CALIFORNIA
INSTITUTE
Hydrodynamics
OF
TECHNOLOGY
laboratories
----------
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OPTIMUM SlENDERNESS RATIO
OF A
STABlE lOW-DRAG BODY
A REPORT ON RESEARCH CONDUCTED UNDER CONTRACT WITH
THE OF.FICE OF NAVAl RESEARCH OF THE DEPARTMENT OF THE NAVY
OOU IDLRfiAL
Navy Department
OFFlCE OF NAVAL RESEARCH
Contract N6onr-24428
OPTIMUM SLENDERNESS RATIO
OF A
STABLE LOW-DRAG BODY
by
F. Barton Drown
Research t.:ngineer
Hydrodynamics Laboratory
California Institute of T ec hnology
Pasadena, California
Robert T. Knapp, Director
September 1949
Report No. N-55.1
Copy No.
r"..cJ;,j
/t?.-?
I~IP P"'...._"'T"t .A
I
C~L
CONTENTS
Page No.
Abstract. . . . . . .
1
Table of Nomenclature
1
Optimum Body Shape .
2
Resistance of an Underwater Body
2
Analytical Determination of the Effect of Change of Slenderness Ratio.
4
Volume and Surface Area in Terms of the Slenderness Ratio
4
Skin Friction of the Body . . . .
5
Total Drag Coefficient of the Body
5
Residual Drag Coefficient .
5
Total Drag Coefficient
. .
10
Velocity of a Body in Terms of Slenderness Ratio.
10
Free Sinking Body . . .
10
Horizontally Moving Body
11
Constant Residual Drag for a Free Sinking Body
11
Constant Residual Drag for a Horizontally Moving Body
11
Constant Total Drag for a Free Sinking Body. . . . .
17
Constant Total Drag Coefficient for Horizontally Moving Body
17
Maximum Point of the Characteristic Curve.
19
Effect of the Residual Drag
20
Some Typical Body Shapes.
20
Effect of Increasing the Surface Area of a Body.
22
Effect of Changing the Volume Reynolds Number
23
Constant Residual Drag and Constant Frictional Resistance
25
Conclusion
.
26
Bibliography.
26
Appendix . .
27
Distribution List .
3'5
CONFIDENTIAL
Fig. 1-Composite Group of Bodies of Various Slenderness Ratios
The nose in this group is the "halfbody ," and the afterbody is from the MK 13-1
torpedo.
rnNI=Ini=NTIAI
CONFIDENTIAL
1
OPTIMUM SLENDERNESS RATIO
OF A
CYLINDRICAL MIDSECTION BODY
ABSTRACT
This paper presents a theoretical and experimental
method of selecting the optimum slenderness ratio of a
body with cylindrical midsection. Whether such a body
is a large submarine or its arch enemy, the depth charg e,
the problem rem a in s to find a slenderness ratio which
will permit the fastest possible velocity consistent
with the power or sinking wei ght available.
Considerable re s earch has been done to determine the
hydrodynamic characteristics of nose shapes both alone
and combined with various afterbodies. In one of our
reports 1 * we have pointed out that any one of several
different nose shapes could be used on a particular
body with little ch a ng e in the total drag coefficient, and
tests for certain afterbody shapes would probably
bring similar results.
Due to practical considerations, most bodies have a
cylindrical mid s ection. Therefore, a typical underwater
body of a xi a l s y mmetry consists of arbitrarily selected
nose and afterbody shapes separated by a cylindrical
midsection.
For dynamic stability, any underwater body must have
fins which increa s e the surface area and, to some degree, the residual drag. We must necessarily consider
the effect of su-.:h fins on slenderness ratio. In the
following discussion of a particular concrete example,
the emphasis has been placed on a body with a nose
and afterbody with fins, of the same geometrical shape
as the MK 13-1 torpedo (less shroud ring). The MK 13-1
torpedo is dynamically stable, has a cylindrical midsection, and has been tested with other nose shapes. 1
After examination of the factors affecting the optimum
slenderness ratio, it is found that a reasonably larg e
variation from the theoretical optimum value will have
little practical effect on the velocity of the body.
Because of practical factors involved in the design
of an undersea body, it may be desirable from the
designer's point of view to have a relatively large
slenderness ratio. This investigation shows that as far
as drag per unit volume is concerned, the designer will
will pay very little, if any, penalty if he disregards the
drag factor and bases his selection of slenderness ratio
entirely on such items as tactical requirements of
maneuverability, structural design and utilization of
internal space.
•See
bibliography at end of this report.
Although this investigation was carried on under the
Office of Naval Research C ontract N6onr-24428 in
the interest of the Bureau of Ships, much of the data
and method of attack was developed previously under
the Bureau of Ordnance Contract NOrd 9612.
TABLE OF NOMENCLATURE
This paper uses the foot-pound (force)-second system
of units. However, any consistent set of units may be
used in the equations.
A
cross-sectional area at maxrmum cross section, sq ft
..,.Lv~e
B
=
C
= total drag coefficient based on cross-sectional
0
a constant, depending on the stufaee area of
the nose and afterbody of a body of given
shape; see Eq. (7), dimensionless
area, A, dimensionless
residual drag coefficient based
sectional area, A, dimensionless
CF
on
cross-
skin friction coefficient based on cross-sectional area, A, dimensionless
= skin friction coefficient based on surface
area, S, dimensionless
D
maximum diameter of a body with cylindrical
midsection which has known nose, afterbody,
and fin shape
the maximum diameter of a particular body for
which the surface area and volume of the nose,
afterbody, and fin surfaces are known, ft
E
a parameter dependent on the slenderness ratio
and skin friction of the body; see Eq. (26),
( lb sec 2 ) per ft 4
F
drag force, lbs
G
=
a parameter dependent on the slenderness ratio
and residual drag of the body; see Eq. ( 27),
(lb sec 2 ) per ft 4
K
any constant
L
over-all length of the body, ft
CONFIDENTIAL
2
length of the afterbody; see Fig. 4, ft
length of the cylindrical center section of the
body; see Fig. 4, ft
.l,.
length of body nose; see Fig. 4, ft
M
the coefficient in the residual drag equation;
see Eq. ( 18), dimensionless
=
m
the exponent in the residual drag equation; see
Eq. ( 18), dimensionless
N
=
n
= the exponent in the flat plate skin friction e-
the coefficient in the flat plate skin friction
equation; see Eq. ( 15), dimensionless
quation; see Eq. ( 15), dimensionless
p
power, ft-lbs per sec
power, horsepower
Q
a constant dependin g on the surface area of
the nose, afterbody, and fins of a body of
given shape; see Eq. (4 ), dimensionless
vL / v
Reynolds number based on over-all
length of body , dimensionless
vV 11 o/v = Reynolds number based on volume
of body, dimensionless
s
= total surface area of body, ft 2
=
v
surface are a of the body's nose, afterbody, and
fin surfaces for a given diameter D0 , ft 2
= volume,
ft 3
= the total volume of the nose and afterbody for
a given diameter, D0 , ft 3
= velocity,
v
ft per sec
yb
specific weight of the body, lbs per ft 3
"~r
pg
specific weight of fluid, lbs per ft 3
=
negative buoyancy, lbs per ft 3
!'Jy
= yb -lj
TJ
= over-all efficiency, per cent
ZJ
= kinematic viscosity , ft 2 per sec
p
mass density of fluid, (lb sec 2 ) per ft 4
f
LI D = slenderness ratio, dimensionless
OPTIMUM BODY SHAPE
There are four hydrodynamic considerations in the
determination of an optimum body shape: nose shape,
afterbody shape, fin design and slenderness ratio, L/ D.
In one of our reports 1, we have shown that any one of
several different nose shapes could be used with little
change in the total drag coefficient (see Figs. 2 and 3).
Also, minor variations in the afterbody shape would
probably cause little change. However, hydrodynamic
considerations must be reconciled with the practical
aspects involved in the design and use of an underwater
body. For a depth charge the following must be included: weight of explosive and propellant charge, dynamic
stability in air and water, favorable cavitation characteristics, and maximum velocity consistent with the
sinking weight of the body.
Some of the practical factors involved in the design
of a submarine include the following : sufficient space
for machinery and personnel, dynamic stability, maneuverability, freedom from cavitation, strength of hull,
and maximum velocity consistent with available power.
