defense documentation center · d = d /(v r m m v w"^ 0(7,r,') algebraic ftmctlon defined by...

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HYDRONAUTICS, incorporated research in hydrodynamics

Research, consulting, and advanced engineering in Ihe fields of NAVAL and INDUSTRIAL HYDRODYNAMICS. Offices and Laboraiory in the Washington, D. C, area: Pindell School Road, Howard County, Laurel, Md.

HYDRONAUTICS, Incorporated

TECHNICAL REPORT 3^9-1

AN ANALYTICAL STUDY OF THE HYDRODYNAMIC PERFORMANCE OF

WINGED, HIGH-DENSITY TORPEDOES

By

C. F. Chen

October 1963

Prepared Under

Office of Naval Research Department of the Navy-

Contract No. Nonr-4083(O0) NR 277-004


-x

TABLE OF CONTENTS

ABSTRACT ,.......,.............••••••••••

INTRODUCTION .............

METHODS OF CALCULATION 2

Page

1

2

A. Drag ......................••••••••••••••••••'■•••••'• (1) Body Drag • • •

( 2) Tal 1 Drag 7

(3 ) Wing Drag • 17 B. Range i

RESULTS AND DISCUSSIONS .................••..•.••••••••••• 19

A. Lift-Drag Ratio 19 22 B. Range ,.,................••-•.•••'••••••••• •'••

CONCLUSIONS ........,...........••.••'

REFERENCES • • • • • •

23

25

ItYDRONAUTICS, Incorporated

-11-

LIST OF FIGURES

1. Effect of Wing-Body Influence on the Total Lift

2. Weight Coefficient as a Function of Forward Speed

3. Lift-Drag Ratio vs Buoyancy Ratio, Rev = 12 x ICf ,

s'z" = 0.1, D = .481 HI

4. Drag Break-downfor C = 0.010,Re = 12 x 107 , s'z = 0.7.

(Optimum Body)

5. Lift-Drag Ratio vs Buoyancy Ratio (Optimum Body Only)

Re = 6 x ICf , s'zf = 0.7 v

6. Lift-Drag Ratio vs Buoyancy Ratio, Rev = 6 x ICf , s'z = 33

7. Effect of D on Lift-Drag Ratio, Re = 6 x l(f , s'z = 33 ' m v

8. Lift-Drag Ratio vs Buoyancy Ratio, Re^ = 4 x 107, s'z = .7

9. Lift-Drag Ratio vs Buoyancy Ratio, Rev = 8 x 107 , s'z" = 0.7

10. Increase in Lift-Drag Ratio Attainable, Wing Aspect Ratio

and Span as Functions of C . Re = 1.2 x ICf , s'z = 0.1,

D =0.481 m


and Span as Functions of C . Rev = 6 x ICf , s'z = 0.7,

D =0.481 m


13

and Span as Functions of C

Effect of Re on A(W/D)

Re = 6 x v

ICf, s'z = 33


-111-

14.

15.

16.

17.

18.

19.

20.

21.

22.

Torpedo Configurations (3000 lb. 45 knots)

The Influence of Buoyancy Ratio on Range. (Fuel Weight

= .5 x displacement) IT = 0.01, Re = 12 x 107 , s'z = 0.7


= .5 x displacement) "C" = 020^0 w

Influence of Buoyancy Ratio on Range, (Fuel Weight

= .5 x displacement) ^ - .04


= .5 x displacement) C" = .06250. W

The Influence of Buoyancy Ratio on Range, (Fuel Weight

= .5 W) Cf = ,01, Re = 12 x 107 , s'z = .7

The Influence of Buoyancy Ratio on Range, (Fuel Weight

= .5 W) H = .02040.,


= .5 W) Ü = .04


= .5 W) C = .06250


-Iv-

NOTATIOM

4(b -Vl - Y3^)' A aspect ratio of wing = ^

w D

A aspect ratio of the ring tall = rt

B buoyancy force

b distance from center of torpedo to tip of wing

I.e. specific fuel consumption: lb-fuel/HP-Hr.

C chord of wing

C. total drag coefficient based on the wetted surface A ö

area of a body of revolution

C drag coefficient

C together with K defines the profile drag of the wing

section

C^ drag coefficient based on (V ) D ■ w

C friction coefficient

C total lift coefficient based on exposed wing area L

C lift coefficient developed on the wing Lw

C lift curve slope

vb C prismatic coefficient =

4 m

C . chord of ring tall Sb C surface coefficient =

S TV D I m


C weight coefficient =

- v -

W

w D total drag

D maximum diameter of torpedo m

D = D /(V r m m v w"^

0(7,R,') algebraic ftmctlon defined by Equation [30]

I moment of Inertia

K, K constants 1

L lift

-t length of torpedo

i. length of central cylindrical portion of the torpedo G

t distance from the e.g. of torpedo to the c,p. of

ring tall

M pitching moment

q dynamic pressure = ip if TZ»

