 10 12
Wattail angle, 9, deg
(d) Mach muiber, 4.5.
Figure 4. - Concluded. Variation of wave-drag coefficient with boattdl
angle.
—..— —-——
—.— —
NAM TN 2972
c!?
-.
.16
.12
.08
.04
0
(a) Area ratio, 0.2~.
.16 l
U.
.12
9
.08 / / ‘
7
/ A
/ /5
.04-
/
-
3
I
0 .2 .4 .6 .8
Mach number reciprocal, l/~
(b) Area ratio, O .4WI.
Figure 5. - Variation of wave-drag coefficient with
reciprocal of Mach nuuiber.
_ —.
v!?
.
%
.3
u
.12
e,
deg
I-1.
.08- A
/
.04
0
(c) Area ratio, 0.600.
.06
11
/9
.04
7
{
5
.02r
o
Figure 5.
ficfent
.2 .4 .6 .8
Mach number reciprocal, l/~
(d) Area ratio, 0.800.
- Concluded. Variation of wave-drag coef-
with reciprocal of Mach number.
— —— —
.-.10
-.08
-.06
-.04
-.02
0
J
‘o
+
f
Tangent parabolic
8ecant parabolic
-Conical
=M
[
(Not to scale) I I I
1
\ / y \
~ Tangent-
parabola
\
//
parabola
I
.2 .4 .6 .8 1.0
Mal station, xiL
Figure 6. - Pressure distributions over three boattail contours.
Free-stream I.kchnumber, 2.5; area ratio, 0.200.
NACA-h@q -7-6-53- 1004
—z. — .—— — ..— —


Theoretical pressure distribution and wave drags for conical boattails

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