The hydrodynamic characteristics are known for many
bodies of different nose and afterbody shapes, but there
is little or no information on the effect of changing the
slenderness ratio. To simplify this problem, let us
consider that the slenderness ratio is changed by varying the length of the cylindrical midsection of a body
for which the hydrodynamic characteristics are known
for some particular slenderness ratio.
RESISTANCE OF AN UNDERWATER BODY
The force required to drive any underwater body can
be predicted by model studies. In the case of ships
where resistance is due to wave, eddy, and frictional
resistance, analysis is made by the Froude method.
Here the wave resistance is a function of Froude's number, and eddy and frictional resistance a function of
Reynolds number.
For an underwater body such as a torpedo (if submergence is assumed sufficient to eliminate any wave
effect) the resistance of the body will be due to eddy
and frictional resistance only.
Therefore, Froude's
number does not enter as a parameter.
If we apply Froude's method to an underwater body,
we assume that the total resistance is equal to the sum
of two separable parts-the frictional resistance and
the residual resistance. Also by Froude's method, we
can calculate the frictional resistance of the body from
the r e sistance of a flat plate. 2 • 3 Therefore, knowing the
total dra g res i stance from water tunnel tests and
the frictional resistance from calculation based on
available data (Schoenherr), 3 the residual resistance
can be determined. As Froude's number 1s not a parameter, the residual resistance will be a function of
Reynolds number only.
CONFIDENTIAL
0.20
0.18
0.16
------- - --
0 .14
0.13
f0.12
c
'-' 0.11
.,.:0.10
z
~"====
7
f
\
r----
-
-= --=---t-===-------- ---- - ----
NO 16- 1.5 I ELLIPSOID
'~ I <'R()n•
NO
'!! 0.09
1-1::::::
- -::::: F
f...J'
'-'
it
.,c_NO 1- HEMISPHERE
-
·-
===--==
~--:.= ~ t...NO 33- MK 14 15
NO 15
-
2.5/1 ELILIPSOID
·=--:.-=:- ~ __,r-.. f!,1_2 t-- SP ~R( GIV
0.08
-- - ---- - ·-=
--
Bom
NO
"'<I~ 0.06
0 .0 5
0-MK
_.;;:;
13~
33.5 K~OTS PROTC TYPE SPE! D 40.5 N<m
0 .04
2
4
3
5
7
8
9 10 7
'2
REYNOLDS N\)MBER, R, (BASED ON \-ENGTH)
6
4
.3
Fig. 2-Drag vs. Reynolds Number For Various Noses
T he noses are shown in Fig. 3. In all runs the afterbody was the MK 13-1 torpedo
shape and the slenderness ratio was 7 .18 .
161'
1 8 4 ~~ CAL. TO RAD, CENTER
1. 8 ~6!!1 CAL TAPER LENGTH- - - - - r 241
CAL-.--
iJ~
kJ~
-
0.'5 CAL-,
NOSE
1.:,2:5
OVERALL
CAL
___
J._
~
I_
~l
1.250 CAL. ________,
NQ I
~
HEMISPHERIC AL
25; 1 ELLI PSOID
l
t_
~
L ~ ll
EL LI P SOIO
196'5
NOSE NO. 27
2 00
a
OUTLINE
65 f
SPH EROGIVE
DRAWINGS OF TORPEDO AND
CAL
NOSE
ST ANOAR O
NOSE
NO. 33
Ml< 14 - MK I '!I
DESIGNS
3 692 M
Fig. 3-Various Noses Used With MK 13-1 Torpedo Afterbody
See Fig. 2 for the resulting drag vs . Reynolds number runs.
5
6
7
8
9
10 8
CONFIDENTIAL
4
ANALYTICAL DETERMINATION OF THE EFFECT
From Fig. 4
OF CHANGE OF SLENDERNESS RATIO
( 2)
Our problem is to find the effect of the slenderness
ratio on the maximum velociry of an undersea body
which has a give n driving force and displacement.
Under these conditions, the velocity depends on the
hydrodynamic resistance. This resistanc e is made up of
two factors, the skin friction and the residual drag. The
problem therefore reduces to expressing these two factors as functions of the slenderness ratio. As we are
considering a body with cylindrical midsection, it will
be a simple matter to express the volume and surface area as functions of the slenderness ratio, a surface
parameter, and a volume parameter. It follows that we
can also express the skin friction and residual drag in
terms of these same parameters.
Now, by adding these two expressions, we have the
total drag coefficient of the body in terms of the slenderness ratio and the shape parameter. An expression
for the total drag coefficient can also be written involving the velocity and frontal area of the body and the
acting force. Therefore, with these two simultaneous
equations, the total drag coefficient can be eliminated,
and we have the desired relation between the velocity
of the body, the force acting on the body, the volume of
the body, and the slenderness ratio.
The surface of the body in Fig. 4 is
( 3)
Let
(4)
or
(5)
The volume of the body in Fig. 4 is
(6)
Let
(7)
or
(8)
To aid in the development of equations to follow, it
VOLUME AND SURFACE AREA IN TERMS
is now convenient to express the reciprocal of the
OF THE SLENDERNESS RATIO
diameter and length in terms of the slenderness ratio
and volume. From Eq. (8)
Fig. 4 shows a hypothetical body shape for which it
is assumed that the proportions of nose, afterbody and
tail fins are already established. The slenderness ratio
can be c han ged only by varying the length , l,., of the
cylindrical midsection.
In the following development there are some tedious
mathematical manipulations. The ratio, L / 0, is awkward to handle and it is more convenient to substitute
a single parameter for the slenderness ratio.
Thus,
tjJ
1
(9)
D
From Eq. ( 1) and Eq. (9)
1
.!_
L
'-/;_
[
rr(B
+ '-/;)
iV
(1)
LI D
1/3
J
D
~n.-----r--------------------~mn-------------------4-----------~~o~--------~
r------------------------------------ L ----------------------------------~
Fig. 4-Hypothetical Body With Cylindrical Midsection.
CONFIDENTIAL
(10)
CONFIDENTIAL
5
SKIN FRICTION OF THE BODY
TOTAL DRAG COEFFICIENT OF THE BODY
Using Froude's method, let us assume that the flow
in the boundary layer is turbulent and that the skin
friction of the body can be calculated from the skin friction of the flat plate. It is conventional to express the
flat plate friction coefficient, C FP' in terms of the
surface area of the body. However, all coefficients in
this paper other than C FP are expressed in terms of the
maximum frontal area of the body, A. Therefore, if CF
is the skin friction coefficient of the body, based on
this frontal area, a relation is needed between C FP
and C F"
The force due to skin friction on the body can be
expressed as
Applying Froude's method again, the total resistance
of the body, or its corresponding coefficient, is equal to
the sum of the frictional resistance and the residual
resistance, or their corresponding coefficients
(11)
The force due to skin friction can also be expressed
as
F
(12)
By combining Eqs. ( ll) and ( 12) we have the desired
relation between C F and C FP"
(S/A) CFP
(13)
By substituting Eq. (5) in Eq. (13)
( 14)
For turbulent flow along a flat plate, the skin friction
coefficient can be determined from Schoenherr's equation.3 For a limited range of Reynolds number, this
equation is practically a straight line when plotted in a
log-log graph and can be expressed as
(15)
By combining Eqs. (14) and (15) we have
CF
= 4 (Q+tj;)N (v/Lv)n
(16)
(17)
RESIDUAL DRAG COEFFICIENT
Fig. 5 shows a typical installation in the High-Speed
Water Tunnel of the model mounted on the threecomponent balances. (For description of balance system
see Ref. 5). Models ranging from a slenderness ratio
of 6.00 to 13.71 were mounted in a similar manner.
Fig. 7 shows the experimental results of determining
the total drag coefficient versus Reynolds number for
five different slenderness ratios of a body with nose
and afterbody shape of the MK 13-1 torpedo (Fig. 6).
From Eqs. (16) and (17) the residual drag coefficient
versus Reynolds number can be calculated and plotted
for each of the slenderness ratios on a log-log graph
(Fig. 8).