R buoyancy ratio = —• W

R.^ radius of body

V =Vb Re. Reynolds number based on -t

Re Reynolds number based on C

Re Reynolds number based on V v w

S range in miles

S, surface area of body


-vi-

w S ' v

wing planform area

ratio of the wetted area of the supporting vanes

to area of ring tail

surface area of ring tall

maximum allowable stress

= sV ^ W w

T

t

U

w V v

W

WI

x

max

a

7

thrust

thickness ratio = D /-t m

uniform stream velocity

volume of the torpedo

volume of water whose weight is W, V w W P S rw

total weight of torpedo

total fuel weight

distance from the nose of a body of revolution to the

transition point

distance from vertical axis to the outermost fiber

nondlmenslonal sectional modulus=

^max angle of attack

yR. Is the distance from the centerllne of the

torpedo to a horizontally mounted wing.

propulsive efficiency


climbing angle

density of body

-vll-

'w density of water

subscripts

b body

r wing root

t tall, or wing tip

w wing


-1-

ABSTRACT

The hydrodynamlc performance of winged torpedoes has been

Investigated. It Is assumed that the wing Is mounted such

that Its center of pressure coincides with the center of gravity

of the body, and that a ring-tall stabilizer is used. The

resulting overall lift-drag ratios are presented as functions

of the buoyancy-weight ratio. The advantages of the winged

torpedoes over those with neutral buoyancy become more pro-

nounced as the nondimenslonal parameter C becomes smaller., and

as the buoyancy-weight ratio decreases, C is defined as w

w/i^U^V )a , where W Is the weight; ^p if, the free stream

dynamic pressure; and V Is the volume of water with weight., W. w

Parametric studies of the lift-drag ratio obtainable have been

carried out with respect to the following quantities: the

, Ä Reynolds number based on (V ) , the nondimenslonal strength

factor of the wing material^ and the maximum allowable diameter.

In conclusion, the ranges obtainable for the ascending,, level,

and descending flights are calculated. It is found that If the

fuel weight is assumed to be always one-half of the displace-

ment, winged torpedoes have longer ranges than those with

neutral buoyancy only for descending missions. If the fuel

weight, however, is assumed to be one-half of the total weight,

the winged torpedo has longer ranges for all missions except

steep ascending ones.


-2-

INTRODUCTION

Up to now, torpedoes have been generally designed with

neutral buoyancy. If one departs from this concept and con-

siders a winged torpedo, for which part of the weight Is

carried by the dynamic lift generated on the wings, a more

efficient torpedo, depending on the design weight and speed,

may be evolved. In this report, a preliminary Investigation

has been made on the overall lift-drag ratio obtainable, the

size of the wing needed, and the range of travel for winged tor-

pedoes at different values of buoyancy to weight ratios.

In the calculations, it is assumed that the wing is mounted

such that its center of pressure coincides in axial position

with that of the center of gravity of the torpedo. The assump-

tion simplifies the tail selection for which considerations of

stability must be made. We also assume that a ring tall is used

to provide the required stability. Two types of bodies are

considered: one is the optimum body for a given volume; the

other Is a diameter-limited body (i.e., limited to the maximum

allowable size for use in existing torpedo tubes). In the

following, the methods by which the component drags and the

range are calculated are developed and the results of such cal-

culations are presented and discussed.

METHODS OP CALCULATION

A. Drag

Let the weight of the torpedo be W; let the drag due to the

body, tall, and wing be denoted by lL$ Dt, and Dw respectively.

The overall lift to drag ratio Is


-3-

W W [1]

The buoyancy to weight ratio, or In other words the ratio of

the density

noted by R,

the-density of water p . to that of the torpedo, p , Is de W

B W !b R B VbPwS P_w

R ^ w = v.rpMg = v ' w vbPbs Pb

[2]

wrw

where B denotes the total buoyancy force. To normalize the

forces, we use the dynamic pressure of the uniform stream.

4D U2 and (V )3, where V Is the volume of water whose weight 2"w ' w W Is W:

w PwS [33

For a constant weight and constant forward speed U, the normal-

izing force, ipwU2 (Vw)

3, is constant. Let

C s w 1

W

2P, if (V. w

[4]

It then follows that

W D

w

+ Cn + C [5]


In the following, we examine each drag component separately.

(1) Body Drag

In the total drag of the body, (skin friction plus form

drag), we use the results of Young (Reference 1), Young's

calculation of total drag coefficients based on wetted surface

area, C , are presented In graphical form. To facilitate compu-

tation by the IBM 1620 digital computer, the process of curve-

fitting has been carried out to give the following formula:

= 10-sx [.035(g - log Re. )s+ .215 - .24 x/l - (.2 - t)Y]. [6]

1 o

Y =

78 - .28(log Re. - 6) + . 12( log Re - 7)3 t > .2, 1 o "k v

.37 - .12(log Re. - 6) + .04(log Re - 7f t < .2,

C7]

in which Re. denotes the Reynolds number based on body length

I; x, distance from the nose to the transition pointj t, the

thickness ratio. In the optimum body, t = 0.2, and the ex-

pression for C is considerably simplified.