Most of the resulting points fall very close to the
straight line drawn through them in Fig. 8. The scatter
represents approximately a plus or minus two percent
variation of the total drag coefficient. Undoubtedly this
scatter includes any variation due to the different
slenderness ratios plus the accumulative error of the
experimental measurements. Therefore any variation of
the residual drag coefficient with a change in slenderness ratio must be much less than two per cent, so we
are well justified in saying that the straight line of
Fig. 6 represents the residual drag independent of the
slenderness ratio. Thus, we can write
and from Fig. 8, we can evaluate the coefficient M and
the exponent m.
The results of determining the residual drag coefficients for other nose and afterbody shapes with fins is
shown in the appendix. It will be noted that the same
result is attained as shown in Fig. 8 for the MK 13-1
torpedo-shape nose and afterbody. Therefore, we conclude that for any one family of bodies of revolution the
residual drag is a function of Reynolds number only as
expressed in Eq. ( 18) and is independent of the slenderness ratio for all useful values. This relationship
between CDF' Re, and L / D is shown in Fig. 9. Fig. 10
shows that this is true even for a blunt body down to a
slenderness ratio of three. For any reasonably well
shaped body the minimum slenderness ratio with no
cylindrical midsection is great enough to insure the
constancy of the residual drag.
6
CONFIDENTIAL
Fig. 5- Typical Installation For Determining Drag vs. Reynolds Number
Body is 'mounted on three-component balance. See Ref. 5.
rnNFinFNTI AI
CONFIDENTIAL
( tj;)
=
( tj;)
= 7 . 18
(tj;)
= 9.70
( tj;)
=
Lt l
= 13.71
6 .00
ll. 7 1
Fig. 6-Sienderness Ratio Study Series
In this series the nose and afterbody are the same as the MK 13-1 torpedo. The s e
bodies were mounted in the High-Speed Water Tunnel at the California Institute of
Technology, as shown in Fig. 5.
7
LUNt-IUt:N IIAL
8
.5
0
0
"' .
!:!.
"': 9 .70
(J
0
of •
II. 71
l.L .3
0' of.
13 .71
.4
0
UJ
~
0
(J
(of
r'i
7 . 18
• L /0)
_r
..n.
<.!)
<t
g;
"': 6 .00
-
.2
1-
_J
<t
10
L
c
0
.I 10 1
n
~
-
n
0
I-
_n
-
r-<-
IU
-
l.J"
~
lu
-
w
IV
u
3
2
.A
5
4
7
6
T
8
9
10
REYNOLDS NUMBER, R6
Fig. 7- Total Drag Coefficient vs. Reynolds Number
Reynolds number is based on the over-all length of the body. The experimental
points were obtained using the two-inch diameter models shown in Fig. 6. The
curves are plotted from Eq. (20) .
.08
... .07
c
0
(J
• .06
l.L
w
0
.05
(J
~
A---
!:!.
of:
of.
of.
0
~. 11.71
0
0
Cf
~
r--
<
~
lJL ~ l.o.
tl~
<.!)
<t .04
a::
0' of:
r::
-a:f'b--
0
g
f
~
[-crQ 1-S:a:
.6.
!:!.
_J
~ .03
t
rll_Q
6.00
7 . 18
9 .70
13.71
1jl • L/0)
IT
p:- ~ 1-~ t-- t-9
Ia
0
<J)
w
a::
2
3
4
5
6
REYNOLDS NUMBER, R 6
Fig. 8-Residual Drag Coefficient vs. Reynolds Number
The experimental points are calculated by using Eqs. (16) and (17).
CONFIDENTIAL
7
8
9
CONFIDENTIAL
9
"'
rn
~4
0
c
I>
r
3
0
"'I>
G>
2
0
0
rn
:n
0
0
0
Fig. 9 - Three-Dimensional Relation of CDFt Re, and
LI D
+0 .0 2
(
-'
~
I
00
::!
0
a::
w
m
-'
I
<t
0/
ci:: -0.02
w
0..
0
0
~
w -0.04
<.!)
z
<t
:r:
0
-0.06
I-
0.4
z
I
w
""w
0
SKIN
0.2
~- f-:::::
0
<.!)
<t
a::
0
-- --+----- -- ~
TOTAL, Co:::=--
0
0
0
-
1---2
,,--,:---
f-.-f-.--
4
ESI~UAL,JoF -
r-
6
8
10
12
SLENDERNESS RATIO-'!'
FOR Re = I, 500,000
Fig. 10-AIIocation of Residual Drag and Skin Friction
for Cylinders with Hemispherical Ends
Col. Gerald B. Robison, at the time of his untimely death, left much
valuable material in his unpublished work. The concept in Fig. 10,
taken from his papers, suggested to the author the presentation
given in this report.
CONFIDENTIAL
10
TOTAL DRAG COEFFICIENT
1
It is now possible to express the total drag coefficient
in terms of the slenderness ratio and the Reynolds number based on the volume of the body. This volume
Reynolds number, Rv, is
1 (B+Ij;)<n-2)/3 (Q+'{;)
FR:;
y;n
=
X
4Nfp/2H77/1) <n+o/3
1
+ FRvm
(19)
X
(8 +f)<m-2 )/ 3
\jim
M(p/ 2)(77/ 4 }tm+ 1 )/ 3 (25)
Eq. (25) can be simplified by letting
By substituting Eqs. (14, (15), and (18) w Eq. (17)
we have
( 26)
CD= 4(Q+Ij.J)N(v/ Lv)n+M(v/ Lv)m
(20)
and
A check on the evaluation of Eq. (18) from Fig. 6 can
now be made by calculating the total drag coefficient
from Eq. (20). The curves in Fig. 7 are calculated from
Eq. (20) and fit very closely to the experimental points.
By substituting Eqs. ( 10) and ( 19) in Eq. (20) we have
G
(B+Ij.J)<m-2) /3
= y;m
M(p/ 2)(77/ 4)<m+1)/3
(27)
After substituting Eqs. (26) and (27) in Eq. (25), and
multiplying through by R:; (remembering that R,~
vnvnt 3j vn), Eq. (25) reduces to
=
F 1/ (2-n) V (n- 2) / 3(2-n)
( 21)
v=
[vn (E+GR <n-m) )]1 / (2-n)
(28)
v
Another relation involving the total drag coefficient
can be obtained by using the following relation with
the drag force, F, acting on the body
(22)
FREE SINKING BODY
Let us assume that the body is sinking vertically in
water and has reached its terminal velocity. The force
acting on the body due to gravity is
or
F
=
6yV
( 29)
(23)
Substitution of Eq. (9) m Eq. (23) gives the following
convenient relation
F(B
where 6y is the negative buoyancy of the body in
water, lbs. per ft3. Substituting Eq. (29) in Eq. (28)
gives the terminal velocity of a free sinking body
+ 1j; }213
=
(24)
VELOCITY OF A BODY IN TERMS OF
THE SLENDERNESS RATIO
By equating Eqs. (21) and (24) the total drag coefficient can be eliminated
V
L:Jy11 (2-n) v<n+l) / 3(2-n)
[vn(E+GR<n-m))]11<2-n)
v
(30)
The effect of changing the slenderness ratio on the
sinking rate can be seen by using a graphical solution
of Eq. (30). The denominator is plotted as the "characteristic curve" in Fig. 11 for an average value of
based on the ran ge of volumes and velocities indicated in the same Fig. 11. For the range of volumes
Rv
I()NFinFNTI AI
CON F I DEt'-H I 1\L
and ne~ative buoyancies indicated on the chart, the
characteristic curve based on an average value of l<v
gives a close approximatio n to the free sinking velocity.
Fig. 11 is based on the ,\JK 13-1 torpedo shape. The
volume and negative buoyancy range in these three
charts were selected to approach that of a depth charge.
The following method is used to determine the velocity of a free sinking body with the MK 13-1 nose and
afterbody shape. Assume the slenderness ratio of the
body is 6.5, the volume of the body is 5 ft3, and the ne g ative buoyancy is 50 lbs. per ft3. Turn to Fig. 11 and
select the slenderness ratio. Move vertically to the
characteristic curve, horizontally to the volume of the
body, vertically to the negative buoyancy, then horizontally to the terminal velocity scale, reading 59 ft.
per sec .