Since the comparisons will be carried out at constant speed

and weight, the Reynolds number based on Vw is considered

known:

UV i Re w LB]

To calculate Re^ from Rev, we first define the (logltudinal)

prismatic coefficient C as


_5_

V,

pr wTfl 4 m

[93

where V Is the volume and D the maximum diameter of the body, b KL

By the definitions of the buoyancy —■ weight ratio and the

thickness ratio, we obtain from [9]

v = ]Lc t3l3

w ^R pr [10]

The Reynolds number based on body length Is then

Re Re. = . r- •

To obtain C we define a surface coefficient C^

V

[11]

TTD I m

[12]

where S Is the wetted area of the body. By simple algebra, we b

obtain

C^ = C.C DL AS b

le-irR2 I*

c 8 t pr [13]

where C Is obtainable from Equations [6] and [?].

For deflnlteness, we shall consider that the optimum body Is

a spheroid of thickness ratio 0.2, and the diameter-limited body


-6-

1s made up of a central cylindrical section of length I and two

semi-spheroidal noses. In such a composite body, the prismatic

and surface coefficients are easily computed to be

2+1/1 c pr

and

C = , s I *„ -t i i

+ 4 [(i - -r (i - t3)"2 sln-^l - t3)2 + t]

[14]

[15]

For a spheroid, -6/6 = 0,

V = 2/3 - i 1

C = ±[(1 - t3)-2 sln-^l - t3)2 + t] s

Cl6a]

[16b]

_ ~i Let D = D V 3 be the nondlmenslonal maximum diameter. By the m m w

definition of the prismatic coefficient, we can write the

thickness ratio t as

TT t = T^ (2 + * A)D.3 . 12 ' c ' m [17]

The volume of the body is

TTD 3

m Vb =^-^(2 + V^

^^slwV^fl2 + V^ [18]


-7-

In the optimum body, t = 0.2, and C = 2/3. Cs may be

calculated according to [l6b]„ Equation [13] together with

[6] gives the required value of C . For a diameter-limited b

body, I /I and D must be specified. From these values, C

and t are calculated according to [14] and [171 respectively.

C may be obtained by Clg], together with the value of CA ob-

tainable from [6] and 111, and "cL can again be calculated, ■ Db

(2) Tall Drag

We assume that the petition of the wing is such chat it

does not contribute to the moment when the entire configura-

tion is pitched. According to experience, the static restoring

moment generated by the tail is usually some fraction of the

static destabilizing moment generated by the body. The con-

figuration must be dynamically stable„ We design the tail such

that

M = K II , K > 0, [19]

where M and M denote moments due to the tail and the body t b

respectively; K is a constant. K > 0 implies that the moments

are opposite in sense. By slender body theory, at a pitch

angle a [20]

w

The moment generated by the tail may be written In terms of the

lift curve slope of the tail, G t, the dynamic pressure q, the


-8-

tall surface area, S , and the distance from the c,

body to the tall center of pressure, l^:

of the

t Lat t t [21]

Equation [19] gives then

For a ring tall

Lat

TTK C t3l3 1 pr

2Stlt

[22]

S^ = TTD C , , m rt

D m

rt [23]

In which C , Is the chord of the ring tall. The diameter of rt

the ring tall Is taken as the maximum diameter of the body.

Using the definition S and A , [22] becomes

Lat i K C l/l. 2 1 pr t [24]

Welsslnger (Reference 2) gives the effectiveness of a ring

wing approximately as

Lat i + i L + tan- i^j . [25]


- 9 "

Solving [24] and C25] simultaneously, we obtain the aspect

ratio of the tall A , thence

L wB 1.2 At + tan ..-^ TT

. .K C l/l. - 1 2 i pr ' t [26]

To determine the frlctlonal drag, we calculate the Reynolds

number of the tall

UC Re, =

t v rt 7— Re, .

At l [27]

The drag of the tall non-dlmenslonallzed with respect to qVw

Is

^STt^l*/1 + SV [28] pr

where S ' Is ratio of the wetted surface area of the supporting v

vanes to that of the tall, and Cf Is the friction coefficient„

(3) Wing Drag.

The drag of the wing can only be determined after the wing

has been sized. Preliminary considerations . show that the wings

may be of small aspect ratio, say near 1, and of small span.

The Interference effects between the wing and the body must now be

considered.

At the design condition, the body is considered to be

at zero angle of attack, while the wing is set so that the desired

HYDRONAUTTCS, Incorporated

-10-

amount of lift Is developed. Lawrence and Flax (Reference 3)

have considered the Interference lift generated on an Infinite

circular cylinder due to a finite lifting line of arbitrary-

lift distribution mounted therein. The problem is solved in

the Trefft* plane by the use of Green's theorem. The Inter-

ference lift generated on the body can be obtained by an

integration along the wing of the product of the lift distribution

and the velocity component normal to the wing due to a vertical

translation of the cylinder. In their paper, Lawrence and Flax

(Reference 3) calculated the Interference lift on the cylinder

due to a constant, elliptic, and optimum lift distribution on a

centrally-mounted,horizontal lifting line. The results do not

differ appreciably. Furthermore, they have shown that the lift

on the body subsides in about two diameters in either direction

from the lifting line; the idealization of an infinite cylinder

for the body is thus not without justification. In the present

case, we have calculated the total lift of the wing-body system

(lift on wing + Interference lift on body) for a horizontal

lifting line of constant lift distribution, whose tip to tip dis-

tance is 2b, mounted yK above the center of the cylinder of

radius R. . Defining the lift coefficient with respect to the

wing area S , we obtain w

CL " qS w

KT*V)

= c 0(7, R: ) , % "b

i - (i - y3 )Rb':

[1 - (1 - Y*)2 Rb'](l + Y'V3)

[29]

, C30]


-11-

where R ' = R^A • For wings of large span, R^' - 0, G(Y,0) = 1,

then

w

The function G is shown in Figure 1 for different values of y.

It is seen that centrally mounted wings are the most efficient

in producing the interference lift on the body. Eowever, for

the present application, the wings will necessarily be stored

within the body prior to and during the launch and a centrally

mounted wing may not be feasible.

Before we select the wing which will carry the required

lift while incurring the least amount of drag for our present

winged torpedo system, it is instructive to consider briefly the

optimum design of an Isolated wing. In general, the drag

coefficient of a wing may be written as

CD = CD +KCL3 +dr ^1 + 5) ' o

[31]

in which the first two terms are the profile drag of the wing

section and the last term is the Induced drag; & Is the factor

Introduced to take into account that the loading may not be

elliptical. Dividing [13] through by CL, and finding the

minimum C /C we obtain .U J-j

"1, mln = 2 C

D K

1+5 irA

[32]


-12-

In which the optimizing components are

L opt [33]

and N opt 2 C D C34] It Is seen that the larger the aspect ratio, the smaller the

resulting drag for the same lift. However, the aspect ratio Is

limited by the bendjlng stress allowable at the mid-span. For

elliptic loading, the bending moment at the mid-span is

^— Lb = Sir y.

[35] max

in which I is the moment of inertia and y is the distance from max

the neutral axis to the outermost fiber, and s is the tensile

stress at the outermost fiber. Rewriting and transposing, as-

suming, a rectangular wing we obtain

CTA3 = STT z - , L q [36]

in which z is the nondlmenslonal sectional modulus

z = y C3 max

[37]

HyDRONAUTICS, Incorporated

-13 •

When the working stress of the wing material is given. Equations

[33] and C36] determine the optimum aspect ratio and C^,

Returning now to the winged torpedo, we follow the same

procedure as above with the added complications that: (a) the

profile drag is only found on the exposed wing, while the Induced

drag may be calculated according to the full span 2b, (b) the

maximum bending stress is at the wing-body juncture, (c) the

lift developed by the wing is some fraction of the total lift

required depending upon the position of the wing with respect to

the center of the body, and (d) we allow the wing to be tapered.

With these considerations, the counterpart of [333 Is

s* K 1+6 irA G(Y,V

)[1 - (! ~ ^)2V]

C38]

and that of C36] is

CT A? w

2%

(1 + ct/cr)£

C39]

\vhere C and C denote the chord at the tip and the root respec-

tivelv, and z is the nondimenslonal sectional modulus of the J r root section. The aspect ratio A is that of the exposed wing.

Let now the working stress be nondlmensionallzed with respect to

WV , and writing it as s', C393 becomes w


-14-

As = 2^Tr z s- C C40]

jw (i + ct/crr

Because of the added variable R. ', another equation Is needed.

This Is obtained by considering the relation between CL and W: w

W(l - R)

-w Giy^H 3^

Oil - R) V. w

2

w [41]

w b ' w

Rewriting the wing area S In terms of aspect ratio and V w

according to [10], we obtain

C (1 - R) w

ir J vllvig^ [42] w G(Y.,Rb') [1 - (1 - ^)

2 Rb']J

Equations [38], [39]. and [42] determine CL , A, and R^ • for w

given values of y, C . t, ^/C^ z^, C^, R, 5, K, and CD .

Now It Is an easy matter to determine the drag contribution

due to the wing:

D w

D w o T ' [43] q V w


By the use of [^l]^ we obtain

-15-

2C

D w

D C (1 - R) O W

w

[^4]

These expressions have been programmed for computation on

the IBM l620 Computer for the following values of the constants:

K = .012

s = „009 0

5 = 0

Ct/Cr = .5

lc/l = .5

X = 0

Y = 0.7 K ä 0.7

S ' = .5 v

The value of K and 0^ determine the profile drag of the wing o

which should be a function of the Reynolds number. In the

present calculation, they are assumed to be constant. Since the

wing drag Is usually one order of magnitude smaller than the body

drag, the simplification should not Incur too much of an error.