As mentioned before, the characteristic curve 1n
Fig. 11 is calculated for an assumed average value of
Rv. This value appears in the denominator of Eq. (30),
therefore if the value of I?.v changes, there is a change
in the terminal velocity. Fig. 12 is a chart similar to
Fig. 11, but it has been constructed to allow for this
variation of Rv. A change in
effects directly the
characteristic curve, therefore in Fig. 12 three curves
have been drawn which cover the range of volumes and
velocities shown on the chart. A trial and error solution
is now necessary. Select the slenderness ratio and the
use the chart in
the volume, and assume a value for
the same manner as indicated for Fig. 11 and obtain a
terminal velocity. Use this terminal velocity to calculate a new value for
Repeat the above process until
the calculated value of
is the same as the assumed
value.
ll
= (PIIY] 550
v
)1/(3-n)
V
(n - 2)/3(3-n >
[vn( f;+ Gl<v(n-m >))
( 3-1)
1/(3-n)
As in the case of th e free si nkin g body, Fig. 13 shows
the effect of changing the slende rne ss ratio. The de nominator i s also plotted as the " chara c t eristic curve"
for an average value of l<v based on th e range of volume s and velocities indicated in Fig. 13.
CONSTANT RESIDUAL DRAG FOR A
FREE SINKING BODY
If the flow patt ern over the body does not vary with a
changi n g Reynolds number, then the residual drag of
the body will be constant and m in Eq. ( 18) w ill be
zero . Therefore M equals CDF and Eq. (27) becomes
c'
= CDF(p/2)(7Tj1)113
(35)
( 13 + tj;) 2/3
Rv
and Eq. (30) becomes
v
Rv;
Rv·
Rv
HORIZONTALLY MOVING BODY
A chart similar to Fig. 11 can be made for a body
moving in a horizontal plane without acceleration. Let
us assume that the input horsepower, Pff, and the overall efficiency, Y], are known. If P is the theoretical
horsepower, we can write
P
=
Fv
( 31)
Rv·
1/(2 -n)
V (n + t) /3(2 -n)
[vn (£ + c' 1<:) )]
1/(2 -n)
(36)
CONSTANT RESIDUAL DRAG FOR A
HORIZONTALLY MOVING BODY
As mentioned for a free sinking body, m equals zero
and M equals CDF when the residual drag is constant
with Reynolds number. Therefore, Eq. (34) becomes
=
(PuYJ 550 ) 1/(3-n) V (n- 2)/3(3-n)
[v n ( E + G'R n )] t/( 3-n
v
(32)
Substitute Eq . (33) in Eq. (28) and we have the velocity
of a horizontally moving body in terms of its slenderness
ratio, volume, input horsepower, over-all efficiency,
and
y
Rv
v
(33)
1::;
Fig . 14 is an example of Eq. ( 36) and is calculated
for an assumed form drag value of 0.013 and the same
a verage value of
as u sed in Fig . 11. If the two charts
are compared, it will be noted th at the terminal velocities are approximately the same for a give n slenderness
ratio, volume of body and ne ga tive buoyancy.
and
or
=
l
(37)
The residual drag coefficient of a body the SIZe of a
submarine is probably constant. If the submarine is
a "clean" shape, we believe that a conservative estimate of this coefficient would be 0.02.
It is shown later in this paper that for normally
shaped bodies the surface area and volume shape parameters have little effect on the optimum slenderness
ratio.
rr.t\1 t:l nt: I-. IT I A I
\,...\..11"11
IVL-1,
Ill\._
12
2
40
3
4
5
6
30 50
7
60
70
9
40
10
(.)
w
(/)
'II.J..
I
50
>-
1(.)
0
...J
lJJ
>
...J
<t
40
60
z
~
0::
w
~
50
70
60
2
3
4
5
6 7 8 9 10
100
VOLUME OF BODY-FT
0
2
4
6
8
3
90
80
70
NEGATIVE BUOYANCY
3
Yb- Yf , L 8 S I F T
10
12
14
16
18
20
SLENDERNESS RATIO- '11
('l'=L/D)
Fig. 11-Sinking Body Velocity Chart for MK 13-1 Torpedo Nose and Afterbody
Shape for Average Value of Volume Reynolds Number
This chart is designed for a free sinking body of a size and velocity range
suitable for a depth charge. The characteristic curve is calculated on the basis
of the data in Fig. 8, and for an average volume Reynolds number based on the
range of velocities and volumes indicated in the chart.
CONFIDENTIAL
22
CONFIDENTIAL
40
13
2
3
4
5
6
7
60
8
70
9
80
40
10
90
100
0
w
(/)
.......
ILL.
50
I
>-
10
0
...J
w
>
...J
<t
40
60
z
~
~
w
1-
50
70
60
2
3
4
5
6 7 8 9 10
VOLUME OF BODY- FT
0
2
4
6
8
100
3
90
80
70
NEGATIVE BUOYANCY
y b- Yf , L 8 S I F T 3
10
12
14
16
18
20
SLENDERNESS RATIO- 'I'
{'I'=L/D)
Fig. 12-Sinking Body Velocity Chart for MK 13-1 Torpedo Nose and Afterbody
Shape for Several Values of the Volume Reynolds Number.
The characteristic curves are calculated for several values of Rv which cover
largely the range of velocities and volumes indicated on the chart. This chart is
otherwise the same as Fig. 11.
22
CONFIDENTIAL
14
VELOCITY- FT /SEC
80
t
I
70
60
50
40
I
80
90
100
0::: 120
w 140
0 160
Q.. 180
w 200
60
50
40
::
30
CJ)
0::: 250
0
300
:I:
I- 350
:::::>
400
~ 450
500
Q..
350
3
..,
I1..1...
I 50
>a
45
0 40
CD 35
LL.. 30
0
w 25
~
:::::> 20
_J
0
>
0
2
4
6
8
10
12
14
16
18
20
22
SLENDERNESS RATI0-'1'
('l'=L/D)
F i g. 13-Horizontally Moving Body Velocity Chart for MK 13-1 Torpedo Nose and
Afterbody Shape for Average Value of Volume Reynolds Number
This chart is desig ned for a hori zo nta ll y mov ing body in the size and horsepower
range of the torpedo. The charac t e ris ti c c ur ve i s calcu la t ed on t he same basis as
the charac t eris t ic c ur ve in Fig. 11.
CONFIDEtHIAL
24
CONFIDENTIAL
40
30
15
2
3
4
5
6
50
7
8
70
9
80
40
10
90
100
(.)
w
(j)
'-..
tl.L.
I
50
>t-
(.)
0
__J
w
>
__J
<t
40
60
z
50
70
60
2
3
4
5
6 7 8 9 10
100
VOLUME OF BODY- FT
0
2
4
6
8
3
90
80
70
NEGATIVE BUOYANCY
r b- Yf , L B S I F T 3
10
12
14
16
18
20
SLENDERNESS RATIO- 'I'
('I'=L/D)
Fig, 14-Sinking Body Velocity Chart for MK 13-1 Torpedo Nose and Afterbody
Shope for Constant Residual Drag Coefficient
The chara cte ri s ti c c urve i s calcul ated for a n a ssumed c o ns t a nt res i dual drag value
of 0 .0 13 a nd the same average va lue o f Rv used i n F ig . 11.
22
CONFIDENTIAL
16
VELOCITY- KNOTS
30
<ooo
a::
w
3 .3ooo
0
0...
w
a:: '~ooo
0
::r:: sooo
1=> 6ooo
(./)
0...
z
"'ooo
eooo
soo
to o
<ooo
,ooo
.3ooo
'~ooo
sooo
,cz;oO
I
1-
z
w(.f)
:?:z
wo
(.)I<!(,!)
-lz
o...o
(./)_J
0
0
oO
oo
">OoO
~o%
rvD<oo
~go
cz;Ooo
,~oo
'~oo
\A 0
0
,~ oo
,o
2
4
6
8
10
12
14
16
18
20
22
SLENDERNESS RATio-'¥
('¥=LID)
F ig. 15-Veloc ity Chart for a Hypothetical Submarine
=
The charact eri s tic curve is calcul a t ed fo r a Rv
10 8 and co n s t a nt res i dual d rag
coeffic ie nt equal to 0 .02.
CONFIDENTIAL
24
CONFIDENTIAL
17
26~--~---r--~----.----.---.----,---.----r---,----~--~
(f)
24
22
20
~
18
~
~--~---+--~~==i===~~~====~~1r===T====~~~--~
16 ~--~----~--~--~~--~--~----4----4----~---t----t----l
;.._ 14
1-(.)