In order to select a suitable range of values for C , we have


-16-

speeds of 10 to 100 knots. The results are shown In Figure 2. It

is seen that the value of C" ranges from 0.01 (at about 100 knots)

to 1 (at about 10 knots). The difference in the weight of the

torpedo has no appreciable effect on the value of 0 . since it

is proportional to W 5.

The coraputation scheme is as follows: For assumed values of

RevJ the product s' z, and B^ the value of C is systematically

varied. For each value of C , the buoyancy ratio R is varied w from 0.2 to 1.0. In each of these cases, the component drags,

the overall llft-to-drag ratio %/B, the lift coefficient C

aspect ratio A and the radius-span ratio R^' are calculated for

both the optimum body (t = 0.2 spheroid) and the diameter limited

body. The following six cases have been computed:

L ' w

1

2

3 4

5

6

Re v

1.2 x itf3

6 x 106

6 x 106

6 x ICP

k x 107

8 x 107

0.7

0.7

33.0

33.0

.7 v

D m

.481

.481

.383

.481

In cases 5 and 6, only optimum bodies are calculated. The re-

sults from cases 1 and 2, together with 5 and 6 give the Reynolds

number effect; cases 2 and 4 give the effect of s' zT; cases

3 and 4 give the effect of D m


-17-

B. Range

Let the torpedo be ascending at an angle 9 with respect

to the horizontal. When In uniform, motion, the thrust balances

the drag and the component of the effective weight. I.e.., weight

less buoyancy, in the direction of flight:

T = (W - B) sin 9 + D [451

Let t.c. be the specific fuel consumption in weight of fuel per

unit power output of the engine, r\ be the propulsive efficiency

The rate at which weight of fuel,W , is being spent is then

dW F dt

•i . c . TU

■n C46]

Rewriting C 46] and using [45], we obtain

t.o. dW Udt F (W B) sin 9 + D [4?]

Assuming now both the specific fuel consumption,^.c., and the

propulsive efficiency, ri, stay essentially constant, the left-hand

side integrates out to be I.e. S/r], where S is the range. In

torpedo applications, the weight of the torpedo is kept constant

by using water to displace the fuel. If the density of the

stored fuel is comparable to that of water, the buoyancy B stays

constant; the total drag of the torpedo D is also a constant.


Then [4?] gives

■18-

l.c. S =

w F W(l-R) sin 6 + D J

[48]

In which W„ is the total fuel weight. Writing in terms of the r

11 ft-drag-ratio., [48] becomes

WT ^■c q "F W/D

w D

(1-1) sin 9 + 1

It is seen from [49], that if 9 is the glide angle

[49]

9 = - sin"1 1

I (-H) [50]

and the torpedo will travel on indefinitely without expending

any fue1.

To facilitate calculation, we use the following units for

the quantities on the left of Equation [49]:

I.e. = lb fuel/horsepower - hour.,

S = range In miles.

Then the range expression becomes

WT ^4^ ^ 375 ^ W/D W § (1-R) sin 9 + 1

miles-lb fuel/horsepower-hour,

[51]

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-19-

Calculatlons of the range factor £.c. S/q have been made

for the optim-um body configuration at Re = 1 „ 2 x 108 and

s' z = 0,7 (corresponding to case l). Two different assumptions

have been made on the fuel weight: in the first case we assume

that the fuel weight equals one-half of the displacement; In

the second case, we assume that it equals one-half of the total

weight.

RESULTS AND DISCUSSIONS

A, Lift-Drag Ratio

In Figure 3) we have presented the results of case 1; the

lift-drag ratio is shown as a function of the buoyancy ratio with

C as a parameter. The results for the torpedo with the optimum

body is shown in solid lines while that for the diameter-limited

body is shown in dash lines. The lift-drag ratio,W/D,is plotted

along a logarithmic scale., since in this way, irrespective of

the values of W/D at R=l> (W/D) , an x-fold increase therefrom i

will appear a constant linear distance (log x) above(W/D) . The i

steeper the curves of W/D, the larger the relative increase of

W/D. The smaller the C , the larger the effect of decreasing w

the buoyancy ratio on the improvement of W/D, The effect of

limiting the body diameter is always detrimental to the value of

W/D obtainable. In Figure k, we have shown the drag breakdown

for this case of C , which is quite typical. It is seen that a w

major portion of the drag is due to the body. If by some means

the drag of the body can be reduced (e,g.3 by boundary layer

suction, by the use of cavity running torpedoes or through the

use of stabilizing additives),the resulting W/D can be further

Increased.


-20

In Figures 5 and 6, only the results from the optimum, body

for cases 2 and 3 are presented. In Figure 5J intermediate

values of C = 0.75 and 0.5 have been calculated and presented. w

It is seen that for C greater than 0.5,, there is no advantage w

to be gained by using wings. However at the lower values of

C- , say, 0.01 («* 3000 pound torpedo at 90 knots, c.f. Figure 2), w

the value of W/D at R = .2 is nearly 3 times that at R = 1.0.

Cross-comparing Figures 5 and 6, it is seen that the effect

of s' z on the resulting W/D is quite small. The larger value of

s' z permits the use of higher aspect-ratio wings with concomitant

savings in the wing drag. However, the drag due to the wing con-

stitutes only a small portion of the total drag, as shown in

Figure 4, and the savings show up significantly only for

C ^ .1, and only at small R's, w

In Figure 7, the effect of the maximum allowable diameter is

shown. The smaller diameter body, having a surface area larger

than that of the bigger diameter body for enclosing the same

volume, incurs larger drag. This explains the difference in W/D

at R=l. However, the improvement on W/D can be made larger for

the smaller diameter body.

Figures 8 and 9 refer to cases 5 and 6,. Cross-comparing

these two figures with Figures 3 and 5, we obtain the effect of

the Reynolds number,, Re . The W/D obtainable are smaller for

smaller Re . However, as we shall see in Figure 13, the Reynolds v

number has little effect on the Improvements attainable on W/D.


-21--

In Figures 10 through 13, we have shown the Improvements on

¥/D attainable as a function of c\ We define the nondlmenslonal

Increase In lift-drag ratio A(W/D) as

A(W/D) = W/D)R^2 - W/D)R=1

W/D)R=1

[52]

.

When A(W/D) = 1, the lift-drag ratio at buoyancy ratio R = 0,2

Is twice that at R = 1. In Figure 10, A(W/D) IS plotted vs_,

logarithm of C for case 1. It Is seen that for C > OA, the

winged torpedo Is less efficient than the neutrally buoyant one.

At low values of c" , the torpedo which consists of the diameter- w

limited body Is not as efficient as that with the optimum body.

In the same Figure, the aspect ratio of the wing Aw and the

span-diameter ratio, or equlvalently half-span-radius ratio,

b/R , for the optimum-body torpedo at R = ,2 are also shown. It

Is seen that at C 01, the wing span as well as the aspect

ratio are quite small; A « 0.5, b/R « 0.8. We note here that

the wing span may be smaller than the body diameter due to the

location of the wing.

In Figure 11, the A(W/D) for case 2 Is shown; the general

trend Is quite similar to that of case 1. In Figure 12, cases

3 and k are shown. For these two cases, the strength factor,

s'z". Is 33 which Is about 50 times that of case 2. The aspect ratio

of the wing Is about six times that of case 2, although the re-

sulting A(W/D) IS not too much different In these two cases.

This Is for the reason which has been stated before, that the


-22-

contribution of the wing to the total drag Is quite small. The

difference In the ^ = 0.383 and 1^ = 0.481 Is principally due

to the drag of the body. The body volume decreases while the

■thickness ratio Increases with decreasing buoyancy ratio. For

\ = •383, the value of the body drag at R = .2 Is about 30^ of

that at R = l, while the value for D = 0.481 Is about 4o^.

This fact accounts for the low values of A(w/D) for the

Dm = 0.481 case.

The Reynolds number effect on A(w/D) Is examined In Figure

12. It Is seen that for Rev range of .6 to 12 x 107 , the differ-

ence Is small for practical C" ^ 0 1 w

It may be of Interest now to show schematically some winged-

torpedo configurations. In Figure 14, we have shown a diameter

limited torpedo {D^ = 21"), and an optimum fineness ratio (5.1)

torpedo both of which weigh' 3000 pounds and have a forward speed

of 45 knots when they are neutrally buoyant, and when the buoyancy

ratio = 0.8 and 0.4. It Is seen that even at R = 0.4, the wings

needed In both cases are quite small. The small size of the

wing panels makes the problem of retracting them when launching

considerably simpler.

B. Range.

The results of the range calculations are shown In Figures

15 through 23. The range factor,-tcS/Ti, In mlles-lb fuel/horsepower-

hour Is shown as a function of buoyancy ratio with the ascending


-23-

angle as a parameter. In the first four Figures (15 through 18),

the fuel weight Is equal to one-half of the displacement, while

In the last four Figures (19 through 22) the fuel weight Is

equal to one-half of the total weight. Only the results for

four practical C 's are presented; they are cT = .01, .0204, w w

.04, .0625. ;

For a torpedo of specified weight. In the case In which the

fuel weight equals one-half of the displacement, since the volume

of the torpedo decreases as the buoyancy ratio decreases, the

fuel weight decreases with the buoyancy ratio. Although it has

been shown that the overall lift-drag ratio increases with de-

creasing buoyancy ratio, the decrease in the amount of fuel

available more than compensates for the effect with a resulting

range which is always less than that obtainable by the neutrally

buoyant torpedo except in descending missions. As the C w

Increases, the advantages of winged-torpedo over those which are

neutrally buoyant becomes more pronounced.