0
_J
w
>
12
10
8
6
4
2
0
0
2
4
6
8
10
12
14
16
SLENDERNESS RATIO-'¥
( '¥= L/0)
18
20
22
24
Fig. 16-Relation of Slenderness Ratio to Velocity for a Submarine of Given
Displacement and Horsepower
Curve is calculated for the same hypothetical shape as 1n Fig. 15 a nd for a
displacement of 2,000 long tons, 2,500 input horsepower and 60 per cent over-all
efficiency.
Therefore, Fig. 15 has been calculated as an example
of Eq. ( 37) for an assumed constant residual drag
coefficient of 0.02 and for the shape of the MK 13-1
torpedo whose shape parameters roughly approximate
those of a submarine.
Fig. 16 is an example of Eq. ( 37) in which the volume
of the body, the horsepower, and the efficiency have
been assumed to have a constant value. We, therefore,
have a picture of the direct effect of changing the
slenderness ratio. As can be seen from Fig. 16, a
rather large change in the slenderness ratio has a small
effect on the velocity of the body.
CONSTANT TOTAL DRAG
( 38)
and Eq. (30) becomes
v
=
/':, 'Y 1/2 v 1/6
(39)
(E ' + C' )vz
Fig. 17 is an example of Eq. (39). Note that Rv does
not enter into Eq . (39) and, therefore, the charac t eristic
curve shown on Fig. 17 is unique for the particular
shape (MK 13-1 or MK 14-1) for which it was calculated.
FOR A FREE SINKING BODY
If the body is sufficie ntly rough, the total drag coefficient may be constant for large values of Reynolds
number. Here the residual drag and the frictional drag
coefficients are constant with Reynolds number. As
previously mentioned, if the residual drag is constant
with Reynolds number, m equals zero and M equals Cvp
It follows in like manner that if the skin friction does
not vary with Reynolds number, n in Eq. ( 15) will be
zero and N will equal CFP· For these conditions Eq.
( 26) be comes
CONSTANT TOTAL DRAG COEFFICIENT
FOR HORI ZONT ALLY MOVING BODY
As mentioned for the free sinking body, n and m equal
zero, M equals CDF• and N equals CFP for a horizontally
moving body when the total drag coefficient is constant
with Reynolds number. Therefore Eq. (34) becomes
v
=
(PH
1) 550) 1/3
(E' + G'
)113
V -219
(40)
Cm.JFIDENTIAL
18
30
40
40
50
60
70
50
(.)
80
w
(/)
........
90
1~
I
100
>-
!::::
(.)
60
0
40
_J
w
>
_.J
<{
z
:::!:
50
0:: 70
w
1-
2
60
3
eo
70
5
6
7 8 9 10
100
VOLUME OF BODY-V
FT 3
0
2
4
6
90
80
NEGATIVE BUOYANCY
f.y=yb-yf, LBS/FT'
8
10
12
14
16
18
20
SLENDERNESS RATIO-'¥
('¥=L/D)
Fig. 17-Sinking Body Velocity Chart for MK 13-1 Torpedo and Afterbody Shape for
Constant Total Drag Coefficient.
The c h aracteri s tic curve is ca lc ul a ted for a n ass umed c o n s t a nt r esidu a l drag
valu e o f 0 .0 13 a nd an assum e d con s t a nt s k in fri c ti o n value of 0 .00229 .
CONFIDENTIAL
22
CONFIDENTIAL
19
MAXIMUM POINT OF THE CHARACTERISTIC CURVE
F'( 1/J)
It will be noted that the maximum velocity is at tained
for a slenderness ratio which passes through the peak
of the characteristic curve. The location of the maximum
of this curve indicates tor a particular body the optimum s lenderne ss ratio. This can be de t ermined with
sufficient accu racy for most uses by visual inspec ti on
of the characteristic curve, due to the fact that the
curve is fairly flat in the immedi a te vicinity of the maxImum.
The tedious work involved in the di ff e renti a tion of
the equation of the characteristic curve therefore is not
justified merely to find the maximum point. However, by
differentiating and setting th e resulting equation equal
to zero, we can observe why different bodies will have
different optimum points. \Xe will then be in a posi tion
to compare directly several different shapes and to
show the effect of varying the residual drag, the volume
Reynolds number and th e surface area of the body.
Examination of ei ther Eq. (30) or Eq. (34) shows that
the determination of the minimum of the following relation will give the maximum of the characteristic curve
for either a free sinking body or a horizontally moving
body.
F(lj;) = E+GRu(n-m>
( 4 1)
Let
(42)
and
( 43)
then Eq . ( 4 1) can be written
F(lj;)
=
y;m(B+t.j;) ( n-
2) /3
(Q +1/J )Kl +y; n(B+Ij;) ( m- 2)/3
y; tn+m)
(44)
In order to find the mtntmum point of Eq. ( 44) we
must differentiate it, set it equal to zero, multiply
through by
3
and combine term s and we have
=0
= y;z + [3B(1-n)-2Q(1+n)] 1/J
(1- 2n)
8 Q
(1-2n)
Rv(n-m)M
+ 4N(7T/4) t n-m)/3(1-
~m-2)lj;tn-m+o
X
2n)
3mlf; tn-m)
l
UB+lj;) tn-m)/3 - (B+ lj;) tn-m- 3)/~ (45)
Let
(46)
and let
or
F' ( lj;)
( 48)
F 1 (lj;) is a dimensionless function which involves
o nl y surface area, volume and slenderness ratio, and is
unaffected by the excellence of the body shape.
F ( lj;) is also dimensionless and is a function of the
2
slenderness ratio, residual drag and Reynolds number,
Ru, and may be considered as a scale of re sidual drag.
By plotting F ( lj;) a nd the ne ga tive of F 2 ( lj;) as shown
1
in Fig. 18, the intersection of these two curves will
give tqe maximum point (i.e., th e solution of Eq. (48) ).
Fig. 18 is plotted for the same values of the coefficients
as Fig. 11 and, therefore, the slenderness value as
determined by the intersection fixes the maximum point
of the characteristic curve in F igs . 11 a nd 13. Thus, we
see that the optimum slenderness ratio for the MK 13-1
torpedo shape when H,; equals 7 x 10 6 is 11. 5 . Fig . 21
likewise g ives th e maximum point of Figs. 31 and 32
for the ~IK 14-1 torpedo shape.
By using the method just described to determine the
maximum point of the characteristic curve, it is possible
to obt ai~ a clear picture of the effect of the various
parameters on the optim um sle nderness ratio. The parameters involved in the de t ermination are:
13, a constant for a give n shape and dependent on
the volume of the nose and afterbody (see
Eq . (7) ).
Q, a constant for a g i ve n shape and dependent on
the s urface area of the nose and afterbody (see
Eq. (4) ).
N, coefficient in the skin friction relation for a
flat plate (see Eq. (1 5)) and assumed to be
independe nt of the shape of the body .
20
CONFIDENTIAL
'/
140
I
:5:'1/
120
-y
100
~
LLN
1
~
/
"""' 1(
80
~ /I !
~
\
/~,o
60
LL-
v\_<(l-~
I
/
I
v
20
/
0
2
4
v
/v
t
I
I
40
0
v
v
;V
F ('¥)• F1 ('¥l+F2 ('¥)•0
a::
0
/
./
!
/
I
I
t
6
8
10
12
14
16
18
SLENDERNESS RATIO-"¥
(\f=LID)
Fig. 18-Maximum Point Chart for MK 13-1 Torpedo Nose and Afterbody Shape
The intersection of the two curves g ives the maximum point of the characteristic
curve in Figs. 11 and 13.
n,
exponent in the skin friction relation for a flat
plate (see E q. ( 15)) and assumed to be independent of the shape of the body. For the case of
a rough body, n is assumed to be zero.
M, coefficient in the residual drag relation (see
Eq. ( 18)) constant for a g iven nose and afterbody shape as indicated in Fig. 8.
m, exponent in the re sidual drag relation (see Eq.
( 18)) and constant for a given nose and afterbody shape as indicated in Fig. 8. If it is ass umed that the body h as constant re s idu a l drag,
then m equals zero. For th e case of a ro ugh
body, it is assumed that m equals zero.
~. Rey nold s number based on the volume of the
body (see Eq. (19)).
EFFECT OF THE RESIDUAL DRAG
In order to determine the effect of different values of
the re s idua l dr ag on the s lendernes s ratio, it is convenient to ass ume that for a given body the re s idua l drag
remains constant with Reynolds num be r, Re.
The abov e assumption means that m equals ze ro and,
therefore, 11 e qu a l s CDF . There will be no chang e in
( 46) but E q. ( 4 7) becomes
(49)
Using the MK 14-1 nose and a fterb ody shape as a n
example, F 1 (lj;) in Fig. 20 is the same as in Fig . 19.
F 2 ( lj;) is c a lculated from Eq. ( 49) for various values of
CDF' We can immedi a tely see th a t the optimum slenderne ss ratio of a body i s very dependent on its residual
drag. Thus for this particular body, the minimum slenderness ratio is approximately 7, where as a re s idual
drag of 0.030 (which is possible with a poor nose)
indic a te s an optimum ratio g reater than 15.
SOME TYPICAL BODY SHAPES
Fig. 21 shows some body shapes which are considered
in this paper. In order to compare their comparative
shapes more ea s ily, the bodies have been superimposed
on each other. Eq. (46) has been evalua ted for several
of these bodies and plotted in Fig . 22 .
Examination of Eq. ( 49) shows that this equation
changes very littl e fo r various body shapes because the
term involvin g the volume parameter B is raised to a
small power. Therefore, in Fig. 22, the various F 2 ( lj;)
curves are th e same as thos e use d in Fig. 20.
We can conclude from this chart that certain bodies,
with a ll other factors th e same, wi ll have lo wer optimum
s lendern ess r atios than o th e r s . We can not tell fr om
thi s chart which s hape is the "bes t." The Lyon Form
"A" 4 i s without fins a nd th e refor e unstable.
Cat-~FIDE~HI/\L
140
21
+ F2 (~)
F' (~)· F1 (~)
I
".... v
0 -
•
v
/
Y/
120
~v
~
"'' I
~,o ~~
100
/ , :_ 1
~
~~
/
Lf'
a::
0
v
60
~
I
I
/
v
I
LL.-
40
/
20
/
/
/
0
l.i.
'-
80
.-/',v
2
0
4
6
8
10
12
14
16
18
SLENDERNESS RATIO-'¥
('¥=LID)
Fig. 19-Maximum Point Chart for tAK 14-1 Torpedo Nose and Afterbody Shape
The intersection of the two curves gives the maximum point of the characteristic
curve in Figs. 31 and 32.
.030
j/
140
~
/ I
120
/
/
100
~
F 1 (~) = F1 (~)
+ F2 (~) •0
/
v
80
LL."'
V/
a::
0
60
~
/
.//
/
/ v~;Y /
v
~ / v: / v ,?--""
/
v
/ v
% ...........
f--/
~ v !--"' .L. v
f...--
LL.-
40
20
-
0
0
2
~
4
,__
6
1/
./'y
vv v v
I
~
v
.........
1--r-
-
II..
0
(.)
(90
v
/
.015
/
<(
X
<.0
0
I'!
a::
10
12
14
>
::>!l:
__.......... .010
...........
v
0
a:
wu.
U)o
a::
~
--
.005
16
18
SLENDERNESS RATIO-'¥
('V•L/D)
Fig. 20-Effect of the Residual Drag
The F 1 ( tj;) curve is calculated for the MK 14·1 torpedo nose and afterbody shape.
rr'\~1 C:l f""\C:lo.ITI
A I
.
L()
_J
<(
.//
8
~
liD
/
/~
4..'
v
/
/ !----""' v
,_...- v
- --
.020
/
/
./ /
v-( /
v
v
/
.025
/v
CONFIDENTIAL
22
TRUNCATED CONICAL NOSE
MARK 13-1
MARK 14-1 TORPEDO AFTERBOD Y
TORPEDO NOSE
MARK 13-1 TORPEDO AFTERBODY
MARK 13-1 TORPEDO FINS
LYON FORM"A" AFTERBODY
Fig. 21-Comparison of Nose and Afterbody Shapes
I
140
I
V lY
030
F('I')=F1 ('I')+F2 ('1') =0
3
I~
~a
120
~
100
~
/
v
v 'I
80
LL."'
/
a::
0
~
vv
60
l/
/
vj 1]Yj/
o/v
'rfj /]//
v/ /
40
20
-
A
r--
/
E;:::/
2
4
6
8
.L"r
/}
.020
u.
cl
v
0
0~
lO
.015
a::
f--
10
- - - -·
..........
0
_J
<I
:::>
010
0
(f)
lD
lD
I{)
">
a::
a::
LJ..J 0
hi
a::
v
L 1--
v
I 0
0
<I X
~
cv·
--
~
/v
j
vv ~
v v/
l7 [. . . J ....-··
f1 /.
v V; v 7
/
/ / /
A v v/ v
~ / v .....~? f-7
~ ~ v ~ 1--
LL.-
00
w~
,' w
E
. 025
LL.
005
I
12
14
16
18
SLENDERNESS RATIO-'!'
('I'= LID)
Fig. 22-Effect of Several Body Shapes on the Maximum Point
EFFECT OF INCREASING THE
SURFACE AREA OF A BODY
As the Lyon Form "A" has a low residual drag, it is
a desirable shape for the nose and afterbody sections
of an underwater body. Therefore, attention is given
here to the effect of increasing its surface area, either
by adding cylindrical midsections or by adding stabilizing fins. Fig. 23 is an outline of the Lyon Form "A"
with imaginary fins drawn in. To have a convenient
means of expressing the surface area percentagewise,
independent of a change in slenderness ratio, imagine
that the body is enclosed by a cy Iinder the same length
as the body.
In Fig. 24 the various F 2 ( t.j; ) curves are again the
same as those used in Fig. 22. The F ( t.j;) curves in
.
I
F 1g. 24 are plotted for a range of 80 to 110 per cent of
the surface area of the enclosing cylinder (ends not included). The limiting curve on the left is for the bare
body without fins. The addition of fins to the bare body
will cause an appreciable increase in the surface area.
For this body to be stable, we conclude that the optimum
slenderness ratio will probably be greater than the
minimum possible value of five.
CONFIDENTIAL
CONFIDENTIAL
23
Fig. 23-Lyon Form "!>:' Body Enclosed by a Cylinder
030
I I I I I I
I- F
120
I-I
1
u.."'
('It), LYON FORM "A"
I I I I I
+ F2 ('¥) =0
v
80
cr
0
~
60
I ~ I ) v; I
1f1/I llj 1 I I A .020
lA 'II If/V; I y /
1/ IJ; 11rll 1/ f"; I .015
/ bVII //1/1r; / 1/ ~
1------
WITHOUT FINS
F 1('¥) = F1('¥)
100
~
1/I/ I l!/ I ]. 025
V/; V; I 1/ A V/
I
140
V/ Vjj II I 7;[7A-V/
/ / j W; 7; /; ./T [7 7
/ /
1Mll j; lj -!; I I v---7 _). /
/ /
/ ~ IJJ V1 l j I v.....t '-1 v
~ A ~ 0 ~ (_J 7 v / v
/ 'A w~ // v/ / / v .L7 ~ ~ V/.: ~ fS7 7 7 /
/
40
~
PER CENT OF
SURFACE AREA
OF ENCLOSING
CYLINDER--------
20
0
0
2
4
~~
0~
I
0
(.9
<[
X
cr
0
_j
<[
.010
A
u.:-
LL.
Ow
lD
lD
.n
">
::::> a:
0
(f)
a:
0
w u..
cr
-
.005
~
~16 r/.~{n ( / / / ,{o·o/.
6
8
10
12
14
16
18
SLENDERNESS RATIO- 'if
('i'=L/D)
Fig. 24-Effect of Varying the Surface Area of a Body
It Is assumed that there is no change in the body shape and that the area IS
increased by the addition of fins, etc.
EFFECT OF CHANGING THE
Rv
VOLUME REYNOLDS NUMBER
If the slope of the residual drag curve is the same as
that of the flat plate curve (m=n) then
~n-m)
=
optimum value of the slenderness ratio. However, if
the slope of the residual drag curve is g reater than that
of the flat plate (m>n), an increase in the value of
decreases the optimum slenderness ratio (see Fig. 25).