When the fuel weight is half of the total weight (Figures

.19 through 22)3 the winged-torpedo has longer ranges than those

with neutral buoyancy except in steep ascending missions. It

is noted that the vertical scales in these Figures are different

so as to exhibit the graphs more clearly.

CONCLUSIONS

1. The overall lift-drag ratios of torpedoes may be im-

proved by the use of wings for small C 's, say less than 0.3.

HYDRONAUTICSj Incorporated

-2k.

2. For C («0.01 (3000 pound torpedo at 90 knots) the w

winged-torpedo of .2 buoyancy ratio may give a lift-drag ratio

2,5-3 times of that obtainable with a neutrally buoyant tor-

pedo.

3. The effect of limiting the body diameter is always

detrimental to the lift-drag ratio obtainable.

k. Due to the small contribution of the wing to the total

drag, an increase in the strength factor s'z does not materially

affect the lift-drag ratio, although it permits wings of larger

aspect ratio and span.

5. The Reynolds number has little effect on the improve-

ments attainable on the lift-drag ratio.

6. Assuming that the fuel weight equals one-half of the

displacement, the winged-torpedo has longer ranges than those

with neutral buoyancy only in descending missions.

7. If the fuel weight equals one-half of the total weight,

the winged-torpedo has longer ranges than those with neutral

buoyancy except In steep ascending missions - 6 > 30 .


-.25-

REFERENCES

3,

Young, A. D., The Calculation of the Total and Skin Friction Drags of Bodies of Revolution at Zero Incidence,, A.R.C. Reports and Memoranda No. 187M-, April 1939»

Welsslnger, J., Einige Ergebnisse aus der Theorie des Ringflügels in Inkompresslbler Strömung, "Advances In Aeronautical Sciences", Volume 2, Fergamon Press, 1959»

Lawrence, H.R., and Flax, A,KL, Wing-Body Interference at Subsonic and Supersonic Speeds - Survey and New Developments, Journal of Aeronautical Sciences, volume 21, No- 5, May 195^.

HYDRONAUTICS, INCORPORATED

XRt

'Lw

/^

^Z^ ^^^

Xr--^

"^

V=o

.5= sin 30°

.707=sin450

.866=3^60«

0.5

R'I =i^F0R Jf^l, R FOR ?= I

FIGURE l-EFFECT OF WING-BODY INFLUENCE ON THE TOTAL LIFT


40 60 80

FORWARD SPEED IN KNOTS

FIGURE 2-WEIGHT COEFFICIENT AS A FUNCTION OF FORWARD SPEED


— ^ .0625

-. .040

-— >. .0278

-. .0204

•^ , .0156

0.2 0.4 0.6

BUOYANCY RATIO, R

0.8

FIGURE 3-LIFT-DRAG RATIO Vs. BUOYANCY RATIO Rev=l2XI08, s:Z = 0.7, Dnf.481


■

1.0

< a: o

o

< cr

0.5

0.5

BUOYANCY-WEIGHT RATIO, R

BODY

TAIL

1.0 WING

FIGURE 4-DRAG BREAK-DOWN FOR Cw = 0.010 Rev=l2XI07, s,Z= 0.7 (OPTIMUM BODY)


60

50

40

30

20

»Q

< tr

< a

10 9 8 7 6

5

4

I .9 .8 .7

.6

.5

.4

0.2 0.4 0.6

BUOYANCY RATIO, R

0.8

.25

.010

1.0

FIGURE 5-LIFT-DRA6 RATIO Vs BUOYANCY RATIO. (OPTIMUM BODY ONLY) Rev = 6 X I06, s'Z = 0.7


0.2 0.4 0.6

BUOYANCY RATIO, R

0.8 1.0

FIGURE 6-LIFT-DRAG RATIO Vs BUOYANCY RATIO Rev=6XI06, sT=33


0.4 0.6

BUOYANCY RATE, R

FIGURE 7-EFFECT OF Dm ON LIFT-DRAG RATIO Rev=6XI061 s'^SS


*V

<

< cc o

ou

40

30

20

10 9 8 7 6

5

4

3

2

^^^^

~-

^^

— ~~

' «^ ^—^

^^^^_

"^ ■ -___

'

\ ^~^-~^.