On the other hand, if the slope of the residual drag
curve is less than that of the flat plate or equal to zero,
an increase in Rv causes an increase in the optimum
slenderness ratio (see Fig. 26).
It is therefore impossible to make a general statement
as to the effect of changing volume Reynolds number on
the optimum value of the slenderness ratio.
1
and the volume Reynolds number has no effect on the
CONFIDENTIAL
24
140
v
/ lv
vA
120
/
/
/
/ v V1~ ~
/ / ~ ~ '?" v
~
LLN
80
v/
/
/
v
~ fq / / ~ ~
V/ ~ ~/ ~ ~ p
~~ / ~~~
~~ ~ ~
~~ /
0
v
~
u..- 60
40
/
/
/
/
2
>
cr
/
:::::-
~ ~~
-;::/
v:
/
v
20
v
/ / v::::
~ ~ '/
v ~b /: ~ v
100
cr
v
V1 v /
F 1 {'it) = F1 ('I')+ F2 {'it)= 0
0
0
2
4
6
8
10
12
14
16
18
SLENDERNESS RATIO-'¥
('¥=LID)
Fig. 25-Effect of Varying Volume Reynolds Number
This chart is based on the MK 13-1 torpedo nose and afterbody. It is important to
note that the residual drag curve (Fig. 8) has a steeper slope than the flat plate
skin friction curve (see Eq. 15) ).
1/
120
F 1{'i')• F1 {'I')+ F2 {'l')= 0
/2
.....
I
1-.
I
100
!/~//
~ ~::tv/v
cr
~
~I8d
v; ~
vv /
60
~ ?1 ~ ~
u..-
.0. ~ ~ ~
/
/
~
6
4
CI
/
/
~~~;;?
(,_'::j~
CJ<fi.. /
80
u.."'
0
~
~'_,-;-;/;~
!:}
Z:
_I/
~§I
~
140
/I
/
_..;;I
r::::::
b:::::::: ~ r-.....-
~ ~ ~ t::::
~
~ v _j. ~ ~ ~ ---- ---/
~ ~ ~ :::::::: r::--- ----
40
:;:;..-
.&
20
t7'"
~ r:::/
/
2
4
6
v
8
10
12
14
16
18
SLENDERNESS RATIO-'¥
('¥= L/ D)
Fig. 26-Effect of Varying Volume Reynolds t~umber
In this chart it is assumed that the residual drag is constant-two different values
being used on the chart.
CONFIDENTIAL
CONFIDENTIAL
25
v
140
I
1/
120
I
11
100
/
~
I.J.."'
~l j
80
0
!k:
0
~
1/
GO
I
I.J._-
1
F ('if)= F, ('if)+ F2('¥) = 0
40
20
r-
v
0
0
2
4
h
v
~
--v
6
I
~
8
-- --~
f \.'¥)
10
v
~
12
14
16
18
SLENDERNESS RATIO-'ll
('I'= L/D)
Fig. 27-Maximum Point Chart for MK 13-1 Torpedo and Afterbody Shape
The chart is calculated for an assumed constant residual drag value of 0.013 and
an assumed constant skin friction value of 0.00229. This chart gives the maximum
point of Fig. 17.
CONSTANT RESIDUAL DRAG AND
CONSTANT FRICTIONAL RESISTANCE
Let us assume that the form drag and the frictional
resistance coefficients are constant as Reynolds number changes. In this case n and m are zero and CDF
equals M, and CFP equals N. Eq. (46) then becomes
(50)
and Eq . (47) becomes
(51)
Evaluation of Eq. (50) for the MK 13-1 shape gives
the F}.(lj;) curve in Fig. 27. Usi ng the same values of
CDF as in Fig.l7, the result is F 2 (lj;) curve in Fig. 27.
in Fig. 27.
Thus, we see that for constant residual and frictional
resistance, the optimum slenderness ratio is still
dependent on their values.
26
CONFIDENTIAL
CONCLUSION
BIBLIOGRAPHY
A body that has a very low drag coefficient must
necessarily have no appendages such as fins, propellers, struts, lugs, etcetera. An example is the Lyon
Form A which has a very low form drag ·c oefficient.
With no cylindrical midsection, its slenderness ratio is
five, and after examining Fig. 24, it is obvious that
this figure would be the optimum value. However, to be
of practical value, an undersea body must be dynamically stable and maneuverable, for which fins, rudders,
and diving planes are required. For propulsion, there
are the struts and propellers. Because of these appendages, the residual drag of the body will be appreciable
and the surface area will be enlarged. As pointed out in
this article, both factors tend to increase the optimum
slenderness ratio. To continue the example, consider
the same Lyon Form with stabilizing and control surfaces and propellers. With these appendages the optimum
slenderness ratio will increase to at least eight or nine.
An examination of a typical characteristic curve (see
Fig. 11 or Fig. 13) of a stable body with fins will show
that in the region of the maximum point the curve is
relatively flat. Therefore, any reasonable variation from
the theoretical optimum value will have little practical
effect on the velocity of the body. To carry the example
one step further, this same body with app.endages could
be constructed with a slenderness ratio as low as 5 or
as high as 13 without reducing the maximum velocity ror
given power more than two per cent.
Because of practical factors involved in the design of
an undersea body, it may be desirable from the designer's point of view to have a relatively large slenderness
ratio. This investigation shows that as far as drag per
unit volume is concerned, the designer will pay very
little if any penalty if he disregards the drag factor and
bases his selection of slenderness ratio entirely on
such items as tactical requirements of maneuverability,
structural design and utilization of internal space.
Doolittle, Harold L., Tests of the MK 13-1 Torpedo with
Various Noses, HML Report NO 15.4, CIT, Feb. 1, 1945.
2 Davidson, Kenneth S.M ., Resistance and Powering,
Chapter II, Volume II of Principles of Naval Architecture,
Rossell and Chapman, 1941.
3 Schoenherr, Karl E., Resistance of Flat Surfaces Moving
Through a Fluid, Transactions of the Society of Naval Architects and Marine Engineers, Vol. 40, pp. 279-313, 1932.
4 Lyon, Hilda M., Effect of Turbul ence o'h Drag of Airs hip
Models, A.R.C. R.&M., No. 1511, 1932.
5 Knapp, Levy, O'Neill, and Brown, The Hydrodynamic s
Laboratory of the California Institute of Technology, Trans.
A.S.M.E., pp. 437 to 457, July, 1948.
CONFIDENTIAL
CONFIDENTIAL
APPENDIX
(tj;)
= 6 .00
( tj;)
=
(tj;)
7.18
= 9.70
( tj; )
=
11. 70
(tj;)
=
13.71
Fig. 28-Sienderness Ratio Study Series
In this series the nose and afterbody are the same as the MK 14-1 torpedo shape.
The two-inch diameter model has a spoiler of 0.005" wire one-half diameter from
the nose. See Fig. 5 for a typical installation of the model.
27
cm~FIDENTIAL
28
.5
.4
0
0
0
"', 6 .00
1:;.
"' , 9 .70
"' . 7 . 18
0
d
"' • 11 .71
u:' . 3
0'
"' . 13.71
w
I--
0
11/f •LID)
-
0
(.!)
<(
K
,..,;
lr<
0:: .2
0
,..,;
lr'l
c.
_J
.r'i
lr<
u
u
<(
'-
1..
n
)-
0
)-
0
-"
.I
10
()
0
0
0
0
v
.
u
A
__()
2
4
3
0
0
()
0
-"'
5
7
6
c
[
-
8
9
REYNOLDS NUMBER, Re
Fig. 29- Total Drag Coefficient vs. Reynolds Number
The experimental points were obtained using the models shown in Fig. 28 .
.09
0
.08
... .07
0
0
u:
w
0
.06
.05
.04
1-'n
\..
0::
0
Q,J
1:;.
"' .
d
"', 11 .71
[Y
"' , 13 .71
9 . 70
n
v
'-'
~0
Fl
_J
~
"' , 6 .00
"', 7. 18
( 1/f •L I D)
auo
(.!)
<(
jJ
lS
0
0
0
8
:d
.03
0
,J
I
0
g
0
·c:,.
'z:
FJ
c
p
c
(
I
c
(f)
w
0::
.02
10°
2
3
4
5
6
REYNOLDS NUMBER, Re
Fig. 3D-Residual Drag Coefficient vs. Reynolds
l~umber
The experimental points are calculated by usin g Eqs. ( 16) and ( 17).
CONFIDENTIAL
7
8
9
29
CONFIDENTIAL
40
3
2
4
5
6
30 50
7
60
8
70
9
80
40
10
90
100
(.)
w
({)
'IJ...
1-
I
>-
50
1(.)
0
_J
w
>
_J
<t
40
60
z
50
70
60
3
4
5
6 7 8 9 10
100
VOLUME OF BOI;)Y- FT
0
2
4
6
8
3
90
80
70
NEGATIVE BUOYANCY
y b - Yt , L 8 S I F T 3
10
14
12
SLENDERNESS
16
18
20
RATIO-~
(~=LID)
Fig. 31-Sinking Body Velocity Chart for IAK 14-1 Torpedo Nose and Afterbody
shape
This chart is designed for a free sinking body of a size and velocity range suitable for a depth charge. The characteristic curve is calculated on the basis of the
data in Fig. 30 and for an average volume Reynolds number based on the rang e
of velocities and volumes indicated in the chart.
22
CONFIDENTIAL
30
VELOCITY- FT/SEC
so
70
I
f
60
50
I
40
I
EFFICIENCYPERCENT
I
100 90 80 70
80
90
100
0:::: 120
w 140
3: 160
0
a.. 180
w 200
60
50
40
30
(f)
0:::: 250
0
:z::. 300
1- 350
::::>
a.. 400
~ 450
500
"'1LL..
I 50
>-
45
0 40
co 35
0
1.1... 30
0
w 25
:::2:
::::> 20
_]
0
>
0
2
4
6
8
10
12
SLENDERNESS
14
16
18
20
22
RATIO-'¥
('l'=L/D)
Fig. 32-Horizontally Moving Body Velocity Chart
Afterbody Shape
for MK 14-1 Torpedo and
This chart is desi gned for a horizontally moving body in the size and horsepower
range of the torpedo. The characteristic c urve is calculated on the same basis
as the characteristic curve in Fi g . 31.
24
CONFIDENTIAL
(~)
= 6 .00
(~)
=
( ~) :-
(~ )
-
(~) =
7.18
9.70
11. 7 1
13.7 1
Fig. 33-Sienderness Ratio Study Series
In this series the nose in a truncated cone faired into the body. The afterbody is
the MK 13-1 torpedo shape. See Fig. 5 for a typical installation of the model.
31
CONFIDENTIAL
32
.5
.4
0
Lc.
(.)
u..:w
h'
.3
k
CJ.
0
<::>
r
0
t.
t.
1:::. t .
cf t .
[]" t .
0
0
_J
~
0
1--
{of
.I
10
...n.
,...,
--v
~ .2
r-(
..r
~
n
1.,-
ri
r
r
~
_n
(.)
1-r
k
LS
~
n
....0.
_n
..n.
[)
_n
r
6 .00
7.1 8
9 . 70
11.71
13.71
•L70)
'
2
3
REYNOLD
4
7
6
5
9
8
10
1
NUMBER, R e
Fig. 34- Total Drag Coefficient vs. Reynolds
t~umber
The experimental points were obtained using the models shown in Fig. 3.1 .
.04
(.)0
....
t.
t .
0
0
1:::.
~
.03
6 .00
7. 18
+•
(.)
0 t .
a' .p .
<::>
(of
w
0
<! .02
9 . 70
11 .7 1
13.71
•L/ 0)
cr
0
_J
<!
::::>
0
y
Q
<fl
w
2':.
cr
.0 1 10'
c
I~
~
kR
~
2
REYN OLDS
C:0;J
~
1:-tR C
~
3
4
J:1
()1" "[,....,
v
l:'.
5
:r:o_
p
6
NUMBER , Re
Fig. 35-Residual Drag Coefficient vs. Reynolds Number
The experime ntal points are calculated by u si ng Eqs . (16) and (17)
CONFIDENTIAL
__IT trc
7
i "rl~
[)~
j
c.
~
8
9
CONFIDENTIAL
40
30
33
2
3
4
5
6
50
7
60
8
9
40
10
(.)
w
CJ)
.......
1I.J...
50
I
>-
1(.)
0
_J
w
>
_J
<t
40
60
z
~
a:::
w
1-
50
70
60
3
4
5
6 7 8 9 10
100
VOLUME OF BODY- FT
0
2
4
6
8
3
90
80
70
NEGATIVE BUOYANCY
yb-yf, LBS/ FT 3
10
12
14
16
18
20
SLENDER NESS RAT 10- 'I'
('I'=L/D)
Fig. 36-Sinking Elody Velocity Chart for Truncated t·~ose and IAK 13-1 Afterbody Shape
This chart is designed for a free sinking body of a size and velocity range
suitable for a depth charge. The characteristic curve is calculated on the basis
of the data in Fig. 35 and for an average volume Reynolds number based on
the range of velocities and volumes indicated on the chart.
22
CONFIDENTIAL
34
VELOCITY- FT /SEC
80
70
I
I
60
50
I
40
I
I
EFFICIENCYPERCENT
f
100 90 80 70
80
90
100
0::: 120
w 140
~
0 160
a.. 180
w 200
60
50
40
30
1- 350
60
70
80
90
100
a.. 400
z 450
140
(/)
0::: 250
0
~
300
:;)
120
500
160
180
50
50
4
0
350
50 45
40
35
500
450
130
25
400
20
,
1LL..
I 50
>Q
45
0 40
(l) 35
LL.. 30
0
w
25
~
:;) 20
_J
0
>
0
2
4
6
8
10
12
SLENDERNESS
14
16
18
20
22
RATIO-'ll
('l'= LID)
Fig. 37-Horizontally Moving Body Velocity Chart for Truncated Nose and
MK 13-1 Afterbody Shape
This ch art is desig ned for a horizonta lly moving body in t h e si ze and horsepo wer
ra nge o f the torp e do . Th e characteris tic curve i s calculated on the same b as i s as
the charac t eri s tic curve i n Fig . 36 .
24
CONFIDENTIAL
DISTRIBUTION LIST
9
Director, Naval Research Laboratory, Washington 25, D.C., Attn: Technical Information
Officer
6
Office of Naval Research, Department of the Navy, Washington 25, D.C., Attn: Mechanics
Branch (Code 438)
1
Commanding Officer, Branch Office, U.S. Navy, Office of Naval Research, 495 Summer
Street, Boston 10, Massachusetts
2
Commanding Officer, Branch Office, U.S. Navy, Office of Naval Research, 50 Church
Street, New York 7, N.Y.
1
Commanding Officer, Branch Office, U.S. Navy, Office of Naval Research, 844 North
Rush Street, Chicago 11, Illinois
1
Commanding Officer, Branch Office, U.S. Navy, Office of Naval Research, 801 Donahue
Street, San Francisco 24, California
2
Commanding Officer, Branch Office, U.S. Navy, Office of Naval Research, 1030 East
Green Street, Pasadena 1, California
2
Assistant Naval Attache for Research, U.S. Navy, Office of Naval Research, American
Embassy, London, England, Navy 100, F.P.O., New York, N.Y.
1
Chief, Bureau of Ships, Department of the Navy, Washington 25, D.C.
1
Bureau of Ships, Department of the Navy, Washington 25, D.C., Attn: Ship Design
Division (Code 410)
1
Bureau of Ships, Department of the Navy, Washington 25, D.C., Attn: Research Division
(Code 330)
2
Bureau of Ships, Department of the Navy, Washington 25, D.C., Attn: Preliminary Design
(Code 420)
1
Director, David W. Taylor Model Basin, Department of the Navy, Washington 7, D.C.
2
David Taylor Model Basin, Department of the Navy, Washington 7, D.C., Attn: Hydromechanics Division
1
Dr. K.S.M. Davidson, Director, Experimental Towing Tank, Stevens Institute of
Technology, Hoboken, New Jersey.
1
Commanding Officer, Office of Naval Research, New York Branch, Bldg. No. 3, Tenth
Floor, New York Naval Shipyard, Brooklyn 1, New York