^ ^^ ̂ ^ '

^^ ̂ -^

>^ ^

^^ r^ .9 .8 .7 .6

.5

.4

■a

V^ " —-^.^ ^v v^ ■ "^-^^

"-—-^^

^~" 1 —-»_,

0.2 0.4 0.6

BUOYANCY RATIO, R

0.8

1.0

.25

III

.0625

.040

.0278

.0204

.OI56

.OIO

1.0

FIGURE 8-LIFT-DRAG RATIO Vs BUOYANCY RATIO Rev=4XI07, *1=.7


5|o

g"

o

0.2 0.4 0.6

BUOYANCY RATIO, R

0.8 1.0

FIGURE 9-UFT-DRAG RATIO Vs BUOYANCY RATIO Rev=8XI07, sZ = 0.7


>- Q O CD

s 3

Q. O cJ" d

lo*

z o

X

<

o X

-I DC \

\\ /

/

\

\ /

/

/

y /

X \

/ ^

/ 1 1 o m S 3 / /

m

— 5 —

t O

/ r i

Ü z

CO <

< a. CO

Q z < g

Ul CO Q. * co d < M

zlQ

-d UJ II _I|MJ CD 'in < - Zoo

IO

CO U> CJ CO d

(O o o

CM

d CJ d o

CD1- d

i

<

X CJ

I- »>

Q:

< o

i

CO

ÜJ

o z I o UJ


>- o o m

a. o

CO d n cr

lo* u. o V) z o

X s < ^Idf

\ yA y /

/

/

>- -y /

/

/ /

o / / \ o / / \ m / > / 1 s / a / 3 1 2 / s / m / 1- / E / I u- / |Q T 1 0 / *ib/ ^t|

,

i r o d

ro O d

o z

V) < < a. in

< o < a:

UJ 00

u>

3

< o 11

2, e Z|Q »<

< zu)' < 9 ^ x 1— < CD

^^ a:

< a: o

to

BO

DY


z 2 1- CL O

CÜ

d II a:

jo

O a> z o

'o X


FIGURE 13-EFFECT OF Re« ON A(-)j)


3000 POUND, 45 KNOT TORPEDOES FIXED MAXIMUM DIAMETER OF 21" OPTIMUM FINENESS RATIO OF 5

C ==t R = 1.0

D = 1490 LBS.

R = 1.0

D = 1350 LBS.

C

R = .8

D = 1270 LBS.

;S

R = .8

D = 1185 LBS.

R =.4 D = 850 LBS.

FIGURE 14

R =4 D = 810 LBS.


o

LLI 3

m

to LLI

S = RANGE IN MILES

Lc = SPECIFIC FUEL CONSUMPTION, LB FUEL/HP-HOUR

V -- PROPULSIVE EFFICIENCY

140 S'- 0.01, REV. = 12 x I07, SZ =0.7 Ö = -30o

\

120 V 1 1 1 100

\ ^^

-20° ^ ̂ ZZ^

80 ^ ' ^^v ̂ ^

y

s' / />1 ^^r

-10° ^ /yt? 60 / A // „ x s y S

0 y // i

40

10° y v ^K 20° / ^ 30° %^ '

20

n

i

0.2 0.4 0.6 0.8

BUOYANCY RATIO. R

1.0

FIGURE 15- THE INFLUENCE OF BUOYANCY RATIO ON RANGE

(FUEL WEIGHT = .5 x DISPLACEMENT)


O X I

a. i

3 h.

CD _l

I w -J

400

360

320

280

240

200

160

120

80

40

Ö = - 20° -30°

C = .02040 W

v

^: ̂ ^

^-^te^r

-10° ^^^

/W r

0 V 0° y 10°

20° 30°

0.2 0.4 0.6 0.8

BUOYANCY RATIO, R

1.0


( FUEL WEIGHT = .5 x DISPLACEMENT)


o

i Q. I

Z) LL.

m _J

i to

(S)\

900

800

700

600

500

400

300

200

100

0.2 0.4 0.6 0.8

BUOYANCY RATIO. R

1.0

FIGURE 17-INFLUENCE OF BUOYANCY RATIO ON RANGE

(FUEL WEIGHT = -5 x DISPLACEMENT)


0.2 0.4 0.6

BUOYANCY RATIO,R

0.8 1.0


( FUEL WEIGHT = . 5 x DISPLACEMENT)


700

600

Z) o I

I a.

LU 3

m _i

i V)

500

400

300

200

100

0.4 0.6 0,8

BUOYANCY RATIO,R


(FUEL WEIGHT = .5W)


700

O I

I Q. I

3

m

600

500

400

300

200

100

1 © = - 1-30°

0 = 0° A \ \ C = .02040

W

10°

20° \

^

30°

^

v. x ^C^

^

0.2 0.4 0.6 0.8 BUOYANCY RATIO, R

1.0


( FUEL WEIGHT = .5W)


O

Q. I

_1 LJ 3

m

1400

1200

1000

800

600

400

200

0.2 0.4 0.6 0.8

BUOYANCY RATIO. R


( FUEL WEIGHT =■ .5W)


3200

2800

2400

o: O

' 2000

\ _i

3

m i

CO

1600

2 1200 to

800

400

0.2 0.4 0.6

BUOYANCY RATIO, R

0.8 1.0

FIGURE 22-THE INFLUENCE OF BUOYANCY RATIO ON RANGE (FUEL WEIGHT = .5 W)

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